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[ [ "Movement Tracks for the Automatic Detection of Fish Behavior in Videos" ], [ "Abstract Global warming is predicted to profoundly impact ocean ecosystems.", "Fish behavior is an important indicator of changes in such marine environments.", "Thus, the automatic identification of key fish behavior in videos represents a much needed tool for marine researchers, enabling them to study climate change-related phenomena.", "We offer a dataset of sablefish (Anoplopoma fimbria) startle behaviors in underwater videos, and investigate the use of deep learning (DL) methods for behavior detection on it.", "Our proposed detection system identifies fish instances using DL-based frameworks, determines trajectory tracks, derives novel behavior-specific features, and employs Long Short-Term Memory (LSTM) networks to identify startle behavior in sablefish.", "Its performance is studied by comparing it with a state-of-the-art DL-based video event detector." ], [ "Introduction", "Among the negative impacts of climate change in marine ecosystems predicted for global warming levels of $1.5$ C to 2C (e.g.", "significant global mean sea level rise , sea-ice-free Artic Oceans , interruption of ocean-based services) are the acidification and temperature rise of waters.", "The behavioral disturbance in fish species resulting from climate change can be studied with the use of underwater optical systems, which have become increasingly prevalent over the last six decades , , .", "However, advancements in automated video processing methodologies have not kept pace with advances in the video technology itself.", "The manual interpretation of visual data requires prohibitive amounts of time, highlighting the necessity of semi- and fully-automated methods for the enhancement , and annotation of marine imagery.", "As a result, the field of automatic interpretation of underwater imagery for biological purposes has experienced a surge of activity in the last decade .", "While numerous works propose the automatic detection and counting of specimen , , , ecological applications require more complex insights.", "Video data provides critical information on fish behavior and interactions such as predation events, aggressive interactions between individuals, activities related to reproduction and startle responses.", "The ability to detect such behavior represents an important shift in the semantic richness of data and scientific value of computer vision-based analysis of underwater videos: from the focused detection and counting of individual specimens, to the context-aware identification of fish behavior.", "Given the heterogeneous visual appearance of diverse behaviors observed in fish, we initially focus our study on a particular target: startle motion patterns observed in sablefish (Anoplopoma fimbria).", "Such behavior is characterized by sudden changes in the speed and trajectory of sablefish movement tracks.", "We propose a novel end-to-end behavior detection framework that considers 4-second clips to 1) detect the presence of sablefish using DL-based object detectors ; 2) uses the Hungarian algorithm to determine trajectory tracks between subsequent frames; 3) measures four handcrafted and behavior-specific features and 4) employs such features in conjunction with LSTM networks to determine the presence of startle behavior and describe it (i.e.", "travelling direction, speed, and trajectory).", "The remainder of this article is structured as follows.", "In sec:prev we discuss works of relevance to the proposed system.", "sec:PA describes the proposed approach.", "In sec:ResultsDisc we present a dataset of sablefish startle behaviors and use it to compare the performance of our system with that of a state-of-the-art event detector .", "sec:con draws conclusions and outlines future work.", "Figure: Computational pipeline of the fish behavior detection system proposed.", "The framework is able to provide clip-wise and movement track-wise classifications alike (see )." ], [ "Related Works", "Related works to our approach include DL-based methods for object detection in images and events in videos.", "Deep learning-based object detection for static images.", "Krizhevsky et al.", "demonstrated the potential of using Convolutional Neural Networks (CNNs) to extract and classify visual features from large datasets.", "Their work motivated the use of CNNs in object detection, where frameworks perform both localization and classification of regions of interest (RoI).", "Girshick et al.", "introduced R-CNN, a system that uses a traditional computer vision-based technique (selective search ) to derive RoIs where individual classification tasks take place.", "Processing times are further reduced in Fast R-CNN and Faster R-CNN .", "A group of frameworks , , referred to as “one-stage” detectors proposed the use of a single trainable network for both creating RoIs and performing classification.", "This reduces processing times, but often leads to a drop in accuracy when compared to two-stage detectors.", "Recent advancements in one-stage object detectors (e.g.", "a loss measure that accounts for extreme class imbalance in training sets ) have resulted in frameworks such as YOLOv3 and RetinaNet , which offer fast inference times and performances comparable with that of two-stage detectors.", "DL-based event detection in videos.", "Although object detectors such as YOLOv3 can be used in each frame of a video, they often ignore important temporal relationships.", "Rather than individual images employed by aforementioned methods, recent works , , , , , , used video's inter-frame temporal relationship to detect relevant events.", "Saha et al.", "use Fast R-CNN to identify motion from RGB and optical flow inputs.", "The outputs from these networks are fused resulting in action tubes that encompass the temporal length of each action.", "Kang et al.", "offered a video querying system that trains specialized models out of larger and more general CNNs to be able to efficiently recognize only specific visual targets under constrained view perspectives with claimed processing speed-ups of up to $340\\times $ .", "Coşar et al.", "combined an object-tracking detector , trajectory- and pixel- based methods to detect abnormal activities.", "Ionescu et al.", "offered a system that not only recognizes motion in videos, but also considers context to differentiate between normal (e.g.", "a truck driving on a road) and abnormal (e.g.", "a truck driving on a pedestrian lane) events.", "Yu et al.", "proposed ReMotENet, a light-weight event detector that leverages spatial-temporal relationships between objects in adjacent frames.", "It uses 3D CNNs (“spatial-temporal attention modules”) to jointly model these video characteristics using the same trainable network.", "A frame differencing process allows for the network to focus exclusively on relevant, motion-triggered regions of the input frames.", "This simple yet effective architecture results in fast processing speeds and reduced model sizes ." ], [ "Proposed approach", "We propose a hybrid method for the automatic detection of context-aware, ecologically relevant behavior in sablefish.", "We first describe our method for tracking sablefish in video, then propose the use of 4 track-wise features to constrain sablefish startle behaviors.", "Finally, we describe a Long Short Term Memory (LSTM) architecture that performs classification using the aforementioned features.", "Object detection and tracking.", "We use the YOLOv3 end-to-end object detector as the first step of this hybrid method.", "The detector was completely re-trained to perform a simplified detection task were only the class fish is targeted.", "We use a novel 600-image dataset (detailed in REF ) of sablefish instances composed of data from Ocean Networks Canada (ONC) to train the object detector.", "The detection of each frame offer a set of bounding boxes and spatial centers.", "In order to track organisms we associate these detection between frames.", "Our association loss value consists simply of the distance between detection centers in two subsequent frames.", "We employ the Hungarian Algorithm to generate a loss minimizing associations between detection in two consecutive frames.", "We then remove any associations where the distance between various detection is greater than 15% of the frame resolution.", "Tracks are terminated if no new detection is associated with them for 5 frames (i.e.", "0.5 seconds—see REF ).", "Figure: Sample movement tracks and object detection confidence scores.", "The bounding boxes highlight the fish detection in the current frame.", "Each color represents an individual track.Behavior Specific Features.", "We propose the use of a series of four domain-specific features that describe the startle behavior of sablefish.", "Each feature conveys independent and complementary information, and the limited number of features (4) prevents over-constraining the behavior detection problem.", "The first two features quantify the direction of travel and speed from a track.", "These track characteristics were selected because often the direction of travel changes and the fish accelerates at the moment of a startle motion.", "Figure: LMCM kernel designed to extract fast changes in sequential images.A third metric considers the aspect ratio of the detection bounding box of a fish instance over time.", "The reasoning behind this feature is the empirical observation that sablefish generally take on a “c” shape when startling, in preparation for moving away from their current location.", "The final Local Momentary Change Metric (LMCM) feature seeks to find fast and unstained motion, or temporal visual impulses, associated with startle events.", "This feature is obtained by convolving the novel 3-dimensional LMCM kernel, depicted in fig:LOG-Kernel, over three temporally adjacent frames.", "This spatially symmetric kernel was designed to produce high output values where impulse changes occur between frames.", "Given its zero-sum and isotropic properties, the kernel outputs zero when none or only constant temporal changes are occurring.", "We observe that the LMCM kernel efficiently detects leading and lagging edges of motion.", "In order to associate this feature with a track we average the LMCM output magnitude inside a region encompassed by a given fish detection bounding box." ], [ "LSTM classifier", "In order to classify an individual track, we first combine its four features as a tensor data structure of dimensions $(40,4)$ ; tensors associated with tracks of less than 40 frames are end-padded with zeros.", "A set of normalization coefficients calculated empirically using the training set (see REF ) is then used to normalize each value in the input time-series to the range $[-1,1]$ .", "A custom-trained long short-term memory (LSTM) classifier receives the normalized tensors as input, and outputs a track classification of non-startle background or startle, as well as a confidence score.", "This is done by considering underlying temporal relationships between their values.", "We chose to use LSTM networks because the temporal progression of values from the features extracted (along 40 frames or 4 seconds) conveys important information for the classification of individual clips/tracks.", "fig:LSTM Classifier details the architecture of the LSTM network employed.", "All convolutional layers employ three-layered 1D kernels." ], [ "Results and Discussion", "We compare our method with a state-of-the-art event detection algorithm .", "Section sec:dataset describes our dataset.", "Our comparison considers standard performance metrics in sec:exresults and discusses the potential advantages of semantically richer data outputs in ecological research." ], [ "Sablefish Startle Dataset", "The data used in this work was acquired with a stationary video camera permanently deployed at 620m of depth at the Barkley Node of Ocean Networks Canada's www.oceannetworks.ca/ NEPTUNE marine cabled observatory infrastructure.", "All videos samples were collected between September 17th and October 24th 2019 because this temporal window contains high sablefish activity.", "The original monitoring videos are first divided into units of 4-second clips (a sablefish startle is expected to last roughly one second) and down-sampled to 10 frames per second for processing reasons.", "An initial filtering is carried out using Gaussian Mixture Models , resulting in a set composed only by clips which contain motion.", "For training purposes, these motion clips are then manually classified as possessing startle or not.", "The Sablefish Startle dataset consists of 446 positive (i.e.", "presenting startle events) clips, as well as 446 randomly selected negative samples (i.e.", "without startle events).", "tab:2-data-split details the dataset usage for training, validation and testing.", "Table: Division of the 4-second clips of the Sablefish Startle Dataset for training, validation and testing purposes.A second dataset composed of 600 images of sabblefish was created to train the YOLOV3 object detector (i.e.", "first step of the proposed approach).", "In order to assess the track-creation performance of the proposed system, we use this custom-trained object detector to derive movement tracks from each of the 892 clips composing the Sablefish Startle Dataset.", "Tracks with length shorter than 2 seconds are discarded.", "The remaining tracks are manually annotated as startle or non-startle (see tab:2-data-split).", "This dual annotation approach (i.e.", "clip- and track-wise) employed with the Sablefish Startle Dataset allows for a two-fold performance analysis: 1) clip-wise classification, where an entire clip is categorized as possessing startle or not, and 2) track-wise classification, which reflects the accuracy in the classification of each candidate track as startle or non-startle." ], [ "Experimental Results", "We calculate the Average Precision (AP), Binary Cross Entropy (BCE) loss and Recall for both track- and clip-wise outputs of the proposed system using only the Test portion of the Sablefish Startle Dataset.", "A threshold of $0.5$ (in a $[0,1]$ range) is set to classify a candidate movement track as positive or negative with respect to all of its constituent points.", "In order to measure the performance of the clip-wise classification and compare it with the baseline method (ReMotENet ), we consider that the “detection score” of a clip is that of its highest-confidence movement track (if any).", "Thus, any clip where at least one positive startle movement track is identified will be classified as positive.", "The conversion from track-wise classification to clip-wise classification is expected to lower the overall accuracy of our proposed approach.", "A “true” startle event might create only short, invalid tracks, or sometimes no tracks at all.", "This situation would lower the clip-wise classification performance, but would not interfere with the track-wise one.", "The track-wise metrics are applicable only to our approach and they mainly reflect the difference between the manual and automatic classification of the tracks created in the dataset by the proposed system, thus evaluating the ability of the LSTM network to classify tracks.", "Table REF shows that the LSTM portion of our method performs well for classifying startle tracks (AP of $0.85$ ).", "Clip-wise, our method outperformed a state-of-the-art DL-based event detector with an AP of $0.67$ .", "Table: Conclusion" ] ]	2011.14070
[ [ "Fluids at the Nanoscale: from continuum to sub-continuum transport" ], [ "Abstract Nanofluidics has firmly established itself as a new field in fluid mechanics, as novel properties have been shown to emerge in fluids at the nanometric scale.", "Thanks to recent developments in fabrication technology, artificial nanofluidic systems are now being designed at the scale of biological nanopores.", "This ultimate step in scale reduction has pushed the development of new experimental techniques and new theoretical tools, bridging fluid mechanics, statistical mechanics and condensed matter physics.", "This review is intended as a toolbox for fluids at the nanometre scale.", "After presenting the basic equations that govern fluid behaviour in the continuum limit, we will show how these equations break down and new properties emerge in molecular scale confinement." ], [ "INTRODUCTION", "Fluid flows at the nanometre scale have been studied indirectly in various disciplines for the last fifty years ([28]).", "However, it is only fifteen years ago that nanofluidics has emerged as a field on its own, first as a natural extension of microfluidics towards smaller scales.", "Back then, it was an issue in itself to establish that nanofluidics deserves its own name, meaning that there are specific effects at the nano-scale that are not present at the micro-scale.", "Indeed, the \"ultimate scale\" for observing specific effects is set by the molecular size of the fluid ; more precisely, a critical confinement $\\ell _c = 1~\\rm nm $ has been generally accepted as the limit of validity for the Navier-Stokes equation ([12], [131]).", "Moreover, it is at the molecular scale that the fluidic functions of biological systems emerge: from the giant permeability and perfect selectivity of the aquaporin ([104]), to the ion specificity of KcsA channels ([90]), to the mechano-sensitivity of Piezo channels ([148]), to name a few.", "However, ten years ago, the exploration of this ultimate scale was hindered by technical challenges, as molecular scale channels could not be fabricated artificially.", "A decade later, nanofluidics has firmly established itself as a field ([10]).", "Indeed, lengthscales associated with the electrostatics and the fluctuations of surfaces may reach up to several tens of nanometres, and their effects may be probed specifically in systems without molecular scale confinement ([124], [12], [131]).", "These lengthscales govern the key nanofluidic phenomena that have been demonstrated over the last ten years, such as, for instance, fast flows in carbon nanotubes ([51]), diffusio-osmotic energy conversion ([128]) or diode-type effects ([144]).", "However, the progress in fabrication technology has now allowed to overcome the challenges that have hindered the development of nanofluidics at the ultimate scales, and artificial devices with confinement down to about one water molecule size (3 $\\overset{\\circ }{\\rm A}$ ) have been achieved, in 0D, 1D or 2D geometry ([33], [139], [43]).", "It is therefore an exciting time for nanofluidics, since it now has the potential to reverse-engineer biological functions: minimal artificial systems that mimic biological processes may be designed and studied.", "Furthermore, nanofluidics is known for its short path from fundamental science to applications and innovation, and developments in single channel fabrication are likely to have direct implications for filtration and membrane science ([10]).", "Figure: An overview of nanofluidic lengthscales.", "Main ingredients of the physics above the continuum limit, and below.At this time of accelerated development of the field, one should realize that the nanoscales under study require new tools for understanding the physics at play.", "The need for \"new physics\" is particularly emphasised in a recent review by [32], which identifies \"critical knowledge gaps in mass transport through single-digit nanopores\" – nanopores that are less than 10 nm in size.", "Those knowledge gaps exist largely because the tools applied for understanding the experiments are built on macroscopic fluid mechanics and continuum electrostatics.", "But consider now a typical nanochannel of radius 1 nm.", "At physiological salt concentration, it contains only a single ion per 50 nm channel length.", "Similarly, the carbon nanotube porins studied by [139] each contain about 30 water molecules.", "These numbers strongly suggest that discrete particle effects may be important, and call for a statistical mechanics description.", "Furthermore, below 1 nm confinement, the length scales associated with the fluid dynamics become comparable to characteristic length scales of the electrons in the confining solids, such as the Thomas-Fermi length ([92]).", "This points to the necessity of describing the confining solids at the level of condensed-matter physics, and not simply as a space impenetrable to the fluid molecules.", "Overall, understanding fluidic phenomena at the nanometer scales requires bridging the gap between fluid mechanics, statistical mechanics and condensed matter physics.", "In the past years, the breakdown of continuum equations has often set a hard limit for fluid mechanics: below the continuum limit was the realm of molecular simulations.", "However, the need for understanding experiments has pushed for the development (or rediscovery) of analytical tools that have allowed to identify some specific phenomena and associated length scales.", "The description of these phenomena is a key part of this review, as summarised in figure 1.", "The review is organised as follows.", "In section 2, we give a brief overview of available nanofluidic systems and fabrication methods.", "In section 3, we focus on continuum modelling of nanofluidic systems, with particular emphasis on the precautions that should be taken when applying it to the smallest scales.", "Finally, in section 4, we go below the continuum limit, and highlight the specific phenomena that emerge along with the theoretical tools to describe them." ], [ "THE TOOLBOX OF EXPERIMENTAL SYSTEMS", "Nanofluidics generally follows a bottom-up approach.", "Elementary phenomena are understood at the well-controlled scale of the individual channel, before eventually being applied to more complex systems.", "Hence, the design of these well-controlled systems is paramount to the development of the field.", "We start this review by going through the systems that have so far been achieved, in order of dimensionality.", "Figure: State-of-the-art nanofluidic systems of various dimensions.", "a. TEM-drilled nanopore in single layer MoS 2 \\rm MoS_2 ().", "b. Boron nitride nanotube inserted into a SiN membrane ().", "c. Schematic and TEM image of the nanoslit device fabricated by ." ], [ "Nanopores", "Nanopores are channels whose length $L$ and diameter $d$ are both in the nanometre range (figure 2a).", "Initial studies focused on solid-state nanopores drilled through membranes made out of ceramics such as SiN of SiC ([67], [25]).", "More recently the advent of 2D materials, such as graphene, hexagonal boron nitride (hBN) or $\\rm MoS_2$ , allowed for the exploration of nanopores in atomically thin membranes ([41], [119]).", "Essentially three types of fabrication pathways have been reported for well-controlled nanopores in 2D materials.", "Drilling with an electron or a focused ion beam (FIB).", "A single 5 nm pore drilled in monolayer graphene was first reported by [41].", "[16] reported FIB drilling of arrays of nanopores in bilayer graphene, ranging from 7.6 nm to 1 $\\mu \\rm m$ in diameter.", "Electrochemical etching.", "[35] reported the opening of pores in monolayer $\\rm MoS_2$ when placed in a salt solution in between two electrodes.", "Applying a potential above the oxydation potential of $\\rm MoS_2$ resulted in the gradual removal of single $\\rm MoS_2$ units, thus creating an opening of controlled size.", "Nanopores down to 0.6 nm in diameter have been reported ([34]).", "The use of intrinsic defects in 2D materials.", "Large area membranes made of graphene or hBN are known to exhibit defects in the form of pores, ranging in size from a few angstroms to 15 nm depending on conditions ([145], [108]), or such defects may be generated on purpose in smaller membranes using ultraviolet-induced oxidative etching ([70]).", "The chemical vapour deposition (CVD) graphene membranes produced by [60] exhibited pores in the sub-2 nm range spaced by 70 to 100 nm.", "Placing the membrane on top of a 30-40 nm pore of a SiN membrane allowed to statistically isolate and study a single pore." ], [ "Nanotubes", "Nanotubes are cylindrical channels of diameter $d$ in the nanometre range, and length $L \\gg d$ (figure 2b).", "They are typically made out of carbon, or the isoelectronic boron nitride.", "The nanotubes themselves, as a product of self-assembly, are readily available, but interfacing a nanotube to a fluidic system and avoiding leakage is still an experimental challenge.", "Three distinct strategies for addressing this challenge have been reported.", "Building a microfluidic system on top of CVD-grown single-walled carbon nanotubes.", "The systems may comprise one or several carbon nanotubes, typically of diameter 1 to 2 nm.", "The tubes generally have very high aspect ratio with lengths up to 0.5 mm ([77], [19], [150]), although similarly built systems with 20 $\\mu \\rm m$ long nanotubes have also been reported ([110]).", "Insertion of a multiwalled nanotube into a solid state membrane.", "[128] reported the fabrication of nanofluidic devices comprising a single boron nitride nanotube, inserted into a hole milled in a SiN membrane by direct nanomanipulation under SEM.", "The hole could be sealed in situ by cracking of naphtalene induced by the electron beam.", "The method was later extended to carbon nanotubes ([125]), of 30 to 100 nm inner diameter and about 1 $\\mu \\rm m$ in length, and recently to smaller, 2 nm inner diameter, multiwall carbon nanotubes.", "Insertion of nanotubes into a lipid membrane.", "[87] reported the insertion of very short (5 to 10 nm) and very narrow (0.8 to 2 nm in diameter) nanotubes into a supported lipid membrane.", "Nanotubes were brought in contact with the lipid bilayer thanks to a microinjection probe.", "Recently, [139] reported the self-assembly of similar nanotubes, which they term carbon nanotube porins, into phospholipid vesicles.", "A single patch of membrane could also be isolated in order to study a single porin." ], [ "Nanoslits down to angström confinements", "Slit-like channels with one dimension below tens of nanometers were first made using micro- and nano- fabrication techniques.", "But recently, [115] reported the manufacturing of two-dimensional channels by van der Waals assembly of 2D materials (figure 2c).", "A few layers of graphene were used as spacers between two crystals of graphite, hBN or $\\rm MoS_2$ , allowing for atomically smooth channels of a few $\\mu \\rm m$ in length, 100 nm in width and down to 7 Å in height, that is the thickness of two graphene layers.", "Very recently ([43]), water transport through one-graphene-layer thick (3.4 Å) channels was reported.", "This brief overview highlights that nanofluidics at the molecular scale is now a reality.", "Not only molecular scale confinement is possible, but the geometry of the confinement and the nature of the confining materials can also be tuned." ], [ "Basic equations", "The two-centuries-old Navier-Stokes equation is remarkably robust at describing fluid flow down to the smallest scale, typically $\\ell _c = 1~\\rm nm$ for water in normal pressure and temperature conditions ([12]).", "This length scale is essentially a lower bound for defining a fluid viscosity $\\eta $ .", "Indeed, in macroscopic fluid mechanics, the kinematic viscosity $\\nu = \\eta /\\rho $ , where $\\rho $ is the mass density, plays the role of a diffusion coefficient for the fluid momentum.", "For such a diffusion coefficient to be defined, the time required for momentum to diffuse across the system, $\\ell _c^2/\\nu $ , must be larger than the timescale of molecular motion, which is the microscopic origin of diffusion.", "A water molecule at a thermal agitation speed of $300~\\rm m\\cdot s^{-1}$ moves by its own size in $\\tau _c = 10^{-12}~\\rm s$ , which defines a molecular time scale.", "Therefore, viscosity may be defined down to a system size $\\ell _c \\sim \\sqrt{\\nu \\tau _c} \\sim 1~\\rm nm.$ Below this length scale, water structuring due to surfaces, memory effects and other sub-continuum phenomena come into play: these will be discussed in section 4.", "For water flow at $10~\\rm nm$ length scales, the Reynolds number remains smaller than 0.1 up to fluid velocities of $10~\\rm m\\cdot s^{-1}$ .", "Hence, in nanofluidic systems, inertial effects may be safely neglected, and the fluid flow is described by the Stokes equation: $\\eta \\Delta v + f = \\nabla p,$ where $p$ is the pressure and $f$ a body force, which may be due, for example, to the application of an electric field (see section 3.3)." ], [ "Boundary conditions", "Stokes flow is often solved with no-slip boundary conditions: the velocity of the liquid is assumed to vanish at a solid-liquid interface.", "This is, however, a limiting case of the more general Navier partial slip boundary condition, which enforces that the viscous stress at the interface should be balancing the solid liquid friction force.", "Within linear response theory, the friction force is proportional to the liquid velocity.", "For a fluid flowing in the direction $x$ along a surface of normal $z$ , the force balance per unit area writes $\\sigma _{xz} = \\lambda v_x,$ with $[\\sigma ]$ the stress tensor and $\\lambda $ the friction coefficient per unit area (expressed in $\\rm N \\cdot s \\cdot m^{-3}$ ).", "For a Newtonian fluid, $\\sigma _{xz} = \\eta \\partial _z v_x$ , which allows to rewrite the Navier boundary condition as $v_x = \\left.", "b \\frac{\\partial v_x}{\\partial z}\\right\|_{\\rm wall},$ introducing the slip length $b = \\eta /\\lambda $ .", "The slip length can be geometrically interpreted as the depth inside the solid where the linearly extrapolated fluid velocity profile vanishes.", "Accordingly, the no-slip boundary condition corresponds to $\\lambda \\rightarrow \\infty $ or $b \\rightarrow 0$ .", "The effect of the partial slip condition is to simply shift the no-slip velocity profile by the slip velocity, which is not negligible roughly within a slip length from the wall.", "Since slip lengths up to tens of nanometers have been measured on atomically flat (and hydrophobic) surfaces, slippage is expected to play a crucial role in nanofluidics, and some of its effects will be discussed in the following sections.", "In the smallest channels, of size $R \\ll b$ , a perfect slip boundary condition may even be appropriate: the flow is then controlled by entrance effects." ], [ "Geometry and entrance effects", "Experimentally, the flow profile inside a nanofluidic channel can hardly be resolved, and one typically measures the total flow rate $Q$ .", "Under a pressure drop $\\Delta P$ and no-slip boundary conditions, the flow rate through a cylindrical channel of radius $R$ is given by the Hagen-Poiseuille formula: $Q_c = \\frac{\\pi R^4}{8 \\eta L} \\Delta P.$ This formula assumes a channel length $L \\gg R$ , and thereby neglects the effect of channel mouths on the flow rate.", "But the transition from a macroscopic reservoir to a nanoscale channel is a source of viscous dissipation, as the streamlines need to be bent in order for the fluid to enter the channel.", "These entrance effects may be examined by considering the flow through an infinitely thin nanopore, which is of interest in itself, given the geometry of certain nanofluidic devices (see section 2.1).", "This problem was addressed by [120].", "For a nanopore of radius $R$ (and vanishing length) under pressure drop $\\Delta P$ , Sampson obtained the expression of the flow rate as $Q_p = \\frac{R^3}{3 \\eta } \\Delta P.$ The scaling in Sampson's formula naturally emerges from a Stokes equation where the only lengthscale is $R$ : $\\eta \\Delta v = \\nabla p \\Rightarrow \\eta v/R^2 \\sim \\Delta P/R$ , and the typical fluid velocity is $v \\sim Q/R^2$ .", "In order to estimate the flow rate through a channel taking into account entrance effects, one may simply add the hydrodynamic resistances of the pore (${\\cal {R}}_p$ ) and the channel (${\\cal {R}}_c$ ).", "If one writes $Q_c =(\\Delta P)_c/ {\\cal {R}}_c$ and $Q_p = (\\Delta P)_p/{\\cal {R}}_p$ , then the entrance-corrected flow rate $Q_{pc}$ is obtained by imposing $Q_p=Q_c=Q_{pc}$ and $\\Delta P=(\\Delta P)_c+(\\Delta P)_p$ , so that $Q_{pc} = \\frac{\\Delta P}{{\\cal {R}}_h + {\\cal {R}}_p} = \\frac{\\pi R^4}{8 \\eta L}\\frac{ \\Delta P}{1+ \\frac{3\\pi }{8}\\frac{R}{L}}.$ An exact computation ([23]) shows that the error made by this a priori crude approximation is less than 1%.", "Equation REF makes a continuous transition between the nanopore and nanochannel regimes, and shows that entrance effects are apparently negligible for channel lengths that exceed a few channel radii.", "Figure: Entrance effects in nanofluidics.", "a. Viscous dissipation rate, and streamlines, for the pressure-driven flow of water across a nanopore, as obtained from a finite elements solution of the Stokes equation (COMSOL).", "The colour scale, from blue to red, encodes the viscous dissipation.", "b. Geometric model of the aquaporin as considered by .", "c. Permeability of the model aquaporin as a function of the cone angle α\\alpha .However, the above discussion has crucially not taken into account slippage, which, as we have highlighted in the previous section, is a strong effect at the nanoscale.", "Introducing a non-zero slip length $b$ , the flow rate though a channel becomes $Q_c = \\frac{\\pi R^4}{8 \\eta L}\\left( 1+ \\frac{4b}{R}\\right) \\Delta P,$ while the flow rate through a pore is not significantly affected ([44]), since the source of dissipation in that case is mostly geometric.", "A full expression can be obtained by gathering previous results, but in the limit where $b \\gg R$ , the entrance-corrected flow rate becomes $Q_{pc} = \\frac{R^3}{3\\eta } \\frac{\\Delta P}{1+ \\frac{2L}{3\\pi b}}.$ Thus, the hydrodynamic resistance is actually dominated by entrance effects as long as the channel is shorter than the slip length, rather than the channel radius (see figure 3a).", "In the presence of significant slippage, one should check whether the low Reynolds number assumption still holds.", "The average velocity through a channel of radius $R = 5~\\rm nm$ and length $L = 1~\\rm \\mu m$ , with slip length $b = 30~\\rm nm$ , under a pressure drop $\\Delta P = 1~\\rm bar$ is $v = 8~\\rm mm \\cdot s^{-1}$ , which is 25 times faster than the no-slip result, but still well below the $1~\\rm m\\cdot s^{-1}$ threshold established in 3.1.1.", "Entrance-effect-dominated transport is particularly relevant for biological nanochannels, due to their relatively small aspect ratio.", "A striking example is aquaporin, which was recently studied from a hydrodynamic point of view by [44].", "Aquaporins are channel proteins that selectively transport water across the cell membrane.", "A simplified geometrical model for the aquaporin consists of two conical vestibules connected by a subnanometric channel where water flows in single file (figure 3b).", "The single-file transport is expected to be nearly frictionless, and therefore the limit to the aquaporin's permeability is set by the entrance effects in the conical vestibules.", "Moreover, since the channel in question is less than 1 nm wide and slip lengths of the order of 10 nm are expected, perfect slip boundary conditions are relevant for the flow, and the viscous dissipation has a purely geometric origin in the curvature of the stream lines.", "With the notations of figure 3b, Gravelle et al.", "express the total hydrodynamic resistance of the aquaporin as $R_{\\rm AQP} = R_1 + R_2 = C_{\\infty } \\frac{\\eta }{a^{\\prime 3}} + C_{\\infty } \\sin \\alpha \\frac{\\eta }{a^3}.$ Both terms are the analogue of Sampson's formula (eq.", "REF ).", "$C_{\\infty } = 3.75$ replaces the factor 3 in the case of perfect slip boundary conditions, and the factor $\\sin \\alpha $ appears because at the cone-cylinder transition, the stream lines turn by an angle $\\alpha $ , as opposed to $\\pi /2$ when the transition is from an infinite reservoir to a pore.", "With that, the permeability $K = R_{\\rm AQP}^{-1}$ turns out to be a non-monotonous function on $\\alpha $ (figure 3c), with the maximal permeability reached for cone angles in the range measured on aquaporin structures obtained by X-ray crystallography.", "It appears, therefore, that the geometry of the aquaporin is an optimum for hydrodynamic entrance effects, highlighting their particular relevance for nanoscale fluid transport." ], [ "Gas transport", "In a manner similar to liquid flow, gas flow can also be used to probe nanofluidic systems.", "Gas flow may display two limiting regimes, depending on the value of the Knudsen number, defined as $Kn = \\frac{\\ell }{R},$ where $\\ell $ is the mean free path and $R$ is the typical system size.", "The mean-free path scales as the inverse density according to $\\ell \\sim (\\rho \\sigma ^2)^{-1}$ , with $\\sigma $ the molecular diameter.", "For $Kn \\ll 1$ , the transport is dominated by intermolecular collisions and is therefore described by hydrodynamics; for example, by the Poiseuille or Sampson formulae introduced above.", "For $Kn \\gg 1$ , the transport is dominated by collisions with the walls and is described by molecular diffusion: this is the so-called Knudsen regime ([79]).", "Consider a cylindrical channel of radius $R$ and length $L \\gg R$ , connecting two gas reservoirs at pressures $P_1$ and $P_2$ and same temperature $T$ (see figure 4a).", "One may then define a diffusion coefficient that relates the molecular flow rate $Q_K$ to the density gradient $\\Delta n/L$ across the channel through a Fick-type law: $\\frac{Q_K}{\\pi R^2} = D_K \\frac{\\Delta n}{L} = \\frac{D_K}{k_BT} \\frac{\\Delta P}{L}.$ The last equality uses that $n = P/k_B T$ through the ideal gas law, and $\\Delta P = P_1 - P_2$ .", "The only parameters involved in the diffusion are the channel radius $R$ and the average thermal velocity $v^* = \\sqrt{8k_B T/\\pi m}$ , where $m$ is the molecular mass of the gas.", "On dimensional grounds, the Knudsen diffusion coefficient should scale as $D_K \\sim R v^$ .", "A kinetic theory computation ([69], [132]) yields $D_K = (2\\pi /3)R v^$ , and the Knudsen formula: $Q_K = \\frac{8}{3} \\frac{\\pi R^3}{\\sqrt{2 \\pi m k_B T}} \\frac{\\Delta P}{L}.$ Now the Knudsen formula relies on the crucial approximation of diffuse reflexion on the channel walls: a molecule that hits a wall has a new velocity randomly picked out of a Boltzmann distribution.", "It was proposed by Maxwell that only a fraction $f$ of reflections should be diffuse, and the rest should be specular, that is correspond to elastic collisions.", "Smoluchowski then derived the corresponding correction to the Knudsen formula ([79], [143]): $Q_K = \\frac{2-f}{f} \\cdot \\frac{8}{3} \\frac{\\pi R^3}{\\sqrt{2 \\pi m k_B T}} \\frac{\\Delta P}{L}.$ There is a divergence of the flow rate in the limit $f \\rightarrow 0$ , that is the channel opposes no more resistance to the gas flow when all the wall reflections are specular.", "This is to be expected, since in that case a molecule that enters the channel necessarily exits at the other end.", "The flow resistance is then dominated by entrance effects; in other words, the flow rate is given by the rate of molecules hitting the channel apertures.", "Gas molecules at a density $n$ hit an aperture of area $\\pi R^2$ at a rate $(1/4)nv^* \\pi R^2$ .", "Hence the effusion flow rate through an opening of radius $R$ is $Q_e = \\pi R^2 \\Delta n \\sqrt{\\frac{k_B T}{2\\pi m}} = \\frac{\\pi R^2}{\\sqrt{2 \\pi m k_B T}} \\Delta P,$ which also sets the flow rate in a long channel with specular reflection at the walls.", "One notes there is a close analogy between the Poiseuille and Sampson formulae for liquids, on the one hand, and the Knudsen and effusion formulae for gases, on the other hand.", "The fraction of specular reflections $1-f$ plays a role similar to the slip length: if it is large, the transport in a long channel may still be dominated by entrance effects.", "The free effusion prediction has been verified in systems of graphene nanopores ([16]), with a transition to the hydrodynamic behaviour (Sampson formula) observed upon reducing the Knudsen number.", "Specular reflections have also been evidenced in longer channels.", "Gas flow exceeding the Knudsen prediction was measured in carbon nanotubes ([51], [93]), and, recently, nearly ballistic transport was evidenced in angstrom-scale slits ([66]).", "These results point out that the tendency to anomalously fast transport in nanoscale confinement exists not only for liquids, but also for gases.", "A striking example of gas-mediated osmotic flow was demonstrated by the group of Karnik.", "[78] fabricated a nanoporous (70 nm pore size) membrane with partially hydrophobic pores, so that a nanobubble is trapped in each pore when the membrane is immersed in water (see figure 4b).", "The nanobubbles are impermeable to salt, but permeable to water through its vapour phase transport.", "They show that the membrane reaches an ion rejection of $99.9\\%$ , while competing with the permeability of state-of-the-art polyamide-based membranes.", "Such high permeability might seem counterintuitive since vapour phase transport is expected to scale with the density of water vapour, 1000 times lower than that of liquid water.", "However, using gas phase transport as the ion rejection mechanism allows for much larger pores (70 nm) then what would be required in the liquid phase ($\\sim 1~\\rm nm$ ) to ensure steric rejection of ions.", "The scaling of the water transport rate with the pore size cubed ([9]) then explains the high transport efficiency in the gas phase.", "As such, nanoscale gas transport shows promise for addressing the permeability-selectivity tradeoff in membrane science." ], [ "Ion transport", "The behaviour of ions in nanofluidic systems is of great practical interest with applications ranging from biological ion channels ([90]) to ionic liquids inside nanoporous electrodes ([18], [99]).", "Ion transport also provides an indirect way of probing fluid transport, which is often useful, since electric currents are much easier to measure than fluid flow.", "However, due to their long range Coulomb interactions and diffusive dynamics, ions in nanochannels give rise to physics of great complexity.", "In this section, we only sketch the main results with particular emphasis on how well-known equations apply to the smallest channels, and we refer refer the reader to other reviews on the subject ([12], [124], [131]) for more details." ], [ "Basic equations", "Consider an aqueous solution of monovalent salt.", "Let $\\rho _+,\\rho _-$ be the densities of positive and negative ions, respectively; $D$ the diffusion coefficient, here assumed to be the same for ions of either sign, and $\\phi $ the electrostatic potential.", "In a mean-field treatment, the convective-diffusive dynamics of ions are described by a Smoluchowski equation: $\\frac{\\partial \\rho _{\\pm }}{\\partial t} = \\nabla \\cdot \\left(D \\nabla \\rho _{\\pm } \\mp \\frac{e D}{k_B T} (\\nabla \\phi ) \\rho _{\\pm } + v \\rho _{\\pm } \\right),$ where $e$ is the unit charge, $\\phi $ the electrostatic potential and $v$ is the fluid velocity field.", "The mean-field assumption implies in particular that correlations between the ions can be neglected: the potential importance of such correlations in nanofluidics will be discussed in section 4.", "Until then, we proceed by specifying the electrostatic potential through Poisson's equation, $\\Delta \\phi = -e \\frac{\\rho _+-\\rho _-}{\\epsilon },$ where $\\epsilon $ is the dielectric permittivity of water.", "For now we assume it to be isotropic, though this assumption may break down for nano-confined water, as we discuss in section 4.", "Lastly, we specify the flow velocity through the Stokes equation, which now includes an electrostatic term: $\\eta \\Delta v - e (\\rho _+-\\rho _-) \\nabla \\phi = \\nabla p.$ We now apply these three coupled equations to a specific geometry, though the discussion that follows could be generalised to channels of any shpae.", "For simplicity, we consider a slit-like channel of height $h$ , width $w$ and length $L$ , with $w, L \\gg h$ , connecting two reservoirs of salt solution at concentration $\\rho _s$ , extending along the direction $x$ , between $z = 0$ and $z = h$ .", "When considering ion transport, it is important to note that most surfaces are charged in water, due either to the dissociation of surface groups or to the adsorption of ions ([112], [47], [101]).", "We hence assume the channel wall carries a surface charge density $-\\Sigma e$ ($\\Sigma $ is expressed in elementary charges per unit area, and we assume here the surface charge to be negative)." ], [ "Ionic conductance", "We first neglect the coupling of ion transport to water transport, and consider the electrophoretic (EP) contribution to the ionic current under an applied electric field $E$ : this means that we start by setting the fluid velocity $v = 0$ .", "In the steady state, the Smoluchowski equation (REF ) reduces to the so-called Nernst-Planck equations for the constant ionic fluxes (along the $x$ direction): $j_{\\pm } = D \\nabla _x \\rho _{\\pm } \\mp \\frac{e D}{k_B T} (\\nabla _x \\phi ) \\rho _{\\pm }.$ Together with the Poisson equation (REF ), these constitute the widely used Poisson-Nernst-Planck (PNP) framework.", "In our geometry, the condition $L \\gg h$ ensures that in the middle of the channel $\\nabla _x \\rho _{\\pm } = 0$ ; moreover to first order in $E$ , $\\nabla _x \\phi = -E$ and the densities reduce to their equilibrium values.", "Hence, the EP contribution to the ionic current writes $I_{\\rm ep} = w\\int _0^h \\mathrm {d}z (j_+-j_-) = w\\frac{e^2D}{k_B T} \\int _0^h \\mathrm {d}z (\\rho _+ + \\rho _-) E.$ In order to compute $I_{\\rm ep}$ , one needs to find the equilibrium solution of the coupled PNP equations for $\\rho _+$ and $\\rho _-$ .", "At equilibrium $j_{\\pm } = 0$ and the Nernst-Planck equations (REF ) can be integrated, imposing that in the reservoirs $\\phi = 0$ and $\\rho _{\\pm } = \\rho _s$ .", "This yields a Boltzmann distribution for the ions in the electrostatic potential $\\rho _{\\pm } = \\rho _s \\exp \\left(\\mp \\frac{e\\phi }{k_BT} \\right) \\equiv \\rho _s e^{\\mp \\psi },$ introducing a dimensionless potential $\\psi $ .", "Combining this with the Poisson equation (REF ) yields the Poisson-Boltzmann (PB) equation: $\\Delta \\psi - \\lambda _D^{-2} \\sinh (\\psi ) = 0,$ which introduces the Debye length $\\lambda _D = (8 \\pi \\rho _s \\ell _B)^{-1/2}$ , with $\\ell _B = e^2/(4\\pi \\epsilon k_B T)$ the Bjerrum length.", "In our geometry, the PB equation has an implicit solution in terms of an elliptic integral ([82], [2]).", "We will not exploit it here, however, and we will instead recover the relevant limiting behaviours from qualitative considerations.", "It is well known that, roughly speaking, the Debye length sets the extension of the diffuse layer of counterions next to a charged surface ([58]).", "Hence, if the channel height $h \\gg \\lambda _D$ , its two opposing walls do not 'see' each other.", "We expect the conductance to be the sum of a bulk term, and a surface term originating in the two Debye layers: $I_{\\rm ep} = 2 w \\frac{e^2D}{k_B T} E(\\rho _s h + \\Sigma ).$ where $\\rho _s h$ and $\\Sigma $ account for the number of charge carriers in the bulk and at surfaces, respectively.", "In the opposite limit where $h \\ll \\lambda _D$ , there is no more distinction between surface and bulk.", "All the quantities may be considered uniform across the channel: this is called the Debye overlap regime.", "However, one may not assume that the channel contains only counterions, and one should go back to the thermodynamic equilibrium with the reservoirs, which in this case bears the name of Donnan equilibrium ([12]).", "One has $\\rho _{\\pm } = \\rho _s e^{\\mp \\psi }$ , which implies a chemical equilibrium $\\rho _+ \\rho _- = \\rho _s^2$ in the channel.", "Going further, in the limit of long channel length, there should be local electroneutrality: $h(\\rho _+-\\rho _-) = 2\\Sigma $ .", "This yields $\\rho _{\\pm } = \\sqrt{\\rho _s^2 + (\\Sigma /h)^2} \\pm \\Sigma /h,$ and the current-voltage relation in the Debye overlap is $I_{\\rm ep} = 2 w \\frac{e^2D}{k_B T} E\\sqrt{(\\rho _s h)^2 + \\Sigma ^2}.$ Equation REF displays the first peculiarity of small channels: one may not simply add the surface and bulk contributions.", "Table 1 (see appendix) lists the values of Debye length for different electrolyte concentrations, showing that the Debye overlap regime is indeed relevant for experimentally accessible nanofluidic systems (see section 2).", "Qualitatively, eqs.", "(REF ) and (REF ) both predict saturation of the conductance at low salt concentrations at a value determined by the surface charge.", "The saturation occurs when $\\rho _s \\sim \\Sigma /h$ , which can be recast in the form $h \\sim \\rho _s/\\Sigma \\equiv \\ell _{Du}$ .", "$\\ell _{Du}$ is called the Dukhin length and quantifies the competition between bulk and surface contributions to the conductance.", "For a channel narrower than $\\ell _{Du}$ , surface contributions dominate, and vice versa.", "The Dukhin length is going to be important in our upcoming discussion of entrance effects.", "[] Bjerrum length$\\ell _B = \\frac{e^2}{ 4\\pi \\epsilon _0k_BT}$ .", "Distance between two unit charges at which their interaction energy is $k_BT$ .", "Debye length$\\lambda _D = (8 \\pi \\rho _s \\ell _B)^{-1/2}$ .", "Thickness of the diffuse layer of counterions next to a charged surface.", "Dukhin length$\\ell _{Du} = \\Sigma /\\rho _s$ .", "Channel width below which surface conductance dominates over bulk conductance.", "Gouy-Chapman length$\\ell _{GC} = (2\\pi \\Sigma \\ell _B)^{-1}$ .", "Distance a unit charge must travel from a charged surface so that its electrostatic energy is reduced by $k_BT$ .", "At this point, a remark should be made concerning the range of validity of equations (REF ) and (REF ).", "Indeed, they have been derived from qualitative considerations, without reference to the exact solution of the PB equation.", "Now, from eqs.", "(REF ) and (REF ), one obtains more generally $I_{\\rm ep} = 2 w \\frac{e^2D}{k_B T} \\left( \\rho _s h \\cosh (\\psi (h/2)) + \\frac{\\mathcal {E}}{2k_BT} \\right),$ where $\\mathcal {E} = (\\epsilon /2) \\int _0^h (\\partial _z \\phi )^2 \\mathrm {d}z$ is the electrostatic energy per unit area.", "The electric double layer can be pictured as a capacitor with charge $\\Sigma $ , hence one would expect its electrostatic energy to scale as $\\Sigma ^2$ ; moreover, this is the prediction of the linearised PB equation, i.e.", "eq.", "(REF ) with the approximation $\\sinh \\psi \\approx \\psi $ .", "This is in contrast to, eq.", "(REF ), which predicts a linear scaling of the conductance with $\\Sigma $ : this scaling must therefore come from the non-linearities of the PB equation.", "The PB equation may be linearised if the potential varies by less than $k_B T$ across the Debye layer (or across the channel if there is Debye overlap).", "This is the case in the high concentration/low surface charge limit and specifically when the Debye length is smaller than the so-called Gouy-Chapman length: $\\ell _{GC} = (2\\pi \\Sigma \\ell _B)^{-1}$ .", "On the other hand, Eq.", "(REF ), valid for the Debye overlap regime, is safe from a condition on $\\ell _{GC}$ , as it predicts both quadratic and linear scalings depending on the value of $\\Sigma $ , as long as there is Debye overlap." ], [ "Entrance effects", "Similarly to what we have discussed for liquid and gas transport, we may now ask, for ion transport, what is the additional electrical resistance due to the channel - reservoir interface.", "Equivalently, we may want to estimate the ionic conductance of a nanopore of small aspect ratio, say of radius $R$ and length $L\\sim R$ .", "The problem has first been considered in the context of biological channels by [50] and [48].", "Hall solved the electrostatic problem with an electrode at infinity and an equipotential disk accounting for the entrance of the pore.", "Translating the solution into scaling arguments, the current through the pore entrance is $I \\sim \\pi R^2 \\kappa _b \\Delta V_{\\rm out}/R$ , where $\\Delta V_{\\rm out}$ is the voltage drop at the entrance of the pore, which is expected to occur over a distance $R$ , and not over the macroscopic distance between the electrodes.", "This defines an electrical access resistance as the ratio $\\Delta V_{\\rm out}/I$ .", "We now specialise to the thin Debye layer regime $\\lambda _D \\ll R$ .", "If one simply sums the access resistance and the channel resistance as given by eq.", "(REF ) (adapted to cylindrical geometry), the current-voltage relation becomes $I_{\\rm ep} = \\kappa _b \\left( \\frac{L}{\\pi R^2} \\frac{1}{1+\\ell _{Du}/R}+ \\frac{1}{\\alpha R} \\right)^{-1} \\Delta V,$ with $\\alpha $ a geometric factor which is 2 in Hall's computation, and introducing the bulk conductivity $\\kappa _b = 2e^2D\\rho _s/(k_BT)$ .", "The above equation predicts vanishing conductance as $\\rho _s \\rightarrow 0$ , since the access resistance becomes infinite in this limit; however, this is not what is observed experimentally in short nanopores ([76], [33]).", "This inconsistency arises because, for a surface-charged pore, the access current has a surface contribution, in addition to the bulk contribution.", "This surface contribution may be evaluated starting from charge conservation at the surface, which imposes a relation between the axial and radial components of the electric field, as pointed out by [68]: $\\kappa _b E_r = \\partial _x [\\kappa _s \\theta (x) E_x],$ with $\\kappa _s$ the surface conductance and $\\theta $ the Heaviside function, accounting for the discontinuity of surface charge at the pore boundary, which leads to subtle consequences.", "Indeed, it reveals that the Dukhin length, $\\ell _{Du} = \\Sigma /\\rho _s \\sim \\kappa _s/\\kappa _b$ , is the relevant lengthscale for the surface contribution to the electric field outside the pore, instead of the channel radius or Debye length.", "The Dukhin length appears here as an electrostatic healing length: feeding the surface conduction at the nanopore mouth disturbs the electric field lines in the bulk over a length $\\ell _{Du}$ .", "This interpretation is supported by the numerical results of [76] as shown in figure 5a.", "The bending of the electric field lines can be estimated by a perturbative approach ([76]) that leads to an analytical expression for the corrected current-voltage relation: $I_{\\rm ep} = \\kappa _b \\left( \\frac{L}{\\pi R^2} \\frac{1}{1+\\ell _{Du}/R}+ \\frac{1}{\\alpha R+\\beta \\ell _{Du}} \\right)^{-1} \\Delta V.$ The surface-charged pore therefore appears, from the perspective of entrance effects, as an uncharged pore of effective size $R + \\ell _{Du}$ , since the geometrical prefactor $\\beta \\approx 2$ .", "In the limit of vanishing salt concentration, $\\rho _s \\rightarrow 0$ , the conductance indeed saturates, and, as expected, the entrance correction disappears in the limit of large aspect ratio pores.", "Lee et al.", "successfully compared the prediction of eq.", "(REF ) to experimental measurements in SiN nanopores in the range $100 - 500~\\rm nm$ .", "More recently, it has been used to describe conductance measurements in $\\rm MoS_2$ nanopores down to 2 nm in diameter ([33]).", "Figure: a. Colour plot of the electrostatic potential around a nanopore immersed in a salt solution and subject to a voltage drop, in the charged and in the uncharged case.", "Around a charged pore, the lengthscale for variation of the potential is set by the Dukhin length.", "Adapted from .", "b. and c. Schematic representation of coupled ion and fluid transport in a nanochannel under electric field driving: in the thin Debye layer limit (b.)", "and in Debye overlap in the presence of slippage (c.).", "The electro-osmotic contribution dominates in the latter case.", "d. Peclet number (defined in the text), as a function of channel width and Gouy-Chapman length.", "The region in parameter space where Pe>1Pe >1 is coloured in dark red in the no-slip case, and in light red when slippage is present." ], [ "Coupling with fluid flow", "So far, we have neglected any coupling of ion transport to fluid transport.", "However, in the presence of charged surfaces, an external electric field exerts a net force on the charged Debye layer, which sets the fluid in motion.", "This interfacially-driven flow is termed electro-osmotic (EO) flow.", "The EO flow in turn drags along the ions in the Debye layer, which makes a supplementary contribution to the ionic current, that we denote $I_{\\rm eo}$ .", "This contribution has a convenient exact expression in the slit-like geometry considered in section 3.3.2.", "In analogy to eq.", "(REF ), $I_{\\rm eo} = 2w\\int _0^{h/2} e(\\rho _+-\\rho _-) v(z) \\mathrm {d}z.$ From the Poisson equation (REF ), we may replace $e(\\rho _+-\\rho _-) = -\\epsilon \\partial ^2_z \\phi $ .", "Then, integrating by parts, $I_{\\rm eo} = 2w\\epsilon \\left[ \\int _0^{h/2} \\partial _z \\phi \\partial _z v \\, \\mathrm {d}z + v(0) \\partial _z \\phi \|_0 \\right].$ Now, we may use the partial slip boundary condition $v(0) = b \\partial _z v\|_0$ , as well as the electrostatic boundary condition $\\partial _z \\phi \|_0 = e \\Sigma /\\epsilon $ .", "This stems from the Gauss theorem applied to the surface, and the assumption that the medium outside the channel has much lower dielectric permittivity than water ([2]).", "Moreover, an integration of the Stokes equation (REF ) allows the replacement $ \\partial _z v = -(\\epsilon E/\\eta ) \\partial _z \\phi $ .", "Altogether, we obtain $I_{\\rm eo} = w\\left[ \\frac{2\\epsilon }{\\eta } \\mathcal {E} + \\frac{2 b e^2 \\Sigma ^2}{\\eta } \\right] E,$ where $\\mathcal {E} = (\\epsilon /2) \\int _0^h (\\partial _z \\phi )^2 \\mathrm {d}z$ is the electrostatic energy per unit surface introduced in 3.3.1.", "It appears here that slippage has an additive contribution to the EO current, which strikingly does not depend on any electrolyte properties.", "We now consider the result in two limiting cases.", "First, in the thin Debye layer limit, by comparing with eq.", "(REF ), one may identify $I_{\\rm eo} = \\frac{k_B T}{2 \\pi \\ell _B \\eta D} I_{\\rm ep}^{\\rm surf} + 2w\\frac{b e^2 \\Sigma ^2}{\\eta } E,$ where $I_{\\rm ep}^{\\rm surf}$ is the surface contribution to the electrophoretic current.", "To quantify the importance of the EO contribution, one may compute the analogue of a Peclet number: $Pe = \\frac{I_{\\rm eo}}{I_{\\rm ep}^{\\rm surf}}= \\frac{3}{2} \\frac{d_i}{\\ell _B}\\left(1+ \\frac{b}{\\ell _{GC}}\\right).$ We have introduced here the ion diameter $d_i$ by using the Einstein relation $D = k_B T/(3\\pi \\eta d_i)$ and we recall that $\\ell _{GC} = (2\\pi \\Sigma \\ell _B)^{-1}$ .", "In water at room temperature, $d_i \\sim \\ell _B \\sim 0.7~\\rm nm$ , therefore, in the absence of slippage, the EO contribution is of the same order as the surface EP contribution, $Pe \\sim 1$ .", "Table in the supplemental appendix lists, for reference, values of the Gouy-Chapman length for typical surface charge values.", "As these are generally in the nanometre range, even in the case of moderate slippage ($b \\sim 10~\\rm nm$ ), there is a strong enhancement of the EO contribution.", "The threshold confinement below which the resulting surface contribution dominates over the bulk contribution is given by a rescaled Dukhin length: $\\ell _{Du}^* = (b/\\ell _{GC}) \\ell _{Du}$ .", "Second, in the Debye overlap regime, the EO current is readily determined from eq.", "(REF ), since the ion densities may then be considered uniform across the channel and are given by eq.", "(REF ).", "Uniform ionic densities also imply that the flow has no longer a surface, but rather a volume driving.", "It is simply a Poiseuille flow, with the pressure gradient $\\Delta P/L$ replaced by the electric driving force $e(\\rho _+-\\rho _-)E$ .", "Altogether one obtains $I_{\\rm eo} \\simeq \\frac{w h}{3 \\eta } e^2 \\Sigma ^2 \\left(1+\\frac{6b}{h}\\right) E.$ Since in Debye overlap it makes no more sense to distinguish a surface and a bulk contribution, we define the Peclet number as the ratio of the EO current to the total EP current: $Pe = \\frac{I_{\\rm eo}}{I_{\\rm ep}} = \\frac{d_i}{\\ell _B} \\frac{h}{4 \\ell _{GC}} \\frac{1+6b/h}{\\sqrt{1+(h/\\ell _{Du})^2}}.$ Figure 5d illustrates the dependence of this Peclet number on $h$ and $\\ell _{GC}$ , with $\\rho _s = 10^{-2}~\\rm M$ so that there is Debye overlap.", "In the absence of slippage, the EO contribution dominates only for high surface charges.", "However, if a small slip length $b = 5~\\rm nm$ is introduced, the Peclet number exceeds 1 for all reasonable surface charges.", "Indeed, the Peclet number is then essentially determined by the ratio $b/\\ell _{GC}$ .", "This highlights that in the Debye overlap regime, ionic conduction should be mostly driven by electro-osmosis.", "If strong EO flows are expected, then the reciprocal effect, streaming current, is expected to be large as well.", "The streaming current results from the application of a pressure gradient $\\Delta P/L$ : in Debye overlap, the charge density $2\\Sigma /h$ is simply dragged along by the Poiseuille flow.", "The current reads $I_{\\rm str} \\simeq hw \\frac{eh\\Sigma }{6\\eta } \\left(1+\\frac{6b}{h}\\right) \\frac{\\Delta P}{L} \\equiv hw \\frac{-\\epsilon \\zeta }{\\eta } \\frac{\\Delta P}{L},$ where we have phenomenologically defined the zeta potential ($\\zeta $ ) via the streaming mobility.", "Independent measurements of the surface charge through the voltage-driven current and of the zeta potential from the pressure-driven current may allow in principle to estimate the slip length $b$ .", "Such an estimate does not replace a direct measurement, however, in particular because the slip length may directly depend on surface charge ([149]); and additional charge-surface coupling effects may also occur, as we discuss below." ], [ "Surface modifications to PNP theory", "A remark is in order at this point concerning the nature of surface charge, that we have so far assumed to be constant, whatever the conditions.", "Usually, surface charge is considered to result from the acid base reactivity of the surface when dipped into water, of the type $\\rm [AH]_s \\rightarrow [A^-]_s + H^+$ , where the negative group $\\rm [A^-]_s$ remains fixed on the surface, while the proton diffuses in solution.", "In the case of the air-water interface, surface charge may result from the adsorption of hydronium ions ([95]), or charged impurities.", "The adsorption of surfactant impurities was also proposed as a charging mechanism for general hydrophobic surfaces ([141]).", "Graphite and hBN, which are of particular relevance in nanofluidics, have a priori no obvious acid-base reactivity in water, but they may develop a surface charge, through chemisorption or physisorption of hydroxyde ions, as inferred from experiments ([126], [128]), and recently confirmed by ab initio simulations ([47]).", "Such a picture of surface charge implies that it may actually depend on electrolyte concentration, which is an example of charge regulation.", "Qualitatively, the salt concentration affects the surface potential, and therefore the concentration of $\\rm H^+$ and $\\rm OH^-$ ions at the surface, which in turn affects the chemical (or the adsorption) equilibrium which governs the surface charge.", "Such charge regulation has been invoked to explain several experiments on carbon nanotubes, where a scaling $I \\propto \\rho _s^{1/2}$ or $I \\propto \\rho _s^{1/3}$ was observed ([126], [86], [110]).", "Various models have been developed ([126], [8], [96], [142]), predicting a rich panorama of different sublinear scalings of conductance with salt concentration, depending on the conditions.", "A further step that may be required to accurately describe the surface charge is to take into account its mobility.", "Surfaces charges may be mobile when embedded in lipid bilayers, or with more relevance to nanofluidics, when resulting from adsorbed ions.", "For instance, the simulations of Grosjean et al.", "showed that physisorbed hydroxyde ions on graphene surfaces retain a high lateral mobility ([47]).", "The effect of a mobile surface charge on ion and fluid transport coefficients (in the thin Debye layer regime) was the subject of several investigations ([91], [127], [102]).", "Using the framework of [102], one may introduce two new friction coefficients: $\\lambda _-$ between the adsorbed (negative) ions and the wall, and $\\xi _-$ between the adsorbed ions and the fluid, in addition to the water-wall friction coefficient $\\lambda $ .", "The force balance on the interfacial fluid layer results in a modified partial slip boundary condition involving the tangential electric field at the surface: $b_{\\rm eff} \\partial _z v\\vert _{z=0} = v\\vert _{z=0} -e \\frac{\\xi _-}{\\xi _-+\\lambda _-} \\frac{b_{\\rm eff}}{\\eta } \\Sigma (- \\partial _x \\phi )\\vert _{z=0},$ where the effective slip length is $b_{\\rm eff} = \\frac{b}{1+\\frac{\\lambda _- \\xi _-}{\\lambda (\\lambda _-+\\xi _-)}\\Sigma }.$ The slip length is therefore reduced by the surface charge mobility, and the boundary condition involves an extra electric term.", "This has in general a moderating effect on the transport coefficients.", "For example, in the case of streaming current, the fluid brings along surfaces charges in addition to the counterions, so that the total current is reduced.", "Notable exceptions are diffusio-osmotic mobility and conductance: the overall ionic conductance increases when surface charges are able to move in response to the electric field.", "We have just introduced ion-wall and ion-water friction for adsorbed ions in the interfacial layer.", "Now, for the smallest accessible channels, which are comparable to the ion size ([115], [77]), such friction needs to be taken into account for all the ions, and should appear at the level of the transport equations.", "This was proposed, for instance, to rationalise the non-linear voltage-pressure couplings observed in angstrom-scale slits by [103].", "They observed that the streaming mobility $\\mu $ , defined by $I_{\\rm str} = hw \\mu \\Delta P/L$ , depends on the applied voltage in a qualitatively different way for graphite and BN slits.", "In the proposed 'surface PNP' model, the ion-wall ($\\lambda _{\\pm }$ ) and ion-water ($\\xi _{\\pm }$ ) friction coefficients are introduced in the same way as above, except they no longer apply to adsorbed 'surface charge' ions, but to all the regular salt ions.", "The equivalent of the Smoluchowski and Stokes equations need then to be rederived by considering the force balance on a single ion and on an element of fluid, respectively, as shown in the Appendix.", "There is a rich phenomenology to be harnessed from the new couplings that appear when all components of the nanofluidic system interact with the surfaces.", "Nevertheless, such surface PNP theory is still in its infancy, and it requires input from more microscopic modelling in order to estimate the various friction coefficients it introduces.", "It even raises mathematical difficulties in the writing of a one-dimensional Poisson equation, whose Green's function has infinite range.", "Finally, the relevance of this phenomenological theory should be assessed in light of the correlation and structuring effects – specific to the smallest scales – that will be described in the next section." ], [ "BEYOND THE CONTINUUM DESCRIPTION", "In this section, we focus on the specific nanoscale effects that are not described in the framework of continuum hydro- and electro-dynamics.", "These effects mainly originate from thermal fluctuations, interparticle correlations and structuring effects, and, as such, they require theoretical tools that bridge the gap between physics of continuous media and statistical mechanics." ], [ "Fluctuating hydrodynamics", "The relative particle number fluctuations in an open system of $N$ particles scale as $1/\\sqrt{N}$ .", "In a nanopore of radius $1~\\rm nm$ and length $10~\\rm nm$ , there are on average about 1000 water molecules and these fluctuations are of a few percent and already not negligible.", "Fundamentally, the description of such fluctuations requires a statistical mechanics framework, such as density functional theory ([4]).", "However, these approaches remain quite formal and difficult to implement, particularly for finite length pores which are non translationally-invariant systems.", "A simpler coarse-grained approach to fluctuations – the so-called fluctuating hydrodynamics (FH) – has been introduced by [74].", "In fluctuating hydrodynamics, a random stress tensor $S$ is added to the Navier-Stokes equation: $\\rho \\partial _t \\mathbf {v} + \\rho (\\mathbf {v} \\cdot \\nabla ) \\mathbf {v} = - \\nabla p + \\eta \\Delta \\mathbf {v} + \\nabla \\cdot S,$ where $S$ satisfies a fluctuation-dissipation theorem: $\\langle S_{ij} (\\mathbf {r},t) S_{kl}(\\mathbf {r}^{\\prime },t^{\\prime }) \\rangle = 2 \\eta k_B T (\\delta _{ij} \\delta _{kl} + \\delta _{il} \\delta _{jk})\\delta (\\mathbf {r}-\\mathbf {r}^{\\prime })\\delta (t-t^{\\prime })$ , similarly to the fluctuating force in a Langevin equation; $\\rho $ is the mass density of the fluid.", "This description introduces thermal fluctuations in a setting where, otherwise, a continuum hydrodynamic description holds.", "FH has been extensively discussed in the literature ([49], [38], [98]) and we only consider here its basic implications for fluid transport in a nanopore.", "In standard hydrodynamics, the fluid inside a nanopore is at rest when subject to no external force.", "The effect of fluctuations is to induce a stochastic flow through the nanopore, which can be pictured as a stochastic motion of the fluid centre of mass (CM).", "[26] solved the FH equations for the velocity correlation function, and found that the CM motion was described by a non-Markovian Langevin equation of the type $m \\frac{\\mathrm {d}v}{\\mathrm {d}t} = - \\int _{-\\infty }^t \\mathrm {d}t^{\\prime }\\xi (t-t^{\\prime }) v(t^{\\prime }) + F(t),$ where the random force $F(t)$ and the memory kernel $\\xi $ are related by $\\langle F(t) F(0) \\rangle = 2 k_B T \\xi (t)$ .", "The memory kernel is indeed non-trivial (i.e.", "not reduced to a $\\delta $ function), as confirmed by molecular dynamics (MD) simulations ([27]).", "Memory effects in the diffusion of fluid inside a nanopore are indeed expected from the analysis of relevant timescales.", "A nanopore of radius $R$ and length $L$ contains a mass $m = \\rho \\pi R^2 L$ of fluid, and its friction coefficient on the wall can be calculated from the Navier-Stokes equation as $\\xi _s \\approx 8 \\pi \\eta L$ .", "The velocity of the fluid CM then relaxes on a timescale $\\tau \\sim m/\\xi _s \\sim R^2/\\nu $ .", "But $R^2/\\nu $ is the time required for fluid momentum to diffuse across the nanopore, that is the time for the wall friction force to establish itself.", "Therefore, the friction force may not adapt instantaneously to the CM velocity, and memory effects are to be expected.", "Properties that are determined by short timescale dynamics may be affected by those memory effects as it was shown to be the case for solute mobility in confinement ([24]).", "More generally, memory effects play an important role in barrier-crossing processes ([63]).", "But other properties only depend on the long timescale dynamics.", "In particular, the fluid centre of mass diffusion coefficient, given by $\\mathcal {D} = k_B T/(2 \\int _0^{\\infty } \\xi (t) \\mathrm {d}t)$ , reduces to the Einstein expression $\\mathcal {D} = k_B T/ \\xi _s$ , that would be expected if the diffusion was Markovian.", "Similarly, the hydrodynamic slip length $b$ is found to satisfy a Green-Kubo relation whatever the form of the memory kernel $\\xi (t)$ ([11]).", "From a practical perspective, one may evaluate the diffusion coefficient $\\cal D$ .", "From the Einstein relation, and the assumption of Poiseuille flow with slip length $b$ , one obtains $\\mathcal {D} = \\frac{k_B T}{\\xi } = \\frac{k_B T}{\\frac{8 \\pi \\eta L}{1+4b/R}+3\\pi ^2 \\eta R},$ where, in the most general case, the friction coefficient $\\xi $ has two contributions, one from the channel interior and one from the entrance effects ([26]).", "Overall, the fluid CM diffusion $\\mathcal {D}$ can be seen as a supplementary contribution to the diffusion coefficient of a particle inside the nanopore.", "If the size of the particle is comparable to that of the nanopore, its self-diffusion is strongly hindered ([117]) and the fluid contribution may actually dominate the particle diffusion ([26])." ], [ "Noise", "Thermal fluctuations are at the origin of the noise in ionic current measurements through a nanopore.", "At zero applied voltage, the total noise amplitude satisfies the Nyquist relation $\\langle I^2 \\rangle = 2k_B T G$ , where $G$ is the nanopore conductance.", "The electro-osmotic term in the conductance makes a supplementary contribution to the noise, which can actually be traced back to the hydrodynamic fluctuations of the fluid CM described in the previous paragraph ([26]).", "When resolved in frequency, numerous experiments on artificial nanochannels ([130], [129], [53], [134], [126]) and biological pores ([146], [7]), have shown that the current spectrum exhibits 'pink noise', that is noise that scales with frequency $f$ as $f^{-\\alpha }$ , with $\\alpha $ in the range $0.5 - 1.5$ .", "Such a spectrum is traditionally described by the empirical Hooge's law ([52]): $\\langle \\delta I^2 \\rangle (f) = A \\frac{\\langle I^2 \\rangle }{f^{\\alpha }},$ with $A$ a coefficient inversely proportional to the number of charge carriers.", "Notably, in experiments, the $f^{-\\alpha }$ scaling is observed down to frequencies below 1 Hz, showing that some correlations in the system under study exist even at such low frequencies.", "Correlations between ions have been put forward as a possible origin for the pink noise ([155]).", "On the other hand, [45] demonstrated in MD simulations that pink noise persisted even with non-interacting ions, if reversible ion adsorption was allowed on the channel wall.", "For simple diffusive dynamics, the longest correlation time that may be expected for adsorption-desorption processes is $R^2/D$ , with $R$ the pore radius and $D$ the ion diffusion coefficient.", "For a 10 nm pore, this would correspond to a frequency cutoff of 100 MHz, below which the noise spectrum should be flat.", "Such a high cutoff frequency is in clear contradiction with experiment.", "However, Gravelle et al.", "showed that in the presence of reservoirs, excursions of ions outside the pore coupled to adsorption-desorption processes on the pore surface result in ionic correlations on much longer time scales, hence predicting pink noise down to very low frequencies." ], [ "Wall fluctuations", "So far, we have only considered thermal fluctuations of the fluid itself.", "However, the confining walls might also be subject to fluctuations.", "Wall fluctuations are known to enhance fluid self-diffusion by inducing flows, through the Taylor-Aris mechanism ([135], [3]).", "On the other hand, static constrictions slow down diffusion in a channel through Fick-Jacobs entropic trapping ([94], [116]).", "Recently, MD simulations have shown diffusion enhancement due to the propagation of phonons in carbon nanotubes ([89], [22]).", "On the theoretical side, a very general framework for evaluating the self-diffusion coefficient for a fluid confined in a fluctuating channel was developed by [97].", "In a slit like-channel of height $h$ , the fluctuation-corrected diffusion coefficient $D$ is formally related to the spectrum of wall fluctuation $S(k,\\omega )$ by $D = D_0 \\left( 1- \\frac{1}{h^2} \\int \\frac{\\mathrm {d}k \\mathrm {d}\\omega }{(2\\pi )^2} \\frac{(D_0k^2)^2-3 \\omega ^2}{(D_0k^2)^2+\\omega ^2} S(k,\\omega )\\right),$ where $D_0$ is the bare diffusion coefficient.", "Equation (REF ) has two limiting regimes, that are governed by a 'fluctuation-related' Peclet number, which may be defined as $Pe = \\ell ^2/(D_0 \\tau )$ , where $\\ell $ and $\\tau $ are the characteristic length and time scales of the fluctuations.", "The diffusion coefficient is enhanced for $Pe >1$ ($D = D_0(1+3\\langle \\delta h^2\\rangle /h^2)$ ), and reduced for $Pe <1$ ($D = D_0(1-\\langle \\delta h^2\\rangle /h^2)$ ) and both tendencies are observed experimentally ([97]).", "Equation (REF ) thus bridges the limiting Taylor-Aris and Fick-Jacobs results.", "The permeability of a fluctuating channel is also expected to be impacted by wall fluctuations.", "However, permeability is related to the diffusion of the fluid centre of mass (see eq.", "(REF )), which is different from the fluid self-diffusion; and extending the above framework to centre of mass diffusion remains to be done." ], [ "Phenomenology", "While a fluid appears disordered at hydrodynamic lengthscales, its molecular nature manifests itself under confinement in the form of structuring effects.", "Even at a non-confined solid-liquid interface, the attractive interactions between the solid and the liquid result in molecular layering, as verified experimentally by x-ray spectroscopy ([17]).", "In between two surfaces, a liquid adopts a layered structure even in the absence of interactions, simply due to geometrical constraints ([59]).", "The onset of structuring clearly represents the transition from continuum to sub-continuum transport.", "The threshold confinement at which this transition occurs has been studied for both planar and cylindrical geometries using MD simulations.", "For water in carbon nanotubes, [137] have found that it retains a bulk-like disordered structure down to a tube diameter of 1.39 nm ((10,10) tube).", "In a (9,9) tube (1.25 nm) water was found to structure in stacked hexagons (see figure 7), and in a single file chain in a 0.83 nm (6,6) tube.", "For water between two graphene sheets, four distinct layers could be observed at a 1.35 nm distance between sheets ([106]), and a single monolayer of water was observed for confinement below 0.8 nm ([105]).", "In should further be noted that the observation of an exotic square ice phase for few-layer water between two graphene sheets has been claimed ([1]).", "In the sub continuum-regime, transport properties show strong qualitative deviations from bulk expectations.", "The MD simulations of [137] showed non-monotonous permeability for carbon nanotubes as a function of diameter, and similar observations were made for the capillary filling velocity of carbon nanotubes, studied by [46].", "On the experimental side, this behaviour can be put in perspective with the recent results by [115] for capillary-driven flow through slit-like graphene channels where a peak in flow rate at around 1.3 nm confinement was observed.", "Figure: MD simulation snapshots of water structure inside carbon nanotubes of different radii, adapted from ." ], [ "Disjoining pressure", "Thermodynamically, the effect of molecular fluid structuring may be described by a supplementary contribution to the pressure, the so-called disjoining pressure.", "For a fluid confined between two walls of area $\\mathcal {A}$ separated by a distance $h$ , in equilibrium with a reservoir at pressure $P_0$ , it is defined in full generality as $\\Pi _d (h) = - \\frac{1}{\\mathcal {A}} \\left( \\frac{\\partial G}{\\partial h} \\right),$ where $G$ is the Gibbs free energy of the whole system ([58], [4]).", "In principle, $G$ varies with $h$ because of distance-dependent interactions between the surfaces; the disjoining pressure can then be seen as the force the fluid needs to exert in order to oppose those interactions.", "These are generally of two types.", "On the one hand, there are van der Waals interactions that scale algebraically as $h^{-2}$ , yielding a contribution $\\Pi _{vdW}=-{A_H\\over 6\\pi h^{3}}$ to the disjoining pressure, with $A_H$ the Hamaker constant (although more complex dependencies may occur in specific geometries).", "On the other hand, electrostatic interactions lead to contributions that scale exponentially with the distance $h$ due to ionic screening.", "The combination of these two interaction terms is the basis for the DLVO (Derjaguin-Landau-Verwey-Overbeek) theory.", "Now, if there is fluid structuring that depends on confinement, it makes a supplementary contribution to the variation of $G$ .", "This contribution exists even for a hard sphere fluid due to excluded volume effects, and is purely entropic in nature.", "It quantifies essentially how much the structure of the fluid in confinement differs from the one in the bulk.", "For hard spheres of diameter $\\sigma $ , the disjoining pressure is usually described by the expression ([58], [73]): $\\Pi _d (h) = - \\rho _{\\infty } k_B T \\cos (2\\pi h /\\sigma ) e^{-h/\\sigma },$ where $\\rho _{\\infty }$ is the bulk density.", "This is an oscillating function of $h$ that decays to 0 at large $h$ , where there is no more fluid structuring.", "It is bounded by $\\Pi _d(0) = \\rho _{\\infty } k_B T$ , which evaluates to about 1000 bar for water at room temperature.", "In practice, for fluid flow measurements where a pressure drop is imposed between two reservoirs, this ( huge) contribution to the pressure does not intervene directly.", "However, in a capillary flow geometry, which involves liquid-vapour interfaces, the disjoining pressure may be much larger than the bare capillary pressure and act as the main driving force.", "Indeed, the capillary pressure drop across a meniscus of radius $R = 1~\\rm nm$ is typically $2\\gamma /R = 140~\\rm bar$ (for water with surface tension $\\gamma = 72~\\rm mN \\cdot m^{-1}$ ).", "Driving by disjoining pressure was observed in simulations of carbon nanotube capillary filling ([46]), where the subtle dependence of the water molecular structure – and therefore of the disjoining pressure – on nanotube radius resulted in the imbibition velocity displaying oscillatory behaviour versus the confinement.", "Experimentally, a similar phenomenology was observed by [115], who measured the capillary flow of water across angstrom scale slits and found a non-monotonous dependence of the evaporation rate on confinement.", "These measurements could be interpreted in terms of disjoining pressure, which was indeed found to be of the order of 1000 bar in MD simulations of the experimental system ([105])." ], [ "Descriptions of fluid transport", "A first understanding of sub-continuum transport may be provided by extending the continuum description with effective, confinement-dependent, values for density, viscosity and slip length.", "In a layered fluid, the density in each layer is generally higher than in the bulk, and a general trend is that viscosity increases ([122]).", "More exotically, an oscillating viscosity as a function of confinement was observed in simulations of water between two graphite slabs ([106]).", "[105] could reproduce the experimental results of [115] by using simulated values of disjoining pressure, along with a Poiseuille formula with effective density and viscosity; however, they assumed a viscosity-independent slip length.", "Now, in general, care should be taken in defining a viscosity for a structured fluid: as its density is non-uniform, a position-dependent viscosity should be introduced in order to describe the details of the flow profile.", "For confinement below five molecular diameters (in the case of a Lennard-Jones fluid) [138] have shown that even a position-dependent viscosity is not sufficient and a non-local viscosity kernel should be used ([153]).", "Ultimately, for confinement below a few molecular layers (typically 1 nm for water), the notion of viscosity itself – which is intrinsically a continuum quantity – looses its fundamental meaning.", "Nevertheless, a simple picture of sub-continuum fluid transport is possible in the case of large slippage, since surface friction then becomes the main mechanism resisting fluid transport.", "This appears indeed when looking at the limit $b\\gg R$ in the Poiseuille formula (REF ), which yields for the average flow velocity: $v = \\frac{R b}{2 \\eta L} \\Delta P = \\frac{ R}{2 \\lambda L} \\Delta P,$ where we have obtained the second equality by relating the slip length to the liquid-wall friction coefficient $\\lambda $ : $b = \\eta /\\lambda $ ; note that, for simplicity, we forget here about entrance.", "The viscosity does not enter the permeability, as expected for surface dominated friction.", "The fluid moves indeed in the channel as a single block, and dissipation only intervenes at the surface.", "The crucial parameter governing the transport is then the solid-liquid friction coefficient $\\lambda $ .", "A model for evaluating $\\lambda $ , tested against MD simulations of water in carbon nanotubes, was proposed by [31].", "They obtain $\\lambda \\approx \\frac{\\tau }{k_B T} f_{q_0}^2 S(q_0),$ where $q_0$ is the wave-vector corresponding to the solid lattice spacing, $f_{q_0}$ is the Fourier component of the fluid-solid interaction potential at that wave-vector, and $S(q)$ is the liquid structure factor; $\\tau $ is the correlation time of the force between the solid and the liquid, which is found to not depend on confinement.", "In this way, $\\lambda $ is found to be directly related to the liquid structure and to the liquid wall interaction.", "Such a friction-dominated approach was applied successfully, for example, to predict the permeability of ultra-confined alcanes in a nanoporous matrix, showing a scaling dependence of the permeance on the alcane length ([30]).", "However, predicting the solid-liquid friction coefficient remains to large extents an open problem.", "As we discuss in the following (see 4.2.6), its determination in MD simulations still remains ambiguous quantitatively, and the role of underlying quantum effects should be assessed." ], [ "Ultimate structuring: single file transport", "The most extreme type of structuring is single-file arrangement of fluid molecules.", "In the case of water transport, this is the realm of biological channels ([55]), where water is conducted through sub-nanometric openings in transmembrane proteins.", "Single file transport may also occur in artificial channels: it was observed in MD simulations for carbon ([57]) and boron nitride ([147]) nanotubes of 0.8 nm in diameter, and experimental observation in carbon nanotubes was recently claimed ([139]).", "Single-file transport is a field of its own and there are dedicated reviews on the subject ([55], [71]); here we will only sketch the main ideas.", "The notion of single file transport and the underlying exclusion transport models are not restricted to the study of confined fluids.", "Indeed, a general feature of particles in one dimension that cannot cross each other is sub-diffusive behaviour: the mean squared displacement of a particle scales with time as $\\langle \\Delta x^2 \\rangle \\propto t^{1/2}$ , as opposed to linear scaling for normal diffusion ([83]).", "Such anomalous diffusion is encountered in a variety of fields ([37]), and may be understood in terms of the normal diffusion of vacancies in the 1D chain; in the case of fluid transport, it was studied by Chou ([21], [20]).", "However, this effect is not expected to play a decisive role in single-file transport through short channels, where vacancies are unlikely to occur ([62]).", "In the case where no vacancies are expected, a simple model for single-file transport was proposed decades ago by [36] for describing water transport in biological channels.", "It stems from a global force balance on the water chain, and is as such equivalent to eq.", "(REF ), which we have derived as a strongly confined limit of the Poiseuille formula.", "In Finkelstein's model, the chain of $N$ water molecules (of total length $L$ ) moving at velocity $v$ is subject to a friction force $- N \\xi v$ , and to a pressure driving $N v_w \\Delta P /L$ , where $v_w$ is the volume of one water molecule.", "The force balance then leads to $v = \\frac{v_w}{\\xi L} \\Delta P,$ which is indeed eq.", "(REF ), granted the identification $\\xi = (2 v_w/R) \\lambda $ .", "In the biophysics literature, the preferred quantity for characterising a channel's transport properties is the unitary channel permeability $p_f$ : $p_f$ is the number of water molecules that crosses the channel per unit time, per unit osmolyte concentration difference applied across the channel, therefore expressed in $\\rm m^3\\cdot s^{-1}$ .", "For an osmolyte concentration difference $\\Delta \\rho _s$ , the osmotic pressure difference is $k_B T \\Delta \\rho _s$ , and hence the unitary permeability is $p_f = \\frac{N}{L} \\frac{v}{\\Delta \\rho _s} = \\frac{v_w}{a L} \\frac{k_B T}{\\xi },$ where we have defined $a = L/N$ the average spacing between water molecules.", "The diffusion coefficient of the water chain centre of mass may be expressed through the Einstein relation as $D = k_B T/(N\\xi )$ , and therefore $p_f = v_w D/a^2$ , hence the diffusion coefficient and the unitary permeability may be used interchangeably.", "A typical value for the permeability in biological channels such, as Aquaporin or Gramicidin A, is in the range of $10^{-14}-10^{-13}~\\rm cm^3 \\cdot s^{-1}$ , which corresponds to 1-10 water molecules crossing the channel per nanosecond under a 1 bar pressure drop.", "In an alternative phenomenological description, transport through the channel is viewed as an activated process with activation energy $\\Delta G^{\\ddagger }$ ([55]).", "The permeability is then expressed in the framework of transition state theory as $p_f=v_w\\,\\nu _0\\, \\exp [-\\Delta G^\\ddagger /k_BT]$ , with $\\nu _0 \\sim 10^{13}$ s$^{-1}$ a molecular attempt frequency.", "For biological channels, this relation is well verified by independent measurements of $p_f$ and $\\Delta G^\\ddagger $ , with $\\Delta G^\\ddagger $ of the order of $ 5~\\rm kcal \\cdot mol^{-1}$ .", "However, values of $\\Delta G^\\ddagger $ remains debated for transport measurements in carbon nanotubes ([54], [140]).", "Although it provides a general guiding line, Finkelstein's formula is challenged both in artificial and biological channels.", "In the case of protein channels, the assumption of uniformly smooth walls breaks down, since there are discrete hydrogen bonding sites ([55]).", "Hence $p_f$ is not found to be inversely proportional to channel length ([121]): it rather has an exponential dependence on the number of hydrogen bonding sites ([56]).", "Such an exponential dependence suggests a collective transport mechanism of the water chain, with bursts requiring the breaking of multiple hydrogen bonds at once.", "There is, however, a notable disagreement between experiments and simulations, as the simulations of water transport through polyalanine channels showed no dependence of $p_f$ on the channel length ([113]).", "In simulations of carbon nanotube channels, all the single-file water molecules were clearly shown to move in a correlated fashion ([57]): water transport occurs when all the molecules simultaneously shift by one molecular diameter.", "These dynamics were successfully described by a continuous time random walk model ([6]), or equivalently by diffusion of a collective coordinate of the water molecules ([154]), which actually echoes our discussion of fluctuations in a more general setting (eq.", "REF and [26]).", "Overall, there is still much to understand about the collective motions and subtle surface interactions involved in single-file transport, and it is an active field of research." ], [ "Structuring and electrostatics: dielectric anomalies", "As confined water becomes structured, its response properties to an external electric field are accordingly modified.", "When an electric field $\\bf E$ is applied in water, the individual molecules reorient and polarise, creating an electric field themselves.", "The total electric field $\\bf E$ is then the sum of the polarization field of the water molecules and of the externally applied field $\\mathbf {D}/\\epsilon _0$ , where $\\bf D$ is called the electric displacement and $\\epsilon _0$ is the vacuum permittivity.", "In bulk water, the dielectric response, that is the relation between $\\bf D$ and $\\bf E$ , may be expressed through a single scalar quantity, the relative permittivity $\\epsilon \\approx 80$ : $\\mathbf {D} = \\epsilon \\epsilon _0 \\mathbf {E}$ .", "However, the most general (static) linear response may be anisotropic, space-dependent and non-local: $D_{\\alpha }(\\mathbf {r}) = \\epsilon _0 \\sum _{\\beta } \\int \\mathrm {d}\\mathbf {r}^{\\prime } \\epsilon _{\\alpha \\beta } (\\mathbf {r},\\mathbf {r^{\\prime }}) E_{\\beta } (\\mathbf {r^{\\prime }}).$ The relative permittivity is then a tensor with components $\\epsilon _{\\alpha \\beta } (\\mathbf {r}, \\mathbf {r}^{\\prime })$ .", "While MD simulations show that the dielectric response in water may be considered local ($\\epsilon _{\\alpha \\beta } (\\mathbf {r}, \\mathbf {r}^{\\prime }) = \\epsilon _{\\alpha \\beta } (\\mathbf {r}) \\delta (\\mathbf {r}-\\mathbf {r^{\\prime }})$ ), it becomes anisotropic and space-dependent in the vicinity of interfaces, as a consequence of the water layering ([14],[13]).", "Qualitatively, the orientations of the water dipoles are anti-correlated in the direction perpendicular to the interface, resulting in a reduced permittivity in that direction, while the permittivity is largely unaffected parallel to the interface.", "In planar confinement, this behaviour could be captured by an effective medium model, in which the water is described by a space-independent, but anisotropic permittivity $(\\epsilon _{\\parallel },\\epsilon _{\\perp })$ (the parallel direction is not confined).", "While $\\epsilon _{\\parallel }$ essentially retains its bulk value, $\\epsilon _{\\perp }$ is reduced by up to an order of magnitude for confinements below 1 nm ([123], [151]).", "Such a reduction of the perpendicular dielectric response was recently observed experimentally for water confined between a graphite and a boron nitride crystal ([39]).", "A deviation from the bulk value was measured up to nearly 100 nm confinement (figure 7).", "The results were well described by assuming each interface carried a 7 Å thick layer of very low permittivity ($\\epsilon = 2.1$ ) \"electrically dead water\".", "An effective medium model based on MD simulations was developed by [88] for cylindrical confinement of water in carbon nanotubes.", "Similarly to the case planar case, the radial permittivity is found to be reduced by up to an order of magnitude for tube radii smaller than 3 nm (figure 7).", "However, for the smallest tubes (below 1 nm radius), the longitudinal permittivity is found to increase with respect to its bulk value, and it skyrockets 1 to 3 orders of magnitude in a 0.4 nm radius tube, where water is in a single file arrangement.", "Knowledge of these modified dielectric properties is important when considering ion transport in strong confinement, as will be discussed in the following (see 4.3.1)." ], [ "Limits of molecular dynamics", "We will close this section on liquid structuring with a word of warning.", "Many of the results we have presented so far have been obtained using classical MD simulations, and like any type of simulation, these come with some underlying assumptions.", "In particular, as we are dealing with strongly confined fluids where all the molecules interact with the confining surface, the modelling of that surface plays an increasingly important role.", "Since the solid is represented by rigid balls interacting with classical force fields, no electron dynamics may be described by the simulation.", "However, some coupling between water and electron dynamics may be expected, based on a lengthscale argument.", "The maximum of the dielectric response of a solid indeed occurs at a lengthscale $k_{\\rm TF}^{-1}$ , where $k_{\\rm TF}$ is the Thomas-Fermi wave-vector ([92]).", "Typically, $k_{TF}^{-1} \\approx 1~\\rm nm$ in graphite, hence the electrons in graphite could expected to respond to the dynamics of individual water molecules.", "There is growing evidence that such electron-water couplings indeed occur.", "A first consequence of taking into account electron dynamics is that a solid should appear polarisable: in simulations, the fluctuations of water next to a graphite surface were found to be strongly affected by the polarisability of carbon atoms ([100]).", "On the experimental side, the induction of an electronic current by a water flow inside a carbon nanotube, as well as the reverse phenomenon, were observed ([42], [114]).", "These results call for caution in the interpretation of MD simulations, which, as we have pointed out, lack electron dynamics.", "They also call for the development of new simulation methods that would take such dynamics into account, while not limited by the very small system size of ab initio simulations." ], [ "Electrostatics in extreme confinement", "In this section, we discuss a few aspects of the behaviour of ions below the continuum limit.", "We will describe how the confinement modifies ionic interactions, and possible consequences on ion transport." ], [ "Ionic interactions and self-energy", "In bulk water, ions interact via a Coulomb potential $\\phi (r) = e/(4\\pi \\epsilon _0 \\epsilon _w r)$ , where $\\epsilon _w$ is the water dielectric permittivity.", "But inside a nanochannel, ions are no longer surrounded by a homogeneous fluid, and their interaction potential may be affected by the dielectric properties of the confining medium.", "In order to assess the importance of this effect, we consider the simple situation where an infinitely long cylindrical channel of radius $R$ is filled with water having isotropic permittivity $\\epsilon _w$ , and the medium outside the channel is a homogeneous dielectric of permittivity $\\epsilon _s$ , with typically $\\epsilon _s \\ll \\epsilon _w$ .", "The electrostatic potential around an ion placed in the middle of the channel can then be determined by solving Poisson's equation in the presence of the dielectric discontinuities, which has been done analytically by several authors ([111], [81], [136], [65]).", "The complicated analytical result may be interpreted in the following way (see figure 8).", "At short distances (much smaller than the channel radius), only the dielectric response of the water is visible and the potential is $\\phi (x) \\sim 1/(\\epsilon _w x)$ .", "At long distances (much larger than the channel radius), it is the dielectric response of the confining medium that matters and $\\phi (x) \\sim 1/(\\epsilon _s x)$ .", "At intermediate distances, the electric field lines are essentially parallel to the channel due to the dielectric contrast $\\epsilon _w \\gg \\epsilon _s$ , and the potential, which is well described by an exponential function, resembles a 1D Coulomb potential: $\\phi (x) \\approx \\frac{e \\alpha }{2 \\pi \\epsilon _0 \\epsilon _w R} e^{-x/(\\alpha R)},$ with $\\alpha $ a numerical coefficient that depends on the ratio $\\epsilon _w/\\epsilon _s$ ; $\\alpha = 6.3$ for $\\epsilon _w/\\epsilon _s = 40$ ([136]).", "It is notable that so far no assumption on the channel radius was made, so that formally this 1D regime exists for a channel of any size.", "However, it is only relevant if it leads to ion-ion interactions stronger than $k_BT$ , that is if $e \\phi (R) > k_B T$ , for a monovalent ion.", "This defines a limiting channel radius, below which ionic interactions are affected by the confining medium: $R_c \\sim 7.5~\\rm nm$ .", "These modified Coulomb interactions therefore may have some effect in any single-digit nanopore, and they have essentially two practical consequences.", "Figure: a. Schematic representation of the electric field lines around an ion confined in a narrow channel.", "b.", "Potential generated along the axis by an ion confined in a channel of radius 2 nm.", "c. Born self-energy of a monovalent ion as a function of channel radius, computed following .First, because ions produce a stronger Coulomb potential in confinement than in the bulk, they have a larger Born self-energy.", "This phenomenon was pointed out decades ago by [111] in the case of ions crossing lipid bilayers, and has since then been commonly considered in the study of biological ion channels.", "For small channels where the 1D contribution to the potential dominates, the additional self-energy due to confinement is ${\\cal E}_s = e\\phi (0)/2$ , with $\\phi (x)$ given by eq.", "(REF ).", "An expression valid for any $R$ was obtained by Teber, and it is plotted in figure 8.", "It shows that ${\\cal E}_s$ is larger than $k_B T$ up to $R \\sim 4~\\rm nm$ .", "${\\cal E}_s$ represents a supplementary energy barrier for entering the channel.", "To illustrate the consequence of this Born self-energy, let us consider a neutral channel in the absence of correlation effects: a barrier ${\\cal E}_s = 4k_BT$ reduces the ionic concentration by a factor $e^{{\\cal E}_s/(k_BT)} = 100$ .", "Second, increased Coulomb interactions with respect to the bulk may actually result in correlation effects.", "To assess the potential importance of correlations, one may introduce a coupling parameter $\\Gamma = e \\phi ((\\pi R^2 \\rho )^{-1})/(k_B T)$ , where $\\rho $ is the ion concentration inside the channel, so that $(\\pi R^2 \\rho )^{-1}$ is the average distance between ions ([12]).", "Let us take as an example a channel of radius 2 nm, where ${\\cal E}_s \\sim 2.5 k_BT$ from eq.", "(REF ).", "An effective Coulomb potential of magnitude $\\sim 5 k_B T$ then extends around an ion over a distance $\\alpha R \\sim 10~\\rm nm$ .", "Therefore, for concentrations above $\\rho = 10^{-2}~\\rm M$ , $\\Gamma > 1$ and correlations are expected to become important.", "Several remarks can be made here.", "A first point is that we have considered monovalent ions so far; for ions of valence $z$ , the self-energy, for example, would be multiplied by a factor $z^2$ .", "Then, we have gone through the discussion assuming for simplicity that water has an isotropic permittivity $\\epsilon _w$ .", "We have highlighted in the previous section that this assumption breaks down in cylindrical channels of radius smaller than 5 nm, and, though it does not affect the qualitative phenomenology, dielectric anisotropy, as predicted by [88], should then be taken into account.", "It mainly affects the behaviour of the potential at short distances $x \\le R$ .", "We should also mention that when entering a channel smaller than its hydrated radius, an ion pays an additional energy penalty due to the shedding of its hydration shell ([118], [29]).", "For chloride, the total hydration energy is as high as $155~k_B T$ , but the hydrated radius is 0.4 nm, so that only partial dehydration arises and only for the smallest pores.", "Finally, we have assumed that the confining medium is a dielectric medium with uniform permittivity.", "This is a good model for a lipid membrane, but artificial confining materials may have a variety of electronic properties (see 4.2.6), and their influence on effective Coulomb interactions has not been thoroughly investigated.", "In the case of confinement by a perfect metal, the ionic interactions were found to be exponentially screened over a distance of order of the channel radius ([72], [88]).", "Such effective interactions were found to lead to like-charge attraction in ionic liquids confined in carbon nanopores ([40])." ], [ "Ion transport beyond the mean-field", "We have highlighted in the previous paragraph that correlations may become important in ion transport through single digit nanopores.", "If it is the case, then mean-field theories, such as the PNP framework, may not be directly applied, and the determination of equilibrium properties, such as ion concentrations inside the nanopore, requires some form of exact statistical mechanics.", "In systems with high aspect ratio (typically nanotubes), the problem can be reduced to a one-dimensional model of the Ising or 1D Coulomb gas type, and the partition function may then be exactly computed through a (functional) transfer matrix formalism ([152], [75], [65]).", "Outside of the 1D geometry, variational methods may be used ([15]), and methods for incorporating ion pairs into Poisson-Boltzmann theory have been developed ([80]).", "A general feature of these calculations ([152], [15]) is that they predict filling transitions: namely, a non-analytic behaviour of ion concentrations in the nanopore as a function of the salt concentration in the reservoir, which strongly deviates from the mean-field Donnan equilibrium result, and may break local electroneutrality ([152]).", "One should note, however, that breakdown of electroneutrality does not require ionic correlations and it has recently been shown to arise in a mean-field setting ([84]).", "If linear response theory applies, then transport properties such as ionic conductance may be determined directly from the concentrations of charge carriers at equilibrium.", "However, this is no longer the case when correlations are very strong, leading to the formation of tightly bound Bjerrum pairs of oppositely charged ions.", "This was first realised by Onsager, who showed that in a three-dimensional weak electrolyte – an electrolyte that forms Bjerrum pairs – there is a quadratic current-voltage relation, a phenomenon known as the second Wien effect ([109], [61]).", "Bjerrum pairing in one-dimensional confinement was studied by [65], and stable ion pairs were shown to arise typically for confinement below 2 nm, similarly to what was observed in MD simulations ([107]).", "The pairing resulted in some very non-linear behaviour governed by discrete particle effects, echoing the phenomenology of Coulomb blockade ([5], [64]).", "As such, ion correlations produced quantised transport behaviour in a purely classical system.", "The coupling of ion transport to fluid transport in the non-mean-field, non linear regime remains largely unexplored, though it promises a rich phenomenology.", "As an example, a recent simulation ([85]) found that pressure driven water flow through a nanochannel could be blocked by an ion tightly bound inside.", "Lastly, we should mention, at the frontier of sub-continuum ion transport, the Grotthus-like translocation of protons, which has been observed in a single file ([139]) and a single plane ([43]) of water, but still remains poorly understood in strongly confined systems." ], [ "CONCLUSION", "This review has explored some defining aspects of fluid transport at the nanoscale.", "We started from continuum theory and reduced the scale down to its limit of applicability; we then explored phenomena that occur below the continuum limit and gave indications of appropriate theoretical descriptions to tackle them.", "The main points summarised below were made.", "Overall, fluids in molecular scale confinement are largely an uncharted territory for theory, and recent experiments urge for the development of theoretical tools beyond those described in this review.", "[SUMMARY POINTS] Experimental systems for studying fluid transport in molecular scale confinement are today within reach.", "Above 10 nm confinement, fluid transport is governed by continuum hydrodynamic equations, with coupling to ion transport and surface effects.", "Below 10 nm –the domain of so-called single-digit nanopores – thermal fluctuations and electrostatic correlations are increasingly important, challenging continuum and mean-field theory.", "In few nanometre confinement, fluid structuring effects and correlations play an overwhelming role." ], [ "DISCLOSURE STATEMENT", "The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review.", "L.B.", "acknowledges funding from the EU H2020 Framework Programme/ERC Advanced Grant agreement number 785911-Shadoks.", "L.B.", "and R.N acknoweldge support from ANR-DFG project Neptune.", "R.N.", "acknowledges funding from the DFG via SFB1078 and NE810/11.", "N.K.", "thanks A. Marcotte for discussions.", "Appendix Table: Electrostatic lengthscales Surface PNP theory Here we reproduce the main equations of the surface PNP-Stokes framework, that was introduced by [103] (see 3.3.5).", "The Nernst-Planck expression for the ion fluxes is modified according to $j_{\\pm } = \\frac{e}{\\xi _{\\pm }+\\lambda _{\\pm }} \\left[ \\frac{k_B T}{e} (-\\partial _x \\rho _{\\pm } \\pm \\rho _{\\pm } (-\\partial _x \\phi ) \\right] + \\alpha _{\\pm }\\rho _{\\pm } v,$ with $\\alpha _{\\pm } = \\frac{\\xi _{\\pm }}{\\xi _{\\pm }+\\lambda _{\\pm }}$ .", "The (integrated) Stokes equation becomes $v = K(\\rho _+,\\rho _-) [ (-\\partial _x p)+ e(\\alpha _+ \\rho _+- \\alpha _-\\rho _-)(-\\partial _x \\phi )],$ with $K(\\rho _+,\\rho _-)$ a concentration-dependent permeability, and these are supplemented by a 1D Poisson equation: $\\partial _x[\\epsilon h (-\\partial _x \\phi )] = h e (\\rho _+-\\rho _-).$ Here we reproduce the main equations of the surface PNP-Stokes framework, that was introduced by [103] (see 3.3.5).", "The Nernst-Planck expression for the ion fluxes is modified according to $j_{\\pm } = \\frac{e}{\\xi _{\\pm }+\\lambda _{\\pm }} \\left[ \\frac{k_B T}{e} (-\\partial _x \\rho _{\\pm } \\pm \\rho _{\\pm } (-\\partial _x \\phi ) \\right] + \\alpha _{\\pm }\\rho _{\\pm } v,$ with $\\alpha _{\\pm } = \\frac{\\xi _{\\pm }}{\\xi _{\\pm }+\\lambda _{\\pm }}$ .", "The (integrated) Stokes equation becomes $v = K(\\rho _+,\\rho _-) [ (-\\partial _x p)+ e(\\alpha _+ \\rho _+- \\alpha _-\\rho _-)(-\\partial _x \\phi )],$ with $K(\\rho _+,\\rho _-)$ a concentration-dependent permeability, and these are supplemented by a 1D Poisson equation: $\\partial _x[\\epsilon h (-\\partial _x \\phi )] = h e (\\rho _+-\\rho _-).$" ] ]	2011.14111
[ [ "Extended full waveform inversion in the time domain by the augmented\n Lagrangian method" ], [ "Abstract Extended full-waveform inversion (FWI) has shown promising results for accurate estimation of subsurface parameters when the initial models are not sufficiently accurate.", "Frequency-domain applications have shown that the augmented Lagrangian (AL) method solves the inverse problem accurately with a minimal effect of the penalty parameter choice.", "Applying this method in the time domain, however, is limited by two main factors: (1) The challenge of data-assimilated wavefield reconstruction due to the lack of an explicit time-stepping and (2) The need to store the Lagrange multipliers, which is not feasible for the field-scale problems.", "We show that these wavefields are efficiently determined from the associated data (projection of the wavefields onto the receivers space) by using explicit time stepping.", "Accordingly, based on the augmented Lagrangian, a new algorithm is proposed which performs in \"data space\" (a lower dimensional subspace of the full space) in which the wavefield reconstruction step is replaced by reconstruction of the associated data, thus requiring optimization in a lower dimensional space (convenient for handling the Lagrange multipliers).", "We show that this new algorithm can be implemented efficiently in the time domain with existing solvers for the FWI and at a cost comparable to that of the FWI while benefiting from the robustness of the extended FWI formulation.", "The results obtained by numerical examples show high-performance of the proposed method for large scale time-domain FWI." ], [ "Introduction", "Full-waveform inversion (FWI) is the state-of-the-art data-fitting based imaging method for subsurface parameters estimation.", "The parameters of interest appear as coefficients in system of hyperbolic partial differential equations (PDE) which serves as the forward modeling engine for simulating seismic waves.", "Sampling these waves at receiver locations gives us synthetic seismograms.", "FWI tries to find the set of parameters from which we can compute synthetic seismograms which best fit (in the least-squares sense) the observed seismograms (data) [18], [30].", "The forward problem is nonlinear thus iterative linearization methods are natural choice to solve the problem [26].", "Traditional formulation of FWI eliminates the intermediate variable (wavefields) from the equations and represents observed data directly in terms of the Green's functions which are nonlinearly related to the model parameters.", "The standard gradient (steepest descent) method solves this nonlinear inverse problem iteratively.", "At each iteration, the parameters are updated along the steepest-descent direction.", "This direction is efficiently determined by using adjoint-state method [19] that is a simple zero-lag cross-correlation between two calculated wavefields: one is the forward wavefield in the current model and the other is a (backward) wavefield obtained by back-propagation of the data residuals acting as if they were sources at receiver locations.", "A more accurate search direction is obtained by using Newton or quasi-Newton methods by including curvature information via the Hessian matrix or an approximation of it [16].", "A simple while effective approximation of the Hessian matrix can also be obtained by using only its diagonal terms that are calculated by zero-lag correlation of the forward wavefield [22].", "Calculated step direction is then scaled by an appropriate step length that can be chosen using a line search method, such as the backtracking line search, and added to the previous model to get the next iterate.", "This process is repeated until the convergence is achieved.", "In this way, the quality of the search direction is the main factor governing the algorithm convergence and it is a directly influenced by the quality of forward and backward wavefields.", "In contrast to the FWI, extended FWI brings flexibilities to fit the data by softening the wave equation constraint at least at early iterations [27], [29], [14], [1].", "The associated objective function is bivariate and besides model parameters we need to also manipulate data-assimilated wavefields [2], specific wavefields that are obtained by solving the data equation and the wave equation simultaneously.", "The objective function is biconvex and the augmented Lagrangian (AL) method provides an excellent framework for its minimization with a minimal sensitivity to the penalty parameter choice [8], [17], [1].", "However, there are two main obstacles that limit the usage of this method in the time domain [31], [21], [28], [2].", "The first is reconstruction of data-assimilated wavefields and the second is the need to store the Lagrange multipliers, which are of the size of the wavefields and thus infeasible for the field-scale problems.", "In this paper, we provide a new formulation for solving the AL-based extended FWI.", "An equivalent formulation of the iteratively-refined wavefield reconstruction inversion [1] is derived in the data space which require manipulating Lagrange multipliers of the data size.", "Furthermore, it replaces the wavefield reconstruction problem in the full space with a residual data reconstruction problem in the data space.", "The resulting reconstruction system is of the data size and thus more tractable compared with the wavefield reconstruction in the original formulation.", "Based on the fact that the AL algorithm is convergent for a wide range of penalty parameters [8], [17], [1], we derive a simplified algorithm which only requires four wave equation solves at each iteration which can be carried out by usual time-stepping.", "An interesting observation is that the AL-based extended FWI updates the model parameters via a normalized adjoint-state formulation similar to the quasi-Newton algorithm for the classical FWI.", "The only difference is in the forward and backward wavefields which are cross-correlated.", "This can be thought of as a generalized algorithm which can be applied to both the extended and reduced FWI and in both the time and frequency domains.", "The contents of this paper are organized as follows: We begin with the general form of the FWI problem as a partial-differential equation (PDE) constrained data-fitting problem.", "We then review the two approaches of classical FWI and the extended FWI for solving this optimization problem.", "A brief review of the augmented Lagrangian (AL) method is presented with its advantages and limitations for time domain FWI.", "Then we derive an algorithm for AL-based extended FWI and show how it can be efficiently applied in the time domain.", "Finally, we will demonstrate the efficiency of the proposed algorithm by using Marmousi II and BP 2004 benchmark velocity models." ], [ "Theory", "FWI seeks the subsurface parameters by solving the following PDE constrained optimization problem [29], [1]: $ & \\min _{{m},{u}} ~~\\frac{1}{2}\\Vert {Pu-d}\\Vert _2^2~~~ \\text{subject to}~ ~~{A(m)u=b},$ where $\\Vert \\cdot \\Vert _2$ denotes the Euclidean norm, ${u}\\in {\\mathbb {R}}^{N\\times 1}$ is the wavefield, ${d} \\in {\\mathbb {R}}^{M \\times 1}$ is the recorded seismic data generated by the sampling operator ${P} \\in {\\mathbb {R}}^{M\\times N}$ (with $M\\ll N$ ), ${A(m)} \\in {\\mathbb {R}}^{N \\times N}$ is the PDE operator, and ${b} \\in {\\mathbb {R}}^{N\\times 1}$ is the source term.", "More specifically, $N=N_x\\times N_z \\times N_t$ and $M=N_t\\times N_r$ with $N_x$ and $N_z$ the number of spatial grid samples of the model parameter ${m}$ (square of slowness) in horizontal and vertical dimensions, $N_t$ the number of time samples, and $N_r$ is the number of receivers." ], [ "FWI", "Eliminating the wavefield from equation REF leads to the conventional (reduced) formulation of FWI as [20] $ \\min _{{m}} ~~\\frac{1}{2}\\Vert {G(m)}{b}-{d}\\Vert _2^2.$ where ${G(m)}={P}{A(m)}^{-1}$ denotes the forward modeling operator.", "This objective function is very hard to solve due to the presence of the inverse operator ${A(m)}^{-1}$ (which includes the Green's functions as its columns).", "The standard gradient-based iterative methods solve this nonlinear inverse problem iteratively [25], [18], [20], [30], in which at iteration $k$ th the parameters are updated as ${m}_{k+1} = {m}_k + \\alpha _k\\delta {m}_k,$ where $\\alpha _k$ is a step length that can be chosen using a line search method, such as the backtracking line search, and $\\delta {m}_k$ is a step direction.", "This step direction is selected as opposite gradient of the objective function (${g}_k$ , in the steepest descent method) or determined by solving a Newton or quasi-Newton system: $ \\delta {m}_k = -{H}_k^{-1}{g}_k,$ where ${H}_k$ is the Hessian matrix or an approximation of it.", "Both the gradient and the Hessian are evaluated at the current model ${m}_k$ .", "The gradient can be computed efficiently by using the adjoint state method [18], [19].", "$ {g}_k = \\int _0^{\\tau } [\\ddot{{u}}^r_k(t)\\circ {v}^r_k(t)] dt,$ where $\\ddot{{u}}^r_k(t)$ is the second time derivative of the forward wavefield induced by the source and ${v}^r_k(t)$ is the adjoint wavefield obtained by back-propagation of the data residuals $\\delta {d}^r_k$ (which acts as a source emitting in a time-reversed manner).", "$ \\delta {d}^r_k={d}-{G}({m}_k){b}.$ In equation REF , both wavefields are propagated in the current model ${m}_k$ and they are recorded over the time interval $(0, \\tau )$ .", "Also, $\\circ $ denotes the element-wise product operator.", "A simple while effective approximation of the Hessian or the pseudo-Hessian includes only the diagonal terms [22], which gives $ \\delta {m}_k = -\\frac{\\int _0^{\\tau } [\\ddot{{u}}^r_k(t)\\circ {v}^r_k(t)] dt}{\\int _0^{\\tau } \|\\ddot{{u}}^r_k(t)\|^2 dt},$ where the division and the square-power are element-wise operations.", "This indeed is the zero-lag crosscorrelation image between the forward and adjoint wavefields (imaging condition) normalized by the square of the source illumination strength [6].", "A more accurate Hessian matrix can also be implement to improve the model update by solving equation REF via an iterative algorithm while computing Hessian-vector products by second-order adjoint state formulas [16].", "However, FWI requires us to provided an initial model of sufficient quality for convergence to the desired solution [30].", "The next section provides a remedy for this issue using extended FWI.", "In the formulas above, superscript $r$ shows that the variable corresponds to the reduce FWI.", "We will use superscript $e$ to refer to the extended FWI." ], [ "Extended FWI", "Extended FWI methods have been developed to decrease the nonlinearity of the objective function and hence increase the robustness of local optimization methods to the initial model [27], [29], [14], [1].", "The simple form of the extended formulation can be obtained by the penalty formulation of the original equation REF : $ \\min _{{m,u}^e} ~\\frac{1}{2}\\Vert {Pu}^e-{d}\\Vert _2^2 + \\frac{\\mu }{2}\\Vert {A(m)}{u}^e-{b}\\Vert _2^2,$ where $\\mu >0$ is the penalty parameter.", "This extended formulation can be thought of as an “error-in-variable\" method which accounts for errors in the wave equation, and it reduces to the FWI objective in equation REF as $\\mu \\rightarrow \\infty $ .", "Equation REF is biconvex and it is usually solved by alternating minimization [3].", "This algorithm however suffers from two major drawbacks which limit its application in a time domain practical setting.", "The first is manipulating the u-subproblem (computation and storage of the wavefield for a given model) and the other is the issue related to the penalty parameter choice.", "The issue of penalty parameter can be overcome by adding a Lagrangian term of the form $\\lambda ^T({A(m)}{u}^e-{b})$ to the penalty function [17], [1]: $ \\min _{{m,u}^e}\\max _{\\lambda } ~\\text{Penalty}({m,u}^e) +\\lambda ^T({A(m)}{u}^e-{b}).$ where superscript $T$ denotes adjoint operator, Penalty denotes the penalty objective function in equation REF and $\\lambda $ is the Lagrange multiplier for the PDE constraint.", "The alternating direction method of multipliers [8], [13], [5], [1] solves the augmented Lagrangian in equation REF iteratively by choosing ${u}$ and $\\lambda $ and then minimizing the objective as a function of ${m}$ .", "The resulting ${m}$ is used to choose a new ${u}$ and $\\lambda $ , and the process repeats until the convergent is achieved.", "${m}_{k}&= \\arg \\min _{{m}} ~\\frac{1}{2}\\Vert {A}({m}){u}^e_{k-1}-{b} - \\lambda _{k-1}\\Vert _2^2, \\\\{u}^e_{k}&= \\arg \\min _{{u}} ~\\frac{1}{2}\\Vert {Pu}-{d}\\Vert _2^2 + \\frac{\\mu }{2}\\Vert {A}_{k}{u}-{b}-\\lambda _{k-1}\\Vert _2^2, \\\\\\lambda _{k} &= \\lambda _{k-1} + {b} - {A}_{k}{u}^e_{k}, $ beginning with initial ${u}_0$ (which is built from an initial model) and $\\lambda _0={0}$ , where ${A}_{k}\\equiv {A}({m}_{k})$ and $k$ is the iteration count.", "Note that [1] first optimize in the wavefield ${u}^e$ and then in the model parameters ${m}$ but for the method to develop in this paper the permuted order in equation REF is more convenient.", "This does not affect the final solution because the order of the optimization in biconvex problems can be permuted.", "The iteration in equation REF without the “add-back-the residual\" term reduces to the wavefield reconstruction inversion (WRI) [29].", "This term accounts for the Lagrangian and besides increasing the convergence rate, reduces the sensitivity of the algorithm to the penalty parameter choice by “adding-back the source residual\".", "The advantage of the augmented Lagrangian method is thus to use a small $\\mu $ (to make the optimization well posed) and “add-back-the-residual\" [13].", "[1] consider this iteration while solving the feasibility form of equation REF in which the data are also updated by adding back the data residual.", "${m}_{k}&= \\arg \\min _{{m}} ~\\frac{1}{2}\\Vert {A}({m}){u}^e_{k-1}-{b}_{k-1}\\Vert _2^2, \\\\{u}^e_{k}&= \\arg \\min _{{u}} ~\\frac{1}{2}\\Vert {Pu}-{d}_{k-1}\\Vert _2^2 + \\frac{\\mu }{2}\\Vert {A}_{k}{u}-{b}_{k-1}\\Vert _2^2, \\\\{d}_{k} &= {d}_{k-1} + {d} - {P}{u}^e_{k}, \\\\{b}_{k} &= {b}_{k-1} + {b} - {A}_{k}{u}^e_{k}, $ beginning with ${b}_0={b}$ and ${d}_0={d}$ which have the role of scaled Lagrange multipliers.", "This algorithm has been successfully applied in the frequency-domain FWI where the most challenging subproblem, equation , has a size of $N_x\\times N_z$ and thus can be solved directly by sparse solvers.", "The Lagrangian multiplier of size $N_x\\times N_z$ can also be conveniently stored during the iterations.", "In the time domain, however, solving equation is more challenging because the size of the system would be prohibitively large ($N_x\\times N_z\\times N_t$ ) due to the extra temporal dimension.", "Furthermore, ${b}_{k}$ of the size $N_x\\times N_z\\times N_t$ would be dense; which causes serious storage problems.", "Several attempts have been make by different authors to solve these issues [31], [21], [28], [2].", "In the next section, we propose an algorithm for efficient treatment of these issues which performs in the data space instead of the full wavefield space." ], [ "Data Reconstruction Inversion", "The general form of data-assimilated wavefield reconstruction, equation , is $ \\min _{{u}^e} ~\\frac{1}{2}\\Vert {Pu}^e-{d}\\Vert _2^2 + \\frac{\\mu }{2}\\Vert {Au}^e-{b}\\Vert _2^2.$ Minimization of equation REF gives ${u}^e$ as the solution of the following system: $ ({P}^T{P} + \\mu {A}^T{A}){u}^e={P}^T{d} + \\mu {A}^T{b},$ This system can be re-expressed by decoupling ${Au}^e$ from other terms, making it suitable for time-stepping: $ {Au}^e={b} + \\frac{1}{\\mu } {G}^T{(d-Pu}^e),$ where ${G}^T\\equiv {A}^{-T}{P}^T= ({A}^{-1})^T{P}^T$ .", "However, since the unknown wavefield ${u}^e$ appears in both sides of this equation, researchers tried different approaches to solve it.", "For example, [32] replaced the wavefield at the right-hand-side with the wavefield obtained by the reduced FWI.", "Later, [2] developed an algorithm which solves equation REF iteratively, beginning with the wavefield constructed by [32].", "In this paper, we solve equation REF in a two step procedure.", "Noting that ${u}^e$ appears in the right-hand side only as (WRI predicted data) ${Pu}^e$ thus if we have an estimate of ${Pu}^e$ then ${u}^e$ can be determined by solving equation REF .", "Accordingly, we rewrite this equation as $ {Au}^e &={b} + \\frac{1}{\\mu } {G}^T\\delta {d}^e, \\\\&={b} + \\delta {b}^e, \\nonumber $ where $ \\delta {b}^e={Au}^e-{b}~~\\text{and}~~\\delta {d}^e={d-Pu}^e,$ are respectively the source residual and data residual (of WRI).", "Note that the source residual is zero for the reduced FWI.", "Up to now we have reduced the optimization dimension from $N$ (directly searching for ${u}^e$ in equation REF or REF ) to $M$ (searching for the residual vector $\\delta {d}^e$ in equation REF ).", "However, the question is now “what is the optimization problem for $\\delta {d}^e$ ?\"", "By eliminating ${u}^e$ from equation REF we get the following relation between the two residual vectors $\\delta {b}^e$ and $\\delta {d}^e$ : $ {G}\\delta {b}^e+\\delta {d}^e=\\delta {d}^r.$ where $\\delta {d}^r$ is the data residual of the FWI, equation REF .", "Furthermore, comparing equations REF and REF shows that $ \\delta {b}^e=\\frac{1}{\\mu }{G}^T\\delta {d}^e.$ Substituting $\\delta {b}^e$ from equation REF into equation REF immediately gives an explicit $M\\times M$ system for $\\delta {d}^e$ in the data space: $ \\left({G}{G}^{T} + \\mu {I}\\right)\\delta {d}^e=\\mu \\delta {d}^r.$ This equation clearly shows that the FWI data residual, $\\delta {d}^r$ , is a blurred version of the WRI data residual, $\\delta {d}^e$ , in which the blurring kernel consists of the correlation of receiver-side Green's functions.", "The solution of equation REF is also the minimizer of the following quadratic objective function: $ \\min _{\\delta {d}^e} ~\\frac{1}{2}\\Vert \\delta {d}^e-\\delta {d}^r\\Vert _2^2 + \\frac{1}{2\\mu }\\Vert {G}^T\\delta {d}^e\\Vert _2^2.$ Plugging $\\delta {b}^e$ and $\\delta {d}^e$ from equations REF and REF into the primal objective in equation REF gives an equivalent objective for the extended FWI which is a weighted norm of the FWI data residual: $\\frac{1}{2\\mu }\\Vert \\delta {d}^e\\Vert _{{Q}({m})}^2=\\frac{\\mu }{2}\\Vert \\delta {d}^r\\Vert _{{Q}({m})^{-1}}^2.$ where $\\Vert {y}\\Vert _{{Q}({m})}^2={y}^T{Q}({m}){y}$ and ${Q}({m})$ is the coefficient matrix in equation REF .", "This equivalence has also been confirmed differently by [28] and [24]." ], [ "IR-WRI Iteration in Data Space", "Using the data space formulation given above the IR-WRI iteration given in equation REF can be performed more efficiently in the data space.", "From equations , and REF we have that $ {\\left\\lbrace \\begin{array}{ll}{d}_{k} ={d}_{k-1} + {d} - {P}{u}_{k} ={d}+\\delta {d}^e_{k},\\\\{b}_{k} = {b}_{k-1} + {b} - {A}_{k}{u}_{k} ={b} -\\delta {b}^e_{k}.\\end{array}\\right.", "}$ Furthermore, from equations REF and REF we have $ {\\left\\lbrace \\begin{array}{ll}\\delta {b}^e_{k}=\\frac{1}{\\mu }{G}_k^{T}\\delta {d}^e_{k},\\\\\\delta {d}^e_{k}=\\mu {Q}_k^{-1}[{d}_{k-1}-{G}_k{b}_{k-1}],\\end{array}\\right.", "}$ where ${Q}_k={G}_k{G}_k^{T}+\\mu {I}.$ Accordingly the wavefield ${u}^e_k$ in equation REF can be written as (see equation REF ) ${u}^e_{k} &={A}_k^{-1}[{b}_{k-1}+ \\delta {b}^e_{k}] \\\\&={A}_k^{-1}[{b} - \\delta {b}^e_{k-1}+ \\delta {b}^e_{k}] \\\\&={A}_k^{-1}[{b} +\\frac{1}{\\mu }{G}_k^{T}(\\delta {d}^e_{k}-\\delta {d}^e_{k-1})],$ (while replacing ${G}_{k-1}$ with its update ${G}_{k}$ ) based on which the model update in equation REF simplifies to $ {m}_{k+1}= \\arg \\min _{{m}} ~\\frac{1}{2}\\Vert {A}({m}){A}_{k}^{-1}{b}_k^+-{b}_k^-\\Vert _2^2,$ in which $ {\\left\\lbrace \\begin{array}{ll}{b}_k^+ ={b} +\\frac{1}{\\mu }{G}_k^{T}(\\delta {d}^e_{k}-\\delta {d}^e_{k-1}), \\\\{b}_k^-={b}- \\frac{1}{\\mu }{G}_k^{T}\\delta {d}^e_k.\\end{array}\\right.", "}$ From equation REF , we see that $\\delta {d}_k^e$ is the minimizer of the following objective function $ f(\\delta {d}^e)=\\frac{1}{2}\\Vert \\delta {d}^e-\\delta {d}^r_{k-1}\\Vert _2^2 + \\frac{1}{2\\mu }\\Vert {G}_k^{T}\\delta {d}^e\\Vert _2^2.$ Differentiating this objective with respect to $\\delta {d}^e$ gives $ \\nabla f(\\delta {d}^e)=\\frac{1}{\\mu }{Q}_k\\delta {d}^e - \\delta {d}^r_{k-1}=\\frac{1}{\\mu }{Q}_k\\delta {d}^e - {d}_{k-1} + {G}_k{b}_{k-1}.$ Substituting the values of ${d}_{k-1}$ and ${b}_{k-1}$ from equations REF and REF into equation REF gives $\\nabla f(\\delta {d}^e)&=\\frac{1}{\\mu }{Q}_k\\delta {d}^e - {d}- \\delta {d}^e_{k-1}+{G}_k({b} -\\delta {b}^e_{k-1}),\\\\&=\\frac{1}{\\mu }{Q}_k\\delta {d}^e-\\delta {d}_k^r-\\delta {d}^e_{k-1}-{G}_k\\delta {b}^e_{k-1},\\\\&=\\frac{1}{\\mu }{Q}_k\\delta {d}^e -\\delta {d}_k^r-\\delta {d}^e_{k-1}-\\frac{1}{\\mu }{G}_k{G}_{k-1}^{T}\\delta {d}^e_{k-1},\\\\&=\\frac{1}{\\mu }{Q}_k\\delta {d}^e-\\delta {d}_k^r-\\frac{1}{\\mu }{Q}_k\\delta {d}^e_{k-1}.$ Note that, in equation REF , we replaced ${G}_{k-1}$ with its update ${G}_{k}$ .", "Setting the gradient, equation , equal to zero and solving for $\\delta {d}^e$ gives the exact minimizer, $\\delta {d}_k^e$ , of the objective $\\delta {d}_k^e=\\delta {d}^e_{k-1}+\\mu {Q}_k^{-1}\\delta {d}_k^r.$ It is seen that in the augmented Lagrangian method the data residual vector is updated via a Newton like algorithm, equation REF , in which the search direction is defined by the FWI data residual $\\delta {d}_k^r$ .", "${Q}_k$ has the role of Hessian matrix.", "After introducing the auxiliary variable ${y}_k=\\delta {d}^e_k/\\mu $ and $\\delta {y}_k={y}_k-{y}_{k-1}$ , equations REF and REF simplify to the following iteration $ {\\left\\lbrace \\begin{array}{ll}\\delta {y}_k={Q}_k^{-1}\\delta {d}^r_k,\\\\{y}_{k} ={y}_{k-1}+\\delta {y}_k,\\\\{b}_k^+ ={b} + {G}_k^{T}\\delta {y}_k, \\\\{b}_k^-={b}- {G}_k^{T}{y}_{k},\\\\{m}_{k+1}= \\arg \\min _{{m}} ~\\frac{1}{2}\\Vert {A}({m}){A}_{k}^{-1}{b}_k^+-{b}_k^-\\Vert _2^2,\\end{array}\\right.", "}$ where ${G}_k^{T}={A}_k^{-T} {P}^T$ .", "As seen, the wavefield reconstruction step in equation REF is replaced by a residual data reconstruction step, $\\delta {y}_k$ in equation REF , and thus we call this new iteration data reconstruction inversion (DRI).", "DRI only needs to store the dual variable ${y}_k$ lying in the data space." ], [ "DRI with Gradient Descent", "A fundamental property of all augmented Lagrangian based algorithms is that they are convergent for an approximate solution of the subproblems [8], [5], [33], [10], [11], [1].", "This motivated us to perform the minimization of objective $f$ in equation REF approximately via a gradient step rather than full minimization given in equation REF .", "$ \\delta {d}^e_{k} = \\delta {d}^e_{k-1}-\\alpha _k\\nabla f(\\delta {d}^e_{k-1}).$ where $\\alpha _{k}$ is a step length and $\\nabla f$ is the gradient of the objective function determined in equation , evaluated at $\\delta {d}^e_{k-1}$ .", "From equation we get that $\\nabla f(\\delta {d}^e_{k-1}) = -\\delta {d}_k^r.$ Accordingly, $\\delta {y}_k$ in equation REF becomes $ \\delta {y}_k \\approx \\alpha _{k} \\delta {d}^r_k,$ which (after replacing ${y}_{k}$ and ${y}_{k-1}$ by ${y}_{k}/\\alpha _k$ and ${y}_{k-1}/\\alpha _k$ ) simplifies the DRI iteration as $ {\\left\\lbrace \\begin{array}{ll}{y}_{k} ={y}_{k-1}+\\delta {d}^r_{k},\\\\{b}_k^+ ={b} + \\alpha _k{G}_k^{T}\\delta {d}^r_k, \\\\{b}_k^-={b}- \\alpha _k{G}_k^{T}{y}_{k},\\\\{m}_{k+1}= \\arg \\min _{{m}} ~\\frac{1}{2}\\Vert {A}({m}){A}_{k}^{-1}{b}_k^+-{b}_k^-\\Vert _2^2.\\end{array}\\right.", "}$" ], [ "Step length determination", "In the DRI algorithm, equation REF , the model parameters are updated by assuming a large penalty parameter and thus more weight given to minimization of the source residuals rather than the data residual.", "Accordingly, we determine the value of $\\alpha _k$ by minimizing the data residuals: $\\alpha _k=\\arg \\min _{\\alpha }\\Vert {P}{u}_k^e(\\alpha )-{d}\\Vert _2^2,$ giving that $ \\alpha _k=\\frac{{q}_k^T\\delta {d}^r_k}{{q}_k^T{q}_k},$ where ${q}_k={G}_k{G}_k^{T}\\delta {d}^r_k$ ." ], [ "Update of the Model Parameters", "We consider constant density acoustic wave equation as the forward modeling operator, given by $ {m}\\circ \\ddot{{u}}(t)-\\nabla ^2{u}(t)={b}(t),$ where ${m}$ is the square of slowness, ${u}(t)$ is the discretized wavefield and $\\ddot{{u}}(t)$ denotes its second time derivative, $\\nabla ^2$ is the Laplacian operator with respect to the spatial coordinates, and ${b}(t)$ is the source term.", "Then, minimization of the source residual, equation REF , leads to $ {m}_{k+1}=\\frac{\\int _0^{\\tau } \\ddot{{u}}^e_k(t)\\circ [\\nabla ^2{u}^e_k(t)+{b}_k^-(t)]dt}{\\int _0^{\\tau } \|\\ddot{{u}}^e_k(t)\|^2 dt},$ where ${{u}}^e_k(t)$ obeys the following equation (see equation REF for ${b}_k^-(t)$ and ${b}_k^+(t)$ ) $ {m}_k\\circ \\ddot{{u}}^e_k(t)-\\nabla ^2{u}^e_k(t)={b}_k^+(t).$ Using the expression for $\\nabla ^2{u}^e_k(t)$ , equation REF can also be written as $ {m}_{k+1} &=\\frac{\\int _0^{\\tau } \\ddot{{u}}^e_k(t)\\circ [{m}_k\\circ \\ddot{{u}}^e_k(t)-\\alpha _k{v}^e_k(t)]dt}{\\int _0^{\\tau } \|\\ddot{{u}}^e_k(t)\|^2 dt},\\\\&= {m}_k + \\alpha _k\\delta {m}_{k}, $ where $ \\delta {m}_{k} = -\\frac{\\int _0^{\\tau } \\ddot{{u}}^e_k(t)\\circ {v}^e_k(t)dt}{\\int _0^{\\tau } \|\\ddot{{u}}^e_k(t)\|^2 dt},$ in which ${v}^e_k(t)$ is the backward-wavefield obtained by back-propagation of the adjoint source ${y}_{k}+ \\delta {d}^r_k$ .", "As seen, the new algorithm simply adds back the residual to improve the estimate." ], [ "Computational Cost", "The computational burden of DRI algorithm lies in the simulation of propagating wavefields using a finite difference method.", "More precisely, two fields ${{u}}^e_k$ and ${{v}}^e_k$ must be simulated and need to be available at the same time level to compute the integral in equation REF .", "One can compute these integrations by first computing the forward wavefield ${{u}}^r_k$ (solution of one forward problem) and storing it in the computer memory.", "This first step also delivers the data residual $\\delta {d}^r_k$ .", "Then the residual field $\\delta {{u}}_k$ is computed by back-propagation of the data residual followed by forward propagation of the resulting backward field (solution of one forward plus one backward problems), during which the step length $\\alpha _k$ , equation REF , and the wavefield ${{u}}^e_k={{u}}^r_k+\\alpha _k\\delta {{u}}_k$ are computed.", "Finally, the backward field ${v}^e_k$ is computed by back-propagating the adjoint source ${y}_{k}+\\delta {d}^r_k$ (solution of one backward problem) and the imaging condition in equation REF is computed simultaneously.", "Therefore, this approach requires one to solve 2 + 2 wave equations at each iteration.", "In practice, storing the wavefield is memory intensive.", "To mitigate this issue, one can use the compression and reconstruction methods to decrease demanded memory [4], [12], [34].", "An alternative method, which is less memory demanding but adds the cost of extra wave equation solves, is to store only the spatial boundary conditions and final time slice and then recomputing the wavefields [9], [23]." ], [ "Camembert model test", "In order to show the robustness of the proposed method with respect to the initial model, we consider the simple while challenging “Camembert\" model [9].", "The subsurface model contains a circular anomaly of velocity 4.6 km/s embedded in a homogeneous background of velocity 4.0 km/s (Figure REF a).", "The dimensions of the model are 4.8 km in distance and 6 km in depth, and the grid spacing is 35.5 m. The crosshole acquisition consists of 14 equally spaced sources of 10 Hz Ricker wavelet in the left side of the model and 170 equally spaced receivers deployed vertically on the opposite side.", "This model was previously used by [7] for an application of FWI based upon an optimal-transport distance.", "Due to the large diameter of the anomaly the inverse problem is nonlinear and thus the initial model plays an important role in the convergence of the local optimization algorithms.", "We start the inversion from the homogeneous background model to meet the nonlinear regime of the FWI and perform one cycle of inversion with the 10 Hz Ricker wavelet.", "The results obtained using FWI and the DRI are presented in Figures REF c and REF d, respectively.", "Figure REF shows the trajectory of the misfit function versus iteration for both methods.", "The result of FWI is quite different from the true one, which indicates that the method suffers from severe cycle skipping due to the rough initial model and 10 Hz frequency inversion.", "In contrast, the DRI mitigates efficiently the cycle skipping.", "This is clearly seen from the reconstructed model which is quite close to the ground truth.", "Figure: Camembert model example.", "(a) True velocity, and inverted velocity models using (b) FWI and (c) DRI.Figure: Trajectory of the misfit function for the Camembert model example in Figure ." ], [ "Checkerboard model test", "We design a checkerboard model composed of a homogeneous background of velocity 1.5 km/s and checkerboard velocity perturbations of velocity 4.0 km/s and dimension 200 m by 200 m (Figure REF a).", "We use a full acquisition with 92 equally-spaced sources placed all around the domain with 50 m distance from the model boundary.", "Each source is associated with a total of 100 equally spaced receivers placed on the model boundary.", "The model is discretized over a 101 by 101 grid with a spatial step of 20 m. The source wavelet is a band-pass filtered Ricker wavelet between 2.5 Hz and 5 Hz with a peak frequency of 3.75 Hz (Figure REF ).", "The initial model is the background velocity.", "Figures REF a and REF b show two data-sets computed with the exact velocity model and with the initial homogeneous-velocity model.", "We compare the inversion results obtained performing 300 iterations (Figure REF ) for both FWI and DRI and the corresponding model estimates are shown in Figures REF b and REF c. The final data residuals corresponding to both methods are also shown in Figures REF c and REF d. For this test, again, unlike FWI, the DRI reconstructed a model which is quite close to the ground truth.", "Figure: A checkerboard model example.", "(a) True velocity, and inverted velocity models using (b) FWI and (c) DRI.Figure: Source wavelet in the time and frequency domains used for the checkerboard example.Figure: A data-set of the checkerboard model.", "Data-set computed in the exact model (a) and in the initial homogeneous model (b).", "Residual data at the convergence point for (c) the FWI and (d) the DRI methods.Figure: Trajectory of the misfit function for the checkerboard model example in Figure ." ], [ "Marmousi II Model test", "We now consider a more realistic multisource experiment using the Marmousi II velocity model (Figure REF ).", "The model is 17 km long and 3.5 km deep.", "The fixed spread acquisition consists of equally-spaced 91 sources and equally-spaced 273 receivers on top side of the model.", "We build reference data using a highpass filter minimum-phase Ricker wavelet with no signal below 2.5 Hz.", "The total recording time is 7.5 seconds with a sample internal 3 ms.", "The initial model is a 1D velocity model linearly increasing in depth from 1.5 km/s to 4.5 km/s.", "Figure REF shows the residual gather for the source at distance 8.5 km.", "Obviously classical FWI will stuck in a local optimum due to severe cycle skipping caused by the 1D initial model and the absence of low-frequencies in the data [1] thus we don't show the corresponding results here.", "The inversion result of the DRI is shown in Figure REF which clearly shows that the method mitigated efficiently the cycle skipping.", "The reconstruction is quite close to the true model in most parts of the model.", "In order to compare the DRI algorithm with the IR-WRI algorithm we performed both algorithm in the frequency domain to invert the data corresponding to the Marmousi II.", "The results obtained by both algorithms are shown in Figures REF b-REF c and as seen the estimated velocity models are similar which confirm the accuracy of the approximation we considered in DRI algorithm, given in equation REF .", "Figure: Marmousi II velocity model (top) and 1D initial model (bottom).Figure: Residual data of Marmousi model for a source at distance 8250 m.Figure: Marmousi II velocity model obtained by (a) time-domain DRI, (b) frequency-domain DRI, (c) frequency-domain IR-WRI." ], [ "2004 BP Model test", "Finally, we assess the DRI method against a target of the challenging 2004 BP salt model when a crude starting model is used (Figure REF ).", "The selected target corresponds to the left part of the 2004 BP salt model and was previously used by [15] for an application of FWI based upon an optimal-transport distance and by [1] for an application of frequency-domain WRI based upon the augmented Lagrangian formulation (IR-WRI).", "The subsurface model is 16.25 km wide and 5.825 km deep.", "The fixed spread acquisition consists of equally-spaced 109 sources and equally-spaced 326 receivers on top side of the model.", "We build reference data using a highpass filter minimum-phase Ricker wavelet with no signal below 2.5 Hz.", "The total recording time is 7.5 seconds with a sample internal 2 ms. We used a smoothed version of the true velocity model without the salt as initial model (Figure REF ).", "A comparison between a common-shot gather computed in the true and initial models shows severe traveltime mismatches (Figure REF ) that drive the classical FWI to a local minimum [1].", "The DRI, however, mitigated efficiently the cycle skipping as it can be seen from the reconstructed model shown in Figure REF .", "As seen from this figure, the precise geometry of the salt body and the subsalt low-velocity anomalies are mostly recovered with a quite high resolution.", "Figure: 2004 BP velocity model (top) and initial model (bottom).Figure: Comparison between a common-shot gather computed in (a) the initial model, (b) the estimated model at first iteration, and (c) the estimated model at final iteration.", "(d) The gather computed in the true 2004 BP velocity model.Figure: 2004 BP velocity model obtained by the proposed algorithm." ], [ "Conclusions", "Time domain full waveform inversion (FWI) is appealing since the wave equation can be solved efficiently by explicit time-stepping but this method require sufficiently accurate initial models which may not be available in practice.", "Extended FWI with wavefield reconstruction is robust with respect to the initial model and it can be solved with a minimal sensitivity to the penalty parameter choice when using the augmented Lagrangian solver.", "However, its application in the time domain is a challenging task due to the absent of explicit time-stepping and because computation and memory burdens grow quickly as the size of model increases due to the inherent high-dimensional nature of the state variable.", "In this paper, we proposed a new augmented Lagrangian algorithm for this task which performs in the data space, a significantly lower dimensional subspace than the dimensionality of the original full space.", "In this algorithm, the wavefield reconstruction appears as a residual data reconstruction and the model parameters are updated by a quasi-Newton like algorithm with a search direction computed by a generalized adjoint state method.", "This latter is computed by zero-lag cross-correlation between the data-assimilated wavefields and adjoint fields coming from back-propagation of all previous data residuals added back to the current residual vector.", "Approximate update of the residual reconstruction by a gradient descent method resulted to an efficient algorithm which only requires four wave equation solves (two forward solves plus two backward solves) at each iteration.", "Different numerical tests including Marmousi II and 2004 BP velocity models confirm the efficiency of this algorithm for time-domain applications similar to the classical FWI while benefiting from the excellent robustness of the extended FWI." ], [ "Acknowledgments", "This study was partially funded by the WIND consortium (https://www.geoazur.fr/WIND), sponsored by Chevron, Shell, and Total.", "This study was granted access to the HPC resources of SIGAMM infrastructure (http://crimson.oca.eu), hosted by Observatoire de la Côte d'Azur and which is supported by the Provence-Alpes Côte d'Azur region, and the HPC resources of CINES/IDRIS/TGCC under the allocation A0050410596 made by GENCI.\"" ], [ " Appendix A", "Consider the acoustic wave equation in a homogeneous 3D medium $ \\frac{1}{c^2}\\frac{\\partial ^2}{\\partial t^2} u(t,{x})-\\nabla ^2u(t,{x})=b(t,{x}),$ in which the acoustic velocity $c$ is a constant.", "The solution of equation REF can be written in the following integral form: $ u(t,{x}) = \\iint g(t-t^{\\prime },{x}-{x}^{\\prime }) b(t^{\\prime },{x}^{\\prime }) dt^{\\prime } d{x}^{\\prime },$ where $g(t,{x}) = \\frac{\\delta (t - \\Vert {x}\\Vert _2/c)}{\\Vert {x}\\Vert _2},$ are the Green's functions and $\\delta $ denotes the delta function.", "Writing equation REF in an operator notation as $u = A^{-1}b$ then the goal is to build $ Q = PA^{-1}A^{-T}P^T.$ where $P$ is the restriction matrix at the receiver locations.", "In order to build $Q$ we need to compute the adjoint wavefield for each column of $P^T$ followed by computing the forward wavefield while using the ajoint field as the source and then sampling the result at receiver location.", "Each column of $P^T$ can be written as $ \\delta (t-\\tau ,{x}-{x}_j)$ in which ${x}_j$ denotes the location of the $j$ th receiver and $\\tau $ denotes the time lag.", "The adjoint wavefield for the delta function in equation REF is $ v(t,{x}) &= \\iint g(t^{\\prime }-t,{x}^{\\prime }-{x}) \\delta (t^{\\prime }-\\tau ,{x}^{\\prime }-{x}_j) dt^{\\prime } d{x}^{\\prime }\\\\&=g(\\tau -t,{x}_j-{x}).$ Then applying the forward operator on this adjoint wavefield gives $u(t,{x}) &= \\iint g(t-t^{\\prime },{x}-{x}^{\\prime }) v(t^{\\prime },{x}^{\\prime }) dt^{\\prime } d{x}^{\\prime }\\\\&= \\iint g(t-t^{\\prime },{x}-{x}^{\\prime }) g(\\tau -t^{\\prime },{x}_j-{x}^{\\prime }) dt^{\\prime } d{x}^{\\prime }.$ Finally, sampling this wavefield at the $i$ th received location, thus counting for the matrix $P$ , gives $Q(t,\\tau ,{x}_i,{x}_j) &= \\iint g(t-t^{\\prime },{x}_i-{x}^{\\prime }) g(\\tau -t^{\\prime },{x}_j-{x}^{\\prime }) dt^{\\prime } d{x}^{\\prime }.$ Substituting for the Green's functions from equation REF and performing the integration over $t^{\\prime }$ gives $ Q(t,\\tau ,{x}_i,{x}_j) &=\\int \\frac{\\delta (t-\\tau + \\frac{1}{c}[\\Vert {x}_j-{x}^{\\prime }\\Vert _2 - \\Vert {x}_i-{x}^{\\prime }\\Vert _2])}{\\Vert {x}_i-{x}^{\\prime }\\Vert _2\\Vert {x}_j-{x}^{\\prime }\\Vert _2}d{x}^{\\prime },$ In order to compute this integral, we note that for each receiver pair $i$ and $j$ and a point ${x}^{\\prime }$ the $Q$ matrix includes a linear event parallel to the main diagonal described by $t = \\tau + \\frac{1}{c}[\\Vert {x}_j-{x}^{\\prime }\\Vert _2 - \\Vert {x}_i-{x}^{\\prime }\\Vert _2],$ in which the second term at right hand side determines the amount of shift from the main diagonal.", "From the backward triangle inequality we have that $\\Vert {x}_j-{x}^{\\prime }\\Vert _2 - \\Vert {x}_i-{x}^{\\prime }\\Vert _2 \\le \\Vert {x}_i-{x}_j\\Vert _2,$ which implies that the maximum shift is $\\frac{1}{c} \\Vert {x}_i-{x}_j\\Vert _2$ .", "Summing over all points ${x}^{\\prime }$ thus leads to a symmetric banded matrix with a band centred on the main diagonal.", "Figure REF shows the Q matrix for a homogeneous model of velocity 2000 m/s for 5 equally spaced receivers each of 251 time samples.", "The distance between receivers is 200 m. The band diagonal structure of each block is clearly seen and the bandwidth follows the theoretical finding.", "Figure: (Left) Q matrix computed for a homogeneous model of velocity 2000 m/s and for 5 receivers each of 251 time samples.", "The distance between receivers is 200 m. (Right) block (1,2) of matrix Q where the dashed lines show the theoretical band." ] ]	2011.14102
[ [ "Minimizing cycles in tournaments and normalized $q$-norms" ], [ "Abstract Akin to the Erd\\H{o}s-Rademacher problem, Linial and Morgenstern made the following conjecture in tournaments: for any $d\\in (0,1]$, among all $n$-vertex tournaments with $d\\binom{n}{3}$ many 3-cycles, the number of 4-cycles is asymptotically minimized by a special random blow-up of a transitive tournament.", "Recently, Chan, Grzesik, Kr\\'al' and Noel introduced spectrum analysis of adjacency matrices of tournaments in this study, and confirmed this for $d\\geq 1/36$.", "In this paper, we investigate the analogous problem of minimizing the number of cycles of a given length.", "We prove that for integers $\\ell\\not\\equiv 2\\mod 4$, there exists some constant $c_\\ell>0$ such that if $d\\geq 1-c_\\ell$, then the number of $\\ell$-cycles is also asymptotically minimized by the same family of extremal examples for $4$-cycles.", "In doing so, we answer a question of Linial and Morgenstern about minimizing the $q$-norm of a probabilistic vector with given $p$-norm for any integers $q>p>1$.", "For integers $\\ell\\equiv 2\\mod 4$, however the same phenomena do not hold for $\\ell$-cycles, for which we can construct an explicit family of tournaments containing fewer $\\ell$-cycles for any given number of $3$-cycles.", "We conclude by proposing two conjectures on the minimization problem for general cycles in tournaments." ], [ "Introduction", "We consider extremal problems on cycles in tournaments.Throughout this paper, by a cycle in a digraph, we always mean a directed one.", "For a cycle $C_{\\ell }$ of length $\\ell $ and a tournament $T$ , a homomorphism of $C_{\\ell }$ to $T$ is a mapping from $V(C_{\\ell })$ to $V(T)$ that preserves edges.", "Let $t(C_{\\ell },T)$ denote the number of homomorphisms of $C_{\\ell }$ to $T$ divided by $\|V(T)\|^{\\ell }$ .", "One of the most natural extremal problems on tournaments is to determine the maximum number of cycles of a given length in tournaments.", "This can be traced back to the work of Kendall and Babington Smith [9] in 1940.", "It is well-known (see [4], [7]) that a tournament $T$ satisfies $t(C_3,T)\\le \\frac{1}{8}$ and it has the maximum number of cycles $C_3$ if and only if it is almost regular.", "Optimal results on 4-cycles and 5-cycles were obtained by Beineke and Harary [1] and by Komarov and Mackey [10], respectively.", "For other cycles, Day conjectured in [5] that the asymptotic maximum of $t(C_{\\ell },T)$ is achieved by a random oriented tournament if and only of $\\ell $ is not divisible by four.", "This was confirmed recently by Grzesik, Král', Lovász and Volec in [8] using algebraic approach, where they also obtained the asymptotic maximum of $t(C_8,T)$ and a very close estimation on the maximum of $t(C_{\\ell },T)$ for any $\\ell $ divisible by four.", "Another natural extremal problem is to study the complementary problem, i.e., the minimum number of cycles of a given length in tournaments.", "Since irregular tournaments (such as transitive tournaments) can contain arbitrary small number of cycles, one has to introduce some extra restriction on tournaments for measuring regularity in this minimizing problem.", "As mentioned the results of [4], [7] earlier, one such good measure could be the density $t(C_3,T)$ of 3-cycles.", "This was indeed addressed by Linial and Morgenstern in [11], where they asked for the asymptotic minimum density $t(C_4,T)$ of 4-cycles in tournaments $T$ with fixed density $t(C_3,T)$ of 3-cycles.", "A random blow-up of a $m$ -vertex transitive tournament is a tournament $T$ with $V(T)=V_1\\cup V_2\\cup ...\\cup V_m$ such that all arcs within each $V_i$ are oriented randomly and for any $i<j$ , all arcs between $V_i$ and $V_j$ are oriented from $V_i$ to $V_j$ .", "Linial and Morgenstern [11] conjectured that a random blow-up $T^$ of a transitive tournament with all but one part of equal size and one smaller part would achieve the asymptotic minimum $t(C_4,T)$ .", "Suppose that $T^$ has $n$ vertices and $t$ parts of equal size $zn$ for some real $z\\in (0,1]$ , then $tz\\le 1< (t+1)z$ , implying that $t=\\lfloor z^{-1}\\rfloor $ .", "Therefore, with high probability, it holds for any $\\ell \\ge 3$ that $t(C_\\ell , T^*)=\\frac{1}{2^\\ell }\\Big (\\lfloor z^{-1}\\rfloor z^\\ell +(1-\\lfloor z^{-1}\\rfloor z)^\\ell \\Big )+o(1), \\mbox{ ~~ where } o(1)\\rightarrow 0 \\mbox{ as } n\\rightarrow \\infty .$ The authors [11] also pointed out that the structure of this conjectured extremal configuration resembles those of the famous Erdős-Rademacher problem [6] on the minimum number of triangles in a graph of a given number of vertices and edges (see [3] for more discussion).", "To state the above conjecture in a precise formula, define a function $g_\\ell :[0,1/8]\\rightarrow [0,1]$ for any integer $\\ell \\ge 4$ as follows: Let $g_\\ell (0)=0$ and for any real $z\\in (0,1],$ let $&g_\\ell \\left(\\frac{1}{8}\\Big (\\lfloor z^{-1}\\rfloor z^3+(1-\\lfloor z^{-1}\\rfloor z)^3\\Big )\\right)= \\frac{1}{2^\\ell }\\Big (\\lfloor z^{-1}\\rfloor z^\\ell +(1-\\lfloor z^{-1}\\rfloor z)^\\ell \\Big ).$ It is worth noting that the function $g_\\ell (\\cdot )$ is continuous and increasing on $[0,1/8]$ .", "Conjecture 1.1 (Linial and Morgenstern [11], Conjecture 2.2) Every tournament $T$ satisfies that $t(C_4,T)\\ge g_4(t(C_3,T))+o(1),$ where the $o(1)$ term goes to zero as $\|V(T)\|$ goes to infinity.", "Linial and Morgenstern [11] proved several other general bounds between the densities of $C_3$ and $C_4$ in tournaments.", "In particular, some of these imply that this conjecture holds for tournaments $T$ with $t(C_3,T)$ asymptotically equal to $0, 1/8$ and $1/32$ .", "They [11] also asked to understand the relationships among the higher densities $t(C_\\ell , T)$ .", "Very recently, Chan, Grzesik, Král' and Noel [3] introduced spectral analysis of adjacency matrices of tournaments in this study.", "They proved the following result, which makes a breakthrough towards Conjecture REF .", "Theorem 1.2 (Chan, Grzesik, Král' and Noel [3]) Conjecture REF holds for $t(C_3,T)\\in [\\frac{1}{72},\\frac{1}{8}]$ .", "Moreover, they [3] developed a limit theory for tournaments and used it to characterize the asymptotic structure of all extremal tournaments $T$ with $t(C_3,T)\\in [1/32, 1/8]$ for Conjecture REF .", "In this paper, we study the minimum number of cycles in tournaments $T$ with fixed density $t(C_3,T)$ of 3-cycles.", "The following is our main result, which exhibits an analog of Theorem REF for a cycle of length not of the form $4k+2$ in tournaments that are “close” to be regular.", "Theorem 1.3 Let $T$ be a tournament and $\\ell \\ge 4$ be an integer with $\\ell \\lnot \\equiv 2\\mod {4}$ .", "If $t(C_3,T)\\ge \\frac{1}{8}-\\frac{1}{10\\ell ^2}$ , then it holds that $t(C_\\ell ,T)\\ge g_\\ell (t(C_3,T))+o(1),$ where the $o(1)$ term goes to zero as $\|V(T)\|$ goes to infinity.", "By the same random blow-up example, we see that the above lower bound for any cycle $C_\\ell $ of length not of the form $4k+2$ is asymptotically tight.", "We conjecture that similarly as in Conjecture REF , the bound in Theorem REF can be extended to any value of $t(C_3,T)$ .", "The same minimizing problem for cycles of length of the form $4k+2$ is much more complicated.", "For instance, it is still an open problem (see [12]) to determine the asymptotic minimum density $t(C_6,T)$ of 6-cycles even in regular tournaments $T$ , i.e., $t(C_3,T)=1/8$ .", "On the other hand, we do know that the analogous bound in Theorem REF for these cycles does not hold for any value of $t(C_3,T)$ (see Lemma REF ).", "For any $\\ell $ of the form $4k+2$ , we will describe an explicit family of tournaments $T$ with $t(C_\\ell , T)< g_\\ell (t(C_3,T))$ for any value of $t(C_3,T)$ , and conjecture that this family provides the optimal value for the minimum $t(C_\\ell , T)$ .", "Using results of [8], one also can give a close estimation on the minimum $t(C_\\ell , T)$ for regular tournaments, which provides evidences for supporting the above conjecture; see Lemma REF .", "We now turn back to Theorem REF .", "The proof of Theorem REF is motivated by the one in [3], which used spectral analysis on adjacency matrices of tournaments.", "Through the spectral analysis, Theorem REF can be reduced to a list of optimization problems (see Subsection REF ), whose constraints are non-linear non-convex polynomials of degree at most three and its objective function is a multinomial of degree $\\ell $ .", "The base case of these optimization problems is a minimization problem on the interplay between $q$ -norms of probability vectors (see Theorem REF below).", "A probability vector $\\vec{w}$ is a vector consisting of entries $w_i$ for each integer $i\\ge 1$ , where all $w_i$ 's are non-negative and add up to exactly one.", "Linial and Morgenstern [11] proved that for any real $0<C<1$ , the minimum of $\\sum w_i^4$ among all probability vectors $\\vec{w}$ satisfying $\\sum w_i^3=C$ is attained by letting $w_1=...=w_{m}> w_{m+1}\\ge 0$ and $w_i=0$ for any $i>m+1$ ,Lemma 2.7 in [11] also requires that $m$ is the smallest integer among all possible such choices.", "However, as we shall explain in Section (in the proof of Theorem REF ) that $\\vec{w}$ satisfying $\\sum w_i=1, \\sum w_i^p=C$ , $w_1=\\cdots =w_{m}> w_{m+1}\\ge 0$ and $w_i=0$ for any $i\\ge m+2$ is unique, which only depends on $p$ and $C$ .", "which was used in the proof of Theorem REF in [3].", "Linial and Morgenstern [11] also raised the following “natural sounding” question: find the smallest $q$ -norm among all probability vectors of given $p$ -norm for any integers $q>p>1$.", "We answer this in the following strengthening, which holds for any reals $q>p>1$ .", "(We should postpone the definition of the function $f_{p,q}$ in Section .)", "Theorem 1.4 For any reals $q>p>1$ and $C\\in (0,1)$ , consider all probability vectors $\\vec{w}$ satisfying $\\sum w_i^p=C$ .", "Then the minimum of $\\sum w_i^q$ among all such vectors equals $f_{p,q}(C)$ , which is attained by letting $w_1=w_2=\\cdots =w_{m}> w_{m+1}\\ge 0$ and $w_i=0$ for any $i\\ge m+2$ .Perhaps it is worth pointing out that it is not trivial to see that there always exists an optimal vector to this infinite-dimensional optimization problem.", "After settling this base case in the list of optimization problems of Theorem REF , we apply mathematical induction to solve other optimization problems.", "The difficulty we face in the inductive step is that unlike in [3], we cannot use the method of Lagrange multipliers because we find that for longer cycles, the Lagrange points of these optimization problems are not unique and their forms are very complicated due to the high degree multinomial objective function.", "To overcome this, we focus on a shorter interval for $t(C_3,T)$ and turn to some analytic method to complete the proof of Theorem REF .", "The rest of the paper is organized as follows.", "In Section , we prove Theorem REF .", "In Section , we prove Theorem REF .", "In Section , we consider cycles $C_\\ell $ for $\\ell =4k+2$ , by giving a family of tournaments $T$ with $t(C_\\ell , T)< g_\\ell (t(C_3,T))$ for every value of $t(C_3,T)$ (see Lemma REF ) and an asymptotic form of the minimum $t(C_\\ell , T)$ as $\\ell \\rightarrow \\infty $ for regular tournaments $T$ (see Lemma REF ).", "We conclude with two conjectures concerning the minimization problem discussed here." ], [ "The optimization problem of Linial-Morgenstern", "In this section we prove Theorem REF , which considers an infinite-dimensional minimization problem.", "We should first prove an analogous finite-dimensional version (see Theorem REF ), and then show how to use this finite-dimensional version to derive Theorem REF .", "We also give a direct corollary of Theorem REF at the end of this section.", "Let $q>p>1$ be two reals.", "We define the function $f_{p,q}(\\cdot ): [0,1]\\rightarrow [0,1]$ as follows: Let $f_{p,q}(0)=0$ and for any $z\\in (0,1]$ , let $f_{p,q}\\left(\\lfloor z^{-1}\\rfloor z^p+(1-\\lfloor z^{-1}\\rfloor z)^p\\right)=\\lfloor z^{-1}\\rfloor z^q+(1-\\lfloor z^{-1}\\rfloor z)^q.$ We point out that for any real $p>1$ , $z\\mapsto \\lfloor z^{-1}\\rfloor z^p+(1-\\lfloor z^{-1}\\rfloor z)^p$ is a strictly increasing continuous bijection from $(0,1]$ to $(0,1]$ .", "From this, it is easy to see that $f_{p,q}(\\cdot )$ is a strictly increasing continuous function on $[0,1]$ .", "Also let us notice that $g_\\ell (s)=f_{3,\\ell }(8s)/2^\\ell $ for any $s\\in [0,1/8]$ ." ], [ "A finite-dimensional version", "Theorem 2.1 For any reals $q>p>1$ , $C\\in (0,1)$ and positive integer $k$ with $\\frac{1}{k^{p-1}}\\le C$ , consider all vectors $\\vec{w}=(w_1,...,w_k)$ satisfying $\\sum _{i=1}^{k}w_i=1, \\sum _{i=1}^{k}w_i^p=C$ and $w_i\\ge 0$ for each $i\\in [k]$ .", "Then the minimum of $\\sum _{i=1}^{k}w_i^q$ among all such vectors equals $f_{p,q}(C)$ , which is attained by letting $w_1=w_2=\\cdots =w_{m}> w_{m+1}\\ge 0$ and $w_i=0$ for any $i\\ge m+2$ .", "First, let us note that optimal solutions to the minimization problem in Theorem REF always exist, by the following two facts: (i) Since the function $(w_1,\\dots ,w_k)\\rightarrow \\sum _{i=1}^{k}w_i^p$ is continuous, and $\\frac{k}{k^p}\\le C< 1$ , there exists a vector $(w_1,w_2,\\cdots ,w_k)$ satisfying $\\sum _{i=1}^{k}w_i=1$ , $\\sum _{i=1}^{k}w_i^p=C$ and $w_i\\ge 0$ for each $i\\in [k]$ .", "Thus, the feasible region is not empty.", "(ii) It is clear that the feasible region for $\\vec{w}$ is compact.", "To prove Theorem REF , we need to establish some lemmas.", "The first one is a generalization of the well-known fact that the determinant of a Vandermonde matrix is non-zero.", "Lemma 2.2 For reals $0<c_1<c_2<...<c_n$ and $s_1<s_2<...<s_n$ , let $a_{i,j}=c_j^{s_i}$ for any $1\\le i, j\\le n$ .", "Then the rank of the matrix $\\lbrace a_{i,j}\\rbrace _{n\\times n}$ is $n$ .", "We prove it by induction on $n$ .", "First, it holds trivially when $n=1$ .", "Now suppose it holds for $n\\le k-1$ for some $k\\ge 2$ and assume by contradiction that when $n=k$ , there exists such a matrix $A=\\lbrace c_j^{s_i}\\rbrace _{k\\times k}$ whose rank is less than $k$ .", "Then there exists a non-zero column vector $\\vec{\\lambda }=(\\lambda _1,...,\\lambda _k)^T$ such that $\\lambda ^T A=(0,0,...,0)$ .", "Thus, the function $\\sum _{i=1}^{k}\\lambda _ix^{s_i}$ has $k$ distinct zero points $c_1, c_2,..., c_k$ .", "This also means that $\\sum _{i=1}^{k}\\lambda _ix^{s_i-s_1}$ has $k$ distinct zero points $c_1, c_2,..., c_k$ .", "By Lagrange's mean value Theorem, the function $\\sum _{i=2}^{k}\\lambda _i(s_i-s_1)x^{s_i-s_1-1}$ has $k-1$ distinct zero points $d_1,..., d_{k-1}$ , where $d_i\\in (c_i,c_{i+1})$ .", "Let $\\mu _i=\\lambda _{i+1}(s_{i+1}-s_1)$ and $t_i=s_{i+1}-s_1-1$ for each $i\\in [k-1]$ .", "Then for the matrix $B=\\lbrace d_j^{t_i} \\rbrace _{(k-1)\\times (k-1)}$ and the vector $\\vec{\\mu }=(\\mu _1,...,\\mu _{k-1})^T$ , we have $\\vec{\\mu }^T B=(0,0,...,0)$ .", "If $\\vec{\\mu }=\\vec{0}$ , then we see $\\lambda _i=0$ for any $i\\in \\lbrace 2,3,...k \\rbrace $ .", "As $\\sum _{i=1}^{k}\\lambda _ix^{s_i}$ has $k$ distinct zero points, it forces $\\lambda _1=0$ and thus $\\vec{\\lambda }=\\vec{0}$ , which is a contradiction.", "Therefore $\\vec{\\mu }\\ne \\vec{0}$ , which implies that the rank of $B$ is less than $k-1$ .", "This is contradictory to our induction hypothesis.", "Let $\\vec{w}=(w_1,...,w_k)$ denote an optimal vector to the minimization problem in Theorem REF such that $w_1\\ge w_2\\ge \\cdots \\ge w_k\\ge 0$ .", "The next lemma says that there are at most two distinct positive real numbers that the entries of $\\vec{w}$ can take.", "Lemma 2.3 There exist two reals $1>a>b>0$ such that for any $i\\in [k]$ , $w_i\\in \\lbrace 0,b,a\\rbrace $ Without loss of generality, we may assume by contradiction that there are $w_1>w_2>w_3>0$ .", "The method of Lagrange multipliers says that, at the optimal vector, the gradient of the objective function is perpendicular to the tangent plane of the feasible region.", "This implies that the rank of the following matrix $ \\begin{pmatrix}1&1&1\\\\w_1^{p-1}&w_2^{p-1}&w_3^{p-1}\\\\w_1^{q-1}&w_2^{q-1}&w_3^{q-1}\\end{pmatrix}$ is less than 3.", "By Lemma REF , we get a contradiction, completing this proof.", "The following lemma says more about the optimal vector $\\vec{w}$ and is key for Theorem REF .", "Lemma 2.4 There exist no reals $1>a>b>0$ and integers $i,j,\\ell \\in [k]$ such that $w_i=a,w_j=b$ and $w_\\ell =b$ .", "We assume by contradiction that $w_i=a, w_j=b$ and $w_\\ell =b$ .", "Our proof strategy is to find a new vector in the feasible region which reduces the objective function.", "We consider two cases separately, depending on whether $2(\\frac{a+2b}{2})^p\\ge a^p+2b^p$ or not.", "Case 1.", "$2(\\frac{a+2b}{2})^p\\ge a^p+2b^p$ .", "Let $f:[0,1)\\rightarrow \\mathbf {R}$ be such that $f(x)=(a-2x)^p+2(b+x)^p$ .", "We have $f^{\\prime }(x)=2p(b+x)^{p-1}-2p(a-2x)^{p-1}$ .", "So $f^{\\prime }(x)<0$ on $[0,\\frac{a-b}{3})$ and $f^{\\prime }(x)>0$ on $(\\frac{a-b}{3},\\frac{a}{2}]$ .", "We also have $f(0)=a^p+2b^p$ and $f(\\frac{a}{2})=2(\\frac{a+2b}{2})^p\\ge a^p+2b^p$ .", "Therefore, there exists one and only one $x_0\\in (\\frac{a-b}{3},\\frac{a}{2}]$ such that $f(x_0)=a^p+2b^p$ .", "Let $c=a-2x_0$ , $d=b+x_0$ .", "Then we have $0\\le c<d, ~ c+2d=2b+a \\mbox{ ~and~ } c^p+2d^p=a^p+2b^p.$ We claim that $0\\le c<b<d<a$ .", "To see this, first note that by the strict convexity of $x\\mapsto x^p$ , we have $f(\\frac{a-b}{2})=b^p+2(\\frac{a+b}{2})^p<b^p+(b^p+a^p)=f(0)$ .", "Then $f(x)<a^p+2b^p=f(x_0)$ on $(0,\\frac{a-b}{2}]$ .", "This implies $x_0>\\frac{a-b}{2}$ and thus $c<b<d$ .", "It remains to show $d<a$ .", "If $\\frac{a+2b}{2}<a$ , then $d=b+x_0\\le b+\\frac{a}{2}<a$ , as desired.", "So we may assume $\\frac{a+2b}{2}\\ge a$ , i.e., $a-b\\le \\frac{a}{2}$ .", "Using the strict convexity of $x\\mapsto x^p$ again, $f(a-b)=(2b-a)^p+2a^p> 2b^p+a^p=f(x_0)$ .", "Because $f^{\\prime }(x)>0$ on $[a-b,\\frac{a}{2}]$ , we have $f(x)\\ge f(a-b)>f(x_0)$ for all $x\\in [a-b,\\frac{a}{2}]$ .", "This implies that $x_0<a-b$ and thus $d=b+x_0<a$ , proving the claim.", "We want to replace $(a,b,b)$ with $(d,d,c)$ in $\\vec{w}$ while reducing the objective function.", "In order to do this, we consider the following function $h(x)$ : $\\mbox{Let } h(x)=c^x+2d^x-a^x-2b^x \\mbox{ if } c>0, \\mbox{ and let } h(x)=2d^x-a^x-2b^x \\mbox{ if } c=0.$ Now we argue that $h(q)<0$ .", "Assume by contradiction that $h(q)\\ge 0$ .", "Using the fact that $0\\le c<b<d<a$ , when $x$ is sufficiently large, we have $h(x)<0$ .", "So there exists some real $s\\ge q$ such that $h(s)=0$ .", "By (REF ), we also have $h(p)=h(1)=0$ .", "If $c>0$ , then $h(0)=0$ and thus $h(x)$ has four zero points $0<1<p<s$ .", "This implies that $\\begin{pmatrix}1&1&1&1\\\\c&b&d&a\\\\c^p&b^p&d^p&a^p\\\\c^s&b^s&d^s&a^s\\end{pmatrix}\\begin{pmatrix}1\\\\-2\\\\2\\\\-1\\end{pmatrix}=\\begin{pmatrix}0\\\\0\\\\0\\\\0\\end{pmatrix}.$ Then the rank of the matrix above is less than 4, which is contradictory to Lemma REF .", "So $c=0$ .", "Then $h(x)$ has three zero points $1<p<s$ , which imply that $\\begin{pmatrix}b&d&a\\\\b^p&d^p&a^p\\\\b^s&d^s&a^s\\end{pmatrix}\\begin{pmatrix}-2\\\\2\\\\-1\\end{pmatrix}=\\begin{pmatrix}0\\\\0\\\\0\\end{pmatrix}.$ Again, this is contradictory to Lemma REF .", "We now complete the proof of $h(q)<0$ .", "Therefore, $c^q+2d^q<a^q+2b^q$ .", "Let $\\vec{w}^{\\prime }$ be obtained from $\\vec{w}$ by replacing $(a,b,b)$ with $(c,d,d)$ .", "Clearly, $\\vec{w}^{\\prime }$ is still in the feasible region, but it reduces the objective function.", "This contradicts that $\\vec{w}$ is an optimal vector, completing the proof for Case 1.", "Case 2.", "$2(\\frac{a+2b}{2})^p<a^p+2b^p$ .", "We first prove that $a>2b$ .", "Let $y(x)=x^p+2-2(\\frac{x}{2}+1)^p$ .", "So $y^{\\prime }(x)=px^{p-1}-p(\\frac{x}{2}+1)^{p-1}$ .", "This shows that $y(x)$ is monotone decreasing on $(0,2)$ and monotone increasing on $(2,+\\infty )$ .", "So $y(x)<y(0)=0$ for any $x\\in (0,2]$ .", "Using the condition of Case 2, $y(\\frac{a}{b})=(\\frac{a}{b})^p+2-2(\\frac{a}{2b}+1)^p>0$ , which implies that $\\frac{a}{b}>2$ .", "So $a>2b$ and moreover, $a+2b>a>\\frac{a+2b}{2}$ .", "Let $g(x)=x^p+(a+2b-x)^p-a^p-2b^p$ .", "Since $g^{\\prime }(x)=px^{p-1}-p(a+2b-x)^{p-1}$ , it is easy to see that $g(x)$ is monotone increasing on $[\\frac{a+2b}{2},a+2b]$ .", "Because $g(a)=a^p+(2b)^p-a^p-2b^p>0$ and $g(\\frac{a+2b}{2})=2(\\frac{a+2b}{2})^p-a^p-2b^p<0$ , there exists a unique $x_0\\in (\\frac{a+2b}{2},a)$ such that $g(x_0)=0$ .", "Let $c=a+2b-x_0$ and $d=x_0$ .", "Then one can derive that $0<b<c<d<a, ~ c+d=2b+a \\mbox{ ~and~ } c^p+d^p=a^p+2b^p.$ In this case, we want to replace $(a,b,b)$ with $(d,c,0)$ in $\\vec{w}$ , while reducing the objective function.", "For this purpose, we consider the function $h(x)=(\\frac{c}{a})^x+(\\frac{d}{a})^x-1-2(\\frac{b}{a})^x.$ We claim that $h(x)>0$ holds for any $x\\in (1,p)$ .", "Assume by contradiction that there exists $s\\in (1,p)$ such that $h(s)\\le 0$ .", "By (REF ), $h(1)=h(p)=0$ .", "Let $\\beta $ be a minimum point of $h(x)$ in $[1,p]$ .", "Because there exists $s\\in (1,p)$ with $h(s)\\le 0$ , we can choose $\\beta \\in (1,p)$ with $h(\\beta )\\le 0$ and $h^{\\prime }(\\beta )=0$ .", "Let $\\alpha \\in [0,\\beta ]$ be a maximum point of $h(x)$ on $[0,\\beta ]$ .", "As $h(0)=-1<0$ and there exists some $s^{\\prime }\\in (0,\\beta )$ with $h(s^{\\prime })\\ge 0\\ge \\max \\lbrace h(0), h(\\beta )\\rbrace $ ,Clearly, here one can take $s^{\\prime }=1$ .", "we can choose $\\alpha \\in (0,\\beta )$ with $h^{\\prime }(\\alpha )=0$ .", "As $\\lim _{x\\rightarrow +\\infty }h(x)=-1<0$ , there exists $t>p$ such that $h(t)<0$ .", "Let $\\gamma $ be a maximum point of $h(x)$ on $[\\beta ,t]$ .", "Since $h(p)\\ge 0\\ge \\max \\lbrace h(\\beta ), h(t)\\rbrace $ where $p\\in (\\beta , t)$ , we can choose $\\gamma \\in (\\beta ,t)$ with $h^{\\prime }(\\gamma )=0$ .", "For $0<\\alpha <\\beta <\\gamma $ , we have $h^{\\prime }(\\alpha )=h^{\\prime }(\\beta )=h^{\\prime }(\\gamma )=0$ , implying that $\\begin{pmatrix}(\\frac{b}{a})^{\\alpha }&(\\frac{c}{a})^{\\alpha }&(\\frac{d}{a})^{\\alpha }\\\\(\\frac{b}{a})^{\\beta }&(\\frac{c}{a})^{\\beta }&(\\frac{d}{a})^{\\beta }\\\\(\\frac{b}{a})^{\\gamma }&(\\frac{c}{a})^{\\gamma }&(\\frac{d}{a})^{\\gamma }\\end{pmatrix}\\begin{pmatrix}-2\\log \\frac{b}{a}\\\\\\log \\frac{c}{a}\\\\\\log \\frac{d}{a}\\end{pmatrix}=\\begin{pmatrix}0\\\\0\\\\0\\end{pmatrix}.$ Then the rank of the matrix above is less than 3, a contradiction to Lemma REF .", "Next we show that $h(x)<0$ holds for any $x\\in (p,+\\infty )$ .", "Assume by contradiction that there exists some $s\\in (p,+\\infty )$ such that $h(s)\\ge 0$ .", "Because $\\lim _{x\\rightarrow +\\infty }h(x)=-1$ , there exists $t\\in (s,+\\infty )$ such that $h(t)<0$ .", "Let $\\alpha $ be a maximum point of $h(x)$ on $[1,p]$ .", "By the previous paragraph, we can choose $\\alpha \\in (1,p)$ with $h(\\alpha )>0$ and $h^{\\prime }(\\alpha )=0$ .", "Let $\\gamma $ be a maximum point of $h(x)$ on $[p,t]$ .", "Because $h(s)\\ge 0\\ge \\max \\lbrace h(p),h(t)\\rbrace $ where $s\\in (p,t)$ , we can choose $\\gamma \\in (p,t)$ with $h(\\gamma )\\ge 0$ and $h^{\\prime }(\\gamma )=0$ .", "Lastly, let $\\beta $ be a minimum point of $h(x)$ on $[\\alpha ,\\gamma ]$ .", "Similarly, we can choose $\\beta \\in (\\alpha ,\\gamma )$ with $h^{\\prime }(\\beta )=0$ .", "For $0<\\alpha <\\beta <\\gamma $ , we have $h^{\\prime }(\\alpha )=h^{\\prime }(\\beta )=h^{\\prime }(\\gamma )=0$ .", "This implies that $\\begin{pmatrix}(\\frac{b}{a})^{\\alpha }&(\\frac{c}{a})^{\\alpha }&(\\frac{d}{a})^{\\alpha }\\\\ (\\frac{b}{a})^{\\beta }&(\\frac{c}{a})^{\\beta }&(\\frac{d}{a})^{\\beta }\\\\(\\frac{b}{a})^{\\gamma }&(\\frac{c}{a})^{\\gamma }&(\\frac{d}{a})^{\\gamma }\\end{pmatrix}\\begin{pmatrix}-2\\log \\frac{b}{a}\\\\\\log \\frac{c}{a}\\\\\\log \\frac{d}{a}\\end{pmatrix}=\\begin{pmatrix}0\\\\0\\\\0\\end{pmatrix}.$ By Lemma REF again, we get a contradiction.", "In particular, $h(q)<0$ , which means $c^q+d^q<a^q+2b^q$ .", "Then we can replace $(a,b,b)$ with $(d,c,0)$ in $\\vec{w}$ .", "The new vector is still in the feasible region but reduces the objective function.", "Thus we get a contradiction.", "This completes the proof of Lemma REF .", "Now we are ready to prove Theorem REF .", "[Proof of Theorem REF .]", "Let $\\vec{w}=(w_1,...,w_k)$ be an optimal vector to the minimization problem in Theorem REF such that $\\sum _{i=1}^k w_i=1, \\sum _{i=1}^k w_i^p=C$ and $w_1\\ge w_2\\ge \\cdots \\ge w_k\\ge 0$ .", "By Lemmas REF and REF , we may assume that there exists some $m$ such that $w_1=\\cdots =w_{m}> w_{m+1}\\ge 0$ and $w_i=0$ for any $i\\ge m+2$ .", "We show that $\\vec{w}$ satisfying the above additional condition is uniquely determined by $p$ and $C$ .", "Set $z=w_1$ .", "Since $0\\le w_{m+1}<z$ , we have $mz\\le 1=\\sum w_i<(m+1)z$ , implying that $z^{-1}-1<m\\le z^{-1}$ , i.e., $m=\\lfloor z^{-1}\\rfloor $ .", "So $w_{m+1}=1-\\lfloor z^{-1}\\rfloor z$ .", "Using the restriction $\\sum _i w_i^p=C$ , we get that $\\lfloor z^{-1}\\rfloor z^p+(1-\\lfloor z^{-1}\\rfloor z)^p=C.$ Note that $z\\mapsto \\lfloor z^{-1}\\rfloor z^p+(1-\\lfloor z^{-1}\\rfloor z)^p$ is a strictly increasing continuous bijection from $(0,1]$ to $(0,1]$ .", "So there exists a unique solution $z\\in (0,1)$ to the above equation.", "This proves that such an optimal vector $\\vec{w}$ is unique, which gives the minimum of $\\sum _{i=1}^k w_i^q$ to be $\\lfloor z^{-1}\\rfloor z^q+(1-\\lfloor z^{-1}\\rfloor z)^q=f_{p,q}(C)$ , completing the proof of Theorem REF ." ], [ "Proof of Theorem ", "We move on to prove Theorem REF .", "The proof will use the fact that the minimum value $f_{p,q}(C)$ of Theorem REF is an increasing continuous function with the variable $C$ on $[0,1]$ .", "[Proof of Theorem REF .]", "Consider any probability vector $\\vec{w}$ satisfying $\\sum w_i^p=C$ .", "Take any small constant $\\epsilon >0$ and choose $k\\in {\\bf N}^+$ to be sufficiently large such that $t:=\\sum _{i=1}^{k}w_i\\ge 1-\\epsilon $ and $\\frac{1}{k^{p-1}}\\le C/2\\le C-\\epsilon $ .", "Then we deduce that $\\sum _{i=k+1}^{\\infty }w_i\\le \\epsilon $ and $\\sum _{i=k+1}^{\\infty }w_i^p\\le \\epsilon $ , implying that $\\sum _{i=1}^{k}w_i^p\\ge C-\\epsilon $ .", "Now we normalize $w_i$ by letting $v_i:=w_i/t$ for each $i\\in [k]$ .", "Then $\\sum _{i=1}^{k}v_i=1$ and as $t\\le 1$ , $\\sum _{i=1}^{k}v_i^p=\\left(\\sum _{i=1}^{k}w_i^p \\right)/t^p\\ge C-\\epsilon $ .", "By Theorem REF , we have $\\sum _{i=1}^{\\infty }w_i^q\\ge \\sum _{i=1}^{k}w_i^q=t^q\\cdot \\sum _{i=1}^{k}v_i^q\\ge t^q\\cdot f_{p,q}\\left(\\sum _{i=1}^{k}v_i^p\\right)\\ge (1-\\epsilon )^q \\cdot f_{p,q}\\left(C-\\epsilon \\right).$ Using the continuity of $f_{p,q}$ , by letting $\\epsilon \\rightarrow 0$ we can get $\\sum _{i=1}^{\\infty }w_i^q\\ge f_{p,q}(C)$ .", "Now let $\\vec{w}_0$ be a probability vector with $z=w_1=\\cdots =w_{m}> w_{m+1}\\ge 0$ and $w_i=0$ for any $i\\ge m+2$ , where $m=\\lfloor z^{-1}\\rfloor $ and $z$ is from (REF ).", "Then such $\\vec{w}$ satisfies that $\\sum _{i=1}^{\\infty }w_i^p=C$ and $\\sum _{i=1}^{\\infty }w_i^q= f_{p,q}(C)$ .", "So the minimum value in Theorem REF exists and equals $f_{p,q}(C)$ .", "Now we present a corollary of Theorem REF , which will be used frequently in Section .", "Corollary 2.5 Let $n, \\ell $ be positive integers and $x_1,\\dots ,x_n$ be any non-negative reals such that $\\ell \\ge 4$ and $\\sum _{i=1}^n x_i=\\frac{1}{2}$ .", "Then $\\sum _{i=1}^{n} x_i^\\ell \\ge g_\\ell \\Big (\\sum _{i=1}^{n}x_i^3\\Big ).$ By definition of $g_\\ell (\\cdot )$ , together with Theorem REF , we have that for any $s\\in [0,1/8]$ , $g_\\ell (s)&=\\frac{f_{3,\\ell }(8s)}{2^\\ell }=\\min \\left\\lbrace \\frac{1}{2^\\ell }\\sum _i w_i^{\\ell }\\ \|\\ \\sum _i w_i^3=8s,\\ \\sum _i w_i=1,\\ w_i\\ge 0 \\ \\mbox{for any}\\ i\\ge 1 \\ \\right\\rbrace \\\\&= \\min \\left\\lbrace \\sum _i x_i^{\\ell }\\ \|\\ \\sum _i x_i^3=s,\\ \\sum _i x_i=\\frac{1}{2},\\ x_i\\ge 0 \\ \\mbox{for any}\\ i\\ge 1 \\ \\right\\rbrace .$ This implies the desired inequality." ], [ "Cycles of given length in tournaments", "In this section, we prove Theorem REF by using the approach based on the spectrum analysis on the adjacency matrix of a tournament, which was initialed in [3], [8]." ], [ "Some preliminaries", "First we give some algebraic notation on a tournament $T$ .", "Let $V(T)=[n]$ .", "For any $i,j\\in [n]$ , we write $ij$ for an arc with head $i$ and tail $j$ .", "The adjacency matrix of $T$ is an $n\\times n$ matrix $M=(m_{i,j})$ , where $m_{i,j}=1$ if $ij\\in E(T)$ and $m_{i,j}=0$ otherwise.", "It is easy to see that $t(C_\\ell ,T)=tr(M^\\ell )/n^\\ell ,$ where $tr(X)$ denotes the trace of the matrix $X$ .", "Let $I$ denote the $n\\times n$ identity matrix.", "We call $A:=(\\frac{1}{2}I+M)/n$ the tournament matrix of $T$ .", "It follows that $A+A^T=J$ , where $J$ is the $n\\times n$ matrix with each entry being $\\frac{1}{n}$ .", "It is not difficult to see that $t(C_\\ell ,T)=tr(M^\\ell )/n^\\ell =tr(A^\\ell )+o(1),$ where $\\lim _{n\\rightarrow +\\infty }o(1)=0$ .", "The following fact will be crucial (see [2], or Lemma 2 in [3]).", "Lemma 3.1 Every eigenvalue of a tournament matrix has nonnegetive real part.", "Besides Corollary REF , we also need to use the following optimization.", "The proof of the case $p=3$ was given in [3] (see Lemma 9), which can be easily transformed to general cases.", "Lemma 3.2 For integers $p\\ge 2$ , $n\\ge 1$ and any real $1>t>0$ such that $nt\\ge 1$ , consider all vectors $(w_1,w_2,...,w_n) $ satisfying $\\sum _{i=1}^{n}w_i=1$ and $w_i\\in [0,t]$ for every $i\\in [n]$ .", "Then the maximum of $\\sum _{i=1}^{n}w_i^p$ is attained by letting $w_1=w_2=...=w_{\\lfloor t^{-1}\\rfloor }=t$ , $w_{\\lfloor t^{-1}\\rfloor +1}=1-t\\lfloor t^{-1}\\rfloor $ and $w_j=0$ for any $j\\ge \\lfloor t^{-1}\\rfloor +2$ .", "For a complex number $z$ , we denote its real part by $\\operatorname{Re~}z$ and its absolute value by $\|z\|$ .", "We write the imaginary unit by $\\textbf {i}$ .", "The following inequality was explicitly given in [8] (see the inequality $(9)$ therein).", "For completeness, we present a proof here.", "Lemma 3.3 For any odd integer $\\ell >0$ and complex number $z$ with $\\operatorname{Re~}z\\ge 0$ , it holds that $\\operatorname{Re~}z^\\ell \\le \\ell \|z\|^{\\ell -1} \\operatorname{Re~}z$ .", "Let $z=a+b\\textbf {i}$ for reals $a, b$ with $a\\ge 0$ .", "It suffices to consider $a>0$ and $b\\ge 0$ .", "Write $b/a=\\tan (\\alpha )$ for $\\alpha \\in [0,\\frac{\\pi }{2})$ .", "Then $\\operatorname{Re~}z^\\ell =\|z\|^{\\ell }\\cdot \\cos (\\ell \\alpha )$ .", "Since $\\ell $ is odd, it follows that $\|\\operatorname{Re~}z^\\ell \|=\|z\|^{\\ell }\\cdot \|\\sin \\ell (\\pi /2-\\alpha )\|\\le \|z\|^{\\ell }\\cdot \\ell \\cdot \|\\sin (\\pi /2-\\alpha )\|=\|z\|^{\\ell }\\cdot \\ell \\cdot \|\\cos \\alpha \|=\\ell \|z\|^{\\ell -1}a$ ." ], [ "Proof of Theorem ", "In view of Theorem REF , we may assume that $\\ell \\ge 5$ is an integer with $\\ell \\lnot \\equiv 2 \\mod {4}$ .", "Let $\\ell =4k+\\mu $ , where $\\mu \\in \\lbrace -1,0,1\\rbrace $ and let $T$ be any tournament satisfying $t(C_3,T)\\ge \\frac{1}{8}-\\frac{1}{10\\ell ^2}.$ Our goal is to show $t(C_\\ell ,T)\\ge g_\\ell (t(C_3,T))+o(1),$ where $\\lim _{\|V(T)\|\\rightarrow +\\infty }o(1)=0$ .", "Denote $A$ by its tournament matrix and let $\\sigma =tr(A^3)$ .", "By the Perron-Frobenius theorem, we may assume that the eigenvalues of $A$ are $\\rho ,r_1,\\dots ,r_t,a_1+b_1\\textbf {i},\\dots ,a_s+b_s\\textbf {i}$ , where $\\rho $ is the spectral radius of $A$ and $r_i, a_j, b_j$ are reals for each $i\\in [t]$ and $j\\in [s]$ .", "We now collect some properties on these eigenvalues.", "First, using Lemma REF together with the fact that $\\rho $ is the spectral radius, we see that $r_i\\in [0,\\rho ]$ for each $i\\in [t]$ and $a_j\\ge 0$ and $a_j^2+b_j^2\\le \\rho ^2$ for each $j\\in [s]$ .", "Since each diagonal element of $A$ is $\\frac{1}{2n}$ , we have $\\rho +\\sum _{i=1}^{t}r_i+\\sum _{j=1}^{s}a_j=\\operatorname{Re~}tr(A)=tr(A)=\\frac{1}{2}$ .", "Similarly, we have $\\rho ^3+\\sum _{i=1}^{t}r_i^3+\\sum _{j=1}^{s}(a_j^3-3a_jb_j^2)=\\operatorname{Re~}tr(A^3)=tr(A^3)=\\sigma ,$ $\\rho ^2+\\sum _{i=1}^{t}r_i^2+\\sum _{j=1}^{s}(a_j^2-b_j^2)=\\operatorname{Re~}tr(A^2)=tr(A^2)\\ge 0,$ where the second inequality implies that $\\sum _{j=1}^{s}b_j^2\\le \\rho ^2+\\sum _{i=1}^{t}r_i^2+\\sum _{j=1}^{s}a_j^2\\le (\\rho +\\sum _{i=1}^{t}r_i+\\sum _{j=1}^{s}a_j)^2=\\frac{1}{4}.$ Since $A$ is a real matrix, if $a_j+b_j\\textbf {i}$ is an eigenvalue of $A$ , then so is $a_j-b_j\\textbf {i}$ .", "Hence we have $2b_j^2\\le \\frac{1}{4}$ , implying that $b_j^2\\le \\frac{1}{8}$ for each $j\\in [s]$ .", "Table: NO_CAPTIONWe define the optimization problem $\\operatorname{OPT}_{\\ell }^\\sigma (s,t,\\rho )$ , for which the optimal solution $\\Phi _{\\ell }^{\\sigma }(s,t,\\rho )$ exists.", "Since the eigenvalues of $A$ satisfy the constraints in $\\operatorname{OPT}_{\\ell }^\\sigma (s,t,\\rho )$ , using (REF ) we have $t(C_\\ell ,T)=tr(A^\\ell )+o(1)=\\rho ^\\ell +\\sum _{i=1}^{t}r_i^\\ell +\\sum _{j=1}^{s}{\\operatorname{Re~}}(a_j+b_j\\textbf {i})^\\ell +o(1)\\ge \\Phi _{\\ell }^{\\sigma }(s,t,\\rho )+o(1),$ where $\\lim _{\|V(T)\|\\rightarrow +\\infty }o(1)=0$ .", "Also by (REF ), we have that $\\sigma =tr(A^3)=t(C_3,T)+o(1)\\ge \\frac{1}{8}-\\frac{1}{10\\ell ^2}+o(1)\\ge \\big (\\frac{1}{2}-\\frac{1}{80k^2}\\big )^3+\\big (\\frac{1}{80k^2}\\big )^3.$ We now devote the rest of this section to show the following statement: [itemindent=58pt] Statement ($\\star $ ).", "For any real $\\rho \\in [0,\\frac{1}{2}]$ and nonnegative integers $\\ell , t, s$ with $\\ell \\ge 5$ , $\\ell \\lnot \\equiv 2 \\mod {4}$ and $t+s\\ge 1$ , if $\\sigma \\ge (\\frac{1}{2}-\\frac{1}{80k^2})^3+(\\frac{1}{80k^2})^3$ , then $\\Phi _{\\ell }^{\\sigma }(s,t,\\rho )\\ge g_{\\ell }(\\sigma )$ .", "Before we process, we point out that to finish the proof of Theorem REF , it suffices to prove ($\\star $ ).", "Indeed, if ($\\star $ ) holds, then using (REF ) and the fact that $g_\\ell (\\cdot )$ is a continuous function, we can deduce the desired inequality as follows: $t(C_\\ell ,T)-o(1)\\ge \\Phi _{\\ell }^{\\sigma }(s,t,\\rho )\\ge g_\\ell (\\sigma )=g_\\ell \\big (t(C_3,T)+o(1)\\big )\\ge g_\\ell (t(C_3,T))+o(1).$ We prove the statement ($\\star $ ) by induction on $s$ .", "If $s=0$ , then applying Corollary REF directly, we can derive $\\Phi _{\\ell }^{\\sigma }(s,t,\\rho )\\ge g_{\\ell }(\\sigma )$ .", "So we may assume that $s\\ge 1$ and there exists some integer $m$ such that ($\\star $ ) holds for any $s<m$ .", "Now we consider when $s=m$ .", "Assume that $(r_1,\\dots ,r_t,a_1,b_1,\\dots ,a_m,b_m)$ is an optimal vector of $\\operatorname{OPT}_{\\ell }^\\sigma (m,t,\\rho )$ .", "Recall $\\ell =4k+\\mu $ , where $\\mu \\in \\lbrace -1, 0, 1\\rbrace $ .", "Suppose that $a_m=0$ .", "If $\\mu =0$ , then $\\operatorname{Re~}(a_m+b_m\\textbf {i})^{4k}=b_m^{4k}$ is minimized in the objective function when $b_m=0$ .", "If $\\mu \\in \\lbrace -1,1\\rbrace $ , then ${\\operatorname{Re~}}(a_m+b_m\\textbf {i})^{\\ell }={\\operatorname{Re~}}(b_m\\textbf {i})^{\\ell }=0$ , so by setting $b_m=0$ the new vector still satisfies all the constraints and the objective value remains unchanged.", "Therefore, when $a_m=0$ , we can always set $b_m=0$ and use induction directly.", "From now on, we may assume $a_m>0$ .", "Since $\\rho +\\sum _{i=1}^t r_i+\\sum _{j=1}^m a_j=\\frac{1}{2}$ and $0\\le r_i, a_j\\le \\rho $ for each $i\\in [t],j\\in [m]$ , we can deduce from Lemma REF that $\\Big (\\frac{1}{2}-\\frac{1}{80k^2}\\Big )^3+\\Big (\\frac{1}{80k^2}\\Big )^3\\le \\sigma \\le \\rho ^3+\\sum _{i=1}^{t}r_i^3+\\sum _{j=1}^{m}a_j^3 \\le \\left\\lfloor \\frac{1}{2\\rho }\\right\\rfloor \\rho ^3+\\Big (\\frac{1}{2}-\\rho \\left\\lfloor \\frac{1}{2\\rho }\\right\\rfloor \\Big )^3.$ This implies that $\\rho \\ge 1/2-1/80k^2 \\mbox{ and thus } a_j\\le 1/80k^2 \\mbox{ for each } j\\in [m].$ Let $\\sigma ^{\\prime }:=\\sigma +3a_mb_m^2$ .", "Shifting $\\sigma $ to $\\sigma ^{\\prime }$ and viewing $a_m$ as a new variable $r_{t+1}$ , we can obtain a new optimization problem $\\operatorname{OPT}_{\\ell }^{\\sigma ^{\\prime }}(m-1,t+1,\\rho )$ .", "Our proof idea in what follows is to compare the optimal values of the two optimization problems $\\operatorname{OPT}_{\\ell }^\\sigma (m,t,\\rho )$ and $\\operatorname{OPT}_{\\ell }^{\\sigma ^{\\prime }}(m-1,t+1,\\rho )$ by using analytic arguments.", "For this purpose, we introduce some definitions as below.", "For $\|x\|\\le \|b_m\|$ , let $\\sigma (x)=\\sigma +3a_mx^2 \\mbox{~~ and ~~} \\lambda (x)=\\frac{1}{3}\\sqrt{6\\sigma (x)-\\frac{3}{16}}.$ By the first two constraints of $\\operatorname{OPT}_{\\ell }^\\sigma (m,t,\\rho )$ , we have $\\frac{1}{32}\\le \\sigma \\le \\sigma (x)\\le \\sigma (b_m)\\le \\rho ^3+\\sum _{i=1}^{t}r_i^3+\\sum _{j=1}^{m}a_j^3\\le \\left(\\rho +\\sum _{i=1}^{t}r_i+\\sum _{j=1}^{m}a_j\\right)^3=\\frac{1}{8}.$ Solving $\\left\\lfloor z^{-1}\\right\\rfloor z^{3}+\\left(1-\\left\\lfloor z^{-1}\\right\\rfloor z\\right)^{3}=8\\sigma (x)$ , we get $\\left\\lfloor z^{-1}\\right\\rfloor =1$ and $z=\\frac{1}{2}+\\frac{2}{3}\\sqrt{6\\sigma (x)-\\frac{3}{16}}=\\frac{1}{2}+2\\lambda (x)$ .", "By definition of $g_{\\ell }$ , we have that for $\|x\|\\le \|b_m\|$ , $g_{\\ell }(\\sigma (x))=\\Big (\\frac{1}{4}-\\lambda (x)\\Big )^{\\ell }+\\Big (\\frac{1}{4}+\\lambda (x)\\Big )^{\\ell }.$ Now we apply induction hypothesis to the optimization problem $\\operatorname{OPT}_{\\ell }^{\\sigma ^{\\prime }}(m-1,t+1,\\rho )$ with variables $r_1,\\ldots ,r_t,r_{t+1},a_1,b_1,\\ldots ,a_{m-1},b_{m-1}$ , where $r_{t+1}:=a_m$ .", "Then it follows that $\\rho ^{\\ell }+\\sum _{i=1}^{t}r_i^{\\ell }+a_m^{\\ell }+\\sum _{j=1}^{m-1}\\operatorname{Re~}(a_j+b_j\\textbf {i})^{\\ell }\\ge g_{\\ell }(\\sigma ^{\\prime })=g_{\\ell }(\\sigma (b_m)).$ By considering a new function $F_{\\ell }(x):=g_{\\ell }(\\sigma (x))+\\operatorname{Re~}(a_m+x\\textbf {i})^\\ell -a_m^\\ell ,$ we see from the previous inequality that the objective value of $\\operatorname{OPT}_{\\ell }^{\\sigma }(m,t,\\rho )$ satisfies $\\Phi _{\\ell }^{\\sigma }(m,t,\\rho )=\\rho ^{\\ell }+\\sum _{i=1}^{t}r_i^{\\ell }+\\sum _{j=1}^{m}\\operatorname{Re~}(a_j+b_j\\textbf {i})^{\\ell }\\ge F_{\\ell }(b_m).$ To prove ($\\star $ ), it further reduces to show $F_{\\ell }(b_m)\\ge F_{\\ell }(0)=g_\\ell (\\sigma (0))=g_\\ell (\\sigma )$ .", "First, we consider the case when $\|b_m\|\\le \\frac{1}{5}$ .", "Note that we have the derivative $F^{\\prime }_{\\ell }(x)&=\\frac{2\\ell a_m x}{\\lambda (x)}\\left[ \\left(\\frac{1}{4}+\\lambda (x)\\right)^{\\ell -1}-\\left(\\frac{1}{4}-\\lambda (x)\\right)^{\\ell -1} \\right]+\\frac{d}{dx}\\operatorname{Re~}(a_m+x\\textbf {i})^{\\ell }\\\\&=\\frac{4\\ell a_m x}{\\lambda (x)}\\sum \\limits _{j=1}^{2k}\\dbinom{\\ell -1}{2j-1}\\left(\\frac{1}{4}\\right)^{\\ell -2j}\\lambda (x)^{2j-1}+\\sum \\limits _{j=1}^{2k}(-1)^j\\dbinom{\\ell }{2j}(2j)x^{2j-1}a_m^{\\ell -2j},$ where the index $j$ is at most $2k$ as $\\ell -1\\le 4k$ .", "Then it follows that for $x\\ge 0$ , $F^{\\prime }_{\\ell }(x)&\\ge \\frac{4\\ell a_m x}{\\lambda (x)}\\sum \\limits _{j=1}^{2k-1}\\dbinom{\\ell -1}{2j-1}\\left(\\frac{1}{4}\\right)^{\\ell -2j}\\lambda (x)^{2j-1}+\\sum \\limits _{j=1}^{2k-1}(-1)^j\\dbinom{\\ell }{2j}(2j)x^{2j-1}a_m^{\\ell -2j}\\\\&=\\ell \\cdot \\sum \\limits _{j=1}^{2k-1}\\dbinom{\\ell -1}{2j-1}x^{2j-1}a_m^{\\ell -2j}\\left[\\left(\\frac{1}{4a_m}\\right)^{\\ell -2j-1}\\left(\\frac{\\lambda (x)}{x}\\right)^{2j-2}+(-1)^j\\right].$ Using (REF ) we have $0<a_m\\le \\frac{1}{80k^2}\\le \\frac{1}{4}$ , implying $\\left(\\frac{1}{4a_m}\\right)^{\\ell -2j-1}\\ge 1$ , where each $\\ell -2j-1\\ge 4k-2-2j\\ge 0$ .", "Also since $\\sigma (x)\\ge \\sigma \\ge \\frac{73}{800}$ , this implies that $\\lambda (x)\\ge \\frac{1}{5}$ and thus for $\|x\|\\le \|b_m\|\\le \\frac{1}{5}$ , we have $\|\\frac{\\lambda (x)}{x}\|\\ge 1$ .", "Putting everything together, we can derive that $F^{\\prime }_{\\ell }(x)\\ge 0$ for all $\|x\|\\le \|b_m\|$ .", "Therefore, $\\Phi _{\\ell }^{\\sigma }(m,t,\\rho )\\ge F_\\ell (b_m)\\ge F_\\ell (0)=g_\\ell (\\sigma )$ , as wanted.", "It remains to consider $\|b_m\|>\\frac{1}{5}$ .", "As a fact we may assume $b_m>\\frac{1}{5}$ (because reversing the sign of $b_m$ would still satisfy all constrictions of $\\operatorname{OPT}_{\\ell }^\\sigma (m,t,\\rho )$ ).", "We claim that $F_{\\ell }(b_m)\\ge g_{\\ell }(\\sigma (b_m))-(\\ell +1)a_mb_m^{\\ell -1}.$ To see this, first consider $\\mu \\in \\lbrace -1,1\\rbrace $ , that is, $\\ell $ is odd.", "By Lemma REF , we have $F_{\\ell }(b_m)\\ge g_{\\ell }(\\sigma (b_m))-a_m^{\\ell }-\\ell \\cdot a_m\\left(\\sqrt{a_m^2+b_m^2}\\right)^{\\ell -1}\\ge g_{\\ell }(\\sigma (b_m))-(\\ell +1)a_mb_m^{\\ell -1},$ where the last inequality holds because $b_m>\\frac{1}{5}, a_m\\le \\frac{1}{80k^2}$ and $\\ell \\ge 5$ .", "Now consider $\\mu =0$ (that is $\\ell =4k$ ).", "Let $b_m/a_m=\\tan \\alpha $ for some $\\alpha \\in (0,\\frac{\\pi }{2})$ .", "Then we have $\\tan \\alpha >16k^2>\\frac{8k}{\\pi }\\ge \\frac{1}{\\tan (\\pi /8k)}=\\tan \\left(\\frac{\\pi }{2}-\\frac{\\pi }{8k}\\right).$ So $\\alpha \\in (\\frac{\\pi }{2}-\\frac{\\pi }{8k},\\frac{\\pi }{2})$ and $\\ell \\alpha =4k\\alpha \\in (2k\\pi -\\frac{\\pi }{2},2k\\pi )$ .", "This implies $\\operatorname{Re~}(a_m+b_m\\textbf {i})^\\ell =(\\sqrt{a_m^2+b_m^2})^{\\ell }\\cos (\\ell \\alpha )\\ge 0$ .", "Together with (REF ), we also can derive that $F_{\\ell }(b_m)\\ge g_\\ell (\\sigma (b_m))-a_m^\\ell \\ge g_{\\ell }(\\sigma (b_m))-(\\ell +1)a_mb_m^{\\ell -1},$ where the last inequality holds since $(b_m/a_m)^{\\ell -1}>16k^2>\\ell +1$ .", "This proves (REF ).", "By (REF ), we have $\\frac{d}{dx}g_{\\ell }(\\sigma (x))=\\frac{2\\ell a_m x}{\\lambda (x)}\\left[ \\left(\\frac{1}{4}+\\lambda (x)\\right)^{\\ell -1}-\\left(\\frac{1}{4}-\\lambda (x)\\right)^{\\ell -1} \\right]$ For any $x\\in [0,b_m]$ , we have $\\lambda (x)\\ge \\frac{1}{5}$ and thus $\\left(\\frac{1}{4}+\\lambda (x)\\right)^{\\ell -1}\\ge \\ell \\left(\\frac{1}{4}-\\lambda (x)\\right)^{\\ell -1}.$ Hence, $\\frac{d}{dx}g_{\\ell }(\\sigma (x))\\ge \\frac{2(\\ell -1) a_m x}{\\lambda (x)}\\left(\\frac{1}{4}+\\lambda (x)\\right)^{\\ell -1}.$ Since $w=\\frac{1}{4}+\\lambda (x)$ is the larger root of the equation $(\\frac{1}{2}-w)^3+w^3=\\sigma (x)$ , where $\\sigma (x)\\ge \\sigma \\ge (\\frac{1}{2}-\\frac{1}{80k^2})^3+(\\frac{1}{80k^2})^3$ , we can deduce that $\\frac{1}{2}>\\frac{1}{4}+\\lambda (x)\\ge \\frac{1}{2}-\\frac{1}{80k^2}.$ Then $&\\frac{d}{dx}g_{\\ell }(\\sigma (x))\\ge 8(\\ell -1)a_mx \\left(\\frac{1}{2}-\\frac{1}{80k^2}\\right)^{\\ell -1}$ for any $x\\in [0,b_m]$ .", "By integration, we have $g_{\\ell }(\\sigma (b_m))-g_{\\ell }(\\sigma )\\ge 4(\\ell -1)a_mb_m^2\\left(\\frac{1}{2}-\\frac{1}{80k^2}\\right)^{\\ell -1}.$ This together with (REF ) and (REF ) gives that $\\Phi _{\\ell }^{\\sigma }(m,t,\\rho )&\\ge F_{\\ell }(b_m)\\ge g_{\\ell }(\\sigma )+(\\ell -1)a_m\\left[4b_m^2\\left(\\frac{1}{2}-\\frac{1}{80k^2}\\right)^{\\ell -1}-\\frac{\\ell +1}{\\ell -1}b_m^{\\ell -1}\\right]\\\\&\\ge g_{\\ell }(\\sigma )+(\\ell -1)a_mb_m^{\\ell -1}\\left[\\frac{4}{b_m^{\\ell -3}} \\left(\\frac{39}{80}\\right)^{\\ell -1}-\\frac{\\ell +1}{\\ell -1}\\right]\\\\&\\ge g_{\\ell }(\\sigma )+(\\ell -1)a_mb_m^{\\ell -1}\\left[\\left(\\frac{39}{40}\\right)^2\\left(\\frac{39\\sqrt{2}}{40}\\right)^{\\ell -3}-\\frac{\\ell +1}{\\ell -1}\\right]>g_{\\ell }(\\sigma ),$ where the second last inequality follows by that $b_m^2\\le \\frac{1}{8}$ and the last inequality holds because $\\left(\\frac{39}{40}\\right)^2\\left(\\frac{39\\sqrt{2}}{40}\\right)^{\\ell -3}-\\frac{\\ell +1}{\\ell -1}$ increases for $\\ell \\ge 5$ and is at least $2\\left(\\frac{39}{40}\\right)^4-\\frac{3}{2}>0$ .", "This finishes the proof of the statement ($\\star $ ) and thus of Theorem REF ." ], [ "Cycles of length $4k+2$", "In this section, we consider cycles $C_\\ell $ , where $\\ell =4k+2$ for some integer $k\\ge 1$ .", "First, we construct a family of tournaments $T$ with $t(C_\\ell , T)< g_\\ell (t(C_3,T))$ for any value of $t(C_3,T)$ .", "We need to introduce some of the limit theory of tournaments established in [8].", "A tournamenton is a measurable function $W: [0,1]^2\\rightarrow [0,1]$ such that $W(x,y)+W(y,x)=1$ for all $(x,y)\\in [0,1]^2$ .", "For a tournamenton $W$ , define $C(W,\\ell )=2^\\ell \\int _{x_1,...,x_\\ell \\in [0,1]} W(x_1,x_2)W(x_2,x_3)\\dots W(x_{\\ell -1},x_\\ell )W(x_\\ell ,x_1)~dx_1\\dots x_\\ell .$ One can also define the spectrum $\\sigma (W)$ (for its precise definition we refer to Section 2.2 of [8]).", "By Proposition 4 of [8], $C(W,\\ell )=2^\\ell \\cdot \\sum _{x\\in \\sigma (W)} x^\\ell .$ A carousel tournament $T_n$ is a tournament with vertex set $\\lbrace 0,1,...,2n\\rbrace $ , where $i\\rightarrow j$ for every $i, j$ satisfying $i+1\\le j\\le i+n$ (computations module $2n+1$ ).", "Note that $T_n$ is regular, implying that $t(C_3,T_n)=1/8+o(1)$ where $o(1)\\rightarrow 0$ as $n\\rightarrow \\infty $ .", "Let $W_C$ be the tournamenton as follows: for $x,y\\in [0,1]$ , let $W_C(x,x)=1/2, W_C(x,y)=1$ if $y\\in (x-1,x-1/2)\\cup (x,x+1/2]$ , and $W_C(x,y)=0$ otherwise.", "Then $W_C$ is the limit object of carousel tournaments $T_n$ satisfying that $\\lim _{n\\rightarrow \\infty }t(C_\\ell ,T_n)=C(W_C,\\ell )/2^\\ell $ .", "One can deduce from [8] that $\\sigma (W_C)=\\lbrace 1/2,0\\rbrace \\cup \\lbrace \\pm \\textbf {i}/(2k-1)\\pi : \\forall k\\in \\mathbb {N}\\rbrace $ and as $\\ell =4k+2$ , $\\lim _{n\\rightarrow \\infty }t(C_\\ell ,T_n)=\\frac{C(W_C,\\ell )}{2^\\ell }=\\sum _{x\\in \\sigma (W_C)}x^\\ell =\\frac{1}{2^\\ell }-2\\sum _{k=1}^\\infty \\Big (\\frac{1}{(2k-1)\\pi }\\Big )^\\ell \\triangleq \\alpha _\\ell .$ Note that $\\alpha _\\ell <1/2^\\ell $ .", "We are ready to construct the desired family of tournaments.", "Definition 4.1 A carousel blow-up of a $m$ -vertex transitive tournament is a tournament $T$ with $V(T)=V_1\\cup V_2\\cup ...\\cup V_m$ such that each $V_i$ induces a carousel tournament and for any $i<j$ , all arcs between $V_i$ and $V_j$ are oriented from $V_i$ to $V_j$ .", "A carousel blow-up of a $m$ -vertex transitive tournament is called balanced if $\|V_1\|=\|V_2\|=\\dots =\|V_{m-1}\|\\ge \|V_m\|$ .", "Now consider a balanced carousel blow-up $T^\\star $ of a transitive tournament, and compute the density $t(C_\\ell , T^\\star )$ .", "Suppose that $T^\\star $ has $n$ vertices and $t$ parts of equal size $zn$ for some $z\\in (0,1]$ .", "Then $tz\\le 1< (t+1)z$ , which implies that $t=\\lfloor z^{-1}\\rfloor $ .", "By (REF ), if $n$ is sufficiently large, then $t(C_\\ell , T^\\star )=\\alpha _\\ell \\cdot \\Big (\\lfloor z^{-1}\\rfloor z^\\ell +(1-\\lfloor z^{-1}\\rfloor z)^\\ell \\Big )+o(1), \\mbox{ ~~ where } o(1)\\rightarrow 0 \\mbox{ as } n\\rightarrow \\infty .$ On the other hand, we have $t(C_3,T^\\star )=\\frac{1}{8}\\Big (\\lfloor z^{-1}\\rfloor z^3+(1-\\lfloor z^{-1}\\rfloor z)^3\\Big )+o(1)$ .", "Note that $z$ can be any real in $(0,1]$ .", "We can summarize this construction as following.", "Lemma 4.2 For any $\\ell =4k+2$ for some integer $k\\ge 1$ , there exist tournaments $T$ with arbitrary $t(C_3,T)$ such that $t(C_\\ell , T)=2^\\ell \\alpha _\\ell \\cdot g_\\ell (t(C_3,T))+o(1)$ , which is strictly less than $g_\\ell (t(C_3,T)).$ Next, we apply results of [8] to obtain a lower bound on $t(C_\\ell ,T)$ for all regular tournaments $T$ .", "We need the following lemma in [8].", "Let $D_n$ denote the $n\\times n$ skew-symmetric matrix with all entries above the diagonal equal to 1 and all entries below the diagonal equal to $-1$ .", "Lemma 4.3 ([8], Lemma 11) For any $n\\in \\mathbb {N}$ , the spectral radius of any skew-symmetric matrix in $[-1,1]^{n\\times n}$ is at most the spectral radius of $D_n$ .", "Let $A$ be the tournament matrix of an $n$ -vertex regular tournament $T$ .", "Let $J$ be the $n\\times n$ matrix with every entry equal to 1 and let $B:=A-J/2n$ .", "Then $B$ is a skew-symmetric matrix with all entries in $\\lbrace \\frac{-1}{2n},\\frac{1}{2n},0 \\rbrace $ .", "Because $T$ is regular, the sum of entries in each column (or row) of $B$ is 0, which shows that $JB=BJ=0$ .", "Therefore by (REF ), we have $t(C^\\ell ,T)=tr(A^{\\ell })+o(1)=tr\\left((J/2n+B)^\\ell \\right)+o(1)=tr((J/2n)^\\ell )+tr(B^\\ell )+o(1).$ It is known that the spectral radiuses $\\rho _n$ of the matrices $D_n$ divided by $n$ converge to $\\frac{2}{\\pi }$ (see the proof of Lemma 12 in [8]).", "Let the spectral radius of $B$ be $\\rho $ and the eigenvalues of $B$ be $\\lambda _1,\\lambda _2,\\ldots ,\\lambda _n$ .", "Then we have that by Lemma REF , $\|\\lambda _i\|\\le \\rho \\le \\frac{\\rho _n}{2n}=\\frac{1}{\\pi }+o(1)$ for any $i\\in [n]$ , and $\\sigma \\triangleq \\sum \\limits _{i=1}^{n}\|\\lambda _i\|^2\\le tr(BB^T)=\\frac{n-1}{4n}$ , where the last equality follows from the fact that every non-diagonal entry of $B$ is $1/2n$ or $-1/2n$ and every diagonal entry of $B$ is 0.", "Applying Lemma REF , as $\\ell $ is even, we can obtain that $\|tr(B^\\ell )\|\\le \\sum \\limits _{i=1}^{n}\|\\lambda _i\|^{\\ell }=\\left\\lfloor \\frac{\\sigma }{\\rho ^2}\\right\\rfloor \\rho ^{\\ell }+\\left(\\sigma -\\left\\lfloor \\frac{\\sigma }{\\rho ^2}\\right\\rfloor \\rho ^2\\right)^{\\ell /2}.$ Note that $\\lfloor \\frac{\\sigma }{\\rho ^2}\\rfloor \\rho ^{\\ell }+(\\sigma -\\lfloor \\frac{\\sigma }{\\rho ^2}\\rfloor \\rho ^2)^{\\ell /2}$ increases as $\\sigma $ and $\\rho $ increase.", "By plugging $\\sigma \\le \\frac{n-1}{4n}$ and $\\rho \\le \\frac{1}{\\pi }+o(1)$ in the above inequality, we obtain $\|tr(B^\\ell )\|\\le 2\\left(\\frac{1}{\\pi }\\right)^\\ell +\\left(\\frac{1}{4}-\\frac{2}{\\pi ^2}\\right)^{\\ell /2}+o(1)$ and thus $t(C^\\ell ,T)\\ge tr\\left((J/2n)^\\ell \\right)-\|tr(B^\\ell )\|+o(1)\\ge \\frac{1}{2^\\ell }- 2\\left(\\frac{1}{\\pi }\\right)^\\ell -\\left(\\frac{1}{4}-\\frac{2}{\\pi ^2}\\right)^{\\ell /2}+o(1),$ where $\\left(\\frac{1}{4}-\\frac{2}{\\pi ^2}\\right)^{1/2}\\approx 0.218 <\\frac{1}{\\pi }$ .", "Together with (REF ), we have the following lemma.", "Lemma 4.4 For every $\\epsilon >0$ , there exists $\\ell _0$ such that for every $\\ell \\ge \\ell _0$ with $\\ell \\equiv 2\\mod {4}$ , $\\frac{1}{2^\\ell }-(2+\\epsilon )\\left(\\frac{1}{\\pi }\\right)^\\ell \\le \\min _{T} t(C_\\ell , T)\\le \\frac{1}{2^\\ell }-2\\left(\\frac{1}{\\pi }\\right)^\\ell ,$ where the minimum is over all $n$ -vertex regular tournaments $T$ for large $n$ .", "It seems plausible to believe that for every such $\\ell \\ge 6$ , $\\lim _{n\\rightarrow \\infty }\\min _{T} t(C_\\ell , T)=\\alpha _\\ell $ where the minimum is over all $n$ -vertex regular tournaments $T$ ." ], [ "Concluding remarks", "We prove the statement of Theorem REF whenever $t(C_3,T)\\ge \\frac{1}{8}-O(\\frac{1}{\\ell ^2})$ .", "It is possible to lower the condition on $t(C_3,T)$ .", "A special case is $\\ell =5$ .", "Using Theorem REF and a similar argument as in [3] (via the method of Lagrange multiplier), we also can obtain the same bound as in Theorem REF that $t(C_5,T)\\ge g_5(t(C_3,T))+o(1)$ whenever $t(C_3,T)\\in [\\frac{1}{72},\\frac{1}{8}]$ .", "For integers $\\ell $ with $\\ell \\equiv 0,1 \\mod {4}$ , using more careful calculation, we can prove the same conclusion of Theorem REF whenever $t(C_3,T)\\ge \\frac{1}{8}-O(\\frac{1}{\\ell })$ .", "However, the approach we used has its obvious limit, so we choose to present a unified, less complicated proof for Theorem REF .", "We now conclude this paper by the following two conjectures.", "The first one is a direct generalization of Conjecture REF for cycles $C_\\ell $ with $\\ell \\lnot \\equiv 2\\mod {4}$ , which is supported by Theorem REF .", "Conjecture 5.1 For any integer $\\ell \\ge 4$ with $\\ell \\lnot \\equiv 2\\mod {4}$ , every tournament $T$ satisfies that $t(C_\\ell ,T)\\ge g_\\ell (t(C_3,T))+o(1),$ where the $o(1)$ term goes to zero as $\|V(T)\|$ goes to infinity.", "Conjecture 5.2 For any integer $\\ell \\ge 6$ with $\\ell \\equiv 2\\mod {4}$ , every tournament $T$ satisfies that $t(C_\\ell ,T)\\ge 2^\\ell \\alpha _\\ell \\cdot g_\\ell (t(C_3,T))+o(1),$ where the $o(1)$ term goes to zero as $\|V(T)\|$ goes to infinity.", "If true, as demonstrated by Lemma REF , the above conjecture would be sharp.", "Acknowledgement.", "The authors would like to thank Prof. Kimchuan Toh for many helpful discussions and valuable comments which greatly improve the presentation." ] ]	2011.14142
[ [ "Machine Intelligent Techniques for Ramp Event Prediction in Offshore and\n Onshore Wind Farms" ], [ "Abstract Globally, wind energy has lessened the burden on conventional fossil fuel based power generation.", "Wind resource assessment for onshore and offshore wind farms aids in accurate forecasting and analyzing nature of ramp events.", "From an industrial point of view, a large ramp event in a short time duration is likely to cause damage to the wind farm connected to the utility grid.", "In this manuscript, ramp events are predicted using hybrid machine intelligent techniques such as Support vector regression (SVR) and its variants, random forest regression and gradient boosted machines for onshore and offshore wind farm sites.", "Wavelet transform based signal processing technique is used to extract features from wind speed.", "Results reveal that SVR based prediction models gives the best forecasting performance out of all models.", "In addition, gradient boosted machines (GBM) predicts ramp events closer to Twin support vector regression (TSVR) model.", "Furthermore, the randomness in ramp power is evaluated for onshore and offshore wind farms by calculating log energy entropy of features obtained from wavelet decomposition and empirical model decomposition." ], [ "Introduction", "With wind energy being a driving force, both, onshore and offshore technologies have attracted big investments globally [1].", "Onshore wind farms have an advantage of proximity to the utility grid whereas offshore wind farms need long transmission cables to transmit power from the sea to grid.", "However, with a strong wind field in the offshore areas compared to onshore, the offshore wind farm installations have risen drastically.", "Further, with lesser turbulence and more uniform wind speed, the lifetime of wind turbines in offshore scenario increases [2].", "Offshore wind farms have a disadvantage of higher installation and engineering costs accompanied by the wear and tear of generator and mechanical equipment.", "Onshore wind farms, on the other hand face the challenge in their acceptance from limited land availability, visual intrusion and damage to wildlife.", "The defining factor of operation between onshore and offshore wind farms is the surface roughness length.", "It is observed that, larger the surface length higher the reduction in wind velocity and theoretically the wind speed at ground is zero [3].", "Wind energy related installation activity, particularly in the offshore areas, is challenged by the transportation costs owing to large blades and support structures to be placed in deep sea water.", "Another issue is the uncontrolled vibrations arising from the combination of wind and wave force [4].", "With a higher wind speed in the sea, the probability of ramp instances increases.", "Wind power growth has led to detailed study of ramp events, particularly categorized as ramp-up and ramp-down events with each characterized by sudden wind speed change in a short period.", "Mathematically, a wind power ramp event [5] is described as the arithmetic difference between wind power at consecutive time instants: $\\Delta P_w^{Ramp}= P_w(t+\\Delta t)-P_w(t).$ The occurrence of a wind power ramp event is ascertained by a threshold value of the total wind farm power.", "Wind power forecasting is an essential market procedure to regulate the electricity markets where the farm operators hold an advantage to derive profit from their generation given an accurate wind forecasting strategy.", "Predicting these ramp events can suitably prevent intermittent grid failures and for that purpose, proper characterization of ramp events is imperative in avoiding potential risks in the management of a power system.", "On the contrary, a poor characterisation of the ramp performance inevitably hinders the identification of the processes that cause ramp events.", "In Horns Rev wind farm, large variation in wind power is seen in a short duration [6].", "In a study carried out by Gjerstad et al., the variation in wind power is triggered by atmospheric properties of boundary layer [7].", "The atmospheric factor that influences the wind speed variations is the mean lapse rate ($\\partial T/\\partial z$ ), that is, rate of change of air temperature ($T$ ) with height $z$ .", "In case of offshore wind farms connected to German power grid, within 8 hrs, the wind power transfer between converters increased from 4 GW to 19 GW as a result of a sudden change in wind speed [8].", "Further, Nissen studies seasonal variations in wind speed over coastal areas of Høvsøre which shows that the wind speed variation are primarily dominant in the spring and winter season [9].", "It is found that the wind power ramp events are characterized based on magnitude error, phase error and location error as pointed out by Potter et al.[10].", "In [11], authors discuss the MesoLAPS and Wind power prediction tool (WPPT) for analyzing and categorizing large wind power ramp events where the problem is modeled as a classification task.", "Root mean squared error (RMSE) is evaluated over 1 year for two wind speed time series with 5-min and 10-min sampling intervals.", "Results reveal that the RMSE values are found to be more optimistic than conventional persistence and climatology methods.", "Gallego-Castillo et al.", "reviewed various wind ramp events and have described preliminary definition of a ramp event and the threshold values that various literature have used to formulate a ramp event [12].", "Bossavy et al.", "proposed a reliable forecast methodology to characterize ramps with a derivative filtering edge detection approach and numerical weather prediction ensembles to make probabilistic forecasts of ramp occurrence [13].", "Understanding the underlying role of the synoptic weather regimes in triggering the wind power ramp events can help improve and complement the current forecast techniques.", "Cuoto et al.", "have identified and classified the weather regimes over mainland Portugal associated with the occurrence of severe wind power ramps [14].", "Cornejo-Bueno et al.", "have described the machine learning techniques like, support vector regression (SVR), gaussian process regression (GPR), multi-layer perceptrons (MLP) and extreme learning machines (ELM) to forecast wind power ramp events [15].", "Cui et al.", "detect ramp event probability characteristics from scenarios captured bands to evaluate the ramp event forecasting method using a modified genetic algorithm with multi-objective fitness functions [16], [17].", "Gallego et al, formulated a ramp function to describe a ramp event characterised by high power output gradients evaluated under different time scales, and is based on discrete wavelet transform to provide a continuous index related to the ramp intensity [18].", "In another work, Cornejo-Bueno et al.", "have considered a neural network based model in tandem with evolutionary algorithm to optimize the parameters of several classifiers such as support vector machines (SVM) and extreme learning machine (ELM) [19], [20], [21].", "Three wind speed datasets from Spain are considered to evaluate the accuracy of hybrid method for classifying wind power ramp events.", "In [22], the authors have presented a feature extraction method based on Gabor filtering where the atmospheric pressure fields are taken into consideration.", "Results are compared with state-of-the-art neural network method and it is found that Gabor method with its change in power as its output gives a better prediction performance.", "In a recent study, Ouyang et al.", "have presented an improved ramp prediction methodology using residual correction through a model that combines the advantages of the auto-regression and Markov chain models [23].", "Classification based forecast is inaccurate and causes class imbalance issue in the machine learning problem since the occurrence of a ramp event is typically rare, and so Takahashi et al.", "have used and experimentally evaluated this sampling approach on a real-world wind power generation dataset [24].", "In another work, Fujimoto et al.", "have presented an altering scheme of an impending ramp event for efficient operational decisions using supervised learning algorithms for better health management and reduced downtime [25].", "Zhang et al.", "decompose wind speed data and analyze each part using different models to make predictions based on wind speed ramp and residual distribution [26].", "In other ramp event prediction related works, Dhiman et al.", "have discussed SVR and its variants [27] for predicting wind power ramp events where five datasets are tested for ramp-up and ramp-down events [28], [29].", "The current work addresses following objectives: Ramp prediction models for onshore and offshore wind farms are discussed.", "The wind powers corresponding to the wind speed time series obtained are calculated for identifying the ramp events.", "The wind power ramp events are predicted using a hybrid method involving wavelet transform decomposition and machine learning methods.", "Thus, the underlying problem here is modeled as a regression task with $\\Delta t=$ 10 min.", "The potential capability of random forest regression and gradient boosted machines are checked.", "Several error metrics for the entire wind speed time series and absolute error for ramp events (ramp-up and ramp-down) are evaluated and further compared with benchmark persistence and SVR models.", "Log energy entropy based randomness is discussed for onshore and offshore wind farm sites.", "The randomness in a ramp signal is an important feature to be dealt with.", "Higher order randomness calls for accurate wind resource assessment and micro-siting.", "This manuscript is organized as follows.", "Section 2 describes machine learning methods, that is, support vector regression and its variants, random forest regression and gradient boosted machines followed by framework of ramp prediction models in Section 3.", "In Section 4 outcomes of the proposed models are presented and Discussions are highlighted in Section 5.", "Section 6 discusses conclusions." ], [ "Methods for Wind Power Ramp Prediction", "Next, we discuss various methods employed to predict ramp events in onshore and offshore wind farms.", "Wavelet decomposition based machine learning methods are used.", "A hybrid model based on Wavelet decomposition serves the purpose of eliminating stochastic trends in wind speed time series.", "Support vector regression and its variants like $\\varepsilon $ -SVR, Least square support vector regression, Twin support vector regression and $\\varepsilon $ -Twin support vector regression are discussed.", "Further, prediction models based on regression trees like Random forest and Gradient boosted machines are also discussed for predicting ramp events." ], [ "Support Vector Regression & its variants", "Support vector regression (SVR) which is coined from support vector machines (SVMs) [30] is a popular machine learning method used in several branches of forecasting like solar radiation forecasting [31], wind forecasting [32] and hydrological time series [33].", "The basic idea of SVR is to map a set of non-linear features to a higher dimension where the data is linearly separable, mathematically expressed as $g(x)=w^Tx+b,$ where $w$ is a weight vector, $x$ represents set of input features and $b$ is the bias term.", "SVR operates on historical data which plays an important role in predicting the future values of the response variable.", "Consider a set $X\\in R^n$ , $X=[x_1, x_2,\\dots ,x_n]$ as the set of input features.", "The weights for the input features are obtained by solving an optimization problem described in Table REF .", "Further, Suykens penned Least square support vector regression (LSSVR), where the objective function for minimization incorporates the squared error term [34].", "The computation time for LSSVR is fast compared to classical SVR owing to its smaller sized lagrangian multiplier matrix.", "In 2010, Peng derived Twin support vector regression (TSVR) where two smaller sized optimization problems are solved to arrive at resultant regressor [35].", "The optimization problem along with resultant regressor $g(x)$ is depicted in Table REF .", "Further, in 2012, Shao et al.", "presented a $\\varepsilon $ -Twin support vector regression that incorporates an extra regularization factor and arrives at the final regressor using successive over relaxation technique which is a fast convergence algorithm [36].", "However, with larger datasets, the regression analysis can be computationally expensive and can lead lead to large errors in predicting ramp events.", "Thus, the ramp event prediction is extended for two more methods, that is, random forest regression (RFR) and gradient boosted machines (GBM) categorically for onshore and offshore wind farm sites, and specifically for offshore wind farms, the dominating strong wind field poses a forecasting challenge for operators.", "Table: ε\\varepsilon -SVR, LS-SVR, TSVR and ε\\varepsilon -TSVR regression models" ], [ "Random forest regression", "Proposed by Brieman, random forest is an ensemble method that generates something, akin to a forest of trees from a given training sample [37].", "The ensemble based models are far more accurate than a single method owing to advantages like capturing linearity and non-linearity of time-series obtained from individual methods.", "Random forest begins with splitting the input features into a group of subsets that essentially form a tree.", "Similar to hyperparameters ($\\sigma $ and $C$ ) in SVR, in random forest regression, the number of trees and number of random features in each tree decomposition are the parameters that decide the performance of regression.", "At each decision tree, a fitting function is created which acts on the random features selected.", "Finally, at the end of the training process, a random forest model is created.", "It is worthwhile to note that during training process, each tree is created from randomly selected input vectors and thus it is called `random' forest.", "The estimated output of a random forest regression is given as $\\hat{A}=\\frac{1}{k}\\sum _{i=1}^{k}\\hat{r}\\Big (X,V_i\\Big ),$ where $\\hat{r}\\Big (X,V\\Big )$ is the representative tree at the end of training process, $X$ is the set of input feature vectors and $T$ is the collective set representing input-output pair $V_i=(x_1,y_1),(x_2,y_2),\\dots (x_n,y_n)$ .", "A particular tree is characterized by a node that leads to number of branches, as depicted in Fig.", "REF .", "Figure: Block diagram for random forest regression modelThe predicted output is averaged over $k$ decision trees.", "An added advantage of random forest regression is its insensitivity to noise due to uncorrelated trees via differential sampling of inputs.", "Figure REF illustrates a generic block diagram of the wavelet based ramp prediction model for onshore and offshore wind farms.", "Figure: Block diagram for ramp event prediction and randomness in ramp signalThe input feature set or predictors are a set of approximation signal (A5) and detail signals (D1, D2, D3, D4 and D5) obtained from wavelet decomposition of wind speed.", "A common problem of over-fitting persists in machine learning regression models when a well trained model captures the noise component as well.", "To reduce the complexities posed by over-fitting, random forest makes a compromise between a flexible and an inflexible model.", "In the training phase, each regression tree draws a sample from the feature set and is drawn repeatedly.", "This ensures that even though the tree might possess a high variance but the overall variance of forest will be low.", "Random forest work on the principle of bagging that combines the predictions from different tree models to give an overall insight to the data under training.", "This also helps to reduce the potential over-fitting caused by supervised machine learning models." ], [ "Gradient boosted machines ", "Gradient boosted machines (GBM) is an ensemble based regression method that takes into account the loss of the previously fit decision tree.", "In a GBM model, various weak learners combine together to arrive at an accurate model.", "Boosting based regression trees are effective in predicting accurate response values.", "A particular boosting algorithm strengthens the tree model by minimizing the inaccuracies of weak models in form of loss function [38].", "The main task is to reduce the error posed by weak learners in each iteration.", "The final predicted response is the linear combination of fit trees.", "The learning rate of each decision tree determines the predictive performance.", "A higher learning rate indicates less number of trees required for prediction and vice versa [39].", "Based on a cross-validation experiment the ideal number of trees required along with learning rate is obtained.", "The aim for a individual model is to reduce the error generated by a loss function.", "Commonly used loss function is $\\mathcal {L}_2$ function that minimizes the sum of squared errors between predicted and actual value.", "Consider $k$ such weak models that account for prediction of a variable $y$ given feature set $x=[x_1,\\dots ,x_n]$ , mathematically it can be expressed as $\\hat{y}=\\sum _{i=1}^{k}f_i(x),$ where $f_i(x)$ represents each weak learner that is collectively trained to improve the prediction.", "The loss function is given as $\\mathcal {L}_2 =\\frac{1}{N}L(y_i,\\hat{y}_i),~~L(y_i,\\hat{y}_i) =\\sum _{i=1}^{N}(y_i-\\hat{y}_i)^2,$ where $L(y_i,\\hat{y}_i)$ is a loss function based on squared errors for $N$ observations.", "The aim of a GBM regression technique is to minimize the $\\mathcal {L}_2$ loss function.", "However, the $\\mathcal {L}_2$ loss function is more sensitive to outliers and can reduce the robustness of the model.", "For optimizing the hyper-parameters in GBM, a gradient descent algorithm is used which minimizes the cost function taking into account the negative of the gradient.", "Let us consider a $\\mathcal {L}_2$ loss function, the gradient of this function with respect to predicted sample $\\hat{y}_i$ is given as $\\frac{\\partial L(y,\\hat{y})}{\\partial \\hat{y}_i}=\\frac{\\partial }{\\partial \\hat{y}_i}\\sum _{i=1}^{N}(y_i-\\hat{y}_i)^2 = -2(y_i-\\hat{y}_i).$ The gradient reflects that while tracking the minimum point of the loss function, the GBM actually tracks the residual vector $(y-\\hat{y})$ .", "Similarly, for $\\mathcal {L}_1$ , the MAE can be tracked by finding the gradient of loss function.", "The essence of a GBM model lies in the boosting technique that combines all the weak models in a stage-wise manner than in a parallel manner as seen in random forest model.", "The GBM model in recursive form is given as $F_m(x)=F_{m-1}(x)+\\eta \\Delta _m(x)$ , where $\\eta $ represents the learning rate and $\\Delta _m$ refers to a regression model fitted to the residuals.", "Figure REF illustrates the XBoost algorithm used for obtaining the best regressor.", "Figure: XBoost algorithm for Gradient boosted machinesWind speed variation depends on the terrain under study.", "For all the wind farms operating in neutral atmospheric boundary layer, the wind speed follows a logarithmic profile with height.", "For onshore wind farms, the surface roughness length ($z_0$ ), is around 0.005 m while for offshore wind farms it is 0.0002 m. The current hybrid prediction model is based on a combination of wavelet transform and a machine learning algorithm.", "Wavelet transform is widely used for defragmentation of a time-series signal into low and high frequency sub-signals called as approximate and detail signals respectively.", "A 5-level db4 wavelet transform is used to decompose the wind speed time series.", "The approximate signal (A5) along with detail signals (D1, D2, ..., D5) form input feature set for all the prediction models.", "The datasets collected for the month of March 2019 are wind speed time-series at a height of 10 m above ground averaged over a time interval of 10 minutes.", "For an accurate wind power ramp event study, it is desirable to transform the available wind speed to a hub height of 90 m using logarithmic law [40].", "In this work, we consider all the wind farms with identical wind turbines having rotor diameter of 120 m with a hub height of 90m.", "The datasets (onshore and offshore) with their site coordinates and statistical parameters are depicted in Table REF .", "Mean represents the mean of the wind speed time-series and SD indicates the standard deviation.", "The variability in wind speed is high as the changes in wind speed on minute to minute scale are likely to cause large power changes.", "Table: Description of wind farm datasets for the month March 2019Ramp events occur when the wind power increases or decreases suddenly in a short duration of time which is typically in the range from 5 minutes to 6 hours [12].", "For a given wind turbine, let us say the ramp threshold power is $\\alpha _{th}\\%$ of the nominal wind power.", "Then we can define two ramp thresholds, that is, $\\Delta P_w^{ramp}&=&\\left\\lbrace \\begin{aligned}&+\\alpha _{th}\\%~~ \\text{of}~~ P_{nom} &=& P^{up},\\\\&-\\alpha _{th}\\%~~ \\text{of}~~ P_{nom} &=& P^{down},\\end{aligned}\\right.$ where $P^{up}$ and $P^{down}$ are the upper and lower ramp thresholds respectively depicting ramp-up and ramp-down events in a short period of time.", "The rated wind speed considered for ramp event study is 12 m/s.", "Further, in order to forecast the ramp events in the wind speed time series, wind power is calculated for the given sample of data.", "The ramp event is identified when the wind power generated exceeds lower ($P^{down}$ ) or upper ($P^{up}$ ) threshold values.", "To evaluate the forecasting performance, several error metrics such as Root mean squared error (RMSE), normalized mean squared error (NMSE), coefficient of performance ($R^2$ ) and Theil's U1 and U2 metrics and are given by (REF )-(e11): $\\text{RMSE} &=& \\Bigg [\\frac{1}{n}\\sum _{i=1}^{n}(\\hat{s_i}-{s_i})^2\\Bigg ]^{1/2} \\\\R^2&=&\\frac{\\sum _{i=1}^{n}(\\hat{s_i}-\\bar{s})^2}{\\sum _{i=1}^{n}({s_i}-\\bar{s_i})^2} \\\\\\text{NMSE}&=&\\frac{\\sum _{i=1}^{n}(\\hat{s_i}-{s_i})^2}{\\sum _{i=1}^{n}({s_i}-\\bar{s})^2} \\\\\\text{Theil U1}&=&\\frac{\\sqrt{\\frac{1}{n} \\sum _{i=1}^{n}\\left(\\hat{s}_{i}-s_{i}\\right)^{2}}}{ \\left(\\sqrt{\\frac{1}{n} \\sum _{i=1}^{n} s_{i}^{2}}+\\sqrt{\\frac{1}{n} \\sum _{i=1}^{n} \\hat{s}_{i}^{2}}\\right)}\\\\\\text{Theil U2}&=&\\frac{\\sqrt{\\frac{1}{n} \\sum _{i=1}^{n}\\left(\\left(s_{i+1}-\\hat{s}_{i+1}\\right) / s_{i}\\right)^{2}}}{\\sqrt{\\frac{1}{n} \\sum _{i=1}^{n}\\left(\\left(s_{i+1}-\\hat{s}_{i}\\right) / s_{i}\\right)^{2}}}$ where $\\hat{s_i},s_i,\\bar{s}$ are the predicted, actual and mean values of the $n$ testing samples." ], [ "Results and Discussions", "In this Section, we discuss the forecasting performance of the prediction models (Persistence, $\\varepsilon $ -SVR, LSSVR, TSVR, $\\varepsilon $ -TSVR, RFR and GBM).", "The ramp-up and ramp-down events are identified for a threshold of 10% of nominal wind power.", "The forecasting process is carried out by splitting the entire dataset into training (80%) and testing (20%) sets.", "For persistence model, the forecast is carried out using two previous dispatch windows.", "Further, for SVR and its variants, the hyperparameters, that is, RBF bandwidth ($\\sigma $ ) and regularization parameter ($C$ ) are tuned from the set $[2^{-10},2^{-9},...2^9,2^{10}]$ .", "For random forest regression (RFR) and gradient boosted machines (GBM), the simulations are carried out in R studio using randomForest package.", "The number of trees used in training phase are 1000 for RFR, while for GBM the learning rate is kept at 0.05 and number of trees are 10000.", "The input feature set to all the prediction models is a matrix consisting of approximation signal A5 and detail signals (D1,D2, $\\dots ,$ D5) obtained from wavelet decomposition of wind speed time series.", "Further, we evaluate, absolute error for ramp-up and ramp-down events as $R^{up}$ and $R^{down}$ respectively.", "For onshore wind farm sites, the datasets labeled are a, b, c, d, e and f. For dataset a, GBM outperforms all the prediction models in terms of RMSE and NMSE.", "For predicting ramp-up and ramp-down events, TSVR outperforms GBM and RFR.", "In terms of $R^2$ , TSVR gives the best fit close to 1.00 whereas $\\varepsilon $ -SVR gives a value greater than 1.00 causing over-fitting.", "The over-fitting in prediction is avoided by using RFR and GBM.", "Similarly, for offshore wind farm datasets (A, B, C, D, E & F), TSVR outperforms all the models in terms of RMSE and NMSE for predicting wind speed, while for predicting ramp events TSVR gives the best result in terms of $R^2$ .", "Since the datasets are large sized, TSVR, $\\varepsilon $ -TSVR, RFR and GBM take less CPU time (which is the time spent by the regression method to predict the ramp events) than conventional $\\varepsilon $ -SVR.", "The detailed performance metrics for onshore and offshore wind farm sites are depicted in Table REF and REF .", "It is worthwhile to note that, ramp prediction performance for GBM is significantly closer to TSVR in case of offshore wind farms.", "Table: Performance metrics for Onshore wind farm sitesTable: Performance metrics for Offshore wind farm sitesResults obtained in aforementioned section depict that SVR based regressors have an upper hand over RFR and GBM based prediction methods.", "While it is worthy to note that, to study the ramp events the time interval between consecutive samples plays an important role in deciding the threshold value.", "In a study carried out by Ouyang et al., the wind speed data collected is sampled every 15 minute for a wind farm site in China and ARIMA model is used to forecast wind power [23].", "Further, swinging door algorithm is used to detect ramp events.", "The RMSE values for the ramp event prediction are found to be in range of 32%.", "Using machine learning techniques such as classical SVR, GPR, MLP and ELM, the ramp events are predicted for time interval 6hr and RMSE values are found to be in range of (5-7 MW).", "However, wind speed time-series of onshore and offshore wind farm has a lot of variability in terms of magnitude.", "Thus, a ramp event study with 10 minute as time interval poses critical impositions in market clearing and day-ahead scenarios.", "The current work deals with predicting wind power ramp events with 10 minute sampling interval for onshore and offshore wind sites.", "With offshore wind farms, the wind speed being high and variable, the probability of ramp event increases.", "Thus, in the current work, we extend the ramp event prediction study by incorporating advanced machine learning algorithms like variants of SVR, random forest regression and gradient boosted machines.", "Results reveal that TSVR based prediction model yield lowest error for ramp-up and ramp-down events." ], [ "Uncertainties in Ramp events", "Ramp event in onshore and offshore wind farms is a challenging issue which could adversely affect the associated power systems.", "The randomness in wind speed stimulated by turbulent air flow can cause unwanted vibrations in turbine blade and tower thus questioning its structural stability.", "Randomness in a signal or time-series can often be expressed by calculating the log energy entropy.", "In order to critically examine a ramp event signal, expressed as (REF ), we decompose the obtained signal by two signal processing techniques such as wavelet transform decomposition (WT) and empirical model decomposition (EMD).", "While wavelet transform decomposes a signal into low-frequency and high-frequency components, EMD on the other hand is capable of extracting oscillatory features from the signal for a dominant frequency.", "The ramp signal is decomposed using db4 wavelet filter using wavelet transform and into 5 intrinsic mode functions (IMFs) using EMD.", "The log energy entropy for a signal $h(t)$ is given as $E\\lbrace h(t)\\rbrace =\\sum _{t=0}^{T}\\log (h(t)^2).$ A typical ramp event signal and its signal decomposition using wavelet transform and EMD is illustrated in Figure REF which shows the low frequency (LF) signals obtained from wavelet transform and EMD and are used to assess the randomness in the ramp event signal by calculating the log energy entropy of predicted ramp signals based on TSVR, RFR and GBM are depicted in Table REF .", "Figure: Ramp event signal and its decomposition using wavelet transform and EMDTable: Log energy entropy for ramp event signalsWe observe that for a particular prediction model, the log energy entropy based on WT is more than that of EMD suggesting higher order randomness in ramp event signal.", "As far as uncertainties are concerned, the decomposition based prediction models yield better ramp prediction than single methods." ], [ "Conclusion", "In the current work, wind power ramp events are studied for 6 onshore and 6 offshore wind farm sites.", "Wind speed data for the month of March 2019 with 10 minute sampling interval is collected.", "The available data is transformed to a hub height of 90m for all the datasets.", "A threshold value for change in wind power is chosen as 10% of nominal wind power and the ramp event points are assessed for absolute error.", "Machine learning based prediction models are compared with benchmark persistence model.", "TSVR based model gives minimum absolute error for both, ramp-up and ramp-down events.", "The coefficient of performance ($R^2$ ) for RFR and GBM in case of offshore wind farm datasets is found to be close to 1.00 which indicates a good agreement between predicted and actual ramp values.", "Further, it is observed that error in predicting ramp-down event is more than in ramp-up event.", "Overall, machine learning based prediction models are found to be good approximators for analyzing 10-min wind ramps.", "The randomness in wind power ramp series in case of TSVR, RFR and GBM is evaluated using signal decomposition techniques and with RFR the randomness is found to be minimum.", "Thus, an accurate machine intelligent model like TSVR and GBM in this case, affirms a stable grid operation in presence of large ramp-up and ramp-down events." ] ]	2011.14220
[ [ "On vanishing near corners of conductive transmission eigenfunctions" ], [ "Abstract In this paper, we consider the transmission eigenvalue problem associated with a general conductive transmission condition and study the geometric structures of the transmission eigenfunctions.", "We prove that under a mild regularity condition in terms of the Herglotz approximations of one of the pair of the transmission eigenfunctions, the eigenfunctions must be vanishing around a corner on the boundary.", "The Herglotz approximation can be regarded as the Fourier transform of the transmission eigenfunction in terms of the plane waves, and the growth rate of the transformed function can be used to characterize the regularity of the underlying wave function.", "The geometric structures derived in this paper include the related results in [5,19] as special cases and verify that the vanishing around corners is a generic local geometric property of the transmission eigenfunctions." ], [ "Background", "In its general form, the transmission eigenvalue problem is given as follows (cf.", "[22]): $\\mathcal {P}_1(\\mathbf {x}, D)u=-\\lambda u, \\quad \\mathcal {P}_2(x, D)v=-\\lambda v\\ \\ \\mbox{in}\\ \\ \\Omega ; \\quad \\mathcal {C}(u)=\\mathcal {C}(v)\\ \\ \\mbox{on}\\ \\ \\partial \\Omega ,$ where $\\Omega $ is a bounded Lipschitz domain in $\\mathbb {R}^n$ , $n=2,3$ , with a connected complement $\\mathbb {R}^n\\backslash \\overline{\\Omega }$ and $P_j(\\mathbf {x}, D)$ are two elliptic partial differential operators (PDOs) with $D$ signifying the differentiations with respect to $\\mathbf {x}=(x_j)_{j=1}^n\\in \\mathbb {R}^n$ , and $\\mathcal {C}$ denotes the Cauchy data set.", "If there exists a nontrivial pair of solutions $(u, v)$ , then $\\lambda \\in \\mathbb {C}$ is called a transmission eigenvalue and $(u, v)$ are the corresponding pair of transmission eigenfunctions.", "Though the PDOs $\\mathcal {P}_j$ , $j=1,2$ , are generally elliptic, selfadjoint and linear, the transmission eigenvalue problems of the form (REF ) are a type of non-elliptic, non-selfadjoint and nonlinear (in terms of the transmission eigenvalue $\\lambda $ ) spectral problems, making the corresponding spectral study highly intriguing and challenging; see [22] for some related discussion.", "The transmission eigenvalue problems arise in the wave scattering theory and connect to many aspects of the wave scattering theory in a delicate way.", "Indeed, many of the spectral results established for the transmission eigenvalue problems in the literature have found important applications in the wave scattering theory, including generating novel wave imaging and sensing schemes, producing important implications to invisibility cloaking and proving new uniqueness results for inverse scattering problems.", "We refer to [10], [11], [16], [22] for historical accounts and surveys on the state-of-the-art developments of the spectral studies for the transmission eigenvalue problems in the literature.", "To a great extent, the spectral properties of the (real) transmission eigenvalues resemble those for the classical Dirichlet/Neumann Laplacian: there are infinitely many real transmission eigenvalues which are discrete and accumulate only at infinity.", "Nevertheless, due to the non-selfadjointness, there are complex transmission eigenvalues; see [10], [16] the references cited therein.", "Recently, several local and global geometric structures of distinct features were discovered for the transmission eigenfunctions [2], [3], [4], [5], [6], [7], [8], [9], [12], [13], [14], [19] and all of them have produced interesting applications of practical importance in the scattering theory.", "In this paper, we are concerned with the vanishing property of the transmission eigenfunctions around a corner on the boundary of the domain, which was first discovered in [5] and further investigated in [19].", "Before discussing our major discoveries, we next specify the transmission eigenvalue problem as well as its vanishing properties in our study.", "Let $\\Omega $ be a bounded Lipschitz domain in $\\mathbb {R}^{n}, n=2,3,$ , with a connected complement $\\mathbb {R}^n\\backslash \\overline{\\Omega }$ , and $V \\in L^{\\infty }(\\Omega )$ and $\\eta \\in L^{\\infty }(\\partial \\Omega )$ be possibly complex-valued functions.", "Consider the following transmission eigenvalue problem for $v, w \\in H^{1}(\\Omega )$ and $\\lambda =k^2$ , $k\\in \\mathbb {R}_+$ : ${\\left\\lbrace \\begin{array}{ll}\\big (\\Delta +k^2(1+V)\\big ) w=0\\ &\\ \\mbox{in}\\ \\ \\Omega ,\\medskip \\\\(\\Delta +k^2) v=0\\ &\\ \\mbox{in}\\ \\ \\Omega ,\\medskip \\\\w=v,\\ \\ \\partial _\\nu w=\\partial _\\nu v+\\eta v\\ &\\ \\mbox{on}\\ \\partial \\Omega ,\\end{array}\\right.", "}$ where $\\nu \\in \\mathbb {S}^{n-1}$ signifies the exterior unit normal to $\\partial \\Omega $ .", "Two remarks concerning the formulation of the transmission eigenvalue problem (REF ) are in order.", "First, we introduce $k^2$ to denote the transmission eigenvalue.", "On the one hand, $k$ signifies a wavenumber in the physical setup and on the other hand, this notation shall ease the exposition of our subsequent mathematical arguments.", "Though only $k\\in \\mathbb {R}_+$ is physically meaningful, some of our subsequent results also hold for the case that $k$ is a complex number, which should be clear from the context.", "Second, the second transmission condition on $\\partial \\Omega $ in (REF ) is known as the conductive transmission condition.", "This type of transmission condition arises in modelling wave interaction with a certain material object and can find important applications in magnetotellurics; see e.g.", "[13], [19] and the references cited therein for more relevant physical backgrounds.", "On the other hand, if one simply takes $\\eta \\equiv 0$ , (REF ) is reduced to the transmission eigenvalue problem that has been more intensively studied in the literature.", "In order to signify such a generalization and extension, we refer to the eigenvalue problem (REF ) as the conductive transmission eigenvalue problem, which includes the conventional transmission eigenvalue problem as a special case.", "Let $\\mathbf {x}_c\\in \\partial \\Omega $ be a corner point, which shall be made more precise in what follows.", "Let $B_\\rho (\\mathbf {x}_c)$ denote a ball of radius $\\rho \\in \\mathbb {R}_+$ centred at $\\mathbf {x}_c$ .", "The vanishing property of the transmission eigenfunction is described as follows: $\\lim _{\\rho \\rightarrow +0} \\frac{1}{{m}\\left(B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega \\right)} \\int _{B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega }\|\\psi (\\mathbf {x})\|\\, \\mathrm {d} \\mathbf {x}=0,\\ \\ \\psi =w\\ \\ \\mbox{or}\\ \\ v,$ where ${m}$ denotes the Lebesgue measure.", "It is noted that $w$ and $v$ are $H^1$ -functions and the vanishing at a boundary point should be understood in the integral sense.", "On the other hand, if $\\psi $ is a continuous function in a neighbourhood of $\\mathbf {x}_c$ , (REF ) clearly implies that $\\psi (\\mathbf {x}_c)=0$ .", "In fact, the regularity of the transmission eigenfunctions $w$ and $v$ in (REF ) is critical for the establishment of the vanishing property (REF ).", "Under the regularity condition that both $w$ and $v$ are additionally Hölder continuous, namely $C^\\alpha $ continuous with $\\alpha \\in (0, 1)$ , it is shown in [5] and [19] that the vanishing property holds respectively in the cases with $\\eta \\equiv 0$ and $\\eta \\ne 0$ .", "By the classical result on the quantitative behaviours of the solutions to elliptic PDEs around a corner (cf.", "[17], [18], [21]), we have the following decompositions $w=w_{\\mathrm {singular}}+w_{\\mathrm {regular}},\\quad v=v_{\\mathrm {singular}}+v_{\\mathrm {regular}},$ where the regular parts belong to $H^2$ and hence by the standard Sobolev embedding, they are Hölder continuous.", "The singular parts may also be Hölder continuous provided the coefficients, namely $V$ , as well as the boundary data of $w$ and $v$ around the corner are sufficiently regular.", "However, in the transmission eigenvalue problem (REF ), the boundary data, namely $(w\|_{\\partial \\Omega }, \\partial _\\nu w\|_{\\partial \\Omega })$ and $(v\|_{\\partial \\Omega }, \\partial _\\nu v\|_{\\partial \\Omega })$ are not specified.", "Hence, it may happen that the transmission eigenfunctions are $H^1$ but not Hölder continuous.", "Clearly, according to our discussion above, the vanishing property may serve as an indicator for such singular behaviours of the transmission eigenfunctions around the corner.", "Indeed, according to the extensive numerical examples in [4], though the transmission eigenfunctions generically vanish around a corner, there are cases that the transmission eigenfunctions are not vanishing and instead they are localizing around a corner, especially when the corner is concave.", "Hence, it is mathematically intriguing and physically significant to thoroughly understand such a singularity formation of the transmission eigenfunctions and its connection to the corresponding vanishing behaviour.", "In [19], [5], a regularity criterion of a different mathematical feature, but more physically related, has been investigated in connection to the vanishing property of the transmission eigenfunction.", "It is given in terms of the Hergoltz approximation of the transmission eigenfunction $v$ in (REF ).", "The Herglotz approximation in a certain sense is the Fourier transform (in terms of the plane waves) of the eigenfunction $v$ who satisfies the homogeneous Helmholtz equation.", "Hence, the growth rate of the transformed function, i.e.", "the density function in the Herglotz wave, can naturally be used to characterize the regularity of the underlying wave function.", "This resembles the classical way of defining the Sobolev space via the Bessel potentials.", "In this paper, we shall explore along this direction and derive much sharper estimates to show that the vanishing property of the transmission eigenfunctions holds for a much broader class of functions in terms of the Herglotz approximation.", "The vanishing property of the transmission eigenfunctions derived in this paper include the corresponding results in [19], [5] as special cases." ], [ "Statement of the main results and discussions", "In order to present a complete and comprehensive study, the statements of our main results are lengthy and technically involved.", "Nevertheless, in order to give the readers a global picture of our study, we briefly summarize the major findings in the following two theorems.", "To that end, we first introduce the Herglotz approximation.", "For $g_{j} \\in L^{2}\\left(\\mathbb {S}^{n-1}\\right)$ , we set $v_{j}(\\mathbf {x})=\\int _{\\mathbb {S}^{n-1}} e^{i k \\xi \\cdot \\mathbf {x}} g_{j}(\\xi ) \\mathrm {d} \\sigma (\\xi ), \\quad \\xi \\in \\mathbb {S}^{n-1}, \\mathbf {x}\\in \\mathbb {R}^{n}.$ $v_j$ is known as a Herglotz wave with kernel $g_j$ .", "It is easy to see from (REF ) that $v_j$ is formed by the superposition of plane waves and it is an entire solution to the Helmholtz equation $\\Delta v_{j}+k^{2} v_{j}=0 $ .", "Hence, $g_j$ can be regarded as the Fourier density of the wave function $v_j$ in terms of the plane waves.", "We have the following denseness property of the Hergoltz waves.", "Lemma 1.1 ([23]) Let $\\Omega \\Subset \\mathbb {R}^n$ be a bounded Lipschitz domain with a connected complement and ${H}_k$ be the space of all the Herglotz wave functions of the form (REF ).", "Define ${S}_k(\\Omega ) = \\lbrace u\\in C^\\infty (\\Omega );\\ \\Delta u+k^2u=0\\rbrace $ and ${H}_k(\\Omega ) = \\lbrace u\|_\\Omega ;\\ u\\in {H}_k\\rbrace .$ Then ${H}_k(\\Omega )$ is dense in ${S}_k(\\Omega ) \\cap L^2 ( \\Omega )$ with respect to the topology induced by the $H^1(\\Omega )$ -norm.", "Theorem 1.2 Consider the transmission eigenvalue problem (REF ) with $\\eta \\lnot \\equiv 0$ .", "Let $\\mathbf {x}_c\\in \\partial \\Omega $ be a corner point in two and three dimensions and $\\mathcal {N}_h$ be a neighbourhood of $\\mathbf {x}_c$ within $\\Omega $ with $h\\in \\mathbb {R}_+$ sufficiently small.", "Suppose that $(1+V) w$ and $\\eta $ are both Hölder continuous on $\\overline{\\mathcal {N}_h}$ and $\\partial \\mathcal {N}_h\\cap \\partial \\Omega $ respectively and $\\eta (\\mathbf {x}_c)\\ne 0$ .", "If there exist constants $C,\\varrho $ and $\\Upsilon $ with $C>0, \\Upsilon >0 \\text{ and } \\varrho <\\Upsilon $ such that the transmission eigenfunction $v$ can be approximated in $H^{1}\\left(\\mathcal {N}_{h}\\right)$ by the Herglotz functions $v_{j}, j=1,2, \\ldots ,$ with kernels $g_{j}$ satisfying $\\left\\Vert v-v_{j}\\right\\Vert _{H^{1}\\left(\\mathcal {N}_{h}\\right)} \\le j^{-\\Upsilon }, \\quad \\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)} \\le C j^{\\varrho }, $ then $w$ and $v$ vanish near $\\mathbf {x}_c$ in the sense of (REF ).", "More detailed results are respectively given in Theorems REF and REF for the two and three dimensions.", "Remark 1.3 As discussed earlier, the vanishing properties were investigated in [19] under a similar setup to Theorem REF .", "Compared to the corresponding results in [19], Theorem REF has two significant improvements in the regularity requirements.", "First, the Herglotz approximation condition in [19] was required to be $\\left\\Vert v-v_{j}\\right\\Vert _{H^{1}\\left(\\mathcal {N}_{h}\\right)} \\le j^{-1-\\Upsilon }, \\quad \\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)} \\le C j^{\\rho },$ where the constants $C>0, \\Upsilon >0 \\text{ and } 0<\\varrho <1$ .", "It is directly verified that the regularity condition (REF ) is included in (REF ) as a special case.", "Second, it was required in [19] that $w-v$ is $H^2$ -regular away from the corner point $\\mathbf {x}_c$ , and we remove this rather artificial regularity requirement in Theorem REF .", "Theorem 1.4 Consider the transmission eigenvalue problem (REF ) with $\\eta \\equiv 0$ .", "Under the same conditions as in Theorem REF , one has $\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega \\right)} \\int _{B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega } V(\\mathbf {x})w(\\mathbf {x}) \\mathrm {d} \\mathbf {x}=0.$ The similar result holds in the three-dimensional case with (REF ) replaced to be (REF ).", "More detailed results are respectively given in Corollaries REF and REF for the two and three dimensions.", "Remark 1.5 If $V(\\mathbf {x})$ is continuous near the corner $\\mathbf {x}_c$ and $V(\\mathbf {x}_c)\\ne 0$ , from the fact that $\\begin{split}&\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega \\right)} \\int _{B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega } V(\\mathbf {x}) w(\\mathbf {x}) \\mathrm {d} \\mathbf {x}\\\\=&V\\left(\\mathbf {x}_{c}\\right) \\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega \\right)} \\int _{B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega } w(\\mathbf {x}) \\mathrm {d} \\mathbf {x},\\end{split}$ one can readily see that $w$ vanishes near $\\mathbf {x}_c$ , which in turn implies the vanishing of $v$ near $\\mathbf {x}_c$ by noting that $w$ and $v$ possess the same traces on $\\partial \\Omega $ .", "Remark 1.6 The vanishing of the transmission eigenfunctions in the case $\\eta \\equiv 0$ was also studied in [5], [19].", "The regularity requirement in [19] is the same as that described in Remark REF , whereas in [5], the Herglotz approximation was required to be $\\left\\Vert v-v_{j}\\right\\Vert _{L^{2}(\\mathcal {N}_h)} \\le e^{-j}, \\quad \\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)} \\le C(\\ln j)^{\\beta },$ where the constants $C>0 \\text{ and } 0<\\beta <1 /(2 n+8),(n=2,3)$ .", "It is directly verified that the corresponding results in [5], [19] are included into Theorems REF and REF as special cases.", "Nevertheless, it is pointed out that in [5], the technical condition $(1+V)w$ being Hölder continuous is not required and instead it is required that $V$ is Hölder continuous.", "Finally, we would like to give two general remarks on the vanishing properties of the transmission eigenfunctions.", "Remark 1.7 The vanishing properties established in Theorems REF and REF as well as those in [5], [19] are of a completely local feature.", "That is, all the results hold for the partial-data transmission eigenvalue problem, namely in (REF ) the transmission boundary conditions on $\\partial \\Omega $ is required to hold only in a small neighbourhood of the corner point.", "It is mentioned that a global rigidity result of the geometric structure of the transmission eigenfunctions was presented in [14].", "Remark 1.8 According to our earlier discussion, if the transmission eigenfunctions $w$ and $v$ are Hölder continuous around the corner, then both of them vanish near the corner.", "Hence, in order to search for the transmission eigenfunctions that are non-vanishing near corners, especially those numerically found in [4] which are actually locally localizing around corners, one should consider transmission eigenfunctions whose regularity lies between $H^1$ and $C^\\alpha $ , $\\alpha \\in (0, 1)$ .", "By using properties of the Herglotz approximation (cf.", "[15]), one can show (though not straightforward) that the regularity criterion (REF ) defines a set of functions which includes some functions that are less regular than $C^\\alpha $ , but also does not include some functions which are more regular than $C^\\alpha $ .", "Hence, the regularity characterization in terms of the Herglotz approximation is of a different feature from the standard Sobolev regularity.", "Nevertheless, Theorems REF and REF indicate that the vanishing near corners is a generic local geometric property of the transmission eigenfunctions.", "In what follows, Sections 2 and 3 are respectively devoted to the vanishing properties of the transmission eigenfunctions in two and three dimensions." ], [ "Vanishing properties in two dimensions", "To facilitate calculation and analysis, we introduce the two-dimensional polar coordinates $(r, \\theta )$ such that $\\mathbf {x}=\\left(x_{1}, x_{2}\\right)=(r \\cos \\theta , r \\sin \\theta ) \\in \\mathbb {R}^{2}$ .", "Set $B_{h}:=B_{h}(\\mathbf {0})$ for $h\\in \\mathbb {R}_+$ .", "Define an open sector in $\\mathbb {R}^{2}$ with the boundary $\\Gamma ^{\\pm }$ as follows, $W=\\left\\lbrace \\mathbf {x}\\in \\mathbb {R}^{2} \|\\ \\mathbf {x}\\ne \\mathbf {0}, \\ \\theta _{m}<\\arg \\left(x_{1}+\\mathbf {i} x_{2}\\right)<\\theta _{M}\\right\\rbrace .$ where $-\\pi <\\theta _{m}<\\theta _{M}<\\pi , \\mathbf {i}:=\\sqrt{-1}$ and $\\Gamma ^{+}$ and $\\Gamma ^{-}$ respectively are $\\left(r, \\theta _{M}\\right)$ and $\\left(r, \\theta _{m}\\right)$ with $r>0 .$ Define that $S_{h}=W \\cap B_{h}, \\Gamma _{h}^{\\pm }=\\Gamma ^{\\pm } \\cap B_{h}, \\bar{S}_{h}=\\overline{W} \\cap B_{h}, \\Lambda _{h}=S_{h} \\cap \\partial B_{h}, \\text{ and } \\Sigma _{\\Lambda _{h}}=S_{h} \\backslash S_{h / 2}.$ We shall make use a particular type of planar complex geometrical optics (CGO) solution, which was first introduced in [2].", "Lemma 2.1 [2] For $\\mathbf {x}\\in \\mathbb {R}^2$ denote $r=\|\\mathbf {x}\|,\\theta =arg(x_1+\\mathbf {i}x_2).$ Define $u_{0}(\\mathbf {x}):=\\exp \\left(\\sqrt{r}\\left(\\cos \\left(\\frac{\\theta }{2}+\\pi \\right)+\\mathbf {i} \\sin \\left(\\frac{\\theta }{2}+\\pi \\right)\\right)\\right),$ then $\\Delta u_0=0 \\text{ in } \\mathbb {R}^2 \\setminus \\mathbb {R}_{0,-}^{2}$ , where $\\mathbb {R}_{0,-}^{2}:=\\left\\lbrace \\mathbf {x}\\in \\mathbb {R}^2\|\\ \\mathbf {x}= (x_1,x_2);\\ x_1<=0, x_2=0\\right\\rbrace $ and $s \\mapsto u_{0}(sx)$ decays exponentially in $R_{+} .$ Let $\\alpha , s>0 .$ Then $\\int _{W}\\left\|u_{0}(s \\mathbf {x}) \\Vert \\mathbf {x}\\right\|^{\\alpha } \\mathrm {d} \\mathbf {x}\\le \\frac{2\\left(\\theta _{M}-\\theta _{m}\\right) \\Gamma (2 \\alpha +4)}{\\delta _{W}^{2 \\alpha +4}} s^{-\\alpha -2}.$ where $\\delta _{W}=-\\max _{\\theta _{m}<\\theta <\\theta _{M}} \\cos (\\theta / 2+\\pi )>0 .", "$ Moreover $\\int _{W} u_{0}(s \\mathbf {x}) \\mathrm {d} x=6 \\mathbf {i}\\left(e^{-2 \\theta _{M} \\mathbf {i}}-e^{-2 \\theta _{m} \\mathbf {i}}\\right) s^{-2}.$ and for $h>0$ $\\int _{W \\backslash B_{h}}\\left\|u_{0}(s \\mathbf {x})\\right\| \\mathrm {d} \\mathbf {x}\\le \\frac{6\\left(\\theta _{M}-\\theta _{m}\\right)}{\\delta _{W}^{4}} s^{-2} e^{-\\delta _{W} \\sqrt{h s} / 2}.$ By direct calculations, one can obtain the following estimates for the CGO solution $u_0(s\\mathbf {x})$ : Corollary 2.1 $u_0 \\notin H^2(B_{\\varepsilon })$ near the origin and $\|u_0(s\\mathbf {x})\|\\le 1$ in $B_{\\varepsilon }$ for sufficiently small $\\varepsilon $ .", "Furthermore, one has the following result: Corollary 2.2 The following estimates hold for the $L^2$ norm of $u_0$ , $\\left\\Vert u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(S_{h}\\right)}^{2} & \\le \\frac{\\left(\\theta _{M}-\\theta _{m}\\right) e^{-2 \\sqrt{s \\Theta } \\delta _{W}} h^{2}}{2}, \\quad \\Theta \\in (0, h),\\\\\\left\\Vert u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(\\Lambda _{h}\\right)} & \\le \\sqrt{h} e^{-\\delta _{W} \\sqrt{s h}} \\sqrt{\\theta _{M}-\\theta _{m}},\\\\\\left\\Vert \\partial _{\\nu } u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(\\Lambda _{h}\\right)} & \\le \\frac{1}{2} \\sqrt{s} e^{-\\delta _{W} \\sqrt{s h}} \\sqrt{\\theta _{M}-\\theta _{m}},\\\\\\left\\Vert \\partial _{\\theta } u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(\\Lambda _{h}\\right)} & \\le \\frac{\\sqrt{s}}{2}h^2 e^{-\\delta _{W} \\sqrt{s h}} \\sqrt{\\theta _{M}-\\theta _{m}},\\\\\\left\\Vert \|\\mathbf {x}\|^{\\alpha } u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(S_{h}\\right)}^{2} & \\le s^{-(2 \\alpha +2)} \\frac{2\\left(\\theta _{M}-\\theta _{m}\\right)}{\\left(4 \\delta _{W}^{2}\\right)^{2 \\alpha +2}} \\Gamma (4 \\alpha +4),$ where $\\delta _{W}$ is defined in (REF ) and $S_{h}$ , $\\Lambda _{h}$ are defined in (REF ).", "Using the integral mean value theorem, one can deduce that $\\begin{split}\\left\\Vert u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(S_{h}\\right)}^{2} &=\\int _{0}^{h} r \\mathrm {d} r \\int _{\\theta _{m}}^{\\theta _{M}} e^{2 \\sqrt{s r} \\cos (\\theta / 2+\\pi )} \\mathrm {d} \\theta \\\\&\\le \\int _{0}^{h} r \\mathrm {d} r \\int _{\\theta _{m}}^{\\theta _{M}} e^{-2 \\sqrt{s r} \\delta _{W}} \\mathrm {d} \\theta \\\\&=\\frac{\\left(\\theta _{M}-\\theta _{m}\\right) e^{-2 \\sqrt{s \\Theta } \\delta _{W}} h^{2}}{2},\\end{split}$ where $\\Theta \\in (0, h)$ .", "On $\\Lambda _{h}$ , it can be seen that $\\begin{split}\\left\|u_{0}(s \\mathbf {x})\\right\| &=e^{\\sqrt{s h} \\cos (\\theta / 2+\\pi )} \\le e^{-\\delta _{W} \\sqrt{s h}} \\\\\\left\|\\partial _{\\nu } u_{0}(s \\mathbf {x})\\right\| &=\\left\|\\frac{\\sqrt{s} e^{i \\cos (\\theta / 2+\\pi )}}{2 \\sqrt{h}} e^{\\sqrt{s h} \\exp (i(\\theta / 2+\\pi ))}\\right\| \\le \\frac{1}{2} \\sqrt{\\frac{s}{h}} e^{-\\delta _{W} \\sqrt{s h}}\\\\\\left\|\\partial _{\\theta } u_{0}(s \\mathbf {x})\\right\| &=\\left\|\\frac{\\sqrt{sh} }{2 } e^{\\sqrt{s h} \\exp (i(\\theta / 2+\\pi ))}\\right\| \\le \\frac{\\sqrt{sh} }{2 } e^{-\\delta _{W} \\sqrt{s h}},\\end{split}$ all of which decay exponentially as $s \\rightarrow \\infty $ .", "By straightforward calculations, one can show that $\\begin{split}\\left\\Vert u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(\\Lambda _{h}\\right)} &\\le \\sqrt{h}e^{-\\delta _{W} \\sqrt{s h}} \\sqrt{\\theta _{M}-\\theta _{m}}, \\\\\\left\\Vert \\partial _{\\nu } u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(\\Lambda _{h}\\right)} &\\le \\frac{1}{2} \\sqrt{s} e^{-\\delta _{W} \\sqrt{s h}} \\sqrt{\\theta _{M}-\\theta _{m}},\\\\\\left\\Vert \\partial _{\\theta } u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(\\Lambda _{h}\\right)} &\\le \\frac{\\sqrt{s}}{2}h^2 e^{-\\delta _{W} \\sqrt{s h}} \\sqrt{\\theta _{M}-\\theta _{m}}.\\end{split}$ Using polar coordinates, we can deduce that $\\begin{split}\\quad \\left\\Vert \|\\mathbf {x}\|^{\\alpha } u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(S_{h}\\right)}^{2}&=\\int _{0}^{h} r \\mathrm {d} r \\int _{\\theta _{m}}^{\\theta _{M}} r^{2 \\alpha } e^{2 \\sqrt{s r} \\cos (\\theta / 2+\\pi )} \\mathrm {d} \\theta \\\\&\\le \\int _{0}^{h} r \\mathrm {d} r \\int _{\\theta _{m}}^{\\theta _{M}} r^{2 \\alpha } e^{-2 \\sqrt{s r} \\delta w} \\mathrm {d} \\theta \\\\&=\\left(\\theta _{M}-\\theta _{m}\\right) \\int _{0}^{h} r^{2 \\alpha +1} e^{-2 \\delta _{W} \\sqrt{s r}} \\mathrm {d} r \\quad \\left(t=2 \\delta _{W} \\sqrt{s r}\\right) \\\\&=s^{-(2 \\alpha +2)} \\frac{2\\left(\\theta _{M}-\\theta _{m}\\right)}{\\left(4 \\delta _{W}^{2}\\right)^{2 \\alpha +2}} \\int _{0}^{2 \\delta _{W} \\sqrt{s h}} t^{4 \\alpha +3} e^{-t} \\mathrm {d} r \\\\&\\le s^{-(2 \\alpha +2)} \\frac{2\\left(\\theta _{M}-\\theta _{m}\\right)}{\\left(4 \\delta _{W}^{2}\\right)^{2 \\alpha +2}}\\Gamma (4 \\alpha +4),\\end{split}$ which completes the proof.", "Next, by direct computations and the compact embeddings of Hölder spaces, one can easily obtain the following result: Lemma 2.2 For the Herglotz wave function $v_j$ defined in (REF ) in two dimensions, $\\begin{split}\\left\\Vert v_{j}\\right\\Vert _{C^{1}} &\\le \\sqrt{2 \\pi }(1+k)\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)},\\\\\\left\\Vert v_{j}\\right\\Vert _{C^{\\alpha }} &\\le \\operatorname{diam}\\left(S_{h}\\right)^{1-\\alpha }\\left\\Vert v_{j}\\right\\Vert _{C^{1}}.\\end{split}$ where $0<\\alpha <1$ and $\\operatorname{diam}\\left(S_{h}\\right)$ is the diameter of $S_{h}$ .", "Furthermore, using the Jacobi-Anger expansion ([15]) in $\\mathbb {R}^2$ , we can obtain the following lemma.", "Lemma 2.3 The Helgloltz wave function $v_j$ defined in (REF ) admits the following asymptotic expansion: $\\begin{split}v_{j}(\\mathbf {x})&=J_{0}(k\|\\mathbf {x}\|)\\int _{\\mathbb {S}^{n-1}} g_{j}(\\theta ) \\mathrm {d} \\sigma (\\theta ) + 2 \\sum _{p=1}^{\\infty } \\mathbf {i}^{p} J_{p}(k\|\\mathbf {x}\|) \\int _{\\mathbb {S}^{n-1}} g_{j}(\\theta ) \\cos (p \\varphi ) \\mathrm {d} \\sigma (\\theta )\\\\&:=v_{j}(0) J_{0}(k\|\\mathbf {x}\|)+2 \\sum _{p=1}^{\\infty } \\mathbf {i}^{p} J_{p}(k\|\\mathbf {x}\|) \\gamma _{p j}, \\quad \\mathbf {x}\\in \\mathbb {R}^{2}.\\end{split}$ where $J_{p}(t)$ is the p-th Bessel function of the first kind.", "Indeed, $J_{p}(t)=\\frac{t^{p}}{2^{p} p !", "}+\\frac{t^{p}}{2^{p}} \\sum _{\\ell =1}^{\\infty } \\frac{(-1)^{\\ell } t^{2 \\ell }}{4^{\\ell }(\\ell !", ")^{2}}, \\quad p=0, 1, 2, \\ldots $ Lemma 2.4 Suppose $f(\\mathbf {x})\\in C^{\\alpha }$ , then the following expansion holds near the origin: $f(\\mathbf {x})=f(\\mathbf {0})+\\delta f(\\mathbf {x}),\\quad \|\\delta f(\\mathbf {x})\|\\le \\left\\Vert f\\right\\Vert _{C^\\alpha } \|\\mathbf {x}\|^{\\alpha }.$ Lemma 2.5 Let $v,w \\in H^{1}(\\Omega )$ be a pair of conductive transmission eigenfunctions to (REF ), $D_{\\varepsilon }=S_{h}\\setminus B_{\\varepsilon }$ for $0<\\varepsilon <h$ , $\\eta \\in C^{\\alpha }\\left(\\bar{\\Gamma }_{h}^{\\pm }\\right)$ for $0<\\alpha <1$ and $\\Gamma ^{\\pm }_{h}$ , $S_{h}$ be defined in (REF ), then $\\begin{split}\\lim _{\\varepsilon \\rightarrow 0} \\int _{D_{\\varepsilon }} \\Delta (v-w)u_{0}(s\\mathbf {x}) \\mathrm {d} \\mathbf {x}= &\\int _{\\Lambda _{h}}\\left(u_{0}(s \\mathbf {x}) \\partial _{\\nu }(v-w)-(v-w) \\partial _{\\nu } u_{0}(s \\mathbf {x})\\right) \\mathrm {d} \\sigma \\\\&-\\int _{\\Gamma _{h}^{\\pm }} \\eta (\\mathbf {x}) u_{0}(s \\mathbf {x}) v(\\mathbf {x}) \\mathrm {d} \\sigma .\\end{split}$ Using Green's formula and the boundary condition in (REF ), we can deduce that $\\begin{split}&\\int _{D_{\\varepsilon }}\\Delta (v-w) u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}\\\\=&\\int _{D_{\\varepsilon }}(v-w) \\Delta u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}+\\int _{\\partial D_{\\varepsilon }}\\left(u_{0}(s \\mathbf {x}) \\partial _{\\nu }(v-w)-(v-w) \\partial _{\\nu } u_{0}(s \\mathbf {x})\\right) \\mathrm {d} \\sigma \\\\=&\\int _{\\Lambda _{h}}\\left(u_{0}(s \\mathbf {x}) \\partial _{\\nu }(v-w)-(v-w) \\partial _{\\nu } u_{0}(s \\mathbf {x})\\right) \\mathrm {d} \\sigma \\\\&+\\int _{\\Lambda _{\\varepsilon }}\\left(u_{0}(s \\mathbf {x}) \\partial _{\\nu }(v-w)-(v-w) \\partial _{\\nu } u_{0}(s \\mathbf {x})\\right) \\mathrm {d} \\sigma \\\\&-\\int _{\\Gamma _{(\\varepsilon ,h)}^{\\pm }} \\eta (\\mathbf {x}) u_{0}(s \\mathbf {x}) v(\\mathbf {x}) \\mathrm {d} \\sigma ,\\end{split}$ where $\\Gamma _{(\\varepsilon , h)}^{\\pm }=\\Gamma ^{\\pm } \\cap \\left(B_{h} \\backslash B_{\\varepsilon }\\right)$ and $\\Lambda _{\\varepsilon }=S_{h}\\cap \\partial B_{\\varepsilon }$ .", "Using the trace theorem, we have $v \\in L^{2}\\left(\\Gamma _{(0, \\varepsilon )}^{\\pm }\\right)$ , where $\\Gamma _{(0, \\varepsilon )}^{\\pm }=\\Gamma ^{\\pm } \\cap B_{\\varepsilon }$ .", "By Corollary REF , one can see that $\\lim _{\\varepsilon \\rightarrow 0} \\int _{\\Lambda _{\\varepsilon }}\\left(u_{0} \\partial _{\\nu }(v-w)-(v-w) \\partial _{\\nu } u_{0}\\right) \\mathrm {d} \\sigma =0,$ and $\\lim _{\\varepsilon \\rightarrow 0} \\int _{\\Gamma _{(0, \\varepsilon )}^{\\pm }} \\eta u_{0} v \\mathrm {d} \\sigma =0.$ Thus (REF ) is obtained by taking the limit on both sides of the equation (REF ).", "Lemma 2.6 Suppose $\\eta \\in C^{\\alpha }\\left(\\bar{\\Gamma }_{h}^{\\pm }\\right)$ for $0<\\alpha <1$ , $\\theta _{M},\\theta _{m}$ are defined in (REF ) and $\\theta _{M}-\\theta _{m}\\ne \\pi $ .", "Define $\\omega (\\theta )=-\\cos (\\theta / 2+\\pi ), \\quad \\mu (\\theta )=-\\cos (\\theta / 2+\\pi )-\\mathbf {i} \\sin (\\theta / 2+\\pi ),$ and $I_{2}^{\\pm }=\\int _{\\Gamma _{h}^{\\pm }} \\eta (\\mathbf {x}) u_{0}(s \\mathbf {x}) v_{j}(\\mathbf {x}) \\mathrm {d} \\sigma .$ Then it holds that $\\begin{split}I_{2}^{-}=& 2 \\eta (\\mathbf {0}) v_{j}(\\mathbf {0}) s^{-1}\\left(\\mu \\left(\\theta _{m}\\right)^{-2}-\\mu \\left(\\theta _{m}\\right)^{-2} e^{-\\sqrt{s h} \\mu \\left(\\theta _{m}\\right)}-\\mu \\left(\\theta _{m}\\right)^{-1} \\sqrt{s h} e^{-\\sqrt{s h} \\mu \\left(\\theta _{m}\\right)}\\right)\\\\&+v_{j}(\\mathbf {0})\\eta (\\mathbf {0})I_{21}^{-} + \\eta (\\mathbf {0})I_{22}^{-}+I_{\\eta }^{-},\\end{split}$ and $\\begin{split}I_{2}^{+}=& 2 \\eta (\\mathbf {0}) v_{j}(\\mathbf {0}) s^{-1}\\left(\\mu \\left(\\theta _{M}\\right)^{-2}-\\mu \\left(\\theta _{M}\\right)^{-2} e^{-\\sqrt{s h} \\mu \\left(\\theta _{M}\\right)}-\\mu \\left(\\theta _{M}\\right)^{-1} \\sqrt{s h} e^{-\\sqrt{s h} \\mu \\left(\\theta _{M}\\right)}\\right)\\\\&+v_{j}(\\mathbf {0})\\eta (\\mathbf {0})I_{21}^{+} + \\eta (\\mathbf {0})I_{22}^{+}+I_{\\eta }^{+},\\end{split}$ where $\\begin{split}I_{21}^{-} &\\le \\mathcal {O}(s^{-3}), \\quad I_{21}^{+} \\le \\mathcal {O}(s^{-3}),\\\\I_{22}^{-} &\\le \\mathcal {O}(\\Vert g_{j}\\Vert _{L^{2}(\\mathbb {S}^{n-1})}s^{-2}),\\quad I_{22}^{+} \\le \\mathcal {O}(\\Vert g_{j}\\Vert _{L^{2}(\\mathbb {S}^{n-1})}s^{-2}),\\\\\|I_{\\eta }^{-}\| &\\le \\Vert \\eta \\Vert _{C^{\\alpha }}\\left(v_{j}(\\mathbf {0})\\mathcal {O}(s^{-1-\\alpha }) + \\mathcal {O}(\\Vert g_{j}\\Vert _{L^{2}(\\mathbb {S}^{n-1})}s^{-2-\\alpha })\\right),\\\\\|I_{\\eta }^{+}\| &\\le \\Vert \\eta \\Vert _{C^{\\alpha }}\\left(v_{j}(\\mathbf {0})\\mathcal {O}(s^{-1-\\alpha }) + \\mathcal {O}(\\Vert g_{j}\\Vert _{L^{2}(\\mathbb {S}^{n-1})}s^{-2-\\alpha })\\right).\\end{split}$ Using Lemma REF , we have $I_{2}^{-}=\\eta (\\mathbf {0}) \\int _{\\Gamma _{h}^{-}} u_{0}(s \\mathbf {x}) v_{j}(\\mathbf {x}) \\mathrm {d} \\sigma + \\int _{\\Gamma _{h}^{-}} \\delta \\eta (\\mathbf {x}) u_{0}(s \\mathbf {x}) v_{j}(\\mathbf {x}) \\mathrm {d} \\sigma .$ Note that $\\omega (\\theta )>0$ for $\\theta _{m} \\le \\theta \\le \\theta _{M}$ .", "By Lemma REF , we can obtain that $\\begin{split}\\int _{\\Gamma _{h}^{-}} u_{0}(s \\mathbf {x}) v_{j}(\\mathbf {x}) \\mathrm {d} \\sigma =&\\int _{0}^{h}\\left(v_{j}(\\mathbf {0}) J_{0}(k r)+2 \\sum _{p=1}^{\\infty } \\gamma _{p j} \\mathbf {i}^{p} J_{p}(k r)\\right) e^{-\\sqrt{s r} \\mu \\left(\\theta _{m}\\right)} \\mathrm {d} r \\\\=&v_{j}(\\mathbf {0})\\left[\\int _{0}^{h} e^{-\\sqrt{s r} \\mu \\left(\\theta _{m}\\right)} \\mathrm {d} r+\\sum _{p=1}^{\\infty } \\frac{(-1)^{p} k^{2 p}}{4^{p}(p !", ")^{2}} \\int _{0}^{h} r^{2 p} e^{-\\sqrt{s r} \\mu \\left(\\theta _{m}\\right)} \\mathrm {d} r\\right]\\\\&+\\int _{0}^{h} 2 \\sum _{p=1}^{\\infty } \\gamma _{p j} \\mathbf {i}^{p} J_{p}(k r) e^{-\\sqrt{s r} \\mu \\left(\\theta _{m}\\right)} \\mathrm {d} r\\\\:=&v_{j}(\\mathbf {0})\\left[\\int _{0}^{h} e^{-\\sqrt{s r} \\mu \\left(\\theta _{m}\\right)} \\mathrm {d} r + I_{21}^{-}\\right] + I_{22}^{-}.\\end{split}$ Furthermore, we have $\\int _{0}^{h} e^{-\\sqrt{s r} \\mu \\left(\\theta _{m}\\right)} \\mathrm {d} r = 2 s^{-1}\\left(\\mu \\left(\\theta _{m}\\right)^{-2}-\\mu \\left(\\theta _{m}\\right)^{-2} e^{-\\sqrt{s h} \\mu \\left(\\theta _{m}\\right)}-\\mu \\left(\\theta _{m}\\right)^{-1} \\sqrt{s h} e^{-\\sqrt{s h} \\mu \\left(\\theta _{m}\\right)}\\right),$ and $\\left\|I_{21}^{-} \\right\| \\le \\sum _{p=1}^{\\infty } \\frac{k^{2 p} h^{2 p-2}}{4^{p}(p !", ")^{2}} \\int _{0}^{h} r^{2} e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r=\\mathcal {O}\\left(s^{-3}\\right).$ By using Lemma REF , one can drive that $\\begin{split}\\left\|I_{22}^{-}\\right\|& \\le 2\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)} \\sum _{p=1}^{\\infty } \\left[\\frac{k^{p}}{2^{p} p !}", "\\int _{0}^{h} r^{p} e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r\\right.\\left.+\\frac{(k h)^{p}}{2^{p}} \\sum _{\\ell =1}^{\\infty } \\frac{k^{2 \\ell } h^{2(\\ell -1)}}{4^{\\ell }(\\ell !", ")^{2}} \\int _{0}^{h} r^{2} e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r\\right] \\\\&\\le 2\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)} \\sum _{p=1}^{\\infty }\\left[\\frac{k^{p} h^{p-1}}{2^{p} p !}", "\\int _{0}^{h} r e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r+\\mathcal {O}\\left(s^{-3}\\right)\\right] \\\\&= \\mathcal {O}\\left(\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)} s^{-2}\\right),\\end{split}$ where we assume that $kh<1$ for sufficiently small $h$ .", "Taking $I_{\\eta }^{-}=\\int _{\\Gamma _{h}^{-}} \\delta \\eta (\\mathbf {x}) u_{0}(s \\mathbf {x}) v_{j}(\\mathbf {x}) \\mathrm {d} \\sigma .$ By Lemmas REF and REF , it holds that $\\begin{split}\\left\|I_{\\eta }^{-}\\right\| \\le &\\Vert \\eta \\Vert _{C^{\\alpha }} \\int _{0}^{h} r^{\\alpha }\\left(v_{j}(0) J_{0}(k r)+2 \\sum _{p=1}^{\\infty }\\left\|\\gamma _{p j}\\right\| J_{p}(k r)\\right) e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r\\\\= & \\Vert \\eta \\Vert _{C^{\\alpha }} v_{j}(0) (\\int _{0}^{h} r^{\\alpha } e^{-\\sqrt{s r \\omega }\\left(\\theta _{m}\\right)} \\mathrm {d} r + \\sum _{p=1}^{\\infty } \\frac{(-1)^{p} k^{2 p}}{4^{p}(p !", ")^{2}} \\int _{0}^{h} r^{\\alpha +2 p} e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r)\\\\&+2 \\Vert \\eta \\Vert _{C^{\\alpha }} \\sum _{p=1}^{\\infty } \\int _{0}^{h} r^{\\alpha } \\gamma _{p j} \\mathbf {i}^{p} J_{p}(k r) e^{-\\sqrt{s r} \\mu \\left(\\theta _{m}\\right)} \\mathrm {d} r.\\end{split}$ Since $\\omega (\\theta _{m})>0$ , as $s\\rightarrow \\infty $ , we have $\\int _{0}^{h} r^{\\alpha } e^{-\\sqrt{s r \\omega }\\left(\\theta _{m}\\right)} \\mathrm {d} r = \\mathcal {O}\\left(s^{-1-{\\alpha }}\\right),$ and $\\left\|\\sum _{p=1}^{\\infty } \\frac{(-1)^{p} k^{2 p}}{4^{p}(p !", ")^{2}} \\int _{0}^{h} r^{\\alpha +2 p} e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r)\\right\| \\le \\sum _{p=1}^{\\infty } \\frac{h^{2 p-2} k^{2 p}}{4^{p}(p !", ")^{2}} \\int _{0}^{h} r^{\\alpha +2} e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r=\\mathcal {O}\\left(s^{-3-\\alpha }\\right),$ for $kh<1$ .", "Furthermore, by direct computations, one can obtain that $\\begin{split}&\\left\|\\sum _{p=1}^{\\infty } \\int _{0}^{h} r^{\\alpha } \\gamma _{p j} \\mathbf {i}^{p} J_{p}(k r) e^{-\\sqrt{s r} \\mu \\left(\\theta _{m}\\right)} \\mathrm {d} r\\right\|\\\\\\le & \\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)} \\sum _{p=1}^{\\infty } \\left[\\frac{k^{p}}{2^{p} p !}", "\\int _{0}^{h} r^{p+\\alpha } e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r\\right.\\\\&\\left.+\\frac{k^{p}}{2^{p}} \\sum _{\\ell =1}^{\\infty } \\frac{k^{2 \\ell } h^{2(\\ell -1)}}{4^{\\ell }(\\ell !", ")^{2}}\\left(\\int _{0}^{h} r^{p+\\alpha +2} e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r\\right)\\right] \\\\\\le & \\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)} \\sum _{p=1}^{\\infty } \\left[\\frac{k^{p}}{2^{p} p !}", "\\int _{0}^{h} r^{p+\\alpha } e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r\\right.\\\\&\\left.\\left.+\\frac{(k h)^{p}}{2^{p}} \\sum _{\\ell =1}^{\\infty } \\frac{k^{2 \\ell } h^{2(\\ell -1)}}{4^{\\ell }(\\ell !", ")^{2}}\\left(\\int _{0}^{h} r^{\\alpha +2} e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r\\right)\\right]\\right] \\\\\\le & \\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)} \\sum _{p=1}^{\\infty }\\left[\\frac{k^{p} h^{p-1}}{2^{p} p !}", "\\int _{0}^{h} r^{\\alpha +1} e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r+\\mathcal {O}\\left(s^{-\\alpha -3}\\right)\\right]\\\\= & \\mathcal {O}\\left(\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)} s^{-\\alpha -2}\\right).\\end{split}$ Combining (REF )-(REF ), one finally obtains $\|I_{\\eta }^{-}\|\\le \\left\\Vert \\eta \\right\\Vert _{C^{\\alpha }}\\left(v_j(\\mathbf {0}) \\mathcal {O}\\left(s^{-1-\\alpha }\\right) + \\mathcal {O}\\left(\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)} s^{-\\alpha -2}\\right)\\right).$ Similarly, one can prove (REF ), which completes the proof.", "Lemma 2.7 Suppose that $\\eta \\in C^{\\alpha }\\left(\\bar{\\Gamma }_{h}^{\\pm }\\right)$ for $0<\\alpha <1$ , $\\Gamma ^{\\pm }_{h}$ is defined in (REF ), $\\theta _{M},\\theta _{m}$ are defined in (REF ) and $\\theta _{M}-\\theta _{m}\\ne \\pi $ .", "Define $\\xi _{j}^{\\pm }(s)=\\int _{\\Gamma _{h}^{\\pm }} \\eta (\\mathbf {x}) u_{0}(s \\mathbf {x})\\left(v(\\mathbf {x})-v_{j}(\\mathbf {x})\\right) \\mathrm {d} \\sigma .$ Then the following estimate holds, $\\begin{split}\\left\|\\xi _{j}^{\\pm }(s)\\right\| \\le & C\\Big (\|\\eta (0)\| \\frac{\\sqrt{\\theta _{M}-\\theta _{m}} e^{-\\sqrt{s \\Theta \\delta _{W}} h}}{\\sqrt{2}}\\\\&+\\Vert \\eta \\Vert _{C^{\\alpha }}s^{-(\\alpha +1)} \\frac{\\sqrt{2\\left(\\theta _{M}-\\theta _{m}\\right) \\Gamma (4 \\alpha +4)}}{\\left(2 \\delta _{W}\\right)^{2 \\alpha +2}}\\Big ) \\left\\Vert v-v_{j}\\right\\Vert _{H^{1}\\left(S_{h}\\right)},\\end{split}$ where $\\delta _{w}$ is defined in (REF ).", "By Lemma REF , the trace theorem and Cauchy-Schwarz inequality, one has $\\begin{split}\\left\|\\xi _{j}^{\\pm }(s)\\right\| & \\le \|\\eta (\\mathbf {0})\| \\int _{\\Gamma _{h}^{\\pm }}\\left\|u_{0}(s \\mathbf {x})\\right\| \\left\|v(\\mathbf {x})-v_{j}(\\mathbf {x})\\right\| \\mathrm {d} \\sigma +\\Vert \\eta \\Vert _{C^{\\alpha }} \\int _{\\Gamma _{h}^{\\pm }} \\left\| \\mathbf {x}\\right\|^{\\alpha }\\left\|u_{0}(s \\mathbf {x})\\right\| \\left\| v(\\mathbf {x})-v_{j}(\\mathbf {x})\\right\| \\mathrm {d} \\sigma \\\\& \\le \|\\eta (\\mathbf {0})\|\\left\\Vert v-v_{j}\\right\\Vert _{H^{1 / 2}\\left(\\Gamma _{h}^{\\pm }\\right)}\\left\\Vert u_{0}(s \\mathbf {x})\\right\\Vert _{H^{-1 / 2}\\left(\\Gamma _{h}^{\\pm }\\right)} \\\\&+\\Vert \\eta \\Vert _{C^{\\alpha }}\\left\\Vert v-v_{j}\\right\\Vert _{H^{1 / 2}\\left(\\Gamma _{h}^{\\pm }\\right)}\\left\\Vert \\left\|\\mathbf {x}\\right\|^{\\alpha } u_{0}(s \\mathbf {x})\\right\\Vert _{H^{-1 / 2}\\left(\\Gamma _{h}^{\\pm }\\right)} \\\\& \\le \|\\eta (\\mathbf {0})\|\\left\\Vert v-v_{j}\\right\\Vert _{H^{1}\\left(S_{h}\\right)}\\left\\Vert u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(S_{h}\\right)}+\\Vert \\eta \\Vert _{C^{\\alpha }}\\left\\Vert v-v_{j}\\right\\Vert _{H^{1}\\left(S_{h}\\right)}\\left\\Vert \\left.\|\\mathbf {x}\\right\|^{\\alpha } u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(S_{h}\\right)}.\\end{split}$ Combining with Corollary REF , one readily obtains (REF ).", "The proof is complete.", "We are in a position to present our first main result on the vanishing properties of the conductive transmission eigenfunctions $(v,w)$ in two dimensions.", "Theorem 2.3 Let $v \\in H^{1}(\\Omega )$ and $w \\in H^{1}(\\Omega )$ be a pair of eigenfunctions to (REF ) associated with $k \\in \\mathbb {R}_{+} .$ Assume that the domain $\\Omega \\subset \\mathbb {R}^{2}$ contains a corner $\\Omega \\cap W,$ where $\\mathbf {x}_{c}$ is the vertex of $\\Omega \\cap W$ and $W$ is a sector defined in (REF ).", "Moreover, there exits a sufficiently small neighbourhood $S_{h}$ (i.e.", "$h>0$ is sufficiently small) of $\\mathbf {x}_{c}$ in $\\Omega ,$ where $S_{h}$ is defined in (REF ), such that $q w \\in C^{\\alpha }\\left(\\bar{S}_{h}\\right)$ with $q:=1+V$ and $\\eta \\in C^{\\alpha }\\left(\\bar{\\Gamma }_{h}^{\\pm }\\right)$ for $0<\\alpha <1,$ .", "If the following conditions are fulfilled: the transmission eigenfunction $v$ can be approximated in $H^{1}\\left(S_{h}\\right)$ by the Herglotz functions $v_{j}, j=1,2, \\ldots ,$ with kernels $g_{j}$ satisfying $\\left\\Vert v-v_{j}\\right\\Vert _{H^{1}\\left(S_{h}\\right)} \\le j^{-\\Upsilon }, \\quad \\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)} \\le C j^{\\varrho }.$ for some constants $C,\\varrho \\text{ and }\\Upsilon $ with $C>0, \\Upsilon >0 \\text{ and } \\varrho <\\Upsilon $ .", "the function $\\eta (\\mathbf {x})$ doest not vanish at the corner, i.e., $\\eta \\left(\\mathbf {x}_{c}\\right) \\ne 0.", "$ the angles $\\theta _{m}$ and $\\theta _{M}$ of the sector $W$ satisfy $-\\pi <\\theta _{m}<\\theta _{M}<\\pi \\ \\text{ and }\\ \\theta _{M}-\\theta _{m} \\ne \\pi .", "$ Then it holds the following vanishing property: $\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega \\right)} \\int _{B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega }\|v(\\mathbf {x})\| \\mathrm {d} \\mathbf {x}=0.", "$ It is known that $\\Delta +k^{2}$ is invariant under rigid motions, we assume without loss of generality that $\\mathbf {x}_{c}$ is the origin.", "From (REF ), define $\\Delta v=-k^{2} v:=f_{1}, \\quad \\Delta w=-k^{2} q w:=f_{2}.$ which in combination with the boundary conditions in (REF ), yields that $\\Delta (v-w)=f_{1}-f_{2} \\text{ in } S_{h}, \\quad v-w=0, \\quad \\partial _{\\nu }(v-w)=-\\eta v \\text{ on } \\Gamma _{h}^{\\pm }.$ For national simplicity, we define $f_{1j}(\\mathbf {x})=-k^2v_j$ .", "Consider the following integral: $\\begin{split}\\int _{S_h} \\Delta (v-w)u_{0}(s\\mathbf {x})\\mathrm {d} \\mathbf {x}&=\\int _{S_h} (k^{2}qw-k^{2}v)u_{0}(s\\mathbf {x})\\mathrm {d} \\mathbf {x}\\\\&=\\int _{S_h} u_{0}(s\\mathbf {x})(f_{1j}-f_2)\\mathrm {d} \\mathbf {x}-k^2\\int _{S_h}(v-v_j)u_{0}(s\\mathbf {x})\\mathrm {d} \\mathbf {x}\\\\&:=I_1+\\Delta _{j}(s).\\end{split}$ By Corollary REF , one has that $u_0\\notin H^2$ near the origin.", "Consider the domain $D_{\\varepsilon }=S_h\\backslash B_{\\varepsilon }$ for $0<\\varepsilon <h$ .", "Using the fact that $\\int _{S_h} \\Delta (v-w)u_{0}(s\\mathbf {x})\\mathrm {d} \\mathbf {x}=\\lim _{\\varepsilon \\rightarrow 0} \\int _{D_{\\varepsilon }}\\Delta (v-w) u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x},$ and by Lemma REF , one can drive that $I_{1}+\\Delta _{j}(s)=I_{3}-I_{2}^{\\pm }-\\xi _{j}^{\\pm }(s),$ where $I_1$ and $\\Delta _{j}(s)$ are defined in (REF ), and $\\begin{split}I_{2}^{\\pm }=&\\int _{\\Gamma _{h}^{\\pm }} \\eta (\\mathbf {x}) u_{0}(s \\mathbf {x}) v_{j}(\\mathbf {x}) \\mathrm {d} \\sigma , \\\\I_{3}=&\\int _{\\Lambda _{h}}\\left(u_{0}(s \\mathbf {x}) \\partial _{\\nu }(v-w)-(v-w) \\partial _{\\nu } u_{0}(s \\mathbf {x})\\right) \\mathrm {d} \\sigma ,\\\\\\xi _{j}^{\\pm }(s)=&\\int _{\\Gamma _{h}^{\\pm }} \\eta (\\mathbf {x}) u_{0}(s \\mathbf {x})\\left(v(\\mathbf {x})-v_{j}(\\mathbf {x})\\right) \\mathrm {d} \\sigma .\\end{split}$ Recalling that $I_1$ is defined in (REF ), by Lemma REF and the compact embedding, one can deduce that $I_{1}=\\left({f}_{1 j}(0)-f_{2}(0)\\right) \\int _{S_{h}} u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}+\\int _{S_{h}} \\delta f_{1 j}(\\mathbf {x}) u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}-\\int _{S_{h}} \\delta f_{2}(\\mathbf {x}) u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}.$ where $\\delta f_{1j}(\\mathbf {x})$ and $\\delta f_{2}(\\mathbf {x})$ are deduced by Lemma REF .", "Using the fact that $\\int _{S_{h}} u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}=\\int _{W} u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}-\\int _{W \\backslash S_{h}} u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x},$ and combining (REF ) and (REF ) in the Lemma REF , it can be derived that $\\left\|\\int _{S_{h}} u_{0}\\left(s \\mathbf {x}\\right) \\mathrm {d} \\mathbf {x}\\right\| \\le 6\\left\|e^{-2 \\theta _{M} \\mathbf {i}}-e^{-2 \\theta _{m} \\mathbf {i}}\\right\| s^{-2}+\\frac{6\\left(\\theta _{M}-\\theta _{m}\\right)}{\\delta _{W}^{4}} s^{-2} e^{-\\frac{\\delta _{W} \\sqrt{h s}}{2}}.$ From Lemma REF , Corollary REF , Lemmas REF and REF , one can deduce that $\\begin{split}\\left\|\\int _{S_{h}} \\delta {f}_{1 j}(\\mathbf {x}) u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}\\right\| \\le &\\int _{S_{h}}\\left\|\\delta {f}_{1 j}(\\mathbf {x})\\right\|\\left\|u_{0}(s \\mathbf {x})\\right\| \\mathrm {d} \\mathbf {x}\\\\\\le &\\left\\Vert {f}_{1 j}\\right\\Vert _{C^{\\alpha }} \\int _{W}\\left\|u_{0}(s \\mathbf {x}) \\Vert \\mathbf {x}\\right\|^{\\alpha } \\mathrm {d} \\mathbf {x}\\\\\\le & \\frac{2\\left(\\theta _{M}-\\theta _{m}\\right) \\Gamma (2 \\alpha +4)}{\\delta _{W}^{2 \\alpha +4}}\\left\\Vert {f}_{1 j}\\right\\Vert _{C^{\\alpha }} s^{-\\alpha -2}\\\\\\le & \\frac{2 \\sqrt{2 \\pi }\\left(\\theta _{M}-\\theta _{m}\\right) \\Gamma (2 \\alpha +4)}{\\delta _{W}^{2 \\alpha +4}} k^{2} \\operatorname{diam}\\left(S_{h}\\right)^{1-\\alpha } \\\\&(1+k)Cj^{\\varrho } s^{-\\alpha -2},\\end{split}$ where $\\delta f_{1j}(\\mathbf {x})$ is deduced by Lemma REF .", "Using the assumption (REF ), we further otain that $\\left\|\\int _{S_{h}} \\delta {f}_{1 j}(\\mathbf {x}) u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}\\right\|\\le C^{\\prime } j^{\\varrho } s^{-\\alpha -2},$ where $C^{\\prime }=(1+k)C \\frac{2 \\sqrt{2 \\pi }\\left(\\theta _{M}-\\theta _{m}\\right) \\Gamma (2 \\alpha +4)}{\\delta _{W}^{2 \\alpha +4}} k^{2} \\operatorname{diam}\\left(S_{h}\\right)^{1-\\alpha } $ and C is defined in (REF ).", "Similarly, we have $\\left\|\\int _{S_{h}} \\delta f_{2}(\\mathbf {x}) u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}\\right\| \\le \\frac{2\\left(\\theta _{M}-\\theta _{m}\\right) \\Gamma (2 \\alpha +4)}{\\delta _{W}^{2 \\alpha +4}}\\left\\Vert f_{2}\\right\\Vert _{C^{\\alpha }} s^{-\\alpha -2}.$ As for the $\\Delta _{j}(s)$ defined in (REF ), using the Cauthy-Schwarz inequality, Corollary REF and the assumption (REF ), we can drive that $\\begin{split}\\left\|\\Delta _{j}(s)\\right\| &\\le k^{2}\\left\\Vert v-v_{j}\\right\\Vert _{L^{2}\\left(S_{h}\\right)}\\left\\Vert u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(S_{h}\\right)}\\\\& \\le \\frac{\\sqrt{\\theta _{M}-\\theta _{m}} k^{2} e^{-\\sqrt{s \\Theta } \\delta _{w}} h}{\\sqrt{2}} j^{-\\Upsilon }.\\end{split}$ By using the H$\\ddot{\\mbox{o}}$ lder inequality, Corollary REF , and the trace theorem, one can prove that $\\begin{split}\\left\|I_{3}\\right\| & \\le \\left\\Vert u_{0}(s \\mathbf {x})\\right\\Vert _{H^{1/2}\\left(\\Lambda _{h}\\right)}\\left\\Vert \\partial _{\\nu }(v-w)\\right\\Vert _{H^{-1/2}\\left(\\Lambda _{h}\\right)}+\\left\\Vert \\partial _{\\nu } u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(\\Lambda _{h}\\right)}\\Vert v-w\\Vert _{L^{2}\\left(\\Lambda _{h}\\right)} \\\\& \\le \\left(\\left\\Vert u_{0}(s \\mathbf {x})\\right\\Vert _{H^1\\left(\\Lambda _{h}\\right)}+\\left\\Vert \\partial _{\\nu } u_{0}(s \\mathbf {x})\\right\\Vert _{L^{2}\\left(\\Lambda _{h}\\right)}\\right)\\Vert v-w\\Vert _{H^{1}\\left(\\Sigma _{\\Lambda _{h}}\\right)} \\le C e^{-c^{\\prime } \\sqrt{s}},\\end{split}$ where $c^{\\prime }>0$ as $s \\rightarrow \\infty $ .", "By Lemma REF and (REF ), multiplying s on the both sides of (REF ), we can deduce that $\\begin{split}2 v_{j}(\\mathbf {0}) \\eta (\\mathbf {0})&\\left[ \\left(\\mu \\left(\\theta _{M}\\right)^{-2}-\\mu \\left(\\theta _{M}\\right)^{-2} e^{-\\sqrt{s h}\\left(\\theta _{M}\\right)}-\\mu \\left(\\theta _{M}\\right)^{-1} \\sqrt{s h} e^{-\\sqrt{s h} \\mu \\left(\\theta _{M}\\right)}\\right)\\right.\\\\&\\left.+\\left(\\mu \\left(\\theta _{m}\\right)^{-2}-\\mu \\left(\\theta _{m}\\right)^{-2} e^{-\\sqrt{s h} \\mu \\left(\\theta _{m}\\right)}-\\mu \\left(\\theta _{m}\\right)^{-1} \\sqrt{s h} e^{-\\sqrt{s h} \\mu \\left(\\theta _{m}\\right)}\\right)\\right] \\\\=&s\\left[I_{3}-\\left({f}_{1 j}(\\mathbf {0})-f_{2}(\\mathbf {0})\\right) \\int _{S_{h}} u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}-\\Delta _{j}(s)\\right.\\\\&\\left.-v_{j}(\\mathbf {0}) \\eta (\\mathbf {0})\\left(I_{21}^{+}+I_{21}^{-}\\right)-\\eta (\\mathbf {0})\\left(I_{22}^{+}+I_{22}^{-}\\right)\\right.\\\\&\\left.-I_{\\eta }^{+}-I_{\\eta }^{-}-\\int _{S_{h}} \\delta {f}_{1 j}(\\mathbf {x}) u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}+\\int _{S_{h}} \\delta f_{2}(\\mathbf {x}) u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}-\\xi _{j}^{\\pm }(s)\\right].\\end{split}$ Taking $s=j^{\\beta }$ , where $max\\lbrace \\varrho ,0\\rbrace <\\beta <\\Upsilon $ , and letting $j\\rightarrow \\infty $ , by (REF ), (REF ), (REF ), (REF ), Lemmas REF and REF , we can deduce that $\\eta (\\mathbf {0})\\left(\\mu \\left(\\theta _{m}\\right)^{-2}+\\mu \\left(\\theta _{M}\\right)^{-2}\\right) \\lim _{j \\rightarrow \\infty } v_{j}(\\mathbf {0})=0.$ Using the assumption of (REF ), one has that $\\mu \\left(\\theta _{m}\\right)^{-2}+\\mu \\left(\\theta _{M}\\right)^{-2}=\\frac{\\left(\\cos \\theta _{m}+\\cos \\theta _{M}\\right)+\\mathbf {i}\\left(\\sin \\theta _{m}+\\sin \\theta _{M}\\right)}{\\left(\\cos \\theta _{m}+\\mathbf {i} \\sin \\theta _{m}\\right)\\left(\\cos \\theta _{M}+\\mathbf {i} \\sin \\theta _{M}\\right)}\\ne 0.$ By assumption (REF ), it holds that $\\eta (0)\\ne 0$ .", "Combining (REF ) with (REF ), we know that $\\lim _{j \\rightarrow \\infty } v_{j}(\\mathbf {0})=0.$ Thus, it is an easy consequence that $\\lim _{j \\rightarrow \\infty } \|v_{j}(\\mathbf {0})\|=\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega \\right)} \\int _{B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega }\\left\|v_{j}(\\mathbf {x})\\right\| \\mathrm {d} \\mathbf {x}.$ Combining with the triangular inequality $\\begin{split}&\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega \\right)} \\int _{B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega }\|v(\\mathbf {x})\| \\mathrm {d} \\mathbf {x}\\\\\\le & \\lim _{j \\rightarrow \\infty }\\left(\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega \\right)} \\int _{B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega }\\left\|v(\\mathbf {x})-v_{j}(\\mathbf {x})\\right\| \\mathrm {d} \\mathbf {x}\\right.\\\\&\\left.+\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega \\right)} \\int _{B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega }\\left\|v_{j}(\\mathbf {x})\\right\| \\mathrm {d} \\mathbf {x}\\right),\\end{split}$ one finally arrives at $\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega \\right)} \\int _{B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega }\|v(\\mathbf {x})\| \\mathrm {d} \\mathbf {x}=0.$ The proof is complete.", "We next consider the degenerate case of Theorem REF with $\\eta \\equiv 0$ .", "By slightly modifying our proof of Theorem REF , we can show the following result.", "Corollary 2.4 Let $v \\in H^{1}(\\Omega )$ and $w \\in H^{1}(\\Omega )$ be a pair of eigenfunctions to (REF ) with $\\eta \\equiv 0$ and $k \\in \\mathbb {R}_{+} $ .", "Let $W$ and $S_h$ be the same as described in Theorem REF .", "Assume that $q w \\in C^{\\alpha }\\left(\\bar{S}_{h}\\right)$ for $0<\\alpha <1$ .", "Under the condition (REF ) and that the transmission eigenfunction $v$ can be approximated in $H^{1}(S_h)$ by the Herglotz functions $v_j$ with kernels $g_j$ satisfying $\\left\\Vert v-v_{j}\\right\\Vert _{H^{1}\\left(S_{h}\\right)} \\le j^{-\\Upsilon }, \\quad \\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)} \\le C j^{\\varrho },$ for some constants $C,\\varrho \\text{ and }\\Upsilon $ with $C>0, \\Upsilon >0 \\text{ and } \\varrho <\\alpha \\Upsilon /2$ , one has $\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega \\right)} \\int _{B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap \\Omega } V(\\mathbf {x})w(\\mathbf {x}) \\mathrm {d} \\mathbf {x}=0.$ Since the proof is similar to that of Theorem REF , we only outline some necessary modifications in the following.", "Without loss of generality, we assume that $\\mathbf {x}_c=0$ .", "Since $\\eta (\\mathbf {x})\\equiv 0$ , from (REF ), (REF ) and (REF ), we have the following integral indentity $\\left({f}_{1 j}(0)-f_{2}(0)\\right) \\int _{S_{h}} u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}+\\Delta _{j}(s) =I_3 - \\int _{S_{h}} \\delta f_{1 j}(\\mathbf {x}) u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x}+\\int _{S_{h}} \\delta f_{2}(\\mathbf {x}) u_{0}(s \\mathbf {x}) \\mathrm {d} \\mathbf {x},$ where $\\Delta _{j}(s)$ and $I_{3}$ are defined in (REF ) and (REF ), $\\delta f_{1j}(\\mathbf {x})$ and $\\delta f_{2}(\\mathbf {x})$ are deduced by Lemma REF with $f_{2}(\\mathbf {x})$ defined in (REF ) and $f_{1j}(\\mathbf {x})=-k^{2}v_{j}$ .", "Multiplying $s^{2}$ on both sides of (REF ), taking $s=J^{\\beta }$ where $\\max \\lbrace \\varrho /\\alpha ,0\\rbrace <\\beta <\\Upsilon /2$ , using the assumptions (REF ) and (REF ), and by letting $j \\rightarrow \\infty $ , from (REF ), (REF ), (REF ), (REF ) and (REF ), we can prove that $\\lim _{j\\rightarrow \\infty } v_{j}(\\mathbf {0})= \\frac{f_{2}(\\mathbf {0})}{-k^2}.$ Since $\\begin{split}&\\lim _{j \\rightarrow \\infty } v_{j}(\\mathbf {0})=\\lim _{j \\rightarrow \\infty } \\lim _{\\rho \\rightarrow +0} \\frac{1}{m(B(\\mathbf {0}, \\rho )\\cap \\Omega )} \\int _{B(\\mathbf {0}, \\rho )\\cap \\Omega } v_{j}(\\mathbf {x}) \\mathrm {d} \\mathbf {x}\\\\=&\\lim _{\\rho \\rightarrow +0} \\frac{1}{m(B(\\mathbf {0}, \\rho )\\cap \\Omega )} \\int _{B(\\mathbf {0}, \\rho )\\cap \\Omega } v(\\mathbf {x}) \\mathrm {d} \\mathbf {x},\\end{split}$ and $\\frac{f_{2}(\\mathbf {0})}{-k^{2}}=\\lim _{\\rho \\rightarrow +0} \\frac{1}{m(B(\\mathbf {0}, \\rho )\\cap \\Omega )} \\int _{B(\\mathbf {0}, \\rho )\\cap \\Omega } q w(\\mathbf {x}) \\mathrm {d} \\mathbf {x},$ which together with the fact that $\\lim _{\\rho \\rightarrow +0} \\frac{1}{m(B(\\mathbf {0}, \\rho )\\cap \\Omega )} \\int _{B(\\mathbf {0}, \\rho )\\cap \\Omega } v(\\mathbf {x}) \\mathrm {d} \\mathbf {x}=\\lim _{\\rho \\rightarrow +0} \\frac{1}{m(B(\\mathbf {0}, \\rho )\\cap \\Omega )} \\int _{B(\\mathbf {0}, \\rho )\\cap \\Omega } w(\\mathbf {x}) \\mathrm {d} \\mathbf {x},$ readily implies (REF ).", "The proof is complete." ], [ "Vanishing properties in three dimensions", "In this section, we consider the vanishing properties of the transmission eigenfunctions in the three-dimensional case.", "We first introduce the (edge) corner geometry in the three-dimensional setting.", "It is described by $W \\times (-M, M),$ where $W$ is a sector defined in (REF ) and $M \\in \\mathbb {R}_{+} .$ It is readily seen that $W \\times (-M, M)$ actually describes an edge singularity and we call it a $3 \\mathrm {D}$ corner for notational unification.", "Suppose that the domain $\\Omega \\subset \\mathbb {R}^{3}$ possesses a $3 \\mathrm {D}$ corner.", "Let $\\mathbf {x}_{c} \\in \\mathbb {R}^{2}$ be the vertex of $W$ and $x_{3}^{c} \\in (-M, M) .$ Then $\\left(\\mathbf {x}_{c}, x_{3}^{c}\\right)$ is defined as an edge point of $W \\times (-M, M)$ .", "In what follows, we use $\\mathbf {x}=(\\mathbf {x}^{\\prime }, x_3)\\in W\\times (-M, M)$ to signify a point in the edge corner.", "Consider the following conductive transmission eigenvalue problem: ${\\left\\lbrace \\begin{array}{ll}\\big (\\Delta +k^2(1+V)\\big ) w=0\\ &\\ \\mbox{in}\\ \\ W\\times (-M,M),\\medskip \\\\(\\Delta +k^2) v=0\\ &\\ \\mbox{in}\\ \\ W\\times (-M, M),\\medskip \\\\w=v,\\ \\ \\partial _\\nu w=\\partial _\\nu v+\\eta v\\ &\\ \\mbox{on}\\ \\Gamma ^\\pm \\times (-M, M),\\end{array}\\right.", "}$ where $\\Gamma ^{\\pm }$ are the two boundary pieces of $W$ .", "For the subsequent use, we introduce the following dimension reduction operator.", "Definition 3.1 [19] Let $W \\subset \\mathbb {R}^{2}$ be defined in (REF ), $M>0 .$ For a given function $g$ with the domain $W \\times (-M, M) .$ Pick up any point $x_{3}^{c} \\in (-M, M) .$ Suppose $\\psi \\in C_{0}^{\\infty }\\left(\\left(x_{3}^{c}-L, x_{3}^{c}+L\\right)\\right)$ is a nonnegative function and $\\psi \\ne 0,$ where $L$ is sufficiently small such that $\\left(x_{3}^{c}-L, x_{3}^{c}+L\\right) \\subset (-M, M),$ and write $\\mathbf {x}=\\left(\\mathbf {x}^{\\prime }, x_{3}\\right) \\in \\mathbb {R}^{3}, \\mathbf {x}^{\\prime } \\in \\mathbb {R}^{2}$ .", "The dimension reduction operator $\\mathcal {R}$ is defined by $\\mathcal {R}(g)\\left(\\mathbf {x}^{\\prime }\\right)=\\int _{x_{3}^{c}-L}^{x_{3}^{c}+L} \\psi \\left(x_{3}\\right) g\\left(\\mathbf {x}^{\\prime }, x_{3}\\right) d x_{3}.$ where $\\mathbf {x}^{\\prime } \\in W$ .", "The following lemma shows the regularity of the functions after applying the dimension reduction operator.", "Lemma 3.1 Let $g \\in H^{1}(W \\times (-M, M)) \\cap C^{\\alpha }(\\bar{W} \\times [-M, M]),$ where $0<\\alpha <1$ .", "Then $\\mathcal {R}(g)\\left(\\mathbf {x}^{\\prime }\\right) \\in H^{1}(W) \\cap C^{\\alpha }(\\bar{W}).$ We first show that $\\mathcal {R}:H^{1}(W\\times (-M,M))\\rightarrow H^{1}(W)$ is a bounded operator.", "Let $g\\in H^{1}(W\\times (-M,M))$ , by the dominated convergence theorem, we know that $\\partial _{\\mathbf {x}^{\\prime }}^{\\alpha }\\mathcal {R}(g)(\\mathbf {x}^{\\prime })= \\mathcal {R}(\\partial _{\\mathbf {x}^{\\prime }}^{\\alpha }g)(\\mathbf {x}^{\\prime })$ for $\\alpha =(i,j)$ with $ i,j=0,1$ and $i+j\\le 1$ , so $\|\\partial _{\\mathbf {x}^{\\prime }}^{\\alpha }\\mathcal {R}(g)(\\mathbf {x}^{\\prime })\| \\le \\int _{x_{3}^{c}-L}^{x_{3}^{c}+L} \\Vert \\psi \\Vert _{\\infty }\\left\|\\partial _{\\mathbf {x}^{\\prime }}^{\\alpha }g(\\mathbf {x}^{\\prime },x_{3}) \\right\| dx_{3}.$ Furthermore, by Minkowski integral, we have $\\Vert \\mathcal {R}(g)(\\mathbf {x}^{\\prime })\\Vert _{H^{1}(W)} \\le \\Vert \\psi \\Vert _{\\infty } \\Vert g(\\mathbf {x}^{\\prime },x_{3})\\Vert _{H^{1}(W\\times (-M,M))}.$ When $g\\in C^{\\alpha }(\\bar{W}\\times [-M, M])$ , it can be easily drived that $\|\\mathcal {R}(g)(\\mathbf {x}^{\\prime }) - \\mathcal {R}(g)(\\mathbf {y}^{\\prime })\| \\le 2 \\Vert \\psi \\Vert _{\\infty } \\Vert g\\Vert _{C^{\\alpha }}\|\\mathbf {x}^{\\prime } - \\mathbf {y}^{\\prime }\|^{\\alpha }.$ which means that $\\mathcal {R}(g)(\\mathbf {x}^{\\prime }) \\in C^{\\alpha }(\\bar{W})$ .", "Similar to Lemma REF , we have the following lemma.", "Lemma 3.2 For the Herglotz wave function $v_j$ defined in (REF ) in three dimensions, it holds that $\\begin{split}\\Vert \\mathcal {R}(v_{j})\\Vert _{C^{1}} & \\le 4L\\sqrt{\\pi }\\Vert \\psi \\Vert _{C^{\\infty }}(1+k)\\Vert g_j\\Vert _{L^{2}(\\mathbb {S}^{2})},\\\\\\Vert \\mathcal {R}(v_{j})\\Vert _{C^{\\alpha }} & \\le \\operatorname{diam}(S_{h})^{1-\\alpha }\\Vert \\mathcal {R}(v_{j})\\Vert _{C^{1}}.\\end{split}$ where $0<\\alpha <1$ and $\\operatorname{diam}(S_{h})$ is the diameter of $S_h$ .", "Using the Jacobi-Anger expansion ([15]) in $\\mathbb {R}^{3}$ , one can derive the following result: Lemma 3.3 The Herglotz function $v_j$ has the expansion in three dimensions as follows: $v_{j}(\\mathbf {x})=v_{j}(\\mathbf {0}) j_{0}(k\|\\mathbf {x}\|)+\\sum _{\\ell =1}^{\\infty } \\gamma _{\\ell j} \\mathbf {i}^{\\ell }(2 \\ell +1) j_{\\ell }(k\|\\mathbf {x}\|), \\quad \\mathbf {x}\\in \\mathbb {R}^{3},$ where $v_{j}(\\mathbf {0})=\\int _{\\mathbb {S}^{2}} g_{j}(\\mathbf {d}) \\mathrm {d} \\sigma (\\mathbf {d}), \\quad \\gamma _{\\ell j}:=\\int _{\\mathbb {S}^{2}} g_{j}(\\mathbf {d}) P_{\\ell }(\\cos (\\varphi )) \\mathrm {d} \\sigma (\\mathbf {d}), \\quad \\mathbf {d} \\in \\mathbb {S}^{2}.$ Here $j_{\\ell }(t)$ is the $\\ell $ -th spherical Bessel function [1], and $\\varphi $ is the angle between $\\mathbf {x}$ and $\\mathbf {d}$ Moreover, we have the explicit expression of $j_{\\ell }(t)$ as $j_{\\ell }(t)=\\frac{t^{\\ell }}{(2 \\ell +1) !", "!", "}\\left(1-\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} t^{2 l}}{2^{l} l !", "N_{\\ell , l}}\\right),\\ \\ell =0,1,2,\\dots ,$ where $N_{\\ell , l}=(2 \\ell +3) \\cdots (2 \\ell +2 l+1)$ .", "Applying the dimension reduction operator to the above spherical Bessel function, we can obtain the following lemma.", "Lemma 3.4 $\\mathcal {R}(j_0)(\\mathbf {x}^{\\prime })$ and $\\mathcal {R}\\left(j_{\\ell }\\right)\\left(\\mathbf {x}^{\\prime }\\right)$ have the deformation as following: $\\begin{split}\\mathcal {R}(j_0)(\\mathbf {x}^{\\prime })&=C(\\psi )\\left[1-\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}}{2^{l} l !", "(2 l+1) !", "!", "}\\left(\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}+a_{0, l}^{2}\\right)^{l}\\right],\\\\\\mathcal {R}\\left(j_{\\ell }\\right)\\left(\\mathbf {x}^{\\prime }\\right)&=\\frac{k^{\\ell }\\left(\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}+a_{\\ell }^{2}\\right)^{(\\ell -1) / 2}}{(2 \\ell +1) !", "!", "}\\left[1-\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}\\left(\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}+a_{\\ell , l}^{2}\\right)^{l}}{2^{l} l !", "N_{\\ell , l}}\\right] C_{1}(\\psi )\\left\|\\mathbf {x}^{\\prime }\\right\|^{2},\\end{split}$ where $N_{\\ell , l}=(2 \\ell +3) \\cdots (2 \\ell +2 l+1),\\quad a_{0, l},a_{\\ell },a_{\\ell ,l} \\in [-L, L], $ and $C(\\psi )=\\int _{-L}^{L} \\psi \\left(x_{3}\\right) \\mathrm {d} x_{3}, \\quad C_{1}(\\psi )=\\int _{-\\arctan L /\\left\|\\mathbf {x}^{\\prime }\\right\|}^{\\arctan L /\\left\|\\mathbf {x}^{\\prime }\\right\|} \\psi \\left(\\left\|\\mathbf {x}^{\\prime }\\right\| \\tan \\varpi \\right) \\sec ^{3} \\varpi \\mathrm {d} \\varpi , $ with $\\varpi \\in \\left[-\\arctan L /\\left\|\\mathbf {x}^{\\prime }\\right\|, \\arctan L /\\left\|\\mathbf {x}^{\\prime }\\right\|\\right].$ By the mean value theorem, we have that $\\begin{split}\\mathcal {R}\\left(j_{0}\\right)\\left(\\mathbf {x}^{\\prime }\\right) &=\\int _{-L}^{L} \\psi \\left(x_{3}\\right) j_{0}(k\|\\mathbf {x}\|) \\mathrm {d} x_{3} \\\\&=\\int _{-L}^{L} \\psi \\left(x_{3}\\right) \\mathrm {d} x_{3}-\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}}{2^{l} l !", "(2 l+1) !", "!}", "\\int _{-L}^{L} \\psi \\left(x_{3}\\right)\\left(\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}+x_{3}^{2}\\right)^{l} \\mathrm {d} x_{3} \\\\&=C(\\psi )\\left[1-\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}}{2^{l} l !", "(2 l+1) !", "!", "}\\left(\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}+a_{0, l}^{2}\\right)^{l}\\right].\\end{split}$ where $C(\\psi )=\\int _{-L}^{L} \\psi \\left(x_{3}\\right) \\mathrm {d} x_{3}$ and $a_{0, l} \\in [-L, L]$ .", "For $\\mathcal {R}\\left(j_{\\ell }\\right)\\left(\\mathbf {x}^{\\prime }\\right),$ using the integral mean value theorem, it can be drived that for $\\ell =1,2, \\ldots $ $\\begin{split}&\\int _{-L}^{L} \\psi \\left(x_{3}\\right)\\left(\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}+x_{3}^{2}\\right)^{\\ell / 2} \\mathrm {d} x_{3} \\\\=&\\left(\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}+a_{\\ell }^{2}\\right)^{(\\ell -1) / 2} \\int _{-L}^{L} \\psi \\left(x_{3}\\right)\\left(\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}+x_{3}^{2}\\right)^{1 / 2} \\mathrm {d} x_{3} \\\\=&\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}\\left(\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}+a_{\\ell }^{2}\\right)^{(\\ell -1) / 2} \\int _{-\\arctan L /\\left\|\\mathbf {x}^{\\prime }\\right\|}^{\\arctan L /\\left\|\\mathbf {x}^{\\prime }\\right\|} \\psi \\left(\\left\|\\mathbf {x}^{\\prime }\\right\| \\tan \\varpi \\right) \\sec ^{3} \\varpi \\mathrm {d} \\varpi \\\\:=&C_{1}(\\psi )\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}\\left(\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}+a_{\\ell }^{2}\\right)^{(\\ell -1) / 2},\\end{split}$ where $a_{\\ell } \\in [-L, L]$ .", "If $L<\\left\|\\mathbf {x}^{\\prime }\\right\|$ , we know that $0<\\sec \\varpi <\\sqrt{\\frac{L^{2}}{\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}}+1}$ , where $\\varpi \\in \\left[-\\arctan L /\\left\|\\mathbf {x}^{\\prime }\\right\|, \\arctan L /\\left\|\\mathbf {x}^{\\prime }\\right\|\\right]$ .", "Hence, it can be derived that $0<C_{1}(\\psi )<2^{1 / 2} \\pi \\Vert \\psi \\Vert _{\\infty }.$ Thus, for $l=1,2,\\ldots $ , one has that $\\begin{split}\\mathcal {R}\\left(j_{\\ell }\\right)\\left(\\mathbf {x}^{\\prime }\\right) &=\\int _{-L}^{L} \\psi \\left(x_{3}\\right) j_{\\ell }(k\|\\mathbf {x}\|)\\, \\mathrm {d} x_{3} \\\\&=\\frac{k^{\\ell }}{(2 \\ell +1) !", "!}", "\\int _{-L}^{L} \\psi \\left(x_{3}\\right)\\left(\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}+x_{3}^{2}\\right)^{\\ell / 2}\\left(1-\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}\\left(\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}+x_{3}^{2}\\right)^{l}}{2^{l} l !", "N_{\\ell , l}}\\right)\\, \\mathrm {d} x_{3} \\\\&=\\frac{k^{\\ell }\\left(\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}+a_{\\ell }^{2}\\right)^{(\\ell -1) / 2}}{(2 \\ell +1) !", "!", "}\\left[1-\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}\\left(\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}+a_{\\ell , l}^{2}\\right)^{l}}{2^{l} l !", "N_{\\ell , l}}\\right] C_{1}(\\psi )\\left\|\\mathbf {x}^{\\prime }\\right\|^{2}.\\end{split}$ where $a_{\\ell }, a_{\\ell , l} \\in [-L, L]$ .", "We next derive several critical auxiliary lemmas.", "Lemma 3.5 Let $v,w\\in H^{1}(W\\times (-M,M))$ be a pair of conductive transmission eigenfunctions to (REF ) and $D_{\\varepsilon }=S_{h}\\setminus B_{\\varepsilon }$ for $0<\\varepsilon <h$ , $\\eta \\in C^{\\alpha }(\\bar{\\Gamma }^{\\pm }_{h} \\times [-M,M])$ for $0<\\alpha <1$ and $\\eta =\\eta (\\mathbf {x}^{\\prime })$ is independent of $x_3$ .", "Then it holds that $\\begin{split}&\\lim _{\\varepsilon \\rightarrow \\infty } \\int _{D_{\\varepsilon }}u_{0}(s\\mathbf {x}^{\\prime })\\left(\\Delta _{\\mathbf {x}^{\\prime }}\\mathcal {R}(v) - \\Delta _{\\mathbf {x}^{\\prime }}\\mathcal {R}(w) \\right) \\mathrm {d} \\mathbf {x}^{\\prime }\\\\=&\\int _{\\Lambda _{h}}\\left(u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\partial _{\\nu } \\mathcal {R}(v-w)\\left(\\mathbf {x}^{\\prime }\\right)-\\mathcal {R}(v-w)\\left(\\mathbf {x}^{\\prime }\\right) \\partial _{\\nu } u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\right) \\mathrm {d} \\sigma \\\\&-\\int _{\\Gamma _{ h}^{\\pm }} \\eta \\left(\\mathbf {x}^{\\prime }\\right) \\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma ,\\end{split}$ where $\\Lambda _{h}=S_{h} \\cap \\partial B_{h}$ and $\\Gamma _{ h}^{\\pm }=\\Gamma ^{\\pm }\\cap B_{h}$ .", "Since $w\\left(\\mathbf {x}^{\\prime }, x_{3}\\right)=v\\left(\\mathbf {x}^{\\prime }, x_{3}\\right)$ when $\\mathbf {x}^{\\prime } \\in \\Gamma $ and $-L<x_{3}<L$ , we have $\\mathcal {R}(w)\\left(\\mathbf {x}^{\\prime }\\right)=\\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right) \\text{ on } \\Gamma .$ Similarly, using the fact that $\\eta $ is independent of $x_3$ , we can obtain that $\\partial _{\\nu } \\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right)+\\eta \\left(\\mathbf {x}^{\\prime }\\right) \\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right)=\\partial _{\\nu } \\mathcal {R}(w)\\left(\\mathbf {x}^{\\prime }\\right) \\text{ on } \\Gamma .$ Therefore, by Green's formula, we have $\\begin{split}&\\int _{D_{\\varepsilon }} \\Delta _{\\mathbf {x}^{\\prime }}\\left(\\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right)-\\mathcal {R}(w)\\left(\\mathbf {x}^{\\prime }\\right)\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime } \\\\&=\\int _{\\partial D_{\\varepsilon }}\\left(u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\partial _{\\nu } \\mathcal {R}(v-w)\\left(\\mathbf {x}^{\\prime }\\right)-\\mathcal {R}(v-w)\\left(\\mathbf {x}^{\\prime }\\right) \\partial _{\\nu } u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\right) \\mathrm {d} \\sigma \\\\&=\\int _{\\Lambda _{h}}\\left(u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\partial _{\\nu } \\mathcal {R}(v-w)\\left(\\mathbf {x}^{\\prime }\\right)-\\mathcal {R}(v-w)\\left(\\mathbf {x}^{\\prime }\\right) \\partial _{\\nu } u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\right) \\mathrm {d} \\sigma \\\\&\\quad +\\int _{\\Lambda _{\\varepsilon }}\\left(u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\partial _{\\nu } \\mathcal {R}(v-w)\\left(\\mathbf {x}^{\\prime }\\right)-\\mathcal {R}(v-w)\\left(\\mathbf {x}^{\\prime }\\right) \\partial _{\\nu } u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\right) \\mathrm {d} \\sigma \\\\&\\quad -\\int _{\\Gamma _{(\\varepsilon , h)}^{\\pm }} \\eta \\left(\\mathbf {x}^{\\prime }\\right) \\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma ,\\end{split}$ $\\text{ where } \\Lambda _{h}=S_{h} \\cap \\partial B_{h}, \\Lambda _{\\varepsilon }=S_{h} \\cap \\partial B_{\\varepsilon } \\text{ and } \\Gamma _{(\\varepsilon , h)}^{\\pm }=\\Gamma ^{\\pm } \\cap \\left(B_{h} \\backslash B_{\\varepsilon }\\right)$ .", "Since $v,w \\in H^{1}\\left(S_{h} \\times (-L, L)\\right),$ from Lemma REF we know that $\\mathcal {R}(v-w) \\in H^1(S_h)$ , and it can be derived that $\\lim _{\\varepsilon \\rightarrow 0} \\int _{\\Lambda _{c}}\\left(u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\partial _{\\nu } \\mathcal {R}(v-w)\\left(\\mathbf {x}^{\\prime }\\right)-\\mathcal {R}(v-w)\\left(\\mathbf {x}^{\\prime }\\right) \\partial _{\\nu } u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\right) \\mathrm {d} \\sigma =0$ Since $v \\in H^{1}\\left(\\left(S_{h} \\cap B_{\\varepsilon }\\right) \\times (-L, L)\\right)$ , from Lemma REF , we also know $\\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right) \\in H^{1}\\left(S_{h} \\cap B_{\\varepsilon }\\right)$ .", "Using the trace theorem, we see that $\\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right) \\in L^{2}\\left(\\Gamma _{(0, e)}^{\\pm }\\right) $ where $\\Gamma _{(0, c)}^{\\pm }=\\Gamma ^{\\pm } \\cap B_{\\varepsilon }$ .", "For sufficiently small $\\varepsilon $ , we further see that $\\left\|u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\right\| \\le 1$ and $\\eta \\in C^{\\alpha }\\left(\\bar{\\Gamma }_{h}^{\\pm } \\times [-M, M]\\right)$ .", "Hence, we have $\\lim _{\\varepsilon \\rightarrow 0} \\int _{\\Gamma _{(0, c)}^{\\pm }} \\eta \\left(\\mathbf {x}^{\\prime }\\right) \\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma =0.$ The proof is complete.", "Lemma 3.6 Suppose that $\\eta \\in C^{\\alpha }\\left(\\bar{\\Gamma }_{h}^{\\pm } \\times [-M,M]\\right)$ for $0<\\alpha <1$ and $\\eta =\\eta (\\mathbf {x}^{\\prime })$ is independent of $x_3$ , and $\\theta _{M},\\theta _{m}$ are defined in (REF ) and $\\theta _{M}-\\theta _{m}\\ne \\pi $ .", "Define $I_{2}^{\\pm }=\\int _{\\Gamma _{h}^{\\pm }} \\eta \\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathcal {R}\\left(v_{j}\\right)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma .$ Then it hods that $\\begin{split}I_{2}^{-}=&2 \\eta (\\mathbf {0}) v_{j}(\\mathbf {0}) s^{-1}\\left(\\mu \\left(\\theta _{m}\\right)^{-2}-\\mu \\left(\\theta _{m}\\right)^{-2} e^{-\\sqrt{s h} \\mu \\left(\\theta _{m}\\right)}-\\mu \\left(\\theta _{m}\\right)^{-1} \\sqrt{s h} e^{-\\sqrt{s h} \\mu \\left(\\theta _{m}\\right)}\\right) C_{2}^{-} \\\\&+v_{j}(\\mathbf {0}) \\eta (\\mathbf {0}) I_{21}^{-}+\\eta (\\mathbf {0}) I_{22}^{-}+I_{\\eta }^{-},\\end{split}$ and $\\begin{split}I_{2}^{+}=&2 \\eta (\\mathbf {0}) v_{j}(\\mathbf {0}) s^{-1}\\left(\\mu \\left(\\theta _{M}\\right)^{-2}-\\mu \\left(\\theta _{M}\\right)^{-2} e^{-\\sqrt{s h} \\mu \\left(\\theta _{M}\\right)}-\\mu \\left(\\theta _{M}\\right)^{-1} \\sqrt{s h} e^{-\\sqrt{s h} \\mu \\left(\\theta _{M}\\right)}\\right) C_{2}^{+} \\\\&+v_{j}(\\mathbf {0}) \\eta (\\mathbf {0}) I_{21}^{+}+\\eta (\\mathbf {0}) I_{22}^{+}+I_{\\eta }^{+}.\\end{split}$ where $C_{2}^{\\pm }$ are positive constants, and $\\begin{split}I_{21}^{-} \\le & \\mathcal {O}\\left(s^{-3}\\right), \\quad I_{22}^{-} \\le \\mathcal {O}\\left(\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{2}\\right)} s^{-3}\\right),\\\\\\left\|I_{\\eta }^{-}\\right\| \\le &\\Vert \\eta \\Vert _{C^{\\alpha }}\\left(v_{j}(\\mathbf {0}) \\mathcal {O}\\left(s^{-1-\\alpha }\\right)+\\mathcal {O}\\left(\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{2}\\right)} s^{-3-\\alpha }\\right)\\right),\\\\I_{21}^{+} \\le & \\mathcal {O}\\left(s^{-3}\\right), \\quad I_{22}^{+} \\le \\mathcal {O}\\left(\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{2}\\right)} s^{-3}\\right),\\\\\\left\|I_{\\eta }^{+}\\right\| \\le &\\Vert \\eta \\Vert _{C^{\\alpha }}\\left(v_{j}(\\mathbf {0}) \\mathcal {O}\\left(s^{-1-\\alpha }\\right)+\\mathcal {O}\\left(\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{2}\\right)} s^{-3-\\alpha }\\right)\\right).\\end{split}$ Using Lemma REF , we have $I_{2}^{-}=\\eta (\\mathbf {0}) \\int _{\\Gamma _{h}^{-}} u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathcal {R}\\left(v_{j}\\right)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma + \\int _{\\Gamma _{h}^{-}} \\delta \\eta \\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathcal {R}\\left(v_{j}\\right)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma .$ Recall the definition in (REF ).", "Combining Lemmas REF and REF , we can obtain that $\\begin{split}&\\int _{\\Gamma _{h}^{-}} u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathcal {R}\\left(v_{j}\\right)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma \\\\=&v_{j}(\\mathbf {0}) \\int _{\\Gamma _{h}^{-}} u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathcal {R}\\left(j_{0}\\right)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma +\\sum _{\\ell =1}^{\\infty } \\gamma _{\\ell j} \\mathbf {i}^{\\ell }(2 \\ell +1) \\int _{\\Gamma _{h}^{-}} u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathcal {R}\\left(j_{\\ell }\\right)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma .\\end{split}$ Using Newton's Binomial expansion (see also Lemma REF ), we have $\\begin{split}&\\int _{\\Gamma _{h}^{-}} u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathcal {R}\\left(j_{0}\\right)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma \\\\=&C(\\psi ) \\int _{0}^{h}\\left[1-\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}}{(2 l+1) !", "!", "}\\left(r^{2}+a_{0, l}^{2}\\right)^{l}\\right] e^{-\\sqrt{s r} \\mu \\left(\\theta _{m}\\right)} \\mathrm {d} r\\\\=& C(\\psi )\\left[1-\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}}{(2 l+1) !", "!}", "a_{0, l}^{2 l}\\right] \\int _{0}^{h} e^{-\\sqrt{s r} \\mu \\left(\\theta _{m}\\right)} \\mathrm {d} r \\\\&-C(\\psi ) \\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}}{(2 l+1) !", "!", "}\\left(\\sum _{i_{1}=1}^{l} C\\left(l, i_{1}\\right) a_{0, l}^{2\\left(l-i_{1}\\right)} \\int _{0}^{h} r^{2 i_{1}} e^{-\\sqrt{s r} \\mu \\left(\\theta _{m}\\right)} \\mathrm {d} r\\right)\\\\:=&2 s^{-1}\\left(\\mu \\left(\\theta _{m}\\right)^{-2}-\\mu \\left(\\theta _{m}\\right)^{-2} e^{-\\sqrt{s h} \\mu \\left(\\theta _{m}\\right)}-\\mu \\left(\\theta _{m}\\right)^{-1} \\sqrt{s h} e^{-\\sqrt{s h} \\mu \\left(\\theta _{m}\\right)}\\right)C_{2}^{-} + I_{21}^{-},\\end{split}$ where $C(\\psi )=\\int _{-L}^{L} \\psi \\left(x_{3}\\right) \\mathrm {d} x_{3}>0$ , $C\\left(l, i_{1}\\right)=\\frac{l !", "}{i_{1} !\\left(l-i_{1}\\right) !", "}$ is the combinatorial number of order $l$ and $C_{2}^{-}=C(\\psi )\\left[1-\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}}{(2 l+1) !", "!}", "a_{0, l}^{2 l}\\right]$ .", "By choosing $L$ such that $kL<1$ , we have that $\\left\|\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}}{(2 l+1) !", "!}", "a_{0, l}^{2 l}\\right\| \\le \\sum _{l=1}^{\\infty }(k L)^{2 l}=\\frac{(k L)^{2}}{1-(k L)^{2}}.$ Therefore, we can deduce that $0<\\frac{C(\\psi )\\left(1-2(k L)^{2}\\right)}{1-(k L)^{2}} \\le C_{2}^{-} \\le \\frac{C(\\psi )}{1-(k L)^{2}}.$ For $I_{21}^{-}$ , choosing $h$ and $L$ such that $k^{2}(h^{2}+L^{2})<1$ , we can deduce that $\\begin{split}\|I_{21}^{-}\| & \\le \|C(\\psi )\| \\sum _{l=1}^{\\infty } \\frac{k^{2 l}}{(2 l+1) !", "!}", "\\sum _{i_{1}=1}^{l} C\\left(l, i_{1}\\right) h^{2\\left(i_{1}-1\\right)} L^{2\\left(l-i_{1}\\right)} \\int _{0}^{h} r^{2} e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r \\\\&=\|C(\\psi )\| \\sum _{l=1}^{\\infty } \\frac{k^{2 l}}{(2 l+1) !", "!", "h^{2}} \\sum _{i_{1}=1}^{l} C\\left(l, i_{1}\\right) h^{2 i_{1}} L^{2\\left(l-i_{1}\\right)} \\int _{0}^{h} r^{2} e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r \\\\&=\|C(\\psi )\| \\sum _{l=1}^{\\infty } \\frac{k^{2 l}}{(2 l+1) !", "!", "h^{2}}\\left(\\left(h^{2}+L^{2}\\right)^{l}-L^{2 l}\\right) \\int _{0}^{h} r^{2} e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r \\\\& \\le 2 L\\Vert \\psi \\Vert _{\\infty } \\sum _{l=1}^{\\infty } \\frac{k^{2 l}}{(2 l+1) !", "!", "h^{2}}\\left(\\left(h^{2}+L^{2}\\right)^{l}-L^{2 l}\\right) \\mathcal {O}\\left(s^{-3}\\right) \\\\&=\\mathcal {O}\\left(s^{-3}\\right).\\end{split}$ Taking $I_{22}^{-}=\\sum _{\\ell =1}^{\\infty } \\gamma _{\\ell j} \\mathbf {i}^{\\ell }(2 \\ell +1) \\int _{\\Gamma _{h}^{-}} u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathcal {R}\\left(j_{\\ell }\\right)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma ,$ we then have $\\begin{split}\\left\|I_{22}^{-}\\right\| \\le & C_{1}(\\psi )\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{2}\\right)} \\\\& \\cdot \\sum _{\\ell =1}^{\\infty } \\int _{0}^{h} r^{2} e^{-\\sqrt{s r \\omega }\\left(\\theta _{m}\\right)} \\frac{k^{\\ell }\\left(\|r\|^{2}+a_{\\ell }^{2}\\right)^{(\\ell -1) / 2}}{(2 \\ell -1) !", "!", "}\\left\|1-\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}\\left(\|r\|^{2}+a_{\\ell , l}^{2}\\right)^{l}}{2^{l} l !", "N_{\\ell , l}}\\right\| \\mathrm {d} r \\\\=& C_{1}(\\psi )\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{2}\\right)} \\sum _{\\ell =1}^{\\infty } \\frac{k^{\\ell }\\left(\\left\|\\beta _{\\ell }\\right\|^{2}+a_{\\ell }^{2}\\right)^{(\\ell -1) / 2}}{(2 \\ell -1) !", "!", "}\\left\|1-\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}\\left(\\left\|\\beta _{\\ell , l}\\right\|^{2}+a_{\\ell , l}^{2}\\right)^{l}}{2^{l} l !", "N_{\\ell , l}}\\right\| \\\\& \\cdot \\int _{0}^{h} r^{2} e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r \\\\=& \\mathcal {O}\\left(\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{2}\\right)} s^{-3}\\right),\\end{split}$ where $C_1(\\psi )$ is defined in Lemma REF .", "Taking $I_{\\eta }^{-}=\\int _{\\Gamma _{h}^{-}} \\delta \\eta \\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathcal {R}\\left(v_{j}\\right)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma ,$ and by Lemmas REF and REF , we can obtain that $\\begin{split}I_{\\eta }^-&=v_j(0)\\int _{\\Gamma _{h}^{-}} \\delta \\eta \\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathcal {R}\\left(j_{0}\\right)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma \\\\&\\quad + \\sum _{\\ell =1}^{\\infty } \\gamma _{\\ell j} \\mathbf {i}^{\\ell }(2 \\ell +1) \\int _{\\Gamma _{h}^{-}} \\delta \\eta \\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathcal {R}\\left(j_{\\ell }\\right)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma .\\end{split}$ Furthermore, we have $\\begin{split}&\\left\|\\int _{\\Gamma _{h}^{-}} \\delta \\eta \\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathcal {R}\\left(j_{0}\\right)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma \\right\|\\\\& \\le \|C(\\psi )\|\\Vert \\eta \\Vert _{C^{\\alpha }} \\int _{0}^{h} r^{\\alpha }\\left\|1-\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}}{(2 l+1) !", "!", "}\\left(r^{2}+a_{0, l}^{2}\\right)^{l}\\right\| e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r \\\\&=2 L\\Vert \\psi \\Vert _{\\infty }\\Vert \\eta \\Vert _{C^{\\alpha }}\\left\|1-\\sum _{l=1}^{\\infty } \\frac{(-1)^{l} k^{2 l}}{(2 l+1) !", "!", "}\\left(\\beta _{0, l}^{2}+a_{0, l}^{2}\\right)^{l}\\right\| \\int _{0}^{h} r^{\\alpha } e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r\\\\&\\le \\mathcal {O}\\left(s^{-\\alpha -1}\\right),\\end{split}$ where $\\beta _{0, l} \\in [0, h]$ such that $k^{2}\\left(\\beta _{0, l}^{2}+a_{0, l}^{2}\\right) \\le k^{2}\\left(h^{2}+L^{2}\\right)<1$ for sufficiently small $h$ and $L$ .", "Next, we can deduce that $\\begin{split}&\\left\|\\sum _{\\ell =1}^{\\infty } \\gamma _{\\ell j} \\mathbf {i}^{\\ell }(2 \\ell +1) \\int _{\\Gamma _{h}^{-}} \\delta \\eta \\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathcal {R}\\left(j_{\\ell }\\right)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\sigma \\right\|\\\\\\le & C_{1}(\\psi )\\Vert \\eta \\Vert _{C^{\\alpha }} \\\\& \\cdot \\sum _{\\ell =1}^{\\infty }\\left\|\\gamma _{\\ell j}\\right\| \\int _{0}^{h} r^{\\alpha } \\frac{k^{\\ell }\\left(r^{2}+a_{\\ell }^{2}\\right)^{(\\ell -1) / 2}}{(2 \\ell -1) !", "!", "}\\left\|1-\\sum _{l=1}^{\\infty } \\frac{k^{2 l}\\left(r^{2}+a_{\\ell , l}^{2}\\right)^{l}}{2^{l} l !", "N_{\\ell , l}}\\right\| r^{2} e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r \\\\\\le & 2C_{1}(\\psi )\\Vert \\eta \\Vert _{C^{\\alpha }}\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{2}\\right)} \\int _{0}^{h} r^{2+\\alpha } e^{-\\sqrt{s r} \\omega \\left(\\theta _{m}\\right)} \\mathrm {d} r \\\\& \\cdot \\sum _{\\ell =1}^{\\infty } \\frac{k^{\\ell }\\left(\\beta _{\\ell }^{2}+a_{\\ell }^{2}\\right)^{(\\ell -1) / 2}}{(2 \\ell -1) !", "!", "}\\left\|1-\\sum _{l=1}^{\\infty } \\frac{k^{2 l}\\left(\\beta _{\\ell , l}^{2}+a_{\\ell , l}^{2}\\right)^{l}}{2^{l} l !", "N_{\\ell , l}}\\right\| \\\\=& \\mathcal {O}\\left(\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{2}\\right)} s^{-\\alpha -3}\\right),\\end{split}$ where we have used the estimates $k^{2}\\left(\\beta _{\\ell }^{2}+a_{\\ell }^{2}\\right) \\le k^{2}\\left(h^{2}+L^{2}\\right)<1$ , $k^{2}\\left(\\beta _{\\ell , l}^{2}+a_{\\ell , l}^{2}\\right) \\le $ $k^{2}\\left(h^{2}+L^{2}\\right)<1$ for sufficiently small $h$ and $L,$ as well as the estimate $\\left\|\\gamma _{\\ell j}\\right\|=\\Big \|\\int _{\\mathbb {S}^{2}} g_{j}(\\mathbf {d}) P_{\\ell }(\\hat{\\mathbf {x}}\|_{\\Gamma _h^-}\\cdot \\mathbf {d}) \\mathrm {d} \\sigma (\\mathbf {d})\\Big \| \\le \\sqrt{2\\pi }\\sqrt{\\frac{2}{2l+1}}\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{2}\\right)}\\le 2\\sqrt{\\pi }\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{2}\\right)}.$ Finally, by combining (REF ), (REF ), (REF ), (REF ) and (REF ), we can prove (REF ).", "Using a similar argument, one can prove (REF ).", "The proof is complete.", "Lemma 3.7 Let $\\eta \\in C^{\\alpha }\\left(\\bar{\\Gamma }_{h}^{\\pm } \\times [-M,M]\\right)$ for $0<\\alpha <1$ and $\\eta =\\eta (\\mathbf {x}^{\\prime })$ be independent of $x_3$ , $\\theta _{M},\\theta _{m}$ be defined in (REF ) and $\\theta _{M}-\\theta _{m}\\ne \\pi $ .", "Set $\\xi _{j}^{\\pm }(s) =\\int _{\\Gamma _{h}^{\\pm }} \\eta \\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathcal {R}\\left(v\\left(\\mathbf {x}^{\\prime }, x_3\\right)-v_{j}\\left(\\mathbf {x}^{\\prime }, x_3\\right)\\right) \\mathrm {d} \\sigma .$ Then it holds that $& \\left\|\\xi _{j}^{\\pm }(s)\\right\| \\\\\\le & C\\Vert \\psi \\Vert _{\\infty }\\left(\|\\eta (\\mathbf {0})\|\\left\\Vert u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\right\\Vert _{L^{2}\\left(S_{h}\\right)}+\\Vert \\eta \\Vert _{C^{\\alpha }}\\left\\Vert \\left\|\\mathbf {x}^{\\prime }\\right\|^{\\alpha } u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\right\\Vert _{L^{2}\\left(S_{h}\\right)}\\right) \\left\\Vert v-v_{j}\\right\\Vert _{H^{1}\\left(S_{h} \\times (-L, L)\\right)},$ where $C$ is a positive constant.", "Using Cauchy-Schwarz inequality and the trace theorem, we can deduce as follows $\\begin{split}& \\left\|\\xi _{j}^{\\pm }(s)\\right\| \\le \|\\eta (\\mathbf {0})\| \\int _{\\Gamma _{h}^{\\pm }}\\left\|u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\Vert \\mathcal {R}\\left(v\\left(\\mathbf {x}^{\\prime }, x_3\\right)-v_{j}\\left(\\mathbf {x}^{\\prime }, x_3\\right)\\right)\\right\| \\mathrm {d} \\sigma \\\\&+\\Vert \\eta \\Vert _{C^{\\alpha }} \\int _{\\Gamma _{h}^{\\pm }}\\left\|\\mathbf {x}^{\\prime }\\right\|^{\\alpha }\\left\|u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\Vert \\mathcal {R}\\left(v\\left(\\mathbf {x}^{\\prime }, x_3\\right)-v_{j}\\left(\\mathbf {x}^{\\prime }, x_3\\right)\\right)\\right\| \\mathrm {d} \\sigma \\\\\\le &\|\\eta (\\mathbf {0})\|\\left\\Vert \\mathcal {R}\\left(v-v_{j}\\right)\\right\\Vert _{H^{1 / 2}\\left(\\Gamma _{h}^{\\pm }\\right)}\\left\\Vert u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\right\\Vert _{H^{-1 / 2}\\left(\\Gamma _{h}^{\\pm }\\right)} \\\\&+\\left.\\Vert \\eta \\Vert _{C^{\\alpha }}\\left\\Vert \\mathcal {R}\\left(v-v_{j}\\right)\\right\\Vert _{H^{1 / 2}\\left(\\Gamma _{h}^{\\pm }\\right)}\\Vert \\Vert \\mathbf {x}^{\\prime }\\right\|^{\\alpha } u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\Vert _{H^{-1 / 2}\\left(\\Gamma _{h}^{\\pm }\\right)} \\\\\\le & \| \\eta (\\mathbf {0})\\left\\Vert \\mathcal {R}\\left(v-v_{j}\\right)\\right\\Vert _{H^{1}\\left(S_{h}\\right)}\\left\\Vert u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\right\\Vert _{L^{2}\\left(S_{h}\\right)} \\\\&+\\left.\\Vert \\eta \\Vert _{C^{\\alpha }}\\left\\Vert \\mathcal {R}\\left(v-v_{j}\\right)\\right\\Vert _{H^{1}\\left(S_{h}\\right)}\\Vert \\Vert \\mathbf {x}^{\\prime }\\right\|^{\\alpha } u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\Vert _{L^{2}\\left(S_{h}\\right)} \\\\\\le & C\\Vert \\psi \\Vert _{\\infty }\\left\\Vert v-v_{j}\\right\\Vert _{H^{1}\\left(S_{h} \\times (-L, L)\\right)}\\left(\|\\eta (\\mathbf {0})\|\\left\\Vert u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\right\\Vert _{L^{2}\\left(S_{h}\\right)}+\\Vert \\eta \\Vert _{C^{\\alpha }}\\left\\Vert \\left\|\\mathbf {x}^{\\prime }\\right\|^{\\alpha } u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\right\\Vert _{L^{2}\\left(S_{h}\\right)}\\right),\\end{split}$ which readily completes the proof.", "We are now in a position of to the vanishing properties of the conductive transmission eigenfunctions $(v,w)$ in the three-dimensional case, and we have the following theorem.", "Theorem 3.1 Let $v, w \\in H^{1}(W \\times (-M, M))$ be a pair of eigenfunctions to (REF ) associated with $k\\in \\mathbb {R}_{+}$ , where $W \\subset \\mathbb {R}^{2}$ is defined in (REF ) and $M>0.$ For any fixed $x_{3}^{c} \\in (-M, M)$ and $L>0$ defined in Definition REF , we suppose that $L$ is sufficiently small such that $\\left(x_{3}^{c}-L, x_{3}^{c}+L\\right) \\subset (-M, M)$ .", "Moreover, there exists a sufficiently small neighbourhood $S_{h}$ of $\\mathbf {x}_{c} \\in \\mathbb {R}^{2}$ such that $q w \\in C^{\\alpha }\\left(\\bar{S}_{h} \\times [-M, M]\\right)$ and $\\eta \\in C^{\\alpha }\\left(\\bar{\\Gamma }_{h}^{\\pm } \\times [-M, M]\\right)$ for $0<\\alpha <1$ , where $q:=1+V$ .", "If the following conditions are fulfilled: the transmission eigenfunction $v$ can be approximated in $H^{1}\\left(S_{h} \\times (-M, M)\\right) b y$ the Herglotz functions $v_{j}, j=1,2, \\ldots ,$ with kernels $g_{j}$ satisfying $\\left\\Vert v-v_{j}\\right\\Vert _{H^{1}\\left(S_{h} \\times (-M, M)\\right)} \\le j^{-\\Upsilon }, \\quad \\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{2}\\right)} \\le C j^{\\varrho },$ for some constants $C,\\varrho \\text{ and }\\Upsilon $ with $C>0, \\Upsilon >0 \\text{ and } \\varrho <(1+\\alpha )\\Upsilon $ ; the function $\\eta =\\eta \\left(\\mathbf {x}^{\\prime }\\right)$ is independent of $x_{3}$ and $\\eta \\left(\\mathbf {x}_{c}\\right) \\ne 0; $ the angles $\\theta _{m}$ and $\\theta _{M}$ of the sector $W$ satisfy $-\\pi <\\theta _{m}<\\theta _{M}<\\pi \\text{ and } \\theta _{M}-\\theta _{m} \\ne \\pi ; $ then for every edge point $\\left(\\mathbf {x}_{c}, x_3^{c}\\right) \\in \\mathbb {R}^{3}$ of $W \\times (-M, M)$ where $x_3^{c} \\in (-M, M),$ one has $\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(B\\left(\\left(\\mathbf {x}_{c}, x_3^{c}\\right), \\rho \\right)\\cap (W\\times (-M,M))\\right)} \\int _{B\\left(\\left(\\mathbf {x}_{c}, x_3\\right), \\rho \\right)\\cap (W\\times (-M,M))}\|v(\\mathbf {x})\| \\mathrm {d} \\mathbf {x}=0.$ For an edge point $\\left(\\mathbf {x}_{c}, x_{3}^{c}\\right) \\in W \\times (-M, M),$ without loss of generality, we assume that the vertex $\\mathbf {x}_{c}$ of the sector $W \\subset \\mathbb {R}^{2}$ is located at the origin of $\\mathbb {R}^{2}$ and $x_{3}^{c}=0$ .", "By direct calculations, we have $\\begin{split}\\Delta _{\\mathbf {x}^{\\prime }} \\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right)&=\\Delta _{\\mathbf {x}^{\\prime }} \\int _{-L}^{L} \\psi (x_3)v(\\mathbf {x}^{\\prime },x_n) \\mathrm {d}x_3\\\\&=\\int _{-L}^{L} \\psi (x_3) \\left(-k^{2}v(\\mathbf {x}^{\\prime },x_n)-\\partial _{x_3}^{2}v(\\mathbf {x}^{\\prime },x_n)\\right) \\mathrm {d}x_3\\\\&=-\\int _{-L}^{L} \\psi (x_3)\\partial _{x_3}^{2}v(\\mathbf {x}^{\\prime },x_n)\\mathrm {d}x_3 - k^{2}\\mathcal {R}(v)(\\mathbf {x}^{\\prime })\\\\&=\\int _{-L}^{L} \\psi ^{\\prime \\prime }\\left(x_3\\right) v\\left(\\mathbf {x}^{\\prime }, x_3\\right) d x_3-k^{2} \\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right).\\end{split}$ Similarly, we can obtain that $\\Delta _{\\mathbf {x}^{\\prime }} \\mathcal {R}(w)\\left(\\mathbf {x}^{\\prime }\\right) =\\int _{-L}^{L} \\psi ^{\\prime \\prime }\\left(x_3\\right) w\\left(\\mathbf {x}^{\\prime }, x_3\\right) d x_3-k^{2} \\mathcal {R}(q w)\\left(\\mathbf {x}^{\\prime }\\right).$ Therefore, we have $\\begin{split}&\\Delta _{\\mathbf {x}^{\\prime }} \\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right) - \\Delta _{\\mathbf {x}^{\\prime }} \\mathcal {R}(w)\\left(\\mathbf {x}^{\\prime }\\right) \\\\=& \\int _{-L}^{L} \\psi ^{\\prime \\prime }\\left(x_3\\right)\\left(v\\left(\\mathbf {x}^{\\prime }, x_3\\right)-w\\left(\\mathbf {x}^{\\prime }, x_3\\right)\\right) \\mathrm {d} x_3+k^{2} \\mathcal {R}(q w)\\left(\\mathbf {x}^{\\prime }\\right)-k^{2} \\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right)\\\\:=& F_{1}\\left(\\mathbf {x}^{\\prime }\\right)+ F_{2}\\left(\\mathbf {x}^{\\prime }\\right)+ F_{3}\\left(\\mathbf {x}^{\\prime }\\right).\\end{split}$ Next, we set $F_{3 j}\\left(\\mathbf {x}^{\\prime }\\right)=-k^{2} \\mathcal {R}\\left(v_{j}\\right)\\left(\\mathbf {x}^{\\prime }\\right), $ and consider the following integral $\\begin{split}&\\int _{S_{h}}\\left(\\Delta _{\\mathbf {x}^{\\prime }} \\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right) - \\Delta _{\\mathbf {x}^{\\prime }} \\mathcal {R}(w)\\left(\\mathbf {x}^{\\prime }\\right)\\right)u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\mathrm {d} \\mathbf {x}^{\\prime } \\\\= &\\int _{S_{h}}\\left(F_{1}\\left(\\mathbf {x}^{\\prime }\\right)+F_{2}\\left(\\mathbf {x}^{\\prime }\\right)+F_{3 j}\\left(\\mathbf {x}^{\\prime }\\right)\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime } + \\int _{S_{h}}\\left(F_{3}\\left(\\mathbf {x}^{\\prime }\\right)-F_{3 j}\\left(\\mathbf {x}^{\\prime }\\right)\\right)\\mathrm {d} \\mathbf {x}^{\\prime }\\\\:=& I_{1} + \\Delta _{j}(s).\\end{split}$ Using the fact that $\\begin{split}&\\int _{S_{h}}\\left(\\Delta _{\\mathbf {x}^{\\prime }} \\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right) - \\Delta _{\\mathbf {x}^{\\prime }} \\mathcal {R}(w)\\left(\\mathbf {x}^{\\prime }\\right)\\right)u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\mathrm {d} \\mathbf {x}^{\\prime }\\\\=& \\lim _{\\varepsilon \\rightarrow 0}\\int _{D_{\\varepsilon }} \\left(\\Delta _{\\mathbf {x}^{\\prime }} \\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right) - \\Delta _{\\mathbf {x}^{\\prime }} \\mathcal {R}(w)\\left(\\mathbf {x}^{\\prime }\\right)\\right)u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\mathrm {d} \\mathbf {x}^{\\prime },\\end{split}$ where $D_{\\varepsilon }=S_{h} \\backslash B_{\\varepsilon } \\text{ for } 0<\\varepsilon <h$ , and by Lemma REF , we can deduce that $I_{1}+\\Delta _{j}(s)=I_{3}-I_{2}^{\\pm }-\\xi _{j}^{\\pm }(s), $ where $\\xi _{j}^{\\pm }(s)$ is defined in Lemma REF , $I_{2}^{\\pm }$ is defined in Lemma REF , and $I_{3}=\\int _{\\Lambda _{h}}\\left(u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\partial _{\\nu } \\mathcal {R}(v-w)-\\mathcal {R}(v-w) \\partial _{\\nu } u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\right) \\mathrm {d} \\sigma ,$ with $\\Lambda _{h}=S_{h}\\cap \\partial B_{h}$ .", "Since $v-w\\in H^1(S_h \\times (-L,L))$ and $qw \\in C^\\alpha (\\bar{S}_h \\times [-L,L]), \\alpha \\in (0,1)$ , from Lemma REF we know that $F_{1}\\left(\\mathbf {x}^{\\prime }\\right) \\in C^{\\alpha }(\\bar{S}_h)$ and $F_{2}\\left(\\mathbf {x}^{\\prime }\\right) \\in C^\\alpha (\\bar{S}_h)$ .", "In addition, we have $\\mathcal {R}(v_{j})(\\mathbf {x}^{\\prime }) \\in C^\\alpha (\\bar{S}_h)$ .", "Therefore, by Lemma REF , we have $\\begin{split}I_{1}=&\\left(F_{1}(\\mathbf {0})+F_{2}(\\mathbf {0})+F_{3 j}(\\mathbf {0})\\right) \\int _{S_{h}} u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }+\\int _{S_{h}} \\delta F_{1}\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime } \\\\&+\\int _{S_{h}} \\delta F_{2}\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }+\\int _{S_{h}} \\delta F_{3 j}\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }.\\end{split}$ where $\\delta F_{1}(\\mathbf {x}^{\\prime })$ , $\\delta F_{2}(\\mathbf {x}^{\\prime })$ and $\\delta F_{3j}(\\mathbf {x}^{\\prime })$ are deduced by Lemma REF with $F_{1}(\\mathbf {x}^{\\prime })$ , $F_{2}(\\mathbf {x}^{\\prime })$ and $F_{3j}(\\mathbf {x}^{\\prime })$ defined in (REF ) and (REF ).", "By Lemmas REF and REF , we can deduce that $\\begin{split}\\left\|\\int _{S_{h}} \\delta F_{1}\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }\\right\| \\le & \\frac{2\\left\\Vert F_{1}\\right\\Vert _{C^{\\alpha }}\\left(\\theta _{M}-\\theta _{m}\\right) \\Gamma (2 \\alpha +4)}{\\delta _{W}^{2 \\alpha +4}} s^{-\\alpha -2}, \\\\\\left\|\\int _{S_{h}} \\delta F_{2}\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }\\right\| \\le & \\frac{2\\left\\Vert F_{2}\\right\\Vert _{C^{\\alpha }}\\left(\\theta _{M}-\\theta _{m}\\right) \\Gamma (2 \\alpha +4)}{\\delta _{W}^{2 \\alpha +4}} s^{-\\alpha -2},\\\\\\left\|\\int _{S_{h}} \\delta F_{3 j}\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }\\right\| \\le & \\frac{8 L \\sqrt{\\pi }\\Vert \\psi \\Vert _{C^{\\infty }}\\left(\\theta _{M}-\\theta _{m}\\right) \\Gamma (2 \\alpha +4)}{\\delta _{W}^{2 \\alpha +4}} k^{2} \\operatorname{diam}\\left(S_{h}\\right)^{1-\\alpha } \\\\& \\times (1+k)\\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{n-1}\\right)} s^{-\\alpha -2}.\\end{split}$ For $\\Delta _{j}(s)$ , using Cauthy-Schwarz inequality, Corollary REF and the assumption (REF ), we can drive that $\\begin{split}\\left\|\\Delta _{j}(s)\\right\| & \\le k^{2}\\left\\Vert \\mathcal {R}(v)-\\mathcal {R}\\left(v_{j}\\right)\\right\\Vert _{L^{2}\\left(S_{h}\\right)}\\left\\Vert u_{0}\\left(s \\mathbf {x}^{\\prime }\\right)\\right\\Vert _{L^{2}\\left(S_{h}\\right)} \\\\& \\le \\frac{k^{2}\\Vert \\psi \\Vert _{\\infty } \\sqrt{C(L, h)\\left(\\theta _{M}-\\theta _{m}\\right)} e^{-\\sqrt{s \\Theta } \\delta _{W}} h}{\\sqrt{2}} j^{-\\Upsilon },\\end{split}$ where $C(L, h)$ is a positive constant depending on $L$ and $h$ and $\\Theta \\in [0, h]$ .", "By Lemma 3.1, and the same arguments in (REF ), we have $\\left\|I_{3}\\right\| \\le C e^{-c^{\\prime } \\sqrt{s}},$ where $c^{\\prime }>0$ as $s \\rightarrow \\infty $ .", "By Lemma REF and (REF ), multiplying $s$ on the both sides of (REF ), we can deduce that $&&2 v_{j}(\\mathbf {0}) \\eta (\\mathbf {0})\\left[\\left(\\mu \\left(\\theta _{M}\\right)^{-2}-\\mu \\left(\\theta _{M}\\right)^{-2} e^{-\\sqrt{s h} \\mu \\left(\\theta _{M}\\right)}-\\mu \\left(\\theta _{M}\\right)^{-1} \\sqrt{s h} e^{-\\sqrt{s h} \\mu \\left(\\theta _{M}\\right)}\\right) C_{2}^{+}\\right.", "\\\\&&\\left.+\\left(\\mu \\left(\\theta _{m}\\right)^{-2}-\\mu \\left(\\theta _{m}\\right)^{-2} e^{-\\sqrt{s h} \\mu \\left(\\theta _{m}\\right)}-\\mu \\left(\\theta _{m}\\right)^{-1} \\sqrt{s h} e^{-\\sqrt{s h} \\mu \\left(\\theta _{m}\\right)}\\right) C_{2}^{-}\\right] \\\\&=&s\\left[I_{3}-\\left(F_{1}(\\mathbf {0})+F_{2}(\\mathbf {0})+F_{3 j}(\\mathbf {0})\\right) \\int _{S_{h}} u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }-\\Delta _{j}(s)\\right.", "\\\\&&-\\eta (\\mathbf {0})\\left(I_{22}^{+}+I_{22}^{-}\\right)-I_{\\eta }^{+}-I_{\\eta }^{-}-\\int _{S_{h}} \\delta F_{1}\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }-\\int _{S_{h}} \\delta F_{2}\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime } \\\\&&\\left.-\\int _{S_{h}} \\delta F_{3 j}\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }-v_{j}(\\mathbf {0}) \\eta (\\mathbf {0})\\left(I_{21}^{-}+I_{21}^{+}\\right)-\\xi _{j}^{\\pm }(s)\\right].$ Taking $s=j^\\beta ,\\text{ with }\\max \\left\\lbrace 0, \\frac{\\varrho }{1+\\alpha }\\right\\rbrace <\\beta <\\Upsilon $ and letting $j\\rightarrow \\infty $ , together with the use of (REF ), (REF ),(REF ), (REF ), Lemmas REF and REF , we can deduce that $\\lim _{j \\rightarrow \\infty } \\eta (\\mathbf {0})\\left(C_{2}^{-} \\mu \\left(\\theta _{m}\\right)^{-2}+C_{2}^{+} \\mu \\left(\\theta _{M}\\right)^{-2}\\right) v_{j}(\\mathbf {0})=0.$ Moreover, by straightforward calculations, we know that $\\begin{split}&C_{2}^{-} \\mu \\left(\\theta _{m}\\right)^{-2}+C_{2}^{+} \\mu \\left(\\theta _{M}\\right)^{-2} \\\\=&\\frac{\\left(C_{2}^{+} \\cos \\theta _{m}+C_{2}^{-} \\cos \\theta _{M}\\right)+\\mathbf {i}\\left(C_{2}^{+} \\sin \\theta _{m}+C_{2}^{-} \\sin \\theta _{M}\\right)}{\\left(\\cos \\theta _{m}+\\mathbf {i} \\sin \\theta _{m}\\right)\\left(\\cos \\theta _{M}+\\mathbf {i} \\sin \\theta _{M}\\right)}\\end{split}$ Because of the assumption (REF ), one can directly verify that $\\big ( \\cos \\theta _{m}+\\cos \\theta _{M}\\big ) \\text{ and } \\big (\\sin \\theta _{m}+\\sin \\theta _{M}\\big )$ can not be zero simultaneously.", "Without loss of generality, we assume that $\\sin \\theta _{m}+\\sin \\theta _{M} \\ne 0$ .", "Then we split our arguments into the following two cases: $\\sin \\theta _{m}+\\sin \\theta _{M}>0$ ; $\\sin \\theta _{m}+\\sin \\theta _{M}<0$ .", "For the first case, if $\\sin \\theta _m$ and $\\sin \\theta _M$ have the same sign, then from (REF ), we know that $C_{2}^{+} \\sin \\theta _{m}+C_{2}^{-} \\sin \\theta _{M}\\ne 0$ which means that $C_{2}^{-} \\mu \\left(\\theta _{m}\\right)^{-2}+C_{2}^{+} \\mu \\left(\\theta _{M}\\right)^{-2} \\ne 0.", "$ If $\\sin \\theta _m$ and $\\sin \\theta _M$ have different signs, under the assumption (REF ) we know that $\\sin \\theta _m<0$ and $\\sin \\theta _M>0$ .", "From (REF ), we have $\\begin{split}\\frac{C(\\psi )}{1-(k L)^{2}}\\left(\\sin \\theta _{m}+\\left(1-2(k L)^{2}\\right) \\sin \\theta _{M}\\right) & \\le C_{2}^{+} \\sin \\theta _{m}+C_{2}^{-} \\sin \\theta _{M} \\\\& \\le \\frac{C(\\psi )}{1-(k L)^{2}}\\left(\\left(1-2(k L)^{2}\\right) \\sin \\theta _{m}+\\sin \\theta _{M}\\right).\\end{split}$ For a given $0<\\varepsilon <1$ , we can choose an appropriate $L$ such that $0<k L<\\sqrt{\\varepsilon / 2},$ from which we can derive the bounds as follows $\\begin{split}\\frac{C(\\psi )}{1-(k L)^{2}}\\left(\\sin \\theta _{m}+(1-\\varepsilon ) \\sin \\theta _{M}\\right) & \\le C_{2}^{+} \\sin \\theta _{m}+C_{2}^{-} \\sin \\theta _{M} \\\\& \\le \\frac{C(\\psi )}{1-(k L)^{2}}\\left((1-\\varepsilon ) \\sin \\theta _{m}+\\sin \\theta _{M}\\right).\\end{split}$ Since $\\sin \\theta _{m}+\\sin \\theta _{M}>0,$ we can establish the lower bound in (REF ) as follows.", "Denote $\\varepsilon _{0}=\\min \\left\\lbrace \\frac{\\sin \\theta _{m}+\\sin \\theta _{M}}{2 \\sin \\theta _{M}}, 1\\right\\rbrace $ and choose $\\varepsilon \\in \\left(0, \\varepsilon _{0}\\right) .$ It can be verified that $C_{2}^{+} \\sin \\theta _{m}+C_{2}^{-} \\sin \\theta _{M} \\ge \\frac{C(\\psi )}{1-(k L)^{2}}\\left(\\sin \\theta _{m}+(1-\\varepsilon ) \\sin \\theta _{M}\\right)>0$ which indicates that (REF ) holds as well.", "For the second case, if $\\sin \\theta _m=0$ or $\\sin \\theta _M=0$ is satisfied, from (REF ) we know that $C_{2}^{+} \\sin \\theta _{m}+C_{2}^{-} \\sin \\theta _{M}<0.$ Otherwise, if $\\left\|\\sin \\theta _{m}\\right\| \\le \\left\|\\sin \\theta _{M}\\right\|,$ from the fact that $(1-\\varepsilon )\\left\|\\sin \\theta _{m}\\right\| \\le \\left\|\\sin \\theta _{M}\\right\|,$ we know that (REF ) still holds from the upper bound of (REF ).", "If $\\left\|\\sin \\theta _{m}\\right\|>\\left\|\\sin \\theta _{M}\\right\|$ we can choose $\\varepsilon $ with $\\varepsilon >1-\\left\|\\sin \\theta _{M}\\right\| /\\left\|\\sin \\theta _{m}\\right\|>0$ such that (REF ) is also fulfilled from the upper bound of (REF ).", "Therefore, for the second case, we know that (REF ) is always fulfilled.", "Therefore, by (REF ) and (REF ) we know that $\\lim _{j \\rightarrow \\infty } v_{j}(\\mathbf {0})=0.$ Next, in order to simplify the notations, we define $\\kappa :=B((\\mathbf {x}_{c}, x_n^c ),\\rho )\\cap (W\\times (-M,M)).$ Then by using the fact that $\\begin{split}&\\lim _{\\rho \\rightarrow +0} \\frac{1}{m(\\kappa )} \\int _{\\kappa }\|v(\\mathbf {x})\| \\mathrm {d} \\mathbf {x}\\\\\\le & \\lim _{j \\rightarrow \\infty }\\left(\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(\\kappa \\right)} \\int _{\\kappa }\\left\|v(\\mathbf {x})-v_{j}(\\mathbf {x})\\right\| \\mathrm {d} \\mathbf {x}+\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(\\kappa \\right)} \\int _{\\kappa }\\left\|v_{j}(\\mathbf {x})\\right\| \\mathrm {d} \\mathbf {x}\\right),\\end{split}$ we finally have $\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(\\kappa \\right)} \\int _{\\kappa }\|v(\\mathbf {x})\| \\mathrm {d} \\mathbf {x}=0.$ The proof is complete.", "Similar to Corollary REF , we consider the vanishing property of the transmission eigenfunctions in the case $\\eta \\equiv 0$ in three dimensions.", "Corollary 3.2 Let $v, w \\in H^{1}(W \\times (-M, M))$ be a pair of eigenfunctions to (REF ) associated with $\\eta \\equiv 0, k\\in \\mathbb {R}_{+}$ and $W \\subset \\mathbb {R}^{2}$ being defined in (REF ), and $M>0.$ For any fixed $x_3^{c} \\in (-M, M)$ and $L>0$ defined in Definition REF , we suppose that $L$ is sufficiently small such that $\\left(x_3^{c}-L, x_3^{c}+L\\right) \\subset (-M, M)$ .", "Moreover, there exists a sufficiently small neighbourhood $S_{h}$ of $\\mathbf {x}_{c} \\in \\mathbb {R}^{2}$ such that $q w \\in C^{\\alpha }\\left(\\bar{S}_{h} \\times [-M, M]\\right)$ for $0<\\alpha <1$ .", "If the following conditions are fulfilled: the transmission eigenfunction $v$ can be approximated in $H^{1}\\left(S_{h} \\times (-M, M)\\right) b y$ the Herglotz functions $v_{j}, j=1,2, \\ldots ,$ with kernels $g_{j}$ satisfying $\\left\\Vert v-v_{j}\\right\\Vert _{H^{1}\\left(S_{h} \\times (-M, M)\\right)} \\le j^{-\\Upsilon }, \\quad \\left\\Vert g_{j}\\right\\Vert _{L^{2}\\left(\\mathbb {S}^{2}\\right)} \\le C j^{\\varrho },$ for some constants $C,\\varrho \\text{ and }\\Upsilon $ with $C>0, \\Upsilon >0 \\text{ and } \\varrho <\\alpha \\Upsilon /2$ ; the angles $\\theta _{m}$ and $\\theta _{M}$ of the sector $W$ satisfy $-\\pi <\\theta _{m}<\\theta _{M}<\\pi \\text{ and } \\theta _{M}-\\theta _{m} \\ne \\pi ;$ then it holds that $\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap W\\right)} \\int _{B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap W} \\mathcal {R}(V w)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }=0,$ where $q\\left(\\mathbf {x}^{\\prime }, x_3\\right)=1+V\\left(\\mathbf {x}^{\\prime }, x_3\\right)$ .", "Without loss of generality, we assume that $\\mathbf {x}_c=0$ .", "Since $\\eta \\equiv 0$ , from (REF ), (REF ), we can obtain that $\\begin{split}&\\left(F_{1}(\\mathbf {0})+F_{2}(\\mathbf {0})+F_{3 j}(\\mathbf {0})\\right) \\int _{S_{h}} u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }+\\Delta _{j}(s)\\\\&= I_{3}-\\int _{S_{h}} \\delta F_{1}\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }-\\int _{S_{h}} \\delta F_{2}\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }-\\int _{S_{h}} \\delta F_{3 j}\\left(\\mathbf {x}^{\\prime }\\right) u_{0}\\left(s \\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime },\\end{split}$ where $\\Delta _{j}(s)$ and $I_{3}$ are defined in (REF ) and (REF ), and $\\delta F_{1}(\\mathbf {x}^{\\prime })$ , $\\delta F_{2}(\\mathbf {x}^{\\prime })$ and $\\delta F_{3j}(\\mathbf {x}^{\\prime })$ are deduced by Lemma REF with $F_{1}(\\mathbf {x}^{\\prime })$ , $F_{2}(\\mathbf {x}^{\\prime })$ and $F_{3j}(\\mathbf {x}^{\\prime })$ defined in (REF ) and (REF ).", "Since $v=w \\text{ on } \\Gamma ^{\\pm } \\times (-M, M)$ , it easy to see that $F_{1}(\\mathbf {0})=\\int _{-L}^{L} \\psi ^{\\prime \\prime }\\left(x_3\\right)\\left(v\\left(\\mathbf {0}, x_3\\right)-w\\left(\\mathbf {0}, x_3\\right)\\right) \\mathrm {d} x_3=0.$ Multiplying $s^{2}$ on both sides of (REF ), taking $s=j^{\\beta }$ with $\\max \\lbrace \\varrho /\\alpha ,0\\rbrace <\\beta <\\Upsilon /2$ , using the assumptions (REF ) and (REF ), and by letting $j\\rightarrow \\infty $ , from (REF ), (REF ), (REF ), (REF ) and (REF ), we can prove that $\\lim _{j \\rightarrow \\infty } F_{3 j}(\\mathbf {0})=-F_{2}(\\mathbf {0}),$ which in turn implies that $\\lim _{j \\rightarrow \\infty } \\mathcal {R}\\left(v_{j}\\right)(\\mathbf {0})=\\mathcal {R}(q w)(\\mathbf {0}).$ Using the boundary condition in (REF ) and Definition REF , we have that $\\mathcal {R}(w)\\left(\\mathbf {x}^{\\prime }\\right)=\\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right) \\text{ on } \\Gamma $ .", "Hence, we have $\\begin{split}& \\lim _{\\rho \\rightarrow +0} \\frac{1}{m(B(\\mathbf {0}, \\rho )\\cap W)} \\int _{B(\\mathbf {0}, \\rho )\\cap W} \\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }\\\\=& \\lim _{\\rho \\rightarrow +0} \\frac{1}{m(B(\\mathbf {0}, \\rho )\\cap W)} \\int _{B(\\mathbf {0}, \\rho )\\cap W} \\mathcal {R}(w)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }.\\end{split}$ which together with the facts that $\\begin{split}\\lim _{j \\rightarrow \\infty } \\mathcal {R}\\left(v_{j}\\right)(\\mathbf {0}) &=\\lim _{j \\rightarrow \\infty } \\lim _{\\rho \\rightarrow +0} \\frac{1}{m(B(\\mathbf {0}, \\rho )\\cap W)} \\int _{B(\\mathbf {0}, \\rho )\\cap W} \\mathcal {R}\\left(v_{j}\\right)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime } \\\\&=\\lim _{\\rho \\rightarrow +0} \\frac{1}{m(B(\\mathbf {0}, \\rho )\\cap W)} \\int _{B(\\mathbf {0}, \\rho )\\cap W} \\mathcal {R}(v)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }, \\\\\\mathcal {R}(q w)(\\mathbf {0}) &=\\lim _{\\rho \\rightarrow +0} \\frac{1}{m(B(\\mathbf {0}, \\rho )\\cap W)} \\int _{B(\\mathbf {0}, \\rho )\\cap W} \\mathcal {R}(q w)\\left(\\mathbf {x}^{\\prime }\\right) \\mathrm {d} \\mathbf {x}^{\\prime }.\\end{split}$ readily completes the proof of the corollary.", "Remark 3.3 If $V(\\mathbf {x}^{\\prime },x_n)$ is continuous near the edge point $(\\mathbf {x}_{c},x_3^{c})$ and $V(\\mathbf {x}_{c},x_3^{c})\\ne 0$ , by the dominant convergent theorem and Definition REF , we can prove that $\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap W\\right)} \\int _{B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap W} \\int _{x_3^{c}-L}^{x_3^{c}+L} \\psi \\left(x_3\\right) w\\left(\\mathbf {x}^{\\prime }, x_3\\right) \\mathrm {d} \\mathbf {x}^{\\prime } \\mathrm {d} x_3=0.$ Furthermore, if $\\psi (x_3^{c})\\ne 0$ , we can show that $\\lim _{\\rho \\rightarrow +0} \\frac{1}{m\\left(B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap W\\right)} \\int _{B\\left(\\mathbf {x}_{c}, \\rho \\right)\\cap W} \\int _{x_3^{c}-L}^{x_3^{c}+L} w\\left(\\mathbf {x}^{\\prime }, x_3\\right) \\mathrm {d} \\mathbf {x}^{\\prime } \\mathrm {d} x_3=0.$ which describes the vanishing property of the transmission eigenfunctions near the edge corner in three dimensions." ], [ "Acknowledgement", "The work of Y. 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[ [ "The Brown measure of unbounded variables with free semicircular\n imaginary part" ], [ "Abstract Let $x_0$ be an unbounded self-adjoint operator such that the Brown measure of $x_0$ exists in the sense of Haagerup and Schultz.", "Also let $\\tilde\\sigma_\\alpha$ and $\\sigma_\\beta$ be semicircular variables with variances $\\alpha\\geq 0$ and $\\beta>0$ respectively.", "Suppose $x_0$, $\\sigma_\\alpha$, and $\\tilde\\sigma_\\beta$ are all freely independent.", "We compute the Brown measure of $x_0+\\tilde\\sigma_\\alpha+i\\sigma_\\beta$, extending the recent work which assume $x_0$ is a bounded self-adjoint random variable.", "We use the PDE method introduced by Driver, Hall and Kemp to compute the Brown measure.", "The computation of the PDE relies on a charaterization of the class of operators where the Brown measure exists.", "The Brown measure in this unbounded case has the same structure as in the bounded case; it has connections to the free convolution $x_0+\\sigma_{\\alpha+\\beta}$.", "We also compute the example where $x_0$ is Cauchy-distributed." ], [ "Introduction", "An elliptic variable $c_{\\alpha ,\\beta }$ is an element in a $W^$ -probability space $({A},\\tau )$ of the form $\\tilde{\\sigma }_\\alpha +i\\sigma _\\beta $ where $\\tilde{\\sigma }_\\alpha $ and $\\sigma _\\beta $ are free semicircular random variables with variances $\\alpha > 0$ and $\\beta >0$ respectively.", "In this paper, we also consider the case $\\alpha = 0$ ; in this case, $c_{\\alpha ,\\beta }$ is a “degenerate” elliptic variable, and is just a imaginary multiple of semicircular variable.", "(We do not consider the case $\\alpha >0$ and $\\beta =0$ in this paper.", "In this case, $c_{\\alpha ,\\beta }$ is also a degenerate elliptic variable — a semicircular variable.)", "The elliptic variable $c_{\\alpha ,\\beta }$ is the limit in $\\ast $ -distribution of the Gaussian random matrix $\\sqrt{\\alpha }\\tilde{X}_N+i\\sqrt{\\beta }X_N$ where $\\tilde{X}_N$ and $X_N$ are independent Gaussian unitary ensembles (GUEs); see [26] and the book [19].", "Also, the empirical eigenvalue distribution of the random matrix $\\sqrt{\\alpha }\\tilde{X}_N+i\\sqrt{\\beta }X_N$ converges to the Brown measure of the elliptic variable $c_{\\alpha ,\\beta }$ [11].", "The Brown measure of $c_{\\alpha ,\\beta }$ has a simple structure; it is the uniform measure in a region bounded by a certain ellipse [5].", "Let $x_0$ be a self-adjoint noncommutative random variable in the $W^$ -probability space ${A}$ containing $c_{\\alpha ,\\beta }$ , and that $x_0$ is freely independent from $c_{\\alpha ,\\beta }$ .", "In particular, $x_0$ is a bounded random variable.", "The author computed the Brown measure of $x_0+c_{\\alpha ,\\beta }$ in [15].", "In this paper, we extend the computation to an unbounded self-adjoint random variable $x_0$ affiliated with ${A}$ .", "When $x_0$ is an unbounded variable, the Brown measure of $x_0+c_{\\alpha ,\\beta }$ has the same structure as in the bounded case; in particuar, it has direct connections to the Brown measure of $x_0+c_{s}$ , where $c_s= c_{s/2,s/2}$ is the circular variable with $s=\\alpha +\\beta $ , and the law of $x_0+\\sigma _{s}$ .", "We also compute, as an example, the Brown measure of $x_0+c_{\\alpha ,\\beta }$ for $\\alpha \\ge 0$ and $\\beta >0$ when $x_0$ has a Cauchy distribution.", "When $x_0$ is a bounded random variable and is the limit in distribution of a sequence $A_N$ of Hermitian random matrix independent from the GUEs $\\tilde{X}_N$ and $X_N$ , Śniady [23] shows that the empirical eigenvalue distribution of $A_N+\\sqrt{\\alpha }\\tilde{X}_N+i\\sqrt{\\beta }X_N,\\quad \\alpha >0, \\beta >0$ converges to the Brown measure of $x_0+c_{\\alpha ,\\beta }$ , as $N\\rightarrow \\infty $ .", "When $x_0$ is unbounded, computer simulations show that the Brown measure of $x_0+c_{\\alpha ,\\beta }$ is a reasonable candidate of the limiting eigenvalue distribution of the random matrix model $A_N+\\sqrt{\\alpha }\\tilde{X}_N+i\\sqrt{\\beta }X_N$ , where the empirical eigenvalue distribution of $A_N$ converges to the law of $x_0$ .", "The computation of the Brown measure in this paper is based on the PDE method introduced by Driver, Hall and Kemp in [8], and noncommutative integration theory introduced by Segal in [22] (also see the books [17], [24] by Jajte and Takesaki respectively).", "In [8], Driver, Hall and Kemp computed the Brown measure of the free multiplicative Brownian motion using the Hamilton–Jacobi method.", "Based on this PDE method, Zhong and the author [16] computed the Brown measures of the circular Brownian motion with arbitrary bounded self-adjoint initial condition $x_0$ and the free multiplicative Brownian motion with arbitrary unitary initial condition.", "Later, Hall and the author [13] computed the Brown measure of the sum of an arbitrary bounded self-adjoint random variable $x_0$ and an imaginary multiple of semicircular variable $\\sigma _t$ , freely independent from $x_0$ .", "Using the results in [13], the author [15] computed the Brown measure of the sum of a bounded self-adjoint random variable $x_0$ and an elliptic variable $c_{\\alpha ,\\beta }$ that is freely independent from $x_0$ .", "The results in [13], [15], [16] assume the boundedness of $x_0$ in the process of computing the PDE.", "The key observation in this paper is to show that the function $S(t,\\lambda ,\\varepsilon ) = \\tau [\\log ((x_0+i\\sigma _t-\\lambda )^(x_0+i\\sigma _t-\\lambda )+\\varepsilon )], \\quad t>0, \\lambda \\in \\mathbb {C}, \\varepsilon >0$ satisfies the same PDE derived in [13] when $x_0$ is an unbounded random variable affiliated with the $W^$ -probability space ${A}$ , using noncommutative integration theory.", "The solution of the PDE yields the Brown measure of $x_0+i\\sigma _t$ .", "By considering the free additive convolution $x_0+\\tilde{\\sigma }_{\\alpha }$ in place of $x_0$ , we derive the family of the Brown measures of $x_0+c_{\\alpha ,\\beta }$ for $\\alpha \\ge 0$ and $\\beta >0$ .", "In particular, by taking $\\alpha =\\beta >0$ , we extend the results in [16] to unbounded self-adjoint $x_0$ .", "The paper is organized as follows.", "Section REF lists out all the Brown measures computed in this paper.", "These Brown measures are not listed in the order of being proved in this paper; they are listed according to the complexity of their structures.", "Section contains some basic definitions and background from free probability theory and noncommutative measure theory.", "It also contains the PDE computed in [13].", "Section contains the crucial theorem of this paper — showing that we get the same PDE computed in [13] for the function $S$ defined in (REF ), even when $x_0$ is unbounded.", "In the same section, we analyze the PDE using the Hamilton–Jacobi method.", "The PDE is already solved in [13]; the solution of the PDE is given without proof.", "In Section , we use the solution of the PDE to compute the Brown measure of $x_0+i\\sigma _t$ .", "The Brown measure has full measure on an open set $\\Omega _{x_0,t}$ .", "Since the law of $x_0$ now possibly has unbounded support, we provide a proof whenever the argument of the corresponding statement in [13] only works for bounded random variable $x_0$ .", "In Section , we replace the random variable $x_0$ in the results in Section by the free convolution $x_0+\\tilde{\\sigma }_\\alpha $ to compute the Brown measure of $x_0+c_{\\alpha ,\\beta }$ .", "Finally, in Section , we compute the Brown measure of $x_0+c_{\\alpha ,\\beta }$ when $x_0$ has a Cauchy distribution." ], [ "Statement of results", "In this subsection, we list out the Brown measures computed in this paper according to the complexity of their structures.", "Let $({A},\\tau )$ be a $W^$ -probability space and $\\tilde{{A}}$ be the algebra of closed densely defined operators affiliated with ${A}$ .", "Let $x_0\\in \\tilde{{A}}$ be self-adjoint satisfying the following standing assumption throughout the paper.", "The law $\\mu _{x_0}$ of $x_0$ is not a Dirac mass at a point on $\\mathbb {R}$ , and satisfies $\\int _1^\\infty \\log x\\,d\\mu _{x_0}(x)<\\infty .$ If $\\mu _{x_0}$ is a Dirac mass, then the Brown measure of $x_0+c_{\\alpha ,\\beta }$ is just the translation of the elliptic law in the case $\\alpha >0$ , or just the translation of the semicircular law on the imaginary axis in the case $\\alpha = 0$ .", "We list out the results for $x_0$ satisfying Assumption REF below.", "We start by Brown measure of $x_0+c_t$ , where $c_t = c_{t/2,t/2}$ is the circular variable, freely independent from $x_0$ .", "Recall that $\\sigma _t$ denotes a semicircular variable of variance $t$ , freely independent from $x_0$ .", "The following theorem is proved in Corollaries and (2).", "When $x_0$ is bounded, this theorem is established in Theorems 3.13 and 3.14 in [16].", "The following statements about the Brown measure of $x_0+c_t$ hold.", "The Brown measure of $x_0+c_t$ has full measure on an open set $\\Lambda _{x_0,t}$ of the form $\\Lambda _{x_0,t} = \\lbrace \\left.u+iv\\in \\mathbb {C}\\right\|\|v\|<v_{x_0,t}(u)\\rbrace $ for a certain function $v_{x_0,t}$ that appears in the study of the free convolution $x_0+\\sigma _t$ .", "Then the density of the Brown measure on $\\Lambda _{x_0,t}$ has the special form $w_t(u+iv) = \\frac{1}{\\pi t}\\left(1-\\frac{t}{2}\\frac{d}{du}\\int \\frac{x\\,d\\mu _{x_0}(x)}{(u-x)^2+v_{x_0,t}(u)^2}\\right), \\quad u+iv\\in \\Lambda _{x_0,t};$ in particular, the density is strictly positive and constant along vertical segments in $\\Lambda _{x_0,t}$ .", "Consider the function $\\psi _{x_0,t}(u) = u+t\\int \\frac{(u-x)\\,d\\mu _{x_0}(x)}{(u-x)^2+v_{x_0,t}(u)^2},\\quad u\\in \\mathbb {R}$ that appears in the study of the free convolution $x_0+\\sigma _t$ .", "Then the push-forward of the Brown measure of $x_0+c_t$ by the map $u+iv\\mapsto \\psi _{x_0,u}(u)$ is the law of $x_0+\\sigma _t$ .", "The Brown measure of $x_0+c_s$ “generates” the family of the Brown measures of $x_0+c_{\\alpha ,\\beta }$ with $\\alpha +\\beta =s$ in the sense of push-forward measure.", "The following theorem is established in Theorem and Corollary .", "This result extends Theorems 3.7 and 4.1 in [15] to possibly-unbounded $x_0$ affiliated with ${A}$ .", "The following about the Brown measure of $x_0+c_{\\alpha ,\\beta }$ , for $\\alpha \\ge 0$ and $\\beta >0$ holds.", "The Brown measure of $x_0+c_{\\alpha ,\\beta }$ has full measure on an open set $\\Omega _{x_0,t}$ of the form $\\Omega _{x_0,\\alpha ,\\beta } = \\lbrace \\left.u+iv\\in \\mathbb {C}\\right\|\|v\|<\\varphi _{x_0,\\alpha ,\\beta }(u)\\rbrace $ for a certain function $\\varphi _{x_0,\\alpha ,\\beta }$ .", "There exists a strictly increasing homeomorphism $f_{x_0,\\alpha ,\\beta }$ on $\\mathbb {R}$ such that, by writing $u=f_{x_0,\\alpha ,\\beta }(u_0)$ , the density of the Brown measure on $\\Omega _{x_0,\\alpha ,\\beta }$ has the special form $w_{\\alpha ,\\beta }(u+iv) = \\frac{1}{4\\pi \\beta }\\left(1+2\\beta \\frac{d}{du}\\int \\frac{(u_0-x)\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v_{x_0,s}(u_0)^2}\\right), \\quad u+iv\\in \\Omega _{x_0,\\alpha ,\\beta },$ where $s=\\alpha +\\beta $ .", "The density is strictly positive and constant along vertical segments in $\\Omega _{x_0,\\alpha ,\\beta }$ .", "Furthermore, by writing $u=f_{x_0,\\alpha ,\\beta }(u_0)$ again, the function $\\varphi _{x_0,\\alpha ,\\beta }$ in (REF ) is given by $\\varphi _{x_0,\\alpha ,\\beta }(u)=v_{x_0,s}(u_0)$ where $v_{x_0,s}$ is defined as in Theorem REF (1) Write $s=\\alpha +\\beta $ .", "Let $U_{\\alpha ,\\beta }:\\overline{\\Lambda }_{x_0,s}\\rightarrow \\overline{\\Omega }_{x_0,\\alpha ,\\beta }$ be defined by $U_{\\alpha ,\\beta }(u+iv) = f_{x_0,\\alpha ,\\beta }(u)+i\\frac{2\\beta v}{s}.$ Then $U_{\\alpha ,\\beta }$ is a homeomorphism, and the push-forward of the Brown measure of $x_0+c_s$ under the map $U_{\\alpha ,\\beta }$ is the Brown measure of $x_0+c_{\\alpha ,\\beta }$ .", "Let $s=\\alpha =\\beta $ .", "Recall the function $\\psi _{x_0,s}$ is defined in (REF ).", "The push-forward of the Brown measure of $x_0+c_{\\alpha ,\\beta }$ by the map $Q_{\\alpha ,\\beta }(u+iv) = {\\left\\lbrace \\begin{array}{ll}\\frac{1}{\\alpha -\\beta }[su-2\\beta f_{x_0,\\alpha ,\\beta }^{-1}(u)]\\quad &\\textrm {if \\alpha \\ne \\beta }\\\\\\psi _{x_0,s}(u)\\quad &\\textrm {if \\alpha = \\beta =s/2}.\\end{array}\\right.", "}$ is the law of $x_0+\\sigma _s$ .", "By putting $\\alpha = 0$ and $\\beta = t$ , the above theorem recovers Theorem REF in this paper which computes the Brown measure of $x_0+i\\sigma _t$ .", "We close this section by a matrix simulation.", "Let $A_N$ be an $N\\times N$ diagonal random matrix whose diagonal entries are independent Cauchy-distributed random variables; that is, each diagonal entry of $A_N$ is a random variable with density $\\frac{1}{\\pi }\\frac{1}{x^2+1},\\quad x\\in \\mathbb {R}.$ Also, let $X_N$ be an $N\\times N$ GUE, independent from $A_N$ .", "Figure REF plots a computer simulation of the eigenvalues of $A_N+iX_N$ with $N=5000$ and the density of the Brown measure of $x_0+i\\sigma _1$ where $x_0$ has the Cauchy distribution.", "This simulation suggests that the Brown measure is a reasonable candidate of the limiting eigenvalue distribution even in the unbounded case.", "Figure: 5000×50005000\\times 5000 matrix simulation of A 5000 +iX 5000 A_{5000}+iX_{5000} and the Brown measure densty of x 0 +iσ 1 x_0+i\\sigma _1." ], [ "Free probability", "We first introduce noncommutative random variables and freeness (or free independence).", "A $W^$ -probability space $({A},\\tau )$ is a finite von Neumann algebra ${A}$ with a normal, faithful, tracial state $\\tau $ .", "The algebra $\\tilde{{A}}$ affiliated with ${A}$ is the set of closed, densely defined operators such that every $a\\in {A}$ has a polar decomposition $a = u\|a\| = u\\int _0^\\infty \\lambda \\,dE_{\|a\|}(\\lambda )$ where $u\\in {A}$ is unitary and the projection-valued spectral measure $E_{\|a\|}$ takes values in ${A}$ .", "The elements in $\\tilde{{A}}$ are called (noncommutative) random variables.", "When we distinguish elements from $\\tilde{{A}}$ and ${A}$ , any $a\\in {A}$ is called a bounded random variable and any $a\\in \\tilde{{A}}\\setminus {A}$ is called an unbounded random variable.", "Given any self-adjoint random variable $a\\in \\tilde{{A}}$ with spectral measure $E_a$ , the law $\\mu _a$ of $a$ is the probability measure on $\\mathbb {R}$ defined by $\\mu _a(B) = \\tau (E_a(B))$ for any Borel set $B$ .", "The von Neumann subalgebras ${A}_1,\\ldots ,{A}_n\\subset {A}$ are said to be free, or freely independent, if given any $i_1,\\ldots ,i_m\\in \\lbrace 1,\\ldots ,n\\rbrace $ with $i_k\\ne i_{k+1}$ and $a_{i_k}\\in {A}_{i_k}$ for all $k$ satisfying $\\tau (a_{i_k}) = 0$ for all $1\\le k\\le m$ , we have $\\tau (a_{i_1}\\cdots a_{i_m}) = 0.$ The random variables $a_1,\\ldots ,a_m\\in \\tilde{{A}}$ are free, or freely independent, if the von Neumann subalgebras ${A}_k$ generated by all the spectral projections of $a_{k}$ are free.", "In this paper, we denote $\\sigma _t$ (or $\\tilde{\\sigma }_t$ ) by the semicircular variable with variance $t$ ; that is, the law $\\mu _{\\sigma _t}$ is given by $\\frac{1}{2\\pi t}\\sqrt{4t-x^2}\\mathbb {1}_{\|x\|\\le 2\\sqrt{t}}\\,dx.$ One way to study the law of $a\\in \\tilde{{A}}$ is to use the Cauchy transform $G_a$ of $a$ defined by $G_a(z) = \\tau [(z-a)^{-1}] = \\int \\frac{d\\mu _a(x)}{z-x},\\quad z\\in \\mathbb {C}^+,$ where $\\mathbb {C}^+$ is the upper half plane.", "The Cauchy transform is determined by the law $\\mu _a$ of $a$ .", "Given any finite measure $\\mu $ on $\\mathbb {R}$ , one can define the Cauchy transform $G_\\mu $ of $\\mu $ by replacing $\\mu _a$ in the right hand side of the above equation by $\\mu $ .", "We can recover the law $\\mu _a$ from $G_a$ by the Stieltjes inversion formula $d\\mu _a(x)=-\\frac{1}{\\pi }\\lim _{\\varepsilon \\rightarrow 0^+}\\mathrm {Im}\\,G_a(x+i\\varepsilon )\\,dx$ where the limit is in weak sense.", "The Cauchy transform $G_a$ of $a$ is invertible on some truncated Stolz angle $\\Gamma _{\\alpha ,\\beta }=\\lbrace \\left.x+iy\\in \\mathbb {C}\\right\|y>\\beta \\textrm { and } x<\\alpha y\\rbrace $ for positive $\\alpha $ and $\\beta $ .", "The inverse of $G_a$ is called the $K$ -transform of $a$ and is denoted by $K_a$ .", "The $R$ -transform $R_a$ is defined as $R_a(z) = K_a(z) - \\frac{1}{z}.$ The following fact about $R$ -transform was first discovered by Voiculescu [25] for bounded random variables, then by Maassen [18] for possibly unbounded random variables with finite variance, and by Bercovici and Voiculescu [2] for arbitrary unbounded random variables.", "If $a,b\\in \\tilde{{A}}$ are self-adjoint freely independent random variables, then, on the common domain where the $R$ -transforms $R_a$ , $R_b$ , and $R_{a+b}$ are defined, $R_{a+b} = R_a+R_b.$" ], [ "Free addition convolution", "When $x_0\\in {A}$ is a bounded self-adjoint random variable free from $\\sigma _t$ , the Brown measure of $x_0+i\\sigma _t$ has direct connections to $x_0+\\sigma _t$ , by [13], [16].", "In this paper, we will show that this is also the case for unbounded self-adjoint random variable $x_0$ .", "In this section, we review the free additive convolution of $x_0$ and $\\sigma _t$ , where $x_0$ can be taken to be unbounded.", "Biane [3] computed the free additive convolution of a self-adjoint random variable $x_0\\in \\tilde{{A}}$ and the semicircular variable $\\sigma _t$ .", "Before we state the results, we first introduce some notations.", "Denote by $\\mu _{x_0}$ the law of the self-adjoint random variable $x_0\\in \\tilde{{A}}$ .", "Let $H_{x_0,t}$ be defined by $H_{x_0,t}(z) = z + t\\int \\frac{d\\mu _{x_0}(x)}{z-x},\\quad z\\notin \\mathrm {supp}(\\mu _{x_0})$ where $\\mathrm {supp}(\\mu _{x_0})$ denotes the support of $\\mu _{x_0}$ .", "Define a nonnegative function on $\\mathbb {R}$ by $v_{x_0,t}(u) = \\inf \\left\\lbrace v > 0\\left\|\\int \\frac{d\\mu _{x_0}(x)}{(u-x)^2+v^2}\\le \\frac{1}{t}\\right.\\right\\rbrace .$ This means, for a given $u\\in \\mathbb {R}$ , $v_{x_0,t}(u)$ is the unique positive number (if exists) such that $\\int \\frac{d\\mu _{x_0}(x)}{(u-x)^2+v_{x_0,t}(u)^2}=\\frac{1}{t}$ ; if such a number does not exist, $v_{x_0,t}(u) = 0$ .", "Let $\\Gamma _{x_0,t} = \\lbrace u+iv\\in \\mathbb {C}\\left\|v>v_{x_0,t}(u)\\right.\\rbrace $ be the region in the upper half plane $\\mathbb {C}^+$ that is above the graph of $v_{x_0,t}$ .", "Recall that $G_a$ denotes the Cauchy transform of $a\\in \\tilde{{A}}$ .", "The following theorem, due to Biane [3], computes the free additive convolution of a self-adjoint random variable $x\\in \\tilde{{A}}$ and the semicircular variable $\\sigma _t$ .", "[[3]] The function $H_{x_0,t}$ is an injective conformal map from $\\Gamma _{x_0,t}$ onto the upper half plane $\\mathbb {C}^+$ .", "The function extends to a homeomorphism from $\\overline{\\Gamma }_{x_0,t}$ onto $\\mathbb {C}^+\\cup \\mathbb {R}$ .", "Hence, for any $u\\in \\mathbb {R}$ , $H_{x_0,t}(u+iv_{x_0,t}(u))$ is real.", "The function $H_{x_0,t}$ satisfies $G_{x_0+\\sigma _t}(H_{x_0,t}(z)) = G_{x_0}(z).$ The law of $x_0+\\sigma _t$ is absolutely continuous with respect to the Lebesgue measure; its density $p_{x_0,t}$ can be computed using the function $\\psi _{x_0,t}(u) = H_{x_0,t}(u+iv_{x_0,t}(u))$ .", "The function $\\psi _{x_0,t}:\\mathbb {R}\\rightarrow \\mathbb {R}$ is a homeomorphism, and $p_{x_0,t}(\\psi _{x_0,t}(u)) = \\frac{v_{x_0,t}(u)}{\\pi t}.$ When $a$ and $b$ are arbitrary self-adjoint random variables, the free additive convolution $a+b$ can be studied using Cauchy transforms in a way similar to (REF ) (the method of subordination functions).", "The subordination functions are first discovered by Voiculescu [27] for bounded random variables $a$ and $b$ .", "Then it is further developed by Biane [4], and again by Voiculescu [28] to a even more general setting.", "Belinschi and Bercovici [1] use a complex analysis approach to show the existence of subordination functions (where they do not assume the random variables to be bounded)." ], [ "The Brown measure", "In this section, we review the definition of the Brown measure, first introduced by Brown [7] for operators in a von Neumann algebra with a faithful, semifinite, normal trace, then extended to a class ${A}^\\Delta $ of unbounded operators by Haagerup and Schultz [12].", "We first introduce the class ${A}^\\Delta $ of unbounded operators.", "We denote by ${A}^\\Delta $ the set of operators $a$ in $\\tilde{{A}}$ such that $\\tau (\\log ^+\|a\|) = \\int _0^\\infty \\log ^+(\\lambda )\\,d\\mu _{\|a\|}(\\lambda )$ where $\\log ^+$ is defined to be $\\log ^+(x) = \\max (\\log x, 0)$ for $x\\ge 0$ .", "Let $a\\in {A}^{\\Delta }$ .", "Define a function $S$ by $S(\\lambda ,\\varepsilon ) = \\tau [\\log (\|a-\\lambda \|^2+\\varepsilon )], \\lambda \\in \\mathbb {C}, \\varepsilon >0.$ Then $s(\\lambda ) = \\lim _{\\varepsilon \\rightarrow 0^+} S(\\lambda ,\\varepsilon ), \\quad \\lambda \\in \\mathbb {C}$ is a subharmonic function, with value in $\\mathbb {R}\\cup \\lbrace -\\infty \\rbrace $ .", "The exponential of $s(\\lambda )/2$ is the Fuglede-Kadison determinant of $a-\\lambda $ .", "The Fuglede-Kadison determinant for operators in a finite factor was introduced by Fuglede and Kadison [10].", "The Brown measure of $a$ is defined to be $\\mathrm {Brown}(a) = \\frac{1}{4\\pi }\\Delta s(\\lambda )$ where the Laplacian is in distributional sense.", "The first statement of the following theorem characterizes the operators in ${A}^\\Delta $ , and is proved in Lemma 2.4 of [12].", "The second statement is an important property of ${A}^\\Delta $ , and is proved in Propositions 2.5 and 2.6 of [12].", "[[12]] Let $a\\in \\tilde{{A}}$ .", "Then $a\\in {A}^\\Delta $ is equivalent to any of the following.", "There exist $x_1\\in {A}$ and $y_1\\in {A}$ such that $a = x_1y_1^{-1}$ and $\\tau [\\log \|a\|] = \\tau [\\log \|x_1\|]-\\tau [\\log \|y_1\|].$ There exist $x_2\\in {A}$ and $y_2\\in {A}$ such that $a = x_2^{-1}y_2$ and $\\tau [\\log \|a\|] = \\tau [\\log \|y_2\|]-\\tau [\\log \|x_2\|]$ The set ${A}^\\Delta $ is an algebra; that is, it is a subspace in $\\tilde{{A}}$ and is closed under multiplication.", "An immediate consequence from Theorem REF is that if $x_0\\in {A}^\\Delta $ , then $x_0+c_{\\alpha ,\\beta }\\in {A}^\\Delta $ , for all $\\alpha \\ge 0$ and $\\beta >0$ , so that the Brown measure of $x_0+c_{\\alpha ,\\beta }$ is defined in the sense of Haagerup and Schultz." ], [ "A PDE for the Brown measure of $x_0+i\\sigma _t$ with bounded {{formula:47c5a589-1c26-44cb-9119-bda9c3e3aac9}}", "In [13], Hall and the author computed the Brown measure of $x_t = x_0+i\\sigma _t$ where $x_0\\in {A}$ is a (bounded) self-adjoint random variable with law $\\mu _{x_0}$ , and $\\sigma _t\\in {A}$ is a semicircular variable, freely independent from $x_0$ .", "Define $S(t, \\lambda ,\\varepsilon ) = \\tau [\\log ((x_t-\\lambda )^(x_t-\\lambda )+\\varepsilon )], \\quad t\\ge 0, \\lambda \\in \\mathbb {C}, \\varepsilon >0.$ Then the function $S$ satisfies the following PDE of Hamilton–Jacobi type.", "Write $\\lambda = u+iv$ .", "Then $S$ satisfies the PDE $\\frac{\\partial S}{\\partial t} = \\varepsilon \\left(\\frac{\\partial S}{\\partial \\varepsilon }\\right)^2+\\frac{1}{4}\\left(\\left(\\frac{\\partial S}{\\partial u}\\right)^2-\\left(\\frac{\\partial S}{\\partial v}\\right)^2\\right)$ with initial condition $S(0,\\lambda ,\\varepsilon ) = \\int \\log ((x-\\lambda )^(x-\\lambda )+\\varepsilon )\\,d\\mu _{x_0}(x).$ Hall and the author [13] analyze the function $S$ using the above PDE to compute the Brown measure of $x_t$ .", "The above PDE is obtained by the free Itô formula developed by Biane and Speicher [6], which only applies to bounded variable $x_0$ .", "The key observation in this paper is to extend Theorem REF to self-adjoint unbounded $x_0\\in {A}^\\Delta $ .", "The solution of this PDE yields the Brown measure of $x_0+i\\sigma _t$ .", "The proof from some of the other results in [13] will also need to modified for $\\mu _{x_0}$ with unbounded support on $\\mathbb {R}$ .", "We do not list all the results of [13] here; rather, we will state the results in the process of computing the Brown measure, with modified proof whenever necessary." ], [ "Noncommutative measure theory", "Given a finite factor ${A}$ (which is a subalgebra of the bounded linear operators of some Hilbert space $H$ ) with a faithful, normal, tracial state $\\tau $ , Segal [22] suggested that we can think of $x\\mapsto \\tau (x)$ as a noncommtative version of integration, with the analogy that ${A}$ is playing the role of $L^\\infty $ of a measure space, and $\\tau $ is playing the role of an integral with respect to some measure.", "In the paper [22], Segal introduced “pointwise” convergence in a von Neumann algebra (which is called convergence in strong operator topology) and the theory of noncommutative integration.", "The noncommutative version of Egoroff's theorem that we introduce in this section is due to Saitô [21].", "A more complete introduction of noncommtative integration can be found, for example, in the books [17], [24].", "We start by defining projections and measure topology on ${A}$ with respect to $\\tau $ .", "The definition of measure topology can be found in Definition 2.1 of [24].", "A operator $p\\in {A}$ is called a (self-adjoint) projection if $p^=p$ and $p^2=p$ .", "The measure topology of ${A}$ with respect to $\\tau $ is the uniform topology given by a neighborhood system $\\lbrace x+N(\\varepsilon ,\\delta ): \\varepsilon ,\\delta >0\\rbrace $ for $x\\in {A}$ , where $N(\\varepsilon ,\\delta )$ is the set of $a\\in {A}$ such that $\\Vert ap\\Vert <\\varepsilon \\quad \\textrm {and}\\quad \\tau (1-p)<\\delta $ for some projection $p$ .", "The completion of ${A}$ with respect to this measure topology is denoted by $\\mathfrak {M}({A})$ .", "The following is a special case in Theorem 2.13(ii) of [24].", "For any $x\\in \\mathfrak {M}(M)$ , set $\\Vert x\\Vert _1 = \\tau (\|x\|)$ and $L^1({A},\\tau ) = \\lbrace \\left.x\\in \\mathfrak {M}(M)\\right\|\\Vert x\\Vert _1<\\infty \\rbrace .$ Then ${A}$ can be embedded continuously into $L^1({A},\\tau )$ , and $L^1({A},\\tau )$ is a Banach space in which ${A}\\cap L^1({A},\\tau )$ is dense.", "Furthermore, $L^1({A},\\tau )$ is invariant under the multiplications of ${A}$ from both sides, and $\\Vert ax\\Vert _1\\le \\Vert a\\Vert \\Vert x\\Vert _1,\\quad \\Vert xa\\Vert _1\\le \\Vert a\\Vert \\Vert x\\Vert _1$ for each $a\\in {A}$ and $x\\in L^1({A},\\tau )$ .", "We now define the strong operator topology on ${A}$ , which can be thought as a noncommutative version of pointwise convergence of measurable functions.", "Recall that we assume ${A}$ is a von Neumann subalgebra of the bounded linear operators of a Hilbert space $H$ .", "Let $x_n$ be a sequence in ${A}$ and $x\\in {A}$ .", "The sequence $x_n$ is said to be convergent to $x$ in the strong operator topology if, given any $h\\in H$ , $\\lim _{n\\rightarrow \\infty } x_n h = xh.$ Recall that, in measure theory, the Egoroff's theorem states that, on a finite measure space, if a sequence of measurable functions $f_n$ converges to a measurable function $f$ pointwise almost everywhere, then, given any $\\varepsilon >0$ , the sequence $f_n$ converges to $f$ uniformly except on a measurable set $E$ with measure less than $\\varepsilon $ .", "The following noncommutative version of the Egoroff's theorem is a weaker version of Theorem 1.3.2 of [17].", "[Noncommutative Egoroff's theorem] Let $x_n$ be a sequence in ${A}$ and let $x\\in {A}$ .", "Suppose that $x_n\\rightarrow x$ in the strong operator topology.", "Then, for any $\\varepsilon >0$ , there exists a projection $p$ in ${A}$ and a subsequence $x_{n_k}$ of $x_n$ such that $\\tau (1-p)<\\varepsilon $ and $\\lim _{k\\rightarrow \\infty }\\Vert (x_{n_k}-x)p\\Vert = 0.$ In analogy to measure theory, the projection $p$ in the above theorem plays the role of an indicator function of a measurable set that is almost of full measure, and the norm convergence plays the role of uniform convergence." ], [ "The PDE", "Let $x_0\\in {A}^\\Delta $ (See Definition REF ) be self-adjoint and write $x_t = x_0+i\\sigma _t$ where $\\sigma _t$ is a semicircular variable, freely independent from $x_0$ .", "By Remark REF , the Brown measure of $x_0+i\\sigma _t$ exists in the sense of Haagerup and Schultz [12].", "Define $S(t,\\lambda ,\\varepsilon )=\\tau [\\log ((x_t-\\lambda )^(x_t-\\lambda )+\\varepsilon )].$ The PDE in Theorem REF was proved in [13] under the assumption that $x_0$ is a bounded self-adjoint random variable in ${A}$ .", "In this section, we extend Theorem REF to arbitrary self-adjoint random variable $x_0\\in {A}^\\Delta $ .", "Write $\\lambda = u+iv$ .", "Then $S$ satisfies the PDE $\\frac{\\partial S}{\\partial t} = \\varepsilon \\left(\\frac{\\partial S}{\\partial \\varepsilon }\\right)^2+\\frac{1}{4}\\left(\\left(\\frac{\\partial S}{\\partial u}\\right)^2-\\left(\\frac{\\partial S}{\\partial v}\\right)^2\\right)$ with initial condition $S(0,\\lambda ,\\varepsilon ) = \\int \\log ((x-\\lambda )^(x-\\lambda )+\\varepsilon )\\,d\\mu _{x_0}(x).$ Since $x_0\\in {A}^\\Delta $ is self-adjoint, it has a spectral decomposition $x_0 = \\int x\\,dE(x)$ where $E$ is the spectral measure of $x_0$ .", "In this section, we denote $p_n =E([-n,n])$ as a spectral projection of $x_0$ , so that $x_0p_n$ is a bounded random variable.", "We first prove that $S$ is the pointwise limit of a functional on a sequene of bounded random variables at $t=0$ .", "For any $\\lambda \\in \\mathbb {C}$ , and $\\varepsilon >0$ , we have $S(0,\\lambda ,\\varepsilon ) = \\lim _{n\\rightarrow \\infty }\\tau [\\log ((x_0p_n-\\lambda )^(x_0p_n-\\lambda )+\\varepsilon )].$ The convergence is uniform for $(\\lambda ,\\varepsilon )$ in every compact set in $\\mathbb {C}\\times (0,\\infty )$ .", "First, $x_0p_n$ is self-adjoint because $x_0$ and $p_n$ commute, and the spectral decomposition of $x_0p_n$ is $x_0p_n = \\int _{[-n,n]}x\\,dE(x)$ where the spectral measure $E$ is as in (REF ).", "So, we have $\\tau [\\log ((x_0p_n-\\bar{\\lambda })(x_0p_n-\\lambda )+\\varepsilon )] = \\int _{[-n,n]} \\log (\|x-\\lambda \|^2+\\varepsilon )\\,d\\mu _{x_0}(x).$ Fix a compact set $K\\subset \\mathbb {C}\\times (0,\\infty )$ .", "Choose $n_0$ such that $\|x-\\lambda \|^2+\\varepsilon >1$ for all $\|x\|>n_0$ whenever $(\\lambda ,\\varepsilon )\\in K$ .", "Then $\\lim _{n\\rightarrow \\infty }\\int _{n\\ge \|x\|>n_0} \\log (\|x-\\lambda \|^2+\\varepsilon )\\,d\\mu _{x_0}(x) = \\int _{\|x\|>n_0} \\log (\|x-\\lambda \|^2+\\varepsilon )\\,d\\mu _{x_0}(x)$ by the monotone convergence theorem, and $&\\quad \\lim _{n\\rightarrow \\infty }\\tau [\\log ((x_0p_n-\\lambda )^(x_0p_n-\\lambda )+\\varepsilon )] \\\\&= \\int _{[-n_0,n_0]} \\log (\|x-\\lambda \|^2+\\varepsilon )\\,d\\mu _{x_0}(x)+\\lim _{n\\rightarrow \\infty }\\int _{n\\ge \|x\|>n_0} \\log (\|x-\\lambda \|^2+\\varepsilon )\\,d\\mu _{x_0}(x)\\\\&= \\int _\\mathbb {R}\\log (\|x-\\lambda \|^2+\\varepsilon )\\,d\\mu _{x_0}(x)\\\\&=\\tau [\\log (\|x_0-\\lambda \|^2+\\varepsilon )],$ completing the proof.", "We try to compute the partial derivatives of $S$ through several functionals of $x_0p_n$ .", "We first introduce some notations.", "We will use the following notations throughout this section.", "Recall that $x_t = x_0+i\\sigma _t$ .", "For any $t\\ge 0$ and $\\lambda \\in \\mathbb {C}$ , we write $x_{n, t,\\lambda } = x_0p_n+i\\sigma _t-\\lambda $ and $x_{t,\\lambda } = x_t-\\lambda $ .", "Define $S_n(t,\\lambda ,\\varepsilon ) = \\tau [\\log (x_{n, t,\\lambda }^x_{n, t,\\lambda }+\\varepsilon )],\\quad t\\ge 0, \\lambda \\in \\mathbb {C}, \\varepsilon >0.$ Define the sequence of functions $f_n(t,\\lambda ,\\varepsilon ) &= \\tau [(x_{n,t,\\lambda }^x_{n,t,\\lambda }+\\varepsilon )^{-1}]\\\\g_n(t,\\lambda ,\\varepsilon ) &= \\tau [x_{n,t,\\lambda }(x_{n,t,\\lambda }^x_{n,t,\\lambda }+\\varepsilon )^{-1}]\\\\h_n(t,\\lambda ,\\varepsilon ) &= \\tau [x_{n,t,\\lambda }^(x_{n,t,\\lambda }^x_{n,t,\\lambda }+\\varepsilon )^{-1}]$ for $t\\ge 0, \\lambda \\in \\mathbb {C}, \\varepsilon >0$ .", "We also define the functions $f(t,\\lambda ,\\varepsilon ) &= \\tau [(x_{t,\\lambda }^x_{t,\\lambda }+\\varepsilon )^{-1}]\\\\g(t,\\lambda ,\\varepsilon ) &= \\tau [x_{t,\\lambda }(x_{t,\\lambda }^x_{t,\\lambda }+\\varepsilon )^{-1}]\\\\h(t,\\lambda ,\\varepsilon ) &= \\tau [x_{t,\\lambda }^(x_{t,\\lambda }^x_{t,\\lambda }+\\varepsilon )^{-1}]$ for $t\\ge 0$ , $\\lambda \\in \\mathbb {C}$ and $\\varepsilon >0$ .", "We will see that $f_n$ and $f$ will be used to compute the $\\varepsilon $ -derivative of $S$ ; the pairs $g_n$ , $g$ and $h_n$ , $h$ will be used to compute the $\\bar{\\lambda }$ - and $\\lambda $ - derivatives of $S$ respectively; furthermore, we will show that the function $S$ is well-approximated by a subsequence of $S_n$ .", "In the following, we first establish that $f$ , $g$ , and $h$ are the limits of some subsequences $f_{n_k}$ , $g_{n_k}$ , and $h_{n_k}$ respectively.", "For any subsequence $f_{n_k}$ and any compact set $K\\subset [0,\\infty )\\times \\mathbb {C}\\times (0,\\infty )$ , we can further extract a subsequence, still labeled by $f_{n_k}$ , such that $\\lim _{k\\rightarrow \\infty }f_{n_k}(t,\\lambda ,\\varepsilon ) =f(t,\\lambda ,\\varepsilon );$ the convergence is uniform for $(t,\\lambda ,\\varepsilon )\\in K$ .", "We only prove for the full sequence $f_n$ for simpler notations.", "Let $K$ be a compact set in $[0,\\infty )\\times \\mathbb {C}\\times (0,\\infty )$ .", "The function $\\varphi _n(t,\\lambda ,\\varepsilon )=\\left\|f_n(t,\\lambda ,\\varepsilon ) - f(t,\\lambda ,\\varepsilon )\\right\|$ is continuous on $K$ .", "There exists $(t_n,\\lambda _n,\\varepsilon _n)\\in K$ such that $\\varphi _n(t_n,\\lambda _n,\\varepsilon _n) = \\max _{K}\\varphi _n(t,\\lambda ,\\varepsilon ).$ By replacing $f_n$ by a subsequence of $f_n$ if necessary, we may assume $(t_n,\\lambda _n,\\varepsilon _n)$ converges to a point $(t_0,\\lambda _0,\\varepsilon _0)\\in K$ .", "The operators $(x_{n,t_n,\\lambda _n}^x_{n,t_n,\\lambda _n}+\\varepsilon _n)^{-1}$ and $(x_{t_0,\\lambda _0}^x_{t_0,\\lambda _0}+\\varepsilon _0)^{-1}$ are in ${A}$ .", "Also, $(x_{n,t_n,\\lambda _n}^x_{n,t_n,\\lambda _n}+\\varepsilon _n)^{-1}$ converges to $(x_{t_0,\\lambda _0}^x_{t_0,\\lambda _0}+\\varepsilon _0)^{-1}$ in the strong operator topology.", "By the noncommutative Egoroff's theorem (Theorem REF ), given any $\\delta >0$ , there exists a projection $p$ such that $\\tau (1-p)<\\delta $ and there exists a subsequence $(x_{n_k,t_{n_k},\\lambda _{n_k}}^x_{n_k,t_{n_k},\\lambda _{n_k}}+\\varepsilon _{n_k})^{-1}$ such that $\\lim _{k\\rightarrow \\infty }\\Vert \\lbrace x_{n_k,t_{n_k},\\lambda _{n_k}}(x_{n_k,t_{n_k},\\lambda _{n_k}}^x_{n_k,t_{n_k},\\lambda _{n_k}}+\\varepsilon _{n_k})^{-1}-x_{t_0,\\lambda _0}(x_{t_0,\\lambda _0}^x_{t_0,\\lambda _0}+\\varepsilon _0)^{-1}\\rbrace p\\Vert =0,$ so that $\\limsup _{k\\rightarrow \\infty }\\left\|\\tau \\big [\\big \\lbrace (x_{n_k,t_{n_k},\\lambda _{n_k}}^x_{n_k,t_{n_k},\\lambda _{n_k}}+\\varepsilon _{n_k})^{-1}-(x_{t_0,\\lambda _0}^x_{t,\\lambda _0}+\\varepsilon _0)^{-1}\\big \\rbrace p\\big ]\\right\|= 0.$ Hence, by Theorem REF , for any $(t,\\lambda ,\\varepsilon )\\in K$ , $\\limsup _{k\\rightarrow \\infty }\\varphi _n(t,\\lambda ,\\varepsilon )&\\le \\limsup _{k\\rightarrow \\infty }\\varphi _{n_k}(t_{n_k},\\lambda _{n_k},\\varepsilon _{n_k})\\\\&=\\limsup _{k\\rightarrow \\infty }\\left\|\\tau \\big [(x_{n_k,t_{n_k},\\lambda _{n_k}}^x_{n_k,t_{n_k},\\lambda _{n_k}}+\\varepsilon _{n_k})^{-1}-(x_{t_0,\\lambda _0}^x_{t,\\lambda _0}+\\varepsilon _0)^{-1}\\big ] \\right\|\\\\&\\le \\tau (1-p)\\cdot \\left(\\Vert (x_{n_k,t_{n_k},\\lambda _{n_k}}^x_{n_k,t_{n_k},\\lambda _{n_k}}+\\varepsilon _{n_k})^{-1}\\Vert +\\Vert (x_{t_0,\\lambda _0}^x_{t_0,\\lambda _0}+\\varepsilon _0)^{-1}\\Vert \\right)\\\\&\\le \\delta M,$ where $M = \\max _{(t,\\lambda ,\\varepsilon )\\in K}(1/\\varepsilon )<\\infty $ , since $K$ is compact.", "Since $\\delta >0$ is arbitrary, uniform convergence (REF ) for $f_{n_k}$ on $K$ is established.", "Given any compact set $K\\subset [0,\\infty )\\times \\mathbb {C}\\times (0,\\infty )$ and subsequences $g_{n_k}$ and $h_{n_k}$ , we can further extract subsequences, still labeled by $g_{n_k}$ and $h_{n_k}$ (with the same $n_k$ ), such that $\\lim _{k\\rightarrow \\infty }g_{n_k}(t,\\lambda ,\\varepsilon ) = g(t,\\lambda ,\\varepsilon )$ and $\\lim _{k\\rightarrow \\infty }h_{n_k}(t,\\lambda ,\\varepsilon ) = h(t,\\lambda ,\\varepsilon ),$ where the convergeces are uniform for $\\lambda \\in K$ .", "The indices $n_k$ of the subsequence can be chosen to be the same for Lemmas REF and REF , by applying Lemma REF to $h_{n_k}$ and $g_{n_k}$ , where $n_k$ is chosen from Lemma REF for $f_n$ .", "Then we can get the same indices $n_{k_j}$ for the sequences of $f_n$ , $g_n$ , and $h_n$ such that the conclusions of Lemmas REF and REF hold.", "We only prove for the full sequences $g_n$ and $h_n$ , for simpler notations.", "Let $K$ be a compact set in $[0,\\infty )\\times \\mathbb {C}\\times (0,\\infty )$ .", "The function $\\phi _n(t,\\lambda ,\\varepsilon )=\\left\|g_n(t,\\lambda ,\\varepsilon ) - g(t,\\lambda ,\\varepsilon )\\right\|$ is continuous on $K$ .", "There exists $(t_n,\\lambda _n,\\varepsilon _n)\\in K$ such that $\\phi _n(t_n,\\lambda _n,\\varepsilon _n) = \\max _{K}\\phi _n(t,\\lambda ,\\varepsilon ).$ By replacing $g_n$ by a subsequence of $g_n$ if necessary, we may assume $(t_n,\\lambda _n,\\varepsilon _n)$ converges to a point $(t_0,\\lambda _0,\\varepsilon _0)\\in K$ .", "The operators $x_{n,t_n,\\lambda _n}(x_{n,t_n,\\lambda _n}^x_{n,t_0,\\lambda _n}+\\varepsilon _n)^{-1}$ and $x_{t_0,\\lambda _0}(x_{t_0,\\lambda _0}^x_{t,\\lambda _0}+\\varepsilon _0)^{-1}$ are in ${A}$ ; furthermore, the sequence $x_{n,t_n,\\lambda _n}(x_{n,t_n,\\lambda _n}^x_{n,t_n,\\lambda _n}+\\varepsilon _n)^{-1}$ converges to $x_{t_0,\\lambda _0}(x_{t_0,\\lambda _0}^x_{t_0,\\lambda _0}+\\varepsilon _0)^{-1}$ in the strong operator topology.", "By the noncommutative Egoroff's theorem (Theorem REF ), given any $\\delta >0$ , there exists a projection $p$ such that $\\tau (1-p)<\\delta $ and there exists a subsequence $x_{n_k,t_{n_k},\\lambda _{n_k}}(x_{n_k,t_{n_k},\\lambda _{n_k}}^x_{n_k,t_{n_k},\\lambda _{n_k}}+\\varepsilon _{n_k})^{-1}$ such that $\\lim _{k\\rightarrow \\infty }\\Vert \\lbrace x_{n_k,t_{n_k},\\lambda _{n_k}}(x_{n_k,t_{n_k},\\lambda _{n_k}}^x_{n_k,t,\\lambda _{n_k}}+\\varepsilon _{n_k})^{-1}-x_{t_0,\\lambda _0}(x_{t_0,\\lambda _0}^x_{t_0,\\lambda _0}+\\varepsilon _0)^{-1}\\rbrace p\\Vert =0.$ Hence, by Theorem REF , for any $(t,\\lambda ,\\varepsilon )\\in K$ , $\\limsup _{k\\rightarrow \\infty }\\phi _n(t,\\lambda ,\\varepsilon )&\\le \\limsup _{k\\rightarrow \\infty }\\phi _{n_k}(t_{n_k},\\lambda _{n_k},\\varepsilon _{n_k})\\\\&\\le \\limsup _{k\\rightarrow \\infty }\\left\|\\tau \\big [\\lbrace x_{n_k,t_{n_k},\\lambda _{n_k}}(x_{n_k,t_{n_k},\\lambda _{n_k}}^x_{n_k,t_{n_k},\\lambda _{n_k}}+\\varepsilon _{n_k})^{-1}-x_{t_0,\\lambda _0}(x_{t_0,\\lambda _0}^x_{t_0,\\lambda _0}+\\varepsilon _0)^{-1}\\rbrace (1-p)\\big ] \\right\|\\\\&\\le \\tau (1-p)\\cdot (\\Vert x_{n_k,t,\\lambda }(x_{n_k,t_{n_k},\\lambda _{n_k}}^x_{n_k,t_{n_k},\\lambda _{n_k}}+\\varepsilon _{n_k})^{-1}\\Vert +\\Vert x_{t_0,\\lambda _0}(x_{t_0,\\lambda _0}^x_{t_0,\\lambda _0}+\\varepsilon _0)^{-1}\\Vert )\\\\&\\le \\delta M,$ where $M = \\max _{(t,\\lambda ,\\varepsilon )\\in K}(1/\\sqrt{\\varepsilon })<\\infty $ , since $K$ is compact.", "Since $\\delta >0$ is arbitrary, uniform convergence (REF ) for $g_{n_k}$ on $K$ holds.", "The proof for $h_n$ is the similar.", "To obtain the same $n_k$ for $g_n$ and $h_n$ , we prove for $h_{n_k}$ instead of $h_n$ , and relabel the subsequences of $g_{n_k}$ and $h_{n_k}$ again by $g_{n_k}$ and $h_{n_k}$ .", "The following elementary theorem about uniform convergence of derivatives will be useful to analyze the function $S$ .", "A proof of the theorem can be found in Theorem 7.17 of [20].", "[[20]] If $u_n$ is a sequence of differentiable functions on $[a,b]$ such that $\\lim _{n\\rightarrow \\infty } u_n(x_0)$ exists (in $\\mathbb {R}$ ) for some $x_0\\in [a,b]$ and the sequence $u_n^{\\prime }$ converges uniformly on $[a,b]$ , then $u_n$ converges uniformly to a function $u$ on $[a,b]$ , and $u^{\\prime }(x) = \\lim _{n\\rightarrow \\infty } u_n^{\\prime }(x)$ for $x\\in [a,b]$ .", "We have $S(t,\\lambda ,\\varepsilon ) = \\lim _{k\\rightarrow \\infty }S_{n_k}(t,\\lambda ,\\varepsilon )$ for some subsequence $S_{n_k}$ .", "Furthermore, the convergence is uniform for $\\lambda $ in any compact set $K\\subset \\mathbb {C}$ and $\\varepsilon $ in any compact set $I\\subset (0,\\infty )$ ; the subsequences $f_{n_k}$ , $g_{n_k}$ and $h_{n_k}$ converge uniformly on $[0,t]\\times K\\times I$ .", "Note that $x_0p_n\\in {A}$ .", "We can compute the partial derivatives of $S_n$ by applying Lemma 1.1 in Brown's paper [7], where Brown proves that the derivative of the trace of the logarithm is given by $\\frac{d}{du}\\tau [\\log f(u)]=\\tau \\left[f(u)^{-1}\\frac{df}{du}\\right].$ Recall that $f_n$ and $g_n$ are defined in Notation REF .", "We must have $\\frac{\\partial S_n}{\\partial \\lambda }&=-\\tau [x_{n,t,\\lambda }^(x_{n,t,\\lambda }^x_{n,t,\\lambda }+\\varepsilon )^{-1}] = -h_n(t,\\lambda ,\\varepsilon )\\\\\\frac{\\partial S_n}{\\partial \\bar{\\lambda }}&=-\\tau [x_{n,t,\\lambda }(x_{n,t,\\lambda }^x_{n,t,\\lambda }+\\varepsilon )^{-1}] = -g_n(t,\\lambda ,\\varepsilon )\\\\\\frac{\\partial S_n}{\\partial \\varepsilon }&=-\\tau [(x_{n,t,\\lambda }^x_{n,t,\\lambda }+\\varepsilon )^{-1}] = f_n(t,\\lambda ,\\varepsilon ).$ By Theorem REF , we have $\\frac{\\partial S_n}{\\partial t}(s,\\lambda ,\\varepsilon ) = \\varepsilon \\,f_n(s,\\lambda ,\\varepsilon )^2+\\mathrm {Re}\\left[h_n(s,\\lambda ,\\varepsilon )^2\\right].$ By Lemmas REF and REF , there exists a subsequence $S_{n_k}$ such that $\\frac{\\partial S_{n_k}}{\\partial t}$ converges uniformly on $[0,t]\\times K\\times I$ .", "Also, $S_n(0,\\lambda ,\\varepsilon )$ converges to $S(0,\\lambda ,\\varepsilon )$ uniformly for $(\\lambda ,\\varepsilon )$ in any compact set as $n\\rightarrow \\infty $ (Lemma REF ); we must have $S_{n_k}(s,\\lambda ,\\varepsilon )$ converges uniformly to $S(s,\\lambda ,\\varepsilon )$ for $(s,\\lambda ,\\varepsilon )\\in [0,t]\\times K\\times I$ , by Theorem REF .", "Now, we are ready to compute the partial derivatives of $S$ .", "The partial derivatives of $S$ with respect to $\\lambda $ , $\\bar{\\lambda }$ and $\\varepsilon $ are given by $\\frac{\\partial S}{\\partial \\lambda }&=-\\tau [x_{t,\\lambda }^(x_{t,\\lambda }^x_{t,\\lambda }+\\varepsilon )^{-1}]\\\\\\frac{\\partial S}{\\partial \\bar{\\lambda }}&=-\\tau [x_{t,\\lambda }(x_{t,\\lambda }^x_{t,\\lambda }+\\varepsilon )^{-1}]\\\\\\frac{\\partial S}{\\partial \\varepsilon }&=\\tau [(x_{t,\\lambda }^x_{t,\\lambda }+\\varepsilon )^{-1}].$ As in the proof of Proposition REF , we must have $\\frac{\\partial S_n}{\\partial \\lambda }&=-\\tau [x_{n,t,\\lambda }^(x_{n,t,\\lambda }^x_{n,t,\\lambda }+\\varepsilon )^{-1}] =- h_n(t,\\lambda ,\\varepsilon )\\\\\\frac{\\partial S_n}{\\partial \\bar{\\lambda }}&=-\\tau [x_{n,t,\\lambda }(x_{n,t,\\lambda }^x_{n,t,\\lambda }+\\varepsilon )^{-1}] = -g_n(t,\\lambda ,\\varepsilon )\\\\\\frac{\\partial S_n}{\\partial \\varepsilon }&=-\\tau [(x_{n,t,\\lambda }^x_{n,t,\\lambda }+\\varepsilon )^{-1}] = f_n(t,\\lambda ,\\varepsilon ).$ Fix $\\lambda \\in \\mathbb {C}$ and $\\varepsilon >0$ .", "Let $K$ be a compact set in $\\mathbb {C}\\times (0,\\infty )$ such that $(\\lambda ,\\varepsilon )$ is in the interior of $K$ .", "By Proposition REF , there exist subsequences such that $f_{n_k}$ , $g_{n_k}$ and $h_{n_k}$ (with the same $n_k$ ) converge uniformly to $f$ , $g$ and $h$ on $[0,t]\\times K$ , in particular, on $\\lbrace t\\rbrace \\times K$ .", "Since $f_{n_k}$ , $g_{n_k}$ and $h_{n_k}$ are $-\\partial S_{n_k}/\\partial \\varepsilon $ , $-\\partial S_{n_k}/\\partial \\bar{\\lambda }$ , and $-\\partial S_{n_k}/\\partial \\lambda $ respectively, the partial derivatives of $S_{n_k}(t, \\cdot , \\cdot )$ converges uniformly.", "Since Proposition REF also concludes $S_{n_k}$ converges to $S$ uniformly at $(t,\\lambda ,\\varepsilon )$ for $(\\lambda ,\\varepsilon )\\in K$ .", "By Theorem REF , the partial derivatives of $S_{n_k}$ converge to the partial derivatives of $S$ ; that is, $-f$ , $-g$ , and $-h$ are the partial derivatives of $S$ with respect to $\\varepsilon $ , $\\bar{\\lambda }$ , and $\\lambda $ respectively.", "[Proof of Theorem REF ] By (REF ), we have, for any subsequence of $S$ , $\\frac{\\partial S_{n_k}}{\\partial t}(s,\\lambda ,\\varepsilon ) = \\varepsilon \\,f_{n_k}(s,\\lambda ,\\varepsilon )^2+\\mathrm {Re}\\left[h_{n_k}(s,\\lambda ,\\varepsilon )^2\\right].$ If we write $\\lambda = u+iv$ , then, by Propositions REF and REF , $\\frac{\\partial S}{\\partial t} &= \\varepsilon f(t,\\lambda ,\\varepsilon )^2+\\mathrm {Re}[h(t,\\lambda ,\\varepsilon )]\\\\&=\\varepsilon \\left(\\frac{\\partial S}{\\partial \\varepsilon }\\right)^2+\\frac{1}{4}\\left[\\left(\\frac{\\partial S}{\\partial u}\\right)^2 - \\left(\\frac{\\partial S}{\\partial v}\\right)^2\\right],$ which is the desired PDE.", "The statement about the initial condition is straightforward." ], [ "Solving the PDE", "As in [13], we use the Hamilton–Jacobi method to analyze the function $S$ through the PDE in Theorem REF .", "We define the Hamiltonian function $H:\\mathbb {R}^6\\rightarrow \\mathbb {R}$ by replacing the partial derivatives of $S$ in the PDE by the corresponding momentum variables and putting an overall minus sign.", "That is, $H(u,v,\\varepsilon , p_u, p_v, p_\\varepsilon )=-\\frac{1}{4}(p_u^2-p_v^2)-\\varepsilon p_\\varepsilon ^2.$ Then, we introduce the Hamiltonian system (the Hamilton's equations) for $H$ : $\\begin{array}{lll}\\displaystyle \\frac{du}{dt} = \\frac{\\partial H}{\\partial p_u}; & \\displaystyle \\frac{dv}{dt} = \\frac{\\partial H}{\\partial p_v}; & \\displaystyle \\frac{d\\varepsilon }{dt} = \\frac{\\partial H}{\\partial p_\\varepsilon };\\\\[10pt]\\displaystyle \\frac{dp_u}{dt} = -\\frac{\\partial H}{\\partial u}; & \\displaystyle \\frac{dp_v}{dt} = -\\frac{\\partial H}{\\partial v}; & \\displaystyle \\frac{dp_\\varepsilon }{dt} = -\\frac{\\partial H}{\\partial \\varepsilon }.\\end{array}$ We note that the Hamiltonian function and the system of ODEs are the same as the one in [13] because the PDE is the same.", "Write $\\lambda =u+iv$ .", "The initial conditions of $u$ , $v$ , $\\varepsilon $ for the ODEs (REF ) can be chosen arbitrarily, and these initial conditions are denoted by $u_0$ , $v_0$ , $\\varepsilon _0$ .", "The initial conditions of $p_u$ , $p_v$ , $p_\\varepsilon $ are denoted by $p_{u,0}$ , $p_{v,0}$ , and $p_0$ respectively, and are determined by $\\begin{split}p_{u,0} &= \\frac{\\partial }{\\partial u_0}S(0, \\lambda _0, \\varepsilon _0)=\\int \\frac{2(u_0-x)\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v_0^2+\\varepsilon _0};\\\\p_{v,0} &= \\frac{\\partial }{\\partial v_0}S(0, \\lambda _0, \\varepsilon _0)=\\int \\frac{2v_0\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v_0^2+\\varepsilon _0};\\\\p_{0} &= \\frac{\\partial }{\\partial \\varepsilon _0}S(0, \\lambda _0, \\varepsilon _0)=\\int \\frac{d\\mu _{x_0}(x)}{(u_0-x)^2+v_0^2+\\varepsilon _0}.\\end{split}$ We have written $\\lambda _0=u_0+iv_0$ .", "Note that $S$ is defined with $\\varepsilon >0$ .", "The $\\varepsilon _0$ that we choose here is also positive.", "Using the Hamilton–Jacobi method, the solution $S$ of the PDE can be expressed in terms of the initial conditions, as stated in the following proposition.", "For the proof for the proposition, readers are referred to [13].", "For a more general statements, see [8] and [9].", "Suppose that the solution to the Hamiltonian system (REF ) exists on a time interval $[0,T)$ such that $\\varepsilon (t)>0$ for all $t\\in [0,T)$ .", "Then we have, for all $t\\in [0,T)$ , $S(t,\\lambda (t),\\varepsilon (t)) = S(0,\\lambda _0,\\varepsilon _0)+tH_0$ where $H_0=H(u_0,v_0,\\varepsilon _0,p_{u,0}, p_{v,0}, p_0)$ .", "Also, we have $\\frac{\\partial S}{\\partial y}(t,\\lambda (t),\\varepsilon (t)) = p_{y}(t)$ for all $y\\in \\lbrace u, v, \\varepsilon \\rbrace $ and $t\\in [0,T)$ ." ], [ "Solving the ODEs", "The PDE in our case (Theorem REF ) is the same as the one in [13].", "In this section, we briefly summarize the results from [13] about the solutions of the ODEs (REF ).", "The following proposition combines [13] and [13].", "Suppose that the initial conditions $\\lambda _0=u_0+iv_0$ , $\\varepsilon _0$ are chosen, and initial momenta $p_{u,0}$ , $p_{v,0}$ , $p_0$ are determined by (REF ).", "Then the solution of the Hamiltonian system (REF ) exists up to $t_\\ast (\\lambda _0,\\varepsilon _0) = \\frac{1}{p_0} = \\left(\\int \\frac{d\\mu _{x_0}(x)}{(u_0-x)^2+v_0^2+\\varepsilon _0}\\right)^{-1}.$ Up until $t_\\ast (\\lambda _0,\\varepsilon _0)$ , we have $\\begin{array}{ll}\\displaystyle u(t) = u_0-\\frac{t}{2}p_{u,0}; &\\displaystyle p_u(t) = p_{u,0};\\\\[10pt]\\displaystyle v(t) = v_0+\\frac{t}{2}p_{v,0}; &\\displaystyle p_v(t) = p_{v,0};\\\\[10pt]\\displaystyle \\varepsilon (t) = \\varepsilon _0(1-tp_0)^2; & \\displaystyle p_{\\varepsilon }(t) = \\frac{p_0}{1-tp_0}.\\end{array}$ For a fixed $\\lambda _0\\in \\mathbb {C}$ , $t_\\ast (\\lambda _0,\\varepsilon _0)$ defined in (REF ) is strictly decreasing in $\\varepsilon _0$ .", "For each $\\lambda _0\\in \\mathbb {C}$ , the lifetime of the solution path in the limit $\\varepsilon _0\\rightarrow 0^+$ is given by $T(\\lambda _0) = \\lim _{\\varepsilon _0\\rightarrow 0^+} \\left(\\int \\frac{d\\mu _{x_0}(x)}{(u_0-x)^2+v_0^2}\\right)^{-1}.$ By Proposition REF , informally there are two ways to make $\\varepsilon (t) = 0$ .", "The first one is to take $\\varepsilon _0 = 0$ ; the second one is to choose $\\varepsilon _0$ such that $p_0=1/t$ .", "The first scheme works only if the lifetime of the solution path remains greater than $t$ in the limit $\\varepsilon _0\\rightarrow 0^+$ ; that is, if $T(\\lambda _0)\\ge t$ .", "If the first scheme does not work (that is, if $T(\\lambda _0)< t$ ), one needs to use the second scheme to achieve $\\varepsilon (t) = 0$ .", "We now identify the set of points $\\lambda _0\\in \\mathbb {C}$ such that $T(\\lambda _0)<t$ .", "This set of points plays a crucial role in [13], [16].", "For proof, see [13].", "The set $\\Lambda _{x_0,t} =\\lbrace \\lambda _0\\in \\mathbb {C}\\left\|T(\\lambda _0)<t\\right.\\rbrace $ can be identified as $\\Lambda _{x_0,t}=\\left.\\lbrace u_0+iv_0\\in \\mathbb {C}\\right\|\\left\|v_0\\right\|<v_{x_0,t}(u_0)\\rbrace $ where $v_{x_0,t}$ is defined in Definition REF (2).", "In particular, $\\Lambda _{x_0,t}\\cap \\mathbb {R}= \\left.\\lbrace u_0\\in \\mathbb {R}\\right\|v_{x_0,t}(u_0)>0\\rbrace $ .", "Furthermore, we have $\\mathrm {supp}\\,\\mu _{x_0}\\subset \\overline{\\Lambda _{x_0,t}}$ .", "For notational convenience, we also define $\\begin{split}\\varepsilon _0^t(u_0+iv_0) &= v_{x_0,t}(u_0)^2-v_0^2\\\\p_1 &= \\int \\frac{x\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v_0^2+\\varepsilon _0}.\\end{split}$ Note that $\\varepsilon _0^t(\\lambda _0)>0$ for $\\lambda _0\\in \\Lambda _{x_0,t}$ and $\\varepsilon _0^t(\\lambda _0)=0$ for $\\lambda _0\\in \\partial \\Lambda _{x_0,t}$ .", "Then we have the following solution at time $t$ .", "For initial conditions $\\lambda _0\\in \\mathbb {C}$ and $\\varepsilon _0>0$ , denote by $\\lambda (t;\\lambda _0,\\varepsilon _0)$ and $\\varepsilon (t;\\lambda _0,\\varepsilon _0)$ the solution to system (REF ) at time $t$ .", "If $\\lambda _0\\in \\Lambda _{x_0,t}^c$ , then $t_\\ast (\\lambda _0,\\varepsilon _0)>t$ for all $\\varepsilon _0>0$ , and $\\lim _{\\varepsilon _0\\rightarrow 0^+}\\lambda (t;\\lambda _0,\\varepsilon _0) &= \\lambda _0-t\\int \\frac{d\\mu _{x_0}(x)}{\\lambda _0-x}\\\\\\lim _{\\varepsilon _0\\rightarrow 0^+}\\varepsilon (t;\\lambda _0,\\varepsilon _0)&=0.$ If $\\lambda _0\\in \\Lambda _{x_0,t}$ , then $\\varepsilon _0^t(\\lambda _0)>0$ and $t_\\ast (\\lambda _0,\\varepsilon _0^t(\\lambda _0))=t$ .", "Furthermore, we have $\\lim _{\\varepsilon _0\\rightarrow \\varepsilon _0^t(\\lambda _0)^+} \\lambda (t;\\lambda _0,\\varepsilon _0)& = tp_1+2iv_0\\\\\\lim _{\\varepsilon _0\\rightarrow \\varepsilon _0^t(\\lambda _0)^+} \\varepsilon (t;\\lambda _0,\\varepsilon _0) &= 0.$ Points 1 and 2 are Propositions 4.7 and 4.5 in [13] respectively.", "The choice $\\varepsilon _0^t(\\lambda _0)$ can be found in Section 7.3 of [13], right before Lemma 7.7." ], [ "The domain", "Define the holomorphic map $H_{x_0,-t}(z) = z - t G_{x_0}(\\lambda _0),\\quad \\lambda _0\\notin \\mathrm {supp}\\,\\mu _{x_0}$ and $\\Omega _{x_0,t} = [H_{x_0,-t}(\\Lambda _t^c)]^c.$ We note that the notation $H_{x_0,-t}$ is consistent with the one in Definition REF (1).", "We will show in the later section that, as in [13], the Brown measure of $x_0+i\\sigma _t$ has mass 1 on the open set $\\Omega _{x_0,t}$ .", "Most of the statements in the following theorem follow the same proof in [13].", "We only supply the proof for the statements that requires a new proof.", "The following statements hold.", "The function $H_{x_0,-t}$ is continuous and injective on $\\overline{\\Gamma }_t$ .", "(See Definition REF for the definition of $\\Gamma _t$ .)", "Define the function $f_{x_0,t}:\\mathbb {R}\\rightarrow \\mathbb {R}$ by $f_{x_0,t}(u_0) = \\mathrm {Re}[H_{x_0,-t}(u_0+iv_{x_0,t}(u_0))].$ Then at any point $u_0$ such that $v_{x_0,t}(u_0)>0$ , the function $f_{x_0,t}$ is differentiable and $0<\\frac{df_{x_0,t}}{du_0}<2.$ The function $f_{x_0,t}$ is continuous and strictly increasing.", "It maps $\\mathbb {R}$ onto $\\mathbb {R}$ .", "In particular, the inverse $f_{x_0,t}^{-1}$ of $f_{x_0,t}$ exists.", "Define the function $\\varphi _{x_0,t}(u) = 2v_{x_0,t}(f_{x_0,t}^{-1}(u)),\\quad u\\in \\mathbb {R}.$ Then $\\varphi _{x_0,t}(u)\\rightarrow 0$ as $\|u\|\\rightarrow \\infty $ and the map $H_{x_0,-t}$ maps the graph of $v_{x_0,t}$ to the graph of $\\varphi _{x_0,t}$ .", "The map $H_{x_0,-t}$ takes the region above the graph of $v_{x_0,t}$ onto the region above the graph of $\\varphi _{x_0,t}$ .", "The set $\\Omega _{x_0,t}$ can be identified as $\\Omega _{x_0,t} = \\left.\\lbrace u+iv\\in \\mathbb {C}\\right\|\\left\|v\\right\|< \\varphi _{x_0,t}(u)\\rbrace .$ We only need to prove that $f_{x_0,t}$ maps $\\mathbb {R}$ onto $\\mathbb {R}$ , $H_{x_0,-t}$ takes the region above the graph of $v_{x_0,t}$ onto the region above the graph of $\\varphi _{x_0,t}$ , and $\\varphi _{x_0,t}(u)\\rightarrow 0$ as $\|u\|\\rightarrow \\infty $ .", "All the other parts follow the same proof as in [13].", "Note that, since $f_{x_0,t}$ is continuous and strictly increasing, by the same argument in [13], $H_{x_0,-t}$ maps the graph of $v_{x_0,t}$ to the graph of a function $\\varphi _{x_0,t}$ defined on the range of $f_{x_0,t}$ using the same formula in (REF ) (except that we define $\\varphi _{x_0,t}$ only on the range of $f_{x_0,t}$ instead of $\\mathbb {R}$ ).", "We first show that $v_{x_0,t}(u_0)\\rightarrow 0$ as $\|u_0\|\\rightarrow \\infty $ .", "Suppose, on the contrary, that there exist $\\delta >0$ and a sequence $u_n\\in \\mathbb {R}$ , such that $\|u_n\|\\rightarrow \\infty $ but $v_{x_0,t}(u_n)>\\delta $ for all $n$ .", "For all of these $u_n$ , since $v_{x_0,t}(u_n)>\\delta >0$ , we must have $\\int \\frac{d\\mu _{x_0}(x)}{(u_n-x)^2+\\delta ^2}\\ge \\int \\frac{d\\mu _{x_0}(x)}{(u_n-x)^2+v_{x_0,t}(u_n)^2}=\\frac{1}{t}.$ Letting $n\\rightarrow \\infty $ , by dominated convergence theorem, we have $0\\ge 1/t>0$ , which is a contradiction.", "Also, we have $\\lim _{y\\rightarrow \\infty }H_{x_0,-t}(iy) = \\infty .$ Let $a = \\lim _{u_0\\rightarrow -\\infty }f_{x_0,t}(u_0)$ and $b=\\lim _{u_0\\rightarrow \\infty }f_{x_0,t}(u_0)$ .", "Then $\\varphi _{x_0,t}$ can be defined on $(a,b)$ using the formula in (REF ).", "The map $H_{x_0,-t}$ takes the graph of $v_{x_0,t}$ to the graph of $\\varphi _{x_0,t}$ .", "The region above the graph of $v_{x_0,t}$ is bounded by the Jordan curve formed by the graph of $v_{x_0,t}$ and the point $\\infty $ on the Riemann sphere.", "Furthermore, the map $H_{x_0,-t}$ is an injective conformal map on the region above the graph of $v_{x_0,t}$ and is extended to a homoemorphism on the closed region bounded by the graph of $v_{x_0,t}$ and the point $\\infty $ on the Riemann sphere.", "Since $\\lim _{y\\rightarrow \\infty }H_{x_0,-t}(iy) = \\infty $ , the point $\\infty $ is on the boundary of the image of the region above the graph of $v_{x_0,t}$ under $H_{x_0,-t}$ .", "Thus, the graph of $\\varphi _{x_0,t}$ and the point $\\infty $ form the boundary of the image of the region above $v_{x_0,t}$ under the map $H_{x_0,-t}$ .", "If $a\\ne -\\infty $ or $b\\ne \\infty $ , by the preceding paragraph, $\\varphi _{x_0,t}(u)\\rightarrow 0$ as $u\\rightarrow a^+$ or $u\\rightarrow b^-$ .", "In this case, the union of the graph of $\\varphi _{x_0,t}$ and any point on the Riemann sphere does not enclose a region in the Riemann sphere.", "This contradicts the preceding paragraph that the graph of $\\varphi _{x_0,t}$ and the point $\\infty $ form the boundary of the image of $H_{x_0,-t}$ .", "Thus, $f_{x_0,t}(u_0)\\rightarrow \\pm \\infty $ as $u_0\\rightarrow \\pm \\infty $ .", "Finally, $\\varphi _{x_0,t}(u)\\rightarrow 0$ as $\|u\|\\rightarrow \\infty $ follows from the fact that $f_{x_0,t}^{-1}(u)\\rightarrow \\pm \\infty $ as $u\\rightarrow \\infty $ , and that $v_{x_0,t}(u_0)\\rightarrow 0$ as $\|u_0\|\\rightarrow \\infty $ ." ], [ "Surjectivity", "We can write $f_{x_0,t}$ into the form $f_{x_0,t}(u_0)=\\mathrm {Re}[H_{x_0,-t}(u_0+iv_{x_0,t}(u_0)] = t\\int \\frac{x\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v_{x_0,t}(u_0)^2}.$ Denote by $\\lambda (t;\\lambda _0,\\varepsilon _0)$ and $\\varepsilon (t;\\lambda _0,\\varepsilon _0)$ be the solution of (REF ), with initial conditions $\\lambda _0, \\varepsilon _0$ .", "By (REF ) and Proposition REF , for any $\\lambda _0=u_0+iv_0\\in \\Lambda _{x_0,t}$ , $\\lim _{\\varepsilon _0\\rightarrow \\varepsilon _0^t(\\lambda _0)^+} \\lambda (t;\\lambda _0,\\varepsilon _0)= f_{x_0,t}(u_0)+2iv_0,$ which is in $\\Omega _{x_0,t}$ since $2\|v_0\|< \\varphi _{x_0,t}(f_{x_0,t}(u_0))$ , by Point 4 of Theorem REF .", "The following theorem shows that every point in $\\Omega _{x_0,t}$ can be obtained from $\\Lambda _{x_0,t}$ through the formula (REF ).", "The proof in [13] still works for our case.", "Let $U_t(u_0+iv_0) = f_{x_0,t}(u_0)+2iv_0, \\quad u_0+iv_0\\in \\Lambda _{x_0,t}$ which is the same as the formula (REF ).", "Then we have the following results.", "The map $U_t$ extends continuously to $\\overline{\\Lambda }_{x_0,t}$ .", "This extension is the unique continuous map of $\\overline{\\Lambda }_{x_0,t}$ into $\\overline{\\Omega }_{x_0,t}$ that agrees with $H_{x_0,-t}$ on $\\partial \\Lambda _{x_0,t}$ and maps each vertical segment in $\\overline{\\Lambda }_{x_0,t}$ linearly to a vertical segment in $\\overline{\\Omega }_{x_0,t}$ .", "The map $U_t$ is a homeomorphism from $\\Lambda _{x_0,t}$ onto $\\Omega _{x_0,t}$ .", "If $x_0\\in {A}^\\Delta $ is an unbounded variable, the support of the law $\\mu _{x_0}$ of $x_0$ is unbounded.", "The support of $\\mu _{x_0}$ can be bounded from below but unbounded from above, bounded from above but unbounded from below, or unbounded from above and below.", "The following proposition gives bounds on the real parts of points in $\\Omega _{x_0,t}$ .", "Let $M = \\sup \\mathrm {supp}(\\mu )$ and $m =\\inf \\mathrm {supp}(\\mu )$ .", "Then every point $\\lambda \\in \\overline{\\Omega }_{x_0,t}$ satisfies $m<\\mathrm {Re}\\lambda <M.$ The proof of [13] works for this proposition.", "The difference from [13] is that $M$ and $m$ may not be both finite — could be one infinite and one finite, or both infinite." ], [ "The Brown measure computation", "In this section, we compute the Brown measure of $x_0+i\\sigma _t$ by taking the Laplacian of $S$ in $\\Omega _{x_0,t}$ .", "By the definition of Brown measure (See Section REF ), the Laplacian is in distributional sense.", "The following regularity result tells us that we can indeed take the ordinary Laplacian to compute the density of the Brown measure.", "Define the function $\\tilde{S}$ by $\\tilde{S}(t,\\lambda ,z) = S(t,\\lambda ,z^2)$ for $z>0$ .", "Then for any $t>0$ and $\\lambda ^\\in \\Omega _{x_0,t}$ , the function $(\\lambda ,z)\\mapsto \\tilde{S}(t,\\lambda ,z)$ extends to a real analytic function in a neighborhood of $(\\lambda ^,0)$ in $\\mathbb {C}\\times \\mathbb {R}$ .", "Let $s_t(\\lambda ) = \\lim _{\\varepsilon \\rightarrow 0^+}S(t,\\lambda ,\\varepsilon ) = \\lim _{\\varepsilon \\rightarrow 0^+}\\tilde{S}(t,\\lambda ,\\sqrt{\\varepsilon })$ where $\\tilde{S}$ is defined in (REF ).", "The computation of the Laplacian $s_t$ is the same as in [13].", "Since the computation of the Laplacian is crucial for the computation of the Brown measure, for completeness, we also include the computation of the Laplacian here.", "Write $\\lambda (t;\\lambda _0,\\varepsilon _0)$ be the solution of the system (REF ) with initial conditions $\\lambda _0$ and $\\varepsilon _0$ .", "By (REF ) and Theorem REF , for any $\\lambda =u+iv\\in \\Lambda _{x_0,t}$ , $\\lambda _0 =u_0+iv_0 = U_t^{-1}(\\lambda )$ satisfies $\\lim _{\\varepsilon _0\\rightarrow \\lambda _0^t(\\lambda _0)^+}\\lambda (t;\\lambda _0,\\varepsilon _0) = \\lambda .$ Then, by Propositions REF and REF , for any $\\lambda =u+iv\\in \\Omega _{x_0,t}$ , $\\frac{\\partial }{\\partial u}s_t(\\lambda ) = \\lim _{\\varepsilon \\rightarrow 0^+} \\frac{\\partial }{\\partial u}\\tilde{S}(t,\\lambda , \\varepsilon )=\\lim _{\\varepsilon _0\\rightarrow \\varepsilon _0^t(\\lambda _0)^+}p_{u}(t) = \\lim _{\\varepsilon _0\\rightarrow \\varepsilon _0^t(\\lambda _0)^+}p_{u,0}.$ Using (REF ), (REF ), and Proposition REF (2).", "$\\lim _{\\varepsilon _0\\rightarrow \\varepsilon _0^t(\\lambda _0)^+}p_{u,0} = \\int \\frac{2(u_0-x)\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v_{x_0,t}(u_0)^2}=\\frac{2u_0}{t}-\\frac{2}{t}\\cdot t\\int \\frac{x\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v_{x_0,t}(u_0)^2} = \\frac{2f_{x_0,t}^{-1}(u)}{t}-\\frac{2u}{t}.$ Hence, we have $\\frac{\\partial s_t}{\\partial u} = \\frac{2f_{x_0,t}^{-1}(u)}{t}-\\frac{2u}{t}, \\quad \\lambda =u+iv\\in \\Omega _{x_0,t}.$ Similarly, since $v = 2v_0$ by (REF ), $\\frac{\\partial }{\\partial v}s_t(\\lambda ) = \\lim _{\\varepsilon _0\\rightarrow \\varepsilon _0^t(\\lambda _0)^+}p_{v,0} = \\int \\frac{2v_0\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v_{x_0,t}(u_0)^2} = \\frac{v}{t}, \\quad \\lambda =u+iv\\in \\Omega _{x_0,t}.$ Combining (REF ) and (REF ), we have $\\Delta s_t(\\lambda ) =\\frac{2}{t}\\frac{d}{du}f_{x_0,t}^{-1}(u)-\\frac{1}{t}=\\frac{2}{t}\\left(\\frac{d}{du}f_{x_0,t}^{-1}(u)-\\frac{1}{2}\\right)$ We arrive at the Brown measure of $x_0+i\\sigma _t$ as well as a push-forward result of the Brown measure.", "The open set $\\Omega _{x_0,t}$ is a set of full measure for the Brown measure of $x_0+i\\sigma _t$ .", "The Brown measure is absolutely continuous with a strictly positive density $w_t$ on the open set $\\Omega _{x_0,t}$ by $w_t(\\lambda ) = \\frac{1}{2\\pi t}\\left(\\frac{d}{du}f_{x_0,t}^{-1}(u)-\\frac{1}{2}\\right),\\quad \\lambda =u+iv\\in \\Omega _{x_0,t}.$ In particular, the density $w_t$ is constant along vertical segments inside $\\Omega _{x_0,t}$ .", "Moreover, the push-forward of the Brown measure of $x_0+i\\sigma _t$ by $Q_t(u+iv) = 2f_{x_0,t}^{-1}(u)-u,\\quad u+iv\\in \\Omega _{x_0,t}$ is the law of $x_0+\\sigma _t$ .", "These results follow from the same proof as for Theorems 7.9 and 8.2 of [13] since Biane's Theorem (Theorem REF ) on the free convolution with a semicircular variable holds for unbounded self-adjoint random variable $x_0\\in {A}^\\Delta $ .", "Theorem 8.2 of [13] also establishes a push-forward property of the Brown measure of $x_0+i\\sigma _t$ to the Brown measure of $x_0+c_t$ where $c_t$ is the circular variable with variance $t$ .", "We have not computed the Brown measure of $x_0+c_t$ for unbounded $x_0$ .", "We will establish this analogous push-forward result to $x_0+c_t$ in the next section." ], [ "The Brown measure of $x_0+c_{\\alpha ,\\beta }$", "Recall that an elliptic variable has the form $c_{\\alpha ,\\beta } = \\tilde{\\sigma }_\\alpha +i\\sigma _\\beta $ where $\\tilde{\\sigma }_\\alpha $ and $\\sigma _\\beta $ are free semicircular random variables with variances $\\alpha $ and $\\beta $ respectively.", "Given any self-adjoint $a\\in \\tilde{{A}}$ , let $\\Lambda _{a,s} = \\lbrace \\left.u+iv\\in \\mathbb {C}\\right\|\|v\|<v_{a,s}(u)\\rbrace ,$ where $v_{a,s}$ is defined in Definition REF (2).", "We note that this definition of $\\Lambda _{a,s}$ is consistent with the notation in the previous section by Proposition REF .", "In this section, we study the Brown measure of $x_0+c_{\\alpha ,\\beta }$ , where $x_0\\in {A}^\\Delta $ is self-adjoint, and $\\alpha \\ge 0$ (the case $\\alpha =0$ agrees with the one computed in Theorem REF ).", "The computation in [15] relies only on the computations of holomorphic functions; the compactness of the support of $\\mu _{x_0}$ in [15] does not play a role (since the computation of the holomorphic maps can be restricted to a Stolz angle).", "The Brown measure computation of [15] can be automatically carried over to the unbounded self-adjoint $x_0$ .", "We start by a few notations; these notations are well-defined for $a\\in \\tilde{{A}}$ .", "Later we will apply these notations to $x_0\\in {A}^\\Delta $ since the Brown measure of $x_0+c_{\\alpha ,\\beta }$ is defined only for $x_0\\in {A}^\\Delta $ but not for $x_0\\in \\tilde{{A}}$ .", "More discussions can be found in Proposition 3.6 of [15].", "Given any self-adjoint $a\\in \\tilde{{A}}$ and $r\\in \\mathbb {R}$ .", "Define $H_{a,r}(z) = z + r G_a(z),\\quad z\\notin \\sigma (a).$ This definition is consistent with (REF ), when $r>0$ , and consistent with (REF ), when $r<0$ .", "Furthermore, $H_{a,r}$ is an injective conformal map on $\\Lambda _{a,\|r\|}^c$ .", "Let $a\\in \\tilde{{A}}$ be self-adjoint.", "Also let $\\alpha \\ge 0$ and $\\beta >0$ and write $s=\\alpha +\\beta $ .", "Define $f_{a,\\alpha ,\\beta }:\\mathbb {R}\\rightarrow \\mathbb {R}$ by $f_{a,\\alpha ,\\beta }(u) = \\mathrm {Re}[H_{\\alpha -\\beta }(u+iv_{x_0,s}(u))].$ This function is strictly increasing and is a homeomorphism onto $\\mathbb {R}$ .", "Furthermore, $f_{a,\\alpha ,\\beta }^{\\prime }(u)>0$ for all $u\\in \\Lambda _{a,s}\\cap \\mathbb {R}$ .", "Let $\\alpha > 0$ and $\\beta >0$ and write $s=\\alpha +\\beta $ .", "Define $\\varphi _{x_0,\\alpha ,\\beta }(u) = 2\\beta v_{a,s}(f_{x_0,\\alpha ,\\beta }^{-1}(u))/s$ and $\\Omega _{x_0,\\alpha ,\\beta } = \\left.\\lbrace u+iv\\in \\mathbb {C}\\right\|\|v\|<\\varphi _{x_0,\\alpha ,\\beta }(u)\\rbrace .$ By Definition 3.1 and Proposition 3.6 of [15], $\\Omega _{x_0,\\alpha ,\\beta }=[H_{x_0,\\alpha -\\beta }(\\Lambda _{x_0,s}^c)]^c$ .", "Remark that $\\varphi _{x_0,0,t} = \\varphi _{x_0,t}$ , which is defined in Theorem REF (4).", "We will see that $\\Omega _{x_0,\\alpha ,\\beta }$ is an open set of full measure with respect to the Brown measure of $x_0+c_{\\alpha ,\\beta }$ .", "The author uses a different parametrization for the elliptic variable in [15].", "In [15] the elliptic variable is parametrized using $s$ and $t$ by $\\tilde{\\sigma }_{s-t/2}+i\\sigma _{t/2}$ .", "The parametrization in [15] is motivated by [14].", "In this paper, the parametrization is a linear transformation of the one in [15], by $\\alpha = s-t/2$ and $\\beta = t/2$ .", "If we apply Theorem REF to compute the Brown measure of $x_0+c_{\\alpha ,\\beta }=x_0+\\tilde{\\sigma }_\\alpha +i\\sigma _\\beta $ , the Brown measure is written in terms of the law of $x_0+\\tilde{\\sigma }_\\alpha $ .", "The key to write the Brown measure in terms of $x_0$ is the following proposition in [15].", "[Theorem 3.4 of [15]] Let $a = x_0+\\tilde{\\sigma }_\\alpha $ .", "Then, by writing $f_{x_0,\\alpha ,\\beta }(u_0)=u$ , we have $f_{a,\\beta }^{-1}(u) = u+\\beta \\int \\frac{(u_0-x)\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v_{x_0,s}(u_0)^2},\\quad u\\in \\mathbb {R}.$ We are ready to state the main result in this case.", "The Brown measure of $x_0+c_{\\alpha ,\\beta }$ is absolutely continuous with respect to the Lebesgue measure on the plane.", "The open set $\\Omega _{\\alpha ,\\beta }$ is a set of full measure of the Brown measure.", "The density of the Brown measure on $\\Omega _{\\alpha ,\\beta }$ is given by $w_{\\alpha ,\\beta }(u+iv) = \\frac{1}{4\\pi \\beta }\\left(1+2\\beta \\frac{d}{du}\\int \\frac{(u_0-x)\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v_{x_0,s}(u_0)^2}\\right),\\quad u+iv\\in \\Omega _{\\alpha ,\\beta },$ where $f_{x_0,\\alpha ,\\beta }(u_0)=u$ .", "The density is strictly positive and constant along the vertical segments in $\\Omega _{x_0,\\alpha ,\\beta }$ .", "If $\\alpha = 0$ and $\\beta = t$ , then Proposition implies that $w_{\\alpha ,\\beta }(u+iv) = \\frac{1}{4\\pi t}\\left(1+2\\frac{d}{du}(f_{x_0,t}^{-1}(u)-u)\\right)=\\frac{1}{2\\pi t}\\left(\\frac{d}{du}f_{x_0,\\beta }^{-1}(u)-\\frac{1}{2}\\right),$ giving the density of the Brown measure of $x_0+i\\sigma _t$ .", "When $x_0\\in {A}$ is a bounded self-adjoint random variable, the Brown measure of $x_0+c_t$ is computed in [16].", "The following corollary computes the Brown measure of $x_0+c_t$ for possibly-unbounded self-adjoint $x_0\\in {A}^\\Delta $ .", "The highlights are, as in [16], the Brown measure has full measure on the open set $\\Lambda _{x_0,t}$ , and the density can be written as terms of the derivative of the function $\\psi _{x_0,t}$ in Theorem REF (3).", "The Brown measure of $x_0+c_t$ is absolutely continuous with respect to the Lebesgue measure on the plane.", "The open set $\\Lambda _{x_0,t}$ is a set of full measure of the Brown measure.", "The density on $\\Lambda _{x_0,t}$ has the form $w_{t/2,t/2}(u+iv) =\\frac{1}{2\\pi t}\\frac{d\\psi _{x_0,t}}{du}= \\frac{1}{\\pi t}\\left(1-\\frac{t}{2}\\frac{d}{du}\\int \\frac{x\\,d\\mu _{x_0}(x)}{(u-x)^2+v_{x_0,t}(u)^2}\\right),\\quad u+iv\\in \\Lambda _{x_0,t},$ where $\\psi _{x_0,t}$ is defined in Theorem REF .", "The density is strictly positive and constant along the vertical segments in $\\Lambda _{x_0,t}$ .", "Take $\\alpha = \\beta =t/2$ in Theorem .", "Then, $\\alpha +\\beta =t$ and $2\\beta /(\\alpha +\\beta )=1$ , so that $f_{x_0,\\alpha ,\\beta }$ is the identity map, and $\\Omega _{x_0,t/2,t/2} = \\left.\\lbrace u+iv\\in \\mathbb {C}\\right\|\|v\|<v_{x_0,t}(u)\\rbrace = \\Lambda _{x_0,t}.$ Then we can compute the density using the formula in Theorem $w_{t/2,t/2}(u+iv) &= \\frac{1}{2\\pi t}\\left(1+t\\frac{d}{du}\\int \\frac{(u-x)\\,d\\mu _{x_0}(x)}{(u-x)^2+v_{x_0,t}(u)^2}\\right)\\\\&=\\frac{1}{2\\pi t}\\frac{d\\psi _{x_0,t}}{du},\\quad u+iv\\in \\Lambda _{x_0,t};$ the last equality follows from the definition of $\\psi _{x_0,t}(u) = H_{x_0,t}(u+iv_{x_0,t}(u))$ .", "By Definiton REF (2) of $v_{x_0,t}$ , we have $\\int \\frac{(u-x)\\,d\\mu _{x_0}(x)}{(u-x)^2+v_{x_0,t}(u)^2} = \\frac{u}{t}-\\int \\frac{x\\,d\\mu _{x_0}(x)}{(u-x)^2+v_{x_0,t}(u)^2}.$ The other claimed formula of $w_t$ follows from a straightforward algebraic computation.", "The following corollary establishes the push-forward properties between the Brown measure of $x_0+c_{\\alpha ,\\beta }$ , the Brown measure of $x_0+c_s$ , and the law of $x_0+\\sigma _t$ , where $s=\\alpha +\\beta $ is the total variance of the elliptic variable $c_{\\alpha ,\\beta }$ .", "Write $s=\\alpha +\\beta $ .", "Let $U_{\\alpha ,\\beta }:\\Lambda _{x_0,s}\\rightarrow \\Omega _{x_0,\\alpha ,\\beta }$ be defined by $U_{\\alpha ,\\beta }(u+iv) = f_{x_0,\\alpha ,\\beta }(u)+i\\frac{2\\beta v}{s}.$ Then $U_{\\alpha ,\\beta }$ extends to a homeomorphism from $\\overline{\\Lambda }_{x_0,s}$ to $\\overline{\\Omega }_{\\alpha ,\\beta }$ and agrees with $H_{x_0,\\alpha -\\beta }$ on the boundary of $\\Lambda _{x_0,s}$ .", "Furthermore, the following push-forward properties hold.", "The push-forward of the Brown measure of $x_0+c_s$ under the map $U_{\\alpha ,\\beta }$ is the Brown measure of $x_0+c_{\\alpha ,\\beta }$ .", "The push-forward of the Brown measure of $x_0+c_{\\alpha ,\\beta }$ by the map $Q_{\\alpha ,\\beta }(u+iv) = {\\left\\lbrace \\begin{array}{ll}\\frac{1}{\\alpha -\\beta }[su-2\\beta f_{x_0,\\alpha ,\\beta }^{-1}(u)],\\quad &\\textrm {if \\alpha \\ne \\beta }\\\\H_{x_0,s}(u+iv_{x_0,s}(u)),\\quad &\\textrm {if \\alpha = \\beta =s/2}.\\end{array}\\right.", "}$ is the law of $x_0+\\sigma _s$ .", "Write $u = f_{x_0,\\alpha ,\\beta }(u_0)$ .", "When $\\alpha \\ne \\beta $ , we can compute that $Q_{\\alpha ,\\beta }(u+iv) =u_0+s\\int \\frac{(u_0-x)\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v_{x_0,s}(u_0)^2}=H_{x_0,s}(u_0+iv_{x_0,s}(u)).$ If $\\alpha =\\beta =s/2$ , then $u=u_0$ and the above equation reduces to $H_{x_0,s}(u+iv_{x_0,s}(u))$ , which is the definition of $Q_{\\alpha ,\\beta }$ when $\\alpha =\\beta =s/2$ .", "That $U_{\\alpha ,\\beta }$ extends to a homeomorphism from $\\overline{\\Lambda }_{x_0,s}$ to $\\overline{\\Omega _{\\alpha ,\\beta }}$ and agrees with $H_{x_0,\\alpha -\\beta }$ on the boundary of $\\Lambda _{x_0,s}$ follows from the proof of Proposition 4.2 of [15].", "For $\\alpha \\ne \\beta $ , the proof follows from the one given for Theorem 4.1 of [15].", "For the case $\\alpha =\\beta =s/2$ , the proof follows from the one given for Theorem 3.13 of [16].", "Using the push-forward property in Point 1 of Corollary , we can express the density of the Brown measure of $x_0+c_{\\alpha ,\\beta }$ in terms of the density of the Brown measure of $x_0+c_{s}$ , if $\\alpha +\\beta =s$ .", "For any $r=2\\beta /s$ , by writing $u+iv=U_{\\alpha ,\\beta }(u_0+iv_0)$ for all $u_0+iv_0\\in \\Lambda _{x_0,s}$ , we have $w_{\\alpha ,\\beta }(u+iv) = \\frac{1}{r}\\frac{w_{s/2,s/2}(u_0+iv_0)}{r+2\\pi (1-r)s\\cdot w_{s/2,s/2}(u_0+iv_0)}.$ This follows from a direct computation of integration of substitution using Corollary .", "Interested readers are referred to the proof of Corollary 4.3 of [15].", "Figure: Visualization of the map U α,β U_{\\alpha ,\\beta } with α=1/8\\alpha = 1/8 and β=7/8\\beta =7/8.Figure REF shows a visualization of the map $U_{\\alpha ,\\beta }$ , with $\\mu _{x_0} = \\frac{1}{3}\\delta _{-1}+\\frac{2}{3}\\delta _1$ , $\\alpha = 1/8$ and $\\beta = 7/8$ .", "The blue equally-spaced vertical lines on the left hand side of the figure are mapped to the corresponding vertical lines on the right hand side of the figure by $U_{\\alpha ,\\beta }$ .", "The blue vertical lines on the right hand side are no longer equally-spaced.", "In the next proposition, we investigate the spacing of the vertical lines on the right hand side by looking at the second derivative of $f_{x_0,\\alpha ,\\beta }$ .", "If $f_{x_0,\\alpha ,\\beta }^{\\prime \\prime }>0$ on an interval $I$ , then the spacing between the image of the vertical lines in $\\Lambda _{x_0,s}$ is increasing on $I$ .", "Since the push-forward of the Brown measure of $x_0+c_s$ by $U_{\\alpha ,\\beta }$ is the Brown measure of $x_0+c_{\\alpha ,\\beta }$ , this proposition describes how mass is transferred under this push-forward map — whether the mass is “squeezed” or “stretched” (relative to the total mass on $\\Omega _{x_0,\\alpha ,\\beta }$ ).", "More precisely, the mass of $\\lbrace \\left.u_0+iv_0\\in \\Lambda _{x_0,s}\\right\|a\\le u_0\\le b\\rbrace $ with respect to the Brown measure $x_0+c_s$ is transferred to the set $\\lbrace \\left.u+iv\\in \\Omega _{x_0,\\alpha ,\\beta }\\right\|f_{x_0,\\alpha ,\\beta }(a)\\le u\\le f_{x_0,\\alpha ,\\beta }(b)\\rbrace .$ If $f_{x_0,\\alpha ,\\beta }^{\\prime \\prime }<0$ on $(a,b)$ and $f_{x_0,\\alpha ,\\beta }^{\\prime \\prime }>0$ on $(b,c)$ , then the push-forward map $U_{\\alpha ,\\beta }$ “squeezes” the mass towards the vertical line intersecting $b$ .", "Similarly, if $f_{x_0,\\alpha ,\\beta }^{\\prime \\prime }>0$ on $(a,b)$ and $f_{x_0,\\alpha ,\\beta }^{\\prime \\prime }<0$ on $(b,c)$ , then the push-forward map $U_{\\alpha ,\\beta }$ “stretches” the mass away from the vertical line intersecting $b$ .", "Writing $s=\\alpha +\\beta $ , we have $f_{x_0,\\alpha ,\\beta }^{\\prime \\prime }(u_0) =2\\pi (\\alpha -\\beta )w_{s/2,s/2}^{\\prime }(u_0)= \\frac{\\alpha -\\beta }{s}\\psi _{x_0,s}^{\\prime \\prime }(u_0)$ for all $u_0\\in \\Lambda _{x_0,s}\\cap \\mathbb {R}$ , where $\\psi _{x_0,s}(u_0) = H_{x_0,s}(u_0+iv_{x_0,s}(u_0))$ is defined in Theorem REF (3).", "Using the formula $f_{x_0,\\alpha ,\\beta }(u_0) = u_0+(\\alpha -\\beta )\\int \\frac{(u_0-x)\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v_{x_0,s}(u_0)^2}=u_0+\\frac{\\alpha -\\beta }{s}(\\psi _{x_0,s}(u_0)-u_0),$ we have $f_{x_0,\\alpha ,\\beta }^{\\prime \\prime }(u_0) = \\frac{\\alpha -\\beta }{s}\\psi _{x_0,s}^{\\prime \\prime }(u_0).$ The other formula for $f_{x_0,\\alpha ,\\beta }^{\\prime \\prime }$ follows from $\\psi _{x_0,s}^{\\prime }(u_0) = 2\\pi s w_{s/2,s/2}(u_0)$ by Corollary ." ], [ "Examples: Cauchy case", "In this section, we compute the Brown measures of $x_0+c_{\\alpha ,\\beta }$ where $x_0$ has the Cauchy distribution $d\\mu _{x_0}(x) = \\frac{1}{\\pi }\\frac{dx}{1+x^2}.$ Since the density of $\\mu _{x_0}$ has polynomial decay at $\\pm \\infty $ , $x_0\\in {A}^\\Delta $ , so does $x_0+c_{\\alpha ,\\beta }\\in {A}^\\Delta $ .", "We first compute the Brown measure of $x_0+c_t$ ; the computation in the process is also useful for the computation of the Brown measure of $x_0+c_{\\alpha ,\\beta }$ .", "Lastly, we compute the Brown measure of $x_0+i\\sigma _t$ by putting $\\alpha = 0$ and $\\beta = t$ to the Brown measure of $x_0+c_{\\alpha ,\\beta }$ ." ], [ "Adding a circular variable", "In this section, we compute the Brown measure of $x_0+c_t$ as in the following theorem.", "Some of the computations will be used again when we compute the Brown measure of $x_0+c_{\\alpha ,\\beta }$ .", "When $x_0$ is Cauchy distributed, the boundary of the domain $\\Lambda _{x_0,t}$ has the form $\\partial \\Lambda _{x_0,t} = \\left\\lbrace u\\pm iv\\left\|u^2=\\frac{1}{v}(1+v)(t-v-v^2), v>0\\right.\\right\\rbrace .$ The upper boundary $(\\partial \\Lambda _{x_0,t})\\cap (\\mathbb {C}^+\\cup \\mathbb {R})$ of $\\Lambda _{x_0,t}$ is the graph of a positive unimodal function with peak $\\frac{-1+\\sqrt{1+4t}}{2}$ at 0.", "The Brown measure of $x_0+c_t$ has full measure on $\\Lambda _{x_0,t}$ , with density $w_{t/2,t/2}(\\lambda ) = \\frac{1}{2\\pi t}\\frac{t+4v^2(1+v)^2}{(1+v)(t+2v^2(1+v))}$ where $v = v_{x_0,t}(u)$ .", "As $\|u\|\\rightarrow \\infty $ , the density $w_t(u)$ does not approach 0.", "In fact, as $\|u\|\\rightarrow \\infty $ , $v_{x_0,t}(u)\\rightarrow 0$ , so $w_t(u)$ approaches $1/(2\\pi t)$ .", "The density $w_t(\\lambda )$ on $\\Lambda _{x_0,t}$ still defines a probability measure because the function $v_{x_0,t}(u)\\approx t/u^2$ as $\|u\|\\rightarrow \\infty $ .", "Figure REF plots the eigenvalue simulation of $x_0+c_t$ , the density of the Brown measure of $x_0+c_t$ , as well as the function $w_t(u)$ for $u\\in \\mathbb {R}$ at $t=1$ .", "Figure: 5000×50005000\\times 5000 matrix simulation of eigenvalues of x 0 +c t x_0+c_t (top), density plot of the Brown measure of x 0 +c t x_0+c_t (middle), and w t/2,t/2 (u)w_{t/2,t/2}(u) for u∈ℝu\\in \\mathbb {R} (bottom), at t=1t=1.", "Top two plotted with boundary of Λ x 0 ,t \\Lambda _{x_0,t}.We start by computing the function $v_{x_0,t}(u)$ , then we compute the derivative of $\\psi _{x_0,t}$ (See Theorem REF for definition of $\\psi _{x_0,t}$ ).", "For each $u\\in \\mathbb {R}$ , $v_{x_0,t}(u)$ is the unique positive number $v$ satisfying $u^2=\\frac{1}{v}(1+v)(t-v-v^2).$ Thus, $u$ and $dv_{x_0,t}/du$ have opposite sign; in particular, $v_{x_0,t}$ is unimodal with peak at 0, and $v_{x_0,t}(0)=\\frac{-1+\\sqrt{1+4t}}{2}$ .", "Let $u\\in \\mathbb {R}$ .", "We can compute (computer software such as Mathematica could be helpful) that $\\frac{1}{t} = \\int \\frac{d\\mu _{x_0}(x)}{(u-x)^2+v_{x_0,t}(u)^2} = \\frac{1+v_{x_0,t}(u)}{v_{x_0,t}(u)(u^2+(1+v_{x_0,t}(u))^2)},$ so that $u^2=\\frac{1}{v_{x_0,t}(u)}[1+v_{x_0,t}(u)][t-v_{x_0,t}(u)-v_{x_0,t}(u)^2].$ This proves (REF ).", "Differentiating both sides of (REF ) with respect to $u$ , we have $2u = \\left(-\\frac{t}{v_{x_0,t}(u)^2}-2(1+v_{x_0,t}(u))\\right)\\frac{dv_{x_0,t}}{du}$ which shows $u$ and $dv_{x_0,t}/du$ have oppposite sign since we must have $-t/v_{x_0,t}(u)-2(1+v_{x_0,t}(u))<0$ .", "It follows that $dv_{x_0,t}/du<0$ for $u>0$ and $dv_{x_0,t}/du>0$ for $u<0$ , proving $v_{x_0,t}$ is unimodal with peak at 0.", "The value $v_{x_0,t}(0)$ comes from solving the equation (REF ) at $u=0$ .", "We have $\\frac{d\\psi _{x_0,t}}{du}=\\frac{t+4v^2(1+v)^2}{(1+v)(t+2v^2(1+v))},$ where $v = v_{x_0,t}(u)$ .", "Since $\\frac{1+v}{v(u^2+(1+v)^2)}=\\int \\frac{d\\mu _{x_0}(x)}{(u-x)^2+v_{x_0,t}(u)^2}=\\frac{1}{t},$ we can compute (again, computer software could be useful) $\\psi _{x_0,t}(u) = u+t\\int \\frac{(u-x)\\,d\\mu _{x_0}(x)}{(u-x)^2+v_{x_0,t}(u)^2} = u+t\\frac{u}{u^2+(1+v)^2} = u+\\frac{uv}{1+v}.$ Using (REF ) and (REF ), we can compute the derivative $\\frac{d}{du}\\frac{uv}{1+v} = \\frac{v}{1+v}+\\frac{u}{(1+v)^2}\\frac{dv}{du}=\\frac{v(-t+2v(1+v)^2)}{(1+v)(t+2v^2(1+v))}$ and so $\\frac{d\\psi _{x_0,t}}{du}=\\frac{t+4v^2(1+v)^2}{(1+v)(t+2v^2(1+v))},$ completing the proof.", "[Proof of Theorem REF ] By Proposition REF , $\\Lambda _{x_0,t} = \\lbrace \\left.u+iv\\in \\mathbb {C}\\right\|\|v\|<v_{x_0,t}(u)\\rbrace $ ; thus, (REF ) and the description of the boundary of $\\Lambda _{x_0,t}$ follow from (REF ).", "The formula of the density of the Brown measure follows from Corollary and Proposition REF ." ], [ "Adding an elliptic variable", "The main result about the Brown measure of $x_0+c_{\\alpha ,\\beta }$ is as follows.", "When $x_0$ is Cauchy distributed, the function $\\varphi _{x_0,\\alpha ,\\beta }$ in Definition (3) is unimodal with peak $\\frac{\\beta (-1+\\sqrt{1+4s})}{s}$ at $u = 0$ .", "The boundary of the corresponding $\\Omega _{x_0,\\alpha ,\\beta }$ (See Definition (3)) has the form $\\partial \\Omega _{x_0,\\alpha ,\\beta }=\\left\\lbrace u\\pm ib\\in \\mathbb {C}\\left\|u^2 = \\frac{(b\\alpha +\\beta )^2(4\\beta ^2-2b\\beta -b^2s)}{b\\beta ^2(bs+2\\beta )}\\right.\\right\\rbrace .$ The function $b=\\varphi _{x_0,\\alpha ,\\beta }(u)$ has a decay of order $1/u^2$ as $\|u\|\\rightarrow \\infty $ .", "The density of the Brown measure has the form $w_{\\alpha ,\\beta }(\\lambda ) = \\frac{1}{4\\pi \\beta }\\frac{b^4 s^3+4b^3s^2\\beta +4b^2 s\\beta ^2+4\\beta ^4}{b^4 s^2\\alpha +4b^3s\\alpha \\beta +4b^2\\alpha \\beta ^2+4b\\beta ^4+4\\beta ^4},\\quad \\lambda \\in \\Omega _{x_0,\\alpha ,\\beta }$ where $b = \\varphi _{x_0,\\alpha ,\\beta }(u)$ .", "The density $w_{\\alpha ,\\beta }$ does not approach 0 as $u=\\mathrm {Re}(\\lambda )$ approaches infinity.", "In fact, as $\|u\|\\rightarrow \\infty $ , $\\varphi _{x_0,\\alpha ,\\beta }(u)\\rightarrow 0$ and so $\\lim _{\|u\|\\rightarrow \\infty }w_{\\alpha ,\\beta }(u) = \\frac{1}{4\\pi \\beta }.$ Figure REF plots an eigenvalue simulation of $x_0+c_{\\alpha ,\\beta }$ , the density of the Brown measure of $x_0+c_{\\alpha ,\\beta }$ , and the graph of the function $w_{\\alpha ,\\beta }(u)$ for $u\\in \\mathbb {R}$ at $\\alpha =1/8$ and $\\beta =7/8$ .", "Figure: 5000×50005000\\times 5000 matrix simulation of eigenvalues of x 0 +c α,β x_0+c_{\\alpha ,\\beta } (top), density plot of the Brown measure of x 0 +c α,β x_0+c_{\\alpha ,\\beta } (middle), and w α,β (u)w_{\\alpha ,\\beta }(u) for u∈ℝu\\in \\mathbb {R} (bottom), at α=1/8\\alpha =1/8 and β=7/8\\beta =7/8.", "Top two plotted with boundary of Ω x 0 ,α,β \\Omega _{x_0,\\alpha ,\\beta }.We start proving Theorem REF by the following lemma which concerns the derivative of $f_{x_0,\\alpha ,\\beta }$ .", "Let $u = f_{x_0,\\alpha ,\\beta }(u_0)$ for $u_0\\in \\mathbb {R}$ .", "Then, by writing $s=\\alpha +\\beta $ and $v=v_{x_0,s}(u_0)$ , $\\frac{du}{du_0} = \\frac{(1+v)(s^2+4\\alpha v^2(1+v))-(\\alpha -\\beta )sv}{s(1+v)(s+2v^2(1+v))}.$ Using a computation similar to (REF ), we have $\\begin{split}u=f_{x_0,\\alpha ,\\beta }(u_0) &= u_0+(\\alpha -\\beta )\\int \\frac{(u_0-x)\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v^2}\\\\&=u_0+\\frac{\\alpha -\\beta }{s}\\frac{u_0v}{1+v}.\\end{split}$ Then using (REF ) with $s$ in place of $t$ and $u_0$ in place of $u$ , we have $\\frac{du}{du_0} = 1+\\frac{\\alpha -\\beta }{s}\\frac{v(-s+2v(1+v)^2)}{(1+v)(s+2v^2(1+v))}.$ The conclusion now follows from an algebraic computation.", "Let $u = f_{x_0,\\alpha ,\\beta }(u_0)$ for $u_0\\in \\mathbb {R}$ .", "Then, by writing $s=\\alpha +\\beta $ and $v=v_{x_0,s}(u_0)$ , we have $\\frac{d}{du}\\int \\frac{(u_0-x)\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v} = \\frac{v(-s+2v(1+v)^2)}{(1+v)(s^2+4\\alpha v^2(1+v))-(\\alpha -\\beta )sv}.$ Using a computation similar to (REF ), we have $\\int \\frac{(u_0-x)\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v} = \\frac{1}{s}\\frac{u_0v}{1+v}.$ Thus, by a computation similar to (REF ) and Lemma REF , $\\frac{d}{du}\\int \\frac{(u_0-x)\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v}& = \\frac{1}{s}\\frac{v(-s+2v(1+v)^2)}{(1+v)(s+2v^2(1+v))}\\frac{du_0}{du}\\\\&= \\frac{v(-s+2v(1+v)^2)}{(1+v)(s^2+4\\alpha v^2(1+v))-(\\alpha -\\beta )sv},$ completing the proof.", "[Proof of Theorem REF ] First, the boundary of $\\Omega _{x_0,\\alpha ,\\beta }$ is the image of the graph $\\lbrace (u_0, v_{x_0,t}(u_0))\\vert \\,u_0\\in \\mathbb {R}\\rbrace $ under the map $U_{\\alpha ,\\beta }$ (See (REF )).", "By (REF ) and (REF ), if we write $u=f_{x_0,\\alpha \\beta }(u_0)$ , then $u^2 = \\frac{(1+v)(s-v-v^2)}{v}\\left(1+\\frac{\\alpha -\\beta }{s}\\frac{v}{1+v}\\right)^2=\\frac{1}{s^2}\\frac{s-v-v^2}{v}\\frac{(s+2\\alpha v)^2}{1+v}$ where $v = v_{x_0,s}(u_0)$ .", "The claimed formula of $\\partial \\Omega _{x_0,\\alpha ,\\beta }$ follows from the fact that $b = \\varphi _{x_0,\\alpha ,\\beta }(u) = 2\\beta v/s$ .", "It is clear that $\\varphi _{x_0,\\alpha ,\\beta }(u)$ has a decay of order $1/u^2$ as $\|u\|\\rightarrow \\infty $ , because $v\\rightarrow 0$ as $\|u\|\\rightarrow \\infty $ .", "Recall from Theorem REF that $v_{x_0,s}(0) = \\frac{-1+\\sqrt{1+4s}}{2}$ .", "Since $f_{x_0,\\alpha ,\\beta }(0) = 0$ by (REF ) and $f_{x_0,\\alpha ,\\beta }$ is strictly increasing, the function $\\varphi _{x_0,\\alpha ,\\beta }(u) = \\frac{2\\beta v_{x_0,s}(u_0)}{s}$ is unimodal with peak $\\frac{\\beta (-1+\\sqrt{1+4s})}{s}$ at 0.", "By Theorem and Lemma REF , we have $w_{\\alpha ,\\beta }(u+iv) &= \\frac{1}{4\\pi \\beta }\\left(1+2\\beta \\frac{d}{du}\\int \\frac{(u_0-x)\\,d\\mu _{x_0}(x)}{(u_0-x)^2+v_{x_0,s}(u_0)^2}\\right)\\\\&= \\frac{1}{4\\pi \\beta }\\left(1+2\\beta \\frac{v(-s+2v(1+v)^2)}{(1+v)(s^2+4\\alpha v^2(1+v))-(\\alpha -\\beta )sv}\\right)\\\\&= \\frac{1}{4\\pi \\beta }\\frac{b^4 s^3+4b^3s^2\\beta +4b^2 s\\beta ^2+4\\beta ^4}{b^4 s^2\\alpha +4b^3s\\alpha \\beta +4b^2\\alpha \\beta ^2+4b\\beta ^4+4\\beta ^4}$ where we have used $b = \\varphi _{x_0,\\alpha ,\\beta }(u) = (2\\beta /s)v$ .", "We close this section by an example illustrating how mass is transformed by the push-forward under $U_{\\alpha ,\\beta }$ (See (REF )), as proved in Proposition .", "By Proposition and Theorem REF , $f_{x_0,\\alpha ,\\beta }^{\\prime \\prime }(u_0)=2\\pi (\\alpha -\\beta )w_{s/2,s/2}^{\\prime }(u_0)=(\\alpha -\\beta )\\frac{4v(1+v)(1+3v(1+v))-s}{ s(1+v)^2(1+2v^2(1+v))^2}\\frac{dv}{du_0}$ where $v=v_{x_0,s}(u_0)$ .", "Recall that $v_{x_0,s}(u_0)$ is unimodal, $dv/du_0=0$ if and only if $u_0=0$ .", "As an example, when $s=1$ , we can solve, by the relation of $u_0$ and $v_{x_0,s}(u_0)$ in Lemma REF , that $f_{x_0,\\alpha ,\\beta }^{\\prime \\prime }(u_0) = 0$ if and only if $u_0 = 0$ or $u_0=\\pm \\frac{-3+\\sqrt{15}}{6}\\approx \\pm 2.56141$ .", "The top diagram of Figure REF shows vertical blue line segments inside $\\Lambda _{x_0,1}$ .", "The spacing between the blue line segments is $0.25$ .", "The middle diagram of Figure REF shows the corresponding vertical blue line segments after being mapped by $U_{1/8,7/8}$ : each blue line segment intersecting $u_0\\in \\mathbb {R}$ in the top diagram is mapped to a vertical line segment intersecting $f_{x_0,1/8,7/8}(u_0)$ in the middle diagram.", "The bottom diagram plots the differences $[f_{x_0,1/8,7/8}(u_0+0.05)-f_{x_0,1/8,7/8}(u_0)]-[ f_{x_0,1/8,7/8}(u_0)-f_{x_0,1/8,7/8}(u_0-0.05)]$ for $u_0 =0.05 k$ for $k = 0,1,\\ldots ,120$ .", "Figure REF agrees with the theoretical computation in the preceding paragraph.", "The spacings between the image of the vertical line segments with real part less than or equal $2.5$ in the top diagram are increasing, as shown in the middle diagram, whereas the spacings between the image of the vertical line segments with real part greater than $2.5$ in the top diagram are decreasing, as shown in the middle diagram.", "The bottom diagram also shows a sign change at a value slightly greater than $2.5$ .", "Figure: Equally-spaced blue lines inside Λ x 0 ,1 \\Lambda _{x_0,1} (Top).", "The corresponding blue lines under the map U 1/8,7/8 U_{1/8,7/8} (Middle).", "Differences between the lines: f x 0 ,1/8,7/8 (u 0 +0.05)-f x 0 ,7/8,7/8 (u 0 )f_{x_0,1/8,7/8}(u_0+0.05)-f_{x_0,7/8,7/8}(u_0) and f x 0 ,1/8,7/8 (u 0 )-f x 0 ,1/8,7/8 (u 0 -0.05)f_{x_0,1/8,7/8}(u_0)-f_{x_0,1/8,7/8}(u_0-0.05) (Bottom)." ], [ "Adding an imaginary multiple of semicircular variable", "In this section, we take $\\alpha =0$ and $\\beta = t$ in Theorem REF to obtain the Brown measure of $x_0+i\\sigma _t$ .", "One significant simplification over the previous cases is that the function $\\varphi _{x_0,t}$ , whose graph is the (upper) boundary of $\\Omega _{x_0,t}$ , and the density $w_t(u+iv)$ can be written explicitly in terms of $u$ , instead of $v_{x_0,t}(u)$ .", "When $x_0$ has the Cauchy distribution, the domain $\\Omega _{x_0,t}$ has the form $\\Omega _{x_0,t} = \\left\\lbrace u+iv\\in \\mathbb {C}\\left\|\|v\|\\le \\frac{4t}{\\sqrt{u^2+1}(\\sqrt{u^2+1}+\\sqrt{u^2+1+4t})}\\right.\\right\\rbrace .$ The density of the Brown measure of $x_0+i\\sigma _t$ is given by $w_{t}(\\lambda )=\\frac{1}{4\\pi t}\\frac{4t+(1+u^2)^2}{(1+u^2)^{3/2}\\sqrt{u^2+1+4t}},\\quad \\lambda =u+iv\\in \\Omega _{x_0,t}.$ When $x_0$ has the Cauchy distribution, the function $\\varphi _{x_0,t}$ defined in Theorem REF (4) can be computed as $\\varphi _{x_0,t}(u) = \\frac{4t}{\\sqrt{u^2+1}(\\sqrt{u^2+1}+\\sqrt{u^2+1+4t})},\\quad u\\in \\mathbb {R}.$ By putting $\\alpha = 0$ and $\\beta =t$ in (REF ), we get $u^2 = \\frac{t^2(4t^2-2bt-b^2t)}{bt^2(bt+2t)} = \\frac{(4t-2b-b^2)}{b(b+2)}$ where $b = \\varphi _{x_0,0,t}(u) = \\varphi _{x_0,t}$ , as indicated in Definition .", "Thus, since $\\varphi _{x_0,t}(u)>0$ , we can solve the above equation and get $\\varphi _{x_0,t}(u) = \\frac{-\\sqrt{u^2+1}+\\sqrt{u^2+1+4t}}{\\sqrt{u^2+1}} = \\frac{4t}{\\sqrt{u^2+1}(\\sqrt{u^2+1}+\\sqrt{u^2+1+4t})},$ completing the proof.", "[Proof of Theorem REF ] The formula for $\\Omega _{x_0,t}$ follows directly from Lemma REF .", "To compute the density of the Brown measure, by Remark , we again put $\\alpha = 0$ and $\\beta = t$ in Theorem REF , and get $w_{t}(\\lambda ) = \\frac{1}{4\\pi t}\\frac{4t^4+tb^2(bt+2t)^2}{4t^3(bt+t)}= \\frac{1}{4\\pi t}\\frac{4t+b^2(b+2)^2}{4t(b+1)},\\quad \\lambda \\in \\Omega _{x_0,t}.$ where $b = \\varphi _{x_0,t}(u)$ .", "By putting in the formula of $\\varphi _{x_0,t}(u)$ in Lemma REF , we have $w_{t}(\\lambda )=\\frac{1}{4\\pi t}\\frac{4t+(1+u^2)^2}{(1+u^2)^{3/2}\\sqrt{u^2+1+4t}},\\quad \\lambda =u+iv\\in \\Omega _{x_0,t}.$ The theorem is established.", "In the case $x_0+i\\sigma _t$ , the density $w_t(\\lambda )$ is very explicit, in terms of $u=\\mathrm {Re}(\\lambda )$ .", "It is not hard to see that $\\lim _{\|u\|\\rightarrow \\infty }w_t(u) = \\frac{1}{4\\pi t}.$ Recall that Figure REF plots an eigenvalue simulation of $x_0+i\\sigma _t$ , the density of the Brown measure of $x_0+i\\sigma _t$ and the function $w_{t}(u)$ for $u\\in \\mathbb {R}$ at $t=1$ .", "Figure REF shows the plots of the Brown measure densities of $x_0+c_t$ ($t=1$ ), $x_0+c_{\\alpha ,\\beta }$ ($\\alpha = 1/8$ , $\\beta = 7/8$ ), and $x_0++i\\sigma _t$ ($t=1$ ).", "It also shows the graphs of $w_{\\alpha ,\\beta }(u)$ , $u\\in \\mathbb {R}$ , for comparison.", "Note that although we observe the trend from Figure REF that $w_{\\alpha ,\\beta }(0)$ decreases as $\\beta $ increases while keeping $\\alpha +\\beta = 1$ , we do not lose mass around vertical strips around the imaginary axis.", "The map $U_{\\alpha ,\\beta }$ indeed pushes mass towards the imaginary axis in a vertical strip including the origin, by the discussion in the last paragraph of Section REF .", "Figure: Left: Brown measure densities of x 0 +c 1 x_0+c_1 (top), x 0 +c 1/8,7/8 x_0+c_{1/8,7/8} (middle), and x 0 +iσ 1 x_0+i\\sigma _1 (bottom).", "Right: Graphs of w 1/2,1/2 w_{1/2,1/2} (top), w 1/8,7/8 w_{1/8,7/8} (middle), and w 1 w_1 (bottom)." ], [ "Acknowledgments", "The author would like to thank Hari Bercovici and Roland Speicher who asked the author in two different seminars whether one can extend the results in [13], [15] to unbounded random variables.", "Their question was the starting point of this paper, and the author had useful discussions with them.", "The author would also like to thank Marek Bożejko, Brian Hall, Eugene Lytvynov for useful conversations.", "The computer simulation and the Brown measure plot in Figure REF was plotted by Hall." ] ]	2011.14222
[ [ "Geometric weight-shifting operators on Hilbert modular forms in\n characteristic p" ], [ "Abstract We carry out a thorough study of weight-shifting operators on Hilbert modular forms in characteristic $p$, generalizing the author's prior work with Sasaki to the case where $p$ is ramified in the totally real field $F$.", "In particular we use the partial Hasse invariants and Kodaira-Spencer filtrations defined by Reduzzi and Xiao to improve on Andreatta and Goren's construction of partial $\\Theta$-operators, obtaining ones whose effect on weights is optimal from the point of view of geometric Serre weight conjectures.", "Furthermore we describe the kernels of partial $\\Theta$-operators in terms of images of geometrically constructed partial Frobenius operators.", "Finally we apply our results to prove a partial positivity result for minimal weights of mod $p$ Hilbert modular forms." ], [ "Introduction", "The study of weight-shifting operations on modular forms has a rich and fruitful history.", "Besides those naively obtained from the graded algebra structure on the space of classical modular forms of all weights, there is a deeper construction due to Ramanujan [30] which shifts the weight by two using differentiation, leading to a more general theory of Maass–Shimura operators.", "Analogous weight-shifting operations in characteristic $p$ , first studied by Swinnerton-Dyer and Serre [34], take on special significance in the context of congruences between modular forms and the implications for associated Galois representations.", "In particular one has the following linear maps from the space of mod $p$ modular forms of weight $k$ and some fixed level $N$ prime to $p$ : multiplication by a Hasse invariant $H$ , to forms of weight $k+p-1$ ; a differential operator $\\Theta $ , to forms of weight $k+p+1$ ; a linearized $p$ -power map $V$ , to forms of weight $pk$ .", "These maps all have simple descriptions in terms of associated $q$ -expansions: if $f$ has $q$ -expansion $\\sum a_n q^n$ , then that of $Hf$ (resp.", "$\\Theta f$ , $V f$ ) is $\\sum a_n q^n$ (resp.", "$\\sum n a_n q^n$ , $\\sum a_n q^{pn}$ ).", "Following the work of Swinnerton-Dyer and Serre, there were further significant developments to the theory due to Katz [23], [24] (interpreting the constructions more geometrically), Jochnowitz [21], [22] (on the weight filtration and Tate's $\\Theta $ -cycles) and Gross [20] (in the study of companion forms), providing crucial ingredients for Edixhoven's proof of the weight part of Serre's Conjecture in [14].", "Suppose now that $F$ is a totally real field of degree $d = [F:{\\mathbb {Q}}]$ and consider spaces of Hilbert modular forms of weight $\\mathbf {k}\\in \\mathbb {Z}^\\Sigma $ and fixed level prime to $p$ , where $\\Sigma $ denotes the set of embeddings $\\lbrace \\sigma : F \\hookrightarrow \\overline{{\\mathbb {Q}}}_p\\rbrace $ .", "For such spaces of $p$ -adic modular forms, Katz [25] constructed a family of commuting differential operators $\\Theta _\\sigma $ , indexed by the $d$ -embeddings $\\sigma \\in \\Sigma $ .", "The theory was further developed by Andreatta and Goren [1] who, building on Katz's work and Goren's definition of partial Hasse invariants in [18], [19] (if $p$ is unramified in $F$ ), defined partial $\\Theta $ -operators on spaces of mod $p$ Hilbert modular forms.", "Under the assumption that $p$ is unramified in $F$ , some aspects of the construction of partial $\\Theta $ -operators in [1] were simplified in [12], which also went on to define partial Frobenius operators (generalizing $V$ ) geometrically and use their image to describe kernels of partial $\\Theta $ -operators.", "When $p$ is ramified in $F$ , the effectiveness of the approach in [1] was limited by the singularities of the available (Deligne–Pappas) model for the Hilbert modular variety.", "Since then however, a smooth integral model was constructed by Pappas and Rapoport [29], and the theory of partial Hasse invariants was further developed in this context by Reduzzi and Xiao in [32].", "The theory of partial $\\Theta $ -operators was revisited in that light by Deo, Dimitrov and Wiese in [8], where they closely follow [1].", "Here we instead exploit the observations and techniques introduced in [12], applying them directly to the special fibre of the Pappas–Rapoport model to construct and relate partial $\\Theta $ and Frobenius operators.", "In particular this eliminates extraneous multiples of partial Hasse invariants that appear in [8], and yields results whose implications for minimal weights are motivated by the forthcoming generalization to the ramified case of the geometric Serre weight conjectures of [12].", "The main contributions of this paper may be summarized as follows: a construction of operators $\\Theta _\\tau $ with optimal effect on weight (Theorem REF ); a geometric construction of partial Frobenius operators $V_{\\mathfrak {p}}$ (see §REF ); a description of the kernel of $\\Theta _\\tau $ in terms of the image of $V_{\\mathfrak {p}}$ (see §REF ); an application to positivity of minimal weights (Theorem REF ).", "We should emphasize that the focus of this paper is entirely on Hilbert modular forms in characteristic $p$ .", "There is also a rich theory of $\\Theta $ -operators on $p$ -adic automorphic forms which has seen major progress recently in the work of de Shalit and Goren [7], and Eischen, Fintzen, Mantovan and Varma [15], which in turn has implications in the characteristic $p$ setting [6], [7], [16], [17].", "Another advance in characteristic $p$ has been Yamauchi's construction [36] of $\\Theta $ -operators for mod $p$ Siegel modular forms of degree two.", "We remark however that all of the work just mentioned only considers automorphic forms on reductive groups which are unramified at $p$ ; the novelty of this paper is largely in the treatment of ramification at $p$ .", "We now describe the contents in more detail.", "We first set out some basic notation and constructions in §REF .", "In particular we fix a prime $p$ , a totally real field $F$ of degreeIncluding the case $F = \\mathbb {Q}$ would introduce different complications in the treatment of cusps and provide no new results.", "$d = [F:\\mathbb {Q}] > 1$ , and let ${\\mathcal {O}}_F$ denote the ring of integers of $F$ and $S_p$ the set of prime ideals of ${\\mathcal {O}}_F$ dividing $p$ .", "For each ${\\mathfrak {p}}\\in S_p$ , let $\\Sigma _{{\\mathfrak {p}},0}$ denote the set of $f_{\\mathfrak {p}}$ embeddings ${\\mathcal {O}}_F/{\\mathfrak {p}}\\rightarrow \\overline{{\\mathbb {F}}}_p$ and $\\Sigma _{\\mathfrak {p}}$ the set of $e_{\\mathfrak {p}}f_{\\mathfrak {p}}$ embeddings $F_{\\mathfrak {p}}\\rightarrow \\overline{{\\mathbb {Q}}}_p$ , where $f_{\\mathfrak {p}}$ (resp.", "$e_{\\mathfrak {p}}$ ) is the residual (resp.", "ramification) degree of ${\\mathfrak {p}}$ .", "We let $\\begin{array}{rcccl} \\Sigma _0 & = & \\coprod _{{\\mathfrak {p}}\\in S_p} \\Sigma _{{\\mathfrak {p}},0} & = & \\lbrace \\,\\tau _{{\\mathfrak {p}},i}\\,\|\\, {\\mathfrak {p}}\\in S_p, i \\in \\mathbb {Z}/f_{\\mathfrak {p}}\\mathbb {Z}\\,\\rbrace \\\\\\mbox{and} \\qquad \\Sigma & = & \\coprod _{{\\mathfrak {p}}\\in S_p} \\Sigma _{\\mathfrak {p}}& = & \\lbrace \\,\\theta _{{\\mathfrak {p}},i,j}\\,\|\\, {\\mathfrak {p}}\\in S_p, i \\in \\mathbb {Z}/f_{\\mathfrak {p}}\\mathbb {Z}, j = 1,\\ldots ,e_{\\mathfrak {p}}\\,\\rbrace ,\\end{array}$ where each $\\tau _{{\\mathfrak {p}},0} \\in \\Sigma _{{\\mathfrak {p}},0}$ is chosen arbitrarily, $\\tau _{{\\mathfrak {p}},i} = \\tau _{{\\mathfrak {p}},0}^{p^i}$ and $\\theta _{{\\mathfrak {p}},i,1},\\ldots ,\\theta _{{\\mathfrak {p}},i,e_{\\mathfrak {p}}}$ is any ordering of the lifts of $\\tau _{{\\mathfrak {p}},i}$ to $\\Sigma _{\\mathfrak {p}}$ .", "We also define a “right-shift” permutation $\\sigma $ of $\\Sigma $ by $\\sigma (\\theta _{{\\mathfrak {p}},i,j}) = \\left\\lbrace \\begin{array}{ll} \\theta _{{\\mathfrak {p}},i,j-1},&\\mbox{if $j > 1$};\\\\ \\theta _{{\\mathfrak {p}},i-1,e_{\\mathfrak {p}}},&\\mbox{if $j=1$.", "}\\end{array}\\right.$ In §REF we recall the definition of the Pappas–Rapoport model $Y_U$ for the Hilbert modular variety of level $U$ , where $U$ is any sufficiently small open compact subgroup of $\\operatorname{GL}_2(\\mathbb {A}_{F,) of level prime to p. This may beviewed as a coarse moduli space for Hilbert-Blumenthal abelian varieties with additional structure, where this additionalstructure includes a suitable collection of filtrations on direct summands of its sheaf of invariant differentials.", "The schemeY_U is then smooth of relative dimension d over {\\mathcal {O}}, where {\\mathcal {O}} is the ring of integers of a finite extensionof {\\mathbb {Q}}_p in \\overline{{\\mathbb {Q}}}_p.", "Since the main results of the paper concern Hilbert modular forms in characteristic p, we willrestrict our attention to this setting for the remainder of the Introduction, and let \\overline{Y}_U = Y_{U,\\mathbb {F}} where \\mathbb {F}is the residue field of {\\mathcal {O}}.", "}$ In §§–REF we construct the automorphic line bundles $\\overline{{\\mathcal {A}}}_{\\mathbf {k},\\mathbf {l}}$ on $\\overline{Y}_{U}$ for all $\\mathbf {k}$ , $\\mathbf {l}\\in \\mathbb {Z}^\\Sigma $ and sufficiently small $U$ (of level prime to $p$ ), and define the space of Hilbert modular forms of weight $(\\mathbf {k},\\mathbf {l})$ and level $U$ over $\\mathbb {F}$ to be $M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) = H^0(\\overline{Y}_{U},\\overline{{\\mathcal {A}}}_{\\mathbf {k},\\mathbf {l}}).$ The spaces are equipped with a natural Hecke action making $M_{\\mathbf {k},\\mathbf {l}}(\\mathbb {F}) := \\varinjlim _U M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F})$ a smooth admissible representation of $\\operatorname{GL}_2(\\mathbb {A}_{F,) over \\mathbb {F}.A key point, as already observed in \\cite {DS} in the unramified case, is that the parity condition on \\mathbf {k} imposedin the definition of Hilbert modular forms in characteristic zero (for the group {\\operatorname{Res}}_{F/\\mathbb {Q}} \\operatorname{GL}_2) disappears in characteristic p.We remark also that the effect of the weight parameter \\mathbf {l} (in characteristic p) is to introduce twists by torsion bundlesthat make various constructions, in particular that of partial \\Theta -operators, compatible with the natural Hecke action.", "}In §§\\ref {sec:KS}--\\ref {sec:stratification} we recall results of Reduzzi and Xiao~\\cite {RX} that will underpinour construction of partial $$-operators.", "Firstly there is a natural Kodaira--Spencer filtration on directsummands of $ 1YU/F$ whose graded pieces are isomorphic tothe automorphic line bundles $A2e,-e$ (where $ e$ denotes the basis elementof $ Z$ indexed by $$).", "Secondly for each $ = p,i,j $, there is a partial Hasse invariant$$H_\\theta \\in M_{\\mathbf {h}_\\theta ,\\bf {0}}(U;\\mathbb {F}),\\qquad \\mbox{where \\,\\,$ \\mathbf {h}_\\theta = n_\\theta {\\mathbf {e}}_{\\sigma ^{-1}\\theta } - {\\mathbf {e}}_\\theta $}$$with $ n= p$ if $ p,i,1$ for some $ p,i$, and $ n= 1$ otherwise.", "Multiplication by the partialHasse invariant $ H$ thus defines a map$$\\cdot H_\\theta : M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) \\longrightarrow M_{\\mathbf {k}+\\mathbf {h}_\\theta ,\\mathbf {l}}(U;\\mathbb {F})$$which is easily seen to be Hecke-equivariant.We also analogously define invariants $ GM0,h(U;F)$ which trivialize the bundles$A0,h$.", "The partial Hasse invariants $ H$ refine the onesdefined by Andreatta and Goren in~\\cite {AG} and give rise to a natural stratification on $YU$and a notion of minimal weight $ k(f)$ for non-zero $ f Mk,l(U;F)$, which the mainresult of \\cite {DK2} shows lies in a certain cone $ Z0$ (see (\\ref {eqn:Ximin})).$ We then follow the approach of [12] to define partial $\\Theta $ -operators in §.", "For each $\\tau = \\tau _{{\\mathfrak {p}},i} \\in \\Sigma _0$ , this gives a Hecke-equivariant operator $\\Theta _\\tau : M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) \\longrightarrow M_{\\mathbf {k}+ \\mathbf {h}_\\theta + 2{\\mathbf {e}}_\\theta ,\\mathbf {l}- {\\mathbf {e}}_\\theta }(U;\\mathbb {F})$ where $\\theta = \\theta _{{\\mathfrak {p}},i,e_{\\mathfrak {p}}}$ .", "Note in particular that if ${\\mathfrak {p}}$ is ramified, then the shiftThis precise shift is predictable from the point of view of forthcoming work with Sasaki generalizing the geometric Serre weight conjectures of [12] to the ramified case.", "in the weight parameter $\\mathbf {k}$ is by ${\\mathbf {e}}_{\\sigma ^{-1}\\theta } + {\\mathbf {e}}_{\\theta }$.", "The idea of the construction, inspired by the one in [1], is to divide by fundamental Hasse invariants to get a rational function on the Igusa cover of $\\overline{Y}_U$ , differentiate, project to the top graded piece of the $\\tau $ -component of the Kodaira–Spencer filtration, and finally multiply by fundamental Hasse invariants to descend to $\\overline{Y}_U$ and eliminate poles.", "The argument also gives a direct (albeit local) definition of the $\\Theta $ -operator without reference to the Igusa cover in (REF ), and establishes the following result (Theorem REF ) generalizing [12]: Theorem A Let $\\tau = \\tau _{{\\mathfrak {p}},i}$ and $\\theta = \\theta _{{\\mathfrak {p}},i,e_{\\mathfrak {p}}}$ .", "Then $\\Theta _{\\tau }(f)$ is divisible by $H_{\\theta }$ if and only if either $f$ is divisible by $H_{\\theta }$ or $p\|k_{\\theta }$ .", "We turn to the construction of partial Frobenius operators $V_{\\mathfrak {p}}$ in §.", "This essentially generalizes a definition in [12], but requires significantly more work to actualize if ${\\mathfrak {p}}$ is ramified.", "We do this using Dieudonné theory to define a partial Frobenius endomorphism $\\Phi _{\\mathfrak {p}}$ of $\\overline{Y}_U$ and an isomorphism $\\Phi _{\\mathfrak {p}}^\\overline{{\\mathcal {A}}}_{\\mathbf {k},\\mathbf {l}} \\cong \\overline{{\\mathcal {A}}}_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime }}$ , where $\\mathbf {k}^{\\prime \\prime } = \\mathbf {k}+ \\sum _{\\theta \\in \\Sigma _{\\mathfrak {p}}} k_\\theta \\mathbf {h}_\\theta \\quad \\mbox{and} \\quad \\mathbf {l}^{\\prime \\prime } = \\mathbf {l}+ \\sum _{\\theta \\in \\Sigma _{\\mathfrak {p}}} l_\\theta \\mathbf {h}_\\theta ,$ in order to obtain, for ${\\mathfrak {p}}\\in S_p$ , commuting Hecke-equivariant operators $V_{\\mathfrak {p}}: M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) \\longrightarrow M_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime }}(U;\\mathbb {F}).$ We will use $q$ -expansions to relate the kernel of $\\Theta _\\tau $ to the image of $V_{\\mathfrak {p}}$ for $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ , so we recall the theory in §.", "This is a straightforward adaptation to our setting of results and methods developed in [31], [4], [13], [5].", "In § we compute the (constant) $q$ -expansions of the invariants $H_\\theta $ and $G_\\theta $ at each cusp of $\\overline{Y}_U$ , and we obtain formulas generalizing the classical ones for the effect of the operators $\\Theta _\\tau $ and $V_{\\mathfrak {p}}$ on all $q$ -expansions.", "In particular this shows that the operators $\\Theta _\\tau $ for varying $\\tau $ commute.", "In §REF we turn our attention to the description of the kernel of $\\Theta _\\tau $ .", "The $q$ -expansion formulas also show that $\\Theta _\\tau \\circ V_{\\mathfrak {p}}= 0$ if $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ , and that $\\ker (\\Theta _\\tau )$ is the same for all $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ .", "Theorem REF then reduces the study of the kernel to the case of weights of the form $(\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime })$ where $\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime }$ are as in the definition of $V_{\\mathfrak {p}}$ , for which the argument proving [12] gives the following:This is a slight reformulation of Theorem REF .", "Theorem B If $\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma $ and $\\tau = \\tau _{{\\mathfrak {p}},i}$ and $\\theta = \\theta _{{\\mathfrak {p}},i,e_{\\mathfrak {p}}}$ , then the sequence $0 \\longrightarrow M_{\\mathbf {k},\\mathbf {l}}(U,\\mathbb {F}) \\stackrel{V_{\\mathfrak {p}}}{\\longrightarrow } M_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime }}(U;\\mathbb {F})\\stackrel{\\Theta _\\tau }{\\longrightarrow } M_{\\mathbf {k}^{\\prime \\prime } + \\mathbf {h}_\\theta + 2{\\mathbf {e}}_\\theta ,\\mathbf {l}^{\\prime \\prime } - {\\mathbf {e}}_\\theta }(U;\\mathbb {F})$ is exact.", "Before discussing the application to positivity of minimal weights, we remark that a less precise relation among the weight-shifting operations can be neatly encapsulated in terms of the algebra of modular forms of all weights $M_{\\operatorname{tot}}(U;\\mathbb {F}) := \\bigoplus _{\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma } M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}),$ or even its direct limit $M_{\\operatorname{tot}}(\\mathbb {F}) := \\varinjlim _U M_{\\operatorname{tot}}(U;\\mathbb {F})$ (over all sufficiently small levels prime to $p$ ).", "It follows from its definition that the operator $V_{\\mathfrak {p}}$ (resp.", "$\\Theta _\\tau $ ) on the direct sum is an $\\mathbb {F}$ -algebra endomorphism (resp.", "$\\mathbb {F}$ -derivation) of $M_{\\operatorname{tot}}(\\mathbb {F})$ .", "One also finds that $V_{\\mathfrak {p}}$ maps the ideal ${\\mathfrak {I}}= \\langle H_\\theta ^{\\prime } - 1, G_\\theta ^{\\prime } - 1\\rangle _{\\theta \\in \\Sigma }$ to itself.The $H_\\theta ^{\\prime }$ and $G_\\theta ^{\\prime }$ are slight modifications of the $H_\\theta $ and $G_\\theta $ obtained by rescaling those for which $j=1$ ; see §REF .", "Furthermore $\\Theta _\\tau (H_\\theta ^{\\prime }) = \\Theta _\\tau (G_\\theta ^{\\prime }) = 0$ for all $\\theta \\in \\Sigma $ , so $V_{\\mathfrak {p}}$ (resp.", "$\\Theta _\\tau $ ) induces an $\\mathbb {F}$ -algebra endomorphism (resp.", "$\\mathbb {F}$ -derivation) of the quotient $M_{\\operatorname{tot}}(\\mathbb {F})/{\\mathfrak {I}}$ .", "We then have the following consequence of Theorem REF (see Theorem REF ): Theorem C If $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ , then the sequence $0 \\longrightarrow M_{\\operatorname{tot}}(\\mathbb {F})/{\\mathfrak {I}}\\stackrel{V_{\\mathfrak {p}}}{\\longrightarrow }M_{\\operatorname{tot}}(\\mathbb {F})/{\\mathfrak {I}}\\stackrel{\\Theta _\\tau }{\\longrightarrow }M_{\\operatorname{tot}}(\\mathbb {F})/{\\mathfrak {I}}$ is exact.", "In §REF we apply our results to refine the main result of [10], which we recall states that minimal weights of non-zero forms always lie in $ \\Xi ^{\\min }$ .", "The geometric Serre weight conjectures of [12] (and its forthcoming generalization to the ramified case) predict that if $f$ is a mod $p$ Hecke eigenform which is non-Eisenstein (in the sense that the associated Galois representation is irreducible), then $\\mathbf {k}_{\\min }(f)$ should be totally positive.", "We use Theorems REF and REF to prove a partial result in this direction (Theorem REF ): Theorem D Suppose that ${\\mathfrak {p}}\\in S_p$ is such that $F_{\\mathfrak {p}}\\ne {\\mathbb {Q}}_p$ and $p^{f_{\\mathfrak {p}}} > 3$ .", "Suppose that $f \\in M_{\\mathbf {k},\\mathbf {l}}(U;{\\mathbb {F}})$ is non-zero and $\\mathbf {k}= \\mathbf {k}_{\\min }(f)$ .", "If $k_\\theta = 0$ for some $\\theta \\in \\Sigma _{\\mathfrak {p}}$ , then $\\mathbf {k}= \\bf {0}$ .", "Since the Hecke action on forms of weight $(\\bf {0},\\mathbf {l})$ is Eisenstein (see Proposition ), the theorem implies the total positivity of minimal weights of non-Eisenstein eigenforms in many situations, for example if $p>3$ and there are no primes ${\\mathfrak {p}}\\in S_p$ such that $F_{\\mathfrak {p}}= {\\mathbb {Q}}_p$ .", "We remark that the hypothesis $F_{\\mathfrak {p}}\\ne {\\mathbb {Q}}_p$ cannot be removed from Theorem REF : if $\\Sigma _{\\mathfrak {p}}= \\lbrace \\theta \\rbrace $ , then there are non-zero forms whose minimal weight $\\mathbf {k}$ satisfies $k_\\theta = 0$ and $k_{\\theta ^{\\prime }} > 0$ for some $\\theta ^{\\prime } \\ne \\theta $ .", "However forthcoming work with Kassaei will show that the Hecke action on such forms is Eisenstein; like in [9], [10], the case of split primes seems to require a completely different method.", "Unfortunately the case of $p \\le 3$ , $f_{\\mathfrak {p}}= 1$ , $e_{\\mathfrak {p}}> 1$ slips through the crack between the two methods.", "We do not know whether Theorem REF should hold in this case, but we still at least conjecture the failure is Eisenstein.", "Acknowledgements: The author would like to thank Payman Kassaei and Shu Sasaki for numerous helpful conversations, particularly in the context of collaboration on forthcoming work that motivates the results in this paper.", "It is also a pleasure to acknowledge the evident debt this work owes to that of Reduzzi and Xiao in [32].", "The author is also grateful to Deo, Dimitrov and Wiese, whose interest in partial $\\Theta $ -operators (and their kernels) encouraged him to explore the ramifications of the approach in [12], and to the referee for pointing out a number of typos and useful references." ], [ "Embeddings and decompositions", "We first set out notation and conventions for various constructions associated to the set of embeddings of a totally real field $F$ , which together with a prime $p$ , will be fixed throughout the paper.", "We assume that $F$ has degree $d=[F:\\mathbb {Q}] > 1$ , let ${\\mathcal {O}}_F$ denote its ring of integers, ${\\mathfrak {d}}$ its different, and $\\Sigma $ the set of embeddings $F \\rightarrow {\\overline{{\\mathbb {Q}}}}$ , where ${\\overline{{\\mathbb {Q}}}}$ is the algebraic closure of ${\\mathbb {Q}}$ in $.$ We also fix an embedding ${\\overline{{\\mathbb {Q}}}}\\rightarrow \\overline{{\\mathbb {Q}}}_p$ .", "We let $S_p$ denote the set of primes of ${\\mathcal {O}}_F$ dividing $p$ , and identify $\\Sigma $ with $\\coprod _{{\\mathfrak {p}}\\in S_p} \\Sigma _{{\\mathfrak {p}}}$ under the natural bijection, where $\\Sigma _{{\\mathfrak {p}}}$ denotes the set of embeddings $F_{{\\mathfrak {p}}} \\rightarrow \\overline{{\\mathbb {Q}}}_p$ .", "For each ${\\mathfrak {p}}\\in S_p$ , we let $F_{{\\mathfrak {p}},0}$ denote the maximal unramified subextension of $F_{\\mathfrak {p}}$ , which we identify with the field of fractions of $W({\\mathcal {O}}_F/{\\mathfrak {p}})$ .", "We also let $f_{\\mathfrak {p}}$ denote the residue degree $[F_{{\\mathfrak {p}},0}:{\\mathbb {Q}}_p]$ , $e_{\\mathfrak {p}}$ the ramification index $[F_{\\mathfrak {p}}:F_{{\\mathfrak {p}},0}]$ , and $\\Sigma _{{\\mathfrak {p}},0}$ the set of embeddings $F_{{\\mathfrak {p}},0} \\rightarrow \\overline{{\\mathbb {Q}}}_p$ , which we may identify with the set of embeddings ${\\mathcal {O}}_F/{\\mathfrak {p}}\\rightarrow \\overline{{\\mathbb {F}}}_p$ , or homomorphisms $W({\\mathcal {O}}_F/{\\mathfrak {p}}) \\rightarrow W(\\overline{{\\mathbb {F}}}_p)$ .", "For each ${\\mathfrak {p}}\\in S_p$ , we fix a choice of embedding $\\tau _{{\\mathfrak {p}},0} \\in \\Sigma _{{\\mathfrak {p}},0}$ , and for $i \\in {\\mathbb {Z}}/f_{\\mathfrak {p}}{\\mathbb {Z}}$ , we let $\\tau _{{\\mathfrak {p}},i} = \\phi ^i \\circ \\tau _{{\\mathfrak {p}},0}$ where $\\phi $ is the Frobenius automorphism of $\\overline{{\\mathbb {F}}}_p$ (or $W(\\overline{{\\mathbb {F}}}_p)$ or its field of fractions), so that $\\Sigma _{{\\mathfrak {p}},0} = \\lbrace \\tau _{{\\mathfrak {p}},1},\\tau _{{\\mathfrak {p}},2},\\ldots ,\\tau _{{\\mathfrak {p}},f_{{\\mathfrak {p}}}}\\rbrace $ .", "We also let $\\Sigma _0 = \\coprod _{{\\mathfrak {p}}\\in S_p} \\Sigma _{{\\mathfrak {p}},0}$ .", "Letting ${\\mathfrak {q}}= \\prod _{{\\mathfrak {p}}\\in S_p}{\\mathfrak {p}}$ denote the radical of $p$ in ${\\mathcal {O}}_F$ , note that $\\Sigma _0$ may also be identified with the set of ring homomorphisms ${\\mathcal {O}}_F/{\\mathfrak {q}}\\rightarrow \\overline{{\\mathbb {F}}}_p$ (or indeed ${\\mathcal {O}}_F \\rightarrow \\overline{{\\mathbb {F}}}_p$ ).", "For each $\\tau = \\tau _{{\\mathfrak {p}},i} \\in \\Sigma _0$ , we let $\\Sigma _\\tau \\subset \\Sigma _{\\mathfrak {p}}$ denote the set of embeddings restricting to $\\tau $ , for which we choose an ordering $\\theta _{{\\mathfrak {p}},i,1},\\theta _{{\\mathfrak {p}},i,2},\\ldots ,\\theta _{{\\mathfrak {p}},i,e_{\\mathfrak {p}}}$ , so that $\\Sigma = \\coprod _{\\tau \\in \\Sigma _0} \\Sigma _\\tau = \\lbrace \\,\\theta _{{\\mathfrak {p}},i,j}\\,\|\\,{\\mathfrak {p}}\\in S_p, i \\in \\mathbb {Z}/f_{\\mathfrak {p}}\\mathbb {Z}, 1 \\le j \\le e_{\\mathfrak {p}}\\,\\rbrace .$ We also define a permutation $\\sigma $ of $\\Sigma $ whose restriction to each $\\Sigma _{\\mathfrak {p}}$ is the $e_{\\mathfrak {p}}f _{\\mathfrak {p}}$ -cycle corresponding to the right shift of indices with respect to the lexicographic ordering, i.e., $\\begin{array}{ccccccccc}(1,1)& \\mapsto &(1,2) &\\mapsto &\\cdots &\\mapsto &(1,e_{\\mathfrak {p}}) & \\mapsto &\\\\(2,1) & \\mapsto & (2,2) & \\mapsto & \\cdots &\\mapsto &(2,e_{\\mathfrak {p}}) &\\mapsto & \\\\&&&&\\vdots &&&&\\\\(f_{\\mathfrak {p}},1)& \\mapsto &(f_{\\mathfrak {p}},2)&\\mapsto & \\cdots &\\mapsto &(f_{\\mathfrak {p}},e_{\\mathfrak {p}}) &\\mapsto &(1,1) .\\end{array}$ Let $E \\subset {\\overline{{\\mathbb {Q}}}}$ be a number field containing the image of $\\theta $ for all $\\theta \\in \\Sigma $ , let ${\\mathcal {O}}$ be the completion of ${\\mathcal {O}}_E$ at the prime determined by the choice of ${\\overline{{\\mathbb {Q}}}}\\rightarrow \\overline{{\\mathbb {Q}}}_p$ , and let $\\mathbb {F}$ be its residue field.", "For any ${\\mathcal {O}}_{F,p} = {\\mathcal {O}}_F \\otimes \\mathbb {Z}_p$ -module $M$ , we write $M = \\bigoplus _{{\\mathfrak {p}}\\in S_p} M_{\\mathfrak {p}}$ for the decomposition obtained from that of ${\\mathcal {O}}_{F,p} \\cong \\prod _{{\\mathfrak {p}}\\in S_p} {\\mathcal {O}}_{F,{\\mathfrak {p}}}.$ Similarly for any $W({\\mathcal {O}}_F/{\\mathfrak {p}}) \\otimes _{\\mathbb {Z}_p} {\\mathcal {O}}$ -module $N$ , we have a decomposition $N = \\bigoplus _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}} N_\\tau $ obtained from $W({\\mathcal {O}}_F/{\\mathfrak {p}}) \\otimes _{\\mathbb {Z}_p} {\\mathcal {O}}\\cong \\prod _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}} {\\mathcal {O}}.$ In particular, for any ${\\mathcal {O}}_F\\otimes {\\mathcal {O}}$ -module $M$ , we have the decomposition $M = \\bigoplus _{{\\mathfrak {p}}\\in S_p} M_{\\mathfrak {p}}= \\bigoplus _{\\tau \\in \\Sigma _0} M_\\tau ,$ where we simply write $M_\\tau $ for $M_{{\\mathfrak {p}},\\tau }$ .", "We also write $M_{{\\mathfrak {p}},i}$ for $M_\\tau $ if $\\tau = \\tau _{{\\mathfrak {p}},i}$ ; thus $M_{{\\mathfrak {p}},i}$ is the summand of the ${\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _{\\mathbb {Z}_p} {\\mathcal {O}}$ -module $M_{\\mathfrak {p}}$ on which $W({\\mathcal {O}}_F/{\\mathfrak {p}})$ acts via $\\tau _{{\\mathfrak {p}},i}$ .", "We also fix a choice of uniformizer $\\varpi _{\\mathfrak {p}}$ for each ${\\mathfrak {p}}\\in S_p$ .", "We let $f_{\\mathfrak {p}}(u)$ denote the minimal polynomial of $\\varpi _{\\mathfrak {p}}$ over $W({\\mathcal {O}}_F/{\\mathfrak {p}})$ , and let $f_\\tau $ denote its image in ${\\mathcal {O}}[u]$ for each $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ ; thus $u \\mapsto \\varpi _{\\mathfrak {p}}\\otimes 1$ induces an isomorphism ${\\mathcal {O}}[u]/(f_\\tau (u)) \\stackrel{\\sim }{\\longrightarrow } {\\mathcal {O}}_{F,{\\mathfrak {p}}} \\otimes _{W({\\mathcal {O}}_F/{\\mathfrak {p}}),\\tau } {\\mathcal {O}}.$ Furthermore we have $f_\\tau (u) = \\prod _{\\theta \\in \\Sigma _\\tau } (u - \\theta (\\varpi _{\\mathfrak {p}}))$ , and we define elements $ \\begin{array}{rcc}s_{\\tau ,j} &=& (u-\\theta _{{\\mathfrak {p}},i,1}(\\varpi _{\\mathfrak {p}}))\\cdots (u-\\theta _{{\\mathfrak {p}},i,j}(\\varpi _{\\mathfrak {p}}))\\\\\\mbox{and}\\quad t_{\\tau ,j} &=& (u-\\theta _{{\\mathfrak {p}},i,j+1}(\\varpi _{\\mathfrak {p}}))\\cdots (u-\\theta _{{\\mathfrak {p}},i,e_{\\mathfrak {p}}}(\\varpi _{\\mathfrak {p}}))\\end{array}$ of ${\\mathcal {O}}[u]/(f_\\tau (u))$ for $j=0,\\ldots ,e_{\\mathfrak {p}}$ (with the obvious convention that $s_{\\tau ,0} = t_{\\tau ,e_{\\mathfrak {p}}} = 1$ ).", "Note that each of the ideals $(s_{\\tau ,j})$ and $(t_{\\tau ,j})$ is the other's annihilator; furthermore the quotients of ${\\mathcal {O}}[u]/(f_\\tau (u))$ by these ideals are free over ${\\mathcal {O}}$ , and the corresponding ideals in ${\\mathcal {O}}_{F,{\\mathfrak {p}}} \\otimes _{W({\\mathcal {O}}_F/{\\mathfrak {p}}),\\tau } {\\mathcal {O}}$ may be described as kernels of projection maps to products of copies of ${\\mathcal {O}}$ , hence depend only on $j$ and the ordering of embeddings, and not on the choice of uniformizer $\\varpi _{\\mathfrak {p}}$ .", "For an invertible ${\\mathcal {O}}_F$ -module $L$ and an embedding $\\theta = \\theta _{{\\mathfrak {p}},i,j} \\in \\Sigma _\\tau $ , we define $L_\\theta $ to be the free rank one ${\\mathcal {O}}$ -module $ L_\\theta = t_{\\tau ,j}(L\\otimes {\\mathcal {O}})_\\tau \\otimes _{{\\mathcal {O}}[u],\\theta } {\\mathcal {O}}.", "$ Note that $L_\\theta $ is not to be identified with $L\\otimes _{{\\mathcal {O}}_F,\\theta } {\\mathcal {O}}$ ; rather there is a canonical map $L_\\theta \\rightarrow L\\otimes _{{\\mathcal {O}}_F,\\theta } {\\mathcal {O}}$ which is an isomorphism if and only if $j = e_{\\mathfrak {p}}$ .", "If $L$ and $L^{\\prime }$ are invertible ${\\mathcal {O}}_F$ -modules, we will write $LL^{\\prime }$ for $L\\otimes _{{\\mathcal {O}}_F}L^{\\prime }$ and $L^{-1}$ for ${\\operatorname{Hom}\\,}_{{\\mathcal {O}}_F}(L,{\\mathcal {O}}_F)$ .", "Note that there are natural maps $L_\\theta \\otimes _{{\\mathcal {O}}} L^{\\prime }_\\theta \\rightarrow (LL^{\\prime })_\\theta $ and $(L^{-1})_\\theta \\rightarrow {\\operatorname{Hom}\\,}_{{\\mathcal {O}}}(L_\\theta ,{\\mathcal {O}})$ , but again these are isomorphisms if and only if $j = e_{\\mathfrak {p}}$ ." ], [ "Pappas–Rapoport models", "In this section we recall the description of the Hilbert modular variety as a coarse moduli space for abelian varieties with additional structure, along with the construction by Pappas and Rapoport of a smooth integral model (see [29] and [33]).", "Let $G = {\\operatorname{Res}}_{F/\\mathbb {Q}} \\operatorname{GL}_2$ and let $U$ be an open compact subgroup of $\\operatorname{GL}_2(\\widehat{{\\mathcal {O}}}_F) \\subset \\operatorname{GL}_2({\\mathbb {A}}_{F,)=G({\\mathbb {A}}_ of the form U_pU^p, where U_p = \\operatorname{GL}_2({\\mathcal {O}}_{F,p}) and U^p \\subset \\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)}) is sufficiently small,in a sense to be specified below.", "}We consider the functor which associates, to a locally Noetherian {\\mathcal {O}}-scheme S, the set of isomorphism classes of data(A,\\iota ,\\lambda ,\\eta ,{\\mathcal {F}}^\\bullet ), where:\\begin{itemize}\\item s:A \\rightarrow S is an abelian scheme of relative dimension d;\\item \\iota : {\\mathcal {O}}_F\\rightarrow {\\operatorname{End}\\,}_S(A) is an embedding such that (s_\\Omega ^1_{A/S})_{\\mathfrak {p}}is, locally on S, free of rank e_{\\mathfrak {p}} over W({\\mathcal {O}}_F/{\\mathfrak {p}})\\otimes _{\\mathbb {Z}_p} {\\mathcal {O}}_S for each {\\mathfrak {p}}\\in S_p;\\item \\lambda is an {\\mathcal {O}}_F-linear quasi-polarization of A such that for each connected componentS_i of S, \\lambda induces an isomorphism {\\mathfrak {c}}_i{\\mathfrak {d}}\\otimes _{{\\mathcal {O}}_F} A_{S_i} \\rightarrow A_{S_i}^\\vee for some fractional ideal {\\mathfrak {c}}_i of F prime to p;\\item \\eta is a level U^p structure on A,i.e., for a choice of geometric point \\overline{s}_i on each connected component S_i of S,the data of a \\pi _1(S_i,\\overline{s}_i)-invariant U^p-orbit of\\widehat{{\\mathcal {O}}}_F^{(p)} = {\\mathcal {O}}_F\\otimes \\widehat{{\\mathbb {Z}}}^{(p)}-linear isomorphisms\\footnote {Note theconventions in place with respect to the different, which are motivated by the point of view thatwe wish to systematically trivialize modules defined by cohomological constructions.", "}\\eta _i :(\\widehat{{\\mathcal {O}}}_F^{(p)})^2 \\rightarrow {\\mathfrak {d}}\\otimes _{{\\mathcal {O}}_F} T^{(p)}(A_{\\overline{s}_i}),where T^{(p)} denotes the product over \\ell \\ne p of the \\ell -adic Tate modules,and g \\in U^p acts on \\eta _i by pre-composing with right multiplication by g^{-1};\\item {\\mathcal {F}}^\\bullet is a collection of Pappas--Rapoport filtrations, i.e., foreach \\tau = \\tau _{{\\mathfrak {p}},i} \\in \\Sigma _0, an increasing filtration of{\\mathcal {O}}_{F,{\\mathfrak {p}}} \\otimes _{W({\\mathcal {O}}_F/{\\mathfrak {p}}),\\tau } {\\mathcal {O}}_S-modules0 = {\\mathcal {F}}_\\tau ^{(0)} \\subset {\\mathcal {F}}_\\tau ^{(1)} \\subset \\cdots \\subset {\\mathcal {F}}_\\tau ^{(e_{\\mathfrak {p}}- 1)} \\subset {\\mathcal {F}}_\\tau ^{(e_{\\mathfrak {p}})}= (s_\\Omega _{A/S}^1)_\\tau such that for j=1,\\ldots ,e_{{\\mathfrak {p}}}, the quotient{{\\mathcal {L}}}_{{\\mathfrak {p}},i,j} := {\\mathcal {F}}_\\tau ^{(j)}/{\\mathcal {F}}_\\tau ^{(j-1)}is a line bundle on S on which {\\mathcal {O}}_F acts via \\theta _{{\\mathfrak {p}},i,j}.\\end{itemize}}$ The proof of [12] does not assume $p$ is unramified in $F$ , and shows that if $U^p$ is sufficiently small and $\\alpha $ is an automorphism of a triple $(A,\\iota ,\\eta )$ over a connected scheme $S$ , then $\\alpha = \\iota (\\mu )$ for some $\\mu \\in U \\cap {\\mathcal {O}}_F^\\times $ .", "If we assume further that $-1 \\notin U\\cap {\\mathcal {O}}_F^\\times $ , then it follows from standard arguments that the functor above is representable by an infinite disjoint union of quasi-projective schemes over ${\\mathcal {O}}$ , which we denote by $\\widetilde{Y}_U$ , and the argument in the proof of [33] shows that $\\widetilde{Y}_U$ is smooth of relative dimension $d$ over ${\\mathcal {O}}$ .", "Furthermore defining an action of ${\\mathcal {O}}_{F,(p),+}^\\times $ on $\\widetilde{Y}_U$ by $\\nu \\cdot (A,\\iota ,\\lambda ,\\eta ,{\\mathcal {F}}^\\bullet ) = (A,\\iota ,\\nu \\lambda ,\\eta ,{\\mathcal {F}}^\\bullet )$ (as in [11]), we see that the resulting action of ${\\mathcal {O}}_{F,(p),+}^\\times /(U\\cap {\\mathcal {O}}_F^\\times )^2$ is free and the quotient is representable by a smooth quasi-projective scheme over ${\\mathcal {O}}$ , which we denote by $Y_U$ .", "We also have a natural right action of $\\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)}) on the inverse system of schemes Y_Uinduced by pre-composing the level structure \\eta with right-multiplication by g^{-1}.", "More precisely suppose thatU_1 and U_2 are as above (with U_1^p and U_2^p sufficiently small) and g \\in \\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)})is such that g^{-1}U_1g \\subset U_2.", "Letting (A,\\iota ,\\lambda ,\\eta ,{\\mathcal {F}}^\\bullet ) denote the universal objectover \\widetilde{Y}_{U_1}, there is a prime-to-p quasi-isogeny A \\rightarrow A^{\\prime } of abelian varietieswith {\\mathcal {O}}_F-action inducing isomorphisms {\\mathfrak {d}}\\otimes _{{\\mathcal {O}}_F} T^{(p)}(A^{\\prime }_{\\overline{s}_i}) \\cong \\eta _i((\\widehat{{\\mathcal {O}}}_{F}^{(p)})^2g^{-1})for each i, from which we obtain a level U_2-structure \\eta ^{\\prime } = \\eta \\circ r_{g^{-1}} on A^{\\prime } (where r_{g^{-1}} denotesright multiplication by g^{-1}).", "Together with the otherdata inherited from A, we obtain an object (A^{\\prime },\\iota ^{\\prime },\\lambda ^{\\prime },\\eta ^{\\prime },{\\mathcal {F}}^{\\prime \\bullet }) corresponding to amorphism \\widetilde{\\rho }_g: \\widetilde{Y}_{U_1} \\rightarrow \\widetilde{Y}_{U_2} and descending to a morphismY_{U_1} \\rightarrow Y_{U_2} which we denote \\rho _g.", "These morphisms satisfy the evident compatibility\\rho _{g_2}\\circ \\rho _{g_1} = \\rho _{g_1g_2} whenever g_1^{-1}U_1g_1 \\subset U_2and g_2^{-1}U_2g_2 \\subset U_3.", "}Finally we remark that the schemes Y_U define smooth integral modelsover {\\mathcal {O}} for the Hilbert modular varieties associated to the group G (with the usual choice of Shimura datum),and their generic fibres and resulting \\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)})-action may be identified with those obtained from asystem of canonical models.", "In particular for any {\\mathcal {O}}\\rightarrow \\mathbb {C}, we have isomorphismsY_U(\\mathbb {C}) \\cong \\operatorname{GL}_2(F)_+\\backslash (\\mathfrak {H}^\\Sigma \\times \\operatorname{GL}_2({\\mathbb {A}}_{F,)/U)\\cong \\operatorname{GL}_2({\\mathcal {O}}_{F,(p)})_+\\backslash (\\mathfrak {H}^\\Sigma \\times \\operatorname{GL}_2({\\mathbb {A}}^{(p)}_{F,)/U^p)compatible with the right action of \\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)}) on the inverse system,and inducing a bijection between the set of geometric components of Y_U and{\\mathbb {A}}_{F,^\\times /F_+^\\times \\det (U) \\cong ({\\mathbb {A}}_{F,^{(p)})^\\times /{\\mathcal {O}}_{F,(p),+}^\\times \\det (U^p)These isomorphisms arise in turn from ones of the form\\widetilde{Y}_U(\\mathbb {C}) \\cong \\operatorname{SL}_2({\\mathcal {O}}_{F,(p)})\\backslash (\\mathfrak {H}^\\Sigma \\times \\operatorname{GL}_2({\\mathbb {A}}^{(p)}_{F,)/U^p),under which the set of geometric components of \\widetilde{Y}_U is described by({\\mathbb {A}}_{F,^{(p)})^\\times /\\det (U^p).", "}}}}}\\section {Automorphic bundles} \\subsection {Pairings and duality} Before introducing the line bundles whose sections define the automorphic forms of interest in the paper, wepresent a plethora of perfect pairings provided by Poincaré duality.", "}We fix a sufficiently small U as in §\\ref {sec:models} andconsider the de Rham cohomology sheaves {\\mathcal {H}}^1_{\\operatorname{dR}}(A/S) = \\mathbb {R}^1s_\\Omega ^\\bullet _{A/S} onthe universal abelian scheme A over S = \\widetilde{Y}_U.", "Recall that these sheaves are locallyfree of rank two over {\\mathcal {O}}_F\\otimes {\\mathcal {O}}_S.", "Furthermore Poincaré duality and the polarization \\lambda induce an{\\mathcal {O}}_F\\otimes {\\mathcal {O}}_S-linear isomorphism\\begin{array}{ccccc}{\\mathcal {H}}^1_{\\operatorname{dR}}(A/S)& \\stackrel{\\sim }{\\longrightarrow } & {\\mathcal {H}om}_{{\\mathcal {O}}_S}({\\mathcal {H}}^1_{\\operatorname{dR}}(A^\\vee /S),{\\mathcal {O}}_S)&\\stackrel{\\sim }{\\longleftarrow }& {\\mathcal {H}om}_{{\\mathcal {O}}_S}({\\mathcal {H}}^1_{\\operatorname{dR}}({\\mathfrak {c}}{\\mathfrak {d}}\\otimes _{{\\mathcal {O}}_F} A)/S),{\\mathcal {O}}_S)\\\\&&&& \\wr \\parallel \\\\&&&& {\\mathcal {H}om}_{{\\mathcal {O}}_S}({\\mathfrak {d}}^{-1} \\otimes _{{\\mathcal {O}}_F}{\\mathcal {H}}^1_{\\operatorname{dR}}(A/S),{\\mathcal {O}}_S)\\end{array}(where {\\mathfrak {c}} depends on the connected component of S and disappears from the last expression since itis prime to p).", "This in turn induces {\\mathcal {O}}_{F,{\\mathfrak {p}}} \\otimes _{W({\\mathcal {O}}_F/{\\mathfrak {p}}),\\tau } {\\mathcal {O}}_S-linear isomorphisms\\begin{equation} {\\mathcal {H}}^1_{\\operatorname{dR}}(A/S)_\\tau \\cong {\\mathcal {H}om}_{{\\mathcal {O}}_S}({\\mathfrak {d}}_{\\mathfrak {p}}^{-1} \\otimes _{{\\mathcal {O}}_{F,{\\mathfrak {p}}}} {\\mathcal {H}}^1_{\\operatorname{dR}}(A/S)_\\tau ,{\\mathcal {O}}_S) \\end{equation}which we view as defining a perfect {\\mathcal {O}}_S-bilinear pairing\\langle \\cdot ,\\cdot \\rangle _\\tau ^0 between {\\mathcal {H}}_\\tau := {\\mathcal {H}}^1_{\\operatorname{dR}}(A/S)_\\tau and {\\mathfrak {d}}_{{\\mathfrak {p}}}^{-1}\\otimes _{{\\mathcal {O}}_{F,{\\mathfrak {p}}}} {\\mathcal {H}}_\\tau for \\tau \\in \\Sigma _{{\\mathfrak {p}},0}.Furthermore the pairing is alternating in the sense that \\langle x, c \\otimes y \\rangle _\\tau ^0 = - \\langle y, c \\otimes x\\rangle _\\tau ^0on sections.", "Alternatively, we may apply the canonical {\\mathcal {O}}_F\\otimes R-linear isomorphism{\\operatorname{Hom}\\,}_{{\\mathcal {O}}_F\\otimes R} (M, {\\mathcal {O}}_F \\otimes R) \\stackrel{\\sim }{\\longrightarrow } {\\operatorname{Hom}\\,}_R({\\mathfrak {d}}^{-1} \\otimes _{{\\mathcal {O}}_F} M,R)induced by the trace for any {\\mathcal {O}}_F \\otimes R-module M to obtain an {\\mathcal {O}}_F \\otimes {\\mathcal {O}}_S-linear isomorphism\\begin{equation} {\\mathcal {H}}^1_{\\operatorname{dR}}(A/S)\\stackrel{\\sim }{\\longrightarrow } {\\mathcal {H}om}_{{\\mathcal {O}}_F\\otimes {\\mathcal {O}}_S}({\\mathfrak {c}}^{-1} \\otimes _{{\\mathcal {O}}_F}{\\mathcal {H}}^1_{\\operatorname{dR}}(A/S),{\\mathcal {O}}_F \\otimes {\\mathcal {O}}_S),\\end{equation}and hence a perfect alternating {\\mathcal {O}}_{F,{\\mathfrak {p}}} \\otimes _{W({\\mathcal {O}}_F/{\\mathfrak {p}}),\\tau } {\\mathcal {O}}_S-bilinear pairing \\langle \\cdot ,\\cdot \\rangle _\\tau on {\\mathcal {H}}_\\tau .", "}Note that {\\mathcal {H}}_\\tau is locally free of rank two over {\\mathcal {O}}_{F,{\\mathfrak {p}}} \\otimes _{W({\\mathcal {O}}_F/{\\mathfrak {p}}),\\tau } {\\mathcal {O}}_S, hence avector bundle of rank 2e_{\\mathfrak {p}} over {\\mathcal {O}}_S.", "Furthermore (s_\\Omega ^1_{A/S})_\\tau is a subbundle of {\\mathcal {H}}_\\tau of rank e_{\\mathfrak {p}}, but is not locally free over {\\mathcal {O}}_{F,{\\mathfrak {p}}} \\otimes _{W({\\mathcal {O}}_F/{\\mathfrak {p}}),\\tau } {\\mathcal {O}}_S if e_{\\mathfrak {p}}> 1(in which case the failure is on a closed subscheme of codimension one), and more generally {\\mathcal {F}}_\\tau ^{(j)}is a subbundle of rank j for j=0,1,\\ldots ,e_{\\mathfrak {p}}.", "}Recall that for j=0,1,\\ldots ,e_{\\mathfrak {p}}, we defined (see (\\ref {eqn:st})) elements s_{\\tau ,j} and t_{\\tau ,j}of the ring {\\mathcal {O}}_{F,{\\mathfrak {p}}} \\otimes _{W({\\mathcal {O}}_F/{\\mathfrak {p}}),\\tau } {\\mathcal {O}}\\cong {\\mathcal {O}}[u]/(f_\\tau (u)), wheref_\\tau is the image under \\tau of the Eisenstein polynomial associated to our choice of uniformizer \\varpi _{\\mathfrak {p}}.In the following, we shall fix \\tau and omit the subscripts \\tau , {\\mathfrak {p}}and i to disencumber the notation; we also write simply W for W({\\mathcal {O}}_F/{\\mathfrak {p}}).Note that since {\\mathcal {H}} is locally free over {\\mathcal {O}}_{F,{\\mathfrak {p}}} \\otimes _W {\\mathcal {O}}_S and{\\mathcal {F}}^{(j)} is annihilated by s_j, we have that {\\mathcal {F}}^{(j)} is in fact a subbundleof t_j{\\mathcal {H}}.", "}For a subsheaf $ EH$ of $ OF,p W OS$-submodules,we define $ E$ to be its orthogonal complement under the pairing $ ,$,i.e., the kernel of the morphism:$${\\mathcal {H}}\\stackrel{\\sim }{\\longrightarrow } {\\mathcal {H}om}_{{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S}({\\mathcal {H}},{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S)\\longrightarrow {\\mathcal {H}om}_{{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S}({\\mathcal {E}},{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S),$$or equivalently the orthogonal complement of $ dp-1 OF,p E$ underthe pairing $ ,0$.", "Note from the latter descriptionthat if $ E$ is an $ OS$-subbundle of $ H$, then so is $ E$.$ Lemma 2.2.1 We have the equality $({\\mathcal {F}}^{(j)})^\\perp = t_j^{-1} {\\mathcal {F}}^{(j)}$ for $j=0,1,\\ldots ,e$ , where $t_j^{-1} {\\mathcal {F}}^{(j)} = \\ker ({\\mathcal {H}}\\stackrel{t_j}{\\longrightarrow } t_j {\\mathcal {H}}\\longrightarrow t_j{\\mathcal {H}}/{\\mathcal {F}}^{(j)})$ is the preimage sheaf of ${\\mathcal {F}}^{(j)}$ under $t_j$ .", "Proof.We prove the lemma by induction on $j$ , the case of $j=0$ being obvious.", "Suppose then that $1 \\le j \\le e$ and that $({\\mathcal {F}}^{(j-1)})^\\perp = t_{j-1}^{-1} {\\mathcal {F}}^{(j-1)}$ .", "Note that $({\\mathcal {F}}^{(j)})^\\perp $ and $t_j^{-1} {\\mathcal {F}}^{(j)}$ are both kernels of surjective morphisms from ${\\mathcal {H}}$ to vector bundles of rank $j$ on $S$ , so each is a subbundle of rank $2e-j$ , and hence it suffices to prove the inclusion $t_j^{-1}{\\mathcal {F}}^{(j)} \\subset ({\\mathcal {F}}^{(j)})^\\perp $ .", "To do so, we may work locally on $S$ , and assume that ${\\mathcal {F}}^{(j)}(V) = {\\mathcal {F}}^{(j-1)}(V) \\oplus R x_j$ where $V = {\\operatorname{Spec}\\,}R$ is a Noetherian open subscheme of $\\widetilde{Y}_U$ such that ${\\mathcal {H}}\|_V$ is free over ${\\mathcal {O}}_{F,{\\mathfrak {p}}} \\otimes _W {\\mathcal {O}}_V$ and $x_j \\in {\\mathcal {H}}(V)$ satisfies $(u-\\theta _j(\\varpi ))x_j \\in {\\mathcal {F}}^{(j-1)}(V)$ .", "In particular $x_j = t_jy_j$ for some $y_j \\in t_{j-1}^{-1}{\\mathcal {F}}^{(j-1)}(V)$ , so that $t_j^{-1}{\\mathcal {F}}^{(j)}(V) = t_j^{-1}{\\mathcal {F}}^{(j-1)}(V) \\oplus Ry_j\\quad \\mbox{and}\\quad ({\\mathcal {F}}^{(j)}(V))^\\perp = ({\\mathcal {F}}^{(j-1)}(V))^\\perp \\cap (Rx_j)^\\perp .$ Note that if $w \\in t_j^{-1}{\\mathcal {F}}^{(j)}(V)$ , then $t_{j-1}w = (u-\\theta _j(\\varpi )) t_j w \\in (u-\\theta _j(\\varpi )){\\mathcal {F}}^{(j)}(V) \\subset {\\mathcal {F}}^{(j-1)}(V),$ so that $w \\in t_{j-1}^{-1} {\\mathcal {F}}^{(j-1)}(V) = ({\\mathcal {F}}^{(j-1)})(V))^\\perp $ .", "Furthermore if $w \\in t_j^{-1}{\\mathcal {F}}^{(j-1)}(V)$ , then $\\langle w,x_j \\rangle = \\langle w, t_jy_j \\rangle = \\langle t_jw, y_j \\rangle $ since $t_jw \\in {\\mathcal {F}}^{(j-1)}$ and $y_j \\in t_{j-1}^{-1}{\\mathcal {F}}^{(j-1)}(V) = ({\\mathcal {F}}^{(j-1)}(V))^\\perp $ .", "Finally since the pairing is alternating, we have $\\langle y_j , x_j \\rangle = \\langle y_j,t_jy_j \\rangle = \\langle t_jy_j,y_j \\rangle = \\langle x_j,y_j \\rangle = - \\langle y_j,x_j \\rangle ,$ which implies that $\\langle y_j,x_j \\rangle = 0$ (since $\\widetilde{Y}_U$ is flat over $\\mathbb {Z}_2$ if $p=2$ ).", "We have now shown that $t_j^{-1}{\\mathcal {F}}^{(j)}(V) \\subset ({\\mathcal {F}}^{(j-1)}(V))^\\perp \\cap (Rx_j)^\\perp = ({\\mathcal {F}}^{(j)}(V))^\\perp $ , as required.", "$\\square $ We now define ${\\mathcal {G}}^{(j)} = (u-\\theta _j(\\varpi ))^{-1}{\\mathcal {F}}^{(j-1)}$ for $j=1,\\ldots ,e$ .", "Thus ${\\mathcal {G}}^{(j)}$ is a rank $j+1$ subbundle of ${\\mathcal {H}}$ , and we have inclusions of subbundles ${\\mathcal {F}}^{(j-1)} \\subset {\\mathcal {F}}^{(j)} \\subset {\\mathcal {G}}^{(j)}$ , so that ${{\\mathcal {L}}}_j := {\\mathcal {F}}^{(j)}/{\\mathcal {F}}^{(j-1)}$ is a rank one subbundle of the rank two vector bundle ${{\\mathcal {P}}}_j := {\\mathcal {G}}^{(j)}/{\\mathcal {F}}^{(j-1)}$ .", "Furthermore all the inclusions are morphisms of ${\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W {\\mathcal {O}}_S$ -modules, and ${\\mathcal {O}}_F$ acts on ${{\\mathcal {P}}}_j$ via $\\theta _j$ .", "Note that ${\\mathcal {G}}^{(j)}$ is annihilated by $s_j$ , so that ${\\mathcal {G}}^{(j)} \\subset t_j{\\mathcal {H}}$ , and we have $t_j^{-1}{\\mathcal {G}}^{(j)} = t_{j-1}^{-1}{\\mathcal {F}}^{(j-1)} = ({\\mathcal {F}}^{j-1})^\\perp $ by Lemma REF , from which it follows also that $t_j^{-1}{\\mathcal {F}}^{(j-1)} = t_j^{-1}(t_j^{-1}{\\mathcal {G}}^{(j)})^\\perp = ({\\mathcal {G}}^{(j)})^\\perp .$ (The last equality can be seen by arguing locally on sections, or by noting that the diagram ${ {\\mathcal {H}}[r][d] & {\\mathcal {H}om}_{{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S} ({\\mathcal {H}},{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S) [r][d] &{\\mathcal {H}om}_{{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S}({\\mathcal {G}}^{(j)},{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S) [d] \\\\{\\mathcal {H}}[r]& {\\mathcal {H}om}_{{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S} ({\\mathcal {H}},{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S) [r]&{\\mathcal {H}om}_{{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S}(t_j^{-1}{\\mathcal {G}}^{(j)},{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S)}$ commutes, where the left horizontal morphisms are defined by the pairing, the right by restriction, and all the vertical morphisms by $t_j$ .", "The kernel of the composite along the top is $({\\mathcal {G}}^{(j)})^\\perp $ , whereas $t_j^{-1}(t_j^{-1}{\\mathcal {G}}^{(j)})^\\perp $ is the kernel of the composite along the left and bottom.", "Since $t_j:t_j^{-1}{\\mathcal {G}}^{(j)} \\rightarrow {\\mathcal {G}}^{(j)}$ is a surjective morphism of vector bundles, the leftmost vertical arrow is injective, so these kernels coincide.)", "Therefore multiplication by $t_j$ defines an isomorphism $({\\mathcal {F}}^{(j-1)})^\\perp /({\\mathcal {G}}^{(j)})^\\perp \\stackrel{\\sim }{\\longrightarrow }{\\mathcal {G}}^{(j)}/{\\mathcal {F}}^{(j-1)}$ , and composing its inverse with the isomorphism $({\\mathcal {F}}^{(j-1)})^\\perp /({\\mathcal {G}}^{(j)})^\\perp \\stackrel{\\sim }{\\longrightarrow } {\\mathcal {H}om}_{{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S}({\\mathcal {G}}^{(j)}/{\\mathcal {F}}^{(j-1)},{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S)$ induced by the pairing on ${\\mathcal {H}}$ , we obtain an alternating $({\\mathcal {O}}_{F,{\\mathfrak {p}}} \\otimes _W {\\mathcal {O}}_S)$ -valued pairing $\\langle \\cdot , \\cdot \\rangle _j$ on ${{\\mathcal {P}}}_j := {\\mathcal {G}}^{(j)}/{\\mathcal {F}}^{(j-1)}$ , whose description on sections is given in terms of () by $\\langle t_j x, t_j y \\rangle _j = \\langle x, t_j y \\rangle = \\langle t_j x , y \\rangle .$ Note that since ${\\mathcal {O}}_{F,{\\mathfrak {p}}}$ acts via $\\tau _j$ on ${{\\mathcal {P}}}_j$ , we in fact have the identification ${\\mathcal {H}om}_{{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S}({{\\mathcal {P}}}_j,{\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S) ={\\mathcal {H}om}_{{\\mathcal {O}}_S}({{\\mathcal {P}}}_j,{\\mathcal {I}}_j)$ where ${\\mathcal {I}}_j$ is the sheaf of ideals, and trivial rank one ${\\mathcal {O}}_S$ -subbundle, of ${\\mathcal {O}}_{F,{\\mathfrak {p}}}\\otimes _W{\\mathcal {O}}_S$ generated by the global section $\\prod _{j^{\\prime }\\ne j}(u-\\tau _{j^{\\prime }}(\\varpi _{\\mathfrak {p}}))$ .", "We thus obtain a trivialization of $\\wedge ^2_{{\\mathcal {O}}_S} {{\\mathcal {P}}}_j$ corresponding to a perfect ${\\mathcal {O}}_S$ -valued pairing $\\langle \\cdot , \\cdot \\rangle _j^0$ , which an unravelling of definitions shows is given in terms of the original pairing $\\langle \\cdot , \\cdot \\rangle ^0$ of () by the formula $\\langle t_j x , y \\rangle _j^0 = \\langle x, f^{\\prime }(\\varpi _{\\pi _{\\mathfrak {p}}})^{-1} \\otimes y \\rangle ^0$ on sections (where $f^{\\prime }$ is the derivative of the Eisenstein polynomial $f$ )." ], [ "Automorphic line bundles", "Recall that the Pappas–Rapoport model $\\widetilde{Y}_U$ is equipped with line bundlesFor the moment, we continue to suppress the fixed $\\tau = \\tau _{{\\mathfrak {p}},i} \\in \\Sigma _0$ from the notation.", "${\\mathcal {L}}_j$ , which we described in § as sub-bundles of the rank two vector bundles ${\\mathcal {P}}_j$ .", "It is natural and convenient to consider also the twists of ${\\mathcal {L}}_j$ by powers of the determinant bundle of ${\\mathcal {P}}_j$ : ${{\\mathcal {N}}}_j = \\wedge ^2_{{\\mathcal {O}}_S}{{\\mathcal {P}}}_j \\cong {{\\mathcal {L}}}_j \\otimes _{{\\mathcal {O}}_S} {{\\mathcal {M}}}_j,$ where ${{\\mathcal {M}}}_j$ is the line bundle ${{\\mathcal {P}}}_j/{{\\mathcal {L}}}_j$ .", "Note that the pairing $\\langle \\cdot , \\cdot \\rangle ^0_j$ defines an isomorphism ${{\\mathcal {M}}}_j\\stackrel{\\sim }{\\longrightarrow } {{\\mathcal {L}}}_j^{-1}$ and a trivialization ${\\mathcal {O}}_S \\stackrel{\\sim }{\\longrightarrow } {{\\mathcal {N}}}_j$ (which depends on the choice of $\\varpi $ ).", "As we will now consider these bundles for varying $\\tau $ , we resume writing the indicative subscripts; thus for $\\tau = \\tau _{{\\mathfrak {p}},i}$ , we will denote ${\\mathcal {G}}^{(j)}$ by ${\\mathcal {G}}_\\tau ^{(j)}$ , ${{\\mathcal {P}}}_j$ by ${{\\mathcal {P}}}_{{\\mathfrak {p}},i,j}$ , and similarly for ${{\\mathcal {M}}}_j$ and ${{\\mathcal {N}}}_j$ .", "We also freely replace the subscript “${\\mathfrak {p}},i,j$ ” by $\\theta $ , where $\\theta = \\theta _{{\\mathfrak {p}},i,j}$ , so that for each $\\theta \\in \\Sigma $ , we have now defined a rank two vector bundle ${{\\mathcal {P}}}_\\theta $ and line bundles ${{\\mathcal {L}}}_\\theta $ , ${{\\mathcal {M}}}_\\theta $ , ${{\\mathcal {N}}}_\\theta $ on $S = \\widetilde{Y}_U$ , along with exact sequences $ 0 \\longrightarrow {{\\mathcal {L}}}_\\theta \\longrightarrow {{\\mathcal {P}}}_\\theta \\longrightarrow {{\\mathcal {M}}}_\\theta \\longrightarrow 0 $ and a trivialization of ${{\\mathcal {N}}}_\\theta = \\wedge ^2_{{\\mathcal {O}}_S} {{\\mathcal {P}}}_\\theta \\cong {\\mathcal {L}}_\\theta \\otimes _{{\\mathcal {O}}_S} {{\\mathcal {M}}}_\\theta $ .", "Furthermore the bundles ${{\\mathcal {P}}}_\\theta $ , ${{\\mathcal {L}}}_\\theta $ and ${{\\mathcal {M}}}_\\theta $ are ${\\mathcal {O}}_F\\otimes {\\mathcal {O}}_S$ -subquotients of ${\\mathcal {H}}^1_{\\operatorname{dR}}(A/S)$ on which ${\\mathcal {O}}_F$ acts via $\\theta $ .", "Recall that we have an action of ${\\mathcal {O}}_{F,(p),+}^\\times $ on $\\widetilde{Y}_U$ defined by multiplication on the quasi-polarization.", "In particular if $\\nu \\in {\\mathcal {O}}_{F,(p),+}^\\times $ , then the identification of $\\nu ^A$ with $A$ induces an ${\\mathcal {O}}_F\\otimes {\\mathcal {O}}_S$ -linear isomorphism $\\nu ^{\\mathcal {H}}^1_{\\operatorname{dR}}(A/S) \\cong {\\mathcal {H}}^1_{\\operatorname{dR}}(A/S)$ under which $\\nu ^{\\mathcal {F}}^\\bullet $ corresponds to ${\\mathcal {F}}^\\bullet $ , and we thus obtain isomorphisms $\\alpha _\\nu : \\nu ^{{\\mathcal {P}}}_\\theta \\stackrel{\\sim }{\\longrightarrow } {{\\mathcal {P}}}_\\theta $ compatible with (REF ) and satisfying $\\alpha _{\\nu ^{\\prime }\\nu } = \\alpha _\\nu \\circ \\nu ^(\\alpha _{\\nu ^{\\prime }})$ (for $\\nu ,\\nu ^{\\prime } \\in {\\mathcal {O}}_{F,(p),+}^\\times $ ).", "Recall also that the action of ${\\mathcal {O}}_{F,(p),+}^\\times $ on $\\widetilde{Y}_U$ factors through ${\\mathcal {O}}_{F,(p),+}^\\times /(U\\cap {\\mathcal {O}}_F^\\times )^2$ , the isomorphism ${\\underline{A}}\\stackrel{\\sim }{\\longrightarrow } \\nu ^{\\underline{A}}$ being defined by $\\iota (\\mu ^{-1})$ if $\\nu = \\mu ^2$ for $\\mu \\in U\\cap {\\mathcal {O}}_F^\\times $ , and one finds that the automorphism of ${{\\mathcal {P}}}_\\theta $ obtained from $\\alpha _\\nu $ is multiplication by $\\theta (\\mu )$ , so the natural action of ${\\mathcal {O}}_{F,(p),+}^\\times $ on the bundles fails to define descent data with respect to the cover $\\widetilde{Y}_U \\rightarrow Y_U$ .", "We do however obtain descent data after taking suitable tensor products or base-changes of these bundles, which we now consider.", "For any ${\\mathcal {O}}$ -algebra $R$ , we will use $\\cdot _R$ to denote the base-change to $R$ of an ${\\mathcal {O}}$ -scheme $X$ , as well as the pull-back to $X_R$ of a quasi-coherent sheaf on $X$ .", "Let $\\lbrace \\,{\\mathbf {e}}_\\theta \\,\|\\,\\theta \\in \\Sigma \\,\\rbrace $ denote the standard basis of $\\mathbb {Z}^\\Sigma $ .", "For $\\mathbf {k}= \\sum k_\\theta {\\mathbf {e}}_{\\theta }$ and $\\mathbf {l}= \\sum l_\\theta {\\mathbf {e}}_\\theta \\in \\mathbb {Z}^\\Sigma $ , we define the line bundle $\\widetilde{{\\mathcal {A}}}_{\\mathbf {k},\\mathbf {l}} = \\bigotimes _{\\theta \\in \\Sigma } \\left( {{\\mathcal {L}}}_\\theta ^{\\otimes k_\\theta } \\otimes {{\\mathcal {N}}}_\\theta ^{\\otimes l_\\theta } \\right)\\cong \\bigotimes _{\\theta \\in \\Sigma } \\left( {{\\mathcal {L}}}_\\theta ^{\\otimes k_\\theta + l_\\theta } \\otimes {{\\mathcal {M}}}_\\theta ^{\\otimes l_\\theta } \\right)$ on $S = \\widetilde{Y}_U$ , where all tensor products are over ${\\mathcal {O}}_S$ .", "For $\\mathbf {n}= \\sum n_\\theta {\\mathbf {e}}_\\theta \\in \\mathbb {Z}^\\Sigma $ , we let $\\chi _{\\mathbf {n}}:{\\mathcal {O}}_F^\\times \\rightarrow {\\mathcal {O}}^\\times $ denote the character defined by $\\chi _{\\mathbf {n}}(\\mu ) = \\prod _{\\theta } \\theta (\\mu )^{n_\\theta }$ , and we let $\\chi _{\\mathbf {n},R}$ denote the associated $R^\\times $ -valued character.", "If $\\mathbf {k}$ , $\\mathbf {l}$ , $R$ and $U$ are such that $\\chi _{\\mathbf {k}+2\\mathbf {l},R}$ is trivial on ${\\mathcal {O}}_F^\\times \\cap U$ , then the action of ${\\mathcal {O}}_{F,(p),+}^\\times $ on $\\widetilde{{\\mathcal {A}}}_{\\mathbf {k},\\mathbf {l},R}$ (over its action on $\\widetilde{Y}_{U,R}$ ) factors through ${\\mathcal {O}}_{F,(p),+}^\\times /(U\\cap {\\mathcal {O}}_F^\\times )^2$ and hence defines descent data, in which case we denote the resulting line bundle on $Y_{U,R}$ by ${\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},R}$ .", "Definition 2.3.1 For $\\mathbf {k}$ , $\\mathbf {l}$ , $U$ and $R$ as above, we call ${\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},R}$ the automorphic line bundle of weight $(\\mathbf {k},\\mathbf {l})$ on $Y_{U,R}$ , and we define the space of Hilbert modular forms of weight $(\\mathbf {k},\\mathbf {l})$ and level $U$ with coefficients in $R$ to be $M_{\\mathbf {k},\\mathbf {l}}(U;R) : = H^0(Y_{U,R} ,{\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},R}).$ We note some general situations in which this space is defined: The paritious setting: if $w=k_\\theta + 2l_\\theta $ is independent of $\\theta $ , then $\\chi _{\\mathbf {k}+2\\mathbf {l}}(\\mu ) = \\mathrm {Nm}_{F/\\mathbb {Q}}(\\mu )^w = 1$ for all $\\mu \\in U \\cap {\\mathcal {O}}_F^\\times $ (assuming only $U$ is small enough that $U\\cap {\\mathcal {O}}_F^\\times $ has no elements of norm $-1$ if $w$ is odd).", "The mod $p$ setting: if $R$ is any $\\mathbb {F}= {\\mathcal {O}}/{\\mathfrak {m}}_{\\mathcal {O}}$ -algebra and $U$ is sufficiently small that $\\mu \\equiv 1 \\bmod {\\mathfrak {p}}$ for all $\\mu \\in U \\cap {\\mathcal {O}}_F^\\times $ and ${\\mathfrak {p}}\\in S_p$ , then $\\theta (\\mu ) \\equiv 1 \\bmod {\\mathfrak {m}}_{\\mathcal {O}}$ for all $\\theta \\in \\Sigma $ , so $\\chi _{\\mathbf {k}+2\\mathbf {l},\\mathbb {F}}$ is trivial on $U \\cap {\\mathcal {O}}_F^\\times $ , and hence so is $\\chi _{\\mathbf {k}+2\\mathbf {l},R}$ .", "The torsion setting: if $R$ is an ${\\mathcal {O}}/p^N{\\mathcal {O}}$ -algebra and $U$ is sufficiently small that $\\mu \\equiv 1 \\bmod p^N{\\mathcal {O}}_F$ for all $\\mu \\in U \\cap {\\mathcal {O}}_F^\\times $ , so $\\chi _{\\mathbf {k}+2\\mathbf {l},{\\mathcal {O}}/p^N{\\mathcal {O}}}$ is trivial on $U \\cap {\\mathcal {O}}_F^\\times $ , and hence so is $\\chi _{\\mathbf {k}+2\\mathbf {l},R}$ .", "We also have a natural left action of $\\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)}) on the direct limit over U of the spaces M_{\\mathbf {k},\\mathbf {l}}(U;R).More precisely suppose that U_1 and U_2 are as above and g \\in \\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)})is such that g^{-1}U_1g \\subset U_2, in which case recall that in §\\ref {sec:models}we defined a morphism \\widetilde{\\rho }_g: \\widetilde{Y}_{U_1} \\rightarrow \\widetilde{Y}_{U_2}descending to a morphism \\rho _g:Y_{U_1} \\rightarrow Y_{U_2}.", "Furthermore the morphism \\widetilde{\\rho }_g is obtained from aprime-to-p quasi-isogeny A \\rightarrow A^{\\prime } where A is the universal abelian scheme over \\widetilde{Y}_{U_1} and A^{\\prime } is the pull-backof the universal abelian scheme.", "We thus obtain isomorphisms \\widetilde{\\rho }_g^{{\\mathcal {P}}}_{\\theta ,2} \\rightarrow {{\\mathcal {P}}}_{\\theta ,1}compatible with (\\ref {eqn:Hodge1}) and the action of {\\mathcal {O}}_{F,(p),+}^\\times (augmenting the notation for the automorphic bundles on\\widetilde{Y}_{U_i} and Y_{U_i,R} with the subscript i).", "Note that U_1 \\cap {\\mathcal {O}}_F^\\times \\subset U_2 \\cap {\\mathcal {O}}_F^\\times , so if\\widetilde{{\\mathcal {A}}}_{\\mathbf {k},\\mathbf {l},R,2} descends to Y_{U_2,R}, then \\widetilde{{\\mathcal {A}}}_{\\mathbf {k},\\mathbf {l},R,1} descends to Y_{U_1,R}, and we obtainan isomorphism \\rho _g^{\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},R,2} \\cong {\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},R,1}.", "We then define [g] = [g]_{U_1,U_2}: M_{\\mathbf {k},\\mathbf {l}}(U_2;R) \\rightarrow M_{\\mathbf {k},\\mathbf {l}}(U_1;R)as the compositeH^0(Y_{U_2}, {\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},R,2}) \\stackrel{\\rho _g^}{\\longrightarrow } H^0(Y_{U_1}, \\rho _g^{\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},R,2})\\stackrel{\\sim }{\\longrightarrow } H^0(Y_{U_1}, {\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},R,1}).These maps satisfy the obvious compatibility, namely that[g_1]_{U_1,U_2} \\circ [g_2]_{U_2,U_3} = [g_1g_2]_{U_1,U_3}whenever g_1^{-1}U_1g_1 \\subset U_2 and g_2^{-1}U_2g_2 \\subset U_3, and hence define an action of\\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)}) on\\begin{equation} M_{\\mathbf {k},\\mathbf {l}}(R) := \\varinjlim _U M_{\\mathbf {k},\\mathbf {l}}(U;R)\\end{equation}(where the limit is over sufficiently small open compact Uwith respect to the maps [1]_{U_1,U_2}).", "For paritious \\mathbf {k}, \\mathbf {l} and any choice of {\\mathcal {O}}\\rightarrow \\mathbb {C}, we may identify the spacesM_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {C}) with those of holomorphic Hilbert modular forms, compatibly with the usual action(up to normalization by a factor of \|\|\\det g\|\| depending on conventions).", "}Finally we remark that the action of \\nu \\in {\\mathcal {O}}_{F,(p),+}^\\times on the trivialization of {{\\mathcal {N}}}_\\theta is given by multiplicationby \\theta (\\nu ), so their products do not descend to trivializations of line bundles on Y_{U,R}.", "However since the stabilizerof each geometric connected component of \\widetilde{Y}_U is {\\mathcal {O}}_{F,+}^\\times \\cap \\det (U), we can obtain a (non-canonical) trivializationof {\\mathcal {A}}_{{\\mathbf {0}},\\mathbf {l},R} as in \\cite [Prop.~3.6.1]{DS}, provided \\chi _{\\mathbf {l},R} is trivial on {\\mathcal {O}}_{F,+}^\\times \\cap \\det (U)and the geometric connected components of Y_U are defined over R.Furthermore the same argument as in the proof of \\cite [Lemma~4.5.1]{DS} shows the following:\\begin{proposition} If p^N R = 0, then the action of \\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)}) on M_{\\bf {0},\\mathbf {l}}(R)factors through \\det : \\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)}) \\rightarrow ({\\mathbb {A}}_{F,^{(p)})^\\times ; furthermore as a representation of ({\\mathbb {A}}_{F,^{(p)})^\\times ,M_{\\bf {0},\\mathbf {l}}(R) is isomorphic to the smooth induction of the character {\\mathcal {O}}_{F,(p),+}^\\times \\rightarrow R^\\times defined by\\alpha \\mapsto \\prod _\\theta \\theta (\\alpha )^{l_\\theta }.", "}}}}\\end{proposition}}\\subsection {The Kodaira--Spencer filtration} In this section we define a filtration on \\Omega ^1_{Y_U/{\\mathcal {O}}} whose pieces are described by automorphic bundleswith weight components k_\\theta = 2, l_\\theta = -1.", "The construction of the filtration isdue to Reduzzi and Xiao (see~\\cite [§2.8]{RX}), but their presentation is complicated by the fact they wish to provesmoothness simultaneously, and it obscures the fact that the bundles we denoted {\\mathcal {G}}^{(j)} automaticallysatisfy the orthogonality condition appearing in the definition of their counterparts in \\cite {RX}.We will show below that, with smoothness already established, one can give a more directconceptual description of the filtration and its properties.\\footnote {The simultaneous treatment in \\cite {RX}seems natural in view of the inherent overlap in the analysis of deformations needed for both results.However the decision not to appeal to the results in \\cite {Vol} and \\cite {Shu} also makes reference to a perceivedminor gap in the proof of \\cite [Prop~2.11]{Vol}; we found no such gap, nor is that result even needed, but we remarkthat we made implicit use of \\cite [Cor.~2.10]{Vol} when invoking the proof of \\cite [Prop.~6]{Shu} to concludethat \\widetilde{Y}_U is smooth over {\\mathcal {O}}.", "}Furthermore in the case p=2, the argument in \\cite {RX} appeals to a very general flatness assertion for divided powerenvelopes for which we could not find a proof or reference, so it is not used here.", "}\\begin{theorem}[Reduzzi--Xiao] There exists a decomposition\\Omega ^1_{Y_U/{\\mathcal {O}}} = \\bigoplus _{{\\mathfrak {p}}\\in S_p} \\bigoplus _{i = 1}^{f_{\\mathfrak {p}}} \\Omega ^1_{Y_U/{\\mathcal {O}},{\\mathfrak {p}},i},together with an increasing filtration\\begin{array}{rcccl}0 = {\\operatorname{Fil}\\,}^0 (\\Omega ^1_{Y_U/{\\mathcal {O}},{\\mathfrak {p}},i} ) &\\subset & {\\operatorname{Fil}\\,}^1 (\\Omega ^1_{Y_U/{\\mathcal {O}},{\\mathfrak {p}},i}) &\\subset & \\cdots \\\\&\\subset & {\\operatorname{Fil}\\,}^{e_{\\mathfrak {p}}- 1}( \\Omega ^1_{Y_U/{\\mathcal {O}},{\\mathfrak {p}},i} )&\\subset & {\\operatorname{Fil}\\,}^{e_{\\mathfrak {p}}}( \\Omega ^1_{Y_U/{\\mathcal {O}},{\\mathfrak {p}},i} ) = \\Omega ^1_{Y_U/{\\mathcal {O}},{\\mathfrak {p}},i}\\end{array}for each {\\mathfrak {p}}\\in S_p and i=1,\\ldots ,f_{\\mathfrak {p}}, such that for each j=1,\\ldots ,e_{\\mathfrak {p}}, {\\operatorname{gr}\\,}^j (\\Omega ^1_{Y_U/{\\mathcal {O}},{\\mathfrak {p}},i} ) isisomorphic to the automorphic bundle {\\mathcal {A}}_{2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\theta ,{\\mathcal {O}}}, where \\theta = \\theta _{{\\mathfrak {p}},i,j}.\\end{theorem}{\\bf Proof.", "}\\hspace{5.0pt}As usual, we first prove the analogous result for $ S: = YU$ and then descend to $ YU$.$ We let $\\delta _0: S \\stackrel{\\sim }{\\longrightarrow } \\Delta \\hookrightarrow Z_0$ denote the first infinitesimal thickening of the diagonal embedding, and we view $\\Omega ^1_{S/{\\mathcal {O}}}$ as $\\delta _0^{\\mathcal {I}}$ , where ${\\mathcal {I}}$ denotes the sheaf of ideals defining $\\Delta $ in $Z_0$ .", "Letting $s:A \\rightarrow S$ denote the universal abelian scheme, the transition maps for the crystal $R^1s_{{\\operatorname{cris}},}{\\mathcal {O}}_{A/{\\mathbb {Z}}_p}$ and canonical isomorphisms with de Rham cohomology yield an ${\\mathcal {O}}_F \\otimes {\\mathcal {O}}_{Z_0}$ -linear isomorphism $\\alpha : p_0^{\\mathcal {H}}^1_{\\operatorname{dR}}(A/S) \\stackrel{\\sim }{\\longrightarrow } q_0^{\\mathcal {H}}^1_{\\operatorname{dR}}(A/S)$ extending the identity on $S \\cong \\Delta $ , where $p_0,q_0: Z_0 \\rightarrow S$ are the two projection maps $Z_0 \\rightarrow S$ .", "Since $\\alpha $ is ${\\mathcal {O}}_F$ -linear, it follows from the definition of ${\\mathcal {P}}_{\\tau ,1} = {\\mathcal {G}}_\\tau ^{(1)}$ that $\\alpha $ restricts to an isomorphism $\\alpha _{\\tau ,1}: p_0^{\\mathcal {P}}_{\\tau ,1} \\stackrel{\\sim }{\\longrightarrow } q_0^{\\mathcal {P}}_{\\tau ,1}$ for each ${\\mathfrak {p}}\\in S_p$ and $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ .", "Furthermore since the composite $p_0^{\\mathcal {L}}_{\\tau ,1} \\hookrightarrow p_0^{\\mathcal {P}}_{\\tau ,1} \\stackrel{\\sim }{\\longrightarrow } q_0^{\\mathcal {P}}_{\\tau ,1}\\twoheadrightarrow q_0^{\\mathcal {M}}_{\\tau ,1}$ has trivial pull-back to $S \\cong \\Delta $ , it factors through a morphism $\\delta _{0,} {\\mathcal {L}}_{\\tau ,1} = p_0^{\\mathcal {L}}_{\\tau ,1} \\otimes _{{\\mathcal {O}}_{Z_0}} ({\\mathcal {O}}_{Z_0}/{\\mathcal {I}})\\longrightarrow q_0^{\\mathcal {M}}_{\\tau ,1} \\otimes _{{\\mathcal {O}}_{Z_0}} {\\mathcal {I}}= \\delta _{0,} {\\mathcal {M}}_{\\tau ,1}\\otimes _{{\\mathcal {O}}_{Z_0}} {\\mathcal {I}},$ and hence induces a morphism $\\beta _{\\tau ,1}: \\delta _{0,} ({\\mathcal {L}}_{\\tau ,1}\\otimes _{{\\mathcal {O}}_S} {\\mathcal {M}}_{\\tau ,1}^{-1}) \\longrightarrow {\\mathcal {I}}.$ We then define the sheaf of ideals ${\\mathcal {I}}_{\\tau ,1}$ on $Z_0$ to be the image of $\\beta _{\\tau ,1}$ , and we let $Z_{\\tau ,1}$ denote the subscheme of $Z_0$ defined by ${\\mathcal {I}}_{\\tau ,1}$ , and $p_{\\tau ,1}$ and $q_{\\tau ,1}$ the resulting projection maps $Z_{\\tau ,1} \\rightarrow S$ .", "By construction the pull-back of $\\beta _{\\tau ,1}$ to $Z_{\\tau ,1}$ is trivial, and hence so is that of the morphism $p_0^{\\mathcal {L}}_{\\tau ,1} \\rightarrow q_0^{\\mathcal {M}}_{\\tau ,1}$ , which implies that the pull-back of $\\alpha $ maps $p_{\\tau ,1}^{\\mathcal {L}}_{\\tau ,1} = p_{\\tau ,1}^{\\mathcal {F}}_\\tau ^{(1)}$ isomorphically to $q_{\\tau ,1}^{\\mathcal {L}}_{\\tau ,1} = q_{\\tau ,1}^{\\mathcal {F}}_\\tau ^{(1)}$ .", "It follows from ${\\mathcal {O}}_F$ -linearity that $\\alpha $ induces an isomorphism $p_{\\tau ,1}^{\\mathcal {G}}_\\tau ^{(2)} \\stackrel{\\sim }{\\longrightarrow } q_{\\tau ,1}^{\\mathcal {G}}_\\tau ^{(2)}$ (if $e_{\\mathfrak {p}}> 1$ ), and hence an isomorphism $\\alpha _{\\tau ,2}: p_{\\tau ,1}^{\\mathcal {P}}_{\\tau ,2} \\stackrel{\\sim }{\\longrightarrow } q_{\\tau ,1}^{\\mathcal {P}}_{\\tau ,2}.$ The same argument as above now yields a morphism $\\beta _{\\tau ,2}: \\delta _{0,} ({\\mathcal {L}}_{\\tau ,2}\\otimes _{{\\mathcal {O}}_S} {\\mathcal {M}}_{\\tau ,2}^{-1}) \\longrightarrow {\\mathcal {I}}/{\\mathcal {I}}_{\\tau ,1},$ whose image is that of a sheaf of ideals on $Z_0$ we denote by ${\\mathcal {I}}_{\\tau ,2}$ .", "Iterating the above construction thus yields, for each $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ , a chain of sheaves of ideals $0 = {\\mathcal {I}}_{\\tau ,0} \\subset {\\mathcal {I}}_{\\tau ,1} \\subset \\cdots \\subset {\\mathcal {I}}_{\\tau ,e_{\\mathfrak {p}}}$ on $Z_0$ such that $\\alpha $ induces isomorphisms $p_{\\tau ,j}^{\\mathcal {F}}_\\tau ^{(j)} \\stackrel{\\sim }{\\longrightarrow } q_{\\tau ,j}^* {\\mathcal {F}}_\\tau ^{(j)}$ , and surjections $\\delta _{0,} ({\\mathcal {L}}_{\\tau ,j}\\otimes _{{\\mathcal {O}}_S} {\\mathcal {M}}_{\\tau ,j}^{-1}) \\twoheadrightarrow {\\mathcal {I}}_{\\tau ,j}/{\\mathcal {I}}_{\\tau ,j-1}$ , for $j=1,\\ldots ,e_{\\mathfrak {p}}$ , where $Z_{\\tau ,j}$ denotes the closed subscheme of $Z_0$ defined by ${\\mathcal {I}}_{\\tau ,j}$ and $p_{\\tau ,j}$ , $q_{\\tau ,j}$ are the projections $Z_{\\tau ,j} \\rightarrow S$ .", "Furthermore we claim that the map $\\bigoplus _{{\\mathfrak {p}}\\in S_p} \\bigoplus _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}} {\\mathcal {I}}_{\\tau ,e_{\\mathfrak {p}}} \\rightarrow {\\mathcal {I}}$ is surjective.", "Indeed let ${\\mathcal {J}}$ denote the image and let $T$ denote the corresponding closed subscheme of $Z_0$ , so $T$ is the scheme-theoretic intersection of the $Z_{\\tau ,e_{\\mathfrak {p}}}$ , and letWith apologies for the temporary dual use of $p$ .", "$p$ , $q: T \\rightarrow S$ denote the projection maps.", "By construction $\\alpha $ pulls back to an isomorphism $p^{\\mathcal {H}}^1_{{\\operatorname{dR}}}(A/S) \\stackrel{\\sim }{\\longrightarrow } q^{\\mathcal {H}}^1_{\\operatorname{dR}}(A/S)$ under which $p^{\\mathcal {F}}_\\tau ^{(j)} \\stackrel{\\sim }{\\longrightarrow } q^{\\mathcal {F}}_\\tau ^{(j)}$ for all $\\tau $ and $j$ .", "In particular $t_\\Omega ^1_{p^A/T} = p^(s_\\Omega ^1_{A/S} ) \\stackrel{\\sim }{\\longrightarrow }q^(s_\\Omega ^1_{A/S}) = u_\\Omega ^1_{q^A/T}$ (where $t:p^A \\rightarrow T$ and $u:q^A \\rightarrow T$ are the structure morphisms), which the Grothendieck–Messing Theorem implies is induced by an isomorphism $p^A \\cong q^A$ of abelian schemes lifting the identity over $S$ .", "Since the isomorphism respects the filtrations ${\\mathcal {F}}^\\bullet $ , and the lifts of the universal auxiliary structures $\\iota $ , $\\lambda $ and $\\eta $ over $T$ are unique, it follows that $p^\\underline{A} \\cong q^\\underline{A}$ , which means that $p=q \\in S(T)$ , so $T = \\Delta $ .", "Now defining $\\Omega ^1_{S/{\\mathcal {O}},{\\mathfrak {p}},i} = \\delta _0^{\\mathcal {I}}_{\\tau _{{\\mathfrak {p}},i},e_{\\mathfrak {p}}}$ and ${\\operatorname{Fil}\\,}^j( \\Omega ^1_{S/{\\mathcal {O}},{\\mathfrak {p}},i} )= \\delta _0^{\\mathcal {I}}_{\\tau _{{\\mathfrak {p}},i},j}$ for ${\\mathfrak {p}}\\in S_p$ , $1 \\le i \\le f_{\\mathfrak {p}}$ , $1 \\le j \\le e_{\\mathfrak {p}}$ , we obtain surjective morphisms ${\\mathcal {L}}_{\\tau _{{\\mathfrak {p}},i},j}\\otimes _{{\\mathcal {O}}_S} {\\mathcal {M}}_{\\tau _{{\\mathfrak {p}},i},j}^{-1} \\twoheadrightarrow {\\operatorname{gr}\\,}^j (\\Omega ^1_{S/{\\mathcal {O}},{\\mathfrak {p}},i} )\\qquad \\mbox{and}\\qquad \\bigoplus _{{\\mathfrak {p}},i} \\Omega ^1_{S/{\\mathcal {O}},{\\mathfrak {p}},i} \\twoheadrightarrow \\Omega ^1_{S/{\\mathcal {O}}}.$ Since the ${\\mathcal {L}}_{\\tau _{{\\mathfrak {p}},i},j}\\otimes _{{\\mathcal {O}}_S} {\\mathcal {M}}_{\\tau _{{\\mathfrak {p}},i},j}^{-1}$ are line bundles and $\\Omega ^1_{S/{\\mathcal {O}}}$ is locally free of rank $d$ , it follows that all the maps are isomorphisms.", "Finally the constructions above are independent of the polarization $\\lambda $ , hence are compatible with the action of $\\nu \\in {\\mathcal {O}}_{F,(p),+}^\\times $ on $S = \\widetilde{Y}_U$ .", "More precisely, the pull-back of $\\alpha $ via the diagonal map $(\\nu ,\\nu )$ is compatible with the canonical isomorphism $\\nu ^{\\mathcal {H}}^1_{\\operatorname{dR}}(A/S) \\cong {\\mathcal {H}}^1_{\\operatorname{dR}}(A/S)$ induced by the identification of $\\nu ^A$ with $A$ , from which it follows easily that the morphisms in the construction of the filtration are invariant under the action of ${\\mathcal {O}}_{F,(p),+}^\\times $ , hence descend to give the decomposition, filtrations and isomorphisms in the statement of the theorem.", "$\\square $ Let us also note the interpretation of the Kodaira–Spencer filtration in terms of tangent spaces.", "For a closed point $y$ of $S$ corresponding to the data $\\underline{A}_0 = (A_0,\\iota _0,\\lambda _0,\\eta _0,{\\mathcal {F}}_0^\\bullet )$ over a finite extension $k$ of the residue field of ${\\mathcal {O}}$ , the fibre $T_y(S)$ of ${\\mathcal {H}om}_{{\\mathcal {O}}_S}(\\Omega ^1_{S/{\\mathcal {O}}},{\\mathcal {O}}_S)$ is canonically identified with the set of isomorphism classes of data $\\underline{A}_1$ over $k[\\epsilon ]$ lifting $\\underline{A}_0$ , and the decomposition and filtrations of the theorem yield dual decompositions of $T_y(S)$ into components $T_y(S)_{\\tau }$ with decreasing filtrations ${\\operatorname{Fil}\\,}^j( T_y(S)_{\\tau })$ .", "From the proof of the theorem one sees immediately that $\\bigoplus _{\\tau } {\\operatorname{Fil}\\,}^{j_\\tau }( T_y(S)_\\tau )$ corresponds to the set of $(A_1,\\iota _1,\\lambda _1,\\eta _1,{\\mathcal {F}}_1^\\bullet )$ such that ${\\mathcal {F}}_{1,\\tau }^{(j)}$ is the image of ${\\mathcal {F}}_{0,\\tau }^{(j)} \\otimes _k k[\\epsilon ]$ for all $\\tau $ and $j \\le j_\\tau $ under the canonical isomorphism $H^1_{\\operatorname{dR}}(A_1/k[\\epsilon ]) \\cong H^1_{\\operatorname{cris}}(A_0/k[\\epsilon ]) \\cong H^1_{\\operatorname{dR}}(A_0/k) \\otimes _k k[\\epsilon ].$ We note also that the theorem yields a canonical (Kodaira–Spencer) isomorphism $\\Omega _{Y_U/{\\mathcal {O}}}^d \\cong \\wedge _{{\\mathcal {O}}_{Y_U}}^d \\Omega ^1_{Y_U/{\\mathcal {O}}} \\cong {\\mathcal {A}}_{{\\bf {2}},{\\bf {-1}},{\\mathcal {O}}}$ (writing $\\sum m{\\mathbf {e}}_\\theta $ as $\\mathbf {$ } for $m \\in \\mathbb {Z}$ ).", "Furthermore the decomposition, filtrations and isomorphisms of the theorem (and hence also the Kodaira–Spencer isomorphism) are Hecke-equivariant in the obvious sense.", "More precisely the same argument as for the compatibility with the ${\\mathcal {O}}_{F,(p),+}^\\times $ -action, but using the quasi-isogeny in the construction of $\\rho _g$ , shows that if $U_1$ , $U_2$ and $g \\in \\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)}) are such that g^{-1}U_1g \\subset U_2,then \\rho _g^ {\\operatorname{Fil}\\,}^j (\\Omega ^1_{Y_{U_2}/{\\mathcal {O}},{\\mathfrak {p}},i}) corresponds to {\\operatorname{Fil}\\,}^j (\\Omega ^1_{Y_{U_1}/{\\mathcal {O}},{\\mathfrak {p}},i}) for all {\\mathfrak {p}},i,junder the canonical isomorphism \\rho _g^\\Omega ^1_{Y_{U_2}/{\\mathcal {O}}} \\stackrel{\\sim }{\\longrightarrow } \\Omega ^1_{Y_{U_1}/{\\mathcal {O}}},and the resulting diagrams{ \\rho _g^{\\mathcal {A}}_{2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\theta ,{\\mathcal {O}},2} [r]^{\\sim }[d]^{\\wr } & {\\mathcal {A}}_{2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\theta ,{\\mathcal {O}},1} [d]^{\\wr } \\\\\\rho _g^* {\\operatorname{gr}\\,}^j (\\Omega ^1_{Y_{U_2}/{\\mathcal {O}},{\\mathfrak {p}},i}) [r]^{\\sim } & {\\operatorname{gr}\\,}^j(\\Omega ^1_{Y_{U_1}/{\\mathcal {O}},{\\mathfrak {p}},i}) }commute (where the top arrow is defined in the discussion preceding (\\ref {eqn:inflevel})).", "}$" ], [ "Construction of $H_\\theta $ and {{formula:ec2eaa62-501e-4b2d-a820-c6d832d3e86a}}", "We now recall the definition, due to Reduzzi and Xiao [32], of generalized partial Hasse invariants on Pappas–Rapoport models.", "These will be, for each $\\theta = \\theta _{{\\mathfrak {p}},i,j} \\in \\Sigma $ , a Hilbert modular form $H_\\theta $ of weight $(\\mathbf {h}_\\theta ,\\bf {0})$ with coefficients in $\\mathbb {F}= {\\mathcal {O}}/{\\mathfrak {m}}_{\\mathcal {O}}$ , where $\\mathbf {h}_\\theta : = n_\\theta {\\mathbf {e}}_{\\sigma ^{-1}\\theta } - {\\mathbf {e}}_\\theta $ , with $n_\\theta = p$ if $j=1$ and $n_\\theta = 1$ if $j > 1$ .", "We also define below a(n in)variant $G_\\theta $ of weight $(\\bf {0},\\mathbf {h}_\\theta )$ .", "We will now be working in the mod $p$ setting, so until further notice $S$ will denote $\\widetilde{Y}_{U,\\mathbb {F}}$ , and $s:A \\rightarrow S$ the universal abelian scheme over it.", "Thus ${\\mathcal {H}}^1_{\\operatorname{dR}}(A/S)$ is a locally free sheaf of rank two over ${\\mathcal {O}}_F\\otimes {\\mathcal {O}}_S \\cong \\bigoplus _{{\\mathfrak {p}}\\in S_p} \\bigoplus _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}} {\\mathcal {O}}_S [u] / (u^{e_{\\mathfrak {p}}}),$ where $u$ acts via $\\iota (\\varpi _{\\mathfrak {p}})^$ on the ${\\mathfrak {p}}$ -component of ${\\mathcal {H}}^1_{\\operatorname{dR}}(A/S) = \\bigoplus _{{\\mathfrak {p}}\\in S_p} {\\mathcal {H}}_{\\mathfrak {p}}= \\bigoplus _{{\\mathfrak {p}}\\in S_p} \\bigoplus _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}} {\\mathcal {H}}_\\tau .$ We will also now be working with a fixed ${\\mathfrak {p}}$ and omit the subscript from the notation, so that ${\\mathcal {H}}= {\\mathcal {H}}^1_{\\operatorname{dR}}(A/S)_{\\mathfrak {p}}= \\bigoplus _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}} {\\mathcal {H}}_\\tau = \\bigoplus _{i \\in \\mathbb {Z}/f\\mathbb {Z}} {\\mathcal {H}}_i $ with each ${\\mathcal {H}}_i$ locally free of rank two over ${\\mathcal {O}}_S[u]/(u^e)$ (where we have also abbreviated the subscript $\\tau _{{\\mathfrak {p}},i}$ by $i$ ).", "Furthermore for each $i\\in \\mathbb {Z}/f\\mathbb {Z}$ , we have a filtration $0 = {\\mathcal {F}}_i^{(0)} \\subset {\\mathcal {F}}_i^{(1)}\\subset \\cdots \\subset {\\mathcal {F}}_i^{(e-1)} \\subset {\\mathcal {F}}_i^{(e)} = (s_\\Omega ^1_{A/S})_i$ by sheaves of ${\\mathcal {O}}_S[u]/(u^e)$ -modules such that the quotients ${\\mathcal {L}}_{i,j} = {\\mathcal {F}}_i^{(j)}/{\\mathcal {F}}_i^{(j-1)}$ are line bundles annihilated by $u$ .", "Firstly note that if $j > 1$ , then $u: {\\mathcal {F}}_i^{(j)} \\rightarrow {\\mathcal {F}}_i^{(j-1)}$ induces a morphism ${\\mathcal {L}}_{i,j} \\rightarrow {\\mathcal {L}}_{i,j-1}$ .", "On the other hand if $j=1$ , then the Verschiebung morphism $\\phi _S^A \\rightarrow A$ over $S$ induces ${\\mathcal {O}}_S[u]/(u^e)$ -linear morphisms $\\operatorname{Ver}_i^: {\\mathcal {H}}_i = {\\mathcal {H}}_{\\tau _i} \\longrightarrow {\\mathcal {H}}^1_{\\operatorname{dR}}(\\phi _S^(A)/S)_{\\tau _i} \\cong \\phi _S^({\\mathcal {H}}_{\\tau _{i-1}}) = \\phi _S^({\\mathcal {H}}_{i-1})$ with image $\\phi _S^({\\mathcal {F}}_{i-1}^{(e)})$ (where $\\phi _S$ denotes the absolute Frobenius on $S$ ).", "Note that ${\\mathcal {L}}_{i,1} = {\\mathcal {F}}_i^{(1)} \\subset u^{e-1} {\\mathcal {H}}_i$ , so that $u^{e-1}$ defines an isomorphism $u^{1-e}{\\mathcal {L}}_{i,1}/u {\\mathcal {H}}_i \\stackrel{\\sim }{\\longrightarrow } {\\mathcal {F}}_i^{(1)},$ and that $\\operatorname{Ver}_i^(u{\\mathcal {H}}_i) = u\\phi _S^({\\mathcal {F}}_{i-1}^{(e)}) \\subset \\phi _S^({\\mathcal {F}}_{i-1}^{(e-1)})$ , so we obtain a well-defined ${\\mathcal {O}}_S$ -linear morphism “$\\operatorname{Ver}_i^\\circ u^{1-e}$ ” ${\\mathcal {L}}_{i,1} \\stackrel{\\sim }{\\longleftarrow } u^{1-e}{\\mathcal {L}}_{i,1} /u {\\mathcal {H}}_i \\longrightarrow \\phi _S^( {\\mathcal {L}}_{i-1,e}) \\cong {\\mathcal {L}}_{i-1,e}^{\\otimes p}.$ We have now defined a morphism ${\\mathcal {L}}_\\theta \\longrightarrow {\\mathcal {L}}_{\\sigma ^{-1}\\theta }^{\\otimes n_\\theta }$ for all $\\theta $ , and hence a section of $\\widetilde{{\\mathcal {A}}}_{\\mathbf {h}_\\theta ,\\bf {0},\\mathbb {F}}$ over $S$ .", "Furthermore it is straightforward to check that the section is invariant under the action of ${\\mathcal {O}}_{F,(p),+}^\\times $ and therefore descends to an element $ H_\\theta \\in M_{\\mathbf {h}_\\theta ,\\bf {0}} (U; \\mathbb {F}) = H^0(Y_{U,\\mathbb {F}} , {\\mathcal {A}}_{\\mathbf {h}_\\theta ,\\bf {0},\\mathbb {F}}),$ which we call the partial Hasse invariant (indexed by $\\theta $ ).", "Furthermore the partial Hasse invariants are stable under the Hecke action, in the sense that if $U_1$ , $U_2$ and $g \\in \\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)}) are such that g^{-1}U_1g \\subset U_2,then [g]H_{\\theta ,2} = H_{\\theta ,1}.", "(Note alsothat the partial Hasse invariants are dependent on the choice of uniformizer \\varpi = \\varpi _{\\mathfrak {p}}only up to a scalar in \\mathbb {F}^\\times : if \\varpi is replaced by a \\varpi for some a \\in {\\mathcal {O}}_{F,{\\mathfrak {p}}}^\\times ,then H_\\theta is replaced by \\tau (a) H_\\theta if j >1 and by \\tau (a)^{1-e} H_\\theta if j=1.", ")}$ We remark also that the line bundles ${\\mathcal {A}}_{\\bf {0},\\mathbf {h}_\\theta ,\\mathbb {F}}$ have canonical trivializations.", "Indeed for each $i \\in \\mathbb {Z}/f\\mathbb {Z}$ and $j=2,\\ldots ,e$ , we have the exact sequence $0 \\rightarrow {\\mathcal {G}}_i^{(j-1)}/{\\mathcal {F}}_i^{(j-1)} \\longrightarrow {\\mathcal {G}}_i^{(j)}/{\\mathcal {F}}_i^{(j-1)} \\stackrel{u}{\\longrightarrow } {\\mathcal {F}}_i^{(j-1)}/{\\mathcal {F}}_i^{(j-2)} \\rightarrow 0$ over $S$ , i.e., $0 \\rightarrow {\\mathcal {M}}_{i,j-1} \\rightarrow {\\mathcal {P}}_{i,j} \\rightarrow {\\mathcal {N}}_{i,j-1} \\rightarrow 0$ , inducing an isomorphism $ {\\mathcal {N}}_{i,j-1} = {\\mathcal {L}}_{i,j-1}\\otimes _{{\\mathcal {O}}_S} {\\mathcal {M}}_{i,j-1} \\cong \\wedge ^2_{{\\mathcal {O}}_S} {\\mathcal {P}}_{i,j} = {\\mathcal {N}}_{i,j}$ and hence ${\\mathcal {O}}_S \\simeq {\\mathcal {N}}_{i,j-1}^{-1}\\otimes _{{\\mathcal {O}}_S}{\\mathcal {N}}_{i,j} = \\widetilde{{\\mathcal {A}}}_{\\bf {0},\\mathbf {h}_\\theta ,\\mathbb {F}}$ for $\\theta = \\theta _{{\\mathfrak {p}},i,j}$ , which it is straightforward to check descends to $Y_{U,\\mathbb {F}}$ .", "Similarly we have the exact sequence $0 \\rightarrow \\phi _S^ ({\\mathcal {G}}_{i-1}^{(e)}/{\\mathcal {F}}_{i-1}^{(e)}) \\stackrel{{\\operatorname{Frob}}_A^}{\\longrightarrow } {\\mathcal {G}}_{i}^{(1)}\\stackrel{\\operatorname{Ver}_A^u^{1-e}}{\\longrightarrow } \\phi _S^({\\mathcal {F}}_{i-1}^{(e)}/{\\mathcal {F}}_{i-1}^{(e-1)}) \\rightarrow 0$ inducing an isomorphism $\\phi _S^({\\mathcal {N}}_{i-1,e}) \\cong {\\mathcal {N}}_{i,1}$ and hence ${\\mathcal {O}}_S \\simeq \\widetilde{{\\mathcal {A}}}_{\\bf {0},\\mathbf {h}_\\theta ,\\mathbb {F}}$ for $\\theta = \\theta _{{\\mathfrak {p}},i,1}$ descending to $Y_{U,\\mathbb {F}}$ .", "Furthermore these isomorphisms are Hecke-equivariant in the usual sense, but note that they depend via $u$ on the choice of $\\varpi _{\\mathfrak {p}}$ .", "For each $\\theta $ , we let $G_\\theta \\in M_{\\bf {0},\\mathbf {h}_\\theta }(U;\\mathbb {F})$ denote the canonical trivializing section." ], [ "Stratification", "We also recall how the partial Hasse invariants define a stratification of the Hilbert modular variety in characteristic $p$ .", "For any $\\theta \\in \\Theta $ , we define $\\widetilde{Z}_\\theta $ (resp.", "$Z_\\theta $ ) to be the closed subscheme of $S = \\widetilde{Y}_{U,\\mathbb {F}}$ (resp.", "$Y_{U,\\mathbb {F}}$ ) defined by the vanishing of $H_\\theta $ , and for any subset $T \\subset \\Sigma $ , we let $\\widetilde{Z}_T = \\bigcap _{\\theta \\in T} \\widetilde{Z}_\\theta \\quad \\mbox{and} \\quad Z_T = \\bigcap _{\\theta \\in T} Z_\\theta .$ Note that the schemes $Z_T$ are stable under the Hecke action, in the strong sense that $Z_{T,1}$ is the pull-back of $Z_{T,2}$ under $\\rho _g:Y_{U_1} \\rightarrow Y_{U_2}$ .", "We then have the following consequence ([32]) of the description of the Kodaira–Spencer filtration on tangent spaces at closed points; we give a proof here as some of the details are relevant to the construction of $\\Theta $ -operators in §REF .", "Proposition 3.2.1 The schemes $\\widetilde{Z}_T$ and $Z_T$ are smooth over $\\mathbb {F}$ of dimension $\|\\Sigma - T\|$ .", "Proof.We prove the result for $\\widetilde{Z}_T$ , from which the result for $Z_T$ is immediate.", "Let $y$ be a closed point of $S$ with local ring $R = {\\mathcal {O}}_{S,y}$ , maximal ideal ${\\mathfrak {m}}$ and residue field $k = R/{\\mathfrak {m}}$ .", "For each $\\theta \\in \\Sigma $ , choose a basis $b_\\theta $ for ${{\\mathcal {L}}}_{\\theta ,y}$ over $R$ and write $H_{\\theta ,y} b_\\theta = x_\\theta b_{\\sigma ^{-1}\\theta }^{n_\\theta }$ .", "Thus if $y \\in \\widetilde{Z}_\\theta $ , then $x_\\theta \\in {\\mathfrak {m}}$ , and we let $\\overline{x}_\\theta $ denote its image in ${\\mathfrak {m}}/{\\mathfrak {m}}^2$ .", "Identifying ${\\mathfrak {m}}/{\\mathfrak {m}}^2$ with the fibre of $\\Omega ^1_{\\widetilde{Y}_U/{\\mathcal {O}}}$ at $y$ and writing ${\\operatorname{Fil}\\,}^j({\\mathfrak {m}}/{\\mathfrak {m}}^2)_\\tau $ for the subspaces obtained from the Kodaira–Spencer filtration, we claim that if $y \\in Z_\\theta $ , then $ {\\operatorname{Fil}\\,}^j({\\mathfrak {m}}/{\\mathfrak {m}}^2)_\\tau = k\\overline{x}_\\theta + {\\operatorname{Fil}\\,}^{j-1}({\\mathfrak {m}}/{\\mathfrak {m}}^2)_\\tau , $ where $\\tau = \\tau _{{\\mathfrak {p}},i}$ and $\\theta = \\theta _{{\\mathfrak {p}},i,j}$ .", "Comparing dimensions, we see it suffices to prove the inclusion of the left-hand side in the right, or equivalently that if $ v \\in T_y(S) = \\bigoplus _{\\tau ^{\\prime } \\in \\Sigma _0} T_y(S)_{\\tau ^{\\prime }}$ is such that its $\\tau $ -component $v_\\tau $ lies in ${\\operatorname{Fil}\\,}^{j-1} T_y(S)_\\tau $ and $v$ is orthogonal to $\\overline{x}_\\theta $ , then in fact $v_\\tau \\in {\\operatorname{Fil}\\,}^j(T_y(S)_\\tau )$ (using the notation of the discussion following the proof of Theorem ).", "Let $\\underline{A}_0 = (A_0,\\iota _0,\\lambda _0,\\eta _0,{\\mathcal {F}}_0^\\bullet )$ denote the data corresponding to the point $y \\in S(k)$ and $\\underline{A}_1 = (A_1,\\iota _1,\\lambda _1,\\eta _1,{\\mathcal {F}}_1^\\bullet )$ that of its lift $v \\in S(k[\\epsilon ])$ .", "With $\\tau = \\tau _{{\\mathfrak {p}},i}$ fixed for now, we will suppress ${\\mathfrak {p}}$ from the notation and replace the subscript $\\tau _{p,i^{\\prime }}$ by $i^{\\prime }$ (for $i^{\\prime } = i,i-1$ ).", "Recall the assumption that $v_i \\in {\\operatorname{Fil}\\,}^{j-1}( T_y(S)_i)$ means that ${\\mathcal {F}}_{1,i}^{(j^{\\prime })}$ corresponds to ${\\mathcal {F}}_{0,i}^{(j^{\\prime })} \\otimes _k k[\\epsilon ]$ for $j^{\\prime } = 1,\\ldots ,j-1$ under the canonical isomorphism $H^1_{\\operatorname{dR}}(A_1/k[\\epsilon ]) \\cong H^1_{\\operatorname{cris}}(A_0/k[\\epsilon ]) \\cong H^1_{\\operatorname{dR}}(A_0/k) \\otimes _k k[\\epsilon ].$ For $v_i$ to be orthogonal to $\\overline{x}_\\theta $ means that the morphism ${\\mathcal {L}}_{1,\\theta } \\longrightarrow {\\mathcal {L}}_{1,\\sigma ^{-1}\\theta }^{\\otimes n_\\theta }$ induced by $H_\\theta $ vanishes, and we need to show this implies that ${\\mathcal {F}}_{1,i}^{(j)}$ is the image of ${\\mathcal {F}}_{0,i}^{(j)} \\otimes _k k[\\epsilon ]$ .", "Suppose first that $j > 1$ .", "Then (REF ) is simply $u: {\\mathcal {F}}_{1,i}^{(j)}/{\\mathcal {F}}_{1,i}^{(j-1)} \\longrightarrow {\\mathcal {F}}_{1,i}^{(j-1)}/{\\mathcal {F}}_{1,i}^{(j-2)},$ whose vanishing means ${\\mathcal {F}}_{1,i}^{(j)} = u^{-1}{\\mathcal {F}}_{1,i}^{(j-2)}$ .", "Since (REF ) sends ${\\mathcal {F}}_{1,i}^{(j-2)}$ to ${\\mathcal {F}}_{0,i}^{(j-2)} \\otimes _k k[\\epsilon ]$ and is compatible with $u$ , it follows that it also sends ${\\mathcal {F}}_{1,i}^{(j)}$ to ${\\mathcal {F}}_{0,i}^{(j)} \\otimes _k k[\\epsilon ]$ .", "On the other hand if $j=1$ , then the vanishing of (REF ) means that $u^{1-e}{\\mathcal {F}}_{1,i}^{(1)}$ is the preimage of $\\phi _1^({\\mathcal {F}}_{1,i-1}^{(e-1)})$ under $\\operatorname{Ver}_i^$ (where $\\phi _1$ is the absolute Frobenius on $k[\\epsilon ]$ , and $\\phi _0$ will denote the absolute Frobenius on $k$ ).", "Since the diagram ${H^1_{\\operatorname{dR}}(A_1/k[\\epsilon ]) [r] [d] & H^1_{\\operatorname{dR}}(A_0/k) \\otimes _k k[\\epsilon ] [d]\\\\\\phi _1^H^1_{\\operatorname{dR}}(A_1/k[\\epsilon ]) [r] & \\phi _0^H^1_{\\operatorname{dR}}(A_0/k) \\otimes _k k[\\epsilon ]}$ commutes, where the vertical maps are induced by Verschiebung, the top arrow is (REF ) and the bottom one is given by the identification of $\\phi _1^A_1$ with $\\phi _0^A_0 \\otimes _k k[\\epsilon ]$ , so in particular identifies $\\phi _1^({\\mathcal {F}}_{1,i-1}^{(e-1)})$ with $\\phi _0^({\\mathcal {F}}_{0,i-1}^{(e-1)}) \\otimes _k k[\\epsilon ]$ , it follows that the top arrow sends $u^{1-e}{\\mathcal {F}}_{1,i}^{(1)}$ to $u^{1-e}{\\mathcal {F}}_{0,i}^{(1)} \\otimes _k k[\\epsilon ]$ , and hence ${\\mathcal {F}}_{1,i}^{(1)}$ to ${\\mathcal {F}}_{0,i}^{(1)} \\otimes _k k[\\epsilon ]$ .", "This completes the proof of the claim.", "Now note that if $y \\in \\widetilde{Z}_T$ , then (REF ) implies that the elements $\\overline{x}_\\theta $ for $\\theta \\in T$ can be extended to a basis for ${\\mathfrak {m}}/{\\mathfrak {m}}^2$ over $k$ , hence are linearly independent.", "Since $R$ is regular of dimension $d = \|\\Sigma \|$ , it follows that ${\\mathcal {O}}_{\\widetilde{Z}_T,y} = R/\\langle x_\\theta \\rangle _{\\theta \\in T}$ is regular of dimension of $d - \|T\|$ , and hence that $\\widetilde{Z}_T$ is smooth over $\\mathbb {F}$ of dimension $d - \|T\|$ .", "$\\square $ Finally we recall the definition of the minimal weight of a non-zero mod $p$ Hilbert modular form.", "If $f \\in M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F})$ , then $\\mathbf {k}_{\\min }(f)$ is defined to be $\\mathbf {k}- \\sum _{\\theta } m_\\theta \\mathbf {h}_\\theta $ where $\\sum _\\theta m_\\theta {\\mathbf {e}}_\\theta $ is the unique maximal element of the set $\\left\\lbrace \\,\\left.", "\\sum _\\theta m_\\theta {\\mathbf {e}}_\\theta \\in \\mathbb {Z}_{\\ge 0}^\\Sigma \\,\\right\|\\,f = g\\prod _{\\theta \\in \\Sigma } H_\\theta ^{m_\\theta } \\mbox{\\,for some\\,}g \\in M_{\\mathbf {k}- \\sum _\\theta m_\\theta \\mathbf {h}_\\theta ,\\mathbf {l}}(U;\\mathbb {F})\\,\\right\\rbrace .$ By the main result of [10], the minimal weight of $f$ always lies in the cone: $\\Xi ^{\\min } := \\left\\lbrace \\, \\left.\\sum _\\theta k_\\theta {\\mathbf {e}}_\\theta \\,\\right\|\\, \\mbox{$n_\\theta k_\\theta \\ge k_{\\sigma ^{-1}\\theta }$ for all $\\theta \\in \\Sigma $}\\,\\right\\rbrace .$ Note that the result stated in [10] applies to forms on a finite étale cover of $Y_{U,\\mathbb {F}}$ , from which the analogous result for forms on $Y_{U,\\mathbb {F}}$ is immediate." ], [ "Fundamental Hasse invariants", "In order to define the partial $\\Theta $ -operators (in §REF below), we first define a canonical factorization of the partial Hasse invariants over a finite flat (Igusa) cover of the Hilbert modular variety over $\\mathbb {F}$ .", "We fix a sufficiently small $U$ that the line bundles ${\\mathcal {L}}_\\theta $ , ${\\mathcal {M}}_\\theta $ , ${\\mathcal {N}}_\\theta $ (and hence $\\widetilde{{\\mathcal {A}}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}}$ ) on $\\widetilde{Y}_{U,\\mathbb {F}}$ descend to $Y_{U,\\mathbb {F}}$ for all $\\theta \\in \\Sigma $ (and all $\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma $ ), and we write simply $\\overline{Y}$ for $Y_{U,\\mathbb {F}}$ , and $\\overline{{\\mathcal {L}}}_{\\tau ,j}$ , $\\overline{{\\mathcal {M}}}_{\\tau ,j}$ and $\\overline{{\\mathcal {N}}}_{\\tau ,j}$ for the line bundles on $\\overline{Y}$ .", "For each ${\\mathfrak {p}}\\in S_p$ and $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ , we let $H_\\tau = \\prod _{j=1}^{e_{\\mathfrak {p}}} H_{\\tau ,j} \\in H^0(\\overline{Y}, \\overline{{\\mathcal {L}}}_{\\tau ,e_{\\mathfrak {p}}}^{-1}\\otimes _{{\\mathcal {O}}_{\\overline{Y}}} \\overline{{\\mathcal {L}}}_{\\phi ^{-1}\\circ \\tau ,e_{\\mathfrak {p}}}^{\\otimes p}).$ Viewing each $H_\\tau $ as a morphism $(\\overline{{\\mathcal {L}}}_{\\phi ^{-1}\\circ \\tau ,e_{\\mathfrak {p}}}^{-1})^{\\otimes p} \\rightarrow \\overline{{\\mathcal {L}}}_{\\tau ,e_{\\mathfrak {p}}}^{-1}$ and $H_{\\mathfrak {p}}:= \\prod _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}} H_\\tau $ as a morphism $\\otimes _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}} (\\overline{{\\mathcal {L}}}_{\\tau ,e_{\\mathfrak {p}}}^{-1})^{\\otimes (p-1)} \\rightarrow {\\mathcal {O}}_{\\overline{Y}}$ , we define the Igusa coverThe cover has a natural moduli-theoretic interpretation in terms of $A[{\\mathfrak {q}}]$ , but we will not need this here.", "of $\\overline{Y}$ (of level ${\\mathfrak {q}}= \\prod {\\mathfrak {p}}$ ) to be ${Y}^{\\mathrm {Ig}} = \\mathbf {Spec} \\left(\\mathrm {Sym}_{\\mathcal {O}_{\\overline{Y}}}(\\bigoplus _{\\tau \\in \\Sigma _0} \\overline{{\\mathcal {L}}}_{\\tau ,e_{\\mathfrak {p}}}^{-1})/\\mathcal {I}\\right),$ where ${\\mathcal {I}}$ is the sheaf of ideals generated by the ${\\mathcal {O}}_{\\overline{Y}}$ -submodules $\\mbox{$(H_\\tau - 1) \\overline{{\\mathcal {L}}}_{\\tau ,e_{\\mathfrak {p}}}^{-1}$ for $\\tau \\in \\Sigma _{0}$, and$(H_{\\mathfrak {p}}-1)\\left(\\bigotimes _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}} (\\overline{{\\mathcal {L}}}_{\\tau ,e_{\\mathfrak {p}}}^{-1})^{\\otimes (p-1)}\\right)$ for ${\\mathfrak {p}}\\in S_p$}$ (where all tensor products are over ${\\mathcal {O}}_{\\overline{Y}}$ ).", "We then define an action of $({\\mathcal {O}}_F/{\\mathfrak {q}})^\\times $ on ${Y}^{{\\mathrm {Ig}}}$ over $\\overline{Y}$ by letting $\\alpha \\in ({\\mathcal {O}}_F/{\\mathfrak {q}})^\\times $ act on the structure sheaf by the automorphism of sheaves of ${\\mathcal {O}}_{\\overline{Y}}$ -algebras induced by multiplication by $\\tau ({\\alpha })^{-1}$ on $\\overline{{\\mathcal {L}}}_{\\tau ,e_{\\mathfrak {p}}}^{-1}$ for each $\\tau $ .", "We then see, exactly as in the proof of parts (1) and (2) of [12], that the projection $\\pi :{Y}^{{\\mathrm {Ig}}} \\rightarrow \\overline{Y}$ is finite flat, generically étale, and identifies $\\overline{Y}$ with the quotient of ${Y}^{{\\mathrm {Ig}}}$ by the action of $({\\mathcal {O}}_F/{\\mathfrak {q}})^\\times $ .", "For each $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ , we let $h_{\\tau ,e_{\\mathfrak {p}}}$ denote the tautological section of $\\pi ^\\overline{{\\mathcal {L}}}_{\\tau ,e_{\\mathfrak {p}}}$ induced by the inclusion $\\overline{{\\mathcal {L}}}_{\\tau ,e_{\\mathfrak {p}}}^{-1} \\hookrightarrow \\pi _{\\mathcal {O}}_{{Y}^{\\mathrm {Ig}}}$ .", "We also define the section $h_{\\tau ,j} = \\pi ^(H_{\\tau ,j+1}\\cdots H_{\\tau ,e_{\\mathfrak {p}}})h_{\\tau ,e_{\\mathfrak {p}}}$ of $\\pi ^\\overline{{\\mathcal {L}}}_{\\tau ,j}$ for $j=1,\\ldots ,e_{{\\mathfrak {p}}}-1$ .", "Note that since ${Y}^{\\mathrm {Ig}}$ is reduced (or since $\\prod _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}} h_{\\tau ,e_{\\mathfrak {p}}}^{p-1} = \\pi ^H_{\\mathfrak {p}}$ by construction), the sections $h_{\\tau ,e_{\\mathfrak {p}}}$ are injective, and hence so are the $h_{\\tau ,j}$ for all $\\tau $ and $j$ .", "We write simply $h_\\theta $ for the section $h_{\\tau ,j}$ of $\\pi ^\\overline{{\\mathcal {L}}}_\\theta = \\pi ^\\overline{{\\mathcal {L}}}_{\\tau ,j}$ , and we call the $h_\\theta $ the fundamental Hasse invariant (indexed by $\\theta $ )." ], [ "Construction of $\\Theta _\\tau $", "We now explain how the construction of $\\Theta $ -operators in [12] directly generalizes to the case where $p$ is ramified in $F$ , yielding an operator that shifts the weight $\\mathbf {k}$ by $(1,1)$ in the final two components corresponding to embeddings with the same reduction, i.e., $\\theta _{{\\mathfrak {p}},i,e_{{\\mathfrak {p}}}-1}$ , $\\theta _{{\\mathfrak {p}},i,e_{\\mathfrak {p}}}$ (and hence, by composing with multiplication by partial Hasse invariants, one can shift weights by $+1$ for any pair of embeddings with the same reduction).", "Indeed for each $\\tau \\in \\Sigma _0$ , we define the operator $\\Theta _{\\tau }$ exactly as in [12], but using the morphism $KS_{\\tau }: \\Omega ^1_{\\overline{Y}/{\\mathbb {F}}} \\longrightarrow {\\operatorname{gr}\\,}^{e_{{\\mathfrak {p}}}}(\\Omega ^1_{\\overline{Y}/{\\mathbb {F}}})_{\\tau } \\stackrel{\\sim }{\\longrightarrow }\\overline{{\\mathcal {L}}}_{\\tau ,e_{\\mathfrak {p}}}\\otimes _{{\\mathcal {O}}_{\\overline{Y}}} \\overline{{\\mathcal {M}}}^{-1}_{\\tau ,e_{\\mathfrak {p}}}$ provided by Theorem via projection to the top graded piece of the filtration of the $\\tau $ -component of $\\Omega ^1_{Y/{\\mathbb {F}}}$ .", "More precisely, fix ${\\mathfrak {p}}_0 \\in S_p$ and $\\tau _0 = \\tau _{{\\mathfrak {p}}_0,i}$ , let $\\theta _0 = \\theta _{{\\mathfrak {p}}_0,i,e_{{\\mathfrak {p}}_0}}$ , and consider the morphism $KS_{\\tau _0}^{{\\mathrm {Ig}}}: \\Omega ^1_{{Y}^{\\mathrm {Ig}}/{\\mathbb {F}}}\\otimes _{\\mathcal {O}_{{Y}^{\\mathrm {Ig}}}}\\mathcal {F}^{\\mathrm {Ig}}\\cong \\pi ^\\Omega ^1_{\\overline{Y}/{\\mathbb {F}}}\\otimes _{\\mathcal {O}_{{Y}^{\\mathrm {Ig}}}}\\mathcal {F}^{\\mathrm {Ig}} \\longrightarrow \\pi ^* ({\\mathcal {A}}_{2{\\mathbf {e}}_{\\theta _0},-{\\mathbf {e}}_{\\theta _0},\\mathbb {F}})\\otimes _{\\mathcal {O}_{Y^{\\mathrm {Ig}}}} \\mathcal {F}^{\\mathrm {Ig}}$ induced by $KS_{\\tau _0}$ , where ${\\mathcal {F}}^{\\mathrm {Ig}}$ is the sheaf of total fractions on ${Y}^{\\mathrm {Ig}}$ .", "Suppose now that $f \\in M_{\\mathbf {k},\\mathbf {l}}(U;{\\mathbb {F}})$ , and write $h^{\\mathbf {k}} = \\prod _{\\theta \\in \\Sigma } h_\\theta ^{k_\\theta }$ and $g^{\\mathbf {l}} = \\prod _{\\theta \\in \\Sigma } g_\\theta ^{l_\\theta }$ for any choice of trivializations $g_\\theta $ of the line bundles $\\overline{{\\mathcal {N}}}_\\theta $ on $\\overline{Y}$ .", "We then define the section $\\Theta _{\\tau _0}^{\\mathrm {Ig}}(f) := h^{\\mathbf {k}} \\pi ^(g^{\\mathbf {l}} H_{\\theta _0}) KS_{\\tau _0}^{\\mathrm {Ig}}(d(h^{-\\mathbf {k}} \\pi ^(g^{-\\mathbf {l}}f))),$ where $ \\mathbf {k}^{\\prime } = \\mathbf {k}+ n_{\\theta _0} {\\mathbf {e}}_{\\sigma ^{-1}\\theta _0} + {\\mathbf {e}}_{\\theta _0}\\quad \\mbox{and}\\quad \\mathbf {l}^{\\prime } = \\mathbf {l}+ {\\mathbf {e}}_{\\theta _0}.$ Furthermore, the section is independent of the choices of $g_\\theta $ and invariant under the action of $({\\mathcal {O}}_F/{\\mathfrak {q}})^\\times $ , hence descends to a section of ${\\mathcal {A}}_{\\mathbf {k}^{\\prime },\\mathbf {l}^{\\prime },{\\mathbb {F}}} \\otimes _{{\\mathcal {O}}_{\\overline{Y}}} {\\mathcal {F}}$ , where ${\\mathcal {F}}$ is the sheaf of total fractions on $\\overline{Y}$ .", "Denoting the section $\\Theta _{\\tau _0}(f)$ , we have the following generalization of [12]: Theorem 4.2.1 If $f \\in M_{\\mathbf {k},\\mathbf {l}}(U;{\\mathbb {F}})$ , then $\\Theta _{\\tau _0}(f) \\in M_{\\mathbf {k}^{\\prime },\\mathbf {l}^{\\prime }}(U;{\\mathbb {F}})$ .", "Moreover $\\Theta _{\\tau _0}(f)$ is divisible by $H_{\\theta _0}$ if and only if either $f$ is divisible by $H_{\\theta _0}$ or $p\|k_{\\theta _0}$ .", "Proof.We see exactly as in [12] that $\\Theta _{\\tau _0}(f)$ is regular on the ordinary locus of $\\overline{Y}$ , i.e., the complement of the divisor $\\cup _{\\theta \\in \\Sigma } Z_\\theta $ (where $Z_\\theta $ was defined in §REF ), so the theorem reduces to proving that if $z$ is the generic point of an irreducible component of $Z_{\\theta _1}$ for some $\\theta _1 \\in \\Sigma $ , then ${\\operatorname{ord}}_z(\\Theta _{\\tau _0}(f)) \\ge 0$ , if $\\theta _1 = \\theta _0$ , then ${\\operatorname{ord}}_z(\\Theta _{\\tau _0}(f)) > 0$ if and only if $p\|k_{\\theta _0}$ or ${\\operatorname{ord}}_z(f) > 0$ .", "Let $R$ denote the discrete valuation ring ${\\mathcal {O}}_{\\overline{Y},z}$ , and for each $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ and $\\theta \\in \\Sigma _\\tau $ , let $y_\\theta = y_{\\tau ,j}$ be a basis for the stalk $\\overline{{\\mathcal {L}}}_{\\theta ,z} = \\overline{{\\mathcal {L}}}_{\\tau ,j,z}$ over $R$ (for $j = 1,\\ldots ,e_{\\mathfrak {p}}$ ).", "For each $\\theta \\in \\Sigma $ , we may then write $H_\\theta y_\\theta = r_\\theta y_{\\sigma ^{-1}\\theta }^{n_\\theta }$ for some $r_\\theta = r_{\\tau ,j} \\in R$ , and we let $r_\\tau = \\prod _{j=1}^{e_{\\mathfrak {p}}} r_{\\tau ,j}$ .", "By construction, we have $T: = (\\pi _{\\mathcal {O}}_{{Y}^{{\\mathrm {Ig}}}})_z = R[x_\\tau ]_{\\tau \\in \\Sigma _0} / I$ , where $I$ is the ideal generated by $x_{\\phi ^{-1}\\circ \\tau }^p - r_\\tau x_\\tau \\mbox{\\,\\,for $\\tau \\in \\Sigma _0$,}\\quad \\mbox{and}\\quad \\prod _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}} x_\\tau ^{p-1} - \\prod _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}}r_{\\tau }^{p-1} \\mbox{\\\\,,for ${\\mathfrak {p}}\\in S_p$,}$ where each $x_\\tau $ is the dual basis of $y_{\\tau ,e_{\\mathfrak {p}}}$ .", "We then have $h_{\\tau ,e_{\\mathfrak {p}}} = x_\\tau y_{\\tau ,e_{\\mathfrak {p}}}$ (in $(\\pi _\\pi ^{\\mathcal {L}}_{\\tau ,e_{\\mathfrak {p}}})_z$ ), from which it follows that $h_{\\tau ,j} = r_{\\tau ,j+1} r_{\\tau ,j+2} \\cdots r_{\\tau ,e_{\\mathfrak {p}}} x_\\tau y_{\\tau ,j}$ for $j=1,\\ldots ,e_{{\\mathfrak {p}}} - 1$ , and hence that $h^{\\mathbf {k}} = \\varphi _{\\mathbf {k}} y^{\\mathbf {k}}$ , where $y^{\\mathbf {k}} = \\prod _{\\theta \\in \\Sigma } y_\\theta ^{k_\\theta }$ and $ \\varphi _{\\mathbf {k}} = \\prod _{{\\mathfrak {p}}\\in S_p} \\prod _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}} \\left( (r_\\tau x_\\tau )^{\\sum _{\\theta \\in \\Sigma _\\tau } k_\\theta }\\prod _{j=1}^{e_{\\mathfrak {p}}} r_{\\tau ,j}^{-\\sum _{j^{\\prime } = j}^{e_{\\mathfrak {p}}} k_{\\tau ,j^{\\prime }}} \\right)$ (writing $k_{\\tau _{{\\mathfrak {p}},i},j}$ for $k_{\\theta _{{\\mathfrak {p}},i,j}}$ as usual, and working over the field of fractions of $T$ ).", "Writing $f = \\varphi _f y^{\\mathbf {k}} g^{\\mathbf {l}}$ , we see that $\\Theta _{\\tau _0}^{\\mathrm {Ig}}(f) = KS_{\\tau _0}^{\\mathrm {Ig}}( r_{\\theta _0} \\varphi _{\\mathbf {k}} d (\\varphi _f\\varphi _{\\mathbf {k}}^{-1} ) ) y_{\\theta _0}^{-1} y_{\\sigma ^{-1} \\theta _0}^{n_{\\theta _0}} y^{\\mathbf {k}} g^{\\mathbf {l}}.$ Since $r_\\tau x_\\tau = x_{\\phi ^{-1}\\circ \\tau }^p $ , we have $d(r_\\tau x_\\tau ) = 0$ and $\\Theta _{\\tau _0}(f) = KS_{\\tau _0} \\left(r_{\\theta _0} d\\varphi _f + r_{\\theta _0} \\varphi _f \\sum _{\\theta \\in \\Sigma } k_\\theta ^{\\prime } \\frac{dr_\\theta }{r_\\theta }\\right)y_{\\theta _0}^{-1} y_{\\sigma ^{-1} \\theta _0}^{n_{\\theta _0}} y^{\\mathbf {k}} g^{\\mathbf {l}},$ where $k^{\\prime }_\\theta = k_{\\tau ,j} + k_{\\tau ,j+1} + \\cdots k_{\\tau ,e_{\\mathfrak {p}}}$ if $\\tau = \\tau _{{\\mathfrak {p}},i}$ and $\\theta = \\theta _{{\\mathfrak {p}},i,j}$ .", "We are therefore reduced to showing that ${\\operatorname{ord}}_z KS_{\\tau _0} (dr_{\\theta _1}) > 0$ if and only if $\\theta _1 = \\theta _0$ .", "However the proof of Proposition REF shows that if $y$ is a closed point of $Z_{\\theta _1}$ , then $KS_{\\tau _0}(dr_{\\theta _1})$ vanishes at $y$ if and only if $\\theta _1 = \\theta _0$ .", "$\\square $ Remark 4.2.2 The Kodaira–Spencer isomorphism is defined in [12] using the Gauss–Manin connection.", "Much of the work in [12] amounts to an explicit translation of this to the context of deformation theory.", "Here however we defined the morphism $KS_{\\tau _0}$ more directly using deformation theory, so the analogue of [12] was not needed here.", "Remark 4.2.3 It is straightforward to check directly that the right-hand side of (REF ) is independent of the choice of local trivializations $y_\\tau $ and $g_\\tau $ , and can therefore be used to define the partial $\\Theta $ -operator without reference to the Igusa cover ${Y}^{\\mathrm {Ig}}$ .", "We call $\\Theta _{\\tau _0}$ the partial $\\Theta $ -operator (indexed by $\\tau _0$ ).", "It is immediate from its definition that the resulting map on $\\mathbb {F}$ -algebras $\\bigoplus _{\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma } M_{\\mathbf {k},\\mathbf {l}}(U;{\\mathbb {F}}) \\longrightarrow \\bigoplus _{\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma } M_{\\mathbf {k},\\mathbf {l}}(U;{\\mathbb {F}}),$ given by the direct sum over all weights of the operators $\\Theta _{\\tau _0}$ , is an $\\mathbb {F}$ -linear derivation, i.e.", "that $\\Theta _{\\tau _0}(f_1f_2) = f_1 \\Theta _{\\tau _0}(f_2) + \\Theta _{\\tau _0}(f_1)f_2$ for all $f_1,f_2$ in $\\oplus M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F})$ .", "It is also clear that $\\Theta _{\\tau _0}(H_\\theta ) = 0$ for all $\\theta \\in \\Sigma $ , and hence that $\\Theta _{\\tau _0}$ commutes with multiplication by partial Hasse invariants.", "It is also straightforward to check that the operator $\\Theta _{\\tau _0}$ commutes with the Hecke action in the obvious sense, and hence induces a $\\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)})-equivariant map\\Theta _\\tau : M_{\\mathbf {k},\\mathbf {l}}({\\mathbb {F}}) \\longrightarrow M_{\\mathbf {k}^{\\prime },\\mathbf {l}^{\\prime }}({\\mathbb {F}})where M_{\\mathbf {k},\\mathbf {l}}({\\mathbb {F}}) (and M_{\\mathbf {k}^{\\prime },\\mathbf {l}^{\\prime }}({\\mathbb {F}}), with their \\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)}) actions) aredefined in (\\ref {eqn:inflevel}) as direct limits over suitable open compact U.", "}Let us also make the effect of \\Theta _{\\tau _0} on the weight \\mathbf {k} more explicit.", "Note thatif \\mathbf {k}= \\sum _\\theta k_\\theta {\\mathbf {e}}_\\theta and \\tau _0 = \\tau _{{\\mathfrak {p}}_0,i_0}, then \\mathbf {k}^{\\prime } = \\sum _\\theta k^{\\prime }_\\theta {\\mathbf {e}}_\\theta , where\\begin{itemize}\\item if e_{{\\mathfrak {p}}_0} = f_{{\\mathfrak {p}}_0} = 1, thenk^{\\prime }_\\theta = \\left\\lbrace \\begin{array}{ll}k_\\theta + p + 1, & \\mbox{if $\\theta = \\theta _0 = \\theta _{{\\mathfrak {p}}_0,1,1}$,}\\\\k_\\theta ,&{otherwise;}\\end{array}\\right.\\item if e_{{\\mathfrak {p}}_0} = 1 and f_{{\\mathfrak {p}}_0} > 1, thenk^{\\prime }_\\theta = \\left\\lbrace \\begin{array}{ll}k_\\theta + 1, & \\mbox{if $\\theta = \\theta _0 = \\theta _{{\\mathfrak {p}}_0,i_0,1},$}\\\\k_\\theta + p, & \\mbox{if $\\theta = \\sigma ^{-1}\\theta _0 = \\theta _{{\\mathfrak {p}}_0,i_0-1,1}$,}\\\\k_\\theta ,&{otherwise;}\\end{array}\\right.\\item if e_{{\\mathfrak {p}}_0} > 1, thenk^{\\prime }_\\theta = \\left\\lbrace \\begin{array}{ll}k_\\theta + 1, & \\mbox{if $\\theta = \\theta _0 = \\theta _{{\\mathfrak {p}}_0,i_0,e_{{\\mathfrak {p}}_0}}$ or $\\theta = \\sigma ^{-1}\\theta _0 = \\theta _{{\\mathfrak {p}}_0,i_0,e_{{\\mathfrak {p}}_0}-1}$,}\\\\k_\\theta ,&{otherwise.", "}\\end{array}\\right.\\end{itemize}}\\begin{remark} Considerations from the theory of Serre weights from the point of \\cite {DS} suggest that the above weight shiftsare in a certain sense optimal.", "One can also define cruder partial \\Theta -operators by composingwith multiplication by (products of) partial Hasse invariants.", "For example, the operatorH_{\\tau _0,1}H_{\\tau _0,2}\\cdots H_{\\tau _0,e_{{\\mathfrak {p}}_0} - 1}\\Theta _{\\tau _0} is the one constructed in \\cite {DDW},and for any j=1,\\ldots ,e_{{\\mathfrak {p}}_0}, the operatorH_{\\tau _0,j}H^2_{\\tau _0,j+1}\\cdots H^2_{\\tau _0,e_{{\\mathfrak {p}}_0}-1}H_{\\tau _0,e_{{\\mathfrak {p}}_0}}\\Theta _{\\tau _0}shifts the weight \\mathbf {k} by {\\mathbf {e}}_\\theta + n_{\\sigma ^{-1}\\theta }{\\mathbf {e}}_{\\sigma ^{-1}\\theta }, where \\theta = \\theta _{{\\mathfrak {p}}_0,i_0,j}.\\end{remark}$" ], [ "Partial Frobenius endomorphisms", "In order to define partial Frobenius operators on Hilbert modular forms (in §REF below), we first need to define partial Frobenius endomorphisms of Hilbert modular varieties over $\\mathbb {F}$ .", "Fix a prime ${\\mathfrak {p}}$ dividing $p$ , and a level $U$ , assumed as usual to be sufficiently small and prime to $p$ .", "We will draw on ideas from [11] to construct an isogeny on the universal abelian variety $s: A \\rightarrow S$ , where $S = \\widetilde{Y}_{U,\\mathbb {F}}$ .", "We begin by associating Raynaud data to the line bundles ${{\\mathcal {L}}}_{{\\mathfrak {p}},i,e_{\\mathfrak {p}}}$ over $S$ , which we write simply as ${{\\mathcal {L}}}_i$ for $i \\in \\mathbb {Z}/f\\mathbb {Z}= \\mathbb {Z}/f_{\\mathfrak {p}}\\mathbb {Z}$ (omitting the subscripts for the fixed ${\\mathfrak {p}}$ and $j = e = e_{\\mathfrak {p}}$ ).", "We define $f_i: {{\\mathcal {L}}}^{\\otimes p}_i \\rightarrow {{\\mathcal {L}}}_{i+1}$ to be zero, and we define $v_i:{{\\mathcal {L}}}_{i+1}\\rightarrow {{\\mathcal {L}}}^{\\otimes p}_i$ to be the morphism induced by the restriction of $\\operatorname{Ver}^_A: {\\mathcal {H}}= {\\mathcal {H}}^1_{\\operatorname{dR}}(A/S)_{\\mathfrak {p}}\\rightarrow {\\mathcal {H}}^1_{\\operatorname{dR}}((\\phi _S^A)/S)_{\\mathfrak {p}}= \\phi _S^{\\mathcal {H}}^1_{\\operatorname{dR}}(A/S)_{\\mathfrak {p}}= \\phi _S^{\\mathcal {H}}$ to ${\\mathcal {F}}_{i}^{(e)} = (s_\\Omega ^1_{A/S})_{i}$ (abbreviating subscripts $\\tau _{{\\mathfrak {p}},i}$ by $i$ ).", "Note that since the image of ${\\mathcal {H}}_{i+1}$ under $\\operatorname{Ver}_A^$ is $\\phi _S^({\\mathcal {F}}_{i}^{(e)})$ , the inclusions ${\\mathcal {F}}_{i+1}^{(e-1)} \\subset u {\\mathcal {H}}_{i+1}$ and $u{\\mathcal {F}}_i^{(e)} \\subset {\\mathcal {F}}_i^{(e-1)}$ ensure that $\\operatorname{Ver}_A^({\\mathcal {F}}_{i+1}^{(e-1)}) \\subset \\phi _S^({\\mathcal {F}}_i^{(e-1)})$ , so the morphism $v_i$ is well-defined.", "We then let $H$ denote the finite flat $({\\mathcal {O}}_F/{\\mathfrak {p}})$ -vector space scheme over $S$ associated to the Raynaud data $({{\\mathcal {L}}}_i, f_i, v_i)_{i \\in \\mathbb {Z}/f\\mathbb {Z}}$ .", "Recall that the Dieudonné crystal of $\\ker ({\\operatorname{Frob}}_A)$ is canonically isomorphic to $\\Phi ^(s_\\Omega ^1_{A/S})$ , with $F = 0$ and $V$ induced by $\\Phi ^(\\operatorname{Ver}_A^)$ (in the notation of [2]).", "On the other hand the Dieudonné crystal of $H$ is identified with $\\Phi ^(\\oplus _i {{\\mathcal {L}}}_i)$ with $F = \\Phi ^(\\oplus _i f_i) = 0$ and $V = \\Phi ^(\\oplus _i v_i)$ (as a simple special case of [11]).", "Therefore the canonical projection $s_\\Omega ^1_{A/S} \\rightarrow \\oplus _{i} {{\\mathcal {L}}}_i$ induces a surjective morphism of Dieudonné crystals $\\ker ({\\operatorname{Frob}}_A)) \\rightarrow H)$ .", "As the base $S$ is smooth over $\\mathbb {F}$ , the exact contravariant functor $ is fully faithful on finite flat $ p$-group schemes over $ S$ (\\cite [Thm.~4.1.1]{BM}),so the surjection arises from a closed immersion $ H (FrobA)$, and we let$$\\alpha : A \\longrightarrow A^{\\prime } := A/H$$denote the resulting isogeny of abelian varieties over $ S$.", "Note that $ A'$ naturallyinherits an $ OF$-action $ '$ from the action $$ on $ A$.$ Let ${\\mathcal {I}}$ denote the image of the morphism $\\alpha ^: {\\mathcal {H}}^1_{\\operatorname{dR}}(A^{\\prime }/\\widetilde{Y}_{U,\\mathbb {F}})_{\\mathfrak {p}}\\rightarrow {\\mathcal {H}}$ .", "By construction, we have the exact sequence $\\begin{array}{ccccccc} A^{\\prime }[p])_{S} & \\longrightarrow & A[p])_{S}& \\longrightarrow & H)_{S} & \\longrightarrow & 0\\\\\\parallel \\wr &&\\parallel \\wr && \\parallel \\wr &&\\\\{\\mathcal {H}}^1_{\\operatorname{dR}}(A^{\\prime }/S) &\\stackrel{\\alpha ^}{\\longrightarrow } &{\\mathcal {H}}^1_{\\operatorname{dR}}(A/S) & \\stackrel{\\operatorname{Ver}_A^}{\\longrightarrow }& \\bigoplus _{i} \\phi _S^({\\mathcal {F}}_\\tau ^{(e)}/{\\mathcal {F}}_\\tau ^{(e-1)}) & \\longrightarrow & 0,\\end{array}$ showing that ${\\mathcal {I}}= \\oplus _{i} {\\mathcal {I}}_i$ , where ${\\mathcal {I}}_i$ is the preimage of $\\phi _S^({\\mathcal {F}}_{i-1}^{(e)})$ under $\\operatorname{Ver}_{A,i}^:{\\mathcal {H}}_i \\rightarrow \\phi _S^({\\mathcal {H}}_{i-1})$ .", "Note in particular that $u{\\mathcal {H}}_i \\subset {\\mathcal {I}}_i$ for all $i$ , so that $H \\subset A[{\\mathfrak {p}}]$ and there is a unique isogeny $\\beta : {\\mathfrak {p}}\\otimes _{{\\mathcal {O}}_F} A^{\\prime } \\longrightarrow A$ such that $\\alpha \\circ \\beta $ is the canonical isogeny ${\\mathfrak {p}}\\otimes _{{\\mathcal {O}}_F} A^{\\prime } \\rightarrow A^{\\prime }$ .", "We now equip $A^{\\prime }$ with auxiliary data corresponding to an element of $\\widetilde{Y}_{U,\\mathbb {F}}(S)$ .", "Since $\\alpha $ induces isomorphisms $T^{(p)}(A_{\\overline{s}}) \\stackrel{\\sim }{\\longrightarrow } T^{(p)}(A^{\\prime }_{\\overline{s}})$ for all geometric points $\\overline{s}$ of $S$ , we immediately have a level $U^p$ structure $\\eta ^{\\prime }$ on $A^{\\prime }$ inherited from $A$ .", "Next we claim that the quasi-polarization $\\lambda $ on $A$ induces an isomorphismHere ${\\mathfrak {c}}$ depends on the connected component of $S$ .", "${\\mathfrak {p}}{\\mathfrak {c}}{\\mathfrak {d}}\\otimes _{{\\mathcal {O}}_F} A^{\\prime } \\rightarrow (A^{\\prime })^\\vee $ , or equivalently $A^{\\prime } \\rightarrow {\\mathfrak {p}}^{-1}{\\mathfrak {c}}^{-1}{\\mathfrak {d}}^{-1} \\otimes _{{\\mathcal {O}}_F} (A^{\\prime })^\\vee $ , which amounts to the claim that $H$ corresponds to the kernel of ${\\mathfrak {c}}^{-1}{\\mathfrak {d}}^{-1} \\otimes \\beta ^\\vee : {\\mathfrak {c}}^{-1}{\\mathfrak {d}}^{-1} \\otimes _{{\\mathcal {O}}_F} A^\\vee \\longrightarrow {\\mathfrak {p}}^{-1}{\\mathfrak {c}}^{-1}{\\mathfrak {d}}^{-1} \\otimes _{{\\mathcal {O}}_F}( A^{\\prime })^\\vee $ under the isomorphism induced by $\\lambda $ .", "Denoting this kernel by $I$ , we have that $H$ and $I$ are finite flat group schemes over $S$ of the same rank, so it suffices to prove that the composite $I \\longrightarrow {\\mathfrak {c}}^{-1}{\\mathfrak {d}}^{-1} \\otimes _{{\\mathcal {O}}_F} A^\\vee [p] \\stackrel{\\sim }{\\longrightarrow } A[p] \\longrightarrow A^{\\prime }[p]$ is trivial.", "Taking Dieudonné modules, this in turn amounts to the vanishing of the composite $A^{\\prime }[p])_{S} \\longrightarrow A[p])_{S} \\longrightarrow {\\mathfrak {c}}^{-1}{\\mathfrak {d}}^{-1} \\otimes _{{\\mathcal {O}}_F} A^\\vee [p])_{S} \\longrightarrow I)_{S}.$ We have already noted that the image of the first map has ${\\mathfrak {p}}$ -component $\\oplus _{i} {\\mathcal {I}}_i$ ; on the other hand the kernel of the last map is the image of the map $ {\\mathfrak {p}}^{-1}{\\mathfrak {c}}^{-1}{\\mathfrak {d}}^{-1} \\otimes _{{\\mathcal {O}}_F} (A^{\\prime })^\\vee [p])_{S} \\longrightarrow {\\mathfrak {c}}^{-1}{\\mathfrak {d}}^{-1} \\otimes _{{\\mathcal {O}}_F} A^\\vee [p])_{S} $ corresponding to the adjoint of $\\beta ^: {\\mathcal {H}}^1_{\\operatorname{dR}}(A/S) \\rightarrow {\\mathcal {H}}^1_{\\operatorname{dR}}(({\\mathfrak {p}}\\otimes _{{\\mathcal {O}}_F} A^{\\prime })/S) \\cong {\\mathfrak {p}}^{-1} \\otimes _{{\\mathcal {O}}_F} {\\mathcal {H}}^1_{\\operatorname{dR}}(A^{\\prime }/S)$ under the canonical isomorphisms $\\begin{array}{rl} {\\mathfrak {c}}^{-1}{\\mathfrak {d}}^{-1} \\otimes _{{\\mathcal {O}}_F} A^\\vee [p])_S\\cong {\\mathcal {H}}^1_{\\operatorname{dR}}(({\\mathfrak {c}}^{-1}{\\mathfrak {d}}^{-1} \\otimes _{{\\mathcal {O}}_F} A^\\vee / S) &\\\\\\cong {\\mathcal {H}om}_{{\\mathcal {O}}_S}({\\mathfrak {d}}^{-1}\\otimes _{{\\mathcal {O}}_F} {\\mathcal {H}}^1_{\\operatorname{dR}}(A/ S), {\\mathcal {O}}_{S})&\\cong {\\mathcal {H}om}_{{\\mathcal {O}}_F\\otimes {\\mathcal {O}}_{S}}({\\mathcal {H}}^1_{\\operatorname{dR}}(A/ S), {\\mathcal {O}}_F \\otimes {\\mathcal {O}}_{S})\\end{array}$ and similarly $ {\\mathfrak {p}}^{-1}{\\mathfrak {c}}^{-1}{\\mathfrak {d}}^{-1} \\otimes _{{\\mathcal {O}}_F} (A^{\\prime })^\\vee [p])_{S }\\cong {\\mathcal {H}om}_{{\\mathcal {O}}_F\\otimes {\\mathcal {O}}_{S}}({\\mathfrak {p}}^{-1} \\otimes _{{\\mathcal {O}}_F}{\\mathcal {H}}^1_{\\operatorname{dR}}(A^{\\prime }/S), {\\mathcal {O}}_F \\otimes {\\mathcal {O}}_S)$ obtained from duality.", "We are therefore reduced to proving that ${\\mathcal {I}}_i$ is orthogonal to the kernel of $\\beta _i^$ for each $i \\in \\mathbb {Z}/f\\mathbb {Z}$ under the pairing $\\langle \\cdot ,\\cdot \\rangle _i$ defined by ().", "Note however that the kernel of $\\beta _i^$ is $u^{e-1} {\\mathcal {I}}_i$ , as can be seen for example from the commutative diagram ${ H^1_\\mathrm {crys}(A_{{\\overline{s}}}/W(\\overline{{\\mathbb {F}}}_p))_i @{^{(}->}[r]@{->>}[d] & ({\\mathfrak {p}}^{-1} \\otimes _{{\\mathcal {O}}_F} H^1_\\mathrm {crys}(A^{\\prime }_{{\\overline{s}}}/W(\\overline{{\\mathbb {F}}}_p)))_i @{->>}[d] \\\\H^1_{\\operatorname{dR}}(A_{{\\overline{s}}}/\\overline{{\\mathbb {F}}}_p))_i [r] & ({\\mathfrak {p}}^{-1} \\otimes _{{\\mathcal {O}}_F} H^1_{\\operatorname{dR}}(A^{\\prime }_{{\\overline{s}}}/\\overline{{\\mathbb {F}}}_p))_i }$ of $W(\\overline{{\\mathbb {F}}}_p)[u]$ -modules for ${\\overline{s}}\\in S(\\overline{{\\mathbb {F}}}_p)$ .", "Finally the orthogonality of ${\\mathcal {I}}_i$ and $u^{e-1}{\\mathcal {I}}_i$ is immediate from that of ${\\mathcal {F}}_{i-1}^{(e-1)}$ and $u^{-1}{\\mathcal {F}}_{i-1}^{(e-1)}$ provided by Lemma REF , completing the proof of the claim.", "We may then define the quasi-polarization on $A^{\\prime }$ by $\\alpha ^(\\lambda ^{\\prime }) = \\delta \\lambda $ for any totally positive generator $\\delta = \\delta _{\\mathfrak {p}}$ of ${\\mathfrak {p}}{\\mathcal {O}}_{F,(p)}$ , so that $\\lambda ^{\\prime }$ induces an isomorphism ${\\mathfrak {c}}^{\\prime }{\\mathfrak {d}}\\otimes _{{\\mathcal {O}}_F} A^{\\prime } \\stackrel{\\sim }{\\longrightarrow } (A^{\\prime })^\\vee $ where ${\\mathfrak {c}}^{\\prime } = \\delta ^{-1}{\\mathfrak {p}}{\\mathfrak {c}}$ .", "Finally we define a Pappas–Rapoport filtration on ${\\mathcal {F}}^{\\prime }_\\tau := (s^{\\prime }_\\Omega ^1_{A^{\\prime }/S})_\\tau $ for all $\\tau \\in \\Sigma _0$ .", "First note that if $\\tau \\notin \\Sigma _{{\\mathfrak {p}},0}$ , then $\\alpha ^$ induces an isomorphism ${\\mathcal {F}}^{\\prime }_\\tau \\simeq {\\mathcal {F}}_\\tau ^{(e)}$ , and we define $({\\mathcal {F}}_\\tau ^{\\prime })^{(j)}$ as the pre-image of ${\\mathcal {F}}_\\tau ^{(j)}$ .", "Suppose now that $\\tau = \\tau _{{\\mathfrak {p}},i}$ .", "Recall from the construction of $A^{\\prime } = A/H$ that $\\operatorname{Ver}_A^({\\mathcal {F}}_{i}^{(e-1)}) \\subset \\phi _S^({\\mathcal {F}}_{i-1}^{(e-1)})$ , so we have ${\\mathcal {F}}_i^{(e-1)} \\subset {\\mathcal {I}}_i$ .", "It follows that $(\\alpha _i^)^{-1}({\\mathcal {F}}_i^{(e-1)})$ is a subbundle of ${\\mathcal {H}}^{\\prime }_i := {\\mathcal {H}}^1_{\\operatorname{dR}}(A^{\\prime })_i$ of the same rank as ${\\mathcal {F}}^{\\prime }_i$ , namely $e$ .", "Furthermore we have $\\begin{array}{rl} \\phi _S^(\\alpha ^({\\mathcal {F}}_i^{\\prime })) = (\\phi _S^(\\alpha ))^ ({\\mathcal {F}}_i^{\\prime }) = &(\\phi _S^(\\alpha ))^(\\operatorname{Ver}_{A^{\\prime }}^({\\mathcal {H}}_{i+1}^{\\prime }))\\\\= &\\operatorname{Ver}_A^(\\alpha ^({\\mathcal {H}}_{i+1}^{\\prime })) = \\operatorname{Ver}_A^({\\mathcal {I}}_{i+1}) \\subset \\phi _S^({\\mathcal {F}}_i^{(e+1)}),\\end{array}$ so in fact ${\\mathcal {F}}_i^{\\prime } \\subset (\\alpha _i^)^{-1}({\\mathcal {F}}_i^{(e-1)})$ , and hence equality holds.", "We thus obtain an exact sequence $0 \\rightarrow \\ker (\\alpha _i^) \\longrightarrow {\\mathcal {F}}^{\\prime }_i \\stackrel{\\alpha _i^}{\\longrightarrow } {\\mathcal {F}}_i^{(e-1)} \\rightarrow 0.$ We may thus define a Pappas–Rapoport filtration on ${\\mathcal {F}}_i^{\\prime }$ by setting $({\\mathcal {F}}_i^{\\prime })^{(j)} = (\\alpha _i^)^{-1}({\\mathcal {F}}_i^{(j-1)})$ for $j=1,\\ldots ,e$ , so in particular $({\\mathcal {F}}_i^{\\prime })^{(1)} = \\ker (\\alpha _i^)$ .", "We now define $\\widetilde{\\Phi }_{\\mathfrak {p}}:\\widetilde{Y}_{U,\\mathbb {F}} = S \\rightarrow \\widetilde{Y}_{U,\\mathbb {F}}$ to be the endomorphism corresponding to the data $(A^{\\prime },\\iota ^{\\prime },\\lambda ^{\\prime },\\eta ^{\\prime },({\\mathcal {F}}^{\\prime })^\\bullet )$ .", "Note that $\\widetilde{\\Phi }_{\\mathfrak {p}}$ depends on the choice of $\\delta $ in the definition of $\\lambda ^{\\prime }$ ; however it is straightforward to check that $\\widetilde{\\Phi }_{\\mathfrak {p}}$ is compatible with the ${\\mathcal {O}}_{F,(p),+}^\\times $ -action on $S$ and descends to an endomorphism $\\Phi _{\\mathfrak {p}}$ of $\\overline{Y}_U$ which is independent of this choice.", "We call $\\widetilde{\\Phi }_{\\mathfrak {p}}$ (resp.", "$\\Phi _{\\mathfrak {p}}$ ) the partial Frobenius endomorphism (indexed by ${\\mathfrak {p}}$ ) of $\\widetilde{Y}_{U,\\mathbb {F}}$ (resp.", "$\\overline{Y}_U$ ); the terminology is justified by the next proposition.", "For the statement of the proposition, we also define the endomorphism $\\widetilde{\\Phi }$ of $S = \\widetilde{Y}_{U,\\mathbb {F}}$ corresponding to the data $\\phi _S^({\\underline{A}}) = (\\phi _S^A, \\phi _S^\\iota ,\\phi _S^\\lambda ,\\phi _S^\\eta ,(\\phi _S^{\\mathcal {F}})^\\bullet )$ , where $(\\phi _S^{\\mathcal {F}})^\\bullet $ is the collection of filtrations on the vector bundles $((\\phi _S^s)_(\\Omega ^1_{(\\phi _S^A)/S}))_\\tau = (\\phi _S^(s_\\Omega ^1_{A/S}))_\\tau = \\phi _S^((s_\\Omega ^1_{A/S})_{\\phi ^{-1}\\circ \\tau })$ given by $(\\phi _S^{\\mathcal {F}})_\\tau ^{(j)} = \\phi _S^({\\mathcal {F}}_{\\phi ^{-1}\\circ \\tau }^{(j)})$ .", "Note that $\\widetilde{\\Phi }$ is not the absolute Frobenius $\\phi _S$ on $S$ (unless $\\mathbb {F}= \\mathbb {F}_p$ ), but we may write $\\phi _S = \\widetilde{\\Phi }\\circ \\widetilde{\\epsilon }$ where $\\widetilde{\\epsilon }$ is the isomorphism defined by the commutative diagram ${S [drr]^{\\widetilde{\\epsilon }} [dr]^{\\varepsilon } [ddr] &&\\\\& \\phi ^S [r]^{\\sim } [d] & S [d] \\\\& {\\operatorname{Spec}\\,}\\mathbb {F}[r]^{\\sim }_{\\phi } & {\\operatorname{Spec}\\,}\\mathbb {F}},$ where the square is Cartesian and $\\varepsilon $ is the inverse of the isomorphism associated to $\\phi ^A = A \\times _{\\mathbb {F},\\phi } \\mathbb {F}$ with the evident auxiliary data.", "We thus have an isomorphism $\\widetilde{\\epsilon }^A \\cong A$ compatible with $\\iota $ , $\\lambda $ and $\\eta $ , and inducing $\\widetilde{\\epsilon }^{\\mathcal {F}}_\\tau ^{(j)} \\cong {\\mathcal {F}}_{\\phi \\circ \\tau }^{(j)}$ for all $\\tau $ and $j$ .", "(Note also that $\\widetilde{\\Phi }$ may be viewed as the base-change of the absolute Frobenius on the descent of $S$ to $\\mathbb {F}_p$ defined by the diagram.)", "The endomorphism $\\widetilde{\\Phi }$ is compatible with the ${\\mathcal {O}}_{F,(p),+}$ -action on $S = \\widetilde{Y}_{U,\\mathbb {F}}$ , and we let $\\Phi $ denote the resulting endomorphism of $\\overline{Y}_U$ .", "Similarly $\\widetilde{\\epsilon }$ descends to a $\\phi $ -linear automorphism $\\epsilon $ of $\\overline{Y}_U$ such that the absolute Frobenius on $\\overline{Y}_U$ is $\\Phi \\circ \\epsilon $ .", "Proposition 5.1.1 The morphisms $\\Phi _{\\mathfrak {p}}$ are finite flat of (constant) degree $\\mathrm {Nm}_{F/\\mathbb {Q}}({\\mathfrak {p}})$ , commute with each other, and satisfy the formula $\\prod _{{\\mathfrak {p}}\\in S_p} \\Phi _{\\mathfrak {p}}^{e_{\\mathfrak {p}}} = \\Phi .$ Proof.We first prove the commutativity and analogous formula for the maps $\\widetilde{\\Phi }_{\\mathfrak {p}}$ on $S = \\widetilde{Y}_{U,\\mathbb {F}}$ , from which the corresponding assertions for $\\Phi _{\\mathfrak {p}}$ follow.", "To that end it suffices to consider the maps on geometric closed points $\\overline{s}\\in S(\\overline{{\\mathbb {F}}}_p)$ , which we will do in order to facilitate computations on Dieudonné modules.", "Let ${\\underline{A}}_0$ denote the data corresponding to $\\overline{s} \\in S(\\overline{{\\mathbb {F}}}_p)$ and ${\\underline{A}}_0^{\\prime } = {\\underline{A}}_{0,{\\mathfrak {p}}}^{\\prime }$ denote the data corresponding to $\\widetilde{\\Phi }_{\\mathfrak {p}}(\\overline{s})$ .", "Let $D = H^1_\\mathrm {crys}(A_0/W(\\overline{{\\mathbb {F}}}_p))$ and $D^{\\prime } = H^1_\\mathrm {crys}(A_0^{\\prime }/W(\\overline{{\\mathbb {F}}}_p))$ , so we may decompose the ${\\mathcal {O}}_F\\otimes W(\\overline{{\\mathbb {F}}}_p)$ -modules $D = \\oplus _{\\tau \\in \\Sigma } D_\\tau $ and $D^{\\prime } = \\oplus _{\\tau \\in \\Sigma ^{\\prime }} D_\\tau ^{\\prime }$ where $D_\\tau $ and $D_\\tau ^{\\prime }$ are free $W(\\overline{{\\mathbb {F}}}_p)[u]/(f^\\tau (u))$ -modules of rank two.", "Furthermore the canonical isogeny $\\alpha : A_0 \\rightarrow A_0^{\\prime }$ induces an injective $W(\\overline{{\\mathbb {F}}}_p)[u]/(f^\\tau (u))$ -linear map $\\alpha _\\tau ^: D_\\tau ^{\\prime } \\rightarrow D_\\tau $ for each $\\tau $ , compatible in the obvious sense with the maps ${\\operatorname{Frob}}_{A_0,\\tau }^: \\phi ^(D_{\\phi ^{-1}\\circ \\tau }) \\rightarrow D_\\tau \\quad \\mbox{and}\\quad {\\operatorname{Frob}}_{A_0^{\\prime },\\tau }^:\\phi ^* (D_{\\phi ^{-1}\\circ \\tau }^{\\prime }) \\rightarrow D^{\\prime }_\\tau ,$ as well as $\\operatorname{Ver}_{A_0,\\tau }^* = p({\\operatorname{Frob}}_{A_0,\\tau }^)^{-1}$ and $\\operatorname{Ver}_{A_0^{\\prime },\\tau }^ = p({\\operatorname{Frob}}_{A_0^{\\prime },\\tau }^)^{-1}$ .", "Letting $F_\\tau ^{(j)} \\subset D_\\tau $ denote the pre-image of ${\\mathcal {F}}_\\tau ^{(j)}$ under the canonical surjection $D_\\tau \\longrightarrow (D/pD)_\\tau \\cong H^1_{\\operatorname{dR}}(A_0/\\overline{{\\mathbb {F}}}_p)_\\tau ,$ we have $\\alpha _\\tau ^(D_\\tau ^{\\prime }) = D_\\tau $ if $\\tau \\notin \\Sigma _{{\\mathfrak {p}},0}$ and $\\alpha _\\tau ^(D_\\tau ^{\\prime }) = (\\operatorname{Ver}_{A_0,\\tau }^)^{-1}(\\phi ^(F_{\\phi ^{-1}\\circ \\tau }^{(e_{\\mathfrak {p}}-1)}))$ if $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ (by the construction of $\\Phi _{\\mathfrak {p}}$ ).", "Furthermore writing $F^{\\prime (j)}_\\tau $ for the submodules of $D_\\tau ^{\\prime }$ similarly defined by the Pappas–Rapoport filtration on $H^0(A_0^{\\prime },\\Omega ^1_{A_0^{\\prime }/\\overline{{\\mathbb {F}}}_p})_\\tau $ , we have $\\alpha _\\tau ^(F^{\\prime (j)}_\\tau ) = F^{(j)}_\\tau $ if $\\tau \\notin \\Sigma _{{\\mathfrak {p}},0}$ , $\\alpha _\\tau ^(F^{\\prime (j)}_\\tau ) = F^{(j-1)}_\\tau $ for $j=2,\\ldots ,e_{\\mathfrak {p}}$ if $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ , and $\\alpha _\\tau ^(F^{\\prime (1)}_\\tau ) = pD_\\tau = {\\operatorname{Frob}}_{A_0,\\tau }^(\\phi ^(F^{(e_{\\mathfrak {p}})}_{\\phi ^{-1}\\circ \\tau }))$ if $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ .", "Thus if ${\\mathfrak {p}}_1$ and ${\\mathfrak {p}}_2$ are distinct elements of $S_p$ , then $\\widetilde{\\Phi }_{{\\mathfrak {p}}_1}(\\widetilde{\\Phi }_{{\\mathfrak {p}}_2}(\\overline{s}))$ corresponds to the data ${\\underline{A}}_0^{\\prime \\prime }$ for which we have an isogeny $\\alpha ^{\\prime }:A_0 \\rightarrow A_0^{\\prime \\prime }$ such that if $\\tau \\notin \\Sigma _{{\\mathfrak {p}}_1,0} \\cup \\Sigma _{{\\mathfrak {p}}_2,0}$ then $(\\alpha ^{\\prime })_\\tau ^(D_\\tau ^{\\prime \\prime }) = D_\\tau $ and $(\\alpha ^{\\prime })_\\tau ^(F^{\\prime \\prime (j)}_\\tau ) = F^{(j)}_\\tau $ (with the obvious notation), but if $\\tau \\in \\Sigma _{{\\mathfrak {p}}_i,0}$ for $i=1$ or 2, then $(\\alpha ^{\\prime })_\\tau ^(D_\\tau ^{\\prime \\prime }) = (\\operatorname{Ver}_{A_0,\\tau }^)^{-1}(\\phi ^(F_{\\phi ^{-1}\\circ \\tau }^{(e_{{\\mathfrak {p}}_i}-1)})),$ $(\\alpha ^{\\prime })_\\tau ^(F^{\\prime \\prime (1)}_\\tau ) = pD_\\tau $ and $(\\alpha ^{\\prime })_\\tau ^(F^{\\prime \\prime (j)}_\\tau ) = F^{(j-1)}_\\tau $ for $j=2,\\ldots ,e_{{\\mathfrak {p}}_i}$ .", "Furthermore we have $\\eta ^{\\prime \\prime } = \\alpha ^{\\prime }\\circ \\eta $ and $(\\alpha ^{\\prime })^(\\lambda ^{\\prime \\prime }) = \\delta _{{\\mathfrak {p}}_2}\\delta _{{\\mathfrak {p}}_1}\\lambda $ , from which it follows that the isomorphism class of the data ${\\underline{A}}_0^{\\prime \\prime }$ also corresponds to $\\widetilde{\\Phi }_{{\\mathfrak {p}}_2}(\\widetilde{\\Phi }_{{\\mathfrak {p}}_1}(\\overline{s}))$ .", "Now consider the data ${\\underline{A}}_0^{(r)}$ associated to $\\widetilde{\\Phi }_{\\mathfrak {p}}^r(\\overline{s})$ for $r=1,\\ldots ,e_{\\mathfrak {p}}$ , and write $D_r = \\oplus D_{r,\\tau }$ for $H^1_\\mathrm {crys}(A_0^{(r)}/W(\\overline{{\\mathbb {F}}}_p))$ , $F_{r,\\tau }$ for the submodule of $D_{r,\\tau }$ determined as above by the Pappas–Rapoport filtration, and $\\alpha _r$ for the composite isogeny $A_0 \\rightarrow A_0^{(1)} \\rightarrow \\cdots \\rightarrow A_0^{(r)}$ .", "By induction on $r$ , we find that if $\\tau \\notin \\Sigma _{{\\mathfrak {p}},0}$ , then $\\alpha _{r,\\tau }^(D_{r,\\tau }) = D_\\tau $ and $\\alpha _{r,\\tau }^(F_{r,\\tau }^{(j)}) = F^{(j)}_\\tau $ , but if $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ , then $\\alpha _{r,\\tau }^(D_{r,\\tau }) = (\\operatorname{Ver}_{A_0,\\tau }^)^{-1}(\\phi ^(F_{\\phi ^{-1}\\circ \\tau }^{(e_{{\\mathfrak {p}}}-r)}))$ , $\\alpha _{r,\\tau }^(F^{(j)}_{r,\\tau }) = F^{(j-r)}_\\tau $ for $j=r+1,\\ldots ,e_{\\mathfrak {p}}$ , $\\alpha _{r,\\tau }^(F^{(j)}_{r,\\tau }) = {\\operatorname{Frob}}_{A_0,\\tau }^(\\phi ^(F^{(e_{\\mathfrak {p}}+j-r)}_{\\phi ^{-1}\\circ \\tau }))$ for $j=1,\\ldots ,r$ .", "In particular taking $r=e_{\\mathfrak {p}}$ gives $\\alpha _{e_{\\mathfrak {p}},\\tau }^(D_{e_{\\mathfrak {p}},\\tau }) = (\\operatorname{Ver}_{A_0,\\tau }^)^{-1}(p\\phi ^(D_{\\phi ^{-1}\\circ \\tau })) = {\\operatorname{Frob}}_{A_0,\\tau }^(\\phi ^(D_{\\phi ^{-1}\\circ \\tau }))$ and $\\alpha _{e_{\\mathfrak {p}},\\tau }^(F^{(j)}_{e_{\\mathfrak {p}},\\tau }) = {\\operatorname{Frob}}_{A_0,\\tau }^(\\phi ^(F^{(j)}_{\\phi ^{-1}\\circ \\tau }))$ for $j=1,\\ldots ,e_{\\mathfrak {p}}$ and $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ .", "It then follows that $\\prod _{{\\mathfrak {p}}\\in S_p}\\widetilde{\\Phi }_{\\mathfrak {p}}^{e_{\\mathfrak {p}}}(\\overline{s})$ corresponds to ${\\underline{A}}_0^{\\prime \\prime }$ with $\\alpha ^{\\prime }:A_0\\rightarrow A_0^{\\prime \\prime }$ satisfying $(\\alpha ^{\\prime })_\\tau ^(D_\\tau ^{\\prime \\prime }) = {\\operatorname{Frob}}_{A_0,\\tau }^(\\phi ^(D_{\\phi ^{-1}\\circ \\tau }))$ and $(\\alpha ^{\\prime })_\\tau ^(F^{\\prime \\prime (j)}_{\\tau }) = {\\operatorname{Frob}}_{A_0,\\tau }^(\\phi ^(F^{(j)}_{\\phi ^{-1}\\circ \\tau }))$ for all $\\tau $ and $j$ .", "Furthermore we have $\\eta ^{\\prime \\prime } = \\alpha ^{\\prime }\\circ \\eta $ and $(\\alpha ^{\\prime })^(\\lambda ^{\\prime \\prime }) = \\prod _{{\\mathfrak {p}}} \\delta _{{\\mathfrak {p}}}^{e_{\\mathfrak {p}}}\\lambda $ , from which it follows that ${\\underline{A}}_0^{\\prime \\prime }$ is isomorphic to $(\\phi ^A_0,\\phi ^\\iota ,\\nu \\phi ^\\lambda ,(\\phi ^{\\mathcal {F}})^\\bullet )$ with $\\nu = p^{-1}\\prod _{{\\mathfrak {p}}} \\delta _{{\\mathfrak {p}}}^{e_{\\mathfrak {p}}} \\in {\\mathcal {O}}_{F,(p),+}^\\times $ .", "This proves that $\\prod _{{\\mathfrak {p}}\\in S_p} \\widetilde{\\Phi }_{\\mathfrak {p}}^{e_{\\mathfrak {p}}}= \\nu \\cdot \\widetilde{\\Phi },$ which in turn implies the desired formula.", "Since $\\Phi $ is finite (and $\\overline{Y}_U$ is separated), it follows that $\\Phi _{\\mathfrak {p}}$ is finite, and therefore also flat since $\\overline{Y}_U$ regular.", "Note furthermore that $\\Phi _{\\mathfrak {p}}$ is therefore bijective on closed points and induces isomorphisms on their residue fields, so the degree of $\\Phi _{\\mathfrak {p}}$ in a neighborhood of any closed point $x$ of $\\overline{Y}_U$ is that of the extension of completed regular local rings $\\Phi _{{\\mathfrak {p}},x}^: {\\mathcal {O}}_{\\overline{Y}_U,y}^\\wedge \\rightarrow {\\mathcal {O}}_{\\overline{Y}_U,x}^\\wedge $ , where $y = \\Phi _{\\mathfrak {p}}(x)$ .", "Since $\\Phi $ factors through $\\Phi _{\\mathfrak {p}}$ , so does the absolute Frobenius on $\\overline{Y}_U$ , and hence the absolute Frobenius on ${\\mathcal {O}}_{\\overline{Y}_U,x}^\\wedge $ factors through $\\Phi _{{\\mathfrak {p}},x}^$ .", "Therefore it follows from [26] that $\\deg (\\Phi _{{\\mathfrak {p}},x}^) = p^n$ where $n = n_{\\mathfrak {p}}$ is the dimension of the kernel of the induced map on tangent spaces $T_x(\\overline{Y}_U) \\rightarrow T_y(\\overline{Y}_U)$ .", "Furthermore since $\\prod _{{\\mathfrak {p}}\\in S_p} \\Phi _{\\mathfrak {p}}^{e_{\\mathfrak {p}}} = \\Phi $ has degree $p^{[F:\\mathbb {Q}]} = \\prod _{{\\mathfrak {p}}\\in S_p}p^{ e_{\\mathfrak {p}}f_{\\mathfrak {p}}}$ , it suffices to prove that $n_{\\mathfrak {p}}\\ge f_{\\mathfrak {p}}= Nm_{F/\\mathbb {Q}}({\\mathfrak {p}})$ for each ${\\mathfrak {p}}$ .", "Note also that we may replace $\\overline{Y}_U$ by $S=\\widetilde{Y}_{U,\\mathbb {F}}$ , $x$ by any point in its pre-image in $S$ and $\\Phi _{\\mathfrak {p}}$ by $\\widetilde{\\Phi }_{{\\mathfrak {p}}}$ .", "Suppose then that $x$ corresponds to the data $(A_0,\\iota _0,\\lambda _0,\\eta _0,{\\mathcal {F}}_0^\\bullet )$ over the residue field $k$ , and its image $y =\\widetilde{\\Phi }_{\\mathfrak {p}}(x)$ corresponds to the data $(A^{\\prime }_0,\\iota ^{\\prime }_0,\\lambda ^{\\prime }_0,\\eta ^{\\prime }_0,{\\mathcal {F}}^{\\prime \\bullet }_0)$ .", "Recall that the Kodaira–Spencer filtration on the fibre of $\\Omega ^1_{S/\\overline{{\\mathbb {F}}}_p}$ at $x$ is dual to one on $T_x(S)$ which was described using Grothendieck–Messing deformation theory (see the discussion following the proof of Theorem ).", "In particular, we have a decomposition $T_x(S) = \\oplus _{\\tau \\in \\Sigma _0} T_x(S)_\\tau $ and a decreasing filtration of length $e_{{\\mathfrak {p}}^{\\prime }}$ on $T_x(S)_\\tau $ for each $\\tau \\in \\Sigma _{{\\mathfrak {p}}^{\\prime },0}$ (where ${\\mathfrak {p}}^{\\prime } \\in S_p$ ) such that $\\bigoplus _{\\tau \\in \\Sigma _0} {\\operatorname{Fil}\\,}^{j_\\tau } T_x(S)_\\tau $ corresponds to the set of lifts of ${\\underline{A}}_0$ to ${\\underline{A}}_1 = (A_1,\\iota _1,\\lambda _1,\\eta _1,{\\mathcal {F}}_1^\\bullet ) \\in S(k[\\epsilon ])$ such that ${\\mathcal {F}}_{1,\\tau }^{(j)}$ is the image of ${\\mathcal {F}}_{0,\\tau }^{(j)} \\otimes _k k[\\epsilon ]$ for all $\\tau $ and $j \\le j_\\tau $ under the canonical isomorphism $H^1_{\\operatorname{dR}}(A_1/k[\\epsilon ]) \\cong H^1_{\\operatorname{cris}}(A_0/k[\\epsilon ]) \\cong H^1_{\\operatorname{dR}}(A_0/k) \\otimes _k k[\\epsilon ].$ We claim that the $f_{\\mathfrak {p}}$ -dimensional subspace $\\oplus _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}} {\\operatorname{Fil}\\,}^{e_{\\mathfrak {p}}-1} T_x(S)_\\tau $ is contained in the kernel of $T_x(S) \\rightarrow T_y(S)$ .", "Indeed if ${\\underline{A}}_1$ is a lift corresponding to an element of this subspace and ${\\underline{A}}_1^{\\prime }$ is its image in $T_y(S)$ and $\\alpha _i:A_i \\rightarrow A_i^{\\prime }$ are the specializations of the universal isogeny $\\alpha :A \\rightarrow A^{\\prime }$ , then the commutativity of the diagram ${ H^1_{\\operatorname{dR}}(A_1^{\\prime }/k[\\epsilon ]) [r]^{\\alpha _1^} [d]^{\\wr } & H^1_{\\operatorname{dR}}(A_1/k[\\epsilon ]) [d]^{\\wr } \\\\H^1_{\\operatorname{dR}}(A^{\\prime }_0/k) \\otimes _k k[\\epsilon ] [r]^{\\alpha _0^\\otimes 1} & H^1_{\\operatorname{dR}}(A_0/k) \\otimes _k k[\\epsilon ]}$ and the definition of $\\widetilde{\\Phi }_{\\mathfrak {p}}$ imply that ${\\mathcal {F}}^{\\prime (j)}_{1,\\tau }$ corresponds to ${\\mathcal {F}}^{\\prime (j)}_{0,\\tau } \\otimes _k k[\\epsilon ]$ for all $\\tau $ and $j$ .", "(Note in particular that ${\\mathcal {F}}_{1,\\tau }^{(1)} = \\ker (\\alpha _{1,\\tau }^)$ for all $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ , and that $H^0(A_1^{\\prime },\\Omega _{A_1^{\\prime }/k[\\epsilon ]})_\\tau $ corresponds to $H^0(A_0,\\Omega _{A_0^{\\prime }/k})_\\tau \\otimes _k k[\\epsilon ]$ for all $\\tau \\in \\Sigma _0$ .)", "It follows that ${\\underline{A}}_1^{\\prime }$ is the trivial deformation of ${\\underline{A}}_0^{\\prime }$ , so the kernel of $T_x(S) \\rightarrow T_y(S)$ has dimension $n \\ge f_{\\mathfrak {p}}$ as required.", "$\\square $ Remark 5.1.2 The final part of the proof of the proposition shows that the kernel of the pull-back map $\\Phi _{\\mathfrak {p}}^\\Omega ^1_{\\overline{Y}_U/\\mathbb {F}} \\rightarrow \\Omega ^1_{\\overline{Y}_U/\\mathbb {F}}$ is precisely $\\Phi _{\\mathfrak {p}}^\\left(\\oplus _{\\tau \\in \\Sigma _{{\\mathfrak {p}},0}} {\\operatorname{Fil}\\,}^1(\\Omega ^1_{\\overline{Y}_U/\\mathbb {F}})_\\tau \\right).$ Furthermore a similar argument shows that the map preserves the Kodaira–Spencer decomposition and filtration, in the obvious sense, and induces isomorphisms $\\begin{array}{cc}\\Phi _{\\mathfrak {p}}^\\left( {\\operatorname{Fil}\\,}^j(\\Omega ^1_{\\overline{Y}_U/\\mathbb {F}})_\\tau /{\\operatorname{Fil}\\,}^1(\\Omega ^!_{\\overline{Y}_U/\\mathbb {F}})_\\tau \\right)\\stackrel{\\sim }{\\longrightarrow }{\\operatorname{Fil}\\,}^{j-1}(\\Omega ^1_{\\overline{Y}_U/\\mathbb {F}})_\\tau , &\\mbox{if $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}, j = 1,\\ldots , e_{\\mathfrak {p}}$,}\\\\\\Phi _{\\mathfrak {p}}^\\left( {\\operatorname{Fil}\\,}^j(\\Omega ^1_{\\overline{Y}_U/\\mathbb {F}})_\\tau \\right)\\stackrel{\\sim }{\\longrightarrow }{\\operatorname{Fil}\\,}^j(\\Omega ^1_{\\overline{Y}_U/\\mathbb {F}})_\\tau , &\\mbox{if $\\tau \\notin \\Sigma _{{\\mathfrak {p}},0}$.", "}\\end{array}$" ], [ "Construction of $V_{\\mathfrak {p}}$", "In this section we generalize the constructionThe operators defined here differ slightly from the ones defined in [12] in the unramified case.", "The construction there is tailored to be compatible with the classical case and to be simply interpreted on $q$ -expansions at cusps at $\\infty $ .", "Doing this in the general ramified case would introduce complications that make it seem not worthwhile.", "of [12] to define partial Frobenius operators, similar to the $V_p$ -operator on classical modular forms.", "We maintain the notation of §REF , so that $\\widetilde{\\Phi }_{\\mathfrak {p}}$ is an endomorphism of $S = \\widetilde{Y}_{U,\\mathbb {F}}$ corresponding to the data $(A^{\\prime },\\iota ^{\\prime },\\lambda ^{\\prime },\\eta ^{\\prime },({\\mathcal {F}}^{\\prime })^\\bullet )$ , where $A^{\\prime } = A/H$ for a certain finite flat subgroup scheme $H \\subset A[{\\mathfrak {p}}]$ , and $\\alpha $ is the projection $A \\rightarrow A^{\\prime }$ .", "It is immediate from the definition of ${\\mathcal {F}}^{\\prime (j)}_\\tau $ that $\\alpha _\\tau ^$ induces an isomorphism ${\\mathcal {L}}_{\\tau ,j}^{\\prime } \\stackrel{\\sim }{\\longrightarrow } {\\mathcal {L}}_{\\tau ,j}$ for all $j$ if $\\tau \\notin \\Sigma _{{\\mathfrak {p}},0}$ , as well as ${\\mathcal {L}}_{\\tau ,j}^{\\prime } = {\\mathcal {F}}^{\\prime (j)}_\\tau /{\\mathcal {F}}^{\\prime (j-1)}_\\tau \\stackrel{\\sim }{\\longrightarrow }{\\mathcal {F}}^{(j-1)}_\\tau /{\\mathcal {F}}^{(j-2)}_\\tau = {\\mathcal {L}}_{\\tau ,j-1}$ for $j=2,\\ldots ,e_{\\mathfrak {p}}$ if $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ .", "To describe ${\\mathcal {L}}_{\\tau ,1}^{\\prime }$ for $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ , note that since $H \\subset \\ker ({\\operatorname{Frob}}_A)$ , there is an isogeny $\\gamma :A^{\\prime } \\rightarrow \\phi _S^A$ such that $\\gamma \\circ \\alpha = {\\operatorname{Frob}}_A$ .", "We thus obtain a surjection $\\gamma _\\tau ^: \\phi _S^({\\mathcal {F}}_{\\phi ^{-1}\\circ \\tau }^{(e_{\\mathfrak {p}})}) = \\ker ({\\operatorname{Frob}}_A)_\\tau ^\\longrightarrow \\ker (\\alpha _\\tau ^) = {\\mathcal {L}}^{\\prime }_{\\tau ,1}.$ Furthermore $\\phi _S^({\\mathcal {F}}_{\\phi ^{-1}\\circ \\tau }^{(e_{\\mathfrak {p}}-1)}) = \\ker (\\gamma _\\tau ^)$ , as can be seen on closed points, so we obtain a surjection, hence isomorphism, of line bundles ${\\mathcal {L}}_{\\phi ^{-1}\\circ \\tau ,e_{\\mathfrak {p}}}^{\\otimes p} \\cong \\phi _S^({\\mathcal {L}}_{\\phi ^{-1}\\circ \\tau ,e_{\\mathfrak {p}}}) =\\phi _S^({\\mathcal {F}}_{\\phi ^{-1}\\circ \\tau }^{(e_{\\mathfrak {p}})} )/ \\phi _S^({\\mathcal {F}}_{\\phi ^{-1}\\circ \\tau }^{(e_{\\mathfrak {p}}-1)})\\stackrel{\\gamma _\\tau ^}{\\longrightarrow } {\\mathcal {L}}^{\\prime }_{\\tau ,1}.$ By construction, we have $\\widetilde{\\Phi }_{\\mathfrak {p}}^{\\mathcal {L}}_{\\tau ,j} = {\\mathcal {L}}^{\\prime }_{\\tau ,j}$ , so we have now defined isomorphisms $\\widetilde{\\Phi }_{\\mathfrak {p}}^{\\mathcal {L}}_{\\theta } \\cong \\left\\lbrace \\begin{array}{ll} {\\mathcal {L}}_{\\sigma ^{-1}\\theta }^{\\otimes n_\\theta },&\\mbox{if $\\theta \\in \\Sigma _{{\\mathfrak {p}}}$;}\\\\{\\mathcal {L}}_{\\theta },&\\mbox{if $\\theta \\notin \\Sigma _{{\\mathfrak {p}}}$.", "}\\end{array}\\right.$ Similarly we find that if $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ , then $\\gamma ^$ induces $\\phi _S^({\\mathcal {P}}_{\\phi ^{-1}\\circ \\tau ,e_{\\mathfrak {p}}}) \\cong {\\mathcal {P}}^{\\prime }_{\\tau ,1}$ and $\\alpha ^$ induces ${\\mathcal {P}}^{\\prime }_{\\tau ,j} \\cong {\\mathcal {P}}_{\\tau ,j-1}$ for $j=2,\\ldots ,e_{\\mathfrak {p}}$ , so that $\\widetilde{\\Phi }_{\\mathfrak {p}}^ {\\mathcal {N}}_\\theta \\cong {\\mathcal {N}}_{\\sigma ^{-1}\\theta }^{\\otimes n_\\theta }$ if $\\theta \\in \\Sigma _{\\mathfrak {p}}$ , while $\\widetilde{\\Phi }_{\\mathfrak {p}}^{\\mathcal {N}}_\\theta \\cong {\\mathcal {N}}_\\theta $ if $\\theta \\notin \\Sigma _{\\mathfrak {p}}$ .", "Taking tensor products, we thus obtain isomorphisms $\\Phi _{\\mathfrak {p}}^\\widetilde{{\\mathcal {A}}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}} \\cong \\widetilde{{\\mathcal {A}}}_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime },\\mathbb {F}}$ for all $\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma $ , where $k^{\\prime \\prime }_\\theta = k_\\theta $ and $l_\\theta ^{\\prime \\prime } = l_\\theta $ if $\\theta \\notin \\Sigma _{\\mathfrak {p}}$ , and $k^{\\prime \\prime }_\\theta = n_{\\sigma \\theta } k_{\\sigma \\theta }$ and $l^{\\prime \\prime }_\\theta = n_{\\sigma \\theta } l_{\\sigma \\theta }$ if $\\theta \\in \\Sigma _{\\mathfrak {p}}$ .", "Furthermore it is straightforward to check that the isomorphisms are compatible with the descent data relative to $S=\\widetilde{Y}_{U,\\mathbb {F}} \\rightarrow Y_{U,\\mathbb {F}}$ , so we obtain isomorphisms $ \\Phi _{\\mathfrak {p}}^{\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}} \\cong {\\mathcal {A}}_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime },\\mathbb {F}} $ for all $\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma $ and sufficiently small $U$ .", "Note that $ \\mathbf {k}^{\\prime \\prime } = \\mathbf {k}+ \\sum _{\\theta \\in \\Sigma _{\\mathfrak {p}}}k_\\theta \\mathbf {h}_\\theta \\quad \\mbox{and}\\quad \\mathbf {l}^{\\prime \\prime } = \\mathbf {l}+ \\sum _{\\theta \\in \\Sigma _{\\mathfrak {p}}} l_\\theta \\mathbf {h}_\\theta .$ Remark 5.2.1 One can check that the resulting isomorphisms $\\Phi _{\\mathfrak {p}}^{\\mathcal {A}}_{2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\theta ,\\mathbb {F}} \\cong {\\mathcal {A}}_{2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\theta ,\\mathbb {F}}$ (for $\\theta \\notin \\Sigma _{\\mathfrak {p}}$ ), and $\\Phi _{\\mathfrak {p}}^{\\mathcal {A}}_{2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\tau ,\\mathbb {F}} \\cong {\\mathcal {A}}_{2{\\mathbf {e}}_{\\sigma ^{-1}\\theta },-{\\mathbf {e}}_{\\sigma ^{-1}\\theta },\\mathbb {F}}$ (for $\\theta = \\theta _{{\\mathfrak {p}},i,j}$ , $j=2,\\ldots ,e_{\\mathfrak {p}}$ ) are compatible via the Kodaira–Spencer isomorphisms of Theorem with the corresponding isomorphisms $\\Phi _{\\mathfrak {p}}^({\\operatorname{gr}\\,}^j(\\Omega ^1_{\\overline{Y}_U/\\mathbb {F}})_\\tau ) \\cong {\\operatorname{gr}\\,}^{j^{\\prime }}(\\Omega ^1_{\\overline{Y}_U/\\mathbb {F}})_\\tau $ given by Remark REF , where $j^{\\prime } = j$ if $\\tau \\notin \\Sigma _{{\\mathfrak {p}},0}$ and $j^{\\prime } = j-1$ if $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ .", "We are now ready to define the partial Frobenius operator (indexed by ${\\mathfrak {p}}$ ) $V_{\\mathfrak {p}}: M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) \\longrightarrow M_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime }}(U,\\mathbb {F})$ as the composite $H^0(Y_{U,\\mathbb {F}},{\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}}) \\stackrel{\\Phi _{\\mathfrak {p}}^}{\\longrightarrow } H^0(Y_{U,\\mathbb {F}},\\Phi _{\\mathfrak {p}}^{\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}})\\stackrel{\\sim }{\\longrightarrow }H^0(Y_{U,\\mathbb {F}},{\\mathcal {A}}_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime },\\mathbb {F}}),$ where the second map is the isomorphism (REF ).", "It is immediate from the definition that $V_{\\mathfrak {p}}$ is injective, and that taking the direct sum over all weights yields an $\\mathbb {F}$ -algebra homomorphism $\\bigoplus _{\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma } M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) \\longrightarrow \\bigoplus _{\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma } M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F})$ for all sufficiently small $U$ containing $\\operatorname{GL}_2({\\mathcal {O}}_{F,p})$ .", "It is also straightforward to check that $V_{\\mathfrak {p}}$ is compatible with the Hecke action in the usual sense, and hence defines a $\\operatorname{GL}_2(\\mathbb {A}_{F,^{(p)})-equivariant mapM_{\\mathbf {k},\\mathbf {l}}(\\mathbb {F}) \\longrightarrow M_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime }}(\\mathbb {F}),where the spaces are defined in (\\ref {eqn:inflevel}) as direct limits over sufficiently small U containing \\operatorname{GL}_2({\\mathcal {O}}_{F,p}).", "}$ It will also be convenient at times to consider instead the operator $ V^0_{\\mathfrak {p}}: M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) \\longrightarrow M_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}}(U,\\mathbb {F})$ defined by $V^0_{\\mathfrak {p}}(f) = V_{\\mathfrak {p}}(f) \\prod _{\\theta \\in \\Sigma _{\\mathfrak {p}}} G_\\theta ^{-l_\\theta }$ , where $G_\\theta $ is the trivialization of ${\\mathcal {A}}_{\\bf {0},\\mathbf {h}_\\theta ,\\mathbb {F}}$ defined at the end of §REF .", "Thus $V_{\\mathfrak {p}}^0$ is also Hecke-equivariant, but depends on the choice of uniformizer $\\varpi _{\\mathfrak {p}}$ .", "We also record the relation between the partial Frobenius operators and the $p$ -power map.", "First note that the identification $\\widetilde{\\Phi }^({\\mathcal {F}}_\\tau ^{(j)}) = \\phi _S^({\\mathcal {F}}_{\\phi ^{-1}\\circ \\tau }^{(j)})$ arising from the definition of $\\widetilde{\\Phi }$ yields isomorphisms $\\widetilde{\\Phi }^{\\mathcal {L}}_{\\tau ,j} \\cong \\phi _S^({\\mathcal {L}}_{\\phi ^{-1}\\circ \\tau ,j} ) \\cong {\\mathcal {L}}_{\\phi ^{-1}\\circ \\tau ,j}^{\\otimes p}$ for all $\\tau $ and $j$ .", "We similarly have $\\widetilde{\\Phi }^{\\mathcal {N}}_{\\tau ,j} \\cong {\\mathcal {N}}_{\\phi ^{-1}\\circ \\tau ,j}^{\\otimes p}$ , and taking tensor products and descending to $\\overline{Y}$ yields isomorphisms $\\Phi ^{\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}} \\cong {\\mathcal {A}}_{p\\mathbf {k}^\\phi ,p\\mathbf {l}^\\phi ,\\mathbb {F}}$ for all $\\mathbf {k},\\mathbf {l}$ , where $k^\\phi _{\\theta _{{\\mathfrak {p}},i,j}} = k_{\\theta _{{\\mathfrak {p}},i+1,j}}$ , and hence an operator $V_p: M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) \\rightarrow M_{p\\mathbf {k}^\\phi ,p\\mathbf {l}^\\phi }(U;\\mathbb {F})$ .", "Similarly the isomorphisms $\\widetilde{\\epsilon }^{\\mathcal {F}}_{\\phi ^{-1}\\circ \\tau }^{(j)} \\cong {\\mathcal {F}}_{\\tau }^{(j)}$ yield $\\widetilde{\\epsilon }^{\\mathcal {L}}_{\\phi ^{-1}\\circ \\tau ,j} \\cong {\\mathcal {L}}_{\\tau ,j}$ and $\\widetilde{\\epsilon }^{\\mathcal {N}}_{\\phi ^{-1}\\circ \\tau ,j} \\cong {\\mathcal {N}}_{\\tau ,j}$ whose tensor products descend to isomorphisms $\\epsilon ^{\\mathcal {A}}_{\\mathbf {k}^\\phi ,\\mathbf {l}^\\phi ,\\mathbb {F}} \\stackrel{\\sim }{\\longrightarrow }{\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}}$ , yielding a $\\phi $ -linear isomorphism $M_{\\mathbf {k}^\\phi ,\\mathbf {l}^\\phi }(U;\\mathbb {F}) \\rightarrow M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F})$ which we denote $\\epsilon _{\\mathbf {k},\\mathbf {l}}$ .", "Furthermore the above isomorphisms of line bundles on $S$ are compatible in the sense that the resulting diagram ${\\widetilde{\\epsilon }^\\widetilde{\\Phi }^{\\mathcal {L}}_{\\tau ,j} [r]^{\\sim }[d]_{\\wr } &\\widetilde{\\epsilon }^{\\mathcal {L}}_{\\phi ^{-1}\\circ \\tau ,j}^{\\otimes p} [d]^{\\wr } \\\\\\phi _S^{\\mathcal {L}}_{\\tau ,j} [r]^{\\sim } & {\\mathcal {L}}_{\\tau ,j}^{\\otimes p} }$ commutes, as does its analogue for the ${\\mathcal {N}}_\\theta $ , from which it follows that the composite $M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) \\stackrel{V_p}{\\longrightarrow } M_{p\\mathbf {k}^\\phi ,p\\mathbf {l}^\\phi }(U;\\mathbb {F})\\stackrel{\\epsilon _{p\\mathbf {k},p\\mathbf {l}}}{\\longrightarrow } M_{p\\mathbf {k},p\\mathbf {l}}(U;\\mathbb {F})$ is the $p$ -power map.", "Returning to the partial Frobenius operators, the isomorphisms between $\\widetilde{\\Phi }_{\\mathfrak {p}}^* {\\mathcal {L}}_\\theta $ and ${\\mathcal {L}}_{\\sigma ^{-1}\\theta }^{\\otimes n_\\theta }$ (resp.", "${\\mathcal {L}}_\\theta $ ) for $\\theta \\in \\Sigma _{\\mathfrak {p}}$ (resp.", "$\\theta \\notin \\Sigma _{\\mathfrak {p}}$ ) for different ${\\mathfrak {p}}\\in S_p$ are compatible with each other in the obvious sense, and taken together with the formula $\\prod _{{\\mathfrak {p}}\\in S_p} \\widetilde{\\Phi }_{\\mathfrak {p}}^{e_{\\mathfrak {p}}}= \\nu \\cdot \\widetilde{\\Phi }$ and the canonical isomorphism $\\nu ^{\\mathcal {L}}_\\theta \\cong {\\mathcal {L}}_\\theta $ yield the isomorphisms $\\widetilde{\\Phi }^{\\mathcal {L}}_{\\tau ,j} \\cong {\\mathcal {L}}_{\\phi ^{-1}\\circ \\tau ,j}^{\\otimes p}$ defined above.", "A similar assertion holds for the line bundles ${\\mathcal {N}}_\\theta $ , and it follows that the operators $V_{\\mathfrak {p}}$ for ${\\mathfrak {p}}\\in S_p$ commute with each other and that $\\prod _{\\mathfrak {p}}V_{\\mathfrak {p}}^{e_{\\mathfrak {p}}} = V_p$ , so that $ \\left(\\epsilon _{p\\mathbf {k},p\\mathbf {l}} \\prod _{{\\mathfrak {p}}\\in S_p} V_{\\mathfrak {p}}^{e_{\\mathfrak {p}}} \\right) (f) = f^p.", "$" ], [ "Toroidal compactifications", "We next recall how $q$ -expansions of Hilbert modular forms are obtained using compactifications of Hilbert modular varieties.", "In this section we review properties of the toroidal compactification constructed by Rapoport [31] (see also [4] and [13]).", "We will consider toroidal compactifications only in the case $U = U(N)$ , but we first describe the set of cusps adelically for any $U$ of level prime to $p$ .", "For an arbitrary open compact subgroup $U$ of $\\operatorname{GL}_2({\\mathbb {A}}_{F,) containing \\operatorname{GL}_2({\\mathcal {O}}_{F,p}),we define the set of {\\em cusps} of Y_U to beY_U^\\infty = B({\\mathcal {O}}_{F,(p)})_+ \\backslash \\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)}) / U^p = B(F)_+\\backslash \\operatorname{GL}_2({\\mathbb {A}}_{F,) / U,where B denotes the subgroup of \\operatorname{GL}_2 consisting of upper-triangular matrices.", "Similarly wedefine the set of cusps of \\widetilde{Y}_U to be\\widetilde{Y}_U^\\infty = B_1({\\mathcal {O}}_{F,(p)})_+ \\backslash \\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)}) / U^pNote that the natural surjection \\widetilde{Y}^\\infty _U\\rightarrow Y^\\infty _U identifies Y^\\infty _Uwith the quotient of \\widetilde{Y}^\\infty _U by the left action of B({\\mathcal {O}}_{F,(p)})_+/B_1({\\mathcal {O}}_{F,(p)}) \\cong {\\mathcal {O}}_{F,(p),+}^\\times .Furthermore the subgroup ({\\mathcal {O}}_F^\\times \\cap U)^2 acts trivially on \\widetilde{Y}^\\infty _U, but the quotient{\\mathcal {O}}_{F,(p),+}^\\times /({\\mathcal {O}}_F^\\times \\cap U)^2 need not act freely; more precisely, thestabilizer of the cusp B_1({\\mathcal {O}}_{F,(p)}) g U of \\widetilde{Y}_U is the group\\det (gUg^{-1} \\cap B(F))_+ \\subset {\\mathcal {O}}_{F,+}^\\times , in which ({\\mathcal {O}}_F^\\times \\cap U)^2has finite index.", "}We also have a natural bijection between Y^\\infty _U and the set of isomorphism classes of data(H, I, [\\lambda ], [\\eta ]) where\\begin{itemize}\\item H is a projective {\\mathcal {O}}_F-module of rank two;\\item I is an invertible submodule of H such that J:=H/I is invertible;\\item [\\lambda ] is a prime-to-p orientation of \\wedge ^2_{{\\mathcal {O}}_F}H \\cong I \\otimes _{{\\mathcal {O}}_F}J = IJ,by which we mean an {\\mathcal {O}}_{F,(p),+}^\\times -orbit of isomorphisms\\lambda : \\wedge ^2_{{\\mathcal {O}}_{F,(p)}} H_{(p)} \\stackrel{\\sim }{\\longrightarrow }{\\mathcal {O}}_{F,(p)}.\\item [\\eta ] is a level U^p-structure on H, i.e., a U^p-orbitof \\widehat{{\\mathcal {O}}}_F^{(p)}-linear isomorphisms\\eta : (\\widehat{{\\mathcal {O}}}_{F}^{(p)})^2 \\stackrel{\\sim }{\\longrightarrow } \\widehat{{\\mathcal {O}}}_F^{(p)} \\otimes _{{\\mathcal {O}}_F} H.\\end{itemize}The bijection is defined by associating the data (H_g, I_g, [\\lambda _g], [\\eta _g]) to the cosetB({\\mathcal {O}}_{F,(p)})_+ g U^p, where H_g = \\widehat{{\\mathcal {O}}}_F^2 g^{-1} \\cap F^2, I_gis its intersection with the subspace \\lbrace 0\\rbrace \\times F, \\lambda _g is induced by the determinant,and \\eta _g is induced by right pre-multiplication by g^{-1}.", "}Note that to give a prime-to-p orientation of \\wedge ^2_{{\\mathcal {O}}_F}H is equivalent to givingan F_+^\\times -orbit of isomorphisms \\wedge _F^2 (\\mathbb {Q}\\otimes H) \\stackrel{\\sim }{\\longrightarrow } F,but the integrality condition is imposed for consistency with the fact that we have a bijection between\\widetilde{Y}^\\infty _U and the set of isomorphism classes of data of the form (H, I, \\lambda , [\\eta ]),which is similarly defined, {\\mathcal {O}}_{F,(p),+}^\\times -equivariant, and compatible in the obvious sensewith the bijection describing Y^\\infty _U.", "In particular if U = U(1) = \\operatorname{GL}_2(\\widehat{{\\mathcal {O}}}_F),then the map sending (H, I, [\\lambda ], [\\eta ]) to the pair (\\wedge ^2_{{\\mathcal {O}}_F}H,I) defines a bijectionbetween Y^\\infty _U and C_F^+ \\times C_F, where C_F^{(+)} denotes the (strict) class group of F.For each such cusp we choose a polyhedral cone decomposition as in \\cite [Lemme~4.2]{rap} (with U^{\\prime } = U_+in the notation there) for (M^\\otimes \\mathbb {R})_+ \\cup \\lbrace 0\\rbrace , where\\begin{equation} \\begin{array}{l}M = {\\operatorname{Hom}\\,}_{{\\mathcal {O}}_F}(I,{\\mathfrak {d}}^{-1} J) = {\\mathfrak {d}}^{-1}I^{-1}J,\\\\M^ = {\\operatorname{Hom}\\,}(M,\\mathbb {Z}) \\cong {\\operatorname{Hom}\\,}_{{\\mathcal {O}}_F}(J,I) = J^{-1}I,\\end{array}\\end{equation}and the positivity is induced by the orientation ofI\\otimes _{{\\mathcal {O}}_F} J.", "}Suppose now that U = U(N) for some N \\ge 3 (not divisible by p) and that {\\mathcal {O}} contains the Nth roots of unity.The above choice of cone decomposition (for the image of each cusp of \\widetilde{Y}^\\infty _U in Y^\\infty _{U(1)}) yieldsa toroidal compactification\\footnote {Compactified in the sense that its (infinitely many) connected components are proper over {\\mathcal {O}}.", "}\\widetilde{Y}_U \\hookrightarrow \\widetilde{Y}_U^{\\operatorname{tor}} such that the set of (geometrically) connected components of its (reduced) closedsubscheme \\widetilde{Z}_U^{\\operatorname{tor}}:= \\widetilde{Y}_U^{\\operatorname{tor}}- \\widetilde{Y}_U is identified with \\widetilde{Y}^\\infty _U.The construction of\\widetilde{Y}_U^{\\operatorname{tor}} identifies its completion along the component corresponding to a cusp \\widetilde{{\\mathcal {C}}} represented by(H,I,\\lambda ,[\\eta ]) with the quotient of a formal scheme\\footnote {The formal scheme depends on the chosen cone decomposition\\lbrace \\sigma _\\alpha ^{\\widetilde{{\\mathcal {C}}}}\\rbrace and is denoted S_N(\\lbrace \\sigma _\\alpha ^{\\widetilde{{\\mathcal {C}}}}\\rbrace )^\\wedge in \\cite [3.4.2]{chai}.", "}\\widehat{S}_{\\widetilde{{\\mathcal {C}}}} by an action of V_N^2, where V_N = \\ker ({\\mathcal {O}}_F^\\times \\rightarrow ({\\mathcal {O}}_F/N{\\mathcal {O}}_F)^\\times ).Furthermore this extends to an action of {\\mathcal {O}}_{F,+}^\\times on \\widehat{S} = \\widehat{S}_{\\widetilde{{\\mathcal {C}}}}, and we havean isomorphism\\begin{equation} \\Gamma (\\widehat{S}, {\\mathcal {O}}_{\\widehat{S}}) \\cong {\\mathcal {O}}[[q^m]]_{m \\in N^{-1}M_+ \\cup \\lbrace 0\\rbrace }\\end{equation}compatible with the obvious action of {\\mathcal {O}}_{F,+}^\\times on the target.", "(The isomorphism depends on a choice of splitting ofthe exact sequence0 \\rightarrow I \\rightarrow H \\rightarrow J \\rightarrow 0of {\\mathcal {O}}_F-modules; modifying the splitting by an element \\beta \\in J^{-1}I \\cong M^alters it by composition with the automorphism defined by q^m \\mapsto \\zeta _N^{-\\beta (m)}q^m form \\in N^{-1}M.) We let \\xi : {\\widehat{S}}\\rightarrow \\widetilde{Y}_U denote the natural morphismof formal schemes, and we write F_{\\widehat{S}} for the field of fractions of \\Gamma ({\\widehat{S}},{\\mathcal {O}}_{\\widehat{S}})and \\mu _\\nu for the automorphism of {\\widehat{S}} defined by \\nu \\in {\\mathcal {O}}_{F,+}^\\times .", "}$ The construction of the toroidal compactification also extends the universal abelian scheme $A$ to a semi-abelian scheme $A^{\\operatorname{tor}}$ whose pull-back to $\\widehat{S}$ is identified with that of the Tate semi-abelian schemeMore precisely, the formal scheme $\\widehat{S}$ has an open cover by affine formal subschemes ${\\operatorname{Spf}\\,}R_\\sigma $ (indexed by cones $\\sigma $ ) such that ${\\operatorname{Spec}\\,}R_\\sigma \\times _{\\widetilde{Y}_U^{\\operatorname{tor}}} A^{\\operatorname{tor}}$ is identified with the semi-abelian scheme $T_{I,J}$ over ${\\operatorname{Spec}\\,}R_\\sigma $ .", "The compatibilities in the discussion that follows are then systematically checked by verifying them over the open subschemes ${\\operatorname{Spec}\\,}R_\\sigma ^0 ={\\operatorname{Spec}\\,}R_\\sigma \\times _{\\widetilde{Y}_U^{\\operatorname{tor}}} \\widetilde{Y}_U$ .", "associated to a quotient of the form $ T_{I,J} := ({\\mathfrak {d}}^{-1}I \\otimes {\\mathbb {G}}_m) /\\widetilde{q}^{{\\mathfrak {d}}^{-1}J},$ where $\\widetilde{q}^\\cdot :{\\mathfrak {d}}^{-1}J \\rightarrow F_{\\widehat{S}}^\\times \\otimes {\\mathfrak {d}}^{-1}I$ is the homomorphism corresponding to the tautological element under the canonical isomorphism ${\\operatorname{Hom}\\,}(M,F_{\\widehat{S}}^\\times ) = {\\operatorname{Hom}\\,}_{{\\mathcal {O}}_F}({\\mathfrak {d}}^{-1}J, {\\mathfrak {d}}^{-1}I \\otimes F_{\\widehat{S}}^\\times ).$ Similarly its dual $A^\\vee $ extends to a Tate semi-abelian scheme $(A^\\vee )^{\\operatorname{tor}}$ whose pull-back via $\\xi $ is associated to $T_{{\\mathfrak {d}}J^{-1},{\\mathfrak {d}}I^{-1}}$ , with the isomorphism ${\\mathfrak {c}}{\\mathfrak {d}}\\otimes _{{\\mathcal {O}}_F}A^{\\operatorname{tor}}\\rightarrow (A^\\vee )^{\\operatorname{tor}}$ defined by the quasi-polarization pulling back to the composite ${\\mathfrak {c}}{\\mathfrak {d}}\\otimes _{{\\mathcal {O}}_F} T_{I,J} \\stackrel{\\sim }{\\longrightarrow }{\\mathfrak {d}}(IJ)^{-1} \\otimes _{{\\mathcal {O}}_F} T_{I,J} \\stackrel{\\sim }{\\longrightarrow }T_{{\\mathfrak {d}}J^{-1},{\\mathfrak {d}}I^{-1}},$ where ${\\mathfrak {c}}= \\lbrace \\,\\alpha \\in F\\,\|\\,\\alpha \\lambda (IJ)\\subset {\\mathcal {O}}_F\\,\\rbrace $ , the first morphism is the isomorphism induced by $\\lambda $ , and the second is the canonical one.", "The subschemes $Z_\\theta $ of $\\widetilde{Y}_U$ (defined in §REF by the vanishing of the partial Hasse invariants $H_\\theta $ ) are closed in $\\widetilde{Y}_U^{\\operatorname{tor}}$ , and we let $\\widetilde{Y}_U^{\\operatorname{ord}}$ (resp.", "$\\widetilde{Y}_U^{\\operatorname{tord}}$ ) denote the complement of their union, i.e., the ordinary locus, in $\\widetilde{Y}_U$ (resp.", "$\\widetilde{Y}_U^{\\operatorname{tor}}$ ), and we use similar notation for the restrictions of $A^{\\operatorname{tor}}$ and $(A^\\vee )^{\\operatorname{tor}}$ .", "Since the sheaf ${\\mathcal {L}ie}(A^{\\operatorname{tord}}/\\widetilde{Y}_U^{\\operatorname{tord}})$ is locally free over ${\\mathcal {O}}_F \\otimes {\\mathcal {O}}_{\\widetilde{Y}_U^{\\operatorname{tord}}}$ , the universal filtration ${\\mathcal {F}}_\\tau ^{(j)}$ on $(s_\\Omega _{A/\\widetilde{Y}_U}^1)_\\tau \\cong {\\mathcal {H}om}_{{\\mathcal {O}}_{\\widetilde{Y}_U}}({\\mathcal {L}ie}(A/\\widetilde{Y}_U)_\\tau ,{\\mathcal {O}}_{\\widetilde{Y}_U})$ extends canonically to one on ${\\mathcal {H}om}_{{\\mathcal {O}}_{\\widetilde{Y}^{\\operatorname{tor}}_U}}({\\mathcal {L}ie}(A^{\\operatorname{tor}}/\\widetilde{Y}^{\\operatorname{tor}}_U)_\\tau ,{\\mathcal {O}}_{\\widetilde{Y}^{\\operatorname{tor}}_U})$ for each $\\tau \\in \\Sigma _0$ .", "Furthermore its pull-back to ${\\widehat{S}}$ is identified (in the notation of (REF )) with $0 \\subset t_{\\tau ,1}(I^{-1}\\otimes {\\mathcal {O}}_{\\widehat{S}})_\\tau \\subset t_{\\tau ,2}(I^{-1}\\otimes {\\mathcal {O}}_{\\widehat{S}}) \\subset \\cdots \\subset t_{\\tau ,e_{\\mathfrak {p}}}(I^{-1} \\otimes {\\mathcal {O}}_{\\widehat{S}})_\\tau = (I^{-1}\\otimes {\\mathcal {O}}_{\\widehat{S}})_\\tau $ under the canonical isomorphism ${\\mathcal {H}om}_{{\\mathcal {O}}_{{\\widehat{S}}}}({\\mathcal {L}ie}(T_{I,J}/{\\widehat{S}}),{\\mathcal {O}}_{\\widehat{S}}) \\cong {\\operatorname{Hom}\\,}({\\mathfrak {d}}^{-1}I, {\\mathcal {O}}_{\\widehat{S}}) \\cong I^{-1}\\otimes {\\mathcal {O}}_{\\widehat{S}}.$ We thus obtain extensions ${\\widetilde{{\\mathcal {L}}}}_\\theta ^{\\operatorname{tor}}$ of the line bundles ${\\widetilde{{\\mathcal {L}}}}_\\theta = {\\widetilde{{\\mathcal {L}}}}_{{\\mathfrak {p}},i,j}$ to $\\widetilde{Y}_U^{\\operatorname{tor}}$ whose pull-back to ${\\widehat{S}}$ is identified with $(I^{-1})_\\theta \\otimes _{{\\mathcal {O}}}{\\mathcal {O}}_{\\widehat{S}}$ , where $(I^{-1})_\\theta $ is defined by (REF ).", "Similarly ${\\mathcal {L}ie}((A^\\vee )^{\\operatorname{tord}}/\\widetilde{Y}_U^{\\operatorname{tord}})$ is locally free over ${\\mathcal {O}}_F \\otimes {\\mathcal {O}}_{\\widetilde{Y}_U^{\\operatorname{tord}}}$ , but the line bundles ${\\widetilde{{\\mathcal {M}}}}_\\theta = {\\widetilde{{\\mathcal {M}}}}_{{\\mathfrak {p}},i,j}$ over $\\widetilde{Y}_U^{\\operatorname{ord}}$ are canonically identified with $(R^1s_{\\mathcal {O}}_{A^{\\operatorname{ord}}})_\\tau [u-\\theta (\\varpi _{\\mathfrak {p}})] \\cong {\\mathcal {L}ie}((A^\\vee )^{\\operatorname{ord}}/\\widetilde{Y}_U^{\\operatorname{ord}})_\\tau [u-\\theta (\\varpi _{\\mathfrak {p}})].$ It follows that each ${\\widetilde{{\\mathcal {M}}}}_\\theta $ extends to a line bundle ${\\widetilde{{\\mathcal {M}}}}_\\theta ^{\\operatorname{tor}}$ on $\\widetilde{Y}_U^{\\operatorname{tor}}$ such that the identification ${\\mathcal {L}ie}(T_{{\\mathfrak {d}}J^{-1},{\\mathfrak {d}}I^{-1}}/{\\widehat{S}}) = J^{-1}\\otimes {\\mathcal {O}}_{\\widehat{S}}$ induces an isomorphism $\\xi ^{\\widetilde{{\\mathcal {M}}}}_\\theta ^{\\operatorname{tor}}\\cong (J^{-1}\\otimes {\\mathcal {O}}_{{\\widehat{S}}})_\\tau [u-\\theta (\\varpi _{\\mathfrak {p}})] \\cong ({\\mathfrak {d}}J^{-1} \\otimes {\\mathcal {O}}_{\\widehat{S}})_\\tau \\otimes _{{\\mathcal {O}}[u],\\theta } {\\mathcal {O}}.$ We can thus identify the pull-back $\\xi ^{\\widetilde{{\\mathcal {N}}}}_\\theta ^{\\operatorname{tor}}$ of the line bundle ${\\widetilde{{\\mathcal {N}}}}_\\theta ^{\\operatorname{tor}}= {\\widetilde{{\\mathcal {L}}}}_\\theta ^{\\operatorname{tor}}\\otimes _{{\\mathcal {O}}_{\\widetilde{Y}_U^{\\operatorname{tor}}}} {\\widetilde{{\\mathcal {M}}}}_\\theta ^{\\operatorname{tor}}$ with $({\\mathfrak {d}}(IJ)^{-1})_\\theta \\otimes _{{\\mathcal {O}}} {\\mathcal {O}}_{\\widehat{S}}$ , which the polarization in turn identifies with $({\\mathfrak {c}}{\\mathfrak {d}})_\\theta \\otimes _{{\\mathcal {O}}} {\\mathcal {O}}_{\\widehat{S}}$ in the notation of (REF ).", "Finally it follows that the automorphic bundles ${\\widetilde{{\\mathcal {A}}}}_{\\mathbf {k},\\mathbf {l}}$ extend to line bundles ${\\widetilde{{\\mathcal {A}}}}_{\\mathbf {k},\\mathbf {l}}^{\\operatorname{tor}}$ on $\\widetilde{Y}_U^{\\operatorname{tor}}$ such that $ \\xi ^{\\widetilde{{\\mathcal {A}}}}_{\\mathbf {k},\\mathbf {l}}^{\\operatorname{tor}}\\cong D_{\\mathbf {k},\\mathbf {l}} \\otimes _{{\\mathcal {O}}} {\\mathcal {O}}_{\\widehat{S}}, \\quad \\mbox{where\\,\\,}D_{\\mathbf {k},\\mathbf {l}} := \\bigotimes _{\\theta \\in \\Sigma } \\left( (I^{-1})^{\\otimes k_\\theta }_\\theta \\otimes ({\\mathfrak {d}}(IJ) ^{-1})^{\\otimes l_\\theta }_\\theta \\right)$ (the tensor products being over ${\\mathcal {O}}$ ).", "We refer to this isomorphism as the canonical trivialization of $\\xi ^{\\widetilde{{\\mathcal {A}}}}_{\\mathbf {k},\\mathbf {l}}^{\\operatorname{tor}}$ .", "Next we consider the completion of $\\widetilde{Y}_U^{\\operatorname{tor}}$ along the component corresponding to the cusp $\\widetilde{{\\mathcal {C}}}$ represented by $(H,I,\\lambda ,[\\eta ])$ , which we denote $(\\widetilde{Y}_{U^{\\operatorname{tor}}})_{\\widetilde{{\\mathcal {C}}}}^\\wedge $ .", "We now describe the global sections of the completions of the line bundles ${\\widetilde{{\\mathcal {A}}}}_{\\mathbf {k},\\mathbf {l}}^{\\operatorname{tor}}$ using the identification $(\\widetilde{Y}_{U}^{\\operatorname{tor}})_{\\widetilde{{\\mathcal {C}}}}^\\wedge = {\\widehat{S}}/V_N^2$ and taking invariants under the action of $V_N^2$ on their trivializations over ${\\widehat{S}}$ .", "Note firstly that $\\Gamma ( (\\widetilde{Y}_U^{\\operatorname{tor}})_{\\widetilde{{\\mathcal {C}}}}^\\wedge , {\\mathcal {O}}_{(\\widetilde{Y}_U^{\\operatorname{tor}})_{\\widetilde{{\\mathcal {C}}}}^\\wedge })= \\Gamma ({\\widehat{S}},{\\mathcal {O}}_{\\widehat{S}})^{V_N^2}$ corresponds to $ \\left\\lbrace \\, \\left.\\sum r_mq^m \\in {\\mathcal {O}}[[q^m]]_{m \\in N^{-1}M_+ \\cup \\lbrace 0\\rbrace } \\,\\right\|\\, r_{\\alpha ^2m} = r_m \\,\\,\\forall \\alpha \\in V_N, m \\in N^{-1}M_+\\,\\right\\rbrace $ under the isomorphism of ().", "One then finds that the descent data for $\\xi ^A^{\\operatorname{tor}}$ is provided by the isomorphisms $T_{I,J} \\stackrel{\\sim }{\\longrightarrow } \\mu _{\\alpha ^2}^T_{I,J}$ induced by $\\alpha \\otimes 1$ on ${\\mathfrak {d}}^{-1}I \\otimes {\\mathbb {G}}_m$ , from which it follows that the descent data for $\\xi ^{\\widetilde{{\\mathcal {L}}}}_{\\theta }^{\\operatorname{tor}}$ is provided on the trivialization by the isomorphisms $(I^{-1})_\\theta \\otimes _{\\mathcal {O}}\\mu ^_{\\alpha ^2} {\\mathcal {O}}_{{\\widehat{S}}} \\stackrel{\\sim }{\\longrightarrow } (I^{-1})_\\theta \\otimes _{\\mathcal {O}}{\\mathcal {O}}_{{\\widehat{S}}}$ induced by $\\theta (\\alpha )$ on $(I^{-1})_\\theta $ .", "On the other hand the descent data for $\\xi ^{\\widetilde{{\\mathcal {M}}}}_\\theta ^{\\operatorname{tor}}$ is similarly induced on the canonical trivialization by $\\theta (\\alpha )^{-1}$ , so that the resulting trivialization of $\\xi ^{\\widetilde{{\\mathcal {N}}}}_\\theta ^{\\operatorname{tor}}$ descends to $(\\widetilde{Y}_{U^{\\operatorname{tor}}})_{\\widetilde{{\\mathcal {C}}}}^\\wedge $ (in fact extending the one already defined over $\\widetilde{Y}_U$ via the choice of generator $t_{\\tau ,j}(f^{\\prime }(\\varpi _{\\mathfrak {p}}) \\otimes 1)$ of $({\\mathfrak {c}}{\\mathfrak {d}})_\\theta $ ).", "Since $\\Gamma ( (\\widetilde{Y}_U^{\\operatorname{tor}})_{\\widetilde{{\\mathcal {C}}}}, ({\\widetilde{{\\mathcal {A}}}}^{\\operatorname{tor}}_{\\mathbf {k},\\mathbf {l}})_{\\widetilde{{\\mathcal {C}}}}^\\wedge ) =\\Gamma ({\\widehat{S}},\\xi ^{\\widetilde{{\\mathcal {A}}}}^{\\operatorname{tor}}_{\\mathbf {k},\\mathbf {l}})^{V_N^2}$ , we conclude the following: Proposition 6.1.1 Suppose that $U=U(N)$ and ${\\mathcal {O}}$ contains the $N$ th roots of unity.", "Then the isomorphism () and the canonical trivialization (REF ) identify $\\Gamma ( (\\widetilde{Y}_U^{\\operatorname{tor}})_{\\widetilde{{\\mathcal {C}}}}, ({\\widetilde{{\\mathcal {A}}}}^{\\operatorname{tor}}_{\\mathbf {k},\\mathbf {l}})_{\\widetilde{{\\mathcal {C}}}}^\\wedge )$ with $\\left\\lbrace \\, \\left.\\sum _{m \\in N^{-1}M_+ \\cup \\lbrace 0\\rbrace } b \\otimes r_mq^m \\,\\right\|\\,r_{\\alpha ^2m} = \\chi _{\\mathbf {k}}(\\alpha ) r_m\\,\\, \\forall \\alpha \\in V_N, m \\in N^{-1}M_+\\,\\right\\rbrace ,$ where $b$ is any choice of basis for $D_{\\mathbf {k},\\mathbf {l}}$ ." ], [ "Minimal compactifications", "We now recall the construction due to Chai [4] of minimal compactifications of Hilbert modular varieties.", "The presentation in [4] is very concise with numerous typos, but a more detailed treatment of the construction can be found in [13] in the case of $U_1({\\mathfrak {n}})$ (with different conventions than ours), and of the descriptions of $q$ -expansions in that case in [5].", "We continue to assume for the moment that $U = U(N)$ for some sufficiently large $N$ prime to $p$ .", "The minimal compactification $\\widetilde{Y}_U \\hookrightarrow \\widetilde{Y}_U^{\\min }$ is then constructed as in [4] or [13].", "More precisely, letting $\\mathbf {t}= \\sum {\\mathbf {e}}_\\theta $ and taking the global sections of $\\oplus _{k \\ge 0} {\\widetilde{{\\mathcal {A}}}}^{\\operatorname{tor}}_{k\\mathbf {t},\\bf {0}}$ over each component of $\\widetilde{Y}_U^{\\operatorname{tor}}$ yields a projective scheme over ${\\mathcal {O}}$ containing the corresponding component of the Deligne-Pappas model as an open subscheme.", "Gluing their ordinary loci to $\\widetilde{Y}_U$ along $\\widetilde{Y}_U^{\\operatorname{ord}}$ yields a scheme $\\widetilde{Y}_U^{\\min }$ and a proper morphism $\\pi : \\widetilde{Y}_U^{\\operatorname{tor}}\\rightarrow \\widetilde{Y}_U^{\\min }$ such that ${\\widetilde{\\iota }}: \\widetilde{Y}_U \\rightarrow \\widetilde{Y}_U^{\\min }$ is an open immersion.", "Furthermore the (reduced) complement $\\widetilde{Y}_U^{\\min } - \\widetilde{Y}_U$ is an infinite disjoint union of copies of ${\\operatorname{Spec}\\,}{\\mathcal {O}}$ indexed by $\\widetilde{Y}^\\infty _U$ , the preimage of each in $\\widetilde{Y}_U^{\\operatorname{tor}}$ being the corresponding connected component of $\\widetilde{Z}_U^{\\operatorname{tor}}$ , and the scheme $\\widetilde{Y}_U^{\\min }$ is independent of the choice of cone decomposition in the construction of $\\widetilde{Y}_U^{\\operatorname{tor}}$ .", "Now recall that the Koecher Principle implies that ${\\mathcal {O}}_{\\widetilde{Y}_U^{\\min }} = \\pi _{\\mathcal {O}}_{\\widetilde{Y}_U^{\\operatorname{tor}}} = {\\widetilde{\\iota }}_{\\mathcal {O}}_{\\widetilde{Y}_U}$ , so that ${\\mathcal {O}}^\\wedge _{\\widetilde{Y}_U^{\\min },\\widetilde{{\\mathcal {C}}}} =\\Gamma ( (\\widetilde{Y}_U^{\\operatorname{tor}})_{\\widetilde{{\\mathcal {C}}}}^\\wedge , {\\mathcal {O}}_{(\\widetilde{Y}_U^{\\operatorname{tor}})_{\\widetilde{{\\mathcal {C}}}}^\\wedge })$ is the ring described by (REF ), where we have written $\\widetilde{{\\mathcal {C}}}$ for the corresponding point of $\\widetilde{Y}_U^{\\min }$ .", "Furthermore the argument of [31] shows that ${\\widetilde{\\iota }}_{\\widetilde{{\\mathcal {A}}}}_{\\mathbf {k},\\mathbf {l}} = \\pi _{\\widetilde{{\\mathcal {A}}}}_{\\mathbf {k},\\mathbf {l}}^{\\operatorname{tor}}$ (see the discussion following [31], or view $\\widetilde{Y}_U$ as a disjoint union of PEL Shimura varieties and apply [28]), so the Theorem on Formal Functions gives that $(\\iota _{\\widetilde{{\\mathcal {A}}}}_{\\mathbf {k},\\mathbf {l}})^\\wedge _{\\widetilde{{\\mathcal {C}}}} =\\Gamma ( (\\widetilde{Y}_{U^{\\operatorname{tor}}})_{\\widetilde{{\\mathcal {C}}}}^\\wedge , ({\\widetilde{{\\mathcal {A}}}}^{\\operatorname{tor}}_{\\mathbf {k},\\mathbf {l}})_{\\widetilde{{\\mathcal {C}}}}^\\wedge ))$ is the ${\\mathcal {O}}^\\wedge _{\\widetilde{Y}_U^{\\min },\\widetilde{{\\mathcal {C}}}}$ -module described in Proposition REF .", "(Note that ${\\widetilde{\\iota }}_{\\widetilde{{\\mathcal {A}}}}_{\\mathbf {k},\\mathbf {l}}$ is coherent, but not necessarily invertible.)", "Similarly for any ${\\mathcal {O}}$ -algebra $R$ , we may identify $({\\widetilde{\\iota }}_{R,}{\\widetilde{{\\mathcal {A}}}}_{\\mathbf {k},\\mathbf {l},R})^\\wedge _{\\widetilde{{\\mathcal {C}}}}$ with $ \\left\\lbrace \\,\\left.", "\\sum _{m \\in N^{-1}M_+ \\cup \\lbrace 0\\rbrace } b \\otimes r_mq^m \\,\\right\|\\,r_{\\alpha ^2m} = \\chi _{\\mathbf {k},R}(\\alpha ) r_m \\,\\,\\forall \\alpha \\in V_N, m \\in N^{-1}M_+\\,\\right\\rbrace $ as a module over ${\\mathcal {O}}^\\wedge _{\\widetilde{Y}_{U,R}^{\\min },\\widetilde{{\\mathcal {C}}}}$ , which the Koecher Principle and () identify with $ \\left\\lbrace \\, \\left.\\sum r_mq^m \\in R[[q^m]]_{m \\in N^{-1}M_+ \\cup \\lbrace 0\\rbrace } \\,\\right\|\\, r_{\\alpha ^2m} = r_m\\,\\, \\forall \\alpha \\in V_N, m \\in N^{-1}M_+\\,\\right\\rbrace ,$ where ${\\widetilde{\\iota }}_R: \\widetilde{Y}_{U,R} \\rightarrow \\widetilde{Y}_{U,R}^{\\min }$ is the base-change of ${\\widetilde{\\iota }}$ to $R$ , the completions are at the fibre over $\\widetilde{{\\mathcal {C}}}$ , and $b$ is any basis for $D_{\\mathbf {k},\\mathbf {l}}$ .", "The compatibility of the choices of polyhedral cone decompositions ensures that the natural action of ${\\mathcal {O}}_{F,(p),+}$ on $\\widetilde{Y}_U$ extends (uniquely) to one on $\\widetilde{Y}_U^{\\operatorname{tor}}$ .", "Furthermore the stabilizer of each component of $\\widetilde{Z}_U^{\\operatorname{tor}}$ is $V_{N,+}$ , and the action of $V_{N,+}$ on each completion $(\\widetilde{Y}_{U^{\\operatorname{tor}}})_{\\widetilde{{\\mathcal {C}}}}^\\wedge = {\\widehat{S}}/V_N^2$ is induced by an action of $V_{N,+}$ on ${\\widehat{S}}$ such that the effect of $\\nu \\in V_{N,+}$ on global sections of ${\\mathcal {O}}_{{\\widehat{S}}}$ is induced by multiplication by $\\nu ^{-1}$ on $M$ .", "We see also that the canonical isomorphism $A \\rightarrow \\nu ^A$ extends to an isomorphism $A^{\\operatorname{tor}}\\rightarrow \\nu ^A^{\\operatorname{tor}}$ whose pull-back via $\\xi $ is induced by the identity on ${\\mathfrak {d}}^{-1}I \\otimes {\\mathbb {G}}_m$ , from which it follows that the action of $\\nu $ is compatible with the canonical trivialization of the line bundle ${\\widetilde{{\\mathcal {L}}}}_\\theta ^{\\operatorname{tor}}$ over ${\\widehat{S}}$ .", "On the other hand the induced isomorphisms $\\nu ^{\\widetilde{{\\mathcal {M}}}}_\\theta ^{\\operatorname{tor}}\\rightarrow {\\mathcal {M}}_\\theta ^{\\operatorname{tor}}$ and $\\nu ^{\\widetilde{{\\mathcal {N}}}}_\\theta ^{\\operatorname{tor}}\\rightarrow {\\widetilde{{\\mathcal {N}}}}_\\theta ^{\\operatorname{tor}}$ pull back to ones corresponding to multiplication by $\\theta (\\nu )$ .", "Since $\\widetilde{Y}_U^{\\min }$ is a disjoint union of projective schemes over ${\\mathcal {O}}$ on which ${\\mathcal {O}}_{F,(p),+}^\\times /V_N^2$ acts with finite stabilizers, the quotient scheme exists, and we define this to be the minimal compactification $Y_U^{\\min }$ of $Y_U$ .", "We thus obtain an open immersion $\\iota :Y_U \\rightarrow Y_U^{\\min }$ such that $Y_U^{\\min }$ is projective over ${\\mathcal {O}}$ and the (reduced) complement of $Y_U$ is a disjoint union of copies of ${\\operatorname{Spec}\\,}{\\mathcal {O}}$ in canonical bijection with the set of cusps $Y_U^\\infty $ .", "Furthermore we again have that $\\iota _{\\mathcal {O}}_{Y_U} = {\\mathcal {O}}_{Y_U^{\\min }}$ , and its completion ${\\mathcal {O}}^\\wedge _{Y_U^{\\min },{\\mathcal {C}}}$ at the cusp ${\\mathcal {C}}$ represented by $(H,I,[\\lambda ],[\\eta ])$ is identified under (REF ) with $\\left\\lbrace \\, \\left.\\sum r_mq^m \\in {\\mathcal {O}}[[q^m]]_{m \\in N^{-1}M_+ \\cup \\lbrace 0\\rbrace } \\,\\right\|\\, r_{\\nu m} = r_m\\,\\, \\forall \\nu \\in V_{N,+}, m \\in N^{-1}M_+\\,\\right\\rbrace $ (where the identification depends as in () on a choice of splitting of the exact sequence $0 \\rightarrow I \\rightarrow H \\rightarrow J \\rightarrow 0$ ).", "Now suppose that $R$ is an ${\\mathcal {O}}$ -algebra such that $\\chi _{\\mathbf {k}+2\\mathbf {l},R}$ is trivial on $V_N$ , so that the line bundle $\\widetilde{{\\mathcal {A}}}_{\\mathbf {k},\\mathbf {l},R}$ descends to one over $Y_{U,R}$ which we denote by ${\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},R}$ .", "We then see that $\\iota _{\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},R}$ is a coherent sheaf on $Y_{U,R}^{\\min }$ whose completion at the (base-change to $R$ of the) cusp ${\\mathcal {C}}$ is identified under (REF ) with $ \\left\\lbrace \\, \\left.\\sum _{m \\in N^{-1}M_+ \\cup \\lbrace 0\\rbrace } b \\otimes r_mq^m \\,\\right\|\\,r_{\\nu ^{-1} m} = \\chi _{\\mathbf {l},R}(\\nu ) r_m\\,\\, \\forall \\nu \\in V_{N,+}, m \\in N^{-1}M_+\\,\\right\\rbrace .$ In particular $\\iota _{\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},R}$ is a line bundle if $\\chi _{\\mathbf {l},R}$ is trivial on $V_{N,+}$ .", "Suppose now that $U^{\\prime }$ is any sufficiently small open compact subgroup of $\\operatorname{GL}_2(\\widehat{{\\mathcal {O}}}_F)$ containing $\\operatorname{GL}_2({\\mathcal {O}}_{F,p})$ .", "One can then carry out a construction similar to the one above to obtain the minimal compactification, or choose an $N$ prime to $p$ such that $U(N) \\subset U^{\\prime }$ , extend the natural (right) action of $U^{\\prime }/U(N)$ on $Y_{U(N)}$ to $Y_{U(N)}^{\\min }$ and take the quotient; we do the latter (see [13] for the former in the case of $U^{\\prime } = U_1({\\mathfrak {n}})$ ).", "Firstly our choice of polyhedral cone decompositions ensures that the natural right action of $U^{\\prime }/U$ on $\\widetilde{Y}_{U}$ extends to $\\widetilde{Y}_{U}^{\\operatorname{tor}}$ , where $U = U(N)$ for some choice of $N$ as above.", "Denoting the resulting automorphism of $\\widetilde{Y}_{U}^{\\operatorname{tor}}$ by $\\widetilde{\\rho }_g$ for $g \\in U^{\\prime }$ , the canonical identification of the universal $A$ over $\\widetilde{Y}_{U}$ with its pull-back extends to an identification $A^{\\operatorname{tor}}= \\widetilde{\\rho }_g^A^{\\operatorname{tor}}$ , giving rise to canonical isomorphisms $\\widetilde{\\rho }_g^{\\widetilde{{\\mathcal {A}}}}^{\\operatorname{tor}}_{k\\mathbf {t},\\bf {0}} = {\\widetilde{{\\mathcal {A}}}}^{\\operatorname{tor}}_{k\\mathbf {t},\\bf {0}}$ , and hence to an action of $U^{\\prime }/U$ on $\\widetilde{Y}_{U}^{\\min }$ extending its action on $\\widetilde{Y}_{U}$ .", "Moreover the action commutes with the natural action of ${\\mathcal {O}}_{F,(p),+}$ , so it descends to an action on $Y_{U}^{\\min }$ extending the action on $Y_{U}$ .", "We denote the resulting automorphisms of $Y_U^{\\min }$ by $\\rho _g$ , and define $Y_{U^{\\prime }}^{\\min }$ to be the quotient of $Y_{U}^{\\min }$ by the action of $U^{\\prime }/U$ (which we will shortly see is independent of the choice of $N$ in its definition).", "Identifying the set of components of $Y_{U}^{\\min } - Y_{U}$ with $Y^\\infty _{U}$ , the resulting action of $g \\in U^{\\prime }$ is given by pre-composing $\\eta $ with right multiplication by $g^{-1}$ , so the set of components of $Y_{U^{\\prime }}^{\\min } - Y_{U^{\\prime }}$ may be identified with $Y^\\infty _{U^{\\prime }}$ .", "For each cusp ${\\mathcal {C}}^{\\prime } \\in Y^\\infty _{U^{\\prime }}$ , the completion ${\\mathcal {O}}^\\wedge _{Y_{U^{\\prime }}^{\\min },{\\mathcal {C}}^{\\prime }}$ is identified with the subring of ${\\mathcal {O}}^\\wedge _{Y_{U}^{\\min },{\\mathcal {C}}}$ invariant under the stabilizer of ${\\mathcal {C}}$ in $U^{\\prime }/U$ , where ${\\mathcal {C}}$ is any cusp of $Y_{U}$ in the preimage of ${\\mathcal {C}}^{\\prime }$ .", "Choose such an $(H,I,[\\lambda ],[\\eta ])$ representing ${\\mathcal {C}}$ and a splitting $\\sigma : H \\stackrel{\\sim }{\\longrightarrow } J \\times I$ , and let $ \\Gamma _{{\\mathcal {C}}} = \\left\\lbrace \\, \\left.\\left(\\begin{array}{cc}\\alpha &\\beta \\\\0&\\delta \\end{array}\\right)\\,\\right\|\\,\\alpha \\in {\\mathcal {O}}_F^\\times , \\beta \\in J^{-1}I, \\delta \\in \\alpha {\\mathcal {O}}_{F,+}^\\times \\,\\right\\rbrace ,$ which we view as acting on $J\\times I$ by right multiplication.", "The stabilizer of ${\\mathcal {C}}$ is then the set of classes $Ug = gU \\in U^{\\prime }/U$ such that $g \\equiv \\eta ^{-1}\\sigma ^{-1}\\gamma \\sigma \\eta \\bmod N\\widehat{{\\mathcal {O}}}_F\\mbox{\\,\\,for some\\,\\,}\\gamma \\in \\Gamma _{{\\mathcal {C}}},$ and we let $\\Gamma _{{\\mathcal {C}},U^{\\prime }} = \\Gamma _{{\\mathcal {C}}} \\cap \\sigma \\eta U^{\\prime } \\eta ^{-1}\\sigma ^{-1}$ .", "Thus the stabilizer of ${\\mathcal {C}}$ is the image of the homomorphism $s: \\Gamma _{{\\mathcal {C}},U^{\\prime }} \\longrightarrow U^{\\prime }/U$ defined by $\\gamma \\mapsto \\eta ^{-1}\\sigma ^{-1}\\gamma \\sigma \\eta U$ .", "We claim that if $g = s(\\gamma )$ , then $\\rho _g^$ on ${\\mathcal {O}}^\\wedge _{Y_{U}^{\\min },{\\mathcal {C}}} = H^0({\\widehat{S}},{\\mathcal {O}}_{{\\widehat{S}}})^{V_{N,+}}$ is induced by an automorphism $\\psi _\\gamma $ of $\\widehat{S}$ whose effect on global sections corresponds to the map defined by $ \\psi _\\gamma ^: q^m \\mapsto \\zeta _N^{-\\beta (\\alpha ^{-1}Nm)}q^{\\alpha ^{-1}\\delta m} $ under () and the identification $M^* = J^{-1}I$ of ().", "Indeed letting $\\nu $ denote $\\alpha \\delta $ (as well as the automorphism of $\\widetilde{Y}_U^{\\operatorname{tor}}$ defined by its effect on the universal polarization), we see that $\\delta \\otimes 1$ on $T_{I,J}$ defines an isomorphism $\\xi ^\\rho _g^\\nu ^A^{\\operatorname{tor}}\\stackrel{\\sim }{\\longrightarrow } \\psi _\\gamma ^\\xi ^* A^{\\operatorname{tor}}$ compatible with all auxiliary data, from which one deduces that $\\nu \\circ \\rho _g\\circ \\xi = \\xi \\circ \\psi _\\gamma $ .", "Note also that (REF ) defines an action of $\\Gamma _{{\\mathcal {C}}}$ on $\\Gamma ({\\widehat{S}},{\\mathcal {O}}_{{\\widehat{S}}})$ which factors through the surjection $\\begin{array}{ccc} \\Gamma _{{\\mathcal {C}}} &\\longrightarrow & (J^{-1}I\\otimes \\mathbb {Z}/N\\mathbb {Z})\\rtimes {\\mathcal {O}}_{F,+}^\\times \\\\\\left(\\begin{array}{cc}\\alpha &\\beta \\\\0&\\delta \\end{array}\\right)&\\mapsto & (-\\alpha ^{-1}\\beta ,\\alpha ^{-1}\\delta ),\\end{array}$ and the latter group acts on $\\Gamma ({\\widehat{S}},{\\mathcal {O}}_{{\\widehat{S}}})^{V_{N,+}}$ via its quotient $(J^{-1}I\\otimes \\mathbb {Z}/N\\mathbb {Z}) \\rtimes ({\\mathcal {O}}_{F,+}^\\times /V_{N,+}).$ We conclude that ${\\mathcal {O}}^\\wedge _{Y_{U^{\\prime }}^{\\min },{\\mathcal {C}}^{\\prime }} = H^0({\\widehat{S}},{\\mathcal {O}}_{{\\widehat{S}}})^{\\Gamma _{{\\mathcal {C}},U^{\\prime }}}$ is identified with $\\left\\lbrace \\, \\sum _{m \\in N^{-1}M_+ \\cup \\lbrace 0\\rbrace } r_mq^m \\,\\left\|\\, r_{\\alpha ^{-1}\\delta m} = \\zeta _N^{-\\beta (\\alpha ^{-1}Nm)}r_m\\,\\, \\forall m \\in N^{-1}M_+, \\bigl ({\\begin{matrix}{\\alpha }&{\\beta }\\\\{0}&{\\delta }\\end{matrix}}\\bigr ) \\in \\Gamma _{{\\mathcal {C}},U^{\\prime }}\\,\\right.\\right\\rbrace ,$ where we recall that the isomorphism may depend on the choice of the splitting $\\sigma $ and that we view $\\beta $ as an element of $M^$ .", "We note in particular if $U^{\\prime } = U(N^{\\prime })$ for some $N^{\\prime }\|N$ , then the resulting description of ${\\mathcal {O}}^\\wedge _{Y_{U^{\\prime }}^{\\min },{\\mathcal {C}}^{\\prime }}$ coincides with the one previously obtained, from which it follows that the same holds for the scheme $Y_{U^{\\prime }}^{\\min }$ , and hence that $Y_{U^{\\prime }}^{\\min }$ is independent of the choice of $N$ in its definition (for any sufficiently small $U^{\\prime }$ containing $\\operatorname{GL}_2({\\mathcal {O}}_{F,p})$ .)", "Suppose now that $\\mathbf {k}, \\mathbf {l}\\in \\mathbb {Z}^\\Sigma $ and $R$ is an ${\\mathcal {O}}$ -algebra such that $\\chi _{\\mathbf {k}+2\\mathbf {l},R}$ is trivial on $U^{\\prime } \\cap {\\mathcal {O}}_F^\\times $ , and consider the automorphic bundle ${\\mathcal {A}}^{\\prime }_{\\mathbf {k},\\mathbf {l},R}$ on $Y_{U^{\\prime },R}$ .", "Letting $\\iota _R^{\\prime }$ denote the open immersion of $Y_{U^{\\prime },R}$ in $Y_{U^{\\prime },R}^{\\min }$ , similar considerations to those above show that $\\iota ^{\\prime }_{R,}{\\mathcal {A}}^{\\prime }_{\\mathbf {k},\\mathbf {l},R}$ is a coherent sheaf on $Y_{U,R}^{\\min }$ whose completion at ${\\mathcal {C}}^{\\prime }$ is identified with the ${\\mathcal {O}}^\\wedge _{Y_{U^{\\prime },R}^{\\min },{\\mathcal {C}}^{\\prime }}$ -module of $\\Gamma _{{\\mathcal {C}},U^{\\prime }}$ -invariants in $({\\widetilde{\\iota }}_{R,}{\\widetilde{{\\mathcal {A}}}}_{\\mathbf {k},\\mathbf {l},R})^\\wedge _{{\\mathcal {C}}}$ .", "Using that the isomorphism $\\xi ^A^{\\operatorname{tor}}\\stackrel{\\sim }{\\longrightarrow } \\psi _\\gamma ^\\xi ^ A^{\\operatorname{tor}}$ is induced by $\\delta \\otimes 1$ on the Tate semi-abelian scheme, we find that the resulting automorphism multiplies the canonical trivialization (REF ) of $\\xi ^\\widetilde{{\\mathcal {A}}}^{\\operatorname{tor}}_{\\mathbf {k},\\mathbf {l},R}$ by $\\chi _{\\mathbf {l},R} (\\alpha )\\chi _{\\mathbf {k}+\\mathbf {l},R}(\\delta )$ .", "We therefore conclude: Proposition 6.2.1 If $\\chi _{\\mathbf {k}+2\\mathbf {l},R}$ is trivial on $U^{\\prime } \\cap {\\mathcal {O}}_F^\\times $ , then $\\iota ^{\\prime }_{R,}{\\mathcal {A}}^{\\prime }_{\\mathbf {k},\\mathbf {l},R}$ is a coherent sheaf on $Y_{U^{\\prime },R}^{\\min }$ whose completion at (the fibre over) ${\\mathcal {C}}^{\\prime }$ is identified by the Koecher Principle and Proposition REF with $\\left\\lbrace \\, \\sum _{m \\in N^{-1}M_+ \\cup \\lbrace 0\\rbrace } b \\otimes r_mq^m \\,\\left\|\\, \\begin{array}{c}r_{\\alpha ^{-1}\\delta m} = \\zeta _N^{-\\beta (\\alpha ^{-1}Nm)}\\chi _{\\mathbf {l},R} (\\alpha )\\chi _{\\mathbf {k}+\\mathbf {l},R}(\\delta ) r_m\\\\ \\mbox{\\rm for all $m \\in N^{-1}M_+,\\bigl ({\\begin{matrix}{\\alpha }&{\\beta }\\\\{0}&{\\delta }\\end{matrix}}\\bigr )\\in \\Gamma _{{\\mathcal {C}},U^{\\prime }}$} \\end{array}\\,\\right.\\right\\rbrace .$ Note that the description of ${\\mathcal {O}}^\\wedge _{Y_{U^{\\prime },R}^{\\min },{\\mathcal {C}}^{\\prime }} = (\\iota ^{\\prime }_{R,} {\\mathcal {O}}_{Y_{U^{\\prime },R}^{\\min }})^\\wedge _{{\\mathcal {C}}^{\\prime }}$ may be viewed as a special case (with $\\mathbf {k}= \\mathbf {l}= \\bf {0}$ ), as can the prior formula for $U = U(N)$ .", "Furthermore the identifications are compatible in the obvious senses with base changes $R \\rightarrow R^{\\prime }$ , inclusions $U^{\\prime \\prime } \\subset U^{\\prime }$ (provided the splittings $\\sigma $ are chosen compatibly), and the natural algebra structure on $\\bigoplus _{\\mathbf {k},\\mathbf {l}} {\\mathcal {A}}^{\\prime }_{\\mathbf {k},\\mathbf {l},R}$ (taking the direct sum over $\\mathbf {k},\\mathbf {l}$ as in the statement).", "Recall that the $q$ -expansion Principle allows one to characterize Hilbert modular forms in terms of their $q$ -expansions: Proposition 6.2.2 If $C \\subset Y^\\infty _{U^{\\prime }}$ is any set of cusps containing at least one on each component of $Y_{U^{\\prime }}$ , then the natural map $M_{\\mathbf {k},\\mathbf {l}}(U^{\\prime };R) = H^0(Y_{U^{\\prime }}^{\\min }, \\iota ^{\\prime }_{R,}{\\mathcal {A}}^{\\prime }_{\\mathbf {k},\\mathbf {l},R}) \\longrightarrow \\bigoplus _{{\\mathcal {C}}^{\\prime } \\in C}(\\iota ^{\\prime }_{R,}{{\\mathcal {A}}^{\\prime }}_{\\mathbf {k},\\mathbf {l},R})^\\wedge _{{\\mathcal {C}}^{\\prime }}$ is injective.", "Note also that we may replace $D_{\\mathbf {k},\\mathbf {l}} \\otimes _{{\\mathcal {O}}}\\cdot $ with $D_{\\mathbf {k},\\mathbf {l},R} \\otimes _{R}\\cdot $ in the description of $q$ -expansions over $R$ .", "In particular if $R$ is an $\\mathbb {F}$ -algebra, the identification $\\begin{array}{rl} (I^{-1})_\\theta \\otimes _{{\\mathcal {O}}} \\mathbb {F}& = (I^{-1}\\otimes {\\mathcal {O}})_\\tau \\otimes _{{\\mathcal {O}}[u],\\theta } \\mathbb {F}=t_{\\tau ,j}(I_{{\\mathfrak {p}}}^{-1}\\otimes _{W({\\mathcal {O}}_F/{\\mathfrak {p}}),\\tau }{\\mathcal {O}})\\otimes _{{\\mathcal {O}}[u],\\theta } \\mathbb {F}\\\\&= t_{\\tau ,j}(I_{{\\mathfrak {p}}}^{-1}\\otimes _{W({\\mathcal {O}}_F/{\\mathfrak {p}}),\\tau }{\\mathcal {O}})/(u, {\\mathfrak {m}}_{\\mathcal {O}})t_{\\tau ,j}(I_{{\\mathfrak {p}}}^{-1}\\otimes _{W({\\mathcal {O}}_F/{\\mathfrak {p}}),\\tau }{\\mathcal {O}})\\\\&= u^{e_{\\mathfrak {p}}- j}(I_{{\\mathfrak {p}}}^{-1}\\otimes _{W({\\mathcal {O}}_F/{\\mathfrak {p}}),\\tau }\\mathbb {F})/u^{e_{\\mathfrak {p}}- j+1}(I_{{\\mathfrak {p}}}^{-1}\\otimes _{W({\\mathcal {O}}_F/{\\mathfrak {p}}),\\tau }\\mathbb {F})\\end{array}$ yields a canonical isomorphism $(I^{-1})_\\theta \\otimes _{{\\mathcal {O}}} R = ({\\mathfrak {p}}^{e_{\\mathfrak {p}}- j}I^{-1}/{\\mathfrak {p}}^{e_{\\mathfrak {p}}- j+1}I^{-1}) \\otimes _{{\\mathcal {O}}_F/{\\mathfrak {p}},\\tau } R.$ The analogous formula holds for the factors $({\\mathfrak {d}}(IJ) ^{-1})_\\theta $ appearing in the definition of $D_{\\mathbf {k},\\mathbf {l}}$ .", "The condition on the $q$ -expansion coefficients in the description of the completions in Proposition REF simplifies for certain standard level structures and cusps, as in [12].", "Suppose that ${\\mathfrak {n}}$ is an ideal of ${\\mathcal {O}}_F$ such that $\\chi _{\\mathbf {k}+2\\mathbf {l},R}$ is trivial on $V_{{\\mathfrak {n}}} = \\ker ({\\mathcal {O}}_F^\\times \\rightarrow ({\\mathcal {O}}_F/{\\mathfrak {n}})^\\times )$ .", "Letting $U^{\\prime } = U({\\mathfrak {n}})$ , we have $\\Gamma _{{\\mathcal {C}},U^{\\prime }} = \\left\\lbrace \\, \\left.\\left(\\begin{array}{cc}\\alpha &\\beta \\\\0&\\delta \\end{array}\\right)\\,\\right\|\\,\\alpha \\in V_{\\mathfrak {n}}, \\beta \\in {\\mathfrak {n}}M^, \\delta \\in \\alpha V_{{\\mathfrak {n}},+}\\,\\right\\rbrace $ for every cusp ${\\mathcal {C}}$ of $Y_U$ .", "Note that $m \\in {\\mathfrak {n}}^{-1} M$ if and only if $\\beta (N m) \\in N\\mathbb {Z}$ for all $\\beta \\in {\\mathfrak {n}}M^$ , and that $\\alpha ,\\delta \\in V_{\\mathfrak {n}}$ implies that $\\chi _{\\mathbf {l},R}(\\alpha )\\chi _{\\mathbf {k}+2\\mathbf {l}}(\\delta ) =\\chi _{\\mathbf {l},R}(\\alpha \\delta ^{-1})$ , so we see that $(\\iota ^{\\prime }_{R,}{{\\mathcal {A}}^{\\prime }}_{\\mathbf {k},\\mathbf {l},R})^\\wedge _{{\\mathcal {C}}^{\\prime }}\\simeq \\left\\lbrace \\left.\\, \\sum _{m\\in (\\mathfrak {n}^{-1}M)_+\\cup \\lbrace 0\\rbrace } b \\otimes r_mq^m \\,\\right\|\\,r_{\\nu ^{-1} m} = \\chi _{\\mathbf {l},R}(\\nu ) r_m \\ \\mbox{for all $\\nu \\in V_{\\mathfrak {n},+}$}\\,\\right\\rbrace $ for every cusp ${\\mathcal {C}}^{\\prime }$ of $Y_{U^{\\prime }}$ .", "Keep the same assumption on ${\\mathfrak {n}}$ , but now let $U^{\\prime } = U_1({\\mathfrak {n}})$ and suppose that ${\\mathcal {C}}^{\\prime }$ is a cusp of $Y_{U^{\\prime }}$ “at $\\infty $ ” in the sense that $\\eta (0,1) \\in I + {\\mathfrak {n}}\\widehat{H}^{(p)}$ .", "We then find that $\\Gamma _{{\\mathcal {C}},U^{\\prime }} = \\left\\lbrace \\, \\left.\\left(\\begin{array}{cc}\\alpha &\\beta \\\\0&\\delta \\end{array}\\right)\\,\\right\|\\,\\alpha \\in \\delta {\\mathcal {O}}_{F,+}^\\times , \\beta \\in M^, \\delta \\in V_{{\\mathfrak {n}}}\\,\\right\\rbrace ,$ and we similarly conclude that $(\\iota ^{\\prime }_{R,}{{\\mathcal {A}}^{\\prime }}_{\\mathbf {k},\\mathbf {l},R})^\\wedge _{{\\mathcal {C}}^{\\prime }}\\simeq \\left\\lbrace \\left.\\, \\sum _{m\\in M_+\\cup \\lbrace 0\\rbrace }b \\otimes r_mq^m \\,\\right\|\\,r_{\\nu ^{-1} m} = \\chi _{\\mathbf {l},R}(\\nu ) r_m \\ \\mbox{for all $\\nu \\in {\\mathcal {O}}_{F,+}$}\\,\\right\\rbrace .$ We remark that every component of $Y_{U^{\\prime }}$ contains such cusps (in the obvious sense), and that in this case the isomorphism is independent of the choice of splitting $\\sigma $ ." ], [ "Kodaira–Spencer filtration", "We next explain how the Kodaira–Spencer filtration on differentials extends to compactifications.", "We maintain the notation from the preceding section.", "In particular, we first assume $U = U(N)$ for some $N$ prime to $p$ before deducing results for more general level structures.", "The construction of $\\widetilde{Y}_U^{\\operatorname{tor}}$ via torus embeddings then yields a canonical isomorphism $ \\xi ^(\\Omega ^1_{\\widetilde{Y}_U^{\\operatorname{tor}}/{\\mathcal {O}}}(\\log \\widetilde{Z}_U^{\\operatorname{tor}})) \\cong N^{-1}M \\otimes {\\mathcal {O}}_{{\\widehat{S}}}$ for each cusp ${\\widetilde{{\\mathcal {C}}}}$ of $\\widetilde{Y}_U$ under which the descent data relative to the quotient map ${\\widehat{S}}= {\\widehat{S}}_{{\\widetilde{{\\mathcal {C}}}}} \\rightarrow (\\widetilde{Y}_U^{\\operatorname{tor}})_{\\widetilde{{\\mathcal {C}}}}^\\wedge $ corresponds to that induced by the obvious action of $V_N^2$ on $N^{-1}M$ , and the completion of the canonical derivation $d: {\\mathcal {O}}_{\\widetilde{Y}_U^{\\operatorname{tor}}} \\rightarrow \\Omega ^1_{\\widetilde{Y}_U^{\\operatorname{tor}}/{\\mathcal {O}}}(\\log \\widetilde{Z}_U^{\\operatorname{tor}})$ pulls back to a derivation ${\\mathcal {O}}_{{\\widehat{S}}} \\rightarrow N^{-1}M \\otimes {\\mathcal {O}}_{{\\widehat{S}}}$ whose effect on global sections corresponds under () to the map defined by $\\sum _{m \\in N^{-1}M_+ \\cup \\lbrace 0\\rbrace } r_mq^m \\mapsto \\sum _{m \\in N^{-1}M_+ \\cup \\lbrace 0\\rbrace } m \\otimes r_m q^m.$ Recall also that $\\begin{array}{rcl}s_\\Omega ^1_{A^{\\operatorname{ord}}/\\widetilde{Y}_U^{\\operatorname{ord}}} &\\cong &{\\mathcal {H}om}_{{\\mathcal {O}}_{\\widetilde{Y}_U^{\\operatorname{ord}}}}({\\mathcal {L}ie}(A^{\\operatorname{ord}}/\\widetilde{Y}_U^{\\operatorname{ord}}), {\\mathcal {O}}_{\\widetilde{Y}_U^{\\operatorname{ord}}}) \\\\\\mbox{and}\\quad R^1s_{\\mathcal {O}}_{A^{\\operatorname{ord}}} &\\cong & {\\mathcal {L}ie}((A^{\\operatorname{ord}})^\\vee /\\widetilde{Y}_U^{\\operatorname{ord}}) \\end{array}$ are locally free sheaves of ${\\mathcal {O}}_F \\otimes {\\mathcal {O}}_{\\widetilde{Y}_U^{\\operatorname{ord}}}$ -modules over $\\widetilde{Y}_U^{\\operatorname{ord}}$ , and therefore so is $s_\\Omega ^1_{A^{\\operatorname{ord}}/\\widetilde{Y}_U^{\\operatorname{ord}}} \\otimes _{{\\mathcal {O}}_F\\otimes {\\mathcal {O}}_{\\widetilde{Y}_U^{\\operatorname{ord}}}}s_\\Omega ^1_{(A^{\\operatorname{ord}})^\\vee /\\widetilde{Y}_U^{\\operatorname{ord}}}$ Decomposing this sheaf over embeddings $\\tau \\in \\Sigma _0$ and equipping it with the filtration defined by the images of the endomorphisms $t_{\\tau ,j}$ defined by (REF ), we see that the successive quotients $\\begin{array}{l} t_j\\left(s_\\Omega ^1_{A^{\\operatorname{ord}}/\\widetilde{Y}_U^{\\operatorname{ord}}} \\otimes _{{\\mathcal {O}}_F\\otimes {\\mathcal {O}}_{\\widetilde{Y}_U^{\\operatorname{ord}}}}s_\\Omega ^1_{(A^{\\operatorname{ord}})^\\vee /\\widetilde{Y}_U^{\\operatorname{ord}}}\\right)_\\tau \\otimes _{{\\mathcal {O}}[u],\\theta } {\\mathcal {O}}\\\\ \\qquad \\cong {\\mathcal {H}om}_{{\\mathcal {O}}_S}(R^1s_{\\mathcal {O}}_{A^{\\operatorname{ord}}}[u-\\theta (\\varpi )],t_j(s_\\Omega ^1_{(A^{\\operatorname{ord}})^\\vee /\\widetilde{Y}_U^{\\operatorname{ord}}})_ \\tau \\otimes _{{\\mathcal {O}}[u],\\theta } {\\mathcal {O}})\\end{array}$ (where $\\tau = \\tau _i$ , $t_j = t_{\\tau ,j}$ and $\\theta = \\theta _{{\\mathfrak {p}},i,j}$ ) are canonically identified with the automorphic bundles $\\widetilde{{\\mathcal {A}}}_{2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\theta }$ over $\\widetilde{Y}_U^{\\operatorname{ord}}$ .", "Furthermore the proof of Theorem shows that the natural map $s_\\Omega ^1_{A^{\\operatorname{ord}}/\\widetilde{Y}_U^{\\operatorname{ord}}} \\otimes _{{\\mathcal {O}}_F\\otimes {\\mathcal {O}}_{\\widetilde{Y}_U^{\\operatorname{ord}}}}s_\\Omega ^1_{(A^{\\operatorname{ord}})^\\vee /\\widetilde{Y}_U^{\\operatorname{ord}}} \\longrightarrow \\Omega ^1_{\\widetilde{Y}_U^{\\operatorname{ord}}/{\\mathcal {O}}}$ arising from Grothendieck–Messing theory, or equivalently the Gauss–Manin connection on ${\\mathcal {H}}^1_{\\operatorname{dR}}(A^{\\operatorname{ord}}/\\widetilde{Y}_U^{\\operatorname{ord}})$ (see [27]), is an isomorphism.", "In particular the Kodaira–Spencer filtration on $\\Omega ^1_{\\widetilde{Y}_U^{\\operatorname{ord}}/{\\mathcal {O}}}$ corresponds under (REF ) to the one defined by the images of the endomorphisms $t_{\\tau ,j}$ .", "Furthermore (REF ) extends over $\\widetilde{Y}_U^{\\operatorname{tord}}$ to an isomorphism $\\begin{array}{c} {\\mathcal {H}om}_{{\\mathcal {O}}_{\\widetilde{Y}_U^{\\operatorname{tord}}}}({\\mathfrak {d}}\\otimes _{{\\mathcal {O}}_F} {\\mathcal {L}ie}(A^{\\operatorname{tord}}/\\widetilde{Y}_U^{\\operatorname{tord}}) \\otimes _{{\\mathcal {O}}_F\\otimes {\\mathcal {O}}_{\\widetilde{Y}_U^{\\operatorname{tord}}}}{\\mathcal {L}ie}((A^{\\operatorname{tord}})^\\vee /\\widetilde{Y}_U^{\\operatorname{tord}}), {\\mathcal {O}}_{\\widetilde{Y}_U^{\\operatorname{tord}}})\\\\ \\stackrel{\\sim }{\\longrightarrow }\\,\\,\\Omega ^1_{\\widetilde{Y}_U^{\\operatorname{tord}}/{\\mathcal {O}}}(\\log \\widetilde{Z}_U^{\\operatorname{tor}})\\end{array}$ whose pull-back via $\\xi = \\xi _{\\widetilde{{\\mathcal {C}}}}$ for each cusp ${\\widetilde{{\\mathcal {C}}}}$ of $\\widetilde{Y}_U$ is compatible with the canonical isomorphisms of the pull-back of each with $M \\otimes {\\mathcal {O}}_{{\\widehat{S}}} = N^{-1}M\\otimes {\\mathcal {O}}_{{\\widehat{S}}}$ (the latter via (REF )).", "Indeed the existence of the extension and the claimed compatibility follow from the analogous well-known result after base-change to $\\mathbb {C}$ .", "We therefore conclude that the Kodaira–Spencer filtration on $\\Omega ^1_{\\widetilde{Y}_U/{\\mathcal {O}}}$ extends over $\\widetilde{Y}_U^{\\operatorname{tor}}$ in the form of a decomposition $\\Omega ^1_{\\widetilde{Y}^{\\operatorname{tor}}_U/{\\mathcal {O}}}(\\log \\widetilde{Z}_U^{\\operatorname{tor}}) = \\bigoplus _{{\\mathfrak {p}}\\in S_p} \\bigoplus _{i \\in \\mathbb {Z}/f_{\\mathfrak {p}}\\mathbb {Z}}\\left( \\Omega ^1_{\\widetilde{Y}^{\\operatorname{tor}}_U/{\\mathcal {O}}}(\\log \\widetilde{Z}_U^{\\operatorname{tor}})\\right)_{{\\mathfrak {p}},i},$ together with an increasing filtration of length $e_{\\mathfrak {p}}$ on each component $(\\Omega ^1_{\\widetilde{Y}^{\\operatorname{tor}}_U/{\\mathcal {O}}}(\\log \\widetilde{Z}_U^{\\operatorname{tor}}))_{{\\mathfrak {p}},i}$ , and isomorphisms $\\widetilde{{\\mathcal {A}}}^{\\operatorname{tor}}_{2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\theta } \\stackrel{\\sim }{\\longrightarrow } {\\operatorname{gr}\\,}^j \\left(\\Omega ^1_{\\widetilde{Y}^{\\operatorname{tor}}_U/{\\mathcal {O}}}(\\log \\widetilde{Z}_U^{\\operatorname{tor}})\\right)_{{\\mathfrak {p}},i}.$ Furthermore for each cusp ${\\widetilde{{\\mathcal {C}}}}$ of $\\widetilde{Y}_U$ and embeddings $\\tau = \\tau _{{\\mathfrak {p}},i}$ and $\\theta = \\theta _{{\\mathfrak {p}},i,j}$ , the pull-back via $\\xi = \\xi _{{\\widetilde{{\\mathcal {C}}}}}$ of ${\\operatorname{Fil}\\,}^j (\\Omega ^1_{\\widetilde{Y}^{\\operatorname{tor}}_U/{\\mathcal {O}}}(\\log \\widetilde{Z}_U^{\\operatorname{tor}}))_{{\\mathfrak {p}},i}$ corresponds to $t_j(N^{-1}M\\otimes {\\mathcal {O}})_\\tau \\otimes _{\\mathcal {O}}{\\mathcal {O}}_{{\\widehat{S}}} = t_j({\\mathfrak {d}}^{-1}I^{-1}J \\otimes {\\mathcal {O}})_\\tau \\otimes _{\\mathcal {O}}{\\mathcal {O}}_{{\\widehat{S}}}$ under (REF ), and the resulting isomorphism $\\xi ^\\widetilde{{\\mathcal {A}}}^{\\operatorname{tor}}_{2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\theta } \\stackrel{\\sim }{\\longrightarrow } {\\operatorname{gr}\\,}^j \\left(\\xi ^(\\Omega ^1_{\\widetilde{Y}^{\\operatorname{tor}}_U/{\\mathcal {O}}}(\\log \\widetilde{Z}_U^{\\operatorname{tor}}))_{{\\mathfrak {p}},i}\\right)\\cong ({\\mathfrak {d}}^{-1}I^{-1}J \\otimes {\\mathcal {O}})_\\theta \\otimes _{\\mathcal {O}}{\\mathcal {O}}_{{\\widehat{S}}}$ coincides with the canonical trivialization of (REF ).", "We now interpret this in the context of minimal compactifications.", "First we note that the argument of [31] yields a Koecher Principle for $\\Omega ^1_{\\widetilde{Y}_U^{\\operatorname{tor}}/{\\mathcal {O}}}(\\log \\widetilde{Z}_U^{\\operatorname{tor}})$ , so that $\\pi _(\\Omega ^1_{\\widetilde{Y}_U^{\\operatorname{tor}}/{\\mathcal {O}}}(\\log \\widetilde{Z}_U^{\\operatorname{tor}})) \\longrightarrow \\widetilde{\\iota }_(\\Omega ^1_{\\widetilde{Y}_U/{\\mathcal {O}}})$ is an isomorphism of coherent sheaves on $\\widetilde{Y}_U^{\\min }$ whose completion at the cusp ${\\widetilde{{\\mathcal {C}}}}$ is identified with $\\left(M\\otimes {\\mathcal {O}}[[q^m]]_{m \\in N^{-1}M_+\\cup \\lbrace 0\\rbrace }\\right)^{V_N^2}$ $ = \\left\\lbrace \\left.\\, \\sum _{m\\in N^{-1}M_+} c_m \\otimes q^m \\,\\right\|\\,c_{\\nu m} = \\nu c_m \\,\\,\\forall \\nu \\in V_N^2, m \\in N^{-1}M_+\\,\\right\\rbrace .$ (Note that $c_0 \\in M^{V_N^2} = 0$ .)", "Furthermore the completion at ${\\widetilde{{\\mathcal {C}}}}$ of the canonical derivation ${\\mathcal {O}}_{\\widetilde{Y}_U^{\\min }} \\rightarrow \\widetilde{\\iota }_ \\Omega ^1_{\\widetilde{Y}_U/{\\mathcal {O}}}$ is given by $\\sum r_mq^m \\mapsto \\sum m \\otimes r_m q^m$ .", "We see also from the description of the extension of the Kodaira–Spencer filtration to $\\Omega ^1_{\\widetilde{Y}_U^{\\operatorname{tor}}/{\\mathcal {O}}}(\\log \\widetilde{Z}_U^{\\operatorname{tor}})$ in terms of $q$ -expansions that $\\pi _({\\operatorname{Fil}\\,}^j(\\Omega ^1_{\\widetilde{Y}_U^{\\operatorname{tor}}/{\\mathcal {O}}}(\\log \\widetilde{Z}_U^{\\operatorname{tor}}))_{{\\mathfrak {p}},i})={\\widetilde{\\iota }}_ ({\\operatorname{Fil}\\,}^j(\\Omega ^1_{\\widetilde{Y}_U/{\\mathcal {O}}})_{{\\mathfrak {p}},i}),$ with completion at ${\\widetilde{{\\mathcal {C}}}}$ given by $\\left(t_j(M\\otimes {\\mathcal {O}})_\\tau \\otimes _{\\mathcal {O}}{\\mathcal {O}}[[q^m]]_{m \\in N^{-1}M_+\\cup \\lbrace 0\\rbrace }\\right)^{V_N^2}$ $ = \\left\\lbrace \\left.\\, \\sum _{m\\in N^{-1}M_+} c_m \\otimes q^m \\,\\right\|\\,c_{\\nu m} = (\\nu \\otimes 1)c_m\\,\\, \\forall \\nu \\in V_N^2, m \\in N^{-1}M_+\\,\\right\\rbrace .$ Furthermore since $V_N^2$ acts freely on $N^{-1}M_+$ (and $M_\\theta ^{V_N^2} = 0$ ), the morphisms $\\widetilde{\\iota }_* ({\\operatorname{Fil}\\,}^j(\\Omega ^1_{\\widetilde{Y}_U/{\\mathcal {O}}})_{{\\mathfrak {p}},i}) \\rightarrow \\widetilde{\\iota }_* \\widetilde{{\\mathcal {A}}}_{2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\theta }$ are surjective on completions at cusps, and hence surjective.", "It follows that the graded pieces of (the obvious extension of) the Kodaira–Spencer filtration on $\\widetilde{\\iota }_\\Omega ^1_{\\widetilde{Y}_U/{\\mathcal {O}}}$ are canonically isomorphic to $\\widetilde{\\iota }_ \\widetilde{{\\mathcal {A}}}_{2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\theta }$ .", "The constructions above are compatible with the natural actions of ${\\mathcal {O}}_{F,(p),+}^\\times $ on $\\widetilde{Y}_U^{\\operatorname{tor}}$ , $\\widetilde{Y}_U^{\\min }$ and $M$ , so that the resulting descriptions carry over to $Y_U^{\\min }$ .", "More precisely, $\\iota _(\\Omega ^1_{Y_U/{\\mathcal {O}}})$ is a coherent sheaf on ${Y}_U^{\\min }$ whose completion at the cusp ${\\mathcal {C}}$ is identified with $\\left(M\\otimes {\\mathcal {O}}[[q^m]]_{m \\in N^{-1}M_+\\cup \\lbrace 0\\rbrace }\\right)^{V_{N,+}}$ $= \\left\\lbrace \\left.\\, \\sum _{m\\in N^{-1}M_+} c_m \\otimes q^m \\,\\right\|\\,c_{\\nu m} = \\nu c_m\\,\\, \\forall \\nu \\in V_{N,+}, m \\in N^{-1}M_+\\,\\right\\rbrace ,$ in terms of which the canonical derivation is $\\sum r_mq^m \\mapsto \\sum m \\otimes r_m q^m$ .", "Furthermore the completion at ${\\mathcal {C}}$ of ${\\operatorname{Fil}\\,}^j(\\iota _(\\Omega ^1_{Y_U/{\\mathcal {O}}})_{{\\mathfrak {p}},i}) : = \\iota _* ({\\operatorname{Fil}\\,}^j(\\Omega ^1_{Y_U/{\\mathcal {O}}})_{{\\mathfrak {p}},i})$ is identified with $\\left(t_j(M\\otimes {\\mathcal {O}})_\\tau \\otimes _{\\mathcal {O}}{\\mathcal {O}}[[q^m]]_{m \\in N^{-1}M_+\\cup \\lbrace 0\\rbrace }\\right)^{V_{N,+}}$ , and the natural maps $ {\\operatorname{gr}\\,}^j(\\iota _* (\\Omega ^1_{Y_U/{\\mathcal {O}}})_{{\\mathfrak {p}},i})\\hookrightarrow \\iota _* ({\\operatorname{gr}\\,}^j(\\Omega ^1_{Y_U/{\\mathcal {O}}})_{{\\mathfrak {p}},i}) \\stackrel{\\sim }{\\longrightarrow }\\iota _{\\mathcal {A}}_{2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\theta } $ are isomorphisms whose completions at the cusps are induced by the surjections $t_j(M\\otimes {\\mathcal {O}})_\\tau \\rightarrow M_\\theta $ .", "Suppose now that $U^{\\prime }$ is an arbitrary sufficiently small open compact subgroup of $\\operatorname{GL}_2(\\mathbb {A}_{F,) of level prime to p, and choose N so that U(N) \\subset U^{\\prime }.The constructions above are then also compatible with the natural actions of U^{\\prime }, so wearrive at similar conclusions with minor modifications to the descriptions of completionsthat result from taking invariants under \\Gamma _{{\\mathcal {C}},U^{\\prime }}.", "We omit the details, butwe remark that letting L (resp.~V) denote the kernel (resp.~image) of the homomorphism\\begin{array}{ccc} \\Gamma _{{\\mathcal {C}},U^{\\prime }}/({\\mathcal {O}}_F^\\times \\cap U^{\\prime }) & \\rightarrow & {\\mathcal {O}}_{F,+}^\\times \\\\\\left(\\begin{array}{cc}\\alpha &\\beta \\\\0&\\delta \\end{array}\\right)\\cdot ({\\mathcal {O}}_F^\\times \\cap U^{\\prime }) & \\mapsto & \\alpha ^{-1}\\delta ,\\end{array}the coefficients r_m of q-expansions in {\\mathcal {O}}_{Y_{U^{\\prime }}^{\\min },{\\mathcal {C}}^{\\prime }}^\\wedge are indexed bym \\in (L^)_+\\cup \\lbrace 0\\rbrace (where L is identified with a finite index subgroup of M).Since V acts freely on L^* (twisting q-expansion coefficientsby a possibly non-trivial cocycle valued in L\\otimes \\mu _N({\\mathcal {O}})), we still obtain the isomorphismof (\\ref {eqn:KScusps}) with U replaced by U^{\\prime }, identifying the graded piecesof the Kodaira--Spencer filtration on \\iota _\\Omega ^1_{Y_{U^{\\prime }}/{\\mathcal {O}}} with the sheaves\\iota _{\\mathcal {A}}_{2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\theta }.", "}$ The description of the extension of the Kodaira–Spencer filtration over compactifications also applies after base-change to an arbitrary ${\\mathcal {O}}$ -algebra $R$ , with one significant difference.", "If $R$ is not flat over ${\\mathcal {O}}$ , then the modules $M\\otimes R$ (and their subquotients) may have invariants under the action of the unit groups $V_{N,+}$ (or more generally the isotropy groups $\\Gamma _{{\\mathcal {C}},U^{\\prime }}$ ), so that $q$ -expansions of meromorphic differentials on $Y_{U,R}$ (and forms of weight $(2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\theta )$ ) may have non-zero constant terms, and the morphism analogous to (REF ) may fail to be an isomorphism.", "(Note that in this case the relevant base-change morphisms $(\\iota _{\\mathcal {F}})_R \\rightarrow \\iota _{R,}({\\mathcal {F}}_R)$ fail to be surjective at the cusps.)", "We can however simplify matters by placing ourselves in the situation when this fails in the extreme.", "Suppose then that $p^n R = 0$ for some $n > 0$ , and that $N$ is sufficiently large that $\\nu \\equiv 1 \\bmod p^n{\\mathcal {O}}_F$ for all $\\nu \\in V_{N,+}$ .", "Arguing exactly as above, we find that $\\iota _{R,}(\\Omega ^1_{Y_{U,R}/R})$ is now a vector bundle over ${Y}_{U,R}^{\\min }$ whose completion at ${\\mathcal {C}}$ is identified with $\\left(M\\otimes R[[q^m]]_{m \\in N^{-1}M_+\\cup \\lbrace 0\\rbrace }\\right)^{V_{N,+}} =M \\otimes {\\mathcal {O}}_{Y^{\\min }_{U,R},{\\mathcal {C}}}^\\wedge $ $ = \\left\\lbrace \\left.\\, \\sum _{m\\in N^{-1}M_+ \\cup \\lbrace 0\\rbrace } c_m \\otimes q^m \\,\\right\|\\,c_{\\nu m} = \\nu c_m\\,\\, \\forall \\nu \\in V_{N,+}, m \\in N^{-1}M_+\\,\\right\\rbrace ,$ with the canonical derivation given by $\\sum r_mq^m \\mapsto \\sum m \\otimes r_m q^m$ .", "Furthermore each ${\\operatorname{Fil}\\,}^j(\\iota _{R,}(\\Omega ^1_{Y_{U,R}/R})_{{\\mathfrak {p}},i}) = \\iota _{R,} ({\\operatorname{Fil}\\,}^j(\\Omega ^1_{Y_{U,R}/R})_{{\\mathfrak {p}},i})$ is a sub-bundle whose completion at ${\\mathcal {C}}$ is identified with $t_j(M\\otimes {\\mathcal {O}})_\\tau \\otimes _{\\mathcal {O}}{\\mathcal {O}}_{Y^{\\min }_{U,R},{\\mathcal {C}}}^\\wedge $ and the natural maps $ {\\operatorname{gr}\\,}^j(\\iota _{R,}(\\Omega ^1_{Y_{U,R}/R})_{{\\mathfrak {p}},i})\\hookrightarrow \\iota _{R,} ({\\operatorname{gr}\\,}^j(\\Omega ^1_{Y_{U,R}/R})_{{\\mathfrak {p}},i}) \\stackrel{\\sim }{\\longrightarrow }\\iota _{R,}{\\mathcal {A}}_{2{\\mathbf {e}}_\\theta ,-{\\mathbf {e}}_\\theta ,R} $ are isomorphisms of line bundles over $Y_{U,R}^{\\min }$ whose completions at the cusps are induced by the surjections $t_j(M\\otimes {\\mathcal {O}})_\\tau \\rightarrow M_\\theta $ .", "We remark also that this carries over with $U$ replaced by arbitrary $U^{\\prime }$ , provided $U^{\\prime }$ is sufficiently small that (in additional to the usual hypotheses) $\\alpha \\equiv \\delta \\mod {p}^n{\\mathcal {O}}_F$ for all $\\left(\\begin{array}{cc}\\alpha &\\beta \\\\0&\\delta \\end{array}\\right)\\in \\Gamma _{{\\mathcal {C}},U^{\\prime }}$ and cusps ${\\mathcal {C}}^{\\prime }$ of $Y_{U^{\\prime }}$ (the condition being independent of the choice of $N$ and ${\\mathcal {C}}$ in the definition of $\\Gamma _{{\\mathcal {C}},U^{\\prime }}$ )." ], [ "Partial Hasse invariants", "We next describe the effect of the various weight-shifting operators on $q$ -expansions, beginning with the simplest case of (multiplication by) partial Hasse invariants.", "We will now only be working in the setting of $R = \\mathbb {F}$ , and we will use $\\overline{\\cdot }$ to denote base-changes from ${\\mathcal {O}}$ to $\\mathbb {F}$ .", "Since the formation of $q$ -expansions is compatible in the obvious sense with pull-back under the projections $\\overline{Y}_U \\rightarrow \\overline{Y}_{U^{\\prime }}$ , it will suffice to consider the case $U = U(N)$ .", "Recall that in §REF we defined the partial Hasse invariants as certain elements $ H_\\theta \\in M_{\\mathbf {h}_\\theta ,\\bf {0}} (U; \\mathbb {F}) = H^0(\\overline{Y}_{U} , \\overline{{\\mathcal {A}}}_{\\mathbf {h}_\\theta ,\\bf {0}}),$ where $\\mathbf {h}_\\theta : = n_\\theta {\\mathbf {e}}_{\\sigma ^{-1}\\theta } - {\\mathbf {e}}_\\theta $ , with $n_\\theta = p$ if $j=1$ and $n_\\theta = 1$ if $j > 1$ .", "In particular if $j > 1$ , then $H_\\theta $ is defined by the morphism $u:{{\\mathcal {L}}}_{\\tau ,j} \\rightarrow {{\\mathcal {L}}}_{\\tau ,j-1}$ induced by $\\varpi _{\\mathfrak {p}}$ on the universal abelian variety over $\\widetilde{Y}_{U,\\mathbb {F}}$ , which evidently extends to the endomorphism $\\varpi _{\\mathfrak {p}}$ of $A^{\\operatorname{tor}}$ over $S:= \\widetilde{Y}_{U,\\mathbb {F}}^{\\operatorname{tor}}$ .", "Since its pull-back via $\\overline{\\xi }$ is defined by $\\varpi _{\\mathfrak {p}}$ on $\\overline{T}_{I,J}$ , the resulting morphism of line bundles $\\overline{\\xi }^{{{\\mathcal {L}}}}_{\\tau ,j}^{\\operatorname{tor}}\\longrightarrow \\overline{\\xi }^{{{\\mathcal {L}}}}_{\\tau ,j-1}^{\\operatorname{tor}}$ is compatible with their canonical trivializations, and more precisely with the morphism $(I^{-1})_\\theta \\otimes _{{\\mathcal {O}}} \\mathbb {F}\\rightarrow (I^{-1})_{\\sigma ^{-1}\\theta } \\otimes _{{\\mathcal {O}}} \\mathbb {F}$ induced by $u = \\varpi \\otimes 1$ on $(I^{-1} \\otimes {\\mathcal {O}})_\\tau $ .", "It follows that $H_\\theta $ has constant $q$ -expansion, where the constant corresponds to the basis element $\\varpi _{\\mathfrak {p}}$ under the identification $\\overline{D}_{\\mathbf {h}_\\theta ,\\bf {0}} \\cong {\\mathfrak {p}}\\otimes _{{\\mathcal {O}}_F,\\theta } \\mathbb {F}.$ provided by (REF ).", "For $j=1$ we use also that the morphism of line bundles $\\overline{\\xi }^{{{\\mathcal {L}}}}_{\\tau ,1}^{\\operatorname{tor}}\\longrightarrow \\overline{\\xi }^\\phi _S^({{{\\mathcal {L}}}}_{\\phi ^{-1}\\circ \\tau ,e_{\\mathfrak {p}}}^{\\operatorname{tor}})$ induced by the Verschiebung $\\phi _S^\\overline{T}_{I,J} \\rightarrow \\overline{T}_{I,J}$ is compatible via the canonical trivializations with the canonical isomorphism $(I^{-1} \\otimes \\mathbb {F})_\\tau \\stackrel{\\sim }{\\longrightarrow } \\phi _S^* (I^{-1} \\otimes \\mathbb {F})_{\\phi ^{-1}\\circ \\tau }.$ So in this case we again find that $H_\\theta $ has constant $q$ -expansion, the constant now corresponding to the basis element $\\varpi _{\\mathfrak {p}}^{1-e_{\\mathfrak {p}}}$ under the identification $\\overline{D}_{\\mathbf {h}_\\theta ,\\bf {0}} \\cong {\\mathfrak {p}}^{1-e_{\\mathfrak {p}}} \\otimes _{{\\mathcal {O}}_F,\\theta } \\mathbb {F}$ given by (REF ).", "The $q$ -expansions of the canonical sections $G_\\theta \\in M_{\\bf {0},\\mathbf {h}_\\theta } (U; \\mathbb {F})$ may be described similarly.", "Indeed for $j > 1$ , the composites $\\begin{array}{cccccc}&{{\\mathcal {L}}}_{\\tau ,j} & \\longrightarrow &{{\\mathcal {P}}}_{\\tau ,j} & \\stackrel{u}{\\longrightarrow } & {{\\mathcal {L}}}_{\\tau ,j-1} \\\\\\mbox{and}& {{\\mathcal {M}}}_{\\tau ,j-1} &\\longrightarrow &{{\\mathcal {P}}}_{\\tau ,j} & \\stackrel{u}{\\longrightarrow } & {{\\mathcal {M}}}_{\\tau ,j} \\end{array}$ on $\\widetilde{Y}^{\\operatorname{ord}}_{U,\\mathbb {F}}$ are isomorphisms whose tensor product defines ${{\\mathcal {N}}}_\\theta \\cong {{\\mathcal {N}}}_{\\sigma ^{-1}\\theta }$ .", "Its unique extension to $\\widetilde{Y}^{\\operatorname{tord}}_{U,\\mathbb {F}}$ , and hence to $\\widetilde{Y}^{\\operatorname{tor}}_{U,\\mathbb {F}}$ , is therefore the isomorphism whose pull-back via $\\overline{\\xi }$ is the tensor product of the isomorphisms defined on canonical trivializations by $(I^{-1})_\\theta \\otimes _{{\\mathcal {O}}} \\mathbb {F}\\stackrel{u}{\\rightarrow } (I^{-1})_{\\sigma ^{-1}\\theta } \\otimes _{{\\mathcal {O}}} \\mathbb {F}$ and by the identity on $({\\mathfrak {d}}J^{-1} \\otimes \\mathbb {F})_\\tau /u ({\\mathfrak {d}}J^{-1} \\otimes \\mathbb {F})_\\tau $ , and hence corresponds to $({\\mathfrak {d}}(IJ)^{-1})_\\theta \\otimes _{{\\mathcal {O}}} \\mathbb {F}\\stackrel{u}{\\rightarrow } ({\\mathfrak {d}}(IJ)^{-1})_{\\sigma ^{-1}\\theta } \\otimes _{{\\mathcal {O}}} \\mathbb {F}.$ Therefore $G_\\theta $ has constant $q$ -expansion, with constant corresponding to the basis element $\\varpi _{\\mathfrak {p}}$ under the identification $\\overline{D}_{\\bf {0},\\mathbf {h}_\\theta } \\cong {\\mathfrak {p}}\\otimes _{{\\mathcal {O}}_F,\\theta } \\mathbb {F}$ provided by the analogue of (REF ) for $({\\mathfrak {d}}(IJ) ^{-1})_\\theta $ .", "Similarly if $j=1$ , we find that ${{\\mathcal {N}}}_{\\tau ,1} \\cong \\phi _S^({{\\mathcal {N}}}_{\\phi ^{-1}\\circ \\tau ,e_{\\mathfrak {p}}})$ over $S = \\widetilde{Y}^{\\operatorname{ord}}_{U,\\mathbb {F}}$ is the tensor product of the isomorphism ${{\\mathcal {L}}}_{\\tau ,1} \\rightarrow \\phi _S^ ({{\\mathcal {L}}}_{\\phi ^{-1}\\circ \\tau ,e_{\\mathfrak {p}}})$ defining the Hasse invariant and the isomorphism $\\phi _S^({{\\mathcal {M}}}_{\\phi ^{-1}\\circ \\tau ,e_{\\mathfrak {p}}}) \\rightarrow {{\\mathcal {M}}}_{\\tau ,1}$ induced by ${\\operatorname{Frob}}_A$ .", "The extensions to $\\widetilde{Y}^{\\operatorname{tord}}_{U,\\mathbb {F}}$ are again compatible with the canonical trivializations, now corresponding to maps whose tensor product is the inverse of the isomorphism $(({\\mathfrak {d}}(IJ)^{-1})_{\\sigma ^{-1}\\theta } \\otimes _{{\\mathcal {O}}} \\mathbb {F})^{\\otimes p} \\stackrel{\\sim }{\\longrightarrow }({\\mathfrak {d}}(IJ)^{-1})_{\\theta } \\otimes _{{\\mathcal {O}}} \\mathbb {F}$ induced by $u^{e_{\\mathfrak {p}}-1}$ .", "So again $G_\\theta $ has constant $q$ -expansion, with constant corresponding to $\\varpi _{\\mathfrak {p}}^{1-e_{\\mathfrak {p}}}$ under the identification $\\overline{D}_{\\bf {0},\\mathbf {h}_\\theta } \\cong {\\mathfrak {p}}^{1-e_{\\mathfrak {p}}} \\otimes _{{\\mathcal {O}}_F,\\theta } \\mathbb {F}$ given by (REF ) for $({\\mathfrak {d}}(IJ) ^{-1})_\\theta $ ." ], [ "Partial $\\Theta $ -operators", "We now compute the effect of $\\Theta $ -operators on $q$ -expansions exactly as in [12].", "Recall from §REF that for each $\\tau _0 = \\tau _{{\\mathfrak {p}},i} \\in \\Sigma _0$ , the associated partial $\\Theta $ -operator is a map $\\Theta _{\\tau _0}: M_{\\mathbf {k},\\mathbf {l}}(U;{\\mathbb {F}}) \\rightarrow M_{\\mathbf {k}^{\\prime },\\mathbf {l}^{\\prime }}(U;{\\mathbb {F}})$ where $\\mathbf {k}^{\\prime } = \\mathbf {k}+ n_{\\theta _0} {\\mathbf {e}}_{\\sigma ^{-1}\\theta _0} + {\\mathbf {e}}_{\\theta _0}$ , $\\mathbf {l}^{\\prime } = \\mathbf {l}+ {\\mathbf {e}}_{\\theta _0}$ and $\\theta _0 = \\theta _{{\\mathfrak {p}},i,e_{\\mathfrak {p}}}$ .", "It is defined for all sufficiently small $U$ of level prime to $p$ , and is Hecke-equivariant.", "In particular it is compatible with restriction for $U \\subset U^{\\prime }$ , so we may assume $U = U(N)$ for some $N$ sufficiently large that $\\nu \\equiv 1 \\mod {p}{\\mathcal {O}}_F$ for all $\\nu \\in V_{N,+}$ .", "Recall from the proof of Theorem REF that $\\Theta _{\\tau _0}$ is defined by a morphism ${\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}} \\rightarrow {\\mathcal {A}}_{\\mathbf {k}^{\\prime },\\mathbf {l}^{\\prime },\\mathbb {F}}$ given locally on sections by formula (REF ).", "Our assumptions on $U$ imply that $(\\iota _{\\mathbb {F},}{\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}})_{{\\mathcal {C}}}^\\wedge =D_{\\mathbf {k},\\mathbf {l},\\mathbb {F}} \\otimes _{\\mathbb {F}}\\cdot {\\mathcal {O}}_{Y_{U,\\mathbb {F}}^{\\min },{\\mathcal {C}}}^\\wedge $ is free of rank one over ${\\mathcal {O}}_{Y_{U,\\mathbb {F}}^{\\min },{\\mathcal {C}}}^\\wedge $ for all weights $\\mathbf {k},\\mathbf {l}$ and cusps ${\\mathcal {C}}$ , so the completion at ${\\mathcal {C}}$ of $\\iota _{\\mathbb {F},}\\Theta _{\\tau _0}$ is the map $(\\iota _{\\mathbb {F},}{\\mathcal {A}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}})_{{\\mathcal {C}}}^\\wedge \\rightarrow (\\iota _{\\mathbb {F},}{\\mathcal {A}}_{\\mathbf {k}^{\\prime },\\mathbf {l}^{\\prime },\\mathbb {F}})_{{\\mathcal {C}}}^\\wedge $ defined by (REF ), where $y_\\theta $ is any basis for $(\\iota _{\\mathbb {F},}{{\\mathcal {L}}}_\\theta )_{{\\mathcal {C}}}^\\wedge $ and $y^{\\mathbf {k}} = \\prod _\\theta y_\\theta ^{k_\\theta }$ .", "In particular we may choose $y_\\theta = b_\\theta \\otimes 1$ where $b_\\theta $ is a basis for $D_{{\\mathbf {e}}_\\theta ,\\bf {0},\\mathbb {F}} = (I^{-1})_\\theta \\otimes _{\\mathcal {O}}\\mathbb {F}$ .", "The fact that $H_\\theta $ has (non-zero) constant $q$ -expansion at ${\\mathcal {C}}$ then means the same holds for the element $r_\\theta \\in {\\mathcal {O}}_{Y_{U,\\mathbb {F}}^{\\min },{\\mathcal {C}}}^\\wedge $ , i.e., $r_\\theta \\in \\mathbb {F}^\\times $ .", "We can even select the bases $b_\\theta $ so that $r_\\theta = 1$ for all $\\theta $ by choosing any basis $b_{\\mathfrak {p}}$ for $(I_{\\mathfrak {p}}\\otimes \\mathbb {F})_{{\\mathfrak {p}},0}$ over $\\mathbb {F}[u]/u^{e_{\\mathfrak {p}}}$ for each ${\\mathfrak {p}}$ , letting $b_{{\\mathfrak {p}},i}$ correspond to $(\\phi ^i)^(b_{\\mathfrak {p}})$ under the canonical isomorphism $(I_{\\mathfrak {p}}\\otimes \\mathbb {F})_{{\\mathfrak {p}},i} \\cong (\\phi ^i)^(I_{\\mathfrak {p}}\\otimes \\mathbb {F})_{{\\mathfrak {p}},0}$ , and defining $b_{{\\mathfrak {p}},i,j}$ to be the image of $u^{e_{\\mathfrak {p}}-j}b_{{\\mathfrak {p}},i}$ in $(I^{-1})_\\theta \\otimes _{\\mathcal {O}}\\mathbb {F}$ Recall also that $g^{\\mathbf {l}} = \\prod _\\theta g_\\theta ^{l_\\theta }$ in (REF ), where each $g_\\theta $ is a trivialization of ${{\\mathcal {N}}}_\\theta $ over $Y_{U,\\mathbb {F}}$ .", "Therefore $g_\\theta $ trivializes $\\iota _{\\mathbb {F},}{{\\mathcal {N}}}_\\theta $ over $Y^{\\min }_{U,\\mathbb {F}}$ , from which it follows that $g_\\theta = c_\\theta \\otimes 1$ for some basis $c_\\theta $ of $D_{\\bf {0},{\\mathbf {e}}_\\theta ,\\mathbb {F}} = ({\\mathfrak {d}}(IJ)^{-1})_\\theta \\otimes _{\\mathcal {O}}\\mathbb {F}$ .", "The formula (REF ) therefore takes the form $\\Theta _{\\tau _0} (b^{\\mathbf {k}} c^{\\mathbf {l}} \\otimes \\varphi _f ) = H_{\\theta _0} b^{\\mathbf {k}}c^{\\mathbf {l}} \\otimes KS_{\\tau _0} (d\\varphi _f) =b^{\\mathbf {k}+\\mathbf {h}_{\\theta _0}} c^{\\mathbf {l}} \\otimes KS_{\\tau _0} (d\\varphi _f),$ for $\\varphi _f \\in {\\mathcal {O}}_{Y_{U,\\mathbb {F}}^{\\min },{\\mathcal {C}}}^\\wedge $ .", "Finally the descriptions in §REF of the canonical derivation, the Kodaira–Spencer filtration and the isomorphism (REF ) in terms of $q$ -expansions yield the formula $ \\Theta _{\\tau _0}\\left( \\sum _{m\\in N^{-1}M_+\\cup \\lbrace 0\\rbrace }(b^{\\mathbf {k}} c^{\\mathbf {l}} \\otimes r_m) q^m \\right) =\\sum _{m\\in N^{-1}M_+} (b^{\\mathbf {k}+\\mathbf {h}_{\\theta _0}} c^{\\mathbf {l}} \\overline{\\tau }_0(m) \\otimes r_m) q^m $ where $\\overline{\\tau }_0$ is the canonical projection $\\begin{array}{rcl}N^{-1}M &\\rightarrow &(N^{-1}M \\otimes \\mathbb {F})_{\\tau _0} = (M\\otimes \\mathbb {F})_{\\tau _0} \\\\& \\rightarrow &M\\otimes _{{\\mathcal {O}}_F,\\overline{\\theta }_0} \\mathbb {F}= ({\\mathfrak {d}}^{-1} I^{-1} J) \\otimes _{{\\mathcal {O}}_F,\\overline{\\theta }_0} \\mathbb {F}= D_{2{\\mathbf {e}}_{\\theta _0},-{\\mathbf {e}}_{\\theta _0},\\mathbb {F}}\\end{array}$ (writing $\\overline{\\theta }_0$ for the composite of ${\\mathcal {O}}_F \\stackrel{\\theta _0}{\\rightarrow } {\\mathcal {O}}\\rightarrow \\mathbb {F}$ ).", "As noted above, it follows that (REF ) holds with $U = U(N)$ replaced by any sufficiently small open compact $U^{\\prime }$ of level prime to $p$ and ${\\mathcal {C}}$ replaced by any cusp of $Y_{U^{\\prime }}$ .", "In this case the $q$ -expansions are necessarily invariant under the natural action of $\\Gamma _{{\\mathcal {C}},U^{\\prime }}$ (whose compatibility with (REF ) is a consequence of the construction, but is straightforward to check directly).", "We see immediately from (REF ) that the operators $\\Theta _\\tau $ for varying $\\tau $ commute.", "We see also that $\\Theta _{\\tau _0}^p(f) = \\Theta _{\\tau _1}(f) H_{\\theta _0}^p H_{\\theta _1} G_{\\theta _1}^{-1}\\prod _{j=1}^{e_{\\mathfrak {p}}-1} \\left(H_{\\sigma ^j\\theta _0} G^{-1}_{\\sigma ^j\\theta _0}\\right),$ where $\\tau _1 = \\tau _0\\circ \\phi = \\tau _{{\\mathfrak {p}},i+1}$ and $\\theta _1 = \\sigma ^{e_{\\mathfrak {p}}}\\theta _0 = \\theta _{{\\mathfrak {p}},i+1,e_{\\mathfrak {p}}}$ .", "Indeed this follows from (REF ) together with the fact that the $q$ -expansions of $H_{\\tau _1} = \\prod _{j=1}^{e_{\\mathfrak {p}}} H_{\\sigma ^j\\theta _0} \\quad \\mbox{and}\\quad G_{\\tau _1} = \\prod _{j=1}^{e_{\\mathfrak {p}}} G_{\\sigma ^j\\theta _0}$ are constants given by the canonical isomorphisms $((I^{-1})_{\\theta _0} \\otimes _{{\\mathcal {O}}} \\mathbb {F})^{\\otimes p} \\cong (I^{-1})_{\\theta _1} \\otimes _{{\\mathcal {O}}} \\mathbb {F}$ and $(({\\mathfrak {d}}(IJ)^{-1})_{\\theta _0} \\otimes _{{\\mathcal {O}}} \\mathbb {F})^{\\otimes p} \\cong ({\\mathfrak {d}}(IJ)^{-1})_{\\theta _1} \\otimes _{{\\mathcal {O}}} \\mathbb {F}$ , so we get a commutative diagram ${ & D_{2{\\mathbf {e}}_{\\theta _1},-{\\mathbf {e}}_{\\theta _1},\\mathbb {F}} [dd]\\\\N^{-1}M [ur]^{\\overline{\\tau }_1} [dr]_{\\overline{\\tau _0}^{\\otimes p}} & \\\\& D_{2{\\mathbf {e}}_{\\theta _0},-{\\mathbf {e}}_{\\theta _0},\\mathbb {F}}^{\\otimes p}}$ where the downward arrow is multiplication by the (constant) $q$ -expansion of $H_{\\tau _1}^2G_{\\tau _1}^{-1}$ ." ], [ "Partial Frobenius operators", "Finally we compute the effect on $q$ -expansions of the partial Frobenius operators $V_{\\mathfrak {p}}$ defined in §REF .", "We must first extend the partial Frobenius endomorphisms $\\widetilde{\\Phi }_{\\mathfrak {p}}$ (and $\\Phi _{\\mathfrak {p}}$ ) defined in §REF to compactifications.", "To that end let $h_{\\mathfrak {p}}$ denote the matrix $\\bigl ({\\begin{matrix}{1}&{0}\\\\{0}&{\\delta }\\end{matrix}}\\bigr )$ , viewed as an element of $\\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)}), where \\delta \\in F^\\times is as in the definition fo \\widetilde{\\Phi }_{\\mathfrak {p}}.We let \\widetilde{\\Phi }_{{\\mathfrak {p}}}^\\infty :\\widetilde{Y}^\\infty _U \\rightarrow \\widetilde{Y}^\\infty _U be the permutation defined on doublecosets by B_1({\\mathcal {O}}_{F,(p)})gU^p \\mapsto B_1({\\mathcal {O}}_{F,(p)})h_{\\mathfrak {p}}^{-1}gU^p, and similarlylet \\Phi _{{\\mathfrak {p}}}^\\infty denote the induced permutation of Y^\\infty _U.", "Then \\widetilde{\\Phi }^\\infty _{\\mathfrak {p}} translates to themap on corresponding data sending (H,I,\\lambda ,[\\eta ]) to (H^{\\prime },I^{\\prime },\\lambda ^{\\prime },[\\eta ^{\\prime }]),where\\begin{itemize}\\item I^{\\prime } = {\\mathfrak {p}}^{-1}\\otimes _{{\\mathcal {O}}_F}I = {\\mathfrak {p}}^{-1}I,\\item H^{\\prime } is the push-out of H with respect to I \\rightarrow I^{\\prime };\\item \\lambda ^{\\prime } = \\delta \\otimes \\lambda (identifying \\wedge ^2_{{\\mathcal {O}}_F}H^{\\prime } = {\\mathfrak {p}}^{-1}\\otimes _{{\\mathcal {O}}_F} \\wedge ^2_{{\\mathcal {O}}_F}H),\\item and \\eta ^{\\prime } = \\eta (identifying \\widehat{{\\mathcal {O}}}_F^{(p)} \\otimes _{{\\mathcal {O}}_F} H^{\\prime } = \\widehat{{\\mathcal {O}}}_F^{(p)} \\otimes _{{\\mathcal {O}}_F} H).\\end{itemize}}Suppose now that $ U = U(N)$ for some sufficiently large $ N$.", "One thenchecks that the morphism $ p: YU,F YU,F$ extends toa morphism $ ptor: YU,Ftor'YU,Ftor$ (where the$ '$ indicates the choice of cone decomposition need not be the same) under which thecomponent corresponding to a cusp $ C$ represented by $ (H,I,,[])$ is sent to the onecorresponding to $ C' = p(C)$ represented by $ (H',I',',['])$, and the resultingmap on completions pulls back to a morphism $ S'C,F SC',F$whose effect on global sections corresponds under the isomorphisms of (\\ref {eqn:bigcompletion})to the homomorphism induced by the canonical inclusion$$M^{\\prime } := {\\mathfrak {d}}^{-1}(I^{\\prime })^{-1}J = {\\mathfrak {p}}{\\mathfrak {d}}^{-1}I^{-1}J \\hookrightarrow {\\mathfrak {d}}^{-1}I^{-1}J.$$Furthermore the pull-back of $ A'tor= (torp)Ator$ to $ S'C,F$ isidentified with the Tate semi-abelian variety $ Tp-1I,J$, and the isomorphismsdefined in §\\ref {sec:V} relating the line bundles $ pL$to $ L$ or $ L-1$ extend to isomorphisms$$\\mbox{$(\\widetilde{\\Phi }^{\\operatorname{tor}}_{\\mathfrak {p}})^ {\\mathcal {L}}^{\\operatorname{tor}}_\\theta \\cong {\\mathcal {L}}^{{\\operatorname{tor}}^{\\prime }}_\\theta $ for $\\theta \\notin \\Sigma _{\\mathfrak {p}}$,and $(\\widetilde{\\Phi }^{\\operatorname{tor}}_{\\mathfrak {p}})^* {\\mathcal {L}}_\\theta ^{\\operatorname{tor}}\\cong ({\\mathcal {L}}^{{\\operatorname{tor}}^{\\prime }}_{\\sigma ^{-1}\\theta })^{\\otimes n_\\theta }$ for $\\theta \\in \\Sigma _{\\mathfrak {p}}$}$$over $ YU,Ftor'$ whose pull-backs are compatible via their canonical trivializations with isomorphisms induced bythe canonical $ (OFF)$-equivariant maps$$(pI^{-1} \\otimes \\mathbb {F}) \\longrightarrow ({\\mathfrak {p}}I^{-1} \\otimes \\mathbb {F})_\\tau \\longrightarrow (I^{-1} \\otimes \\mathbb {F})_\\tau .$$More precisely if $ p,0$, then the second map is an isomorphism identifying$ (pI-1)OF$ with $ (I-1)OF$, and if$ = p,i$ and $ = p,i,j$, then this map also induces the desired isomorphisms$$\\begin{array}{rcl}({\\mathfrak {p}}I^{-1})_\\theta \\otimes _{\\mathcal {O}}\\mathbb {F}&=& u^{e_{\\mathfrak {p}}- j}({\\mathfrak {p}}I^{-1} \\otimes \\mathbb {F})_\\tau \\otimes _{\\mathbb {F}[u]/u^{e_{\\mathfrak {p}}}} \\mathbb {F}\\\\& \\stackrel{\\sim }{\\longrightarrow }& u^{e_{\\mathfrak {p}}- j+1}(I^{-1} \\otimes \\mathbb {F})_\\tau \\otimes _{\\mathbb {F}[u]/u^{e_{\\mathfrak {p}}}} \\mathbb {F}= (I^{-1})_{\\sigma ^{-1}\\theta }\\otimes _{\\mathcal {O}}\\mathbb {F}\\end{array}$$for $ j=2,...,ep$.", "On the other hand if $ = p,i$ and $ = p,i,1$, thenthe first map induces an isomorphism$$(pI^{-1} \\otimes \\mathbb {F})_\\tau \\otimes _{\\mathbb {F}[u]/u^{e_{\\mathfrak {p}}}} \\mathbb {F}\\stackrel{\\sim }{\\longrightarrow }u^{e_{\\mathfrak {p}}- j}({\\mathfrak {p}}I^{-1} \\otimes \\mathbb {F})_\\tau \\otimes _{\\mathbb {F}[u]/u^{e_{\\mathfrak {p}}}} \\mathbb {F}= ({\\mathfrak {p}}I^{-1})_\\theta \\otimes _{\\mathcal {O}}\\mathbb {F}$$whose composite with the ones induced by$$\\phi ^(I^{-1}\\otimes \\mathbb {F})_{\\phi ^{-1}\\circ \\tau }\\stackrel{\\sim }{\\longrightarrow }(I^{-1}\\otimes \\mathbb {F})_\\tau \\stackrel{p\\otimes 1}{\\longrightarrow }(pI^{-1}\\otimes \\mathbb {F})_\\tau $$yields the desired isomorphism$$((I^{-1})_{\\sigma ^{-1}\\theta } \\otimes _{\\mathcal {O}}\\mathbb {F})^{\\otimes p} \\stackrel{\\sim }{\\longrightarrow }(I^{-1})_{\\theta } \\otimes _{\\mathcal {O}}\\mathbb {F}.$$The relations between the line bundles $ pN$and $ N$ or $ N-1$ extend similarly over $ YU,Ftor'$, so for$ k”,l”$ as in (\\ref {eqn:Phiweight}) we obtainisomorphisms$$(\\widetilde{\\Phi }^{\\operatorname{tor}}_{\\mathfrak {p}})^* \\widetilde{{\\mathcal {A}}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}}^{\\operatorname{tor}}\\cong \\widetilde{{\\mathcal {A}}}_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime },\\mathbb {F}}^{{\\operatorname{tor}}^{\\prime }}$$whose pull-backs to $ S'C,F$ are compatible via their canonical trivializations with the isomorphisms\\begin{equation} \\overline{D}^{\\prime }_{\\mathbf {k},\\mathbf {l}} \\cong \\overline{D}_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime }} \\end{equation}obtained as the tensor products of the ones just defined(where $D'k,l$ is associated to the data for the cusp $ C'$, and $Dk”,l”$to the data for $ C$).$ It follows from the above description of $\\widetilde{\\Phi }_{\\mathfrak {p}}^{\\operatorname{tor}}$ that $\\widetilde{\\Phi }_{\\mathfrak {p}}$ extends to the morphism $\\widetilde{\\Phi }_{\\mathfrak {p}}^{\\min }:\\widetilde{Y}_{U,\\mathbb {F}}^{\\min } \\rightarrow \\widetilde{Y}_{U,\\mathbb {F}}^{\\min }$ restricting to $\\widetilde{\\Phi }^\\infty _{{\\mathfrak {p}}}$ on the set of cusps, with the induced maps on completed local ringsNote that we have implicitly chosen different representatives $(H,I,\\lambda ,[\\eta ])$ for each cusp ${\\widetilde{{\\mathcal {C}}}}$ according to whether $\\widetilde{Y}_{U,\\mathbb {F}}$ is viewed as the source or target of $\\widetilde{\\Phi }_{\\mathfrak {p}}$ , but the rings and modules arising from the two different descriptions of completions at ${\\widetilde{{\\mathcal {C}}}}$ are canonically isomorphic.", "being the restriction to $V_N^2$ -invariants of the canonical inclusion $ \\mathbb {F}[[q^m]]_{m \\in (N^{-1}M^{\\prime })_+\\cup \\lbrace 0\\rbrace } \\longrightarrow \\mathbb {F}[[q^m]]_{m \\in (N^{-1}M)_+\\cup \\lbrace 0\\rbrace },$ where $M = {\\mathfrak {d}}^{-1}I^{-1}J$ and $M^{\\prime } = {\\mathfrak {p}}M$ .", "Furthermore the commutativity of the diagram ${({\\widetilde{\\iota }}_{\\mathbb {F},}{\\widetilde{{\\mathcal {A}}}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}})^\\wedge _{{\\widetilde{{\\mathcal {C}}}}^{\\prime }}@{^{(}->}[d]& \\Gamma ( (\\widetilde{Y}_{U,\\mathbb {F}}^{\\operatorname{tor}})_{{\\widetilde{{\\mathcal {C}}}}^{\\prime }}^\\wedge , ({\\widetilde{{\\mathcal {A}}}}^{\\operatorname{tor}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}})_{{\\widetilde{{\\mathcal {C}}}}^{\\prime }}^\\wedge )@{^{(}->}[d] [l]_-{\\sim } [r]^-{\\sim }& ( \\overline{D}^{\\prime }_{\\mathbf {k},\\mathbf {l}} \\otimes _{\\mathbb {F}} \\Gamma ({\\widehat{S}}_{{\\widetilde{{\\mathcal {C}}}}^{\\prime },\\mathbb {F}}, {\\mathcal {O}}_{{\\widehat{S}}_{{\\widetilde{{\\mathcal {C}}}}^{\\prime },\\mathbb {F}}}) )^{V_N^2} @{^{(}->}[d]\\\\({\\widetilde{\\iota }}_{\\mathbb {F},}{\\widetilde{\\Phi }_{\\mathfrak {p}}^\\widetilde{{\\mathcal {A}}}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}})^\\wedge _{\\widetilde{{\\mathcal {C}}}}[d]^{\\wr }& \\Gamma ( (\\widetilde{Y}_{U,\\mathbb {F}}^{{\\operatorname{tor}}^{\\prime }})_{{\\widetilde{{\\mathcal {C}}}}}^\\wedge , (\\widetilde{\\Phi }_{\\mathfrak {p}}^{{\\operatorname{tor}},}({\\widetilde{{\\mathcal {A}}}}^{\\operatorname{tor}}_{\\mathbf {k},\\mathbf {l},\\mathbb {F}}))_{{\\widetilde{{\\mathcal {C}}}}}^\\wedge )[d]^{\\wr } [l]_-{\\sim } [r]^-{\\sim }& ( \\overline{D}^{\\prime }_{\\mathbf {k},\\mathbf {l}} \\otimes _{\\mathbb {F}} \\Gamma ({\\widehat{S}}^{\\prime }_{{\\widetilde{{\\mathcal {C}}}},\\mathbb {F}}, {\\mathcal {O}}_{{\\widehat{S}}^{\\prime }_{{\\widetilde{{\\mathcal {C}}}},\\mathbb {F}}}) )^{V_N^2} [d]^{\\wr }\\\\ ({\\widetilde{\\iota }}_{\\mathbb {F},}{\\widetilde{{\\mathcal {A}}}}_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime },\\mathbb {F}})^\\wedge _{\\widetilde{{\\mathcal {C}}}}& \\Gamma ( (\\widetilde{Y}_{U,\\mathbb {F}}^{{\\operatorname{tor}}^{\\prime }})_{{\\widetilde{{\\mathcal {C}}}}}^\\wedge , ({\\widetilde{{\\mathcal {A}}}}^{{\\operatorname{tor}}^{\\prime }}_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime },\\mathbb {F}})_{{\\widetilde{{\\mathcal {C}}}}}^\\wedge )[l]_-{\\sim } [r]^-{\\sim }& ( \\overline{D}_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime }} \\otimes _{\\mathbb {F}} \\Gamma ({\\widehat{S}}^{\\prime }_{{\\widetilde{{\\mathcal {C}}}},\\mathbb {F}}, {\\mathcal {O}}_{{\\widehat{S}}^{\\prime }_{{\\widetilde{{\\mathcal {C}}}},\\mathbb {F}}}) )^{V_N^2}}$ (where the top vertical arrows are defined by pulling back via $\\widetilde{\\Phi }_{\\mathfrak {p}}^{\\min }$ , $\\widetilde{\\Phi }_{\\mathfrak {p}}^{{\\operatorname{tor}}}$ and the map ${\\widehat{S}}^{\\prime }_{{\\widetilde{{\\mathcal {C}}}},\\mathbb {F}} \\rightarrow {\\widehat{S}}_{{\\widetilde{{\\mathcal {C}}}}^{\\prime },\\mathbb {F}}$ ) shows that the resulting map on $q$ -expansions is the restriction to $V_N^2$ -invariants of the map $ \\overline{D}^{\\prime }_{\\mathbf {k},\\mathbf {l}} \\otimes _\\mathbb {F}\\mathbb {F}[[q^m]]_{m \\in (N^{-1}M^{\\prime })_+\\cup \\lbrace 0\\rbrace } \\longrightarrow \\overline{D}_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime }} \\otimes _{\\mathbb {F}} \\mathbb {F}[[q^m]]_{m \\in (N^{-1}M)_+\\cup \\lbrace 0\\rbrace },$ obtained as the tensor product of () and (REF ).", "(Note that the isomorphism () is $V_N^2$ -equivariant, but we can also choose $N$ sufficiently large that the action is trivial.)", "The constructions above are all compatible with the natural action of ${\\mathcal {O}}_{F,(p),+}^\\times $ , so the morphism $\\widetilde{\\Phi }_{\\mathfrak {p}}^{\\min }$ induces a morphism $\\Phi _{{\\mathfrak {p}}}^{\\min } : \\overline{Y}_U^{\\min } \\rightarrow \\overline{Y}_U^{\\min }$ extending $\\Phi _{\\mathfrak {p}}$ by the map $\\Phi ^\\infty _{{\\mathfrak {p}}}$ on cusps, and its effect on completed local rings is given by the $V_{N,+}$ -invariants of (REF ).", "Furthermore the map $V_{\\mathfrak {p}}: M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) \\longrightarrow M_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime }}(U,\\mathbb {F})$ is described on $q$ -expansions by taking the $V_{N,+}$ -invariants of the tensor product of () and (REF ).", "(Note that $\\Phi _{\\mathfrak {p}}^{\\min }$ is proper and quasi-finite, hence finite, but not necessarily flat at the cusps.)", "Similarly for any sufficiently small $U^{\\prime }$ , we may choose $N$ so $U = U(N) \\subset U^{\\prime }$ and take invariants under the natural action of $U^{\\prime }/U$ , with which the above constructions are also easily seen to be compatible.", "We thus obtain the description of $V_{\\mathfrak {p}}$ on $q$ -expansions (under the identifications of Proposition REF ) as the resulting map $ \\left( \\overline{D}^{\\prime }_{\\mathbf {k},\\mathbf {l}} \\otimes _\\mathbb {F}\\mathbb {F}[[q^m]]_{m \\in (N^{-1}M^{\\prime })_+\\cup \\lbrace 0\\rbrace } \\right)^{\\Gamma _{{\\mathcal {C}}^{\\prime },U^{\\prime }}}\\longrightarrow \\left( \\overline{D}_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime }} \\otimes _\\mathbb {F}\\mathbb {F}[[q^m]]_{m \\in (N^{-1}M)_+\\cup \\lbrace 0\\rbrace } \\right)^{\\Gamma _{{\\mathcal {C}},U^{\\prime }}}.$ (Note that the maps () and (REF ) are in fact $\\Gamma _{{\\mathcal {C}}}$ -equivariant, where $\\Gamma _{{\\mathcal {C}}}$ is defined in (REF ) and its action on the target is via the natural inclusion in $\\Gamma _{{\\mathcal {C}}^{\\prime }}$ .)", "Finally we note that the effect of the operator $V_{\\mathfrak {p}}^0: M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) \\longrightarrow M_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}}(U,\\mathbb {F})$ on $q$ -expansions has the same description, but with () replaced by its composite with the isomorphism $\\overline{D}_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime }} \\cong \\overline{D}_{\\bf {0},\\mathbf {l}-\\mathbf {l}^{\\prime \\prime }} \\otimes _{\\mathbb {F}} \\overline{D}_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime }}= \\overline{D}_{\\mathbf {k}^{\\prime \\prime },\\mathbf {l}}$ given by choosing the basis element of $\\overline{D}_{\\bf {0},\\mathbf {l}-\\mathbf {l}^{\\prime \\prime }}$ to be the (constant) $q$ -expansion of $\\prod _{\\theta \\in \\Sigma } G_\\theta ^{-l_\\theta }$ ." ], [ "Determination of the kernel", "In this section we analyze the kernel of the partial $\\Theta $ -operator $\\Theta _{\\tau }: M_{\\mathbf {k},\\mathbf {l}}(U;{\\mathbb {F}}) \\rightarrow M_{\\mathbf {k}^{\\prime },\\mathbf {l}^{\\prime }}(U;{\\mathbb {F}})$ for $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ and relate it to the image of a partial Frobenius operator.", "We allow $U$ to be any sufficiently small open compact subgroup of $\\operatorname{GL}_2({\\mathbb {A}}_{F,) oflevel prime to p, and (\\mathbf {k},\\mathbf {l}) any weight such that \\chi _{\\mathbf {k}+2\\mathbf {l},\\mathbb {F}} is trivial onU \\cap {\\mathcal {O}}_F^\\times .", "First note that by (\\ref {eqn:Thetaq}) and the q-expansion Principle(Proposition~\\ref {prop:qexp}), the kernel of\\Theta _\\tau consists precisely of those f whose q-expansions\\sum _{m\\in N^{-1}M_+\\cup \\lbrace 0\\rbrace }(b^{\\mathbf {k}} c^{\\mathbf {l}} \\otimes r_m) q^mat all cusps, or even a cusp on each connected component, satisfyr_m = 0 for all m \\notin {\\mathfrak {p}}N^{-1}M. (Recall that M = {\\mathfrak {d}}^{-1}I^{-1}J if the cusp {\\mathcal {C}}if represented by (H,I,[\\lambda ],[\\eta ]), and note that the condition depends only on the isomorphismclass of (H,I,[\\lambda ],[\\eta ]) and in particular is independent of N prime to p such that U(N) \\subset U.", ")Note that the condition is the same for all \\tau \\in \\Sigma _{{\\mathfrak {p}},0}, so that\\ker (\\Theta _\\tau ) = \\ker (\\Theta _{\\tau ^{\\prime }}) \\quad \\mbox{for all $\\tau ,\\tau ^{\\prime } \\in \\Sigma _{{\\mathfrak {p}},0}$}.Note also that the condition is invariant under multiplicationby the Hasse invariants H_\\theta (and of course the forms G_\\theta ) for all \\theta , sothat \\Theta _\\tau (f) = 0 if and only if\\Theta _\\tau (f \\prod _{\\theta \\in \\Sigma } G_\\theta ^{m_\\theta } H_\\theta ^{n_\\theta }) = 0for all \\mathbf {\\in }\\mathbb {Z}^\\Sigma , \\mathbf {n}\\in \\mathbb {Z}_{\\ge 0}^\\Sigma , if and only if\\Theta _\\tau (f \\prod _{\\theta \\in \\Sigma } G_\\theta ^{m_\\theta } H_\\theta ^{n_\\theta }) = 0for some \\mathbf {\\in }\\mathbb {Z}^\\Sigma , \\mathbf {n}\\in \\mathbb {Z}_{\\ge 0}^\\Sigma .", "(Alternatively notethat this follows from the fact the partial \\Theta -operators commute with multiplicationby the G_\\theta and H_\\theta , as can be seen directly from their definition.", ")}Suppose now that $ k= k(f)$, so that $ f$ is not divisible by any partial Hasse invariants (see §\\ref {sec:stratification}).Then if $ f ()$, and hence $ f (p,i)$ for all $ i Z/fpZ$,then Theorem~\\ref {thm:theta} implies that $ p\| kp,i,ep$ for all $ i Z/fpZ$.Therefore $ k$ is of the form $ k0”$ for some $ k0$, where $ k0”$ is as in the definition of $ Vp$in §\\ref {sec:V}, or equivalently $ Vp0$.", "Furthermore it is immediate from the description of the effect of $ Vp0$on $ q$-expansions in (\\ref {eqn:Vonq}) that its image is contained in the kernel of $$.", "We now use themethod of \\cite [Thm.~9.8.2]{DS} to prove the kernel is precisely the image of $ Vp0$.$ Theorem 8.1.1 Suppose that $f \\in M_{\\mathbf {k}_0^{\\prime \\prime },\\mathbf {l}}(U;{\\mathbb {F}})$ and $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ .", "If $\\Theta _\\tau (f) = 0$ , then $f = V_{\\mathfrak {p}}^0(g)$ for some $g \\in M_{\\mathbf {k}_0,\\mathbf {l}}(U;{\\mathbb {F}})$ .", "Proof.Let $\\overline{\\iota }$ denote the embedding $\\overline{Y}_U \\hookrightarrow \\overline{Y}_U^{\\min }$ , and choose a set of cusps ${\\mathcal {S}}\\subset Y^\\infty _U$ consisting of precisely one on each connected component of $\\overline{Y}_U$ .", "Note that since $\\Phi _{\\mathfrak {p}}^{\\min }$ (defined in §REF ) is bijective on cusps as well as connected components, the set ${\\mathcal {S}}^{\\prime } : =\\Phi ^\\infty _{{\\mathfrak {p}}}({\\mathcal {S}})$ also includes exactly one cusp on each connected component.", "Recall from Proposition REF that the sheaves $\\overline{\\iota }_{\\mathcal {A}}_{\\mathbf {k}_0,\\mathbf {l},\\mathbb {F}}$ and $\\overline{\\iota }_{\\mathcal {A}}_{\\mathbf {k}^{\\prime \\prime }_0,\\mathbf {l},\\mathbb {F}}$ are coherent, as is $\\overline{\\iota }_\\Phi _{{\\mathfrak {p}},}{\\mathcal {A}}_{\\mathbf {k}_0^{\\prime \\prime },\\mathbf {l}^{\\prime \\prime },\\mathbb {F}} = \\Phi _{{\\mathfrak {p}},}^{\\min }\\overline{\\iota }_{\\mathcal {A}}_{\\mathbf {k}_0^{\\prime \\prime },\\mathbf {l},\\mathbb {F}}$ since $\\Phi _{{\\mathfrak {p}}}^{\\min }$ is finite.", "For each ${\\mathcal {C}}\\in {\\mathcal {S}}$ , let ${\\mathcal {C}}^{\\prime } = \\Phi _{{\\mathfrak {p}}}^\\infty ({\\mathcal {C}})$ , so that $\\Phi _{{\\mathfrak {p}}}^{{\\min },}$ defines a finite extension ${\\mathcal {O}}_{\\overline{Y}_U^{\\min },{\\mathcal {C}}^{\\prime }} \\hookrightarrow {\\mathcal {O}}_{\\overline{Y}_U^{\\min },{\\mathcal {C}}}$ of local rings.", "We let $N_{{\\mathcal {C}}^{\\prime }} = (\\overline{\\iota }_{\\mathcal {A}}_{\\mathbf {k}_0,\\mathbf {l},\\mathbb {F}})_{{\\mathcal {C}}^{\\prime }}$ denote the stalk at ${\\mathcal {C}}^{\\prime }$ of $\\overline{\\iota }_{\\mathcal {A}}_{\\mathbf {k}_0,\\mathbf {l},\\mathbb {F}}$ , and similarly let $N^{\\prime \\prime }_{{\\mathcal {C}}} = (\\overline{\\iota }_{\\mathcal {A}}_{\\mathbf {k}_0,\\mathbf {l},\\mathbb {F}})_{{\\mathcal {C}}} = (\\overline{\\iota }_\\Phi _{{\\mathfrak {p}},}{\\mathcal {A}}_{\\mathbf {k}_0^{\\prime \\prime },\\mathbf {l},\\mathbb {F}} )_{{\\mathcal {C}}^{\\prime }}$ .", "The stalk at ${\\mathcal {C}}^{\\prime }$ of $\\overline{\\iota }_$ of the adjoint of $\\Phi _{{\\mathfrak {p}}}^*{\\mathcal {A}}_{\\mathbf {k}_0,\\mathbf {l},\\mathbb {F}} \\stackrel{\\sim }{\\longrightarrow } {\\mathcal {A}}_{\\mathbf {k}_0^{\\prime \\prime },\\mathbf {l},\\mathbb {F}}$ then defines an injective homomorphism $N_{{\\mathcal {C}}^{\\prime }} \\rightarrow N^{\\prime \\prime }_{{\\mathcal {C}}}$ of finitely generated ${\\mathcal {O}}_{\\overline{Y}_U^{\\min },{\\mathcal {C}}^{\\prime }}$ -modules, extending $V_{\\mathfrak {p}}^0$ to a map $\\bigoplus _{{\\mathcal {C}}^{\\prime }\\in {\\mathcal {S}}^{\\prime }} N_{{\\mathcal {C}}^{\\prime }} \\rightarrow \\bigoplus _{{\\mathcal {C}}\\in {\\mathcal {S}}} N^{\\prime \\prime }_{{\\mathcal {C}}}.$ Similarly localizing at the generic points of $\\overline{Y}_U$ (or equivalently $\\overline{Y}_U^{\\min }$ ) extends $V_{\\mathfrak {p}}^0$ to a map $H^0(\\overline{Y}_U, {\\mathcal {A}}_{\\mathbf {k}_0,\\mathbf {l},\\mathbb {F}}\\otimes _{\\overline{Y}_U} {\\mathcal {F}}_U) \\rightarrow H^0(\\overline{Y}_U, {\\mathcal {A}}_{\\mathbf {k}^{\\prime \\prime }_0,\\mathbf {l},\\mathbb {F}}\\otimes _{\\overline{Y}_U} {\\mathcal {F}}_U)$ , so we obtain a commutative diagram of injective maps $ {M_{\\mathbf {k}_0,\\mathbf {l}}(U;\\mathbb {F}) [r][d] & \\bigoplus _{{\\mathcal {C}}^{\\prime }\\in {\\mathcal {S}}^{\\prime }} N_{{\\mathcal {C}}^{\\prime }} [r][d] & H^0(\\overline{Y}_U, {\\mathcal {A}}_{\\mathbf {k}_0,\\mathbf {l},\\mathbb {F}} \\otimes _{\\overline{Y}_U} {\\mathcal {F}}_U) [d] \\\\M_{\\mathbf {k}^{\\prime \\prime }_0,\\mathbf {l}}(U;\\mathbb {F}) [r]& \\bigoplus _{{\\mathcal {C}}\\in {\\mathcal {S}}} N^{\\prime \\prime }_{{\\mathcal {C}}} [r] & H^0(\\overline{Y}_U, {\\mathcal {A}}_{\\mathbf {k}^{\\prime \\prime }_0,\\mathbf {l},\\mathbb {F}} \\otimes _{\\overline{Y}_U} {\\mathcal {F}}_U).", "}$ (Note that the horizontal maps, defined by localization, are injective since ${\\mathcal {S}}$ and ${\\mathcal {S}}^{\\prime }$ each contain a unique cusp on each component of $\\overline{Y}_U$ .)", "Let $N_{{\\mathcal {C}}^{\\prime }}^\\wedge $ denote the completion of $N_{{\\mathcal {C}}^{\\prime }}$ with respect to the maximal ideal of ${\\mathcal {O}}_{\\overline{Y}_U^{\\min },{\\mathcal {C}}^{\\prime }}$ , and similarly let $N^{\\prime \\prime \\wedge }_{{\\mathcal {C}}}$ denote the completion of $N^{\\prime \\prime }_{{\\mathcal {C}}}$ with respect to the maximal ideal of ${\\mathcal {O}}_{\\overline{Y}_U^{\\min },{\\mathcal {C}}}$ , or equivalently ${\\mathcal {O}}_{\\overline{Y}_U^{\\min },{\\mathcal {C}}^{\\prime }}$ .", "Note that the map $N_{{\\mathcal {C}}^{\\prime }}^\\wedge \\rightarrow N^{\\prime \\prime \\wedge }_{{\\mathcal {C}}}$ is the one described by (REF ), or more precisely its variant for $V_{\\mathfrak {p}}^0$ .", "Recall from (REF ) that if $f \\in \\ker \\Theta _\\tau $ , then for each ${\\mathcal {C}}\\in {\\mathcal {S}}$ , the $q$ -expansion of $f$ : $\\sum _{m\\in N^{-1}M_+\\cup \\lbrace 0\\rbrace }(b^{\\mathbf {k}_0^{\\prime \\prime }} c^{\\mathbf {l}} \\otimes r_m) q^m \\in N^{\\prime \\prime \\wedge }_{{\\mathcal {C}}}$ satisfies $r_m = 0$ for all $m \\notin {\\mathfrak {p}}N^{-1}M = N^{-1}M^{\\prime }_+$ , where $M^{\\prime } = {\\mathfrak {p}}M$ , and is therefore in the image of $N_{{\\mathcal {C}}^{\\prime }}^\\wedge $ .", "Since the completion ${\\mathcal {O}}^\\wedge _{\\overline{Y}_U^{\\min },{\\mathcal {C}}^{\\prime }}$ is faithfully flat over ${\\mathcal {O}}_{\\overline{Y}_U^{\\min },{\\mathcal {C}}^{\\prime }}$ , it follows that the image of $f$ in $\\bigoplus _{{\\mathcal {C}}\\in {\\mathcal {S}}} N^{\\prime \\prime }_{{\\mathcal {C}}}$ is of the form $V_{\\mathfrak {p}}^0(g)$ for some $g \\in \\bigoplus _{{\\mathcal {C}}^{\\prime }\\in {\\mathcal {S}}^{\\prime }} N_{{\\mathcal {C}}^{\\prime }}$ , and hence that its image in $H^0(\\overline{Y}_U, {\\mathcal {A}}_{\\mathbf {k}^{\\prime \\prime }_0,\\mathbf {l},\\mathbb {F}} \\otimes _{\\overline{Y}_U} {\\mathcal {F}}_U)$ is of the form $V_{\\mathfrak {p}}^0(g)$ for some $g \\in H^0(\\overline{Y}_U, {\\mathcal {A}}_{\\mathbf {k}^{\\prime \\prime }_0,\\mathbf {l},\\mathbb {F}} \\otimes _{\\overline{Y}_U} {\\mathcal {F}}_U)$ .", "It just remains to prove that $g \\in M_{\\mathbf {k}_0,\\mathbf {l}}(U;\\mathbb {F})$ , or equivalently that ${\\operatorname{ord}}_z(g) \\ge 0$ for all prime divisors $z$ on $\\overline{Y}_U$ .", "To that end, note that the operators $V_{{\\mathfrak {p}}^{\\prime }}$ for all ${\\mathfrak {p}}^{\\prime } \\in S_p$ , and $\\epsilon _{\\mathbf {k},\\mathbf {l}}$ for all $\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma $ (see §REF )) similarly extend to maps on stalks at generic points satisfying (REF ), so that $V_{\\mathfrak {p}}(g) = f \\prod _{\\theta \\in \\Sigma } G_\\theta ^{l_\\theta }$ and $g^p = \\left(\\epsilon _{p\\mathbf {k},p\\mathbf {l}} \\prod _{{\\mathfrak {p}}^{\\prime }\\in S_p} V_{\\mathfrak {q}}^{e_{\\mathfrak {q}}} \\right) (g) =\\left(\\delta _{p\\mathbf {k},p\\mathbf {l}} \\prod _{{\\mathfrak {p}}^{\\prime } \\ne {\\mathfrak {p}}} V_{{\\mathfrak {p}}^{\\prime }}^{e_{\\mathfrak {p}}} \\right) V_{\\mathfrak {p}}^{e_{\\mathfrak {p}}-1}(f).$ Therefore $p\\cdot {\\operatorname{ord}}_z(g) \\ge 0$ , and hence ${\\operatorname{ord}}_z(g) \\ge 0$ .", "$\\square $ For the following corollary, recall that $\\Xi ^{\\min }$ is defined by (REF ) and that the main result of [10] states that if $f$ is a non-zero form in $M_{\\mathbf {k},\\mathbf {l}}(U;{\\mathbb {F}})$ , then $\\mathbf {k}_{\\min }(f) \\in \\Xi ^{\\min }$ .", "Corollary 8.1.2 Suppose that $f \\in M_{\\mathbf {k},\\mathbf {l}}(U;{\\mathbb {F}})$ and $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ .", "Then $\\Theta _\\tau (f) = 0$ if and only if there exist $\\mathbf {k}_0 \\in \\Xi ^{\\min }$ , $\\mathbf {n}\\in \\mathbb {Z}^\\Sigma _{\\ge 0}$ and $g \\in M_{\\mathbf {k}_0,\\mathbf {l}}(U;{\\mathbb {F}})$ such that $\\mathbf {k}= \\mathbf {k}_0^{\\prime \\prime } + \\sum _\\theta n_\\theta \\mathbf {h}_\\theta $ and $f = V_{\\mathfrak {p}}^0(g) \\prod _\\theta H_\\theta ^{n_\\theta }.$ Proof.We have already seen that if $f = V_{\\mathfrak {p}}^0(g) \\prod _\\theta H_\\theta ^{n_\\theta }$ , then $\\Theta _\\tau (f) = 0$ .", "For the converse, note that we may assume $\\mathbf {k}= \\mathbf {k}_{\\min }(f)$ , so that $\\mathbf {k}\\in \\Xi ^{\\min }$ and $\\mathbf {k}= \\mathbf {k}_0^{\\prime \\prime }$ for some $\\mathbf {k}_0 \\in \\mathbb {Z}^\\Sigma $ .", "Therefore the theorem implies that $f= V_{\\mathfrak {p}}^0(g)$ for some $g \\in M_{\\mathbf {k}_0,\\mathbf {l}}(U;{\\mathbb {F}})$ .", "Finally it is immediate from the definitions of $\\Xi ^{\\min }$ and $\\mathbf {k}_0^{\\prime \\prime }$ that $\\mathbf {k}_0 \\in \\Xi ^{\\min }$ if (and only if) $\\mathbf {k}_0^{\\prime \\prime } \\in \\Xi ^{\\min }$ .", "$\\square $" ], [ "Forms of partial weight 0", "We now apply our results on partial $\\Theta $ -operators to prove a partial positivity result for minimal weights of Hilbert modular forms.", "Recall the main result of [10] proves that minimal weights $\\mathbf {k}= \\sum k_\\theta {\\mathbf {e}}_\\theta $ of Hilbert modular forms necessarily lie in the cone $\\Xi ^{\\min }$ , and hence satisfy $k_\\theta \\ge 0$ for all $\\theta $ , and that forms with $\\mathbf {k}= \\bf {0}$ are easily described by Proposition .", "We prove the following restriction on possible minimal weights $\\mathbf {k}$ with $k_\\theta = 0$ for some (but not all) $\\theta \\in \\Sigma $ .", "Theorem 8.2.1 Suppose that ${\\mathfrak {p}}\\in S_p$ is such that $f_{\\mathfrak {p}}> 1$ , or $e_{\\mathfrak {p}}> 1$ and $p > 3$ .", "Suppose that $f \\in M_{\\mathbf {k},\\mathbf {l}}(U;{\\mathbb {F}})$ is non-zero and $\\mathbf {k}= \\mathbf {k}_{\\min }(f)$ .", "If $k_\\theta = 0$ for some $\\theta \\in \\Sigma _{\\mathfrak {p}}$ , then $\\mathbf {k}= \\bf {0}$ .", "Proof.Writing simply $f = f_{\\mathfrak {p}}$ and $e=e_{\\mathfrak {p}}$ , note that the hypotheses mean that $ef > 1$ and $p^f > 3$ .", "Choose any $\\tau = \\tau _{{\\mathfrak {p}},i} \\in \\Sigma _{{\\mathfrak {p}},0}$ and let $\\theta _0 = \\theta _{{\\mathfrak {p}},i,e}$ .", "We will first prove that $\\Theta _\\tau (f) = 0$ .", "Note that since $\\mathbf {k}\\in \\Xi ^{\\min }$ and $k_\\theta = 0$ for some $\\theta \\in \\Sigma _{\\mathfrak {p}}$ , we in fact have $k_\\theta = 0$ for all $\\theta \\in \\Sigma _{\\mathfrak {p}}$ .", "In particular $p\|k_{\\theta _0}$ , so Theorem REF implies that $H_{\\theta _0}\|\\Theta _\\tau (f)$ .", "Therefore if $\\Theta _\\tau (f) \\ne 0$ , then $\\mathbf {k}_{\\min }(\\Theta _\\tau (f)) \\le _{{\\mathrm {Ha}}} \\mathbf {k}+ 2{\\mathbf {e}}_{\\theta _0}$ , i.e., $ \\mathbf {k}+ 2{\\mathbf {e}}_{\\theta _0} - \\sum _{\\theta \\in \\Sigma } m_\\theta \\mathbf {h}_\\theta \\in \\Xi ^{\\min }$ for some integers $m_\\theta \\ge 0$ .", "Letting $m_r = m_{\\sigma ^r\\theta _0}$ for $r=1,\\ldots ,ef$ , this implies that $ \\begin{array}{rl}& m_1 - m_2 \\le m_2 - m_3 \\le \\cdots \\le m_{e-1} - m_e \\le m_e - pm_{e+1} \\\\\\le &p(m_{e+1} - m_{e+2}) \\le p(m_{e+2} - m_{e+3}) \\le \\cdots \\le p(m_{2e-1} - m_{2e}) \\le p(m_{2e} - pm_{2e+1}) \\\\& \\vdots \\\\\\le &p^{f-1}(m_{(f-1)e+1} - m_{(f-1)e+2}) \\le \\cdots \\le p^{f-1}(m_{ef-1} - m_{ef}) \\le p^{f-1}(2+m_{ef} - pm_1) \\\\\\le &p^f(m_1-m_2),\\end{array}$ (with the obvious collapsing here and in subsequent inequalities if $e$ or $f=1$ ).", "In particular all the expressions in (REF ) are non-negative, so we have $m_1 \\ge m_2 \\ge \\cdots m_e \\ge pm_{e+1} \\ge pm_{e+2} \\ge \\cdots \\ge p^{f-1}m_{ef-1} \\ge p^{f-1}m_{ef}$ and $2 + m_{ef} - pm_1 \\ge 0$ , which implies that $(p^f-1)m_{ef} \\le 2$ .", "Since $p^f > 3$ , it follows that $m_{ef} = 0$ , so $pm_1 \\le 2$ , which implies that either $m_1 = 0$ , or $m_1 =1$ and $p=2$ .", "If $m_1 = 0$ , then $m_r = 0$ for all $r$ , which contradicts the final inequality in (REF ).", "On the other hand if $m_1 = 1$ and $p=2$ , then all the expressions in (REF ) are zero, which in turn implies that $m_1 = p^{f-1}m_{ef}$ , which again yields a contradiction.", "We have now shown that $\\Theta _\\tau (f) = 0$ .", "Note that $\\mathbf {k}^{\\prime \\prime } = \\mathbf {k}$ since $k_\\theta = 0$ for all $\\theta \\in \\Sigma _{\\mathfrak {p}}$ , so Theorem REF implies that $f = V_{\\mathfrak {p}}^0(f_1)$ for some $f_1 \\in M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F})$ .", "We may therefore iterate the above argument to conclude that $f_1 = V_{\\mathfrak {p}}^0(f_2)$ for some $f_2 \\in M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F})$ , and by induction that for all $n \\ge 1$ , we have $f = (V_{\\mathfrak {p}}^0)^n(f_n)$ for some $f_n \\in M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F})$ .", "It follows that for all $n \\ge 1$ , the $q$ -expansion of $f$ at every cusp of $\\overline{Y}_U$ satisfies $r_m = 0$ for all $m \\notin {\\mathfrak {p}}^n M$ , so in fact the $q$ -expansion of $f$ at every cusp is constant.", "To prove that $\\mathbf {k}= \\bf {0}$ , recall that $\\Xi ^{\\min }$ is contained in the cone spanned by the partial Hasse invariants, so $\\mathbf {k}= \\sum _{\\theta \\in \\Sigma } s_\\theta \\mathbf {h}_\\theta $ for some $s_\\theta \\in \\mathbb {Q}_{\\ge 0}$ .", "Furthermore the denominators are divisors of $M = \\operatorname{lcm}\\lbrace \\, p^{f_{\\mathfrak {q}}} - 1 \\,\|\\, {\\mathfrak {q}}\\in S_p\\,\\rbrace $ , so that $M\\mathbf {k}= \\sum m_\\theta \\mathbf {h}_\\theta $ for some $m_\\theta \\in \\mathbb {Z}_{\\ge 0}$ .", "Similarly $M\\mathbf {l}= \\sum n_\\theta \\mathbf {h}_\\theta $ for some $n_\\theta \\in \\mathbb {Z}$ .", "Since $f$ has constant $q$ -expansions, so does $f^M$ , and therefore $f^M = h \\prod _{\\theta \\in \\Sigma } (H_\\theta ^{m_\\theta } G_\\theta ^{n_\\theta })$ for some $h \\in H^0(\\overline{Y}_U, {\\mathcal {O}}_{\\overline{Y}_U})$ .", "For each $\\theta \\in \\Sigma $ , the assumption that $\\mathbf {k}= \\mathbf {k}_{\\min }(f)$ means that $f$ is not divisible by $H_\\theta $ , so ${\\operatorname{ord}}_z(f) = 0$ for some irreducible component $z$ of $Z_\\theta $ .", "On the other hand we have $M{\\operatorname{ord}}_z(f) = {\\operatorname{ord}}_z(f^M) \\ge m_\\theta $ , so $m_\\theta = 0$ .", "As this holds for all $\\theta \\in \\Sigma $ , we conclude that $\\mathbf {k}= \\bf {0}$ .", "$\\square $" ], [ "The kernel revisited", "Finally we present a cleaner, but less explicit, variant of Corollary REF describing the kernels of partial $\\Theta $ -operators.", "We first record the effect of $V_{\\mathfrak {p}}$ on the partial Hasse invariants $H_\\theta $ .", "For each prime ${\\mathfrak {p}}\\in S_p$ , we let $\\beta _{\\mathfrak {p}}= p^{-1}\\varpi _{\\mathfrak {p}}^{e_{\\mathfrak {p}}} \\in {\\mathcal {O}}_{F,{\\mathfrak {p}}}^\\times $ .", "It is straightforward to check, directly from the definition of $V_{\\mathfrak {p}}$ or from the description (REF ) of its effect on $q$ -expansions (and those of the $H_\\theta $ in §REF ), that if $\\theta \\notin \\Sigma _{\\mathfrak {p}}$ then $V_{\\mathfrak {p}}(H_\\theta ) = H_\\theta $ , but if $\\theta = \\theta _{{\\mathfrak {p}},i,j}$ then $V_{\\mathfrak {p}}(H_\\theta ) = \\left\\lbrace \\begin{array}{ll}H_{\\sigma ^{-1}\\theta }^p, & \\mbox{if $e_{\\mathfrak {p}}= 1$;}\\\\\\overline{\\theta }(\\beta _{\\mathfrak {p}})^{-1} H_{\\sigma ^{-1}\\theta }^p, & \\mbox{if $e_{\\mathfrak {p}}> 1$ and $j=1$;}\\\\\\overline{\\theta }(\\beta _{\\mathfrak {p}}) H_{\\sigma ^{-1}\\theta }, & \\mbox{if $e_{\\mathfrak {p}}> 1$ and $j=2$;}\\\\H_{\\sigma ^{-1}\\theta },& \\mbox{otherwise.", "}\\end{array}\\right.$ Therefore we define the modified partial Hasse invariant to be $H^{\\prime }_\\theta = \\overline{\\theta }(\\beta _{{\\mathfrak {p}}^{\\prime }})H_\\theta $ if $\\theta = \\theta _{{\\mathfrak {p}}^{\\prime },i,1}$ for some ${\\mathfrak {p}}^{\\prime } \\in S_p$ and $i \\in \\mathbb {Z}/f_{{\\mathfrak {p}}^{\\prime }}\\mathbb {Z}$ , and $H^{\\prime }_\\theta = H_\\theta $ otherwise, so that $V_{\\mathfrak {p}}(H^{\\prime }_\\theta ) = \\left\\lbrace \\begin{array}{ll} H^{\\prime n_\\theta }_{\\sigma ^{-1}\\theta } & \\mbox{if $\\theta \\in \\Sigma _{{\\mathfrak {p}},0}$;}\\\\H^{\\prime }_\\theta & \\mbox{if $\\theta \\notin \\Sigma _{{\\mathfrak {p}},0}$.", "}\\end{array}\\right.$ Similarly letting $G^{\\prime }_\\theta = \\overline{\\theta }(\\beta _{{\\mathfrak {p}}^{\\prime }})G_\\theta $ if $\\theta = \\theta _{{\\mathfrak {p}}^{\\prime },i,1}$ and $G^{\\prime }_\\theta = G_\\theta $ otherwise, we have $V_{\\mathfrak {p}}(G^{\\prime }_\\theta ) = G^{\\prime n_\\theta }_{\\sigma ^{-1}\\theta }$ if $\\theta \\in \\Sigma _{{\\mathfrak {p}}}$ and $V_{\\mathfrak {p}}(G^{\\prime }_\\theta ) = G_\\theta $ if $\\theta \\notin \\Sigma _{\\mathfrak {p}}$ .", "Now for any sufficiently small $U$ of level prime to $p$ , consider the $\\mathbb {F}$ -algebra $M_{{\\operatorname{tot}}}(U;\\mathbb {F}) = \\bigoplus _{\\mathbf {k},\\mathbf {l}\\in (\\mathbb {Z}^\\Sigma )^2} M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F})$ of Hilbert modular forms of all weights and level $U$ (where we let $M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) =0$ if $\\chi _{\\mathbf {k}+2\\mathbf {l},\\mathbb {F}}$ is non-trivial on $U\\cap {\\mathcal {O}}_F^\\times $ ).", "We may then consider $V_{\\mathfrak {p}}$ (resp.", "$\\Theta _\\tau $ ) as an $\\mathbb {F}$ -algebra homomorphism (resp.", "$\\mathbb {F}$ -linear derivation) $M_{{\\operatorname{tot}}}(U;\\mathbb {F}) \\rightarrow M_{{\\operatorname{tot}}}(U;\\mathbb {F})$ for any ${\\mathfrak {p}}\\in S_p$ and $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ .", "Furthermore letting ${\\mathfrak {I}}$ denote the ideal $\\langle \\,H_\\theta ^{\\prime } - 1, G_\\theta ^{\\prime }-1\\,\\rangle _{\\theta \\in \\Sigma }$ in $M_{{\\operatorname{tot}}}(U;\\mathbb {F})$ and $R_U = M^{{\\operatorname{tot}}}(U;\\mathbb {F})/{\\mathfrak {I}}$ , we see that $V_{\\mathfrak {p}}({\\mathfrak {I}}) \\subset {\\mathfrak {I}}$ and $\\Theta _\\tau ({\\mathfrak {I}}) \\subset {\\mathfrak {I}}$ , so we obtain an $\\mathbb {F}$ -algebra homomorphism $V_{\\mathfrak {p}}$ and derivation $\\Theta _\\tau $ such that the composite $R_U \\stackrel{V_{\\mathfrak {p}}}{\\longrightarrow } R_U \\stackrel{\\Theta _\\tau }{\\longrightarrow } R_U$ is zero for any ${\\mathfrak {p}}\\in S_p$ , $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ .", "Let $\\Lambda $ denote the subgroup $\\bigoplus _{\\theta \\in \\Sigma } \\mathbb {Z}\\mathbf {h}_\\theta $ of $\\mathbb {Z}^\\Sigma = \\bigoplus _{\\theta \\in \\Sigma } \\mathbb {Z}{\\mathbf {e}}_\\theta $ , so $\\Lambda $ is the image of the image of the endomorphism of $\\mathbb {Z}^\\Sigma $ defined by $\\sum _\\theta m_\\theta {\\mathbf {e}}_\\theta \\mapsto \\sum _\\theta m_\\theta \\mathbf {h}_\\theta $ .", "Writing $\\mathbf {h}_\\theta = \\sum _{\\theta ^{\\prime }} n_{\\theta ,\\theta ^{\\prime }} {\\mathbf {e}}_{\\theta ^{\\prime }}$ , it is straightforward to check that the matrix $(n_{\\theta ,\\theta ^{\\prime }})$ has determinant $\\prod _{{\\mathfrak {p}}\\in S_p}(p^{f_{\\mathfrak {p}}} - 1)$ , so this is the index of $\\Lambda $ in $\\mathbb {Z}^\\Sigma $ .", "On the other hand, let $\\Psi $ denote the group of characters $\\psi : ({\\mathcal {O}}_F/{\\mathfrak {q}})^\\times = \\bigoplus _{{\\mathfrak {p}}\\in S_p} ({\\mathcal {O}}_F/{\\mathfrak {p}})^\\times \\longrightarrow \\mathbb {F}^\\times ,$ and consider the surjective homomorphism $\\varrho : \\mathbb {Z}^\\Sigma \\rightarrow \\Psi $ defined by $\\varrho (\\mathbf {k}) = \\prod _{\\theta \\in \\Sigma } \\overline{\\theta }^{k_\\theta } = \\bigoplus _{{\\mathfrak {p}}\\in S_p} ( \\prod _{\\theta \\in \\Sigma _{\\mathfrak {p}}} \\overline{\\theta }^{k_\\theta } ).$ Note that $\\varrho (\\mathbf {h}_\\theta )$ is trivial for all $\\theta \\in \\Sigma $ , so $\\Lambda \\subset \\ker (\\varrho )$ .", "Since $\\mathbb {Z}^\\Sigma /\\Lambda $ and $\\Psi $ each have order $\\prod _{{\\mathfrak {p}}\\in S_p}(p^{f_{\\mathfrak {p}}} - 1)$ , it follows that $\\Lambda = \\ker (\\varrho ) = \\left\\lbrace \\,\\sum k_\\theta {\\mathbf {e}}_\\theta \\,\\,\\left\|\\,\\, \\sum _{i=1}^{f_{\\mathfrak {p}}} \\sum _{j=1}^{e_{\\mathfrak {p}}} k_{\\theta _{{\\mathfrak {p}},i,j}} p^i \\equiv 1 \\bmod (p^{f_{\\mathfrak {p}}} - 1)\\,\\,\\forall {\\mathfrak {p}}\\in S_p\\,\\right.\\right\\rbrace .$ Remark 8.3.1 Recall that the $G_\\theta $ , and hence $G^{\\prime }_\\theta $ , are invertible in $M^{\\operatorname{tot}}(U;\\mathbb {F})$ , so if $\\mathbf {l}^{\\prime } - \\mathbf {l}= \\sum _\\theta m_\\theta \\mathbf {h}_\\theta \\in \\Lambda $ , then multiplication by $\\prod _{\\theta } G^{\\prime m_\\theta }_\\theta $ defines an isomorphism $M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) \\stackrel{\\sim }{\\rightarrow } M_{\\mathbf {k},\\mathbf {l}^{\\prime }}(U;\\mathbb {F})$ .", "We may therefore write $R_U$ as the quotient of $\\displaystyle \\bigoplus _{\\mathbf {k}\\in \\mathbb {Z}^\\Sigma ,\\psi \\in \\Psi } M_{\\mathbf {k},\\psi }(U;\\mathbb {F})$ by the ideal $\\langle \\,H^{\\prime }_\\theta - 1 \\,\\rangle _{\\theta \\in \\Sigma }$ , where $M_{\\mathbf {k},\\varrho (\\mathbf {l})}(U;\\mathbb {F})$ is canonically isomorphic to $M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F})$ for each $\\mathbf {l}\\in \\mathbb {Z}^\\Sigma $ .", "Furthermore the main result of [10] immediately implies the natural map $\\bigoplus _{\\mathbf {k}\\in \\Xi ^{\\min }, \\psi \\in \\Psi } M_{\\mathbf {k},\\psi }(U;\\mathbb {F}) \\longrightarrow R_U$ is surjective, so we may also replace $\\mathbb {Z}^\\Sigma $ by the submonoid $\\Xi ^{\\min } + \\Lambda $ as the index set for $\\mathbf {k}$ in the definition of $R_U$ .", "We will now describe the ideal ${\\mathfrak {I}}$ in terms of $q$ -expansions.", "For each cusp ${\\mathcal {C}}\\in Y_U^\\infty $ we choose a representative $(H,I,[\\lambda ],[\\eta ])$ , and for each $\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma $ , we let $\\overline{D}_{\\mathbf {k},\\mathbf {l}}^{\\mathcal {C}}$ denote the one-dimensional vector space $\\mathbb {F}\\otimes _{{\\mathcal {O}}} D_{\\mathbf {k},\\mathbf {l}}$ over $\\mathbb {F}$ (where $D_{\\mathbf {k},\\mathbf {l}}$ is defined by (REF )).", "We then let $\\overline{D}_{\\operatorname{tot}}^{{\\mathcal {C}}}$ denote $\\bigoplus _{\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma } \\overline{D}_{\\mathbf {k},\\mathbf {l}}$ with its natural $\\mathbb {F}$ -algebra structure.", "For each $\\theta \\in \\Sigma $ , let $c_\\theta \\in \\overline{D}^{{\\mathcal {C}}}_{\\mathbf {h}_\\theta ,\\bf {0}}$ denote the (constant) $q$ -expansion of $H_\\theta ^{\\prime }$ at ${\\mathcal {C}}$ .", "(Recall that the $q$ -expansions of $H_\\theta $ were explicitly described in §REF , from which one gets a description of $c_\\theta $ by multiplying by $\\overline{\\theta }(b_{\\mathfrak {p}})$ if $\\theta = \\theta _{{\\mathfrak {p}},i,1}$ .)", "Similarly let $d_\\theta \\in \\overline{D}_{\\bf {0},\\mathbf {h}_\\theta }^{\\mathcal {C}}$ denote the (constant) $q$ -expansion of $G^{\\prime }_\\theta $ at ${\\mathcal {C}}$ , and define ${\\mathfrak {I}}^{{\\mathcal {C}}}$ to be the ideal $\\langle \\, c_\\theta -1,d_\\theta -1\\,\\rangle _{\\theta \\in \\Sigma }$ of $\\overline{D}_{\\operatorname{tot}}^{{\\mathcal {C}}}$ .", "We may then view the quotient $\\overline{D}_{\\operatorname{tot}}^{{\\mathcal {C}}} / {\\mathfrak {I}}^{\\mathcal {C}}$ as the space of $\\Lambda ^2$ -coinvariants of the free $\\mathbb {F}[(\\mathbb {Z}^\\Sigma )^2]$ -module $\\overline{D}_{\\operatorname{tot}}^{{\\mathcal {C}}}$ and decompose $\\overline{D}_{\\operatorname{tot}}^{{\\mathcal {C}}} / {\\mathfrak {I}}^{\\mathcal {C}}= \\bigoplus _{\\chi ,\\psi \\in \\Psi } \\overline{D}^{\\mathcal {C}}_{\\chi ,\\psi },$ so that the natural projection map $\\overline{D}_{\\mathbf {k},\\mathbf {l}}^{{\\mathcal {C}}} \\rightarrow \\overline{D}^{\\mathcal {C}}_{\\varrho (\\mathbf {k}),\\varrho (\\mathbf {l})}$ is an isomorphism for all $\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma $ .", "Now observe that the collection of $q$ -expansion maps $M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) \\rightarrow \\overline{D}_{\\mathbf {k},\\mathbf {l}}^{{\\mathcal {C}}} \\otimes _\\mathbb {F}\\mathbb {F}[[q^m]]_{m \\in N^{-1}M^{\\mathcal {C}}_+ \\cup \\lbrace 0\\rbrace }$ (where $M^{\\mathcal {C}}= {\\mathfrak {d}}^{-1}I^{-1}J$ and $U(N) \\subset U$ ) induces an $\\mathbb {F}$ -algebra homomorphism $\\overline{q}: M_{\\operatorname{tot}}(U;\\mathbb {F}) \\longrightarrow \\bigoplus _{{\\mathcal {C}}\\in Y_U^\\infty } \\bigoplus _{\\chi ,\\psi \\in \\Psi }\\overline{D}_{\\chi ,\\psi }^{{\\mathcal {C}}} \\otimes _\\mathbb {F}\\mathbb {F}[[q^m]]_{m \\in N^{-1}M^{\\mathcal {C}}_+ \\cup \\lbrace 0\\rbrace }.$ Lemma 8.3.2 The kernel of $\\overline{q}$ is ${\\mathfrak {I}}$ .", "Proof.The inclusion ${\\mathfrak {I}}\\subset \\ker (\\overline{q})$ is clear from the definitions.", "Suppose then that $\\overline{q}(f) = 0$ and write $f = \\sum _{\\mathbf {k},\\mathbf {l}\\in W} f_{\\mathbf {k},\\mathbf {l}}$ for some finite subset $W$ of $\\mathbb {Z}^\\Sigma $ .", "For each $\\chi \\in \\Psi $ , choose $\\mathbf {k}_\\chi \\in \\varrho ^{-1}(\\chi )$ sufficiently large that $\\mathbf {k}\\le _{{\\mathrm {Ha}}} \\mathbf {k}_\\chi $ for all $\\mathbf {k}\\in \\varrho ^{-1}(\\chi ) \\cap W$ .", "Thus for each $\\mathbf {k}\\in W$ , there is a unique $\\mathbf {_}{\\mathbf {k}} = \\sum _{\\theta } m_{\\mathbf {k},\\theta } {\\mathbf {e}}_\\theta \\in \\mathbb {Z}^\\Sigma _{\\ge 0}$ such that $\\mathbf {k}_{\\varrho (\\mathbf {k})} = \\mathbf {k}+ \\sum _{\\theta } m_{\\mathbf {k},\\theta } \\mathbf {h}_\\theta $ .", "Now note that $ g := \\sum _{\\mathbf {k},\\mathbf {l}\\in W} H^{\\prime \\mathbf {_}{\\mathbf {k}}} G^{\\prime \\mathbf {_}{\\mathbf {l}}} f_{\\mathbf {k},\\mathbf {l}} \\in \\bigoplus _{\\chi ,\\psi \\in \\Psi } M_{\\mathbf {k}_\\chi ,\\mathbf {k}_\\psi }(U;\\mathbb {F})$ and that $f-g \\in {\\mathfrak {I}}$ (where $H^{\\prime \\mathbf {_}\\mathbf {k}} = \\prod _\\theta H_\\theta ^{\\prime m_{\\mathbf {k},\\theta }}$ and $G^{\\prime \\mathbf {_}\\mathbf {l}} = \\prod _\\theta G_\\theta ^{\\prime m_{\\mathbf {l},\\theta }}$ ).", "Since ${\\mathfrak {I}}\\subset \\ker (\\overline{q})$ , it follows that $\\overline{q}(g) = 0$ .", "However $\\overline{q}$ restricts to the $q$ -expansion map $\\bigoplus _{\\chi ,\\psi \\in \\Psi } M_{\\mathbf {k}_\\chi ,\\mathbf {k}_\\psi }(U;\\mathbb {F}) \\longrightarrow \\bigoplus _{{\\mathcal {C}}\\in Y_U^\\infty } \\bigoplus _{\\chi ,\\psi \\in \\Psi }\\overline{D}_{\\chi ,\\psi }^{{\\mathcal {C}}} \\otimes _\\mathbb {F}\\mathbb {F}[[q^m]]_{m \\in N^{-1}M^{\\mathcal {C}}_+ \\cup \\lbrace 0\\rbrace },$ which is injective by Proposition REF , so $g = 0$ , and hence $f = f-g \\in {\\mathfrak {I}}$ .", "$\\square $ We also extract the following observation from the proof of the lemma: Lemma 8.3.3 If $W \\subset (\\mathbb {Z}^\\Sigma )^2$ is such that $(\\varrho ,\\varrho ):W \\rightarrow \\Psi ^2$ is injective, then $ {\\mathfrak {I}}\\cap \\bigoplus _{(\\mathbf {k},\\mathbf {l}) \\in W} M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) = 0.$ We are now ready to interpret the description of the kernel of the partial $\\Theta $ -operator in terms of the algebra $R_U$ .", "Theorem 8.3.4 If ${\\mathfrak {p}}\\in S_p$ , $\\tau \\in \\Sigma _{{\\mathfrak {p}},0}$ and $U$ is any sufficiently small open compact subgroup of $\\operatorname{GL}_2(\\mathbb {A}_{F,) containing \\operatorname{GL}_2({\\mathcal {O}}_{F,p}), then the sequence0 \\longrightarrow R_U \\stackrel{V_{\\mathfrak {p}}}{\\longrightarrow } R_U \\stackrel{\\Theta _\\tau }{\\longrightarrow } R_Uis exact.", "}{\\bf Proof.", "}\\hspace{5.0pt}The injectivity of $ Vp: RU RU$ follows\\footnote {Alternatively, one canappeal to Lemma~\\ref {lem:qbar2} instead of Lemma~\\ref {lem:qbar} and argue similarly to the forthcomingproof of the other exactness assertion.}", "from Lemma~\\ref {lem:qbar} and the commutativity of the diagram$$\\begin{array}{ccc} M_{\\operatorname{tot}}(U;\\mathbb {F})& \\stackrel{\\overline{q}^{\\prime }}{\\longrightarrow }& \\displaystyle {\\bigoplus _{{\\mathcal {C}}^{\\prime } \\in Y_U^\\infty } \\bigoplus _{\\chi ,\\psi \\in \\Psi }}\\overline{D}_{\\chi ,\\psi }^{{\\mathcal {C}}^{\\prime }} \\otimes _\\mathbb {F}\\mathbb {F}[[q^m]]_{m \\in N^{-1}M^{{\\mathcal {C}}^{\\prime }}_+ \\cup \\lbrace 0\\rbrace }\\\\{\\scriptstyle {V_{\\mathfrak {p}}}}\\downarrow && \\downarrow \\\\M_{\\operatorname{tot}}(U;\\mathbb {F})& \\stackrel{\\overline{q}}{\\longrightarrow }&\\displaystyle { \\bigoplus _{{\\mathcal {C}}\\in Y_U^\\infty } \\bigoplus _{\\chi ,\\psi \\in \\Psi }}\\overline{D}_{\\chi ,\\psi }^{{\\mathcal {C}}} \\otimes _\\mathbb {F}\\mathbb {F}[[q^m]]_{m \\in N^{-1}M^{{\\mathcal {C}}}_+ \\cup \\lbrace 0\\rbrace },\\end{array}$$where the right downward arrow is the direct sum over $ (C,,)$ of the tensor product ofthe isomorphism $D,C' D,C$ induced by(\\ref {eqn:Phibres}) and the injective $ F$-algebra homomorphism (\\ref {eqn:Phat}).", "(Note that$ ((k),(l)) = ((k”),(l”))$, and recall that the data $ (H,I,[],[])$representing each cusp of $ YU$ were implicitly chosen differently for the source and target of$ p$ to simplify the resulting description of $ Vp$ on $ q$-expansions in §\\ref {sec:V2}.", ")$ Since $\\Theta _\\tau \\circ V_{\\mathfrak {p}}= 0$ , it just remains to prove that $\\ker (\\Theta _\\tau ) \\subset {\\operatorname{im}}(V_{\\mathfrak {p}})$ .", "Suppose then that $f \\in M_{\\operatorname{tot}}(U;\\mathbb {F})$ is such that $\\Theta _\\tau (f) \\in {\\mathfrak {I}}$ .", "As in the proof of Lemma REF , write $f = \\sum _{\\mathbf {k},\\mathbf {l}\\in W} f_{\\mathbf {k},\\mathbf {l}}$ for some finite subset $W$ of $\\mathbb {Z}^\\Sigma $ , and for each $\\chi \\in \\Psi $ choose $\\mathbf {k}_\\chi \\in \\varrho ^{-1}(\\chi )$ so that $\\mathbf {k}\\le _{{\\mathrm {Ha}}} \\mathbf {k}_\\chi $ for all $\\mathbf {k}\\in \\varrho ^{-1}(\\chi ) \\cap W$ , and consider $ g := \\sum _{\\mathbf {k},\\mathbf {l}\\in W} H^{\\prime \\mathbf {_}{\\mathbf {k}}} G^{\\prime \\mathbf {_}{\\mathbf {l}}} f_{\\mathbf {k},\\mathbf {l}} \\in \\bigoplus _{\\chi ,\\psi \\in \\Psi } M_{\\mathbf {k}_\\chi ,\\mathbf {k}_\\psi }(U;\\mathbb {F}).$ Since $f-g \\in {\\mathfrak {I}}$ and $\\Theta _\\tau (f) \\in {\\mathfrak {I}}$ , we have $\\Theta _\\tau (g) \\in {\\mathfrak {I}}$ .", "Note however that $\\Theta _\\tau (g) \\in \\bigoplus _{\\chi ,\\psi \\in \\Psi } M_{\\mathbf {k}_\\chi ^{\\prime },\\mathbf {l}_\\psi ^{\\prime }}(U;\\mathbb {F})$ where $\\mathbf {k}_\\chi ^{\\prime } = \\mathbf {k}_\\chi + n_{\\theta _0}{\\mathbf {e}}_{\\sigma ^{-1}\\theta _0} + {\\mathbf {e}}_{\\theta _0}$ and $\\mathbf {l}_\\psi ^{\\prime } = \\mathbf {k}_\\psi - {\\mathbf {e}}_{\\theta _0}$ , so Lemma REF implies that $\\Theta _\\tau (g) = 0$ .", "Writing $g = \\sum _{\\chi ,\\psi } g_{\\chi ,\\psi }$ with $g_{\\chi ,\\psi } \\in M_{\\mathbf {k}_\\chi ,\\mathbf {k}_\\psi }$ , this means that $\\Theta _\\tau (g_{\\chi ,\\psi }) = 0$ for all $\\chi ,\\psi \\in \\Psi $ .", "Corollary REF now implies that for each $\\chi ,\\psi \\in \\Psi $ , we have $g_{\\chi ,\\psi } = G^{\\prime \\mathbf {}H^{\\prime \\mathbf {n}} V_{\\mathfrak {p}}(h_{\\chi ,\\psi })for some h_{\\chi ,\\psi } \\in M_{\\mathbf {k}_0,\\mathbf {l}_0}(U;\\mathbb {F}), where \\mathbf {k}_0,\\mathbf {l}_0,\\mathbf {\\in }\\mathbb {Z}^\\Sigma and \\mathbf {n}\\in \\mathbb {Z}^\\Sigma _{\\ge 0}depend on \\chi and \\psi .", "It follows that g_{\\chi ,\\psi } - V_{\\mathfrak {p}}(h_{\\chi ,\\psi }) \\in {\\mathfrak {I}}, so setting h = \\sum _{\\chi ,\\psi } h_{\\chi ,\\psi },we conclude that f - V_{\\mathfrak {p}}(h) = (f-g) + (g-V_{\\mathfrak {p}}(h)) \\in {\\mathfrak {I}}, as required.", "{\\mbox{}\\hfil \\mbox{$\\square $} }\\vspace{5.0pt}}$ Finally consider $R = \\varinjlim _U R_U$ , where the direct limit is over all sufficiently small open compact subgroups $U$ containing $\\operatorname{GL}_2({\\mathcal {O}}_{F,p})$ .", "(Note that this is the same as the quotient of $M_{\\operatorname{tot}}(\\mathbb {F}) := \\varinjlim _U M_{\\operatorname{tot}}(U;\\mathbb {F}) \\cong \\bigoplus _{\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma } \\left(\\varinjlim _U M_{\\mathbf {k},\\mathbf {l}}(U;\\mathbb {F}) \\right)= \\bigoplus _{\\mathbf {k},\\mathbf {l}\\in \\mathbb {Z}^\\Sigma } M_{\\mathbf {k},\\mathbf {l}}(\\mathbb {F})$ by the ideal $\\langle \\,H_\\theta ^{\\prime } - 1, G_\\theta ^{\\prime }-1\\,\\rangle _{\\theta \\in \\Sigma }$ .)", "Since the maps $V_{\\mathfrak {p}}$ and $\\Theta _\\tau $ are Hecke-equivariant in the obvious sense, we obtain an $\\mathbb {F}$ -algebra endomorphism $V_{{\\mathfrak {p}}}$ of $R$ , and an $\\mathbb {F}$ -linear derivation $\\Theta _\\tau $ on $R$ , each of which is $\\operatorname{GL}_2({\\mathbb {A}}_{F,^{(p)})-equivariant, andsuch that the sequence0 \\longrightarrow R \\stackrel{V_{\\mathfrak {p}}}{\\longrightarrow } R \\stackrel{\\Theta _\\tau }{\\longrightarrow } Ris exact.", "}\\bibliographystyle {abbrv}$" ] ]	2011.14128
[ [ "A Data-Driven Study of Commonsense Knowledge using the ConceptNet\n Knowledge Base" ], [ "Abstract Acquiring commonsense knowledge and reasoning is recognized as an important frontier in achieving general Artificial Intelligence (AI).", "Recent research in the Natural Language Processing (NLP) community has demonstrated significant progress in this problem setting.", "Despite this progress, which is mainly on multiple-choice question answering tasks in limited settings, there is still a lack of understanding (especially at scale) of the nature of commonsense knowledge itself.", "In this paper, we propose and conduct a systematic study to enable a deeper understanding of commonsense knowledge by doing an empirical and structural analysis of the ConceptNet knowledge base.", "ConceptNet is a freely available knowledge base containing millions of commonsense assertions presented in natural language.", "Detailed experimental results on three carefully designed research questions, using state-of-the-art unsupervised graph representation learning ('embedding') and clustering techniques, reveal deep substructures in ConceptNet relations, allowing us to make data-driven and computational claims about the meaning of phenomena such as 'context' that are traditionally discussed only in qualitative terms.", "Furthermore, our methodology provides a case study in how to use data-science and computational methodologies for understanding the nature of an everyday (yet complex) psychological phenomenon that is an essential feature of human intelligence." ], [ "Introduction", "Despite the ubiquity of intelligent agents such as Alexa and Siri in modern life, these agents have yet to capture the human element in natural conversations.", "Even with advances in Natural Language Processing (NLP), deep learning, and knowledge graphs [1], [2], such agents are merely good at answering questions with a clear structure and `factual need' (e.g.,`What is the weather today in Los Angeles?').", "They are still not capable of answering questions with incomplete information or under-specified needs (such as `Should I put my spare change in a piggy bank?')", "that require more contextual and implicit knowledge that humans often take for granted when navigating daily situations.", "Among other things, such agents are limited in their commonsense reasoning abilities.", "Commonsense reasoning is the process that involves processing information about a scenario in the world, and making inferences and decisions by using not only the `explicit' information available to the senses and conscious mind, but also context and implicit information that is based on our `commonsense knowledge'.", "Commonsense knowledge is difficult to define precisely (not unlike `intelligence') but it may be assumed to be a broad body of knowledge of how the `world' works [3].", "Generally, such knowledge is essential for navigating social situations and interactions, physical reasoning (e.g., the simple knowledge that when an object on the table is `picked up', it is not on the table anymore) and (more controversially), knowledge that may rely on local culture and milieu [4], [5].", "Commonsense knowledge and reasoning have both been recognized as essential for building more advanced `general' AI systems that have human-like capabilities and reasoning ability, even when facing uncertain, implicit (or potentially contradictory) information.", "Recognizing its importance, researchers in several communities have increasingly engaged in researching and evaluating commonsense reasoning on tasks pertaining to question answering and abductive reasoning [6], [7], [4], [8].", "Figure: A fragment of the ConceptNet Knowledge Graph (KG).Knowledge in ConceptNet is represented as `triples' or 3-tuples, with each triple containing two entities connected by a relation.", "This fragment contains 21 triples (entities are shown as circles and relations are directed, labeled edges).", "Unlike certain other KGs (primarily in the Semantic Web community, such as DBpedia), typical versions of ConceptNet do not syntactically distinguish between an entity and the attribute of an entity (e.g., `democratic').ConceptNethttps://conceptnet.io/.", "is a large-scale, freely available knowledge base that describes commonsense knowledge as a set of assertions [9].", "It is designed to represent the common knowledge needed to help machines better understand the meaning of words and phrases people use.", "The graph structure that represents knowledge in ConceptNet in patterns of connected word/phrase nodes is particularly useful for textual reasoning over natural language documents.", "An example of how such knowledge is organized in ConceptNet is illustrated in Figure REF .", "ConceptNet originated from the Open Mind Common Sense [10] project, itself launched in 1999 at the MIT Media Lab.", "It is continuously updated to include new knowledge from crowdsourced resources, expert-created resources, and games designed specificallyDescribed on the website as `games with a purpose'.", "to elicit certain kinds of commonsense annotations, such as knowledge about people's intuitive word associations.", "ConceptNet has some notable differences from other popular knowledge bases.", "In comparison with WordNet [11], which focuses on preserving lexicographic information and the relationship between words and their senses, ConceptNet maintains a semantic network structure that is designed to capture commonsense assertions, such as shown in Figure REF .", "In particular, ConceptNet contains more kinds of relationships than WordNet.", "Another comparable knowledge base is Cyc [5], which focuses on standardization of commonsense for efficient logical reasoning.", "However, ConceptNet is optimized for making inferences over natural language texts, and unlike Cyc, is not proprietary.", "ConceptNet captures a wide range of commonsense knowledge (such as in Cyc); however, it expresses knowledge in an easy-to-use format (i.e.", "as sets of 3-tuple assertions, such as illustrated in Figure REF ) rather than higher-order logic notation.", "Because of these advantages, ConceptNet has rapidly emerged as a practical dataset and resource for various kinds of machine learning and NLP in the last ten years in particular [12], [13], [14].", "While the utility of ConceptNet as a resource in algorithmic pipelines has been demonstrated, we hypothesize that it can be used as a fundamental resource to understand the nature of commonsense knowledge more precisely.", "Note that we are not trying to reverse-engineer cognitive processes in the human brain that led to the ingestion and formation of such knowledge, but rather, to yield key findings into some central properties of the knowledge itself.", "Despite being blended into diverse aspects of daily life, we are not aware of a systematic explanation about the nature of commonsense as a `belief system about the world' that informs most adults' everyday thoughts, behaviors and actions [15].", "Researchers have also differed on ideas concerning commonsense knowledge and reasoning [16], [17].", "Therefore, there is strong motivation to gain new insights into commonsense by using a systematic methodology, and easily available and trustworthy data.", "In this article, we take up this challenge both by presenting a methodology and by showing how the methodology yields useful and novel results when applied to relations and entities in ConceptNet.", "Specific contributions include: * We embed nodes and relations in ConceptNet into a continuous, low-dimension embedding space by a state-of-art graph embedding technique PyTorch-BigGraph(PBG) [18] to enable more robust computational analysis in vector space.", "Using both information theory and standard correlational analysis (as well as a judicious set of visualizations), we show that the embeddings reveal some interesting aspects of ConceptNet relations and their similarities to each other.", "Relations in ConceptNet are clearly not independent from one another and share unique degrees of overlap and information content.", "* For specific high-volume and coarse-grained relations (such as `HasContextDefined by ConceptNet as `A is a word used in the context of B, which could be a topic area, technical field, or regional dialect'.", "', `SymbolOf`A symbolically represents B'.", "', and `FormOf`A is an inflected form of B; B is the root word of A'.'", "that are known to be significant in commonsense reasoning, we discover and explore substructures within these relations by using a combination of clustering and qualitative analysis.", "* We shed light into another important phenomenon (`negation') by contrast the triple structures between an important relation `Desires' and its negation `NotDesires'.", "In commonsense knowledge, we show that negation does not obey logical intuitions, and that in the embedding space, triples with the negated relation are virtually undistinguishable from triples with the non-negated relation.", "However, we find that a machine learning model can be trained to recognize these differences.", "This finding reveals the nature of “negation” relation in ConceptNet and the structure properties between a relation and its negation.", "The rest of this article is structured as follows.", "In Section 2, we describe relevant lines of research related to this work.", "In Section 3, we present our experimental study and methodology.", "Section 4 summarizes and discusses the key findings, and the article concludes in Section 5.", "Although both Wikipedia and the (relatively) recent DARPA Machine Common Sense (MCS) programhttps://www.darpa.mil/program/machine-common-sense; https://en.wikipedia.org/wiki/Common_sense.", "define commonsense reasoning as `Äúthe basic ability to perceive, understand, and judge things that are shared by nearly all people and can be reasonably expected of nearly all people without need for debate'Äù, there is no official, sufficiently broad definition that we can cite outside of the psychology context.", "Within psychology, [19] has defined commonsense as `the system of implications shared by the competent users of a language'.", "One important commonality that is shared, no matter the definition or field considered, is that commonsense knowledge is (at least to a degree) implicit.", "It is also not controversial that commonsense knowledge is key to humans being able to interact with the world (and other humans) around them without needing extensive training for every situation, as machines seem to require (e.g., face recognition or information extraction).", "Despite its seeming simplicity, commonsense reasoning in machines involves several research issues, including but not limited to the representation of given scenario, the representation of commonsense knowledge, context-sensitive effects of events and the representation (and reasoning) of concurrent events [3].", "There is very little work on the nature of commonsense knowledge: one rare example is a recent book [20] attempts to provide a `theory' of commonsense by breaking down commonsense knowledge into various categories, and present formalisms for those categories.", "Other similar work along those lines include [21], [22], [23].", "Unlike those works, we take an inductive, data-driven approach.", "Our work is potentially complementary to [20], since some of our findings may be used in the future to provide evidence for (or against) some of their purely theoretical claims.", "Progress has been achieved in specific kinds of commonsense reasoning, especially in reasoning about time and internal relations [24], [25], reasoning about actions and change [26], and the sign calculus [6].", "Semantics have played an important role in some of these successes [27]; in fact, ConceptNet itself has been described as a `semantic network' [28].", "WordNet [11] and Cyc [5] are two useful resources that allow us, in different ways, to obtain more insight into the semantics of sentences.", "WordNet is a semantic network whose entries are organized in terms of semantic inter-relationships.", "The easy-to-use network structure lends it to being frequently applied in multiple reasoning systems [29], [30], [31].", "Cyc focuses on designing a universal schema (a higher-order logic) to represent commonsense assertions, which also supports reasoning systems [32], [33] conduct richer logic inferences.", "The increasing enrichment of commonsense knowledge graphs fits with one of the approaches proposed in the aforementioned DARPA MCS program for developing flexible machine commonsense services.", "Commonsense reasoning systems are typically measured against benchmark tests, some well known examples of such datasets including aNLI [34], HellaSwag [35] and Physical IQA [36].", "The other approach strongly supported by DARPA is to develop systems that think and learn as humans do in the very early stages (the first 18 months).", "Performers in that task aim to develop systems that can provide empirical and theoretical guidance on various aspects of commonsense reasoning abilities in humans by mimicking the core domains of child cognition for objects (intuitive physics), agents (intentional actors), and places (spatial navigation).", "Our work is more strongly related to the former approach, since ConceptNet primarily comprises assertions of commonsense knowledge that adults, and not necessarily children, would be expected to have.", "Potentially, our methodology could be applied to other knowledge graphs that are more representative of the knowledge that children have, but we are not aware of any such knowledge graphs currently in existence." ], [ "Knowledge Graphs", "A knowledge graph (KG), of which ConceptNet is an example, is a graph-theoretic way to represent knowledge modeled according to an ontology, typically with the help of subject-matter experts, automatic data extraction, and machine learning algorithms.Rapid advances in KG construction and application have been witnessed in recent years.", "KGs such as Freebase [37], WordNet [11], DBpedia [38], and Yago [39] have been successfully created and applied to a range of tasks [40], [41], [42].", "Using these KGs, machine learning and representation leaning algorithms have also advanced with every passing year, in areas as wide ranging as knowledge graph `embeddings', question answering and distant supervision [43], [44], [45], [46].", "Concerning commonsense reasoning, ConceptNet and Cyc [5] (mentioned in the introduction as a proprietary KG that precedes ConceptNet) are the only two examples that we are aware of.", "Owned by Cycorp Inc., Cyc's knowledge base claimshttps://www.cyc.com/.", "to be the `broadest, deepest, and most complete repository ever developed', but given its proprietary nature, the claim is difficult to validate.", "It is not known how similar Cyc is to ConceptNet, but potentially, the same methodology proposed in this paper could be applied to Cyc to replicate, strengthen or refute some of our findings.", "In contrast, ConceptNet is a freely available multilingual KG that connects everyday entities using a rich set of relations [47].", "As mentioned in the introduction, ConceptNet serves as an important background resource for commonsense reasoning and question answering.", "However, it has not been studied directly for gaining insights into commonsense knowledge, even though there is precedent.", "For example, studies on DBpedia and YAGO have been conducted specifically to understand their relational structure and the structural properties of the encyclopedic knowledge that these KGs are known for [48], [49], [50].", "We attempt to do the same, but with commonsense knowledge as the focus.", "Another line of work highly related to this one is representation learning, especially as they pertain to KGs.", "Such algorithms take as input the KG, including entities and relations, and embed them into continuous vector spaces, as surveyed by [51], models such as RESCAL [52], TransE [53], TransH [54], KG2E [55], and RotateE [56] all achieve good performance on various tasks, such as KG completion [53] and relation extraction [57], which allows for further improvement of the KG.", "In this article, we use a graph embedding package that builds on the ones above (especially TransE) and is especially designed for graphs with many millions of triples." ], [ "Empirical Study", "Our guiding principle in this paper is that a commonsense KG such as ConceptNet could be used as the basis for understanding the nature of commonsense knowledge and (potentially) commonsense reasoning.", "Since this is a broad area of study, we define a specific set of research questions that could be used to shed more light on this subject matter.", "In this section, we detail critical aspects of the ConceptNet dataset and methodology that are relevant for our empirical investigations.", "Subsequently, we also investigate each question as separate but related studies, with accompanying descriptions of methodology, results, discussion and implications.", "Towards the end of the section, we collect these findings and synthesize them into a cohesive set of insights." ], [ "Preliminaries and Setup", "Conceptually, ConceptNet may be described as a multi-relational graph $G = (V, R, E)$ , where V is the set of entities or nodes, R is the set of 49 relations and E is the set of triples or edges where each triple $e = (h, r, t) \\in E \\subseteq V \\times R \\times V$ .", "$h$ and $t$ are respectively referred to as the source (or the head) and the destination (or the tail) entity, and $r$ is the relation.", "For the purposes of maintaining consistent terminology, we use the terms triple, head, tail, and relation to refer to the concepts $e$ , $h$ , $t$ and $r$ respectively.", "Furthermore, these head and tail entities are collectively referred to as entities rather than nodes.", "Note that $G$ may be thought of in a non graph-theoretic way as a set of triples as has been common in the natural language community when discussing these datasets as knowledge bases rather than (multi-relational) knowledge graphs (where triples are usually interpreted as directed edges [42]).", "The distinction is not relevant for the purposes of this paper, but we do note that it is more natural to think about ConceptNet as a graph due to its structural properties.", "Earlier, Figure REF expressed a fragment of ConceptNet as a multi-relational graph (with 21 triples).", "Entities and relations are projected into a continuous vector space by PyTorch-BigGraph (PBG), an efficient and state-of-the-art embedding system for large-scale graphsIn particular, PBG supports multi-entity and multi-relation graph embeddings, unlike network embeddings such as DeepWalk and node2vec [58], [59], and its embedding quality has been found to be comparable with (or even exceed) existing `knowledge graph' embedding systems, evaluated on the Freebase [37], LiveJournal [60] and YouTube graphs [61]., for further computation and analysis [18].", "Next, we describe the raw data and our sampling mechanism for studying the raw data, as well as the setup and training of PBG.", "In our empirical study, we use the latest version of ConceptNet (version 5.7) that was available at the time of writing.", "One important aspect of ConceptNet is the ratio of the number of unique entities to the total number of triples, which is much higher in ConceptNet (0.833) than in other similarly-sized KGs such as Freebase (0.055) or WordNetFor the interested reader, we are specifically referring to the FB15k and WN18 datasets, as designated in multiple papers on knowledge graph embeddings [62], [63], [64].", "(0.289).", "Additionally, while ConceptNet contains far more entities than the other KGs, it contains fewer unique relations (the `labels' on the edges, or the size of the set $R$ ) than the other KGs.", "In practice, these significant deviations in terms of basic entity/relation statistics can cause problems for KG representation learning packages (even those designed for large-scale KGs) including PBG, which is used in this article for experiments.", "In embedding the full ConceptNet knowledge base, we found that, during the training process, the PBG algorithm fails with a `bus error' message if the number of input triples exceeds 4 million.", "This seems to be occurring due to the algorithm running out of shared memory (despite the fact that we execute the algorithm on a machine with 61 GB memory).", "Table: The number of triples, entities (including separate head and tail entity counts, and their overlap) in the sample of ConceptNet considered in this study.Table: The numbers of triples and entities corresponding to each of the 49 relations in the ConceptNet sample studied in this article.", "Note that an entity could occur in multiple triples, each with a different relation.", "The `RelatedTo' relation, shown in bold, has the most number of triples and entities associated with it.To address the memory issue and ensure that our results can be extended or replicated in the future using reasonable computation resources, we randomly sampled 4 million triples from ConceptNet for all studies described herein.", "Key statistics are tabulated in Table REF .", "The head entities-set is twice the size of the tail entities-set and their overlap is approximately 1/20 of the total entities.", "Cursory analysis also showed that the head entity `/c/en/person' and tail entity `/c/fr/francais' were respectively found to have the most triples associated with them.", "Other relation-specific statistics are tabulated in Table REF .", "`/r/RelatedTo' was found to be the most frequent relation, occurring in more than 1 million triples.", "We input these 4 million triples into the PBG algorithm for representation learning.", "We partition the 4 million sampled triples into train, validation and test splits, containing 75%, 12.5%, 12.5% of the total triples, respectively.", "Before doing the sampling, we also remove triples with the `ExternalURL' relation.", "ExternalURL is a `non-semantic' relation that is only referring to a URL identifier and cannot be used for structural analysis of the kind proposed in this paper.", "We conduct training on a single server in the Amazon cloud with 4 Intel Xeon cores (one socket) and 61 GB of RAM.", "At the end of training, the algorithm outputs a vector for every unique relation and entity in the training dataset." ], [ "Validating Quality of Embeddings", "As mentioned earlier, due to the limits on the memory requirements of PBG, we randomly sampled 4 million triples from the raw ConceptNet knowledge base for representation learning and for the subsequently described studies.", "Due to the sampling, a reasonable suspicion arises as to whether the quality of the learned representations or embeddings can be trusted.", "To validate the quality of these embeddings, we designed a series of quantitative measures to analyze the effectiveness of the PBG outputs.", "Specifically, having obtained the embeddings output by PBG, we first compute a centroid vector for each relation.", "Recall that we denoted the graph using the symbol $G=(V,R,E)$ , where $E$ was the set of triples or `edges' in the graph.", "In a slight abuse of notation, we use the symbol $G_E$ to represent the set $E$ associated with $G$ .", "Given a relation $r \\in R$ , let $G_r \\subseteq G_E$ be the subset of triples in $G_E$ that have relation $r$ .", "For each such triple $(h,r,t)$ in $G_r$ , we define the translation vector $\\vec{v} = \\vec{t} - \\vec{h}$ , where $t$ and $h$ are the embeddings learned by the (previously described) PBG algorithm for entities $t$ and $h$ respectively.", "The `centroid' of $r$ is then simply the mean of the translation vectors in $G_r$ .", "Formally, the centroid vector $\\vec{r_c}$ of $r$ is given by the equation below: $\\vec{r_c} = \\frac{1}{\|G_r\|}\\sum _{(h,r,t) \\in G_r}(\\vec{t} - \\vec{h})$ Note that this yields two distinct vectors for $r$ : the vector directly output by the graph embedding (denoted as $\\vec{r}$ ) and the centroid vector $\\vec{r_c}$ .", "We use the symbol $\\mathcal {R}$ to indicate the set of direct embeddings for all 49 relations and the symbol $\\mathcal {R}_c$ to indicate the set of centroid vectors.", "The latter is expected to be more robust and interpretable since it is directly derived using translation (between the tail and head entity embeddings, averaged over all the triples where it occurs).", "With this technical machinery in place, we validate our 4 million-triples sample as follows.", "First, we calculate two similarity lists, $SL_r$ and $SL_r^{^{\\prime }}$ , per relation, using each of these two notions of embedding $r$ .", "$SL_r$ denotes the list containing the cosine similarities between $\\vec{r}$ and each translation vector in $G_r$ (since in the general case, each triple could yield a different translation vector, even with $r$ fixed), while $SL_r^{^{\\prime }}$ denotes the list of cosine similarities between the centroid vector $\\vec{r_c}$ and each translation vectorSince there is only one $\\vec{r_c}$ and relation vector $r$ per relation, the two lists are aligned (by virtue of using the same set of translation vectors computed over the triples in $G_r$ )..", "Given these two per-relation lists and using established statistical and information-theoretic concepts such as Spearman's rank correlation and the Kullback-Leibler (KL) Divergence, we show in the subsequent experiment that the two lists are highly correlated and have low `divergence'.", "Since $\\vec{r_c}$ is a function of the entities in the triples (and never directly uses the relation embedding output by PBG), while $r$ was directly output by PBG, this result shows that the embeddings learned on our sample are highly self-consistent and conform closely to the notion of translationIn contrast, if $r$ had showed great divergence (or no correlation) compared to $r_c$ , it would have raised questions about whether the vectors had truly been learned by PBG in a sufficiently non-random way that (at least approximately) resemble the translation operation.", "that such neural graph embeddings have been designed to capture in vector space [53].", "Table: The Spearman's rank correlation score between SL r SL_r and SL r ' SL_r^{^{\\prime }}.", "The methodology for constructing these two (aligned) lists is described in the text.Spearman's rank correlation was designed to measure both the strength and direction of association between two ranked variables and ranges from -1 (perfect negative correlation) to 1 (perfect positive correlation).", "Because of the geometric features of the embedding space, we are interested in whether there is correlation (i.e.", "the strength), rather than the direction of the correlation.", "For this reason, given the two aligned lists per relation ($SL_r$ and $SL_r^{^{\\prime }}$ ), we computed the absolute value of the Spearman's rank correlation for each relation in Table REF As expected, some of the correlations were indeed negative.", "Of the 49 relations, 24 relations have a Spearman's rank correlation greater than 0.6; however, 25 other relations have negative correlations.", "In no case is the absolute value less than 0.4..", "The consistency of the embeddings are further supported by the information-theoretic KL-Divergence measure, detailed in Appendix A.", "A key contribution of our study is to classify and study the relational structure of ConceptNet by using graph embeddings.", "Our goal is to use such a study to understand the relational structure of commonsense knowledge itself, since ConceptNet primarily contains commonsense facts and relations.", "Given that ConceptNet is a large, crowdsourced commonsense knowledge base with semantically diffuseBy semantically diffuse, we intuitively mean that some relations (e.g., FormOf) are very broad, and relations may also overlap with each other in terms of usage, intent or semantics.", "relations, a specific question that arises is: how similar are two relations?", "Since there are multiple ways to quantify or even define `similarity', we explore three broad empirical methodologies for answering this question.", "While one of the methodologies is based on the graph embeddings that we learned earlier, two others are more conservative and traditional.", "While there is some overlap across the findings yielded by applying all three methodologies, there are also clear differences.", "In fact, even when using the embedding-based methodology, we show that the choice between using embeddings directly output by PBG and the `centroid' relation embeddings computed using Equation REF , can yield nuanced results.", "It is important to note that a methodology should not be thought of (in the context of defining and quantifying `similarity') as `correct' or `incorrect'; rather, our goal is to show the different aspects of relation similarity that the methodologies can collectively reveal.", "Next, we detail the three methodologies and the corresponding findings." ], [ "Using Explicit Relation Definitions", "Perhaps the most obvious way of comparing relations to one another is to do a `textual' comparison of their explicitly declared definitions.", "Taking relations HasContext and PartOf as examples, HasContext is defined as “A is a word used in the context of BConceptNet relation definitions can be found in https://github.com/commonsense/conceptnet5/wiki/Relations.", "Additionally, we reproduce them in Appendix B.” and PartOf is defined as “A is a part of B”.", "We use a traditional bag-of-words technique, the TF/IDF embedding with cosine similarity, to compare two definitions.", "With the `vocabulary' consisting of all words that occur in all definitions of relations in ConceptNet, we compute the TF/IDF vector by treating each definition as a `document', followed by the calculation of the cosine similarity between two vectors (corresponding to two definitions) to derive a conservative measure of the similarity between two relations.", "For the example above (HasContext and PartOf), the cosine similarity between the TF/IDF vectors of HasContext and PartOf is 0.178.", "For each relation, we can compute the relation to which it has the strongest similarity using the methodology above.", "The results are tabulated for 33 mainOn the ConceptNet5 website in the previous footnote, these so-called `main' relations are the ones with definitions.", "relations in Table REF .", "We find that the closest relation of HasContext is UsedFor, even though the definition of UsedFor states that “A is used for B\", which is different semantically than saying that “A is a word used in the context of B\" (though the two are not contradictory).", "This simple procedure illustrates why it is important to use the underlying data in the knowledge base to truly understand the empirical connections between relations; merely considering the surface form of definitions yields a very incomplete (if not misleading) picture.", "While word embeddings and other sophisticated methods could address some of the limitations of TF/IDF (including synonyms), the basic problem still remains unresolved i.e., explicit comparison of relational definitions tells us nothing about the way in which they have been asserted in triples, in practice.", "Table: The most similar relation for each relation, obtained using the TF/IDF methodology described in Section ." ], [ "Using Overlap of Head and Tail Entities", "The previous method is restricted by the combination of words chosen to express the relation's definition (which may not capture actual usage or `practical' semantics of the relations).", "In order to capture usage-based semantics of relations (and their corresponding similarity to one another), the triples that assert that relation, rather than the surface form of the relation definition, can be expected to describe the relation more comprehensively.", "An alternative approach then is to evaluate the similarity between two relations by (intuitively) measuring the overlap between their participating entity sets.", "Specifically, given the graph $G_E$ (represented as a set of triples, as noted earlier) and two relations $r_1$ and $r_2$ between which we wish to compute the similarity $sim(r_1,r_2)$ , let $H_{r_1}$ be defined as the set of all head entities such that the head entity occurs in some triple in $G_E$ with relation $r_1$ .", "Similarly, $H_{r_2}$ can be defined.", "Table: The most similar relation corresponding to each relation, obtained by choosing the one that has the highest Jaccard similarity when comparing the head entity sets of the two relations.Given the two sets $H_{r_1}$ and $H_{r_2}$ above, we compute $sim(r_1,r_2)$ as the Jaccard similarityDefined as $\\frac{H_{r_1} \\cap H_{r_2}}{H_{r_1} \\cup H_{r_2}}$ between $H_{r_1}$ and $H_{r_2}$ .", "We tabulate the results in Table REF , wherein, for each relation, we note the relation to which it is most similar, along with the Jaccard similarity score as defined above.", "We conduct a symmetric exercise, but using the tail entities instead of the head entities (tabulated in Table REF in the Appendix).", "Taken together, the two tables yield the following insights: We find that a few relations have the same most similar relations regardless of whether we choose the closest relation according to their head-entities overlap, or using the TF/IDF methodology in the previous section.", "For example, AtLocation has UsedFor as its most similar relation in both Tables REF and REF , and ReceivesAction is most similar to HasProperty Tables REF and REF .", "Therefore, it is not the case that the two methodologies are necessarily in conflict with one another; sometimes, usage and surface semantics coincide.", "In most cases, however, they do not.", "Taking again the example of HasContext, we find that in Table REF it is most similar to RelatedTo, as we would intuitively expect, while in Table REF (as explained earlier) it is most similar to UsedFor, with which is has higher surface (but not necessarily semantic) overlap.", "When we compare the differences between head-entities and tail-entities overlap methodologies, we find that, while there is some agreement on relations such as FormOf and Synonym, there are differences in other relations such as SimilarTo and MadeOf.", "In looking at the varying results between Tables REF and REF , we find that the former is more accurate and makes the most sense.", "For example, according to the former, the closest relation to MadeOf is HasA, while the latter declares it to be Desires, which is not intuitively plausible.", "Also, the frequent occurrence of RelatedTo in Table REF is a noteworthy example of a `jack of all trades' relation that is the closest relation to several more specific relations.", "One reason is that RelatedTo is used in very diverse contexts, and this diversity leads to a larger overlap between entity sets.", "Methodologically, it may be apt to treat these more general and diverse relations differently from more specific ones (such as LocatedNear)" ], [ "Using Graph Embeddings", "While the previous methodologies for measuring relation similarity had utility, their robustness was clearly limited.", "Graph embeddings are an efficient means to transform nodes (entities) and edges (relations) in the knowledge graph into vectors.", "Those vectors maximally preserve information and properties of entities and relations, especially the `structural properties' of information–entities or relations close in the graph are mapped to vectors that are also close (in the vector space).", "We claim that graph embeddings provide us with an effective empirical construct for quantifying the similarity of two relations while addressing robustness issues.", "Earlier, we presented our methodology for obtaining the embeddings from a sampled set of ConceptNet assertions using the PBG algorithm.", "We also showed that there are two ways to get a valid vector for every relation in $R$ (the set of relations in ConceptNet).", "One option is to use the `direct' relation vectors as output by the PBG algorithm, but we are also interested in interpreting the relation embeddings in the context of the triples that the relation occurs in.", "To this end, we also employ $\\vec{r_c}$ (the centroid vector defined in Equation 1) in the analysis below.", "One way to analyze the embeddings is to visualize them using dimensionality reduction techniques such as t-Distributed Stochastic Neighbor Embedding (t-SNE) algorithm [65], followed by visual analysis and inspection.", "While this has some benefits, including obtaining an `intuitive' feel of the embeddings by seeing their dispersion in a 2D space, visualization methods such as t-SNE do not allow us to formally quantify the similarities between relations in a systematic wayNevertheless, we do provide auxiliary analyses using t-SNE visualizations in Appendix C. Therein, we also show that, while t-SNE does reveal some interesting aspects of the relations, it is unable to avoid the issue of information loss compared to cosine-similarity heatmaps.. Additionally, dimensionality reduction can lose information due to projection in lower dimensions.", "Therefore, we chose to quantify inter-relation similarities by computing the cosine similarity between each pair of centroid embeddings and (separately), direct embeddings, and plotting them on a square (and symmetric) heatmap.", "Figures REF and REF respectively illustrate these results for the two different embedding methodologies.", "The heatmaps reveal some intriguing, though intuitively plausible, insights that we summarize below: Figure: The heatmap of cosine similarities between 49 relations' centroid vectorThe centroid vector for each relation has been described in the text.", "In the heatmap, a redder shade implies that the corresponding relation pair's centroid vectors' cosine similarity is low (tending to -1.0-1.0) while yellower shades represent high cosine similarities.", "The cells in the main diagonal represent the cosine similarity between a relation's centroid vector to itself, which are all shown in yellow, since they equal 1.0 by definition.Figure: The heatmap of cosine similarities between 49 relations' relation embeddingThe relation embeddings are directly yielded by PBG.", "The methodology is consistent with that of Figure .", "Some relations have high similarity with their negations.", "For example, Desires and NotDesires, UsedFor and NotUsedFor, and Synonym and Antonym.", "Their closeness can be explained by a number of reasons, including high overlap in their participating entity-sets.", "We will provide a detailed analysis about a relation example and its negation in Study 3.", "A relation generally possesses a high similarity to its `inclusion' relation.", "An inclusion relation $r_2$ of relation $r_1$ implies that the triples-set in $r_2$ is expected (to a large degree) to be a subset of the triples-set of $r_1$ , according to relation definitions.", "For example, HasFirstSubEvent and HasLastSubEvent are inclusion relations of HasSubEvent, and CausesDesire is an inclusion relation of Causes.", "We find that such relations show high similarities with their entailed relation in the heatmap, even though an auxiliary analysis using t-SNE visualizations suggest otherwise (Appendix C).", "It is also instructive to understand these results in the context of the two previous methodologies described in Sections REF and REF .", "In some respects, the results are not very different qualitatively.", "An example is that, as demonstrated in the heatmap, relations tend to be close to their inclusion relations due to the overlap between participating entity sets, regardless of whether we use overlap directly (as in the previous section) or a graph embedding methodology as in this section.", "In Table REF , for example, HasSubEvent was inferred as the closest relation to HasLastSubEvent.", "Certain relations in ConceptNet are deliberately designed to be broad.", "A good example is the HasContext relation, which is defined on the ConceptNet website as: A HasContext B is declared in the knowledge base if `A is a word used in the context of B, which could be a topic area, technical field, or regional dialect'.", "We hypothesize that, despite being originally defined so broadly, there is considerable substructure in these relations.", "In considering the definition of HasContext above, multiple contexts are suggested (e.g., technical field, regional dialect, and presumably, other contexts that may be similar to these explicit cases).", "Another example is a relation such as FormOf, where a triple A FormOf B may be declared if `A is an inflected form of B; B is the root word of A'.", "Even the basic official definition suggests breadth, since A could either be an `inflected' form of B, or the `root word' of B.", "Furthermore, there is nothing in the definition that places a strict constraint on such triples, either in theory or in practice.", "Since ConceptNet is crowdsourced (at least to a great extent), it is quite likely that many people have interpreted these relations at `face value' i.e.", "in accordance with what one would understand their `everyday' meaning to be.", "Therefore, the set of experiments described in this section is to measure and describe the empirical substructure, if any, of three specific relations (HasContext, FormOf and SymbolOf) using a systematic methodology.", "The SymbolOf relation is succinctly defined as: the triple A SymbolOf B is asserted in the knowledge base if `A symbolically represents B'.", "An important aspect of these three relations is not just that they are defined broadly (i.e.", "are coarse-grained) but are also (relatively) high-volume.", "Within our sample of 4 million triples, HasContext, FormOf and SymbolOf are asserted in 133,038, 630,914 and 63,785 triples respectively.", "This provides an added incentive to study these relations further, since they are clearly central to the knowledge base and its purpose of capturing commonsense knowledge as sets of assertions.", "An established unsupervised methodology for discovering structure in large collections of data points is clustering [66].", "The relations in ConceptNet were meant to capture common, informative patterns from various data sources that fed into ConceptNet, as well as crowdsourcing.", "If well-defined clusters exist, it proves that these coarse-grained relations are not as coarse-grained as the volume of triples suggests.", "By studying the clusters (including the consistency and subjective nature of data within each cluster, as well as the approximate number of `optimal' clusters revealed through application of an algorithm such as k-Means), we can start gaining insight into each relation.", "These insights allow us to gain an empirical understanding of words such as `context' and `form' that are important in commonsense reasoning and communication, beyond their formally stated (and rather broad) definitions.", "Before diving into the results for each of these relations, the question of methodology still remains.", "Most established clustering algorithms require the collection and representation of data points to be described in advance.", "In this case, since we are interested in studying each independent assertion (and thereby, use) of (each of) the three relations, we would ideally like to have a data point for each triple where the relation of interest occurs.", "Our treatment in earlier sections already provided a mechanism for doing so i.e.", "obtaining a data point for each relation $r$ per triple $(h,r,t)$ where that relation is asserted by using the translation vector $\\vec{t}-\\vec{h}$ , and where the entity embeddings $\\vec{t}$ and $\\vec{h}$ are obtained using the PBG package, just as in Study 1.", "Since HasContext, FormOf and SymbolOf are respectively asserted in 133,038, 630,914 and 63,785 triples within our sample, we obtain the same (respective) numbers of data points (as translation vectors) for each of the three relations.", "Furthermore, because of the large numbers of data points and the lack of a task-specific objective function (or alternatively, training labels), as well as the preference for using an established and reasonably robust clustering algorithm, we chose to use the classic k-Means algorithm [67].", "Briefly, k-Means works iteratively to partition the dataset into k clusters, each of which is disjoint (since the clusters constitute a partition).", "Note that $k$ is a hyperparameter that must be predefined prior to executing the algorithm.", "There are several ways to obtain the `best' value of $k$ given a collection of points.", "The underlying commonality between these methods is to compute a `score' (typically an error score, implying that lower is better) for each value of $k$ (over some predetermined range and increment over the range) after executing k-means for that $k$ .", "Next, we plot the score vs. $k$ and look for sudden shifts in the second derivative of the curveAlthough the curve can be monotonic for some methods, it is not always guaranteed.", "Hence, it is incorrect to look for a `minimum'..", "Intuitively, this looks like a clear `bend' in the curve.", "One such candidate method is the elbow method, which computes an error-based score based on the `dispersion' of points within each cluster.", "However, as the first subplot in Figure REF showsIn Appendix D, we reproduce analogous subplots (Figures REF and REF ) for the SymbolOf and FormOf relations., there is no visible decline in the second derivative with $k$ (i.e.", "the bend, if it exists, is far too subtle to be useful).", "In considering three alternative established methods–namely, the Silhouette Coefficient method, the Davies-Bouldin Index, and the Calinski-Harabasz IndexThe Silhouette Coefficient value measures how similar a point is to its own cluster's centroid (cohesion) compared to other clusters' centroids (separation).", "The Index measures compute their scores in slightly different ways, but with the same underlying philosophy that clusters should be cohesive and well-separated.", "Specific details and formulae may be found in the cited works.", "[68], [69], we find that $k=20$ is a robust choice for all the three relations.", "Figure: Selection of the parameter k for k-means clustering of HasContext triples.", "The line chart describes the trend in the HasContext triples' clustering scores with different values of k. The clustering score used in the subplots (from left to right), is measured by the elbow method, Silhouette Coefficient, Davies-Bouldin index, and Calinski-Harabasz index, respectively (with the last two in the same subplot).", "The x-axis shows the k value and the score is shown on the y-axis.With this value of $k$ ($=20$ ) in place, we conduct k-Means clustering for each of the three relations, and perform dimensionality reduction using t-SNE to obtain a visualization of all the clusters and triplesUnlike the previous study, a heatmap-style study is not possible here, due to the large numbers of triples and data points involved.", "Nevertheless, we do quantify the clusters using separation and cohesion measures in Tables REF and REF ..", "In describing the discovered substructures (`clusters') for each relation below, we define and quantitatively analyze the cohesion and separation of these sub-clusters in Table REF and REF ; we also sample some triples and study them qualitatively in Appendix D to comment on the nature of triples within the substructures and provide some subjective validation of the k-Means output.", "Table: The average cohesion of FormOf, HasContext and SymbolOf clusters, along with their mean and standard deviation.", "Cohesion is defined in the text (Study 2).", "Note that cluster IDs are assigned arbitrarily and independently across relations, and not `aligned' in any way.", "The IDs are re-used again in Figures , , , and , as well as in Tables and to report on further results.Table: The average separation between clusters in FormOf, HasContext and SymbolOf, along with their mean and standard deviation.", "Separation is measured by the distances from the center of a cluster to all other clusters' centers.", "The cluster IDs in this Table are aligned with those in the previous table, for any given relation (e.g., FormOf Cluster 1 in this table is the same as FormOf Cluster 1 in Table .As the name suggests, cohesion measures the extent to which the points in each cluster are tightly grouped together.", "A cluster with high dispersion would have low cohesion.", "While this intuitive measure could be quantitatively measured in several conceivable ways, we use a relatively simple formula.", "First, for each cluster, we calculate the average Euclidean distance between the normalized points in the cluster and the cluster's centroid (also normalizedSince the graph embeddings are not normalized to lie on a unit-radius hypersphere, we normalize the vectors before computing the distance to enable cross-cluster comparisons, and also comparisons with the (subsequently described) separation measures.).", "Since smaller distances indicate greater cohesion, we subtract the average from 1 to obtain a cohesion on a scale of 0.0 to 1.0, with 1.0 indicating perfect cohesion (i.e., all points inside the cluster coincide after normalization).", "In Table REF , we report the cohesion per cluster for each of the relations.", "Based on the table, we find that the average cohesion for FormOf, HasContext and SymbolOf is 4.439, 4.562 and 4.831 respectively.", "Note that the absolute values here are not meaningful, and must be interpreted relative to each other.", "We find that, while the average cohesion scores of clusters in these three relations are close in value, the standard deviations show significant differences.", "The standard deviations of HasContext cluster cohesion scores are generally higher than the standard deviations of the other two relations' cohesion scores.", "This simple result suggests the hypothesis that HasContext is more diverse (and hence, more dispersed in embedding space) than the other two relations.", "It is interesting that the deviation is inversely related to the number of triples corresponding to each relation (but not linearly).", "While cohesion is a good measure for computing the quality of clustering, it is not adequate by itself.", "An optimal cohesion can be obtained by assigning each point to its own cluster (in which case, the point becomes the centroid of the cluster).", "An additional metric, even after controlling for $k$ , is the separation of the clusters i.e., how `far apart' the different clusters are in the embedding space.", "Similar to cohesion, there are multiple mathematical ways to capture this qualitative notion.", "We employ a simple method that is analogous to the cohesion measure–namely, for a given cluster, we compute its separation by computing the average (normed Euclidean) distance from its centroid to each of the other 19 centroids.", "Table REF reports the results, along with the mean and standard deviation.", "We find that, once again, HasContext has highest average separation which is 4.622.", "This suggests that the `contexts' represented by these clusters are well-separated.", "FormOf and SymbolOf obtain similar average separation (2.985 and 2.970, respectively).", "In contrast to cohesion, the standard deviation of cluster separation scores is highest for the SymbolOf relation.", "In comparing the cohesion and the separation of clusters for all three relations in Tables REF and REF , we find that the HasContext average separation is close to the average cohesion.", "In other words, the average distance from one cluster center to a within-cluster node is close to the average distance from that cluster center to other cluster centers.", "FormOf and SymbolOf's average separation is lower than average cohesion.", "It suggests that (some) nodes in these two relations' clusters might be closer to other cluster centers than their own centers.", "Substructures in these two relations are less independent than in HasContext.", "Figure: The visualization of clusters in HasContext by t-SNEHere, 20 HasContext triples-clusters, discovered using k-Means, are represented in different colors and the ID of each cluster is shown in the color bar on the right side.", "The cluster IDs are consistent with those introduced and used in Tables (in Appendix D), , and .", "The t-SNE `dimensions' merely work for visualization and do not have intrinsic meaning.While these results demonstrate evidence of some substructure for all three relations, it is also instructive to study the substructure by further analyzing triples within these clusters.", "While a full qualitative study of this nature for all three relations is beyond the scope of this article, we perform a limited study for the HasContext relation, which has particularly `pure' clusters (Figure REF ), by randomly sampling five triples per cluster and determining if we can deduce the `theme' of the cluster from these five triples.", "We also create the visualization for the substructures of FormOf and SymbolOf in Figures REF and REF , and sample 5 example triples for each cluster in Tables REF and Figure REF in Appendix D, but we do not do an analysis for the sampled triples (for each cluster) of these two relations here.", "Figure: The visualization of clusters in FormOf by t-SNEThe same methodology applies as in Figure .Figure: The visualization of clusters in SymbolOf by t-SNEThe same methodology applies as in Figure .As illustrated in Table REF in Appendix D (which lists our sampled triples per cluster), the triples show that some of the clusters clearly embody scientific `domains' of study such as geography (Cluster 14), chemistry (Cluster 6), medicine (Cluster 17), mathematics (Cluster 18), and physics (Cluster 19).", "However, there are also `mixed' clusters that seem to be related to more than one theme (at least on the surface).", "For example, Cluster 3 contains some `locality' triples, even though Cluster 15 is predominantly concerned with localities, and Cluster 3 is mainly concerned with zoology.", "We believe that there could be two causes for such `confusion'.", "The first is due to the automatic and unsupervised nature of the embedding and the second is due to ConceptNet itself, both in terms of the noise within ConceptNet and also because some tail entities (such as /c/fr/localites) may be imbalanced in terms of the head entities associated with them.", "Some other clusters also include some interesting combinations, such as Cluster 12 which contains triples corresponding to both `computing' and `slang'.", "In the embedding and clustering process, `computing' and `slang'-related triples are thought to be close to each other in embedding space, even though the semantic similarity may not be evident.", "Yet other clusters (such as Cluster 0) seem to encapsulate the broad notion of HasContext and do not have an evident thematic classification that we can determine.", "In using visualization to investigate the inter-relationships between these clusters in Figure REF , we uncover other interesting findings.", "For example, the cluster focused on chemistry-related triples seems to overlap with the cluster containing biology (and also astronomy-related triples).", "While some of the overlap in the figure is exaggerated due to dimensionality reduction, it is nonetheless indicative of the low separation between these two clusters in high-dimensional space and an indirect acknowledgement of the shared lineage of some of these scientific disciplines.", "An interesting avenue for future exploration could be to quantify and explain such topical overlap between certain clusters by using techniques such as hierarchical clustering." ], [ "Study 3: Correlation between a specific relation and its negation", "While Study 1 focused on the `macroscopic' structure of ConceptNet through a study of relational structure, and Study 2 attempted to map specific substructures in three coarse-grained relations, they provided less insight on the relations between relations.", "One such relation is negation.", "Negation is an important everyday relation that also has a distinctive role in formal and logical paradigms (including both propositional and first-order logic).", "In a mathematical framework, a set $S$ and its negation $\\lnot S$ are considered to be as different as two sets could get, since they are guaranteed to be disjoint.", "Another way to think about negation is as a meta-relation.", "While there is a wide body of work on such meta-relations in formal paradigms such as logic [x], not much is known about them with respect to commonsense reasoning.", "In this section, we attempt to derive more structural insights on the nature of the negation relation by studying two specific relations in ConceptNet: Desires and NotDesires.", "Since both relations can be explicitly asserted in facts in ConceptNet, we can use these collections of assertions to detect high-level patterns in usage.", "One question that arises, for example, is: in embedding space, do the Desires and NotDesires translation embeddingsRecall, from previous sections, that a translation embedding can be generated per fact or assertion.", "neatly separate into clusters?", "Also, when it is `unknown' whether there is a Desires or NotDesires relation between two entities $A$ and $B$ , how does a hypothetical `Unknown' relation compare to Desires or NotDesires?", "To answer these questions, we set up the experiment as follows.", "Let us define $\\mathcal {T}$ as the set of triples containing either the Desires or NotDesires relation.", "Let $H$ and $T$ be the sets of head and tail entities in $\\mathcal {T}$ .", "Furthermore, let us define the set $U$ as the subset of $H \\times T$ such that neither $(h, Desires, t)$ nor $(h, NotDesires, t)$ is in $\\mathcal {T}$ .", "$U$ is precisely the set of `unknown' head-tail entity pairs we intuited about earlier i.e.", "the combinations of head and tail entities that individually participate in at least one Desires/NotDesires relation as a head or tail entity, but that have never been asserted together in a single triple (either with the Desires or the NotDesires relation).", "Given these notions, let us define $H_U$ and $T_U$ as the respective sets of all head and tail entities in $U$ .", "Note that, empirically, we found that $H_U$ is actually equal to $H$ and $T_U$ is equal to $T$ in ConceptNet for the reason that for any given head entity in $\\mathcal {T}$ , there is at least one tail entity in $\\mathcal {T}$ such that we do not know the relationship between $H$ and $T$ (and analogously for the tail entities $T$ ).", "With these notations and definitions in place, we investigate the differences between pairs of entities that are related through Unknown, Desires and NotDesires relations.", "Each of these three relations yields a set of mutually disjointWe found that there were 16 triples where a head-tail entity pair simultaneously participated in both a Desires or NotDesires relation.", "We remove these contradictory triples, which are small in number to the total number of Desires and NotDesires triples.", "We also remove duplicate triples, where applicable.", "entity pairs.", "One issue that arises due to the Unknown class is the number of $(h,t)$ pairs in $U$ .", "Since $U = \|H \\times T\| - \|\\mathcal {T}\|$ , and since $\|H\|=2,683, \|T\|=5,549$ , $\|\\mathcal {T}\|=8,352$ , $\|U\|=14,879,615$ .", "Clearly, visualizing all triples in $U$ is not feasible, nor desired, since $U$ dwarfs $\\mathcal {T}$ in size (intuitively, much more is unknown about what the entities desire or not desire, than is known).", "To compare all three relations on a relatively balanced footing, we sample relation triples from $U$ as follows.", "To obtain roughly the same number of samples as in $\\mathcal {T}$ , we set a tail sampling ratio, $r=\\lceil \|\\mathcal {T}\|/2 * \|H_U\|\\rceil $ rounded down to $r^{\\prime }= 2$ i.e.", "for each head entity $h$ in $H_U$ , we randomly sample $r^{\\prime }$ tail entities from the set $\\lbrace t\|(h,t) \\in U \\rbrace $ , if $r^{\\prime }$ is less than the cardinality of this set (otherwise, we sample the whole set).", "After sampling, we get a sampled set $U^{\\prime } \\subset U$ , with $\|U^{\\prime }\|=5,366$ .", "Next, we embed $\\mathcal {T} \\cup U^{\\prime }$ following the translational methodology that we described earlier for the previous research questions, and assign one of three colors to each embedding, based on whether the embedding represents a pair in $U^{\\prime }$ , or (if the pair is in $\\mathcal {T}$ ), whether it represents a Desires/NotDesires triple.", "Contrary to expectation, the triples in Desires and NotDesires do not separate in the t-SNE visualization.", "To better observe the relationship, we render the same visualization in two different ways (Figures REF and REF ).", "In one of these figures, the Desires translational embeddings are `overlaid', while in the other, the NotDesires embeddings are overlaid.", "Regardless of how we visualize the triples, we find that there is high overlap between them.", "The primary reason clearly is that the same semantic categories of head and tail entities tend to participate in both Desires and NotDesires relations.", "For example, a person may Desire a movie, but may NotDesire another movie, which would lead to the kind of conflation that we observe in Figures REF and REF .", "Hence, the nuances between these two relations are lost in such a plot.", "Figure: The visualization of Desires, NotDesires, and Unknown triples using t-SNEThe methodology for constructing Unknown triples is described in the text.", "Data points corresponding to Desires, NotDesires and Unknown triples are shown in green, red and gray, respectively.", "Similar to previous t-SNE visualizations, the dimensions are only for visualization and do not have intrinsic meaning.", "Since the points nearly coincide in this 2D space, we `overlay' the Desires points on the NotDesires points where there is a conflict (with the opposite result shown in the next figure)Figure: The visualization of NotDesires, Desires, and Unknown triples using t-SNEThe methodology is the same as in Figure , but in this figure, we `overlay' the NotDesires points when there is a conflict.The Unknown embeddings also yield interesting conclusions.", "While these embeddings also have high overlap with the other embeddings (the `middle' portion of Figures REF and REF ), they also occupy portions of the embedding space not occupied by either Desires or NotDesires.", "For example, there is a long grey lobe both at the top and the bottom of Figures REF and REF .", "These are precisely the types of spaces that link prediction (and other recommendation-style) algorithms attempt to map using a set of semantic and structural features [70], [71].", "Nevertheless, there are alternative ways to detect such nuances, if they exist.", "As proof-of-concept, we trained ordinary machine learning classifiers (both random forest and logistic regression) to determine whether the embeddings still contain latent information (that may not easily be visualized) about the Desires and NotDesires relations.", "Specifically, we used 10-fold cross-validation to train and evaluate a classifier on predicting whether a translational embedding could be correctly labeled as Desires or NotDesires.", "The best performance was achieved with a tuned random forest model that obtained an accuracy of 76% on this task.", "Hence, the negation meta-relation does leave behind a semantic trace that a machine learning model, appropriately trained, is able to learn, but that is not discernible in an unsupervised framework such as clustering or visualizationThe t-SNE visualization algorithm may itself be thought of as a kind of clustering based on optimizing relative distances in a `visual' i.e.", "2-D space..", "Hence, there are semantic differences between the two relations, but it is an open question whether there is a structural difference, structure being defined similarly as in Study 2.", "To conclude, this study clearly refutes the hypothesis that a relation like Desires should have a `clear' separation from its negated version.", "Note the difference from Study 2, where we found clear separations and substructures upon analyzing triples that belonged to a single relation that had one definition (at least in principle).", "The qualitative and structural differences between Unknown and the other two relations provides evidence of the somewhat intuitive notion that unknown relations constitute their own relation class in such commonsense knowledge bases.", "This is an important aspect of ConceptNet that commonsense reasoners need to account for when using it as a distant supervision resource for question answering tasks.", "In fact, a key problem that commonsense reasoners need to successfully address is accurate inference over `unknown'-relation spaces (not only for Desires, but other relations of a similar nature) when presented with questions or scenarios that require such inference.", "The results in Sections REF , REF , and REF reveal some insights into commonsense knowledge through an empirical and systematic study of ConceptNet's relational structure (and substructures of certain relations), as well as the characteristics of the ubiquitous negation relation.", "Except the experiment results which have been detailed presented above, there are still some findings needed to be mentioned.", "We summarize these findings below: By using a methodology based on centroid relation vectors (Section REF ), we were able to magnify the (sometimes, subtle) differences between relations using a symmetric heatmap.", "In contrast, using `direct' relation vectors was found to lead to more uniformity.", "Similarly, in considering other methodologies for detecting relational similarity, we found that comparing explicit textual definitions had limited use, and when considering overlap between entities (rather than graph embeddings), overlap between head entities-sets of two relations had more utility and intuitive plausibility than overlap between the relations' tail entities-sets.", "We also proposed and used a methodology for studying a relation (Desires) in the context of its negation (NotDesires), and a novel `Unknown' relation that expressed (given a head entity and a tail entity) that we did not know whether the relation or its negation applied to the two entities.", "We found that, while the Unknown relation seems to occupy a distinct portion of the embedding space, the Desires and NotDesires embeddings overlap significantly.", "In terms of usage in assertions (in the ConceptNet KG), the Desires and NotDesires relations are not very different.", "Yet, the two are also not indistinguishable, since machine learning methods like random forest are able to distinguish between them (when trained on some labeled assertions).", "We believe that a similar finding could apply to other such relation `pairs', such as causal relations and sub-relations (e.g., HasSubevent and HasLastSubevent).", "We have also empirically observed that, while `super-class' semantics tend to be associated with the definition of HasContext (i.e., when head entity A is a word that is used in the context of tail entity B, B tends to be a more general, abstract `super-class' of A, such as a topic area, technical field, or regional dialect, as is also mentioned in the official definition of HasContext), there are significant substructures that can't be uniformly explained by an umbrella term like HasContext.", "These substructures may help us better understand what the different contexts are in which people interpret pairs of words or entities.", "Understanding context is critical for building systems that have commonsense, such as chatbots and conversational agents that are able to understand sentences in the specific context in which the sentences are uttered.", "A particularly interesting relation is SymbolOf.", "The ratio of the size of the head entity set of SymbolOf to the size of tail entity set is 0.011, almost 1/200 of the average value observed for all relations.", "This indicates that the number of words or phrases which could be used to `explain' the symbols is far greater than the number of symbols themselves.", "Symbols (and arguably, by extension, the emojis used on social media) are necessarily under-determined, and the semantics assigned to symbols vary in different contexts.", "While not qualitatively unsurprising, our results suggest that the `diffusion' of symbol-semantics is far greater than one might have thought.", "There are other interesting findings about the ratio of the number of unique head entities (participating in a relation) to the number of tail entities, although these might be more specific to ConceptNet than to commonsense knowledge per se.", "For example, the ratio for both HasFirstSubevent and HasLastSubevent relations is around 0.3, which indicates that, on average (in ConceptNet), a `subevent' has the potential to be the first in three ($\\approx 1/0.3$ ) sequences of subevents (that each constitute an `event').", "A similar conclusion applies for the terminal subevent in the sequence.", "Whether this is a general statistical truth or not is left for future work to explore." ], [ "Conclusion", "In this article, we explored a set of data-driven methodologies, using the ConceptNet knowledge graph, to derive insights at scale on the nature of commonsense knowledge.", "To this end, we conducted a set of three carefully designed studies, and a range of experiments.", "Our results show that there is significant structure (and substructure) in ConceptNet (Study 1), especially with respect to relations such as FormOf, SymbolOf and HasContext.", "Study 2, in particular, shows that there are at least (and possibly more than) 20 different kinds of context that can be discovered within ConceptNet, some of them very well-defined (such as a scientific field of study, or even slang), with others being more diffuse.", "In some cases, there are unusual but non-random degrees of overlap and association (e.g., computer science and slang).", "In Study 3, we apply a different, but related, methodology, to study negation, and find that `unknown'-type relations occupy unique regions of the embedding space, whereas the two relations (Desires and NotDesires) significantly overlap in the embedding space, even though a trained machine learning model can distinguish between them on the basis of their features.", "Several avenues of future research remain.", "For example, it may be worthwhile going even deeper into a relation like HasContext to discover if there are hierarchical substructures, rather than a single layer of substructures.", "Hierarchical clustering algorithms in the graph embedding space could be used to achieve this goal methodologically.", "Similar to Study 3, a relation and its causal equivalent, could be studied together to understand their connection in an embedding space.", "A more ambitious line of study would be to try and connect these empirical results to the more theoretical results on commonsense reasoning (e.g., by [20])." ], [ "Competing interests", "The authors declare that they have no competing interests.", "Shen performed the experimental study and contributed significantly to writing and study design.", "Kejriwal primarily contributed to the writing and project ideation, as well as formulation of the specific research questions.", "This project was funded under MOWGLI, a project in the Defense Advanced Research Projects Agency (DARPA) Machine Common Sense program, supported by United States Office Of Naval Research under Contract No.", "N660011924033.", "The views expressed in this paper are those of the authors.", "...", "Given two probability distributions $p$ and $q$ with the same support (intuitively, the same event space or `x-axis' over which the probability density/mass function is computed), the KL-Divergence $D_{KL}(p\|\|q)$ of $p$ with respect to $q$ is given by: $D_{KL}(p\|\|q) = \\sum ^N_{i=1}p(x_i) \\cdot (\\log p(x_i) - \\log q(x_i))$ Table: The KL-Divergence of 49 relations when using the per-relation similarity lists and methodology described in the text.", "The two probability distributions used in the KL-divergence calculation for relation rr (Equation 2) are the frequencies of cosine similarity scores in the two lists (SL r SL_r and SL r ' SL_r^{^{\\prime }}).The higher the KL-Divergence, the more different the two distributions are.", "Note also that the measure is asymmetric in the general case i.e.", "$D_{KL}(p\|\|q) \\ne D_{KL}(q\|\|p)$ .", "In our particular setting, we want to use KL-Divergence to measure how much information is `lost' when we use $\\vec{r_c}$ as a replacement for $\\vec{r}$ .", "If not much information is lost (i.e.", "the KL-Divergence is low), then $\\vec{r_c}$ may be said to be a robust replacement for $\\vec{r}$ .", "Specifically, given the lists $SL_r$ and $SL_r^{^{\\prime }}$ for relation $r$ , we first compute an empirical probability distribution for each list by using frequencies of scores in the lists.", "Next, we compute the KL-Divergence by using the probability distribution of $SL_r$ as the reference distribution $q$ .", "For each relation, the KL-Divergence is tabulated in Table REF and we show two examples in Figure REF .", "Both the table and figure show that divergence (between using the centroid and the `directly output' vector) is generally low, attesting to the previous result the PBG embeddings over the 4 million triples-sample, and our methodology for constructing the centroid embedding per relation, are consistent and robust.", "Figure: The cosine similarity frequency line chartThe frequency distribution of cosine similarities for two relations Antonym (left) and Causes (right), where the similarity is calculated between the translational embedding of a triple (containing the relation) and one of two vectors used to represent the relation (centroid vector, or direct relation embedding output by PBG).", "For each of the two relations, and for each choice of the relation representation, a `cosine similarity' distribution is thereby obtained with each triple containing the relation contributing a single data point to the distribution.We reproduce the definitions of the 33 `main' relations (except ExternalURL) in ConceptNet5 from the original project pagehttps://github.com/commonsense/conceptnet5/wiki/Relations, for the sake of completeness: RelatedTo: The most general relation.", "There is some positive relationship between A and B, but ConceptNet can't determine what that relationship is based on the data.", "This was called \"ConceptuallyRelatedTo\" in ConceptNet 2 through 4.", "Symmetric.", "FormOf: A is an inflected form of B; B is the root word of A. IsA: A is a subtype or a specific instance of B; every A is a B.", "This can include specific instances; the distinction between subtypes and instances is often blurry in language.", "This is the hyponym relation in WordNet.", "PartOf: A is a part of B.", "This is the part meronym relation in WordNet.", "HasA: B belongs to A, either as an inherent part or due to a social construct of possession.", "HasA is often the reverse of PartOf.", "UsedFor: A is used for B; the purpose of A is B. CapableOf: Something that A can typically do is B. AtLocation: A is a typical location for B, or A is the inherent location of B.", "Some instances of this would be considered meronyms in WordNet.", "Causes: A and B are events, and it is typical for A to cause B. HasSubevent: A and B are events, and B happens as a subevent of A. HasFirstSubevent: A is an event that begins with subevent B. HasLastSubevent: A is an event that concludes with subevent B. HasPrerequisite: In order for A to happen, B needs to happen; B is a dependency of A. HasProperty: A has B as a property; A can be described as B. MotivatedByGoal: Someone does A because they want result B; A is a step toward accomplishing the goal B. ObstructedBy: A is a goal that can be prevented by B; B is an obstacle in the way of A. Desires: A is a conscious entity that typically wants B.", "Many assertions of this type use the appropriate language's word for \"person\" as A. CreatedBy: B is a process or agent that creates A. Synonym: A and B have very similar meanings.", "They may be translations of each other in different languages.", "This is the synonym relation in WordNet as well.", "Symmetric.", "Antonym: A and B are opposites in some relevant way, such as being opposite ends of a scale, or fundamentally similar things with a key difference between them.", "Counterintuitively, two concepts must be quite similar before people consider them antonyms.", "This is the antonym relation in WordNet as well.", "Symmetric.", "DistinctFrom: A and B are distinct member of a set; something that is A is not B. Symmetric.", "DerivedFrom: A is a word or phrase that appears within B and contributes to B's meaning.", "SymbolOf: A symbolically represents B. DefinedAs: A and B overlap considerably in meaning, and B is a more explanatory version of A. MannerOf: A is a specific way to do B.", "Similar to \"IsA\", but for verbs.", "LocatedNear: A and B are typically found near each other.", "Symmetric.", "HasContext: A is a word used in the context of B, which could be a topic area, technical field, or regional dialect.", "SimilarTo: A is similar to B. Symmetric.", "EtymologicallyRelatedTo: A and B have a common origin.", "Symmetric.", "EtymologicallyDerivedFrom: A is derived from B. CausesDesire: A makes someone want B. MadeOf: A is made of B. ReceivesAction: B can be done to A.", "In Study 1, we used the KL-Divergence to define `heatmaps' between relation embeddings (using both centroid and direct relation embeddings); however, a more informal analysis could also be undertaken using the t-Distributed Stochastic Neighbor Embedding (t-SNE) algorithm [65], which is an established neural dimensionality-reduction algorithm that has been shown to be superior to other dimensionality-reduction techniques (e.g., Principle Components Analysis) for visualizing the outputs of representation learning algorithms such as PBG.", "Our methodology is similar to that in Study 1 for generating KL-Divergence heatmaps.", "Once the centroid vectors are obtained, one for each relation, we visualize the similarities between the relations in two dimensions by using t-SNE (Figure REF ).", "Figure: The visualization of 49 relation centroid vectors using t-SNE.The centroid vector for each relation is calculated using Equation .", "Note that t-SNE dimensions (or axes) are only used for visualization and do not have intrinsic meaning.", "We manually mark the three obvious clusters by the rectangle for further analysis.The figure shows that, while many relations are reasonably uniformly distributed (visualized in the figure as a `circle' with relatively uniform spacing between relations), some relations do congregate in clusters, suggesting significantly higher inter-similarities in the embedding space.", "These findings are largely consistent with those reported in the main text.", "Three of the most prominent clusters are manually demarcated in the figure, and are described below: The cluster in the upper left contains the relations Entails, Antonym, Desires, ParOf, LocatedNear, dbpedia/capital, dbpedia/genre, dbpedia/genus, and dbpedia/influencedBy.", "Although the dictionary semantics of these relations are quite different, the similarities in embedding space shows that (relatively speaking) there is likely high overlap between the sets of entity pairs that are related using each of these relations.", "For example, LocatedNear is described as `A and B are typically found near each other'; dbpedia/influencedBy states that `A is influenced by B'.", "If two objects were close to each other, it is quite possible that one object is influenced by another.", "These kinds of complex interplays seem to be captured, at least in part, in the embedding space and visualization.", "The cluster in the upper right contains the relations HasA, RelatedTo, EtymologicallyDerivedFrom, HasSubevent, dbpedia/field, DefinedAs, ObstructedBy, MannerOf, and Causes.", "The large degree of overlap between the corresponding entity sets (the union set of head and tail entity set) is the main reason why relations are grouped in this cluster.", "The closeness between ObstructedBy and MannerOf is an obvious instance.", "ObstructedBy indicates that “B is an obstacle to achieve the goal action A” and MannerOf emphasizes that “A is a specific way to do goal action B\".", "The overlap of “action”-type entities directly results in the proximity of ObstructedBy to MannerOf.", "The cluster in the bottom contains the relations NotDesires, Synonym, SymbolOf, MadeOf, HasPrerequisite, HasContext, SimilarTo, CreatedBy, dbpedia/product, dbpedia/occupation, dbpedia/language, and dbpedia/KnownFor.", "This cluster is similar to the first cluster, but contains some negations of relations in the first cluster.", "One interpretation of this cluster is that it may be the converse or negation of the first cluster, even though the actual relations within this cluster do not seem to be closely semantically related to one another.", "An interesting question that arises is whether similar conclusions can be drawn from the relation embeddings that were directly output by the PBG algorithm.", "These `direct' relation embeddings are visualized in Figure REF , using the same methodology as for the centroid embeddings.", "While there are a few similarities between Figures REF and REF , there are also significant differences.", "For example, the distribution is more uniform, suggesting that direct relation embeddings may not (hypothetically) capture `usage' semantics as effectively as the centroid embeddings.", "Again, these findings are consistent with the heatmaps illustrated in the main text for Study 1.", "In the clusters that are observed, we find that the relations have a more obvious semantic connection to each other (e.g., EtymologicallyDerivedFrom and DerivedFrom, Desires and CausesDesire).", "Other less obvious examples also point to a similar conclusion.", "Figure: The visualization of 49 relation embeddings using t-SNEThe relation embeddings are directly yielded by PBG.", "The methodology is the same as ; note that the t-SNE dimensions do not have intrinsic meaning and are only useful for 2D visualization.Despite their differences, the two sets of results (`centroid' and `direct') illustrate that all relations are not equal (i.e.", "uniformly distributed), and that some structure does exist in the relation embedding space.", "In the case of centroid embeddings, usage semantics seem to be captured more effectively in the visualization, while in the case of direct embeddings, a more direct semantic connection (learned by the PBG in the course of training) seems to be necessary for relations to be clustered together.", "However, there is clear information loss in projecting the embeddings down to 2D.", "A proper analysis should therefore rely on the information-theoretic methodology in the main text, and not on informal visual analysis, though the visualizations are undoubtedly useful for gaining a `sense' of how the relations differ from each other depending on the embedding methodology employed.", "[h!]", "\|c\|c\|c\| 1\|c\| cluster ID 1c\| head 1c\| tail 3c 9 – continued from previous page 1\|c\| cluster ID 1c\| head 1c\| tail 3\|r\|Continued on next page 50 /c/en/chifferobe/n /c/en/chifforobe /c/en/fluoroscans/n /c/en/fluoroscan /c/en/woodifying/v /c/en/woodify /c/pt/nintendinhos/n /c/pt/nintendinho /c/lb/maach/vn /c/lb/maachen 51 /c/es/ondearíais/v /c/es/ondear /c/it/istallerà/v /c/it/istallare /c/la/dextrarum/a /c/la/dexter /c/es/envasáramos/v /c/es/envasar /c/de/gestatte/v /c/de/gestatten 52 /c/la/redeatis/v /c/la/redeo /c/la/ruminemur/v /c/la/ruminor /c/la/protrusi/v /c/la/protrudo /c/la/normabere/v /c/la/normo /c/es/campean/v /c/es/campear 53 /c/hu/hátsók/n /c/hu/hátsó /c/la/perspeculati /c/la/perspeculatus /c/de/iq tests /c/de/iq_test/n /c/en/duck_dives/n /c/en/duck_dive /c/fr/argenteurs/n /c/fr/argenteur 54 /c/la/insutasv /c/la/insutus /c/fr/icosanoïde /c/fr/eicosanoïde/n /c/it/parlamenteremo/v /c/it/parlamentare /c/es/galardonase/v /c/es/galardonar /c/it/metterti/v /c/it/mettersi 55 /c/la/renidebam/v /c/la/renideo /c/it/cogestivano/v /c/it/cogestire /c/la/configurabimus/v /c/la/configuro /c/it/disincanterebbero/v /c/it/disincantare /c/la/fundassent/v /c/la/fundo 56 /c/es/ventilé/v /c/es/ventilar /c/es/embargases/v /c/es/embargar /c/es/vinculando/v /c/es/vincular /c/fr/cylindrasses/v /c/fr/cylindrer /c/fr/classai/v /c/fr/classer 57 /c/it/ratificasti/v /c/it/ratificare /c/es/pichicateé/v /c/es/pichicatear /c/es/blandías/v /c/es/blandir /c/it/girandoti/v /c/it/girarsi /c/it/sabbierò/v /c/it/sabbiare 58 /c/la/insultasset/v /c/la/insulto /c/la/concutitote/v /c/la/concutio /c/la/adprobaremus/v /c/la/adprobo /c/la/consepsissemus/v /c/la/consepio /c/la/variegarentur/v /c/la/variego 59 /c/de/liebhabers /c/de/liebhaber/n /c/de/kalbst /c/de/kalben/v /c/de/emblemen /c/de/emblem/n /c/de/unseres/n /c/de/unser /c/de/absurdem/a /c/de/absurd 510 /c/es/retasteis/v /c/es/retar /c/es/canalizarlas/v /c/es/canalizar /c/es/abombes/v /c/es/abombar /c/es/transportada/v /c/es/transportar /c/fr/brancarderait/v /c/fr/brancarder 511 /c/nrf/boulets_contraceptifs/n /c/nrf/boulet_contraceptif /c/de/schwammerls/n /c/de/schwammerl/n /c/is/álfu /c/is/álfa/n /c/es/mosqueándola/v /c/es/mosqueando /c/it/disgregandoti/v /c/it/disgregarsi 512 /c/en/pennames/n /c/en/penname /c/en/falcades /c/en/falcade/n /c/en/electrical_engineers /c/en/electrical_engineer /c/ku/transitiv /c/ku/berdan/v /c/sv/klassrums/n /c/sv/klassrum 513 /c/la/propinaremur/v /c/la/propino /c/la/friarat/v /c/la/frio /c/la/lucubrere/v /c/la/lucubro /c/la/labascimur/v /c/la/labasco /c/la/interimenti /c/la/interimens 514 /c/es/progresás/v /c/es/progresar /c/es/tintáremos/v /c/es/tintar /c/es/resentíamos/v /c/es/resentirse /c/la/centesimabere/v /c/la/centesimo /c/la/perviguerunt/v /c/la/pervigeo 515 /c/fr/entradmirera/v /c/fr/entradmirer /c/es/coagularé/v /c/es/coagular /c/frp/itèila /c/frp/esèila/n /c/la/furcillas/n /c/la/furcilla /c/en/emanations/n /c/en/emanation 516 /c/no/dådyret/n /c/no/dådyr /c/nl/oppakt/v /c/nl/oppakken /c/de/anwaltsgehilfin /c/de/anwaltsgehilfe/n /c/fi/kaverisektoriin/n /c/fi/kaverisektori /c/it/chicchiriai/v /c/it/chicchiriare 517 /c/it/miracoleremmo/v /c/it/miracolare /c/la/inaudias/v /c/la/inaudio /c/la/extergamus/v /c/la/extergo /c/la/allegavisse/v /c/la/allego /c/de/anarchischeres/a /c/de/anarchisch 518 /c/fr/remailles/v /c/fr/remailler /c/en/ebbtide/n /c/en/ebb_tide /c/fr/froufrouterait/v /c/fr/froufrouter /c/fr/attireront/v /c/fr/attirer /c/en/languaging/v /c/en/language 519 /c/sv/folkminnens/n /c/sv/folkminne /c/la/eluctari/v /c/la/eluctor /c/la/adpares/v /c/la/adparo /c/es/calcinarme/v /c/es/calcinar /c/pt/alocaríeis/v /c/pt/alocar Example triples in 20 FormOf subclusters.", "The cluster IDs are consistent with those used in Figure REF .", "[h!]", "\|c\|c\|c\| 1\|c\| cluster ID 1c\| head 1c\| tail 3l 9 – continued from previous page 1\|c\| cluster ID 1c\| head 1c\| tail 3\|r\|Continued on next page 50 /c/en/immunosenescent/a /c/en/pathology /c/en/handball/v /c/en/soccer /c/en/screenplay/n /c/fr/cinéma /c/en/crivvens /c/en/scotland /c/en/rhizomatic/a /c/en/philosophy 51 /c/fr/sapide/a /c/en/literary /c/hu/szirn/n /c/en/literary /c/ga/eo/n/wikt/en_3 /c/en/literary /c/af/elk/n /c/en/literary /c/ga/gair/v/wikt/en_1 /c/en/literary 52 /c/it/vena_cava/n /c/en/anatomy /c/et/fluor/n /c/fr/chimie /c/fr/saksaoul/n /c/fr/botanique /c/mul/raw /c/fr/linguistique /c/fr/tagbanoua/n /c/fr/linguistique 53 /c/fr/schipluiden/n /c/fr/localités /c/en/brontotherid/n /c/en/zoology /c/fr/de_hem/n /c/fr/localités /c/fr/brozolo/n /c/fr/localités /c/en/onychoteuthid/n /c/en/zoology 54 /c/ha/umra/n /c/en/islam /c/no/jordakse/n /c/en/geometry /c/en/tsar/n /c/en/historical /c/lij/dîsètte /c/en/cardinal /c/mi/iwa /c/en/cardinal 55 /c/fr/scheelite/n /c/en/mineral /c/de/natriumdichromat/n /c/en/inorganic_compound /c/en/oxazepane/n /c/en/organic_compound /c/en/gelsemine/n /c/en/organic_compound /c/en/conductin/n /c/en/protein 56 /c/en/azodicarbonamide/n /c/en/chemistry /c/en/ricinoleate/n /c/en/chemistry /c/fi/rikkiyhdiste/n /c/en/chemistry /c/en/test/v/wikt/en_1 /c/en/chemistry /c/en/vinyl_acetate/n /c/en/chemistry 57 /c/en/business/n /c/en/los_angeles /c/sq/shkretëroj/v /c/en/tosk /c/en/hooklet/n /c/en/natural_history /c/da/femten /c/en/cardinal /c/it/un/a /c/en/sometimes_before_vowel_or_h 58 /c/fr/déontologie/n /c/en/philosophy /c/en/cap_cloud/n /c/en/meteorology /c/en/back_ganging/n /c/en/uk /c/en/syringic/a /c/en/organic_chemistry /c/en/diethenoid/a /c/en/organic_chemistry 59 /c/en/meteor/n /c/en/martial_arts /c/fr/pause/n /c/fr/musique /c/cs/moderátor/n /c/en/uk /c/en/lin/v/wikt/en_1 /c/en/uk /c/ms/kata_benda/n /c/en/grammar 510 /c/en/neurodegeneration/n /c/fr/neurologie /c/no/oppholde/v /c/en/somewhere /c/scn/lu /c/en/definite_article /c/en/monotypy/n /c/en/conservation_biology /c/nl/wao/n /c/en/netherlands 511 /c/sl/kriptozoologija/n /c/fr/biologie /c/nl/neptunus/n /c/fr/astronomie /c/fr/l1/n/wikt/fr_2 /c/fr/astronomie /c/en/freedom_rider/n /c/en/politics /c/fr/corps/n /c/fr/numismatique 512 /c/en/vamptastic/a /c/en/slang /c/de/funzen/v/wikt/en_1 /c/en/slang /c/fi/filu/n /c/en/computing /c/en/dep/n /c/en/computing /c/en/nonserver/a /c/en/computing 513 /c/en/fiscal/n/wikt/en_1 /c/en/legal /c/de/silver_goal/n /c/en/football /c/en/shitcan/v /c/en/vulgar /c/en/eicosanoid/n /c/fr/biochimie /c/rm/mel/n /c/en/rumantsch_grischun 514 /c/fr/océan_atlantique/n /c/fr/géographie /c/sl/balkanski/a /c/fr/géographie /c/fr/riviera/n /c/fr/géographie /c/alt/cyy/n /c/fr/géographie /c/fr/sapouy/n /c/fr/géographie 515 /c/fr/lapedona/n /c/fr/localités /c/fr/tour_de_faure/n /c/fr/localités /c/fr/amendeuix_oneix/n /c/fr/localités /c/fr/espédaillac/n /c/fr/localités /c/fr/rye/n /c/fr/localités 516 /c/fro/voleir/n /c/en/anglo_norman /c/en/stoater/n /c/en/horse_racing /c/nrf/malon/n /c/en/jersey /c/en/antieczema/a /c/en/pharmacology /c/ga/heitribhéascna/n /c/en/linguistics 517 /c/fr/humoral/a /c/fr/médecine /c/en/fasciculatory/a /c/en/medicine /c/fr/antiépileptique/a /c/fr/médecine /c/de/tropf/n /c/fr/médecine /c/fr/sida/n/wikt/fr_1 /c/fr/médecine 518 /c/en/polymodality/n /c/en/mathematics /c/en/biplanar/a /c/en/mathematics /c/it/esaedro/n /c/en/mathematics /c/de/divergieren/v /c/en/mathematics /c/en/local_maximum/n /c/en/mathematics 519 /c/en/thermoelasticity/n /c/en/physics /c/pt/hidrostático/a /c/en/physics /c/en/remanence/n /c/en/physics /c/en/specific/a /c/en/physics /c/en/microelectronvolt/n /c/en/physics Example triples in 20 HasContext subclusters.", "The cluster IDs are consistent with those used in Figure REF .", "Figure: Example triples in SymbolOf's 20 subclusters Because most entities in SymbolOf triples are emojis, we take a screenshot of triples-examples in 20 SymbolOf clusters rather than show these examples in a formal table.", "For each cluster, we randomly select 5 examples; the index in the first column indicates the triple's cluster ID.", "These cluster IDs are consistent with those in Figure , Table , and Table" ] ]	2011.14084
[ [ "PCPs: Patient Cardiac Prototypes" ], [ "Abstract Many clinical deep learning algorithms are population-based and difficult to interpret.", "Such properties limit their clinical utility as population-based findings may not generalize to individual patients and physicians are reluctant to incorporate opaque models into their clinical workflow.", "To overcome these obstacles, we propose to learn patient-specific embeddings, entitled patient cardiac prototypes (PCPs), that efficiently summarize the cardiac state of the patient.", "To do so, we attract representations of multiple cardiac signals from the same patient to the corresponding PCP via supervised contrastive learning.", "We show that the utility of PCPs is multifold.", "First, they allow for the discovery of similar patients both within and across datasets.", "Second, such similarity can be leveraged in conjunction with a hypernetwork to generate patient-specific parameters, and in turn, patient-specific diagnoses.", "Third, we find that PCPs act as a compact substitute for the original dataset, allowing for dataset distillation." ], [ "Introduction", "Modern medical research is arguably anchored around the gold standard of evidence provided by randomized control trials (RCTs) [9].", "However, RCT-derived conclusions are population-based and fail to capture nuances at the individual patient level [1].", "This is primarily due to the complex mosaic that characterizes a patient from demographics, to physiological state, and treatment outcomes.", "Similarly, despite the success of deep learning algorithms in automating clinical diagnoses [14], [4], [5], [20], network-generated predictions remain population-based and difficult to interpret.", "Such properties are a consequence of a network's failure to incorporate patient-specific structure during training or inference.", "As a result, physicians are reluctant to integrate such systems into their clinical workflow.", "In contrast to such reluctance, personalized medicine, the ability to deliver the right treatment to the right patient at the right time, is increasingly viewed as a critical component of medical diagnosis [18].", "The medical diagnosis of cardiac signals such as the electrocardiogram (ECG) is of utmost importance in a clinical setting [31].", "For example, such signals, which convey information about potential abnormalities in a patent's heart, also known as cardiac arrhythmias, are used to guide medical treatment both within and beyond the cardiovascular department [8].", "In this paper, we conceptually borrow insight from the field of personalized medicine in order to learn patient representations which allow for a high level of network interpretability.", "Such representations have several potential clinical applications.", "First, they allow clinicians to quantify the similarity of patients.", "By doing so, network-generated predictions for a pair of patients can be traced back to this similarity, and in turn, their corresponding ECG recordings.", "Allowing for this inspection of ECG recordings aligns well with the existing clinical workflow.", "An additional application of patient similarity is the exploration of previously unidentified patient relationships, those which may lead to the discovery of novel patient sub-cohorts.", "Such discoveries can lend insight into particular diseases and appropriate medical treatments.", "In contrast to existing patient representation learning methods [38], [33], we concurrently optimize for a predictive task (cardiac arrhythmia classification), leverage patient similarity, and design a system specifically for 12-lead ECG signals.", "Contributions.", "Our contributions are the following: Patient cardiac prototypes (PCPs) - we learn representations that efficiently summarize the cardiac state of a patient in an end-to-end manner via contrastive learning.", "Patient similarity quantification - we show that, by measuring the Euclidean distance between PCPs and representations, we can identify similar patients across different datasets.", "Dataset distillation - we show that PCPs can be used to train a network, in lieu of the original dataset, and maintain strong generalization performance." ], [ "Related Work", "Contrastive learning is a self-supervised method that encourages representations of instances with commonalities to be similar to one another.", "This is performed for each instance and its perturbed counterpart [23], [10], [11], [16] and for different visual modalities (views) of the same instance [34].", "Such approaches are overly-reliant on the choice of perturbations and necessitate a large number of comparisons.", "Instead, [7] propose to learn cluster prototypes.", "Most similar to our work is that of [12] and CLOCS [19] which both show the benefit of encouraging patient-specific representations to be similar to one another.", "Although DROPS [3] leverages contrastive learning, it does so at the patient-attribute level.", "In contrast to existing methods, we learn patient-specific representations, PCPs, in an end-to-end manner Meta-learning designs learning paradigms that allow for the fast adaptation of networks.", "Prototypical Networks [29] average representations to obtain class-specific prototypes.", "During inference, the similarity of representations to these prototypes determines the classification.", "Relational Networks [32] build on this idea by learning the similarity of representations to prototypes through a parametric function.", "[15] and [27] exploit hypernetworks [17] and propose to generate the parameters of the final linear layer of a network for few-shot learning on visual tasks.", "In contrast, during inference only, we compute the cosine similarity between representations and PCPs and use the latter as the input to a hypernetwork.", "Patient similarity aims at discovering relationships between patient data [28].", "To quantify these relationships, [24] and [25] propose Patient Similarity Networks for cancer survival classification.", "Exploiting electronic health record data, [38] use Word2Vec to learn patient representations, and [33] propose to exploit patient similarity to guide the re-training of models, an approach which is computationally expensive.", "Instead, our work naturally learns PCPs as efficient descriptors of the cardiac state of a patient." ], [ "Learning Patient Cardiac Prototypes via Contrastive Learning", "We assume the presence of a dataset, $\\mathcal {D}=\\lbrace x_{i},y_{i}\\rbrace _{i=1}^{N}$ , comprising $N$ ECG recordings, $x$ , and cardiac arrhythmia labels, $y$ , for a total of $P_{\\mathrm {tot}}$ patients.", "Typically, multiple recordings are associated with a single patient, $p$ .", "This could be due to multiple recordings within the same hospital visit or multiple visits to a hospital.", "Therefore, each patient is associated with $N/P_{\\mathrm {tot}}$ recordings.", "We learn a feature extractor $f_{\\theta }:x \\in \\mathbb {R}^{D} \\xrightarrow{} h \\in \\mathbb {R}^{E}$ , parameterized by $\\theta $ , that maps a $D$ -dimensional recording, $x$ , to an $E$ -dimensional representation, $h$ .", "In the quest to learn patient-specific representations, we associate each patient, $p$ , out of a total of $P$ patients in the training set with a unique and learnable embedding, $v \\in \\mathbb {R}^{E}$ , in a set of embeddings, $V$ , where $\|V\| = P \\ll N$ .", "Such embeddings are designed to be efficient descriptors of the cardiac state of a patient, and we thus refer to them as patient cardiac prototypes or PCPs.", "We propose to learn PCPs in an end-to-end manner via contrastive learning.", "More specifically, given an instance, $x_{i}$ , that belongs to a particular patient, $k$ , we encourage its representation, $h_{i}=f_{\\theta }(x_{i})$ , to be similar to the same patient's PCP, $v_{k}$ , and dissimilar to the remaining PCPs, $v_{j}$ , $j \\ne k$ .", "We quantify this similarity, $s(h_{i},v_{k})$ , by using the cosine similarity with a temperature parameter, $\\tau $ .", "The intuition is that each PCP, in being attracted to a diverse set of representations that belong to the same patient, should become invariant to insidious intra-patient differences.", "For a mini-batch of size, $B$ , the contrastive loss is as follows.", "$\\mathcal {L}_{contrastive} = - \\sum _{i}^{B} \\log \\left[ \\frac{e^{s(h_{i},v_{k})}}{\\sum _{j}^{P} e^{s(h_{i},v_{j})}} \\right]$ $s(h_{i},v_{j}) = \\frac{f_{\\theta }(x_{i}) \\cdot v_{j}}{\\Vert f_{\\theta }(x_{i}) \\Vert \\Vert v_{j} \\Vert } \\cdot \\frac{1}{\\tau }$ Figure: Training and inference stages of the personalized diagnosis pipeline.", "During training, instance representations, hh, have a dual role.", "They are fed into a hypernetwork, g φ g_{\\phi }, to generate parameters for a linear classification layer, p ω p_{\\omega }, that outputs a prediction, y ^\\hat{y}.", "They are also encouraged to be similar to their corresponding patient cardiac prototype (PCP), vv.", "During inference, the nearest PCP, v k v_{k}, to the representation is fed into the hypernetwork, thus generating patient-specific parameters for classification." ], [ "Generating Patient-specific Parameters via Hypernetworks", "Network parameters are typically updated during training and fixed during inference.", "This allows the parameters to exploit population-based information in order to learn high-level features useful for solving the task at hand.", "Such an approach, however, means that all instances are exposed to the same set of parameters during inference, regardless of instance-specific information.", "Such information can be related to any meta-label including, but not limited to, patient ID, geographical location, and even temporal period.", "As an exemplar, and motivated by the desire to generate patient-specific diagnoses, we focus on patient-specific information.", "We are essentially converting a traditional classification task to one that is conditioned on patient-specific information.", "To perform such conditioning, we propose to exploit both PCPs and hypernetworks, as explained next.", "We assume the presence of a hypernetwork, $g_{\\phi }: h \\in \\mathbb {R}^{E} \\xrightarrow{} \\omega \\in \\mathbb {R}^{E \\times C}$ , parameterized by $\\phi $ , that maps an $E$ -dimensional representation, $h$ , to a matrix of classification parameters, $\\omega $ , where $C$ is the number of class labels.", "During training, we feed a representation, $h_{i}$ , to the hypernetwork and generate instance-specific parameters, $\\omega _{i}$ (see Fig.", "REF left).", "During inference, however, we retrieve, and feed into the hypernetwork, the most similar PCP, $v_{k}$ , to the current representation, $h_{i}$ , (based on similarity metric, $s$ ).", "We chose this strategy after having experimented with several of them (see Sec.", "REF ).", "It is worthwhile to note that although this approach bears some resemblance to clustering, it is distinct from it.", "In a clustering scenario, we would have assigned labels to instances based on their proximity to PCPs.", "In contrast, we are leveraging this proximity to determine the input of a hypernetwork (see Fig.", "REF right).", "$\\omega _{i} = {\\left\\lbrace \\begin{array}{ll} g_{\\phi }(h_{i}) &\\mbox{for training}\\\\g_{\\phi }(v_{k}) &\\mbox{for inference, } v_{k} = \\underset{v_{j}}{\\operatornamewithlimits{arg\\,max}} \\: s(h_{i},v_{j})\\\\\\end{array}\\right.", "}$ By performing this retrieval, we exploit the similarity between patients in the training and inference set.", "As a result, the hypernetwork generates patient-specific parameters that parameterize the linear classifier, $p_{\\omega }: h \\in \\mathbb {R}^E \\xrightarrow{} y \\in \\mathbb {R}^C$ , which maps a representation, $h$ , to a posterior class distribution, $y$ .", "We train the entire network in an end-to-end manner using a combined contrastive and supervised loss.", "$\\mathcal {L}_{supervised} = - \\sum _{i}^{B} \\log p_{\\omega _{i}}(y_{i}=c\|h_{i})$ $\\mathcal {L}_{combined} = \\mathcal {L}_{contrastive} + \\mathcal {L}_{supervised}$" ], [ "Datasets", "We conduct experiments using PyTorch [26] on three large-scale ECG datasets that contain a significant number of patients.", "PhysioNet 2020 ECG consists of 12-lead ECG recordings from 6,877 patients alongside labels corresponding to 9 different classes of cardiac arrhythmia.", "Each recording can be associated with multiple labels.", "Chapman ECG [37] consists of 12-lead ECG recordings from 10,646 patients alongside labels corresponding to 11 different classes of cardiac arrhythmia.", "As is suggested by [37], we group these labels into 4 major classes.", "PTB-XL ECG [35] consists of 12-lead ECG recordings from 18,885 patients alongside 71 different types of annotations provided by two cardiologists.", "We follow the training and evaluation protocol presented by [30] where we leverage the 5 diagnostic class labels.", "We alter the original setup to only consider ECG segments with one label assigned to them and convert the task into a binary classification problem.", "Further details can be found in Appendix REF .", "Unless otherwise mentioned, datasets were split into training, validation, and test sets according to patient ID using a 60, 20, 20 configuration.", "In other words, patients appeared in only one of the sets.", "Further details about the dataset splits can be found in Appendix REF ." ], [ "Hyperparameters", "When calculating the contrastive loss, we chose $\\tau =0.1$ as in [19].", "We also use the same neural network architecture for all experiments.", "Further implementation details can be found in Appendix ." ], [ "Patient Cardiac Prototypes are Discriminative", "During training, we optimize an objective function that consists of both a supervised and contrastive loss term (see Eq.", "REF ).", "Based on the former, we expect representations to exhibit discriminative behaviour for the task at hand.", "The latter term encourages these representations to be similar to PCPs, and thus we also expect PCPs to be discriminative.", "In Fig.", "REF , we illustrate the representations of instances in the training set and the PCPs after being projected to a 2-dimensional subspace using t-SNE and colour-coded according to their class label.", "We find that both training representations, $h$ , and PCPs, $v$ , are class-discriminative.", "This can be seen by the high separability of the projected representations along class boundaries.", "Based on this finding alone, one could make the argument that PCPs are trivially detecting class label differences between patients.", "Figure: t-SNE projection of (a) representations, h∈ℝ 128 h \\in \\mathbb {R}^{128}, of instances in the training set of the Chapman dataset and (b) PCPs, v∈ℝ 128 v \\in \\mathbb {R}^{128}, learned on the training set, colour-coded according to the arrhythmia label assigned to each patient.", "Learned PCPs are also class-discriminative." ], [ "Effect of Hypernetwork Input Strategy on Performance", "As described, our pipeline uses the PCP nearest to each representation as input to the hypernetwork.", "This approach places a substantial dependency on that single chosen PCP.", "Therefore, we explore three additional input strategies that incorporate PCPs differently.", "Nearest 10 searches for, and takes the average of, the 10 PCPs that are nearest to the representation.", "Mean simply takes the average of all PCPs.", "Similarity-Weighted Mean takes a linear combination of all PCPs, weighted according to their cosine similarity to the representation.", "In Fig.", "REF , we show the effect of these strategies on the test set AUC as the embedding dimension, $E$ , is changed.", "Figure: AUC on test set of Chapman dataset as a function of hypernetwork input strategies and embedding dimension, EE.", "Bars are averaged across five seeds and the error bars illustrate one standard deviation.", "The Nearest 10\\mathrm {Nearest\\:10} input strategy outperforms its counterparts and is unaffected by changes to the embedding dimension.We find that exploiting the similarity of representations during inference benefits the generalization performance of the network.", "This is shown by the inferiority of the mean strategy relative to the remaining strategies.", "For example, at $E=256$ , the $\\mathrm {Mean}$ strategy achieves an $\\mathrm {AUC} \\approx 0.50$ , equivalent to a random guess.", "However, simply weighting those PCPs according to their similarity to representations, as exemplified by $\\mathrm {Similarity\\:Weighted\\:Mean}$ achieves an $\\mathrm {AUC} \\approx 0.65$ .", "This implies that representations are capturing patient-specific information.", "We also find that it is more advantageous to exploit similarity to identify the nearest PCPs than to weight many PCPs.", "In Fig.", "REF , the $\\mathrm {Nearest}$ and $\\mathrm {Nearest\\:10}$ input strategies perform best, with the latter achieving an $\\mathrm {AUC} \\approx 0.90$ , regardless of the embedding dimension.", "We hypothesize that such behaviour can be attributed to the notion that fewer PCPs are less likely to overwhelm the hypernetwork.", "This, in turn, allows the hypernetwork to generate reasonable parameters for the linear classification layer.", "Moreover, the strong performance of these strategies despite their high dependence on so few PCPs reaffirms the utility of the learned representations." ], [ "Patient Cardiac Prototypes are Patient-Specific", "So far, we have shown that PCPs are class-discriminative and can assist in achieving strong generalization performance.", "In this section, we aim to validate our initial claim that PCPs are patient-specific.", "To do so, we calculate the Euclidean distance between each PCP and two sets of representations.", "The first includes representations corresponding to the same patient as that of the PCP (PCP to Same Training Patient).", "The second includes representations that correspond to all remaining patients (PCP to Different Training Patients).", "In Fig.", "REF , we illustrate the distribution of these distances.", "Figure: Distribution of pairwise (left) Euclidean and (right) cosine distance from the learned PCPs on the Chapman dataset to three sets of representations: those in the training set that belong to the same patient (blue), those in the training set that belong to different patients (orange), and those in the validation set (purple).", "PCPs are patient-specific since they are closer to representations belonging to the same patient than they are to representations belonging to different patients.We find that PCPs are indeed patient-specific.", "This can be seen by the smaller average distance between PCPs and representations of the same patient ($\\approx 4.5$ ) than between PCPs and representations of different patients ($\\approx 9.5$ ).", "Such a finding implies that PCPs are, on average, a factor of two more similar to representations from the same patient than they are to those from other patients.", "We also set out to investigate whether computing their similarity to representations of instances in the validation set (as is done in Fig.", "REF ) was appropriate.", "In Fig.", "REF , we overlay the distribution of distances between the PCPs and representations of instances from the validation set (PCP to Validation Patients).", "We find that comparing PCPs to representations of instances in the validation set is appropriate.", "This is emphasized by how the average Euclidean distance between these two entities ($\\approx 9$ ) is on the same order of the average Euclidean distance between PCPs and representations of instances in the training set ($\\approx 4$ ).", "Based on the distributions in Fig.", "REF , we can also confirm that patients in the validation set are not present in the training set, as was enforced by design.", "This can be seen by the minimal overlap between the blue and purple distributions.", "Such a finding suggests that PCPs can be deployed to detect out-of-distribution patients or distribution shift.", "We also illustrate the generalizability of these findings by arriving at the same conclusion on the PTB-XL and PhysioNet 2020 datasets (see Appendix )." ], [ "Discovery of Similar Patients via Patient Cardiac Prototypes", "Having established that PCPs are patient-specific and class-discriminative, we now investigate whether they can be exploited to quantify patient similarity.", "From a clinical perspective, such information can allow physicians to discover similar patient sub-cohorts and guide medical treatment.", "This is particularly consequential when patients are located across different healthcare institutions.", "Patient similarity quantification also has the potential to add a layer of interpretability to exigent network-generated diagnoses.", "In this section, we exploit, and validate the ability of, PCPs to identify similar (and dissimilar) patients.", "To quantify patient similarity, we compute the pairwise distance (e.g., Euclidean) between each PCP and each representation in the validation set.", "The distribution of these distances can be found in Fig.", "REF (top).", "We average these distances across representations that belong to the same patient, and generate a matrix of distances between pairs of patients (see Fig.", "REF centre for a subset of that matrix).", "Validating our findings, however, is non-trivial since similarity can be based on patient demographics, physiology, or treatment outcomes.", "With this in mind, we decide to validate our findings both qualitatively and quantitatively.", "For the former, we locate the cell in the distance matrix with the lowest distance, and in turn, identify the most similar pair of patients.", "We then visualize their corresponding 12-lead ECG recordings (Fig.", "REF bottom).", "Figure: (Top) Distribution of all pairwise Euclidean distances between PCPs and representations in the validation set.", "(Centre) Matrix illustrating average pairwise distances between a subset of PCPs and representations of patients in the validation set.", "(Bottom) Visualization of the 12-lead ECG recordings of the two patients identified as being most similar by our method.", "Both recordings are similar and correspond to the same arrhythmia, supra-ventricular tachycardia, thus lending support to PCPs as a reliable patient-similarity tool.We find that PCPs are able to sufficiently distinguish between unseen patients and thus act as reasonable patient-similarity tools.", "In Fig.", "REF (centre), we see that there exists a large range of distance values for any chosen PCP (row).", "In other words, it is closer to some representations than to others, implying that a chosen PCP is not trivially equidistant to all other representations.", "However, distinguishing between patients is not sufficient for a patient-similarity tool.", "We show that PCPs can also correctly capture the relative similarity to these patients.", "In Fig.", "REF (bottom), we show that the two patients identified as being most similar to one another, using our method, have ECG recordings with a similar morphology and arrhythmia label, supra-ventricular tachycardia.", "We hypothesize that this behaviour arises due to the ability of PCPs to efficiently summarize the cardiac state of a patient.", "Such a finding reaffirms the potential of PCPs as patient-similarity tools.", "We also repeat the above procedure in attempt to discover similar and dissimilar patients across different datasets.", "In doing so, we arrive at positive results and similar conclusions to those above (see Appendix ).", "Figure: Percentage of retrieved pairs of patients with a matching cardiac arrhythmia label.", "At a precision of 0.900.90, we can identify a threshold distance between patients (e.g., d E ≈6.2d_{E} \\approx 6.2).We now move on to quantitatively validate the PCP-derived patient similarity values.", "Conceptually, we build on our qualitative findings and assume that a pair of patients, identified as being similar by our method, are in fact similar if they happen to be associated with the same cardiac arrhythmia label.", "More specifically, we retrieve all pairs of patients that are more similar than some threshold distance, $d_{E}$ , and determine what percentage of such retrieved pairs consist of patients with a matching cardiac arrhythmia label ($\\mathrm {Precision}$ ).", "In Fig.", "REF , we illustrate this precision as a function of different threshold distances.", "We find that PCP-derived similarity values are able to identify patients with matching cardiac arrhythmia labels.", "For example, $>90\\%$ of the pairs of patients that are deemed very similar to one another (i.e., $d_{E}<6.0$ ) exhibit a perfect cardiac arrhythmia label match.", "As we increase the threshold distance, $d_{E} \\rightarrow 8.5$ , we see that $\\mathrm {Precision} \\rightarrow 0$ .", "Such a decay is expected of a reasonable similarity metric where patients that are deemed dissimilar do not match according to their cardiac arrhythmia labels.", "Moreover, based on an acceptable level of precision, (e.g., $0.90$ ), we can identify an appropriate threshold distance (e.g., $d_{E} \\approx 6.2$ ).", "It is worthwhile to note that this specific threshold coincides with the region of minimal distribution overlap we observed in Fig.", "REF .", "This suggests that a simple threshold can be derived from those distributions." ], [ "Dataset Distillation via Patient Cardiac Prototypes", "Our interpretation and the growing evidence we have presented in support of PCPs as efficient descriptors of the cardiac state of a patient led us to investigate the following question: could PCPs be sufficient for training classification tasks, in lieu of the original dataset?", "This idea is analogous to dataset distillation which focuses on obtaining a coreset of instances that do not compromise the generalization capabilities of a model [13], [36].", "To investigate the role of PCPs as dataset distillers, we train a Support Vector Machine (SVM) on them and evaluate the model on representations of held-out instances.", "We compare PCPs to three coreset construction methods: 1) Lucic [21], 2) Lightweight [6], and 3) Archetypal [22].", "In constructing the coreset, these methods generate a categorical proposal distribution over all instances in the dataset before sampling $k$ instances and assigning them weights.", "For a fair comparison to our method, we chose $k=P$ where $P$ is the number of PCPs.", "In addition to exploring the effect of these coreset construction methods based on raw instances, we do so based on representations of instances learned via our network.", "In Table REF , we illustrate the performance of these methods on various datasets.", "Table: Validation AUC after training an SVM on a different fraction of available PCPs (E=32E=32).", "By training on 100%100\\% of PCPs (N=6387N=6387), we achieve similar performance to when training on the full training set (N=76644N=76644), illustrating the dataset distillation capabilities of PCPs.", "Results are shown across 5 seeds.In Table REF , we find that coresets of raw instances generated by traditional coreset construction methods are insufficient for achieving strong generalization performance.", "For example, the $\\mathrm {Archetypal}$ method achieves an $\\mathrm {AUC}=54.8$ on Chapman.", "Such poor performance is likely attributed to the poor class separability of the input features.", "Nonetheless, we show that the exact same set of methods still perform poorly, albeit slightly better, even when constructing coresets from network-derived representations that have been shown to be separable (see Fig.", "REF ).", "For example, the $\\mathrm {Archetypal}$ method now achieves an $\\mathrm {AUC}=58.1$ on Chapman.", "In contrast, we show that PCPs are relatively more effective coresets than those constructed by the remaining methods.", "On Chapman, for example, PCPs achieve an $\\mathrm {AUC}=88.7$ .", "These findings suggest that PCPs have the potential to effectively summarize datasets in a compact manner and act as dataset distillation tools.", "Having shown the utility of PCPs as dataset distillers, we wanted to investigate the extent to which further distillation was possible.", "In Fig.", "REF , we illustrate the generalization performance of models trained on a different fraction of the available PCPs.", "For comparison's sake, we also show the AUC of our network when trained on all instances in the training set (Full Training Set), which is several folds larger than the number of PCPs.", "We find that PCPs do indeed act as effective dataset distillers.", "In Fig.", "REF , we show that training on $100\\%$ of the PCPs ($N=6,387$ ) achieves an $\\mathrm {AUC} \\approx 0.89$ which is similar to that achieved when training on the full training set ($N=76,614$ ).", "In other words, we achieve similar generalization performance despite a 12-fold decrease in the number of training instances.", "We also show that further reducing the number of PCPs, by selecting a random subset for training, does not significantly hurt performance.", "For example, training with only $5\\%$ of available PCPs ($N=319$ ) achieves an $\\mathrm {AUC} \\approx 0.82$ .", "Concisely, this corresponds to a $7\\%$ reduction in performance despite a 240-fold decrease in the number of training instances relative to that found in the standard training procedure.", "We arrive at similar conclusions when changing the embedding dimension, $E$ (see Appendix ).", "Such a finding supports the potential of PCPs at dataset distillers.", "We hypothesize that this behaviour arises due to our patient-centric contrastive learning approach.", "By encouraging each PCP to be similar to several representations of instances that belong to the same patient, it is able to capture the most pertinent information and shed that which is least useful." ], [ "Discussion and Future Work", "In this paper, we proposed to learn efficient representations of the cardiac state of a patient, entitled patient cardiac prototypes, using a combination of contrastive and supervised learning.", "We showed that patient cardiac prototypes are both patient-specific and discriminative for the task at hand.", "We successfully deployed PCPs for the discovery of similar patients within the same dataset and across different datasets.", "This opens the door to leveraging clinical information that is available in disparate healthcare institutions.", "Lastly, we illustrated the potential of PCPs as dataset distillers, where they can be used to train models in lieu of larger datasets without compromising generalization performance.", "We now elucidate several future avenues worth exploring.", "Obtaining multi-modal summary of cardiac state of patient.", "Although our approach was validated on multiple, large, time-series datasets, it was limited to a single modality, the ECG.", "Incorporating additional modalities to our approach such as coronary angiograms, cardiac MRI, and cardiac CT, may provide a more reliable summary of the cardiac state of the patient.", "This could ultimately lead to more reliable patient similarity quantification.", "Guiding design of graph neural networks.", "Arriving at ground-truth values for the similarity of a pair of patients is non-trivial.", "Recently, graph neural networks have been relatively successful at discovering and quantifying the similarity of instances, but most necessitate a pre-defined graph structure, which may be difficult to design in the context of physiological signals.", "We believe that designing this graph structure can be facilitated by our patient-similarity scores, for instance, by using them as an initialization of the edge weights between nodes." ], [ "Data Preprocessing", "For all of the datasets, frames consisted of 2500 samples and consecutive frames had no overlap with one another.", "Data splits were always performed at the patient-level.", "PhysioNet 2020 [2].", "Each ECG recording varied in duration from 6 seconds to 60 seconds with a sampling rate of 500Hz.", "Each ECG frame in our setup consisted of 2500 samples (5 seconds).", "We assign multiple labels to each ECG recording as provided by the original authors.", "These labels are: AF, I-AVB, LBBB, Normal, PAC, PVC, RBBB, STD, and STE.", "The ECG frames were normalized in amplitude between the values of 0 and 1.", "Chapman [37].", "Each ECG recording was originally 10 seconds with a sampling rate of 500Hz.", "We downsample the recording to 250Hz and therefore each ECG frame in our setup consisted of 2500 samples.", "We follow the labelling setup suggested by [37] which resulted in four classes: Atrial Fibrillation, GSVT, Sudden Bradychardia, Sinus Rhythm.", "The ECG frames were normalized in amplitude between the values of 0 and 1.", "PTB-XL [35].", "Each ECG recording was originally 10 seconds with a sampling rate of 500Hz.", "We extract 5-second non-overlapping segments of each recording generating frames of length 2500 samples.", "We follow the diagnostic class labelling setup suggested by [35] which resulted in five classes: Conduction Disturbance (CD), Hypertrophy (HYP), Myocardial Infarction (MI), Normal (NORM), and Ischemic ST-T Changes (STTC).", "We alter the original setup in two main ways.", "Firstly, we only consider ECG segments with one label assigned to them.", "Secondly, we convert the task into a binary classification problem of NORM vs. (CD, HYP, MI, STTC) from above.", "The ECG frames were normalized in amplitude between the values of 0 and 1." ], [ "Data Samples", "In this section, we outline the number of instances used during training.", "Table: Number of instances (number of patients) used during training.", "These represent sample sizes for all 12 leads." ], [ "Network Architecture", "In this section, we outline the architecture of the neural network used for all experiments.", "Table: Network architecture used for all experiments.", "K, Cin, and Cout represent the kernel size, number of input channels, and number of output channels, respectively.", "A stride of 3 was used for all convolutional layers.", "EE represents the dimension of the final representation." ], [ "Patient Cardiac Prototypes are Patient-Specific", "We claim that the patient cardiac prototypes that we learn in an end-to-end manner are also patient-specific.", "In the main manuscript, we provided evidence for this in the form of Euclidean distances between representations.", "We showed that PCPs are much closer to representations of instances that correspond to the same patient than they are to representations of instances from different patients.", "To determine the generalizability of these claims to other datasets, we reproduce Fig.", "REF (in the main manuscript) for two additional datasets, PTB-XL and PhysioNet 2020.", "In Figs.", "REF and REF , we illustrate the distribution of Euclidean distances between the aforementioned representations for the two datasets, respectively." ], [ "PTB-XL", "Patient cardiac prototypes are indeed patient-specific.", "In Fig.", "REF , this is supported by the smaller average Euclidean distances between PCPs and representations of instances of the same patient than between PCPs and representations of instances from different patients (average Euclidean distance $\\approx 7$ vs. 10, respectively.", "The same conclusion was arrived at in the main manuscript.", "Furthermore, PCPs have the potential to be used for the detection of out-of-distribution data.", "The high overlap between the PCP to Validation Patients and PCP to Different Training Patients has a twofold implication.", "First, it suggests that instances in the validation set belong to patients not found in the training set (by design).", "Second, that patients in the validation set, on average, belong to the same overall distribution of patients.", "Figure: Distribution of pairwise Euclidean distance from the learned PCPs on the PTB-XL dataset to three sets of representations: those in the training set that belong to the same patient (blue), those in the training set that belong to different patients (orange), and those in the validation set (purple).", "PCPs are patient-specific since they are closer to representations belonging to the same patient than they are to representations belonging to different patients." ], [ "PhysioNet 2020", "As shown for the Chapman and PTB-XL datasets, we also show that PCPs are patient-specific when trained on the PhysioNet 2020 dataset.", "This is emphasized by the high degree of separability between the PCP to Same Training Patient and PCP to Different Training Patients distributions.", "We claim that the instances in the PTB-XL dataset exhibit a higher degree of diversity relative to those in other datasets.", "We see this by comparing the PCP to Same Training Patient distribution which has a larger mean ($\\approx 8$ ) compared to $\\approx 4$ (Fig.", "REF ) and 7 (Fig.", "REF ) for the Chapman and PTB-XL datasets, respectively.", "Such a finding implies that the PCPs had a more difficult time summarizing the representations that belong to the same patient.", "Figure: Distribution of pairwise Euclidean distance from the learned PCPs on the PhysioNet 2020 dataset to three sets of representations: those in the training set that belong to the same patient (blue), those in the training set that belong to different patients (orange), and those in the validation set (purple).", "PCPs are patient-specific since they are closer to representations belonging to the same patient than they are to representations belonging to different patients." ], [ "Discovering (Dis)similar Patients", "In the main manuscript, we showed that patients identified as being similar, based on the pairwise Euclidean distance between PCPs and representations of instances in the validation set, are indeed similar.", "In this section, we use additional datasets to further validate the role of PCPs as tools for quantifying patient similarity (Secs.", "REF and REF ) and dissimilarity (Secs.", "REF and REF ).", "We quantify patient similarity by calculating the pairwise Euclidean distance between their representations.", "More specifically, when computing the similarity between patients in the training set, we calculate the Euclidean distance between the PCPs.", "In contrast, when comparing patients in the validation set to those in the training set, we calculate the Euclidean distance between the latter's PCP and the former's representations.", "Such pairwise distances are averaged across the multiple instances that may exist for the same patient in the validation set." ], [ "Quantification of Patient Similarity Across Datasets (Chapman $\\xrightarrow{}$ PTB-XL)", "In this section, we attempt to discover similar patients across datasets.", "To do so, we compute the pairwise Euclidean distance between the PCPs pf each dataset.", "The distribution of all these distances are shown in Fig.", "REF (top).", "We also illustrate the pairwise distances for a subset of the PCPs in the Chapman and PTB-XL dataset Fig.", "REF (centre).", "From a clinical perspective, such a matrix provides physicians with the ability to identify patients that are most similar to the one they are currently diagnosing or treating.", "This information can help guide future clinical intervention.", "By locating the pair of PCPs with the lowest Euclidean distance, we identify the pair of patients from each dataset that are most similar to one another.", "We visualize their corresponding 12-Lead ECG recordings in Fig.", "REF (bottom).", "Figure: Identification of the two most similar patients in the training sets of the Chapman and PTB-XL dataset.", "(top) Distribution of all pairwise Euclidean distances between PCPs and representations in the validation set.", "(centre) Matrix illustrating pairwise distances between a subset of PCPs and representations of patients in the validation set.", "(bottom) Visualization of the 12-Lead ECG recordings of the two most similar patients.", "Both recordings are similar and correspond to the same morphology, sinus rhythm, which is considered normal." ], [ "Chapman", "To further validate the PCPs and their ability to discern between patients, we use the complete version of Fig.", "REF to identify two patients deemed dissimilar according to their Euclidean distance.", "In Fig.", "REF , we illustrate the 12-Lead ECG segments corresponding to these two patients.", "The different morphology of the ECG segments between patients and the different arrhythmia labels (Sinus Rhythm vs. Atrial Fibrillation) show that these patients are indeed dissimilar.", "Such a finding reaffirms the notion that PCPs both capture the cardiac state of the patient and allow for reliably patient similarity quantification.", "Figure: 12-Lead ECG segments corresponding to two dissimilar patients in the training and validation set of the Chapman dataset.", "Similarity is defined as low Euclidean distance between patient cardiac prototypes (PCPs) and representations of instances in the validation set.", "ECG segments between patients exhibit different morphology and correspond to the different arrhythmia labels (Sinus Rhythm vs. Atrial Fibrillation)." ], [ "Visualization of Dissimilar Patients Across Different Datasets (Chapman $\\xrightarrow{}$ PTB-XL)", "In this section, we leverage the distance matrix visualized in Fig.", "REF to identify the two most dissimilar patients across the Chapman and PTB-XL datasets.", "In Fig.", "REF , we visualize the 12-Lead ECG recordings of this pair of patients.", "PCPs are able to reliably identify two patients that are dissimilar.", "This can be seen the drastically different ECG morphology present in Fig.", "REF .", "The patient in the Chapman dataset is suffering from sudden bradycardia, a decrease in the rate at which the heart beats.", "In contrast, the patient from the PTB-XL dataset is suffering from changes to the ST segment of the ECG recording.", "Such a finding reaffirms our interpretation of PCPs as reliable descriptors of the cardiac state of a patient.", "Figure: 12-Lead ECG segments corresponding to two dissimilar patients in the training set of the Chapman and PTB-XL datasets, respectively.", "Similarity is defined as low Euclidean distance between patient cardiac prototypes (PCPs) and representations of instances in the validation set.", "ECG segments between patients exhibit different morphology and correspond to the different arrhythmia labels (Sudden Bradycardia vs. Ischemic ST-T Changes)." ], [ "Effect of Number of Leads on Performance", "In the main manuscript, we conducted experiments with all 12 leads of an ECG.", "However, the availability of all 12 leads for training is not always guaranteed.", "This can be the case, for instance, in low-resource clinical settings where medical infrastructure is lacking or in the context of home-monitoring where wearable sensors are used.", "To investigate the robustness of our method to such scenarios, we repeat a subset of the experiments in the presence of only 4 leads (II, V2, aVR, aVL), whose results can be found in Table.", "REF .", "Table: AUC on test set of Chapman dataset in the presence of a different number of leads.", "The inference strategy involves using the nearest PCP (E=128E=128) as input to the hypernetwork.", "Results are shown for five seeds." ], [ "Dataset Distillation - Training with Patient Cardiac Prototypes", "In the main manuscript, we illustrated the potential applicability of PCPs as dataset distillers.", "Namely, PCPs can act as a compact core-set that does not compromise the generalization performance of a model.", "In this section, we build on those results and illustrate the utility of PCPs as dataset distillers as we change the dimension of the representation, $E=[64,128,256]$ ." ], [ "Chapman", "We show that the embedding dimension, $E$ , has a significant effect on the dataset distillation capabilities of PCPs.", "In Fig.", "REF , at an embedding dimension, $E=128$ , the performance drop due to training with $100\\%$ of PCPs relative to training with the full training set is minimal, $\\Delta \\mathrm {AUC} \\approx 0.90 - 0.89 = 0.01$ .", "In contrast, at $E=256$ , this performance drop is more substantial $\\Delta \\mathrm {AUC} \\approx 0.905 - 0.86 = 0.045$ .", "Such a finding suggests that more attention should be given to the embedding dimension when designing dataset distillation methods.", "Figure: Validation AUC after training an SVM on a different fraction of PCPs available.", "The generalization performance when training on the full training set is also shown.", "(a)-(c) illustrate the effect of changing the embedding dimension, EE, on the generalization performance.", "Results are shown across 5 seeds." ] ]	2011.14227
[ [ "Sharp Convergence Rates for Darcy's Law" ], [ "Abstract This paper is concerned with Darcy's law for an incompressible viscous fluid flowing in a porous medium.", "We establish the sharp $O(\\sqrt{\\e})$ convergence rate in a periodically perforated and bounded domain in $R^d$ for $d\\ge 2$, where $\\e$ represents the size of solid obstacles.", "This is achieved by constructing two boundary correctors to control the boundary layers created by the incompressibility condition and the discrepancy of boundary values between the solution and the leading term in its asymptotic expansion.", "One of the correctors deals with the tangential boundary data, while the other handles the normal boundary data." ], [ "This paper is concerned with Darcy's law for an incompressible viscous fluid in a porous medium.", "More precisely, we consider the Dirichlet problem for the steady Stokes equations, $\\left\\lbrace \\aligned -\\varepsilon ^2 \\mu \\Delta u_\\varepsilon + \\nabla p_\\varepsilon & = f & \\quad & \\text{ in } \\Omega _\\varepsilon ,\\\\\\text{\\rm div} (u_\\varepsilon ) & =0 & \\quad & \\text{ in } \\Omega _\\varepsilon ,\\\\u_\\varepsilon & = 0 & \\quad & \\text{ on } \\partial \\Omega _\\varepsilon ,\\endaligned \\right.$ where $\\mu >0$ is the viscosity constant, $0< \\varepsilon < 1$ , and $\\Omega _\\varepsilon $ is a periodically perforated and bounded domain in $\\mathbb {R}^d$ , $d\\ge 2$ .", "In (REF ) we have normalized the velocity vector by a factor $\\varepsilon ^2$ , where $\\varepsilon $ is the period.", "To describe the porous domain $\\Omega _\\varepsilon $ , we let $Y=[0, 1]^d$ be a closed unit cube and $Y_s $ (solid part) an open subset of $Y$ with Lipschitz boundary.", "Throughout the paper we shall assume that dist$(\\partial Y, \\partial Y_s)>0$ and that $Y_f =Y\\setminus \\overline{Y_s}$ (the fluid part) is connected.", "Let $\\Omega $ be a bounded domain in $\\mathbb {R}^d$ with Lipschitz boundary.", "For $0< \\varepsilon <1$ , define $\\Omega _\\varepsilon = \\Omega \\setminus \\bigcup _k \\varepsilon \\left(\\overline{Y_s} +z_k\\right),$ where $z_k \\in \\mathbb {Z}^d$ and the union is taken over those $k$ 's for which $\\varepsilon (Y+z_k) \\subset \\Omega $ .", "For $f\\in L^2(\\Omega ; \\mathbb {R}^d)$ , let $(u_\\varepsilon , p_\\varepsilon )\\in H_0^1(\\Omega _\\varepsilon ; \\mathbb {R}^d) \\times L^2(\\Omega _\\varepsilon )$ be the weak solution of (REF ) with $\\int _{\\Omega _\\varepsilon } p_\\varepsilon \\, dx =0$ .", "We extend $u_\\varepsilon $ to the whole domain $\\Omega $ by zero and still denote the extension by $u_\\varepsilon $ .", "Let $P_\\varepsilon $ be the extension of $p_\\varepsilon $ to $\\Omega $ , defined by (REF ).", "It has been known since late 1970's that as $\\varepsilon \\rightarrow 0$ , $u_\\varepsilon \\rightarrow u_0$ weakly in $L^2(\\Omega ; \\mathbb {R}^d)$ and $P_\\varepsilon \\rightarrow p_0$ strongly in $L^2(\\Omega )$ , where $(u_0, p_0)$ is given by a Darcy law, $\\left\\lbrace \\aligned & u_0= \\mu ^{-1} K (f-\\nabla p_0) & \\quad & \\text{ in } \\Omega ,\\\\& \\text{\\rm div} ( u_0) =0 & \\quad & \\text{ in } \\Omega ,\\\\& u_0\\cdot n =0 & \\quad &\\text{ on } \\partial \\Omega ,\\endaligned \\right.$ with $\\int _\\Omega p_0\\, dx =0$ .", "In (REF ) the permeability matrix $K=(K_j^i) $ is a $d\\times d$ positive definite and symmetric matrix defined by (REF ), and $n$ denotes the outward unit normal to $\\partial \\Omega $ .", "Furthermore, it was observed in [3] by G. Allaire that as $\\varepsilon \\rightarrow 0$ , $u_\\varepsilon -\\mu ^{-1} W(x/\\varepsilon ) ( f- \\nabla p_0) \\rightarrow 0 \\quad \\text{ strongly in } L^2(\\Omega ; \\mathbb {R}^d),$ where $W(y)=(W_j^i (y) ) $ is an 1-periodic $d\\times d$ matrix defined by the cell problem (REF ) and $_Y W(y)\\, dy =K$ .", "For an excellent exposition on Darcy's law and closely related topics, we refer the reader to [4] by G. Allaire.", "The purpose of this paper is to study the convergence rates for $u_\\varepsilon - \\mu ^{-1} W(x/\\varepsilon ) (f-\\nabla p_0)$ and $P_\\varepsilon -p_0$ in $L^2(\\Omega )$ .", "The following is the main result of the paper.", "The $O(\\sqrt{\\varepsilon })$ rate in (REF ) is sharp.", "Theorem 1.1 Let $\\Omega $ be a bounded $C^{2, \\alpha }$ domain in $\\mathbb {R}^d$ , $d\\ge 2$ for some $\\alpha >0$ .", "Also assume that $Y_s$ is an open subset of $Y=[0, 1]^d$ with $C^{1, \\alpha }$ boundary.", "Let $(u_\\varepsilon , p_\\varepsilon )\\in H_0^1(\\Omega _\\varepsilon ; \\mathbb {R}^d) \\times L^2(\\Omega _\\varepsilon )$ be a weak solution of (REF ), where $f\\in C^{1, 1/2}(\\overline{\\Omega }; \\mathbb {R}^d)$ and $\\int _{\\Omega _\\varepsilon } p_\\varepsilon \\, dx =0$ .", "Then $\\aligned & \\Vert u_\\varepsilon - \\mu ^{-1}W (x/\\varepsilon ) (f -\\nabla p_0)\\Vert _{L^2(\\Omega )}+\\Vert P_\\varepsilon - p_0 \\Vert _{L^2(\\Omega )} \\\\& \\qquad \\qquad + \\Vert \\varepsilon \\nabla u_\\varepsilon -\\mu ^{-1} \\nabla W(x/\\varepsilon ) (f-\\nabla p_0)\\Vert _{L^2(\\Omega )}\\le C \\sqrt{\\varepsilon }\\, \\Vert f\\Vert _{C^{1, 1/2} (\\Omega )},\\endaligned $ where $C$ depends only on $d$ , $\\mu $ , $\\Omega $ , and $Y_s$ .", "The first rigorous proof of Darcy's law by homogenization was given by L. Tartar in an appendix of [22], using an energy method.", "We refer the reader to [4] for references on earlier work on the formal derivation of Darcy's law, using two-scale asymptotic expansions.", "In [2], [3], the strong convergence of $(u_\\varepsilon , P_\\varepsilon )$ in $L^2(\\Omega )$ was established by the method of two-scale convergence.", "Also see related work in [15], [19], [8], [17], [18], [16].", "Regarding the rate of convergence for $(u_\\varepsilon , P_\\varepsilon )$ in $L^2(\\Omega )$ , to the best of the author's knowledge, the only previous result for a bounded domain with the Dirichlet condition was obtained by E. Marušić-Paloka and A. Mikelić in [17], where a rate $O(\\varepsilon ^{1/6})$ was established for the case $d=2$ .", "See [8] for an earlier result for a unbounded domain $\\Omega =(0, L) \\times \\mathbb {R}_+$ .", "We remark that for Laplace's equation and systems of linear elasticity, quantitative error estimates have been established in [14], [11], [21], [10], [9].", "As pointed out in [17], the simple cut-off argument, which seems to work well for standard elliptic equations and systems, does not yield any convergence rate for the Stokes equations because of the incompressibility condition.", "In [17], using a stream function from [23], a boundary corrector was constructed in the case $d=2$ to control the boundary layer near $\\partial \\Omega $ created by the incompressibility condition.", "We mention that [17] also treated the case of nonlinear stationary Navier-Stokes equations.", "We now describe our approach to the problem of convergence rates and error estimates, which is based on energy estimates.", "Let $u(x, x/\\varepsilon ) = \\mu ^{-1} W(x/\\varepsilon ) \\big ( f(x) -\\nabla p_0(x)\\big ).$ To address the discrepancy of boundary values between $u_\\varepsilon $ and $u(x, x/\\varepsilon )$ as well as the incompressibility condition, we introduce two boundary correctors $(\\Psi _t, q_t) $ and $(\\Psi _n, q_n) $ .", "Let $\\partial \\Omega _\\varepsilon =\\partial \\Omega \\cup \\Gamma _\\varepsilon $ .", "The tangential boundary corrector $(\\Psi _t, q_t)$ is a weak solution of $\\left\\lbrace \\aligned -\\varepsilon ^2 \\mu \\Delta \\Psi _t +\\nabla q_t & =0 & \\quad & \\text{ in } \\Omega _\\varepsilon ,\\\\\\text{\\rm div}(\\Psi _t) & =0& \\quad & \\text{ in }\\ \\Omega _\\varepsilon ,\\endaligned \\right.$ with boundary data $\\Psi _t =0$ on $\\Gamma _\\varepsilon $ , and $\\Psi _t = - u(x, x/\\varepsilon ) + [ u(x, x/\\varepsilon ) \\cdot n] n \\quad \\text{ on } \\partial \\Omega .$ Note that $\\Psi _t \\cdot n=0$ on $\\partial \\Omega $ .", "This gives, $\\Vert \\nabla \\Psi _t \\Vert ^2_{L^2(\\Omega _\\varepsilon )} \\le \\Vert \\nabla \\Psi _t \\Vert _{L^2(\\partial \\Omega )} \\Vert \\Psi _t\\Vert _{L^2(\\partial \\Omega )}.$ We use a localized Rellich estimate in a Lipschitz domain to show that $\\Vert \\nabla \\Psi _t \\Vert _{L^2(\\partial \\Omega )}\\le C\\left\\lbrace \\Vert \\nabla _{\\tan } \\Psi _t \\Vert _{L^2(\\partial \\Omega )}+ \\varepsilon ^{-1/2} \\Vert \\nabla \\Psi _t \\Vert _{L^2(\\Omega _\\varepsilon )} \\right\\rbrace ,$ where $\\nabla _{\\tan } \\Psi _t$ denotes the tangential gradient of $\\Psi _t$ on the boundary $\\partial \\Omega $ .", "The desired $O(\\sqrt{\\varepsilon })$ bound for $\\varepsilon \\Vert \\nabla \\Psi _t \\Vert _{L^2(\\Omega _\\varepsilon )}$ follows from (REF ) and (REF ).", "See Section for details.", "The normal boundary corrector $(\\Psi _n, q_n)$ is defined as the solution of the Stokes equations (REF ) in $\\Omega _\\varepsilon $ , with the boundary conditions $\\Psi _n =0$ on $\\Gamma _\\varepsilon $ , and $\\Psi _n =- \\big [ u(x, x/\\varepsilon ) \\cdot n -\\gamma \\big ] n \\quad \\text{ on } \\partial \\Omega ,$ where $\\gamma =_{\\partial \\Omega } u(x, x/\\varepsilon ) \\cdot n\\, d\\sigma .$ Thanks to (REF ), we may write $u(x, x/\\varepsilon )\\cdot n = \\mu ^{-1} n_i\\big [ W_j^i (x/\\varepsilon ) -K_j^i \\big ] \\Big ( f_j -\\frac{\\partial p_0}{\\partial x_j} \\Big ) \\quad \\text{ on } \\partial \\Omega $ (the repeated indices are summed from 1 to $d$ ).", "Furthermore, there exists an 1-periodic tensor $(\\phi _{\\ell j}^i)$ such that $\\phi _{\\ell j}^i =-\\phi _{ij}^\\ell \\quad \\text{ and } \\quad W_j^i (y) -K_j^i =\\frac{\\partial }{\\partial y_\\ell } \\phi _{\\ell j}^i (y).$ It follows from (REF ) and (REF ) that $u(x, x/\\varepsilon )\\cdot n=\\varepsilon \\mu ^{-1} \\Big ( n_i \\frac{\\partial }{\\partial x_\\ell } - n_\\ell \\frac{\\partial }{\\partial x_i} \\Big ) \\Big ( \\phi _{\\ell j}^i (x/\\varepsilon ) \\Big )\\cdot \\Big ( f_j -\\frac{\\partial p_0}{\\partial x_j} \\Big ) \\quad \\text{ on } \\partial \\Omega .$ Since $n_i \\frac{\\partial }{\\partial x_\\ell } - n_\\ell \\frac{\\partial }{\\partial x_i}$ is a tangential derivative, the formula (REF ) allows us to use an integration by parts on $\\partial \\Omega $ (see (REF )), which generates the needed decay factor ${\\varepsilon }$ .", "In order to carry out this argument, we use an energy estimate to reduce the problem to the $L^2$ estimate for the Stokes equations in $\\Omega $ , whose solutions are then represented by integrals on $\\partial \\Omega $ , using the Poisson kernels.", "See Section for details.", "The paper is organized as follows.", "In Section we introduce some notations and collect several known results that will be used in later sections.", "In Section we establish an energy estimate for the Stokes equations in $\\Omega _\\varepsilon $ .", "The tangential boundary corrector $(\\Psi _t, q_t)$ is constructed in Section , while the normal boundary corrector $(\\Psi _n, q_n)$ and its estimates are given in Section .", "The proof of Theorem REF is contained in Section , where an interior corrector is constructed.", "In fact, a more general case is treated in Section , where we assume $u_\\varepsilon =b \\in H^1(\\partial \\Omega ; \\mathbb {R}^d)$ on $\\partial \\Omega $ .", "See Theorem REF .", "Due to the discrepancy of $u_\\varepsilon $ and $u(x, x/\\varepsilon )$ on $\\partial \\Omega $ , the $O(\\sqrt{\\varepsilon }) $ rate in Theorem REF is sharp.", "See Remark REF Throughout the paper, the repeated indices are summed from 1 to $d$ .", "We will use $C$ and $c$ to denote positive constants that depend at most on $d$ , $\\mu $ , $\\Omega $ , and $Y_s$ .", "Since the value of $\\mu $ is not relevant in this study, we will assume $\\mu =1$ in the rest of the paper for simplicity.", "Acknowledgement.", "The author thanks Jinping Zhuge for several valuable comments." ], [ "Preliminaries", "Let $Y=[0, 1]^d$ and $Y_s$ (solid part) be an open subset of $Y$ with Lipschitz boundary.", "We assume that dist$(\\partial Y, \\partial Y_s)>0$ and that (the fluid part) $Y_f= Y \\setminus \\overline{Y_s}$ is connected.", "Let $\\omega =\\bigcup _{z\\in \\mathbb {Z}^d} \\big ( Y_f +z\\big )$ be the periodic repetition of $Y_f$ .", "It is easy to see that the unbounded domain $\\omega $ is connected, 1-periodic, and that $\\partial \\omega $ is locally Lipschitz.", "For $1\\le j \\le d$ , let $ (W_j (y), \\pi _j (y)) = (W_j^1(y), \\dots , W_j^d (y), \\pi _j (y))\\in H^1_{\\text{loc}}(\\omega ; \\mathbb {R}^d) \\times L^2_{\\text{loc}}(\\omega )$ be the 1-periodic solution of $\\left\\lbrace \\aligned -\\Delta W_j +\\nabla \\pi _j & =e_j & \\quad & \\text{ in } \\omega , \\\\\\text{\\rm div} (W_j) & =0 & \\quad & \\text{ in }\\omega ,\\\\W_j & =0& \\quad & \\text{ on } \\partial \\omega ,\\\\\\endaligned \\right.$ with $\\int _{Y_f} \\pi _j \\, dy=0$ , where $e_j=(0, \\dots , 1, \\dots , 0)$ with 1 in the $j^{th}$ place.", "We extend $W_j$ to $\\mathbb {R}^d$ by zero and define $K_j^i =\\int _{Y} W^i_j (y)\\, dy.$ Using $K_j^i =\\int _Y \\nabla W^\\ell _j\\cdot \\nabla W^\\ell _i\\, dy,$ it is not hard to show that the $d\\times d$ constant matrix $(K_j^i)$ is symmetric and positive definite.", "Thanks to the assumption dist$(\\partial Y, \\partial Y_s)>0$ , we have $\\partial \\Omega _\\varepsilon =\\partial \\Omega \\cup \\Gamma _\\varepsilon $ and dist$(\\partial \\Omega , \\Gamma _\\varepsilon ) \\ge c \\varepsilon $ , where $\\Gamma _\\varepsilon = \\Omega \\cap \\partial \\Omega _\\varepsilon \\subset \\partial (\\varepsilon \\omega ).$ For $f\\in L^2(\\Omega ; \\mathbb {R}^d)$ , let $(u_\\varepsilon , p_\\varepsilon )$ be a weak solution in $H_0^1(\\Omega _\\varepsilon ; \\mathbb {R}^d) \\times L^2(\\Omega _\\varepsilon )$ of the Dirichlet problem, $\\left\\lbrace \\aligned -\\varepsilon ^2 \\Delta u_\\varepsilon + \\nabla p_\\varepsilon & = f & \\quad & \\text{ in } \\Omega _\\varepsilon ,\\\\\\text{\\rm div} (u_\\varepsilon ) & =0 & \\quad & \\text{ in } \\Omega _\\varepsilon ,\\\\u_\\varepsilon & =0 & \\quad & \\text{ on } \\partial \\Omega _\\varepsilon ,\\endaligned \\right.$ with $\\int _{\\Omega _\\varepsilon } p_\\varepsilon \\, dx=0$ .", "We extend $u_\\varepsilon $ to $\\Omega $ by zero and still denote the extension by $u_\\varepsilon $ .", "Let $P_\\varepsilon $ be the extension of $p_\\varepsilon $ , defined by $P_\\varepsilon (x) =\\left\\lbrace \\aligned & p_\\varepsilon (x) & \\quad & \\text{ if } x\\in \\Omega _\\varepsilon , \\\\& _{\\varepsilon ( Y_f +z_k)} p_\\varepsilon & \\quad & \\text{ if } x\\in \\varepsilon (Y_s + z_k) \\text{ and } \\varepsilon (Y+z_k) \\subset \\Omega \\text{ for some }z_k \\in \\mathbb {Z}^d.\\endaligned \\right.$ Theorem 2.1 Let $\\Omega $ be a bounded Lipschitz domain in $\\mathbb {R}^d$ , $d\\ge 2$ .", "Let $p_0\\in H^1(\\Omega )$ be a weak solution of the Neumann problem, $\\left\\lbrace \\aligned \\frac{\\partial }{\\partial x_i} K_j^i \\Big (f_j -\\frac{\\partial p_0}{\\partial x_j}\\Big ) & =0 & \\quad & \\text{ in } \\Omega ,\\\\n_i K_j^i \\Big (f_j -\\frac{\\partial p_0}{\\partial x_j} \\Big ) & =0 & \\quad & \\text{ on } \\partial \\Omega ,\\endaligned \\right.$ with $\\int _{\\Omega } p_0\\, dx=0$ , where $n=(n_1, \\dots , n_d)$ denotes the outward unit normal to $\\partial \\Omega $ .", "Then, as $\\varepsilon \\rightarrow 0$ , $\\left\\lbrace \\aligned {u}_\\varepsilon - W_j (x/\\varepsilon ) \\Big ( f_j - \\frac{\\partial p_0}{\\partial x_j}\\Big ) & \\rightarrow 0 & \\quad & \\text{ in }L^2(\\Omega ; \\mathbb {R}^d), \\\\P_\\varepsilon -p_0 & \\rightarrow 0 & \\quad & \\text{ in } L^2(\\Omega ).\\endaligned \\right.$ As indicated in Introduction, a proof of Theorem REF , using the method of two-scale convergence, may be found in [2], [3].", "We do not use the theorem in this paper.", "However, we will need several other known results stated below.", "Theorem 2.2 Let $\\Omega $ be a bounded Lipschitz domain in $\\mathbb {R}^d$ , $d\\ge 2$ .", "Assume that $\\Gamma _\\varepsilon \\ne \\emptyset $ .", "Let $u \\in H^1(\\Omega _\\varepsilon )$ with $u=0$ on $\\Gamma _\\varepsilon $ .", "Then $\\Vert u \\Vert _{L^2(\\Omega _\\varepsilon )} \\le C\\varepsilon \\Vert \\nabla u \\Vert _{L^2(\\Omega _\\varepsilon )}.$ The case $u\\in H_0^1(\\Omega _\\varepsilon )$ is more or less well known.", "See e.g.", "[5].", "The proof for the case $u\\in H^1(\\Omega _\\varepsilon )$ with $u_\\varepsilon =0$ on $\\Gamma _\\varepsilon $ is the same.", "We sketch a proof here for the reader's convenience.", "Suppose $\\varepsilon (Y+z_k)\\subset \\Omega $ for some $z_k\\in \\mathbb {Z}^d$ .", "Since $u=0$ on $\\Gamma _\\varepsilon $ , it follows by Poincaré's inequality that $\\int _{\\varepsilon (Y_f +z_k)} \|u\|^2\\, dx \\le C \\varepsilon ^2 \\int _{\\varepsilon (Y_f +z_k)} \|\\nabla u\|^2\\, dx.$ Similarly, $\\int _{B(x_0, C \\varepsilon )\\cap \\Omega _\\varepsilon } \|u\|^2\\, dx\\le C \\varepsilon ^2 \\int _{B(x_0, C\\varepsilon )\\cap \\Omega _\\varepsilon } \|\\nabla u\|^2\\, dx,$ if $x_0\\in \\partial \\Omega $ and $\\varepsilon (Y+ z) \\subset B(x_0, C\\varepsilon ) \\cap \\Omega $ for some $z\\in \\mathbb {Z}^d$ .", "The estimate (REF ) follows from (REF ) -(REF ) by a covering argument.", "Theorem 2.3 Let $\\Omega $ be a bounded Lipschitz domain in $\\mathbb {R}^d$ , $d\\ge 2$ .", "There exists a bounded linear operator $R_\\varepsilon : H^1(\\Omega ; \\mathbb {R}^d) \\rightarrow H^1(\\Omega _\\varepsilon , \\mathbb {R}^d),$ such that $\\left\\lbrace \\aligned & R_\\varepsilon (u)=0 \\quad \\text{ on } \\Gamma _\\varepsilon \\quad \\text{ and } \\quad R_\\varepsilon (u)=u \\quad \\text{ on } \\partial \\Omega ,\\\\& R_\\varepsilon (u)\\in H^1_0(\\Omega _\\varepsilon ; \\mathbb {R}^d) \\quad \\text{ if } u\\in H^1_0(\\Omega ; \\mathbb {R}^d),\\\\&R_\\varepsilon (u) =u \\quad \\text{ in } \\Omega _\\varepsilon \\ \\text{ if } \\ u=0 \\text{ on } \\Gamma _\\varepsilon ,\\\\& \\text{\\rm div}(R_\\varepsilon (u) ) =0 \\quad \\text{ in } \\Omega _\\varepsilon \\quad \\text{ if } \\ \\text{\\rm div}(u) =0\\text{ in }\\Omega ,\\endaligned \\right.$ and $\\varepsilon \\, \\Vert \\nabla R_\\varepsilon (u)\\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert R_\\varepsilon (u)\\Vert _{L^2(\\Omega _\\varepsilon )}\\le C \\Big \\lbrace \\varepsilon \\Vert \\nabla u\\Vert _{L^2(\\Omega )} + \\Vert u\\Vert _{L^2(\\Omega )} \\Big \\rbrace ,$ where $C$ depends only on $\\Omega $ and $Y_s$ .", "Moreover, $\\Vert \\text{\\rm div} (R_\\varepsilon (u) ) \\Vert _{L^2(\\Omega _\\varepsilon )}\\le C\\, \\Vert \\text{\\rm div} (u)\\Vert _{L^2(\\Omega )}.$ The proof is the similar to that of a lemma due to Tartar (in an appendix of [22], also see Lemma 1.7 in [4]).", "Let $u\\in H^1(\\Omega ; \\mathbb {R}^d)$ .", "For each $\\varepsilon (Y+z) \\subset \\Omega $ , where $z\\in \\mathbb {Z}^d$ , we define $R_\\varepsilon (u)$ on $\\varepsilon (Y_f+z)$ by the Dirichlet problem, $\\left\\lbrace \\aligned -\\varepsilon ^2 \\Delta R_\\varepsilon (u) +\\nabla q & =-\\varepsilon ^2 \\Delta u & \\quad & \\text{ in } \\varepsilon (Y_f +z),\\\\\\text{\\rm div} (R_\\varepsilon (u)) & = \\text{\\rm div} (u)+ \\frac{1}{\|\\varepsilon (Y_f + z)\|}\\int _{\\varepsilon (Y_s +z)} \\text{\\rm div} (u)\\, dx &\\quad & \\text{ in } \\varepsilon (Y_f +z),\\\\R_\\varepsilon (u) & =0& \\quad & \\text{ on } \\partial ( \\varepsilon (Y_s +z)),\\\\R_\\varepsilon (u) & = u & \\quad & \\text{ on } \\partial (\\varepsilon (Y+z)).\\endaligned \\right.$ If $x\\in \\Omega _\\varepsilon $ and $x \\notin \\varepsilon (Y_f +z) $ for any $\\varepsilon (Y+z)\\subset \\Omega $ , we let $R_\\varepsilon (u)(x) =u(x)$ .", "It is not hard to show that $R_\\varepsilon (u)\\in H^1(\\Omega _\\varepsilon ; \\mathbb {R}^d)$ satisfies the conditions in (REF )-(REF ).", "Theorem 2.4 Let $\\Omega $ be a bounded Lipschitz domain in $\\mathbb {R}^d$ , $d\\ge 2$ .", "Suppose that $g \\in L^2(\\Omega _\\varepsilon )$ and $\\int _{\\Omega _\\varepsilon }g\\, dx =0$ .", "Then there exists $v_\\varepsilon \\in H_0^1(\\Omega _\\varepsilon ; \\mathbb {R}^d)$ such that $\\text{\\rm div} (v_\\varepsilon ) = g \\text{ in } \\Omega _\\varepsilon $ and $\\varepsilon \\Vert \\nabla v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} + \\Vert v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}\\le C \\Vert g\\Vert _{L^2(\\Omega _\\varepsilon )},$ where $C$ depends only on $\\Omega $ and $Y_s$ .", "See e.g.", "[5]." ], [ "Energy estimates", "In this section we establish the energy estimates for the Dirichlet problem, $\\left\\lbrace \\aligned -\\varepsilon ^2 \\Delta u_\\varepsilon + \\nabla p_\\varepsilon & = f + \\varepsilon \\, \\text{\\rm div} (F) & \\quad & \\text{ in } \\Omega _\\varepsilon ,\\\\\\text{\\rm div}(u_\\varepsilon ) & =g & \\quad & \\text{ in } \\Omega _\\varepsilon ,\\\\u_\\varepsilon & =0 & \\quad & \\text{ on } \\Gamma _\\varepsilon , \\\\u_\\varepsilon & =h & \\quad & \\text{ on } \\partial \\Omega ,\\endaligned \\right.$ where $(g, h)$ satisfies the compatibility condition, $\\int _\\Omega g\\, dx =\\int _{\\partial \\Omega } h \\cdot n\\, d\\sigma .$ Theorem 3.1 Let $\\Omega $ be a bounded domain in $\\mathbb {R}^d$ , $d\\ge 2$ with Lipschitz boundary.", "Let $(u_\\varepsilon , p_\\varepsilon ) \\in H^1(\\Omega _\\varepsilon ; \\mathbb {R}^d) \\times L^2(\\Omega _\\varepsilon )$ be a weak solution of (REF ) with $\\int _{\\Omega _\\varepsilon } p_\\varepsilon \\, dx=0$ .", "Then $\\aligned & \\varepsilon \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert p_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}\\\\&\\le C \\left\\lbrace \\Vert f\\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert F\\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert g\\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert h\\Vert _{L^2(\\partial \\Omega )}+ \\varepsilon \\Vert h\\Vert _{H^{1/2}(\\partial \\Omega )}\\right\\rbrace ,\\endaligned $ for any $0< \\varepsilon < 1$ , where $C$ depends only on $\\Omega $ and $Y_s$ .", "We divide the proof into several steps.", "Step 1.", "By Theorem REF , there exists $v_\\varepsilon \\in H_0^1(\\Omega _\\varepsilon ; \\mathbb {R}^d)$ such that $\\text{div} (v_\\varepsilon ) =p_\\varepsilon $ in $\\Omega _\\varepsilon $ and $\\varepsilon \\Vert \\nabla v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} + \\Vert v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}\\le C \\Vert p_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}.$ By using $v_\\varepsilon $ as a test function we see that $\\aligned \\Vert p_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}^2&\\le \\varepsilon ^2 \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} \\Vert \\nabla v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert f \\Vert _{L^2(\\Omega _\\varepsilon )} \\Vert v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\varepsilon \\Vert F \\Vert _{L^2(\\Omega _\\varepsilon )} \\Vert \\nabla v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}\\\\&\\le C \\Vert p_\\varepsilon \\Vert _{L^2(\\Omega )}\\left\\lbrace \\varepsilon \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert f\\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert F \\Vert _{L^2(\\Omega _\\varepsilon )} \\right\\rbrace ,\\endaligned $ where we have used (REF ) for the last inequality.", "This gives $\\Vert p_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}\\le C \\left\\lbrace \\varepsilon \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert f\\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert F \\Vert _{L^2(\\Omega _\\varepsilon )} \\right\\rbrace .$ Step 2.", "We consider the case $h=0$ on $\\partial \\Omega $ .", "This allows us to use the test function $u_\\varepsilon \\in H^1_0(\\Omega _\\varepsilon ; \\mathbb {R}^d)$ to obtain $\\varepsilon ^2 \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}^2\\le \\Vert p_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} \\Vert g\\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert f\\Vert _{L^2(\\Omega _\\varepsilon )} \\Vert u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\varepsilon \\Vert F \\Vert _{L^2(\\Omega _\\varepsilon )} \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega )},$ where we have also used the Cauchy inequality.", "It follows from the inequality $\\Vert u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}\\le C \\varepsilon \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}$ as well as the Cauchy inequality that $\\varepsilon ^2 \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}^2\\le C\\left\\lbrace \\Vert p_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} \\Vert g \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert F \\Vert _{L^2(\\Omega _\\varepsilon )}^2 + \\Vert f\\Vert _{L^2(\\Omega _\\varepsilon )}^2 \\right\\rbrace .$ This, together with (REF ), yields (REF ) by the Cauchy inequality.", "Step 3.", "In the general case, we let $(H, q)\\in H^1(\\Omega ; \\mathbb {R}^d) \\times L^2(\\Omega )$ be a weak solution of $-\\Delta H +\\nabla q =0 \\quad \\text{ and } \\quad \\text{\\rm div}(H) =\\gamma \\quad \\text{ in } \\Omega ,$ with boundary data $H=h$ on $\\partial \\Omega $ , where $\\gamma = \\frac{1}{\|\\Omega \|} \\int _{\\partial \\Omega } h \\cdot n \\, d\\sigma .$ Let $w_\\varepsilon = R_\\varepsilon (H)$ , where $R_\\varepsilon $ is the operator given by Theorem REF .", "Note that $w_\\varepsilon =0 $ on $\\Gamma _\\varepsilon $ , $w_\\varepsilon =h$ on $\\partial \\Omega $ , $\\aligned \\varepsilon \\Vert \\nabla w_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} + \\Vert w_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}& \\le C \\left\\lbrace \\varepsilon \\Vert \\nabla H\\Vert _{L^2(\\Omega )} + \\Vert H\\Vert _{L^2(\\Omega )} \\right\\rbrace ,\\\\\\endaligned $ and $\\Vert \\text{\\rm div}(w_\\varepsilon )\\Vert _{L^2(\\Omega _\\varepsilon )} \\le C \|\\gamma \|.$ Thus, $u_\\varepsilon -w_\\varepsilon \\in H^1_0(\\Omega _\\varepsilon ; \\mathbb {R}^d)$ , and $-\\varepsilon ^2 \\Delta (u_\\varepsilon -w_\\varepsilon ) +\\nabla p_\\varepsilon = f + \\varepsilon \\, \\text{\\rm div} (F) +\\varepsilon ^2 \\Delta w_\\varepsilon \\quad \\text{ in } \\Omega _\\varepsilon .$ Hence, by Step 2, we obtain $\\aligned & \\varepsilon \\Vert \\nabla (u_\\varepsilon -w_\\varepsilon ) \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert u_\\varepsilon -w_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert p_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}\\\\&\\le C \\big \\lbrace \\Vert f\\Vert _{L^2(\\Omega _\\varepsilon )} + \\Vert F\\Vert _{L^2(\\Omega _\\varepsilon )} + \\varepsilon \\Vert \\nabla w_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert g\\Vert _{L^2(\\Omega _\\varepsilon )} + \|\\gamma \| \\big \\rbrace ,\\endaligned $ where we have used (REF ).", "It follows from (REF ) that $\\aligned & \\varepsilon \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} + \\Vert u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} + \\Vert p_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}\\\\&\\le C \\big \\lbrace \\Vert f\\Vert _{L^2(\\Omega _\\varepsilon )} + \\Vert F\\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert g\\Vert _{L^2(\\Omega _\\varepsilon )} + \|\\gamma \|+ \\varepsilon \\Vert \\nabla H \\Vert _{L^2(\\Omega )} + \\Vert H \\Vert _{L^2(\\Omega )} \\big \\rbrace .\\endaligned $ Step 4.", "To estimate $\\Vert \\nabla H\\Vert _{L^2(\\Omega )}$ and $\\Vert H \\Vert _{L^2(\\Omega )}$ , we let $\\widetilde{H}= H - \\gamma d^{-1} (x-x_0),$ where $x_0\\in \\Omega $ is fixed.", "Note that $-\\Delta \\widetilde{H} +\\nabla q =0 \\quad \\text{ and } \\quad \\text{\\rm div} (\\widetilde{H} )=0 \\quad \\text{ in } \\Omega .$ By the energy estimates, $\\Vert \\nabla \\widetilde{H}\\Vert _{L^2(\\Omega )}\\le C \\Vert \\widetilde{H}\\Vert _{H^{1/2}(\\partial \\Omega )}\\le C \\Vert h \\Vert _{H^{1/2}(\\partial \\Omega )},$ and by the nontangential-maximal-function estimates for the Stokes equations in [6], $\\Vert \\widetilde{H}\\Vert _{L^2(\\Omega )} \\le C \\Vert \\widetilde{H} \\Vert _{L^2(\\partial \\Omega )}\\le C \\Vert h\\Vert _{L^2(\\partial \\Omega )}.$ It follows that $\\varepsilon \\Vert \\nabla H \\Vert _{L^2(\\Omega )}+ \\Vert H \\Vert _{L^2(\\Omega )}\\le C \\big \\lbrace \\varepsilon \\Vert h\\Vert _{H^{1/2}(\\partial \\Omega )} + \\Vert h\\Vert _{L^2(\\partial \\Omega )} \\big \\rbrace .$ This, together with (REF ), completes the proof." ], [ "Correctors for tangential boundary data", "Consider the Dirichlet problem, $\\left\\lbrace \\aligned -\\varepsilon ^2 \\Delta u_\\varepsilon + \\nabla p_\\varepsilon & = 0 & \\quad & \\text{ in } \\Omega _\\varepsilon ,\\\\\\text{\\rm div} (u_\\varepsilon ) & =0 & \\quad & \\text{ in } \\Omega _\\varepsilon ,\\\\u_\\varepsilon & =0 & \\quad & \\text{ on } \\Gamma _\\varepsilon ,\\\\u_\\varepsilon & = h & \\quad & \\text{ on } \\partial \\Omega ,\\endaligned \\right.$ with boundary data $h$ satisfying the condition $h \\cdot n =0 \\quad \\text{ on } \\partial \\Omega .$ The goal of this section is to prove the following.", "Theorem 4.1 Let $\\Omega $ be a bounded Lipschitz domain in $\\mathbb {R}^d$ , $d\\ge 2$ .", "Let $(u_\\varepsilon , p_\\varepsilon )$ be a weak solution in $H^1(\\Omega _\\varepsilon ; \\mathbb {R}^d) \\times L^2(\\Omega _\\varepsilon )$ of (REF ) with $\\int _{\\Omega _\\varepsilon } p_\\varepsilon \\, dx =0$ , where $h\\in H^1(\\partial \\Omega ; \\mathbb {R}^d)$ satisfies (REF ).", "Then $\\varepsilon \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}+\\Vert u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}+\\Vert p_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}\\le C \\sqrt{\\varepsilon }\\left\\lbrace \\Vert h \\Vert _{L^2(\\partial \\Omega )}+\\varepsilon \\Vert \\nabla _{\\tan } h \\Vert _{L^2(\\partial \\Omega )} \\right\\rbrace ,$ where $\\nabla _{\\tan } h$ denotes the tangential gradient of $h$ on $\\partial \\Omega $ .", "Let $\\aligned D_r & =\\big \\lbrace (x^\\prime , x_d)\\in \\mathbb {R}^d: \\ \|x^\\prime \|< r \\ \\ \\text{ and } \\ \\ \\psi (x^\\prime )< x_d < 100 d(M+1)r \\big \\rbrace ,\\\\I_r & =\\big \\lbrace (x^\\prime , \\psi (x^\\prime ) )\\in \\mathbb {R}^d: \\ \|x^\\prime \|< r \\big \\rbrace ,\\endaligned $ where $\\psi : \\mathbb {R}^{d-1} \\rightarrow \\mathbb {R}$ is a Lipschitz function such that $\\psi (0)=0$ and $\\Vert \\nabla \\psi \\Vert _\\infty \\le M$ .", "Lemma 4.2 Let $(v, q)$ be a weak solution in $H^1(D_r; \\mathbb {R}^d)\\times L^2(D_r)$ of the Dirichlet problem, $\\left\\lbrace \\aligned -\\Delta v +\\nabla q & =0&\\quad &\\text{ in } D_r,\\\\\\text{\\rm div} (v) & =0 & \\quad & \\text{ in } D_r,\\\\v & =g & \\quad & \\text{ on } \\partial D_r,\\endaligned \\right.$ where $0< r<\\infty $ and $g\\in H^1(\\partial D_r; \\mathbb {R}^d)$ satisfies the condition $\\int _{\\partial D_r} g \\cdot n\\, d\\sigma =0$ .", "Then $\\int _{\\partial D_r} \|\\nabla v\|^2\\, d\\sigma \\le C \\int _{\\partial D_r} \|\\nabla _{\\tan } v\|^2\\, d\\sigma ,$ where $C$ depends only on $d$ and $M$ .", "By dilation we may assume $r=1$ , in which case the Rellich estimate (REF ) was proved in [6].", "Lemma 4.3 Let $(v, q)$ be a weak solution of (REF ) with $r=2$ .", "Then $\\int _{I_1} \|\\nabla v\|^2\\, d\\sigma \\le C\\int _{I_2} \|\\nabla _{\\tan } g \|^2\\, d\\sigma + C \\int _{D_2} \|\\nabla v\|^2\\, dx,$ where $C$ depends only on $d$ and $M$ .", "It follows from (REF ) that for $1< r< 2$ , $\\int _{I_1} \|\\nabla v\|^2\\, d\\sigma \\le C \\int _{\\partial D_r} \|\\nabla v\|^2\\, d\\sigma \\le C \\int _{\\partial D_r} \|\\nabla _{\\tan } v\|^2\\, d\\sigma .$ Hence, $\\int _{I_1} \|\\nabla v\|^2\\, d\\sigma \\le C \\int _{I_2} \|\\nabla _{\\tan } g \|^2\\, d\\sigma + C \\int _{D_2 \\cap \\partial D_r} \|\\nabla v\|^2\\, d\\sigma .$ By integrating the inequality above in $r$ over the interval $(1, 2)$ , we obtain (REF ).", "[Proof of Theorem REF ] We start with the observation, $\\varepsilon ^2 \\int _{\\Omega _\\varepsilon } \|\\nabla u_\\varepsilon \|^2\\, dx = \\varepsilon ^2 \\int _{\\partial \\Omega } \\frac{\\partial u_\\varepsilon }{\\partial n} \\cdot u_\\varepsilon \\, d\\sigma ,$ where we have used (REF ) and (REF ).", "It follows by the Cauchy inequality that $\\varepsilon ^2 \\int _{\\Omega _\\varepsilon } \|\\nabla u_\\varepsilon \|^2\\, dx\\le \\varepsilon ^2 \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\partial \\Omega )} \\Vert h\\Vert _{L^2(\\partial \\Omega )}.$ We will show that $\\int _{\\partial \\Omega } \|\\nabla u_\\varepsilon \|^2\\, d\\sigma \\le C \\int _{\\partial \\Omega } \|\\nabla _{\\tan } h\|^2\\, d\\sigma + \\frac{C}{\\varepsilon } \\int _{\\Sigma _{c\\varepsilon }} \|\\nabla u_\\varepsilon \|^2\\, dx,$ where $\\Sigma _{c\\varepsilon } =\\lbrace x\\in \\Omega : \\ \\text{dist}(x, \\partial \\Omega )< c\\varepsilon \\rbrace \\subset \\Omega _\\varepsilon $ .", "Assume (REF ) for a moment.", "Then $\\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\partial \\Omega )}\\le C\\Vert \\nabla _{\\tan } h\\Vert _{L^2(\\partial \\Omega )} + C \\varepsilon ^{-1/2} \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}.$ This, together with (REF ) and the Cauchy inequality, gives $\\aligned \\varepsilon ^2 \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}^2&\\le C \\varepsilon ^2\\Vert h\\Vert _{L^2(\\partial \\Omega )} \\Vert \\nabla _{\\tan } h\\Vert _{L^2(\\partial \\Omega )}+ C \\varepsilon ^{3/2} \\Vert h\\Vert _{L^2(\\partial \\Omega )} \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}\\\\&\\le C \\varepsilon ^2 \\Vert h\\Vert _{L^2(\\partial \\Omega )} \\Vert \\nabla _{\\tan } h\\Vert _{L^2(\\partial \\Omega )}+ C \\varepsilon \\Vert h\\Vert _{L^2(\\partial \\Omega )}^2+ (1/2) \\varepsilon ^2 \\Vert \\nabla u_\\varepsilon \\Vert ^2_{L^2(\\Omega _e)},\\endaligned $ which yields the estimate for $\\varepsilon \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} $ in (REF ).", "The estimate for $\\Vert u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}$ follows by (REF ), while the bound for $\\Vert p_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} $ follows from (REF ), It remains to prove (REF ).", "To this end, we claim that $\\int _{B(x_0, c_0\\varepsilon )\\cap \\partial \\Omega } \|\\nabla u_\\varepsilon \|^2\\, d\\sigma \\le C \\int _{B(x_0, c_1 \\varepsilon )\\cap \\partial \\Omega } \|\\nabla _{\\tan } h\|^2\\, d\\sigma + \\frac{C}{\\varepsilon } \\int _{B(x_0, c_1\\varepsilon )\\cap \\Omega } \|\\nabla u_\\varepsilon \|^2\\, dx$ for any $x_0\\in \\partial \\Omega $ , where $0<c_0< c_1$ are sufficiently small.", "The desired estimate (REF ) follows from (REF ) by covering $\\partial \\Omega $ with a finite number of balls $\\lbrace B(x_k, c_0 \\varepsilon )\\rbrace $ centered on $\\partial \\Omega $ .", "Finally, we note that if $v (x)=u_\\varepsilon (\\varepsilon x)$ and $q(x)=\\varepsilon ^{-1} p_\\varepsilon (\\varepsilon x)$ , then $-\\Delta v +\\nabla q=0$ and div$(v)=0$ .", "As a result, the estimate (REF ) follows from (REF ) by a translation and rotation of the coordinate system.", "We point out that since the constant $C$ in (REF ) depends only on $d$ and $M$ , the constant $C$ in (REF ) depends only on $d$ and the Lipschitz character of $\\Omega $ .", "In particular, $C$ does not depend on $\\varepsilon $ .", "As a corollary of Theorem REF , we are able to construct a tangential boundary corrector.", "Theorem 4.4 Let $\\Omega $ be bounded domain with $C^{2, \\alpha }$ boundary for some $\\alpha >0$ .", "Also assume that $\\partial Y_s$ is $C^{1, \\alpha }$ .", "Let $(\\Psi _t, q_t)$ be a weak solution in $H^1(\\Omega _\\varepsilon ; \\mathbb {R}^d) \\times L^2(\\Omega _\\varepsilon )$ of the Dirichlet problem (REF ) with $\\int _{\\Omega _\\varepsilon } q_t \\, dx=0$ , where the boundary data $h$ is given by $h= b -W_j(x/\\varepsilon ) \\Big (f_j -\\frac{\\partial p_0}{\\partial x_j}\\Big )+ \\Big [-b\\cdot n + n_i W^i_j (x/\\varepsilon ) \\Big (f_j -\\frac{\\partial p_0}{\\partial x_j}\\Big ) \\Big ] n,$ $ f\\in C^{1, 1/2} (\\overline{\\Omega }; \\mathbb {R}^d)$ , $b\\in H^1(\\partial \\Omega ; \\mathbb {R}^d)$ satisfies $\\int _{\\partial \\Omega } b \\cdot n \\, d \\sigma =0$ , and $p_0$ is the solution of the Neumann problem, $\\left\\lbrace \\aligned K_j^i \\frac{\\partial }{\\partial x_i} \\Big ( f_j -\\frac{\\partial p_0}{\\partial x_j} \\Big ) & = 0 & \\quad & \\text{ in } \\Omega ,\\\\n_i K_j^i \\Big (f_j -\\frac{\\partial p_0}{\\partial x_j}\\Big ) & =b\\cdot n & \\quad & \\text{ on } \\partial \\Omega ,\\endaligned \\right.$ with $\\int _\\Omega p_0\\, dx=0$ .", "Then $\\aligned & \\varepsilon \\Vert \\nabla \\Psi _t \\Vert _{L^2 (\\Omega _\\varepsilon )} + \\Vert \\Psi _t \\Vert _{L^2(\\Omega _\\varepsilon )} + \\Vert q_t \\Vert _{L^2(\\Omega _\\varepsilon )} \\\\& \\le C \\sqrt{\\varepsilon } \\left\\lbrace \\Vert f-\\nabla p_0\\Vert _{L^2(\\partial \\Omega )} + \\Vert b \\Vert _{L^2(\\partial \\Omega )}+ \\varepsilon \\Vert \\nabla _{\\tan } ( f-\\nabla p_0)\\Vert _{L^2(\\partial \\Omega )} + \\varepsilon \\Vert \\nabla _{\\tan } b \\Vert _{L^2(\\partial \\Omega )} \\right\\rbrace \\endaligned $ for any $0< \\varepsilon < 1$ .", "Note that $h\\cdot n=0$ on $\\partial \\Omega $ .", "Also, under the assumption that $\\partial Y_s$ is $C^{1, \\alpha }$ , we have $W_j=W_j(y) \\in C^1(\\overline{\\omega })$ .", "It follows that $\\Vert h\\Vert _{L^2(\\partial \\Omega )}\\le C \\big \\lbrace \\Vert f -\\nabla p_0\\Vert _{L^2(\\partial \\Omega )} + \\Vert b\\Vert _{L^2(\\partial \\Omega )} \\big \\rbrace ,$ and $\\aligned & \\Vert \\nabla _{\\tan } h \\Vert _{L^2(\\partial \\Omega )}\\\\& \\le C \\big \\lbrace \\varepsilon ^{-1} \\Vert f-\\nabla p_0 \\Vert _{L^2(\\partial \\Omega )} + \\Vert \\nabla _{\\tan } ( f-\\nabla p_0) \\Vert _{L^2(\\partial \\Omega )}+ \\Vert \\nabla _{\\tan } b \\Vert _{L^2(\\partial \\Omega )} +\\Vert b\\Vert _{L^2(\\partial \\Omega )} \\big \\rbrace .\\endaligned $ As a result, the estimate (REF ) follows readily from (REF )." ], [ "Correctors for normal boundary data", "In this section we consider the Dirichlet problem (REF ), where the boundary data $h$ is given by $h=\\big \\lbrace n_i [ W_j^i (x/\\varepsilon ) - K_j^i ] g_j -\\gamma \\big \\rbrace n,$ where $g=(g_1, g_2, \\dots , g_d) \\in H^1(\\partial \\Omega ; \\mathbb {R}^d)$ , and $\\gamma \\in \\mathbb {R}$ is chosen so that $\\int _{\\partial \\Omega } h \\cdot n \\, d\\sigma =0$ , i.e., $\\gamma =\\frac{1}{\|\\partial \\Omega \|}\\int _{\\partial \\Omega }n_i [ W_j^i (x/\\varepsilon ) - K_j^i ] g_j\\, d\\sigma .$ The goal of this section is to prove the following.", "Theorem 5.1 Let $\\Omega $ be a bounded $C^{2, \\alpha }$ domain in $\\mathbb {R}^d$ for some $\\alpha >0$ .", "Also assume that $\\partial Y_s$ is $C^{1, \\alpha }$ .", "Let $(u_\\varepsilon , p_\\varepsilon )$ be a weak solution in $H^1(\\Omega _\\varepsilon ; \\mathbb {R}^d) \\times L^2(\\Omega _\\varepsilon )$ of (REF ) with $\\int _{\\Omega _\\varepsilon } p_\\varepsilon \\, dx =0$ , where $h$ is given by (REF ).", "Then $\\varepsilon \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} +\\Vert u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert p_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}\\le C \\sqrt{\\varepsilon } \\,\\left\\lbrace \\Vert g \\Vert _{L^2 (\\partial \\Omega )}+ \\sqrt{\\varepsilon }\\, \\Vert \\nabla _{\\tan } g\\Vert _{L^2 (\\partial \\Omega )} \\right\\rbrace ,$ for any $0< \\varepsilon <1$ .", "We will prove a series of lemmas before we give the proof of Theorem REF .", "We begin with an estimate for $\\Vert h\\Vert _{H^{1/2}(\\partial \\Omega )}$ .", "Lemma 5.2 Let $h$ be given by (REF ).", "Then $\\Vert h\\Vert _{H^{1/2}(\\partial \\Omega )}\\le C \\varepsilon ^{-1/2} \\big \\lbrace \\Vert g\\Vert _{L^2 (\\partial \\Omega )}+ \\varepsilon \\Vert \\nabla _{\\tan } g \\Vert _{L^2(\\partial \\Omega )} \\big \\rbrace $ for any $0< \\varepsilon <1$ .", "Note that $\\aligned \\Vert h \\Vert _{H^{1/2}(\\partial \\Omega )}&\\le C \\Vert h \\Vert _{L^2(\\partial \\Omega )}^{1/2} \\Vert h \\Vert ^{1/2} _{H^1(\\partial \\Omega )}\\\\& \\le C \\left\\lbrace \\varepsilon ^{-1/2} \\Vert h \\Vert _{L^2(\\partial \\Omega )} + \\varepsilon ^{1/2} \\Vert \\nabla _{\\tan } h \\Vert _{L^2(\\partial \\Omega )}\\right\\rbrace ,\\endaligned $ where we have used the Cauchy inequality.", "It is easy to see that $\\Vert h \\Vert _{L^2(\\partial \\Omega )}\\le C \\Vert g\\Vert _{L^2(\\partial \\Omega )},$ and $\\Vert \\nabla _{\\tan } h \\Vert _{L^2(\\partial \\Omega )}\\le C\\left\\lbrace \\varepsilon ^{-1} \\Vert g\\Vert _{L^2(\\partial \\Omega )}+ \\Vert \\nabla _{\\tan } g \\Vert _{L^2(\\partial \\Omega )} \\right\\rbrace ,$ where we have used the fact $W_j =W_j(y) \\in C^1(\\overline{\\omega }; \\mathbb {R}^d)$ .", "It follows that $\\aligned \\Vert h \\Vert _{H^{1/2}(\\partial \\Omega )}&\\le C \\varepsilon ^{-1/2} \\left\\lbrace \\Vert g\\Vert _{L^2(\\partial \\Omega )}+ \\varepsilon \\Vert \\nabla _{\\tan } g \\Vert _{L^2(\\partial \\Omega )}\\right\\rbrace \\endaligned $ for any $0< \\varepsilon < 1$ .", "Lemma 5.3 There exist 1-periodic functions $\\phi _{ij}^\\ell \\in H^1_{\\text{loc}} (\\mathbb {R}^d)$ , where $1\\le i, j, \\ell \\le d$ , such that $\\frac{\\partial \\phi _{ij}^\\ell }{\\partial y_i} = W_j^\\ell (y) - K_j ^\\ell \\quad \\text{ and } \\quad \\phi _{ij}^\\ell =- \\phi _{\\ell j}^i,$ where the index $i$ is summed from 1 to $d$ .", "The proof is the same as Lemma 3.1 in [13].", "Since $\\int _{Y} \\left( W_j^\\ell (y) - K_j^\\ell \\right) dy =0,$ one may solve the periodic boundary value problem, $\\left\\lbrace \\aligned & \\Delta f_j^\\ell = W_j^\\ell - K_j^\\ell \\quad \\text{ in } Y, \\\\& f_j^\\ell \\text{ is 1-periodic.", "}\\endaligned \\right.$ Let $\\phi _{ij}^\\ell =\\frac{\\partial f_j^\\ell }{\\partial y_i} -\\frac{\\partial f_j^i}{\\partial y_\\ell }.$ Then $ \\phi _{ij}^\\ell =- \\phi _{\\ell j}^i$ .", "Using $\\frac{\\partial }{\\partial y_\\ell } W_j^\\ell = 0,$ we obtain the first equation in (REF ).", "Remark 5.4 Using (REF ), for $1\\le j \\le d$ , we may write $\\aligned n_\\ell [ W_j^\\ell (x/\\varepsilon ) -K_j^\\ell ]&= \\varepsilon n_\\ell \\frac{\\partial }{\\partial x_i} \\left\\lbrace \\phi _{ij}^\\ell (x/ \\varepsilon ) \\right\\rbrace \\\\&= \\frac{\\varepsilon }{2}\\Big \\lbrace n_\\ell \\frac{\\partial }{\\partial x_i} -n_i \\frac{\\partial }{\\partial x_\\ell } \\Big \\rbrace \\left\\lbrace \\phi _{ij}^\\ell (x/ \\varepsilon ) \\right\\rbrace ,\\endaligned $ where the skew-symmetric property is used for the last step.", "It follows from an integration by parts on $\\partial \\Omega $ that $\\int _{\\partial \\Omega }n_\\ell [ W_j^\\ell (x /\\varepsilon ) -K_j^\\ell ] \\psi \\, d\\sigma (x)=-\\frac{\\varepsilon }{2} \\int _{\\partial \\Omega }\\phi _{ij}^\\ell (x/ \\varepsilon )\\left\\lbrace n_\\ell \\frac{\\partial }{\\partial x_i} -n_i \\frac{\\partial }{\\partial x_\\ell } \\right\\rbrace \\psi \\, d\\sigma (x).$ This, in particular, implies that $\\aligned \|\\gamma \|&\\le C \\varepsilon \\Vert \\nabla _{\\tan } g \\Vert _{L^2(\\partial \\Omega )},\\endaligned $ where $\\gamma $ is given by (REF ), assuming that $ (\\phi _{ij}^\\ell )$ are bounded.", "Let $\\Sigma _\\rho =\\big \\lbrace x\\in \\Omega : \\ \\text{dist}(x, \\partial \\Omega )< \\rho \\big \\rbrace .$ Lemma 5.5 Let $T$ be the operator defined by $T(f) (x)=\\int _{\\partial \\Omega } \\frac{f(y)}{\|x-y\|^d}\\, d\\sigma (y).$ Then $\\int _{\\Omega \\setminus \\Sigma _\\varepsilon } \| T(f)\|^2\\, dx\\le C \\varepsilon ^{-1} \\int _{\\partial \\Omega } \|f\|^2\\, d\\sigma $ for any $0<\\varepsilon <1$ , and $\\int _\\Omega [\\text{\\rm dist}(x, \\partial \\Omega )]^\\delta \|T(f)\|^2\\, dx\\le C_\\delta \\int _{\\partial \\Omega } \|f\|^2\\, d\\sigma $ for any $\\delta >1$ .", "It is not hard to see that $\\int _{\\partial \\Omega } \\frac{d\\sigma (y)}{\|x-y\|^d} \\le \\frac{C}{\\text{dist}(x, \\partial \\Omega )}$ for any $x\\in \\Omega $ .", "By the Cauchy inequality, $\|T(f)(x)\|^2\\le \\frac{C}{\\text{dist}(x, \\partial \\Omega )}\\int _{\\partial \\Omega }\\frac{\|f(y)\|^2}{\|x-y\|^{d}} \\, d\\sigma (y).$ The estimate (REF ) follows by integrating the inequality above and using Fubini's Theorem.", "A similar argument gives REF .", "Lemma 5.6 Let $(H, q)$ be a weak solution in $H^1(\\Omega ; \\mathbb {R}^d)\\times L^2(\\Omega )$ of the Dirichlet problem, $\\left\\lbrace \\aligned -\\Delta H +\\nabla q & =0 & \\quad & \\text{ in } \\Omega ,\\\\\\text{\\rm div} (H) & =0 & \\quad & \\text{ in } \\Omega ,\\\\H & = h & \\quad & \\text{ on } \\partial \\Omega ,\\endaligned \\right.$ where $h$ is given by (REF ).", "Then $\\aligned \\varepsilon \\, \\Vert \\nabla H \\Vert _{L^2(\\Omega )}+\\Vert H \\Vert _{L^2(\\Omega )}\\le C \\sqrt{\\varepsilon } \\, \\Big \\lbrace \\Vert g\\Vert _{L^2(\\partial \\Omega )}+ \\sqrt{\\varepsilon }\\, \\Vert \\nabla _{\\tan } g \\Vert _{L^2(\\partial \\Omega )} \\Big \\rbrace ,\\endaligned $ for any $0< \\varepsilon <1$ .", "We first point out that by the standard energy estimates for the Stokes equations, $\\Vert \\nabla H \\Vert _{L^2(\\Omega )}\\le C\\, \\Vert h \\Vert _{H^{1/2}(\\partial \\Omega )}.$ In view of (REF ), this gives $\\varepsilon \\, \\Vert \\nabla H\\Vert _{L^2(\\Omega )}\\le C \\sqrt{\\varepsilon } \\, \\left\\lbrace \\Vert g\\Vert _{L^2 (\\partial \\Omega )}+ \\varepsilon \\, \\Vert \\nabla _{\\tan } g \\Vert _{L^2 (\\partial \\Omega )} \\right\\rbrace .$ Next, we use the nontangential-maximal-function estimate, $\\Vert (H)^* \\Vert _{L^2(\\partial \\Omega )}\\le C\\, \\Vert h \\Vert _{L^2(\\partial \\Omega )},$ to bound $H$ on $\\Sigma _\\varepsilon $ .", "The estimate (REF ) was proved in [6] for a Lipschitz domain $\\Omega $ , where the nontangential maximal function $(H)^$ is defined by $(H)^ (x) =\\sup \\big \\lbrace \|H(y)\|: \\ y\\in \\Omega \\text{ and } \|y-x\|< C_0\\, \\text{dist}(y, \\partial \\Omega ) \\big \\rbrace $ for $x\\in \\partial \\Omega $ .", "It follows that $\\aligned \\Vert H \\Vert _{L^2(\\Sigma _\\varepsilon )}&\\le C \\varepsilon ^{1/2}\\, \\Vert (H)^* \\Vert _{L^2(\\partial \\Omega )}\\le C \\varepsilon ^{1/2} \\, \\Vert h \\Vert _{L^2(\\partial \\Omega )}\\\\&\\le C \\varepsilon ^{1/2} \\, \\Vert g\\Vert _{L^2(\\partial \\Omega )}.\\endaligned $ It remains to bound $H$ on $\\Omega \\setminus \\Sigma _\\varepsilon $ .", "To this end, we let $(G(x, y), \\Pi (x, y))$ denote the matrix of Green functions for the Stokes equation (REF ) in $\\Omega $ .", "That is, for each fixed $x\\in \\Omega $ , $G(x, y)=(G^{ij} (x, y) ) \\in H^2_{\\text{loc}}(\\Omega \\setminus \\lbrace x \\rbrace ; \\mathbb {R}^{d\\times d})$ and $\\Pi (x, y) =(\\Pi ^i (x, y)) \\in L^2_{\\text{loc}} (\\Omega \\setminus \\lbrace x\\rbrace ; \\mathbb {R}^d)$ satisfy $\\left\\lbrace \\aligned -\\Delta _ y G^{ij} (x, y) +\\frac{\\partial }{\\partial y_j} \\Pi ^i (x, y) & =\\delta _x \\delta _{ij}& \\quad & \\text{ in } \\Omega \\setminus \\lbrace x \\rbrace ,\\\\\\frac{\\partial }{\\partial y_j} (G^{ij} (x, y)) & =0& \\quad & \\text{ in } \\Omega \\setminus \\lbrace x \\rbrace ,\\\\G^{ij} (x, y) & =0 & \\quad & \\text{ for } y \\in \\partial \\Omega ,\\endaligned \\right.$ in the sense of distribution.", "We also require that $\\Pi (x, \\cdot ) \\in L^1(\\Omega ; \\mathbb {R}^d) \\quad \\text{ and } \\quad \\int _\\Omega \\Pi (x, y)\\, dy =0.$ Under the assumption that $\\Omega $ is a bounded $C^{2,\\alpha }$ domain for some $\\alpha >0$ , solutions of the Stokes equations (REF ) satisfy the $C^{1, 1}$ estimate for $H$ and $C^{0, 1}$ estimate for $q$ , up to the boundary.", "It follows that $\\aligned \|\\nabla _x G(x, y)\| +\|\\nabla _y G(x, y)\| & \\le C \|x-y\|^{1-d},\\\\\|\\nabla _y G(x, y)\| & \\le C \\text{\\rm dist}(x, \\partial \\Omega ) \|x-y\|^{-d},\\\\\|\\nabla ^2_xG(x, y)\| + \|\\nabla _y^2 G(x, y)\| +\|\\nabla _x\\nabla _y G(x, y)\|& \\le C \|x-y\|^{-d},\\endaligned $ and that $\\aligned \|\\Pi (x, y)\| &\\le C \|x-y\|^{1-d},\\\\\|\\Pi (x, y)-\\Pi (x, z)\|&\\le \\text{\\rm dist}(x, \\partial \\Omega )\\lbrace \|x-y\|^{-d} + \|x-z\|^{-d}\\rbrace ,\\\\\|\\nabla _y \\Pi (x, y)\| & \\le C \|x-y\|^{-d},\\endaligned $ for any $x, y\\in \\Omega $ and $x\\ne y$ , $x\\ne z$ .", "See e.g.", "[7], [20].", "This allows us to represent the solution $H(x)$ by $H^i (x)=-\\int _{\\partial \\Omega }\\Big \\lbrace n_k(y) \\frac{\\partial }{\\partial y_k} G^{ij}(x, y) - \\big [ \\Pi ^i (x, y)-\\Pi ^i (x, z) \\big ] n_j (y) \\Big \\rbrace h^j (y) \\, d\\sigma (y)$ for any $x\\in \\Omega $ , where $z\\in \\Omega $ and $z\\ne x$ (due to the compatibility condition for $h$ , the choice of $z$ is arbitrary).", "Using (REF ), we may write $h = h^{(1)} + h^{(2)}$ , where $\\aligned h^{(1), k}& =\\frac{\\varepsilon }{2}\\Big ( n_\\ell \\frac{\\partial }{\\partial x_i} - n_i \\frac{\\partial }{\\partial x_\\ell } \\Big )\\Big \\lbrace \\phi _{ij}^\\ell (x/\\varepsilon ) g_j n_k \\Big \\rbrace , \\\\h^{(2), k}&=-\\frac{\\varepsilon }{2} \\phi _{ij}^\\ell (x/\\varepsilon ) \\Big ( n_\\ell \\frac{\\partial }{\\partial x_i} - n_i \\frac{\\partial }{\\partial x_\\ell } \\Big ) \\Big ( g_j n_k\\Big )-\\gamma n_k,\\endaligned $ for $1\\le k \\le d$ .", "Let $H^{(1)}(x)$ , $H^{(2)} (x)$ be given by (REF ), with $h$ being replaced by $h^{(1)}$ , $h^{(2)}$ , respectively.", "Observe that by the divergence theorem, $\\int _{\\partial \\Omega }\\Big ( n_\\ell \\frac{\\partial }{\\partial x_i} - n_i \\frac{\\partial }{\\partial x_\\ell } \\Big ) v \\cdot w\\, d\\sigma =- \\int _{\\partial \\Omega } v \\cdot \\Big ( n_\\ell \\frac{\\partial }{\\partial x_i} - n_i \\frac{\\partial }{\\partial x_\\ell } \\Big ) w\\, d\\sigma $ for $1\\le i, \\ell \\le d$ .", "It follows that $\\aligned & \|H^{(1)}(x)\|\\\\& \\le C \\varepsilon \\int _{\\partial \\Omega }\\Big \\lbrace \|\\nabla _y G(x, y)\| + \|\\nabla _y^2 G (x, y)\| + \|\\nabla _y \\Pi (x, y)\|+ \|\\Pi (x, y)-\\Pi (x, z)\| \\Big \\rbrace \| g(y) \|\\, d\\sigma (y)\\\\&\\le C \\varepsilon \\int _{\\partial \\Omega } \\frac{\| g(y) \|}{\|x-y\|^d}\\, d\\sigma (y),\\endaligned $ where we have used the estimates in (REF ) and (REF ).", "In view of Lemma REF , we obtain $\\aligned \\Vert H^{(1)} \\Vert _{L^2(\\Omega \\setminus \\Sigma _\\varepsilon )}& \\le C \\varepsilon ^{1/2} \\Vert g \\Vert _{L^2(\\partial \\Omega )}.\\endaligned $ Finally, note that $\\aligned \|H^{(2)} (x)\|& \\le C \\varepsilon \\Vert \\nabla _{\\tan } g\\Vert _{L^2 (\\partial \\Omega )}\\\\& + C \\varepsilon \\int _{\\partial \\Omega }\\Big \\lbrace \|\\nabla _y G(x, y)\| + \|\\Pi (x, y) -\\Pi (x, z)\| \\Big \\rbrace \\left( \|g(y)\| + \|\\nabla _{\\tan } g (y)\| \\right) d\\sigma (y),\\endaligned $ where we have used (REF ).", "Using estimates in (REF ) and (REF ), we may deduce from (REF ) that $\\Vert H^{(2)} \\Vert _{L^2(\\Omega )}\\le C \\varepsilon \\big \\lbrace \\Vert \\nabla _{\\tan } g \\Vert _{L^2 (\\partial \\Omega )}+\\Vert g\\Vert _{L^2 (\\partial \\Omega )} \\big \\rbrace ,$ which completes the proof.", "We are now ready to give the proof of Theorem REF [Proof of Theorem REF ] Let $(u_\\varepsilon , p_\\varepsilon )$ be a weak solution in $H^1(\\Omega _\\varepsilon ; \\mathbb {R}^d) \\times L^2(\\Omega _\\varepsilon )$ of (REF ) with $\\int _{\\Omega _\\varepsilon } p_\\varepsilon \\, dx=0$ , where $h$ is given by (REF ).", "Let $(H, q)$ be a solution of (REF ) with boundary data $h$ .", "It follows from (REF ) that $\\aligned \\varepsilon \\Vert \\nabla u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} +\\Vert u_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} + \\Vert p_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}& \\le C \\Big \\lbrace \\varepsilon \\Vert \\nabla H \\Vert _{L^2(\\Omega )} + \\Vert H\\Vert _{L^2(\\Omega ) } \\Big \\rbrace \\\\&\\le C \\sqrt{\\varepsilon }\\Big \\lbrace \\Vert g\\Vert _{L^2 (\\partial \\Omega )} + \\sqrt{\\varepsilon } \\Vert \\nabla _{\\tan } g \\Vert _{L^2(\\partial \\Omega )} \\Big \\rbrace ,\\endaligned $ where we have used (REF ) for the last inequality.", "As a corollary of Theorem REF , we construct a normal boundary corrector.", "Theorem 5.7 Let $\\Omega $ be a bounded $C^{2, \\alpha }$ domain for some $\\alpha >0$ .", "Also assume that $\\partial Y_s$ is $C^{1, \\alpha }$ .", "Let $( \\Psi _n, q_n )$ be a weak solution of (REF ) with $\\int _{\\Omega _\\varepsilon } q_n\\, dx =0$ , where the boundary data $h$ is given by $h= \\Big \\lbrace - n_i W_j^i (x/\\varepsilon ) \\Big ( f_j -\\frac{\\partial p_0}{\\partial x_j} \\Big ) + b\\cdot n - \\gamma \\Big \\rbrace n,$ $p_0$ is defined by (REF ), and $\\gamma \\in \\mathbb {R}$ is such that $\\int _{\\partial \\Omega } h\\cdot n\\, d\\sigma =0$ .", "Then $ \\aligned & \\varepsilon \\Vert \\nabla \\Psi _n \\Vert _{L^2(\\Omega _\\varepsilon )} + \\Vert \\Psi _n\\Vert _{L^2(\\Omega _\\varepsilon )} + \\Vert q_n \\Vert _{L^2(\\Omega _\\varepsilon )} \\\\& \\qquad \\qquad \\le C \\sqrt{\\varepsilon }\\Big \\lbrace \\Vert f -\\nabla p_0\\Vert _{L^2(\\partial \\Omega )}+ \\sqrt{\\varepsilon } \\Vert \\nabla _{\\tan } (f -\\nabla p_0) \\Vert _{L^2(\\partial \\Omega )} \\Big \\rbrace \\endaligned $ for any $0< \\varepsilon <1$ .", "Moreover, $\|\\gamma \| \\le C\\varepsilon \\Vert \\nabla _{\\tan } ( f- \\nabla p_0)\\Vert _{L^2(\\partial \\Omega )}.$ Note that by the boundary condition in (REF ), $h= \\Big \\lbrace - n_\\ell \\Big [ W_j^\\ell (x/\\varepsilon ) -K_j^\\ell \\Big ] \\Big ( f_j -\\frac{\\partial p_0}{\\partial x_j} \\Big ) -\\gamma \\Big \\rbrace n\\quad \\text{ on } \\partial \\Omega .$ As a result, the estimate (REF ) follows readily from Theorem REF with $g= -(f-\\nabla p_0)$ ." ], [ "Convergence rates", "In this section we prove the following theorem, which contains Theorem REF .", "Theorem 6.1 Let $\\Omega $ be a bounded $C^{2, \\alpha }$ domain in $\\mathbb {R}^d$ , $d\\ge 2$ for some $\\alpha >0$ .", "Also assume that $\\partial Y_s$ is $C^{1, \\alpha }$ .", "Let $(u_\\varepsilon , p_\\varepsilon )\\in H^1(\\Omega _\\varepsilon ; \\mathbb {R}^d) \\times L^2(\\Omega _\\varepsilon ; \\mathbb {R}^d)$ be a weak solution of the Dirichlet problem, $\\left\\lbrace \\aligned -\\varepsilon ^2 \\Delta u_\\varepsilon +\\nabla p_\\varepsilon & = f & \\quad & \\text{ in } \\Omega _\\varepsilon ,\\\\\\text{\\rm div} (u_\\varepsilon ) &=0& \\quad & \\text{ in } \\Omega _\\varepsilon , \\\\u_\\varepsilon &=0 & \\quad & \\text{ on } \\Gamma _\\varepsilon , \\\\u_\\varepsilon & = b & \\quad & \\text{ on } \\partial \\Omega ,\\endaligned \\right.$ where $f\\in C^{1, 1/2} (\\overline{\\Omega }; \\mathbb {R}^d)$ and $b \\in H^1(\\partial \\Omega ; \\mathbb {R}^d)$ satisfies the compatibility condition $\\int _{\\partial \\Omega } b\\cdot n\\, d\\sigma =0$ .", "Assume that $\\int _{\\Omega _\\varepsilon } p_\\varepsilon \\, dx =0$ .", "Then for $0< \\varepsilon <1$ , $\\aligned & \\Vert u_\\varepsilon - W (x/\\varepsilon ) ( f -\\nabla p_0) \\Vert _{L^2(\\Omega )}+\\Vert P_\\varepsilon -p_0 \\Vert _{L^2(\\Omega )}\\\\&\\qquad +\\Vert \\varepsilon \\nabla u_\\varepsilon -\\nabla W(x/\\varepsilon ) ( f-\\nabla p_0)\\Vert _{L^2(\\Omega )}\\\\& \\qquad \\qquad \\le C \\sqrt{\\varepsilon }\\Big \\lbrace \\Vert f\\Vert _{C^{1, 1/2}(\\Omega )}+ \\Vert b\\Vert _{H^1(\\partial \\Omega )} \\Big \\rbrace ,\\endaligned $ where $p_0$ is defined by (REF ), $P_\\varepsilon $ is given by (REF ), and $C$ depends only on $\\Omega $ and $Y_s$ .", "We begin by introducing a corrector for the divergence operator.", "For $1\\le i, k \\le d$ , let $ (\\chi ^1 _{ik} (y), \\dots , \\chi ^d_{ik}(y), \\pi _{2, ik} (y) ) \\in H^1_{\\text{loc}}(\\omega ; \\mathbb {R}^d) \\times L^2_{\\text{loc}}(\\omega )$ be an 1-periodic solution of $\\left\\lbrace \\aligned -\\Delta \\chi ^j _{ik} +\\frac{\\partial }{\\partial y_j} \\pi _{2, ik} & =0 & \\quad & \\text{ in } \\omega ,\\\\\\frac{\\partial }{\\partial y_j} \\chi ^j _{ik} & =-W_k^i + \|Y\\setminus Y_s\|^{-1} K_k^i & \\quad & \\text{ in } \\omega ,\\\\\\chi _{ik} ^j & =0 & \\quad & \\text{ on } \\partial \\omega .\\endaligned \\right.$ Since the compatibility condition, $\\int _{Y\\setminus Y_s} \\big \\lbrace -W_k^i + \|Y\\setminus Y_s \|^{-1} K_k^i \\big \\rbrace \\, dy=0,$ is satisfied, the 1-periodic solutions of (REF ) exist.", "Moreover, under the assumption that $\\partial Y_s$ is $C^{1, \\alpha }$ , the functions $\\nabla \\chi _{ik}^j$ and $\\pi _{2, ik}$ are bounded.", "As usual, we extend $\\chi _{ik}^j$ from $\\omega $ to $\\mathbb {R}^d$ by zero.", "Fix a function $\\varphi \\in C_0^\\infty (B(0, 1/8))$ with the properties that $\\varphi \\ge 0$ and $\\int _{\\mathbb {R}^d} \\varphi \\, dx=1$ .", "Let $S_\\varepsilon (\\psi ) (x) = \\psi * \\varphi _\\varepsilon (x) =\\int _{\\mathbb {R}^d} \\psi (y) \\varphi _\\varepsilon (x-y)\\, dy,$ where $\\varphi _\\varepsilon (x)= \\varepsilon ^{-d} \\varphi (x/\\varepsilon )$ .", "Define $\\Phi _\\varepsilon (x) = (\\Phi ^1_\\varepsilon (x), \\Phi _\\varepsilon ^2 (x), \\dots , \\Phi _\\varepsilon ^d (x)) $ , where $\\Phi _\\varepsilon ^j (x) = \\varepsilon \\eta _\\varepsilon (x) \\chi ^j _{k\\ell } (x/\\varepsilon ) \\frac{\\partial }{\\partial x_\\ell } S_\\varepsilon \\Big ( f_k -\\frac{\\partial p_0}{\\partial x_k} \\Big ),$ $p_0$ is a solution of the Neumann problem (REF ), and $\\eta _\\varepsilon $ is a cut-off function in $C_0^1( \\Omega )$ such that $0\\le \\eta _\\varepsilon \\le 1$ , $\\eta _\\varepsilon =1$ in $\\Omega \\setminus \\Sigma _{3\\varepsilon }$ , $\\eta _\\varepsilon =0$ in $\\Sigma _{2\\varepsilon }$ , and $\|\\eta _\\varepsilon \|\\le C \\varepsilon ^{-1}$ .", "The use of the $\\varepsilon $ -smoothing operator $S_\\varepsilon $ in (REF ) allows us to trade excessive powers of $\\varepsilon $ for lowering derivatives of $f-\\nabla p_0$ .", "The following lemma will be useful to us.", "Lemma 6.2 Let $S_\\varepsilon $ be defined by (REF ).", "Then $\\Vert \\psi - \\eta _\\varepsilon S_\\varepsilon (\\psi ) \\Vert _{L^2(\\Omega )}\\le C \\Vert \\psi \\Vert _{L^2(\\Sigma _{3\\varepsilon })}+ C \\varepsilon \\Vert \\nabla \\psi \\Vert _{L^2(\\Omega \\setminus \\Sigma _\\varepsilon )}$ for $0<\\varepsilon <1$ .", "Note that $\\Vert \\psi -\\eta _\\varepsilon S_\\varepsilon (\\psi )\\Vert _{L^2(\\Omega )}\\le \\Vert (1-\\eta _\\varepsilon )\\psi \\Vert _{L^2(\\Omega )}+ \\Vert \\eta _\\varepsilon (\\psi -S_\\varepsilon (\\psi )\\Vert _{L^2(\\Omega )}.$ Clearly, the first term in the right-side hand is bounded by $\\Vert \\psi \\Vert _{L^2(\\Sigma _{3\\varepsilon })}$ .", "To bound the second term, we use $\\psi (x) -S_\\varepsilon (\\psi )(x)=\\int _{\\mathbb {R}^d} \\varphi _\\varepsilon (y) [ \\psi (x-y) -\\psi (x)]\\, dy$ and $\\psi (x-y) -\\psi (x) =\\int _0^1 (-y) \\cdot \\nabla \\psi (x-ty)\\, dt.$ It follows that $\\aligned \\Vert \\eta _\\varepsilon (\\psi -S_\\varepsilon (\\psi )\\Vert _{L^2(\\Omega )}&\\le \\int _{\\mathbb {R}^d} \\varphi _\\varepsilon (y) \\Vert \\psi (\\cdot -y) -\\psi (\\cdot ) \\Vert _{L^2(\\Omega \\setminus \\Sigma _{2\\varepsilon })}\\, dy\\\\& \\le C\\int _{\\mathbb {R}^d} \\varphi _\\varepsilon (y) \|y\|\\, dy\\, \\Vert \\nabla \\psi \\Vert _{L^2(\\Omega \\setminus \\Sigma _\\varepsilon )}\\\\& \\le C \\varepsilon \\Vert \\nabla \\psi \\Vert _{L^2(\\Omega \\setminus \\Sigma _\\varepsilon )},\\endaligned $ where we have used Minkowski's inequality.", "Note that for $x\\in \\Omega _\\varepsilon $ , $\\aligned \\text{\\rm div} ( \\Phi _\\varepsilon )&=\\text{\\rm div} (\\chi _{k \\ell } ) (x/\\varepsilon ) \\Big [ \\eta _\\varepsilon \\frac{\\partial }{\\partial x_\\ell } S_\\varepsilon \\Big ( f_k -\\frac{\\partial p_0}{\\partial x_k} \\Big ) \\Big ]+ \\varepsilon \\chi _{k\\ell }^j (x/\\varepsilon ) \\frac{\\partial }{\\partial x_j}\\Big [ \\eta _\\varepsilon \\frac{\\partial }{\\partial x_\\ell } S_\\varepsilon \\Big ( f_k -\\frac{\\partial p_0}{\\partial x_k} \\Big ) \\Big ]\\\\&=-\\Big [ W_k^\\ell (x/\\varepsilon ) -_{Y\\setminus Y_s} W_k^\\ell \\Big ]\\Big [ \\eta _\\varepsilon \\frac{\\partial }{\\partial x_\\ell } S_\\varepsilon \\Big ( f_k -\\frac{\\partial p_0}{\\partial x_k} \\Big ) \\Big ]\\\\&\\qquad \\qquad + \\varepsilon \\chi _{k\\ell }^j (x/\\varepsilon ) \\frac{\\partial }{\\partial x_j}\\Big [ \\eta _\\varepsilon \\frac{\\partial }{\\partial x_\\ell } S_\\varepsilon \\Big ( f_k -\\frac{\\partial p_0}{\\partial x_k} \\Big ) \\Big ].\\endaligned $ Since $\\aligned \\text{\\rm div} \\Big ( W(x/\\varepsilon ) (f -\\nabla p_0) \\Big )& = W_k^\\ell (x/\\varepsilon ) \\frac{\\partial }{\\partial x_\\ell } \\Big ( f_k -\\frac{\\partial p_0}{\\partial x_k} \\Big )\\\\& = \\Big [ W_k^\\ell (x/\\varepsilon ) -_{Y\\setminus Y_s} W_k^\\ell \\Big ]\\frac{\\partial }{\\partial x_\\ell } \\Big ( f_k -\\frac{\\partial p_0}{\\partial x_k} \\Big ),\\endaligned $ where we have used the equation in (REF ), it follows that $\\aligned & \\Vert \\text{\\rm div} \\Big ( \\Phi _\\varepsilon + W(x/\\varepsilon ) (f -\\nabla p_0)\\Big )\\Vert _{L^2(\\Omega _\\varepsilon )}\\\\&\\quad \\le C \\Vert \\nabla (f-\\nabla p_0)\\Vert _{L^2(\\Sigma _{3\\varepsilon })}+ C \\Vert \\nabla [ ( f-\\nabla p_0) -S_\\varepsilon (f-\\nabla p_0) \\big ] \\Vert _{L^2(\\Omega \\setminus \\Sigma _{2\\varepsilon } )}\\\\&\\qquad \\qquad +C \\varepsilon \\Vert \\nabla ^2 S_\\varepsilon (f -\\nabla p_0)\\Vert _{L^2(\\Omega \\setminus \\Sigma _{2\\varepsilon })}.\\endaligned $ Let $(u_\\varepsilon , p_\\varepsilon )$ be a weak solution of (REF ) with $\\int _{\\Omega _\\varepsilon } p_\\varepsilon \\, dx =0$ .", "Let $v_\\varepsilon = u_\\varepsilon - \\Big \\lbrace W(x/\\varepsilon ) (f-\\nabla p_0) +\\Phi _\\varepsilon + \\Psi _t +\\Psi _n \\Big \\rbrace ,$ where $\\Phi _\\varepsilon $ is defined by (REF ), and $\\Psi _t, \\Psi _n $ are given by Theorems REF and REF , respectively.", "A direct computation shows that $\\aligned & -\\varepsilon ^2 \\Delta \\Big \\lbrace W(x/\\varepsilon ) (f-\\nabla p_0)\\Big \\rbrace + \\nabla \\Big \\lbrace p_0 + \\varepsilon \\pi (x/\\varepsilon ) (f-\\nabla p_0) \\Big \\rbrace \\\\& =f -\\varepsilon ^2 \\nabla \\big ( W(x/\\varepsilon )\\nabla (f-\\nabla p_0)\\big )- \\varepsilon (\\nabla W)(x/\\varepsilon ) \\cdot \\nabla ( f-\\nabla p_0)+ \\varepsilon \\pi (x/\\varepsilon ) \\nabla (f-\\nabla p_0)\\endaligned $ in $\\Omega _\\varepsilon $ .", "It follows that $\\aligned & -\\varepsilon ^2 \\Delta v_\\varepsilon +\\nabla \\big \\lbrace p_\\varepsilon -p_0 - p_t -p_n -\\varepsilon \\pi (x/\\varepsilon ) (f-\\nabla p_0) \\big \\rbrace \\\\&= \\varepsilon ^2 \\Delta \\Phi _\\varepsilon -\\varepsilon ^2 \\nabla \\big ( W(x/\\varepsilon )\\nabla (f-\\nabla p_0) \\big ) \\\\&\\qquad \\qquad - \\varepsilon (\\nabla W)(x/\\varepsilon ) \\cdot \\nabla ( f-\\nabla p_0)+ \\varepsilon \\pi (x/\\varepsilon ) \\nabla (f-\\nabla p_0)\\endaligned $ in $\\Omega _\\varepsilon $ .", "Also, observe that $\\text{\\rm div}(v_\\varepsilon ) = -\\text{\\rm div} \\Big ( \\Phi _\\varepsilon + W(x/\\varepsilon ) (f-\\nabla p_0) \\Big ) \\quad \\text{ in } \\Omega _\\varepsilon ,$ $ v_\\varepsilon =0$ on $\\Gamma _\\varepsilon $ , and that $v_\\varepsilon = \\gamma n \\quad \\text{ on } \\partial \\Omega ,$ where $\\gamma $ is a constant satisfying (REF ).", "Hence, by Theorem REF , $\\aligned & \\varepsilon \\Vert \\nabla v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} + \\Vert v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}\\\\&\\le C \\Big \\lbrace \\varepsilon \\Vert \\nabla \\Phi _\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\varepsilon \\Vert \\nabla ( f-\\nabla p_0) \\Vert _{L^2(\\Omega )}+ \\Vert \\text{\\rm div} (v_\\varepsilon )\\Vert _{L^2(\\Omega _\\varepsilon )}+ \|\\gamma \| \\Big \\rbrace \\\\& \\le C \\Big \\lbrace \\varepsilon \\Vert \\nabla ( f-\\nabla p_0)\\Vert _{L^2(\\Omega )}+ \\Vert \\nabla (f-\\nabla p_0) \\Vert _{L^2(\\Sigma _{3\\varepsilon })}\\\\&\\qquad \\qquad +C \\Vert \\nabla [ ( f-\\nabla p_0) -S_\\varepsilon (f-\\nabla p_0) \\big ] \\Vert _{L^2(\\Omega \\setminus \\Sigma _{2\\varepsilon } )}\\\\&\\qquad \\qquad + \\varepsilon \\Vert \\nabla ^2 S_\\varepsilon (f-\\nabla p_0)\\Vert _{L^2(\\Omega \\setminus \\Sigma _{2\\varepsilon })}+ \\varepsilon \\Vert \\nabla _{\\tan } ( f-\\nabla p_0)\\Vert _{L^2(\\partial \\Omega )}\\Big \\rbrace ,\\endaligned $ where we have used (REF ) and (REF ).", "Let $q_\\varepsilon = p_\\varepsilon - p_0 - q_t -q_n -\\varepsilon \\pi (x/\\varepsilon ) (f-\\nabla p_0).$ Note that Theorem REF also gives $\\aligned \\Vert q_\\varepsilon - _{\\Omega _\\varepsilon } q_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}& \\le C \\Big \\lbrace \\varepsilon \\Vert \\nabla ( f-\\nabla p_0)\\Vert _{L^2(\\Omega )}+ \\Vert \\nabla (f-\\nabla p_0) \\Vert _{L^2(\\Sigma _{3\\varepsilon })}\\\\&\\qquad +C \\Vert \\nabla [ ( f-\\nabla p_0) -S_\\varepsilon (f-\\nabla p_0) \\big ] \\Vert _{L^2(\\Omega \\setminus \\Sigma _{2\\varepsilon } )}\\\\&\\qquad + \\varepsilon \\Vert \\nabla ^2 S_\\varepsilon (f-\\nabla p_0)\\Vert _{L^2(\\Omega \\setminus \\Sigma _{2\\varepsilon })}+ \\varepsilon \\Vert \\nabla _{\\tan } ( f-\\nabla p_0)\\Vert _{L^2(\\partial \\Omega )}\\Big \\rbrace .\\endaligned $ Lemma 6.3 Let $(u_\\varepsilon , p_\\varepsilon )$ be a weak solution of (REF ) with $\\int _{\\Omega _\\varepsilon } p_\\varepsilon \\, dx=0$ .", "Then $\\aligned & \\varepsilon \\Vert \\nabla \\big ( u_\\varepsilon - W(x/\\varepsilon ) (f-\\nabla p_0) \\big ) \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert p_\\varepsilon - p_0\\Vert _{L^2(\\Omega _\\varepsilon )}\\\\& \\le C \\varepsilon ^{1/2}\\Big \\lbrace \\varepsilon ^{1/2} \\Vert \\nabla ( f-\\nabla p_0)\\Vert _{L^2(\\Omega )}+\\varepsilon ^{-1/2} \\Vert \\nabla (f-\\nabla p_0) \\Vert _{L^2(\\Sigma _{3\\varepsilon })}\\\\&\\qquad \\qquad \\quad +\\varepsilon ^{-1/2} \\Vert \\nabla [ ( f-\\nabla p_0) -S_\\varepsilon (f-\\nabla p_0) \\big ] \\Vert _{L^2(\\Omega \\setminus \\Sigma _{2\\varepsilon } )}\\\\&\\qquad \\qquad \\quad + \\varepsilon ^{1/2} \\Vert \\nabla ^2S_\\varepsilon (f-\\nabla p_0)\\Vert _{L^2(\\Omega \\setminus \\Sigma _{2\\varepsilon })}+ \\Vert f-\\nabla p_0 \\Vert _{L^2(\\partial \\Omega )}\\\\&\\qquad \\qquad \\quad + \\Vert b\\Vert _{L^2(\\partial \\Omega )} + \\varepsilon \\Vert \\nabla _{\\tan } b \\Vert _{L^2(\\partial \\Omega )}+ \\sqrt{\\varepsilon } \\Vert \\nabla _{\\tan } ( f-\\nabla p_0) \\Vert _{L^2(\\partial \\Omega )}\\Big \\rbrace \\endaligned $ for $0< \\varepsilon < 1$ .", "The estimate (REF ) follows readily from (REF ), (REF ), (REF ), and (REF ).", "To bound the right-hand side of (REF ), we let $p_0=p_0^{(1)} + p_0^{(2)}$ , where $p_0^{(1)}$ and $p_0^{(2)}$ are solutions of the Neumann problems, $\\left\\lbrace \\aligned K_j^i \\frac{\\partial }{\\partial x_i} \\Big ( f_j -\\frac{\\partial p_0^{(1)}}{\\partial x_j} \\Big )&=0 & \\quad & \\text{ in } \\Omega ,\\\\n_i K_j^i \\Big ( f_j -\\frac{\\partial p_0^{(1)} }{\\partial x_j} \\Big ) & =0& \\quad & \\text{ on } \\partial \\Omega ,\\endaligned \\right.$ and $\\left\\lbrace \\aligned K_j^i \\frac{\\partial ^2 p_0^{(2)} }{\\partial x_i \\partial x_j} & =0 & \\quad & \\text{ in }\\Omega ,\\\\n_i K_j^i \\frac{\\partial p_0^{(2)}}{\\partial x_j} & =- b\\cdot n & \\quad & \\text{ on } \\partial \\Omega ,\\endaligned \\right.$ respectively, with $\\int _\\Omega p_0^{(1)}\\, dx =\\int _\\Omega p_0^{(2)}\\, dx =0$ .", "Lemma 6.4 Let $p_0^{(1)}$ be a solution of (REF ) for some $f\\in C^{1, 1/2}(\\overline{\\Omega }, \\mathbb {R}^d)$ .", "Then $\\aligned \\Vert \\nabla p_0^{(1)} \\Vert _{L^\\infty (\\Omega )}+ \\Vert \\nabla ^2 p_0^{(1)} \\Vert _{L^\\infty (\\Omega )}& \\le C \\Vert f\\Vert _{C^{1, 1/2}(\\overline{\\Omega })}, \\\\\\Vert \\nabla ^2 S_\\varepsilon (f) \\Vert _{L^\\infty (\\Omega \\setminus \\Sigma _\\varepsilon )}+ \\Vert \\nabla ^2 S_\\varepsilon (\\nabla p^{(1)} _0) \\Vert _{L^\\infty (\\Omega \\setminus \\Sigma _\\varepsilon )}& \\le C \\varepsilon ^{-1/2} \\Vert f\\Vert _{C^{1, 1/2}(\\overline{\\Omega })},\\endaligned $ and $\\Vert \\nabla f-S_\\varepsilon (\\nabla f) \\Vert _{L^\\infty (\\Omega \\setminus \\Sigma _\\varepsilon )}+ \\Vert \\nabla ^2 p^{(1)} _0 - S_\\varepsilon (\\nabla ^2 p^{(1)} _0) \\Vert _{L^\\infty (\\Omega \\setminus \\Sigma _{\\varepsilon })}\\le C \\varepsilon ^{1/2} \\Vert f\\Vert _{C^{1, 1/2}(\\overline{\\Omega })}.$ Since $\\Omega $ is a bounded $C^{2, \\alpha }$ domain, the first inequality in (REF ) follows from the classical $C^2$ estimates, up to the boundary, for second-order elliptic equations with constant coefficients.", "Next, note that for $x\\in \\Omega \\setminus \\Sigma _\\varepsilon $ , $\\aligned \\frac{\\partial }{\\partial x_i} S_\\varepsilon (\\nabla f) (x)&=\\varepsilon ^{-1-d}\\int _{\\mathbb {R}^d} \\frac{\\partial \\varphi }{\\partial y_i} (y/\\varepsilon ) \\nabla f (x-y)\\, dy\\\\&=\\varepsilon ^{-1-d}\\int _{\\mathbb {R}^d} \\frac{\\partial \\varphi }{\\partial y_i} (y/\\varepsilon ) \\big [ \\nabla f(x-y) -\\nabla f(x)]\\, dy.\\endaligned $ It follows that $\\aligned \\Vert \\nabla ^2 S_\\varepsilon (f)\\Vert _{L^\\infty (\\Omega \\setminus \\Sigma _\\varepsilon )}& \\le C \\varepsilon ^{-1-d} \\int _{B(0, \\varepsilon /4)} \|\\nabla \\varphi (y/\\varepsilon )\| \|y\|^{1/2}\\, dy\\, \\Vert f\\Vert _{C^{1, 1/2}(\\overline{\\Omega })}\\\\& \\le C\\varepsilon ^{-1/2} \\Vert f\\Vert _{C^{1, 1/2}(\\overline{\\Omega })}.\\endaligned $ By the interior $C^{2, 1/2}$ estimates, $\|\\nabla ^2 p^{(1)} _0(x-y) -\\nabla ^2 p^{(1)} _0(y)\|\\le C \| y\|^{1/2} \\Big \\lbrace \\Vert f\\Vert _{C^{1, 1/2}(\\overline{\\Omega })} + \\Vert p^{(1)} _0\\Vert _{C^2(\\overline{\\Omega })} \\Big \\rbrace $ for any $x\\in \\Omega \\setminus \\Sigma _{\\varepsilon }$ and $\|y\|\\le \\frac{1}{4} \\varepsilon $ .", "As in the case of $\\nabla ^2 S_\\varepsilon (f)$ , this implies that $\|\\nabla ^2 S_\\varepsilon (\\nabla p_0^{(1)}) (x)\|\\le C \\varepsilon ^{-1/2} \\Vert f\\Vert _{C^{1, 1/2}(\\overline{\\Omega })}$ for any $x\\in \\Omega \\setminus \\Sigma _\\varepsilon $ .", "Finally, to see (REF ), we write $S_\\varepsilon (\\nabla f) (x)-\\nabla f (x) =\\int _{\\mathbb {R}^d} \\varphi _ \\varepsilon (x-y) [ \\nabla f(x-y) -\\nabla f(x)]\\, dy$ and proceed as in the previous estimates.", "Lemma 6.5 Let $p_0^{(2)}$ be a solution of (REF ).", "Then $\\Vert \\nabla p_0^{(2)} \\Vert _{L^2(\\Omega )}+ \\Vert \\nabla p_0^{(2)} \\Vert _{L^2(\\partial \\Omega )}& \\le C \\Vert b\\cdot n \\Vert _{L^2(\\partial \\Omega )}, \\\\\\Vert \\nabla ^2 p_0^{(2)}\\Vert _{L^2(\\Omega )}+ \\Vert \\nabla ^2 p_0^{(2)} \\Vert _{L^2(\\partial \\Omega )}+ \\varepsilon ^{-1/2} \\Vert \\nabla ^2 p_0^{(2)} \\Vert _{L^2(\\Sigma _{3\\varepsilon })} & \\le C \\Vert b\\cdot n \\Vert _{H^1(\\partial \\Omega )}, \\\\\\varepsilon ^{-1/2} \\Vert \\nabla ^2 p_0^{(2)} -S_\\varepsilon (\\nabla ^2 p_0^{(2)} ) \\Vert _{L^2(\\Omega \\setminus \\Sigma _{2\\varepsilon })}& \\le C \\Vert b\\cdot n \\Vert _{H^1(\\partial \\Omega )}, \\\\\\varepsilon ^{1/2} \\Vert \\nabla ^2 S_\\varepsilon (\\nabla p_0^{(2)}) \\Vert _{L^2(\\Omega \\setminus \\Sigma _{3\\varepsilon })}&\\le C \\Vert b\\cdot n \\Vert _{H^1(\\partial \\Omega )}, $ for $0< \\varepsilon <1$ .", "The estimates (REF )-() follow from the nontangential-maximal-function and square-function estimates for the Neumann problems, $\\Vert (\\nabla p_0^{(2)} )^\\Vert _{L^2(\\partial \\Omega )}+ \\left( \\int _{\\Omega } \\text{\\rm dist}(x, \\partial \\Omega )\|\\nabla ^2 p_0^{(2)} (x)\|^2\\, dx \\right)^{1/2}& \\le C \\Vert b\\cdot n \\Vert _{L^2(\\partial \\Omega )},\\\\\\Vert (\\nabla ^2 p_0^{(2)} )^\\Vert _{L^2(\\partial \\Omega )}+ \\left( \\int _{\\Omega } \\text{\\rm dist}(x, \\partial \\Omega )\|\\nabla ^3 p_0^{(2)} (x)\|^2\\, dx \\right)^{1/2}& \\le C \\Vert b\\cdot n \\Vert _{H^1(\\partial \\Omega )},$ where $(u)^*$ denotes the nontangential maximal function of $u$ , defined by (REF ).", "We remark that the estimate (REF ) hols if $\\Omega $ is a bounded Lipschitz domain [12], while () holds for $C^{2, \\alpha }$ domains.", "We only give the proof of (); the others follow readily from (REF )-().", "Choose $\\widetilde{\\eta }_\\varepsilon \\in C_0^1(\\Omega )$ such that $\\widetilde{\\eta }_\\varepsilon =1$ in $\\Omega \\setminus \\Sigma _{2\\varepsilon }$ , $\\widetilde{\\eta }_\\varepsilon =0$ in $\\Sigma _\\varepsilon $ , and $\|\\nabla \\widetilde{\\eta }_\\varepsilon \| \\le C \\varepsilon ^{-1}$ .", "Then the left-hand side of () is bounded by $\\varepsilon ^{-1/2} \\Vert \\nabla ^2 p_0^{(2)} -\\widetilde{\\eta }_\\varepsilon S_\\varepsilon ( \\nabla ^2 p_0^{(2)}) \\Vert _{L^2(\\Omega )}.$ Using the same argument as in the proof of (REF ), we may show that (REF ) is bounded by $C \\varepsilon ^{-1/2} \\Vert \\nabla ^2 p_0^{(2)} \\Vert _{L^2(\\Sigma _{3\\varepsilon })}+ C \\varepsilon ^{1/2} \\Vert \\nabla ^3 p_0^{(2)}\\Vert _{L^2(\\Omega \\setminus \\Sigma _\\varepsilon )}\\le C \\Vert b\\cdot n \\Vert _{H^1(\\partial \\Omega )},$ where we have used () for the last step.", "We are now in a position to give the proof of Theorem REF .", "[Proof of Theorem REF ] Using Lemmas REF and REF , it is not hard to see that the right-hand side of (REF ) is bounded by $C \\sqrt{\\varepsilon } \\big \\lbrace \\Vert f\\Vert _{C^{1, 1/2}(\\overline{\\Omega })} + \\Vert b\\Vert _{H^1(\\partial \\Omega )} \\big \\rbrace .$ As a result, we have proved that $\\aligned & \\varepsilon \\Vert \\nabla \\big ( u_\\varepsilon - W (x/\\varepsilon ) ( f -\\nabla p_0) \\big ) \\Vert _{L^2(\\Omega _\\varepsilon )}+\\Vert p_\\varepsilon -p_0 \\Vert _{L^2(\\Omega _\\varepsilon )}\\\\& \\qquad \\qquad \\le C \\sqrt{\\varepsilon }\\left\\lbrace \\Vert f\\Vert _{C^{1, 1/2}(\\Omega )}+ \\Vert b\\Vert _{H^{1} (\\partial \\Omega )} \\right\\rbrace .\\endaligned $ In view of Theorem REF , it remains to show that $\\Vert P_\\varepsilon -p_0\\Vert _{L^2(\\Omega )}\\le C \\sqrt{\\varepsilon }\\left\\lbrace \\Vert f\\Vert _{C^{1, 1/2}(\\Omega )}+ \\Vert b\\Vert _{H^{1} (\\partial \\Omega )} \\right\\rbrace ,$ where $P_\\varepsilon $ is an extension of $p_\\varepsilon $ to $\\Omega $ , defined by (REF ).", "To this end, we define $p_0^\\varepsilon =\\left\\lbrace \\aligned & p_0 & \\quad & \\text{ if } x\\in \\Omega _\\varepsilon ,\\\\& _{\\varepsilon (Y_f + z_k) } p_0 & \\quad & \\text{ if } x\\in \\varepsilon (Y_s + z_k) \\text{ and } \\varepsilon (Y+ z_k) \\subset \\Omega \\text{ for some } z_k \\in \\mathbb {Z}^d,\\endaligned \\right.$ i.e., we extend $p_0\|_{\\Omega _\\varepsilon }$ to $\\Omega $ in the same manner as we do $p_\\varepsilon $ from $\\Omega _\\varepsilon $ to $\\Omega $ .", "Then, $\\aligned \\Vert P_\\varepsilon -p_0\\Vert _{L^2(\\Omega )}&\\le \\Vert P_\\varepsilon - p_0^\\varepsilon \\Vert _{L^2(\\Omega )} + \\Vert p_0^\\varepsilon - p_0\\Vert _{L^2(\\Omega )}\\\\& = \\Vert p_\\varepsilon -p_0\\Vert _{L^2(\\Omega _\\varepsilon )}+ \\Vert P_\\varepsilon -p_0^\\varepsilon \\Vert _{L^2(\\Omega \\setminus \\Omega _\\varepsilon )}+ \\Vert p_0^\\varepsilon -p_0\\Vert _{L^2(\\Omega \\setminus \\Omega _\\varepsilon )}.\\endaligned $ Note that $\\Vert P_\\varepsilon -p_0^\\varepsilon \\Vert _{L^2(\\Omega \\setminus \\Omega _\\varepsilon )}\\le C \\Vert p_\\varepsilon -p_0\\Vert _{L^2(\\Omega _\\varepsilon )}.$ Using Poincaré's inequality on each cell $\\varepsilon (Y_f +z_k)$ , we may show that $\\aligned \\Vert p_0^\\varepsilon -p_0\\Vert _{L^2(\\Omega \\setminus \\Omega _\\varepsilon )}& \\le C \\varepsilon \\Vert \\nabla p_0 \\Vert _{L^2(\\Omega )}\\le C \\varepsilon \\left\\lbrace \\Vert f\\Vert _{L^2(\\Omega )} + \\Vert b \\Vert _{L^2(\\partial \\Omega )} \\right\\rbrace .\\endaligned $ As a result, we have proved that $\\Vert P_\\varepsilon -p_0\\Vert _{L^2(\\Omega )}\\le C \\Vert p_\\varepsilon -p_0\\Vert _{L^2(\\Omega _\\varepsilon )}+ C \\varepsilon \\left\\lbrace \\Vert f\\Vert _{L^2(\\Omega )} + \\Vert b \\Vert _{L^2(\\partial \\Omega )} \\right\\rbrace .$ This completes the proof.", "Remark 6.6 Let $u(x, x/\\varepsilon )$ be given by (REF ).", "Due to the discrepancy of $u_\\varepsilon $ and $u(x, x/\\varepsilon )$ on $\\partial \\Omega $ , the $O(\\sqrt{\\varepsilon })$ rate in Theorem REF is sharp.", "Indeed, by applying the following trace inequality to the function $v= v_\\varepsilon =u_\\varepsilon - u(x, x/\\varepsilon )$ , $\\Vert v\\Vert _{L^2(\\partial \\Omega )} \\le C\\left\\lbrace \\varepsilon ^{-1/2} \\Vert v\\Vert _{L^2(\\Sigma _{c\\varepsilon })} +\\Vert v\\Vert _{L^2(\\Sigma _{c\\varepsilon })}^{1/2} \\Vert \\nabla v\\Vert _{L^2(\\Sigma _{c\\varepsilon })}^{1/2} \\right\\rbrace ,$ we obtain $\\aligned \\sqrt{\\varepsilon } \\Vert v_\\varepsilon \\Vert _{L^2(\\partial \\Omega )}& \\le C \\big \\lbrace \\Vert v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} + \\varepsilon \\Vert \\nabla v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}\\big \\rbrace \\\\& \\le C \\varepsilon \\Vert \\nabla v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )},\\endaligned $ where we have used the Cauchy inequality for the first inequality and (REF ) for the second.", "It follows that the error estimate $\\varepsilon \\Vert \\nabla v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} =o(\\sqrt{\\varepsilon }) \\quad \\text{ as } \\varepsilon \\rightarrow 0,$ cannot hold in general.", "In fact, if $\\Omega $ is smooth and uniformly convex, then $\\aligned \\lim _{\\varepsilon \\rightarrow 0} _{\\partial \\Omega } \|v_\\varepsilon \|^2\\, d\\sigma & =\\lim _{\\varepsilon \\rightarrow 0} _{\\partial \\Omega } \| W(x/\\varepsilon ) (f-\\nabla p_0)\|^2\\, d\\sigma \\\\&=_{\\partial \\Omega } \| K (f-\\nabla p_0)\|^2\\, d\\sigma .\\endaligned $ See the proof of Lemma 3.2 in [1].", "Also, note that by Theorem REF , $\\Vert \\nabla v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}\\le C \\varepsilon ^{-1/2} \\Vert f\\Vert _{C^{1, 1/2}(\\Omega )}.$ This, together with (REF ), yields $\\Vert v_\\varepsilon \\Vert _{L^2(\\partial \\Omega )}\\le C \\left\\lbrace \\varepsilon ^{- 1/2} \\Vert v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )}+ \\big ( \\varepsilon ^{-1/2} \\Vert v_\\varepsilon \\Vert _{^2(\\Omega _\\varepsilon )} \\big )^{1/2} \\Vert f\\Vert _{C^{1, 1/2} (\\Omega )}^{1/2} \\right\\rbrace .$ As a result, it is not possible to have $\\Vert v_\\varepsilon \\Vert _{L^2(\\Omega _\\varepsilon )} =o(\\sqrt{\\varepsilon }) \\quad \\text{ as } \\varepsilon \\rightarrow 0,$ unless $f=\\nabla p_0$ in $\\Omega $ , in which case, $v_\\varepsilon \\equiv 0$ in $\\Omega _\\varepsilon $ .", "Finally, to see (REF ), choose a function $\\beta \\in C^1(\\mathbb {R}^d, \\mathbb {R}^d)$ such that $ \\beta \\cdot n \\ge c_0>0$ on $\\partial \\Omega $ , supp$(\\beta ) \\subset \\lbrace x\\in \\mathbb {R}^d: \\text{dist}(x, \\partial \\Omega )\\le c \\varepsilon \\rbrace $ , and $\| \\nabla \\beta \| \\le C \\varepsilon ^{-1}$ .", "It follows by the divergence theorem that $\\aligned c_0 \\int _{\\partial \\Omega } \|v\|^2\\, d\\sigma &\\le \\int _{\\partial \\Omega } \|v\|^2\\, \\beta \\cdot n \\, d\\sigma \\le \\int _\\Omega \|v\|^2 \\text{div} (\\beta )\\, dx+ 2 \\int _{\\Omega } \|v\| \|\\nabla v\| \|\\beta \|\\, dx\\\\&\\le C \\varepsilon ^{-1} \\int _{\\Sigma _{c\\varepsilon }} \|v\|^2\\, dx+ C \\Vert v\\Vert _{L^2(\\Sigma _{c\\varepsilon })} \\Vert \\nabla v\\Vert _{L^2(\\Sigma _{c\\varepsilon })},\\endaligned $ where we have used the Cauchy inequality for the last step.", "Zhongwei Shen, Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506, USA.", "E-mail: [email protected]" ] ]	2011.14169
[ [ "Semi-Supervised Learning for Sparsely-Labeled Sequential Data:\n Application to Healthcare Video Processing" ], [ "Abstract Labeled data is a critical resource for training and evaluating machine learning models.", "However, many real-life datasets are only partially labeled.", "We propose a semi-supervised machine learning training strategy to improve event detection performance on sequential data, such as video recordings, when only sparse labels are available, such as event start times without their corresponding end times.", "Our method uses noisy guesses of the events' end times to train event detection models.", "Depending on how conservative these guesses are, mislabeled samples may be introduced into the training set.", "We further propose a mathematical model for explaining and estimating the evolution of the classification performance for increasingly noisier end time estimates.", "We show that neural networks can improve their detection performance by leveraging more training data with less conservative approximations despite the higher proportion of incorrect labels.", "We adapt sequential versions of CIFAR-10 and MNIST, and use the Berkeley MHAD and HMBD51 video datasets to empirically evaluate our method, and find that our risk-tolerant strategy outperforms conservative estimates by 3.5 points of mean average precision for CIFAR, 30 points for MNIST, 3 points for MHAD, and 14 points for HMBD51.", "Then, we leverage the proposed training strategy to tackle a real-life application: processing continuous video recordings of epilepsy patients, and show that our method outperforms baseline labeling methods by 17 points of average precision, and reaches a classification performance similar to that of fully supervised models.", "We share part of the code for this article." ], [ "Introduction", "Labeled image and video datasets are crucial for training and evaluating machine learning models.", "As a result, computer vision researchers have compiled a number of labeled benchmark datasets, such as MNIST [17], ImageNet [7], MSCOCO [18], Kinetics [15], CIFAR [16], and Cityscapes [6].", "However, many application areas still remain poorly covered, such as medical imaging data, despite recent initiatives such as the UK Biobank [24].", "Although medical institutions often possess large amounts of data, most of it remains unlabeled and underutilized.", "Weakly-supervised learning aims to leverage datasets with either incomplete or incorrect labels.", "Zhou et al.", "[26] identified two subtypes of weak supervision schemes: incomplete and inaccurate supervision.", "Incomplete supervision applies when only a portion of the training samples are labeled.", "For example, semi-supervised learning methods are designed to leverage unlabeled samples next to labeled samples.", "Inaccurate supervision applies when the given labels are not necessarily correct (e.g., crowd-sourcing [19], [5]).", "The works of Hao et al.", "[10] on mammograms and Karimi et al.", "[14] on brain MRIs are also examples of inaccurate supervision with deep learning for medical data.", "In this work, we propose a method which combines semi-supervised learning and inaccurate supervision to leverage sparsely-labeled sequential data.", "The main task is to detect sequences of events, given only sparse training labels, i.e., the start times of these events.", "The end times and the duration of these events remain unknown, which prevents sampling any positives events with certainty (Figure REF ).", "For example, in a cooking videos dataset, sparse training labels could indicate when cooking an ingredient started at time T, but without any information about when the cooking of that ingredient stopped.", "To address this problem, we propose making a noisy approximation of event end times.", "For each sparse label, we choose a fixed number of consecutive elements in sequence that follow the sparse label, and use them as positive training samples (essentially providing a noisy estimate of duration).", "In the above example with cooking videos, we could guess that the cooking of the ingredient lasts one minute, or 1500 frames at 25 frames per second, and use all 1500 frames as positive samples.", "The longer the estimated guess, the more likely it is that we introduce potential incorrectly-labeled samples (false positives in the training set).", "We further propose a mathematical model to estimate the number of incorrect labels introduced using the proposed training strategy.", "We empirically evaluate our method on sparsely-labeled sequences of MNIST and CIFAR-10 images and show an improvement from 12 points of mean average precision for MNIST and 3.5 points for CIFAR over the baseline method.", "Finally, we demonstrate our method on a real-life sequential analysis task—video monitoring of epileptic hospital patients.", "Electroencephalography (EEG) is a common modality for recording brain activity and monitoring patients.", "Automated methods have been developed to automatically detect seizures from EEG activity [9], [20] but can fail to discern seizures from artifacts caused by disturbances in EEG measurement (e.g., patting on the back or rocking neonatal patients can trigger false positive seizure detections).", "We address EEG artifacts by automatically detecting an examplar artifact–patting of neonates–from continuous video recordings acquired during clinical routine.", "In our dataset, most patting events are annotated with sparse labels (only start times, no end times), which is common practice for the labeling of continuous recordings in clinical routine [22].", "Our method for learning from sparsely-labeled sequences can leverage those sparse labels, outperforming baseline methods by 10 points of average precision.", "To summarize, our main contributions are: A training strategy for semi-supervised learning with sparsely-labeled sequential data.", "A mathematical model to estimate the exposure of models to incorrectly labeled training samples when trained with such a strategy.", "A method that automatically detects patting events from sparsely-labeled continuous video recordings of hospital neonates." ], [ "Related Work", "Semi-supervised training strategies have been developed for a myriad of computer vision tasks.", "In image classification, most state-of-the-art semi-supervised methods are based on self-supervision and use contrastive learning approaches [11], [4].", "MoCo [11] encodes and matches query images to keys of a dynamic dictionary.", "SimCLR [4] improves upon MoCo by removing the need of specialised architecture.", "The authors of SimCLR claimed that the composition of data augmentation is crucial in achieving a high performance.", "Earlier, MixMatch [1] had already advocated the importance of data augmentation for semi-supervised learning for image classification.", "Given unlabeled images, MixMatch generated a set of augmented images, passed the images through the network and guessed the label using the mean of the model's predictions.", "The distribution of predictions was then sharpened using the guessed label.", "Semi-supervised learning has also been proposed for semantic image segmentation [2], [3].", "Bortsova et al [2] proposed augmenting unlabeled images, and applying reverse transformations to the output segmentations.", "The transformed segmentations were then forced to be similar to the originals using a consistency loss term.", "Jing et al.", "[13] proposed a semi-supervised learning method for video classification, using pseudo-labels and normalization probabilities of unlabeled videos to improve the classification performance.", "Sibechi et al.", "[21] proposed a semi-supervised method for the segmentation of sparsely-labeled video data.", "Our work leverages the sequentiality of the data through sampling, while Sibechi et al.", "leverages the sequentiality by directly including it as a architectural component of their model.", "Overall, sparsely-labeled video data is less studied than imaging data, which opens an avenue for novel semi-supervised learning methods that can leverage the sequentiality of frames." ], [ "Training Strategy", "We consider an infinite sequence $(x_{n})_{n\\in \\mathbb {N}}$ , and a sparse label $l$ indicating the start of an event, i.e., a subsequence of consecutive positive elements $(x_{n})_{n\\in [l,l+M-1]}$ , with $M$ being the duration of the event.", "The remaining elements outside of this positive subsequence are considered negative elements.", "Typically, if the length of positive subsequence, $M$ is known, we can create a sequence of corresponding labels $(y_{n})_{n\\in \\mathbb {N}}$ , where $y_{n} = 1$ if $n \\in [l,l+M-1]$ and $y_{n} = 0$ otherwise.", "Together, both sequences $(x_{n})_{n\\in \\mathbb {N}}$ and $(y_{n})_{n\\in \\mathbb {N}}$ can be used to train a machine learning model to detect positive events.", "In this work, we would like to train such models under the conditions that the length $M$ is unknown and only sparse labels $l$ indicating the start of the subsequence are known during training time.", "To address this problem, we propose making a noisy estimate of $M$ , using parameter $N$ , which in turn sets the risk associated with this estimate.", "We assign $N$ elements $(x_{n})_{n\\in [l,l+N-1]}$ that follow the labeled positive element $x_{l}$ to be positives.", "This results in potentially inaccurate labels $(\\hat{y}_{n})_{n\\in \\mathbb {N}}$ , with $\\hat{y}_{n} = 1$ if $n \\in [l,l+N-1]$ and $\\hat{y}_{n} = 0$ otherwise.", "When $N \\le M$ , the $N$ selected elements $\\lbrace x_{l},...,x_{l+N-1}\\rbrace $ are true positives, and none of the training elements are mislabeled.", "But when $N<M$ , our estimate misses $M-N$ true positive elements $\\lbrace x_{l+N},...,x_{l+M-1}\\rbrace $ that will not be used for training.", "This can be suboptimal, especially in datasets where positives are rare.", "On the other hand, when $N>M$ , this method introduces $N-M$ false positive elements $\\lbrace x_{l+M},...,x_{l+N-1}\\rbrace $ that are incorrectly labeled as positive for training.", "When $N \\ge M$ , the higher the value of $N$ is, the higher the proportion of incorrectly labeled samples in the training set and hence, higher the risk.", "Negative sampling is simpler.", "Negatives can be sampled in sequences that do not have positive sparse labels.", "Assuming only one positive event–and label–exists in the sequence, negative training elements can also be safely sampled before the labeled time $l$ as $(x_{n})_{n\\in [0,l-1]}$ , because $(y_{n})_{n\\in [0,l-1]} = (\\hat{y}_{n})_{n\\in [0,l-1]}$ .", "Other negative elements can be reasonably safely sampled far away from the labeled time as $(x_{n})_{n\\in [P,\\infty ]}$ , provided $P \\gg N$ .", "We train neural networks using sets of incorrectly labeled sequences, with a fixed risk level $N$ for all sequences, and evaluate the detection performance on independent sequences where the length $M$ is known." ], [ "Inaccuracy Exposure Estimation", "The further away an element is from the sparse label, the higher its probability is to be negative, and according to our sampling strategy, a false positive.", "To model this, we estimate the probability of the label of an element in a sequence to be incorrect using a standard exponential decay: $P(x_{n}) = 1 - e^{-\\alpha (n-l)}$ defined for $n \\in \\mathbb {N} \\cap [l,l+N-1]$ , and where $\\alpha $ is a calibration factor (the exponential decay constant).", "We recommend $\\alpha =\\frac{\\ln (2)}{\\mathbb {E}(M)}$ , where $\\mathbb {E}(M)$ is the estimated average length of positive subsequences (Supplementary Materials).", "The probability of having an incorrect label in a sequence of N elements sampled from a single labeled time $l$ is consequently $\\frac{1}{N} \\sum _{n=l}^{l+N-1} P(x_{n}) = \\frac{1}{N} \\sum _{n=l}^{l+N-1} (1 - e^{-\\alpha (n-l)})$ $\\qquad \\qquad \\quad = \\frac{1}{N} \\sum _{n=0}^{N-1} (1 - e^{-\\alpha n})$ For a set of $T$ labeled times $l_{t}$ , from $t$ independent sequences, the probability of having an incorrect label becomes $\\frac{1}{T} \\sum _{t=1}^{T} \\frac{1}{N_t} \\sum _{n=0}^{N_{t}-1} (1 - e^{-\\alpha n})$ where $N_t$ is the number of elements sampled for time $t$ .", "If the same number of elements $N$ is sampled for each labeled times, as in our experiments, this can be simplified as $P_{\\alpha }(N) = \\frac{1}{TN} \\sum _{t=1}^{T} \\sum _{n=0}^{N-1} (1 - e^{-\\alpha n})$ $P_{\\alpha }(N) = 1 - \\frac{1}{N} \\sum _{n=0}^{N-1} e^{-\\alpha n}$ The probability $P_{\\alpha }(N)$ can be used to estimate the level of exposure of a model–trained with such a dataset–to incorrect labels." ], [ "Inaccuracy Exposure Estimation for Video Segments", "We consider a video to be a sequence of frames.", "A first approach for estimating the inaccuracy exposure for videos would be to consider each frame of the video as an individual element $x_{n}$ .", "However, in video processing, a single frame is often not enough to accurately detect an event.", "Instead, sequences of frames—video segments—need to be processed.", "Consequently, for video processing, the inaccuracy exposure estimation model in Equation REF needs to be parameterized using the video segment length $L$ and becomes $P_{\\alpha ,L}(N) = 1 - \\frac{1}{N} \\sum _{n=0}^{N-1} e^{-\\alpha nL}$ Figure REF compares levels of inaccuracy exposure as a function of the risk level $N$ with varying calibration levels $\\alpha $ and segment lengths $L$ .", "Figure: Comparison of inaccuracy exposure levels P α,L (N)P_{\\alpha ,L}(N) as a function of the risk level NN with varying calibration levels α\\alpha and segment lengths LL.", "As expected, under a fixed calibration, sampling shorter segments allows the gathering of more segments without changing estimated exposure to incorrect labels." ], [ "Network Architecture", "We use 2D convolutional neural networks that take a 2D matrix as input, and output a single logit for binary classification.", "The architecture is adopted from a small ResNet [12]–two 3×3 convolutional layers, followed by a 2×2 max-pooling layer, again two 3×3 convolutional layers, a global average pooling layer, and a fully connected layer followed by a sigmoid activation function, combining the contribution of the different features into a single output in $[0,1]$ .", "The first two convolutional layers has 32 filters each, and the last two convolutional layers, 64 filters each.", "The convolutions are zero-padded and followed by ReLU activations.", "We use skip connections between the input and output of two successive convolutional layers." ], [ "Conservative versus Risk-tolerant Models", "We call conservative a model trained using the original sparse labeling, i.e.", "using only the first element following the sparse labels $l$ , which correspond to using a risk level $N=1$ .", "We consider this model to be the baseline.", "We call risk-tolerant a model trained using risk-tolerant labeling, i.e.", "using more than the first element following the sparse labels $l$ .", "In the MNIST and CIFAR experiments, we experiment with risk levels in $[1,9]$ .", "In the video experiments, the risk-tolerant model is trained using a risk level $N=3$ .", "The architecture, initialization and optimizer are the same for all models." ], [ "Experiments on Image Sequences", "To study the proposed training strategy in a controlled setting, we create a toy dataset using MNIST and other toy datasets using CIFAR-10 images.", "We find that risk-tolerant labeling outperforms sparse labeling by up to 12 points of mean average precision for MNIST, and 3.5 points for CIFAR.", "Experiments on image sequences are designed according to a similar scheme.", "One of the classes is selected as positive – e.g.", "the digit 1 in MNIST – and another class as negative – e.g.", "the digit 0 in MNIST.", "All images of the training set are equally split into training and validation sets, and the testing images are kept aside.", "We arrange the training images into 50 sequences of 10 images each by drawing images at random from the two target classes.", "Each sequence is parameterized by an integer duration $M$ (drawn uniformly from $0 \\le M \\le 10$ ) of true positive elements, with the first $M$ images drawn from the positive class and the remaining $10-M$ images drawn from the negative class (Figures REF and REF ).", "As we want to model a scenario where the duration $M$ is not known during all the optimization steps, sequence creation is performed similarly for the validation set.", "To construct our risk-dependent training labels, we select a risk level $1 \\le N \\le 9$ over all 50 sequences and sample the $N$ first images of each sequence as positives, independent of their true class.", "If $N>M$ , this results in using $N-M$ incorrectly labeled images per sequence for training (false positives in Figure REF ).", "Next to this, 50 negative training images are directly sampled from the negative class.", "For preprocessing, image intensity values are rescaled in $[0,1]$ using the the image-wise minimum and maximum to facilitate the training.", "We use this dataset to train a convolutional network with the Adadelta optimizer [25] and optimize the binary cross-entropy.", "For every epoch, the same proportion of negative and positive training images are shown to the network to avoid rebalancing the loss function.", "Training is stopped after the validation loss diverges, and the best model is selected as the one minimizing the validation loss.", "The trained model's performance is evaluated on the left-out test set and measured using recall, precision, F1-score, average precision and AUC.", "All experiments described above are repeated using varying levels of risk $N \\in [1,9]$ .", "Figure: Sequences of MNIST images (first row) and consecutive patting video segments (second row).", "On the first row, the risk level is N=5N = 5, and the true length of the positive subsequence is M=3M = 3, which results in N-M=2N-M=2 incorrectly labeled training samples (FP).", "On the second row, the risk level is N=3N = 3, and the true length of the positive subsequence is M=2M = 2, which results in N-M=1N-M=1 incorrectly labeled training sample.", "TP indicates true positives, FP false positives and TN true negatives.Figure: Examples of image sequences for MNIST and CIFAR-10.", "1 vs. 0 is shown on the first two rows, automobile (car) vs. airplane on rows 3 and 4, and automobile vs. bird on rows 5 and 6." ], [ "Results on the MNIST dataset", "The digit 1 is chosen as positive class and the digit 0 as negative class (Figure REF , row 1).", "The other digits are ignored in all the experiments.", "To complexify the task, Gaussian noise–with a mean of 1 and standard deviation of 2–is added to the images.", "Each experiment is repeated 10 times using different random initializations of the network's weights.", "This results in a total of 9 risk levels times 10 runs equals 90 experiments.", "The recall improves with higher levels of risk (Figure REF ), while the precision does not significantly vary (Supplementary Materials).", "Additional results are shown in Supplementary Materials.", "Overall, risk-tolerant labeling outperforms conservative labeling ($N=1$ ) by up to 12 points of mean average precision with risk level $N=3$ .", "Figure: Recall averaged over 10 runs of the 1 vs. 0 MNIST experiements, with varying risk levels NN.", "95% confidence interval are computed with bootstrapping.", "The baseline method, which does not leverage unlabeled data, is for risk N=1N=1." ], [ "Results on CIFAR-10 datasets", "The automobile class is chosen as the positive class, and 9 series of experiments are realized using the 9 other classes as negatives, respectively.", "In the first series of experiments, we create sequences of automobiles and airplanes (Figure REF , rows 3 and 4).", "In the second series of experiments, we create sequences of automobiles and birds (Figure REF , rows 5 and 6), and so forth.", "Each experiment is repeated 10 times using different random initializations of the network's weights.", "This results in a total of 9 risk levels times 9 classes times 10 runs equals 810 experiments.", "Detailed results are shown in Tables REF , Figure REF and in supplementary materials.", "Overall, risk-tolerant labeling outperforms conservative labeling ($N=1$ ) by up to 3.5 points of mean average precision with risk level $N=6$ .", "Figure: Recall averaged over all 810 the CIFAR-10 experiements, with varying risk levels NN.", "95% confidence interval are computed with bootstrapping.", "The baseline method, which does not leverage unlabeled data, is for risk N=1N=1.Table: Average Precision for the CIFAR-10 dataset.", "Means (standard deviations) are computed for 10 repetitions of the experiments with different random initialization of the weights.", "Best statistically significant (p-value << 0.05) results are highlighted in bold.", "The baseline method, which does not leverage unlabeled data, is highlighted in grey." ], [ "Experiments on Videos of Hospital Patients", "We evaluate our method on a real-world task: the detection of patting events from continuous video recordings of hospital patients.", "We present results comparing two levels of risk: a conservative model with $N=1$ , and a risk-tolerant model with $N=3$ .", "We find that our risk-tolerant model outperforms the conservative model by 10 points of average precision." ], [ "Dataset", "The dataset consists of 294 video recordings acquired during continuous monitoring of neonate epileptic patients as part of the clinical routine of a hospital.", "The recording time per video lasts between 99 seconds and 720 minutes with a median of 33 minutes.", "EEG monitoring of the patients is also continuously acquired to detect seizures, though this data is not used in the experiments.", "The objective is to detect the clinicians' patting of neonatal patients in the video, as it can create brain activity wave-forms that mislead seizure detection in the patient's EEG.", "Consequently, patting is selected as the positive class and all other events as negatives.", "As part of the clinical routine, these video recordings are labeled by multiple clinicians to indicate the occurrence of patting events with the corresponding start time only.", "We call these labels weak labels because the end frames of the patting events are unknown.", "28 of the 294 videos, recorded from 26 patients, are reviewed a second time, and both the beginning and the end of the patting segments are labeled.", "We call these labels strong labels.", "The strong labels encompass 2214 seconds of patting in total.", "We include strongly-labeled data along with our weakly labeled data in our training to boost the performance.", "The set of 28 videos with strong labels is split randomly into three subsets using stratified sampling over the number of patting seconds: the training set, 14 videos, the validation set, 4 videos, the test set, 10 videos.", "The training, validation and test videos are also separated by patient.", "The rest of the dataset, 399 weak labels from 266 other videos from 224 other patients, are used in the training set only.", "The models are optimized using the training and validation set, while the test set is set aside for independent evaluation." ], [ "Sampling of Video Segments", "Five second long, non-overlapping, video segments, sampled at 15 frames per seconds, are extracted from the videos.", "For the strongly-labeled training and validation datasets, patting segments are sampled to cover the patting time indicated by the strong labels.", "Negative segments are sampled from the rest of the recording at random.", "For the weak labels, $N$ consecutive positives segments are sampled after the labeled times (Figure in page 1).", "Weak negative segments are sampled at random, far away (5 minutes) from the weak label times.", "To evaluate the models on the test set, consecutive non-overlapping video segments are extracted to span the full video time.", "We choose the strong labels as ground truth for those segments." ], [ "Preprocessing", "Instead of feeding image sequences directly to the models, we take inspiration from Dwibedi et al.", "[8] and computed 75x75 frame-to-frame normalized cross-correlation matrices over all the frames within each video segments to highlight repetitive motions.", "While Dwibedi et al.", "[8] used the cross-correlation matrices to count the number of repetitions in preselected video segments displaying repetitive events, we use the cross-correlation matrices to detect the repetitive events in longer video recordings.", "Examples of such matrices are shown in Figure REF .", "The diagonal from the top left corner to the bottom right corner indicates cross-correlation of the current frame with itself, and displays consequently the highest values.", "The top right half of the cross-correlation matrices are set to 0 to avoid repetitive information due to the symmetry of the cross-correlation function.", "To facilitate training, these matrices are normalized in $[0,1]$ using their 1st and 99th percentiles." ], [ "Training", "The networks are trained using a weighted-cross entropy loss and Adam optimizer.", "To ease and accelerate the training, we pretrain the models using the strong labels only.", "For pretraining, a single model is trained with random weight initialization with as many random negatives as available positives for both the training and validation sets.", "The conservative model is fine-tuned from this model using 10$\\times $ more negatives than positives–to simulate the proportion of positives to negatives in full video recordings.", "All available strong positives are used, plus 399 ($N=1$ ) weak positives from the weak labels.", "The risk-tolerant model is fine-tuned like the conservative model, with 1197 ($N=3$ ) weak positives instead of 399.", "The training is stopped after the validation F1-score has not increased for 400 epochs.", "The best model is selected as the one with maximum F1-score on the validation set." ], [ "Results", "We compute evaluation metrics segment-wise on the test set.", "The risk-tolerant model achieves a better performance (average precision of 0.49) than the conservative model (average precision of 0.39).", "The difference in average precision mostly results from the risk-tolerant model having a higher recall than the conservation model (Figure REF ).", "The F1 score is highly correlated to length of patting event the segment is extracted from: 0.91 Pearson correlation coefficient for the conservative model and 0.87 for the risk-tolerant model.", "Shorter patting sequences are much more difficult to accurately detect and account for a large portion of the dataset.", "For videos with longer patting sequence, results are as high as 0.90 recall and 0.94 precision, while poor results are achieved on videos with short patting sequences with 0.29 recall, and 0.40 precision.", "Figure: Comparison of precision-recall curves for patting detection from videos of hospital patients.", "The conservative model is displayed in blue and the risk-tolerant model in orange." ], [ "Inspection of Network Attention", "To verify that the networks focuses on the repetitive portions (checkerboard) of the cross-correlation matrices, we compute attention maps highlighting areas of the image used for the models' predictions.", "Attention maps are computed using guided-backpropagation [23], which computes the derivative of the output with respect to the input.", "Pixels with high absolute values in the attention map correspond to regions used by the network for its prediction.", "Qualitative inspection of the attention maps (Figure REF ) reveals that the risk-tolerant model focuses more on elements of the cross-correlation matrix that are close to the diagonal, i.e.", "frames close to each other, while the conservative model seems to search for checkerboard patterns randomly in the cross-correlation matrices, e.g.", "at the bottom left corner, which encodes information of frames far apart in time.", "Figure: Comparison of the attention maps of the risk-tolerant and conservative models for two random positive video segments (row 1 and row 2).", "The face is masked and blurred for privacy." ], [ "Discussion", "For all datasets–MNIST, CIFAR, and hospital videos–using the proposed semi-supervised training strategy improves detection performance over the conservative model baseline, which uses only sparse labeling.", "The networks leverage the additional–yet partially incorrect–positive training samples to significantly improve recall without substantially changing the precision, leading to a significantly higher mean average precision.", "Sampling more positive training elements from sparse labels–increasing the risk level $N$ –exposes the models to more incorrectly labeled samples.", "For lower levels of risk ($N < 4$ ), the performance improves linearly with the level of risk on both MNIST and CIFAR-10 datasets.", "However, with higher levels of risk, the performance does not improve further.", "The performance stagnates on the MNIST dataset, while it linearly decreases on the CIFAR-10 dataset, until being similar to that of $N=1$ for $N=9$ .", "In the experiments on sequences of images, risk levels higher than $N=5$ means that on average, more than 50% of training samples are incorrectly labeled.", "Following these observations, we recommend avoiding the use of risk levels that lead to having on average more than 50% incorrectly labeled positive training samples.", "Using a higher risk level can lead to a deterioration of the performance.", "The modelling of the inaccuracy exposure estimation for video segments in Figure REF shows that, with a fixed calibration, sampling shorter segments would increase the number of video segments without changing the estimated exposure to incorrect labels.", "Consequently, we recommend splitting videos into segments that are as short as possible – to sample a higher number of segments for a fixed level of inaccuracy exposure – while being long enough for the models to detect the target event in each individual segments.", "In our experiments, we also experimented with shorter segments (two seconds long), but could not reach a satisfactory performance on the validation set.", "Next to the methodological contributions, we develop a model to detect repetitive motion in continuous video recordings of hospital patients, basing our analysis on frame-to-frame correlation matrices.", "Detecting repetitive motions allows the discarding of false positive seizure detections on concomitant EEG recordings.", "The performance is high for the detection of long patting events, and low when the events are shorter.", "Shorter patting sequences, e.g.", "with interruptions, are expected to be more difficult to detect but are less relevant to the application of seizure detection.", "Only longer sequences are expected to create EEG signals that may resemble that of seizures and mislead the automated methods, and consequently need to be detected to filter out potential false positives.", "On the CIFAR-10 experiments, despite running the experiments multiple times–10 runs for each parameter setting–the evolution of the performance sometimes abruptly varies for similar neighboring risk levels $N$ .", "We suspect that this is a consequence of the underlying 50 sequences being sampled only once for every run.", "An unlucky sampling could lead to a worse performance overall, and vice-versa.", "In one-vs-all preliminary experiments on MNIST, risk-conservative models did not perform significantly better than the conservative models, which suggest that the performance gain depends on the dataset.", "One of the limitations of the proposed model is that the sparse labels $l$ are always considered to be correct.", "In our experiments with continuous video recordings, the reviewing of a subset of such labels reveal that some are incorrect: no patting is found in the video.", "In that case, according to the inaccuracy exposure estimation model, the conservative model has an inaccuracy exposure $P_{\\alpha ,L}(1)$ equal to 0, while in reality the model uses some incorrect training labels.", "To improve the inaccuracy exposure estimation model, one could incorporate a term that accounts for the risk of the sparse labels of being incorrect.", "In this work, only non-overlapping video segments are considered.", "Sampling overlapping videos could act as data augmentation, regularize the models, and improve the performance further.", "The inaccuracy exposure estimation model would consequently need to be adapted.", "Finally, next to the MNIST and CIFAR datasets, the proposed method is only evaluated on patting hospital videos.", "We expect similar results for other types of repetitive motions such a rocking of neonates.", "The generalization to other videos and events still needs to be studied." ], [ "Conclusion", "We proposed a semi-supervised training strategy for sparsely-labeled sequential data and showed that it can improve detection performance.", "Convolutional neural networks were able to leverage the additional–yet partially incorrectly labeled–positive training samples to significantly improve recall without substantially changing the precision, leading to a significantly higher average precision.", "We demonstrated this improvement on image sequences from benchmark datasets and detection of video events.", "We also proposed an inaccuracy exposure estimation model to help choose the hyperparameters of inaccurately supervised video models, and empirically found that the proportion of incorrectly labeled sampled should not exceed 50% for best performance.", "We also recommended sampling the shortest possible video segments to reduce number of incorrectly labeled training samples.", "Finally, we also contributed a patting detection method based on frame-to-frame cross-correlation matrices and that has the potential to improve seizure detection." ], [ "Acknowledgments", "We thank the Wu Tsai Neurosciences Institute Seed Grant program for their support." ] ]	2011.14101
[ [ "Fusion reaction of a weakly-bound nucleus with a deformed target" ], [ "Abstract We discuss the role of deformation of the target nucleus in the fusion reaction of the $^{15}$C + $^{232}$Th system at energies around the Coulomb barrier, for which $^{15}$C is a well-known one-neutron halo nucleus.", "To this end, we construct the potential between $^{15}$C and $^{232}$Th with the double folding procedure, assuming that the projectile nucleus is composed of the core nucleus, $^{14}$C, and a valance neutron.", "By taking into account the halo nature of the projectile nucleus as well as the deformation of the target nucleus, we simultaneously reproduce the fusion cross sections for the $^{14}$C + $^{232}$Th and the $^{15}$C + $^{232}$Th systems.", "Our calculation indicates that the net effect of the breakup and the transfer channels is small for this system." ], [ "Introduction", "One of the most important discoveries in nuclei near the neutron drip line is the halo phenomenon [2], [1].", "It is characterized by a spatially extended density distribution, originated from the weakly-bound property of neutron-rich nuclei.", "Starting from $^{11}$ Li [2], several other halo nuclei have also been observed successively.", "For example, $^{6}$ He, $^{14}$ Be, and $^{17}$ B are regarded as two-neutron halo nuclei, while $^{11}$ Be, $^{15}$ C and $^{19}$ C are categorized as one-neutron halo nuclei [3], [4].", "Recently, heavier halo nuclei, such as $^{19}$ C[5], $^{22}$ C [6], $^{31}$ Ne [7], and $^{37}$ Mg [8] have also been found at Radioactive Ion Beam Facility (RIBF) in RIKEN.", "Fusion reactions of halo nuclei have attracted lots of attention [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].", "It is generally known that fusion cross sections at energies around the Coulomb barrier are sensitive to the structure of colliding nuclei [20], [21], [22], [23], [24], and it is thus likely that the halo structure significantly affects fusion reactions, both in a static and a dynamical ways.", "With the development of the radio-isotope technology, a large number of experimental data for fusion of halo nuclei have been accumulated.", "For instance, fusion cross sections for the $^{11}$ Li + $^{208}$ Pb [25], $^{\\text{6}}$ He + $^{238}$ U [26], $^{6}$ He + $^{209}$ Bi [27], [28] , $^{11}$ Be + $^{209}$ Bi [29], and $^{15}$ C + $^{232}$ Th [30] systems have been reported.", "Interestingly, it has been reported that fusion cross sections for the $^{\\text{6}}$ He + $^{238}$ U system [26] do not show any significant influence of the halo structure of $^{6}$ He albeit that $^{6}$ He is a well-known halo nucleus.", "This is in contrast to fusion cross sections for the $^{11}$ Li + $^{208}$ Pb [25] system, which show an enhancement with respect to the fusion cross sections for the $^{9}$ Li + $^{208}$ Pb system.", "The $^{6}$ He + $^{209}$ Bi system also shows a similar trend as in the $^{11}$ Li + $^{208}$ Pb system [28].", "In the case one-neutron halo nuclei, cross sections for the $^{11}$ Be+$^{209}$ Bi system are reported to be similar to those for the $^{10}$ Be+$^{209}$ Bi system [29].", "Origins for this apparent difference among these systems have not yet been understood completely, even though the fissile nature of the $^{238}$ U may play some role.", "In this regard, it is interesting to notice that $^{238}$ U is a well deformed nucleus while $^{208}$ Pb and $^{209}$ Bi are spherical nuclei.", "The aim of this paper is to investigate the role of deformation of the target nucleus in fusion of a halo nucleus.", "To this end, we shall discuss the fusion reaction of the $^{15}$ C+$^{232}$ Th system.", "The $^{15}$ C nucleus is a one-neutron halo nucleus [4], and its structure is much simpler than the structure of the two-neutron halo nuclei $^{11}$ Li and $^{6}$ He.", "The $^{15}$ C+$^{232}$ Th system thus provides an ideal opportunity to disentangle the deformation and the halo effects.", "Moreover, $^{15}$ C is heavier than $^{11}$ Li and $^{6}$ He, and more significant effects of the target deformation can be expected for the $^{15}$ C+$^{232}$ Th system as compared to the $^{11}$ Li, $^{6}$ He+$^{238}$ U systems.", "Notice that the previous calculation for this system used a very simple spectator model and did not take into account the halo structure of $^{15}$ C [30].", "It has yet to be clarified how much the measured fusion enhancement can be accounted for by taking into account the halo structure of $^{15}$ C. The paper is organized as follows.", "In Sec.", ", we first analyze the fusion of the $^{14}$ C + $^{232}$ Th system by using a deformed Woods-Saxon potential.", "In Sec.", ", we analyze the fusion of the $^{15}$ C + $^{232}$ Th system and discuss the role of the halo structure and the deformation effect.", "To this end, we construct the potential between $^{15}$ C and $^{232}$ Th with the double folding formalism accounting for the halo structure of the $^{15}$ C nucleus.", "We finally summarize the paper in Sec." ], [ "Fusion reaction of the $^{14}$ C+{{formula:e1782bef-4fc9-4167-8a10-21efcc7eb79d}} Th system", "Before we discuss the fusion cross sections for the $^{15}$ C+$^{232}$ Th system, we first analyze the $^{14}$ C+$^{232}$ Th system.", "In order to take into account the deformation effect of the target nucleus $^{232}$ Th, we employ a deformed Woods-Saxon (WS) potential for the relative motion between the target and the projectile nuclei [22], [31], [32]: $V_{^{14}{\\rm C}-T}(r,\\theta ) = - \\frac{V_0}{1+ \\exp \\left[ \\left( r-R_0 - R_T\\sum _{\\lambda } \\beta _{\\lambda T} Y_{\\lambda 0 }(\\theta ) \\right)/a\\right]},$ where $V_{0}$ , $R_{0}$ , and $a$ are the depth, the radius, and the diffuseness parameters, respectively.", "$R_T$ and $\\beta _{\\lambda T}$ are the radius and the deformation parameters of the target nucleus $^{232}$ Th.", "The Coulomb potential also has a deformed form given by [22], [31] $V_{C}(r,\\theta ) &=& \\frac{Z_{P}Z_{T}e^{2}}{r} \\nonumber \\\\& &+\\frac{3Z_{P}Z_{T}e^{2}}{5}\\frac{R_{T}^{2}}{r^{3}}\\left(\\beta _{2T}+\\frac{2}{7} \\sqrt{\\frac{5}{\\pi }}\\beta _{2T}^{2} \\right)Y_{20}(\\theta )\\nonumber \\\\& &+\\frac{3Z_{P}Z_{T}e^{2}}{9}\\frac{R_{T}^{4}}{r^{5}}\\left(\\beta _{4T}+\\frac{9}{7\\sqrt{\\pi }}\\beta _{2T}^{2} \\right)Y_{40}(\\theta ),$ with the second order in the quadrupole deformation parameter, $\\beta _{2T}$ , and the first order in the hexadecapole deformation parameter, $\\beta _{4T}$ .", "$Z_P$ and $Z_T$ are the atomic number of the projectile and the target nuclei, respectively.", "Fusion is simulated with the incoming wave boundary condition [22], [31].", "For simplicity, in this paper we assume that $^{14}$ C is inert.", "The fusion cross sections for the $^{14}$ C+$^{232}$ Th system so obtained are presented in Fig.", "REF .", "The actual values for the parameters in the WS potential employed in this calculation are given in Table REF .", "We use the same values for the radius and the diffuseness parameters as those of the global type of Akÿz-Winther (AW) potential [33], while we adjust the depth parameter, $V_0$ to fit the measured fusion cross sections.", "For the deformation parameters, we employ $\\beta _{2T} = 0.233$ and $\\beta _{4T}= 0.0946$ [34].", "In the figure, the blue dashed line shows the cross sections in the absence of the deformation effect, while the black solid line shows the cross sections with the deformation effect.", "One can clearly see that the enhancement of the fusion cross sections below the Coulomb barrier region can be well accounted for by taking into account the deformation of $^{232}$ Th.", "It is apparent that the deformation plays an important role in this system.", "Table: The depth, V 0 V_0, the radius, r 0 r_0, and the diffuseness, aa, parametersfor the deformed Woods-Saxon potential for the 14 ^{14}C + 232 ^{232}Th reaction.Here, the radius parameter r 0 r_0 is defined as R 0 =r 0 (A P 1/3 +A T 1/3 R_0=r_0(A_P^{1/3}+A_T^{1/3}), whereA P A_P and A T A_T are the mass numbers of the projectile and the target nuclei, respectively.", "The resultant barrier height, V b V_{b}, the barrier position, R b R_{b}, and the barrier curvature, ℏΩ\\hbar \\Omega are also shown.Figure: Fusion cross sections for the 14 ^{14}C + 232 ^{232}Th system.The dashed line denotes cross sections in the absence of the deformationeffect of the target nucleus 232 ^{232}Th while the solid line is obtained by takinginto account the deformation effect with a deformed Woods-Saxon potential.The experimental data are taken from Ref.", "." ], [ "The internuclear potential", "Let us now discuss the fusion reaction of the $^{15}$ C + $^{232}$ Th system.", "We first construct the potential between $^{15}$ C and $^{232}$ Th taking into account the deformation effect of the target nucleus as well as the halo structure of the projectile.", "To this end, we employ the double folding approach and construct the potential as $V_{^{15}{\\rm C}-T}(\\mbox{$r$};\\mbox{$r$}_{\\rm d})&=& \\int d\\mbox{$r$}_p \\int d\\mbox{$r$}_T \\rho _p(r_p)\\rho _T(\\mbox{$r$}_T;\\mbox{$r$}_{\\rm d})\\nonumber \\\\&&\\times V_{NN} (\\mbox{$r$}-\\mbox{$r$}_p+\\mbox{$r$}_T),$ where $V_{NN}$ is an effective nucleon-nucleon interaction, while $\\rho _p(r_p)$ and $\\rho _T(\\mbox{$r$}_T;\\mbox{$r$}_{\\rm d})$ are the density profiles for the projectile and the target nuclei, respectively.", "Here, the density of the deformed target is defined with respect to the orientation angle, $\\mbox{$r$}_{\\rm d}$ , in the space fixed frame.", "In this paper, we employ the deformed Woods-Saxon density given by $\\rho _T(\\mbox{$r$};\\mbox{$r$}_{\\rm d})=\\frac{\\rho _0}{1+\\exp \\left[\\left(\\frac{r-c(1+\\sum _{\\lambda } \\beta _{\\lambda T} Y_{\\lambda 0 }(\\theta _{rd}))}{z}\\right)\\right]},$ where $\\theta _{rd}$ is the angle between $\\mbox{$r$}$ and $\\mbox{$r$}_{\\rm d}$ , and $\\beta _{\\lambda T}$ are the deformation parameters.", "We take $c$ =6.851 fm, $z$ =0.518 fm and $\\rho _{\\text{0}}$ =0.162 fm$^{-3}$ for the $^{232}$ Th nucleus [35].", "For the nucleon-nucleon interaction, $V_{NN}$ , we use the M3Y interaction [36] given by $V_{NN}(r)=-2134\\frac{e^{-2.5r}}{2.5r}+7999\\frac{e^{-4r}}{4r}-275.81\\delta (r),$ where the energy and the length are given in units of MeV and fm, respectively.", "Notice that this interaction also includes the knock-on exchange effect in the zero-range approximation.", "To evaluate the double folding potential, the target density (REF ) is expanded as $\\rho _T(\\mbox{$r$};\\mbox{$r$}_{\\rm d})&=&\\sum _{\\lambda } \\rho _{T \\lambda }(r) Y_{\\lambda 0}(\\theta _{rd}) \\\\&=&\\sum _{\\lambda ,\\mu } \\rho _{T \\lambda }(r) \\sqrt{\\frac{4\\pi }{2\\lambda +1}}Y_{\\lambda \\mu }(\\hat{\\mbox{$r$}})Y^_{\\lambda \\mu }(\\hat{\\mbox{$r$}}_{\\rm d}).$ Substituting this into Eq.", "(REF ), one obtains the potential in a form of $V_{^{15}{\\rm C}-T}(\\mbox{$r$};\\mbox{$r$}_{\\rm def})=\\sum _{\\lambda ,\\mu } V_{\\lambda }(r) \\sqrt{\\frac{4\\pi }{2\\lambda +1}}Y_{\\lambda \\mu }(\\hat{\\mbox{$r$}})Y^_{\\lambda \\mu }(\\hat{\\mbox{$r$}}_{\\rm d}),$ with $V_{\\lambda }(r) = \\int d\\textbf {r}_p \\int d\\textbf {r}_T \\rho _n(r_p) \\rho _{T \\lambda } (r_T)V_{NN} (\\mbox{$r$}-\\mbox{$r$}_p+\\mbox{$r$}_T).$ In the isocentrifugal approximation, one then sets $\\hat{\\mbox{$r$}}_{\\rm d}=0$ and finally obtains [22] $V_{^{15}{\\rm C}-T}(r,\\theta )=\\sum _{\\lambda } V_{\\lambda }(r) Y_{\\lambda 0}(\\theta ).$ We assume that the projectile nucleus $^{15}$ C takes the two-body structure, with the spherical core nucleus $^{14}$ C and a valence neutron.", "The density of the projectile is then given as $\\rho _p(\\mbox{$r$})=\\rho _c(r)+\\rho _n(r),$ where $\\rho _c(r)$ and $\\rho _n(r)$ are the density for the core nucleus and the valence neutron, respectively.", "If one uses this density, the folding potential of Eq.", "(REF ) is also separated into two parts: $V_{^{15}{\\rm C}-T}(r,\\theta ) = V_{^{14}{\\rm C}-T}(r,\\theta )+V_{n-T}(r,\\theta ) .$ For simplicity, we replace the interaction between the core and the target nuclei, $V_{^{14}{\\rm C}-T}(r,\\theta )$ , by the deformed Woods-Saxon potential determined in the previous section.", "For the density for the valence neutron, we construct it using a $2s_{1/2}$ neutron wave function in a Woods-Saxon potential as $\\rho _n(r)=\\frac{1}{4\\pi }\\,\\left[R_{2s_{1/2}}(r)\\right]^2,$ where $R_{2s_{1/2}}(r)$ is the radial part of the wave function.", "To this end, we use the Woods-Saxon potential with set C in Ref.", "[37], which reproduces the empirical neutron separation energy for this state, $\\epsilon _{2s_{1/2}}=-1.21$ MeV.", "Figure REF shows the projectile density thus obtained.", "The blue dashed line shows the density for the core nucleus, $^{14}$ C, while the red dot-dashed line denotes the valence neutron density.", "For the description of the core density, we use the modified harmonic-oscillator model, whose parameters can be found in Ref. [35].", "One can see that the valence neutron density has a long tail, reflecting the halo structure of the $^{15}$ C nucleus.", "Figure: The density distribution of the 15 ^{15}C nucleus (the solid line).The dashed and the dot-dashed lines denote the contribution of the core nucleusand the valence neutron, respectively.Figure: The potential between the valence neutron in 15 ^{15}Cand the target nucleus 232 ^{232}Th for the 15 ^{15}C + 232 ^{232}Th reaction.The top, the middle, and the bottom panels are forV 0 (r)V_{0}(r), V 2 (r)V_{2}(r), and V 4 (r)V_{4}(r) in Eq.", "(), respectively.The black solid lines show the results of the double folding potential,while the red dashed lines shows fits with the Woods-Saxon function.The solid lines in Fig.", "REF shows the neutron-target potential obtained with the double folding procedure.", "The top, the middle, and the bottom panels show the monopole, the quadrupole, and the hexadecapole components, respectively.", "In order to discuss properties of these potentials, we fit them with a Woods-Saxon function and its first derivative.", "That is, $V_\\lambda (r)&=&\\frac{-V_0}{1+ \\exp \\left[ \\left( r-R_0 \\right)/a\\right]}~~~(\\lambda =0), \\\\&=&\\frac{-V_0 \\exp \\left[ \\left( r-R_0 \\right)/a\\right]}{\\left(1+\\exp \\left[ \\left( r-R_0 \\right)/a\\right]\\right)^2}~~~(\\lambda =2,4), \\nonumber \\\\$ The results of the fitting are shown in Fig.", "REF by the dashed lines (see Table REF for the parameters).", "Since the region around the position of the Coulomb barrier is most important for fusion cross sections, the fitting are performed mainly in the surface region, $r >$ 8 fm.", "In Fig.", "REF , one can see that the folding potential can be well fitted with the Woods-Saxon function.", "The hexadecapole component, $V_{4}(r)$ , has some deviation from the Woods-Saxon function, but its contribution to the total potential is much smaller than the monopole and the quadrupole components.", "We also find that the contribution of $\\lambda =6$ , that is, $V_{6}(r)$ , is negligible, with a small depth size of about $-$ 0.1 MeV.", "Note that the Coulomb barrier parameters, obtained by setting the deformation parameters $\\beta _{T\\lambda }$ to be zero in the folding procedure, are $V_{b}$ =58.81 MeV, $R_{b}$ =12.33 fm, and the $\\hbar \\Omega $ =4.10 MeV, which can be compared to those for the $^{14}$ C+$^{232}$ Th system listed in Table REF .", "The Coulomb barrier height is lowered by 1.53 MeV owing to the weakly bound valence neutron in $^{15}$ C. Table: Parameters for the neutron-target part of thenuclear potential for the 15 ^{15}C+ 232 ^{232}Th system.Those are obtained by fitting the double folding potential tothe Woods-Saxon function and its derivative.The depth, V 0 V_{0}, the radius, R 0 R_{0}, and the diffuseness, aa, parametersare shown for each multipole component.Figure REF shows the total potential (that is, the sum of the nuclear and the Coulomb potentials).", "The red dashed and the green dot-dashed lines show the potential for the $^{14}$ C+$^{232}$ Th reaction for $\\theta = 0^o$ and at $\\theta = 90^o$ , respectively.", "Note that the case of $\\theta =0^o$ is referred to as a tip collision while that of $\\theta =90^o$ is referred to as a side collision.", "The black solid and the blue dotted lines show the corresponding potentials for the $^{15}$ C+$^{232}$ Th reaction, obtained with the Woods-Saxon fit to the double folding potential between the valence neutron and the core nucleus.", "For both the cases, the Coulomb barrier is significantly lowered due to the addition of the valence neutron in $^{15}$ C. It is well known that lowering of the Coulomb barrier leads to an increase of the penetration probability for fusion cross sections.", "Figure: The total potentials (the sum of the nuclearand the Coulomb potentials) for the 14 ^{14}C + 232 ^{232}Th and the 15 ^{15}C + 232 ^{232}Th systems forθ=0 o \\theta = 0^o and θ=90 o \\theta = 90^o.The solid and the dotted lines are for the 15 ^{15}C + 232 ^{232}Th system,for which the solid line corresponds to θ=0 o \\theta = 0^o and thedashed line corresponds to θ=90 o \\theta = 90^o.The dashed and the dot-dashed lines are the same as the solid and thedotted lines, but for the 14 ^{14}C + 232 ^{232}Th system." ], [ "Fusion cross sections", "Let us now calculate fusion cross sections for the $^{15}$ C + $^{232}$ Th system using the potential constructed in the previous subsection.", "For simplicity, we include up to $\\lambda =4$ in the multipole expansion of the double folding potential between the valence neutron in $^{15}$ C and the target nucleus.", "Figure: Fusion cross sections for the 15 ^{15}C + 232 ^{232}Th system.The dot-dashed line shows fusion cross sections obtained by scalingthe potential for the 14 ^{14}C + 232 ^{232}Th system with the mass numberof the colliding nuclei.The dotted line shows fusion cross sections with the double folding potential,which takes into account the halo structure of the 15 ^{15}C nucleus, whilethe solid line showsthe result of the coupled-channels calculationwith the transfer coupling.For comparison, the figure also shows thetheoretical fusion cross sections for the 14 ^{14}C + 232 ^{232}Th systemby the dashed line.The experimental data are taken from Ref.", ".Figure REF presents a comparison between the calculated fusion cross sections and the experimental data.", "The red circles and the blue stars show the experimental data for the $^{15}$ C + $^{232}$ Th and $^{14}$ C + $^{232}$ Th systems, respectively.", "The violet dashed curve shows the results for the $^{14}$ C + $^{232}$ Th system, which is the same as the solid line in Fig.REF .", "The coral dot-dashed curve shows the results for the $^{15}$ C + $^{232}$ Th system, obtained using the potential which is simply scaled from that for the $^{14}$ C + $^{232}$ Th system.", "That is, the radius parameter in the Woods-Saxon potential for the $^{14}$ C + $^{232}$ Th system is changed from $r_{0}\\left(14^{1/3} + 232^{1/3}\\right)$ to $r_{0}\\left(15^{1/3} + 232^{1/3}\\right)$ .", "This calculation does not take into account the weakly bound nature of the $^{15}$ C projectile, and corresponds to the calculation presented in Ref.", "[30].", "In fact, the dashed and the dot-dashed lines show similar fusion cross sections to each other, as has been argued in Ref.", "[30].", "Even though this calculation reproduces the experimental data at energies above the Coulomb barrier, $E_{\\rm c.m.", "}$ $\\geqq $ 60 MeV, it considerably underestimates fusion cross sections in the energy region below the Coulomb barrier.", "Fusion cross sections evaluated with the halo nature of the $^{15}$ C nucleus are shown by the brown dotted line.", "This result clearly shows an enhancement of fusion cross sections with respect to the dot-dashed line, and reproduce the experimental data at the two lowest energies.", "However, the fusion cross sections in the region of 60 MeV $\\lesssim E_{\\rm c.m.}", "\\lesssim $ 65 MeV are clearly overestimated.", "In order to investigate a possible origin for the discrepancy, we follow Ref.", "[18] and consider a transfer coupling to the $^{14}$ C+$^{233}$ Th channel.", "That is, we consider a transfer to a single effective channel [38] and solve the coupled-channels equations [22], [31] of $&&\\left(\\begin{array}{cc}K+V_1(r, \\theta ) & F_{ 1 \\rightarrow 2 }(r) \\\\F_{ 1 \\rightarrow 2 }(r) & K+V_2(r, \\theta )-Q \\\\\\end{array}\\right)\\left(\\begin{array}{c}\\psi _1(r) \\\\\\psi _2(r) \\\\\\end{array}\\right)\\nonumber \\\\&&\\hspace*{128.0374pt}=E\\left(\\begin{array}{c}\\psi _1(r) \\\\\\psi _2(r) \\\\\\end{array}\\right).$ Here, the channels 1 and 2 denote the $^{15}$ C+$^{232}$ Th and the $^{14}$ C+$^{233}$ Th systems, respectively.", "$K$ is the kinetic energy (with the centrifugal potential) and $V_i(r,\\theta )$ ($i$ =1,2) is the inter-nucleus potential for each partition.", "$Q$ is the effective $Q$ -value for the one-neutron transfer process, while $F_{ 1 \\rightarrow 2 }(r)$ is the coupling form factor.", "Notice that the effective transfer channel may mock up also the breakup channel to some extent.", "Table: The parameters for the transfer coupling.Here, those in the coupling form factor, Eq.", "(), are determined by fittingthe results of the coupled-channels calculations to the experimentalfusion cross section for the 15 ^{15}C+ 232 ^{232}Th systemwith Q=0Q=0 for the transfer QQ-value.In the calculation, we assume that the potential $V_2(r, \\theta )$ for the $^{14}$ C + $^{233}$ Th channel is the same as the potential for the $^{14}$ C + $^{232}$ Th presented in Table REF .", "For the coupling form factor, $F_{ 1 \\rightarrow 2 }(r)$ , we employ the derivative form of the Woods-Saxon potential [39], [40] given by $F_{1\\rightarrow 2}(r)=F_t \\,\\frac{d}{dr}\\left( \\frac{1}{1+\\exp ((r-R_{\\rm coup})/a_{\\rm coup})} \\right).$ The parameters are determined by fitting to the experimental data for fusion cross sections (see Table III).", "To this end, we take the transfer $Q$ -value to be zero, $Q=0$ , rather than the ground-state-to-ground-state $Q$ -value, $Q_{\\rm gg}=+3.568$ MeV, taking into account the $Q$ -value matching condition [38].", "The black solid line in Fig.", "REF shows the fusion cross sections so obtained.", "One can see that the experimental data are well reproduced in the whole energy region shown in the figure, indicating an importance of the dynamical effect on fusion of the neutron-rich nucleus, $^{15}$ C." ], [ "Summary", "We have calculated fusion cross sections for the $^{15}$ C+$^{232}$ Th system, for which $^{15}$ C is a well-known one-neutron halo nucleus while $^{232}$ Th is a well deformed nucleus.", "To this end, we have evaluated the cross sections within the double folding formalism taking into account the halo structure of $^{15}$ C and the deformation of the target nucleus.", "In addition, we have also taken into account the coupling to the one-neutron transfer channel to the $^{14}$ C+$^{233}$ Th system with the coupled-channels formalism.", "We have shown that such calculation reproduces simultaneously well the experimental data for the $^{15}$ C+$^{232}$ Th and $^{14}$ C+$^{232}$ Th systems.", "This clearly indicates that all of the halo structure of $^{15}$ C, the deformation of $^{232}$ Th, and the dynamical effect such as transfer and breakup play an important role in fusion of the $^{15}$ C+$^{232}$ Th system.", "In this regard, it would be useful to investigate systematically fusion of a halo nucleus with deformed target nuclei in order to gain a deeper insight into the role of deformation of the target in fusion of neutron-rich nuclei.", "A comparison of the $^{15}$ C + $^{232}$ Th system to the $^{15}$ C + $^{238}$ U and $^{15}$ C+$^{208}$ Pb systems might also provide useful information." ], [ "Acknowledgment", "This work was supported by the National Research Foundation of Korea (Grant No.", "NRF-2016R1C1B1012874, 2018R1D1A1B07045915, NRF-2017R1E1A1A01074023,NRF-2019R1H1A1102164, NRF-2020R1A2C3006177 and NRF-2013M7A1A1075764) and by JSPS KAKENHI (Grant No.", "JP19K03861).", "K. S. Kim's work was supported by MSIT (No.2018R1A5A1025563)." ] ]	2011.14086
[ [ "On semisimple standard compact Clifford-Klein forms" ], [ "Abstract In this paper we give the classification of standard compact Clifford-Klein forms corresponding to triples (g,h,l) such that g = h+l and g is a sum of two absolutely simple ideals.", "The classification is done using Onishchik's results concerning semisimple decompositions of semisimple Lie algebras.", "Using this classification we obtain new examples of reductive homogeneous spaces admitting non-standard compact Clifford-Klein forms." ], [ "Introduction", "Let $G$ be a non-compact real linear semisimple Lie group and H a reductive subgroup.", "We say that the homogeneous space $G/H$ admits a compact Clifford-Klein form if there exists a discrete subgroup $\\Gamma \\in G$ such that $\\Gamma $ acts freely, properly and co-compactly on $G/H.$ Notice that in a case when $H$ and $G/H$ are non-compact such $\\Gamma $ may not exist.", "The problem of finding and classifying homogeneous spaces admitting compact Clifford-Klein forms is in general difficult and is an active area of study (see for example [2], [3], [4], [5], [7], [9], [10], [11], [12]).", "There is basically one known way of constructing non-trivial examples of compact Clifford-Klein forms.", "Assume there exists a reductive connected subgroup $L\\subset G$ such that $L$ acts properly and co-compactly on $G/H.$ Then one can choose a co-compact lattice $\\Gamma $ of $L$ so that $\\Gamma $ acts properly, co-compactly and freely on $G/H.$ In such case we say that $G/H$ admits a standard compact Clifford-Klein form.", "There is a conjecture concerning standard Clifford-Klein forms (see Conjecture 3.3.10 in [5]) stating that if $G/H$ admits a compact Clifford-Klein form then $G/H$ also admits a standard one.", "Denote by $G/H$ a reductive homogeneous space of a real connected linear semisimple Lie group $G$ such that the Lie algebra $\\mathfrak {g}$ of $G$ is a direct sum of two absolutely simple ideals $\\mathfrak {g}_i, \\ i=1,2;$ $\\mathfrak {g}=\\mathfrak {g}_1 \\oplus \\mathfrak {g}_{2} .$ Denote by $\\mathfrak {h}$ the Lie algebra of a connected reductive subgroup $H\\subset G.$ In this paper we list all possible triples $(\\mathfrak {g}, \\mathfrak {h}, \\mathfrak {l})$ such that $\\mathfrak {g}=\\mathfrak {h}+\\mathfrak {l}$ and $\\mathfrak {l}$ is a Lie algebra of a connected subgroup $L\\subset G$ that acts properly and co-compactly on $G/H.$ This gives the list of all homogeneous spaces $G/H$ which admit standard compact Clifford Klein forms such that $G=HL$ and the lie algebra of $\\mathfrak {g}$ is a sum of two absolutely simple ideals.", "Notice that we may assume that $\\mathfrak {g}_{1}, \\mathfrak {g}_{2}$ are of non-compact type.", "Also we may assume that $\\mathfrak {h},$ $\\mathfrak {l}$ are semisimple without ideals of compact type.", "This is because the result of [1] which states that if a reductive homogeneous space of a semisimple Lie group admits a compact Clifford-Klein form then the center of the isotropy subgroup is compact.", "In other words if $\\mathfrak {h}=\\mathfrak {c}\\oplus \\mathfrak {h}_{1}$ where $\\mathfrak {c}$ is an ideal of compact type then $(\\mathfrak {g},\\mathfrak {h},\\mathfrak {l})$ induces a standard Clifford-Klein form if and only if $(\\mathfrak {g},\\mathfrak {h}_{1},\\mathfrak {l})$ induces a standard Clifford-Klein form.", "Main results of the paper are given in Theorem REF , Table REF and in Proposition REF .", "The proof of Theorem REF is based on the classification of semisimple decompositions of simple and semisimple Lie algebras given in [13], [14].", "The proof of Proposition REF is carried out using the ideas and results of Kobayashi [6], [7], [8] and Kassel [4].", "Notice that it is reasonable to expect that the class of standard Clifford-Klein forms of semisimple groups is significantly richer than the class of standard Clifford-Klein forms of simple Lie groups.", "However, in this article we present a rather surprising result that the class of standard compact Clifford-Klein forms of semisimple groups is rather small.", "It suggests that basically one should not expect to greatly enrich the class of compact Clifford-Klein forms beyond the double quotients $\\Gamma \\backslash G/H$ of simple Lie groups $G$ .", "In fact, the only non-trivial examples obtained in this paper are the following $G\\times G/\\textrm {diag}(G),$ $L=G^{\\prime }\\times G^{\\prime \\prime },$ where the triple $(G,G^{\\prime },G^{\\prime \\prime })$ is given by a triple in Table REF .", "$SO(4,4)\\times SO(3,4)/\\Delta (SO(3,4)),$ $L=SO(3,4)\\times SO(1,4), $ where $\\Delta (SO(3,4)):=\\lbrace (\\iota _{1}(g),g) \\ \| \\ g\\in SO(3,4) \\rbrace \\ \\textrm {for} \\ \\iota _{1}:SO(3,4)\\hookrightarrow SO(4,4) .$ $SO(4,4)\\times SO(2,4)/\\Delta (SO(2,4)),$ $L=SO(3,4)\\times SO(1,4), $ where $\\Delta (SO(2,4)):=\\lbrace (\\iota _{2}(g),g) \\ \| \\ g\\in SO(2,4) \\rbrace \\ \\textrm {for} \\ \\iota _{2}:SO(2,4)\\hookrightarrow SO(4,4) .$ $SO(3,4)\\times SO(2,4)/\\Delta (SO(2,4)),$ $L=G_{2(2)}\\times SO(1,4), $ where $\\Delta (SO(2,4)):=\\lbrace (\\iota _{3}(g),g) \\ \| \\ g\\in SO(2,4) \\rbrace \\ \\textrm {for} \\ \\iota _{3}:SO(2,4)\\hookrightarrow SO(3,4) .$ However, one may find in the above list new examples of reductive homogeneous spaces admitting non-standard Clifford-Klein forms.", "For instance Proposition 1 The spaces $SO(4,4)\\times SO(2,4)/\\Delta (SO(2,4)),$ $SO(3,4)\\times SO(2,4)/\\Delta (SO(2,4))$ admit non-standard Clifford Klein-forms.", "Acknowledgment.", "The author acknowledges the support of the National Science Center (grant NCN no.", "2018/31/D/ST1/00083)." ], [ "Results and proofs", "We start we the following observation.", "If $\\mathfrak {g}=\\mathfrak {h}+\\mathfrak {l}$ then (by Theorem 3.1 in [13]) we have $G=HL.$ Therefore $G/H \\cong L/H\\cap L$ so $L$ acts properly (and co-compactly) on $G/H$ if and only if $H\\cap L$ is compact.", "In order to find all triples $(\\mathfrak {g},\\mathfrak {h},\\mathfrak {l})$ corresponding to standard Clifford-Klein forms we have to list all semisimple decompositions $\\mathfrak {g}=\\mathfrak {h}+\\mathfrak {l}$ such that $\\mathfrak {h}\\cap \\mathfrak {l}$ is of compact type.", "Denote by $\\pi _{i}:\\mathfrak {g}\\rightarrow \\mathfrak {g}_{i}, \\ \\ i=1,2, $ the projection on i-th simple factor of $\\mathfrak {g}.$ Theorem 1 Let $\\mathfrak {g}=\\mathfrak {g}_{1}\\oplus \\mathfrak {g}_{2}$ be a semisimple Lie algebra which is a sum of two absolutely simple ideals of non-compact type.", "Let $\\mathfrak {h},\\mathfrak {l}\\subset \\mathfrak {g}$ be two semisimple subalgebras such that $\\mathfrak {g}=\\mathfrak {h}+\\mathfrak {l} .$ Then the triple $(\\mathfrak {g},\\mathfrak {h},\\mathfrak {l})$ gives a standard compact Clifford-Klein form if and only if it is contained in the following list $\\mathfrak {g}_{1},\\mathfrak {g}_{2}$ are arbitrary absolutely simple Lie algebras, $\\mathfrak {h}= \\mathfrak {g}_{1}\\oplus \\lbrace 0 \\rbrace ,$ $\\mathfrak {l}= \\lbrace 0 \\rbrace \\oplus \\mathfrak {g}_{2}$ and $\\pi _{1}(\\mathfrak {h})=\\mathfrak {g}_{1}, \\ \\pi _{2}(\\mathfrak {h})=\\lbrace 0 \\rbrace , \\ \\pi _{1}(\\mathfrak {l})=\\lbrace 0 \\rbrace , \\ \\pi _{2}(\\mathfrak {l}) = \\mathfrak {g}_{2} .$ $\\mathfrak {g}_{2}$ is an arbitrary absolutely simple Lie algebra, $\\mathfrak {h}= \\mathfrak {g}_{1}^{\\prime }\\oplus \\lbrace 0 \\rbrace ,$ $ \\mathfrak {l}= \\mathfrak {g}_{1}^{\\prime \\prime }\\oplus \\mathfrak {g}_{2},$ the triple $(\\mathfrak {g}_{1},\\mathfrak {g}_{1}^{\\prime },\\mathfrak {g}_{1}^{\\prime \\prime })$ is contained in Table REF and $\\pi _{1}(\\mathfrak {h})=\\mathfrak {g}_{1}^{\\prime }, \\ \\pi _{2}(\\mathfrak {h})=\\lbrace 0 \\rbrace , \\ \\pi _{1}(\\mathfrak {l})=\\mathfrak {g}_{1}^{\\prime \\prime } , \\ \\pi _{2}(\\mathfrak {l}) = \\mathfrak {g}_{2} .$ $\\mathfrak {h}= \\mathfrak {g}_{1}^{\\prime }\\oplus \\mathfrak {g}_{2}^{\\prime }$ , $\\mathfrak {l}= \\mathfrak {g}_{1}^{\\prime \\prime }\\oplus \\mathfrak {g}_{2}^{\\prime \\prime }$ where the triples $(\\mathfrak {g}_{1},\\mathfrak {g}_{1}^{\\prime },\\mathfrak {g}_{1}^{\\prime \\prime }),$ $(\\mathfrak {g}_{2},\\mathfrak {g}_{2}^{\\prime },\\mathfrak {g}_{2}^{\\prime \\prime })$ are contained in Table REF and $\\pi _{1}(\\mathfrak {h})=\\mathfrak {g}_{1}^{\\prime }, \\ \\pi _{2}(\\mathfrak {h})=\\mathfrak {g}_{2}^{\\prime } , \\ \\pi _{1}(\\mathfrak {l})= \\mathfrak {g}_{1}^{\\prime \\prime }, \\ \\pi _{2}(\\mathfrak {l}) = \\mathfrak {g}_{2}^{\\prime \\prime } .$ $\\mathfrak {g}_{1},\\mathfrak {g}_{2}$ are arbitrary absolutely simple Lie algebras such that $\\mathfrak {g}_{2}\\subset \\mathfrak {g}_{1}$ , $\\mathfrak {h}= \\mathfrak {g}_{1}\\oplus \\lbrace 0 \\rbrace ,$ $ \\mathfrak {l}\\cong \\mathfrak {g}_{2}$ and $\\pi _{1}(\\mathfrak {h})=\\mathfrak {g}_{1}, \\ \\pi _{2}(\\mathfrak {h})=\\lbrace 0 \\rbrace , \\ \\pi _{1}(\\mathfrak {l})= \\mathfrak {g}_{2}, \\ \\pi _{2}(\\mathfrak {l}) = \\mathfrak {g}_{2} .$ Take a triple $(\\mathfrak {g}_{1},\\mathfrak {g}^{\\prime },\\mathfrak {g}^{\\prime \\prime })$ from Table REF .", "Take any simple subalgebra $\\mathfrak {g}_{2}\\subset \\mathfrak {g}_{1}$ so that $\\mathfrak {g}^{\\prime \\prime }\\subset \\mathfrak {g}_{2}\\subset \\mathfrak {g}_{1}.$ Put $\\mathfrak {h}=\\mathfrak {g}^{\\prime }\\oplus \\mathfrak {g}^{\\prime \\prime }$ and $\\mathfrak {l}=\\mathfrak {g}_{2}$ so that $\\pi _{1}(\\mathfrak {h})=\\mathfrak {g}^{\\prime }, \\ \\pi _{2}(\\mathfrak {h})=\\mathfrak {g}^{\\prime \\prime } , \\ \\pi _{1}(\\mathfrak {l})= \\mathfrak {g}_{2}, \\ \\pi _{2}(\\mathfrak {l}) = \\mathfrak {g}_{2} .$ All triples of that type are listed in Table REF .", "Remark 1 Given a triple $(\\mathfrak {g},\\mathfrak {h},\\mathfrak {l})$ and corresponding groups $G,H,L$ we see that if $G/H$ admits a standard compact Clifford-Klein form then $G/L$ admits a standard compact Clifford-Klein form.", "Therefore one of each triples $(\\mathfrak {g},\\mathfrak {h},\\mathfrak {l}),$ $(\\mathfrak {g},\\mathfrak {l},\\mathfrak {h})$ is described in Theorem REF and given in Tables REF and REF .", "One easily verifies that in each case $\\mathfrak {h}\\cap \\mathfrak {l}$ is of compact type and $\\mathfrak {g}=\\mathfrak {h}+\\mathfrak {l}.$ Therefore we only need to show that there are no other possibilities.", "Cases 1.-3. contain all decomposable triples, that is triples such that $\\mathfrak {g}_{i} = \\mathfrak {g}_{i}\\cap \\mathfrak {h} + \\mathfrak {g}_{i}\\cap \\mathfrak {l}, \\ \\ i=1,2.$ Thus decomposition of $\\mathfrak {g}$ is obtained from decompositions of simple Lie algebras $\\mathfrak {g}_{1}, \\mathfrak {g}_{2}.$ But all possible proper semisimple decomposition of a simple Lie algebras (with $\\mathfrak {h}\\cap \\mathfrak {l}$ of compact type) are given in Table REF (Theorem 4.1 and Table 2 in [13]).", "So the triples $(\\mathfrak {g}_{i}, \\mathfrak {g}_{i}\\cap \\mathfrak {h}, \\mathfrak {g}_{i}\\cap \\mathfrak {l})$ are given in Table REF or are equal to $(\\mathfrak {g}_{i}, \\mathfrak {g}_{i}, \\lbrace 0 \\rbrace ).$ In Case 4. one of subalgebras from the triple $(\\mathfrak {g}, \\mathfrak {h}, \\mathfrak {l})$ contains a non-zero ideal of $\\mathfrak {g} .$ Without loss of generality assume that $\\mathfrak {g}_{1}\\subset \\mathfrak {h}.$ Then $\\pi _{2}(\\mathfrak {h})=\\lbrace 0 \\rbrace $ as $\\mathfrak {h}$ does not contain ideal of compact type and $\\mathfrak {h}\\cap \\mathfrak {l}$ is a subalgebra of compact type.", "Since the triple is indecomposable thus $\\pi _{i}(\\mathfrak {l})\\ne \\lbrace 0 \\rbrace $ for $i=1,2.$ But $\\mathfrak {h}+\\mathfrak {l}=\\mathfrak {g}$ so we have to have $\\pi _{2}(\\mathfrak {l})=\\mathfrak {g}_{2}.$ Case 5. describes all other triples.", "In [13], [14] such triples are called primitive indecomposable and the classification of the triples is done using principal schemes.", "In our setting (where $\\mathfrak {g}$ is a sum of two simple ideals), Theorem 4.3 in [13] can be stated as follows Proposition 2 If the triple $(\\mathfrak {g},\\mathfrak {h},\\mathfrak {l})$ is indecomposable and neither $\\mathfrak {h}$ nor $\\mathfrak {l}$ contains a non-zero ideal of $\\mathfrak {g}$ then there is a triple $(\\mathfrak {g}_{1},\\mathfrak {g}^{\\prime },\\mathfrak {g}^{\\prime \\prime })$ from Table REF and a subalgebra $\\mathfrak {g}_{2} \\subset \\mathfrak {g},$ $\\mathfrak {g}^{\\prime \\prime }\\subset \\mathfrak {g}_{2}\\subset \\mathfrak {g}_{1},$ such that $\\mathfrak {h}=\\mathfrak {g}^{\\prime }\\oplus \\mathfrak {g}^{\\prime \\prime },$ $\\mathfrak {l}=\\mathfrak {g}_{2}$ and $\\pi _{1}(\\mathfrak {l})\\ne \\lbrace 0 \\rbrace , \\ \\pi _{2}(\\mathfrak {l}) = \\mathfrak {g}_{2} .$ This completes the proof of Theorem REF .", "Proof of Proposition REF .", "Consider the space $SO(4,4)\\times SO(2,4)/\\Delta (SO(2,4))$ (The proof for $SO(3,4)\\times SO(2,4)/\\Delta (SO(2,4))$ is similar).", "Fix Iwasawa decompositions $\\mathfrak {so}(4,4)=\\mathfrak {k}_{1}+\\mathfrak {a}_{1}+\\mathfrak {n}_{1},$ $\\mathfrak {so}(2,4)=\\mathfrak {k}_{2}+\\mathfrak {a}_{2}+\\mathfrak {n}_{2},$ $\\mathfrak {so}(3,4)=\\mathfrak {k}_{3}+\\mathfrak {a}_{3}+\\mathfrak {n}_{3},$ $\\mathfrak {so}(1,4)=\\mathfrak {k}_{4}+\\mathfrak {a}_{4}+\\mathfrak {n}_{4}$ with $\\mathfrak {a}_{3}\\subset \\mathfrak {a}_{1}, \\ \\ \\mathfrak {a}_{4}\\subset \\mathfrak {a}_{2}$ Choose positive Weyl chambers $\\mathfrak {a}_{1}^{+}\\subset \\mathfrak {a}_{1}$ and $\\mathfrak {a}_{2}^{+}\\subset \\mathfrak {a}_{2}$ and the corresponding Cartan projections $\\mu _{1}:SO(4,4)\\rightarrow \\mathfrak {a}_{1}^{+}$ and $\\mu _{2}:SO(2,4)\\rightarrow \\mathfrak {a}_{2}^{+}.$ Then $\\mu := \\mu _{1} + \\mu _{2}$ is a Cartan projection of $G$ w.r.t.", "the Weyl chamber $\\mathfrak {a}^{+} := \\lbrace t_{1}A_{1} + t_{2}A_{2} \\ \| \\ A_{1}\\in \\mathfrak {a}_{1}^{+}, \\ A_{2}\\in \\mathfrak {a}_{2}^{+}, \\ t_{1},t_{2}\\ge 0 \\rbrace ,$ of $G.$ Notice that we may conjugate $\\Delta (SO(2,4))$ inside G so that $\\mu (\\Delta (SO(2,4)))$ is a finite sum of subspaces of $\\mathfrak {a} .$ Take the norm $ \\left\\Vert .", "\\right\\Vert $ on $\\mathfrak {a}$ induced by the Killing form B of $\\mathfrak {g}$ .", "Take $L^{\\prime }=SO(1,4),$ $L=SO(3,4)\\times L^{\\prime },$ and let $\\Gamma _{L^{\\prime }}$ be a torsion-free co-compact lattice of $L^{\\prime }.$ Now we follow the line of argument of Section 5.4 in [4].", "By Theorem 1.4 in [4], for any $\\epsilon >0$ there exists a neighborhood $U_{\\epsilon }\\subset \\textrm {Hom}(\\Gamma _{L^{\\prime }}, SO(2,4))$ of the natural inclusion such that for any $\\rho \\in U_{\\epsilon }$ and any $\\gamma \\in \\Gamma _{L^{\\prime }}$ $\\left\\Vert \\mu (\\rho (\\gamma )) - \\mu (\\gamma ) \\right\\Vert \\le \\epsilon \\left\\Vert \\mu (\\gamma ) \\right\\Vert .$ Fix a torsion-free co-compact lattice $\\tilde{\\Gamma }\\subset SO(3,4).$ For any $\\epsilon >0$ and any $\\rho \\in U_{\\epsilon }$ we have for $\\phi := id \\times \\rho $ $\\left\\Vert \\mu (\\phi (\\gamma _{1}, \\gamma _{2})) - \\mu (\\gamma _{1}, \\gamma _{2}) \\right\\Vert =\\left\\Vert \\mu (\\gamma _{1}) + \\mu (\\rho (\\gamma _{2})) - (\\mu (\\gamma _{1}) + \\mu (\\gamma _{2})) \\right\\Vert \\le \\epsilon \\left\\Vert \\mu (\\gamma _{2}) \\right\\Vert $ where $\\gamma _{1}\\in \\tilde{\\Gamma },$ $\\gamma _{2}\\in \\Gamma _{L^{\\prime }}$ are arbitrary.", "By the Kobayashi criterion of proper actions (Theorem 4.1 in [8]), $\\mu (\\Delta (SO(2,4)))$ and $\\mu (L)$ are sums of finite number of subspaces of $\\mathfrak {a}$ which are pairwise disjoint.", "Therefore there are constants $\\epsilon , C$ such that $d\\big ( \\mu (l) , \\mu (\\Delta (SO(2,4))) \\big ) \\ge 2\\epsilon \\left\\Vert \\mu (l) \\right\\Vert - C,$ for all $l\\in L.$ Since $\\mathfrak {a}_{1}, \\mathfrak {a}_{2}$ are $B$ -orthogonal we have $\\left\\Vert \\mu (\\gamma _{1},\\gamma _{2}) \\right\\Vert = \\left\\Vert \\mu (\\gamma _{1}) + \\mu (\\gamma _{2}) \\right\\Vert = \\sqrt{\\left\\Vert \\mu (\\gamma _{1}) \\right\\Vert ^{2} + \\left\\Vert \\mu (\\gamma _{1}) \\right\\Vert ^{2}}$ and therefore $d\\big ( \\mu (\\phi (\\gamma _{1},\\gamma _{2})) , \\mu (\\Delta (SO(2,4))) \\big ) \\ge $ $\\ge d\\big ( \\mu (\\gamma _{1},\\gamma _{2}) , \\mu (\\Delta (SO(2,4))) \\big ) - \\left\\Vert \\mu (\\phi (\\gamma _{1},\\gamma _{2})) - \\mu (\\gamma _{1},\\gamma _{2}) \\right\\Vert \\ge $ $\\ge 2\\epsilon \\sqrt{\\left\\Vert \\mu (\\gamma _{1}) \\right\\Vert ^{2} + \\left\\Vert \\mu (\\gamma _{2}) \\right\\Vert ^{2}} - C - \\epsilon \\left\\Vert \\mu (\\gamma _{2}) \\right\\Vert $ $\\ge 2\\epsilon \\sqrt{\\left\\Vert \\mu (\\gamma _{1}) \\right\\Vert ^{2} + \\left\\Vert \\mu (\\gamma _{2}) \\right\\Vert ^{2}} - C - \\epsilon \\sqrt{\\left\\Vert \\mu (\\gamma _{1}) \\right\\Vert ^{2} + \\left\\Vert \\mu (\\gamma _{2}) \\right\\Vert ^{2}} \\ge $ $\\ge \\epsilon \\left\\Vert \\mu (\\gamma _{1},\\gamma _{2}) \\right\\Vert -C,$ for any $(\\gamma _{1},\\gamma _{2})\\in \\tilde{\\Gamma }\\times \\Gamma _{L^{\\prime }}.$ Since $\\mu $ is a proper map, $\\tilde{\\Gamma }\\times \\rho (\\Gamma _{L^{\\prime }})$ is a discrete subgroup that acts properly on $SO(4,4)\\times SO(2,4)/\\Delta (SO(2,4)).$ Also $\\phi $ is injective (as the kernel of $\\phi $ is finite).", "Thus the action of $\\tilde{\\Gamma }\\times \\rho (\\Gamma _{L^{\\prime }})$ is proper and co-compact for sufficiently small $\\epsilon .$ By Lemma 6.2 in [4] we can chose $\\rho \\in U_{\\epsilon }$ so that $\\rho (\\Gamma _{L^{\\prime }})$ is Zariski dense in $SO(2,4).$ This way we obtain a non-standard compact Clifford-Klein form of $SO(4,4)\\times SO(2,4)/\\Delta (SO(2,4)) .$" ], [ "Tables", "Table REF contains all possible proper semisimple decompositions of absolutely simple real Lie algebras $\\mathfrak {g},$ such that $\\mathfrak {g}=\\mathfrak {g}^{\\prime }+\\mathfrak {g}^{\\prime \\prime }$ and $\\mathfrak {g}^{\\prime }\\cap \\mathfrak {g}^{\\prime \\prime }$ is of compact type (see Table 2 in [13]).", "Table: Proper semisimple decompositions of absolutely simple real Lie algebras inducing standard compact Clifford-Klein forms.Table REF contains triples described in Case 5. of Theorem REF .", "Table: Triples corresponding to Case 5. in Theorem ." ] ]	2011.14106
[ [ "The hard-core model on planar lattices: the disk-packing problem and\n high-density phases" ], [ "Abstract We study dense packings of disks and related Gibbs distributions representing high-density phases in the hard-core model on unit triangular, honeycomb and square lattices.", "The model is characterized by a Euclidean exclusion distance $D>0$ and a value of fugacity $u>0$.", "We use the Pirogov-Sinai theory to study the Gibbs distributions for a general $D$ when $u$ is large: $u>u_0(D)$.", "For infinite sequences of values $D$ we describe a complete high-density phase diagram: it exhibits a multitude of co-existing pure phases, and their number grows as $O(D^2)$.", "For the remaining values of $D$, except for those with sliding, the number of co-existing pure phases is still of the form $E(D)\\geq O(D^2)$; however, the exact identification of the pure phases requires an additional analysis.", "Such an analysis is performed for a number of typical examples, which involves computer-assisted proofs.", "Consequently, for all values $D>0$ where sliding does not occur, we establish the existence of a phase transition.", "The crucial steps in the study are (i) the identification of periodic ground states and (ii) the verification of the Peierls bound.", "This is done by using connections with algebraic number theory.", "In particular, a complete list of so-called sliding values of $D$ has been specified.", "As a by-product, we solve the disk-packing problem on the lattices under consideration.", "The number and structure of maximally-dense packings depend on the disk-diameter $D$, unlike the case of $\\mathbb{R}^2$." ], [ "Introduction", "1.1.", "The hard-core model emerged in XIX Century in an attempt to describe a system of particles viewed as non-overlapping spheres of a non-negligible diameter (Boltzmann, 1872).", "The model became popular in various areas of theoretical and applied science.", "In this work, we focus on aspects of the model which are within the remit of the theory of phase transitions.", "We analyze the hard-core (H-C) model on a planar lattice and study configurations, or packings, of hard-disks of diameter $D$ representing particles with the Euclidean exclusion distance $D$ and centers at lattice sites.", "When the particle density is low, the particle system is in a gaseous phase which is highly disordered, corresponds to low values of fugacity/activity $u$ and is mathematically described in terms of a unique Gibbs distribution (GD).", "An important question is how the system evolves when the density/fugacity increases, e.g., whether it undergoes phase transitions.", "In a high-density regime (where $u$ is large) the H-C model is intrinsically related to the optimal (i.e., maximally-dense) disk-packing problem.", "On a lattice, a high-density H-C particle system is expected to become ordered, i.e., to be in a crystalline/solid phase (one or several).", "This is the principal question addressed in the current paper.", "Mathematically, the above question is about the structure of high-density GDs for the H-C model on a lattice in a large-fugacity regime.", "We focus on unit triangular, honeycomb and square lattices, ${\\mathbb {A}}_2$ , ${\\mathbb {H}}_2$ and ${\\mathbb {Z}}^2$ .", "It turns out that only attainable values of $D$ are of interest, i.e., those that can be realized on the corresponding lattice.", "On ${\\mathbb {A}}_2$ and ${\\mathbb {H}}_2$ the values $D^2$ are of the form $D^2=a^2+b^2+ab$ , while on ${\\mathbb {Z}}^2$ they obey $D^2=a^2+b^2$ , where $a, b$ are non-negative integers.", "(For a non-attainable disk diameter $D^{\\prime }$ one has to apply our results for the smallest attainable $D$ such that $D>D^{\\prime }$ .)", "A complete description of the high-density phase diagram requires the determination of all pure phases; mathematically it means an identification of extreme Gibbs distributions (EGDs).", "A popular tool here is the Pirogov-Sinai (PS) theory based on (i) a specification of periodic ground states (PGSs) and (ii) verification of the Peierls bound for the statistical weight of a deviation of an admissible particle configuration from a PGS.", "Lattice H-C models attracted a considerable interest and have a history in statistical mechanics.", "The existence of multiple EGDs in a high-density regime has been established (i) in [2] on ${\\mathbb {Z}}^d$ ($d>1$ ) for $D=\\sqrt{2}$ , (ii) in [6] on $\\mathbb {A}_2$ for a family of values $D$ and (iii) for a class of non-sliding H-C lattice particle systems in [7].", "In relation to the H-C model on ${\\mathbb {A}}_2$ , ${\\mathbb {H}}_2$ and ${\\mathbb {Z}}^2$ , our work includes these results as partial cases of a general theory covering all attainable values of $D$ .", "The set of attainable values of $D$ is partitioned into groups characterized by different structures of EGDs.", "For some explicitly determined infinite sequences of numbers $D$ we provide a complete picture of the set of the EGDs.", "It yields a first-order phase transition where every PGS generates an EGD, and the number $E(D)$ of the EGDs grows as $O(D^2)$ .", "For the remaining infinite sets of values $D$ , except for 39 particular numbers on ${\\mathbb {Z}}^2$ and 4 on ${\\mathbb {H}}_2$ , there is still a first-order phase transition, and $E(D)\\ge O(D^2)$ , but not all PGSs generate EGDs.", "Here the problem is reduced to an identification of dominant PGSs, which is done for a selection of typical examples.", "The 43 values of $D$ excluded from our considerations exhibit a phenomenon of sliding, and for them the PS theory is not applicable.", "1.2.", "In the course of identifying ground states we solve the disk-packing problem on the lattices under consideration.", "For any attainable $D$ we establish the supremum of the disk-packing density among all packings (both periodic and non-periodic) and show that it is achievable.", "Let $\\mathbb {W}={\\mathbb {A}}_2, {\\mathbb {H}}_2, {\\mathbb {Z}}^2$ and $\\Lambda _l$ be a square of side-length $l$ on $\\mathbb {R}^2$ .", "We define the maximal disk-packing density by $\\delta (D, {\\mathbb {W}}):=\\sup _{\\Phi }\\Bigg \\lbrace \\limsup _{l\\rightarrow \\infty } \\frac{{\\rm Area}(\\Phi \\cap \\Lambda _l)}{{\\rm Area}(\\Lambda _l)}\\Bigg \\rbrace ,$ where $\\Phi $ is a packing of disks of diameter $D$ with centers at sites of ${\\mathbb {W}}$ .", "For any attainable $D$ on ${\\mathbb {A}}_2$ and for any attainable $D$ such that $D^2$ is divisible by 3 on ${\\mathbb {H}}_2$ $\\delta (D, {\\mathbb {A}}_2)=\\delta (D, {\\mathbb {H}}_2)=\\frac{\\pi }{2\\sqrt{3}}$ (which is the maximal disk-packing density on ${\\mathbb {R}}^2$ ).", "For all attainable $D$ with $D^2\\notin {\\mathcal {N}}:=\\lbrace 1, 4, 7, 13, 16, 28, 31, 49, 64, 67, 97, 133, 157, 256 \\rbrace $ and $D^2$ is non-divisible by 3 on ${\\mathbb {H}}_2$ $\\delta (D, {\\mathbb {H}}_2)=\\frac{\\pi D^2}{2\\sqrt{3}(D^{\\ast })^2},$ where $D^>D$ is the closest attainable number with $(D^)^2$ divisible by 3.", "For the case $D^2\\in {\\mathcal {N}}$ on ${\\mathbb {H}}_2$ , we refer the reader to [8].", "For all attainable $D$ on $\\mathbb {Z}^2$ $\\delta (D, {\\mathbb {Z}}^2)=\\frac{\\pi D^2}{4S(D)},$ where $S(D)/2$ is the solution to optimization problem (5); see below.", "Furthermore, for any attainable non-sliding $D$ we describe all periodic optimizers, i.e., periodic packings achieving the maximal density.", "On ${\\mathbb {A}}_2$ these are $D$ -sub-lattices and their shifts (see sub-section 3.2), while on ${\\mathbb {Z}}^2$ they are the so-called MDA-sub-lattices and their shifts (see sub-section 3.5).", "On ${\\mathbb {H}}_2$ the optimizers are $D$ -sub-lattices and their shifts when $D^2$ is divisible by 3, and $D^$ -sub-lattices and their shifts when $D^2\\notin {\\mathcal {N}}$ and $D^2$ is not divisible by 3 (see sub-section 3.4).", "For $D^2\\in \\lbrace 1, 13, 16, 28, 49, 64,67, 97, 157, 256 \\rbrace $ the optimal packings are not necessarily sub-lattices: cf.", "[8].", "For the 43 sliding values $D$ (on ${\\mathbb {Z}}^2$ and ${\\mathbb {H}}_2$ ), we construct infinitely many maximally-dense periodic packings.", "We would like to note that, for each of ${\\mathbb {A}}_2$ , ${\\mathbb {Z}}^2$ and ${\\mathbb {H}}_2$ , there are infinitely many values of $D$ for which the optimizer is unique up to lattice symmetries, and there are also infinitely many values of $D$ with multiple, but finitely many, optimizers up to lattice symmetries.", "Moreover, these alternatives exhausts all attainable non-sliding values of $D$ .", "The number of optimal packings depends upon $D$ ; on ${\\mathbb {Z}}^2$ and ${\\mathbb {H}}_2$ the disks in these packings do not necessarily touch their `neighbors'.", "Dense-packings on ${\\mathbb {A}}_2$ have been also considered in [1]." ], [ "Gibbs distributions", "2.1.", "A $D$ -admissible configuration ($D$ -AC) on ${\\mathbb {W}}$ is a map $\\phi :{\\mathbb {W}}\\rightarrow \\lbrace 0, 1\\rbrace $ such that ${\\rm {dist}}(x,y)\\ge D$ , for all $x, y\\in {\\mathbb {W}}$ with $\\phi (x)=\\phi (y)=1$ .", "Sites $x\\in {\\mathbb {W}}$ with $\\phi (x)=1$ are treated as occupied, those with $\\phi (x)=0$ as vacant.", "The set of $D$ -ACs is denoted by ${\\mathcal {A}}={\\mathcal {A}}(D,{\\mathbb {W}})$ .", "Similar definitions can be introduced for a subset ${\\mathbb {V}}\\subseteq {\\mathbb {W}}$ .", "Cf.", "Fig.", "1.", "Given ${\\mathbb {V}}\\subseteq {\\mathbb {W}}$ , we say that configuration $\\psi \\in {\\mathcal {A}}(D,{\\mathbb {V}}$ ) is compatible with $\\phi \\in {\\mathcal {A}}$ if the concatenation $\\psi \\vee (\\phi \\hspace{-3.0pt}\\upharpoonright _{{\\mathbb {W}}\\setminus {\\mathbb {V}}})\\in {\\mathcal {A}}$ .", "Define the probability distribution $\\mu _{{\\mathbb {V}}} (\\, \\cdot \\, \|\| \\phi )$ on ${\\mathcal {A}}(D,{\\mathbb {V}}$ ): $\\mu _{{\\mathbb {V}}} (\\psi \|\| \\phi ):= {\\left\\lbrace \\begin{array}{ll}\\displaystyle \\frac{u^{\\sharp \\psi }}{Z({\\mathbb {V}}\|\|\\phi )}, &\\hbox{$\\psi $\\, is\\, compatible\\, with\\, $\\phi $,}\\\\0, & \\hbox{otherwise.}\\end{array}\\right.", "}\\qquad \\mathrm {(1)}$ Here $\\sharp \\psi $ stands for the number of occupied sites in $\\psi \\in {\\mathcal {A}}(D,{\\mathbb {V}})$ and $Z(\\mathbb {V} \|\| \\phi )$ is the partition function in $\\mathbb {V}$ with the boundary condition $\\phi \\in \\mathcal {A}$ : $Z(\\mathbb {V} \|\| \\phi ):=\\sum \\limits _{\\psi \\in \\mathcal {A}(D,\\mathbb {V})} u^{\\sharp \\psi }\\mathbf {1}(\\psi \\,\\,\\hbox{is compatible with}\\,\\, \\phi ).\\qquad \\mathrm {(2)}$ $\\mu _{\\mathbb {V}} (\\, \\cdot \\, \|\| \\phi )$ is called an H-C Gibbs distribution in `volume' ${\\mathbb {V}}$ with the boundary condition $\\phi $ at fugacity $u>0$ .", "Figure: An admissible configuration2.2.", "A Gibbs distribution ${\\mbox{${\\mu }$}}$ on ${\\mathbb {W}}$ is defined as a (weak) limiting point of $\\mu _{{\\mathbb {V}}_n}(\\,\\cdot \\,\|\|\\phi )$ as ${\\mathbb {V}}_n\\nearrow {\\mathbb {W}}$ in the Van Hove sense (the boundary condition $\\phi $ may vary with ${\\mathbb {V}}$ ).", "It is a (Borel) probability measure on ${\\mathcal {A}}$ .", "The Gibbs distribution $\\lim _{{\\mathbb {V}}_n\\nearrow {\\mathbb {W}}}\\mu _{{\\mathbb {V}}_n}(\\,\\cdot \\,\|\|\\phi ),\\qquad \\mathrm {(3)}$ if it exists, is denoted by ${\\mbox{${\\mu }$}}_\\phi $ ; we say that ${\\mbox{${\\mu }$}}_\\phi $ is generated by $\\phi \\in {\\mathcal {A}}$ .", "The set of Gibbs distributions ${\\mathcal {G}}(D)={\\mathcal {G}}(D,u,{\\mathbb {W}})$ is a Choquet simplex, so ${\\mathcal {G}}(D)$ is defined by the set ${\\mathcal {E}}(D)={\\mathcal {E}}(D,u,{\\mathbb {W}})$ of its extreme elements.", "An extreme Gibbs distribution ${\\mbox{${\\mu }$}}\\in {\\mathcal {E}}$ has the property that ${\\mbox{${\\mu }$}}$ cannot be written as a non-trivial mixture over ${\\mathcal {G}}(D)$ .", "2.3.", "This work studies the H-C EGDs for values $u$ large enough; such an assumption is in force throughout the paper and not stressed every time again.", "The structure of set ${\\mathcal {E}}(D)$ depends both on the choice of lattice ${\\mathbb {W}}$ and arithmetic properties of $D$ and is studied by using the PS theory based on the notion of a PGS.", "A ground state is defined as a $D$ -AC $\\varphi \\in {\\mathcal {A}}$ with the property that one cannot remove finitely many particles from $\\varphi $ and replace them by a larger number of particles without breaking admissibility.", "A PGS is a ground state invariant under two non-collinear lattice shifts, hence under their convolutions.", "A parallelogram spanned by the corresponding vectors is called a period of a PGS.", "We let ${\\mathcal {P}}(D)={\\mathcal {P}}(D,{\\mathbb {W}})$ denote the set of PGSs for given $D$ and ${\\mathbb {W}}$ .", "The PS theory [10], [11], states that, under certain assumptions, (a) every periodic EGD ${\\mbox{${\\mu }$}}$ is generated by a PGS $\\varphi $ in the sense of Eqn (3): ${\\mbox{${\\mu }$}}={\\mbox{${\\mu }$}}_\\varphi $ , and (b) every periodic EGD ${\\mbox{${\\mu }$}}$ has a list of properties stressing its pure-phase character.", "See items (P1)–(P5) below.", "According to general results from [3], in dimension two non-periodic ground states do not generate EGDs.", "So, identifying the PGSs allows us to describe all EGDs.", "The assumptions that we need to verify are that (I) there are finitely many PGSs, and (II) a deviation from a PGS is controlled by a suitable Peierls bound based on a suitable notion of a contour.", "Verifying these assumptions is the central part of this work." ], [ "Periodic ground states", "3.1.", "Let us consider assumption (I).", "It turns out that, apart from a few exceptional values of $D$ on ${\\mathbb {H}}_2$ and ${\\mathbb {Z}}^2$ which have to be analyzed separately, the PGSs on ${\\mathbb {W}}={\\mathbb {A}}_2, {\\mathbb {H}}_2, {\\mathbb {Z}}^2$ are constructed from maximally-dense admissible (MDA) sub-lattices ${\\mathbb {E}}\\subset {\\mathbb {W}}$ by means of ${\\mathbb {W}}$ -shifts.", "Non-periodic ground states are commented on in sub-section 3.6.", "The ${\\mathbb {W}}$ -symmetries (${\\mathbb {W}}$ -shifts and ${\\mathbb {W}}$ -reflections/rotations) define a partition of set ${\\mathcal {P}}(D,{\\mathbb {W}})$ into PGS-equivalence classes.", "Let $K=K(D,{\\mathbb {W}})$ denote the number of PGS-equivalence classes, then $K<\\infty $ since each class is represented by MDA-sub-lattice(s).", "Next, let $\\sigma $ stand for the discrete area of the fundamental parallelogram of an MDA-sub-lattice and $m_k$ denotes the number of distinct MDA-sub-lattices within a given PGS-equivalence class, $k=1,\\ldots ,K$ .", "Thus, for the cardinality $\\sharp {\\mathcal {P}}(D,{\\mathbb {W}})$ of the set of PGSs ${\\mathcal {P}}(D,{\\mathbb {W}})$ we have $\\sharp {\\mathcal {P}}(D, {\\mathbb {W}})=\\sigma \\sum \\limits _{k=1}^{K}m_k.\\qquad \\mathrm {(4)}$ Further specifications for $K$ , $\\sigma $ and $m_k$ depend on the choice of ${\\mathbb {W}}$ and $D$ and are provided below.", "In particular, dependence upon $D$ is non-monotonic and exhibits connections with algebraic number theory.", "3.2.", "On ${\\mathbb {A}}_2$ , for any $D$ , every MDA sub-lattice is a $D$ -sub-lattice, for which a fundamental parallelogram is formed by two equilateral triangles with side-length $D$ ($D$ -triangles).", "This is due to the fact that a triangular arrangement solves the dense-packing problem in ${\\mathbb {R}}^2$ .", "Consequently, on ${\\mathbb {A}}_2$ the value $\\sigma =D^2$ .", "The number $K$ of PGS-classes is related to the number of solutions $a,b\\in {\\mathbb {Z}}$ , $0\\le a\\le b$ , to the Diophantine equation $D^2=a^2+b^2+ab$ , for a given value $D$ .", "A helpful fact is that a solution exists (i.e., $D^2$ is a Löshian number) iff the prime decomposition of $D^2$ has primes of the from $3v+2$ in even powers.", "We will consider three disjoint groups of values of $D$ (or $D^2$ ) deemed TA1, TA2 and TB, and based on further conditions upon prime factors of $D^2$ .", "The first character in the labels TA1, TA2 and TB reflects the choice of a lattice (T stands for triangular).", "First, take the case TA1: here $D^2$ is of the form $D^2=a^2$ or $D^2=3a^2$ where $a\\in {\\mathbb {N}}$ has only prime factors 3 or prime factors of the form $3v+2$ , i.e., the prime decomposition of $D^2$ has no primes of form $3v+1$ .", "Then the above Diophantine equation has a unique solution, and the $D$ -sub-lattice is ${\\mathbb {A}}_2$ -reflection invariant; consequently, $K=1$ , $m_1=1$ and the number of PGSs is $D^2$ .", "Case TA2 emerges when the equation has a unique solution such that $a< b$ , with $a, b\\in {\\mathbb {N}}$ .", "A value $D^2$ belongs to this group iff its prime decomposition has a single prime of form $3v+1$ .", "In this case $K=1$ , the $D$ -sub-lattice is not ${\\mathbb {A}}_2$ -reflection invariant, $m_1=2$ , and the number of PGSs is $2D^2$ .", "Cf.", "Fig.", "2.", "Figure: PGSs on 𝔸 2 {\\mathbb {A}}_2,for D 2 =D^2= 9 (Case TA1) (a), and D 2 =D^2= 13 (Case TA2) (b).The number of PGSs is 9 and 26, respectively.The 3rd case, TB, covers all remaining values of $D$ ; in this case the equation has multiple solutions.", "Each solution $a,b$ , $0\\le a\\le b$ , generates a PGS-equivalence class, and its cardinality is $D^2$ or $2D^2$ , similarly to the above cases TA1 and TA2.", "Cf.", "Fig.", "3.", "Figure: PGSs on 𝔸 2 {\\mathbb {A}}_2 for D 2 =49D^2=49 (Case TB).There are 49 horizontal PGSs (a) and 98 inclined (b).", "The horizontalPGSs are the only dominant.3.3.", "Now consider ${\\mathbb {W}}={\\mathbb {H}}_2, {\\mathbb {Z}}^2$ .", "As was said, for some values $D^2$ on ${\\mathbb {H}}_2$ and ${\\mathbb {Z}}^2$ we encounter a phenomenon of sliding.", "It occurs when one can pass from one PGS to another without any local loss in `energy' i.e., without decreasing the local particle numbers.", "As a result, the Peierls bound does not hold, and the PS theory does not apply.", "Figure: Sliding on ℍ 2 {\\mathbb {H}}_2 for D 2 =4D^2=4 (a) and on ℤ 2 {\\mathbb {Z}}^2for D 2 =9D^2=9 (b).On ${\\mathbb {H}}_2$ there are just 4 sliding values: $D^2=$ 4, 7, 31, 133; the proof of this fact is computer-assisted.", "Cf.", "Fig.", "4 (a).", "On ${\\mathbb {Z}}^2$ there exist 39 sliding values: $D^2=$ 4, 8, 9, 18, 20, 29, 45, 72, 80, 90, 106, 121, 157, 160, 218, 281, 392, 521, 698, 821, 1042, 1325, 1348, 1517, 1565, 2005, 2792, 3034, 3709, 4453, 4756, 6865, 11449, 12740, 13225, 15488, 22784, 29890, 37970.", "Cf.", "Fig.", "4 (b).", "This was first conjectured by the authors [8], then in [4] it was proved that the sliding list is finite, then finally the completeness of the above list was independently established in [9], [5].", "We conjecture that for the sliding values of $D$ on ${\\mathbb {H}}_2$ and ${\\mathbb {Z}}^2$ , the EGD is unique for all values of fugacity $u$ .", "In what follows we consider the non-sliding values of $D$ only.", "3.4.", "Additionally, on ${\\mathbb {H}}_2$ there are exceptional values $D^2=$ 1, 13, 16, 28, 49, 64, 97, 157, 256 where the PGS-class is unique but is non-lattice.", "Cf.", "Fig.", "5.", "Furthermore, for $D^2=$ 67 there are two classes one of which is non-lattice.", "These values are analyzed via a special approach and not discussed in this article.", "Figure: Non-lattice PGSs on ℍ 2 {\\mathbb {H}}_2 for D 2 =D^2= 13 (a)and D 2 =D^2= 16 (b).Figure: PGSs on ℍ 2 {\\mathbb {H}}_2, for D 2 =D^2= 48 (Case HA1) (a), and D 2 =D^2= 39(Case HA2) (b).", "The number of PGSs is 32 and 52, respectively.Thus, in what follows, on ${\\mathbb {H}}_2$ we refer to non-exceptional values of $D$ : here every PGS is an MDA-sub-lattice.", "More precisely, if $3\|D^2$ then every PGS is a $D$ -sub-lattice since a $D$ -triangle can be inscribed in ${\\mathbb {H}}_2$ .", "Accordingly, we define groups of values $D^2$ deemed HA1, HA2 and HB and formed by the values from cases TA1, TA2 and TB divisible by 3.", "When $D^2$ belongs to one of these H-cases, the theory goes in parallel to the respective T-case.", "In particular, this yields the same values $K$ , $m_k$ while $\\sigma =2/3D^2$ .", "Cf.", "Fig.", "6.", "When $D^2$ is not divisible by 3, we pick the nearest Löschian number $(D^)^2>D^2$ divisible by 3: the fact is that every MDA-sub-lattice is a $D^$ -sub-lattice in ${\\mathbb {H}}_2$ , as triangles with area less than that of the $D^$ -triangle do not generate $D$ -admissible PGSs.", "This defines a group of values $D^2$ called case HC, for which the above theory is repeated with $D$ replaced by $D^$ .", "Cf.", "Fig.", "7.", "Figure: (a) PGSson ℍ 2 {\\mathbb {H}}_2 for D 2 =147D^2=147 (Case HB).", "The `vertical' PGSs (black) are dominant.", "(b) PGSs on ℍ 2 {\\mathbb {H}}_2 forvalues D 2 D^2 from Class HC, withD 2 =D 2 +2{D^}^2=D^2+2, and their associated D {D^}-triangles (black).", "(i) For D 2 =19D^2=19, D 2 =21{D^}^2=21.", "(ii) For D 2 =61D^2=61, D 2 =63{D^}^2=63.", "(iii) For D 2 =217D^2=217, D 2 =219{D^}^2=219.The gray ℍ 2 {\\mathbb {H}}_2-triangles give the minimal area when the side-lengthsare ≥D\\ge D and the angles ≤π/2\\le \\pi /2.", "However, they do not generate PGSs.The PGSs are MDA-lattices constructed from the black D {D^*}-triangles.3.5.", "A different situation emerges on ${\\mathbb {Z}}^2$ , where a $D$ -triangle can never be inscribed.", "Nevertheless, every PGS- equivalence class is constructed from an MDA-sub-lattice.", "The MDA-sub-lattices in ${\\mathbb {Z}}^2$ are defined implicitly, via solutions of a discrete optimization problem $\\begin{array}{c}\\hbox{\\bf minimize the area of a${{\\mathbb {Z}}}^2$-triangle \\ $\\triangle $}\\\\\\hbox{\\bf with side-lengths \\ $\\ell _i\\ge D$ and angles \\ $\\alpha _i\\le \\pi /2$.", "}\\end{array}\\qquad \\mathrm {(5)}$ The term ${\\mathbb {Z}}^2$ -triangle means a triangle with vertices in ${\\mathbb {Z}}^2$ .", "Accordingly, $\\sigma =S$ where $S/2$ is the minimum achieved in (5).", "A minimizing triangle in (5) is referred to as an M-triangle.", "Adjacent pairs of M-triangles form fundamental parallelograms of MDA-sub-lattices generating the PGSs.", "Cf.", "Fig.", "8.", "Problem (5) always has a solution but the M-triangle $\\triangle $ may be non-unique.", "A delicate point is that there are different types of non-uniqueness of an M-triangle $\\triangle $ : (i) there are $N_0>1$ M-triangles, and they are ${\\mathbb {R}}^2$ - but not ${\\mathbb {Z}}^2$ -congruent; (ii) there are $N_1>1$ M-triangles, and they are not ${\\mathbb {R}}^2$ -congruent; (iii) a mixture of (i) and (ii).", "Cf.", "Fig.", "9.", "The value $K$ is the number of non-${\\mathbb {Z}}^2$ -congruent M-triangles.", "Next: (a) $m_k=1$ if $D^2=2$ , and (b) $m_k=2$ or 4 when $D^2>2$ and the M-triangle $\\triangle $ defining the PGS-equivalence class is isosceles or not, respectively.", "Similarly to ${\\mathbb {A}}_2$ and ${\\mathbb {H}}_2$ , the identification of PGSs on ${\\mathbb {Z}}^2$ is intrinsically connected with algebraic number theory.", "It turns out that, for a given $D$ , one can characterize the corresponding M-triangles via solutions to norm equations in ring ${\\mathbb {Z}}[\\@root 6 \\of {-1}]$ .", "Such a connection helps to prove that uniqueness of an M-triangle and each of non-uniqueness types (i)–(iii) occur for infinitely many values $D$ , and the degrees of degeneracy $N_0$ , $N_1$ can be arbitrarily large as $D\\rightarrow \\infty $ .", "Figure: PGSs on ℤ 2 {\\mathbb {Z}}^2 for (a) D 2 =16D^2=16, S=15S=15and (b) D 2 =25D^2=25, S=23S=23.", "In both cases, the M-triangles are ℤ 2 {\\mathbb {Z}}^2-congruent,and there is a single PGS-equivalence class.", "Consequently, K=1K=1.", "ForD 2 =16D^2=16 the M-triangles are isosceles, and there are 2 MDA-sub-lattices.", "For D 2 =25D^2=25the M-triangles are non-isosceles, and there are 4 MDA-sub-lattices.Accordingly, σ=15\\sigma =15, m=2m=2 for D 2 =16D^2=16 andσ=23\\sigma =23, m=4m=4 for D 2 =25D^2=25.", "The number of PGSs is 30 and 92, respectively.Figure: Non-uniqueness of M-triangles on ℤ 2 {\\mathbb {Z}}^2:(a) for D 2 =425D^2=425, S=375S=375 (ℝ 2 {\\mathbb {R}}^2- but not ℤ 2 {\\mathbb {Z}}^2-congruent, N 0 =2N_0=2)and (b) for D 2 =65D^2=65, S=60S=60 (ℝ 2 {\\mathbb {R}}^2-non-congruent, N 1 =2N_1=2).", "Which PGS-classgenerates EGDs is determined by dominance.3.6.", "It is possible to check that any non-periodic ground state on ${\\mathbb {Z}}^2$ contains at least one infinite connected component of non-M-triangles and no finite ones.", "Moreover, the number of non-M-triangles in a ${\\mathbb {Z}}^2$ -square ${\\mathbb {V}}(L)$ of side-length $L$ can only grow at most linearly with $L$ ; this means that in a non-periodic ground state, non-M-triangles form, effectively, a one-dimensional array.", "A similar pattern for non-periodic ground states emerges on ${\\mathbb {A}}_2$ and ${\\mathbb {H}}_2$ .", "Let us repeat once more that non-periodic ground states do not generate EGDs in dimension two [3]." ], [ "The Peierls bound", "4.1.", "Contours.", "Physically speaking, contours describe local perturbations of PGSs.", "They emerge when we (i) remove some particles from a PGS $\\varphi $ and (ii) add some new particles at `inserted' sites, maintaining $D$ -admissibility.", "A formal definition is as follows.", "First, we define a template as a common period of all MDA-sub-lattices and its shifts by the multiples of the spanning vectors.", "Cf.", "Fig.", "10.", "Figure: Templates (gray) and fundamental parallelograms on 𝔸 2 {\\mathbb {A}}_2 for D 2 =7D^2=7(a) and on ℤ 2 {\\mathbb {Z}}^2 for D 2 =25D^2=25 (b).We say that a template $F$ is $\\varphi $ -regular in $\\phi $ if, $\\forall $ $x\\in F$ , we have $\\phi (x)=\\varphi (x)$ .", "A template $F$ is called $\\varphi $ -correct if $F$ and all 8 of its adjacent templates are $\\varphi $ -regular.", "The frustrated set is formed by the union of templates that are not $\\varphi $ -correct $\\forall $ $\\varphi \\in {\\mathcal {P}}(D)$ .", "A contour $\\Gamma $ in a $D$ -AC $\\phi \\in {\\mathcal {A}}(D)$ is defined as a pair $({\\rm S},\\phi \\hspace{-2.0pt}\\upharpoonright _{\\rm S})$ where ${\\rm S}=\\rm {Supp}(\\Gamma )\\subset {\\mathbb {W}}$ is a connected component of the frustrated set.", "We say $\\Gamma $ is finite if the set $\\rm {Supp}(\\Gamma )$ is finite.", "Let $\\Gamma $ be a finite contour in a $D$ -AC $\\phi \\in {\\mathcal {A}}(D)$ .", "The complement ${\\mathbb {W}}\\setminus \\rm {Supp}(\\Gamma )$ has one infinite connected component which we call the exterior of $\\Gamma $ and denote by Ext$\\,(\\Gamma )$ .", "In addition, set ${\\mathbb {W}}\\setminus \\rm {Supp}(\\Gamma )$ may have finitely many interior connected components; they are denoted by Int$_j(\\Gamma )$ , $j=1,\\ldots ,J$ , and we set Int$\\,(\\Gamma )=\\operatornamewithlimits{\\cup }\\limits _{j=1}^J$ Int$_j(\\Gamma )$ .", "Cf.", "Fig.", "11.", "We say that $\\Gamma $ is a $\\varphi $ -contour in $\\phi $ if every template $F\\subset \\rm {Ext}(\\Gamma )$ adjacent to $\\rm {Supp}(\\Gamma )$ is $\\varphi $ -correct in $\\phi $ .", "We say that $\\Gamma $ is an external contour in $\\phi $ if $\\rm {Supp}(\\Gamma )$ does not lie in Int$\\,(\\Gamma ^{\\prime })$ for any other contour $\\Gamma ^{\\prime }$ .", "Figure: Contour supports on 𝔸 2 /ℍ 2 {\\mathbb {A}}_2/{\\mathbb {H}}_2 (a) and ℤ 2 {\\mathbb {Z}}^2 (b).Dark-gray color marks non-correct templates, light-gray templates are their neighbors.An important point is that a contour can be considered without a reference to the AC $\\phi $ : it is enough that we indicate (i) a $D$ -AC $\\psi _\\Gamma $ over set $\\rm {Supp}(\\Gamma )$ , (ii) an external phase $\\varphi $ and the internal phases $\\varphi _j$ , such that every template $F\\subset \\rm {Ext}(\\Gamma )$ adjacent to $\\rm {Supp}(\\Gamma )$ is $\\varphi $ -correct and every template $F\\subset \\rm {Int}_j(\\Gamma )$ adjacent to $\\rm {Supp}(\\Gamma )$ is $\\varphi _j$ -correct, $j=1,\\ldots ,J$ .", "With the above definitions at hand, we write down the contour representation for the partition function: $Z ({\\mathbb {V}}_n\\Vert \\varphi )=u^{\\sharp (\\varphi \\upharpoonright _{{\\mathbb {V}}_n})}\\sum _{\\lbrace \\Gamma _i\\rbrace \\subset {\\mathbb {V}}_n}\\prod \\limits _iw (\\Gamma _i).\\qquad \\mathrm {(6)}$ Here the summation goes over compatible contour collections $\\lbrace \\Gamma _i\\rbrace $ with pair-wise disjoint $\\rm {Supp}(\\Gamma _i)\\subset {\\mathbb {V}}_n$ , while $w(\\Gamma )$ stands for the statistical weight of contour $\\Gamma $ : $w (\\Gamma )=u^{\\sharp (\\psi _\\Gamma )-\\sharp (\\varphi \\upharpoonright _\\Gamma )}.", "\\qquad \\mathrm {(7)}$ Pictorially, compatibility means that if two of contours, $\\Gamma $ and $\\Gamma ^{\\prime }$ , from the collection are not separated by a third contour $\\overline{\\Gamma }$ then their external and/or internal phases are coordinated.", "Formally, it requires two properties.", "(a) If (i) $\\rm {Supp}(\\Gamma ^{\\prime })\\subset \\rm {Int}_j(\\Gamma )$ and (ii) there is no contour $\\overline{\\Gamma }$ in $\\lbrace \\Gamma _i\\rbrace $ with $\\rm {Supp}(\\Gamma ^{\\prime })\\subset \\rm {Int}(\\overline{\\Gamma })$ and $\\rm {Supp}(\\overline{\\Gamma })\\subset \\rm {Int}_j (\\Gamma )$ then the internal phase $\\varphi _j$ of $\\Gamma $ serves as the external phase for $\\Gamma $ .", "(b) If (i) $\\rm {Supp}(\\Gamma ^{\\prime })\\subset \\rm {Ext}(\\Gamma )$ and (ii) there is no contour $\\overline{\\Gamma }$ in $\\lbrace \\Gamma _i\\rbrace $ with $\\rm {Supp}(\\overline{\\Gamma })\\subset \\rm {Ext}(\\Gamma )$ and $\\rm {Supp}(\\Gamma ^{\\prime })\\subset \\rm {Int}(\\overline{\\Gamma })$ then $\\Gamma $ and $\\Gamma ^{\\prime }$ have the same external phase.", "Note that all external contours in a compatible collection $\\lbrace \\Gamma _i\\rbrace $ are $\\varphi $ -contours for the same $\\varphi $ .", "The condition upon ${\\mathbb {V}}_n$ in (6) is that ${\\mathbb {V}}_n$ is a finite union of templates.", "We then can think of a Gibbs distribution $\\mu _{{\\mathbb {V}}_n}(\\,\\cdot \\,\\Vert \\varphi )$ as a probability measure on compatible contour collections $\\lbrace \\Gamma _i\\rbrace $ in ${\\mathbb {V}}_n$ .", "4.2.", "The Peierls constant.", "The next step is to establish a bound $\\sharp (\\psi _\\Gamma )-\\sharp (\\varphi \\hspace{-3.0pt}\\upharpoonright _\\Gamma )\\le -p\\Vert \\rm {Supp}(\\Gamma )\\Vert .\\qquad \\mathrm {(8)}$ Here $\\Vert \\rm {Supp}(\\Gamma )\\Vert $ stands for the number of templates in $\\rm {Supp}(\\Gamma )$ and $p=p(D,{\\mathbb {W}})>0$ is a Peierls constant per a template.", "Then the statistical weight $w (\\Gamma )$ will obey $w (\\Gamma )\\le u^{-p\\Vert \\rm {Supp}(\\Gamma )\\Vert }$ , and a standard Peierls argument will lead to properties (P1-5) below.", "By the definition of a contour, every template $F\\subset \\rm {Supp}(\\Gamma )$ contains some sort of a defect.", "A trivial defect is when at least one particle can be added to configuration $\\psi _\\Gamma \\hspace{-2.0pt}\\upharpoonright _F$ ; in this case we get at least a factor 1 in place of $p$ in inequality (8).", "For a saturated defective template $F\\subset \\rm {Supp}(\\Gamma )$ , the situation is more complex.", "Here we use the Delaunay triangulation for $\\psi _\\Gamma \\hspace{-4.0pt}\\upharpoonright _F$ ; the fact is that for each of the triangles the area is $\\ge \\sigma /2$ , and for at least one triangle the area is $\\ge (1+\\sigma )/2$ .", "(Note that the lattice triangle has a half-integer area.)", "The number of such triangles is $O\\left(\\Vert \\rm {Supp}(\\Gamma )\\Vert \\right)$ .", "On the other hand, the number of particles inside $\\rm {Supp}(\\Gamma )$ equals the doubled number of triangles.", "This ultimately leads to (8).", "The constant $p (D)$ can actually be lover-bounded by $c/D^2$ ." ], [ "Dominance, extreme Gibbs distributions", "5.1.", "If for a given $D$ , the number $K$ of PGS-equivalence classes equals 1, the number of EGDs $\\sharp {\\mathcal {E}}(D, \\mathbb {W})$ equals the number of PGSs $\\sharp {\\mathcal {P}}(D, \\mathbb {W})$ .", "When $K>1$ , i.e., there are multiple PGS-equivalence classes, not all PGS-classes generate EGDs, only the dominant ones.", "The dominant classes are those which maximize a `truncated free energy' for the ensemble of small contours; cf.", "[11].", "Thus, for a given non-sliding $D$ , $\\sharp {\\mathcal {E}}(D, \\mathbb {W})=\\sigma \\sum \\limits _{k=1}^{L} m_k, \\qquad \\mathrm {(9)}$ where $L\\le K$ is the number of dominant PGS-equivalence classes.", "We conjecture that for the H-C model the value $L=1$ , that is, there exist a unique dominant PGS-class.", "5.2.", "As an outcome of the PS theory for the H-C model we have the following properties.", "(P1) Each EGD ${\\mbox{${\\mu }$}}\\in {\\mathcal {E}}(D,\\mathbb {W})$ is generated by a PGS.", "That is, each EGD is of the form ${\\mbox{${\\mu }$}}_\\varphi $ for some $\\varphi \\in {\\mathcal {P}}(D,\\mathbb {W})$ .", "If PGSs $\\varphi _i$ generate EGDs ${\\mbox{${\\mu }$}}_{\\varphi _i}$ , $i=1,2$ , and $\\varphi _1\\ne \\varphi _2$ then ${\\mbox{${\\mu }$}}_{\\varphi _1}\\perp {\\mbox{${\\mu }$}}_{\\varphi _2}$ .", "The EGDs inherit symmetries between the PGSs.", "(P2) Consequently, EGD-generation is a class property: if a PGS $\\varphi \\in {\\mathcal {P}}(D,\\mathbb {W})$ generates an EGD ${\\mbox{${\\mu }$}}_\\varphi $ then every PGS ${\\varphi ^{\\prime }}$ from the same equivalence class generates an EGD ${\\mbox{${\\mu }$}}_{\\varphi ^{\\prime }}$ .", "Such a class is referred to as dominant.", "If a PGS-equivalence class is unique, it is dominant.", "(P3) Each EGD ${\\mbox{${\\mu }$}}_\\varphi $ exhibits the following properties.", "For ${\\mbox{${\\mu }$}}_\\varphi $ -almost all $\\phi \\in {\\mathcal {A}}$ : (i) all contours $\\Gamma $ in $\\phi $ are finite, (ii) for any site $x\\in {\\mathbb {W}}$ there exist only finitely many contours $\\Gamma $ (possibly none) such that $x$ lies in the interior of $\\Gamma $ , (iii) there are countably many disjoint $D$ -connected sets of $\\varphi $ -correct templates one of which is infinite while all remaining $D$ -connected sets are finite, (iv) for every $\\varphi ^{\\prime }\\in {\\mathcal {P}}(D,\\mathbb {W})\\setminus \\lbrace \\varphi \\rbrace $ , there are countably many $D$ -connected sets of $\\varphi ^{\\prime }$ -correct templates all of which are finite.", "(P4) EGD ${\\mbox{${\\mu }$}}_\\varphi $ admits a polymer expansion and has an exponential decay of correlations.", "(P5) As $u\\rightarrow \\infty $ , ${\\mbox{${\\mu }$}}_\\varphi $ converges weakly to a measure sitting on a single $D$ -AC $\\varphi $ .", "Property (P3) establishes a percolation picture for EGD ${\\mbox{${\\mu }$}}_\\varphi $ : there is a `sea' of $\\varphi $ -correct templates with `islands' of non-$\\varphi $ -correct sites inside which there may be `lakes' of templates of different correctness, etc.", "In such a picture, contours mark `coastal/shallow-water strips'.", "5.3.", "Specification of dominant PGS-classes for a general value $D$ remains an open question; we present a complete answer for values $D^2=$ 49, 147, 169 on ${\\mathbb {A}}_2$ or ${\\mathbb {H}}_2$ by classifying the small contours.", "In doing so we develop an approach which could be used for other examples.", "The simplest small contour occurs when we merely remove a particle from a PGS $\\varphi $ .", "Clearly, all PGSs are `equal in rights' relative to this perturbation type, as they are maximally-dense.", "The same is true about removing pairs of particles, etc.", "Then there is a possibility of inserting one and removing three particles, at the vertices of the Delaunay triangle of $\\varphi $ ; these are single insertions.", "Continuing, we can think of double insertions where pairs of inserted particles remove four particles at the vertices of covering parallelogram in $\\varphi $ , and so on, maintaining the difference 2 between the number of removed and inserted particles (and, of course, $D$ -admissibility).", "This yields a finite collection of admissible $u^{-2}$ -insertions where we add $n$ and remove $n+2$ particles.", "Figure: Dominance on 𝔸 2 {\\mathbb {A}}_2, for D 2 =147D^2=147.", "This figure shows admissible u -2 u^{-2}-insertions for D 2 =147D^2=147 on𝔸 2 \\mathbb {A}_2.", "Here we have two PGS-equivalence classes,one `inclined' (a, c) and one `vertical' (b, d).", "Single admissibleu -2 u^{-2}-insertions are marked by brown dots (a, b); gray lensesindicate areas from where a singleinsertion repels four sites in a PGS and hence yields au -2 u^{-2}-insertion.", "The number of single u -2 u^{-2}-insertions is 68per a DD-rhombus for both PGS-classes.", "Double, triple andquadruple u -2 u^{-2}-insertions are shown on frames (c, d)by red, green and blue colors, respectively.", "For the inclinedclass there is no admissible quadruple u -2 u^{-2}-insertion, andfor the neither class there is an admissible nn-particle u -2 u^{-2}-insertion withn≥5n\\ge 5.", "The count of all u -2 u^{-2}-insertions shows that the verticalPGS-class is dominant.In the above-mentioned examples, it is possible to (a) enumerate the $u^{-2}$ -insertions, and (b) verify that higher-order insertions (beginning with $u^{-3}$ ) can be discarded when fugacity $u$ is large enough.", "It allows us to identify the dominant PGS-classes for these values of $D$ .", "The most involved part of the argument (requiring a computer-assisted proof) is to check that the maximal number of added particles in admissible $u^{-2}$ -insertions is $n=4$ : here we remove 6 occupied sites from the boundary of a $2D$ -triangle in a given PGS." ], [ "Conclusions", "In this work we address a construction of the phase diagram for the hard-core model on ${\\mathbb {A}}_2$ , ${\\mathbb {H}}_2$ and ${\\mathbb {Z}}^2$ in a high-density regime.", "One can say that the H-C model admits a straightforward application of the PS theory, but the challenge here lies in determining PGSs and verifying a suitable Peierls bound.", "A zest of the work is that the PGSs have been identified by means of algebraic number theory.", "Since the PGSs coincide with periodic disk-packings of maximal density, we also obtain the description of the latter, which yields a solution to the disk-packing problem on ${\\mathbb {A}}_2$ , ${\\mathbb {H}}_2$ and ${\\mathbb {Z}}^2$ .", "A number of questions have been answered, covering the phenomenon of sliding and the complete structure of the phase diagram when there is a single PGS-equivalence class.", "The only remaining question within the remit of the PS theory is how to identify the dominant class(es) among multiple PGS-equivalence classes.", "The work also casts light on several important open problems beyond the PS theory.", "(1) What is the phase diagram as a function of $u>0$ for non-sliding $D$ , including existence of a Kosterlitz–Thouless transition, presence of amorphous/glassy phases, etc.", "(2) What is the dependence of the critical value(s) of $u$ upon $D$ ?", "If point $u^0=u^0(D)$ separates the uniqueness and non-uniqueness domains, and $u^1(D)$ is the lower threshold for the PS regime, then do we have $u^0(D)=u^1(D)$ for some/all/no $D$ ?", "(3) Is there a phase transition in the presence of sliding, and what is the full phase diagram?", "(4) Could our results provide hints towards understanding the model in ${\\mathbb {R}}^2$ ?", "All assertions in this article are mathematically rigorous.", "The details can be found in [8], [9].", "Acknowledgement.", "IS and YS thank the Math Department, Penn State University, for support during this work.", "YS thanks St John's College, Cambridge, for support." ] ]	2011.14156
[ [ "Contributions of flame thickening and local extinctions to burning rate\n of intensely turbulent premixed flames" ], [ "Abstract Influences of reaction zone thickening and local extinctions on the burning rate of extremely turbulent hydrogen-enriched methane-air flames are investigated using simultaneous planar laser-induced fluorescence of formaldehyde molecule and hydroxyl radical as well as separate stereoscopic particle image velocimetry techniques.", "Karlovitz numbers upto 76 are examined.", "It is shown that, by increasing the turbulence intensity, the preheat and reaction zone thicknesses can increase to values that are, respectively, 6.3 and 4.9 of the corresponding laminar flames.", "Broadening of these zones for intensely turbulent hydrogen-enriched methane-air flames is shown experimentally for the first time.", "Broadening of the reaction zone suggests that the flamelet assumption used for development of the burning rate formulations may not hold.", "Thus, a new formulation, which does not utilize the flamelet assumption, is developed and used to calculate the burning rate of the tested flames.", "It is shown that, at small turbulence intensities, the burning rate values follow those of the local consumption speed, which is developed in the literature based on the flamelet assumption.", "However, at large turbulence intensities, the estimated burning rate features large values, and the ratio of this parameter to the local consumption speed is consistent with the ratio of the global and local consumption speeds reported in the literature.", "It is shown that the ratio of the normalized burning rate to the normalized local consumption speed is correlated with the broadening of reaction zone, suggesting that the disparity between the values of the burning rate and local consumption speed is linked to the reaction zone thickening.", "It is shown, although the flame thickening increase the burning rate, local extinctions decrease this parameter leading to the bending behavior reported in the literature." ], [ "Introduction", "Hydrogen-enrichment of natural gas (which mainly contains methane) will be used as a means to decarbonize several industries that use combustion equipment, such as land-based power generation gas turbine engines in the future [1].", "While this is underway, our understanding related to the speed at which the fuel and air mixture is converted to combustion products (referred to as the turbulent burning velocity) at engine flow-relevant conditions is limited for both pure and hydrogen-enriched methane-air flames to our best of knowledge.", "The burning velocity has been estimated using the flamelet assumption for methane-air flames, see for example the review papers by Driscoll [2], [3].", "This assumption suggests that the reaction zone of the premixed flames is relatively thin [2], [3], [4], [5], [6], [7], [8].", "Although several studies show this assumption holds for a wide range of turbulence conditions [9], [10], results of several other investigations [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] disagree and suggest the flamelet assumption may not hold.", "For these studies [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], the turbulent burning velocity cannot be estimated using the formulations developed based on the flamelet assumption.", "The present investigation is motivated by development and assessment of a formulation that allows for calculation of the burning velocity for turbulent premixed flames whose internal structure do not necessarily follow the prediction of the flamelet assumption for both pure and hydrogen-enriched methane-air turbulent premixed flames.", "In the following, first, literature related to the effect of the turbulent flow on the flame structure, as well as the implication of this effect on the burning velocity of methane-air premixed flames is briefly reviewed.", "Then, background related to the effect of hydrogen-enrichment on the methane-air premixed flames internal structure and burning velocity is presented.", "The internal structure of premixed flames can be highly influenced by the Karlovitz number, $Ka=\\tau _\\mathrm {f}/\\tau _{{\\eta }_\\mathrm {K}}$ , where $\\tau _\\mathrm {f}$ and $\\tau _{{\\eta }_\\mathrm {K}}$ are the flame and Kolmogorov time scales, respectively [22], [23].", "Flames with $Ka \\lesssim 1$ usually feature both thin preheat and reaction zones, and the turbulent eddies only tend to wrinkle the flame surfaces [22], [24], [25], [26].", "For $Ka \\gtrsim 1$ , the results reported in the literature for the preheat zone thickness are controversial.", "On one hand, as predicted by Peters [22], for $1 \\lesssim Ka \\lesssim 100$ , small eddies may penetrate into and broaden the preheat zone.", "This is confirmed through both experimental [27], [28], [9], [12], [29], [30], [31], [11] and Direct Numerical Simulation (DNS) studies [32], [33].", "However, several studies [34], [25], [35], [36], [9] suggest that the preheat zone may not broaden for $Ka \\gtrsim 1$ .", "The reason for this discrepancy is elaborated in [9], [3].", "Driscoll et al.", "[3] and Skiba et al.", "[9] argue that, eddies associated with $Ka \\gtrsim 1$ may feature sufficient turbulent kinetic energy allowing them to penetrate and broaden the preheat zone.", "Skiba et al.", "[9] and Driscoll et al.", "[3] suggested that, in addition to $Ka \\gtrsim 1$ , flames with $u^{\\prime }\\Lambda /(S_\\mathrm {L,0}\\delta _\\mathrm {L}) \\gtrsim 180$ may feature broadened preheat zones, where $\\Lambda $ , $S_\\mathrm {L,0}$ , and $\\delta _\\mathrm {L}$ are the integral length scale, unstretched laminar flame speed, and laminar flame thickness, respectively [3], [9].", "Similar to the preheat zone, the results reported in the literature regarding the turbulent flames reaction zone thickness are also controversial.", "Experimental [9] and DNS studies of [37], [38], [39] show that the reaction zone remains thin upto $Ka \\approx 550$ and 1000, respectively.", "In contrast to the studies of [9], [37], [38], [39], those of [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] suggest that the reaction zone is in fact broadened compared to the laminar flame counterpart.", "For example, in a recent investigation [11], the authors show that for flames with $Ka \\approx 76$ , the reaction zone thickness can increase up to 3.9 times that of the corresponding laminar flame.", "Mohammadnejad et al.", "[11] speculated that relatively small size energetic eddies may penetrate into and broaden the reaction zone.", "It was shown [11] that the probability density function of eddy size distribution and kinetic energy of the eddies can be influenced by the type of the utilized turbulence generating mechanism, and this may be the reason for the discrepancy in the literature regarding the turbulent premixed flames reaction zone thickness.", "The internal structure of turbulent premixed flames has implications for estimation of the turbulent burning velocity.", "This parameter can be estimated globally and locally, which are referred to as the global and local consumption speeds, respectively [2], [3].", "These speeds are estimated using [40], [3], [2] $S_\\mathrm {T,GC}=\\frac{\\dot{m}_\\mathrm {r}}{\\rho _\\mathrm {r} \\overline{A}_\\mathrm {f}}$ $S_\\mathrm {T,LC}=\\frac{S_\\mathrm {L,0} I_0}{L_\\mathrm {\\xi }} \\int _{-\\infty }^{\\infty } \\int _{-\\infty }^{\\infty }\\Sigma (\\eta ,\\xi ) d\\eta d\\xi ,$ where $S_\\mathrm {T,GC}$ and $S_\\mathrm {T,LC}$ are referred to as the global and local consumption speeds, respectively [2].", "$\\dot{m}_\\mathrm {r}$ and $\\rho _\\mathrm {r}$ are the reactants mass flow rate and density.", "$\\overline{A}_\\mathrm {f}$ is the flame surface area estimated based on the mean progress variable ($\\overline{c}$ ).", "In Equation (REF ), $\\Sigma $ is the flame surface density, $L_\\mathrm {\\xi }$ is the length of the mean progress variable contour, $I_0$ is the Bray-Cant [41] stretch factor, and $\\eta $ and $\\xi $ are the curvilinear axes, which are respectively normal and tangent to a given $\\overline{c}$ contour (usually $\\overline{c} = 0.5$ ).", "As reviewed in Driscoll et al.", "[2], [3], there are two ongoing questions regarding the global and local consumption speeds of methane-air turbulent premixed flames in the literature.", "First, the DNS studies of [42], [43], [44] suggest that increasing the turbulence intensity increases the flame surface area such that the local consumption speed nearly equals the global consumption speed.", "However, several other studies, see for example [3], [45], [46], [40], [47], [48], [21], [49], suggest that as the turbulence intensity increases, the global consumption speed increases, but the flame surface area and the local consumption speed plateau.", "For instance, Wabel et al.", "[40] show that, at $u^{\\prime }/S_\\mathrm {L,0} \\approx 150$ , the global consumption speed is about 5 times the local consumption speed.", "Wabel et al.", "[40] speculated that, as the preheat zone broadens with increasing the turbulence intensity, the turbulent diffusivity of the gas in this zone increases, enhancing the transport of reactants and as a result increase of the global consumption speed.", "Recently, Mohammadnejad et al.", "[11] showed that, in addition to the preheat zone, the reaction zone also broadens with increasing the turbulence intensity, which may also contribute to the increase of the gas diffusivity.", "Both Gülder [46] and Nivarti et al.", "[45] performed theoretical calculations and showed that the enhancement of gas diffusivity inside the flame region can allow for elaborating the reason for the increase of the global consumption speed with increase of the turbulence intensity and as a result the difference between the local and global consumption speed values reported in for example [3], [45], [46], [40], [47], [48], [21], [49].", "However, in both Gülder [46] and Nivarti et al.", "[45], the preheat and reaction zones thicknesses are not measured; and as a result, it is not known whether the enhancement of gas diffusivity is due to broadening of only the preheat zone or both the preheat and reaction zones.", "This requires further investigations.", "The second question is related to the behavior of the global consumption speed itself.", "As hypothesized by Damköhler [50], also known as the first Damköhler's hypothesis, increasing $u^{\\prime }/S_\\mathrm {L,0}$ is expected to increase the global consumption speed linearly.", "Although, at relatively small turbulence intensities, the global consumption speed nearly follows a linear relation with the turbulence intensity [3], [51], [22], [50], [40], [21], [49], [52], at relatively intense turbulence conditions, the global consumption speed plotted against the turbulence intensity bends towards the axis of $u^{\\prime }/S_\\mathrm {L,0}$ .", "This is referred to as the bending behavior in the literature, see for example [40], [21], [49], [52], [53], [26].", "Several reasons such as nonlinear variation of gas diffusivity with turbulence intensity [3], [50], [51], [45] as well as local quenching [3] are suggested to contribute to the bending behavior.", "To our best of knowledge, a potential relation between the flame quenching (or local extinction) and the bending behavior has not been studied yet." ], [ "Effect of hydrogen enrichment on turbulent premixed flame internal structure and burning velocity", "The internal structure of hydrogen-enriched turbulent methane-air premixed flames has been recently investigated in Mohammadnejad et al.", "[8] and Zhang et al. [54].", "The study of Mohammadnejad et al.", "[8] shows that, for relatively moderate turbulence conditions ($u^{\\prime }/S_\\mathrm {L,0}$ up to 2.6), the preheat and reaction zone thicknesses of the hydrogen-enriched flames are similar to the laminar premixed flame counterparts.", "Mohammadnejad et al.", "[8] showed that hydrogen enrichment (up to 50%) does not significantly influence the preheat and reaction zone thicknesses normalized by the corresponding laminar flame counterparts.", "Compared to [8], whose tested turbulence intensity was limited to 2.6, Zhang et al.", "[54] investigated both the reaction and preheat zone thicknesses for larger turbulence intensities ($3 \\lesssim u^{\\prime }/S_\\mathrm {L,0} \\lesssim 12.5$ ) with hydrogen-enrichment percentages examined up to 60%.", "They [54] showed that, similar to the methane-air turbulent premixed flames, increasing the turbulence intensity can lead to broadening of the hydrogen-enriched methane-air flames preheat zone.", "However, Zhang et al.", "[54] showed that the reaction zone thickness does not change by increasing the turbulence intensity.", "Although studies of both [8], [54] provide insight into the structure of hydrogen-enriched methane-air premixed flames at relatively small and moderate turbulence intensities, the preheat and reaction zone thicknesses of these flames pertaining to relatively intense turbulence conditions remain to be investigated.", "Few experimental investigations [55], [56] have studied the effect of hydrogen-enrichment on local and global consumption speeds of turbulent premixed flames.", "Both Halter et al.", "[56] and Guo et al.", "[55] showed that increasing the hydrogen-enrichment percentage increases the global consumption speed normalized by the unstretched laminar flame speed.", "Guo et al.", "[55] showed this parameter also increases with increasing the turbulence intensity.", "The values of the local consumption speed are not reported in [56], however, Guo et al.", "[55] showed that increasing both the hydrogen-enrichment percentage and the turbulence intensity increase the local consumption speed normalized by the unstretched laminar flame speed.", "Although these studies [55], [56] elaborated the effect of hydrogen enrichment and turbulence intensity on the local/global consumption speeds, their investigations are performed for $u^{\\prime }/S_\\mathrm {L,0} \\lesssim 5$ .", "To our best of knowledge, values of the local and global consumption speeds for hydrogen-enriched premixed flame are not investigated at larger turbulence intensities.", "This study aims to address three objectives.", "The first objective of the present study is to measure the preheat and reaction zone thicknesses of hydrogen-enriched methane-air premixed flames pertaining to relatively intense turbulence conditions.", "To our best of knowledge, such information is not available in the literature.", "The second objective of this study is to develop a consumption speed estimation framework that does not rely on the flamelet assumption.", "Then, using this framework, we aim to understand the reason for the difference between local and global consumption speeds reported in the literature.", "The third objective of the present study is to characterize the bending behavior of the global consumption speed and to develop a mathematical formulation that allows to estimate the amount of bending.", "The experimental setup, diagnostics, coordinate system, and the tested conditions are presented in this section." ], [ "Setup", "The setup used in this study is a Bunsen burner identical to that utilized in [11], [34].", "This burner consists of an expansion section, a settling chamber (with 5 mesh screens), a contraction section, and a nozzle with exit diameter of 22.2 mm.", "The burner schematic and the burner nozzle technical drawing are shown in Figs.", "REF and REF (a), respectively.", "A mixture of 75% hydrogen and 25% methane (volumetric percentages) is injected through the burner rim, generating a pilot flame, see Fig.", "REF .", "Similar to [11], the volumetric flowrate of the pilot flow is only 2% of the main flow, which is small, and as a result, the pilot flame is not expected to influence the main Bunsen flames.", "In order to generate a relatively wide range of turbulence intensities, three different turbulence generating mechanisms are used similar to [11].", "Specifically, zero, one, or two perforated plates were utilized.", "For the conditions with no perforated plate (which is the first turbulence generating mechanism), turbulence is primarily produced by the mesh screens and the Kelvin-Helmholtz instability that is developed in the jet shear layer, similar to [11], [57], [58], [59].", "For the second turbulence generating mechanism, a perforated plate, with the technical drawing shown in Fig.", "REF (c), is placed 44.4 mm upstream of the nozzle exit plane.", "This plate contains 1.8 mm diameter circular holes arranged in a hexagonal pattern, generating a blockage ratio of 60%.", "For the third turbulence generating mechanism, one perforated plate (identical to that used for the second turbulence generating mechanism) is positioned 44.4 mm, and a second plate is located 30.4 mm upstream of the burner exit plane, see Fig.", "REF (b).", "The second perforated plate has holes positioned on a hexagonal arrangement with a diameter of 1.8 mm.", "The blockage ratio of this plate is 77%.", "Figure: Schematic of the experimental setup and the diagnostics.Figure: Technical drawing of (a) the burner nozzle as well as (b and c) the perforated plates." ], [ "Diagnostics", "Simultaneous Planar Laser-Induced Fluorescence (PLIF) of hydroxyl radical ($\\mathrm {OH}$ ) and formaldehyde molecule ($\\mathrm {CH_2O}$ ) as well as separate Stereoscopic Particle Image Velocimetry (SPIV) were performed in this study.", "The diagnostics used here is similar to that used in Mohammadnejad et al. [11].", "However, compared to [11], the SPIV experiments in the present study are performed only for the non-reacting flow conditions and to characterize the background turbulent flow.", "Both PLIF and SPIV data acquisition rate were 1 Hz.", "A dual-head Nd:YAG pump laser (Quanta Ray PIV400) and a frequency-doubled dye laser (Sirah Precision Scan) are used for generating the PLIF signals.", "Two laser pulses with wavelengths of 532 nm and $355\\pm 3$ nm are produced by the pump laser using harmonic generators.", "As shown in Fig.", "REF , the 532 nm beam is used to pump the dye laser to generate a 283 nm ($282.94\\pm 0.005$ nm) beam.", "This beam is used to excite $\\mathrm {Q_1}$ (6) line of the OH A-X system (1,0) vibrational band [11], [8], [60], [61], [62], [63], [64], [65], [66], [67].", "The 355 nm beam is used to excite the $\\tilde{A}^1A_2-\\tilde{X}^1A_14_0^1$ vibronic manifold of $\\mathrm {CH_2O}$ similar to [11], [8], [60], [62], [63], [64], [68], [69], [70], [67].", "Using mirrors ($\\mathrm {M_1-M_3}$ ) and cylindrical lenses ($\\mathrm {L_1-L_3}$ ) shown in Fig.", "REF , the 282.94 nm and 355 nm beams are converted to coincident laser sheets with the thickness and height of about $250~\\mathrm {\\mu m}$ and 40 mm, respectively.", "Two PIMAX ICCD cameras (see, $\\mathrm {C_1}$ and $\\mathrm {C_2}$ in Fig.", "REF ), each equipped with a Nikkor UV lens and a bandpass filter with center wavelength of $320\\pm 20$ nm (for OH) and Coastal Optics UV lens and a Schott GG 395 longpass filter (for $\\mathrm {CH_2O}$ ), are used to collect the PLIF signals.", "The projected spatial resolution of both cameras is $89~\\mathrm {\\mu m}$ per pixel.", "The knife edge technique [71], [72] was used to obtain the line spread functions of both cameras.", "It was calculated that the optical system effective resolution (the full width at half maximum of the line-spread function) is $263~\\mathrm {\\mu m}$ .", "Two photodiodes, shown in Fig.", "REF , are utilized to collect pulse-to-pulse variations of the 282.94 nm and 355 nm beams energies, which are utilized for reduction of the PLIF data discussed in the next section.", "Further details regarding the utilized PLIF system can be found in [11], [8].", "The SPIV system contains a dual-cavity, double-pulse, Nd:YAG laser (BSL Twins CFR PIV200) as well as $\\mathrm {C_3}$ and $\\mathrm {C_4}$ sCMOS cameras (LaVision Imager sCMOS), which are equipped with Scheimpflug adapters and Tokina lenses, see Fig.", "REF .", "The laser generates pairs of 532 nm beams with a 4–33 $\\mu $ s separation time (depending on the tested mean bulk flow velocity).", "$\\mathrm {M_1-M_3}$ mirrors as well as $\\mathrm {L_1-L_3}$ cylindrical lenses (see Fig.", "REF ) are utilized to convert the PIV laser beams to laser sheets that are coincident with those of the PLIF lasers.", "The field of view and projected spatial resolution of the SPIV cameras are 84 mm$\\times $ 70 mm and 29.3 $\\mu $ m/pixel, respectively.", "Olive oil is atomized using a TSI Six-Jet atomizer and is fed into the burner.", "PLIF and SPIV images are registered and mapped to one physical coordinate system using a three-dimensional LaVision Type 20 target plate.", "LaVision DaVis 8.4 software is used to calculate the velocity vectors.", "The details of the PIV vector calculation is identical to that described in [11]." ], [ "Coordinate system", "A Cartesian coordinate system, presented in Fig.", "REF , is used in the present study.", "The origin of the coordinate system is at the intersection of the burner centerline and the nozzle exit plane.", "The $x-$ and $y-$ axes are at the imaging plane and lie inside the laser sheets.", "The $z-$ axis is normal to the $x-y$ plane.", "The PLIF and SPIV field of views, shown by a green dashed window in Fig.", "REF , are coincident.", "In the figure, the flow characterization region (FCR) highlights the location at which the non-reacting turbulent flow characteristics are averaged, which is similar to [11].", "Figure: The coordinate system, field of view, and flow characterization region (FCR).", "The dimensions are in mm." ], [ "Tested conditions", "In total, 68 experimental conditions are tested, with the corresponding details tabulated in Table REF .", "Mixture of methane (grade 2 with 99% chemical purity) and hydrogen (grade 5 with 99.999% chemical purity) is used as the fuel in this study.", "The fuel-air equivalence ratio ($\\phi $ ) for all tested conditions is 0.7.", "All tested conditions labels are provided in the first column of the table.", "In this column, U$\\#$ and H$\\#$ indicate the tested mean bulk velocity ($U$ ) and the hydrogen-enrichment percentage ($\\mathrm {H_2}$ %).", "Four mean bulk flow velocities of 5, 15, 25, and 35 m/s are examined, which are provided in the second column of the table.", "$\\mathrm {H_2}\\%$ varied from 0 to a maximum of 70%, with the corresponding values provided in the third column of Table REF .", "In the label of each tested condition, T$\\#$ indicates the number of utilized perforated plates.", "Conditions with zero (T0), one (T1), and two (T2) perforated plates are highlighted by the blue, green, and red colors in the table, respectively.", "Cantera simulations [73] with the GRI-Mech 3.0 mechanism are used to estimate the unstretched laminar flame speeds ($S_\\mathrm {L,0}$ ).", "Laminar flame thickness is calculated using $\\delta _\\mathrm {L}=(\\lambda /c_\\mathrm {p})/(\\rho _0S_\\mathrm {L,0})$ , with $\\lambda $ and $c_\\mathrm {p}$ being the gas thermal conductivity and specific heat (both estimated at 1500 K), and $\\rho _0$ being the fuel and air mixture density (estimated at 300 K), similar to [11], [3], [9], [22].", "Root Mean Square (RMS) of the velocity fluctuations along the $y$ ($u^{\\prime }$ ), $x$ ($v^{\\prime }$ ), and $z$ ($w^{\\prime }$ ) directions as well as the integral ($\\Lambda $ ), Taylor ($l_\\mathrm {T}=\\Lambda (u^\\prime \\Lambda / \\nu )^{-1/2}$ ), and Kolmogorov ($\\eta _\\mathrm {K}=(\\nu ^3/\\epsilon )^{1/4}=\\Lambda (u^\\prime \\Lambda / \\nu )^{-3/4}$ ) length scales are averaged within the red dotted-dashed window of Fig.", "REF , and the results are presented in the table.", "In the equations used for calculation of the Taylor and Kolmogorov length scales, $\\nu $ is the fuel and air mixture kinematic viscosity (estimated at 300 K) and $\\epsilon =u^{\\prime 3}/\\Lambda $ [3], [74].", "The integral length scale, $\\Lambda $ , is estimated using the formulations provided in [74] and the procedures discussed in [11], [8].", "Similar to [11], conditions pertaining to the first turbulence generating mechanism feature the Kelvin-Helmholtz instability.", "This instability is generated by the jet shear layer, can influence the values of the RMS velocity and integral length scale, and does not correspond to the background turbulent flow.", "Thus, using the Proper Orthogonal Decomposition technique [75] and similar to [11], the Kelvin-Helmholtz modes are identified and excluded from the non-reacting flow data pertaining to zero perforated plates, following the procedures discussed in [11].", "The Reynolds ($Re_\\mathrm {T}$ ) and Karlovitz ($Ka$ ) numbers are estimated using $Re_\\mathrm {T} =u^\\prime \\Lambda / \\nu $ and $Ka = (u^{\\prime }/S_\\mathrm {L,0})^{3/2}(\\Lambda /\\delta _\\mathrm {L})^{-1/2}$ , with the values provided in Table REF .", "The effective Lewis number ($L_\\mathrm {eff}$ ) of the fuel and air mixture was estimated using the formulations provided in Mohammadnejad et al.", "[8], with the corresponding values presented in the last column of the table.", "All test conditions are overlaid on the Borghi–Peters [22], [76] diagram, as shown in Fig.", "REF .", "The results in the figure suggest that flames with $U=5$ m/s and no perforated plate pertain to the wrinkled flames regime.", "Flames with $U=5$ m/s and one perforated plate as well as Flame U15H70T1 pertain to the corrugated flames regime.", "The rest of the tested conditions are positioned in the thin reaction zones regime.", "In Fig.", "REF , the newly-suggested border of $u^{\\prime }\\Lambda /(S_\\mathrm {L,0}\\delta _\\mathrm {L}) = 180$ [9], [3], which separates the flames with thin and broadened preheat zones, is also presented.", "This border suggests that the conditions pertaining to $U=15, 25$ , and 35 m/s with 2 perforated plates as well as Flame U35H00T0 are expected to feature a broadened preheat zone.", "Flames U25H00T0, U25H40T0, and U35H40T0 are positioned near the border of $u^{\\prime }\\Lambda /(S_\\mathrm {L,0}\\delta _\\mathrm {L}) = 180$ .", "The rest of the tested conditions are expected to feature a relatively thin preheat zone.", "The predictions of the Borghi-Peters diagram related to thin and broadened preheat zones are discussed in detail later in the results section.", "Table: Tested experimental conditions.", "For all conditions, φ=0.7\\phi =0.7.", "UU, S L ,0S_\\mathrm {L,0}, u ' u^{\\prime }, v ' v^{\\prime }, and w ' w^{\\prime } are in m/s and δ L \\delta _\\mathrm {L}, η K \\eta _\\mathrm {K}, l T l_\\mathrm {T} and Λ\\Lambda are in mm.Figure: The tested experimental conditions overlaid on the Borghi–Peters diagram ." ], [ "Data reduction", "For each test condition, 500 pairs of OH and $\\mathrm {CH_2O}$ PLIF images were collected simultaneously.", "The PLIF data is used to study both the internal structure as well the burning rate of the tested flames.", "The processes related to reducing the raw PLIF images is discussed in this section.", "Figures REF (a) and (b) show two representative frames of the raw OH and $\\mathrm {CH_2O}$ PLIF images, respectively.", "These frames are related to condition U5H40T1.", "In order to correct the PLIF images for the effect of the background noise, separate 500 pairs of images were collected for each test condition using the PLIF cameras while the lasers were turned off.", "These images are averaged and subtracted from the corresponding OH and $\\mathrm {CH_2O}$ PLIF data.", "Then, the obtained images are normalized by the corresponding laser profiles.", "The profile of the 283 nm laser sheet is obtained from separate acetone PLIF experiments.", "For these, the ICCD camera $\\mathrm {C_1}$ (in Fig.", "REF ) was equipped with a 305 nm longpass filter.", "Then, 500 acetone PLIF images were collected, averaged, and used as the laser profile correction of the OH PLIF images.", "The 355 nm laser sheet profile, which is used to normalized the $\\mathrm {CH_2O}$ PLIF images, is acquired similarly except that the longpass filter was removed from the camera and Rayleigh scattering images were collected and used instead, which is similar to our previous study [11].", "As discussed in Section , two photodiodes were used to capture the laser pulse energy variations of both 283 and 355 nm beams.", "These are used to normalize both the OH and $\\mathrm {CH_2O}$ PLIF images (after laser profile correction) in order to remove the effect of the shot-to-shot variation of the lasers pulse energies.", "Figures REF (c) and (d) show the representative frames of OH and $\\mathrm {CH_2O}$ PLIF images corrected for the effects of background noise, laser profile, and shot-to-shot laser energy variations.", "These corrections are shown as process (1) in Fig.", "REF .", "After applying the corrections related to process (1), in order to decrease the remaining background noise, similar to [9], [12], [77], [78], [29], [11], [8], $7\\times 7$ and $11\\times 11$ $\\mathrm {pixels^2}$ median-based filters are applied to the OH and $\\mathrm {CH_2O}$ PLIF images, respectively.", "Similar results are obtained using the Wiener [79] and Gaussian [80], [81] filters.", "Care was taken to ensure the utilized filters do not influence the results.", "Finally, a global threshold is applied to the images.", "The application of the median based filters and the global threshold is denoted by process (2) in Fig.", "REF .", "The results in Fig.", "REF (e) and (f) demonstrate the final representative OH and $\\mathrm {CH_2O}$ PLIF images after applying both processes (1) and (2).", "Figure: Procedures for reducing OH (a, c, and e) as well as CH 2 O\\mathrm {CH_2O} (b, d, and f) PLIF images.", "(g) is the estimated heat release rate.", "(h) presents the preheat (cyan), reaction (red), and combustion products (yellow) zones.", "Process (1) corresponds to the background noise, laser profile, and laser pulse energy corrections.", "Process 2 pertains to applying the median-based filters as well as a global threshold.", "Process 3 corresponds to estimation of the heat release rate from the processed OH and CH 2 O\\mathrm {CH_2O} PLIF images.", "The results pertain to U5H40T1 condition.After generating the reduced OH and $\\mathrm {CH_2O}$ PLIF images, the local, normalized, and relative heat release rate (HRR) was obtained, using the model presented in [8].", "This model suggests that the heat release rate is proportional to $(\\mathrm {CH_2O}_\\mathrm {PLIF})^\\alpha \\times (\\mathrm {OH_{PLIF}})^\\beta $ , with $\\alpha $ and $\\beta $ depending on the tested fuel-air equivalence ratio and the hydrogen-enrichment percentage [8].", "In the study of [8], $\\alpha $ and $\\beta $ are reported for $\\mathrm {H_2\\%}$ upto 50%.", "Since, in the present study, tests at $\\mathrm {H_2\\%} =$ 60 and 70% are also performed, and the corresponding values of $\\alpha $ and $\\beta $ are not available in [8], these parameters are calculated following the procedure discussed in [8] and the results are tabulated in Table REF .", "The results correspond to $\\phi =0.7$ .", "Please note that using the conventional method for calculation of HRR (i.e.", "$\\alpha = \\beta = 1$ [65], [9], [82], [83], [12], [61], [77], [60], [84]) leads to similar results and the non unity values of $\\alpha $ and $\\beta $ do not influence the conclusions of the present study.", "Figure REF (g) presents the HRR ($I_\\mathrm {HRR}$ ) pertaining to the raw data shown in Figs.", "REF (a and b).", "Table: Values of α\\alpha and β\\beta used for calculation of heat release rate.", "Results pertain to φ=0.7\\phi =0.7.The PLIF data was used to estimate the preheat and reaction zones thicknesses.", "Similar to [9], [83], [82], [11], pixels that feature $I_\\mathrm {HRR}$ values larger than 50% of the global maximum highlight the reaction zone.", "The leading edge of the preheat zone is where $\\mathrm {CH_2O}$ PLIF signal equals 35% of the maximum $\\mathrm {CH_2O}$ PLIF signal [83], [77].", "The trailing edge of the preheat zone is the leading edge of the reaction zone.", "The preheat and reaction zones for the representative frames presented in Fig.", "REF are shown by the cyan and red colors in Fig.", "REF (h), respectively.", "After removing the regions pertaining to the reaction zones, the combustion products are defined as the regions where $\\mathrm {OH}$ PLIF signal is larger than 15% of the global maximum.", "The combustion products are shown by the yellow color in Fig.", "REF (h).", "The above procedure for obtaining the preheat, reaction, and combustion products zones are identical to that used in [11].", "As discussed in [11], five parameters may potentially lead to the uncertainty in estimation of the preheat and reaction zone thicknesses.", "These parameters are the filtering process, three-dimensional orientation of the flames, imaging resolution, optical blur, and laser sheet thickness.", "Influence of filtering was discussed earlier.", "Effect of three-dimensional orientation of the flame is studied in past investigations [85], [86], [24], [87], [88], [80], [89], [90], [91].", "For example, study of Rosell et al.", "[80] shows that for a flame with $Ka=60$ , the estimated preheat and reaction zone thicknesses can be influenced by three-dimensionality of the flames up to 8% and 11%, respectively.", "Since this Karlovitz number is close to the maximum tested $Ka$ of the present study (which is $Ka=76$ ), the effect of three-dimensionality on the reported results is not expected to be significant.", "The effects of imaging resolution, optical blur, and laser sheet thickness on the reported preheat and reaction zone thicknesses have been investigated and reported in our previous study [11] using the results, guidelines, and procedures presented in [42], [92].", "As discussed in [11], such effects are also not expected to influence the preheat and reaction zone thicknesses reported in the present study." ], [ "Results", "The results are grouped into two subsections.", "In the first subsection, the preheat and reaction zone thicknesses of the hydrogen-enriched methane-air turbulent premixed flames are presented and discussed.", "Then, in the second subsection, the turbulent premixed flames burning rate is investigated." ], [ "Preheat and reaction zones thicknesses", "Representative planar laser-induced fluorescence images pertaining to the test conditions of U5H00T0 ($Ka = 0.3$ ), U5H40T0 ($Ka = 0.1$ ), U35H00T2 ($Ka = 76.0$ ), and U35H40T2 ($Ka = 40.4$ ) are presented in the first, second, third, and fourth rows of Fig.", "REF , respectively.", "The results in the first and second columns are representative hydroxyl and formaldehyde PLIF data.", "The third column is the heat release rate estimated utilizing the procedure discussed in Section .", "The corresponding preheat, reaction, and combustion products zones were obtained using the procedure discussed in Section , and these zones are shown by the cyan, red, and yellow colors in the last column of Fig.", "REF , respectively.", "For both pure and hydrogen-enriched methane-air flames with $Ka = 0.3$ (U5H00T0) and 0.1 (U5H40T0), both preheat and reaction zones remain relatively thin, see the cyan and red regions in Figs.", "REF (d and h).", "However, for both pure and hydrogen-enriched flames, as the Karlovitz number increases to large values, the preheat and reaction zones broaden significantly.", "This observation is consistent with the results presented in [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21].", "Figure: Representative OH (first column) and CH 2 O\\mathrm {CH_2O} (second column) PLIF images as well as the HRR (third column).", "The preheat, reaction, and combustion products zones in the last column are shown by the cyan, red, and yellow colors, respectively.", "The first, second, third, and fourth rows pertain to test conditions of U5H00T0, U5H40T0, U35H00T2, and U35H40T2, respectively.Following the procedure elaborated in [11] and discussed in Section , the thicknesses of both preheat and reaction zones were estimated for each frame of the tested conditions.", "The averaged preheat ($\\delta _\\mathrm {P}$ ) and reaction ($\\delta _\\mathrm {F}$ ) zone thicknesses are presented in Figs.", "REF (a) and (b), respectively.", "The results in Figs.", "REF (a) and (b) normalized by the corresponding laminar flame preheat ($\\delta _\\mathrm {P,L}$ ) and reaction ($\\delta _\\mathrm {F,L}$ ) zone thicknesses for matching fuel-air equivalence ratio and hydrogen-enrichment percentages are also shown in Figs.", "REF (c) and (d), respectively.", "Values of $\\delta _\\mathrm {P,L}$ and $\\delta _\\mathrm {F,L}$ are tabulated in Table REF and correspond to test conditions with no turbulence generator and at the smallest tested mean bulk flow velocity (5 m/s).", "The condition associated with the largest tested turbulence intensity (that is U35H00T2) features the maximum uncertainty for estimation of the preheat and reaction zone thicknesses.", "In order to estimate the maximum uncertainty associated with $\\delta _\\mathrm {P}$ and $\\delta _\\mathrm {F}$ , the test condition of U35H00T2 was repeated three times, the maximum deviation from the mean value was estimated, and the corresponding error bars were overlaid on Figs.", "REF (a) and (b), respectively.", "The probability density functions of the test conditions with the largest instantaneous preheat and reaction zone thicknesses, which correspond to U35H30T2 and U35H10T2 test conditions, are overlaid by the hot color bar contour on Figs.", "REF (a) and (b), respectively.", "Figure: (a and b) are the preheat and reaction zone thicknesses.", "(c and d) are the results in (a and b) normalized by δ P ,L\\delta _\\mathrm {P,L} and δ F ,L\\delta _\\mathrm {F,L}.Table: Preheat and reaction zone thicknesses at the smallest tested mean bulk flow velocity and with no turbulence generator.Overlaid on Fig.", "REF (c) are the normalized preheat zone thickness pertaining to the studies of Skiba et al.", "[9], Wang et al.", "[21], and Mohammadnejad et al.", "[11], which are shown by the dark red dotted-dashed, blue dotted, and black dashed curves respectively.", "The studies of [9], [21], [11] correspond to pure methane-air flames.", "Unlike the present study that the fuel-air equivalence ratio is fixed and equals 0.7, results of [9] pertain to the fuel-air equivalence ratios of 0.65, 0.85, and 1.05, and those of [21] pertain to $\\phi =0.4$ , 0.7, and 1.0.", "Comparison of our results with those of Skiba et al.", "[9] and Wang et al.", "[21] at a similar fuel-air equivalence ratio (0.65 and 0.7) suggests that the values of $\\delta _\\mathrm {P}/\\delta _\\mathrm {P,L}$ nearly collapse.", "The results show that, for both pure and hydrogen-enriched methane-air flames, the preheat zone features broadening, and this becomes more pronounced with increasing the turbulence intensity.", "The broadening of the preheat zone for hydrogen-enriched methane-air turbulent premixed flames has been recently reported in the study of Zhang et al. [54].", "However, the tested conditions in [54] correspond to to a maximum turbulence intensity of about 12.5.", "The results of the present study suggest this broadening extends to $u^{\\prime }/S_\\mathrm {L,0} \\approx 42$ (for 10% hydrogen-enrichment).", "Driscoll et al.", "[3] and Skiba et al.", "[9] suggest the turbulent premixed methane-air flames with $u^{\\prime }\\Lambda /({S_\\mathrm {L,0} \\delta _\\mathrm {L}}) \\gtrsim 180$ may feature preheat zone broadening.", "Variation of the normalized preheat zone thickness versus $u^{\\prime }\\Lambda /({S_\\mathrm {L,0} \\delta _\\mathrm {L}})$ is presented in Fig.", "REF for all tested conditions.", "As can be seen, test conditions with $u^{\\prime }\\Lambda /({S_\\mathrm {L,0} \\delta _\\mathrm {L}}) \\gtrsim 180$ feature $\\delta _\\mathrm {P}/\\delta _\\mathrm {P,L} \\gtrsim 1.5$ , however, those with $u^{\\prime }\\Lambda /({S_\\mathrm {L,0} \\delta _\\mathrm {L}}) \\lesssim 180$ feature $\\delta _\\mathrm {P}/\\delta _\\mathrm {P,L} \\lesssim 1.5$ .", "For pure methane-air turbulent premixed flames, our results agree with those of [9], [3].", "The results presented in Fig.", "REF suggest that the border of $u^{\\prime }\\Lambda /({S_\\mathrm {L,0} \\delta _\\mathrm {L}}) \\approx 180$ can also be used for predicting the preheat zone broadening of hydrogen-enriched methane-air turbulent premixed flames.", "Figure: Variation of the normalized preheat zone thickness versus u ' Λ/(S L ,0δ L ){u^{\\prime }\\Lambda }/({S_\\mathrm {L,0} \\delta _\\mathrm {L}}) for all tested conditions.", "The dashed line of u ' Λ/(S L ,0δ L )=180u^{\\prime }\\Lambda /({S_\\mathrm {L,0} \\delta _\\mathrm {L}}) = 180 is the border proposed in and Driscoll et al.", ".The normalized reaction zone thickness reported in the studies of Skiba et al.", "[9], Zhou et al.", "[12], and Mohammadnejad et al.", "[11] are overlaid on Fig.", "REF (d), which are shown by the dark red dotted-dashed, blue dotted, and black dashed lines respectively.", "Results of the present study show that increasing the turbulence intensity increases the normalized reaction zone thickness as shown in Fig.", "REF (d).", "This increasing trend is similar for all tested hydrogen-enrichment percentages and agrees well with the results of both Zhou et al.", "[12] and Mohammadnejad et al. [11].", "Compared to the results of the present study and those of [11], [12], Skiba et al.", "[9] suggest that the reaction zone thickness remains nearly constant and does not vary by increasing the turbulence intensity.", "For pure methane-air turbulent premixed flames, Mohammadnejad et al.", "[11] experimentally showed that the reason for the broadening of the preheat zone is due to penetration of turbulent eddies into this zone.", "Also, Mohammadnejad et al.", "[11] showed that the probability density function of the turbulent eddy size is significantly influenced by the utilized turbulence generating mechanism.", "As a result, they [11] speculated the reason for the controversial observation reported for the reaction zone thickness in their study and in [12] compared to that in [9] is due to the difference between the utilized turbulence generating mechanisms.", "Nevertheless, the broadening of the reaction zone for hydrogen-enriched methane-air flames shown in Fig.", "REF (b and d) at relatively large turbulence intensities is reported in the present study for the first time to the best knowledge of the authors.", "In essence, the results presented in Fig.", "REF show that both the preheat and reaction zones of the pure and hydrogen-enriched methane-air turbulent premixed flames can feature broadening.", "For example, the preheat and reaction zone thicknesses can feature values that are respectively 6.3 and 4.9 times those of the laminar flame counterparts as shown by the probability density functions in Figs.", "REF (a and b)." ], [ "Turbulent premixed flame burning rate", "The local consumption speed normalized by the unstretched laminar flame speed is estimated from Eq.", "(REF ) and the results are presented in Fig.", "REF (a) for all tested conditions.", "Please note that in Eq.", "(REF ), the stretch factor ($I_0$ ) is assumed to be unity, similar to [40], as the effective Lewis number is close to unity for the tested conditions (please see the last column of Table REF ).", "The maximum uncertainty in estimation of the local consumption speed is related to test condition of U35H00T2, and is shown by the error bar in Fig.", "REF (a).", "The procedure for estimation of the error bar is similar to that discussed in Section 5.1.", "The results in Fig.", "REF (a) are also color coded based on the integral length scale of the test conditions and presented in Fig.", "REF (b).", "Overlaid on both Figs.", "REF (a) and (b) are the results extracted from Wabel et al.", "[40] and Wang et al.", "[21], which are shown by the red dotted-dashed and blue dotted lines, respectively.", "The integral length scale in the study of Wabel et al.", "[40] varies between 6.1 and 41 mm and that in Wang et al.", "[21] is 2.9 mm.", "As can be seen, variation of $S_\\mathrm {T,LC}/S_\\mathrm {L,0}$ is significantly dependent on the integral length scale and follows two trends.", "For $\\Lambda \\lesssim 4$ mm, our results suggest that the normalized local consumption speed plateaus at about 2 following [21]; and, for $\\Lambda \\gtrsim 4$ mm, $S_\\mathrm {T,LC}/S_\\mathrm {L,0}$ plateaus at about 5 following [9].", "Plateau of the normalized local consumption speed with increase of the turbulence intensity (evident in Fig.", "REF ) has been a matter of discussions in the literature.", "Several investigations [40], [21], [45], [3] suggested the reason for this plateau is saturation of the flame surface area with increase of $u^{\\prime }/S_\\mathrm {L,0}$ at large values of this parameter.", "However, the results presented in Fig.", "REF as well as those in [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] suggest that increasing the turbulence intensity broadens both the preheat and reaction zones, and as a result, the flamelet assumption may not hold for such conditions.", "This means that Eq.", "(REF ), which is developed based on the flamelet assumption, may lead to inaccurate estimation of the local consumption speed.", "Instead, here, we propose a new formulation that does not rely on the flamelet assumption, which is discussed in the following.", "Figure: (a) Mean normalized local consumption speed (S T , LC /S L ,0S_\\mathrm {T,LC}/S_\\mathrm {L,0}) versus u ' /S L ,0u^{\\prime }/S_\\mathrm {L,0}.", "(b) presents the results in (a) color coded based on the corresponding tested integral length scale.In order to develop a formulation that does not take into account the flamelet assumption, first, similar to [40], [21], [93], [94], the OH PLIF images are binarized and the mean progress variable field ($\\overline{c}_\\mathrm {OH}$ ) is obtained by averaging the binarized images.", "Please note this binarization is only performed to calculate the mean progress variable fields, is conducted for flames that are reported to be both relatively thin [40] and thick [21], and does not necessarily imply a presumed structure for the reaction zone.", "Then, a curvilinear coordinate system ($\\xi -\\eta $ ), with $\\xi $ and $\\eta $ being respectively tangent and normal to $\\overline{c}_\\mathrm {OH}$ contours are constructed.", "The time-average heat release rate is integrated along $\\eta -$ axis, which is locally normal to the mean progress variable contours of $\\overline{c}_\\mathrm {OH} = 0.5$ .", "This integral depends on $\\xi $ and is given by $B_\\mathrm {T,\\xi }(\\xi )={\\int _{\\eta _\\mathrm {min}}^{\\eta _\\mathrm {max}}\\overline{HRR}_\\mathrm {T}(\\eta ,\\xi ) \\mathrm {d}\\eta },$ where $\\overline{HRR}_\\mathrm {T}(\\eta ,\\xi )$ is the mean heat release rate of turbulent flames estimated locally.", "$\\eta _\\mathrm {min}$ and $\\eta _\\mathrm {max}$ are the extents of the $\\eta -$ axis in the domain of investigation.", "For a relatively large domain of investigation, the left hand side of Eq.", "(REF ) does not change by extending the bounds of integration.", "The turbulent flame burning rate ($B_\\mathrm {T}$ ) is defined as the spatially averaged (along $\\xi $ ) value of $B_\\mathrm {T,\\xi }(\\xi )$ and is given by $B_\\mathrm {T}=\\frac{1}{\\underbrace{\\int _{\\xi _\\mathrm {min}}^{\\xi _\\mathrm {max}}\\mathrm {d}\\xi }_{L_{\\xi ,\\mathrm {T}}}}{\\int _{\\xi _\\mathrm {min}}^{\\xi _\\mathrm {max}}\\underbrace{\\int _{\\eta _\\mathrm {min}}^{\\eta _\\mathrm {max}}\\overline{HRR}_\\mathrm {T}(\\eta ,\\xi ) \\mathrm {d}\\eta }_{B_\\mathrm {T,\\xi }(\\xi )} \\mathrm {d}\\xi }.$ In Eq.", "(REF ), $\\xi _\\mathrm {min}$ and $\\xi _\\mathrm {max}$ correspond to the boundaries of $\\xi -$ axis in the domain of investigation.", "In the denominator of Eq.", "(REF ), $L_{\\xi ,\\mathrm {T}}$ is the length of $\\overline{c}_\\mathrm {OH} = 0.5$ contour and depends on the tested condition.", "Similarly, the laminar flame burning rate can be estimated from $B_\\mathrm {L}=\\frac{1}{L_{\\xi ,\\mathrm {L}}} {\\int _{\\xi _\\mathrm {min}}^{\\xi _\\mathrm {max}}\\int _{\\eta _\\mathrm {min}}^{\\eta _\\mathrm {max}}\\overline{HRR}_\\mathrm {L}(\\eta ,\\xi ) \\mathrm {d}\\eta \\mathrm {d}\\xi },$ where $\\overline{HRR}_\\mathrm {L}(\\eta ,\\xi )$ and $L_{\\xi ,\\mathrm {L}}$ are the mean laminar flame heat release rate estimated locally and the laminar flame length estimated using the length of $\\overline{c}_\\mathrm {OH} = 0.5$ .", "Dividing Eq.", "(REF ) by Eq.", "(REF ) leads to $\\frac{B_\\mathrm {T}}{B_\\mathrm {L}}=\\frac{L_{\\xi ,\\mathrm {L}}}{L_{\\xi ,\\mathrm {T}}}\\frac{\\int _{\\xi _\\mathrm {min}}^{\\xi _\\mathrm {max}}\\int _{\\eta _\\mathrm {min}}^{\\eta _\\mathrm {max}}\\overline{HRR}_\\mathrm {T}(\\eta ,\\xi ) \\mathrm {d}\\eta \\mathrm {d}\\xi }{\\int _{\\xi _\\mathrm {min}}^{\\xi _\\mathrm {max}}\\int _{\\eta _\\mathrm {min}}^{\\eta _\\mathrm {max}}\\overline{HRR}_\\mathrm {L}(\\eta ,\\xi ) \\mathrm {d}\\eta \\mathrm {d}\\xi }.$ It is assumed that the heat release rate is proportional to $I_\\mathrm {HRR}$ estimated using the $\\mathrm {OH}$ and $\\mathrm {CH_2O}$ PLIF signals following the procedures provided in Section .", "Using this assumption, Eq.", "(REF ) can be simplified to $\\frac{B_\\mathrm {T}}{B_\\mathrm {L}}=\\frac{L_{\\xi ,\\mathrm {L}}}{L_{\\xi ,\\mathrm {T}}}\\frac{\\int _{\\xi _\\mathrm {min}}^{\\xi _\\mathrm {max}}\\int _{\\eta _\\mathrm {min}}^{\\eta _\\mathrm {max}}\\overline{I}_\\mathrm {HRR,T}(\\eta ,\\xi ) \\mathrm {d}\\eta \\mathrm {d}\\xi }{\\int _{\\xi _\\mathrm {min}}^{\\xi _\\mathrm {max}}\\int _{\\eta _\\mathrm {min}}^{\\eta _\\mathrm {max}}\\overline{I}_\\mathrm {HRR,L}(\\eta ,\\xi ) \\mathrm {d}\\eta \\mathrm {d}\\xi }$ where $\\overline{I}_\\mathrm {HRR,T}$ and $\\overline{I}_\\mathrm {HRR,L}$ are the mean $I_\\mathrm {HRR}$ corresponding to turbulent and laminar flames, respectively.", "Equation (REF ) allows for estimation of the turbulent flame burning rate normalized by that for the laminar counterpart and is independent of the internal flame structure.", "In fact, in the following, it is shown that for flames whose internal structure follows the flamelet assumption, Eq.", "(REF ) leads to values of burning rate that are similar to those obtained utilizing the formulation proposed by Driscoll [2], i.e.", "Eq.", "(REF ).", "Variation of $B_\\mathrm {T}/B_\\mathrm {L}$ , estimated from Eq.", "(REF ), versus $u^{\\prime }/S_\\mathrm {L,0}$ is presented in Fig.", "REF (a).", "The maximum uncertainty associated with calculation of $B_\\mathrm {T}/B_\\mathrm {L}$ pertains to test condition of U35H00T2, which is shown by the error bar in the figure.", "Among the formulations proposed in the literature, see for example the review paper by Driscoll [2], derivation of the global consumption speed formulation does not necessarily depend on the flamelet assumption, and as a result, $S_\\mathrm {T,GC}/S_\\mathrm {L,0}$ can be compared to $B_\\mathrm {T}/B_\\mathrm {L}$ .", "Since the field of view of the present investigation is relatively small, the normalized global consumption speed values cannot be estimated for the present investigation; however, $S_\\mathrm {T,GC}/S_\\mathrm {L,0}$ associated with the studies of [40], [21] are overlaid on Fig.", "REF by the red dotted-dashed and blue dotted curves.", "Two trends can be observed for the variation of the normalized burning rate.", "In order to highlight this, the data presented in Fig.", "REF (a) is color coded based on the tested integral length scale and presented in Fig.", "REF (b).", "As evident in the figure, for $\\Lambda \\lesssim 4$ mm, the values of $B_\\mathrm {T}/B_\\mathrm {L}$ are close to the normalized global consumption speed pertaining to the study of Wang et al.", "[21]; however, for $\\Lambda \\gtrsim 4$ mm, the results are closer to those of Wabel et al. [40].", "Comparison of the results presented in Fig.", "REF with those in Fig.", "REF suggests that the values of $B_\\mathrm {T}/B_\\mathrm {L}$ and $S_\\mathrm {T,GC}/S_\\mathrm {L,0}$ are similar, and they are both significantly larger than those of $S_\\mathrm {T,LC}/S_\\mathrm {L,0}$ .", "Figure: (a) Variation of the normalized burning rate (B T /B L B_\\mathrm {T}/B_\\mathrm {L}) versus u ' /S L ,0u^{\\prime }/S_\\mathrm {L,0}.", "(b) is the results in (a) color-coded based on the tested integral length scale.In order to study the reason for the disparity between the values of the normalized burning rate and the normalized local consumption speed, the following parameter is defined.", "$J=\\frac{B_\\mathrm {T}/B_\\mathrm {L}}{S_\\mathrm {T,LC}/S_\\mathrm {L,0}}.$ Variation of $J$ versus $u^{\\prime }/S_\\mathrm {L,0}$ for the present study is shown in Fig.", "REF by the open black circular data symbol.", "As can be seen, the values of $J$ are larger than unity, and this parameter increases with increasing $u^{\\prime }/S_\\mathrm {L,0}$ .", "Variation of the normalized global consumption speed divided by the normalized local consumption speed (i.e.", "$S_\\mathrm {T,GC}/S_\\mathrm {T,LC}$ ) is extracted from the studies of Wabel et al.", "[40] and Wang et al.", "[21], and their results are presented in Fig.", "REF by the solid dark red and blue triangular data symbols, respectively.", "Agreeing, with our results, those of past investigations [40], [21] also suggest that the local consumption speed underpredicts the turbulent premixed flames burning velocity.", "Figure: The normalized burning rate predicted by Eq.", "() (or the normalized global consumption speed) divided by the normalized local consumption speed versus turbulence intensity.The potential underlying reason for the underprediction of the burning velocity by the local consumption speed has been a matter of discussion over the past decades, see for example [3].", "Gülder [46] and Nivarti et al.", "[45] suggested that the reason for the observed difference between the local and global consumption speed values can be explained by taking into account the enhanced diffusivity of the reactants due to penetration of small scale eddies into the flame region.", "Using the mathematical formulations proposed by Nivarti et al [45] and Gülder [46] as well as the non-dimensional numbers related to test conditions of the present study ($Ka$ , $u^{\\prime }/S_\\mathrm {L,0}$ , and $Re_\\lambda = u^{\\prime }l_\\mathrm {T}/\\nu $ ), the ratio of the global to local consumption speed was estimated and the results are presented in the Fig.", "REF by the green dot-and-circle as well as pink double circle data symbols, respectively.", "As can be seen, the ratio of the burning rate to local consumption speed in this study ($J$ , the black circular data points) is close to that observed in the literature [40], [21] for the ratio of the global to local consumption speeds ($S_\\mathrm {T,GC}/S_\\mathrm {T,LC}$ ).", "Also, the results estimated based on the mathematical formulation proposed in Nivarti et al.", "[45] for $S_\\mathrm {T,GC}/S_\\mathrm {T,LC}$ agrees well with $J$ obtained from the present study.", "However, values of the global consumption speed divided by the local consumption speed obtained based on the Gülder's formulation [46] deviates from those of the present study and other experimental results [40], [21] at large turbulence intensities.", "The reason for the deviation of the results of the present study and the prediction from the formulation provided by Gülder [46] is speculated to be possibly linked to the assumptions made for derivation of the formulation in [46].", "Specifically, Gülder [46] assumed that the enhanced diffusivity occurs only at the Taylor length scale, while Nivarti et al.", "[45] considered enhancement of diffusivity at all length scales smaller than the laminar flame thickness.", "The underlying reason for the increasing trend pertaining to variations of $J$ and $S_\\mathrm {T,GC}/S_\\mathrm {T,LC}$ with $u^{\\prime }/S_\\mathrm {L,0}$ is hypothesized to be linked to the internal structure of the tested flames.", "In order to investigate this hypothesis, variations of $J$ verses the preheat and reaction zone thicknesses are shown in Figs.", "REF (a) and (b), respectively.", "As can be seen, $\\delta _\\mathrm {P}/\\delta _\\mathrm {P,L}$ and $\\delta _\\mathrm {F}/\\delta _\\mathrm {F,L}$ are positively correlated with $J$ .", "Such correlation is even more pronounced for the reaction zone thickness and follows a linear trend.", "Thus, a line was fit to the data in Fig.", "REF (b) using the least-square technique.", "The equation of the line is give by $J=1.9(\\delta _\\mathrm {F}/\\delta _\\mathrm {F,L})-0.9.$ Equation (REF ) suggests that, at the limit of $\\delta _\\mathrm {F} = \\delta _\\mathrm {F,L}$ , where the flamelet assumption holds, the normalized burning rate estimated from Eq.", "(REF ) equals the normalized consumption speed estimated from Eq.", "(REF ).", "However, increasing the reaction zone thickness leads to larger values of the normalized burning rate compared to the normalized local consumption speed.", "This means that thickening of the turbulent premixed flames is linked to the deviation of the normalized burning rate (and the normalized global consumption speed) from the corresponding normalized local consumption speed values.", "In fact, the reason for this deviation being linked to the internal flame structure was speculated by Wabel et al.", "[40] earlier.", "However, compared to Wabel et al.", "[40] who suggested the reason for the deviation between $S_\\mathrm {T,GC}$ and $S_\\mathrm {T,LC}$ may be due to the broadening of the preheat zone, our results show such deviation is linked to broadening of both the preheat and reaction zones.", "It is experimentally shown that [11], [95], with increasing the turbulence intensity, the eddies can penetrate into the preheat zone.", "This is expected to increase the gas turbulent diffusivity, which increases the burning rate [50], [3], [45], [46], and as a result, the larger than unity values of $J$ presented in Fig.", "REF (a).", "It is speculated that, similar to the preheat zone, the turbulent eddies may also penetrate into the reaction zone, leading to increase of $\\delta _\\mathrm {F}$ (as evident in Fig.", "REF (b) as well as the results presented in [11], [21], [12]), increasing the turbulent diffusivity, and as a result increasing the flame burning rate.", "However, the above speculation (that is penetration of turbulent eddies into the reaction zone) remains to be investigated experimentally.", "It is proposed that Eqs.", "(REF ) and (REF ) can be combined with Eq.", "(REF ) in order to modify the formulation of the local consumption speed suggested by Driscoll et al.", "[2] accounting for the reaction zone thickening.", "This equation is given by $\\frac{B_\\mathrm {T}}{B_\\mathrm {L}}=\\overbrace{[1.9(\\delta _\\mathrm {F}/\\delta _\\mathrm {F,L})-0.9]}^{J}I_0\\frac{\\int _{-\\infty }^{\\infty } \\int _{-\\infty }^{\\infty }\\Sigma (\\eta ,\\xi ) d\\eta d\\xi }{L_\\mathrm {\\xi }} .$ The values of the normalized burning rate estimated from Eq.", "(REF ) along with the prediction of Eq.", "(REF ) are presented by circular and square data symbols in Fig.", "REF for all test conditions.", "The results in this figure suggest that the local consumption speed corrected for the effect of flame thickening (Eq.", "(REF )) follows the values of the burning rate estimated from Eq.", "(REF ).", "This means that the reason for the deviation between the global (here burning rate) and local consumption speeds is the thickening of the reaction zone.", "This is not shown in the past investigations to our best of knowledge and is demonstrated here.", "Figure: The ratio of the normalized burning rate and the normalized local consumption speed versus (a) the preheat and (b) the reaction zone thicknesses normalized by those of the corresponding laminar flame counterparts.", "Overlaid on (b) is the linear fit to the data.Figure: Comparison of the burning rate values estimated from Eqs.", "() and ().Although the above argument suggests the values of normalized burning rate can be reconciled using the reaction zone thickness and the integral of the flame surface density, the reason why the variation of $B_\\mathrm {T}/B_\\mathrm {L}$ versus $u^{\\prime }/S_\\mathrm {L,0}$ bends toward the horizontal axis with increasing the turbulence intensity, see Fig.", "REF , is unclear in the literature, has been a matter of discussions [3], and is investigated here.", "This behavior is referred to as the bending behavior and is reported in past investigations, see for example [40], [21], [49], [52], [53], [26].", "Here, first, the bending behavior is characterized by estimating the difference between the values of the normalized burning rate and the prediction of the Damköhler's first hypothesis [50] (shown by the dashed line in Fig.", "REF ).", "This difference is referred to as $D$ and is estimated by $D=\\left(1+\\frac{u^{\\prime }}{S_\\mathrm {L,0}}\\right)-\\frac{B_\\mathrm {T}}{B_\\mathrm {L}}.$ For example, at $u^{\\prime }/S_\\mathrm {L,0} = 19.4$ and $\\mathrm {H_2}\\% = 50$ %, $D$ is about 4.6, which is shown by the double-sided arrow in Fig.", "REF (a).", "$D$ was estimated for all tested conditions in Table REF , and the results are presented in Fig.", "REF (a).", "Also overlaid on the figure is the difference between the values of the normalized global consumption speed pertaining to the studies of [40], [21] from the dashed line in Fig.", "REF (a).", "The estimated values of $D$ are also color coded based on the integral length scale of the tested conditions and shown in Fig.", "REF (b).", "As can be seen, $D$ is nearly zero for test conditions corresponding to small turbulence intensities, but it increases with increasing $u^{\\prime }/S_\\mathrm {L,0}$ .", "Also, the results in Fig.", "REF (b) show that, at a given turbulence intensity, $D$ nearly decreases with increasing the integral length scale.", "Figure: The difference between the normalized burning rate estimated based on the prediction of Damköhler's first hypothesis and the experimental results of the present study.", "Also overlaid on the figures are the results from the studies of , .The underlying reason for the above observations is hypothesized to be linked to pronounced occurrence of local extinctions and incomplete combustion at relatively intense turbulence conditions.", "The hydroxyl radical is a post flame species and its concentration is expected to be positively related to completion of the combustion process [22], [96].", "Thus, it is expected that generation of OH radical should be inversely related to incomplete combustion as well as the occurrence of local extinction events.", "In fact, occurrence of local extinctions has been investigated using the OH PLIF signal in the past, see for example the studies of [97], [98], [99], [100], [101].", "In the present study, the number of pixels that feature relatively large values of OH PLIF signal (more than 15% of the maximum) were obtained and averaged for all acquired frames of a given test condition, which is referred to as $\\overline{N}_\\mathrm {OH}$ .", "In order to facilitate comparison between different test conditions, $\\overline{N}_\\mathrm {OH}$ was normalized by the number of pixels that are expected to feature relatively large time-averaged OH PLIF signal, which is referred to as $N_\\mathrm {\\overline{OH}}$ .", "Variations of $(\\overline{N}_\\mathrm {OH}/N_\\mathrm {\\overline{OH}})^{-1}$ versus $D$ as well as $u^{\\prime }/S_\\mathrm {L,0}$ are presented in Fig.", "REF (a), and (b), respectively.", "The maximum uncertainty for estimation of $(\\overline{N}_\\mathrm {OH}/N_\\mathrm {\\overline{OH}})^{-1}$ corresponds to the test condition of U35H00T2, is estimated using the procedure discussed in Section 5.1, and is shown by the error bar in Fig.", "REF (a).", "The results in the figure show that $(\\overline{N}_\\mathrm {OH}/N_\\mathrm {\\overline{OH}})^{-1}$ is highly correlated with $D$ .", "This correlation suggests that the occurrence of local extinctions and incomplete combustion are related to the observed bending behavior.", "Figure: Variations (N ¯ OH /N OH ¯ ) -1 (\\overline{N}_\\mathrm {OH}/N_\\mathrm {\\overline{OH}})^{-1} versus (a) DD and (b) u ' /S L ,0u^{\\prime }/S_\\mathrm {L,0}.", "The results in (c) are those in (b) color-coded based on the corresponding tested integral length scale values.The results presented in Fig.", "REF (b) suggest that increasing the turbulence intensity increases $(\\overline{N}_\\mathrm {OH}/N_\\mathrm {\\overline{OH}})^{-1}$ .", "As this parameter increases, the possibility of local extinction occurrence increases, the fuel can potentially escape from the flame region, decreasing the generated heat release rate, and as a result the pronounced deviation of the burning rate from the prediction of the Damköhler's first hypothesis shown in Fig.", "REF .", "It is of interest to develop a model that allows for predicting the effect of governing parameters on the occurrence of local extinctions, the bending behavior, and as a result the burning rate.", "To address these, first, the results presented in Fig.", "REF (b) are color-coded based on the corresponding tested integral length scale and presented in Fig.", "REF (c).", "As evident in the inset of the figure, $(\\overline{N}_\\mathrm {OH}/N_\\mathrm {\\overline{OH}})^{-1}$ does not change by changing the integral length scale at relatively small values of $u^{\\prime }/S_\\mathrm {L,0}$ .", "However, at relatively large values of the turbulence intensity, decreasing the integral length scale increases the values of $(\\overline{N}_\\mathrm {OH}/N_\\mathrm {\\overline{OH}})^{-1}$ (compare the values of this parameters at $u^{\\prime }/S_\\mathrm {L,0} \\approx 20$ and at different integral length scales).", "At relatively small turbulence intensities, the turbulent eddies cannot penetrate into the flame zone, local extinctions rarely occur, and the values of $(\\overline{N}_\\mathrm {OH}/N_\\mathrm {\\overline{OH}})^{-1}$ are relatively small and nearly independent of the tested integral length scales.", "However, at larger values of $u^{\\prime }/S_\\mathrm {L,0}$ , eddies feature relatively large rotational kinetic energy and can penetrate into the flame zone [11].", "At large values of $u^{\\prime }/S_\\mathrm {L,0}$ , decreasing the integral length scale facilitates penetration of relatively small scale eddies into the flame zone, increasing the occurrence of local extinctions, and as a result the larger values of $(\\overline{N}_\\mathrm {OH}/N_\\mathrm {\\overline{OH}})^{-1}$ and $D$ shown in Figs.", "REF and REF (b).", "In essence, the above argument suggests that both $(\\overline{N}_\\mathrm {OH}/N_\\mathrm {\\overline{OH}})^{-1}$ and $D$ should be positively (negatively) correlated with $u^{\\prime }/S_\\mathrm {L,0}$ ($\\Lambda $ ).", "To investigate this, variations of $(\\overline{N}_\\mathrm {OH}/N_\\mathrm {\\overline{OH}})^{-1}$ and $D$ versus $({u^{\\prime }}/{S_\\mathrm {L,0}})({\\Lambda }/{\\delta _\\mathrm {L}})^{-1}$ are presented in Figs.", "REF (a) and (b), respectively.", "The parameter $({u^{\\prime }}/{S_\\mathrm {L,0}})({\\Lambda }/{\\delta _\\mathrm {L}})^{-1}$ is similar to that used for calculation of the Karlovitz number, however, different exponents for ${u^{\\prime }}/{S_\\mathrm {L,0}}$ and ${\\Lambda }/{\\delta _\\mathrm {L}}$ are used here.", "This is because, when presented against $({u^{\\prime }}/{S_\\mathrm {L,0}})({\\Lambda }/{\\delta _\\mathrm {L}})^{-1}$ (compared to the Karlovitz number), both $(\\overline{N}_\\mathrm {OH}/N_\\mathrm {\\overline{OH}})^{-1}$ and $D$ variations collapse.", "Using the least-square technique, a second order polynomial curve was fit to the variation of $D$ versus $({u^{\\prime }}/{S_\\mathrm {L,0}})({\\Lambda }/{\\delta _\\mathrm {L}})^{-1}$ , and it is obtained that $D \\approx 3.1\\left((\\frac{u^{\\prime }}{S_\\mathrm {L,0}})(\\frac{\\Lambda }{\\delta _\\mathrm {L}})^{-1}\\right)^2+3.5\\left((\\frac{u^{\\prime }}{S_\\mathrm {L,0}})(\\frac{\\Lambda }{\\delta _\\mathrm {L}})^{-1}\\right).$ Combining Eqs.", "(REF ) and (REF ), it can be shown that the normalized burning rate is obtained from $\\frac{B_\\mathrm {T}}{B_\\mathrm {L}} \\approx \\underbrace{\\left(1+\\frac{u^{\\prime }}{S_\\mathrm {L,0}}\\right)}_{\\mathrm {Term ~I}}\\underbrace{-3.1\\left((\\frac{u^{\\prime }}{S_\\mathrm {L,0}})(\\frac{\\Lambda }{\\delta _\\mathrm {L}})^{-1}\\right)^2-3.5\\left((\\frac{u^{\\prime }}{S_\\mathrm {L,0}})(\\frac{\\Lambda }{\\delta _\\mathrm {L}})^{-1}\\right)}_{\\mathrm {Term~II}}.$ In Eq.", "(REF ), Term I pertains to the Damköhler's first hypothesis, and Term II highlights the negative contribution of bending to the normalized burning rate of turbulent premixed flames.", "To assess prediction of Eq.", "(REF ), first, values of the normalized burning rate color-coded based on the normalized integral length scale are presented in Fig.", "REF .", "Then, the values of $B_\\mathrm {T}/B_\\mathrm {L}$ obtained from Eq.", "(REF ) are overlaid on the figure for several values of $\\Lambda /\\delta _\\mathrm {L}$ by the dotted-dashed curves.", "The prediction of Damköhler [50] is also shown by the black dashed line.", "The results in Fig.", "REF suggest that the formulation proposed in Eq.", "(REF ) allows for prediction of the normalized burning rate.", "The results also show that, flames with larger $\\Lambda /\\delta _\\mathrm {L}$ feature less pronounced bending behavior; and, at a fixed value of $u^{\\prime }/S_\\mathrm {L,0}$ , increasing $\\Lambda /\\delta _\\mathrm {L}$ increases the normalized burning rate.", "In fact, at the limit of $\\Lambda /\\delta _\\mathrm {L}$ approaching infinity, Eq.", "(REF ) predicts that $B_\\mathrm {T}/B_\\mathrm {L}$ approaches the prediction of the Damköhler's first hypothesis shown by the dashed line in Fig.", "REF .", "Also, for a fixed value of $\\Lambda /\\delta _\\mathrm {L}$ , the variation of the normalized burning rate versus $u^{\\prime }/S_\\mathrm {L,0}$ features a parabolic behavior.", "For extremely large values of $u^{\\prime }/S_\\mathrm {L,0}$ , the effect of Term II in Eq.", "(REF ) becomes dominant, which is equivalent to pronounced occurrences of local extinctions, leading to decrease of $B_\\mathrm {T}/B_\\mathrm {L}$ .", "Figure: Variations of (a) (N ¯ OH /N OH ¯ ) -1 (\\overline{N}_\\mathrm {OH}/N_\\mathrm {\\overline{OH}})^{-1} and (b) DD with (u ' /S L ,0)(Λ/δ L ) -1 ({u^{\\prime }}/{S_\\mathrm {L,0}})({\\Lambda }/{\\delta _\\mathrm {L}})^{-1}.Figure: The normalized burning rate versus turbulence intensity.", "The dotted-dashed curves pertain to the prediction of Eq. ().", "The dashed line is the prediction of the Damköhler's first hypothesis .The bending behavior and decrease of the burning rate due to local extinctions are consistent with the results of past studies, see for example [26], [7], [102], [103], [104].", "The implication of Eq.", "(REF ) prediction and the results presented in Fig.", "REF is that the burning rate cannot increase indefinitely with increasing the turbulence intensity.", "However, at very large normalized integral length scales, the amount of bending is less pronounced.", "This conclusion is of relevance to DNS studies.", "Compared to experimental investigations, the DNS studies tend to utilize a relatively small domain of investigation and as a result a relatively small integral length scale, see for example studies of [32], [33], [44], [43].", "Our results suggest, increasing the size of the domain of investigation and the integral length scale could potentially allow for the DNS investigations to capture relatively larger values of the burning rate." ], [ "Conclusions", "The internal structure and burning rate of extremely turbulent hydrogen-enriched methane-air premixed flames were investigated using simultaneous planar laser-induced fluorescence of $\\mathrm {OH}$ and $\\mathrm {CH_2O}$ as well as separate stereoscopic particle image velocimetry techniques.", "In total, 68 test conditions were examined.", "Fuel-air equivalence ratio was set to 0.7 for all tested conditions, and hydrogen-enrichment percentage was varied from 0% to 70% with steps of 10%.", "Four mean bulk flow velocities of 5, 15, 25, and 35 m/s were examined.", "Three turbulence generating mechanisms corresponding to none, one, or two perforated plates were utilized.", "The turbulence intensity ($u^{\\prime }/S_\\mathrm {L,0}$ ) of the tested conditions varied from 0.3 to 44.6, leading to Reynolds and Karlovitz numbers ranging from 18 to 2729 and 0.1 to 76.0, respectively.", "Preheat and reaction zones of hydrogen-enriched methane-air turbulent premixed flames were estimated.", "It was shown that, at large turbulence intensities, both preheat and reaction zones feature broadening, and their thickness can increase up to 6.3 and 4.9 times those of the laminar flame counterparts, respectively.", "While broadening of the preheat zone at large turbulence intensities has been reported in the literature for methane-air flames, broadening of this zone for hydrogen-enriched extremely turbulent premixed flames is reported in this study for the first time to the best knowledge of the authors.", "Similar to extremely turbulent pure methane-air flames, our results show that turbulent premixed hydrogen-enriched methane-air flames also feature broadening of the reaction zone, which is shown here for the first time to our best of knowledge.", "Broadening of the reaction zone observed for both pure and hydrogen-enriched methane-air flames suggests that the flamelet assumption may not lead to an accurate estimation of the burning rate for these flames.", "A framework that does not utilize the flamelet assumption was developed here and was used to estimated the tested flames burning rate.", "Also, the values of the local consumption speed were estimated for all tested conditions.", "The results show that, at relatively small values of the turbulence intensity, the burning rate follows the local consumption speed.", "However, at larger turbulence intensities, the burning rate follows the global consumption speed reported in the literature.", "Similar to past investigations, our results show that increasing the turbulence intensity increases the difference between the estimated values of the burning rate and the local consumption speed.", "Such disparity has been a matter of discussion in the literature.", "Our results show that the ratio of the normalized burning rate and the normalized local consumption speed is positively correlated with both preheat and reaction zone thicknesses, with the latter correlation being more pronounced.", "This implies that the flamelet assumption used for calculation of the local consumption speed may have caused the reason for the deviation between the values of global and local consumption speeds reported in the literature.", "It is shown that, correcting the formulation of the local consumption speed by a non-dimensional factor that takes into account the reaction zone thickening, the burning rate of the tested flames can be reconciled.", "Similar to past investigations related to the global consumption speed, the results show that increasing the turbulence intensity does not linearly increase the burning rate, and this parameter features a bending behavior.", "First, the amount of bending in variation of the normalized burning rate versus the turbulence intensity was quantified.", "It was hypothesized and shown that the amount of bending is positively related to the occurrence of local extinctions.", "The results show that both the amount of bending and the occurrence of local extinctions are positively (negatively) related to the turbulence intensity (normalized integral length scale).", "Using this, a formulation that allows for quantifying the bending behavior was developed.", "The prediction of the formulation was compared with the experimentally estimated value of the burning rate, and it was shown that both agree well.", "The proposed formulation suggests that, at a given turbulence intensity, increasing the normalized integral length scale increases the normalized burning rate approaching a limiting value predicted by Damköhler's first hypothesis.", "At a fixed value of the integral length scale, increasing the turbulence intensity, first, increases the burning rate.", "However, the proposed formulation (in agreement with past experimental results) predicts that further increasing the turbulence intensity decreases the burning rate, which is due to pronounced occurrence of local extinctions." ], [ "Acknowledgments", "The authors would like to thank financial support from the Mitacs Accelerate program, the Gas Turbine Laboratory, and Fortis BC.", "Sina Kheirkhah and Patrizio Vena thank Professors Ömer L. Gülder (from the University of Toronto), Matthew Johnson (from the Carleton University), and Dr. Greg Smallwood (from NRC) for lending majority of the setup and the diagnostics." ] ]	2011.14042
[ [ "Time Series Change Point Detection with Self-Supervised Contrastive\n Predictive Coding" ], [ "Abstract Change Point Detection (CPD) methods identify the times associated with changes in the trends and properties of time series data in order to describe the underlying behaviour of the system.", "For instance, detecting the changes and anomalies associated with web service usage, application usage or human behaviour can provide valuable insights for downstream modelling tasks.", "We propose a novel approach for self-supervised Time Series Change Point detection method based onContrastivePredictive coding (TS-CP^2).", "TS-CP^2 is the first approach to employ a contrastive learning strategy for CPD by learning an embedded representation that separates pairs of embeddings of time adjacent intervals from pairs of interval embeddings separated across time.", "Through extensive experiments on three diverse, widely used time series datasets, we demonstrate that our method outperforms five state-of-the-art CPD methods, which include unsupervised and semi-supervisedapproaches.", "TS-CP^2 is shown to improve the performance of methods that use either handcrafted statistical or temporal features by 79.4% and deep learning-based methods by 17.0% with respect to the F1-score averaged across the three datasets." ], [ "Introduction", "The ubiquity of digital technologies along with the substantial processing power and storage capacity on offer means we currently have an unprecedented ability to access and analyse data.", "The scale and velocity in which data is being stored and shared, however, means that we often lack the resources to utilise traditional data curation processes.", "For instance, in supervised machine learning approaches, the data annotation process can be an expensive, unwieldy and inaccurate one.", "Consequently, this is why self-supervised and unsupervised learning methods are currently hot topics in the machine learning community where the goal is to maximise the value of raw data.", "Change point detection (CPD), an analytical method to identify the times associated with abrupt transitions of a series can be used to extract meaning from non-annotated data.", "Change points, whether they have been generated from video cameras, microphones, environmental sensors or mobile applications can provide a critical understanding of the underlying behaviour of the system being modelled.", "For instance, change points can represent alterations in the system state that might require human attention, such as a system fault or an upcoming emergency.", "Furthermore, CPD methods can be employed in related problems of temporal segmentation, event detection and temporal anomaly detection.", "Figure: Overview of presented change point detection approach based on predictive representation learning.CPD techniques have been applied to multivariate time series data in a broad range of research areas including network traffic analysis [27], IoT applications and smart homes [3], human activity recognition (HAR) [29], [42], [46], [5], [13], human physiological and emotional analysis [13], factory automation [51], trajectory prediction [37], user authentication [22], life-logging [7], elderly rehabilitation [28], and daily work routine studies [12].", "In addition to time series, CPD can applied to other data modalities with a temporal dimension, such as video, where it has been used for video captioning [14], [16] and video summarising [1], [47] applications.", "Change points are commonly estimated from one of a number of different properties of a time series, including its temporal continuity, distribution or shape.", "Unsupervised CPD methods are generally developed to identify changes based upon one particular property.", "For instance, FLOSS [18] was developed to detect changes in the temporal shape, whilst RuLSIF [30] and aHSIC [52] were developed to identify changes in the statistical distribution.", "Current CPD methods have failed to generalise effectively [13] as the semantic boundaries of different applications will usually be associated with different time series properties.", "For example, abnormalities in the rhythm of the human heart are best characterised by changes in the temporal shape pattern of an electrocardiogram (ECG), whereas changes in human posture (as measured with an RFID sensor system) are best characterised by abrupt statistical changes.", "In this case, the detection performance degrades when a statistical CPD method is applied to the heart beat application, whilst shape based CPD methods will fail in the human posture application.", "Furthermore, for many applications in which data is continuously collected, time series with be characterised by slowly varying temporal shape and statistical properties.", "The change points associated with such time series can be subtle and remain a challenge for CPD methods to address.", "In this work, we propose $TS-CP^{2} $ , a novel approach for self-supervised Time Series Change Point detection method based on Contrastive Predictive coding.", "We pose the question of whether self-supervised learning can be used to provide an effective, general representation for CPD.", "The intuition here is to exploit the local correlation present within a time series by learning a representation that maximises the shared information between contiguous time intervals, whilst minimising the shared information between pairs of time intervals that are separated in time (i.e.", "pairs of time intervals with less correlation).", "It is hypothesised that whenever the learnt representation differs significantly between time adjacent intervals, a change point is more likely to be present.", "We aim to show that this self-supervised representation is capable of detecting a broader range of change points than previous methods that have been specifically designed to exploit a narrow scope of time series properties (i.e.", "commonly either its temporal continuity, distribution or shape patterns).", "Figure REF shows a high-level overview of the approach, which is the first CPD approach based upon contrastive representation learning.", "Furthermore, whilst there are contrastive learning methods for image [8], [44], audio [35], [38] and text [35], this is the first approach utilising contrastive learning on general time series, which in turn, introduces some unique challenges.", "Furthermore, our technique does not rely on any assumptions about the statistical distribution of the data making it applicable to a broad range of real-world applications.", "The main contributions of our paper are as follows: We leverage contrastive learning as an unsupervised objective function for the CPD task.", "To the best of our knowledge, we are the first to employ contrastive learning to the CPD problem.", "We propose a representation learning framework to tackle the problem of self-supervised CPD by capturing compact, latent embeddings that represent historical and future time intervals of the times series.", "We compare our proposed method against five state of the art CPD methods, which include deep learning and non deep learning based methods, investigate the benefits of each through extensive experiments.", "We investigate the performance impact of the hyperparameters used within our self-supervised learning method including batch size, code size, and window size.", "To make $TS-CP^{2} $ reproducible, all the code, data and experiments are available in the project's web page https://github.com/cruiseresearchgroup/TSCP2." ], [ "Related Work and Background", "In this section, we review existing approaches for the CPD problem.", "Since we employ contrastive learning for our time series change point detection method, we also outline recent works on self-supervised contrastive learning.", "We will then review recent representation learning approaches, not only for time series data, but other data modalities as well." ], [ "Time series change point detection", "Although self-supervised learning methods have recently attracted the interest of the deep learning community, current CPD methods are mostly based on non deep learning approaches yet.", "Existing approaches can be categorised based upon the features of the time series that they consider for CPD.", "Statistical methods often compute change points on the basis of identifying statistical differences between adjacent short intervals of a time series.", "The statistical differences between intervals are usually measured with either parametric or non-parametric approaches.", "Parametric methods use a Probability Density Function (PDF) such as [4] or auto-regressive model [53] to represent the time intervals, however, such convenient representations limit the types of statistical changes that can be detected.", "Non-parametric methods offer a greater degree of flexibility to represent the density functions of time intervals by utilising kernel functions.", "Estimating the ratio of time interval PDFs is a simpler problem to address than estimating the individual PDFs of the time intervals.", "The methods in RuLSIF [30], KLIEP [54] and SEP [3] used a non-parametric Gaussian kernel to model the density ratio distribution between subsequent time intervals.", "[52] detected abrupt change points by calculating separability of adjacent intervals based on kernel-based additive Hilbert-Schmidt Independence Criterion (aHSIC).", "Kernel approaches assume there is statistical homogeneity within each interval, which can be problematic for change point detection.", "Furthermore, kernel functions often require parameters to be carefully tuned.", "There is another category of statistical CPD approaches that identify change points as the segment boundaries that optimise a statistical cost function across the segmented time series.", "IGTS [36] and OnlineIGTS [55] estimated change points by proposing top-down and dynamic programming approaches to search for the boundaries that maximised the information gain of the segmented time series.", "GGS [20] proposed an online CPD approach that used a greedy search to identify the boundaries that maximised the regularised likelihood estimate of the segmented Gaussian model.", "Another broad category of CPD methods exploit the temporal shape patterns of time series.", "FLOSS was proposed to detect change points by identifying the positions within the time series associated with a salient change in its shape patterns [18].", "Authors of [51] proposed a motif discovery approach in order to extract rare patterns that can distinguish separate segments [23].", "Recently, ESPRESSO [13] proposed a hybrid CPD approach that exploit both the temporal shape pattern and statistical distribution of time series.", "It was shown that the hybrid model was able to detect change points across a diverse range of time series datasets with greater accuracy than purely statistical or temporal shape based methods.", "Deep learning based CPD methods have also recently been proposed.", "The authors of [11] used an AutoEncoder for CPD by exploiting peaks in the reconstruction error of the encoded representation.", "Kernel Learning Change Point Detection, KL-CPD [6], is a state-of-the-art end-to-end CPD method which solves the problem of parameter tuning in kernel-based methods, by automatically learning the kernel parameters and combining multiple kernels to capture different types of change points.", "KL-CPD utilised a two-sample test for measuring the difference between contiguous sub-sequences.", "KL-CPD was shown to significantly outperform other deep learning and non deep learning CPD methods.", "Figure: Illustration of the overall architecture of our TS-CP 2 TS-CP^{2} .", "Blue dash arrows indicate the back propagation in the training phase.CPD is also useful in video processing applications for summarising video, extracting segments of interest [47], and automatic caption generation and synchronisation [1], [14], [16].", "Existing video segmentation approaches are commonly supervised and benefit from having knowledge of the order of actions.", "In contrast, [1] proposed an auto-regressive model to predict the next video frames based on the most recently seen frames.", "Abrupt increases in the prediction error were then used to detect the segment boundaries." ], [ "Representation Learning", "In recent years, self-supervised representation learning has been used to capture informative and compact representations of video [1], [33], image [8], [21], text[35], and time series [39], [31], [40], [17] data." ], [ "Contrastive Learning", "Contrastive learning is an approach used to formulate what makes the samples in a dataset similar or dissimilar using a set of training instances composed of positive sample pairs (samples considered to be similar in some sense) and negative sample pairs (samples considered to be different).", "A representation is learnt to bring the positive sample pairs closer together and to further separate negative sample pairs within the embedding space.", "Contrastive loss [9] and Triplet loss [48] are the most commonly used loss functions.", "In general, the triplet loss function outperforms the contrastive loss function because it considers the relationship between positive and negative pairs, whereas the positive and negative pairs are considered separately in the contrastive loss function.", "Triplet loss, however, only considers one positive and one negative pair of instances at a time.", "Both functions suffer from slow convergence and require expensive data sampling methods to provide informative instance pairs, or triplets of instances, that accelerate training [44].", "To solve the aforementioned problems, Multiple Negative Learning loss functions have been proposed to consider multiple negative sample pairs simultaneously.", "N-Paired loss [44] and infoNCE based on Noise Contrastive Estimation [19], [32] are examples of recent multiple negative learning loss functions.", "These approaches, however, require computationally expensive sampling approaches to select negative sample instances for training.", "This issue of complexity has been addressed by Hard Negative Instance Mining, which has been shown to play a critical role in ensuring contrastive cost functions are more efficient [15], [49].", "A number of sampling strategies have been proposed, including hard negative sampling [43], semi-hard mining [41], distance weighted sampling [49], hard negative class mining [44], and rank-based negative mining [45]." ], [ "Contrastive-based Representation Learning", "Most existing work on representation learning focus upon natural language processing [35] and computer vision [8], [21] domains.", "Howerver, to the best of our knowledge, it is the first time contrastive learning has been used for change point detection.", "There is a few works that investigates the use of representation learning with multivariate time series.", "The authors of [17] proposed a general-purpose approach to learn representations of variable length time series using a deep dilated convolutional network (WaveNet [34]) and an unsupervised triplet loss function based on negative sampling.", "Contrastive predictive coding, CPC [35], uses auto-regressive models to learn representations within a latent embedding space.", "The aim of CPC is to learn within an abstract, global representation of the signal as opposed to a high dimension, lower level representation.", "The authors demonstrated it could learn effective representations of different data modalities such as images, text and speech for downstream modelling tasks.", "Firstly, a deep network encoder was used to map the signal into a lower dimension latent space before an auto-regressive model was then applied to predict future frames.", "A contrastive loss function maximised the mutual information between the density ratio of the current and future frames.", "CPCv2 [21] replaced the auto-regressive RNN of CPC with a convolutional neural network (CNN) to improve the quality of the learnt representations for image classification tasks." ], [ "Problem Definition", "Given a multivariate time series $\\lbrace X_{1}, X_{2},...,X_{T}\\rbrace $ of $T$ observations, where the vector $X_{i} \\in \\mathbb {R}^{d}$ , we attempt to estimate the times ($t$ ) that are associated with a change in the time series properties.", "We define change points (or segment boundaries) as the time points in future can not be anticipated from the data before this point.", "Hence, the dissimilarity between future representation and anticipated representations can be used as a measure to detect transition to the next segment.", "Figure: The encoder architecture for TS-CP 2 TS-CP^{2} Ẇe use two stacks of TCN with kernel size 4 and dilation sizes of 1, 4, and 16 followed by three Dense layers as the prediction head." ], [ "$TS-CP^{2} $ Overview", "There are change point and temporal anomaly detection methods for video [1] and time series [26] that use an auto-regressive model for prediction.", "In these approaches, change points are detected at samples associated with a salient increase in the prediction error.", "However, since the prediction error is highly dependent upon the distribution of the data, we propose to use representation learning to extract a compact latent representation that is invariant to the original distribution of the data.", "Figure REF illustrates the main idea behind our $TS-CP^{2} $ approach.", "We hypothesize that this approach is much more effective to detect change points because the embedding space that is extracted from contiguous time intervals are likely to be dependent upon the same shared information.", "Here we adopt a similar approach to the CPC [35], [21] method to learn a representation that maximises the mutual information between consecutive time windows.", "Firstly, an auto-regressive deep convolution network, WaveNet [34], was employed to encode each of the time series windows.", "Secondly, a 3-layer fully connected network was employed on top of this encoding to produce a more compact, embedded representation.", "The cosine similarity was computed between the embeddings of consecutive time windows in order to estimate the change points.", "The time intervals associated with smaller similarity values had a higher likelihood of being change points.", "A contrastive learning approach was used to train the encoder by using a single pair of contiguous time windows (positive pair) and a set of window pairs that were separated across time (negative pairs) within each batch.", "We applied the $N-paired loss$ metric [44] (which is described in section REF ) to maximise the mutual information between the positive pairs amongst the set of negative pairs of samples.", "Figure REF shows the overall architecture of the proposed method.", "In the following section we will describe the main modules in the following order: 1) Representation learning, 2) Negative sampling, and 3) Change point detection." ], [ "Representation Learning", "At the core of the $TS-CP^{2} $ approach is an encoder that maps pairs of contiguous time windows into a compact embedding representation.", "This representation was trained to learn about the concept of similarity over short temporal scales by maximising the mutual information between the pairs of adjacent time windows.", "We employ the auto-regressive deep convolution network, WaveNet [34], to learn our encoded representation.", "We do not use an LSTM to encode the time series, given it has been shown that temporal convolutional networks (TCN) can often produce superior prediction performance with sequential data [21] and are generally easier to train.", "Figure REF illustrates the encoder architectures.", "It consists of two blocks of TCN with 64 kernel filters of size 4 and three layers of dilation with respective rates of 1, 4 and 16.", "The TCN is then followed by a simple three-layer projection head with ReLU activation function and batch normalisation.", "The modified illustration of TCN layerWe acknowledge the main illustration and TCN implementation: https://github.com/philipperemy/keras-tcn is shown in the figure.", "Pairs of history and future time windows are fed into an encoder.", "A projection head is used (shown as $g(\\cdot )$ and $g^{\\prime }(\\cdot )$ in Figure REF , respectively) to map each window encoding into a lower dimension space.", "To this end, an MLP neural network with three hidden layers was used.", "Contrastive learning used pairs of history and future windows for training an embedded representation.", "Two different types of time window pairs were contrasted for training.", "Each training instance was comprised of a positive sample pair of contiguous time intervals and a set of negative sample pairs with intervals separated across time.", "In the next subsection, we define the $InfoNCE$ cost function that will be used for representation learning." ], [ "Cost function", "We applied the $InfoNCE$ loss function that is based upon Noise Contrastive Estimation [32], which was originally proposed for natural language representations but has also recently been adopted for image representation learning techniques [21], [35], [8].", "The $InfoNCE$ cost function is defined to maximise the mutual information between consecutive time windows.", "A single positive pair of time adjacent intervals $(h_i,f_i)$ , the history window (${h_i}$ ) and future window (${f_i}$ ), and a set of $K-1$ negative pairs ($(h_i,f_j)_{j\\ne i}$ ) where the intervals ${h_i}$ and ${f_j}$ were well separated in time across the sequence.", "Using the $InfoNCE$ loss function, we calculate the probability $\\rho _{i}$ of the positive sample pair in each batch using the scaled-$Softmax$ function: $\\rho _{i} = {\\frac{exp(Sim(h_i,f_i)/\\tau )}{\\sum _{j=1}^{K}{exp(Sim(h_i,f_j)/\\tau )}}}\\ $ where ${\\tau }$ is a scaling parameter and $Sim$ is the cosine similarity between each pair of data embeddings.", "The final loss is calculated with the binary cross-entropy function over the probabilities of all $K$ positive pairs belonging to the training batch.", "Since the probabilities of the positive sample pairs are computed using the similarity scores of the negative sample pairs in (REF ), the cross entropy loss function can be simplified to: $ \\mathbf {{L}} = - \\sum _{i,j}{y_{ij} log(\\rho _{i}) + (1-y_{ij}) log(1-\\rho _{i})}$ $\\centering {y}_{ij} = {\\left\\lbrace \\begin{array}{ll}1 & \\text{if $i=j$} \\\\0 & \\text{if $i \\ne j$}\\\\\\end{array}\\right.", "}$ $\\mathbf {{L}} = \\sum _{i}{-log(\\rho _{i})}$" ], [ "Negative Sampling", "Following on from the hard negative class mining approach in [44], we propose a simpler sampling strategy where positive sample pairs are randomly sampled and used to construct the negative sample pairs for each batch.", "Figure REF depicts the process of batch construction in our model.", "We choose $K$ random pairs of contiguous windows $(h_i,f_i)$ as the positive pairs in each training batch.", "Each pair must adhere to the constraint of being a minimum temporal distance from the other pairs.", "This minimum temporal distance constraint is used to enable each batch to adopt the future windows of the other $K-1$ positive pairs as negative pairs, given they are guaranteed to be sufficiently separated from the history window in the batch's own positive pair.", "The intuition is that time series are commonly non-stationary, and hence, windows that are temporally separate from one another are likely to exhibit far weaker statistical dependencies than adjacent windows.", "We need to set the threshold of minimum temporal distance based upon the time series application being considered.", "Figure: Batch constructionFigure: Three examples of negative pairs of samples (left, length = 100) and their corresponding embedding (right, length = 16).", "The negative pairs correspond to time intervals of the time series that well separated in time.", "A change point or anomaly has occurred somewhere in either history or future frame of the negative pairs.Consequently, we can select positive and negative sample pairs with a relatively low complexity relative to the other negative mining approaches mentioned in Section .", "Figure REF and REF show examples of time windows belonging to the positive and negative pairs, respectively, and their corresponding embedding vectors." ], [ "Change Point Detection Module", "We hypothesize that when a change point intersects a pair of history and future windows, their associated embeddings will be distributed differently.", "Consequently, in order to detect change points from the time series being tested, we transform pairs of history and future windows into a compact embedding and compute the cosine similarity ($Sim(h_i,f_i)$ ) between the embedding pairs across the time series being tested.", "The difference between the cosine similarity and moving average of the cosine similarity was computed and a peak finding algorithm was applied to find local maxima in the difference function (increase in difference function is associated with decrease in similarity metric).", "The time intervals associated with these local maxima are considered as the change point estimates.", "Figure REF shows an example of the cosine similarity between the latent embeddings of the history and future windows within a time series.", "The green areas show the interval pairs $(h_i,f_i)$ which contain a change point for a subset of the Benchmark-4 of the Yahoo!", "dataset .", "It is clear that the local minima of the difference between the cosine similarity of each interval pair $(h_i,f_i)$ and the average similarity across recent intervals pairs coincide with true change points.", "Figure: An example of detecting change points based upon the cosine similarity between consecutivewindow pairs.", "The black line shows the cosine similarity between subsequent time intervals in Benchmark-4 of the Yahoo!", "dataset .", "The green areas highlight the intervals with change points and the yellow line shows the moving average of the cosine similarity for the previous WW intervals.In this section, we present our evaluation of the proposed $TS-CP^{2} $ method.", "Firstly, we introduce the datasets and outline the baseline CPD methods used in our experiments.", "A sensitivity analysis of the $TS-CP^{2} $ method is presented along with a performance comparison with baseline CPD methods.", "$TS-CP^{2} $ is implemented using Tensorflow 2.2.0 and python 3.7.2." ], [ "Datasets", "We show the effectiveness of our method across a diverse range of applications that include web service traffic analysis, human activity recognition and mobile application usage analysis.", "Yahoo!BenchmarkYahoo Research Webscope dataset, S5 - A Labeled Anomaly Detection Dataset, version 1.0,https://webscope.sandbox.yahoo.com/ .", "The Yahoo!", "benchmark dataset is one of the most widely cited benchmarks for anomaly detection.", "It contains time series with varying trend, seasonality, and noise including random anomaly change points.", "We used all 100 time series of the fourth benchmark, as it is the only portion of the dataset that includes change points.", "HASC http://hasc.jp/hc2011 [24], [25].", "The HASC challenge 2011 dataset provides human activity data collected by multiple sensors including an accelerometer and gyrometer.", "We used a subset of the HASC dataset (The same subset used by recent state-of-the-art method, KL-CPD) including only 3-axis accelerometer recordings.", "The aim of detecting change point detection with this dataset is to find transitions between physical activities such as \"stay\", \"walk\", \"jog\", \"skip\", \"stair up\", \"stair down\".", "USC-HAD http://sipi.usc.edu/had [56].", "USC-HAD dataset includes twelve human activities that were recorded separately across multiple subjects.", "Each human subject was fitted with a 3-axis accelerometer and 3-axis gyrometer that were attached to the front of the right hip and sampled at 100Hz.", "Activities were repeated five times for each subject and consisted of: \"walking forward\", \"walking left\", \"walking right\", \"walking upstairs\", \"walking downstairs\", \"running forward\", \"jumping up\", \"sitting\", \"standing\", \"sleeping\", \"elevator up\", and \"elevator down\".", "We randomly chose 30 activities from the first six participants and stitched the selected recordings together in a random manner.", "In the experiments undertaken in this paper, only the data from the accelerometer was used.", "Table REF outlines the properties of each dataset.", "Table: The properties of the three datasets used in our experiments.T isthe total number of samples, #sequences is the quantity oftime series, #channels represents thetime series dimensionalityand #CP is the total number of change points in each dataset.Figure: A sensitivity analysis of the the Yahoo!Benchmark dataset with respect to the code size, batch size and four window sizes of: (a) 24 samples (b) 50 samples (c) 75 samples and (d) 100 samples." ], [ "Baseline Methods", "The performance of the proposed $TS-CP^{2} $ method was compared against five state-of-the-art unsupervised change point detection techniques that included ESPRESSO [13], FLOSS [18], aHSIC [52], RuLSIF [30], and KL-CPD [6].", "To avoid inconsistencies and implementation errors, and to provide a fair comparison, baseline methods were evaluated using the publicly available source code.", "Kernel specific parameters were used by the RuLSIF and aHSIC methods.", "For RuLSIF, the regularisation constant was set to a value of 0.01, as suggested in [30].", "For aHSIC, the regularisation constant and the kernel bandwidth parameters were set to values of 0.01 and 1, respectively, as specified in [52].", "Detection performance was compared across a range of window sizes that were unique to each dataset based on its sampling rate.", "As a deep learning based method, KL-CPD required several hyper-parameters to be tuned; window size, batch size and learning rate.", "To enable a fair comparison with the other methods, a grid search was performed across the sets of hyper-parameter values.", "Only the hyper-parameter configuration that provided the highest rate of true positives was presented.", "We used the same evaluation approach as undertaken with KL-CPD KL-CPD source code: https://github.com/OctoberChang/KL-CPD_code to calculate the F1-score.", "The remaining parameters were set according to the values specified in [6].", "Although the training process of KL-CPD was unsupervised, the method still required ground truth labels to be used to fine-tune the model hyper-parameters during the validation phase.", "For the FLOSS and ESPRESSO methods, we used the z-normalised euclidean distance as the similarity metric, as suggested by authors of their underlying structure they used in [57]." ], [ "Evaluation Metrics", "The performance of models were evaluated with respect to the F1-score.", "The error margin in which a change point can be detected is an important factor in evaluating the performance of each CPD method [2].", "Hence, we report the F1-scores of each dataset for the three different detection margins specified in Table REF .", "Each change point estimate was defined as a true positive when it was located within the specified error margin of the ground truth change point.", "When multiple change point estimates were located within the error margin of the the ground truth change point, only the closest estimate was considered to be a true positive.", "The remaining estimates were considered to be false positives.", "Ground truth change points without any estimates that fell within the specified error margin were considered to be false negatives." ], [ "Fine-Tuning and Sensitivity Analysis", "In this section, we have done extensive experiments to analyse the sensitivity of our proposed method to: Window size is the length of the history and future intervals.", "We consider a range of values that are dependent upon the particular application and sampling rate of the dataset.", "The window size should be large enough to encapsulate the properties of the time series but not too large to encompass multiple change points.", "We select window sizes of 1,2,3 and 4 days for the Yahoo!Benchmark dataset and 0.5,1,2 and 4 seconds for the HAR dataset.", "Batch Size specifies the number of training instances ($K$ ) that were processed before the model was updated.", "It also specifies the number of negative pairs ($K-1$ ) used in each training instance.", "The batch size was selected to range between 4 and 128 samples.", "Code Size specifies the length of the embedding vector that is extracted from the encoder network.", "The range of code sizes that were selected ranged between 4 to 20 dimensions.", "Window size is the only input parameter to investigate for the ESPRESSO and FLOSS methods and the main parameter for the RulSIF and aHSIC methods.", "For the deep learning based methods, including the proposed method, we also investigate the performance of a number of parameters including the window size, batch size, code size and learning rate.", "Figure REF compares the $TS-CP^{2} $ performance (with respect to the F1-score) for the Yahoo!Benchmark dataset across the different parameter settings." ], [ "Window size", "Since the Yahoo!Benchmark dataset is sampled hourly and the minimum length between two consecutive change points is approximately 160 samples, we consider a range of window sizes between 24 (1 day) to 100 ( 4 days).", "Figure REF shows there was a monotonic increasing relationship between the window size and detection performance averaged across the code size and batch size.", "It was hypothesized that longer windows possess the highest F1 scores, given they encapsulate additional properties of the time series into modelling." ], [ "Batch Size", "We varied the batch size with respect to the set of $\\lbrace 4, 8, 16, 32, 64, 128\\rbrace $ dimensions to investigate its impact upon detection performance.", "Figure REF shows there was a monotonic increasing relation between the batch size and detection performance when averaged across the code size and window size.", "There are particular situations for the smallest code size, however, where the largest batch sizes had inferior detection performance to the smaller batch sizes.", "We hypothesize such situations can occur given larger batches are more likely to generate false negative samples from the time series datasets and smaller code size are too short to represent all informative features of data.", "False pairs of negative samples are pairs of time windows that are considered to be false instances, but are found to be similarly distributed.", "Whilst the negative sample pairs are constrained to be intervals that are temporally separate from one another, time series are often comprised of patterns (and their associated semantic classes) that repeat at different, non-contiguous positions within the sequence.", "Consequently, using contrastive learning to separate these false pairs of negative samples within the embedding space can degrade detection performance." ], [ "Code Size", "In contrast to many representation learning approaches, we investigated how the embedding dimensionality affected the detection performance.", "We varied the code size from 4 to 20 dimensions which equates to representing between 4% to 83% of a window from the experiment.", "As shown in Figure REF , the optimal code size was dependent upon the window size and batch size.", "In general, the smallest code size of 4 showed a relatively weak performance for each of the window sizes, given there was insufficient capacity to represent the key features of the time series to learn an effective representation.", "The relationship between detection performance and code size was not monontonic increasing, however, given the largest code sizes were often shown to be inferior to the more compact embeddings with a code size of between 8 and 12 dimensions.", "Table: The performance of the proposed TS-CP 2 TS-CP^{2} method was compared to the other baselines methods across the Yahoo!Benchmark, HASC, and USC datasets.", "The bold and underlined texts represent the methods with the first and second highest F1-scores, respectively.", "The detection margin is the maximum number of samples that an estimated change point can be from a ground truth change point to still be considered a True Positive.", "We present the highest F1-score of each method (for the best window size) and the F1-score of the methods averaged across all window sizes." ], [ "Baseline Comparison", "The performance of the proposed $TS-CP^{2} $ method was compared to the five baseline methods across the three datasets.", "To enable a fair comparison between the methods, we performed a grid search of the set of parameters associated with each method.", "For each method, the model with the best F1-score and its corresponding parameters were presented in Table REF .", "Figure: Four random time series from the Yahoo!", "Benchmark dataset.", "The yellow vertical lines correspond to the change points.", "The spatial anomalies are not highlighted as they are not the focus of the experiment." ], [ "Yahoo! Benchmark Evaluation", "To compare the ability of each method to detect change points, we set three different detection error margins of 24, 50, and 75 samples.", "If the difference between the actual and estimated change points were less than the specified margin, it was considered to be a true positive.", "Table REF shows the highest F1-score and corresponding window size for each method.", "The Yahoo!", "Benchmark dataset is well-cited (according to [50]) and one of the more complex datasets for temporal anomaly detection given the anomalies are mostly based upon changes in the seasonality, trend and noise.", "Based on the results reported in Table REF , our proposed method $TS-CP^{2} $ strongly outperforms each of the other baseline methods.", "Although all of the baselines are state-of-the-art methods for CPD, this dataset was shown to be challenging for them.", "Four randomly selected sequences of this dataset are illustrated in Figure REF .", "$FLOSS$ and $aHSIC$ were able to detect changes in temporal shape patterns, however, they could not distinguish the change points associated with subtle statistical differences.", "$RuLSIF$ estimate change points based upon the difference in the ratio of the distributions of adjacent time intervals.", "It was clear for some of the change points of the sequences in Figure REF that adjacent segments were similarly distributed and only exhibited clear changes in their temporal shape." ], [ "USC-HAD Dataset Evaluation", "Given the sampling rate for this dataset is 100Hz, the maximum error margins for which a change point estimate was considered to be a true positive was 1, 2, and 4 seconds.", "We investigate different values of the kernel bandwidth for RuLSIF and different kernel sizes (20, 40, and 50) for aHSIC.", "Different window size were investigated for FLOSS, ESPRESSO, KL-CPD, and $TS-CP^{2} $ as they were varied between 100, 200, and 400 samples.", "We also used different learning rates for KL-CPD and $TS-CP^{2} $ of $3\\times 10^{-4}$ and $1\\times 10^{-4}$ , respectively.", "As shown in Table REF , our proposed method outperformed the other baselines across each of the error margins.", "$TS-CP^{2} $ is the only method that delivers a high F1-score for the smallest error (100 samples) meaning it can reliably detect change points within one second of its occurrence.", "Similarly to the Yahoo!", "Benchmark dataset, we compare the effect of batch size across the different window sizes for $TS-CP^{2} $ in Figure REF .", "It was shown that larger batch sizes offered a superior detection performance across the longer windows.", "The shorter batch sizes, however, were shown to offer superior detection performance across the smaller windows.", "Figure: Comparing the effect of batch size across different window sizes for USC-HAD dataset." ], [ "HASC dataset Evaluation", "The HASC dataset was found to be the most challenging dataset for $TS-CP^{2} $ and the other baseline methods.", "Although $TS-CP^{2} $ achieves the second highest performance level with respect to each of the different window sizes, it still achieves the highest average F1-score with a 19.2%, 54.8%, 10.1%, 11.1%, and 3.8% improvement over FLOSS, aHSIC, RulSIF, ESPRESSO, and KL-CPD, respectively.", "The HASC dataset is the smallest in size (around 39K samples) but contains the largest number of change points (65 change points in total).", "Consequently, the relatively high density of change points means there there is a greater likelihood to generate positive sample pairs that encompass change points.", "Such false positive sample pairs will degrade model training.", "Since the model is self-supervised, ground truth labels cannot be used to rectify any such errors with positive sample pairs.", "Consequently, to address this problem, we suggest to enhance the model by injecting a light negative mining.", "We could also generate more positive sample pairs through augmentation." ], [ "Discussion", "We showed that $TS-CP^{2} $ was able to outperform non deep learning based methods, FLOSS, aHSIC, RulSIF and ESPRESSO, by 104.3%, 91.0%, 68.9%, 53.3% and 24.9% improvement with respect to the F1-score averaged over all of the datasets.", "In addition, $TS-CP^{2} $ showed a 17.0% improvement in F1-score over KL-CDP, which is the most recent and competitive deep-learning-based change point detection method.", "Since the baseline CPD methods exploit abrupt changes in one particular property of the time series, they do not effectively generalise to different types of datasets.", "For example, the hybrid ESPRESSO method performs well on the USC-HAD dataset, given its change points are commonly associated with abrupt changes in both its temporal shape and statistical properties.", "But neither ESPRESSO nor the other non deep learning approaches were as effective in estimating change points associated with the Yahoo!Benchmark given its change points were composed of more subtle and slowly evolving transitions in properties.", "In contrast, we showed that our proposed $TS-CP^{2} $ method achieved either the first or second highest result across each of the datasets.", "$TS-CP^{2} $ showed a significant improvement over the five baselines for the Yahoo!", "and USC-HAD datasets, whilst its average performance across the different window sizes was superior to each baseline with the HASC dataset.", "In future work, we will investigate using augmentation and negative mining batch construction to address the problem of high frequency change points that present themselves in some of the datasets discussed in section REF .", "Finally, $TS-CP^{2} $ employs a compact structure with a shared representation to encode its history and future windows.", "This enables faster training convergence compared to its other deep learning counterpart, KL-CPD.", "Furthermore, once the representation model is trained, CPD is very simple to implement given it only involves a comparison between the learnt representations of the history and future windows.", "Consequently it has the potential to be implemented for online operation on low resource devices.", "The baseline methods (other than the FLOSS method which has introduced a streaming version) cannot be applied online as they need to consider a reasonably large batch of data to capture repeated patterns or to optimise the entropy-based loss function." ], [ "Conclusion", "We propose a novel self-supervised CPD method, $TS-CP^{2} $ for time series.", "$TS-CP^{2} $ learns an embedded representation predict a future interval of a times series from historical samples.", "Change points are detected at the times in which the embedded are relatively high.", "Our proposed method is the first CPD method that employs contrastive learning to extract a compact and informative representation vector for every frame and estimate the change point based on the agreement between the learnt representation of subsequent frames.", "We evaluated the ability of $TS-CP^{2} $ in detecting change points against six other well-known state-of-the-art methods across three datasets.", "We have shown that our proposed method significantly outperform other baselines in two dataset and reaches a comparable score for the other dataset.", "Although the pre-trained $TS-CP^{2} $ can detect changes in online applications, we aim to expand this method in our future work to continuously learn changes, anomalies and drifts in data.", "We would like to acknowledge the support from CSIRO Data61 Scholarship program (Grant number 500588), RMIT Research International Tuition Fee Scholarship (RRITFS), and Australian Research Council (ARC) Discovery Project DP190101485." ] ]	2011.14097
[ [ "Quantum entropy model of an integrated QRNG chip" ], [ "Abstract We present the physical model for the entropy source of a quantum random number generator chip based on the quantum fluctuations of the photon number emitted by light-emitting diodes.", "This model, combined with a characterization of the chip, estimates a quantum min-entropy of over 0.98 per bit without post-processing.", "Finally, we show with our model that the performances in terms of security are robust against fluctuations over time." ], [ "Introduction", "Random numbers are used in a wide range of applications such as gambling, numerical simulations and cryptography.", "Lack of a good random number generator (RNG) can have serious consequences on the security of devices and protocols [1], [2], [3].", "Currently, many applications rely on RNGs based on a stochastic process and lack a complete security model.", "In order to have a sequence usable for cryptographic applications, the source of randomness must be completely unpredictable, even if a malicious adversary has a perfect description of the system [4].", "Quantum RNGs (QRNGs) can overcome this problem thanks to the intrinsically probabilistic nature of quantum mechanics.", "One key challenge today is to have a fully integrated QRNG device that reach mass-market deployment.", "Several works have been carried out toward that goal [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].", "One of them is a QNRG implementation based solely on components that are compatible with integrated electronics, namely a light-emitting diode (LED), a CMOS image sensor (CIS) and an analog-to-digital converter (ADC) [7].", "More precisely, this work showed that a CIS-based mobile phone camera could be used as an entropy source providing 10-bit long strings containing 5.7 bits of quantum entropy.", "However, this approach still required software-based randomness extraction to generate bits with close-to-maximal entropy, and a fully integrated implementation remained to be demonstrated.", "In this paper, we present a fully integrated QRNG architecture and chip implementation based on the quantum statistics of light captured by a CIS, and we present a model showing that the quantum entropy of each bit produced is close to unity without the need of randomness extraction.", "This architecture is used to provide small-form factor and low power consumption chips, making them suitable for mobile devices such as smartphones.", "A scheme of the architecture of IDQ's QRNG chips is shown in fig:chipscheme.", "A LED is used as a continuous source of photons.", "As the light field emitted is highly multi-mode, the probability distribution of the photon number is very well approximated by a Poisson distribution with mean $\\mu _\\text{ph}$ [19].", "The probability to have $n$ photons emitted during a fixed time interval is given by : $p(n,\\mu _\\mathrm {ph}) = \\frac{\\mu _\\mathrm {ph}^n}{n!", "}\\mathrm {e}^{-\\mu _\\mathrm {ph}}$ Figure: Schematic representation of the QRNG.", "All the components are embedded on a single chip.Photons are converted into electrons by a CMOS image sensor array during the integration time of the sensor.", "Each pixel of the sensor has an efficiency $\\eta $ (taking into account transmission losses and detection efficiencies) that may vary between them.", "The number of electrons $N_\\mathrm {e}$ is directly correlated to the quantum fluctuations of the LED and follows a Poisson distribution with mean value $\\mu _\\mathrm {e} = \\eta \\mu _\\mathrm {ph}$ .", "We assume that pixels are independent from each other and that there is no correlation from frame to frame (these assumptions are verified in sec:tests).", "After accumulation, the number of electrons is converted into a voltage, then digitized with a 10-bits ADC.", "We define $K$ as the gain between $N_\\mathrm {e}$ and the analog-to-digital unit of the ADC.", "We also define two random variables $X$ and $Z$ .", "$X$ is a continuous random variable representing the voltage value distribution at the input of the ADC and can be written $X = KN_\\text{e} + E$ where $E$ is the random variable associated with the classical noise (see sec:noise).", "$Z$ is the random variable returned by the ADC and is defined as $Z ={\\left\\lbrace \\begin{array}{ll}0 &\\text{ if } X < 0 \\\\\\lfloor X\\rfloor &\\text{ if } X \\in [0;1023] \\\\1023 &\\text{ if } X>1023\\end{array}\\right.", "}$ where $\\lfloor .", "\\rfloor $ is the floor operator.", "sub:Z shows a simulated distribution of $Z$ with $\\mu _\\mathrm {e}=625$ .", "On this graph, we observe a normal distribution of the ADC output values, combined with a series of peaks with twice the probability.", "This “pile-up\" effect is due to the factor $K$ of the chip which is inferior to 1.", "As one electron is not enough to increase the signal by a full ADC step, two electron numbers can lead to the same ADC output making this value twice more probable, with a periodicity that goes roughly like $1/(1-K)$ .", "[figure]style=plain,subcapbesideposition=top Figure: (a) Simulated ADC output distribution in case there is no noise, with K=0.8192K=0.8192 (obtained from the factory-given parameters of the chip).", "(b) Two bits probability distribution simulated from (a) giving a min-entropy per bit H min =0.982H_{\\min }=0.982.To generate entropy bits from the 10-bits ADC output $Z$ , we keep the least significant bits (LSB) 2 and 3, noted $Z_{23}$ .", "Indeed, their entropy is the more robust of all the bits against imperfections of the system.", "This happens because the most significant bits will be biased if $\\mu _\\mathrm {e}$ is not well controlled.", "Moreover, LSB 0 and 1 can be affected by small and uncontrolled fluctuations which are not from quantum origin, but also by the pile-up effect.", "By taking only LSB 2 and 3, we can easily mitigate these effects to obtain bits with a very high min-entropy $H_{\\min }$ without post-processing as it can be seen in sub:Z23.", "We note that this principle can be applied with ADCs of different resolution, with the right choice of bits retained to generate the entropy bits.", "These two bits can be used as entropy bits directly, or can be seeded to a Hash-based deterministic random bit generator (DRBG) as recommended by the National Institute of Standards and Technology (NIST) documentations (SP 800-90A) [20]." ], [ "Noise model", "To complete our model, we need to take into account the classical noise $E$ as it can impact the security of the chip.", "We consider two sources of noise as shown in fig:NoiseSources.", "Figure: Schematic representation of the noise sources in the chip.", "Dark electrons are added to the electrons generated by the LED.", "The total number of electrons is converted into a voltage with a factor KK.", "After conversion, noise from the readout circuit is added before the signal is digitized with the ADC.First, we have a discrete source of dark electrons which are generated by another process than the absorption of a photon emitted by the LED (e.g.", "thermal excitation).", "These follow a Poisson distribution with parameter $\\mu _\\text{dark}$ and are added to the photo-electrons.", "Second, we consider a continuous source due to electronic noise in the readout circuit following a normal probability distribution $\\mathcal {N}$ described by a probability density function $\\Phi _{\\mu _\\mathrm {r},\\sigma _\\mathrm {r}}$ with mean $\\mu _\\text{r}$ and variance $\\sigma ^2_\\text{r}$ [21], [22], [23].", "The probability density function $P_E$ of the classical noise is therefore a convolution of a Poisson and a normal distribution and can be written : $P_E(e) = \\sum _n p(n,\\mu _\\text{dark})\\Phi _{\\mu _\\mathrm {r}+Kn,\\sigma _\\mathrm {r}}(e).$ We assume all sources of classical noise are accessible to an adversary (called Eve).", "We suppose that Eve cannot change it after fabrication and characterization of the chip and that is it uncorrelated to the quantum entropy source.", "We then need to calculate the min-entropy of $Z_{23}$ given $E$ as it is defined in [24] : $H_{\\min } (Z_{23}\|E) = -\\log _2\\left(p_{guess}\\right),$ where $p_{guess} = \\int P_E(e) \\max _{z_{23}}\\left(P_{Z_{23}\|E=e}(z_{23})\\right)\\mathop {}\\!\\mathrm {d}e$ is the optimal guessing probability of $Z_{23}$ given $E$ .", "This gives the quantum min-entropy output of the chip." ], [ "Experimental characterization", "In our model, we made several assumptions (photon number distribution, independence between pixels and between frame).", "In this section, we show results from measurements on a QRNG chip (model IDQ6MC1) to validate these assumptions." ], [ "Light source", "Firstly, we want to characterize our source in order to verify that the number of photons emitted follows a Poisson statistics.", "To achieve that goal, we can measure the distribution of the ADC output $Z$ for various intensities by changing the current inside the LED.", "Results are displayed in sub:pow.", "On the plot, we can observe a pile-up effect similar to the one predicted by our model (see sub:Z).", "Peaks are less prominent than in our simulations; that is due to the presence of the classical noise averaging them out.", "From these acquisitions, we can plot the variance of $Z$ , $\\sigma _Z^2$ , as a function of its expected value $\\left<Z\\right>$ (see sub:var).", "Due to the conversion factor $K$ affecting the mean value and the variance of the number of electrons differently and the offset of the ADC, we do not have $\\left<Z\\right>=\\sigma _Z^2$ as expected from a Poisson distribution.", "Nevertheless, this does not affect the linear relationship between them, as we can see in sub:var, validating the Poissonian nature of the light emitted by the LED and the transfer of this statistics to the electron number distribution.", "[figure]style=plain,subcapbesideposition=top Figure: (a) ADC output distribution ZZ given by one pixel of the array for various light intensity.", "(b) Variance of ZZ versus its mean value for the distributions of (a)." ], [ "Classical noise", "We characterize the noise distribution for 4 different pixels on the array.", "For that, we switch off the LED and measure the distribution $Z_E$ at the output of the ADC with only classical noise.", "As this distribution is centered near 0 in the default settings, we adjust the ADC offset to shift it to the right by 8 ADC steps in order to see the distribution completely.", "The histogram of $Z_E$ is given in fig:noisehist.", "We observe a similar pile-up effect to the one observed with the LED on coming from the discrete component of $E$ .", "We can fit this histogram with eq:noise to extract the different parameters of the classical noise presented in tab:noise.", "The value $\\mu _r$ depends on the ADC offset but we can extrapolate from our measurements in order to find its value for the default settings of the chip.", "Figure: Noise distribution of one of the pixels.Table: Parameters of the noise distribution for 4 pixels of the CMOS image sensor.", "The value for μ r \\mu _r was extrapolated from our measurement to find the value with the default ADC offset.As we can see, classical noise is mainly given by dark electrons ($\\mu _\\mathrm {dark}>>\\sigma _\\mathrm {r}^2$ ).", "Moreover, the noise parameters for the 4 pixels spread across the array are quite close.", "We can therefore assume all the pixels will have similar noise distribution." ], [ "Correlation measurements", "In our model, we supposed that pixels are independent from each over (no crosstalk) and that the result of a pixel in one acquisition frame has no effect on the next frame.", "In order to validate these hypothesis, we acquired 10000 frames and we calculate the Pearson correlation coefficient $\\rho _{ij}$ between pixels $i$ and $j$ and the autocorrelation coefficient $\\rho _{i}(l)$ for pixel $i$ at lag $l$ : $\\begin{aligned}\\rho _{ij} &= \\frac{\\left<\\left(Z_t^{(i)}-\\left<Z^{(i)}\\right>\\right)\\left(Z_{t}^{(j)}-\\left<Z^{(j)}\\right>\\right)\\right>}{\\sigma _i\\sigma _j},\\\\\\rho _{i}(l) &= \\frac{\\left<\\left(Z_t^{(i)}-\\left<Z^{(i)}\\right>\\right)\\left(Z_{t+l}^{(i)}-\\left<Z^{(i)}\\right>\\right)\\right>}{\\sigma _i^2}\\end{aligned}$ where $Z^{(i)}_t$ is the value returned by the pixel $i$ at time $t$ .", "Results are given in fig:correlations.", "As we can see in sub:correlation, the values of $\\rho _{ij}$ are normally distributed around 0 and with a standard deviation of 0.01.", "This corresponds to the expected uncertainty of the measurements with a sample size of 10000.", "On sub:autocorrelation, we plotted the values of $\\rho _{i}(l)$ for 4 pixels on the CMOS array.", "For $l=1$ , the autocorrelation coefficient is already in the uncertainty region due to our samples size and then fluctuates around 0 at all lags.", "These results validate the assumption made in our model that correlations are negligible and will not affect the entropy of the device.", "[figure]style=plain,subcapbesideposition=top Figure: (a) Probability distribution of the Pearson correlation factors measured between all pairs of pixels.", "The standard deviation σ\\sigma on the correlation factor is 0.01 which correspond to the uncertainty expected for the size of our data.", "(b) Autocorrelation of 4 pixels from the array.", "The solid and dashed grey lines represents respectively the confidence intervals of σ\\sigma and 3σ3\\sigma ." ], [ "Quantum entropy estimation", "Following the characterization of the chip, we can calculate the final quantum entropy of our two bits per pixel as a function of $\\mu _\\mathrm {e}$ .", "Results are shown in fig:finalentropy.", "As we can see, the quantum min-entropy is very close to its maximum value for a large range of $\\mu _\\mathrm {e}$ making it robust against fluctuations of the light intensity.", "It is also robust against small variations of the classical noise parameters whose effects only appear on the sharp edges of the curve.", "For $\\mu _\\mathrm {e}\\in [500,750]$ which is the range where the chip normally operates, $H_{\\min }(Z_{23}\|E)$ is over 0.98 per bit which is a significant improvement compared to the 0.57 on average per bit measured in Ref.", "[7] for a specific intensity of the LED.", "However, with this device, we do not have access to the mean photon number arriving on each pixel to ensure we are in the optimal region i.e.", "$\\overline{H}_{\\min }(Z_{23}\|E) \\ge H_{\\min }^l$ where $\\overline{H}_{\\min }(Z_{23}\|E)$ is the average min-entropy per pixel over the array and $H_{\\min }^l$ is a lower bound on the entropy per pixel.", "If no control is implemented, fluctuations of the LED intensity or of the pixel efficiencies could lead to a degradation of the entropy.", "To make sure the chip is always providing the optimal entropy, we can define two thresholds on the ADC output $T^-$ and $T^+$ to record on each frame how many pixel outputs $n^-$ and $n^+$ were out of the interval $[T^-;T^+]$ .", "If $n^\\pm $ exceeds a predefined value $N^\\pm $ , it is registered as a failure and the frame is discarded.", "As we know the distribution of $Z$ for all pixels as a function of $\\mu _\\mathrm {e}$ , we can therefore calculate the probability of failure $p_f = 1-\\epsilon $ and the average min-entropy $\\overline{H}_{\\min }(Z_{23}\|E)$ per pixel of one frame for any distribution of the light intensity over the array.", "For predefined values of $\\epsilon $ and $H_{\\min }^l$ , appropriate parameters $T^\\pm $ and $N^\\pm $ can be found such that : $\\mathrm {Prob}\\left(\\overline{H}_{\\min }(Z_{23}\|E)\\le H_{\\min }^l\\right) \\le \\epsilon $ [figure]style=plain,subcapbesideposition=top Figure: Probability of failure and quantum entropy per bit of an array of 64 pixels uniformly illuminated as a function of the mean photo-electron number.As an example, we consider a chip with 64 pixels uniformly illuminated.", "In fig:pfail is plotted the probability of failure and the entropy per bit if the power of LED is drifting.", "The simulations were ran with $N^\\pm =1$ , $T^-=64$ and $T^+=940$ .", "With this configuration, we can see that the entropy per bit is only dropping in the region were the failure probability is equal to 1.", "Other scenarios (e.g.", "one or several pixels losing efficiency) give similar results.", "This provides a strong indication that the chip can provide long-term robustness against LED failures “in the field\" because it will raise an alarm before the quantum entropy is even impacted." ], [ "NIST tests", "The quality of our entropy source is assessed using the tests suite provided by NIST (details on the procedure can be found in [25].", "The IID (independent and identically distributed) track of the test suite gives an entropy estimation of over 0.998 per bit for 10 Mbytes samples, using MCV (most common value) estimator.", "This value is higher than the 0.98 per bit given in fig:finalentropy because the entropy test takes into account all sources of noise (quantum and classical) without distinction.", "If we run our simulations without considering that the classical is accessible to Eve, we obtain a value for the min-entropy of 0.999 per bit which is very close to the NIST result.", "This highlight an important advantage of our model compared to NIST entropy test.", "We can isolate the quantum contribution from the rest in order to calculate the quantum min-entropy.", "Figure: Typical results for the different entropy estimators on the NIST non-IID tests.", "The tests are carried out on 10 Mbytes samples.We also run the non-IID tests which consists of 10 different entropy estimators.", "Results are presented in fig:nist.", "This approach is more conservative as it takes the lowest value of all the estimators and does not assume that the IID hypothesis is true.", "Nevertheless, this method gives for our chip an entropy value over 0.94 per bit.", "We can note that this value is lower than the one given by our model.", "This difference comes from how the tests are done.", "The entropy estimation is based on some statistical properties of a sample with finite size output by the device.", "Due to statistical fluctuations, the entropy estimated will be slightly different than its true value.", "We ran these tests with other entropy sources and with DRBG and the entropy value we obtained was always around 0.94 which tends to show that it is a limitation of the tests and not of the chip." ], [ "Conclusion", "In this paper, we presented a physical model for the quantum entropy of the architecture on which the quantum random number generator of ID Quantique are based.", "With our model and after characterization of the device, we estimated that our chip can provide a quantum entropy of 0.98 per bit with a simple and low power consuming filtering of the bits.", "Finally, we show that the performances of the chip are robust against fluctuations over time making it suitable for mobile applications.", "This project was funded from the European Union's Horizon 2020 programme (Marie Skłodowska-Curie grant 675662) and from the European Union’s Horizon 2020 research and innovation programme under grant agreement N$^\\mathrm {o}$ 820405.", "We thank Florian Fröwis and Hyoungill Kim for helpful discussions." ] ]	2011.14129
[ [ "An Investigation of Language Model Interpretability via Sentence Editing" ], [ "Abstract Pre-trained language models (PLMs) like BERT are being used for almost all language-related tasks, but interpreting their behavior still remains a significant challenge and many important questions remain largely unanswered.", "In this work, we re-purpose a sentence editing dataset, where faithful high-quality human rationales can be automatically extracted and compared with extracted model rationales, as a new testbed for interpretability.", "This enables us to conduct a systematic investigation on an array of questions regarding PLMs' interpretability, including the role of pre-training procedure, comparison of rationale extraction methods, and different layers in the PLM.", "The investigation generates new insights, for example, contrary to the common understanding, we find that attention weights correlate well with human rationales and work better than gradient-based saliency in extracting model rationales.", "Both the dataset and code are available at https://github.com/samuelstevens/sentence-editing-interpretability to facilitate future interpretability research." ], [ "Introduction", "Pre-trained language models (PLMs) [8], [20], [3] are pervasively used in language-related tasks, but interpreting their predictions is notoriously difficult because of their parameters' complex inter-dependencies.", "Given a specific prediction, we want to know why a model made that decision, both to further improve performance and to use the model in high-stakes scenarios such as healthcare or bank loan approvals where interpretability is important.", "This has motivated efforts in extracting model explanations, typically in the form of rationales, i.e., subsets of the original input that support a decision [33].", "Attention heatmaps [32] and gradient-based saliency maps [27] are common extraction methods.", "Figure: Two “need edit” examples from AESW in the original data format and a human-readable format.The first example (a) has a spelling error “descripted” and the second (b) is edited for concision.There have been efforts on developing datasets for interpretability research, for example, the recent ERASER benchmark [9].", "However, the majority of ERASER tasks use human rationales highlighted by a different annotator after the original labeling process.", "Such rationales are not necessarily faithful; a rationale highlighted by the second annotator may not have been actually used by the first annotator while labeling.", "Manual rationale labeling is also difficult and time-consuming; of the six datasets in the ERASER benchmark, only one has more than 200 examples.", "Our first contribution is the realization that AESW (Automatic Evaluation of Scientific Writing; [6]), a sentence editing dataset, contains thousands of faithful human rationales that can be automatically re-purposed for interpretability research.", "See Figure REF for examples.", "This provides a new, large-scale dataset with truly faithful human rationales for interpretability questions surrounding model rationales.", "Our second contribution is investigating multiple factors in PLM rationale plausibility.", "More plausible rationales are valuable in human-in-the-loop systems where humans use model rationales to make a final decision.", "We compare (1) pre-training procedures, (2) attention weight- and input gradient-based methods of extracting model rationales, (3) correlation between model rationale plausibility and model confidence, and (4) differences in transformer layers.", "While previous work [13], [25] has shown that attention weights are not always faithful, we find that they correlate with human rationales better than gradient-based methods." ], [ "Related Work", "Human rationales (as defined by [33]) are subsets of input highlighted by human annotators as evidence to support a decision.", "The same annotator labeling an example might also highlight their rationale [15], [30].", "In other cases, rationales are collected for an existing dataset by different annotators [34], [4], [22].", "As previously stated, such rationales may not be faithful.", "Rationale length can vary from sub-sentence spans [29] to multiple sentences [19].", "Model rationales can be produced as an explicit training objective [34] or extracted as a post-hoc explanation.", "Post-hoc methods typically assign token-level importance scores: attention weights are often used in attention-based models [1], gradient-based explanations are typical for differentiable models [7], [26], and LIME is a model-agnostic method [23].", "We follow work using BERT's attention [5], [18] to extract rationales.", "A model rationale is evaluated on faithfulness (if it is actually used to make a decision) and plausibility (if it is easily understood by humans).", "Faithfulness can be measured by perturbing inputs marked as evidence and measuring change in outputs [13], [25].", "Plausibility can be measured through user studies, wherein users are given a model rationale and asked either to predict the model's decision [16] or to rate rationale understandability [21], [10], [11], [28].", "Rationale plausibility can also be measured by similarity to human rationales [9], but this requires faithful human rationales.", "We use similarity to evaluate rationale plausibility because we gather faithful human rationales from sentence editing annotations." ], [ "Proposed Task", "We propose re-purposing the AESW classification task for measuring model interpretability.", "We gather examples from AESW from which we can automatically extract faithful and sufficient human rationales, and then use said rationales to investigate factors in PLM rationale plausibility, specifically BERT [8] and its variants.", "It is worth noting that our rationale dataset can be used for other interpretability topics such as training with rationales or evaluating rationale faithfulness; we will focus on plausibility considering the scope of this paper." ], [ "Human Rationales", "Human rationales are substrings used as evidence for a decision [33].", "Faithful and sufficient (enough evidence to justify a decision) human rationales can be used as gold labels for evaluating model rationale plausibility.", "The original AESW task is to predict if a sentence from a scientific paper needs editing.", "[6] extract spans of a sentence before and after professional editingNative English speaking editors working at VTeX.", "as deleted (<del>) or inserted (<ins>) and provide 1.1M training, 140K validation and 140K testing examples.", "Sentences without changes are assumed to not require editing.", "We exploit the data format to automatically extract faithful and sufficient human rationales.", "Delete text (text between <del> tags) is always a faithful rationale (such text is a source of the error).", "For edits with arbitrary <del> text alone and <ins> text, the <del> alone is not always a sufficient rationale to justify “need edit”.", "Consider a sentence where a verb is incorrectly conjugated and replaced with the correctly conjugated verb.", "The incorrectly conjugated verb is not sufficient to decide that the sentence needs editing; the surrounding context is required.", "To find edits where <del> text is always a sufficient rationale, we use two criteria: A spelling error is corrected.", "(spelling error) Text is only deleted, not added.", "(deleted text) Spelling errors are always a sufficient rationale to justify editing a sentence (see Figure REF a).", "In edits with no insertions, removing the <del> text leads to an error-free sentence, so the <del> text is sufficient explanation for editing (see Figure REF b).", "These criteria lead to both a simple, lexical task (spelling error) and a more challenging semantic task (deleted text) and constitute a wider range of challenges for future interpretability research.", "We extract faithful and sufficient human rationales for 1,321 spelling error edits and 6,741 deleted text edits from the validation set of AESW.More details on extracting human rationales, as well as additional examples, can be found in Appendix" ], [ "Model Rationales", "Model rationales are substrings provided by a model as evidence for a decision.", "Given a model, an example $x_i$ and a prediction $y_i$ , we extract three model rationales.", "First, we use attention maps [32], [18] to rank word relevance.", "We measure the total attention weight from BERT's final layer's initial [CLS] token to each token $t_j$ across $H$ attention heads.", "Then we add those totals together for each token $t_j$ in a word $w$ : $\\text{score}(w) = \\sum _{t_j \\in w}\\sum _{h = 1}^{H} \\text{Attn}_{h_j}\\left( \\text{[CLS]} \\rightarrow t_j \\right).$ Second, we use gradient-based saliency (specifically gradient$\\times $ input; [7], [26]) to rank word relevance.", "We calculate a saliency score for each token $t_j$ in $x_i$ .", "First, change in model output with respect to $t_j$ 's input embedding $\\nabla _{e(t_j)}f_{y_i}(x_i)$ captures the sensitivity to token $t$ .", "The dot product with $e(t_j)$ is then a scalar measure of each token's marginal impact on the model prediction [12].", "Finally, we again compute a word-level score by summing over each token $t_j$ in word $w$ : $\\text{score}(w) = \\sum _{t_j \\in w}\\nabla _{e(t_j)}f_{y_i}(x_i) \\cdot e(t_j)$ In contrast to attention, gradient$\\times $ input faithfully measures the marginal effect of each input token on the prediction [2].", "Finally, we extract a third set of rankings using gradient$\\times $ input's magnitude to rank words." ], [ "Evaluation", "We evaluate extracted rationale plausibility using similarity to human rationales [9].", "We use the continuous word scores generated in the previous section to rank relevance, then use mean reciprocal rank as an evaluation metric.We performed our analyses with additional, similar metrics; our main observations remain consistent independent of the choice of metric.", "Appendix contains additional details and complete results." ], [ "Experiments", "To demonstrate the utility of the AESW task for interpretability research, we present four experiments, each with the goal of understanding factors in PLM rationale plausibility.", "For our experiments, we add a linear layer and sigmoid activation function on top of the [CLS] token representation, fine-tune BERT-base end-to-end on the AESW training set using the original training objective (classify a sentence as “need edit” or “no edit”) and use validation loss to tune hyperparameters.", "We do not add any interpretability or rationale-related objectives.", "The classification F1 for the models on spelling error and deleted text edits is 82.0 and 74.0, respectively.Appendix contains more details on fine-tuning procedure and results.", "We extract and evaluate model rationales on all spelling error and deleted text edits, even edits that a model does not correctly predict as “need edit.”" ], [ "Does pre-training procedure affect rationale plausibility?", "We are interested in how pre-training affects plausibility after fine-tuning (to the best of our knowledge, this is a previously unexplored topic).", "We compare BERT and two variants, RoBERTa [20] and SciBERT [3].", "These models have the same architecture but differ in pre-training: RoBERTa is pre-trained for 5x longer with 10x more data than BERT, and SciBERT is pre-trained on a corpus of academic papers.", "We extract model rationales using attention weights and evaluate their plausibility.", "As seen in Figure REF , RoBERTa and BERT generate nearly equally plausible rationales despite differences in pre-training corpus size.", "We hypothesize that SciBERT generates less plausible rationales because it encodes “need edit” representations in earlier layers (rather than in the final layer), then attends to [SEP] as a no-op in later layers, as proposed in [5] and [17] and further confirmed in the subsequent experiments.", "Figure: Left: BERT, RoBERTa and SciBERT's attention weight rationale plausibility.Right: BERT's attention weight, gradient×\\times input and \|gradient×input\|\\vert \\text{gradient}\\times \\text{input}\\vert rationale plausibility (see Section )." ], [ "Do attention weights or input gradients produce better rationales?", "In contrast to attention weights, gradient$\\times $ input scores are naturally faithful with respect to individual feature importance [13], [2].", "However, attention weights can represent word relevance in context, potentially leading to more plausible rationales.", "We extract and evaluate rationales using attention weights and the two gradient$\\times $ input methods described in Section REF .", "Figure REF shows that attention-based rationales are more plausible and that the difference is more pronounced on deleted text edits.", "Using $\\vert \\text{gradient}\\times \\text{input}\\vert $ (right-most) also shows improvements over directional gradient$\\times $ input (middle-right), in contrast to [12]." ], [ "Are plausibility and confidence correlated?", "We are curious if BERT is more confident in its classification decisions when it attends to the original evidence used by editors.", "We quantify BERT's classification confidence by treating the sigmoid activation function's value as the probability of a “need edit” decision.", "We only consider examples that BERT correctly classifies as “need edit” and calculate BERT's mean confidence.", "We find that when human rationales are BERT's most attended-to words, it is 11.7% and 11.9% more confident in its predictions for spelling error and deleted text edits, respectively.", "One possible explanation for such a correlation is that easy-to-classify examples also have easy-to-identify rationales.", "Because agreement with human rationales is not the training objective, however, we believe this correlation suggests that BERT learns to classify edits similarly to humans." ], [ "How does transformer layer affect plausibility?", "It is widely agreed that BERT's early layers encode more lexical and phrasal information than other layers [14], [24].", "We hypothesize that rationales extracted from early layers will be more plausible for spelling error than deleted text edits because it is a more lexical task.", "We extract rationales from each layer's attention weights and measure their plausibility in Figure REF .", "[4] BERT shows a decline in plausibility at layer 11 in both edit types because it attends heavily to periods.", "The results confirm the hypothesis and show that early layers in BERT models are indeed more lexically-oriented.", "We also find that SciBERT strongly attends to spelling errors in earlier layers.", "We believe it is because that, pre-trained on well-formed academic text, SciBERT is not as well exposed to spelling errors as BERT and RoBERTa so it learns to attend to spelling errors in earlier layers during fine-tuning.", "Figure: Mean reciprocal rank for each layer (using the mean strategy) for each model for spelling error and deleted text edits.", "[4]" ], [ "Conclusion", "We re-purpose the AESW task to gather thousands of inherently faithful human rationales and investigate an array of questions regarding PLM interpretability.", "We find, among other new insights, that attention weights correlate well with human rationales and produce more plausible rationales than input gradients, which is different from existing understanding.", "Furthermore, we find that BERT is more confident in its predictions when it attends to the same words that a human did, supporting the idea that while attention is not inherently faithful, attention-based models might rely on the same information as humans when making a prediction.", "Future work might expand the subset of examples for which human rationales can be automatically extracted, include human rationales during training or evaluate faithful-by-design model rationales on this dataset." ], [ "Acknowledgements", "The authors would like to extend their thanks to the anonymous reviewers for their insightful feedback." ], [ "AESW Details", "The original AESW task is to classify a sentence from an academic paper as “need edit” or “no edit”.", "There is no special markup attached to the sentence when it is given as input to a model.", "Task participants receive the training and validation sets with <del> and <ins> tokens for model development, and [6] provided an automatic test set evaluation through an online portal.", "[6] collected these sentences from 9,919 journal articles published by Springer Publishing Company and edited at VTeX.", "Sentences before and after editing were automatically aligned using a modified diff algorithm.More details can be found in [6] After the challenge, [6] released the dataset in its entirety (all three datasets with all tokens/spans included)." ], [ "Training Details", "The AESW task uses scientific papers written in LaTeX, which contains markup characters that impact sentence meaning.", "The original authors ([6]) replace these characters with special tokens, as seen in Table REF .", "We add these four special tokens (_MATH_, _MATHDISP_, _CITE_ and _REF_) to the model vocabulary, fine-tuning the word representations during training.", "Table: Special tokens found in the original AESW data that should not be split further into bytes/tokens.We train all models for a maximum of 30 epochs with a patience of 5 on a single Tesla P100 GPU.", "All models (BERThttps://huggingface.co/transformers/v3.0.2/model_doc/bert.html#bertforsequenceclassification, SciBERThttps://github.com/allenai/scibert#pytorch-huggingface-models, RoBERTAhttps://huggingface.co/transformers/v3.0.2/model_doc/roberta.html#robertaforsequenceclassification) are based on their HuggingFace [31] implementations.", "We list all the key hyperparameters and tuning bounds for reproducibility in Table REF .", "Our final results for all three BERT-based models along with the top three models from [6] can be found in Table REF .", "Additionally, we will release code and instructions for reproducing our results.", "Table: Performance on the original AESW sentence classification task.", "Dev set results are not available for models reported in .Table: Hyperparameter options for each model.", "Note that each model had 4 special tokens added to the vocabulary.", "BERT was fine-tuned first.", "Because of compute limitations, RobERTa and SciBERT were both fine-tuned using the same hyperparameters as the optimal BERT configuration (learning rate of (1×10 -6 1 \\times 10^{-6}))." ], [ "Gathering Human Rationales", "To find spelling error edits, we look for sentences with a deleted, misspelled word followed by an inserted, correctly spelled word.", "The first two examples in Figure REF are examples of spelling error edits, while the third is not.", "To find deleted text edits, we look for sentences where text is removed but not added.", "The fourth and fifth examples in Figure REF are examples of deleted text edits.", "Figure: Additional examples of the two types of edits extracted from the original AESW dataset." ], [ "Evaluating Model Rationales", "Although the main text presents our analyses using mean reciprocal rank, we performed our analyses with multiple metrics.", "Here we provide specific definitions for our metrics and our complete results for all models with every metric." ], [ "Mean reciprocal rank", "Because human rationales can be made up of multiple words, we need to modify mean reciprocal ranking.", "For a single sentence, a model's ordered ranking of words $M$ and an unordered human rationale $H$ : Find the top ranked word $w$ in $M$ from $H$ and record the rank.", "Remove $w$ from $M$ and $H$ .", "Repeat until $H$ is empty.", "Use the reciprocal of the mean rank.", "If we did not remove words from $M$ , a perfect score for a rationale with multiple words would be impossible.", "Consider the sentence “The boy ate the found his ball.” If the rationale ranked `ate' and `the' as most important, the mean rationale rank would be $(1 + 2) / 2 = 1.5$ and the rationale's reciprocal rank would be ${1}{1.5} = 0.66$ , despite a perfect rationale.", "We take the mean reciprocal ranking across all examples to evaluate rationales." ], [ "Mean area under precision-recall curve", "Using the model relevance scores for words in a sentence, we adjust the threshold for classifying a word as part of the rationale, calculate a precision-recall curve and measure the area underneath.", "We take the mean AUPRC across all examples." ], [ "Mean top 1 match", "We score a rationale as 1 if the rationale's top ranked word is the human rationale and 0 otherwise.", "This means that all multiple-word rationales are automatically not a match and scored as 0.", "We take the mean of these scores across all examples.", "Our main observations are consistent across metrics: Table REF contains all results for models on spelling error edits and Table REF contains all results for models on deleted text edits.", "Table: Results for all three models for spelling error edits across all methods of selecting rationales (attention, gradient×\\times input and \|gradient×input\|\\vert \\text{gradient}\\times \\text{input}\\vert ) and all metrics to evaluate rationales (mean reciprocal ranking, AUPRC and mean top 1 match).Table: Results for all three models for deleted text edits across all methods of selecting rationales (attention, gradient×\\times input and \|gradient×input\|\\vert \\text{gradient}\\times \\text{input}\\vert ) and all metrics to evaluate rationales (mean reciprocal ranking, AUPRC and mean top 1 match)." ] ]	2011.14039
[ [ "On linear shifts of finite type and their endomorphisms" ], [ "Abstract Let $G$ be a group and let $A$ be a finite-dimensional vector space over an arbitrary field $K$.", "We study finiteness properties of linear subshifts $\\Sigma \\subset A^G$ and the dynamical behavior of linear cellular automata $\\tau \\colon \\Sigma \\to \\Sigma$.", "We say that $G$ is of $K$-linear Markov type if, for every finite-dimensional vector space $A$ over $K$, all linear subshifts $\\Sigma \\subset A^G$ are of finite type.", "We show that $G$ is of $K$-linear Markov type if and only if the group algebra $K[G]$ is one-sided Noetherian.", "We prove that a linear cellular automaton $\\tau$ is nilpotent if and only if its limit set, i.e., the intersection of the images of its iterates, reduces to the zero configuration.", "If $G$ is infinite, finitely generated, and $\\Sigma$ is topologically mixing, we show that $\\tau$ is nilpotent if and only if its limit set is finite-dimensional.", "A new characterization of the limit set of $\\tau$ in terms of pre-injectivity is also obtained." ], [ "Introduction", "Let $G$ be a group and let $A$ be a set, called the alphabet.", "The set $A^G \\lbrace x \\colon G \\rightarrow A\\rbrace $ , consisting of all maps from $G$ to $A$ , is called the set of configurations over the group $G$ and the alphabet $A$ .", "We equip $A^G = \\prod _{g \\in G} A$ with its prodiscrete uniform structure, i.e., the product uniform structure obtained by taking the discrete uniform structure on each factor $A$ of $A^G$ .", "Thus, two configurations are “close” if they coincide on a “large” subset of $G$ .", "Note that $A^G$ is a totally disconnected Hausdorff space and that $A^G$ is compact if and only if $A$ is finite.", "The shift action of the group $G$ on $A^G$ is the action defined by $(g,x) \\mapsto g x$ , where $gx(h) x(g^{-1}h)$ for all $g,h \\in G$ and $x \\in A^G$ .", "This action is uniformly continuous with respect to the prodiscrete uniform structure.", "A closed $G$ -invariant subset $\\Sigma \\subset A^G$ is called a subshift of $A^G$ .", "Given subsets $D \\subset G$ and $P \\subset A^D$ , the set $\\Sigma (D,P) = \\Sigma (A^G;D,P) \\lbrace x \\in A^G: (g^{-1} x)\\vert _{D} \\in P \\text{ for all } g \\in G\\rbrace $ is a $G$ -invariant subset of $A^G$ (here $(g^{-1}x)\\vert _D \\in A^D$ denotes the restriction of the configuration $g^{-1}x$ to $D$ ).", "When $D$ is finite, $\\Sigma (D,P)$ is also closed in $A^G$ , and therefore is a subshift.", "One then says that $\\Sigma (D,P)$ is the subshift of finite type, briefly SFT, associated with $(D,P)$ and that $D$ (resp.", "$P$ ) is a defining memory set (resp.", "a defining set of admissible patterns) for $\\Sigma $ .", "Note that a defining set of admissible patterns for an SFT is not necessarily finite.", "Let $B$ be another alphabet set.", "A map $\\tau \\colon B^G \\rightarrow A^G$ is called a cellular automaton, briefly a CA, if there exist a finite subset $M \\subset G$ and a map $\\mu \\colon B^M \\rightarrow A$ such that $\\tau (x)(g) = \\mu ((g^{-1}x)\\vert _M) \\quad \\text{for all } x \\in B^G \\text{ and } g \\in G.$ Such a set $M$ is then called a memory set and $\\mu $ is called a local defining map for $\\tau $ .", "It is immediate from the above definition that every CA $\\tau \\colon B^G \\rightarrow A^G$ is uniformly continuous and $G$ -equivariant (cf.", "[6], see also [8]).", "More generally, if $\\Sigma _1 \\subset B^G$ and $\\Sigma _2 \\subset A^G$ are subshifts, a map $\\tau \\colon \\Sigma _1 \\rightarrow \\Sigma _2$ is called a CA if it can be extended to a CA $\\tilde{\\tau } \\colon B^G \\rightarrow A^G$ .", "Suppose now that $A$ and $B$ are vector spaces over a field $K$ .", "Then $A^G$ and $B^G$ inherit a natural $K$ -vector space structure.", "A subshift $\\Sigma \\subset A^G$ which is also a vector subspace of $A^G$ is called a linear subshift.", "A $K$ -linear CA $\\tau \\colon B^G \\rightarrow A^G$ is called a linear CA.", "Note that a CA $\\tau \\colon B^G \\rightarrow A^G$ with memory set $M \\subset G$ is linear if and only if the associated local defining map $\\mu \\colon B^M \\rightarrow A$ is $K$ -linear (see [8]).", "More generally, given linear subshifts $\\Sigma _1 \\subset B^G$ and $\\Sigma _2 \\subset A^G$ , a map $\\tau \\colon \\Sigma _1 \\rightarrow \\Sigma _2$ is called a linear CA if it is the restriction of some linear CA $\\tilde{\\tau } \\colon B^G \\rightarrow A^G$ .", "A linear subshift $\\Sigma \\subset A^G$ is called a linear-sofic subshift provided there exists a vector space $B$ , a linear SFT $\\Sigma ^{\\prime } \\subset B^G$ , and a linear CA $\\tau \\colon B^G \\rightarrow A^G$ such that $\\tau (\\Sigma ^{\\prime }) = \\Sigma $ .", "In our recent papers [12], [26] we introduced the notion of an algebraic sofic subshift $\\Sigma \\subset A^G$ , where $A$ is the set of $K$ -points of an algebraic variety over an algebraically closed field $K$ , and studied cellular automata $\\tau \\colon \\Sigma \\rightarrow \\Sigma $ whose local defining maps are induced by algebraic morphisms.", "When referring to these notions, we shall refer to the “algebraic setting”.", "When the field $K$ is algebraically closed, linear-sofic subshifts and linear cellular automata are algebraic sofic subshifts and algebraic cellular automata, respectively.", "Therefore, several results in [12], [26] hold true in the present setting, even when the field $K$ is not algebraically closed, by a direct adaptation of the proofs given therein.", "However the proofs in the algebraic setting are much more technical and involved, and one of the purposes of this paper is to present simpler and more direct proofs of these results in the linear setting.", "We also obtain several new results and consequences as indicated below.", "Our first result is a linear version of the well known characterization of SFT with finite alphabets by the descending chain condition (see [12] and [26] for a similar result in the algebraic setting, and [26] for the more general admissible group shifts, [26]).", "Theorem 1.1 Let $G$ be a countable group and let $A$ be a finite-dimensional vector space over a field $K$ .", "Let $\\Sigma \\subset A^G$ be a linear subshift.", "Then the following conditions are equivalent: $\\Sigma $ is a SFT; every decreasing sequence of linear subshifts of $A^G$ $\\Sigma _0 \\supset \\Sigma _1 \\supset \\cdots \\supset \\Sigma _n \\supset \\Sigma _{n+1} \\supset \\cdots $ such that $\\Sigma = \\bigcap _{n \\in \\mathbb {N}} \\Sigma _n$ , eventually stabilizes (that is, there exists $n_0 \\in \\mathbb {N}$ such that $\\Sigma _{n_0} = \\Sigma _n$ for all $n \\ge n_0$ ).", "Corollary 1.2 Let $G$ be a countable group and let $A$ be a finite-dimensional vector space over a field $K$ .", "Then the following conditions are equivalent: every linear subshift $\\Sigma \\subset A^G$ is an SFT; $A^G$ satisfies the descending chain condition for linear subshifts, that is, every decreasing sequence of linear subshifts of $A^G$ $\\Sigma _0 \\supset \\Sigma _1 \\supset \\cdots \\supset \\Sigma _n \\supset \\Sigma _{n+1} \\supset \\cdots $ eventually stabilizes.", "Given a field $K$ , we say that a group $G$ is of $K$ -linear Markov type provided that the equivalent conditions in Corollary REF hold for every finite-dimensional vector space $A$ over $K$ .", "Let $G$ be a group and let $K$ be a field.", "Given $\\alpha \\in K[G]$ we define $\\alpha ^* \\in K[G]$ by setting $\\alpha ^(g) \\alpha (g^{-1})$ for all $g \\in G$ .", "It is straightforward to check that the map $\\alpha \\mapsto \\alpha ^$ yields a $K$ -algebra isomorphism of the group algebra $K[G]$ onto the opposite algebra $K[G]^{opp}$ .", "As a consequence, $K[G]$ is left-Noetherian if and only if it is right-Noetherian and, if this is the case, we simply say that $K[G]$ is one-sided Noetherian.", "In the proofs, however, in order to use a working definition at hand, we shall always refer to left Noetherianity.", "We have the following characterization of countable groups of $K$ -linear Markov type.", "Theorem 1.3 Let $G$ be a countable group and let $K$ be a field.", "Then the group algebra $K[G]$ is one-sided Noetherian if and only if $G$ is of $K$ -linear Markov type.", "Recall that a group $G$ is said to be polycyclic if it admits a subnormal series with cyclic factors, that is, a finite sequence $G = G_0 \\supset G_1 \\supset \\cdots \\supset G_n = \\lbrace 1_G\\rbrace $ of subgroups such that $G_{i+1}$ is normal in $G_i$ and $G_i/G_{i+1}$ is (possibly infinite) cyclic group, for $i=0,1,\\ldots , n-1$ .", "More generally, $G$ is said to be polycyclic-by-finite if it admits a polycyclic subgroup of finite index.", "The following result is the linear version of a famous result by Klaus Schmidt [30] (see also [20]).", "The third-named author considered the notion of an admissible Artinian group structure (cf.", "[26]): this includes, for instance, a group structure of finite Morley rank, e.g.", "an algebraic group, an Artinian group, or an Artinian module.", "Since a finite-dimensional vector space over a field $K$ is an Artinian $K$ -module and therefore it is naturally equipped with an admissible Artinian group structure (see [26]), Corollary REF also constitutes a simpler case of the more general results [26] or [26], where the alphabet set can be taken as an admissible Artinian group structure.", "Also note that the proof of [30] heavily relies on the fact that the alphabet set therein is a compact Lie group, so that the configuration space (equipped with the product topology) is itself compact.", "We deduce Corollary REF below from Theorem REF and a result of Philip Hall ([18], see also [24]) extending Hilbert's basis theorem on Noetherian rings.", "An alternative self-contained proof of Corollary REF , using only linear symbolic dynamics, is presented in Remark REF .", "Corollary 1.4 Let $K$ be a field.", "Then all polycyclic-by-finite groups (e.g., the free abelian groups $\\mathbb {Z}^d$ , $d \\ge 1$ ) are of $K$ -linear Markov type.", "The free group $F_2$ (and, more generally, any group which contains a subgroup which is not finitely generated) is not of $K$ -linear Markov type (see Section ).", "Groups of $K$ -linear Markov type, or, more generally, monoids of $K$ -linear Markov type, satisfy interesting topological properties.", "For example, it is shown in [27] that the natural action of every finitely generated abelian monoid of linear CA on any linear subshift satisfies the shadowing property.", "Let now $f \\colon X \\rightarrow X$ be a selfmap of a set $X$ .", "One has $X \\supset f(X) \\supset f^2(X) \\supset \\cdots \\supset f^n(X) \\supset f^{n+1}(X) \\supset \\cdots $ and the set $\\Omega (f) \\bigcap _{n \\ge 1} f^n(X) \\subset X$ is called the limit set of $f$ .", "This is the set of points of $X$ that occur after iterating $f$ arbitrarily many times.", "The notion of a limit set was introduced in the framework of cellular automata by Wolfram [31] and was subsequently investigated for instance in [14], [17], [19], [22], and [12].", "Observe that $f(\\Omega (f)) \\subset \\Omega (f)$ .", "The inclusion may be strict (cf.", "[12] and Example (3) in Section REF ) and equality holds if and only if every $x \\in \\Omega (f)$ admits a backward orbit, i.e., a sequence $(x_i)_{i \\ge 0}$ of points of $X$ such that $x_0 = x$ and $f(x_{i + 1}) = x_i$ for all $i \\ge 0$ .", "Clearly, $f$ is surjective if and only if $\\Omega (f) = X$ .", "Note also that $\\operatorname{Per}(f) \\bigcup _{n \\ge 1} \\lbrace x\\in X: f^n(x)=x\\rbrace $ , the set of $f$ -periodic points, is contained in $\\Omega (f)$ and that $\\Omega (f^n) = \\Omega (f)$ for every $n \\ge 1$ .", "One says that the map $f$ is stable if $f^{n+1}(X)=f^n(X)$ for some $n \\ge 1$ .", "Assume that $X$ is a topological space and $f \\colon X \\rightarrow X$ is a continuous map.", "One says that $x \\in X$ is a recurrent (resp.", "non-wandering) point of $f$ if for every neighborhood $U$ of $x$ , there exists $n \\ge 1$ such that $f^n(x) \\in U$ (resp.", "$f^n(U)$ meets $U$ ).", "Let $\\operatorname{R}(f)$ (resp.", "$\\operatorname{NW}(f)$ ) denote the set of recurrent (resp.", "non-wandering) points of $f$ .", "It is immediate that $\\operatorname{Per}(f) \\subset \\operatorname{R}(f) \\subset \\operatorname{NW}(f)$ and that $\\operatorname{NW}(f)$ is a closed subset of $X$ .", "Suppose now that $X$ is a uniform space and let $f \\colon X \\rightarrow X$ be a uniformly continuous map.", "One says that a point $x \\in X$ is chain-recurrent if for every entourage $E$ of $X$ there exist an integer $n \\ge 1$ and a sequence of points $x_0,x_1,\\dots ,x_n \\in X$ such that $x = x_0 = x_n$ and $(f(x_i),x_{i + 1}) \\in E$ for all $0 \\le i \\le n - 1$ .", "We shall denote by $\\operatorname{CR}(f)$ the set of chain-recurrent points of $f$ .", "Observe that $\\operatorname{CR}(f)$ is always closed in $X$ .", "We shall establish the following result (compare with [12] in the algebraic setting).", "Theorem 1.5 Let $G$ be a finitely generated group and let $A$ be a finite-dimensional vector space over a field $K$ .", "Let $\\Sigma \\subset A^G$ be a linear subshift and let $\\tau \\colon \\Sigma \\rightarrow \\Sigma $ be a linear CA.", "Then the following hold: $\\Omega (\\tau )$ is a linear subshift of $A^G$ ; $\\tau (\\Omega (\\tau )) = \\Omega (\\tau )$ ; $\\operatorname{Per}(\\tau ) \\subset \\operatorname{R}(\\tau ) \\subset \\operatorname{NW}(\\tau ) \\subset \\operatorname{CR}(\\tau ) \\subset \\Omega (\\tau )$ ; if $\\Omega (\\tau )$ is of finite type then $\\tau $ is stable; if $\\Omega (\\tau )$ is finite-dimensional then $\\tau $ is stable.", "In the above theorem, we may relax the condition on $G$ being finitely generated provided we assume in addition that the linear subshift $\\Sigma \\subset A^G$ is linear-sofic.", "We thus have the following.", "Corollary 1.6 Let $G$ be a group and let $A$ be a finite-dimensional vector space over a field $K$ .", "Let $\\Sigma \\subset A^G$ be a linear-sofic subshift (e.g., a linear SFT) and let $\\tau \\colon \\Sigma \\rightarrow \\Sigma $ be a linear CA.", "Then properties (i) – (v) in Theorem REF hold.", "As in [12], the proof relies on the analysis of the so called space-time inverse system associated with a CA (cf.", "Section REF ).", "Let $G$ be a group and let $A$ be a set.", "A CA $\\tau \\colon \\Sigma _1 \\rightarrow \\Sigma _2$ between subshifts of $A^G$ is called pre-injective if whenever $x,y \\in A^G$ are two configurations that coincide outside of a finite subset of $G$ and satisfy $\\tau (x)=\\tau (y)$ , then one has $x=y$ .", "When $A$ is a vector space over a field $K$ and $\\Sigma _1, \\Sigma _2 \\subset A^G$ are linear subshifts, a linear CA $\\tau \\colon \\Sigma _1 \\rightarrow \\Sigma _2$ is pre-injective if and only if the restriction of $\\tau $ to the vector subspace of configurations in $\\Sigma _1$ with finite support is injective.", "A subshift $\\Sigma \\subset A^G$ is called strongly irreducible if there exists a finite subset $\\Delta \\subset G$ such that for all $x, y \\in A^G$ and for all finite subsets $E, F \\subset G$ such that $E \\cap F \\Delta = \\varnothing $ , then there exists $z \\in \\Sigma $ such that $z\\vert _E = x\\vert _E$ and $z\\vert _F = y\\vert _F$ .", "We obtain the following characterization of limit sets of linear cellular automata in terms of pre-injectivity.", "Corollary 1.7 Let $G$ be a polycyclic-by-finite group and let $A$ be a finite-dimensional vector space.", "Let $\\Sigma \\subset A^G$ be a strongly irreducible linear subshift and let $\\tau \\colon \\Sigma \\rightarrow \\Sigma $ be a linear CA.", "Then $\\Omega (\\tau )$ is the largest strongly irreducible linear subshift $\\Lambda \\subset A^G$ contained in $\\Sigma $ such that $\\tau (\\Lambda ) \\subset \\Lambda $ and $\\tau \\vert _{\\Lambda }$ is pre-injective.", "Given a set $X$ , one says that a map $f \\colon X \\rightarrow X$ is nilpotent if there exist a constant map $c \\colon X \\rightarrow X$ and an integer $n_0 \\ge 1$ such that $f^{n_0} = c$ .", "This implies $f^n = c$ for all $n \\ge n_0$ .", "Such a constant map $c$ is then unique and we say that the unique point $x_0 \\in X$ such that $c(x) = x_0$ for all $x \\in X$ is the terminal point of $f$ .", "The terminal point of a nilpotent map is its unique fixed point.", "Observe that if $f \\colon X \\rightarrow X$ is nilpotent with terminal point $x_0$ then $\\Omega (f) = \\lbrace x_0\\rbrace $ is a singleton.", "The converse is not true in general (cf.", "[12] and Example (1) in Section REF ).", "For linear cellular automata we establish the following characterization of nilpotency.", "Theorem 1.8 Let $G$ be a group and let $A$ be a finite-dimensional vector space over a field $K$ .", "Let $\\Sigma \\subset A^G$ be a linear-sofic subshift (e.g., a linear SFT) and let $\\tau \\colon \\Sigma \\rightarrow \\Sigma $ be a linear CA.", "Then the following conditions are equivalent: $\\tau $ is nilpotent; $\\Omega (\\tau ) = \\lbrace 0\\rbrace $ .", "The analog of Theorem REF for classical cellular automata follows from [14].", "In the algebraic setting, this corresponds to [12].", "Given a set $X$ , one says that a map $f \\colon X \\rightarrow X$ is pointwise nilpotent if there exist a point $x_0 \\in X$ such that for each $x \\in X$ there exists an integer $n_x \\ge 1$ such that $f^n(x) = x_0$ for all $n \\ge n_x$ .", "Such a point $x_0$ is clearly unique and it is called the terminal point of $f$ .", "If $f$ is nilpotent then it is also pointwise nilpotent and the terminal points relative to the two notions of nilpotency coincide.", "Let $G$ act on a Hausdorff topological space $X$ .", "One says that the action is topologically mixing provided that given two nonempty open subsets $U, V \\subset X$ there exists a finite subset $F \\subset G$ such that $U \\cap gV \\ne \\varnothing $ for all $g \\in G \\setminus F$ .", "If $A$ is a set, a subshift $\\Sigma \\subset A^G$ is said to be topologically mixing if the restriction to $\\Sigma $ of the $G$ -shift is topologically mixing.", "We obtain the following characterization of nilpotency for linear cellular automata over infinite groups.", "Theorem 1.9 Let $G$ be a finitely generated infinite group and let $A$ be a finite-dimensional vector space over a field $K$ .", "Let $\\Sigma \\subset A^G$ be a topologically mixing linear subshift (e.g., $\\Sigma = A^G$ ) and let $\\tau \\colon \\Sigma \\rightarrow \\Sigma $ be a linear CA.", "Then the following conditions are equivalent: $\\tau $ is nilpotent; $\\tau $ is pointwise nilpotent; there exists $n_0 \\in \\mathbb {N}$ such that $\\tau ^{n_0}(\\Sigma )$ is finite-dimensional; $\\Omega (\\tau )$ is finite-dimensional; $\\Omega (\\tau ) = \\lbrace 0\\rbrace $ .", "In the above theorem, we may relax the condition of being finitely generated on the infinite group $G$ provided we assume that the subshift $\\Sigma \\subset A^G$ is, in addition, linear-sofic.", "We thus have the following.", "Corollary 1.10 Let $G$ be an infinite group and let $A$ be a finite-dimensional vector space over a field $K$ .", "Let $\\Sigma \\subset A^G$ be a topologically mixing linear-sofic subshift (e.g., $\\Sigma = A^G$ ) and let $\\tau \\colon \\Sigma \\rightarrow \\Sigma $ be a linear CA.", "Then conditions (a) – (e) in Theorem REF are all equivalent.", "The analog of Theorem REF for classical cellular automata follows from [14] (see also [17]).", "In the algebraic setting, this corresponds to [12], but the equivalence of (a), (b), (c), and (e) with the point (d) is a new result.", "The paper is organized as follows.", "In Section we fix notation and establish some preliminary results.", "In particular, we study linear SFTs and show that every finite-dimensional linear subshift $\\Sigma \\subset A^G$ is of finite type if the group $G$ is finitely generated (Proposition REF ).", "We introduce the notion of a memory set for a linear-sofic subshift $\\Sigma $ and, given a cellular automaton $\\tau \\colon \\Sigma \\rightarrow \\Sigma $ , we review the properties of the restriction cellular automaton $\\tau _H \\colon \\Sigma _H \\rightarrow \\Sigma _H$ , for any subgroup $H$ containing both a memory set for $\\tau $ and a memory set for $\\Sigma $ .", "As an application, we establish a relation between the limit sets $\\Omega (\\tau )$ and $\\Omega (\\tau _H)$ of $\\tau $ and $\\tau _H$ , respectively, and deduce that $\\tau $ is nilpotent if and only if $\\tau _H$ is nilpotent (Lemma REF ).", "In Section we review from [12] the notion of space-time inverse system, together with its inverse limit, associated with a cellular automaton $\\tau \\colon \\Sigma \\rightarrow \\Sigma $ , where $\\Sigma $ is a linear-sofic subshift in $A^G$ , with $G$ a countable group and $A$ a finite-dimensional vector space over a field $K$ .", "As an application, in the subsequent section we prove the closed image property for linear cellular automata: we show that essentially under the above assumptions, $\\tau (\\Sigma )$ is closed in the prodiscrete topology in $A^G$ (Theorem REF ).", "In Section we present the proofs of all results stated in the Introduction.", "In Section we further investigate the class of groups of $K$ -linear Markov type.", "We show that this class is closed under the operation of taking subgroups, quotients, and extensions by finite and cyclic groups, and that it is contained in the class of Noetherian groups (the latter are the groups satisfying the maximal condition on subgroups).", "Finally, in the last section we present some examples/counterexamples and discuss some further remarks.", "In particular, in Subsection REF we present an example of a linear cellular automaton $\\tau \\colon A^G \\rightarrow A^G$ , where $G$ is any non-periodic group (e.g., $G = \\mathbb {Z}$ ) and $A$ is any infinite-dimensional vector space, which does not satisfy the closed image property.", "At last, in Subsection REF we study nilpotency and pointwise nilpotency for linear cellular automata over infinite-dimensional vector spaces and present some examples of the associated limit sets.", "As a byproduct, we show that the conclusions of Theorem REF may fail to hold, in general, if the finite-dimensionality of the alphabet set $A$ is dropped from the assumptions.", "Acknowledgments.", "We express our deepest gratitude to the anonymous referee for his thorough reading, for pointing out to us a few inaccuracies, and providing most valuable comments and remarks." ], [ "Notation", "We use the symbols $\\mathbb {Z}$ for the integers and $\\mathbb {N}$ for the non-negative integers.", "We write $A^B$ for the set consisting of all maps from a set $B$ into a set $A$ .", "Let $C \\subset B$ .", "If $x \\in A^B$ , we denote by $x\\vert _C$ the restriction of $x$ to $C$ , that is, the map $x\\vert _C \\colon C \\rightarrow A$ given by $x\\vert _C(c) = x(c)$ for all $c \\in C$ .", "If $X \\subset A^B$ , we set $X_C \\lbrace x\\vert _C \\colon x \\in X \\rbrace \\subset A^C$ .", "Let $E,F$ be subsets of a group $G$ .", "We write $E F \\lbrace g h : g \\in E, h \\in F\\rbrace $ and define inductively $E^n$ for all $n \\in \\mathbb {N}$ by setting $E^0 \\lbrace 1_G\\rbrace $ and $E^{n + 1} E^n E$ .", "Let $A$ be a set and let $E$ be a subset of a group $G$ .", "Given $x \\in A^E$ , we define $gx \\in A^{gE}$ by $(gx)(h) x(g^{-1}h)$ for all $h \\in g E$ ." ], [ "Topologically transitive linear subshifts", "An action of a group $G$ on a Hausdorff topological space $X$ is called topologically transitive provided that given any two nonempty open subsets $U, V \\subset X$ there exists $g \\in G$ such that $U \\cap gV \\ne \\varnothing $ .", "Moreover, if $A$ is a set, a subshift $\\Sigma \\subset A^G$ is said to be topologically transitive if the restriction to $\\Sigma $ of the $G$ -shift is topologically transitive.", "It is straightforward that if the acting group $G$ is infinite, then every topologically mixing $G$ -action (e.g., any topologically mixing subshift $\\Sigma \\subset A^G$ ) is topologically transitive.", "The following result, which we shall use in the proof of Theorem REF , has some interest on its own.", "We thank the referee for pointing out a gap in our original argument and providing us with an outline of the present proof.", "Proposition 2.1 Let $G$ be a group and let $A$ be a vector space over a field $K$ .", "Let $\\Sigma \\subset A^G$ be a linear subshift and suppose that $\\Sigma $ is topologically transitive and finite dimensional.", "Then $\\Sigma = \\lbrace 0\\rbrace $ .", "Suppose, by contradiction, that $\\Sigma $ is nontrivial.", "Let $x_0 \\in \\Sigma \\setminus \\lbrace 0\\rbrace $ and let $g_0 \\in G$ such that $a x_0(g_0) \\ne 0_A$ .", "Consider the open subsets $U_1 \\lbrace x \\in \\Sigma : x(g_0) = 0_A\\rbrace $ and $V_1 \\lbrace x \\in \\Sigma : x(g_0) = a\\rbrace $ .", "Note that $U_1 \\ne \\varnothing $ since $0 \\in U_1$ and $V_1 \\ne \\varnothing $ since $x_0 \\in V_1$ .", "By topological transitivity, we can find $h_1 \\in G$ such that $U_1 \\cap h_1V_1 \\ne \\varnothing $ .", "Thus, if $z_1 \\in U_1 \\cap h_1V_1$ and $g_1 h_1g_0$ , we have $z_1(g_0) = 0_A$ and $z_1(g_1) = a$ .", "Consider now the open subsets $U_2 \\lbrace x \\in \\Sigma : x(g_0) = x(g_1) = 0_A\\rbrace $ and $V_1 \\lbrace x \\in \\Sigma : x(g_0) = 0 \\mbox{ and } x(g_1) = a\\rbrace $ .", "Note that $U_2 \\ne \\varnothing $ since $0 \\in U_2$ and $V_2 \\ne \\varnothing $ since $x_1 \\in V_2$ .", "By topological transitivity, we can find $h_2 \\in G$ such that $U_2 \\cap h_2V_2 \\ne \\varnothing $ .", "Thus, if $z_2 \\in U_2 \\cap h_2V_2$ and $g_2 h_2g_1$ , we have $z_2(g_0) = z_2(g_1) = 0_A$ and $z_1(g_2) = a$ .", "Continuing this way, we inductively find $z_1, z_2, \\ldots \\in \\Sigma $ and group elements $g_1, g_2, \\ldots \\in G$ such that $z_n(g_0) = z_n(g_1) = \\cdots = z_n(g_{n-1}) = 0_A \\ \\mbox{ and } \\ z_n(g_n) = a$ for all $n \\ge 1$ .", "As $a \\ne 0_A$ , it is straightforward that the configurations $z_1, z_2, \\ldots $ are linearly independent, contradicting the assumption that $\\Sigma $ is finite dimensional.", "We deduce that $\\Sigma = \\lbrace 0\\rbrace $ ." ], [ "Linear subshifts of finite type", "We begin with two simple useful facts.", "Lemma 2.2 Let $A$ be a set and let $G$ be a group.", "Let $D \\subset G$ and let $P \\subset A^D$ .", "Let $\\Sigma \\Sigma (D, P) \\subset A^G$ .", "Let $E \\subset G$ such that $D \\subset E$ .", "Then one has $\\Sigma = \\Sigma (E, \\Sigma _E)$ .", "In particular, $\\Sigma = \\Sigma (D, \\Sigma _D)$ .", "Let $x \\in \\Sigma $ and let $g \\in G$ .", "Then $(g^{-1}x) \\vert _E \\in \\Sigma _E$ .", "Thus $\\Sigma \\subset \\Sigma (E, \\Sigma _E)$ .", "Conversely, let $x \\in \\Sigma (E, \\Sigma _E)$ .", "Then, for every $g \\in G$ , we have $(g^{-1}x)\\vert _D = ((g^{-1}x)\\vert _E)\\vert _D\\in (\\Sigma _E)_D = \\Sigma _D \\subset P$ , since $D \\subset E$ .", "Therefore, $x \\in \\Sigma (D,P)=\\Sigma $ , and the conclusion follows.", "The following lemma states that every linear SFT admits a defining set of admissible patterns which is a vector space.", "Lemma 2.3 Let $G$ be a group and let $A$ be a vector space over a field $K$ .", "Let $\\Sigma \\subset A^G$ be a linear SFT and let $D \\subset G$ be a memory set for $\\Sigma $ .", "Then there exists a vector subspace $W \\subset A^D$ such that $\\Sigma = \\Sigma (D,W)$ .", "The set $W \\Sigma _D = \\lbrace x\\vert _D: x \\in \\Sigma \\rbrace \\subset A^D$ is a vector subspace of $A^D$ and we have $\\Sigma = \\Sigma (D,W)$ by Lemma REF .", "Proposition 2.4 Let $G$ be a finitely generated group and let $A$ be a vector space over a field $K$ .", "Then every finite-dimensional linear subshift $\\Sigma \\subset A^G$ is of finite type.", "Let $S \\subset G$ be a finite generating subset of $G$ .", "After replacing $S$ by $S \\cup S^{-1} \\cup \\lbrace 1_G\\rbrace $ , we can assume that $S = S^{-1}$ and $1_G \\in S$ .", "Then, given any element $g \\in G$ , there exist $n \\in \\mathbb {N}$ and $s_1, s_2, \\ldots , s_n \\in S$ such that $g = s_1s_2 \\cdots s_n$ .", "The minimal $n \\in \\mathbb {N}$ in such an expression of $g$ is the $S$ -length of $g$ , denoted by $\\ell _S(g)$ .", "For every $n \\in \\mathbb {N}$ we set $B_n \\lbrace g \\in G: \\ell _S(g) \\le n\\rbrace $ .", "Let $\\Sigma \\subset A^G$ be a finite-dimensional linear subshift.", "For every $n \\in \\mathbb {N}$ denote by $\\pi _n \\colon \\Sigma \\rightarrow \\Sigma _{B_n}$ the restriction map.", "Note that $\\pi _n$ is linear and that setting $\\Sigma _n \\ker \\pi _n$ we have that $(\\Sigma _n)_{n \\in \\mathbb {N}}$ is a decreasing sequence of vector subspaces of $\\Sigma $ .", "Now, on the one hand, since $\\bigcup _{n \\in \\mathbb {N}} B_n = G$ , we have $\\bigcap _{n \\in \\mathbb {N}} \\Sigma _n = \\lbrace 0\\rbrace $ .", "On the other hand, since $\\Sigma $ is finite-dimensional, the above sequence eventually stabilizes, i.e., there exists $n_0 \\in \\mathbb {N}$ such that $\\Sigma _n = \\Sigma _{n_0}$ for all $n \\ge n_0$ .", "We deduce that $\\Sigma _{n_0} = \\lbrace 0\\rbrace $ .", "Thus, setting $\\Delta B_{n_0}$ , the restriction map $\\Sigma \\rightarrow \\Sigma _\\Delta $ is injective (in fact bijective).", "Set $D S \\Delta \\subset G$ and $W \\Sigma _D \\subset A^D$ , and let us show that $\\Sigma = \\Sigma (D,W)$ .", "Let $x \\in \\Sigma $ .", "Then for every $g \\in G$ we have $g^{-1}x \\in \\Sigma $ so that $(g^{-1}x)\\vert _D \\in \\Sigma _D = W$ .", "This shows that $x \\in \\Sigma (D,W)$ , and the inclusion $\\Sigma \\subset \\Sigma (D,W)$ follows.", "Conversely, suppose that $x \\in \\Sigma (D,W)$ .", "By definition of $\\Sigma (D,W)$ , for every $g \\in G$ , there exists $x_g \\in \\Sigma $ such that $(g^{-1} x)\\vert _D = (x_g)\\vert _D$ .", "Observe that, given $g \\in G$ , such an $x_g$ is unique since $\\Delta \\subset D$ .", "Let us show, by induction on the $S$ -length of $g$ , that $x_g = g^{-1}x_{1_G}$ for all $g \\in G$ .", "If $\\ell _S(g) = 0$ , then $g = 1_G$ and there is nothing to prove.", "Suppose now that (REF ) is satisfied for all $g \\in G$ such that $\\ell _S(g) = n$ and let $h \\in G$ such that $\\ell _S(h) = n + 1$ .", "Then there exist $g \\in G$ with $\\ell _S(g) = n$ and $s \\in S$ such that $h = g s$ .", "For all $d \\in \\Delta $ , we have $x_h(d) &= (h^{-1} x)(d) && \\text{(since $\\Delta \\subset D$)} \\\\&= (g^{-1} x)(s d) && \\text{(since $h = g s$)} \\\\&= x_g(s d) && \\text{(since $S \\Delta \\subset D$)} \\\\&= (g^{-1} x_{1_G})(s d) && \\text{(by our induction hypothesis)} \\\\&= (h^{-1} x_{1_G})(d) && \\text{(since $h = g s$).", "}$ Thus $x_h$ and $h^{-1} x_{1_G}$ coincide on $\\Delta $ .", "As $x_h, h^{-1} x_{1_G} \\in \\Sigma $ , this implies that $x_h = h^{-1} x_{1_G}$ .", "By induction, we conclude that (REF ) holds for all $g \\in G$ .", "Since $1_G \\in D$ , we deduce that $x(g) = (g^{-1}x)(1_G) = x_g(1_G) = (g^{-1}x_{1_G})(1_G) = x_{1_G}(g)$ for all $g \\in G$ .", "This shows that $x = x_{1_G} \\in \\Sigma $ , and the inclusion $\\Sigma (D,W) \\subset \\Sigma $ follows.", "In conclusion, $\\Sigma = \\Sigma (D,W)$ is a subshift of finite type.", "The condition that $G$ is finitely generated cannot be removed from the assumptions in Proposition REF .", "In fact we have the following (cf.", "[29]; see also Section ).", "Corollary 2.5 Let $G$ be a group and let $A$ be a nontrivial finite dimensional vector space over a field $K$ .", "Consider the subshift $\\Sigma \\subset A^G$ consisting of all constant configurations.", "Then $\\Sigma $ is a SFT if and only if $G$ is finitely generated.", "As the linear subshift $\\Sigma $ satisfies $\\dim _K(\\Sigma ) = \\dim _K(A) < \\infty $ , it is a SFT whenever $G$ is finitely generated by Proposition REF .", "Conversely, suppose that $\\Sigma $ is an SFT.", "Thus there exists a finite subset $D \\subset G$ and $P \\subset A^D$ such that $\\Sigma = \\Sigma (D,P)$ .", "Consider the subgroup $H \\subset G$ generated by $D$ and let $a \\in A$ such that $a \\ne 0$ .", "Then the configuration $x \\in A^G$ such that $x(g) = 0$ if $g \\in H$ and $x(g) = a$ otherwise belongs to $\\Sigma (D,P)$ since, for each $g \\in G$ , either $gD \\subset H$ or $gD \\subset G \\setminus H$ .", "As $\\Sigma (D,P) = \\Sigma $ and every configuration in $\\Sigma $ is constant, we conclude that $H = G$ .", "Therefore $G$ is finitely generated." ], [ "Restriction of linear-sofic subshifts and of linear CAs", "Let $G$ be a group and let $A$ be a vector space over a field $K$ .", "Recall that a linear subshift $\\Sigma \\subset A^G$ is said to be a linear-sofic subshift if there exists a vector space $B$ over $K$ , an SFT $\\Sigma ^{\\prime } \\subset B^G$ , and a linear cellular automaton $\\tau \\colon B^G \\rightarrow A^G$ such that $\\Sigma = \\tau (\\Sigma ^{\\prime })$ .", "We shall refer to a finite subset $M \\subset G$ containing both a memory set for $\\Sigma ^{\\prime }$ as well as a memory set for $\\tau $ as to a memory set for the linear-sofic subshift $\\Sigma $ .", "Let $\\Sigma \\subset A^G$ be a linear-sofic subshift.", "Let $H\\subset G$ be a subgroup of $G$ containing a memory set for $\\Sigma $ .", "Denote by $G/H \\lbrace gH: g \\in G\\rbrace $ the set of all right cosets of $H$ in $G$ .", "As the right cosets of $H$ in $G$ form a partition of $G$ , we have a natural factorization $A^G = \\prod _{c \\in G/H} A^c$ in which each $x \\in A^G$ is identified with $(x\\vert _c)_{c \\in G/H} \\in \\prod _{c \\in G/H} A^c$ .", "The above factorization of $A^G$ induces a factorization (cf.", "[12]) $\\Sigma = \\prod _{c \\in G/H} \\Sigma _c,$ where $\\Sigma _c = \\lbrace x\\vert _c: x \\in \\Sigma \\rbrace $ is a vector subspace of $A^c$ for all $c \\in G/H$ .", "Let $T \\subset G$ be a complete set of representatives for the right cosets of $H$ in $G$ such that $1_G \\in T$ .", "Then, for each $c \\in G/H$ , we have a linear uniform homeomorphism $\\phi _c \\colon \\Sigma _c \\rightarrow \\Sigma _H$ given by $\\phi _c(y)(h) = y(gh)$ for all $y \\in \\Sigma _c$ , where $g \\in T$ represents $c$ .", "Now suppose in addition that $\\tau \\colon \\Sigma \\rightarrow \\Sigma $ is a linear CA which admits a memory set contained in $H$ .", "Then we have $\\tau = \\prod _{c \\in G/H} \\tau _c$ , where $\\tau _c \\colon \\Sigma _c \\rightarrow \\Sigma _c$ is the linear map defined by setting $\\tau _c(y) \\tau (x)\\vert _c$ for all $y \\in \\Sigma _c$ , where $x \\in \\Sigma $ is any configuration extending $y$ .", "Note that for each $c \\in G/H$ , the linear maps $\\tau _c$ and $\\tau _H$ are conjugated by $\\phi _c$ , i.e., we have $\\tau _c = \\phi _c^{-1} \\circ \\tau _H \\circ \\phi _c$ .", "This allows us to identify the action of $\\tau _c$ on $\\Sigma _c$ with that of the restriction cellular automaton $\\tau _H$ on $\\Sigma _H$ .", "The following extends [9] (cf. [12]).", "Lemma 2.6 Let $G$ be a group, let $A$ be a vector space over a field $K$ , and let $\\Sigma \\subset A^G$ be a linear-sofic subshift.", "Let $\\tau \\colon A^G \\rightarrow A^G$ be a cellular automaton.", "Let $H \\subset G$ be a subgroup containing memory sets for both $\\Sigma $ and $\\tau $ .", "Then $\\tau (\\Sigma )$ is closed in $A^G$ if and only if $\\tau _H(\\Sigma _H)$ is closed in $A^H$ .", "With the above notation, we have $\\tau (\\Sigma ) = \\prod _{c \\in G/H} \\tau _c(\\Sigma _c)$ .", "It is immediate that $\\tau _H(\\Sigma _H)$ is closed in $A^H$ if $\\tau (\\Sigma )$ is closed in $A^G$ .", "Conversely, if $\\tau _H(\\Sigma _H)$ is closed in $A^H$ , then so are $\\tau _c(\\Sigma _c) = \\phi _c^{-1}(\\tau _H(\\Sigma _H))$ in $A^c$ for all $c \\in G/H$ , since the $\\phi _c \\colon A^c \\rightarrow A^H$ are uniform homeomorphisms.", "Consequently, $\\tau (\\Sigma )$ is closed in $A^G$ whenever $\\tau _H(\\Sigma _H)$ is closed in $A^H$ since the product of closed subspaces is closed in the product topology." ], [ "Nilpotent linear cellular automata", "Let $G$ be a group, let $A$ be a vector space over a field $K$ , and let $\\tau \\colon \\Sigma \\rightarrow \\Sigma $ be a linear cellular automaton, where $\\Sigma \\subset A^G$ is a linear subshift.", "By linearity, $\\tau $ is nilpotent if and only if there exists and integer $n_0 \\ge 1$ such that $\\tau ^{n_0} = 0$ .", "Moreover, $\\tau ^n(\\Sigma )$ , $n \\in \\mathbb {N}$ , and therefore $\\Omega (\\tau )$ are vector subspaces of $\\Sigma $ .", "The following is the linear version of [12].", "Lemma 2.7 Let $G$ be a group, let $A$ be a vector space over a field $K$ , and let $\\Sigma \\subset A^G$ be a linear-sofic subshift.", "Let $\\tau \\colon \\Sigma \\rightarrow \\Sigma $ be a linear cellular automaton.", "Let $H \\subset G$ be a subgroup containing memory sets of both $\\Sigma $ and $\\tau $ .", "Then the following hold: $\\Omega (\\tau ) = \\prod _{c \\in G/H} \\Omega (\\tau _c)$ ; $\\Omega (\\tau )$ is linearly uniformly homeomorphic to $\\Omega (\\tau _H)^{G/H}$ ; $\\tau $ is nilpotent if and only if $\\tau _H \\colon \\Sigma _H \\rightarrow \\Sigma _H$ is nilpotent.", "We have $\\tau ^n(\\Sigma ) = \\prod _{c \\in G/H} \\tau _c^n(\\Sigma )$ for all $n \\in \\mathbb {N}$ , so that $\\Omega (\\tau ) = \\bigcap _{n \\in \\mathbb {N}} \\tau ^n(\\Sigma ) = \\bigcap _{n \\in \\mathbb {N}}\\prod _{c \\in G/H} \\tau _c^n(\\Sigma _c) =\\prod _{c \\in G/H} \\bigcap _{n \\in \\mathbb {N}} \\tau _c^n(\\Sigma _c) = \\prod _{c \\in G/H} \\Omega (\\tau _c).$ This proves (i).", "It is then clear that $\\phi \\prod _{c \\in G/H} \\phi _c \\colon A^G \\rightarrow (A^H)^{G/H}$ yields, by restriction, a linear uniform homeomorphism $\\Omega (\\tau ) = \\prod _{c \\in G/H} \\Omega (\\tau _c) \\rightarrow \\Omega (\\tau _H)^{G/H}$ .", "This proves (ii).", "We have that $\\tau $ is nilpotent if and only if there exists an integer $n_0 \\ge 1$ such that $\\tau ^{n_0}(\\Sigma ) = \\lbrace 0\\rbrace $ .", "By the above discussion, this is equivalent to $\\tau _H^{n_0}(\\Sigma _H) = \\lbrace 0\\rbrace $ , that is, to $\\tau _H$ being nilpotent.", "Given a group $G$ and a vector space $A$ over a field $K$ , the set $\\operatorname{LCA}(G,A)$ of all linear cellular automata $\\tau \\colon A^G \\rightarrow A^G$ has a natural structure of a $K$ -algebra (cf.", "[8]).", "Indeed, it is a subalgebra of the $K$ -algebra $\\operatorname{End}_{K[G]}A^G$ ([8]).", "In [3] and [4] (see also [5] and [8]) it is shown that if $A$ is finite-dimensional with, say, $\\dim _K(A) = d$ , then, once fixed a vector basis $\\mathcal {B}$ for $A$ , there exists a canonical $K$ -algebra isomorphism $\\tau \\mapsto M_\\mathcal {B}(\\tau )$ of $\\operatorname{LCA}(G,A)$ onto $\\operatorname{Mat}_d(K[G])$ , the $K$ -algebra of $d \\times d$ matrices with coefficients in the group ring $K[G]$ .", "Recall that an element $M$ of a ring $R$ is called nilpotent if there exists an integer $n \\ge 1$ such that $M^n = 0_R$ .", "We then have Proposition 2.8 Let $G$ be a group and let $A$ be a finite-dimensional vector space over a field $K$ .", "Suppose that $\\dim _K(A) = d$ and that $A$ is equipped with a basis $\\mathcal {B}$ .", "Let $\\tau \\colon A^G \\rightarrow A^G$ be a linear CA.", "Then the following conditions are equivalent: $\\tau $ is nilpotent; the matrix $M_\\mathcal {B}(\\tau ) \\in \\operatorname{Mat}_d(K[G])$ is nilpotent.", "The proof follows immediately from the fact that $\\operatorname{LCA}(G,A)$ and $\\operatorname{Mat}_d(K[G])$ are isomorphic as $K$ -algebras, and that nilpotency is preserved under $K$ -algebra homomorphisms." ], [ "Inverse limits of sets", "Let $(I, \\preceq )$ be a directed set, i.e., a partially ordered set in which every pair of elements admits an upper bound.", "An inverse system of sets indexed by $I$ consists of the following data: (1) a set $Z_i$ for each $i \\in I$ ; (2) a transition map $\\varphi _{ij} \\colon Z_j \\rightarrow Z_i$ for all $i,j \\in I$ such that $i \\preceq j$ .", "Furthermore, the transition maps must satisfy the following conditions: $\\varphi _{ii} &= \\operatorname{Id}_{Z_i} \\text{ (the identity map on $Z_i$) for all } i \\in I, \\\\[4pt]\\varphi _{ij} \\circ \\varphi _{jk} &= \\varphi _{ik} \\text{ for all $i,j,k \\in I$ such that } i \\preceq j \\preceq k.$ One then speaks of the inverse system $(Z_i,\\varphi _{ij})$ , or simply $(Z_i)$ if the index set and the transition maps are clear from the context.", "One says that an inverse system $(Z_i,\\varphi _{ij})$ satisfies the Mittag-Leffler condition provided that for each $i \\in I$ there exists $j \\in I$ with $i \\preceq j$ such that $\\varphi _{i k} (X_k) = \\varphi _{i j} (X_j)$ for all $j \\preceq k$ .", "The inverse limit of an inverse system $(Z_i,\\varphi _{i j})$ is the subset $\\varprojlim _{i \\in I} (Z_i,\\varphi _{i j}) = \\varprojlim _{i \\in I} Z_i \\subset \\prod _{i \\in I} Z_i$ consisting of all $(z_i)_{i \\in I}$ such that $\\varphi _{i j}(z_j)= z_i$ for all $i \\preceq j$ .", "The following useful lemma is an application of the classical Mittag-Leffler lemma to affine inverse systems (see, e.g.", "[16], where Grothendieck used it in his study of the cohomology of affine schemes, the general treatment by Bourbaki [2], or [9] for a self-contained proof in the countable case.", "Cf.", "also [25] and [26] for more details).", "Lemma 3.1 Let $K$ be a field.", "Let $(X_i, f_{ij})$ be an inverse system indexed by an index set $I$ , where each $X_i$ is a nonempty finite-dimensional $K$ -affine space and each transition map $f_{i j} \\colon X_j \\rightarrow X_i$ is a $K$ -affine map for all $i \\preceq j$ .", "Then $\\varprojlim _{i \\in I} X_i \\ne \\varnothing $ ." ], [ "Space-time inverse systems", "Let $G$ be a group and let $A$ be a vector space over a field $K$ .", "Let $\\Sigma \\subset A^G$ be a linear subshift and let $\\tau \\colon \\Sigma \\rightarrow \\Sigma $ be a linear CA.", "Let $\\widetilde{\\tau } \\colon A^G \\rightarrow A^G$ be a linear CA extending $\\tau $ and let $M\\subset G$ be a memory set for $\\widetilde{\\tau }$ .", "Since every finite subset of $G$ containing a memory set for $\\widetilde{\\tau }$ is itself a memory set for $\\widetilde{\\tau }$ , we can choose $M$ such that $1_G \\in M$ and $M = M^{-1}$ .", "Let ${\\mathcal {P}}^(G)$ denote the set of all finite subsets of $G$ containing $1_G$ equipped with the ordering given by inclusion.", "Also equip $\\mathbb {N}$ with the natural ordering.", "Equip $I {\\mathcal {P}}^(G) \\times \\mathbb {N}$ with the product ordering $\\preceq $ .", "Thus, given $\\Omega , \\Omega ^{\\prime } \\in {\\mathcal {P}}^(G)$ and $n,n^{\\prime } \\in \\mathbb {N}$ , we have $(\\Omega ,n) \\preceq (\\Omega ^{\\prime },n^{\\prime })$ if and only if $\\Omega \\subset \\Omega ^{\\prime }$ and $n \\le n^{\\prime }$ .", "It is clear that $(I, \\preceq )$ is directed.", "We construct an inverse system $(\\Sigma _{\\Omega ,n})_{(\\Omega ,n) \\in I}$ indexed by $I$ as follows.", "Firstly, given $(\\Omega ,n) \\in I$ , we set $\\Sigma _{\\Omega ,n} \\Sigma _{\\Omega M^n} = \\lbrace x\\vert _{\\Omega M^n} : x \\in \\Sigma \\rbrace \\subset A^{\\Omega M^n}.$ To define the transition maps $\\Sigma _{\\Omega ^{\\prime },n^{\\prime }} \\rightarrow \\Sigma _{\\Omega ,n}$ ($(\\Omega ,n) \\preceq (\\Omega ^{\\prime },n^{\\prime })$ ) of the inverse system $(\\Sigma _{\\Omega ,n})_{(\\Omega ,n) \\in I}$ , it is clearly enough to define, for all $\\Omega , \\Omega ^{\\prime }\\in {\\mathcal {P}}^(G)$ and $n,n^{\\prime } \\in \\mathbb {N}$ , with $\\Omega \\subset \\Omega ^{\\prime }$ and $n \\le n^{\\prime }$ , the horizontal transition map $p_{\\Omega , \\Omega ^{\\prime }; n} \\colon \\Sigma _{\\Omega ^{\\prime }, n} \\rightarrow \\Sigma _{\\Omega , n}$ , the vertical transition map $q_{\\Omega ; n, n^{\\prime }} \\colon \\Sigma _{\\Omega , n^{\\prime }} \\rightarrow \\Sigma _{\\Omega ,n}$ , and verify that the diagram $\\begin{tikzcd}\\Sigma _{\\Omega ,n^{\\prime }} \\ [d, swap, \"q_{\\Omega ; n,n^{\\prime }}\"]& \\ \\Sigma _{\\Omega ^{\\prime },n^{\\prime }} [l, swap, \"p_{\\Omega , \\Omega ^{\\prime }; n^{\\prime }}\"] [d, \"q_{\\Omega ^{\\prime }; n,n^{\\prime }}\"] \\\\\\Sigma _{\\Omega ,n} \\ & \\ \\Sigma _{\\Omega ^{\\prime }, n} [l, \"p_{\\Omega , \\Omega ^{\\prime }; n}\"],\\end{tikzcd}$ is commutative, i.e., $q_{\\Omega ;n,n^{\\prime }} \\circ p_{\\Omega , \\Omega ^{\\prime };n^{\\prime }} = p_{\\Omega , \\Omega ^{\\prime }; n} \\circ q_{\\Omega ^{\\prime };n,n^{\\prime }}$ for all $\\Omega , \\Omega ^{\\prime } \\in {\\mathcal {P}}^(G)$ and $n,n^{\\prime } \\in \\mathbb {N}$ , with $\\Omega \\subset \\Omega ^{\\prime }$ and $n \\le n^{\\prime }$ .", "We define $p_{\\Omega , \\Omega ^{\\prime };n}$ as being the linear map obtained by restriction to $\\Omega M^n \\subset \\Omega ^{\\prime } M^n$ .", "Thus, for all $\\sigma \\in \\Sigma _{\\Omega ^{\\prime },n} = \\Sigma _{\\Omega ^{\\prime } M^n}$ , we have $p_{\\Omega , \\Omega ^{\\prime };n}(\\sigma ) = \\sigma \\vert _{\\Omega M^n}.$ We now define $q_{\\Omega ; n,n^{\\prime }}$ .", "If $n = n^{\\prime }$ , then $q_{\\Omega ; n,n^{\\prime }} = q_{\\Omega ; n,n} \\operatorname{Id}_{\\Omega M^n} \\colon \\Omega M^n \\rightarrow \\Omega M^n$ , is the identity map.", "Suppose now that $n+1 \\le n^{\\prime }$ .", "We first observe that, given $x \\in \\Sigma $ and $g \\in G$ , it follows from (REF ) applied to $\\widetilde{\\tau }$ that $\\tau ^{n^{\\prime }-n}(x)(g)$ only depends on the restriction of $x$ to $gM^{n^{\\prime }-n}$ .", "As $gM^{n^{\\prime }-n} \\subset \\Omega M^n M^{n^{\\prime }-n} = \\Omega M^{n^{\\prime }}$ for all $g \\in \\Omega M^n$ , we deduce from this observation that, given $\\sigma \\in \\Sigma _{\\Omega , n^{\\prime }} = \\Sigma _{\\Omega M^{n^{\\prime }}}$ and $x \\in \\Sigma $ extending $\\sigma $ , the formula $q_{\\Omega ; n, n^{\\prime }}(\\sigma ) \\tau ^{n^{\\prime }-n}(x)\\vert _{\\Omega M^n}$ yields a well-defined element $q_{\\Omega ; n, n^{\\prime }}(\\sigma ) \\in \\Sigma _{\\Omega M^n} = \\Sigma _{\\Omega , n}$ , and hence a linear map $q_{\\Omega ; n,n^{\\prime }} \\colon \\Sigma _{\\Omega , n^{\\prime }} \\rightarrow \\Sigma _{\\Omega , n}$ .", "Definition 3.2 The inverse system $(\\Sigma _{\\Omega ,n})_{(\\Omega ,n) \\in I}$ is called the space-time inverse system associated with the triple $(\\Sigma , \\tau , M)$ .", "When $G$ is countable, we can simplify the above construction by slightly modifying the definitions therein.", "Since $G$ is countable, we can find a sequence $(M_n)_{n \\in \\mathbb {N}}$ of finite subsets of $G$ such that $M_0 = \\lbrace 1_G\\rbrace $ and $M_1 = M$ (the memory set for $\\widetilde{\\tau }$ ), $M_iM_j \\subset M_{i+j}$ for all $i,j \\in \\mathbb {N}$ , $\\bigcup _{n \\in \\mathbb {N}} M_n = G$ .", "For instance, if $G$ is finitely generated and $M$ in addition generates $G$ , then one may take $M_n M^n$ for all $n \\in \\mathbb {N}$ .", "We equip $\\mathbb {N}^2$ with the product ordering $\\preceq $ , that is, given $i,j,k,l \\in \\mathbb {N}$ , we have $(i,j) \\preceq (k,l)$ if and only if $i \\le k$ and $j \\le l$ .", "We then construct an inverse system $(\\Sigma _{i j})_{i,j \\in \\mathbb {N}}$ indexed by the directed set $(\\mathbb {N}^2, \\preceq )$ by setting $\\Sigma _{i j} \\Sigma _{M_{i+j}} = \\lbrace x\\vert _{M_{i + j}} : x \\in \\Sigma \\rbrace \\subset A^{M_{i + j}}$ and defining, for all $i,j \\in \\mathbb {N}$ , the unit-horizontal transition map $p_{i j} \\colon \\Sigma _{i+1, j} \\rightarrow \\Sigma _{i j}$ as being the linear map obtained by restriction to $M_{i + j} \\subset M_{i + j + 1}$ , and the unit-vertical transition map $q_{i j} q_{M_i;M_j;M_{j+1}} \\colon \\Sigma _{i,j+1} \\rightarrow \\Sigma _{i j}$ by setting $q_{i j}(\\sigma ) (\\tau (x))\\vert _{M_{i + j}}$ for all $\\sigma \\in \\Sigma _{i,j + 1}$ and $x \\in \\Sigma $ extending $\\sigma $ (as in the general case, this gives a well-defined element $q_{i j}(\\sigma ) \\in \\Sigma _{i j}$ ).", "Finally, as for (REF ), one checks that $q_{i j} \\circ p_{i,j+1} = p_{i j} \\circ q_{i+1,j}$ for all $i,j \\in \\mathbb {N}$ .", "Definition 3.3 The inverse system $(\\Sigma _{ij})_{i,j \\in \\mathbb {N}}$ is called the space-time inverse system associated with the triple $(\\Sigma , \\tau , (M_n)_{n \\in \\mathbb {N}})$ ." ], [ "Space-time-systems and limit sets", "We keep the assumptions and notation from the above subsection.", "Let us fix $n \\in \\mathbb {N}$ .", "Then, in our space-time inverse system we get an horizontal inverse system $(\\Sigma _{\\Omega ,n})_{\\Omega \\in {\\mathcal {P}}^(G)}$ indexed by ${\\mathcal {P}}^(G)$ whose transition maps are the restriction maps $p_{\\Omega , \\Omega ^{\\prime };n} \\colon \\Sigma _{\\Omega ^{\\prime } M^n} \\rightarrow \\Sigma _{\\Omega M^n}$ , $\\Omega , \\Omega ^{\\prime } \\in {\\mathcal {P}}^(G)$ such that $\\Omega \\subset \\Omega ^{\\prime }$ .", "Note that the horizontal inverse system satisfies the Mittag-Leffler condition and that in fact, as it immediately follows from the closedness of $\\Sigma $ in $A^G$ and the fact that $G M^n = G$ , one has that the limit $\\Sigma _n \\varprojlim _{\\Omega \\in {\\mathcal {P}}^(G)} \\Sigma _{\\Omega ,n}$ can be identified with $\\Sigma $ in a canonical way.", "Moreover, the linear maps $q_{\\Omega ; n,n^{\\prime }} \\colon \\Sigma _{\\Omega ,n^{\\prime }} \\rightarrow \\Sigma _{\\Omega ,n}$ , for $\\Omega \\in {\\mathcal {P}}^(G)$ , define an inverse system linear morphism from the inverse system $(\\Sigma _{\\Omega ,n^{\\prime }})_{\\Omega \\in {\\mathcal {P}}^(G)}$ to the inverse system $(\\Sigma _{\\Omega ,n})_{\\Omega \\in {\\mathcal {P}}^(G)}$ .", "This yields a linear limit map $\\tau _{n,n^{\\prime }} \\colon \\Sigma _{n^{\\prime }} \\rightarrow \\Sigma _n$ .", "Using the identifications $\\Sigma _n = \\Sigma _{n^{\\prime }} = \\Sigma $ , we have $\\tau _{n,n^{\\prime }} = \\tau ^{n^{\\prime }-n}$ for all $n,n^{\\prime } \\in \\mathbb {N}$ such that $n \\le n^{\\prime }$ .", "We deduce that the limit $\\varprojlim _{(\\Omega , n) \\in I} \\Sigma _{\\Omega , n} = \\varprojlim _{n \\in \\mathbb {N}} \\Sigma _n$ is the set of backward orbits of $\\tau $ , that is, the set consisting of all sequences $(x_n)_{n \\in \\mathbb {N}}$ such that $x_n \\in \\Sigma $ and $x_n = \\tau (x_{n+1})$ for all $n \\in \\mathbb {N}$ .", "Each such a sequence satisfies that $x_0 = \\tau ^n(x_n)$ for all $n \\in \\mathbb {N}$ , and hence $x_0 \\in \\Omega (\\tau )$ .", "This determines a canonical linear map $\\Phi \\colon \\varprojlim _{(\\Omega , n) \\in I} \\Sigma _{\\Omega , n} \\rightarrow \\Omega (\\tau ).$ Proposition 3.4 Let $G$ be a group and let $A$ be a finite dimensional vector space over a field $K$ .", "Let $\\Sigma \\subset A^G$ be a linear subshift and let $\\tau \\colon \\Sigma \\rightarrow \\Sigma $ be a linear CA.", "Let $M \\subset G$ be a memory set for $\\tau $ which is symmetric and contains $1_G$ , and consider the space-time inverse system associated with the triple $(\\Sigma ,\\tau , M)$ .", "Then the canonical map $\\Phi $ is surjective.", "Let $y_0 \\in \\Omega (\\tau ) \\subset \\Sigma $ .", "For every $\\Omega \\in {\\mathcal {P}}^(G)$ and $n \\in \\mathbb {N}$ , define a finite dimensional affine subspace $B_{\\Omega , n} \\subset A^{\\Omega M^n}$ by setting $B_{\\Omega , n} \\left(q_{\\Omega ;0,1} \\circ q_{\\Omega ;1,2} \\circ \\dots \\circ q_{\\Omega ; n-1,n}\\right)^{-1} (y_0\\vert _\\Omega ) \\subset \\Sigma _{\\Omega M^n}.$ By definition of $\\Omega (\\tau )$ , for every $n \\in \\mathbb {N}$ there exists an element $y_n \\in \\Sigma $ such that $\\tau ^n (y_n)=y_0$ .", "Hence, it follows from the definition of the transition maps $q_{\\Omega ; j-1,j}$ and of $\\Sigma _{\\Omega , j}$ , $j \\in \\mathbb {N}$ , that $y_n\\vert _{\\Omega M^n} \\in B_{\\Omega ,n}$ .", "In particular, $B_{\\Omega ,n} \\ne \\varnothing $ for every $\\Omega \\in {\\mathcal {P}}^(G)$ and $n \\in \\mathbb {N}$ .", "By restricting the transition maps of the space-time inverse system $(\\Sigma _{\\Omega , n})_{(\\Omega , n) \\in I}$ to the sets $B_{\\Omega , n}$ , we obtain a well-defined inverse subsystem $(B_{\\Omega , n})_{(\\Omega , n) \\in I}$ of finite dimensional affine spaces with affine transition maps.", "By Lemma REF , we can find $x \\in \\varprojlim _{(\\Omega , n) \\in I} B_{\\Omega , n} \\subset \\varprojlim _{(\\Omega , n) \\in I} \\Sigma _{\\Omega , n}.$ It is clear from the constructions of the inverse system $(B_{\\Omega , n})_{(\\Omega , n) \\in I}$ and of the map $\\Phi $ that $\\Phi (x)= y_0$ .", "This shows that $\\Phi $ is surjective." ], [ "The closed-image-property for linear cellular automata", "The closed-image-property Using the space-time inverse system, we give a short proof of the following result extending [9] (see also [8]).", "Theorem 4.1 Let $G$ be a group and let $A$ be a finite-dimensional vector space over a field $K$ .", "Let $\\Sigma \\subset A^G$ be a linear subshift and let $\\tau \\colon A^G \\rightarrow A^G$ be a linear CA.", "Then $\\tau (\\Sigma )$ is a linear subshift of $A^G$ .", "Since the cellular automaton $\\tau $ is linear and $G$ -equivariant, its image $\\tau (\\Sigma )$ is a $G$ -invariant vector subspace of $A^G$ .", "We thus only need to show that $\\tau (\\Sigma )$ is closed in $A^G$ .", "Let $M \\subset G$ be a memory set for $\\tau $ which is symmetric and contains $1_G$ , and consider the space-time inverse system associated with the triple $(\\Sigma , \\tau , M)$ as in Section REF .", "Suppose that $x \\in \\Sigma $ belongs to the closure of $\\tau (\\Sigma )$ .", "We must show that $x \\in \\tau (\\Sigma )$ .", "For every $\\Omega \\in {\\mathcal {P}}^(G)$ , define an affine subspace $Z_\\Omega \\subset A^{\\Omega M}$ by setting $Z_\\Omega (q_{\\Omega ;0,1})^{-1}(x\\vert _\\Omega ) \\cap \\Sigma _{\\Omega ,1}.$ Since $x$ belongs to the closure of $\\tau (\\Sigma )$ , it follows that $Z_\\Omega \\ne \\varnothing $ for all $\\Omega \\in {\\mathcal {P}}^(G)$ .", "By restricting the projections $p_{\\Omega , \\Omega ^{\\prime };1} \\colon A^{\\Omega ^{\\prime }M} \\rightarrow A^{\\Omega M}$ (cf.", "(REF ), $\\Omega , \\Omega ^{\\prime } \\in {\\mathcal {P}}^(G)$ , with $\\Omega \\subset \\Omega ^{\\prime }$ ) to the $Z_\\Omega $ 's, we obtain affine maps $\\pi _{\\Omega , \\Omega ^{\\prime }} \\colon Z_{\\Omega ^{\\prime }} \\rightarrow Z_{\\Omega }$ of the inverse system $(Z_\\Omega )_{\\Omega \\in {\\mathcal {P}}^(G)}$ .", "It then follows from Lemma REF that $\\varprojlim _{\\Omega \\in {\\mathcal {P}}^(G)} (Z_\\Omega ,\\pi _{\\Omega , \\Omega ^{\\prime }}) \\ne \\varnothing $ .", "Therefore, by construction of $Z_\\Omega $ and $\\Sigma _{\\Omega ,1}$ , for every $c \\in \\varprojlim _{\\Omega \\in {\\mathcal {P}}^(G)}Z_\\Omega \\subset \\varprojlim _{\\Omega \\in {\\mathcal {P}}^(G)} \\Sigma _{\\Omega ,1} = \\Sigma $ (cf.", "(REF )) we have $\\tau (c) = x$ .", "This shows that $\\tau (\\Sigma )$ is closed.", "Remark 4.2 We observe that the the hypothesis of finite-dimensionality of the vector space $A$ in Theorem REF cannot be dropped, as the example in Section REF below shows." ], [ "Proof of Theorem ", "Suppose $\\Sigma $ is of finite type.", "Hence $\\Sigma = \\Sigma (D,W) \\subset A^D$ , where $D \\subset G$ is finite and $W \\subset A^D$ is a vector subspace (cf.", "Lemma REF ).", "Let $\\Sigma _0 \\supset \\Sigma _1 \\supset \\cdots $ be a decreasing sequence of linear subshifts of $A^G$ such that $\\bigcap _{n \\ge 0} \\Sigma _n = \\Sigma $ .", "Let $(M_n)_{n \\in \\mathbb {N}}$ be a sequence of finite subsets of $G$ satisfying conditions (M-1)-(M-3) and such that $D \\subset M_1$ .", "Consider the inverse system $(X_{ij})_{i,j \\in \\mathbb {N}}$ defined by setting $X_{ij} (\\Sigma _j)_{M_i} \\subset A^{M_i}$ .", "Observe that $X_{i,j+1} \\subset X_{ij}$ since $\\Sigma _{j+1} \\subset \\Sigma _j$ for all $i,j \\in \\mathbb {N}$ .", "Also, we define the transition maps $p_{ij} \\colon X_{i+1,j} \\rightarrow X_{ij}$ by setting $p_{ij}(x) x\\vert _{M_i}$ for all $x \\in X_{i+1,j} = (\\Sigma _j)_{M_{i+1}}$ and $q_{ij} \\colon X_{i,j+1} \\rightarrow X_{ij}$ as the inclusion maps.", "The decreasing sequence $(X_{1,j})_{j \\in \\mathbb {N}}$ of finite-dimensional vector spaces eventually stabilizes so that there exists $j_0 \\ge 1$ such that $X_{1,j} = X_{1,j_0}$ for all $j \\ge j_0$ .", "Set $W^{\\prime } X_{1,j_0}$ and let us show that $\\Sigma $ equals the linear SFT $\\Sigma ^{\\prime } \\Sigma (M_{1}, W^{\\prime })$ .", "First note that $\\Sigma _{j_0} \\subset \\Sigma ^{\\prime }$ so that $\\Sigma \\subset \\Sigma ^{\\prime }$ .", "Conversely, let $w \\in W^{\\prime }$ .", "We construct an inverse subsystem $(Z_{ij})_{i \\ge 1, j \\ge 0}$ of $(X_{ij})_{i \\ge 1, j \\ge 0}$ as follows.", "For $i \\ge 1$ and $j \\ge 0$ , consider the affine subspace of $X_{ij}$ : $Z_{ij} \\lbrace x \\in X_{ij}: x\\vert _{M_1} = w\\rbrace \\subset X_{ij}.$ The transition maps of $(Z_{ij})_{i \\ge 1, j \\ge 0}$ are well-defined as the restrictions of the transition maps of the system $(X_{ij})_{i \\ge 1, j \\ge 0}$ .", "By our construction, each $Z_{ij}$ is clearly nonempty.", "Hence, Lemma REF implies that there exists $x = (x_{ij})_{i \\ge 1, j \\ge 0} \\in \\varprojlim Z_{ij}$ .", "Let $y \\in A^G$ be defined by $y(g)= x_{i0}(g)$ for every $g \\in G$ and any large enough $i\\ge 1$ such that $g \\in M_{i}$ .", "Observe that $x_{ij} = x_{ik}$ for every $i \\ge 1$ and $0 \\le j \\le k$ since the vertical transition maps $X_{ik} \\rightarrow X_{ij}$ are simply inclusions.", "Consequently, for every $n \\in \\mathbb {N}$ , we have $y \\in \\Sigma _n$ by (REF ).", "Hence $y \\in \\Sigma $ .", "By construction, $y\\vert _{M_{1}} = w$ .", "Since $w$ was arbitrary, this shows that $W^{\\prime } \\subset \\Sigma _{M_{1}}$ .", "Hence, $\\Sigma ^{\\prime } = \\Sigma (M_{1}, W^{\\prime }) \\subset \\Sigma (M_{1}, \\Sigma _{M_{1}}) = \\Sigma $ .", "The last equality follows from Lemma REF as $D \\subset M_{1}$ .", "Therefore, $\\Sigma ^{\\prime } = \\Sigma $ and $\\Sigma _n = \\Sigma $ for all $n \\ge j_0$ .", "This proves the implication (a) $\\Rightarrow $ (b).", "Suppose now that $\\Sigma \\subset A^G$ is a linear subshift which is not of finite type.", "Let $(M_n)_{n \\in \\mathbb {N}}$ be a sequence of finite subsets of $G$ satisfying conditions (M-2)-(M-3).", "For every $n \\in \\mathbb {N}$ , set $W_{n} \\Sigma _{M_n}$ (as in Section REF ).", "Then, $W_n$ is a vector subspace of $A^{M_n}$ .", "For every $n \\in \\mathbb {N}$ we consider the linear SFT $\\Sigma _n \\Sigma (M_n, W_n)$ .", "As $(\\Sigma _{M_{n+1}})_{M_n} = \\Sigma _{M_n}$ , it is clear that $\\Sigma \\subset \\Sigma _{n+1} \\subset \\Sigma _n$ for all $n \\in \\mathbb {N}$ .", "We claim that $\\Sigma = \\bigcap _{n \\in \\mathbb {N}} \\Sigma _n$ .", "We only need to prove that $\\bigcap _{n \\in \\mathbb {N}} \\Sigma _n \\subset \\Sigma $ .", "Let $x \\in \\bigcap _{n \\in \\mathbb {N}} \\Sigma _n$ .", "Then by definition of $\\Sigma _n$ , we find that $x\\vert _{M_n} \\in W_n = \\Sigma _{M_n}$ for every $n \\in \\mathbb {N}$ .", "Thus, since $\\Sigma $ is closed, $x \\in \\varprojlim _{n \\in \\mathbb {N}} \\Sigma _{M_n} = \\Sigma $ (cf.", "(REF )) and hence $\\bigcap _{n \\in \\mathbb {N}} \\Sigma _n \\subset \\Sigma $ .", "However, the decreasing sequence $(\\Sigma _n)_{n \\in \\mathbb {N}}$ cannot stabilize since, otherwise, the subshift $\\Sigma $ would be of finite type.", "This shows that (b)$\\Rightarrow $ (a).", "The proof of Theorem REF is complete.", "$\\Box $ Suppose first that $A^G$ satisfies condition (b) and let $\\Sigma \\subset A^G$ be a linear subshift.", "Let $(D_n)_{n \\in \\mathbb {N}}$ be an increasing sequence of finite subsets of $G$ such that $\\bigcup _{n \\in \\mathbb {N}} D_n = G$ .", "For every $n \\in \\mathbb {N}$ let $W_n \\Sigma _{D_n} \\subset A^{D_n}$ .", "Then $\\Sigma _n \\Sigma (D_n,W_n) \\subset A^G$ is a linear SFT and $\\Sigma _0 \\supset \\Sigma _1 \\supset \\cdots \\Sigma _n \\supset \\Sigma _{n+1} \\cdots $ for all $n \\in \\mathbb {N}$ .", "We claim that $\\bigcap _{n \\in \\mathbb {N}} \\Sigma _n = \\Sigma $ .", "Since $\\Sigma _n \\supset \\Sigma $ for all $n \\in \\mathbb {N}$ , we only need to show that $\\bigcap _{n \\in \\mathbb {N}} \\Sigma _n \\subset \\Sigma $ .", "Let $x \\in \\bigcap _{n \\in \\mathbb {N}} \\Sigma _n$ .", "This means that for each $n \\in \\mathbb {N}$ there exists $x_n \\in \\Sigma $ such that $x\\vert _{D_n} = x_n\\vert _{D_n}$ .", "Since the sequence $(D_n)_{n \\in \\mathbb {N}}$ is exhausting and $\\Sigma $ is closed in the prodiscrete topology, we deduce that $x \\in \\Sigma $ .", "This proves the claim.", "Since $A^G$ satisfies condition (b), there exists $n_0 \\in \\mathbb {N}$ such that $\\Sigma _n = \\Sigma _{n_0}$ for all $n \\ge n_0$ .", "We deduce that $\\Sigma = \\Sigma _{n_0}$ is of finite type.", "Conversely, suppose that every linear subshift $\\Sigma \\subset A^G$ is of finite type and let $(\\Sigma _n)_{n \\in \\mathbb {N}}$ be a decreasing sequence of linear subshifts.", "Set $\\Sigma \\bigcap _{n \\in \\mathbb {N}} \\Sigma _n \\subset A^G$ .", "Then $\\Sigma $ is a linear subshift and, by our assumptions, it is of finite type.", "It follows from Theorem REF that the sequence $(\\Sigma _n)_{n \\in \\mathbb {N}}$ eventually stabilizes.", "The proof of Corollary REF is complete.", "$\\Box $ We first observe that if $G$ is uncountable then, on the one hand $G$ is not finitely generated and thus is not Noetherian (that is, it does not satisfy the maximal condition on subgroups) and therefore the group algebra $K[G]$ is not one-sided Noetherian (cf.", "[24]), and, on the other hand, $G$ is not of $K$ -linear Markov type, since the linear subshift consisting of all constant configurations in $K^G$ is not of finite type (cf.", "Corollary REF ).", "Thus, in order to prove Theorem REF , it is not restrictive to assume that $G$ is countable.", "Recall that $\\operatorname{LCA}(G,A)$ denotes the $K$ -algebra of all linear cellular automata $\\tau \\colon A^G \\rightarrow A^G$ (cf.", "[8]).", "The evaluation map $(\\tau , x) \\mapsto \\tau (x)$ , where $\\tau \\in \\operatorname{LCA}(G,A)$ and $x \\in A^G$ , yields a $K$ -bilinear map $\\operatorname{LCA}(G,A) \\times A^G \\rightarrow A^G$ .", "Given a subset $\\Gamma $ in $\\operatorname{LCA}(G,A)$ , set $\\Gamma ^\\perp \\bigcap _{\\tau \\in \\Gamma } \\ker (\\tau ) \\subset A^G.$ Since every map $\\tau \\in \\operatorname{LCA}(G,A)$ is linear, continuous, and $G$ -equivariant, we deduce immediately that its kernel $\\ker (\\tau )$ is a linear subshift of $A^G$ .", "Moreover, since the set of all linear subshifts in $A^G$ is closed under intersections, we have that $\\Gamma ^\\perp $ is a linear subshift of $A^G$ .", "Given a subset $\\Sigma \\subset A^G$ , set $\\Sigma ^\\perp \\lbrace \\tau \\in \\operatorname{LCA}(G,A): \\Sigma \\subset \\ker (\\tau )\\rbrace \\subset \\operatorname{LCA}(G,A).$ We claim that $\\Sigma ^\\perp $ is a left ideal in $\\operatorname{LCA}(G,A)$ .", "First of all, we clearly have $0 \\in \\Sigma ^\\perp $ , since $\\Sigma \\subset A^G = \\ker (0)$ .", "Suppose that $\\tau _1, \\tau _2 \\in \\Sigma ^\\perp $ .", "Then $(\\tau _1-\\tau _2)(x) = \\tau _1(x) - \\tau _2(x) = 0-0 = 0$ for all $x \\in \\Sigma $ , showing that $\\tau _1 - \\tau _2 \\in \\Sigma ^\\perp $ .", "Finally, if $\\tau \\in \\operatorname{LCA}(G,A)$ , we have $(\\tau \\circ \\tau _1)(x) = \\tau (\\tau _1(x)) = \\tau (0) = 0$ for all $x \\in \\Sigma $ , showing that $\\tau \\circ \\tau _1 \\in \\Sigma ^\\perp $ .", "This proves the claim.", "We note also that if $\\Sigma _1, \\Sigma _2 \\subset A^G$ , then $\\Sigma _1 \\subset \\Sigma _2 \\ \\Rightarrow \\ \\Sigma _2^\\perp \\subset \\Sigma _1^\\perp .$ We have the following key lemmata: Lemma 5.1 Let $G$ be a group, let $A$ be a vector space over a field $K$ , and let $\\Sigma \\subset A^G$ be a linear subshift.", "Then $(\\Sigma ^\\perp )^\\perp = \\Sigma .$ It trivially follows from the definitions that $\\Sigma \\subset (\\Sigma ^\\perp )^\\perp $ .", "In order to show the other inclusion, let $x \\in A^G \\setminus \\Sigma $ and let us show that $x \\notin (\\Sigma ^\\perp )^\\perp $ .", "Since $\\Sigma $ is closed, by the definition of prodiscrete topology we can find a finite subset $\\Omega \\subset G$ such that $x\\vert _\\Omega \\notin \\Sigma _\\Omega $ .", "It is a classical and easy argument in Linear Algebra that there exists a linear map $\\mu \\colon A^\\Omega \\rightarrow A$ such that $\\mu \\vert _{\\Sigma _\\Omega } \\equiv 0$ , that is, $\\Sigma _\\Omega \\subset \\ker (\\mu )$ , and $\\mu (x\\vert _\\Omega ) \\ne 0$ .", "It is then clear that the linear CA $\\tau $ with memory set $\\Omega $ and local defining map $\\mu $ satisfies that $\\Sigma \\subset \\ker (\\tau )$ , that is, $\\tau \\in \\Sigma ^\\perp $ , but $\\tau (x) \\ne 0$ .", "Thus $x \\notin (\\Sigma ^\\perp )^\\perp $ .", "In the proof of the following lemma, we explicitly use the $K$ -algebra isomorphism $\\operatorname{Mat}_d(K[G]) \\cong \\operatorname{LCA}(G,K^d)$ we alluded to above (cf.", "[8]) for $d=1$ .", "This is given by associating with each $\\alpha \\in K[G]$ the linear cellular automaton $\\tau _\\alpha \\colon K^G \\rightarrow K^G$ with memory set $M_\\alpha \\lbrace g \\in G: \\alpha (g) \\ne 0\\rbrace $ , the support of $\\alpha $ , and local defining map $\\mu _\\alpha \\colon K^{M_\\alpha } \\rightarrow K$ defined by setting $\\mu _\\alpha (y) \\sum _{h \\in M_\\alpha }\\alpha (h)y(h)$ for all $y \\in K^{M_\\alpha }$ .", "We shall also make use of the following notation.", "Given $\\alpha \\in K[G]$ , for every finite subset $E \\subset G$ such that $M_\\alpha \\subset E$ we define the linear map $\\mu _{\\alpha , E} \\colon K^E \\rightarrow K$ by setting $\\mu _{\\alpha , E} \\mu _\\alpha \\circ \\pi _{M_\\alpha , E}$ , where $\\pi _{M_\\alpha , E} \\colon K^E \\rightarrow K^{M_\\alpha }$ is the projection map induced by the inclusion $M_\\alpha \\subset E$ .", "Note that $\\mu _{\\alpha , E}$ is the local defining map of $\\tau _{\\alpha }$ associated with the memory set $E$ .", "Lemma 5.2 Let $G$ be a countable group and let $K$ be a field.", "Let $\\Gamma \\subset K[G]$ be a left ideal.", "Suppose that $\\Gamma ^\\perp \\subset K^G$ is a linear SFT.", "Then $\\Gamma $ is a finitely generated left ideal.", "Since $G$ is countable, we can find an increasing sequence $(E_n)_{n \\in \\mathbb {N}}$ of finite subsets of $G$ such that $G = \\bigcup _{n \\in \\mathbb {N}} E_n$ .", "For every $n \\in \\mathbb {N}$ , let $\\Gamma _n \\subset \\Gamma $ be the ideal of $K[G]$ generated by the elements of $\\Gamma $ whose supports are contained in $E_n$ .", "Then $\\Gamma _n \\subset \\Gamma _{n+1}$ for all $n \\in \\mathbb {N}$ and $\\Gamma = \\bigcup _{n \\in \\mathbb {N}} \\Gamma _n$ .", "We thus obtain a decreasing sequence $(\\Gamma _n^\\perp )_{n \\in \\mathbb {N}}$ of linear subshifts of $K^G$ .", "Remark that we can write $\\Gamma ^\\perp = \\bigcap _{\\alpha \\in \\Gamma } \\operatorname{Ker}(\\tau _\\alpha ) = \\bigcap _{n \\in \\mathbb {N}} \\bigcap _{\\alpha \\in \\Gamma _n} \\operatorname{Ker}(\\tau _\\alpha ) = \\bigcap _{n \\in \\mathbb {N}} \\Gamma _n^\\perp .$ Since, by hypothesis, the linear subshift $\\Gamma ^\\perp \\subset K^G$ is of finite type, we deduce from Theorem REF that there exists $n_0 \\in \\mathbb {N}$ such that $\\Gamma _{n}^\\perp = \\Gamma _{n_0}^\\perp $ for every $n \\ge n_0$ , equivalently, $\\Gamma ^\\perp = \\Gamma _{n_0}^\\perp $ .", "Claim.", "$\\Gamma = \\Gamma _{n_0}$ .", "Set $J \\Gamma _{n_0}$ and suppose by contradiction that there exists $\\alpha \\in \\Gamma \\setminus J$ .", "Let $m_0 \\in \\mathbb {N}$ be such that $E_{m_0}$ contains the support $M_\\alpha \\subset G$ of $\\alpha $ .", "For every $m \\in \\mathbb {N}$ , we set $V_m K^{E_m}$ and denote by $V_m^$ the dual $K$ -vector space of $V_m$ .", "Given a vector subspace $W_m \\subset V_m$ (resp.", "$J_m \\subset V_m^$ ) we set $W_m^\\perp \\lbrace v^ \\in V_m^: W_m \\subset \\ker (v^)\\rbrace \\subset V_m^$ (resp.", "$J_m^\\perp \\bigcap _{v^ \\in J_m} \\ker (v^) \\subset V_m$ ).", "Since $V_m$ is finite-dimensional, we have $(J_m^\\perp )^\\perp = J_m$ .", "We then denote by $J_m \\subset J$ the subset containing all elements of $J$ whose supports are contained in $E_m$ .", "Observe that $J_m \\subset J_{m+1}$ and $J = \\bigcup _{m \\ge m_0} J_m$ .", "We regard $J_m$ as a linear subspace of $V_m^$ via the map $\\beta \\mapsto \\mu _{\\beta , E_m}$ .", "This way, setting $W_m \\bigcap _{\\beta \\in J_m} \\operatorname{Ker}(\\mu _{\\beta , E_m}) \\subset V_m$ , we have $W_m = J_m^\\perp $ and therefore $\\lbrace v^* \\in V_m^: W_m \\subset \\ker (v^)\\rbrace = W_m^\\perp = (J_m^\\perp )^\\perp = J_m.$ From this we deduce that for every $m \\ge m_0$ $U_m W_m \\setminus \\operatorname{Ker}(\\mu _{\\alpha , E_m}) \\ne \\varnothing .$ Indeed, otherwise, we would have $W_m \\subset \\operatorname{Ker}(\\mu _{\\alpha , E_m})$ so that, by (REF ), $\\alpha \\in J_m \\subset J$ , a contradiction since $\\alpha \\notin J$ .", "For every $m \\ge n \\ge m_0$ , let $\\pi _{nm} \\colon K^{E_m} \\rightarrow K^{E_n}$ be the projection map induced by the inclusion $E_n \\subset E_m$ .", "It is clear that $\\pi _{nm}(U_m) \\subset U_n$ since $\\operatorname{Ker}(\\mu _{\\alpha , E_m}) = \\operatorname{Ker}(\\mu _\\alpha ) \\times K^{E_m \\setminus M_\\alpha } \\subset K^{E_m}$ and $\\pi _{nm}(W_m) \\subset W_n$ for all $m \\ge n \\ge m_0$ .", "Therefore, we obtain an inverse system $(U_m)_{m \\ge m_0}$ of nonempty sets with transition maps $\\varphi _{nm} \\pi _{nm}\\vert _{U_m} \\colon U_m \\rightarrow U_n$ for $m \\ge n \\ge m_0$ .", "As in Lemma REF , an immediate application of the Mittag-Leffler condition to the inverse system $(U_m)_{m \\ge m_0}$ shows that there exists a configuration $c \\in \\varprojlim _{m \\ge m_0} U_m \\subset \\varprojlim _{m \\ge m_0} W_m$ .", "Let us show that $c \\in J^\\perp = \\bigcap _{\\beta \\in J}\\ker (\\tau _\\beta ) \\subset K^G$ .", "Let $\\beta \\in J$ and let $g \\in G$ .", "Since $J$ is an ideal of $K[G]$ and $J = \\bigcup _{m \\ge m_0} J_m$ , there exists $m \\ge m_0$ such that $g\\beta \\in J_m$ .", "Since $c\\vert _{E_m} \\in W_m$ , it follows from the definition of $W_m$ that $\\tau _\\beta (c)(g) = \\mu _{\\beta , E_m}((g^{-1}c)\\vert _{E_m})= \\mu _{g\\beta , E_m}(c\\vert _{E_m}) = 0.$ Since $g \\in G$ was arbitrary, this shows that $\\tau _\\beta (c) = 0$ .", "Since $\\beta \\in J$ was arbitrary, this shows that $c \\in J^\\perp $ .", "On the other hand, by construction, we have that $\\mu _\\alpha (c\\vert _{M_\\alpha }) \\ne 0$ so that $\\tau _\\alpha (c) \\ne 0$ .", "Since $\\alpha \\in J$ , we deduce that $c \\notin J^\\perp $ , a contradiction.", "The claim is proved.", "We are now in a position to show that $\\Gamma $ is finitely generated as a left ideal.", "With the above notation, $J_{n_0}$ , the subset consisting of all elements in $J = \\Gamma $ whose supports are contained in $E_{n_0}$ is a vector subspace of $V_{n_0} = K^{E_{n_0}}$ , and therefore is finite dimensional.", "It is then clear that any vector basis of $J_{n_0}$ also generates $\\Gamma _{n_0} = \\Gamma $ as a left ideal.", "We conclude that $\\Gamma $ is a finitely generated left ideal of $K[G]$ .", "We are now in a position to prove Theorem REF .", "Recall that we assume that $G$ is countable.", "Suppose first that the group algebra $K[G]$ is one-sided Noetherian.", "Let $A$ be a finite-dimensional vector space over $K$ and let $d = \\dim _K(A)$ .", "We then observe that since $K[G]$ is one-sided Noetherian, so is the finitely generated left $K[G]$ -module $\\operatorname{Mat}_d(K[G])$ , the $K$ -algebra of $d \\times d$ matrices with coefficients in the group ring $K[G]$ .", "Since every left ideal in $\\operatorname{Mat}_d(K[G])$ is trivially a left $K[G]$ -module, we deduce that $\\operatorname{Mat}_d(K[G])$ is one-sided Noetherian as well as a ring.", "As mentioned above (cf.", "[8]), once fixed a vector basis for $A$ , there exists a canonical $K$ -algebra isomorphism of $\\operatorname{LCA}(G,A)$ onto $\\operatorname{Mat}_d(K[G])$ .", "We deduce that $\\operatorname{LCA}(G,A)$ is one-sided Noetherian.", "In order to show that $G$ is of $K$ -linear Markov type, let $(\\Sigma _n)_{n \\in \\mathbb {N}}$ be a decreasing sequence of linear subshifts in $A^G$ and let us show that it stabilizes.", "Setting $\\Gamma _n \\Sigma _n^\\perp $ for all $n \\in \\mathbb {N}$ , we get an increasing sequence $(\\Gamma _n)_{n \\in \\mathbb {N}}$ of left ideals in $\\operatorname{LCA}(G,A)$ .", "Since the latter is left-Noetherian, such a sequence stabilizes, that is, there exists $n_0 \\in \\mathbb {N}$ such that $\\Gamma _n = \\Gamma _{n_0}$ for all $n \\ge n_0$ .", "It then follows from Lemma REF that $\\Sigma _n = \\Gamma _n^\\perp = \\Gamma _{n_0}^\\perp = \\Sigma _{n_0}$ for all $n \\ge n_0$ , that is, $(\\Sigma _n)_{n \\in \\mathbb {N}}$ stabilizes.", "This shows that $G$ is of $K$ -linear Markov type.", "Conversely, suppose that $G$ is of $K$ -linear Markov type and let $\\Gamma \\subset K[G]$ be a left ideal.", "Then the linear subshift $\\Gamma ^\\perp \\subset K^G$ is of finite type.", "Lemma REF implies that $\\Gamma $ is finitely generated.", "This shows that the group algebra $K[G]$ is one-sided Noetherian.", "The proof of Theorem REF is complete.", "$\\Box $ Let $G$ be a polycyclic-by-finite group and let $K$ be a field.", "It follows from a famous result of P. Hall [18] (see also [24]) that $K[G]$ is one-sided Noetherian.", "We then deduce from Theorem REF that $G$ is of $K$ -linear Markov type.", "$\\Box $ Remark.", "(1) At our knowledge, it is not known whether or not there exist groups $G$ , other than the polycyclic-by-finite groups, whose group algebra $K[G]$ is one-sided Noetherian.", "See Section for more on this.", "(2) An alternative and self-contained proof of Corollary REF is obtained from Lemma REF and Lemma REF below combined with an easy induction argument.", "For the details see Remark REF .", "(i) It follows from Theorem REF that $\\tau ^n(\\Sigma )$ is a linear subshift in $A^G$ for all $n \\in \\mathbb {N}$ .", "Since the intersection of any family of linear subshifts is itself a linear subshift, we deduce that $\\Omega (\\tau ) = \\bigcap _{n \\in \\mathbb {N}} \\tau ^n(\\Sigma )$ is a linear subshift.", "(ii) Let $x \\in \\Omega (\\tau )$ , that is, $x \\in \\tau ^n(\\Sigma )$ for every $n \\ge 0$ .", "Thus $\\tau (x) \\in \\tau ^{n+1}(\\Sigma )$ for every $n \\ge 0$ and it follows that $\\tau (x) \\in \\Omega (\\tau )$ .", "Therefore, $\\tau (\\Omega (\\tau )) \\subset \\Omega (\\tau )$ .", "For the converse inclusion, let $y \\in \\Omega (\\tau )$ .", "Let also $M \\subset G$ be a memory set for $\\tau $ such that $1_G \\in M$ and $M = M^{-1}$ .", "Then, by Proposition REF , there exists $x \\in \\varprojlim _{(\\Omega ,n) \\in I} \\Sigma _{\\Omega ,n}$ such that $\\Phi (x)=y$ .", "On the other hand, (REF ) tells us that $\\Phi ^{-1}(y) \\subset \\varprojlim _{(\\Omega ,n) \\in I} \\Sigma _{\\Omega ,n}$ is the set of backward orbits of $y$ under $\\tau $ .", "Hence, we can find $z \\in \\Omega (\\tau )$ such that $\\tau (z)= y$ .", "Thus, $\\Omega (\\tau ) \\subset \\tau (\\Omega (\\tau ))$ and equality follows.", "(iii) As already mentioned in the Introduction, the inclusions $\\operatorname{Per}(\\tau ) \\subset \\operatorname{R}(\\tau ) \\subset \\operatorname{NW}(\\tau )$ are immediate from the definitions.", "In [12] it is shown that if $X$ is a uniform space and $f \\colon X \\rightarrow X$ is a continuous map, then $\\operatorname{NW}(f) \\subset \\operatorname{CR}(f)$ .", "Since every cellular automaton is continuous, we deuce that $\\operatorname{NW}(\\tau ) \\subset \\operatorname{CR}(\\tau )$ .", "In [12] it is shown that if $X$ is a Hausdorff uniform space and $f \\colon X \\rightarrow X$ is a uniformly continuous map such that $f^n(X)$ is closed in $X$ for all $n \\in \\mathbb {N}$ , then $\\operatorname{CR}(f) \\subset \\Omega (f)$ .", "In our setting, uniform continuity of $\\tau $ is a general property of cellular automata already mentioned in the Introduction.", "Moreover, $\\tau ^n(\\Sigma )$ is closed in $\\Sigma $ for all $n \\in \\mathbb {N}$ by Theorem REF .", "We deduce the last inclusion, namely $\\operatorname{CR}(\\tau ) \\subset \\Omega (\\tau )$ .", "(iv) Suppose that $\\Omega (\\tau )$ is of finite type.", "It follows from Theorem REF that the sequence $(\\tau ^n(\\Sigma ))_{n \\in \\mathbb {N}}$ eventually stabilizes, that is, there exists $n_0 \\ge 1$ such that $\\tau ^n(\\Sigma ) = \\tau ^{n_0}(\\Sigma )$ for all $n \\ge n_0$ .", "This shows that $\\tau $ is stable.", "(v) Suppose that $\\Omega (\\tau )$ is finite-dimensional.", "It follows from Proposition REF that $\\Omega (\\tau )$ is of finite type.", "Using (iv) we deduce that $\\tau $ is stable.", "This ends the proof of Theorem REF .", "$\\Box $ We only need to prove the statements for $G$ not finitely generated.", "Let $M \\subset G$ be a finite subset serving as a memory set for both $\\Sigma $ and $\\tau $ , and denote by $H \\subset G$ the subgroup generated by $M$ .", "The proof of Theorem REF .", "(ii) did not use any finite generation assumption on $G$ and therefore holds true in the present setting as well.", "(i) It follows from Theorem REF .", "(i) applied to the restriction cellular automaton $\\tau _H \\colon \\Sigma _H \\rightarrow \\Sigma _H$ that $\\Omega (\\tau _H)$ is a linear subshift.", "As a consequence, $\\Omega (\\tau _c)$ are closed in $A^c$ for all $c \\in G/H$ .", "As products of closed subspaces are closed in the product topology, we deduce from Lemma REF .", "(i) that $\\Omega (\\tau ) = \\prod _{c \\in G/H} \\Omega (\\tau _c)$ is also closed in $A^G$ .", "Since $\\Omega (\\tau )$ is a $K$ -linear and $G$ -invariant subset of $A^G$ , we conclude that it is a linear subshift of $A^G$ .", "(ii) It follows from Theorem REF .", "(ii) applied to the restriction cellular automaton $\\tau _H \\colon \\Sigma _H \\rightarrow \\Sigma _H$ that $\\tau _H(\\Omega (\\tau _H)) = \\Omega (\\tau _H)$ .", "As a consequence, $\\tau _c(\\Omega (\\tau _c)) = \\Omega (\\tau _c)$ for all $c \\in G/H$ .", "We deduce from Lemma REF .", "(i) that $\\tau (\\Omega (\\tau )) = \\prod _{c \\in G/H} \\tau _c(\\Omega (\\tau _c)) = \\prod _{c \\in G/H} \\Omega (\\tau _c) = \\Omega (\\tau )$ .", "(iii) Just note that, by virtue of Theorem REF , $\\tau ^n_H(\\Sigma _H)$ is closed in $A^H$ for all $n \\in \\mathbb {N}$ .", "Hence, by Lemma REF , $\\tau ^n(\\Sigma )$ is closed in $A^G$ for all $n \\in \\mathbb {N}$ , and the proof of Theorem REF .", "(iii) applies verbatim.", "(iv) Up to enlarging $M \\subset G$ , if necessary, we may suppose that $M$ also serves as a memory set for the SFT $\\Omega (\\tau )$ , say $\\Omega (\\tau ) = \\Sigma (M, W) \\subset A^G$ for some $W \\subset A^M$ .", "We have that $\\Omega (\\tau )_H = \\Omega (\\tau _H) = \\Sigma (M, W) \\subset A^H$ is of finite type as well.", "It then follows from Theorem REF .", "(iv) applied to the restriction cellular automaton $\\tau _H$ , that $\\tau _H$ is stable.", "Since stability is invariant under the operation of restriction, we deduce that $\\tau $ is itself stable.", "(v) If $\\Omega (\\tau )$ is finite-dimensional, so is $\\Omega (\\tau _H) = \\Omega (\\tau )_H$ .", "It then follows from Theorem REF .", "(v) applied to the restriction cellular automaton $\\tau _H$ , that $\\tau _H$ is stable.", "Thus $\\tau $ is itself stable.", "This proves Corollary REF .", "$\\Box $ We first observe that every polycyclic-by-finite group is amenable (see, for instance [8]).", "Let $\\Lambda \\subset A^G$ be a strongly irreducible subshift such that $\\Lambda \\subset \\Sigma $ , $\\tau (\\Lambda ) \\subset \\Lambda $ , and such that the restriction linear CA $\\tau \\vert _\\Lambda \\colon \\Lambda \\rightarrow \\Lambda $ is pre-injective.", "Since $G$ is polycyclic-by-finite, Corollary REF ensures that $\\Lambda $ is a linear subshift of finite type.", "Since $G$ is amenable, the implication pre-injectivity $\\Rightarrow $ surjectivity in the Garden of Eden theorem [10] yields the equality $\\tau (\\Lambda ) = \\Lambda $ .", "It follows immediately that $\\Lambda \\subset \\Omega (\\tau )$ .", "Theorem REF .", "(i) and Corollary REF imply that $\\Omega (\\tau )$ is a linear subshift of finite type.", "Thus, by Theorem REF .", "(iv), $\\tau $ is stable and therefore there exists an integer $n \\ge 1$ such that $\\tau ^n(\\Sigma ) = \\Omega (\\tau )$ .", "Since the image of a strongly irreducible subshift under a CA is also strongly irreducible, it follows that $\\Omega (\\tau )$ is a strongly irreducible linear SFT.", "By Theorem REF .", "(ii), $\\tau (\\Omega (\\tau )) \\subset \\Omega (\\tau )$ and the restriction linear CA $\\tau \\vert _{\\Omega (\\tau )} \\colon \\Omega (\\tau ) \\rightarrow \\Omega (\\tau )$ is surjective.", "We can thus conclude from the implication surjectivity $\\Rightarrow $ pre-injectivity in the Garden of Eden theorem [10] that $\\tau \\vert _{\\Omega (\\tau )}$ is pre-injective.", "The proof of Corollary REF is complete.", "$\\Box $ Let $H \\subset G$ be a finitely generated subgroup containing both a memory set for $\\Sigma $ and a memory set for $\\tau $ .", "By virtue of Lemma REF , we have, on the one hand that $\\tau $ is nilpotent if and only if $\\tau _H$ is, and, one the other hand that $\\Omega (\\tau ) = \\lbrace 0\\rbrace $ if and only if $\\Omega (\\tau _H) = \\lbrace 0\\rbrace $ .", "Thus, it is not restrictive to suppose that $G = H$ is finitely generated.", "Suppose that $\\tau $ is nilpotent.", "Then there exists $n_0 \\ge 1$ such that $\\tau ^{n_0}(\\Sigma ) = \\lbrace 0\\rbrace $ .", "It then follows that $\\Omega (\\tau ) = \\lbrace 0\\rbrace $ .", "Conversely, suppose (b).", "Then $\\Omega (\\tau )$ is of finite type.", "By the characterization of linear SFT in Theorem REF , the sequence $(\\tau ^n(\\Sigma ))_{n \\in \\mathbb {N}}$ eventually stabilizes, that is, there exits $n_0 \\ge 1$ such that $\\tau ^{n_0}(\\Sigma ) = \\Omega (\\tau ) = \\lbrace 0\\rbrace $ .", "This shows that $\\tau $ is nilpotent.", "This completes the proof of Theorem REF .", "$\\Box $ We shall prove the implications $\\mbox{(a) $\\iff $ (b) \\ \\ \\ \\ and \\ \\ \\ \\ (a) $\\Rightarrow $ (c) $\\Rightarrow $ (d) $\\Rightarrow $ (e) $\\Rightarrow $ (a).", "}$ The implication (a) $\\Rightarrow $ (b) is trivial.", "Suppose that $\\tau $ is pointwise nilpotent, so that for every $x \\in \\Sigma $ , there exists an integer $n_x \\ge 1$ such that $\\tau ^n(x) = 0$ for all $n \\ge n_x$ .", "Since $G$ is finitely generated, it is countable.", "Then, the configuration space $A^G$ , being a countable product of discrete (and therefore completely metrizable) spaces, it admits a complete metric compatible with its topology, and hence is a Baire space.", "Since $\\Sigma $ is closed in $A^G$ , it is a Baire space as well.", "For each integer $n \\ge 1$ , the set $X_n (\\tau ^n)^{-1}(0) = \\lbrace x \\in \\Sigma : \\tau ^n(x) = 0\\rbrace $ is a linear subshift of $A^G$ contained in $\\Sigma $ .", "We have $\\Sigma = \\bigcup _{n \\ge 1} X_n$ by our hypothesis on $\\tau $ .", "By the Baire category theorem, there is an integer $n_0 \\ge 1$ such that $X_{n_0}$ has a nonempty interior.", "Since $\\Sigma $ is topologically mixing and $G$ is infinite, $\\Sigma $ is topologically transitive.", "It follows from a standard fact (cf.", "[12]) that $X_{n_0} = \\Sigma $ , equivalently, $\\tau ^{n_0}(\\Sigma ) = \\lbrace 0\\rbrace $ .", "The latter is equivalent to $\\tau $ being nilpotent, and the implication (b) $\\Rightarrow $ (a) follows.", "From the first implication we deduce that in fact (a) $\\iff $ (b).", "The implication (a) $\\Rightarrow $ (c) is obvious.", "The implication (c) $\\Rightarrow $ (d) is clear since $\\Omega (\\tau )$ is a vector subspace of $\\tau ^{n_0}(\\Sigma )$ .", "The implication (d) $\\Rightarrow $ (e) follows from Proposition REF since any topologically mixing action of an infinite group is topologically transitive.", "Finally, suppose (e).", "As $\\Omega (\\tau )$ is of finite type, we deduce from Theorem REF that the sequence $(\\tau ^n(\\Sigma ))_{n \\in \\mathbb {N}}$ eventually stabilizes, that is, there exists an $n_0 \\in \\mathbb {N}$ such that $\\tau ^{n_0}(\\Sigma ) = \\Omega (\\tau ) = \\lbrace 0\\rbrace $ .", "Thus $\\tau $ is nilpotent.", "This shows the outstanding implication (e) $\\Rightarrow $ (a), and the proof of Theorem REF is complete.", "$\\Box $ We only need to prove the equivalences for $G$ not finitely generated.", "Let $H \\subset G$ be a finitely generated subgroup containing both a memory set for $\\Sigma $ and a memory set for $\\tau $ .", "Observe that $[G:H] = \\infty $ , since $G$ is not finitely generated.", "Denote by $\\tau _H \\colon \\Sigma _H \\rightarrow \\Sigma _H$ the corresponding restriction cellular automaton.", "The implication (a) $\\Rightarrow $ (b) is trivial.", "Suppose (b).", "It is straightforward that $\\tau _H$ is also pointwise nilpotent.", "It then follows from the finitely generated case (i.e., from the implication (b) $\\Rightarrow $ (a) in Theorem REF ) that $\\tau _H$ is nilpotent.", "We then deduce from Lemma REF .", "(iii) that $\\tau $ is itself nilpotent.", "This shows the implication (b) $\\Rightarrow $ (a).", "Combined with the previous implication, this gives the equivalence (a) $\\iff $ (b).", "The implications (a) $\\Rightarrow $ (c) $\\Rightarrow $ (d) are trivial.", "Suppose (d).", "Recalling that $H$ has infinite index in $G$ , we deduce from Lemma REF .", "(i) that $\\Omega (\\tau _H) = \\lbrace 0\\rbrace $ and $\\Omega (\\tau ) = \\lbrace 0\\rbrace $ .", "This shows the implication (d) $\\Rightarrow $ (e).", "The final implication (e) $\\Rightarrow $ (a) follows from Theorem REF .", "The proof of Corollary REF is complete.", "$\\Box $" ], [ "Groups of $K$ -linear Markov type", "We have seen in Corollary REF that the condition that $G$ be finitely generated cannot be removed from the assumptions in Proposition REF .", "More generally, if a finitely generated group $G$ admits a subgroup $H$ which is not finitely generated (for instance, if $G$ contains a subgroup $K$ isomorphic to $F_2$ , the free group of rank 2, and $H = [K,K] \\subset K$ its commutator subgroup) then the subshift consisting of all configurations $x \\in A^G$ which are constant on each left coset of $H$ in $G$ is not of finite type (note that $H$ has necessarily infinite index in $G$ ).", "Recall that a group $G$ satisfies the maximal condition on subgroups if any ascending sequence $G_0 \\subset G_1 \\subset \\cdots \\subset G_n \\subset G_{n+1} \\subset \\cdots \\subset G$ of subgroups eventually stabilizes, that is, there exists $n_0 \\ge 1$ such that $G_n = G_{n_0}$ for all $n \\ge n_0$ .", "It is immediately verified that a group $G$ satisfies the maximal condition on subgroups if and only if all of its subgroups are finitely generated.", "A group satisfying the maximal condition on subgroups is also called a Noetherian group.", "From the above discussion we immediately deduce the following.", "Corollary 6.1 Let $G$ be a group of $K$ -linear Markov type for some field $K$ .", "Then $G$ is Noetherian.", "In particular, $G$ is finitely generated.", "As remarked above, we don't know whether or not the class of polycyclic-by-finite groups coincides with the class of groups of $K$ -linear Markov type.", "We remark that there exist Noetherian groups constructed by A.Y.", "Olshanskii [23], for which the group algebra is not known to be one-sided Noetherian, equivalently (cf.", "TheoremREF ), it is not known whether or not they are of $K$ -linear Markov type.", "On the other hand, it follows from the work of L. Bartholdi [1] and P. Kropholler and K. Lorensen [21], and Theorem REF , that if $G$ is of $K$ -linear Markov type, then $G$ is necessarily amenable.", "We refer to Mathoverflow [13] for other interesting information.", "In the next two lemmas we show that the class of groups of $K$ -linear Markov type is closed under finite and cyclic extensions.", "Lemma 6.2 Let $G$ be a countable group and let $H \\subset G$ be a normal subgroup of finite index.", "Suppose that $H$ is of linear Markov type.", "Then also $G$ is of linear Markov type.", "Let $A$ be a finite-dimensional vector space and let $\\Sigma \\subset A^G$ be a linear subshift.", "Let $T \\subset G$ be a complete set of representatives for the cosets of $H$ in $G$ , so that $G = HT$ .", "Then $B A^T$ is a finite-dimensional vector space and the map $\\varphi \\colon A^G \\rightarrow B^H$ defined by $\\left(\\varphi (x)(h)\\right)(t) = x(ht)$ for all $x \\in A^G$ , $h \\in H$ , and $t \\in T$ , is a linear isomorphims and uniform homeomorphism.", "Moreover, $\\left(\\varphi (kx)(h)\\right)(t) = (kx)(ht) = x(k^{-1}ht) = \\left(\\varphi (x)(k^{-1}h)\\right)(t) = \\left(k\\varphi (x)(h)\\right)(t)$ for all $x \\in A^G$ , $k,h \\in H$ , and $t \\in T$ , showing that $\\varphi $ is $H$ -equivariant.", "Then $\\Sigma ^{\\prime } \\varphi (\\Sigma )$ is a linear subshift in $B^H$ .", "Since $H$ is of linear Markov type, and $B$ is finite-dimensional, there exists a finite subset $D_H \\subset H$ and a subspace $P_H \\subset B^{D_H}$ such that $\\Sigma ^{\\prime } = \\Sigma (B^H;D_H,P_H)$ .", "Set $D D_HT \\subset G$ and consider the map $\\psi \\colon A^D \\rightarrow B^{D_H}$ defined by $\\left(\\psi (y)(h)\\right)(t) = y(ht)$ for all $y \\in A^D$ , $h \\in D_H$ , and $t \\in T$ .", "Then $\\psi $ is a linear isomorphism and a uniform homeomorphism.", "Let us set $P \\psi ^{-1}(P_H) \\subset A^D$ .", "Note that if $x \\in A^G$ , $h \\in D_H$ , and $t \\in T$ we have $\\left(\\varphi (x)\\vert _{D_H}(h)\\right)(t) = \\left(\\varphi (x)(h)\\right)(t) = x(ht) = x\\vert _D(ht) = \\left(\\psi (x\\vert _D)(h)\\right)(t)$ so that $\\varphi (x)\\vert _{D_H} = \\psi (x\\vert _D).$ It follows that $x \\in \\Sigma & \\iff tx \\in \\Sigma , \\mbox{ for all } t \\in T && \\mbox{(by $G$-invariance of $\\Sigma $)}\\\\& \\iff \\varphi (tx) \\in \\Sigma ^{\\prime }, \\mbox{ for all } t \\in T && \\mbox{(by definition of $\\Sigma ^{\\prime }$ and $\\varphi $ being $1$-$1$)}\\\\& \\iff \\left(h \\varphi (tx)\\right)\\vert _{D_H} \\in P_H, \\mbox{ for all } h \\in H \\mbox{ and } t \\in T && \\mbox{(since $\\Sigma ^{\\prime } = \\Sigma (B^H;D_H,P_H)$)}\\\\& \\iff \\left(\\varphi (htx)\\right)\\vert _{D_H} \\in P_H, \\mbox{ for all } h \\in H \\mbox{ and } t \\in T &&\\mbox{(by $H$-equivariance of $\\varphi $)}\\\\& \\iff \\left(\\varphi (gx)\\right)\\vert _{D_H} \\in P_H, \\mbox{ for all } g \\in G && \\mbox{(since $G = HT$)}\\\\& \\iff \\psi ^{-1}\\left(\\left(\\varphi (gx)\\right)\\vert _{D_H}\\right) \\in P, \\mbox{ for all } g \\in G && \\mbox{(by definition of $P$ and $\\psi $ being $1$-$1$)}\\\\& \\iff (gx)\\vert _D \\in P, \\mbox{ for all } g \\in G && \\mbox{(by (\\ref {e:phi-psi}))}\\\\& \\iff x \\in \\Sigma (A^G; D, P).", "&&$ This shows that $\\Sigma = \\Sigma (A^G; D, P)$ is of finite type.", "We deduce that $G$ is of linear Markov type.", "The following is the linear (and therefore simpler) version of the more general result [26].", "Lemma 6.3 Let $G$ be a countable group and let $H \\subset G$ be a normal subgroup such that $G/H$ is infinite cyclic.", "Suppose that $H$ is of linear Markov type.", "Then also $G$ is of linear Markov type.", "Let $A$ be a finite-dimensional vector space and let $\\Sigma \\subset A^G$ be a linear subshift.", "Let $a \\in G$ such that $aH$ generates $G/H \\cong \\mathbb {Z}$ and set $T^{\\prime } \\lbrace a^n: n \\in \\mathbb {Z}\\rbrace $ .", "Then $T^{\\prime }$ is a complete set of representatives for the cosets of $H$ in $G$ so that $G = HT^{\\prime }$ .", "Since $H$ is also countable, we can find an increasing sequence $(F_m)_{m \\in \\mathbb {N}}$ of finite subsets $F_m \\subset H$ such that $1_H \\in F_0$ and $H = \\bigcup _{m \\in \\mathbb {N}} F_m$ .", "For $i,j \\in \\mathbb {Z}\\cup \\lbrace -\\infty , +\\infty \\rbrace $ and $i \\le j$ let us set $T_i^j \\lbrace a^i, a^{i+1}, \\cdots , a^j\\rbrace \\subset T^{\\prime }$ .", "Claim 1.", "For every $n \\ge 1$ the set $X_n \\lbrace x\\vert _H: x \\in \\Sigma \\mbox{ such that } x(g) = 0_A \\mbox{ for all } g \\in HT_{-n}^{-1}\\rbrace \\subset A^H$ is a linear subshift in $A^H$ .", "Let $n \\ge 1$ .", "The fact that $X_n$ is a vector subspace of $A^H$ is clear.", "Let now $x \\in X_n$ .", "Then, there exists $y \\in \\Sigma $ such that $x = y\\vert _H$ and $y(g) = 0_A$ for all $g \\in HT_{-n}^{-1}$ .", "Given $h \\in H$ , we have $hy \\in \\Sigma $ , because $\\Sigma $ is a subshift in $A^G$ , and $(hy)(g) = y(h^{-1}g) = 0_A$ for all $g \\in HT_{-n}^{-1}$ , since $h^{-1}g \\in hHT_{-n}^{-1} = HT_{-n}^{-1}$ .", "It follows that $hx = (hy)\\vert _H \\in X_n$ , and this shows that $X_n$ is $H$ -invariant.", "We are only left to show that $X_n$ is closed with respect to the prodiscrete topology in $A^H$ .", "For $k \\ge n$ let us set $X_{n,k} \\lbrace x\\vert _{F_kT_{-k}^k}: x \\in \\Sigma \\mbox{ such that } x(g) = 0_A \\mbox{ for all } g \\in HT_{-n}^{-1}\\rbrace \\subset \\Sigma _{F_kT_{-k}^k}.$ Note that $X_{n,k}$ is a finite-dimensional vector space.", "For $m \\ge k \\ge n$ , let $\\pi _{k,m} \\colon A^{F_mT_{-m}^m} \\rightarrow A^{F_kT_{-k}^k}$ denote the projection map.", "Note that if $x \\in A^{F_mT_{-m}^m}$ satisfies that $x(g) = 0_A$ for all $g \\in F_mT_{-n}^{-1}$ , then $\\pi _{k,m}(x)(g^{\\prime }) = 0_A$ for all $g^{\\prime } \\in F_kT_{-n}^{-1}$ .", "Hence, setting $p_{k,m} \\pi _{k,m}\\vert _{X_{n,m}}$ we have $p_{k,m} \\colon X_{n,m} \\rightarrow X_{n,k}$ , and $(X_{n,k}, p_{k,m})_{m \\ge k \\ge n}$ is an inverse system of finite-dimensional vector spaces.", "Let $z \\in A^H$ be a configuration belonging to the closure of $X_n$ in $A^H$ .", "We must show that $z \\in X_n$ .", "By definition, for each $k \\ge n$ there exists a configuration $x_k \\in \\Sigma $ such that $x_k\\vert _{F_k} = z\\vert _{F_k} \\mbox{ \\ and \\ } x_k(g) = 0_A \\mbox{ for all } g \\in HT_{-n}^{-1}.$ Let us set $X_{n,k}(z) \\lbrace x\\vert _{F_kT_{-k}^k}: x \\in \\Sigma \\mbox{ such that } x\\vert _{F_k} = z\\vert _{F_k} \\mbox{ and } x(g) = 0_A \\mbox{ for all } g \\in HT_{-n}^{-1}\\rbrace \\subset X_{n,k}.$ Note that $X_{n,k}(z)$ is an affine subset in $A^{F_kT_{-k}^{k}}$ .", "Moreover, for $i \\le j$ we have that $p_{ij}(X_{n,j}(z)) \\subset X_{n,i}(z)$ , showing that $(X_{n,k}(z))$ is an inverse system (in fact, an inverse subsystem of $(\\Sigma _{F_kT_{-k}^{k}})$ ).", "By Lemma REF , there exists $x \\in \\varprojlim _{k \\ge n} X_{n,k}(z) \\subset \\varprojlim _{k \\ge n} \\Sigma _{F_kT_{-k}^{k}} = \\Sigma $ .", "By construction, we have $x(g) = 0_A \\mbox{ for all } g \\in F_kT_{-n}^{-1} \\mbox{ \\ and \\ } x\\vert _{F_k} = z\\vert _{F_k}, \\mbox{ for all } k \\ge n,$ so that, letting $k \\rightarrow \\infty $ , $x(g) = 0_A \\mbox{ for all } g \\in HT_{-n}^{-1} \\mbox{ \\ and \\ } x\\vert _H = z.$ This shows that $z = x\\vert _H \\in X_n$ .", "The claim follows.", "It is clear that $X_n \\supset X_{n+1}$ for all $n \\ge 1$ .", "Thus, as a consequence of Claim 1, $(X_n)_{n \\in \\mathbb {N}}$ is a decreasing sequence of linear subshifts of $A^H$ .", "Since $A$ is finite-dimensional and $H$ is of linear Markov type, by Corollary REF the above sequence must stabilize: there exists $n_0 \\in \\mathbb {N}$ such that $X_n = X_{n_0}$ for all $n \\ge n_0$ .", "Thus, setting $X \\bigcap _{n \\in \\mathbb {N}} X_n = \\lbrace x\\vert _H: x \\in \\Sigma \\mbox{ such that } x(g) = 0_A \\mbox{ for all } g \\in HT_{-\\infty }^{-1}\\rbrace \\subset A^H$ we have that $X$ is a linear subshift in $A^H$ and, moreover, $X = X_{n_0} = \\lbrace x\\vert _H: x \\in \\Sigma \\mbox{ such that } x(g) = 0_A \\mbox{ for all } g \\in HT_{-n_0}^{-1}\\rbrace .$ Consider the finite set $T T_{-n_0}^{n_0} \\subset T^{\\prime }$ and set $\\Omega HT \\subset G$ .", "The action of $H$ on $\\Omega $ by left multiplication induces an action of $H$ on $A^\\Omega $ : this is given by setting $(hx)(kt) x(h^{-1}kt)$ for all $h,k \\in H$ , $x \\in A^\\Omega $ , and $t \\in T$ .", "Claim 2.", "The subset $\\Sigma _\\Omega \\subset A^\\Omega $ is $H$ -invariant and closed with respect to the prodiscrete topology on $A^\\Omega $ .", "Let $z \\in \\Sigma _\\Omega $ .", "Then there exists $x \\in \\Sigma $ such that $z = x\\vert _\\Omega $ .", "Given $h \\in H$ , we have, for all $k \\in H$ and $t \\in T$ , $(hz)(kt) = z(h^{-1}kt) = x(h^{-1}kt) = (hx)(kt) = (hx)\\vert _\\Omega (kt).$ Since $hx \\in \\Sigma $ , we deduce that $hz = (hx)\\vert _\\Omega \\in \\Sigma _\\Omega $ .", "This shows that $\\Sigma _\\Omega $ is $H$ -invariant.", "Since $G$ is countable, we can find an increasing sequence $(E_n)_{n \\in \\mathbb {N}}$ of finite subsets of $G$ such that $G = \\bigcup _{n \\in \\mathbb {N}} E_n$ .", "Setting $F_n E_n \\cap \\Omega $ for all $n \\in \\mathbb {N}$ , we obtain an increasing sequence $(F_n)_{n \\in \\mathbb {N}}$ of finite subsets of $\\Omega $ such that $\\Omega = \\bigcup _{n \\in \\mathbb {N}} F_n$ .", "Let $d \\in A^\\Omega $ and suppose it belongs to the closure of $\\Sigma _\\Omega $ in $A^\\Omega $ .", "We must show that $d \\in \\Sigma _\\Omega $ .", "For each $n \\in \\mathbb {N}$ there exists $y \\in \\Sigma _\\Omega $ such that $d\\vert _{F_n} = y\\vert _{F_n}$ .", "Since $y \\in \\Sigma _\\Omega $ , there exists $x \\in \\Sigma $ such that $y = x\\vert _\\Omega $ .", "Setting $z x\\vert _{E_n} \\in \\Sigma _{E_n}$ , we have $z\\vert _{F_n} = x\\vert _{F_n} = (x\\vert _\\Omega )\\vert _{F_n} = y\\vert _{F_n} = d\\vert _{F_n}$ , so that the finite-dimensional affine set $Z_n \\lbrace z \\in \\Sigma _{E_n}: z\\vert _{F_n} = d\\vert _{F_n}\\rbrace \\subset \\Sigma _{E_n}$ is nonempty.", "It is clear that for $m,n \\in \\mathbb {N}$ with $m \\ge n$ the restriction map $\\pi _{n,m} \\colon A^{E_m} \\rightarrow A^{E_n}$ induces, by restriction, a well defined linear map $p_{n m} \\colon Z_m \\rightarrow Z_n$ .", "Hence, by applying Lemma REF to the inverse system $(Z_m, p_{n m})$ , there exists $x \\in \\varprojlim _{n \\rightarrow \\infty } Z_n \\subset \\varprojlim _{n \\rightarrow \\infty } \\Sigma _{E_n} = \\Sigma $ .", "By definition, we have $x\\vert _{F_n} = d\\vert _{F_n}$ for every $n \\in \\mathbb {N}$ , so that $x\\vert _\\Omega = d$ .", "This shows that $d = x\\vert _\\Omega \\in \\Sigma _\\Omega $ .", "We deduce that $\\Sigma _\\Omega $ is closed, and the claim follows.", "Claim 3.", "$\\Sigma = \\Sigma (A^G;\\Omega , \\Sigma _\\Omega )$ .", "Let us set $\\widetilde{\\Sigma } \\Sigma (A^G;\\Omega , \\Sigma _\\Omega ) \\subset A^G$ .", "It is clear that $\\Sigma \\subset \\widetilde{\\Sigma }$ .", "To prove the converse inclusion, let $y \\in \\widetilde{\\Sigma }$ .", "Then, there exists $z_0 \\in \\Sigma $ such that $z_0\\vert _\\Omega = y\\vert _\\Omega $ .", "Since also $a^{-1}y \\in \\widetilde{\\Sigma }$ , there exists $y_0 \\in \\Sigma $ such that $y_0\\vert _\\Omega = (a^{-1}y)\\vert _\\Omega $ .", "As a consequence, setting $z_1 ay_0 \\in \\Sigma $ , one has $z_1(a \\omega ) = (a^{-1}z_1)(\\omega ) = y_0(\\omega ) = (a^{-1}y)(\\omega ) = y(a\\omega )$ for all $\\omega \\in \\Omega $ , equivalently, $z_1\\vert _{a\\Omega } = y\\vert _{a\\Omega }$ .", "Note that $a\\Omega = aHT = HaT = HaT_{-n_0}^{n_0} = HT_{-n_0+1}^{n_0+1}$ so that $\\Omega \\cap (a\\Omega ) = HT_{-n_0+1}^{n_0} \\supset HT_{1}^{n_0}$ .", "Thus, for the configuration $z z_0 - z_1 \\in \\Sigma $ we have $z(g) = 0_A$ for all $g \\in HT_{1}^{n_0}$ .", "Moreover, if $g^{\\prime } \\in HT_{-n_0}^{-1}$ , then $g a^{n_0+1}g^{\\prime } \\in HT_{1}^{n_0}$ and therefore $(a^{-n_0-1}z)(g^{\\prime }) = z(a^{n_0+1}g^{\\prime }) = z(g) = 0_A$ .", "As a consequence, the configuration $v (a^{-n_0-1}z)\\vert _H \\in A^H$ is in $X$ (cf.", "(REF )).", "Set $L(v) \\lbrace x \\in \\Sigma : x(g) = 0_A \\mbox{ for all } g \\in HT_{-\\infty }^{-1} \\mbox{ and } x\\vert _H = v\\rbrace \\subset A^G.$ Clearly, $L(v)$ is a nonempty affine subspace: keeping in mind (REF ), there exists $z^{\\prime } \\in \\Sigma $ with $z^{\\prime }\\vert _H = (a^{-n_0-1}z)\\vert _H = v$ and $z^{\\prime }(g) = 0_A$ for all $g \\in HT_{-\\infty }^{-1}$ , so that $z^{\\prime } \\in L(v)$ .", "Let $c \\in L(v)$ and consider the configuration $x z_0 - a^{n_0+1}c \\in \\Sigma $ .", "Let $h \\in H$ and set $h^{\\prime } a^{-n_0-1}ha^{n_0+1} \\in H$ .", "Then, for $1 \\le n \\le n_0$ we have $x(ha^n) = z_0(ha^n) - (a^{n_0+1}c)(ha^n) = z_0(ha^n) - c(h^{\\prime }a^{n-n_0-1}) = z_0(ha_n) = y(ha^n),$ where we used the fact that, on the one hand $-n_0 \\le n-n_0-1 \\le -1$ and $c\\vert _{HT_{-n_0}^{-1}} = 0$ , and, on the other hand, $HT_{-n_0}^{-1} \\subset HT_{-n_0}^{n_0} = \\Omega $ and $z_0\\vert _{\\Omega } = y\\vert _\\Omega $ .", "Moreover, $x(ha^{n_0+1}) & = x(a^{n_0+1}h^{\\prime }) = z_0(a^{n_0+1}h^{\\prime }) - (a^{n_0+1}c)(a^{n_0+1}h^{\\prime }) && \\\\& = z_0(a^{n_0+1}h^{\\prime }) - c(h^{\\prime }) = z_0(a^{n_0+1}h^{\\prime }) - (a^{-n_0-1}z)(h^{\\prime }) && \\mbox{(since $c \\in L(v)$)}\\\\& = z_0(a^{n_0+1}h^{\\prime }) - z(a^{n_0+1}h^{\\prime }) = z_1(a^{n_0+1}h^{\\prime }) && \\mbox{(since $z = z_0 - z_1$)}\\\\& = z_1(a(a^{n_0}h^{\\prime })) = y(a(a^{n_0}h^{\\prime })) && \\mbox{(since $z_1\\vert _{a\\Omega } = y\\vert _{a\\Omega }$)}\\\\& = y(a^{n_0+1}h^{\\prime }) = y(h a^{n+1}).", "&&$ Keeping in mind (REF ), this shows that $x\\vert _{HT_1^{n_0+1}} = y\\vert _{HT_1^{n_0+1}}$ .", "An immediate induction on $m \\ge 1$ yields a sequence $(x_m)_{m \\ge 1}$ in $\\Sigma $ such that $x_m\\vert _{HT_1^m} = y\\vert _{HT_1^m}$ for all $m \\ge 1$ .", "Let now $F \\subset G$ be a finite subset.", "Then we can find $i,j \\in \\mathbb {Z}$ , with $i \\le j$ , such that $F \\subset HT_i^j$ .", "Setting $m j-i+1$ , it follows that $a^{-i+1}F \\subset HT_1^m$ .", "Consider the configuration $y^{\\prime } a^{-i+1}y \\in \\Sigma ^{\\prime }$ .", "Then by using (REF ) applied to $y^{\\prime }$ , we can find $x_m^{\\prime } \\in \\Sigma $ such that $x_m^{\\prime }\\vert _{HT_1^m} = y^{\\prime }\\vert _{HT_1^m}$ .", "Then setting $x_m a^{i-1}x_m^{\\prime } \\in \\Sigma $ , we obtain $x_m\\vert _{HT_i^j} = y\\vert _{HT_i^j}$ so that, in particular, $x_m\\vert _F = y\\vert _F$ .", "Since $\\Sigma $ is closed and $F$ was arbitrary, this shows that $y \\in \\Sigma $ .", "This proves $\\widetilde{\\Sigma } \\subset \\Sigma $ , and the claim follows.", "The remaining of the proof of the lemma follows step by step the end of the proof of Lemma REF , with $G$ replaced by $\\Omega $ and $\\Sigma ^{\\prime }$ replaced by $\\varphi (\\Sigma _\\Omega )$ .", "We thus set $B A^T$ , so that $B$ is a finite-dimensional vector space and the map $\\varphi \\colon A^\\Omega \\rightarrow B^H$ defined by (REF ) is an $H$ -equivariant linear isomorphism and uniform homeomorphism.", "By virtue of Claim 2, we have that $\\Sigma ^{\\prime } \\varphi (\\Sigma _\\Omega ) \\subset B^H$ is a subshift.", "Since $H$ is of linear Markov type, and $B$ is finite-dimensional, there exists a finite subset $D_H \\subset H$ and a subspace $P_H \\subset B^{D_H}$ such that $\\Sigma ^{\\prime } = \\Sigma (B^H;D_H,P_H)$ .", "Then, setting $D D_HT \\subset G$ and $P \\psi ^{-1}(P_H) \\subset A^D$ , where $\\psi \\colon A^D \\rightarrow B^{D_H}$ is as in (REF ), we have that $\\Sigma = \\Sigma (A^G; D, P)$ is of finite type.", "Proposition 6.4 Let $K$ be a field.", "Then the class of $K$ -linear Markov groups is closed under the operations of taking subgroups, quotients, and extensions by finite or cyclic groups.", "Let $G$ be a group, let $H \\subset G$ be a subgroup, and let $A$ be a finite-dimensional vector space over a field $K$ .", "Given a subshift $\\Sigma \\subset A^H$ we set $\\Sigma ^{(G)} \\lbrace x \\in A^G: (gx)\\vert _H \\in \\Sigma \\mbox{ for all } g \\in G\\rbrace \\subset A^G.$ Roughly speaking, $\\Sigma ^{(G)}$ is the set of all configurations in $A^G$ whose restriction to each left coset $c \\in G/H$ yields – modulo the bijection $h \\mapsto gh$ , induced by an element $g \\in c$ , which identifies $H$ and $c$ – and element in $\\Sigma $ .", "It is easy to see that $\\Sigma ^{(G)} \\subset A^G$ is a linear subshift and that it is of finite type if and only if $\\Sigma $ is (cf.", "[8]; see also [29]).", "We deduce that if $G$ is of linear Markov type, so are all of its subgroups.", "Suppose now that $H$ is normal in $G$ and denote by $\\pi \\colon G \\rightarrow K G/H$ the canonical quotient homomorphism.", "Given a subshift $\\Sigma \\subset A^K$ we denote by $\\Sigma (G) \\lbrace x \\circ \\pi : x \\in \\Sigma \\rbrace \\subset A^G.$ Roughly speaking, $\\Sigma (G)$ is the set of all configurations $x \\in A^G$ which are constant on each left coset $c \\in G/H$ and such that, if $T \\subset G$ is a complete set of representatives of the cosets of $H$ in $G$ , then the restriction $x\\vert _T$ yields – modulo the bijection $\\pi \\vert _T \\colon T \\rightarrow K$ – an element in $\\Sigma $ .", "Assume that $G$ is of linear Markov type.", "Once again, it is easy to see that $\\Sigma (G) \\subset A^G$ is a linear subshift and that it is of finite type if and only if $\\Sigma $ is.", "We deduce that $K$ is of linear Markov type as well.", "The fact that the class of groups of linear Markov type is closed under finite or cyclic extensions follows from Lemma REF and Lemma REF , respectively.", "It is a well known fact (see, e.g., [28]) that a solvable group is polycyclic if and only if it is Noetherian.", "Similarly, one has that a virtually solvable group is polycyclic-by-finite if an only if it Noetherian (cf.", "[29]).", "From Corollary REF and Corollary REF we deduce the following (cf.", "[29]): Corollary 6.5 Let $G$ be a virtually solvable group and let $K$ be a field.", "Then the following conditions are equivalent: $G$ is of $K$ -linear Markov type; $G$ is Noetherian; $G$ is polycyclic-by-finite.", "Remark 6.6 As mentioned above, we can directly deduce Corollary REF from Lemma REF and Lemma REF , thus without using P. Hall's theorem.", "For the sake of completeness, we produce here the alternative proof, by induction.", "Thus, suppose that $G$ is a polycyclic-by-finite group.", "Then $G$ admits a subnormal series $G = G_n \\supset G_{n-1} \\supset \\cdots \\supset G_1 \\supset G_0 = \\lbrace 1_G\\rbrace $ whose factors are finite or cyclic groups.", "We first observe that if $G$ is a trivial group, then it is of $K$ -linear Markov type.", "Indeed, let $A$ be a finite-dimensional vector space over a field $K$ .", "Then, setting $D \\lbrace 1_G\\rbrace $ and identifying $A$ with $A^D$ and $A^G$ , we have that the (identity) map $B \\mapsto \\Sigma (A^G,D,B)$ yields a bijection between subspaces $B \\subset A$ and subshifts $\\Sigma \\subset A^G$ .", "Since every descending sequence of vector subspaces of a finite-dimensional vector space eventually stabilizes, it follows from Corollary REF that all subshifts $A^G$ are of finite type.", "This proves the base of induction.", "A recursive application of Lemma REF or Lemma REF then shows that $G_0, G_1, \\ldots , G_{n-1}$ , and $G_n = G$ are all of $K$ -linear Markov type." ], [ "The descending chain condition", "Let $G$ be a group and let $A$ be an infinite-dimensional vector space over a field $K$ .", "Then $A$ admits a strictly decreasing sequence $(A_n)_{n \\in \\mathbb {N}}$ of vector subspaces and the sequence $(\\Sigma _n)_{n \\in \\mathbb {N}}$ , where $\\Sigma _nA_n^G \\subset A^G$ , is a strictly decreasing sequence of linear subshifts of $A^G$ .", "Thus $A^G$ does not satisfy the descending chain condition for linear subshifts." ], [ "The closed image property", "In [9] it is shown that if $A$ is an infinite-dimensional vector space and $G$ is any nonperiodic group, then there exists a linear cellular automaton $\\tau \\colon A^G \\rightarrow A^G$ whose image $\\tau (A^G)$ is not closed in $A^G$ .", "This shows that Theorem REF fails to hold in general if the finite-dimensionality of the alphabet $A$ is dropped.", "Explicitly, the linear cellular automaton $\\tau \\colon A^G \\rightarrow A^G$ we alluded to above can be defined as follows.", "Since $A$ is infinite-dimensional, we can find a sequence $(a_i)_{i \\in \\mathbb {N}}$ of linearly independent vectors in $A$ .", "Let $E$ denote the vector subspace spanned by the $a_i$ 's and let $F$ be a vector subspace such that $A = E \\oplus F$ .", "Let $\\psi \\colon A \\rightarrow A$ denote the linear map defined by setting $\\psi (a_i) = a_{i+1}$ for all $i \\in \\mathbb {N}$ and $\\psi \\vert _F = 0$ .", "Since $G$ is nonperiodic, there exists an element $g \\in G$ of infinite order.", "Then the cellular automaton $\\tau \\colon A^G \\rightarrow A^G$ with memory set $M = \\lbrace 1_G,g\\rbrace \\subset G$ and local defining map $\\mu \\colon A^M \\rightarrow A$ given by $\\mu (y) y(g) - \\psi (y(1_G)),$ for all $y \\in A^M$ , satisfies that $\\tau (A^G)$ is not closed in $A^G$ (cf.", "[9]; see also [8])." ], [ "Nilpotency for linear cellular automata", "Let $G$ be a group and let $A$ be a vector space over a field $K$ .", "Given a linear map $f \\colon A \\rightarrow A$ , we denote by $\\tau _f \\colon A^G \\rightarrow A^G$ the LCA with memory set $M \\lbrace 1_G\\rbrace $ and associated local defining map $\\mu _f f \\colon A = A^M \\rightarrow A$ .", "In other words, $\\tau _f = \\prod _{g \\in G} f$ so that, in particular, $\\tau _f^n(A^G) = \\prod _{g \\in G} f^n(A)$ for all $n \\in \\mathbb {N}$ .", "As a consequence, $\\Omega (\\tau _f) = \\bigcap _{n \\in \\mathbb {N}} \\tau _f^n(A^G) = \\bigcap _{n \\in \\mathbb {N}} \\prod _{g \\in G} f^n(A) =\\prod _{g \\in G} \\bigcap _{n \\in \\mathbb {N}} f^n(A) = \\prod _{g \\in G} \\Omega (f) = \\Omega (f)^G.$ Note that $f$ is nilpotent (resp.", "pointwise nilpotent) if and only if $\\tau _f$ is nilpotent (resp.", "pointwise nilpotent).", "Suppose that $A$ is infinite-dimensional.", "Let $\\lbrace e_n: n \\in \\mathbb {N}\\rbrace \\subset A$ be an independent subset and set $A_1 \\operatorname{span}_K\\lbrace e_n:n \\in \\mathbb {N}\\rbrace $ and $A_2 A \\ominus A_1$ .", "Consider the linear map $f \\colon A \\rightarrow A$ defined by setting $f(e_n) = e_{n+1}$ for all $n \\in \\mathbb {N}$ and $f(a) = 0$ for all $a \\in A_2$ .", "It is then clear that $\\Omega (f) = \\lbrace 0\\rbrace $ so that, by (REF ), $\\Omega (\\tau _f) = \\lbrace 0\\rbrace $ .", "However, $\\tau _f$ is not pointwise nilpotent (and therefore not nilpotent either).", "Consider the linear map $f \\colon A \\rightarrow A$ defined by setting $f(e_0) = 0$ , $f(e_n) = e_{n-1}$ for all $n \\ge 1$ , and $f(a) = 0$ for all $a \\in A_2$ .", "Then $f$ and therefore $\\tau _f$ are surjective so that $\\tau _f$ is not nilpotent, $\\Omega (\\tau _f) = A^G$ .", "However, $f$ and therefore $\\tau _f$ are pointwise nilpotent.", "Consider, for each $n \\ge 1$ , the set $I_n \\lbrace 0,1,\\dots ,n\\rbrace $ and the map $g_n \\colon I_n \\rightarrow I_n$ given by $g_n(k) k - 1$ if $k \\ge 1$ and $g_n(0) = 0$ .", "Let $X$ be the set obtained by taking disjoint copies of the sets $I_n$ , $n \\ge 1$ , and identifying all copies of 0 in a single point $y_0$ and all copies of 1 in a single point $y_1 \\ne y_0$ .", "Then the maps $g_n$ induce a well defined quotient map $g \\colon X \\rightarrow X$ .", "Clearly $\\Omega (g) = \\lbrace y_0,y_1\\rbrace $ and $g(\\Omega (g)) = \\lbrace y_0\\rbrace $ .", "Since $X$ is countable, we can find a bijection $\\varphi \\colon \\mathbb {N}\\rightarrow X$ such that $\\varphi (0) = y_0$ and $\\varphi (1) = y_1$ .", "Setting $h \\colon \\varphi ^{-1} \\circ g \\circ \\varphi \\colon \\mathbb {N}\\rightarrow \\mathbb {N}$ we thus have $\\Omega (h) = \\lbrace 0,1\\rbrace $ and $h(\\Omega (h)) = \\lbrace 0\\rbrace $ .", "Consider the linear map $f \\colon A \\rightarrow A$ defined by setting $f(e_n) e_{h(n)}$ for all $n \\in \\mathbb {N}$ and $f(a) = 0$ for all $a \\in A_2$ .", "Then $\\Omega (f) = \\operatorname{span}_K\\lbrace e_0,e_1\\rbrace = Ke_0 \\oplus Ke_1$ while $f(\\Omega (f)) = \\operatorname{span}_K\\lbrace e_0\\rbrace = Ke_0$ .", "As a consequence, $\\tau _f(\\Omega (\\tau _f)) = (Ke_0)^G \\subsetneq (Ke_0 \\oplus Ke_1)^G = \\Omega (\\tau _f)$ ." ] ]	2011.14191
[ [ "On Generalization of Adaptive Methods for Over-parameterized Linear\n Regression" ], [ "Abstract Over-parameterization and adaptive methods have played a crucial role in the success of deep learning in the last decade.", "The widespread use of over-parameterization has forced us to rethink generalization by bringing forth new phenomena, such as implicit regularization of optimization algorithms and double descent with training progression.", "A series of recent works have started to shed light on these areas in the quest to understand -- why do neural networks generalize well?", "The setting of over-parameterized linear regression has provided key insights into understanding this mysterious behavior of neural networks.", "In this paper, we aim to characterize the performance of adaptive methods in the over-parameterized linear regression setting.", "First, we focus on two sub-classes of adaptive methods depending on their generalization performance.", "For the first class of adaptive methods, the parameter vector remains in the span of the data and converges to the minimum norm solution like gradient descent (GD).", "On the other hand, for the second class of adaptive methods, the gradient rotation caused by the pre-conditioner matrix results in an in-span component of the parameter vector that converges to the minimum norm solution and the out-of-span component that saturates.", "Our experiments on over-parameterized linear regression and deep neural networks support this theory." ], [ "Introduction", "* indicates equal contribution Corresponding Emails: [email protected], [email protected], [email protected], [email protected] The success of deep learning has uncovered a new mystery of benign overfitting [3], [25], i.e., systems with a large number of parameters can not only achieve zero training error but are also able to generalize well.", "Also, over-parameterized systems exhibit a double descent-behavior [3], [4]; as the number of parameters/epochs increases, the test error first decreases, then increases before falling again.", "This goes against the conventional wisdom of overfitting in machine learning, which stems from the classical bias-variance tradeoff [6], [13], [32].", "In the absence of explicit regularization, a typical over-parameterized setting possesses multiple global minima.", "Classical gradient descent based methods can achieve one of these many global minima [1], [11], [10], [15], [34], however not all optima generalize equally.", "[5], [8], [26] suggest many practical approaches to improve generalization; however, there remains a considerable gap between theory and practice [45], [1].", "In this paper, we will focus on two categories of optimization algorithms: pure gradient descent based (non-adaptiveNow onwards, optimization methods that satisfy equation (REF ) will be referred to as non-adaptive purposes.)", "methods and adaptive methods.", "The primary distinguishing factor between these two methods is determined by the update step.", "For the class of non-adaptive methods, the expected gradient update step is given as follows: $\\mathbb {E}\\left[w(t+1)\|w(t)\\right] = w(t) - \\eta \\nabla f(w(t)),$ where $w(t)$ indicates the estimate of the underlying parameter vector, $\\eta $ represents the learning rate and $f(w(t)), \\nabla f(w(t))$ represent the loss function and its gradient, respectively.", "Popular methods like gradient descent, stochastic gradient descent (SGD), batch gradient descent fall under this class.", "Training any model using non-adaptive methods involves tuning over many hyperparameters, of which step size is the most essential one [36], [30].", "The step size could be set as constant, or could be changing per iteration $\\eta (t)$ [8], usually based on a predefined learning rate schedule [7], [43], [31].", "During the past decade, we have also witnessed the rise of a family of algorithms called adaptive methods that argue for automatic hyper-parameter adaptation [28] during training (including step size).", "The list includes AdaGrad [12], Adam [20], AdaDelta [44], RMSProp [39], AdaMax [20], Nadam [9], just to name a few.", "These algorithms utilize current and past gradient information $\\lbrace \\nabla f(w(i))\\rbrace _{i = t}^k$ , for $t < k$ , to design preconditioning matrices $D(t) \\succeq 0$ that better pinpoint the local curvature of the objective function as follows: $\\mathbb {E}\\left[w(t+1)\|w(t)\\right] = w(t) - \\eta D(t) \\nabla f(w(t))$ Usually, the main argument for using adaptive methods is that $D(t)$ eliminates pre-setting a learning rate schedule, or diminishes initial bad step size choices, thus, detaching the time-consuming part of step size tuning from the practitioner [46].", "[14] was one of the first papers to discuss the implicit bias introduced by optimization methods for over-parameterized systems and how the choice of optimization algorithm affects the global minima it attains.", "However, the generalization behavior of these optimization methods remains a mystery.", "As a result, researchers have re-focussed their attention on understanding the most straightforward over-parameterized setting of linear regression [3], [16], [22], [25] as a first step in unraveling the mysterious behavior of neural networks.", "Gradient descent-based methods converge to the minimum norm interpolated solution [25] for over-parameterized linear regression.", "Under certain assumptions on the data distribution, the minimum norm solution achieves near-optimal accuracy for unseen data [3].", "Unlike SGD, the presence of $D(t)$ in adaptive methods can alter the span of the final converged solution in the presence of any non-trivial initialization, which makes the task of commenting on adaptive methods challenging.", "Despite being a key reason behind the success of deep learning, the convergence behavior of adaptive methods is not well understood.", "The convergence bounds for most adaptive methods hold for only a specific pre-conditioner matrix [20], [40], [14].", "Besides, theoretical guarantees for adaptive methods often minimize regret [27], [12], which makes it further challenging to comment on the generalization of adaptive methods.", "As a result, the generalization of adaptive methods for a general $D(t)$ remains an open problem even for an over-parameterized linear regression setting.", "In this paper, we aim to explicitly characterize the sub-class of adaptive methods that mimic the convergence, and generalization behaviors seen in SGD and the sub-class that does not.", "In addition, we observe a double descent like phenomena for a sub-class of adaptive methods as the number of training epochs increases.", "Table: Table illustrating differing generalization guarantees of three distinct Adaptive Methods (AM) with SGD in overparameterized setting, i.e.", "d>nd>n, where nn: number of examples, dd: dimension.", "AM1: Diagonalized Adagrad, AM2: Adagrad (AM1) Variant (where we square the diagonal terms instead of taking the square root), AM3: Projected version of AM1 onto the span of XX.", "The exact expressions for the pre-conditioner matrix are available in Section 3.In this paper, we would like to understand how adaptive methods affect generalization guarantees of over-parameterized problems.", "To motivate this, we consider a toy example for simple linear regression in the under-determined/over-parameterized framework in Table REF .", "As is evident, some adaptive methods have the same generalization as SGD, while others can yield quite different generalization guarantees.", "Key Contributions: For the theoretical contribution, we focus on over-parameterized linear regression.", "Here, plain gradient descent methods converge to the minimum Euclidean norm solution, while adaptive methods may or may not.", "In this paper, we provide explicit conditions on the structure of pre-conditioner matrices, $D(t)$ , which allow us to distinguish between two classes of adaptive methods, the ones which behave similarly to SGD and the ones that do not.", "Based on these conditions, we compare the generalization performance between adaptive and non-adaptive methods.", "For the experimental component, we begin by revisiting the mystery posed by Table REF , and demonstrate that the experimental results are in line with our theoretical guarantees.", "Further, we show using a toy example that the adaptive methods can have a superior generalization performance than SGD.", "The discussion “which method is provably better”, however, is inconclusive and ultimately depends on the problem/application at hand.", "Lastly, we empirically demonstrate the validity of our claims for over-parameterized neural networks as well and recommend exercising caution when proposing or choosing adaptive methods for training, depending on the goal in hand." ], [ "Problem Setup", "Notation.", "For any matrix $A \\in \\mathbb {R}^{m\\times n}$ , $A_{pq}$ indicates the element corresponding to the $p$ -th row and $q$ -th column.", "The $\\text{rank}(A)$ denotes the rank of $A$ .", "For a sequence of matrices $A_0$ to $A_n$ , we have the definition $\\prod _{k=i+m}^{i} A_k = A_{(i+m)} A_{(i+m-1)} \\dots A_{i}$ .", "Note that, $a(t)$ indicates the value of the the function $a(\\cdot )$ after the $t$ -th update.", "Note that $\\lambda $ without any subscript indicates the regularizer, and $\\lambda _i$ with a subscript denotes the $i^{th}$ eigenvalue.", "The subscript $._{(1)}$ will denote the in-span component, subscript $._{(2)}$ will denote the out-of-span component and $.", "(t)$ will indicate the $t^{th}$ iterate.", "We consider an over-parameterized noisy linear regression (possibly with regularization), where the relationship between the data matrix $X \\in \\mathbb {R}^{n\\times d}$ , the noise vector $\\zeta \\in \\mathbb {R}^{n}$ , and the labels $y\\in \\mathbb {R}^{n}$ is as follows: $y = X w^\\star + \\zeta .$ We are concerned with the following optimization problem: $f(w) = \\arg \\min _{w} \\left\\lbrace \\mathbb {E}\\left[\\Vert y - Xw\\Vert ^2\\right] + \\dfrac{\\lambda }{2} \\Vert w\\Vert _2^2\\right\\rbrace .$ In particular, we study the convergence of the following iterative updates $w(t+1) = w(t) - \\eta D(t) \\nabla f(w(t)),$ where the pre-conditioner matrices are bounded, positive definite and hence full rank; i.e., $\\inf _t \\text{rank}(D(t)) = d$ .", "The system is assumed over-parameterized; i.e., $R = \\text{rank}(X) < d$ .", "Before we discuss the generalization of adaptive methods in over-parameterized settings, let us briefly explain their performance on the training set.", "For linear regression with $\\ell _2$ -norm regularization, we observe that adaptive methods with any full rank pre-conditioner matrix $D(t)$ will converge to the same solution as its non-adaptive counterpart and thus mimic their performance.", "However, for unregularized linear regression, adaptive methods can converge to entirely different solutions than SGD.", "Both SGD and adaptive methods can achieve zero training error despite attaining different stationary pointsFor more, refer to the Appendix Sections A.2 and A.3.." ], [ "Performance on Unseen Data", "As a result, our primary focus in this paper is to understand the generalization capabilities of adaptive methods.", "We observe that the generalization depends on two key factors: $i)$ Does $w^\\star $ lie in the span of the data matrix, $X$ ?", "$ii)$ How does pre-multiplying with the pre-conditioner matrix alter the span of final converged $w$ ?" ], [ "Spectral Representation", "The switch to the spectral domain allows us to simplify and understand the relationship between the final converged solution with the span of data matrix $X$ , pre-conditioner matrix $\\tilde{D}(t)$ and the initialization $w(0)$ .", "We express the data matrix using its singular value decomposition (SVD): $X = \\sum _{r= 1}^{R} \\lambda _r u_r v_r^T$ , $\\lambda _r \\ne 0$ for all $r$ where $\\lambda _r, u_r, v_r$ represent the $r^{th}$ largest eigenvalue and the corresponding right and left eigenvectors respectively.", "We complete the basis using the left eigenvectors of the data matrix to form a complete orthogonal spectral basis of $\\mathbb {R}^d$ , $\\lbrace v_r: r= 1\\dots , d\\rbrace $ form the basis vectors and denote it by $V$ .", "Similarly, $U$ forms the complete orthogonal spectral basis of $\\mathbb {R}^n$ using the right eigenvectors of the data matrix as $\\lbrace u_r: r= 1\\dots , n\\rbrace $ .", "The eigenvalue matrix is $\\Lambda $ where $\\Lambda _{rr} = \\lambda _r$ if $1\\le r \\le R$ and 0 otherwise.", "We next express useful quantities in the above bases in Table REF .", "=0ex =0ex The definition of adaptive pre-conditioner matrices in the above table holds since $V$ represents a complete orthogonal spectral basis of $\\mathbb {R}^d$ .", "Additionally, we also have the following property, where we show that pre- and post-multiplication by an orthogonal matrix $V$ does not alter the eigenvalues of the original matrix, i.e., the set of eigenvalues for $\\tilde{D}(t)$ is identical to the set of eigenvalues of $D(t)$ (Appendix Section A.5).", "Table: Notation in spectral domain" ], [ "Closed Form Expression for the Iterates", "Our objective is to understand how the iterates evolve depending on the space spanned by the data matrix.", "First, we establish a closed-form expression for the updates of the vector $\\tilde{w}(t)$ .", "Proposition 1 Consider the over-parameterized linear regression setting with data matrix $X$ , noise $\\zeta $ , and regularizer $\\lambda >0$ .", "If the pre-conditioner matrix $D(t)\\succ 0$ for all $t\\ge 0$ , then, for any $T \\ge 0$ , the iterate $\\tilde{w}(T)$ admits the following closed form expression: $&\\tilde{w}(T) = \\prod _{i=0}^{T-1} \\left(I - \\eta \\tilde{D}(i) (\\Lambda ^2 + \\lambda I)\\right) \\tilde{w}(0) \\\\&+\\sum _{i= 0}^{T-1} \\prod _{j=(i+1)}^{T-1} \\left(I - \\eta \\tilde{D}(j) (\\Lambda ^2 + \\lambda I)\\right) \\eta \\tilde{D}(i)(\\Lambda ^2 \\tilde{w}^\\star +\\Lambda \\zeta ) $ The final expression of $w(T)$ implies that the final solution depends on the initialization point, the span of the data matrix in $\\mathbb {R}^d$ space, and the pre-conditioner matrix.", "Further, the closed-form indicates that the presence of pre-conditioning matrices $\\tilde{D}(j)$ may cause $w(t)$ to lie outside of the span of the data in the complete $\\mathbb {R}^d$ space.", "We observe that the presence or absence of regularizer can significantly alter the stationary points to which adaptive methods converge.", "In the presence of $\\ell _2$ -norm regularization, we observe that the adaptive methods converge to the same solution independent of the initialization or the step-size.", "However, in the absence of regularization, things are not as straight-forward.", "Here, we will try to capture the convergence of over parameterized linear regression using dynamics described by equation (REF )." ], [ "$\\ell _2$ -norm Regularized Linear Regression.", "In presence of $\\ell _2$ -norm regularization, the over-parameterized linear regression problem becomes strongly convex and possesses a unique global optima.", "Proposition 2 serves a sanity check; where we show the convergence to this unique optima for any positive definite pre-conditioner matrix $D(t)$ in the spectral domain.", "We utilize this result as a stepping stone in understanding the convergence behavior of adaptive methods for the unregularized over-parameterized settings.", "Proposition 2 Consider the over-parameterized linear regression setting with data matrix $X$ , noise $\\zeta $ , and regularizer $\\lambda >0$ .", "Suppose for all $t\\ge 0$ , the pre-conditioner matrix $D(t) \\succ 0$ , and the learning rate satisfies $\\eta \\in \\left(0, 2\\left(\\lambda _{\\max }(D(t))(\\lambda _{\\max }^2 (X)+ \\lambda )\\right)^{-1}\\right).$ where $\\lambda _{\\max }(\\cdot )$ indicates the maximum eigenvalue.", "Then, $\\tilde{w}(t)$ converges to the following fixed point $&\\lim _{t\\rightarrow \\infty }\\tilde{w}(t) =(\\Lambda ^2 + \\lambda I)^{-1} (\\Lambda ^2 \\tilde{w}^\\star +\\Lambda \\zeta ).$ Proposition REF states that like the gradient descent based methods, the adaptive methods will perfectly capture the component of the generative $w^$ that lies in the subspace formed by the data matrix.", "In other words, with regularization the parameter vector converges in the span of $X$ .", "In the proof of the proposition presented in the Appendix, we use contraction properties to show convergence where $\\lambda > 0$ plays a significant role.", "Further, as $\\inf _t \\text{rank}(D(t))$ , a simple fixed-point analysis provides us with the in-span component and shows that for $\\lambda > 0$ there is no out-of-span component of the solution.", "Note that this proposition acts as a proof of concept for the well-known result that adaptive methods and non-adaptive methods converge to the same solution in the presence of $\\ell _2$ -norm regularization.", "Lastly, different regularization techniques alter the implicit bias of the final converged solution differently.", "The claims made in this sub-section are only valid for $\\ell _2$ -norm regularization." ], [ "Unregularized Linear Regression.", "Next, we focus on the slightly more interesting problem of unregularized linear regression in the over-parameterized regime.", "The optimization problem with squared loss is no longer strongly convex, and there are infinite solutions that can achieve zero training error.", "In this case, the convergence of unregularized linear regression depends on the initialization $\\tilde{w}(0)$ .", "Further, as $\\lambda _{\\min }(\\tilde{D}(t) \\Lambda ^2) = 0$ we cannot directly prove (using contraction mapping) convergence for general unregularized over-parameterized linear regression.", "However, when the pre-conditioner matrices satisfy a block matrix structure, then we can say something about the converged solution.", "Note that most of the popular adaptive algorithms [12], [20], [27], [42] satisfy the block matrix structure.", "The out-of-span component behavior depends subtly on the interplay of the pre-conditioner matrices and the span of data.", "Now, we establish sufficient conditions on the class of pre-conditioner matrices for which the convergence is guaranteed (for more details refer Appendix).", "We define some notations useful to state our main theorems.", "We use $\\tilde{w}_{(1)}(\\infty ) = \\lim \\limits _{t\\rightarrow \\infty }\\tilde{w}_{(1)}(t)$ to denote the in-span component, and $\\tilde{w}_{(2)}(\\infty ) = \\lim \\limits _{t\\rightarrow \\infty }\\tilde{w}_{(2)}(t)$ to denote the in-span component of the stationary point.Note that $\\tilde{w}_{(2)}(\\infty ) \\in (\\mathbb {R}\\cup \\lbrace \\infty \\rbrace )^d$ for $i=1,2$ , as we can not assume convergence of the iterates a pirori.", "Let, $e_{(1)}(t) = \\Vert \\tilde{w}_{(1)}(\\infty ) - \\tilde{w}_{(1)}(i)\\Vert _2 = \\mathcal {O}\\left(\\dfrac{1}{t^\\beta }\\right)$ be the $\\ell _2$ -norm distance of in-span component of the iterate from the in-span stationary point at time $t\\ge 1$ .", "We further define: Definition 1 For a data matrix $\\mathbf {X}$ and an adaptive method, with preconditioning matrices $\\lbrace \\mathbf {D}(t): t\\ge 1\\rbrace $ , we call the adaptive method $(\\alpha ,\\beta )$ -converging on data, for any $\\alpha , \\beta \\ge 0$ , if and only if: $i)$ the out-of-span component of the pre-condition matrices decays as $\|\\lambda _{\\max }(\\tilde{D}_2(t))\| = \\mathcal {O}\\left(\\dfrac{1}{t^\\alpha }\\right)$ ; $ii)$ the in in-span component of the iterates converges as, $e_{(1)}(t) = \\mathcal {O}\\left(\\frac{1}{t^\\beta }\\right)$ (under Eq.", "(REF )).", "Any adaptive method with a pre-conditioner matrix that lies entirely in the span of the matrix will have $\\alpha $ set to $\\infty $ .", "Full-matrix Adagrad, GD, and Newton all fall under this class of adaptive methods.", "Popular adaptive methods, such as diagonalized Adagrad, RMSProp, and methods with a diagonal pre-conditioner matrix with non-zero entries, the convergence depends on the rate of decay of both the $\\tilde{D}_2(t)$ as well as the rate of decay of the error of the in-span component.", "Theorem 1 Consider the problem of over-parameterized linear regression with data matrix $X$ and noise $\\zeta $ in the absence of regularization $\\lambda = 0$ .", "If the preconditioner matrix $D(t)\\succ 0$ $\\forall t\\ge 0$ , and $\\eta \\in \\left(0, \\dfrac{2}{\\lambda _{\\max }(D(t))\\lambda _{\\max }^2 (X)} \\right),$ then in-span component of $\\tilde{w}(t)$ converges as follows $&\\tilde{w}_{(1)}(\\infty ) =(\\tilde{w}_{(1)}^ +\\Lambda _{(1)}^{-1} \\zeta _{(1)}).$ Furthermore, for an adaptive method (in Eq.", "(REF )) which is $(\\alpha ,\\beta )$ -converging on data, if $\\alpha + \\beta > 1$ the out-of-span component converges to a stationary point that satisfies $\\Vert \\tilde{w}_{(2)}(\\infty ) - \\tilde{w}_{(2)}(0)\\Vert _2 \\le \\mathcal {O}\\left(\\Vert \\tilde{w}_{(1)}(0)\\Vert _2 + \\frac{1}{\\alpha +\\beta -1}\\right).$ Remark on Theorem REF : Let us deconstruct the claims made in Theorem REF .", "Theorem 1 says that if $\\eta $ is set appropriately, then adaptive methods will perfectly fit the noisy training data.", "This is consistent with the claims in [25].", "The convergence of out-of-span component depends on the decay rate of the pre-conditioner matrix $\\tilde{D}_2(t)$ as well as the decay rate of the error term in the in-span component $e_{(1)}(t) = \\Vert (w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}) - \\tilde{w}_{(1)}(i)\\Vert _2$ .", "For the simple case, when $\\beta \\ge 1$ and $\\tilde{D}_{(2)}(t) = 0$ for all $t$ , we have that the out-of-span component converges to $\\tilde{w}_{(2)}(0)$ .", "Next, if $\\lim _{T\\rightarrow \\infty }\\sum _{t=0}^{T} \\max \|\\lambda (\\tilde{D}_{(2)}(t))\| < \\infty $ and $\\alpha + \\beta \\ge 1$ , then the out-of-span component converge.", "For all other cases, it is difficult to comment whether the out-of-span component will converge or diverge.", "Specifically, for $\\alpha + \\beta < 1$ , we may not have divergence as the pre-conditioner matrices may align cancel the cumulative errors.", "Figure: Evolution of upper bound dynamics for adaptive methods with different rates of (α,β)(\\alpha , \\beta ) convergence,with a=1a = 1, b=0.7b = 0.7, and c=0.1c = 0.1.Remark on Dynamics: It is also interesting to note that the error in the in-span component converges to 0 and the error in the out-of-span component increases and saturates to a particular value.", "Our derived upper bound on the training error at time $T$ for an adaptive method that is $(\\alpha ,\\beta )$ converging on data is given as : $a + \\dfrac{b}{(T+1)^{\\beta }}\\left(1 - \\dfrac{c}{\\alpha +\\beta -1} \\dfrac{1}{(T+1)^{\\alpha -1}}\\right)$ for appropriate constants $a,b,c >0$ .", "The dynamics is shown in the Figure REF .", "Depending on the values of the constants, $\\alpha $ , and $\\beta $ , adaptive methods demonstrate variable convergence dynamics." ], [ "Experiments", "Some of the experiments in this paper were also present in an earlier paper by us (Shah, V., Kyrillidis, A., Sanghavi, S. (2018).", "Minimum weight norm models do not always generalize well for over-parameterized problems.", "arXiv preprint arXiv:1811.07055) which had a different flavor of theoretical results.", "The experiments have been reused as they reinforce some of the claims made in this paper In this section, we focus on replicating the theoretical claims made in the previous parts using synthetic experiments for over-parameterized linear regression.", "Next, we show that these observations can be extended to the deep learning setup as well.", "We empirically compare two classes of algorithms: [leftmargin=0.5cm] Plain gradient descent algorithms, including the mini-batch stochastic gradient descent and the accelerated stochastic gradient descent, with constant momentum.", "Adaptive methods like AdaGrad [12], RMSProp [39], and Adam [20], and the AdaGrad variant.", "Adaptive methods can include anything with a time-varying pre-conditioner matrix." ], [ "Linear Regression", "In the first part, we consider a simple linear regression example generated with where the elements of both $X$ and $\\zeta $ are generated using from $\\mathcal {N}(0,1)$ distribution in an i.i.d.", "manner.", "The test data is sampled from the same distribution as training data.", "Here, we show that different adaptive methods can yield better performance in terms of generalization.", "The expressions for AM1, AM2 and AM3 are given as follows: Figure: Synthetic example of over-parameterized linear regression where adaptive methods show better test error performance.", "Notice that adaptive method AM1 not only allows us to achieve faster convergence but also better generalization.", "Estimation error, ∥∥w(t)-w ∥∥\\Vert \\Vert w(t) - w^\\Vert \\Vert is in the semilog scale on the x axis (to highlight the double descent like phenomena in AM2 as predicted by the Remark at the end of Section 2).", "The reported results are the average of 5 runs with different initializations for a given realization of data.Table: Illustrating the varying performances of adaptive methods for over-parameterized linear regression.", "The final values are the average of 5 runs.", "AM1: Diagonalized Adagrad, AM2: Adagrad (AM1) Variant (where we square the diagonal terms instead of taking the square root), AM3: Projected version of AM1 onto the span of X.", "For AM3, D ˜ 2 (t)=0,∀t\\tilde{D}_2(t) = 0,~ \\forall ~ t and consistent with Theorem \\ref {thm:unregularized} it converges to the same point as SGD.", "AM1 and AM2 satisfy the (α,β)(\\alpha , \\beta ) convergence criterion leading to convergence to a different point and different generalization than SGD.Table: Prediction accuracy and distances from the minimum norm solution for plain gradient descent and adaptive gradient descent methods.", "Adagrad variant squares the pre-conditioner matrix values instead of taking the square root in Adagrad.", "The distances shown are median values out of 100 different realizations for each setting; the accuracies are obtained by testing 10 4 10^4 predictions on unseen data.$D_{1}(t) &= \\texttt {diag}\\left( \\dfrac{1}{\\sum _{j = t - J}^t \\nabla f(w(j)) \\odot \\nabla f(w(j)) + \\varepsilon }\\right) \\\\D_{2}(t) &=\\texttt {diag}\\left(\\dfrac{1}{\\sum _{j = t - J}^t \\nabla f(w(j)) \\odot \\nabla f(w(j)) + \\varepsilon }\\right) \\\\D_{3}(t) &= \\mathcal {P}_{X}(D_1(t))$ Here, we assume that $ D_{i}(t) \\succ 0, \\text{for some} ~\\varepsilon > 0, ~~\\text{and}~~ J < t \\in \\mathbb {N}_{+} \\quad \\forall ~i$ .", "In Figure REF , we observe that the conditions described in previous section allow us to predict the performance of adaptive methods relative to gradient descent based methods, i.e.", "we can ascertain whether a given adaptive method will have the same generalization performance as SGD based methods or not." ], [ "Classification:", "In this example, we consider a linear regression problem with binary outputs i.e.", "$y_i \\in \\lbrace -1, 1\\rbrace $ (a variant of the example proposed in [41]).", "$\\left(x_i\\right)_j &={\\left\\lbrace \\begin{array}{ll}y_i \\ell , & \\!\\!j = 1, \\\\1, & \\!\\!j = 2, 3,\\\\1, & \\!\\!j=4+ 5(i-1), \\\\0, & \\!\\!\\text{otherwise}.\\end{array}\\right.}", "~~~~~ \\text{if} ~ y_i = 1,\\\\\\left(x_i\\right)_j &={\\left\\lbrace \\begin{array}{ll}y_i\\ell , & \\!\\!j = 1, \\\\1, & \\!\\!j = 2, 3, \\\\1, & \\!\\!j= 4+ 5(i-1),\\\\& \\hspace{8.5359pt} \\cdots , 8+ 5(i-1), \\\\0, & \\text{otherwise}.\\end{array}\\right.}", "~~~ \\text{if} ~ y_i = -1.", "$ The expressions for the pre-conditioner matrix of Adagrad variant is: $D_{AV}(t) &= \\texttt {diag}\\left(\\dfrac{1}{\\sum _{j = t - J}^t \\left(\\nabla f(w(j)) \\odot \\nabla f(w(j))\\right)^2 + \\varepsilon }\\right)$ Note that $ D_{AV}(t) \\succ 0, \\text{for some} ~\\varepsilon > 0, ~~\\text{and}~~ J < t \\in \\mathbb {N}_{+}$ .", "Table REF depicts that even in terms of test accuracy, adaptive algorithms can yield better generalization performance than SGD.", "This supports the claim made recently in [24] for non-adaptive methods; the testing criterion can play a crucial role in determining the generalization performance.", "We observe that this claim holds for adaptive methods as well.", "Figure: Accuracy results on unseen data, for different NN architectures on CIFAR100.", "Left panel: Accuracy and training loss for PreActResNet18 in ; Left middle panel: Accuracy and training loss for MobileNet in ; Right middle panel: Accuracy and training loss for MobileNetV2 in , Right panel: Accuracy and training loss for GoogleNet in .", "Top row: Weight vectors of the last layer, Middle row: Training Loss, Last row: Test Accuracy." ], [ "Deep Learning", "In this experiment, we observe that the theoretical claims made for the generalization of adaptive methods for over-parameterized linear regression extend over to over-parameterized neural networks.", "We perform extensive experiments on CIFAR-100 in Figure REF .", "For CIFAR-100 datasets we explore four different architectures; PreActResNet18 [18], MobileNet [19], MobileNetV2 [29], GoogleNet [29].", "After a hyper-parameter tuning phase, we selected the best choices among the parameters tested.", "The results show no clear winner once again, which overall support our claims: the superiority depends on the problem/data at hand; also, all algorithms require fine tuning to achieve their best performance.", "An important observation of Figure REF comes from the top row of the panel.", "There, we plot the Euclidean norm $\\Vert \\cdot \\Vert _2$ of all the trainable parameters of the corresponding neural network.", "While such a norm could be considered arbitrary (e.g., someone could argue other types of norms to make more sense, like the spectral norm of layer), we use the Euclidean norm as $i)$ it follows the narrative of algorithms in linear regression, where plain gradient descent algorithms choose minimum $\\ell _2$ -norm solutions, and $ii)$ there is recent work that purposely regularizes training algorithms towards minimum norm solutions [2].", "Our findings support our claims: in particular, for the case of MobileNet and MobileNetV2, Adam, an adaptive method, converges to a solution that has at least as good generalization as plain gradient methods, while having $2 \\times $ larger $\\ell _2$ -norm weights.", "However, this may not always be the trend: in Figure REF , left panel, the plain gradient descent models for the PreActResNet18 architecture [18] show slightly better performance, while preserving low weight norm.", "The same holds for GoogleNet; see Figure REF , right panel.", "Another observation is that like linear regression, here also adaptive methods can be clubbed into different categories based on the evolution of their pre-conditioner matrices, $D(t)$ .", "It is evident that algorithms that have similar weight norms have similar training loss performance; however the other side of the claim need not be true.", "The experiments indicate the potential of adaptive methods to show better generalization w.r.t.", "unseen data.", "The details about experiments on more datasets, hyperparameter tuning, practical issues in implementation are available in the Appendix." ], [ "Conclusions", "In this paper, we consider two class of methods described: non-adaptive methods (Eq.", "(REF )) and adaptive methods (Eq.", "(REF )).", "Switching to a spectral domain allows us to divide adaptive methods into two further categories based on if they will have the same generalization as SGD or not (assuming the same initialization point).", "We obtain that the convergence of adaptive methods completely depends on the structure of pre-conditioner matrices $D(t)$ along with the initialization point and the given data.", "Our theoretical analysis allows us to obtain useful insights into the convergence of adaptive methods, which can be useful while designing new adaptive methods.", "If the aim while designing an adaptive method is faster convergence and similar generalization as SGD, then it is important to ensure that the pre-conditioner matrix lies in the span of the data matrix $D(t) = \\mathcal {P}_{X}(\\cdot )$ .", "Examples of such $D(t)$ include $\\lbrace \\mathbb {I}, (X^\\top X)^{-1}\\rbrace $ .", "However, if the aim is to hope for a different generalization than SGD (if SGD gets stuck on specific bad minima), then it is essential to ensure that the conditions in Theorem REF are satisfied to ensure that $\\tilde{w}(t)$ converges to a different solution.", "Our experimental results on over-parameterized settings for both linear regression and deep learning back our theoretical claims." ], [ "Proof of Proposition ", "From Theorem , we know that a sufficient condition for the convergence under update (REF ) is $\\sup _{t\\ge 1} \|\\lambda \\left(I - \\eta \\tilde{D}(t) (\\Lambda ^2 + \\lambda I)\\right)\| < 1.$ Thus, for $\\lambda > 0$ , the dynamics converges to a bounded weight vector for any $\\eta \\in \\left(0, 2\\left(\\lambda _{max}(D(t))(\\lambda _{max}^2 (X)+ \\lambda )\\right)^{-1}\\right).$ We now characterize the fixed point of the dynamics in REF .", "When the convergence happens, for any fixed point $\\hat{w}$ of the updates in REF $D(t) \\left( \\lambda \\hat{w} + X^TX(\\hat{w}- w^) - X^Tw \\right) = 0.$ Because, $\\inf _t rank(D(t)) = d$ (full rank) we must have ${\\lambda \\hat{w} + X^TX(\\hat{w}- w^) - X^Tw = 0}$ .", "Expanding the l.h.s.", "in terms of the SVD of the data matrix we obtain, $\\sum _{r=1}^{d} \\lambda \\tilde{w}_r v_r + \\sum _{r=1}^{R} \\left(\\lambda _r^2 \\tilde{w}_r - \\lambda _r^2\\tilde{w}^_r - \\lambda _r \\tilde{\\zeta }_r \\right) v_r = 0.$ Therefore, for $\\lambda \\ge 0$ (holds for both regularized and unregularized) we have $v_r^T\\hat{w} = \\tfrac{ \\lambda _r^2 \\tilde{w}^_r + \\lambda _r \\tilde{\\zeta }_r }{ \\lambda + \\lambda _r^2 }$ for $r\\le R$ .", "Further, for $\\lambda > 0$ , $v_r^T\\hat{w} = 0$ for $r\\ge (R+1)$ ." ], [ "Proof of Lemma ", "Using the above structure we obtain the following lemma concerning the closed form expression of the iterates.", "Let us define for any matrix $A \\in \\mathbb {R}^{d\\times d}$ and any vector $b \\in \\mathbb {R}^{d}$ : $&A_{(1)} = \\lbrace A_{ij}: 1\\le i, j \\le R \\rbrace ,\\\\&A_{(2)} = \\lbrace A_{ij}: R+1\\le i \\le d, 1\\le j \\le R \\rbrace ,\\\\&b_{(1)} = \\lbrace b_{i}: 1\\le i \\le R \\rbrace ,\\\\&b_{(2)} = \\lbrace b_{i}: R+1\\le i \\le d \\rbrace ,$ where $R$ is the rank of the data matrix $D$ and $d$ is the dimension of the data.", "Lemma 1 If $D(t)$ is full rank for all $t\\ge 0$ and regularizer $\\lambda = 0$ , then for any $T \\ge 0$ , the closed form of the iterate $\\tilde{w}(T)$ admits the following expression: $&\\tilde{w}_{(1)}(T) = A(T\\mathtt {-}1, 0)\\tilde{w}_1(0)\\\\&+ \\sum _{i=0}^{T-1} A(T\\mathtt {-}1, i\\mathtt {+}1) \\eta \\tilde{D}_{(1)}(i) \\Lambda ^2_{(1)}( w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}),\\\\&\\tilde{w}_{(2)}(T) = B(T\\mathtt {-}1, 0) \\tilde{w}_{(1)}(0) + \\tilde{w}_{(2)}(0)\\\\&+ \\sum _{i=0}^{T-1}\\eta \\left(B(T\\mathtt {-}1, i\\mathtt {+}1)\\tilde{D}_{(1)}(i) \\mathtt {+} \\tilde{D}_{(2)}(i) \\right) \\times \\\\&\\times \\Lambda ^2_{(1)}( w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}),$ where for all $t_2 \\ge t_1 \\ge 0$ , $&A(t_2, t_1) = \\prod _{i=t_1}^{t_2}\\left(I - \\eta \\tilde{D}_{(1)}(i) \\Lambda _{(1)}^2 \\right)\\\\&B(t_2, t_1) \\mathtt {=} - \\eta \\tilde{D}_{(2)}(t_1)\\Lambda _{(1)}^2 \\\\&\\hspace{48.36958pt}\\mathtt {-} \\eta \\sum _{i=t_1\\mathtt {+}1}^{t_2} \\tilde{D}_{(2)}(i)\\Lambda _{(1)}^2A(i\\mathtt {-}1, t_1)$ In the above lemma, the vector $\\tilde{w}_{(1)}(T)$ represents the in-span component of the iterate, where as $\\tilde{w}_{(2)}(T)$ represents the out-of-span component of the iterate.", "We make an important observation in the complex expression in Lemma REF that for appropriate choice of $\\eta $ , we have $\\max \|\\lambda (A(t_2, t_1))\| < 1$ for all $t_2\\ge t_1 \\ge 0$ .", "This is true because, even though $\\lambda _{min}(\\Lambda ^2) = 0$ , when only the $R\\times R$ submatrix $\\Lambda _{(1)}$ is considered, we have $\\lambda _{min}(\\Lambda _{(1)}^2) > 0$ .", "Using this result we prove the convergence of in-span component.", "We will use the following equations regarding the product two specific block matrices.", "$\\begin{bmatrix}A_1 & 0\\\\B_1 & I\\end{bmatrix}\\begin{bmatrix}A_2 & 0\\\\B_2 & I\\end{bmatrix}= \\begin{bmatrix}A_1 A_2 & 0\\\\B_1 A_2 + B_2 & I\\end{bmatrix},\\hspace{28.45274pt}\\begin{bmatrix}A_1 & 0\\\\B_1 & C_1\\end{bmatrix}\\begin{bmatrix}A_2 & 0\\\\0 & 0\\end{bmatrix}= \\begin{bmatrix}A_1 A_2 & 0\\\\B_1 A_2 & 0\\end{bmatrix}$ Firstly, we obtain the block structure shown in the paper.", "$\\left(I - \\eta \\tilde{D}(i) \\Lambda ^2\\right) = I - \\eta \\begin{bmatrix}\\tilde{D}_{(1)}(i) & \\tilde{D}_{(2)}(i)\\\\\\tilde{D}_{(2)}(i) & \\tilde{D}_{(3)}(i)\\\\\\end{bmatrix}\\begin{bmatrix}\\Lambda ^2_{(1)} & 0\\\\0 & 0\\end{bmatrix}= \\begin{bmatrix}\\left(I - \\eta \\tilde{D}_{(1)}(i)\\Lambda ^2[1] \\right) & 0_{R\\times (d-R)}\\\\- \\eta \\tilde{D}_{(2)}(i) \\Lambda ^2[1] & I_{(d-R)\\times (d-R)}\\end{bmatrix}$ The block structure is maintained for the product of these matrices, i.e.", "for all $t_2 \\ge t_1 \\ge 0$ , $&\\prod _{i=t_1}^{t_2}\\left(I - \\eta \\tilde{D}(i) \\Lambda ^2\\right)= \\begin{bmatrix}A(t_2, t_1)& 0_{R\\times (d-R)}\\\\B(t_2, t_1) & I_{(d-R)\\times (d-R)}\\end{bmatrix},\\\\&A(t_2, t_1) = \\prod _{i=t_1}^{t_2}\\left(I - \\eta \\tilde{D}_{(1)}(i) \\Lambda _{(1)}^2 \\right),B(t_2, t_1) \\mathtt {=} - \\eta \\tilde{D}_{(2)}(t_1)\\Lambda _{(1)}^2 \\mathtt {-} \\eta \\sum _{i=t_1\\mathtt {+}1}^{t_2} \\tilde{D}_{(2)}(i)\\Lambda _{(1)}^2A(i\\mathtt {-}1, t_1)$ This can be shown easily using induction and using Equation (REF ).", "Substituting these results in the closed form of the iterates in proposition REF we obtain $&\\tilde{w}(T) = \\begin{bmatrix}A(T\\mathtt {-}1, 0)\\tilde{w}_1(0)\\\\B(T\\mathtt {-}1, 0) \\tilde{w}_1(0) + \\tilde{w}_{(2)}(0)\\end{bmatrix} +\\sum _{i=0}^{T-1} \\begin{bmatrix}A(T\\mathtt {-}1, i\\mathtt {+}1) & 0\\\\B(T\\mathtt {-}1, i\\mathtt {+}1) & I\\end{bmatrix}\\begin{bmatrix}\\eta \\tilde{D}_1(i) \\Lambda ^2_1( w^_1 + \\Lambda ^{-1}_1 \\zeta _1)\\\\\\eta \\tilde{D}_{(2)}(i) \\Lambda ^2_1( w^_1 + \\Lambda ^{-1}_1 \\zeta _1)\\end{bmatrix}$" ], [ "In-span Component:", "Therefore, the component of in the span of data is $\\tilde{w}(T)$ $&\\tilde{w}_{(1)}(T) = A(T\\mathtt {-}1, 0)\\tilde{w}_1(0) + \\sum _{i=0}^{T-1} A(T\\mathtt {-}1, i\\mathtt {+}1) \\eta \\tilde{D}_1(i) \\Lambda ^2_{(1)}( w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}),$ Similar to the regularized case, we have for any $\\eta \\in \\left(0, 2\\left(\\lambda _{max}(D(t))(\\lambda _{max}^2 (X))\\right)^{-1}\\right)$ the in-span component converges.", "Further, from the fixed point argument we know that $v_r^T \\hat{w} =\\tilde{w}^_r + \\lambda _r^{-1} \\tilde{\\zeta }_r$ ." ], [ "Out-of-span Component:", "The component outside the span of the data is $\\tilde{w}_{(2)}(T) &= B(T\\mathtt {-}1, 0) \\tilde{w}_{(1)}(0) + \\tilde{w}_{(2)}(0) \\\\&+ \\sum _{i=0}^{T-1}\\eta \\left(B(T\\mathtt {-}1, i\\mathtt {+}1)\\tilde{D}_{(1)}(i) \\mathtt {+} \\tilde{D}_{(2)}(i) \\right)\\Lambda ^2_{(1)}( w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}).$" ], [ "Proof of Theorem ", "The convergence of the in-span component follows similar to the regularized case.", "In particular, we observe $\\lambda _{max}(I - \\eta \\tilde{D}_{(1)}(t) \\Lambda _{(1)}^2) \\le 1 - \\eta \\lambda _{min}(\\tilde{D}_{(1)}(t)) \\lambda _{min}(\\Lambda _{(1)}^2) < 1.$ The last inequality is true as 1) $\\lambda _{min}(\\tilde{D}_{(1)}(t)) > 0$ due to the positive definiteness of the matrix $\\tilde{D}(t)$ , and 2) $\\lambda _{min}(\\Lambda _{(1)}^2) > 0$ as it considers only the in-span component (i.e.", "the top-left $R\\times R$ sub-matrix of $\\Lambda $ ).", "On the other hand, we have $\\lambda _{min}(I - \\eta \\tilde{D}_{(1)}(t) \\Lambda _{(1)}^2) \\ge 1 - \\eta \\lambda _{max}(\\tilde{D}_{(1)}(t)) \\lambda _{max}(\\Lambda _{(1)}^2)$ .", "Therefore, we obtain $\\lambda _{min}(I - \\eta \\tilde{D}_{(1)}(t) \\Lambda _{(1)}^2) > -1$ for any $0< \\eta < 2/ (\\lambda _{max}(\\tilde{D}_{(1)}(t)) \\lambda _{max}(\\Lambda _{(1)}^2))$ .", "As $\\tilde{D}_{(1)}(t)$ is a principal sub-matrix of $\\tilde{D}(t)$ for each $t\\ge 0$ , we have from Cauchy Interlacing Theorem $\\lambda _{max}(\\tilde{D}_{(1)}(t)) \\le \\lambda _{max}(\\tilde{D}(t)) = \\lambda _{max}(D(t))$ .", "The last equality is due to Proposition .", "The characterization of the fixed point follows the same argument as Proposition REF .", "To prove the second part, we further simplify the out-of-span component using exchange of summation (for finite $T$ ).", "Here, we use the convention $A(t_1, t_2) = I$ for any $t_1 < t_2$ .", "$&\\tilde{w}_{(2)}(T) - \\left(B(T\\mathtt {-}1, 0) \\tilde{w}_{(1)}(0) + \\tilde{w}_{(2)}(0) \\right)\\\\&= \\sum _{i=0}^{T-1}\\eta \\left(\\tilde{D}_{(2)}(i) - \\eta \\tilde{D}_{(2)}(i+1)\\Lambda _{(1)}^2\\tilde{D}_{(1)}(i) \\mathtt {-} \\right.\\\\&\\quad \\quad \\left.\\eta \\sum _{j=i\\mathtt {+}2}^{T-1} \\tilde{D}_{(2)}(i)\\Lambda _{(1)}^2A(j\\mathtt {-}1, i+1)\\tilde{D}_{(1)}(i) \\right) \\Lambda ^2_{(1)}( w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}).\\\\&=\\left(\\sum _{i=0}^{T-1}\\eta \\tilde{D}_{(2)}(i) - \\eta ^2 \\sum _{i=0}^{T-1}\\sum _{j=i+1}^{T-1} \\tilde{D}_{(2)}(j)\\Lambda _{(1)}^2A(j\\mathtt {-}1, i+1)\\tilde{D}_{(1)}(i)\\right) \\Lambda ^2_{(1)}( w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}).", "\\\\& = \\left(\\sum _{i=0}^{T-1} \\eta \\tilde{D}_{(2)}(i) \\left( I - \\eta \\Lambda _{(1)}^2 \\sum _{j=0}^{i-1} A(i\\mathtt {-}1, j+1)\\tilde{D}_{(1)}(j) \\right)\\right) \\Lambda ^2_{(1)}( w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}),\\\\& = \\left(\\sum _{i=0}^{T-1} \\eta \\tilde{D}_{(2)}(i) \\left( \\Lambda ^2_{(1)}( w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}) - \\Lambda _{(1)}^2 \\tilde{w}_{(1)}(i) + \\Lambda _{(1)}^2 A(i\\mathtt {-}1, 0)\\tilde{w}_1(0) \\right)\\right),\\\\& = \\sum _{i=0}^{T-1} \\eta \\tilde{D}_{(2)}(i) \\Lambda ^2_{(1)}\\left( (w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}) - \\tilde{w}_{(1)}(i)\\right)+ \\sum _{i=0}^{T-1} \\eta \\tilde{D}_{(2)}(i)\\Lambda _{(1)}^2A(i\\mathtt {-}1, 0)\\tilde{w}_1(0).$ Therefore, we have $&\\Vert \\tilde{w}_{(2)}(T) - \\tilde{w}_{(2)}(0)\\Vert _2 \\\\&\\le \\sum _{i=0}^{T-1} \\eta \\tilde{D}_{(2)}(i) \\Lambda ^2_{(1)}\\left( (w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}) - \\tilde{w}_{(1)}(i)\\right) + \\left(B(T\\mathtt {-}1, 0) + \\sum _{i=0}^{T-1} \\eta \\tilde{D}_{(2)}(i)\\Lambda _{(1)}^2A(i\\mathtt {-}1, 0)\\right)\\tilde{w}_1(0) \\\\&= \\sum _{i=0}^{T-1} \\eta \\tilde{D}_{(2)}(i) \\Lambda ^2_{(1)}\\left( (w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}) - \\tilde{w}_{(1)}(i)\\right)- \\eta \\tilde{D}_{(2)}(0)\\Lambda _{(1)}^2\\tilde{w}_1(0)$ We have $\|\\lambda \|_{\\max } = \\sup _t \|\\lambda _{\\max }(I- \\eta \\tilde{D_1(t)}\\Lambda _{(1)}^2)\| < 1$ due to appropriate choice of $\\eta $ .", "Also, by assumption of the theorem we have for some $\\alpha \\ge 0$ , $\\beta \\ge 0$ , $\\alpha + \\beta > 1$ , for some universal constants $ 0 < c_{conv}, c_{\\lambda } <\\infty $ , and for all $t\\ge 0$ : (i) the out-of-span pre-conditioner matrix decaying as $O(1/t^\\alpha )$ for some $\\alpha \\ge 0$ , i.e.", "$\|\\lambda _{\\max }(\\tilde{D}_{(2)}(i))\| = \\tfrac{c_{\\lambda }}{(t+1)^\\alpha }$ , and (ii) the convergence rate of the in-span component is $O(1/t^\\beta )$ with iteration $t$ for some $\\beta > 0$ , i.e.", "$\\Vert (w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}) - \\tilde{w}_{(1)}(i)\\Vert _2 \\le \\tfrac{c_{conv}}{(t+1)^\\beta }$ .", "For the first term we have, $&\\Vert \\sum _{i=0}^{T-1} \\eta \\tilde{D}_{(2)}(i) \\Lambda ^2_{(1)}\\left( (w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}) - \\tilde{w}_{(1)}(i)\\right)\\Vert _2\\\\&\\le c_{\\lambda } \\sum _{i=0}^{T-1}\\lambda _{\\max }(\\tilde{D}_{(2)}(i))\\Vert (w^_{(1)} + \\Lambda ^{-1}_{(1)} \\zeta _{(1)}) - \\tilde{w}_{(1)}(i) \\Vert _2\\\\&\\le c_{\\lambda } c_{conv} \\sum _{i=1}^{T-1} \\tfrac{1}{(i+1)^{(\\alpha +\\beta )}}\\le \\tfrac{ c_{\\lambda } c_{conv}}{\\alpha +\\beta -1} \\left(1 - \\tfrac{1}{(i+1)^{(\\alpha +\\beta -1)}}\\right).$ Therefore, the first term saturates to a value at most $\\tfrac{c_{\\lambda } c_{conv}}{\\alpha +\\beta -1}$ .", "For the second term we have, $&\\Vert \\eta \\tilde{D}_{(2)}(0)\\Lambda _{(1)}^2\\tilde{w}_1(0)\\Vert _2\\le \\eta \\lambda _{\\max }(\\tilde{D}_{(2)}(0)) \\lambda ^2_{\\max }(\\Lambda _{(1)})\\Vert \\tilde{w}_1(0)\\Vert _2$" ], [ "Proof of Proposition ", "Proposition 3 The following pre-conditioner matrices have $\\tilde{D}_{(2)}(t) = 0$ .", "$D(t)= I$ , i.e.", "gradient descent, $D(t) = (X^T X + \\epsilon I)^{-1}$ for all $t \\ge 0$ .", "We have $(X^T X + \\epsilon I)^{-1} = \\sum _{r = 1}^{R} (\\lambda _r^2 + \\epsilon )^{-1}v_r v^T_r + \\sum _{r = R+1}^{d} \\epsilon ^{-1}v_r v^T_r$ .", "Further, $I = \\sum _{r = 1}^{d} bs{v}_r v^T_r$ .", "So the proposition is true." ], [ "Proof of Lemma ", "We will prove this using induction.", "Let $Q = diag(\|X^T y\|)$ We will show that $w_k = \\lambda _k Q^{-1} sign(X^T y)$ for some $\\lambda _k$ .", "$w_0 = 0$ is satsified for $\\lambda _0=0$ and so the base case is trivially true.", "$g_k &= X^T (X w_k - y)\\\\&= \\lambda _k X^T X Q^{-1} sign(X^T y) - X^T y\\\\&= (\\lambda _k c - 1)X^T y$ where the last inequality follows from $w_k = \\lambda _k Q^{-1} sign(X^T y)$ .3 $H_k = diag(\\sum _{s=1}^n g_s \\cdot g_s) = \\nu _k diag(\|X^T y\|^2) = \\nu _k Q^2$ $w_{k+1} &= w_{k} - \\alpha _k H_k^{-1} X^T (X w_k - y) \\\\&= w_k - \\alpha _k H_k^{-1} X^T X w_k + \\alpha _k H_k^{-1} X^T y \\\\&= \\lambda _k Q^{-1} sign(X^T y) - \\lambda _k \\alpha _k H_k^{-1} X^T X Q^{-1} X^T y + \\alpha _k H_k^{-1} X^T y \\\\&= \\lambda _k Q^{-1} sign(X^T y) - \\lambda _k \\alpha _k c H_k^{-1} X^T y + \\alpha _k H_k^{-1} X^T y\\\\&= \\left( \\lambda _k - \\frac{\\lambda _k \\alpha _k c}{\\nu _k} + \\frac{\\alpha _k}{\\nu _k}\\right) Q^{-1} sign(X^T y) \\\\&= \\lambda _{k+1} Q^{-1} sign(X^T y)$" ], [ "Generalization with respect to test accuracy for over-parameterized linear regression", "In the first set of experiments, we showed how adaptive methods converging to a different solution might lead to solutions farther from $w^*_b$ , i.e.", "with higher L2-norm .", "Thus, the pre-conditioner matrices satisfying $D_{ij}(t) = 0 , D(t)\\succ 0$ have different generalization than their gradient based counterparts.", "In this section, we empirically demonstrate that pre-conditioner matrices of the form: $D_{ij}(t) = 0 , D(t)\\succ 0$ can guarantee better generalization than gradient based methods depending on the problem in hand.", "As a direct consequence of this, we show that solutions with a minimum norm should not be used as a yardstick to guarantee good generalization.", "We alter the previous counterexample in [41] by slightly changing the problem setting: at first, we reduce the margin between the two classes; the case where we increase the margin is provided in the Appendix.", "We empirically show that gradient-descent methods fail to generalize as well as adaptive methods –with a slightly different $D_k$ than AdaGrad.", "In particular, for the responses, we consider two classes $y_i \\in \\lbrace \\pm \\ell \\rbrace $ for some $\\ell \\in (0, 1)$ ; i.e., we consider a smaller margin between the two classes.$\\ell $ can take different values, and still we get the same performance, as we show in the experiments below.", "$\\left(x_i\\right)_j ={\\left\\lbrace \\begin{array}{ll}y_i \\ell , & \\!\\!j = 1, \\\\1, & \\!\\!j = 2, 3,\\\\1, & \\!\\!j=4+ 5(i-1), \\\\0, & \\!\\!\\text{otherwise}.\\end{array}\\right.}", "~~~~~ \\text{if} ~ y_i = 1,\\qquad \\left(x_i\\right)_j ={\\left\\lbrace \\begin{array}{ll}y_i\\ell , & \\!\\!j = 1, \\\\1, & \\!\\!j = 2, 3, \\\\1, & \\!\\!j= 4+ 5(i-1),\\\\& \\hspace{8.5359pt} \\cdots , 8+ 5(i-1), \\\\0, & \\text{otherwise}.\\end{array}\\right.}", "~~~ \\text{if} ~ y_i = -1.", "$ Given this generative model, we construct $n$ samples $\\lbrace y_i, x_i\\rbrace _{i=1}^n$ , and set $d = 6n$ , for different $n$ values.", "We compare two simple algorithms: $i)$ the plain gradient descent, for $\\eta = {1}{\\lambda _1(X^\\top X)}$ ; $ii)$ the recursion $w(t+1) = w(t) - \\eta D(t) X^\\top \\left(X w(t) - y\\right)$ , where $\\eta $ is set as above, and $D(t)$ follows the rule: $D(t) &= \\texttt {diag}\\left( 1 / \\left(\\sum _{j = t - J}^t \\nabla f(w(j)) \\odot \\nabla f(w(j)) + \\varepsilon \\right)^2\\right) \\\\&\\succ 0, \\quad \\quad \\text{for some} ~\\varepsilon > 0, ~~\\text{and}~~ J < t \\in \\mathbb {N}_{+}$ Observe that $D(t)$ uses the dot product of gradients, squared.", "A variant of this preconditioner is found in [23]; however our purpose is not to recommend a particular preconditioner but to show that there are $D_k$ that lead to better performance than the minimum norm solution.", "We denote as $w^{\\text{ada}}$ , $w^{\\text{adam}}$ and $w^{\\text{GD}}$ the estimates of the adam, adagrad variant and simple gradient descent, respectively.", "The experiment obeys the following steps: $i)$ we train both gradient and adaptive gradient methods on the same training set, $ii)$ we test models on new data $\\lbrace y_i^{\\text{test}}, x_i^{\\text{test}}\\rbrace _{i = 1}^Q$ .", "We define performance in terms of the classification error: for a new sample $\\lbrace y_i^{\\text{test}}, x_i^{\\text{test}}\\rbrace $ and given $w^{\\text{ada}}$ , $w^{\\text{adam}}$ and $w^{\\text{GD}}$ , the only features that are non-zeros in both $x_i^{\\text{test}}$ and $w$ 's are the first 3 entries [41].", "This is due to the fact that, for gradient descent and given the structure in $X$ , only these 3 features affects the performance of gradient descent.", "Thus, the decision rules for both algorithms are: $\\widehat{y}_i^{~\\text{ada}} &=\\texttt {quant}_{\\ell }\\left( w^{\\text{ada}}_1 \\cdot y_i^{\\text{test}} + w^{\\text{ada}}_2 + w^{\\text{ada}}_3 \\right), \\\\ ~\\widehat{y}_i^{~\\text{GD}} &= \\texttt {quant}_{\\ell }\\left( w^{\\text{GD}}_1 \\cdot y_i^{\\text{test}} + w^{\\text{GD}}_2 + w^{\\text{GD}}_3 \\right),\\\\~\\widehat{y}_i^{~\\text{adam}} &= \\texttt {quant}_{\\ell }\\left( w^{\\text{adam}}_1 \\cdot y_i^{\\text{test}} + w^{\\text{adam}}_2 + w^{\\text{adam}}_3 \\right),$ where $\\texttt {quant}_{\\ell }(\\alpha )$ finds the nearest point w.r.t.", "$\\lbrace \\pm \\ell \\rbrace $ .", "With this example, our aim is to show that adaptive methods lead to models that have better generalization than gradient descent.", "Table 2 summarizes the empirical findings.", "In order to cover a wider range of settings, we consider $n = [10, ~50, ~100]$ and set $d = 6n$ , as dictated by [41].", "We generate $X$ as above, where instances in the positive class, $y_i \\in +\\ell $ , are generated with probability $p = 7/8$ ; the cases where $p = 5/8$ and $p = 3/8$ are provided in the appendix, and also convey the same message as in Table .", "Further details on the experiments are provided in the Appendix.", "The proposed AdaGrad variant described in equation REF falls under the broad class of adaptive algorithms with $D_k$ .", "However, for the counter example in [41], the AdaGrad variant neither satisfies the convergence guarantees of Lemma 3.1 there, nor does it converge to the minimum norm solution evidenced by its norm in Table .", "To buttress our claim that the AdaGrad variant in (REF ) converges to a solution different than that of minimum norm (which is the case for plain gradient descent), we provide the following proposition for a specific class of problemsNot the problem proposed in the counter-example 1 on pg 5.; the proof is provided in Appendix.", "Proposition 4 Suppose $X^\\top y$ has no zero components.", "Define $D = \\texttt {diag}(\|X^\\top y\|^{3})$ and assume there exists a scalar $c$ such that $X D^{-1} \\texttt {sign}(X^\\top y) = c y$ .", "Then, when initialized at 0, the AdaGrad variant in (REF ) converges to the unique solution $w \\propto D^{-1} \\texttt {sign}(X^\\top y)$ ." ], [ "More details and experiments for the counter-example", "The simulation is completed as follows: For each setting $(n, p, J)$ , we generate 100 different instances for $(X, y)$ , and for each instance we compute the solutions from gradient descent, AdaGrad variant and Adam (RMSprop is included in the Appendix) and the minimum norm solution $w_{\\text{mn}}$ .", "In the appendix, we have the above table with the Adagrad variant that normalizes the final solution $\\widehat{w}$ (Table ) before calculating the distance w.r.t.", "the minimum norm solution: we observed that this step did not improve or worsen the performance, compared to the unnormalized solution.", "This further indicates that there is an infinite collection of solutions –with different magnitudes– that lead to better performance than plain gradient descent; thus our findings are not a pathological example where adaptive methods work better.", "We record $\\Vert \\widehat{w}- w_{\\text{mn}}\\Vert _2$ , where $\\widehat{w}$ represents the corresponding solutions obtained by the algorithms in the comparison list.", "For each $(X, y)$ instance, we further generate $\\lbrace y_i^{\\text{test}}, x_i^{\\text{test}}\\rbrace _{i = 1}^{100}$ , and we evaluate the performance of both models on predicting $y_i^{\\text{test}}$ , $\\forall i$ .", "Table shows that gradient descent converges to the minimum norm solution, in contrast to the adaptive methods.", "This justifies the fact that the adaptive gradient methods (including the proposed adagrad variant) converge to a different solution than the minimum norm solution.", "Nevertheless, the accuracy on unseen data is higher in the adaptive methods (both our proposed AdaGrad variant and in most instances, Adam), than the plain gradient descent, when $\\ell $ is small: the adaptive method successfully identifies the correct class, while gradient descent only predicts one class (the positive class; this is justified by the fact that the accuracy obtained is approximately close to $p$ , as $n$ increases).", "We first provide the same table in Table but with unnormalized values for distances with respect to Adagrad variant.", "Table: Prediction accuracy and distances from the minimum norm solution for plain gradient descent and adaptive gradient descent methods.", "We set p=7/8p = 7/8 and J=10J=10, as in the main text.", "The adaptive method uses D k D_k according to ().", "The distances shown are median values out of 100 different realizations for each setting; the accuracies are obtained by testing 10 4 10^4 predictions on unseen data.Here, we provide further results on the counterexample in Subsubsection REF .", "Tables REF and REF contains results for $J = 10$ : the purpose of these tables is to show that even if we change the memory use footprint of the AdaGrad variant—by storing fewer or more gradients to compute $D_k$ in (REF )—the results are the same: the AdaGrad variant consistently converges to a solution different than the minimum norm solution, while being more accurate than the latter for small values of $\\ell $ (i.e., smaller margin between the two classes).", "Plain gradient descent methods provably need to rely on the first elements to decide; using the same rule for adaptive methodsWe note that using only the three elements in adaptive methods is not backed up by theory since it assumes that the training and test datasets have no overlap.", "We include this in comparison for completeness..", "The remaining subsection considers the case where we decide based on the $y = \\texttt {sign}(x^\\top w)$ rule, where $w$ is the complete learned model.", "As we show empirically, more often than not adaptive methods outperform plain gradient methods.", "Observing the performance of various optimization techniques for different values of $n$ , $p$ and $\\ell $ , we observed that the best performances are obtained when the dataset is highly imbalanced irrespective of the optimization algorithm chosen.", "When the data is (almost) balanced, it is difficult to comment on how the performance of these algorithms is affected by variations in the levels $\\ell $ and probability $p$ ." ], [ "Deep Learning", "In this section, we will extend the experiments to over-parameterized and under-parameterized neural networks without regularization.", "We begin with a detailed description of the datasets and the architectures we use along with comprehensive set of experiments with hyperparameter tuning.", "Table: Summary of the datasets and the architectures used for experiments.", "CNN stands for convolutional neural network, FF stands for feed forward network.", "More details are given in the main text.Figure: Accuracy results on unseen data, for different NN architectures and datasets.", "Over-parameterized problems.", "Left two panels: Accuracy and training loss for MNIST; Right two panels: Accuracy and training loss for CIFAR10." ], [ "MNIST dataset and the M1 architecture.", "Each experiment for M1 is simulated over 50 epochs and 10 runs for both under- and over-parameterized settings.", "Both the MNIST architectures consisted of two convolutional layers (the second one with dropouts [35]) followed by two fully connected layers.", "The primary difference between the M1-OP ($\\sim 73$ K parameters) and M1-UP ($\\sim 21$ K parameters) architectures was the number of channels in the convolutional networks and $\\#$ of nodes in the last fully connected hidden layer.", "Figure REF , left two columns, reports the results over 10 Monte-Carlo realizations.", "Top row corresponds to the M1-UP case; bottom row to the M1-OP case.", "We plot both training errors and the accuracy results on unseen data.", "For the M1-UP case, despite the grid search, observe that AdaGrad (and its variant) do not perform as well as the rest of the algorithms.", "Nevertheless, adaptive methods (such as Adam and RMSProp) perform similarly to simple SGD variants, supporting our conjecture that each algorithm requires a different configuration, but still can converge to a good local point; also that adaptive methods require the same (if not more) tuning.", "For the M1-OP case, SGD momentum performs less favorably compared to plain SGD, and we conjecture that this is due to non-optimal tuning.", "In this case, all adaptive methods perform similarly to SGD." ], [ "CIFAR10 dataset and the C1 architecture.", "For C1, C1-UP is trained over 350 epochs, while C1-OP was trained over 200 epochs.", "The under-parameterized setting is on-purpose tweaked to ensure that we have fewer parameters than examples ($\\sim 43$ K parameters), and slightly deviates from [21]; our generalization guarantees ($\\sim 76\\%$ ) are in conjunction with the attained test accuracy levels.", "Similarly, for the C1-OP case, we implement a Resnet [17] + dropout architecture ($\\sim 0.25$ million parameters) Adam and RMSProp achieves the best performance than their non-adaptive counterparts for both the under-parameterized and over-parameterized settings.", "Figure REF , right panel, follows the same pattern with the MNIST data; it reports the results over 10 Monte-Carlo realizations.", "Again, we observe that AdaGrad methods do not perform as well as the rest of the algorithms.", "Nevertheless, adaptive methods (such as Adam and RMSProp) perform similarly to simple SGD variants.", "Further experiments on CIFAR-100 for different architecture are provided in the Appendix." ], [ "CIFAR100 and other deep architectures (C{2-5}-OP).", "In this experiment, we focus only on the over-parameterized case: DNNs are usually designed over-parameterized in practice, with ever growing number of layers, and, eventually, a larger number of parameters [38].", "We again completed 10 runs for each of the set up we considered.", "C2-OP corresponds to PreActResNet18 from [18], C3-OP corresponds to MobileNet from [19], C4-OP is MobileNetV2 from [29], and C5-OP is GoogleNet from [37].", "The results are depicted in Figure REF .", "After a similar hyper-parameter tuning phase, we selected the best choices among the parameters tested.", "The results show no clear winner once again, which overall support our claims: the superiority depends on the problem/data at hand; also, all algorithms require fine tuning to achieve their best performance.", "We note that a more comprehensive reasoning requires multiple runs for each network, as other hyper-parameters (such as initialization) might play significant role in closing the gap between different algorithms.", "An important observation of Figure REF comes from the bottom row of the panel.", "There, we plot the Euclidean norm $\\Vert \\cdot \\Vert _2$ of all the trainable parameters of the corresponding neural network.", "While such a norm could be considered arbitrary (e.g., someone could argue other types of norms to make more sense, like the spectral norm of layer), we use the Euclidean norm as $i)$ it follows the narrative of algorithms in linear regression, where plain gradient descent algorithms choose minimum $\\ell _2$ -norm solutions, and $ii)$ there is recent work that purposely regularizes training algorithms towards minimum norm solutions [2].", "Our findings support our claims: in particular, for the case of MobileNet and MobileNetV2, Adam, an adaptive method, converges to a solution that has at least as good generalization as plain gradient methods, while having $2 \\times $ larger $\\ell _2$ -norm weights.", "However, this may not always be the trend: in Figure REF , left panel, the plain gradient descent models for the PreActResNet18 architecture [18] show slightly better performance, while preserving low weight norm.", "The same holds also for the case of GoogleNet; see Figure REF , right panel." ], [ "Hyperparameter tuning", "Both for adaptive and non-adaptive methods, the step size and momentum parameters are key for favorable performance, as also concluded in [41].", "Default values were used for the remaining parameters.", "The step size was tuned over an exponentially-spaced set $\\lbrace 0.0001, 0.001, 0.01, 0.1, 1\\rbrace $ , while the momentum parameter was tuned over the values of $\\lbrace 0, 0.1, 0.25, 0.5, 0.75, 0.9 \\rbrace $ .", "We observed that step sizes and momentum values smaller/bigger than these sets gave worse results.", "Yet, we note that a better step size could be found between the values of the exponentially-spaced set.", "The decay models were similar to the ones used in [41]: no decay and fixed decay.", "We used fixed decay in the over-parameterized cases, using the StepLR implementation in pytorch.", "We experimented with both the decay rate and the decay step in order to ensure fair comparisons with results in [41]." ], [ "Results", "Our main observation is that, both in under- or over-parameterized cases, adaptive and non-adaptive methods converge to solutions with similar testing accuracy: the superiority of simple or adaptive methods depends on the problem/data at hand.", "Further, as already pointed in [41], adaptive methods often require similar parameter tuning.", "Most of the experiments involve using readily available code from GitHub repositories.", "Since increasing/decreasing batch-size affects the convergence [33], all the experiments were simulated on identical batch-sizes.", "Finally, our goal is to show performance results in the purest algorithmic setups: often, our tests did not achieve state of the art performance.", "Overall, despite not necessarily converging to the same solution as gradient descent, adaptive methods generalize as well as their non-adaptive counterparts.", "In M1 and C1-UP settings, we compute standard deviations from all Monte Carlo instances, and plot them with the learning curves (shown in shaded colors is the one-apart standard deviation plots; best illustrated in electronic form).", "For the cases of C{1-5}-OP, we also show the weight norms of the solutions (as in Euclidean distance $\\Vert \\cdot \\Vert _2$ of all the trainable weights in the network).", "Such measure has been in used in practice [2], as a regularization to find minimum Euclidean norm solutions, inspired by the results from support vector machines [4].", "We observe that adaptive methods (such as Adam and RMSProp) perform similarly to simple SGD variants, supporting our conjecture that each algorithm requires a different configuration, but still can converge to a good local point; also that adaptive methods require the same (if not more) tuning.", "Again, we observe that AdaGrad methods do not perform as well as the rest of the algorithms.", "Nevertheless, adaptive methods (such as Adam and RMSProp) perform similarly to simple SGD variants.", "Further experiments on CIFAR-100 for different architecture are provided in the Appendix." ] ]	2011.14066
[ [ "$\\eta$ mesons in hot and dense asymmetric nuclear matter" ], [ "Abstract We study the $\\eta N$ interactions in the hot and dense isospin asymmetric nuclear matter using two different approaches.", "In the first approach, the in-medium mass and optical potential of $\\eta$-meson have been calculated in the chiral SU(3) model, considering the effect of explicit symmetry breaking term and range terms in the $\\eta N$ interaction Lagrangian density.", "In the second scenario, the conjunction of chiral perturbation theory and chiral SU(3) model is employed.", "In this case, the next-to-leading order $\\eta N$ interactions are evaluated from the chiral perturbation theory (ChPT), and the in-medium contribution of scalar densities are taken as input from chiral SU(3) model.", "We observe a larger negative mass-shift in the ChPT+chiral model approach compared to the chiral SU(3) model alone as a function of nuclear density.", "Moreover, the increase in the asymmetry and temperature cause a decrease in the magnitude of mass-shift.", "We have also studied the impact of $\\eta N$ scattering length $a^{\\eta N}$ on the $\\eta$ meson mass $m^_\\eta$ and observed that the $m^_\\eta$ decrease more for increasing the value of scattering length." ], [ "Introduction", "The meson-baryons interactions are very imperative topic of research to study the physics of non-perturbative QCD regime [1], [2], [3], [5], [4], [6], [7], [8], [9], [10], [11], [12].", "The heavy ion-collisions (HICs) are used to study the strong-interaction physics by colliding high energy particles.", "As a byproduct of the collision, the Quark Gluon Plasma (QGP) appears under the utmost conditions of density and temperature [12].", "Afterward with the expansion of fireball the QGP cools down and changes its phase to the hadronic matter through hadronization process [12].", "These two regimes $i.e.$ QGP phase and hadronic phase have different characterization of the respective medium.", "For example, in the former phase quarks and gluons act as a degree of freedom whereas in the latter, mesons and baryons play this role.", "In QGP phase the chiral symmetry is followed ($m_q \\sim 0$ ) but in hadronic phase it is broken explicitly ($m_q \\ne 0$ ) and spontaneously ($\\langle \\bar{q} q \\rangle \\ne 0$ ) [13], [12].", "Furthermore, in the hadronic ensemble, the thermodynamics quantities namely nuclear density (number density of nucleons), isospin asymmetry (number of neutrons vs the number of protons), and temperature also play a crucial role to modify the in-medium properties of the mesons and baryons [2], [1], [12].", "The operation of future experimental facilities such as CBM and PANDA at GSI, Germany, NICA at Dubna, Russia and, J-PARC at Japan may lead to considerable progress in the understanding of meson-baryons interactions [12], [13].", "On the theoretical side, several potential models have been theorized to study the physics of the non-perturbative regime.", "Some of these are: Nambu-Jona-Lasinio (NJL) model [14], the Polyakov loop extended NJL (PNJL) model [15], [16], [17], chiral perturbation theory (ChPT) [10], [5], coupled channel approach [1], [6], [7], [8], [18], chiral $SU(3)$ model [2], [19], [20], [21], [22], [25], [26], [27], [28], [21], [24], [23], Quark-Meson Coupling (QMC) model [29], [30], [31], [32], [33], [34], Polyakov Quark Meson (PQM) model [35], [36], and QCD sum rules [37], [3], [38], [39], [40], [41], etc.", "Various effective models are formulated keeping in view the fundamental QCD properties such as broken scale invariance and spontaneous and explicit breaking of the chiral symmetry.", "For the first time, Haider and Liu anticipated that the $\\eta N$ interactions are attractive and suggested the possibility of $\\eta $ -meson bound states [42], [43].", "The negative mass-shift/optical potential of $\\eta $ -meson has attracted researchers to study the possibilities of $\\eta $ -mesic nuclei formation [5], [10], [11].", "At nuclear saturation density, the optical potential of -20 MeV was anticipated in the chiral coupled channel approach, considering leading order terms [11].", "Using same coupled channel model, Chiang $et.", "al.$ obtained optical potential $U_{\\eta }$ = -34 MeV in the normal nuclear matter, assuming the $\\eta N$ interactions dominated by $N^$ (1535) excitation [44] and anticipated that the attractive potential can produce $\\eta $ -meson bound state with light and heavy nucleus.", "Using the QMC model, authors of Ref.", "[45] obtained optical potential -60 MeV at $\\rho _{N}=\\rho _0$ .", "The chiral unitary approach was also implied to evaluate the $\\eta $ potential and it was observed to be -54 MeV [46].", "A more deep optical potential of -72 MeV was observed in Ref.", "[47].", "In this article the possibility of a bound state with $\\eta $ -meson was also explored.", "In Ref.", "[10], using $\\eta N$ Lagrangian off-shell terms, at normal nuclear density, the in-medium mass of $\\eta $ -meson was found to be (0.84 $\\pm $ 0.015)$m_{\\eta }$ and the corresponding optical potential was observed as -(83 $\\pm $ 5) MeV.", "Furthermore, using the relativistic mean-field theory, Song $et$ $al.$ observed the optical potential by varying the scattering length [48].", "Clearly, the values of $\\eta $ optical potential predicted in various studies varies over large range, $i.e.,$ -20 MeV to -85 MeV and hence, have considerable model dependence.", "In addition to theoretical attempts, there are experimental studies to explore the properties of $\\eta $ mesons [49], [52], [53], [50], [54], [51].", "For example, for different $\\eta $ hadron interactions, the $\\eta $ -meson production has been studied in Refs.", "[49], [50], [51] and the transverse momentum spectra of $\\eta $ -meson is measured in HICs near the free $N$ -$N$ production threshold [51].", "In the current investigation, we present the in-medium mass and optical potential of the $\\eta $ -meson in hot and dense asymmetric nuclear matter using chiral SU(3) model.", "By incorporating the medium induced nucleon scalar densities, we calculate the in-medium mass-shift of $\\eta $ -meson using the $\\eta N$ effective Lagrangian of chiral SU(3) model.", "Furthermore, as discussed earlier, the in-medium mass and optical potential of $\\eta $ -meson have been studied using the unitary approach of ChPT and relativistic mean-field model [10], [48].", "Following this work, as a second part of the current investigation the effective mass of $\\eta $ -meson is also evaluated using the chiral $\\eta N$ Lagrangian of chiral perturbation theory [10].", "In this approach, the nucleon scalar densities are calculated from chiral SU(3) model and plugged in the dispersion relation of $\\eta N$ interactions derived from ChPT Lagrangian.", "The chiral $SU(3)$ model is extensively used to explore the in-medium properties of the mesons and baryons in the hot and dense hadronic matter [21], [55], [19].", "For example, the model was used to study the in-medium mass and optical potential of kaons, antikaons and phi mesons in the nuclear and hyperonic matter [19], [22], [26].", "Furthermore, in the nuclear and hadronic matter the in-medium mass of spin 0, spin 1 $D$ mesons and quarkonia were calculated using the conjunction of chiral SU(3) model and QCD sum rules with [23], [56], [24], [25] and without taking the effect of magnetic field [57], [58], [59], [60], [21].", "The model was extended to $SU(4)$ and $SU(5)$ sector to evaluate the medium induced properties of heavy mesons such as $D$ and $B$ [27], [28], [22].", "On the other hand, the chiral perturbation theory is also a successful theoretical framework to study the baryon-meson interactions.", "The in-medium properties of $K$ meson were first studied by Kaplan and Nelson using chiral perturbation theory (ChPT) [4].", "The same theory was applied to study the $\\eta $ -nucleon interactions via adding leading order terms in the model Lagrangian [5].", "The heavy baryon chiral perturbation theory was also applied to study the kaon condensation which is an imperative property to study the neutron star matter [61], [62], [63].", "The ChPT theory was also improved by the introduction of next-to-leading order terms in the chiral effective Lagrangian.", "By including these off-shell terms, Zhong $et.", "al.$ anticipated appreciable decrease in the in-medium mass of $\\eta $ -meson which is favorable for the formation of $\\eta $ -mesic nuclei [10].", "The layout of the present paper is as follows: In the next section, we will give brief explanation of the formalism used in the present work.", "In section REF , we will derive the $\\eta N$ interactions in the chiral SU(3) model whereas, in section REF , $\\eta N$ methodology will be given in the unified approach of chiral perturbation theory and chiral model.", "In section , we will discuss the in-medium effects on the mass of $\\eta $ -meson, and finally in section , we will present the summary.", "The Lagrangian density of the chiral SU(3) model is written as ${\\cal L}_{\\text{chiral}} = {\\cal L}_{kin} + \\sum _{ M =S,V}{\\cal L}_{NM}+ {\\cal L}_{vec} + {\\cal L}_0 + {\\cal L}_{SB}.", "$ The model preserves the fundamental QCD properties such as the broken scale invariance and non-linear realization of the chiral symmetry [64], [65], [66], [67], [23], [2], [24].", "It is successfully used to explain the nuclear matter, finite nuclei, neutron star, and hypernuclei [64], [65], [66], [67], [23], [2], [24].", "In this model, the nucleons and baryons interact by the exchange of the vector fields $\\omega $ and $\\rho $ along with the scalar fields $\\sigma $ , $\\zeta $ and $\\delta $ in the nuclear medium.", "The vector fields give short-range repulsion or attraction which depends on the type of meson-nucleon interaction whereas the scalar fields give attractive contributions to the medium [26].", "The $\\sigma $ field is a non-strange scalar-isoscalar field which represents the scalar mesons $\\sigma $ ($u\\bar{d}$ ) whereas the $\\zeta $ field is a strange scalar-isoscalar field which represent the scalar meson ($s\\bar{s}$ ) [68].", "Moreover, the scalar-isovector field $\\delta $ $\\sim (\\bar{u} u-\\bar{d} d$ ) is incorporated in the present model to study the effect of the isospin asymmetric matter.", "Further, the glueball field, $\\chi $ is a hypothetical gluon field that contains gluon particles and is introduced in the chiral models to incorporate the scale invariance property of QCD [2], [69].", "We have used mean-field approximation to simplify the model by neglecting the effect of quantum and thermal fluctuations near phase transitions [23], [70].", "In Eq.", "(REF ), the ${\\cal L}_{kin}$ term describes the the kinetic energy term and the second term ${\\cal L}_{NM}$ given by ${\\cal L}_{NM} = - \\sum _{i} \\bar{\\psi _i}\\left[ m_{i}^{} + g_{\\omega i} \\gamma _{0} \\omega + g_{\\rho i} \\gamma _{0} \\rho \\right] \\psi _{i},$ defines the nucleon-meson interactions with in-medium nucleon mass as $m_{i}^{} = -(g_{\\sigma i}\\sigma + g_{\\zeta i}\\zeta + g_{\\delta i}\\tau _3 \\delta ),$ where $\\tau _3$ denotes the 3rd component of isospin and $g_{\\sigma i}$ , $g_{\\zeta i}$ and $g_{\\delta i}$ are the coupling constants of $\\sigma $ , $\\zeta $ and field $\\delta $ with nucleons ($i$ =$p,n$ ), respectively.", "The next term $ {\\cal L}_{vec}$ is given by ${\\cal L} _{vec} & = & \\frac{1}{2} \\left( m_{\\omega }^{2} \\omega ^{2}+ m_{\\rho }^{2} \\rho ^{2} \\right)\\frac{\\chi ^{2}}{\\chi _{0}^{2}}+ g_4 (\\omega ^4 +6\\omega ^2 \\rho ^2+\\rho ^4),$ reproduces the mass of vector mesons through self-interactions.", "The ${\\cal L}_{0}$ defines the spontaneous chiral symmetry breaking by the equation ${\\cal L} _{0} & = & -\\frac{1}{2} k_{0}\\chi ^{2} \\left( \\sigma ^{2} + \\zeta ^{2}+ \\delta ^{2} \\right) + k_{1} \\left( \\sigma ^{2} + \\zeta ^{2} + \\delta ^{2}\\right)^{2} \\nonumber \\\\&+& k_{2} \\left( \\frac{\\sigma ^{4}}{2} + \\frac{\\delta ^{4}}{2} + 3 \\sigma ^{2}\\delta ^{2} + \\zeta ^{4} \\right)+ k_{3}\\chi \\left( \\sigma ^{2} - \\delta ^{2} \\right)\\zeta \\nonumber \\\\&-& k_{4} \\chi ^{4}- \\frac{1}{4} \\chi ^{4} {\\rm {ln}}\\frac{\\chi ^{4}}{\\chi _{0}^{4}}+ \\frac{d}{3} \\chi ^{4} {\\rm {ln}} \\Bigg (\\bigg ( \\frac{\\left( \\sigma ^{2}- \\delta ^{2}\\right) \\zeta }{\\sigma _{0}^{2} \\zeta _{0}} \\bigg )\\bigg (\\frac{\\chi }{\\chi _0}\\bigg )^3 \\Bigg ).$ In this equation, the $\\sigma _0$ , $\\zeta _0$ , $\\delta _0$ and $\\chi _0$ denote the vacuum values of $\\sigma $ , $\\zeta $ , $\\delta $ and $\\chi $ scalar fields, respectively.", "Also, the parameter $d$ =0.064 along with $k_i(i=1$ to $4)$ and other medium parameters are fitted to regenerate the vacuum values of scalar and vector fields, $\\eta $ , $\\eta ^{\\prime }$ mesons and the nucleon mass [2], [21], [24].", "In ccc, we have tabulated the values of various parameters.", "Furthermore, the last term ${\\cal L}_{SB} $ in Eq.", "(REF ) describes the explicit chiral symmetry breaking property and is written as ${\\cal L} _{SB} = -\\left( \\frac{\\chi }{\\chi _{0}}\\right)^{2}\\left[ m_{\\pi }^{2}f_{\\pi } \\sigma + \\big ( \\sqrt{2} m_{K}^{2}f_{K} - \\frac{1}{\\sqrt{2}}m_{\\pi }^{2} f_{\\pi } \\big ) \\zeta \\right].$ In the above equation, $m_\\pi $ , $m_K$ , $f_\\pi $ , and $f_K$ symbolize the masses and decay constants of pions and kaons, respectively.", "The non-linear coupled equations of motion of the scalar and vector fields are deduced by solving the total Lagrangian (Eq.", "(REF )) using the Euler-Lagrange equations [24], [25] and are given as $k_{0}\\chi ^{2}\\sigma -4k_{1}\\left( \\sigma ^{2}+\\zeta ^{2}+\\delta ^{2}\\right)\\sigma -2k_{2}\\left( \\sigma ^{3}+3\\sigma \\delta ^{2}\\right)-2k_{3}\\chi \\sigma \\zeta \\nonumber \\\\-\\frac{d}{3} \\chi ^{4} \\bigg (\\frac{2\\sigma }{\\sigma ^{2}-\\delta ^{2}}\\bigg )+\\left( \\frac{\\chi }{\\chi _{0}}\\right) ^{2}m_{\\pi }^{2}f_{\\pi }=\\sum g_{\\sigma i}\\rho _{i}^{s} ,$ $k_{0}\\chi ^{2}\\zeta -4k_{1}\\left( \\sigma ^{2}+\\zeta ^{2}+\\delta ^{2}\\right)\\zeta -4k_{2}\\zeta ^{3}-k_{3}\\chi \\left( \\sigma ^{2}-\\delta ^{2}\\right)\\nonumber \\\\-\\frac{d}{3}\\frac{\\chi ^{4}}{\\zeta }+\\left(\\frac{\\chi }{\\chi _{0}} \\right)^{2}\\left[ \\sqrt{2}m_{K}^{2}f_{K}-\\frac{1}{\\sqrt{2}} m_{\\pi }^{2}f_{\\pi }\\right]=\\sum g_{\\zeta i}\\rho _{i}^{s} ,$ $k_{0}\\chi ^{2}\\delta -4k_{1}\\left( \\sigma ^{2}+\\zeta ^{2}+\\delta ^{2}\\right)\\delta -2k_{2}\\left( \\delta ^{3}+3\\sigma ^{2}\\delta \\right) +2k_{3}\\chi \\delta \\zeta \\nonumber \\\\+ \\frac{2}{3} d \\chi ^4 \\left( \\frac{\\delta }{\\sigma ^{2}-\\delta ^{2}}\\right)=\\sum g_{\\delta i}\\tau _3\\rho _{i}^{s} ,$ $\\left(\\frac{\\chi }{\\chi _{0}}\\right) ^{2}m_{\\omega }^{2}\\omega +g_{4}\\left(4{\\omega }^{3}+12{\\rho }^2{\\omega }\\right) =\\sum g_{\\omega i}\\rho _{i}^{v} ,$ $\\left(\\frac{\\chi }{\\chi _{0}}\\right) ^{2}m_{\\rho }^{2}\\rho +g_{4}\\left(4{\\rho }^{3}+12{\\omega }^2{\\rho }\\right)=\\sum g_{\\rho i}\\tau _3\\rho _{i}^{v} ,$ and $k_{0}\\chi \\left( \\sigma ^{2}+\\zeta ^{2}+\\delta ^{2}\\right)-k_{3}\\left( \\sigma ^{2}-\\delta ^{2}\\right)\\zeta + \\chi ^{3}\\left[1+{\\rm {ln}}\\left( \\frac{\\chi ^{4}}{\\chi _{0}^{4}}\\right) \\right]+(4k_{4}-d)\\chi ^{3}\\nonumber \\\\-\\frac{4}{3} d \\chi ^{3} {\\rm {ln}} \\Bigg ( \\bigg (\\frac{\\left( \\sigma ^{2}-\\delta ^{2}\\right) \\zeta }{\\sigma _{0}^{2}\\zeta _{0}} \\bigg )\\bigg (\\frac{\\chi }{\\chi _0}\\bigg )^3 \\Bigg )+\\frac{2\\chi }{\\chi _{0}^{2}}\\left[ m_{\\pi }^{2}f_{\\pi }\\sigma +\\left(\\sqrt{2}m_{K}^{2}f_{K}-\\frac{1}{\\sqrt{2}}m_{\\pi }^{2}f_{\\pi } \\right) \\zeta \\right] \\nonumber \\\\-\\frac{\\chi }{{{\\chi _0}^2}}(m_{\\omega }^{2} \\omega ^2+m_{\\rho }^{2}\\rho ^2) = 0 ,$ respectively.", "In above equations, the $\\rho ^{s}_{i}$ and $\\rho ^{v}_{i}$ denote the scalar and vector densities of $i^{th}$ nucleons ($i=n,p$ ) [24], [2] and are given as $\\rho _{i}^{v} = \\gamma _{i}\\int \\frac{d^{3}k}{(2\\pi )^{3}}\\Bigg (\\frac{1}{1+\\exp \\left[\\beta (E^{\\ast }_i(k)-\\mu ^{}_{i}) \\right]}-\\frac{1}{1+\\exp \\left[\\beta (E^{\\ast }_i(k)+\\mu ^{}_{i}) \\right]}\\Bigg ),$ and $\\rho _{i}^{s} = \\gamma _{i}\\int \\frac{d^{3}k}{(2\\pi )^{3}}\\frac{m_{i}^{}}{E^{\\ast }_i(k)} \\Bigg (\\frac{1}{1+\\exp \\left[\\beta (E^{\\ast }_i(k)-\\mu ^{}_{i}) \\right]}+\\frac{1}{1+\\exp \\left[\\beta (E^{\\ast }_i(k)+\\mu ^{}_{i}) \\right]}\\Bigg ),$ respectively, where $\\beta = \\frac{1}{kT}$ , $E^{\\ast }_i(k)=\\sqrt{k^2+{m^_i}^2}$ , $ \\mu ^{}_{i}=\\mu _{i}-g_{\\omega i}\\omega -g_{\\rho i}\\tau _{3}\\rho $ and $\\gamma _i$ is the degeneracy factor.", "Moreover, the isospin effect on the scalar and vector density is measured by the definition, $I = -\\frac{\\Sigma _i \\tau _{3i} \\rho ^{v}_{i}}{2\\rho _{N}}$ .", "In the next section, we calculate the medium-modified mass of $\\eta $ mesons in hot asymmetric nuclear matter.", "The medium modified $\\eta $ meson mass is evaluated from the dispersion relation which is obtained from the $\\eta N$ equation of motion.", "Table: Different constants used in the present work .", "In the chiral SU(3) model, the $\\eta N$ interaction Lagrangian density can be written as $ \\mathcal {L_{\\eta }} &=&\\left( \\frac{1}{2}-\\frac{\\sigma ^\\prime + 4 \\zeta ^\\prime (2 f_K-f_\\pi ) }{\\sqrt{2}f^2} \\right) \\partial ^{\\mu }\\eta \\partial _{\\mu }\\eta \\nonumber \\\\&-&\\frac{1}{2}\\left(m_{\\eta }^2-\\frac{(\\sqrt{2}\\sigma ^\\prime -4 \\zeta ^\\prime )m^2_\\pi f_\\pi + 8 \\zeta ^\\prime m^2_K f_K}{\\sqrt{2} f^2}\\right) \\eta ^2\\nonumber \\\\&&+\\frac{d^{\\prime }}{f^2} \\left( \\frac{\\rho ^s_p+\\rho ^s_n}{4} \\right) \\partial ^{\\mu }\\eta \\partial _{\\mu }\\eta ,$ The above chiral $\\eta N$ Lagrangian consists of three terms.", "First Range Term: The first term in the chiral Lagrangian describes the first range term [22], [2] and is obtained from ${\\mathcal {L}}_{{\\mathrm {1st range term}}} = Tr (u_{\\mu } X u^{\\mu }X +X u_{\\mu } u^{\\mu } X) .$ In the above equation, $u_{\\mu } =-\\frac{i}{2} \\left[u^{\\dagger }(\\partial _{\\mu }u)-u (\\partial _{\\mu }u^\\dagger ) \\right]$ and $u$ =$ \\text{exp}\\left[ \\frac{i}{\\sqrt{2}\\sigma _{0}}P\\gamma _{5}\\right]$ , which is expanded up to second order.", "Here, symbols $X$ and $P$ represent the scalar and pseudoscalar meson matrices [2], respectively and are explicitly given by Eqs.", "(REF ) and (REF ) in the appendix.", "Furthermore, the vacuum values of $\\sigma $ and $\\zeta $ fields are deduced in terms of pions and kaons decay constant by solving the axial current of pions and kaons [2] through relation $\\sigma _0 = -f_{\\pi } \\qquad \\zeta _0 = -\\frac{1}{\\sqrt{2}}(2 f_K - f_{\\pi }).$ Moreover, in the first term of $\\eta N$ Lagrangian $\\sigma ^{\\prime }(=\\sigma -\\sigma _0)$ , $\\zeta ^{\\prime }(=\\zeta -\\zeta _0)$ and $\\delta ^{\\prime }(=\\delta -\\delta _0)$ define the digression of the expectation values of fields from their vacuum expectations.", "Also, $f$ =$\\sqrt{f_\\pi ^2+2(2 f_K - f_\\pi )^2}$ and $d^{\\prime }$ =$3d_1+d_2$ are the constant parameters.", "Mass Term: Further, the mass term of the chiral model gives the second term of $\\eta N$ Lagrangian and is given by ${\\cal L}_{SB} =-\\frac{1}{2} {\\rm Tr}A_p \\left(uXu+u^{\\dagger }Xu^{\\dagger }\\right),$ where $A_p$ is a diagonal matrix given in the Eq.", "(REF ).", "The vacuum mass of $\\eta $ meson, $m_{\\eta }$ , is extracted from the above term and is given by the relation $m_{\\eta }=\\frac{1}{f}\\sqrt{\\left(3 m_\\pi ^2 f_K m_K^2+\\frac{8 f_K^2 m_K^2}{f_\\pi ^2} -\\frac{4f_K m_\\pi ^2}{f_\\pi } \\right)}.$ Substituting the values of various constants in above $m_{\\eta }$ turns out to be 574.374 MeV which is with an accuracy of 4.9 $\\%$ of physical mass $i.e.$ 547.862 MeV [71].", "The vacuum mass of $\\eta $ -meson has model dependency [72] but here in the present work, we are more concerned in the $\\eta $ in-medium mass-shift which is nearly same for both the masses.", "In Ref.", "[72], using Gell-Mann Okubo mass formula under octet approximation in the SU(4) meson multiplets, authors calculated the vacuum mass of $\\eta $ -meson to be 567 MeV which is with an accuracy of 3.6 $\\%$ .", "$d^{\\prime }$ Term: The third term ($i.e.$ $d^{\\prime }$ term) in the $\\eta N$ Lagrangian originates from the baryon-meson interaction Lagrangian densities [27], [20] ${\\cal L }_{d_1}^{BM} =\\frac{d_1}{2} Tr (u_\\mu u ^\\mu )Tr( \\bar{B} B),$ and ${\\cal L }_{d_2}^{BM} =d_2 Tr (\\bar{B} u_\\mu u ^\\mu B).$ In above, $B$ denotes the baryon matrix (see Eq.", "(REF )).", "It should be noted that in case of $\\eta N$ interactions of Eq.", "(REF ), the terms corresponding to vectorial Weinberg-Tomozawa term vanishes.", "On the the other hand, the Weinberg-Tomozawa term plays a crucial role in the determination of $K (\\bar{K})$ and $D (\\bar{D})$ in-medium mass [22], [28].", "Using the $\\eta N$ Lagrangian in the Euler-Lagrange equation for $\\eta $ meson, the equation of motion is evaluated as $&& \\partial ^{\\mu }\\partial _{\\mu } \\eta -\\left(m_{\\eta }^2-\\frac{(\\sqrt{2}\\sigma ^\\prime -4 \\zeta ^\\prime )m^2_\\pi f_\\pi + 8 \\zeta ^\\prime m^2_K f_K}{\\sqrt{2} f^2}\\right)\\eta \\nonumber \\\\&&+\\frac{2d^{\\prime }}{f^2} \\left( \\frac{\\rho ^s_p+\\rho ^s_n}{4} -\\frac{\\sigma ^\\prime + 4 \\zeta ^\\prime (2 f_K-f_\\pi ) }{\\sqrt{2}} \\right) \\partial ^{\\mu } \\partial _{\\mu } \\eta =0.$ Performing the Fourier transformation on the above equation, the dispersion relation for $\\eta $ meson turns out to be $-\\omega ^2+ { \\textbf {k}}^2 + m_\\eta ^2 -\\Pi ^(\\omega , \| \\textbf {k}\|)=0.$ In the above equation, $\\Pi ^$ denotes the effective self-energy of $\\eta $ meson, explicitly given as $\\Pi ^* (\\omega , \| \\textbf {k}\|) &= & -\\frac{(\\sqrt{2}\\sigma ^\\prime -4 \\zeta ^\\prime )m^2_\\pi f_\\pi + 8 \\zeta ^\\prime m^2_K f_K}{\\sqrt{2} f^2}+\\frac{2d^{\\prime }}{f^2} \\left( \\frac{\\rho ^s_p+\\rho ^s_n}{4} \\right)(\\omega ^2 - {\\vec{k}}^2) \\nonumber \\\\&-& \\frac{2}{f^2} \\left( \\frac{\\sigma ^\\prime + 4 \\zeta ^\\prime (2 f_K-f_\\pi ) }{\\sqrt{2}} \\right)(\\omega ^2 - {\\textbf { k}}^2).$ The unknown parameter, $d^{\\prime }$ is approximated from the experimental values of scattering length, $a^{\\eta N}$ [10].", "In the chiral model, the expression of scattering length derived from the scattering amplitude is given by $a^{\\eta N} &=& \\frac{1}{4 \\pi \\left(1+\\frac{m_\\eta }{M_N}\\right)} \\Big [ \\Big ( \\frac{d^{\\prime }}{\\sqrt{2}}-\\frac{g_{\\sigma N}}{m^2_\\sigma }+\\frac{4 (2f_K-f_\\pi ) g_{\\zeta N}}{m^2_\\zeta } \\Big ) \\frac{m_\\eta ^2}{\\sqrt{2}f^2} \\nonumber \\\\&+& \\left( \\frac{\\sqrt{2} g_{\\sigma N}}{m^2_\\sigma }-\\frac{4 g_{\\zeta N}}{m^2_\\zeta } \\right)\\frac{m^2_\\pi f_\\pi }{2\\sqrt{2}f^2}+\\tau _3 \\frac{2\\sqrt{2} g_{\\delta N}}{m^2_\\delta } \\frac{m^2_K f_K}{f^2} \\Big ].$ Rearranging the above for $d^{\\prime }$ gives $d^{\\prime } &=& \\frac{ f^2}{2 \\pi \\left(1+\\frac{m_\\eta }{M_N}\\right)} \\frac{ a^{\\eta N}}{m^2_\\eta } +\\frac{\\sqrt{2}g_{\\sigma N}}{m^2_\\sigma }-\\frac{4 \\sqrt{2} (2f_K-f_\\pi ) g_{\\zeta N}}{m^2_\\zeta } \\nonumber \\\\&-& \\left( \\frac{\\sqrt{2} g_{\\sigma N}}{m^2_\\sigma }-\\frac{4 g_{\\zeta N}}{m^2_\\zeta } \\right)\\frac{m^2_\\pi f_\\pi }{\\sqrt{2} m^2_\\eta }-\\tau _3 \\frac{4\\sqrt{2} g_{\\delta N}m^2_K f_K}{m^2_\\delta m^2_\\eta }.$ Using the condition, $m_{\\eta }^=\\omega (\| \\textbf {k}\|$ =0) in Eq.", "(REF ), we obtain the effective mass of $\\eta $ meson in the nuclear medium.", "Further, the momentum dependent optical potentials are defined through the relation [73], [28] $U^_{\\eta }(\\omega ,\\textbf { k}) = \\omega (\\textbf {k}) -\\sqrt{\\textbf {k}^2 + m^{^2}_{\\eta }}.$ At zero momentum, the above equation gives $U^_{\\eta } =\\Delta m_\\eta ^={m_{\\eta }^}-m_{\\eta }.$" ], [ " UNIFICATION OF CHIRAL PERTURBATION THEORY (ChPT) AND CHIRAL MODEL", "In this section, we discuss the unified approach of ChPT and chiral model to compute the in-medium mass of $\\eta $ mesons.", "The ChPT comprises the underlying chiral symmetry property of QCD and use an effective field theory approach [10].", "The same theory along with Relativistic mean-field model has been used to deduce the eta-nucleon interactions in the symmetric nuclear matter [10], [48].", "The Lagrangian density defining the meson-baryons interactions in this theory is given by ${\\mathcal {L}_{\\text{ChPT}} }={\\mathcal {L}_{P} }+{\\mathcal {L}_{PB} },$ with $P$ representing the pseudoscalar meson multiplet (see Eq.", "(REF )).", "Up to second chiral order, the $\\mathcal {L}_{P}$ term is defined as [10], [4] ${\\mathcal {L}_{P} }&=&\\frac{1}{4}f_\\pi ^{2}\\textrm {Tr}\\partial ^{\\mu }\\Sigma \\partial _{\\mu }\\Sigma ^{\\dagger }+\\frac{1}{2}f_\\pi ^2 B_0\\left\\lbrace \\mbox{Tr} M_{q}(\\Sigma -1)+\\mathrm {h.c.}\\right\\rbrace ,$ where $\\Sigma =\\xi ^2=\\exp {(i\\sqrt{2}P/f_\\pi )}$ and $M_{q}=\\mbox{diag}\\lbrace m_{q}, m_{q}, m_{s}\\rbrace $ is the current quark mass matrix.", "The Lagrangian term, $\\mathcal {L}_{P B}$ =$\\mathcal {L}^L_{P B}$ +$\\mathcal {L}^{NL}_{P B}$ describes the leading and next to leading order contributions [4].", "Jenkins and Manohar developed the next to leading order terms using heavy baryon chiral theory [5].", "In this Lagrangian, the loop contributions are not considered as the higher-order corrections get suppressed for the small momentum scale, $Q^{2}$ [10].", "The different nuclear properties are studied successfully using $\\mathcal {L}^{NL}_{P B}$ [74].", "The $\\eta N$ Lagrangian is obtained by expanding the Eq.", "(REF ) up to the second order of multiplet $P$ [10] $ \\mathcal {L_{\\eta N }} &=&\\frac{1}{2}\\partial ^{\\mu }\\eta \\partial _{\\mu }\\eta -\\frac{1}{2}\\left(m{^\\prime }_{\\eta }^2-\\frac{\\Sigma _{\\eta \\mathrm {N}}}{f_\\pi ^2}\\bar{\\Psi }_{\\mathrm {N}}\\Psi _{\\mathrm {N}}\\right) \\eta ^2+\\frac{1}{2}\\frac{\\kappa }{f_\\pi ^2}\\bar{\\Psi }_{\\mathrm {N}}\\Psi _{\\mathrm {N}}\\partial ^{\\mu }\\eta \\partial _{\\mu }\\eta .$ Here, $m{^\\prime }_{\\eta }=\\sqrt{\\frac{2}{3}B_0 (m_q+2m_s)}$ denotes the vacuum mass of $\\eta $ meson calculated in chiral perturbation theory.", "In the mass expression, $B_0$ symbolize the relation with the order parameter of spontaneously broken chiral symmetry and $m_{q(s)}$ denote the mass of light (strange) quarks [72].", "For consistency with the chiral SU(3) model, we have used the same value of $\\eta $ meson vacuum mass $i.e.$ $m{^\\prime }_{\\eta }$ =$m_{\\eta }$ = 574.374 MeV in the further calculations of ChPT.", "The $\\eta N$ sigma term $\\Sigma _{\\eta \\mathrm {N}}$ , obtained from $a_i$ terms of the next-to-leading order chiral Lagrangian density is given as [10] $ \\Sigma _{\\eta \\mathrm {N}}=-\\frac{2}{3}[a_{1}m_{q}+4a_{2}m_{s}+2a_{3}(m_{q}+2m_{s})].$ The $\\Sigma _{\\eta \\mathrm {N}}$ value is estimated to be 280 $\\pm $ 130 MeV from the different empirical observations of $\\Sigma _{\\mathrm {KN}}$ term having value 380 MeV $\\pm $ 100 MeV [75], [76], [77], [61], [78], [79], [62], [10].", "Also, the parameter $\\kappa $ in the last term of the Eq.", "(REF ) comprises the contributions from the off-shell $d_i$ terms of the next to Leading order Lagrangian [10].", "In the present work, we determined $\\kappa $ using the expression of $\\eta N$ scattering length, $a^{\\eta \\mathrm {N}}$ , calculated from the ChPT matrix amplitude (on-shell constraints) [10] $a^{\\eta \\mathrm {N}} =\\frac{1}{4\\pi f_\\pi ^2(1+m_{\\eta }/M_{\\mathrm {N}})}\\left(\\Sigma _{\\eta \\mathrm {N}}+ \\kappa m_{\\eta }^2\\right),$ and by re-arranging for $\\kappa $ it becomes $ \\kappa =4\\pi f_\\pi ^2\\left(\\frac{1}{m_{\\eta }^2}+\\frac{1}{m_{\\eta }M_{\\mathrm {N}}}\\right)a^{\\eta \\mathrm {N}} -\\frac{\\Sigma _{\\eta \\mathrm {N}}}{m_{\\eta }^2}.$ We have taken the experimentally determined $a^{\\eta \\mathrm {N}}$ values $i.e.$ 0.91 $\\sim $ 1.14 fm in the present investigation [80], [81], [82], [83], [10].", "Furthermore, the $\\eta N$ equation of motion has been derived using the interaction Lagrangian (Eq.", "(REF )) in the Euler Lagrange equation of motion: $\\left(\\partial _{\\mu }\\partial ^{\\mu }+m_{\\eta }^2-\\frac{\\Sigma _{\\eta N}}{2f_\\pi ^2}\\langle \\bar{\\Psi }_{\\mathrm {N}}\\Psi _{\\mathrm {N}} \\rangle +\\frac{\\kappa }{2f_\\pi ^2}\\langle \\bar{\\Psi }_{\\mathrm {N}}\\Psi _{\\mathrm {N}} \\rangle \\partial _{\\mu }\\partial ^{\\mu }\\right)\\eta = 0.$ In above, $ \\langle \\bar{\\Psi }_{\\mathrm {N}}\\Psi _{\\mathrm {N}}\\rangle \\equiv \\rho ^s_{N}$ =$\\left(\\rho ^s_{p}+\\rho ^s_{n} \\right)$ defines the in-medium scalar density of nucleons calculated within the mean-field chiral SU(3) model (see Eqs.", "(REF ) and (REF )).", "The Fourier transformation of Eq.", "(REF ) gives $ -\\omega ^2+\\textbf {k}^2+m_{\\eta }^2-\\frac{\\Sigma _{\\eta \\mathrm {N}}}{2f_\\pi ^2}\\rho ^s_{N}+\\frac{\\kappa }{2f_\\pi ^2}\\rho ^s_{N}\\left(-\\omega ^2+\\textbf {k}^2\\right)=0,$ From the above equation, the effective mass $m_{\\eta }^=\\omega (\| \\textbf {k}\|$ =0) of $\\eta $ meson can be written as $m_{\\eta }^=\\sqrt{\\left(m_{\\eta }^2-\\frac{\\Sigma _{\\eta \\mathrm {N}}}{2f_\\pi ^2}\\rho ^s_{N}\\right)\\Big / \\left(1+\\frac{\\kappa }{2f_\\pi ^2}\\rho ^s_{N}\\right)}.$ Further, the $\\eta $ -meson self-energy derived from Eq.", "(REF ) is given by $\\Pi ^(\\omega ,\\textbf {k})=\\Big (-\\frac{\\Sigma _{\\eta \\mathrm {N}}}{2f_\\pi ^2}+\\frac{\\kappa }{2f_\\pi ^2}(-\\omega ^2+\\textbf {k}^2)\\Big ) \\rho ^s_{N}.$" ], [ "Results and Discussions", "In this section, at first we discuss the behavior of in-medium nucleon scalar densities in the hot asymmetric nuclear matter.", "Further, we discuss the effective mass of $\\eta $ meson which is derived using the chiral SU(3) model alone in section REF and with the unified approach of ChPT and chiral SU(3) model in section REF .", "In both approaches, we show the results for range of scattering length, $a^{\\eta _N}$ =0.91-1.14 fm.", "Various parameters used in the present investigation are mentioned in ccc.", "In chiral model, the scalar densities of nucleons have been calculated through Eq.", "(REF ).", "This equation contains the effect of medium modified scalar and vector fields [2].", "The in-medium behavior of these fields is obtained by solving the coupled equations of motion (Eqs.", "(REF ) to (REF )) [26].", "In rs, we plot the scalar density of proton and neutron as a function of number density for finite values of temperature, $T$ and isospin asymmetry parameter.", "In symmetric nuclear matter as the contribution of $\\delta $ and $\\rho $ field is zero [24], we get the same behavior of neutron and proton scalar densities.", "The $\\delta $ and $\\rho $ field changes the in-medium value of baryon mass $m^_i$ and effective chemical potential $\\mu ^_i$ , respectively which further modifies the nucleon scalar density (see Eq.", "(REF )) [2].", "In figure, at $T=0$ the scalar density increases linearly in the low density regime and becomes non-linear in the high density regime.", "When we move from $I$ =0 to I$\\ne $ 0 region, we observe a gradual increase in the neutron scalar density whereas the proton scalar density decreases.", "This is because of the non-zero contribution of $\\delta $ and $\\rho $ field in the isospin asymmetric nuclear matter which changes the effective mass as well as chemical potential and therefore scalar density [2].", "Another thermodynamic quantity $i.e.$ temperature is also a main property of the nuclear medium and in rs we have shown how the in-medium dynamics changes under non-zero temperature.", "The effect of temperature is observed more in the high density regime as compared to the low density regime.", "For symmetric matter in sub-plots (a) and (c), we anticipate appreciable effect of temperature.", "Here, for a particular value of nuclear density the value of scalar densities decrease as a function of temperature.", "This is because of the Fermi distribution integral, due to the coupled nature of the equations (Eqs.", "(REF ) to (REF )) the value of scalar density in Eq.", "(REF ) decreases when we increase the temperature in integral.", "On the other hand, in the highly asymmetric matter $i.e.$ $I$ =0.5, for the neutron scalar density the temperature effects become more appreciable.", "In addition, we observe a minor contribution to the proton scalar density for higher temperature values.", "This is because at finite temperature the proton condensate ($\\bar{p} p$ ) $i.e.$ proton scalar density still populates in the medium despite the zero value of proton number density $\\rho _p$ .", "The observed behavior of scalar densities in the symmetric nuclear matter is in agreement with the calculations of the Relativistic mean-field model [10], [48].", "Figure: (Color online) The in-medium scalar density of nucleons." ], [ "Optical Potential and Mass of $\\eta $ Meson in Chiral Model", "In masseta, we have illustrated the medium modified mass of $\\eta $ -meson as a function of nuclear density for different values of scattering length.", "In the same figure, we also show the impact of isospin asymmetry and temperature.", "For given value of asymmetry, temperature, scattering length, the in-medium mass of $\\eta $ -meson is observed to decrease as a function of nuclear density.", "The rate of decrease is linear in the low density regime whereas in the high density regime it becomes non-linear.", "This behavior reflects the opposite variation of nucleon scalar density plotted in rs.", "This is because the self-energy of $\\eta $ -meson (see Eq.", "(REF )) has a direct dependence on the sum of scalar densities of nucleons.", "When we change the value of $a^{\\eta N}$ from 0.91 to 1.14 fm, we observe a decrement in the effective mass.", "For example, at $\\rho _N$ =$\\rho _0 (4\\rho _0)$ , $I$ =$T$ =0, the effective mass of $\\eta $ -meson changes from 528 (441) to 512 (423) MeV when we change $a^{\\eta N}$ value from 0.91 to 1.14 fm, respectively.", "This is due to the $d^\\prime $ term in Eq.", "(REF ).", "The $d^\\prime $ term has direct dependence on $a^{\\eta N}$ as shown in Eq.", "(REF ) and therefore increasing the value of scattering length cause an increase in the value of $d^\\prime $ .", "Due to the attractive contribution of the self-energy part corresponding to $d^\\prime $ term the value of effective mass decreases.", "We also observed the substantial impact of the temperature on the in-medium mass in the symmetric nuclear matter which reflects the in-medium behavior of scalar densities.", "However, in the asymmetric nuclear matter, we observe the temperature effects on the mass to be less appreciable which reflects the less contribution of the net scalar density ($\\rho ^s_p+\\rho ^s_n$ ).", "The self-energy expression given by Eq.", "(REF ) contain three terms (i) first range term (ii) mass term and (iii) $d^\\prime $ term.", "To understand the contribution of these individual terms, we illustrated the in-medium mass of $\\eta $ -meson at zero and non-zero value of asymmetry and temperature in chiralterms for these different terms.", "At zero temperature and asymmetry, one can see that the first range term gives an appreciable repulsive contribution to the effective mass whereas the mass and $d^\\prime $ terms give attractive contributions.", "We observe the dominant contribution of $d^\\prime $ term which in turn gets reflected in the net effective mass.", "For non-zero temperature and asymmetry, the variation in the $d^\\prime $ term becomes less hence we get a lower value of effective mass.", "This is due to the effect of scalar density terms present in the $d^\\prime $ term (Eq.", "(REF )).", "For further understanding, in masseta we plot the $\\eta $ -meson effective mass as a function of scattering length $a^{\\eta N}$ at $\\rho _N$ =$\\rho _0,4\\rho _0$ .", "At nuclear saturation density, we observe a linear decrease of effective mass with the increase in scattering length.", "Furthermore, the effective mass decrease more rapidly in the high density regime.", "The observed behaviour emphasizes the importance of scattering length in the $\\eta N$ interactions.", "The decrease in the in-medium mass leads to a negative mass-shift which suggests the bound state formation of $\\eta $ -meson with a nucleus [5], [10].", "To understand the bound state phenomenon, the study of in-medium optical potential is very imperative.", "By using the effective mass in Eq.", "(REF ), we plotted the optical potential of $\\eta $ -meson as a function of momentum $\\vert \\textbf {k}\\vert $ for different values of $\\eta N$ scattering length and other medium parameters in Un0.91,Un1.02,Un1.14.", "In Un0.91, at $\\rho _N$ =$\\rho _0$ we observe a negative value of the optical potential.", "The value of optical potential becomes less negative as we increase the momentum of the $\\eta $ -meson.", "The variation of optical potential reflects the interplay between the effective mass and momentum.", "At high values of the momentum, Eq.", "(REF ) gets dominated by momentum and the contribution of effective mass becomes less.", "A similar phenomenon happens in the high density regime.", "In this region, we observe appreciable values of optical potential which become less as momentum increases.", "Moreover, in the presence of a high density of neutron matter , we anticipate less effect of temperature which reflects the in-medium behavior of $\\eta $ -meson mass.", "In Un1.02,Un1.14 we observe a similar trend of optical potential with $\\eta $ momentum.", "In these figures, we observe a more negative value of optical potential as we increase the scattering length.", "As discussed earlier, the optical potential is directly related to in-medium mass, here it is illustrated to get a clear idea of negative potential.", "In the cold symmetric nuclear matter, at $\\rho _N$ =$\\rho _0(4\\rho _0)$ we observe optical potential to be -54.61 (-146.77) MeV for $a^{\\eta N}$ =1.02 fm, whereas for $I$ =0.5 these values changes to -52.99 (-136.93) MeV.", "For better understanding, we have tabulated the in-medium mass-shift of $\\eta $ -meson at zero momentum in tablems0.", "Figure: (Color online) In-medium η\\eta meson mass in the chiral model.Figure: (Color online) Comparison of the different terms of η\\eta -meson effective mass in chiral model at a ηN =1.14a^{\\eta N}=1.14 fm.Figure: (Color online) The in-medium η\\eta -meson mass as a function of scattering length.Figure: (Color online) The in-medium η\\eta meson optical potential in chiral model at a ηN a^{\\eta N}=0.91 fm.Figure: (Color online) The in-medium η\\eta meson optical potential in chiral model at a ηN a^{\\eta N}=1.02 fm.Figure: (Color online) The in-medium η\\eta meson optical potential in chiral model at a ηN a^{\\eta N}=1.14 fm." ], [ "In-medium Mass of $\\eta $ Meson in Unified Approach of ChPT and Chiral Model", "In this section, we have used the unified approach of chiral SU(3) model and chiral perturbation theory to calculate the medium induced mass of $\\eta $ -meson.", "As discussed in the methodology section, the $\\eta N$ equation of motion is obtained from the Lagrangian density of ChPT.", "Further, the scalar density of nucleons appearing in the ChPT equation of motion is obtained from the chiral SU(3) model.", "In this calculation, we have taken the value of parameter $\\Sigma _{\\eta N}$ to be 280 MeV.", "We have not considered the contribution of uncertainties in the $\\Sigma _{\\eta N}$ parameter because of the less contribution of sigma term in the in-medium mass as compared to $\\kappa $ term which we will see later.", "The value of in-medium $\\eta $ mass-shift calculated using present unified approach are given in tablems.", "In massra, we illustrated the ratio of the in-medium and vacuum mass of the $\\eta $ -meson as a function of nuclear density.", "In this figure, we have also included the effect of $\\eta N$ scattering length, temperature, and medium isospin asymmetry.", "Moreover, we compared the results obtained from two different approaches $i.e.$ (i) chiral model alone (ii) ChPT and chiral model.", "Using the second approach, we observed a substantial decrease in the mass of $\\eta $ -meson.", "We observed the same behavior of the in-medium mass with respect to temperature, asymmetry, and scattering length as was observed in the situation when the chiral model was used alone.", "The main difference is, in the ChPT the $\\eta $ meson gets a more net attractive contribution than the chiral model which is due to the absence of first range term in ChPT Lagrangian.", "In ChPTterms we have plotted the contributions of individual terms to the in-medium mass of $\\eta $ -meson calculated from the unified approach.", "The $\\eta $ -meson in-medium mass given by Eq.", "(REF ) in the ChPT+chiral model approach has two terms (i) $\\Sigma _{\\eta N}$ term and (ii) $\\kappa $ term.", "In this figure, we have shown the individual contribution of these terms with increasing nuclear density and observed a non-appreciable contribution with $\\Sigma _{\\eta N}$ but appreciable with $\\kappa $ term.", "This is because in the $\\eta $ -meson in-medium mass expression (see Eq.", "(REF )), the denominator has a positive contribution of the scalar densities and the increase in scalar density with number density increases the denominator hence the value of effective mass becomes more negative.", "Clearly, there is no first range term with the positive contribution as was observed in the previous chiral model calculations, and therefore in the present case we get substantial attractive mass-shift.", "The present observations can be compared with the $\\eta $ -meson effective mass calculated in the unified approach of ChPT and relativistic mean-field model of Ref.", "[10].", "In this article, authors also considered the effect of scattering length and at nuclear saturation density and $a^{\\eta N}$ =1.02 fm they anticipated the effective mass to be 0.84 $m_{\\eta }$ whereas we observed it to be 0.79 $m_{\\eta }$ .", "At nuclear saturation density, the effective mass equal to 0.95 $m_{\\eta }$ was obtained within the coupled channel approach with scattering length $a^{\\eta N}$ $\\sim $ 0.25 fm [11].", "In this non-diagonal coupled channel approach, there are only leading order contributions and hence, only a small decrement in the in-medium mass is observed.", "Also, in the QMC model at $\\rho _0$ the in-medium mass of $\\eta $ -meson having value 0.88 $m_{\\eta }$ was observed [45].", "The obtained values are comparable with the calculations of the ChPT+chiral model for $a^{\\eta N}$ $\\sim $ 0.50 fm.", "In the cold symmetric nuclear matter, at $\\rho _N$ =$\\rho _0 (4\\rho _0)$ and $\\vert \\textbf {k} \\vert $ =0, we observe optical potential to be -116.83 (-232.28) MeV for $a^{\\eta N}$ =1.02 fm and in the cold isospin asymmetric nuclear matter the values modifies to -114.35 (-220.99) MeV.", "Using the ChPT+chiral model approach, we observed a even deeper optical potential than evaluated in the relativistic mean-field model+ChPT approach of Ref.", "[10].", "This is due to the difference in the in-medium scalar densities obtained in two models.", "In our approach, we have taken the effect of scalar and vector fields under the influence of isospin asymmetry, and finite temperature whereas in the relativistic model approach only cold symmetric medium was considered.", "The $\\eta $ optical potential was also observed in the different theoretical observations [11], [45], [9], [46], [10].", "$U_{\\eta }$ =-34 MeV was observed by studying the $\\eta N$ interactions near the threshold using free space chirally inspired coupled channel approach by considering the contributions of $N^*$ (1535) baryon resonance [9].", "Besides, the optical potential $U_{\\eta }$ =-54, -60, and -83 MeV was observed in the chiral unitary approach [46], QMC model [45], and the ChPT [10], respectively.", "Figure: (Color online) Comparison of in-medium η\\eta meson mass calculated from chiral model and ChPT.Figure: (Color online) Comparison of different terms of the effective mass of η\\eta -meson calculated using unification of ChPT and chiral model at a ηN =1.14a^{\\eta N}=1.14 fm." ], [ "SUMMARY", "We investigated the in-medium mass of $\\eta $ -meson in the asymmetric nucear matter at finite temperature.", "Under these medium conditions, we studied the behavior of the $\\eta $ -meson using two different methodologies.", "In the first methodology, using the chiral model alone we calculated the medium modified mass and optical potential of $\\eta $ -meson by considering the $\\eta N$ interactions up to second order in the Lagrangian and observe a decrease in the effective mass of $\\eta $ -meson as a function of density.", "We find the in-medium effects to be more appreciable in the high density regime.", "In the second, we used the unified approach of chiral perturbation theory (ChPT) and chiral SU(3) model to study the in-medium attributes of $\\eta $ -meson.", "In this approach, we took the next-to-leading order contributions.", "We incorporated the medium effects from the chiral SU(3) model through scalar density which is plugged in the $\\eta N$ equation of motion, which is calculated from the effective $\\eta N$ Lagrangian of ChPT.", "Using this methodology, we find a substantial decrease in the mass of $\\eta $ -meson as a function of nuclear density.", "The temperature and asymmetry effects are also studied and found to be slight repulsive in nature.", "Also, in the both approaches the mass-shift is observed to increase with an increase in the value of scattering length.", "The decrement on the $\\eta $ -meson mass leads to a negative mass-shift/optical potential which further suggests the possibility of $\\eta N$ bound states.", "The optical potential calculated in the present work will be used in future to calculate the spectroscopic state of the $\\eta $ -mesic nuclei [10].", "Also, the momentum dependent optical potential can be used to study the $\\eta $ -meson production rate [49], [50], [51] and its momentum dependence in the asymmetric nuclear medium [85], [84], [52]." ], [ "Acknowledgment", "One of the authors, (R.K) sincerely acknowledge the support towards this work from Ministry of Science and Human Resources Development (MHRD), Government of India via Institute fellowship under National Institute of Technology Jalandhar." ], [ "EXPLICIT REPRESENTATION OF DIFFERENT MATRICES", "Here, we give the matrix representation of meson, baryon and mass matrices which are used in present calculations [2].", "The Scalar Meson Matrix, $X$ : $X=\\frac{1}{\\sqrt{2}}\\sigma ^a \\lambda _a=\\left( \\begin{array}{ccc}(\\delta +\\sigma )/\\sqrt{2} & \\delta ^+ & \\kappa ^+\\\\\\delta ^- & (-\\delta +\\sigma )/\\sqrt{2} & \\kappa ^0 \\\\\\kappa ^- & \\overline{\\kappa ^0}& \\zeta \\end{array} \\right).$ The Pseudoscalar Meson Matrix, $P$ : $P=\\frac{1}{\\sqrt{2}}\\pi _a \\lambda ^a=\\left(\\begin{array}{ccc}\\frac{1}{\\sqrt{2}}\\left( \\pi ^0+{\\frac{\\eta }{\\sqrt{1+2\\,{w}^{2}}}}\\right)&\\pi ^{+}&2\\,{\\frac{K^{+}}{w+1}}\\\\{\\medskip }\\pi ^{-}&\\frac{1}{\\sqrt{2}}\\left(- \\pi ^0+{\\frac{\\eta }{\\sqrt{1+2\\,{w}^{2}}}}\\right)&2\\,{\\frac{ K^0}{w+1}}\\\\{\\medskip }2\\,{\\frac{K^-}{w+1}}&2\\,{\\frac{\\overline{K}^0}{w+1}}&-{\\frac{\\eta \\,\\sqrt{2}}{\\sqrt{1+2\\,{w}^{2}}}}\\end{array}\\right),$ where $w=\\sqrt{2}\\zeta _0/\\sigma _0$ .", "The $A_p$ Matrix: $A_p=\\frac{1}{\\sqrt{2}}\\left( \\begin{array}{ccc}m_{\\pi }^2 f_{\\pi }& 0& 0\\\\0 & m_\\pi ^2 f_\\pi & 0\\\\0 & 0& 2 m_K^2 f_K-m_{\\pi }^2 f_\\pi \\end{array} \\right).$ The Baryon Matrix, $B$ : $B=\\frac{1}{\\sqrt{2}}b^a \\lambda _a=\\left( \\begin{array}{ccc}\\frac{\\Sigma ^0}{\\sqrt{2}} +\\frac{\\Lambda ^0}{\\sqrt{6}}& \\Sigma ^+ & p\\\\\\Sigma ^- & -\\frac{\\Sigma ^0}{\\sqrt{2}} +\\frac{\\Lambda ^0}{\\sqrt{6}} & n \\\\\\Xi ^- & \\Xi ^0& -2 \\frac{\\Lambda ^0}{\\sqrt{6}}\\end{array} \\right).$" ] ]	2011.14072
[ [ "Weak harmonic labeling of graphs and multigraphs" ], [ "Abstract In this article we introduce the notion of weak harmonic labeling of a graph, a generalization of the concept of harmonic labeling defined recently by Benjamini et al.", "that allows extension to finite graphs and graphs with leaves.", "We present various families of examples and provide several constructions that extend a given weak harmonic labeling to larger graphs.", "In particular, we use finite weak models to produce new examples of (strong) harmonic labelings.", "As a main result, we provide a characterization of weakly labeled graphs in terms of harmonic subsets of the integers and use it to compute every such graphs of up to ten vertices.", "In particular, we characterize harmonically labeled graphs as defined by Benjamini et al.", "We further extend the definitions and main results to the case of multigraphs and total labelings." ], [ "Introduction", "The notion of harmonic labeling of an infinite (simple) graph was introduced recently by Benjamini, Cyr, Procaccia and Tessler in [1].", "If $G=(V,E)$ is an infinite graph of bounded degree then an harmonic labeling of $G$ is a bijective function $\\ell :V\\rightarrow \\mathbb {Z}$ such that $\\ell (v)=\\frac{1}{\\deg (v)}\\sum _{\\lbrace v,w\\rbrace \\in E}\\ell (w)$ for every $v\\in V$ .", "In [1] the authors provide some examples of harmonic labelings and prove the existence of such labelings for regular trees and the lattices $\\mathbb {Z}^d$ and the non-existence for cylinders $G\\times \\mathbb {Z}$ for non-trivial $G$ .", "Graph labeling is a widely developed topic and has a broad range of applications (see, e.g., [2], [3], [4]).", "Harmonically labellable graphs seem to have a rather restrictive configuration.", "Particularly, these graphs do not have leaves (pendant vertices) since there are no one to one functions verifying harmonicity on such vertices.", "Actually, this turns out to be the main obstacle for a generalization of this concept to the context of finite graphs, which is a natural extension taking into account the fruitful link between harmonic functions and geometric properties of finite graphs (see e.g.", "[5]).", "Furthermore, finite examples might be useful as local models to produce new harmonically labeled (infinite) graphs.", "In this paper we propose a two-way generalization of the notion of harmonic labeling, introducing the concept of weak harmonic labeling.", "On one hand, we require satisfying equation (REF ) only for $v\\in V\\setminus S$ , where $S$ is the set of leaves of $G$ .", "On the other hand, we let the function $\\ell $ be a bijection with an integer interval $I$ (finite or infinite).", "These conditions permit a straighforward extension of harmonic labelings to the finite setting.", "This results in a more general structure which provides a far wider theory, which was one the of the ambitions in [1].", "We present several examples of weak harmonic labelings and show the non-existence of this type of labelings for various families of (finite and infinite) graphs.", "We further introduce constructions to obtain new examples from given ones.", "In particular, we define the notion of inner cylinder and a way to extend any weakly labeled finite graph into an infinite one.", "We use weak finite models to construct new families of harmonic labelings.", "In particular, we exhibit a non-numerable collection of harmonically labeleled graphs, which additionally contains an infinite number of examples spanned by finite sets of vertices, thus answering a question raised in [1] (see Remark REF ).", "The main result of this article is the characterization of weakly labeled graphs in terms of certain families of collections of finite subsets of $\\mathbb {Z}$ called harmonic subsets.", "Since the statement of this result without many preliminary conventions would be too lengthy, the reader is invited to turn to Lemma REF and Theorem REF for a first impression.", "This characterization provides a way to compute all weakly labeled graphs, thing which we do for graphs of up to ten vertices (see Appendix).", "In particular, we obtain a characterization of harmonically labeled graphs, as defined in [1], in terms of the aforementioned harmonic subsets (Theorem REF ).", "All the definitions and results of weak harmonic labelings can be extended to the case of multigraphs (or total labelings) in a straightforward way.", "We prove the version for multigraphs of Theorem REF and exhibit an algorithm that produces a total weak harmonic labeling from a given admissible labeling (see Algorithm ).", "The paper is organized as follows.", "In Section 2 we introduce the concept of harmonic labeling and exhibit several examples of (families) of weakly labeled (finite and infinite) graphs.", "In Section 3 we present two constructions to obtain a new labelings from a given one and we use finite models of weakly labeled graphs to construct new families of harmonically labeled graphs.", "In Section 4 we prove the characterization of weakly labeled graphs (and, in particular, of harmonic labelings) in terms of families of collections of harmonic subsets of $\\mathbb {Z}$.", "In Section 5 we extend the definitions and main results of the theory to the case of multigraphs and total labelings.", "In the Appendix we have included the list of all possible weakly labeled graphs up to ten vertices." ], [ "Weak Harmonic Labelings of Simple Graphs", "All graphs considered are connected, have bounded degree and at least three vertices.", "For a simple graph $G$ we write $V_G$ for it set of vertices and $E_G$ for its set of edges.", "We put $v \\sim w$ if $v$ and $w$ are adjacent and we let $N_G(v)=\\lbrace v\\rbrace \\cup \\lbrace w\\,:\\, w\\sim v\\rbrace \\subset V_G$ denote the closed neighborhood of $v$ .", "Throughout, $S_G$ will denote the set of leaves (pendant vertices) of $G$ and $I$ will denote a generic integer interval (a set of consecutive integers).", "Remark 1 Note that, for any $G$ , $v\\sim w$ implies $\\lbrace v,w\\rbrace \\cap (V_G\\setminus S_G)\\ne \\emptyset $ .", "Definition 2 A weak harmonic labeling of a graph $G$ (simply weak labeling in this context) is a bijective function $\\ell : V_G\\rightarrow I$ such that $ \\ell (v)=\\frac{1}{deg(v)}\\sum _{w\\sim v}\\ell (w)\\hspace{14.45377pt}\\forall v \\in V_G\\setminus S_G.$ When we want to explicitate the interval of the labeling, we shall say weak harmonic labeling onto $I$.", "As mentioned earlier, the relativeness to $V_G\\setminus S_G$ of the harmonicity property is natural as there cannot be one to one functions with harmonic leaves.", "Harmonic labelings are particular cases of weak harmonic labelings since harmonically labellable infinite graphs have no leaves.", "More precisely, a weak harmonic labeling onto $I$ is an harmonic labeling if and only if $I=\\mathbb {Z}$ and $S_G=\\emptyset $ .", "Remark 3 Since a function $\\ell $ satisfies equation (REF ) if and only if $\\pm \\ell +k$ satifies it for any $k\\in \\mathbb {Z}$ , we shall not distinguish between labelings obtained from translations or invertions.", "Thus, we make the convention that in the case $I\\ne \\mathbb {Z}$ we shall normalize all labelings to the intervals $[0,\|V_G\|-1]=\\lbrace k\\in \\mathbb {Z}\\,:\\, 0\\le k\\le \|V_G\|-1\\rbrace $ or $[0,\\infty ]=\\lbrace k\\in \\mathbb {Z}\\,:\\, k\\ge 0\\rbrace $ .", "The simplest examples of weakly labeled finite graphs are the paths $P_n$ and the stars $K_{1,n}$ for even $n$ (Figure REF ).", "Paths can be extended either to $\\infty $ or to both $-\\infty $ and $\\infty $ to obtain a weak harmonic labeling onto $[0,\\infty ]$ or $\\mathbb {Z}$ respectively.", "In the latter, we obtain the trivial harmonically labeled graph $\\mathbb {Z}$ .", "We invite the reader to check the Appendix for a numerous (concrete) examples of weakly labeled finite graphs, where additionally it can be verified that a given graph can admit more than one weak harmonic labeling.", "Figure: Weak harmonic labeling on P n P_n and K 1,n K_{1,n}.Note that the minimum and maximum values of a weak harmonic labeling over a finite $G$ must take place on leaves, so any finite graph with less than two leaves does not admit weak harmonic labelings.", "This is the analogue result that nonconstant harmonic funtions have at least two poles (see e.g.", "[5]).", "In particular, cycles, complete graphs $K_n$ with $n\\ge 3$ , complete bipartite graphs $K_{n,m}$ with $n,m\\ge 2$ and cylinders $G\\times P_n$ for $n\\ge 2$ and any $G$ do not admit a weak harmonic labeling.", "It is not hard to characterize finite graphs with maximum and minimum number of leaves which admit this type of labeling.", "Lemma 4 Let $G$ be an $n$ -vertex graph which admits a weak harmonic labeling.", "$G$ has two leaves if and only if $G=P_{n}$ .", "$G$ has $n-1$ leaves if and only if $n$ is even and $G=K_{1,n}$ .", "We prove the direct of (1), which is the only non-trivial implication.", "Let $\\ell :V_G\\rightarrow I$ be a weak harmonic labeling of $G$ and denote $v_i$ be the vertex labeled $i$ .", "We may assume $n\\ge 4$ .", "By the previous remarks, $v_0$ and $v_n$ are the leaves of $G$ .", "Since the vertex $v_1\\notin S_G$ then $v_1\\sim v_0$ .", "Now $\\deg (v_1)=\\sum _{w \\sim v_1}\\ell (w)\\ge \\sum _{\\begin{array}{c}w \\sim v_1 \\\\ w\\ne v_0\\end{array}}2=2(\\deg (v_1)-1),$ from where $\\deg (v_1)= 2$ .", "Therefore, $N_{G}(v_1)=\\lbrace v_0,v_2\\rbrace $ .", "The same argument shows that $N_{G}(v_{n-1})=\\lbrace v_{n-2},v_n\\rbrace $ .", "Assume inductively that $N_{G}(v_i)=\\lbrace v_{i-1},v_{i+1}\\rbrace $ for $0<i<k<n-1$ .", "Then $\\deg (v_k)k=\\sum _{w \\sim v_k}\\ell (w)\\ge (k-1)+\\sum _{\\begin{array}{c}w \\sim v_k \\\\ w\\ne v_{k-1}\\end{array}}(k+1)=k-1+(k+1)(\\deg (v_k)-1),$ and $\\deg (v_k)\\le 2$ .", "This proves that $N_{G}(v_k)=\\lbrace v_{k-1},v_{k+1}\\rbrace $ and hence $G=P_n$ .", "More general families of weakly labeled finite graphs are shown in Figure REF .", "Note that $P_n$ and $K_{1,n}$ ($n$ even) are extremal cases of the collection pictured in Figure REF (top).", "The non-acyclic family in Figure REF (bottom), which can be inferred from the examples in the Appendix, can be trivially extended to labelings onto $[0,\\infty ]$ and $\\mathbb {Z}$ .", "In the latter, we obtain again an harmonic labeling.", "Furthermore, another such labeling for this graph can be produced by adding the edges $\\lbrace \\lbrace 2k-1,2k+1\\rbrace \\,\|\\, k\\in \\mathbb {Z}\\rbrace $ .", "These two examples are different from all those present in [1], which evidences how new examples of harmonic labelings can be deduced from finite weakly labeled ones.", "We shall present more examples obtained in this fashion in the next section.", "Remark 5 Recall that the Laplacian of a finite graph $G$ is the operator $L_G= D-A\\in \\mathbb {Z}^{n\\times n}$ where $A$ is the adjacency matrix of $G$ and $D$ is the diagonal degree matrix.", "If we let $\\tilde{L}_G$ denote the operator obtained from $L_G$ by removing the rows corresponding to leaves (the reduced Laplacian of $G$ ) then $G$ admits a weak harmonic labeling if and only if there exists a permutation $\\sigma \\in S_n$ such that $\\sigma (0,\\ldots ,n-1)\\in ker(\\tilde{L}_G)$ .", "Figure: Top: A collection of weakly labeled graphs for m,n,k∈ℤ ≥0 m,n,k\\in \\mathbb {Z}_{\\ge 0}, m≥1m \\ge 1, nn odd and 0≤k≤(n-1)m 20\\le k\\le \\frac{(n-1)m}{2}.", "The graphs P n P_n and K 1,n K_{1,n} (nn even) are extremal cases of this family for m=1m=1.", "Bottom: A family of weakly labeled finite graphs than can additionally be extended to weakly labeled graphs onto [0,∞][0,\\infty ] and ℤ\\mathbb {Z}." ], [ "Harmonic labelings from finite weak models", "More complex weakly labeled (finite and infinite) graphs can be built up from simpler finite examples.", "Some of these graphs can be inferred from the structure of the finite model and some can be constructed by performing unions and considering cylinders on them.", "In many cases, we shall obtain (new) harmonically labeled graphs." ], [ "Coalescence and Inner Cylinders", "We first show two constructions that produce new weakly labeled (finite and infinite) graphs from a finite weak model.", "Particularly, these constructions provide a way to produce infinitely many weak harmonic labelings onto $[0,\\infty ]$ and $\\mathbb {Z}$ .", "For simple graphs $G,H$ and $v\\in V_G$ and $w\\in V_H$ we let $G\\cdot _v^w H$ denote the graph obtained from $G\\cup H$ by identifying the vertex $v$ with the vertex $w$ (this is sometimes referred by some authors as the coalescence between $G$ and $H$ at vertices $v$ and $w$ ).", "Lemma 6 Let $\\ell _G:V_G\\rightarrow [0,n-1]$ and $\\ell _H:V_H\\rightarrow I$ , $I=[0,m-1]$ or $[0,\\infty ]$ be weak harmonic labelings on graphs $G$ and $H$ respectively.", "Let $v_i\\in V_G$ be the vertex labeled $i$ in $G$ ($0\\le i \\le n-1$ ) and $w_j\\in V_H$ be the vertex labeled $j$ in $H$ ($0\\le j \\le m-1$ ).", "If the sole vertex $v$ adjacent to $v_{n-1}$ in $G$ and the sole vertex $w$ adjacent to $w_0$ in $H$ satisfy $\\ell _G(v)+\\ell _H(w)=n-1$ then there exists a weak harmonic labeling of $G\\cdot _{v_{n-1}}^{w_0} H$ .", "The desired weak harmonic labeling $\\ell $ over $G\\cdot _{v_{n-1}}^{w_0} H$ is given $\\ell (u)={\\left\\lbrace \\begin{array}{ll}\\ell _G(u)&u\\in G\\\\ \\ell _H(u)+n-1&u\\in H.\\end{array}\\right.", "}$ The construction of Lemma REF can be iterated to produce infinitely many new examples (both finite and infinite).", "Furthermore, any weakly labeled graph can be extended to a new (finite or infinite) weakly labeled graph since the family of bipartite complete graphs $\\lbrace K_{1,n}\\,:\\,n\\text{ even}\\rbrace $ has a member of average $k$ for each $k\\in \\mathbb {N}$ .", "Figure REF shows a particular example of this situation.", "Figure: Extending weak harmonic labelings through coalescense.The other aforementioned construction, which produces exclusively weak harmonic labelings onto $\\mathbb {Z}$ , is based on the notion of inner cylinder of a graph.", "Definition 7 Given a graph $G$ , we define the inner cylinder of $G$ as the graph $G\\breve{\\times }\\mathbb {Z}$ such that: $V_{G\\breve{\\times }\\mathbb {Z}}=\\lbrace (v,i)\\,:\\, v\\in V_G, i\\in \\mathbb {Z}\\rbrace $ $(v,i)\\sim (w,j)$ if and only if ($i=j$ and $v\\sim w\\in G$ ) or ($v=w\\in V_G\\setminus S_G$ and $i=j+1$ or $i=j-1$ ).", "Interestingly, examples of weak harmonic labelings onto $\\mathbb {Z}$ can be produced from any finite example as the following lemma shows.", "Lemma 8 A weak harmonic labeling on a finite graph $G$ induces a weak harmonic labeling onto $\\mathbb {Z}$ on $G\\breve{\\times }\\mathbb {Z}$ .", "Write $\|V_G\|=n$ and let $\\ell :V_G\\rightarrow [0,n-1]$ be a weak harmonic labeling.", "Then, the claimed labeling $\\ell ^{\\prime }:V_{G\\breve{\\times }\\mathbb {Z}}\\rightarrow \\mathbb {Z}$ over $G\\breve{\\times }\\mathbb {Z}$ is given by $\\ell ^{\\prime }(v,k)=\\ell (v)+kn.$ Figure REF (Top) shows examples of weak harmonic labelings onto $\\mathbb {Z}$ defined using this construction.", "In some cases we can “complete\" these (weak) infinite examples to harmonic labelings.", "For instance, the weak harmonic labeling of $K_{1,2}\\breve{\\times }\\mathbb {Z}$ and $K_{1,4}\\breve{\\times }\\mathbb {Z}$ given in Lemma REF can be extended to an harmonic labeling as it is shown in Figure REF (Bottom).", "Figure: Top.", "The weak harmonic labeling induced in the inner cylinder of K 1,2 K_{1,2} (left) and K 1,4 K_{1,4} (right).", "Bottom.", "Harmonic labeling from the weak labeling of K 1,2 × ˘ℤK_{1,2}\\breve{\\times }\\mathbb {Z} (left) and K 1,4 × ˘ℤK_{1,4}\\breve{\\times }\\mathbb {Z} (right).", "The cyan colored edges represent added edges to the original weak labelings.The weakly labeled graph in Figure REF (bottom) is a particular case of the family portrayed in Figure REF , which we call $C^{k,h}$ .", "We note that this collection can too be extended to $[0,\\infty ]$ and $\\mathbb {Z}$ , and that this last extension produces an harmonically labeled graph, $C^{k,\\infty }$ .", "Formally, $V_{C^{k,\\infty }}=\\mathbb {Z}$ and $E_{C^{k,\\infty }}=\\lbrace \\lbrace a,b\\rbrace \\,:\\,b=a-1,a+1,a+k,a-k\\rbrace $ .", "This new example of harmonic labeling is indeed part of a far more general family.", "Note that for $b\\nsim a$ we can add the edges $(s+1)(b-a)+a\\sim s(b-a)+a$ for each $s\\in \\mathbb {Z}$ and obtain a new harmonically labeled graph (see Figure REF ).", "We can repeat this process to the newly generated example to obtain infinitely many new ones (a different for each edges selected for addition and each $k$ ).", "We make this construction precise next.", "Figure: The family C k,h C^{k,h} of non-acyclic weakly labeled graphs which generalizes the family of Figure (bottom).Figure: New harmonic labeling from C k,∞ C^{k,\\infty }.Let $\\mathcal {B}=\\lbrace (i,k)\\, : \\, k>1 \\text{ and } 0\\le i\\le k-1\\rbrace $ .", "For any (finite or infinite) subset $B$ of $\\mathcal {B}$ we form the graph $P_B$ obtained from (the harmonically labeled graph) $\\mathbb {Z}$ by adding the edges $\\lbrace (s+1)k+i,sk+i\\rbrace $ for every $s\\in \\mathbb {Z}$ for each $(i,k)\\in B$ .", "We call $B$ a base for $P_B$ and we write $P_B=\\langle x\\, : \\, x\\in B\\rangle $ (the elements of $B$ are the spanning edges of $P_B$ ).", "We picture a concrete example in Figure REF .", "Proposition 9 For any $B\\subset \\mathcal {B}$ , $P_B$ is an harmonically labeled graph.", "Furthermore, $P_B=P_{B^{\\prime }}$ if and only if $B=B^{\\prime }$ .", "First of all, we note that the set of edges added by different pairs $(i,k)$ and $(i^{\\prime },k^{\\prime })$ are disjoint.", "Indeed, the system ${\\left\\lbrace \\begin{array}{ll}sk+i=s^{\\prime }k^{\\prime }+i^{\\prime }\\\\(s+1)k+i=(s^{\\prime }+1)k^{\\prime }+i^{\\prime }\\end{array}\\right.", "}$ has unique solution $s=s^{\\prime }$ , $k=k^{\\prime }$ and $i=i^{\\prime }$ for $0\\le i,i^{\\prime }\\le k-1$ .", "So it suffices to show that if a vertex $v$ is harmonically labeled then adding the edges $\\lbrace (s+1)k+i,sk+i\\rbrace $ to a $P_{B^{\\prime }}$ corresponding to a single member $(i,k)\\in B\\setminus B^{\\prime }$ keeps $v$ harmonic.", "This is clear if the vertex $v$ is not incident to any of the added edges.", "Otherwise, $v$ has new adjacent vertices labeled $\\ell (v)-k$ and $\\ell (v)+k$ .", "Therefore $\\sum _{w\\sim v\\in P_{B^{\\prime }}}\\ell (w)+(\\ell (v)-k)+(\\ell (v)+k)=(\\deg (v)+2)\\ell (v),$ which proves the claim.", "Finally, by the previous remarks, every edge is exclusive of a given $(i,k)$ with $k\\ge 2$ and $0\\le i\\le k-1$ .", "Therefore, $P_B=P_{B^{\\prime }}$ if and only if $B=B^{\\prime }$ .", "Corollary 10 The collection $\\mathcal {P}=\\lbrace P_B\\, : \\, B\\subset \\mathcal {B}\\rbrace $ is a non-numerable family of harmonically labeled graphs.", "Figure: The harmonically labeled graph 〈(0,2),(1,3),(3,5)〉\\langle (0,2), (1,3), (3,5)\\rangle (in cyan, the spanning edges).Some of the previously presented examples actually belong to the collection $\\mathcal {P}$ .", "For example, $C^{k,\\infty }=\\langle (0,k) \\rangle $ and $K_{1,2}\\breve{\\times }\\mathbb {Z}=\\langle (1,3) \\rangle $ .", "However, $K_{1,4}\\breve{\\times }\\mathbb {Z}$ is not one of these graphs.", "Remark 11 A set $V^{\\prime }\\subset V_G$ is said to be a labeling spanning set if the values of a labeling $\\ell $ on the vertices of $V^{\\prime }$ completely determines the labeling of $G$ (by the harmonic property).", "In [1] the authors ask which connected graphs other than $\\mathbb {Z}$ admit an harmonic labeling spanned by a finite set.", "We claim that the members $\\langle (0,k) \\rangle $ of $P_{\\mathcal {B}}$ for any $k\\in \\mathbb {Z}$ are finitely spanned by vertices labeled 0 and 1.", "Indeed, these two labels trivially determine all labels from 0 to $k$ .", "The labels $x_{k+1},x_{k+2},\\dots ,x_{2k}$ pictured in Figure REF are solutions of the system $\\left(\\begin{matrix}1&0&0&\\ldots &0&0&1\\\\2&-1&0&\\dots &0&0&0\\\\-1&2&-1&\\ldots &0&0&0\\\\\\vdots &\\vdots &\\vdots &\\ldots &\\vdots &\\vdots &\\vdots \\\\0&0&0&\\ldots &-1&2&-1\\\\\\end{matrix}\\right)\\cdot \\left(\\begin{matrix}x_{k+1}\\\\x_{k+2}\\\\x_{k+3}\\\\\\vdots \\\\x_{2k}\\\\\\end{matrix}\\right)=\\left(\\begin{matrix}3k+1\\\\k\\\\0\\\\\\vdots \\\\0\\\\\\end{matrix}\\right)$ whose matrix is non-singular for every $k\\in \\mathbb {Z}$ .", "The claim is then settled by an inductive argument.", "Figure: The harmonic labeling of 〈(0,k)〉\\langle (0,k)\\rangle is finitely spanned by {0,1}\\lbrace 0,1\\rbrace for every k∈ℤk\\in \\mathbb {Z}." ], [ "A Characterization of weak harmonic labelings", "In this section we characterize weakly labeled graphs in terms of certain collection of sets of integers which we call harmonic subsets of $\\mathbb {Z}$.", "Definition 12 Given a non-empty finite subset $A\\subset \\mathbb {Z}$ we let $av(A)=\\frac{1}{\|A\|}\\sum \\limits _{k\\in A}k.$ Here $\|A\|$ denotes the cardinality of $A$ .", "We say that $A$ is an harmonic subset of $\\mathbb {Z}$ if $av(A)\\in A$ .", "Remark 13 Note that every unit subset of $\\mathbb {Z}$ is harmonic; we call them trivial harmonic subsets.", "Also, there are no two-element harmonic subsets of $\\mathbb {Z}$ .", "Therefore, any non-trivial harmonic subset of $\\mathbb {Z}$ has at least three elements.", "We shall show that certain collections of harmonic subsets of $\\mathbb {Z}$ characterize weakly labeled graphs.", "For this, we consider pairs $(G,\\ell )$ of a graph $G$ and a weak harmonic labeling $\\ell $ over $G$ .", "Define an isomorphism between two weakly labeled graphs $(G,\\ell ) $ and $(G^{\\prime },\\ell ^{\\prime })$ as a graph isomorphism $f:G\\rightarrow G^{\\prime }$ such that $\\ell ^{\\prime }(f(v))=\\ell (v)$ for every $v\\in V_G$ .", "We let $\\mathcal {G}$ denote the quotient set of pairs $(G,\\ell )$ under the isomorphism relation.", "Given $(G,\\ell )\\in \\mathcal {G}$ we consider the collection $\\mathcal {A}_{(G,\\ell )}=\\lbrace A_{v}\\,:\\, v\\in V_G\\setminus S_G\\rbrace $ where $A_{v}=\\lbrace \\ell (w)\\,:\\, w\\in N_G(v)\\rbrace $ .", "It is easy to see that $\\mathcal {A}_{(G,\\ell )}$ is a well-defined collection of non-trivial harmonic subsets of $\\mathbb {Z}$ such that $av(A_{v})=\\ell (v)$ .", "In particular, $A_v\\ne A_u$ if $v\\ne u$ .", "Also, this collection is finite if and only if $G$ is finite.", "Furthermore, the collection $\\mathcal {A}_{(G,\\ell )}$ satisfies the following conditions (whose easy verification are left to the reader).", "Lemma 14 Let $\\mathcal {A}$ be the collection $\\mathcal {A}_{(G,\\ell )}$ of harmonic subsets of $\\mathbb {Z}$ defined as above.", "For $A,B\\in \\mathcal {A}$ , we have: $\\bigcup _{C\\in \\mathcal {A}_{(G,\\ell )}} C$ is an integer interval.", "$av(A)\\ne av(B)$ if $A\\ne B$ If $t\\in A\\cap B$ , $A\\ne B$ , then there exists $C\\in \\mathcal {A}$ such that $av(C)=t$ .", "If $av(A)\\in B$ then $av(B)\\in A$ There exists a sequence $A_{i_1},\\ldots ,A_{i_r}\\subset \\mathcal {A}$ such that $A_{i_1}=A$ , $A_{i_r}=B$ and $av(A_{i_j})\\in A_{i_{j+1}}$ for $1\\le j\\le r-1$ (connectedness condition).", "Note that (P2) implies that the $t$ in (P3) is unique.", "Actually, (P2) is covered by requesting the unicity of $t$ in (P3).", "However, we state it in this form for computational reasons that will become evident later.", "On the other hand, (P5) is a direct consequence of the connectedness of $G$ .", "The main result of this section is that properties (P1) through (P5) of Lemma REF characterize weak harmonic labelings, in the sense that $(G,\\ell )\\mapsto \\mathcal {A}_{(G,\\ell )}$ is a bijection between $\\mathcal {G}$ and the class $\\mathcal {H}$ of collections of non-trivial harmonic subsets of $\\mathbb {Z}$ satisfying (P1) through (P5).", "Furthermore, if $\\mathcal {G}_I\\subset \\mathcal {G}$ is the subset of pairs $(G,\\ell )$ for which $\\ell $ is a weak harmonic labeling onto $I$ and $\\mathcal {H}_I\\subset \\mathcal {H}$ is the class of collections $\\mathcal {A}$ for which $\\bigcup _{C\\in \\mathcal {A}} C=I$ then the bijection takes $\\mathcal {G}_I$ onto $\\mathcal {H}_I$ .", "Note that the map $(G,\\ell ) \\mapsto \\mathcal {A}_{(G,\\ell )}$ sends elements of $\\mathcal {G}_I$ to $\\mathcal {H}_I$ by Remark REF .", "We next build the inverse map $\\mathcal {H}_I\\rightarrow \\mathcal {G}_I$ .", "Let $\\mathcal {A}=\\lbrace A_i\\rbrace _{i\\in J}\\in \\mathcal {H}_I$ .", "We define the associated graph $G_{\\mathcal {A}}$ as follows: $V_{G_{\\mathcal {A}}}=I$ $i\\sim j \\Leftrightarrow (\\exists \\, t / i=av(A_t)\\text{ and }j\\in A_t)\\text{ or }(\\exists \\, t / j=av(A_t)\\text{ and }i\\in A_t$ ).", "Furthermore, we define a vertex labeling $\\ell _{\\mathcal {A}}:V_{G_{\\mathcal {A}}}\\rightarrow I$ by $\\ell _{\\mathcal {A}}(i)=i$ .", "Lemmas REF and REF and Corollary REF below prove that $(G_{\\mathcal {A}},\\ell _{\\mathcal {A}})\\in \\mathcal {G}_I$ .", "Lemma 15 With the notations as above, $G_{\\mathcal {A}}$ is connected.", "Let $p,q\\in I$ .", "By (P1) there exists $A^{\\prime }_{p},A^{\\prime }_{q}\\in \\mathcal {A}$ such that $p\\in A^{\\prime }_{p}$ and $q\\in A^{\\prime }_{q}$ .", "Note that either $p=av(A_{i_p})$ or $p\\sim av(A_{i_p})$ (and analogously with $q$ ).", "By (P5) there exists a sequence $A^{\\prime }_{p}=A_{i_1},A_{i_2},\\ldots ,A_{i_r}=A^{\\prime }_{q}$ such that $av(A_{i_j})\\in A_{i_{j+1}}$ for every $1\\le j\\le r$ .", "In particular, $av(A_{i_j})\\sim av(A_{i_{j+1}})$ for every $1\\le j\\le r$ .", "Hence, the walk $p, av(A_{i_1}),av(A_{i_2}),\\ldots ,av(A_{i_r}),q$ connects $p$ with $q$ .", "Lemma 16 With the notations as above, $i\\in V_{G_{\\mathcal {A}}}\\setminus S_{G_{\\mathcal {A}}}$ if and only if $\\exists \\, t\\in J$ such that $i=av(A_t)$ .", "Furthermore, this $t$ is unique and $N_{G_{\\mathcal {A}}}(i)=A_{t}$ .", "If $i\\in V_{G_{\\mathcal {A}}}\\setminus S_{G_{\\mathcal {A}}}$ then there exist $j_1\\ne j_2$ such that $j_1,j_2\\in N_{G_{\\mathcal {A}}}(i)$ .", "If $i\\ne av(A_t)$ for every $t$ then $\\exists \\, t_1, t_2$ such that $j_1=av(A_{t_1})$ , $j_2=av(A_{t_2})$ and $i\\in A_{t_1}\\cap A_{t_2}$ .", "But then (P3) implies the existence of $t$ such that $i=av(A_t)$ , contradicting our assumption.", "Suppose now that $i=av(A_t)$ for some $t$ .", "In particular, $i\\in A_{t}$ (because $A_t$ is an harmonic subset).", "By Remark REF , there exist $j_1,j_2\\in A_{t}$ non-equal such that $j_1,j_2\\ne av(A_t)$ .", "Thus $j_1,j_2\\in N_{G_{\\mathcal {A}}}(i)$ by the definition of adjacency in $G_{\\mathcal {A}}$ and $i\\in V_{G_{\\mathcal {A}}}\\setminus S_{G_{\\mathcal {A}}}$ .", "The uniqueness of $t$ is a direct consequence of (P2).", "Now, if $j\\in N_{G_{\\mathcal {A}}}(i)$ then either ($\\exists \\, s / i=av(A_s)\\text{ and }j\\in A_s)\\text{ or }(\\exists \\, s / j=av(A_s)\\text{ and }i\\in A_s$ ).", "In the first case $s=t$ by unicity.", "In the latter case, (P4) implies that $j=av(A_s)\\in A_{t}$ .", "In any case $j\\in A_{t}$ , which proves $N_{G_{\\mathcal {A}}}(i)\\subset A_{t}$ .", "Now, if $j\\in A_{t}$ then $i\\sim j$ by the definition of adjacency of $G_{\\mathcal {A}}$ .", "Hence, $j\\in N_{G_{\\mathcal {A}}}(i)$ .", "Corollary 17 With the notations as above, $\\ell _{\\mathcal {A}}$ is a weak harmonic labeling over $G_{\\mathcal {A}}$ .", "If $i\\in V_{\\mathcal {A}}\\setminus S_{\\mathcal {A}}$ , let $t$ be such that $i=av(A_t)$ .", "Then $\\ell _{\\mathcal {A}}(i)=i=av(A_t)=\\frac{1}{\|A_{t}\|}\\sum _{k\\in A_{t}} k=\\frac{1}{\|N_{G_{\\mathcal {A}}}(i)\|}\\sum _{k\\in N_{G_{\\mathcal {A}}}(i)} k=\\frac{1}{\\deg (i)+1}\\sum _{\\begin{array}{c}k\\sim i \\\\ k=i\\end{array}} \\ell _{\\mathcal {A}}(k).$ Theorem 18 The maps $(G,\\ell )\\mapsto \\mathcal {A}_{(G,\\ell )}$ and $\\mathcal {A}\\mapsto (G_{\\mathcal {A}},\\ell _{\\mathcal {A}})$ are mutually inverse.", "Define the function $f:(G,\\ell )\\rightarrow (G_{\\mathcal {A}_{(G,\\ell )}},\\ell _{\\mathcal {A}_{(G,\\ell )}})$ as $f(v)=\\ell (v)$ .", "We will show that $f$ is a graph isomorphism between $G$ and $G_{\\mathcal {A}_{(G,\\ell )}}$ and that $\\ell _{\\mathcal {A}_{(G,\\ell )}}(f(v))=\\ell (v)$ .", "Since $\\ell $ is a weak harmonic labeling then $f$ is a bijection between $V_G$ and $I$ , so it suffices to show that $v\\sim w$ if and only if $f(v)\\sim f(w)$ .", "Now, if $v\\sim w$ then either $v$ or $w$ must belong to the set of non-leaves of $G$ (Remark REF ).", "Assume $v\\in V_G\\setminus S_G$ .", "Then, by definition of $\\mathcal {A}_{(G,\\ell )}$ it exists $A_v$ with $av(A_v)=\\ell (v)$ .", "Also, since $v\\sim w$ then $w\\in N_G(v)$ and hence $\\ell (w)\\in A_v$ .", "Therefore $\\ell (v)\\sim \\ell (w)$ ; that is, $f(v)\\sim f(w)$ .", "Now, suppose $f(v)\\sim f(w)$ .", "Then $\\ell (v)\\sim \\ell (w)$ in $G_{\\mathcal {A}_{(G,\\ell )}}$ .", "Then, either ($\\exists u\\in V_G\\setminus S_G$ such that $ \\ell (v)=av(A_{u})$ and $\\ell (w)\\in A_{u}$ ) or ($\\exists x\\in V_G\\setminus S_G$ such that $\\ell (w)=av(A_{x})$ and $\\ell (v)\\in A_{x}$ ).", "Without loss of generality we may assume the first case happens.", "Since $\\ell $ is a bijection then $w$ must belong to $N_G(v)$ .", "Hence $w\\sim v$ .", "This proves that $G$ is isomorphic to $G_{\\mathcal {A}_{(G,\\ell )}}$ .", "Finally, from the definition of $\\ell _{\\mathcal {A}_{(G,\\ell )}}$ : $\\ell _{\\mathcal {A}_{(G,\\ell )}}(f(v))=f(v)=\\ell (v),$ which finishes proving that $(G,\\ell )\\mapsto \\mathcal {A}_{(G,\\ell )}\\mapsto (G_{\\mathcal {A}_{(G,\\ell )}},\\ell _{\\mathcal {A}_{(G,\\ell )}})$ is the identity.", "We now prove that $\\mathcal {A}\\mapsto (G_{\\mathcal {A}},\\ell _{\\mathcal {A}})\\mapsto \\mathcal {A}_{(G_{\\mathcal {A}},\\ell _{\\mathcal {A}})}$ is the identity.", "Define $g:\\mathcal {A}\\rightarrow \\mathcal {A}_{(G_{\\mathcal {A}},\\ell _{\\mathcal {A}})}$ as follows: $g(A_t)=\\tilde{A}_i$ where $i\\in V_{G_{\\mathcal {A}}}\\setminus S_{G_{\\mathcal {A}}}$ is such that $i=av(A_t)$ (Lemma REF ).", "Note that $g$ is one to one by (P2) and the fact that $i\\ne j$ implies $\\tilde{A}_i\\ne \\tilde{A}_j$ in $\\mathcal {A}_{(G_{\\mathcal {A}},\\ell _{\\mathcal {A}})}$ (see properties of $\\mathcal {A}_{(G,\\ell )}$ before Lemma REF ).", "Also, Lemma REF implies that $g$ is onto.", "Since $\\ell _{\\mathcal {A}}(s)=s$ and $A_t=N_{G_{\\mathcal {A}}}(i)$ (again by Lemma REF ) then $\\tilde{A}_i=\\lbrace \\ell _{\\mathcal {A}}(s)\\,:\\, s\\in N_{G_{\\mathcal {A}}}(i)\\rbrace =N_{G_{\\mathcal {A}}}(i)=A_t$ .", "Theorem REF provides a concrete way to compute weak harmonic labelings of finite graphs.", "A list of all possible weakly labeled graphs up to ten vertices can be found in the Appendix." ], [ "On non-connected graphs.", "In [1], harmonic labelings are defined for general graphs (not necessarily connected ones).", "However, the non-connected case gives rise to many superfluous examples, as the following construction shows.", "Given a graph $G$ and an harmonic labeling $\\ell :V_G\\rightarrow \\mathbb {Z}$ , let $H=\\bigvee _{1\\le i\\le k} G_i$ be the disjoint union of $k\\in \\mathbb {Z}$ copies of $G$ .", "Then, we can define an harmonic labeling $\\ell _H$ over $H$ as follows: $\\ell _H(v)=k\\ell (v)+i-1\\text{, if $v\\in V_{G_i}$}.$ The definitions and results for the connected case can be extended to the non-connected case in a straightforward manner as long as every connected components of $G$ have at least three vertices.", "The case for connected components with less than three vertices give rise to uninteresting examples as these components are “invisible\" to the requirement of harmonicity and can be used to complete partial one to one labelings.", "Even with these requirements, harmonically labeled non-connected graphs are in great amount uninteresting examples, which arise from simply disconnecting connected cases (see Figure REF (Top)).", "The first non-trivial examples appear on 8-vertex graphs and are shown in Figure REF (Bottom).", "Figure: Top.", "Trivial examples of disconnected weakly labeled graphs.", "Bottom.", "Non-trivial examples of disconnected weakly labeled graphs.The same characterization given in Theorem REF also holds for non-connected graphs provided that the condition (P5) is dropped from Lemma REF .", "Actually, it is straightforward to see that $(G,\\ell )$ (resp.", "$G_{\\mathcal {A}}$ ) is connected if and only if $\\mathcal {A}_{(G,\\ell )}$ (resp.", "$\\mathcal {A}$ ) satisfies $(P5)$ .", "Particularly, if we let $\\tilde{\\mathcal {G}}_{\\mathbb {Z}}\\subset \\mathcal {G}_{\\mathbb {Z}}$ denote the set of pairs $(G,\\ell )$ for which $S_G=\\emptyset $ ($G$ not necessarily connected) then $\\tilde{\\mathcal {G}}_{\\mathbb {Z}}$ is the set of harmonically labeled graphs as defined in [1].", "From the above considerations, we obtain the following characterization of harmonic labelings.", "Theorem 19 A (non-necessarily connected) graph $G$ admits an harmonic labeling $\\ell $ if and only if $S_G=\\emptyset $ and $\\mathcal {A}_{(G,\\ell )}$ satisfies: For every $k\\in \\mathbb {Z}$ there exists $C\\in \\mathcal {A}_{(G,\\ell )}$ such that $av(C)=k$ .", "$av(A)\\ne av(B)$ if $A\\ne B$ .", "If $av(A)\\in B$ then $av(B)\\in A$ .", "The result follows from Theorem REF by noting that P1 transforms into ZP1 and that P3 is covered by ZP1." ], [ "Multigraphs and total labelings", "In this section we extend the main definitions and results of weak harmonic labelings to multigraphs and provide a generalization of Theorem REF in this context.", "All multigraphs are connected, loopless and have bounded degree (see Remark REF ).", "Also, since the identity of the edges is indifferent to the theory, we consider all parallel edges to be indistinguishable.", "Recall that a (finite) multiset $\\mathcal {M}$ is a pair $(A,m)$ where $A$ is a (finite) non-empty set and $m:A\\rightarrow \\mathbb {N}$ is a function giving the multiplicity of each element in $A$ (the number of instances of that element).", "The cardinality of $\\mathcal {M}$ is the number $\\vert \\mathcal {M}\\vert =\\sum _{x\\in A} m(x)$ .", "If $A=\\lbrace x_1,x_2,\\cdots x_n\\rbrace $ we shall often write $\\mathcal {M}=\\lbrace x_1^{m(x_1)},x_2^{m(x_2)},\\ldots , x_n^{m(x_n)} \\rbrace $ .", "If $m(x_i)=1$ we simply write $x_i$ .", "Given a multigraph $G$ we let $m_G(v,w)=m_G(w,v)\\in \\mathbb {Z}_{\\ge 0}$ denote the number of edges between vertices $v,w\\in V_G$ , $v\\ne w$ .", "If $m_G(v,w)\\ne 0$ then $v$ and $w$ are adjacent and we write $v\\sim w$ .", "If $m(v,w)=k\\ge 2$ we shall often write $v\\stackrel{k}{\\sim } w$ or $\\lbrace v,w\\rbrace ^k\\in G$ .", "A vertex $v\\in G$ is a leaf if $m_G(v,w)\\ne 0$ for exactly one $w\\ne v$ .", "As in the simple case, we shall denote $S_G$ the set of leaves of the multigraph $G$ .", "The simplification of a multigraph $G$ is the simple graph $sG$ where $V_{sG}=V_G$ and $\\lbrace u,v\\rbrace \\in E_{sG}$ if and only if $m_G(v,w)\\ne 0$ ($v\\ne w$ ).", "We shall call the closed multi neighborhood of $v\\in V_G$ in a multigraph $G$ to the multiset $\\mathcal {N}_G(v)=\\lbrace v\\rbrace \\cup \\lbrace w^{m_G(v,w)}\\,:\\, v\\sim w\\rbrace $ .", "Thus, the closed multi neighborhood of $v$ keeps track of the multiplicities of the vertices adjacent to $v$ as well.", "The (standard) close neighborhood of $v$ is $N_{sG}(v)\\subset V_G.$ Definition 20 A weak harmonic labeling of a multigraph $G$ is a bijective function $\\ell : V_G\\rightarrow I$ such that $\\ell (v)=\\frac{1}{deg(v)}\\sum _{w\\sim v}m_G(v,w)\\phi (w)\\hspace{14.45377pt}\\forall v \\in V_G\\setminus S_G.$ Figure REF shows some examples of harmonic labelings of finite multigraphs.", "Note that the presence of at least two leaves is still a requirement for the existence of a weak harmonic labeling.", "Figure: Examples of weak harmonic labeling on multigraphsWe next show that Theorem REF can be generalized to multigraphs.", "Definition 21 For a multiset $\\mathcal {M}=(A,m)$ with finite non-empty $A\\subset \\mathbb {Z}$ we let $av(\\mathcal {M})=\\frac{1}{\|\\mathcal {M}\|}\\sum \\limits _{k\\in A}m(k)k.$ We say that $\\mathcal {M}$ is an harmonic multiset of $\\mathbb {Z}$ if $av(\\mathcal {M})\\in A$ .", "Remark 22 As for harmonic subsets, the multisets whose underlying set is a unit set of $\\mathbb {Z}$ are (trivial) harmonic multisets of $\\mathbb {Z}$ .", "Also, there are no harmonic multisets of $\\mathbb {Z}$ whose underlying set has two elements.", "Therefore, any non-trivial harmonic multiset of $\\mathbb {Z}$ has an underlying set of at least three elements.", "Analgously to the simple case, we consider pairs $(G,\\ell )$ for a multigraph $G$ and a weak harmonic labeling $\\ell :V_G\\rightarrow I$ and define an isomorphism between two weakly labeled multigraphs $(G,\\ell )$ and $(G^{\\prime },\\ell ^{\\prime })$ as a multigraph isomorphism $f:G\\rightarrow G^{\\prime }$ such that $\\ell (f(v))=\\ell (v)$ for every $v\\in V_G$ .", "We let $\\mathcal {MG}_I$ denote the quotient set of pairs $(G,\\ell )$ , $\\ell :V_G\\rightarrow I$ , under the isomorphism relation.", "Given $(G,\\ell )\\in \\mathcal {MG}_I$ we consider the collection $\\mathcal {MA}_{(G,\\ell )}=\\lbrace \\mathcal {B}_{v}\\,:\\, v\\in V_G\\setminus S_G\\rbrace $ where $\\mathcal {B}_{v}=\\lbrace \\ell (v)\\rbrace \\cup \\lbrace \\ell (w)^{m_G(v,w)}\\,:\\, w\\sim v\\rbrace $ .", "As in the simple case, it is easy to see that $\\mathcal {MA}_{(G,\\ell )}$ is a collection of non-trivial harmonic multisets of $\\mathbb {Z}$ verifying $av(\\mathcal {B}_{v})=\\ell (v)$ that satisfies the (analoguous) conditions than Lemma REF .", "Namely, if $A_{\\mathcal {M}}$ stands for the underlying set of the multiset $\\mathcal {M}$ : Lemma 23 Let $\\mathcal {MA}$ be the collection $\\mathcal {MA}_{(G,\\ell )}$ of harmonic multisets of $\\mathbb {Z}$ defined as above.", "For $\\mathcal {B},\\mathcal {C}\\in \\mathcal {MA}$ , we have: $\\bigcup _{\\mathcal {D}\\in \\mathcal {MA}} A_{\\mathcal {D}}=I$ .", "$av(\\mathcal {B})\\ne av(\\mathcal {C})$ if $\\mathcal {B}\\ne \\mathcal {C}$ .", "If $t\\in A_{\\mathcal {B}}\\cap A_{\\mathcal {C}}$ then there exists $\\mathcal {D}\\in \\mathcal {MA}$ such that $av(\\mathcal {D})=t$ .", "If $av(\\mathcal {B})^k\\in \\mathcal {C}$ then $av(\\mathcal {C})^k\\in \\mathcal {B}$ .", "There exists a sequence $\\mathcal {B}_{i_1},\\ldots ,\\mathcal {B}_{i_r}\\subset \\mathcal {MA}$ such that $\\mathcal {B}_{i_1}=\\mathcal {B}$ , $\\mathcal {B}_{i_r}=\\mathcal {C}$ and $av(\\mathcal {B}_{i_j})\\in \\mathcal {B}_{i_{j+1}}$ for $1\\le j\\le r-1$ (connectedness condition).", "We let $\\mathcal {MH}_I$ stand for the class of collections of non-trivial harmonic multisets of $\\mathbb {Z}$ with $\\bigcup _{\\mathcal {D}\\in \\mathcal {MA}} A_{\\mathcal {D}}=I$ satisfying (MP1) through (MP5) of Lemma REF .", "With the analogous constructions as in the simple case it can be shown that there is a bijection $\\mathcal {MG}_I\\equiv \\mathcal {MH}_I$ .", "Namely, for $\\mathcal {MA}=\\lbrace \\mathcal {B}_i\\rbrace _{i\\in J}\\in \\mathcal {MH}_I$ define the associated multigraph $G_{\\mathcal {MA}}$ as: $V_{\\mathcal {MA}}=I$ $i\\stackrel{k}{\\sim }j\\in G_{\\mathcal {MA}} \\Leftrightarrow (\\exists \\, t / i=av(\\mathcal {B}_t)\\text{ and }j^k\\in \\mathcal {B}_t)\\text{ or }(\\exists \\, t / j=av(\\mathcal {B}_t)\\text{ and }i^k\\in \\mathcal {B}_t) $ Note that, by (MP4), this multigraph is well-defined.", "Finally, we define a vertex labeling $\\ell _{\\mathcal {MA}}$ over $G_{\\mathcal {MA}}$ by $\\ell _{\\mathcal {MA}}(i)=i$ .", "Identical arguments as in the proofs of Lemmas REF and REF , Corollary REF and Theorem REF go through to prove the following analogous results for multigraphs.", "Lemma 24 With the notations as above, $G_{\\mathcal {MA}}$ is connected.", "$i\\in V_{G_{\\mathcal {MA}}}\\setminus S_{G_{\\mathcal {MA}}}$ if and only if $\\exists \\,t\\in J$ such that $i=av(\\mathcal {B}_t)$ .", "Furthermore, this $t$ is unique and $\\mathcal {N}_{G_{\\mathcal {MA}}}(i)=\\mathcal {B}_t$ .", "In particular, $j\\in A_{\\mathcal {B}_t}$ if and only if $j=i$ or $j\\sim i$ in $G_{\\mathcal {MA}}$ .", "$\\ell _{\\mathcal {MA}}$ is a weak harmonic labeling over $G_{\\mathcal {MA}}$ .", "Theorem 25 The maps $(G,\\ell )\\rightarrow \\mathcal {MA}_{(G,\\ell )}$ and $\\mathcal {MA}\\rightarrow (G_{\\mathcal {MA}},\\ell _{\\mathcal {MA}})$ are mutually inverse.", "Remark 26 All the results of this section can be extended in a straighforward manner to multigraphs with loops.", "This is consequence of the fact that a multiset $\\lbrace x_1^{m_1},x_2^{m_{2}},\\dots , x_k,\\ldots , x_n^{m_{n}}\\rbrace $ is a harmonic with average $x_k$ if and only if $\\lbrace x_1^{m_1},x_2^{m_{2}},\\dots , x_k^{m},\\ldots , x_n^{m_{n}}\\rbrace $ is harmonic with average $x_k$ for all $k>0$ ." ], [ "Total weak harmonic labelings", "Since a weak harmonic labeling over a multigraph $G$ is trivially equivalent to a total labeling over $sG$ we can state the theory in terms of total labelings.", "Definition 27 If $G$ is a simple graph, then we call a total weak harmonic labeling of $G$ onto $I$ to a function $\\ell : V_G\\cup E_G\\rightarrow \\mathbb {Z}$ such that $\\ell \|_{V_G}$ is a bijection with $I$ and $\\ell (v)=\\frac{1}{deg(v)}\\sum _{w\\sim v}\\ell (\\lbrace v,w\\rbrace )\\ell (w)\\hspace{14.45377pt}\\forall v \\in V_G\\setminus S_G.$ Note that total weak harmonic labelings have no restriction on the edges.", "Now, given a weak harmonic labeling $\\ell :V\\rightarrow I$ over a multigraph $G$ we have the associated total weak harmonic labeling $\\ell ^: V_{sG}\\cup E_{sG}\\rightarrow \\mathbb {Z}$ over $sG$ defined as ${\\left\\lbrace \\begin{array}{ll}\\ell ^(v)=\\ell (v)& v\\in V_{sG}\\\\ \\ell ^*(\\lbrace v,w\\rbrace )=m_G(v,w)&\\lbrace u,v\\rbrace \\in E_{sG}.\\end{array}\\right.", "}$ Conversely, given a total weak harmonic labeling $\\ell : V_G\\cup E_G\\rightarrow \\mathbb {Z}$ over a simple graph $G$ then we can define a weak harmonic labeling over the multigraph $G_{\\ell }$ where $V_{G_{\\ell }}=V_G$ and $m_{G_{\\ell }}(v,w)=\\ell (\\lbrace v,w\\rbrace )$ .", "View in this fashion, weak harmonic labelings of simple graphs are a particular case of total weak harmonic labelings of simple graphs.", "Total weak harmonicity is naturally much less restrictive than weak harmonicity.", "Any finite simple graph $G$ which admits a weak harmonic labeling in particular admits a bijective vertex-labeling $\\ell :V_G\\rightarrow [0,n-1]$ such that $\\min _{w\\in N_v(G)}\\lbrace \\ell (w)\\rbrace <\\ell (v)<\\max _{u\\in N_v(G)}\\lbrace \\ell (u)\\rbrace $ for every $v\\in V_G\\setminus S_G$ .", "Algorithm produces a total weak harmonic labeling from any labeling $\\phi $ fulfilling (REF ) on a finite simple graph $G$ .", "It makes use of the following Remark 28 If $\\mathcal {M}=(A,m)$ is a finite multiset and $x\\in \\mathcal {M}$ is neither the maximum or minimum of $\\mathcal {M}$ then we can correct the multiplicities of the elements of $\\mathcal {M}$ so $av(\\mathcal {M})=x$ .", "Indeed, if $x>av(\\mathcal {M})$ then letting $s=\\min _{y\\in \\mathcal {M}}\\lbrace y\\rbrace $ and $m^{\\prime }(y)={\\left\\lbrace \\begin{array}{ll}m(y)\\cdot m(s)\\cdot (x-s)& y\\ne s, x \\\\m(s)\\cdot \|\\sum \\limits _{z\\ne s}(x-z)m(z)\|&y=s\\end{array}\\right.", "}$ we readily see that $\\mathcal {M}^{\\prime }=(A,m^{\\prime })$ is an harmonic multiset of $\\mathbb {Z}$ .", "The case $x<av(\\mathcal {M})$ is analogous.", "Additionally, note that multiplying the multiplicities of every element in an harmonic multiset of $\\mathbb {Z}$ by a fixed positive integer does not alter its harmonicity.", "Total weak harmonic labeling [1] $\\phi :V_G\\rightarrow [0,n-1]$ with property (REF ) $\\ell $ a total harmonic labeling on $G$ totalLabelingFrom$\\phi $ Order $V_G\\setminus S_G=\\lbrace v_1,\\ldots ,v_t\\rbrace $ such that $\\phi (v_i)<\\phi (v_j)$ if $i<j$ .", "$\\mathcal {B}_{i} \\leftarrow \\lbrace \\phi (w)\\,\|\\, w\\in N_G(v_i)\\rbrace $ ($1\\le i\\le t$ ).", "$1\\le i\\le t$ $av(\\mathcal {B}_i)\\ne \\phi (v_i)$ [t]@X@ Harmonize $\\mathcal {B}_i$ by conveniently altering the multiplicity of the elements different from $\\phi (v_i)$ (see Remark REF ).", "$\\phi (v_j)\\in \\mathcal {B}_i$ [t]@X@ For $1\\le j<i$ : Correct the multiplicities of the elements of $\\mathcal {B}_j$ so the multiplicity of $\\phi (v_i)\\in \\mathcal {B}_j$ coincides with that of $\\phi (v_j)\\in \\mathcal {B}_i$ [t]@X@ For $i< j\\le t$ : Correct the multiplicity of $\\phi (v_i)\\in \\mathcal {B}_j$ so it coincides with that of $\\phi (v_j)\\in \\mathcal {B}_i$ $\\ell (v)\\leftarrow \\phi (v)$ for every $v\\in V_G$ $\\ell (\\lbrace w,u\\rbrace )\\leftarrow $ multiplicity of $\\phi (u)$ in $\\mathcal {B}_{\\phi (w)}$ Figure REF shows examples of total weak harmonic labelings obtained from Algorithm to some complete graphs with two leaves added.", "Figure: Examples of total weak harmonic labelings obtained from Algorithm .", "The label of the edges appear in parenthesis (labels equal to 1 are omitted).All possible weakly labeled finite graphs up to ten vertices.", "Table: Every possible weakly labeled graph of up to seven vertices." ] ]	2011.14063
[ [ "Exoplanet Detection using Machine Learning" ], [ "Abstract We introduce a new machine learning based technique to detect exoplanets using the transit method.", "Machine learning and deep learning techniques have proven to be broadly applicable in various scientific research areas.", "We aim to exploit some of these methods to improve the conventional algorithm based approaches presently used in astrophysics to detect exoplanets.", "Using the time-series analysis library TSFresh to analyse light curves, we extracted 789 features from each curve, which capture the information about the characteristics of a light curve.", "We then used these features to train a gradient boosting classifier using the machine learning tool lightgbm.", "This approach was tested on simulated data, which showed that is more effective than the conventional box least squares fitting (BLS) method.", "We further found that our method produced comparable results to existing state-of-the-art deep learning models, while being much more computationally efficient and without needing folded and secondary views of the light curves.", "For Kepler data, the method is able to predict a planet with an AUC of 0.948, so that 94.8 per cent of the true planet signals are ranked higher than non-planet signals.", "The resulting recall is 0.96, so that 96 per cent of real planets are classified as planets.", "For the Transiting Exoplanet Survey Satellite (TESS) data, we found our method can classify light curves with an accuracy of 0.98, and is able to identify planets with a recall of 0.82 at a precision of 0.63." ], [ "Introduction", "Since the discovery of the first exoplanets [57], [37], the field of planet detection has become a major research area in astrophysics.", "To date, over 4,000 exoplanets have been discovered with various techniques, such as the radial velocity method [8], [33], astrometry [5], [43], direct imaging [13], [53], and gravirational microlensing [36], [4].", "Yet, the majority of confirmed exoplanets have been discovered with the transit method [11], [44].", "Depending on the position of the observer, a planet may move in front of its host star blocking a part of the star's light, which causes the observed visual brightness of the star to drop by a small amount, depending on the relative sizes of the star and the planet.", "For the transit method, the brightness of a star is thus continually observed, whereupon `dips' in the obtained light curve are sought.", "One drawback of this method is that transits can only be observed when the planet's orbit is perfectly aligned from the location of the observer.", "Another issue is the high rate of false detections, e.g.", "through eclipsing binary systems or transits by planet sized stars, so that discoveries need to be confirmed with alternative methods.", "NASA's Kepler mission [6], and its second survey programme K2 [23], were a major step forward in the creation of a vast catalogue of exoplanet systems, which may give insights into the formation process of planets and the abundance of potentially habitable Earth-like analogues.", "Although Kepler was designed as a statistical mission to investigate the frequency of Earth-size exoplanets in or near habitable zones, most early results focused on individual systems [41], [3], resulting in rather homogeneous catalogues.", "Later studies shifted their focus towards the statistics of the exoplanet population [19], [7].", "However, the validation process of the candidates discovered in the Kepler mission is still an ongoing process, as ruling out false-positive detections is time-consuming and new candidates are still being discovered.", "In April 2018, NASA launched the Transiting Exoplanet Survey Satellite [49] as a successor of Kepler with the primary objective to survey some of the brightest stars.", "It covers 85 per cent of the sky – an area that is 400 times larger than that covered by Kepler – and produces around one million light curves per month.", "TESS observes stars that are $30-100$ times brighter than those selected by the Kepler mission.", "Consequently, it is possible to identify targets that are far easier to follow up for detailed observation with other space-based and ground-based telescopes.", "Some of the early results of TESS include the discovery of a Super-Earth in the Pi Mensae system [24], an ultra-short period planet orbiting a red dwarf [55], and an Earth-sized exoplanet in the habitable-zone [22].", "Figure: Phase-folded light curve of a K dwarf being orbited by a hot Jupiter candidate, observed by the Pan-Planets survey.", "The Box-fitting Least Squares (BLS) method has been used to fit the light curve.", "Shown at the top are the resulting parameters: period (days), transit duration q (in units of phase), transit v shape (0 corresponds to a box, 1 to a V), transit light drop, signal-to-noise (S/N) and number of transits/number of points in the transits.", "Binned data points are shown in red, while the green line shows the best-fit model.", "See for more details.The first step in analysing observed light curves is to search for periodic signals that may be consistent with transiting planets, so-called “threshold crossing events” (TCEs).", "These TCEs are then inspected to eliminate erroneously selected non-signals (e.g.", "instrumental noise or astrophysical variability), either manually by humans, or in an automated way [15].", "A typical phase-folded light curve from the Pan-Planets survey [46] is shown in Figure REF .", "One of the most commonly used methods to detect exoplanet in light curves is Box-fitting Least Squares [30], as it is very effective in detecting periodical signals that can be approximated by a two level system such as transits.", "In practice, a box model is fit to the light curve, which yields parameters such as the period, the transit duration, and the transit light drop.", "The green line in Figure REF shows the BLS fit to the light curve [45].", "Cases with a seemingly good fit can then be manually reviewed.", "However, BLS is limited in terms of signal-to-noise and data cadence, and is vulnerable to false-positive detections created by cosmic rays, random noise patterns, or stellar variability.", "Kepler, TESS, and other similar surveys still rely on manual analysis [60] for a good overview of this process).", "For a typical TESS sector, usually a group of experts manually eliminate obvious false positive cases, a process which alone can take a few days.", "From the remaining cases, each one has to be viewed by at least 3 experts.", "This kind of procedure can lead to disagreements on a particular case, as even experts might not maintain the same definition for the classification.", "A difference of opinion may arise in a team of experts on a particular case due to external factors like the way a case is presented, other TCEs viewed recently, or something as little as the time of day or their mood [15].", "For these reasons, systems are needed that can reliably and repeatably select the most important planet candidates for us, which can then be manually reviewed for confirmation at a later stage.", "Over the past years, there was a growing interest in building automatic vetting systems with machine learning methods, as they provide computers the ability to learn from data without being explicitly programmed, and are instead able to detect patterns automatically.", "Some of the notable earliest attempts include the Robovetter [16], the Autovetter project [38] and SIDRA [39], where tree-based models trained for vetting.", "However, the research moved away from classical machine learning methods recently, and deep learning methods became the new focus.", "Among the most notable work in this area was done by [51].", "They introduced a novel deep learning architecture, Astronet, which produced very accurate results for the Kepler data.", "Their approach and their model architecture was adapted and applied to data from several different surveys such as Kepler's K2 mission [17], next generation transit survey [12] and TESS data [60].", "Since the induction by [51], the community has moved on to deep learning, as deep learning methods tend to produce better results than classical machine learning approaches, especially for more complex problems or data types.", "However, deep learning models are usually computationally expensive and require large amounts of data to train properly.", "Often they are prone to overfitting the data if not regularised sufficiently.", "Moreover, it is usually difficult to understand, which input data (features) have been important to derive the results, which limits the physical interpretability.", "Therefore, in some cases, these models can be superfluous for the given problem and simpler or less computationally expensive approaches can perform equally well or even better, and can allow for a better understanding of the underlying problem.", "On the other hand, it is also important to sometimes move away from the currently best performing methods and explore new alternatives.", "With this motivation, in this paper we propose a new direction to approach the problem of finding exoplanets in light curve data using classical machine learning.", "This paper is organised in five sections.", "Section 2 contains details about our methodology, specifically data preparation, feature extraction and model training.", "Section 3 explains the results achieved on simulated, Kepler and TESS data.", "We also compare our results with some of the currently best performing models.", "In section 4, we compare the main differences how our model differs from the other deep learning-based methods being used in the area.", "We also provide details of our vetting tool which can be used to make real time inferences.", "Finally, we conclude and discuss future steps in section 5.", "Figure: Workflow of our method starting from raw light curves to performing inference on the sample.", "I the first step (i) the raw light curves are processed to remove low frequency variability and noise, and to sample the curve uniformly in time.", "In the second step (ii) features are extracted from the light curves and organised in a feature array.", "In the final third step (iii) the feature arrays are used as the input for a classification algorithm (gradient boosted trees) and the model is trained." ], [ "Methods ", "Machine Learning methods are widely used in scientific research areas to build classifiers i.e.", "an algorithm which separates data into two or multiple classes.", "In our case we are building a binary classifier which will separate each time-series photometry, a so-called light curve, into the classes `planet candidate' and `non-candidate'.", "The state-of-the-art machine learning method for planet detection presented by [51] utilises deep learning, a class of machine learning methods based on artificial neural networks.", "Our method on the other hand is based on classical machine learning.", "One essential difference between our approach and deep learning is that the latter is able to extract features automatically, while for classical machine learning the features have to be calculated beforehand and provided as input to the model.", "To this end, we use Time Series Feature extraction based on scalable hypothesis tests [14], a python-based library for feature extraction.", "The idea behind the method was inspired by methodology used in time-series prediction projects employing feature engineering libraries.", "TSFresh is also used in projects like machine fault prediction [31], diagnosing Alzheimer’s disease [40], identifying epileptic seizures in electroencephalography signals [59], earthquake prediction [29], and time series forecasting for business applications [2].", "Light curves are essentially a time series and so these tools can directly be used for our case.", "We trained and tested our model on three kinds of data sets.", "The first stage used simulated data with K2 photometry as a baseline with additional injected transits.", "We then trained it on Kepler and finally TESS photometry.", "Each stage is split into three parts: processing and labelling the training data; extracting features from each light curve using TSFresh; model training.", "This workflow is shown in Figure REF .", "More details on the individual steps are given in the following sections." ], [ "Simulated Data", "We obtained the K2 photometry from the Mikulski Archive for Space Telescopes (MAST) and used the calibration by [54].", "While the processing of [54] already removed the vast majority of systematic effects, we further cleaned up the data by identifying and removing remaining cosmic ray artefacts and creating a new noise model.", "If a point is an outlier with respect to its neighbours, i.e.", "if it is more than $5\\sigma $ above its previous and following point, it was assumed to be a cosmic ray artefact.", "Then we removed the stellar variability that is common, depending on stellar type, with an iterative process.", "We smoothed the data, binned it, fit cubic splines to the binned data, clipped negative $3\\sigma $ outliers and iterated this process until it converged.", "Transit signals were randomly injected in half of the processed light curves in an approach similar to that introduced by [46].", "We removed known planet systems from the data sample beforehand and then performed a blind search for eclipsing binary systems based on our implementation of the BLS method.", "As the K2 light curves sometimes had gaps, we interpolated between those gaps to sample the time series uniformly.", "Any light curve with a detected signal-to-noise ratio of more than 12 was removed from the sample.", "We then created a randomly filled space of orbital periods, stellar limb darkening and radii, planet radii and orbit inclinations.", "An example of the resulting simulated planet systems can be seen in Figure REF .", "Figure: Sample light curves containing transit signals scaled to arbitrary units.", "All of these curves are taken from our test set of the respective data set.", "It can be observed that the TESS (blue) and Kepler (green) data are a lot more noisy and likely to contain a lower number of transits.", "Simulated data (red) on the other hand is cleaner, as all the curves with S/N over 12 were removed from the data set before the planet signals were injected.", "However, some of the injected transits were generated with low S/N values to imitate the realistic scenarios.", "The second light curve of simulated data is one such case.Each light curve with injected planets was labelled as class `1' and remaining curves were categorised as class `0'.", "These labels were then used to train our classifier with the objective of identifying the injected transit signals.", "In total, our training set consisted of 7,873 light curves out of which 3,937 belong to class `1' (planet candidate).", "This data set was then divided into training-validation set for 10-fold cross validation (90 per cent) and test set (10 per cent)." ], [ "Kepler Data", "We used the publicly available data set that was employed in the work of [51]; the details about data processing can be found in section 3.2 of their paper.", "These light curves were produced by the Kepler pipeline [27], where each light curve contains around 70,000 data points equally spaced out at the interval of 29.4 minutes.", "The light curves were flattened and outliers were removed as described in the previous section.", "The labels for the curves were taken from the Autovetter Planet Candidate Catalog [10].", "The catalog is divided into four light curve categories: planet candidate (PC), astrophysical false positive (AFP), non-transiting phenomenon (NTP) and unknown (UNK).", "All UNK light curves were removed from the data set and all PC light curves were given class label 1 (`planet candidate').", "Both the remaining cases were assigned to class 0 (`non-candidate').", "In total, there are 3,600 PC light curves and 12,137 non-PC light curves.", "The data is already divided into training set (80 per cent), validation set (10 per cent), and test set (10 per cent).", "We used the same test set for our model performance as [51] in order to compare our model performance.", "Apart from this, we combined their training and validation set into a training-validation set for 10-fold cross validation." ], [ "TESS Data", "For the TESS data, we again used publicly available data provided by [60] for their model AstroNet-Vetting.", "Details about their light curve processing can be found in the section 2.1, where they used the MIT Quick Look Pipeline [25] for processing the light curves.", "The QLP is designed to process the full-frame images (FFI) and produces light curves using its internal calibrated images.", "The labelling was done by visually inspecting each light curve, categorising them into 3 classes: Planet Candidates (PC), Eclipsing Binary (EB), and Junk (J), where junk signals were the cases with stellar variability and instrumental noise.", "The data set consists of 2,154 EB, 13,805 J and 492 PC signals, where PCs were labelled as class 1 (`planet candidate') and everything else was labelled as class 0 (`non-candidate').", "These data are also already divided into training set (80 per cent), validation set (10 per cent) and test set (10 per cent).", "Like the previous case, we used the same test set and combined the training and validation set into a training-validation set for 10-fold cross validation.", "Some of the sample light curves containing transit signals are shown in figure REF .", "All of these curves are taken from our test set of the respective data set.", "It can be seen that the TESS (shown in blue) and Kepler (shown in green) data are a lot more noisy and likely to contain a lower number of transits.", "On the other hand, the simulated curves are cleaner as all the curves with S/N over 12 were removed from the data set before the planet signals were injected.", "However, some of the injected transits such as the second light curve of the simulated data were generated with low S/N values to imitate realistic scenarios." ], [ "Feature Extraction", "Having processed the light curves, the next step is to extract features, which capture information about the characteristics of each light curve and are used as the input for our classification model both for training and inference.", "We used the python framework TSFresh [14] to extract features that are typical for time series such as light curves, e.g.", "the absolute energy of the time series (the sum over the squared values), the number of values that are higher than the mean, and the coefficients of the one-dimensional discrete Fourier transform.", "For all used data sets, the light curves are resampled to the frequency of 1 hour, i.e.", "all light curves are reformatted into windows of 1 hour to ensure that data points are uniformly distributed in time, a process commonly referred to as `resampling'.", "Although this resampling process is not mandatory, it is a standard practice and proven to produce a better representation of the data, leading to faster convergence.", "In the resampling process, if multiple data points are inside one window, they are summed up and missing data points on the curves are interpolated.", "The frequency of one hour is chosen to ensure minimum data loss while not increasing the data size significantly.", "These resampled time series can now be directly used with multiple time-series analysis tools.", "We used TSFresh's efficient feature extraction setting which extracted around 790 generalised time series featuresDetails of these features can be found at: https://tsfresh.readthedocs.io/en/latest/text/list_of_features.html.", "After the feature extraction process, we implemented some standard data pre-processing steps for machine learning.", "We removed all irrelevant features, i.e.", "those with values that were constant throughout the data set, and imputed missing values by interpolating.", "Lastly, we scaled the whole data set using a robust scaler.", "This data set was then used for training a tree-based classifier." ], [ "Model Training", "Our model is a binary classifier, i.e.", "it classifies every light curve into two classes, either `planet candidate' or `non-candidate'.", "We employed a gradient boosted tree (GBT) model using the popular machine learning framework lightGBM [28] for our classifier.", "A GBT model is an ensemble of decision trees which are trained in sequence.", "In each iteration, the GBT model is trained to reproduce the results of the decision trees by fitting the negative gradients [20].", "Moreover, we used a 10-fold cross validation (CV) during training.", "This means the training set is split into 10 smaller sets and the model is trained using 9 sets at a time.", "The remaining last set is used to evaluate the model performance.", "This is done iteratively until the model has been evaluated on all 10 sets.", "The final performance is then the average of all the values computed in the iteration.", "We use the following four metrics to evaluate our results: Accuracy: the fraction of samples correctly classified, including both `planet candidates' and `non-candidates'.", "Precision: the fraction of predicted `planet candidates' that are true planets, i.e.", "the accuracy of the `planet candidate' predictions.", "Recall: ratio of the true planets correctly detected by the classifier as `planet candidate', i.e.", "what fraction of true planets is recovered (also called “true positive rate”).", "AUC: the Area Under the receiver operating characteristic (ROC) Curve, called the AUC, gives the probability that a randomly chosen light curve with a true planet is ranked higher than a randomly chosen light curve without a planet.", "The accuracy is generally not a proper metric to judge the performance of the planet detection algorithm, since most of the exoplanet detection data sets are unbalanced, i.e.", "usually there are significantly more light curves without any planet signal than cases with a planet.", "For instance, in the TESS data set only 3 per cent of the light curves are true planet candidates.", "If a classifier simply predicts `non-candidate' for all light curves it will still lead to a `high' accuracy of 97 per cent, even though it has not learned from the data and cannot identify any planet candidates.", "While a high precision ensures that most of the predicted `planet candidates' are true planet candidates, it is also not a very useful metric, as a trivial way to achieve a high precision is to make only few `planet candidate' predictions and ensure they are correct.", "Consequently, many possible planet candidates would be missed.", "As we would rather allow for a higher number of false positives than losing possible planet signals, the recall is a much more important metric to assess the performance of our algorithm.", "A model with a high recall may lead to a lower precision and vice versa, which is commonly known as the precision-recall trade-off.", "Classification decisions are based on a decision threshold, which is another hyperparameter of the model.", "If the predicted model probability surpasses this threshold, the light curve is classified as `planet-candidate'.", "For a typical classification problem a threshold of $p_\\mathrm {threshold}=0.5$ is chosen, so that light curves with a predicted probability of more than 0.5 are considered `planet candidates'.", "If a higher decision threshold is chosen, the model will have a higher precision but a lower recall, and vice-versa, so it can be optimized to yield a high recall.", "Finally, the AUC is independent of the decision threshold, as it gives the integral over all possible decision thresholds.", "Therefore, it can be used to optimise the remaining hyperparameters first to get a model that performs well for any threshold.", "In the second step, the threshold can then be chosen to achieve a high recall.", "We argue that out of the above four metrics AUC and recall are the most important as the main imperative is to design a method that is able to recover a high percentage of light curves containing planet signals.", "In other words, we aim to optimise our model for not missing light curves that contain a true planet signal.", "Due to these reasons, we first optimised all hyperparameters of our model to maximize the AUC, and then optimised the decision threshold to obtain a high recall while maintaining precision as high as possible.", "In practice, once the all hyperparameters except for the threshold are fixed, we check whether it is possible to get both precision and recall in the upper percentile ($> 90$ per cent), and if so select the highest possible recall while maintaining a precision in that percentile.", "If this is not possible, we allow the precision to drop to the next percentile and select the highest possible recall." ], [ "Simulated data", "In order to attain a proof of concept the method was first applied to our simulated data that was obtained as described in section REF and consists of 7,873 light curves.", "Extracting features for each of the light curves and using them as inputs for the GBTs, the model outputs the probability for a light curve to contain a transit.", "Given a decision threshold, the light curve is then either classified as `planet-candidate', or `non-candidate'.", "Figure REF shows the precision, recall, and the F1 score (the harmonic mean between precision and recall), as a function of the decision threshold.", "All curves have been computed with the model that was optimized for a high AUC as described above.", "Choosing a decision threshold of $p_\\mathrm {threshold}=0.13$ , we get a recall of 0.94 and precision of 0.92.", "This result is preferred over the one with the standard threshold of 0.5 as for planet detection, it is preferred to extract the maximum possible light curves at the cost of a minor increase in false positives.", "The results on our validation set are shown in the Table REF .", "Table: Results on simulated dataWe then used the BLS algorithm on our simulated data set and were able to detect only around 84 per cent of the true planets as opposed to 94 per cent with our machine learning based method.", "Overall our machine learning model failed to identify around 63 (out of 970) cases, while BLS failed to detect 155 cases.", "We also found an overlap in the cases that were not detected by our method and BLS.", "We investigated these cases manually and found that in this sample the transits were injected randomly with random parameters which resulted in many non-detectable transits such as: Cases with a relatively low inclination angle for the given star, which resulted in very small planet signals.", "Cases with low S/N ratio where the injected transit signal was weaker than the noise, and hence it was not detected either by our method or by BLS.", "Figure: Precision, recall and F1 score (harmonic mean of precision and recall) as function of the decision threshold for the test set of our simulated data.", "The typical threshold of 0.5 for classification problems can be adapted to increase or decrease the model sensitivity.", "For the simulated planet data set, choosing a threshold of 0.13 results in a recall of 0.94 and a precision of 0.92.", "As the objective is to retrieve the maximum number of possible light curves containing transit signals, this result is preferred over the ones produced by a default threshold of 0.5.", "The AUC of the model is 0.92.The majority of these cases had a signal-to-noise ratio of $\\mathrm {S/N} < 12$ , so that the signal was simply too weak.", "These results demonstrate that our machine learning based method can detect planets more accurately and efficiently than BLS, especially in cases with low signal-to-noise ratio.", "Similar results were also shown by [47], who compared various machine learning methods with BLS.", "Their machine learning models were able to detect planet signals with higher signals and much lower false positive rate compared to BLS.", "Therefore, our next step was to apply our methodology on realistic data sets, i.e.", "the Kepler and TESS light curves." ], [ "Kepler data", "Using the Kepler data set, [51] produced very promising results with a deep learning model.", "To date, these are the best current results on this data set.", "They shared their training dataset publicly, which we used to evaluate our method, and we used their results as a benchmark for our results.", "The data set consists of 15,737 light curves that were labelled into 3 classes by the Autovetter and combined into a `planet candidate' class and a `non-candidate' class, as described in section REF .", "We again extracted features from the light curves with TSFresh and used them along with the class labels to train a GBT model.", "Optimizing the hyperparameters, we found the best model to yield an AUC of 0.948.", "The resulting precision, recall, and F1 score as a function of the decision threshold are shown in Figure REF .", "As discussed in section REF , we decided to optimize the threshold to obtain the highest possible recall, while maintaining a good precision.", "Choosing a threshold of $p_\\mathrm {threshold}=0.46$ results in a recall of 0.96 and a precision of 0.82.", "A comparison of the results of our method with the results by [51] is summarised in Table REF .", "It can be seen that the results of both the methods are comparable.", "While our model has a somewhat lower AUC, it has a slightly higher recall, albeit at a considerably lower precision.", "However, as discussed before, we specifically aimed at getting a high recall instead of precision, as on major criterion is to identify as many planet candidates as possible.", "Figure: As Figure , but for the Kepler data set.", "Choosing a threshold of 0.46 results in a recall of 0.96 and a precision of 0.82.", "The AUC of the model is 0.948.Table: Results on Kepler dataFigure: t-SNE visualisation of the Kepler data.", "Light curves with planet transit signal(s) are shown in green.", "It can be observed that these curves are clustered in one region of the space.To further analyse the Kepler data set and to visualize how different light curves cluster, we embedded the data in a low-dimensional space of two dimensions with the t-distributed stochastic neighbor embedding [35] algorithm.", "To this end, we extracted time series features from each light curves as described in section REF .", "These features were then scaled using a robust scaler, and any missing values were filled using the mean of that column, which resulted in around 700 processed features.", "The top 70 per cent of those features were selected, based on the feature importance found by the GBT model (as using only the top 70 per cent of the features resulted in an optimized AUC), and used as the input for the t-SNE algorithm for dimensionality reduction.", "The results are presented in the Figure REF .", "As the t-SNE algorithm retains the local structure of the data, similar light curves are modelled by nearby points with high probability, i.e.", "the points closer in the 2D representation are also closer in the high dimensional space.", "However, a point far away in the 2D space might not be far away in high dimensional space.", "As Figure REF shows, most of the light curves containing a transit are clustered in one region of the plot, which further validates our pre-processing and feature extraction process.", "Broadly there are 3 main clusters in the plot, and samples from these clusters can be analysed to understand how these clusters vary from each other.", "However, we left this analysis for future investigations." ], [ "TESS data", "TESS light curves are in many way different from the Kepler data set.", "Kepler observed a fixed field of view in its 4 year timeline and K2 observed each of its 19 campaigns for 80 days.", "On the other hand, TESS observed each sector for 27 days.", "Longer baselines contain more details and lead to a much higher signal to noise in terms of detections, hence the extracted features are also more rich in information.", "The short time span of TESS implies that each light curve (and features extracted from it) tends to contain less data points with transits.", "Consequently it becomes more difficult for a classifier to differentiate a case with a planet signal from a case without it.", "Moreover, the short length of the light curves makes the presence of multiple-periodic transit signals less likely to be observed.", "For longer exoplanet transits, this problem is further compounded if only a single transit is recorded.", "In contrast, multiple transiting planets in the Kepler data may lead to an automatic confirmation due to the low probability of any fitting false-positive scenario.", "The first machine leaning classifier to be trained and tested on real TESS data was introduced by [60].", "Their model is an adapted version of [51] as shown in Figure 2 of their paper.", "We again trained our model on the data set publicly shared by [60] and used their results as benchmark.", "The data consists of 16,500 light curves, with only 490 planet candidates.", "Applying TSFresh and a GBT model, we optimized the hyperparameters to get an AUC of 0.81.", "The resulting precision, recall, and F1 score as a function of the decision threshold are shown in Figure REF .", "We again optimized the decision threshold to get a high recall, which resulted in a threshold of $p_\\mathrm {threshold}=0.12$ , a recall of 0.82, and a precision of 0.63 for the `planet candidate' class.", "A comparison of the results of both the methods with default and optimal thresholds is shown in Tables REF and REF , respectively.", "Figure: As Figure , but for the TESS data set.", "With a threshold of 0.12 the recall is 0.82 and the precision is 0.63.", "The AUC of the model is 0.80.Table: Results on TESS data with default thresholdTable: Results on TESS data with optimized thresholdWith the optimized threshold, our model was able to find 40 out of 49 curves with planet transit in our test set as opposed to 44 samples identified by [60] on this test set.", "On the other hand, the precision of our model is significantly higher, i.e.", "it will result in almost half of the false positives as encountered by [60].", "With the default threshold, the deep learning method by [60] resulted in a much higher AUC than our model, but only had a recall of 0.57, so only about 57 per cent of all planets were detected by the model.", "On the other hand, our machine learning method is able to obtain a recall of 0.62.", "Of course, a recall of 0.82 is still not ideal and there is a lot more work to be done before such system can be used in production, both in preparing the data set and tuning the hyperparameters.", "For planet detection with both the Kepler and the TESS data sets, class imbalance is one of the biggest problems the model is facing.", "Now that more and more planets are identified in the TESS data, more light curves with planets can be gathered and used to train machine learning models, which is expected to improve the results (see the discussion in section ).", "Even though our machine learning algorithm is not yet fit for production with the TESS data, it is certainly possible to use the model for reliably eliminating false positives, so that the amount of human vetting can be greatly reduced on cases that are a clear false positive.", "The performance for all three data sets is compared in Figure REF , where we plot the precision as a function of recall for each data set.", "We can see that the performance for the simulated and the Kepler data is similar.", "There is a slight increase in the number of false positives in the Kepler data set, as it is more noisy than the simulated one.", "On the other hand, the performance for the TESS data set is significantly worse, with the high class imbalance being the primary cause of this.", "With a more balanced data set, the performance for the TESS data is expected to approach the other two cases.", "Figure: Precision as a function of recall for the test sets of TESS, Kepler and the simulated data, demonstrating the trade-off between precision and recall for different decision thresholds.", "The performance for the simulated and the Kepler data is similar, with a slight increase in the number of false positives for the Kepler data set, as it is more noisy.", "The performance for the TESS data set is worse as a result of a high class imbalance." ], [ "Classical Machine Learning vs Deep Learning", "One of the primary differences between classical machine learning and deep learning methods is the fact that deep learning models are able to automatically extract useful features from raw data, while classical machine learning methods usually cannot deal with raw data and need extracted features.", "Consequently, the classical machine learning approach does not work well when problems become very complex, such as language translation or image classification, where instead deep learning is pushing new frontiers.", "However, classical machine learning methods should not be disregarded, as they are much more efficient and can still produce valuable results for several problems, such as classifying quasars [9], or identifying asteroids [52].", "Our classical machine learning approach has the following advantages over state-of-the-art deep learning methods: It can work with only a global view of the light curve, whereas deep learning additionally requires folded or secondary views.", "Training is less time consuming and takes less than 5 minutes to train on a 2 CPU system.", "This also indicates that it can be quickly adapted to a new data source, while deep learning can take more than 5 hours for training and much longer for tuning hyperparameters.", "The exact same model setup and code can be used for data from different sources such as Kepler, K2 and TESS.", "Hyperparameters need to be optimized only once, whereas deep learning almost always needs re-tuning when the data set changes.", "The most important features can be automatically identified, which allows for a better understanding of the data and the underlying physical processes.", "By contrast, deep learning models are usually black boxes, so that it is hard to interpret the results.", "No special hardware, such as a GPU, is required for training.", "However, our approach has the following disadvantages compared to state-of-the-art deep learning methods: Generally, the performance is lower compared to deep learning, which when properly trained can often achieve better results.", "Time series data such as the global view, and folded view can be directly used as an input for deep learning, while a classical machine learning model requires extracted features from the time series data." ], [ "The Vetting Tool", "Machine learning methods are still relatively new in astrophysics and often met with scepticism.", "While there is a new paradigm of building classification techniques with machine learning methods, it still has a long way to go before being used in production pipelines.", "One major factor is the non-deterministic or black-box nature of such methods.", "Unlike conventional algorithms, the outcome of such methods cannot always be understood due to the non-linear and stochastic nature of such methods, so that it is harder for the user to explain the reason behind predictions made by the model.", "Even though these models can outperform conventional methods like BLS, they do not always make correct predictions which is another contributing factor for the scepticism.", "Hence, it is important to note that they cannot completely replace human vetting experts.", "However, if they are supported to be used alongside the domain knowledge, these techniques can help to automate processes like planet detection and enable us to deal with the growing data size in astronomy.", "These tools are not widely used yet but for these systems to progress further, it is important to take feedback and expertise from the community on the methods.", "For this reason, we have developed a vetting tool where a few results from our model can be explored in an interactive way without any knowledge of the underlying machine learning method or the feature calculation technique.", "The tool is hosted on GitHubhttps://github.com/abhmalik/Exoplanet-Vetting-Tool.", "The vetting tool allows the user to explore cases where the model assigns a high probability of planet candidacy.", "It allows the user to view the light curve in the global view and the folded view (also called local view).", "It further gives a list of features that were important in the model prediction for that case.", "It can be easily adapted to make model inference on any new light curve." ], [ "Future steps", "With this paper, we tried to provide a new direction for a light and efficient automatic vetting system.", "However, there is a lot more that can be done in this direction.", "Since our method is able to work with data from different sources such as Kepler, K2, and TESS, it is possible to construct a global classifier that can process light curves of any length from any data source directly.", "This would enable us to continue using the same model in case the data format changes, as happened for K2 when light curves had a time gap.", "As we saw, the performance of our model was worse for the TESS data set compared to the other two data sets.", "The primary reason for this was the high class imbalance in TESS data, i.e.", "less than 3 per cent of the light curves contain transit signals, and most of them are just single transits.", "Now that more and more planets are being confirmed from TESS data, it will be possible to construct a data set with better class balance, which might help us to reach the performance levels we are reaching with the Kepler and the simulated data.", "As shown in the t-SNE plot in Figure REF , light curves are clustered in different regions.", "We can extract samples from these groups and analyse how the light curves in different group differ.", "This information can further be used to optimize the model performance for that particular data set.", "Another possibility would be to forego the supervised learning approach and turn towards a semi-supervised approach.", "To this end the data would first have to be embedded in a lower-dimensional space (e.g.", "using an autoencoder, or principal component analysis).", "An unsupervised clustering algorithm could then find clusters of data in the second step.", "Finally, a limited amount of well-understood and labeled light curves could be used to label the individual clusters.", "If a new light curve would then be determined to be a member of a specific cluster, the cluster label can thus be used for the light curve.", "We leave such an approach for future work." ], [ "Conclusions", "Machine learning methods have seen very active development in the last decade and now they are an essential part of our way of working.", "In fact, for everyone who interacts with a computer or smartphone, it is highly likely to interact with some sort of machine learning program.", "They are also widely used in sciences for several use cases such as detecting diseases [50] and finding string models in a string theory landscape [18].", "Within astronomy, there have been many applications such as classifying galaxy morphology [58], creating new large-scale structure samples without long simulations [42], and identifying gravitational waves [21] and gravitational lenses [32].", "With new and advanced telescopes, data in astronomy are growing at a fast pace.", "Conventional methods that involve human judgements are not efficient and prone to variability depending on the investigating expert.", "For example, the commonly used method for exoplanet detection, BLS, produces large number of false positives in case of noisy data, which have to be reviewed manually.", "In this paper, we proposed a novel planet detection method based on classical machine learning.", "Our method consists of automatically extracting time series features from light curves which are then used as input to a gradient boosted trees model.", "We were able to demonstrate that with our machine learning method, we could identify light curves with planet signals more accurately as compared to BLS while significantly reducing the number of false positives.", "Pioneering work in the area has been done by [51] using deep learning.", "Although deep learning methods often outperform classical machine learning approaches, specifically in complex problems such as machine translation and object detection, they tend to be harder to optimize and usually require a long training time on special hardware (GPUs).", "Moreover, they typically rely on large sets of training data and are more difficult to hypertune as they have to navigate a much bigger parameter space as compared to classical machine learning models.", "Hence, in this work, we attempted to move away from the standard deep learning approach and introduced a new direction that provides a light-weight model.", "Our approach consists of three main steps.", "First, the raw light curves were processed, so that systematic effects, cosmic rays, and stellar variability were removed to get flat curves with a uniform time sampling.", "In the second step, the processed light curves were then analysed by the TSFresh library, which extracted features that are typical for time series (e.g.", "absolute energy and number of values above mean).", "Finally, the third step was to use these features to train a gradient boosted tree model to predict the class (`planet candidate' or `non-candidate') for each light curve.", "In this way, our model is able to distinguish important features from unimportant ones, and there is no need for manually selecting features.", "Moreover, our approach does not require special hardware like GPUs and it takes about 2 minutes to train our classical machine learning model on a dual core CPU system.", "This means it can be easily trained even on low end computers and can be revised quickly in case it is required (e.g.", "when new data is available).", "We found that our machine learning method is able to outperform the classical method to identify planet signals, BLS.", "For our simulated data set, BLS was able to identify 84 per cent of true planets, while the machine learning model could identify 94 per cent.", "We also compared the performance of our results with other state-of-the-art models based on deep learning, and found that, for the Kepler data, our model is able to achieve comparable results (a recall of 0.96 at a precision of 0.82).", "For the TESS data our model is able to achieve a recall comparable to the deep learning models (82 vs 89 per cent), but at a precision that is significantly higher (81 per cent vs 44 per cent), so that fewer false positives have to be removed manually.", "While these results are very encouraging, such systems are not yet robust enough to be broadly applicable.", "Even if a machine learning model works well in the development environment, it is prone to make mistakes on unseen data.", "Hence, such methods should be used alongside with human supervision.", "Nonetheless, at the current stage, these models can provide a very reliable system to rule out large number of false positives and can drastically reduce the number of cases requiring manual reviews." ], [ "Acknowledgements", "We are grateful to Arno Riffeser and Jana Steuer for useful discussions and comments.", "We would also like to thank Christopher Shallue and Liang Yu for publicly sharing their code and training data.", "This enabled us to quickly test our model on different data sets and provided a benchmark to compare our results.", "We thank the developers of Lightkurve [34], NumPy [56], Matplotlib [26], Scikit-learn [48] and TensorFlow [1] for their very useful free software.", "The Astrophysics Data Service (ADS) and arXiv preprint repository were used intensively in this work.", "BPM acknowledges an Emmy Noether grant funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – MO 2979/1-1.", "The derived data in this article will be shared on reasonable request to the corresponding author." ] ]	2011.14135
[ [ "Characterizing Bipartite Consensus on Signed Matrix-Weighted Networks\n via Balancing Set" ], [ "Abstract In contrast with the scalar-weighted networks, where bipartite consensus can be achieved if and only if the underlying signed network is structurally balanced, the structural balance property is no longer a graph-theoretic equivalence to the bipartite consensus in the case of signed matrix-weighted networks.", "To re-establish the relationship between the network structure and the bipartite consensus solution, the non-trivial balancing set is introduced which is a set of edges whose sign negation can transform a structurally imbalanced network into a structurally balanced one and the weight matrices associated with edges in this set have a non-trivial intersection of null spaces.", "We show that necessary and/or sufficient conditions for bipartite consensus on matrix-weighted networks can be characterized by the uniqueness of the non-trivial balancing set, while the contribution of the associated non-trivial intersection of null spaces to the steady-state of the matrix-weighted network is examined.", "Moreover, for matrix-weighted networks with a positive-negative spanning tree, necessary and sufficient condition for bipartite consensus using the non-trivial balancing set is obtained.", "Simulation examples are provided to demonstrate the theoretical results." ], [ "Introduction", "The consensus problem of multi-agent networks has been extensively studied in the last two decades ([8], [11], [15], [10]).", "The analysis of multi-agent networks, from a graph-theoretic perspective, emerges from the well-established algebraic graph theory [6].", "In [11], [16], [8], it was shown that a systematical unity is guaranteed under the consensus protocol whenever the communication graph with positive scalar-valued weights is (strongly) connected.", "An alteration was further made by [1] on the protocol which allows the scalar-valued weights to be either positive or negative while guaranteeing asymptotic stability of the network.", "Recently, the consensus protocol has been examined in a broader context which in turn calls for the possibility of matrices as edge weights.", "A common practice is to adopt real symmetric matrices that are either positive (semi-)definite or negative (semi-)definite as edge weights.", "In fact, the involvement of matrix-valued weights arises naturally when characterizing the inter-dimensional communication amongst multi-dimensional agents, the scenarios being, for instance, graph effective resistance and its applications in distributed control and estimation [20], [3], opinion dynamics on multiple interdependent topics [5], [22], bearing-based formation control [23], coupled oscillators dynamics [21], and consensus and synchronization problems [19], [18], [12].", "The weight matrices inflict drastic change on the graph Laplacian thus urging many of the old topics to be re-investigated like the controllability of matrix-weighted networks [13] and its $H_{2}$ performance [4], [2].", "In retrospect of the consensus algorithm on matrix-weighted networks, [18] examined multi-agent networks that involve positive (semi-)definite matrices as edge weights.", "In this setting they have, among other things, proposed the positive spanning tree as a sufficient graph condition for the network consensus.", "Antagonistic interaction represented by negative (semi-)definite matrices was soon extended to both undirected and directed networks [12], [14].", "It was shown that for the matrix-weighted network with a positive-negative spanning tree, it being structurally balanced is equivalent to admitting a bipartite consensus solution.", "Nevertheless, a missing correspondence between the network structure and its steady-state was pointed out in [12].", "It was stated that structural balance is not sufficient in admitting the bipartite consensus in the presence of positive/negative semi-definite weight matrices; while [17] affirmed that the structural balance property is not a necessary condition either.", "Up to now, most of the research is done on sufficient graph-theoretic conditions for the bipartite consensus by ruling out the ramification of semi-definite weight matrices on the network.", "To the best of our knowledge, the bipartite consensus of general matrix-weighted networks is still deficient in any consistent graph-theoretic interpretation.", "In this paper, we propose the non-trivial balancing set (NBS) as a tentative step to re-establish the relationship between the network structure and the bipartite consensus of matrix-weighted networks.", "The NBS defines a set of edges with non-trivial intersecting null spaces and, by their negation, restore the potential structural balance of the network.", "With the non-trivial balancing set, we would first study the matrix-weighted network in general, with or without structural balance or positive-negative spanning trees.", "The uniqueness of the non-trivial balancing set turns out to be a necessary yet insufficient condition for the bipartite consensus in this case.", "Inflicting a stronger precondition, the uniqueness of the NBS becomes both necessary and sufficient to achieve bipartite consensus for networks with positive-negative spanning trees.", "We extend from this well-defined case and discuss the sufficient condition to have the agents converge bipartitely in a more general setting.", "The remainder of this paper is arranged as follows.", "Basic notations and definitions of graph theory and matrix theory are introduced in §2.", "In §3, we formulate the dynamical protocol and provide a simulation example to motivate our work, before we formally introduce the definition of the non-trivial balancing set in §4.", "§5 incorporates the main results in terms of the uniqueness of the non-trivial balancing set and its contribution to the bipartite consensus.", "§6 presents simulation results on the constructed graph that support the derived theories.", "Some concluding remarks are given in §7.", "Let $\\mathbb {R}$ , $\\mathbb {N}$ and $\\mathbb {Z}_{+}$ be the set of real numbers, natural numbers and positive integers, respectively.", "For $n\\in \\mathbb {Z}_{+}$ , denote $\\underline{n}=\\left\\lbrace 1,2\\text{,}\\cdots ,n\\right\\rbrace $ .", "We note specifically that for sets, the notation $\|\\cdot \|$ is used for cardinality.", "The symmetric matrix $Q\\in \\mathbb {R}^{n\\times n}$ is positive(negative) definite if $z^{T}Qz>0$ ($z^{T}Qz<0$ ) for all $z\\in \\mathbb {\\mathbb {R}}^{n}$ and $z\\ne 0$ , in which case it is denoted by $Q\\succ 0$ ($Q\\prec 0$ ).", "While it is positive (negative) semi-definite, denoted by $Q\\succeq 0$ ($Q\\preceq 0$ ), if $z^{T}Qz\\ge 0$ ($z^{T}Qz\\le 0$ ) for all $z\\in \\mathbb {\\mathbb {R}}^{n}$ and $z\\ne 0$ .", "We adopt an extra matrix-valued sign function ${\\bf sgn}(\\cdot ):\\mathbb {R}^{n\\times n}\\mapsto \\left\\lbrace 0,-1,1\\right\\rbrace $ to express this positive/negative (semi-)definiteness of a symmetric matrix $Q$ , it is defined such that $\\text{{\\bf sgn}}(Q)=1$ if $Q\\succeq 0$ and $Q\\ne 0$ or $Q\\succ 0$ , $\\text{{\\bf sgn}}(Q)=-1$ if $Q\\preceq 0$ and $Q\\ne 0$ or $Q\\prec 0$ , and $\\text{{\\bf sgn}}(Q)=0$ if $Q=0$ .", "We shall employ $\|\\cdot \|$ for such symmetric matrices to denote the operation ${\\bf sgn}(Q)\\cdot Q$ , namely, $\|Q\|=Q$ if $Q\\succ 0$ or $Q\\succeq 0$ , $\|Q\|=-Q$ if $Q\\prec 0$ or $Q\\preceq 0$ , and $\|Q\|=Q=-Q$ when $Q=0$ .", "Denote the null space of a matrix $Q\\in \\mathbb {R}^{n\\times n}$ as $\\text{{\\bf null}}(Q)=\\left\\lbrace z\\in \\mathbb {R}^{n}\|Qz=0\\right\\rbrace $ .", "The notation $B={\\bf blk}\\lbrace \\cdot \\rbrace $ is used for the block matrix $B$ that is partitioned into the blocks in $\\lbrace \\cdot \\rbrace $ , and there is further ${\\bf blkdiag}\\lbrace \\cdot \\rbrace $ to denote when all the non-zero blocks in $\\lbrace \\cdot \\rbrace $ are on the diagonal of $B$ ; while ${\\bf blk}_{ij}(B)$ refers to the intersection of the $i$ th row block and the $j$ th column block of $B$ ." ], [ "Graph Theory ", "A multi-agent network can be characterized by a graph $\\mathcal {G}$ with node set $\\mathcal {V}=\\underline{n}$ and edge set $\\mathcal {E}\\subseteq \\mathcal {V}\\times \\mathcal {V}$ , for which $e_{ij}=(i,j)\\in \\mathcal {E}$ if there is a connection between node $i$ and $j$ for $\\forall i,j\\in \\mathcal {V}$ .", "Define the matrix-weighted graph (network) $\\mathcal {G}$ as a triplet $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathcal {A})$ , where $\\mathcal {A}$ is the set of all weight matrices.", "A subgraph of $\\mathcal {G}$ is a graph $\\overline{\\mathcal {G}}=(\\overline{\\mathcal {V}},\\overline{\\mathcal {E}},\\overline{\\mathcal {A}})$ such that $\\overline{\\mathcal {V}}\\subseteq \\mathcal {V},\\overline{\\mathcal {E}}\\subseteq \\mathcal {E},\\overline{\\mathcal {A}}\\subseteq \\mathcal {A}$ .", "Let $\\mathcal {W}(e_{ij})$ denote the weight matrix assigned to edge $e_{ij}$ such that $\\mathcal {W}(e_{ij})=A_{ij}\\in \\mathcal {A}\\subset \\mathbb {R}^{d\\times d}$ .", "We shall refer to a matrix-weighted network $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathcal {A})$ with $n$ nodes and $d\\times d$ weight matrices as $(n,d)-$ matrix-weighted network.", "Reversely, $\\mathcal {W}^{-1}(A_{ij})=e_{ij}$ maps from the weight matrix to the corresponding edge.", "In this paper, we use symmetric matrices for all edges in $\\mathcal {G}$ , which are $A_{ij}\\in \\mathbb {R}^{d\\times d}$ such that $\|A_{ij}\|\\succeq 0$ or $\|A_{ij}\|\\succ 0$ if $(i,j)\\in \\mathcal {E}$ and $A_{ij}=0$ otherwise for all $i,j\\in \\mathcal {V}$ .", "Thereby the adjacency matrix for a matrix-weighted graph $A=[A_{ij}]\\in \\mathbb {R}^{dn\\times dn}$ is a block matrix such that the block on the $i$ -th row and the $j$ -th column is $A_{ij}$ .", "We say an edge $(i,j)\\in \\mathcal {E}$ is positive(negative) definite or positive(negative) semi-definite if the corresponding weight matrix $A_{ij}$ is positive(negative) definite or positive(negative) semi-definite.", "Since the graphs considered are simple and undirected, we assume that $A_{ij}=A_{ji}$ for all $i\\lnot \\ne j\\in \\mathcal {V}$ and $A_{ii}=0$ for all $i\\in \\mathcal {V}$ .", "Let $\\mathcal {N}_{i}=\\left\\lbrace j\\in \\mathcal {V}\\,\|\\,(i,j)\\in \\mathcal {E}\\right\\rbrace $ be the neighbor set of an agent $i\\in \\mathcal {V}$ .", "We use $C=\\text{{\\bf blkdiag}}\\left\\lbrace C_{1},C_{2},\\cdots ,C_{n}\\right\\rbrace \\in \\mathbb {R}^{dn}$ to represent the matrix-weighted degree matrix of a graph where $C_{i}=\\sum _{j\\in \\mathcal {N}_{i}}\|A_{ij}\|\\in \\mathbb {R}^{d\\times d}$ .", "The matrix-valued Laplacian matrix of a matrix-weighted graph is defined as $L(\\mathcal {G})=C-A$ , which is real and symmetric.", "The gauge transformation for $\\mathcal {G}$ is performed by the diagonal block matrix $D=\\text{{\\bf blkdiag}}\\left\\lbrace \\sigma _{1},\\sigma _{2},\\ldots ,\\sigma _{n}\\right\\rbrace $ where $\\sigma _{i}=I_{d}$ or $\\sigma _{i}=-I_{d}$ .", "A gauge transformed Laplacian is a matrix $\\bar{L}$ such that $\\bar{L}=DLD$ .", "A path $\\mathcal {P}$ in a matrix-weighted graph $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathcal {A})$ is defined as a sequence of edges in the form of $\\lbrace (i_{1},i_{2}),(i_{2},i_{3}),\\ldots ,(i_{p-1},i_{p})\\rbrace $ where nodes $i_{1},i_{2},\\ldots ,i_{p}\\in \\mathcal {V}$ are all distinct and it is said that $i_{1}$ is reachable from $i_{p}$ .", "A matrix-weighted graph $\\mathcal {G}$ is connected if any two distinct nodes in $\\mathcal {G}$ are reachable from each other.", "All graphs mentioned in this paper, unless stated otherwise, are assumed to be connected.", "The sign of a path $\\text{{\\bf sgn}}(\\mathcal {P})$ is defined as $\\text{{\\bf sgn}}(A_{i_{1}i_{2}})\\cdots \\text{{\\bf sgn}}(A_{i_{\|\\mathcal {P}\|}i_{\|\\mathcal {P}\|+1}})$ , while the null space of the path $\\text{{\\bf null}}(\\mathcal {P})$ refers to ${\\displaystyle \\bigcup _{k=1}^{\|\\mathcal {P}\|}}\\text{{\\bf null}}(A_{i_{k}i_{k+1}})$ .", "A path is said to be positive/negative definite if there is no semi-definite weight matrix on the path, i.e., ${\\bf null}(\\mathcal {P})={\\bf span}\\lbrace {\\bf 0}\\rbrace $ ; otherwise, if ${\\bf null}(\\mathcal {P})\\ne {\\bf span}\\lbrace {\\bf 0}\\rbrace $ , the path is positive/negative semi-definite.", "A positive-negative tree in a matrix-weighted graph is a tree such that every edge in this tree is either positive definite or negative definite.", "A positive-negative spanning tree of a matrix-weighted graph $\\mathcal {G}$ is a positive-negative tree containing all nodes in $\\mathcal {G}$ .", "A cycle $\\mathcal {C}$ of $\\mathcal {G}$ is a path that starts and ends with the same node, i.e., $\\mathcal {C}=\\lbrace (i_{1},i_{2}),(i_{2},i_{3}),\\ldots ,(i_{p-1},i_{1})\\rbrace $ .", "Note that a spanning-tree does not contain any circle.", "The sign of the cycle ${\\bf sgn}(\\mathcal {C})$ is defined similarly as that of the path, we say the cycle is negative if it contains an odd number of negative (semi-)definite weight matrices (${\\bf sgn}(\\mathcal {C})<0$ ), and it is positive if the negative connections are of even number (${\\bf sgn}(\\mathcal {C})>0$ ).", "It is well-known that the structural balance of signed networks is a paramount graph-theoretic condition for achieving (bipartite) consensus.", "For matrix-weighted networks, there is an analogous definition as follows.", "Definition 1 [12] A matrix-weighted network $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathcal {A})$ is $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ structurally balanced if there exists a bipartition of nodes $\\mathcal {V}=\\mathcal {V}_{1}\\cup \\mathcal {V}_{2},\\mathcal {V}_{1}\\cap \\mathcal {V}_{2}=\\emptyset $ , such that the matrix-valued weight between any two nodes within each subset is positive (semi-)definite, but negative (semi-)definite for edges connecting nodes of different subsets.", "A matrix-weighted network is structurally imbalanced if it is not structurally balanced.", "By indexing the edges into $\\mathcal {E}=\\lbrace e_{1},...,e_{\|\\mathcal {E}\|}\\rbrace $ along with their weight matrices $\\mathcal {A}=\\lbrace A_{1},...,A_{\|\\mathcal {E}\|}\\rbrace $ , we have the following definition of signed incidence matrix for matrix-weighted networks.", "Definition 2 A signed incidence matrix $H={\\bf blk}\\lbrace I_{d},-I_{d},{\\bf 0}_{d\\times d}\\rbrace $ of a matrix-weighted network $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathcal {A})$ is an $\|\\mathcal {E}\|d\\times nd$ block matrix for which, the $k$ -th $d\\times dn$ row block $H^{k},k\\in \\underline{\|\\mathcal {E}\|}$ , corresponds to the edge $e_{k}$ with weight matrix $A_{ij}$ between agent $i$ and $j$ .", "The $i$ -th and $j$ -th blocks of $H^{k}$ are $I_{d}$ and $-I_{d}$ respectively if $A_{ij}\\succ 0(A_{ij}\\succeq 0)$ , while let them be $I_{d}$ and $I_{d}$ if $A_{ij}\\prec 0(A_{ij}\\preceq 0)$ ; any other block would be ${\\bf 0}_{d\\times d}$ .", "Lemma 1 Let $H$ be the signed incidence matrix of a matrix-weighted network $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathcal {A})$ .", "Then the matrix-valued Laplacian of $\\mathcal {G}$ can be characterized by $L=H^{T}\\text{{\\bf blkdiag}}\\lbrace \|A_{k}\|\\rbrace H,$ where the $k$ -th $d\\times dn$ row block of $H$ corresponds to the edge whose matrix weight is the $k$ -th block in $\\text{{\\bf blkdiag}}\\lbrace \|A_{k}\|\\rbrace $ .", "Proof The proof is straightforward thus is omitted.", "Problem Formulation and Motivation Consider a multi-agent network consisted of $n\\in \\mathbb {N}$ agents.", "The states of each agent $i\\in \\mathcal {V}$ is denoted by $x_{i}(t)=\\mathbb {R}^{d}$ where $d\\in \\mathbb {N}$ .", "The interaction protocol reads $\\dot{x}_{i}(t)=-\\sum _{j\\in \\mathcal {N}_{i}}\|A_{ij}\|(x_{i}(t)-\\text{{\\bf sgn}}(A_{ij})x_{j}(t)),i\\in \\mathcal {V},$ where $A_{ij}\\in \\mathbb {R}^{d\\times d}$ denotes the weight matrix on edge $(i,j)$ .", "The collective dynamics of the multi-agent network (REF ) can be characterized by $\\dot{x}(t)=-Lx(t),$ where $x(t)=[x_{1}^{T}(t),x_{2}^{T}(t),\\ldots ,x_{n}^{T}(t)]^{T}\\in \\mathbb {R}^{dn}$ and $L$ is the matrix-valued graph Laplacian.", "Definition 3 (Bipartite Consensus) The multi-agent network (REF ) is said to admit a bipartite consensus solution if there exists a solution $x$ such that $\\lim {}_{t\\rightarrow \\infty }x_{i}(t)=\\lim {}_{t\\rightarrow \\infty }x_{j}(t)\\ne 0$ or $\\lim _{t\\rightarrow \\infty }x_{i}(t)=-\\lim _{t\\rightarrow \\infty }x_{j}(t)\\ne {\\bf 0}$ for any $\\lbrace i,j\\rbrace \\subset \\mathcal {V}$ .", "When $\\lim {}_{t\\rightarrow \\infty }x_{i}(t)=\\lim {}_{t\\rightarrow \\infty }x_{j}(t)={\\bf 0}$ for all $\\lbrace i,j\\rbrace \\subset \\mathcal {V}$ , the network admits a trivial consensus.", "We employ the following example to motivate our work in this paper.", "Example 1 Consider the network $\\mathcal {G}_{1}$ in Figure REF which is structurally imbalanced with one negative circle.", "One may obtain a structurally balanced network from it by negating the sign of $e_{23}$ or alternatively, by negating the sign of $e_{34}$ .", "Figure: A structurally imbalanced matrix-weighted network 𝒢 1 \\mathcal {G}_{1}.The red solid (resp., dashed) lines denote edges weighted by positivedefinite (resp., semi-definite) matrices; the blue solid (resp., dashed)lines denote edges weighted by negative definite (resp., semi-definite)matrices.The edges are endowed with matrix-valued weights $A_{23} & =\\begin{bmatrix}-2 & 2 & 0\\\\2 & -2 & 0\\\\0 & 0 & -1\\end{bmatrix}\\preceq 0,\\\\A_{24} & =\\begin{bmatrix}1 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 1\\end{bmatrix}\\succeq 0,\\\\A_{12} & =\\begin{bmatrix}-2 & 0 & 0\\\\0 & -1 & 0\\\\0 & 0 & -1\\end{bmatrix}\\prec 0,$ and $A_{15}=-A_{23},A_{45}=A_{12},A_{34}=-A_{12}$ .", "Note that ${\\bf null}(A_{23})={\\bf null}(A_{15})={\\bf span}\\lbrace [\\begin{array}{ccc}1 & 1 & 0\\end{array}]^{T}\\rbrace $ and ${\\bf null}(A_{24})={\\bf span}\\lbrace [\\begin{array}{ccc}0 & 1 & 0\\end{array}]^{T}\\rbrace $ .", "The null space of the weight matrices of the remaining edges are trivially spanned by the zero vector.", "Under the above selection of edge weights, we examine the evolution of multi-agent system (REF ) on $\\mathcal {G}_{1}$ , yielding the state trajectories of each agent shown in Figure REF .", "Despite the fact that $\\mathcal {G}_{1}$ is structurally imbalanced, a bipartition of agents emerges by their steady-states, namely the agents of $\\mathcal {V}_{a}=\\lbrace 2,3,4\\rbrace $ converge to $[\\begin{array}{ccc}1.6525 & 1.6525 & 0\\end{array}]^{T}$ , others of $\\mathcal {V}_{b}=\\lbrace 1,5\\rbrace $ converge to $[\\begin{array}{ccc}-1.6525 & -1.6525 & 0\\end{array}]^{T}$ , and both of the steady states are spanned by $[\\begin{array}{ccc}1 & 1 & 0\\end{array}]^{T}$ .", "What we have noticed is that $[\\begin{array}{ccc}1 & 1 & 0\\end{array}]^{T}$ happens to span the null space of $A_{23}$ ; interestingly, by negating the sign of $A_{23}$ , the resulting network becomes structurally balanced and the structurally balanced partition of nodes is precisely $\\mathcal {V}_{a}=\\lbrace 2,3,4\\rbrace $ and $\\mathcal {V}_{b}=\\lbrace 1,5\\rbrace $ .", "Figure: State trajectories of multi-agent dynamics ()on 𝒢 1 \\mathcal {G}_{1}.The edge $e_{23}$ is, therefore, deemed highly correlated with the bipartite consensus solution of the network in Figure REF .", "A comparison with edges $e_{24}$ and $e_{34}$ elaborates on how $e_{23}$ is being essential in realizing the bipartite consensus.", "First, we notice that both $e_{23}$ and $e_{24}$ are of non-trivial null space, yet the dynamics did not converge to ${\\bf null}(A_{24})={\\bf span}\\lbrace [\\begin{array}{ccc}0 & 1 & 0\\end{array}]^{T}\\rbrace $ , presumably because the negation of $e_{24}$ does not yield any structurally balanced partition of the graph, which means $e_{24}$ is structurally unimportant.", "As for $e_{23}$ and $e_{34}$ , both being structurally critical in the above sense (structurally balanced partition $\\mathcal {V}_{a}=\\lbrace 1,3,5\\rbrace ,\\mathcal {V}_{b}=\\lbrace 2,4\\rbrace $ for $e_{34}$ ), we notice that the bipartition of the convergence has abided by the grouping of $e_{23}$ instead of that of $e_{34}$ , presumably because ${\\bf null}(A_{34})$ is merely ${\\bf span}\\lbrace {\\bf 0}\\rbrace $ and algebraically trivial.", "The edge $e_{23}$ is thus intuitively significant since it is the only edge in the graph that is non-trivial in both senses; it is where the structural and algebraic properties of the network intersect.", "Motivated by the above observations, we make the assumption that the bipartite consensus of the matrix-weighted network indicates the existence of a set of edges with non-trivial intersecting null spaces and, by their negation, restore the potential structural balance of the network.", "For the rest of this paper, we attempt to make a valid definition of these edges, and to examine the role they ought to play in the bipartite consensus of multi-agent systems on matrix-weighted networks.", "Balancing Set of Matrix-weighted Networks A prominent structural feature of the edge we studied in Example REF concerns its negation of the sign, by which the structurally imbalanced network is rendered structurally balanced.", "This method was studied in [9], [7] which suggested that given any structurally imbalanced network, one can always transform it into a structurally balanced one with any preassigned node partition by negating the signs of the relevant edges.", "We refer to this approach in the literature and introduce the following concept to embed it in the context of the matrix-weighted multi-agent networks.", "Definition 4 (Balancing set) Let $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ denote a bipartition of node set $\\mathcal {V}$ in a network $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathcal {A})$ .", "Define the $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ as a set of edges such that a $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ structurally balanced network can be obtained if the sign of each edge in $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ is negated.", "Example 2 In Figure REF we have constructed a matrix-weighted network $\\mathcal {G}$ that is structurally imbalanced.", "Given a node partition $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ of $\\mathcal {G}$ , its balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ is consisted of the negative connections within $\\mathcal {V}_{1}$ and $\\mathcal {V}_{2}$ and the positive connections between them (edges colored in red).", "By negating the signs of the edges in $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ , we derive a graph $\\mathcal {G}^{\\prime }$ that is structurally balanced between $\\mathcal {V}_{1}$ and $\\mathcal {V}_{2}$ .", "Remark 1 The $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ is empty if and only if the network is $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ structurally balanced.", "For instance, the balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ of the structurally balanced $\\mathcal {G}^{\\prime }$ in Figure REF is empty since there is no edge to be negated.", "Figure: The negation operation in Example .Note that the matrix-valued weight plays a role in shaping the null space of the matrix-valued graph Laplacian, thus more constraints on the balancing set are needed to complete the definition.", "We proceed to quantitatively characterize the contribution of matrix-valued weights to the null space of the matrix-valued graph Laplacian.", "Definition 5 A set of matrices $A_{i}\\in \\mathbb {R}^{n\\times n},i\\in \\underline{l},$ are said to have non-trivial intersection of null spaces, or, to have non-trivial intersecting null space if $\\bigcap _{i=1}^{l}{\\bf null}(A_{i})\\ne \\left\\lbrace {\\bf 0}\\right\\rbrace .$ Definition 6 (Non-trivial balancing set) A $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ of a matrix-weighted network $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathcal {A})$ is a non-trivial $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ balancing set (NBS), denoted by $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ , if the weight matrices associated with edges in $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ have non-trivial intersection of null spaces.", "Remark 2 The non-trivial balancing set $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})=\\emptyset $ if and only if the corresponding balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})=\\emptyset $ .", "In this case, define $\\mathcal {W}(\\mathcal {E}^{nb})={\\bf 0}$ where ${\\bf 0}$ is the $d\\times d$ zero matrix.", "A matrix-weighted network $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathcal {A})$ has a unique non-trivial balancing set if there is only one bipartition $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ of $\\mathcal {V}$ such that the corresponding $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ -balancing set meets $\\bigcap _{e\\in \\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})}{\\bf null}(\\mathcal {W}(e))\\ne \\left\\lbrace {\\bf 0}\\right\\rbrace .$ In this case, we shall denote $\\text{{\\bf null}}(\\mathcal {E}^{nb})=\\bigcap _{e\\in \\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})}{\\bf null}(\\mathcal {W}(e))$ for brevity.", "Remark 3 It is noteworthy that to have only an empty non-trivial balancing set does not suggest there is no NBS in the graph; a graph without NBS is a structurally imbalanced graph for which any partition of nodes has a balancing set whose weight matrices share a trivial intersecting null space.", "Main Results General Networks In this section, we set to examine the validity of the concept termed as the non-trivial balancing set through its correlation with the network steady-state, bearing in mind the question of to what extent is the non-trivial balancing set a satisfactory interpretation of the numerical solutions.", "We shall recall some facts about the algebraic structure of the Laplacian null space when bipartite consensus is achieved on matrix-weighted networks.", "Also, technical preparations Lemma REF and Lemma REF are presented for the proof of Theorem REF and Theorem REF , one is referred to the Appendix for their proofs.", "Lemma 2 [17] The matrix-weighted multi-agent network (REF ) achieves bipartite consensus if and only if there exists a gauge transformation $D$ such that $\\text{{\\bf null}}(L)=\\mathcal {S}=\\text{{\\bf span}}\\lbrace D(\\text{\\textbf {1}}_{n}\\otimes \\Psi )\\rbrace $ , where $D$ is a gauge transformation, $\\Psi =[\\psi _{1},\\psi _{2},...,\\psi _{s}]$ , and $\\psi _{i}$ , $i\\in \\underline{s}$ , $s\\le d$ , are orthogonal basis vectors in $\\mathbb {R}^{d}$ .", "Remark 4 The space $\\mathcal {S}$ in Lemma REF is defined as the bipartite consensus subspace in [17].", "And in [12], when $s=d$ , $\\mathcal {S}$ is proved to be the bipartite consensus subspace of a structurally balanced matrix-weighted network.", "Lemma 3 Let $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathcal {A})$ be a matrix-weighted network with $n$ agents, each of dimension $d$ .", "If $\\mathcal {G}$ is structurally balanced with partition $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ , then there exists a gauge transformation $D\\in \\mathbb {R}^{nd\\times nd}$ such that ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes I_{d})\\rbrace \\subset \\text{{\\bf null}}(L(\\mathcal {G}))$ , and ${\\bf blk}_{ii}(D)=I_{d}$ for $i\\in \\mathcal {V}_{1}$ and ${\\bf blk}_{ii}(D)=-I_{d}$ for $i\\in \\mathcal {V}_{2}$ , where $L(\\mathcal {G})$ is the network Laplacian.", "Lemma 4 For any nonzero $v_{1},v_{2},v\\in \\mathbb {R}^{d}$ , the linear combination of $D_{1}(\\text{\\textbf {1}}_{n}\\otimes v_{1})$ and $D_{2}(\\text{\\textbf {1}}_{n}\\otimes v_{2})$ do not yield $D(\\text{\\textbf {1}}_{n}\\otimes v)$ whether or not $v_{1}$ and $v_{2}$ are linearly independent, where $D$ is a gauge transformation, $D_{1},D_{2}$ are gauge transformations with $D_{1}\\ne D_{2}$ and $D_{1}\\ne -D_{2}$ .", "In the Preliminary section, the gauge transformation is introduced as a block matrix $D={\\bf blkdiag}\\lbrace I_{d},-I_{d}\\rbrace $ for which the identity blocks $I_{d},-I_{d}$ are sequenced on the diagonal with a certain pattern of signs.", "Since the gauge matrix $D$ has $N$ $Nd$ -by-$d$ column blocks, the non-zero diagonal blocks could be considered to have a one-to-one correspondence with the well-indexed $N$ agents, reflecting either the relative positivity/negativity of their steady states, or their bipartition by structure.", "To exert this correspondence, consider a matrix-weighted network $\\mathcal {G}$ for which a partition $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ of the node set $\\mathcal {V}$ defines a balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ .", "We map this partition onto a gauge matrix $D$ such that ${\\bf blk}_{ii}(D)=I_{d}$ for $i\\in \\mathcal {V}_{1}$ and ${\\bf blk}_{ii}(D)=-I_{d}$ for $i\\in \\mathcal {V}_{2}$ , suppose the nodes are properly indexed.", "The node partition is then fully described by the pattern of the signs of the diagonal blocks, and we phrase $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ as a balancing set with division $D$ or of $D$ -division to notify how the nodes are actually partitioned given the edge set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ .", "We are ready to establish how the non-trivial balancing set is related to the null space of the matrix-valued Laplacians.", "Theorem 1 For a matrix-weighted network $\\mathcal {G}$ , the following properties are equivalent: 1) there exists a non-trivial balancing set $\\mathcal {E}^{nb}$ in $\\mathcal {G}$ with division $D$ , such that ${\\bf span}\\lbrace \\Xi \\rbrace \\subset {\\bf null}(\\mathcal {E}^{nb})$ , 2) ${\\bf span}\\lbrace D(\\text{\\textbf {1}}_{n}\\otimes \\Xi )\\rbrace \\subset \\text{{\\bf null}}(L(\\mathcal {G}))$ , where $D$ is a gauge transformation and $\\Xi =[\\xi _{1},...,\\xi _{r}]$ where $\\xi _{i}\\in \\mathbb {R}^{d},i\\in \\underline{r},0<r\\leqslant d$ are linearly independent.", "Proof For any matrix-weighted network the graph Laplacian can be expressed as $L=H^{T}\\text{{\\bf blkdiag}}\\lbrace \|A_{k}\|\\rbrace H,$ where $H$ is the signed incidence matrix and the blocks in $\\text{{\\bf blkdiag}}\\lbrace \|A_{k}\|\\rbrace $ are ordered the same as their appearances in $H$ .", "Thus, $Lx={\\bf 0}$ if and only if $\\text{{\\bf blkdiag}}\\lbrace \|A_{k}\|\\rbrace ^{\\frac{1}{2}}Hx={\\bf 0}$ , namely, $\|A_{ij}\|^{\\frac{1}{2}}(x_{i}-\\text{{\\bf sgn}}(A_{ij})x_{j})={\\bf 0},\\forall (i,j)\\in \\mathcal {E}.$ Note that $(x_{i}-\\text{{\\bf sgn}}(A_{ij})x_{j})^{T}\|A_{ij}\|(x_{i}-\\text{{\\bf sgn}}(A_{ij})x_{j})\\\\=\|\|\|A_{ij}\|^{\\frac{1}{2}}(x_{i}-\\text{{\\bf sgn}}(A_{ij})x_{j})\|\|^{2} & = & {\\bf 0},$ then $\|A_{ij}\|^{\\frac{1}{2}}(x_{i}-\\text{{\\bf sgn}}(A_{ij})x_{j})={\\bf 0}$ if and only if $A_{ij}(x_{i}-\\text{{\\bf sgn}}(A_{ij})x_{j})={\\bf 0},$ which implies $Hx\\in \\text{{\\bf null}}(\\text{{\\bf blkdiag}}\\lbrace A_{k}\\rbrace )$ where the weight matrices $A_{ij}$ are relabelled as $A_{k},k\\in \\underline{\|\\mathcal {E}\|}$ .", "1) → 2): Consider when the network has a non-trivial balancing set $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ with division $D$ .", "Without loss of generality, we block the signed incidence matrix $H$ as $H=[H_{1}^{T}H_{2}^{T}]^{T}$ where $H_{2}$ corresponds to edges in $\\mathcal {E}^{nb}$ whose weights have intersecting null space that is non-trivial.", "We know from Definition REF that the edges in $H_{1}$ constructs a structurally balanced subgraph, and since $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ is with division $D$ , the blocks of $D$ are assigned as ${\\bf blk}_{ii}(D)=I_{d}$ for $i\\in \\mathcal {V}_{1}$ and ${\\bf blk}_{ii}(D)=-I_{d}$ for $i\\in \\mathcal {V}_{2}$ .", "From Lemma REF the gauge transformation $D$ satisfies ${\\color {black}{\\color {black}{\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes I_{d})\\rbrace \\subset {\\bf null}(H_{1}^{T}H_{1})}}$ because $H_{1}^{T}H_{1}$ is the Laplacian matrix of a structurally balanced network with the same topology as the $H_{1}$ subgraph except the absolute weights are all identity matrices.", "It is then derived that ${\\bf span}\\lbrace H_{1}D(\\text{\\textbf {1}}_{n}\\otimes I_{d})\\rbrace =\\lbrace {\\bf 0}\\rbrace $ .", "Now consider $H_{2}D(\\text{\\textbf {1}}_{n}\\otimes \\xi _{p})$ where $\\xi _{p}\\in {\\bf span}\\lbrace \\xi _{1},...,\\xi _{r}\\rbrace & \\subset & \\text{{\\bf null}}(\\mathcal {E}^{nb})\\\\& = & \\bigcap _{A_{j}\\in \\mathcal {W}(\\mathcal {E}^{nb})}\\text{{\\bf null}}(A_{j}),p\\in \\underline{r}.$ For any row block $H_{2}^{j}\\in \\mathbb {R}^{d\\times nd}$ of $H_{2}$ , it is composed of either $\\pm I_{d}$ or 0, and the corresponding weight matrix $A_{j}$ breaks the structural balance of the $H_{1}$ subgraph.", "That means, if $H_{2}^{j}$ has two $I_{d}$ matrices, the corresponding weight matrix is negative (semi-)definite, and the $H_{1}$ subgraph puts the connected vertices in the same partition (e.g., both are in $\\mathcal {V}_{1}$ ), therefore $H_{2}^{j}D$ has two $I_{d}$ matrices or two $-I_{d}$ matrices.", "If $H_{2}^{j}$ has $I_{d}$ and $-I_{d}$ , the corresponding weight matrix is positive (semi-)definite and the $H_{1}$ subgraph puts the connected vertices in different partitions (one in $\\mathcal {V}_{1}$ and the other in $\\mathcal {V}_{2}$ ), then $H_{2}^{j}D$ has two $I_{d}$ matrices or two $-I_{d}$ matrices.", "Thus $H_{2}D({\\bf 1}_{n}\\otimes \\xi _{p})=2[\\pm \\xi _{p}^{T},...,\\pm \\xi _{p}^{T}]^{T}\\in \\mathbb {R}^{\|\\mathcal {E}^{nb}\|d\\times 1}$ and since $\\text{{\\bf span}}\\lbrace {\\bf 1}_{n}\\otimes \\xi _{p}\\rbrace \\subset \\text{{\\bf span}}\\lbrace \\text{\\textbf {1}}_{n}\\otimes I_{d}\\rbrace $ , $HD(\\text{\\textbf {1}}_{n}\\otimes \\xi _{p}) & =\\left[\\begin{array}{c}H_{1}D({\\bf 1}_{n}\\otimes \\xi _{p})\\\\H_{2}D({\\bf 1}_{n}\\otimes \\xi _{p})\\end{array}\\right]\\nonumber \\\\& =\\left[\\begin{array}{c}{\\bf 0}_{(\|\\mathcal {E}\|-\|\\mathcal {E}^{nb}\|)d}\\\\\\pm 2\\xi _{p}\\\\\\vdots \\\\\\pm 2\\xi _{p}\\end{array}\\right].$ Block the matrix $A$ as $A={\\bf blkdiag}\\lbrace \\mathcal {A}_{1},\\mathcal {A}_{2}\\rbrace $ , where matrix $\\mathcal {A}_{1}$ has the weight matrices of the $H_{1}$ subgraph as its diagonal blocks, and matrix $\\mathcal {A}_{2}$ , the $H_{2}$ subgraph.", "Then we have $\\text{{\\bf blkdiag}}\\lbrace \\mathcal {A}_{1},\\mathcal {A}_{2}\\rbrace HD(\\text{\\textbf {1}}_{n}\\otimes \\xi _{p})={\\bf 0}_{\|\\mathcal {E}\|d}.$ Because the $H_{2}$ subgraph contains edges in $\\mathcal {E}^{nb}$ and $\\xi _{p}\\in \\bigcap _{A_{j}\\in \\mathcal {W}(\\mathcal {E}^{nb})}\\text{{\\bf null}}(A_{j})$ , therefore $\\mathcal {A}_{2}H_{2}D({\\bf 1}_{n}\\otimes \\xi _{p})={\\bf 0}$ holds, for $p\\in \\underline{r}$ .", "Hence we have proved that ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes \\Xi )\\rbrace \\subset {\\bf null}(L)$ .", "2) → 1): Consider when the Laplacian has $D(\\text{\\textbf {1}}_{n}\\otimes \\xi _{p})\\subset \\text{{\\bf null}}(L)$ for $p\\in \\underline{r}$ , which means to assign the nodes with $+\\xi _{p}$ or $-\\xi _{p}$ according to the sign pattern of $D$ satisfies $A_{ij}(x_{i}-{\\bf sgn}(A_{ij})x_{j})={\\bf 0},\\forall (i,j)\\in \\mathcal {E}$ .", "The sign pattern of $D$ corresponds to a partition $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ which defines a balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ on the graph.", "For any edge $e_{lm}\\in \\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ , it either a) connects within $\\mathcal {V}_{1}$ or $\\mathcal {V}_{2}$ and has $A_{lm}\\prec 0(A_{lm}\\preceq 0)$ , or b) connects between $\\mathcal {V}_{1}$ and $\\mathcal {V}_{2}$ and has $A_{lm}\\succ 0(A_{lm}\\succeq 0)$ .", "Note that those in $\\mathcal {V}_{1}$ are assigned with $+\\xi _{p}$ by $D(\\text{\\textbf {1}}_{n}\\otimes \\xi _{p})$ while those in $\\mathcal {V}_{2}$ are assigned with $-\\xi _{p}$ .", "Therefore for a), eqn.", "(REF ) gives $A_{lm}(\\xi _{p}+\\xi _{p})={\\bf 0}$ and $\\xi _{p}\\in {\\bf null}(A_{lm})$ ; for b), eqn.", "(REF ) gives $A_{lm}(\\xi _{p}-(-\\xi _{p}))={\\bf 0}$ and $\\xi _{p}\\in {\\bf null}(A_{lm})$ .", "We now have for all $e_{lm}\\in \\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ , there is $\\xi _{p}\\in {\\bf null}(A_{lm}),p\\in \\underline{r}$ , thus the balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ with division $D$ has non-trivial intersecting null space, and ${\\bf span}\\lbrace \\Xi \\rbrace \\subset {\\bf null}(\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2}))$ .", "Corollary 1 For a non-trivial balancing set $\\mathcal {E}^{nb}$ in $\\mathcal {G}$ with division $D$ , it holds that $D({\\bf 1}_{n}\\otimes {\\bf null}(\\mathcal {E}^{nb}))\\subset {\\bf null}(L)$ .", "Proof This is a direct inference from the proposition 1) → 2) in Theorem REF by considering the bases that span ${\\bf null}(\\mathcal {E}^{nb})$ as $\\Xi $ .", "Theorem REF has illustrated how the existence of a non-trivial balancing set in the network interchanges with a set of vectors ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes \\Xi )\\rbrace $ in the Laplacian null space.", "Particularly, proposition 2) → 1) shows that as long as ${\\bf null}(L)$ includes vectors of the form ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes \\Xi )\\rbrace $ , they indicate the existence of a non-trivial balancing set $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ in $\\mathcal {G}$ whose partition follows the sign pattern of $D$ , and the columns of $\\Xi $ are included in the non-trivially intersecting null space ${\\bf null}(\\mathcal {E}^{nb})$ .", "The correlation plays a central role in establishing the fact that when bipartite consensus is admitted, there is at least one NBS in the matrix-weighted network.", "We are now able to derive a necessary condition on the bipartite consensus for matrix-weighted networks in general.", "Theorem 2 If the multi-agent system (REF ) admits a bipartite consensus solution with a steady state $\\bar{x}\\ne {\\bf 0}$ , then there exists a unique non-trivial balancing set $\\mathcal {E}^{nb}$ in $\\mathcal {G}$ such that $\\bar{x}_{i}\\in {\\bf null}(\\mathcal {E}^{nb})$ for all $i\\in \\mathcal {V}$ .", "Proof If bipartite consensus is achieved on the matrix-weighted network $\\mathcal {G}$ , then by Lemma REF there is ${\\bf null}(L)=\\text{{\\bf span}}\\lbrace D(\\text{\\textbf {1}}_{n}\\otimes \\Psi )\\rbrace $ where $\\psi _{1},...,\\psi _{r}$ are orthogonal vectors, and for the steady state we have $\\bar{x}\\in {\\bf span}\\lbrace D(\\text{\\textbf {1}}_{n}\\otimes \\Psi )\\rbrace $ , i.e., $\\bar{x}_{i}\\in {\\bf span}\\lbrace \\Psi \\rbrace $ .", "According to Theorem REF , ${\\bf span}\\lbrace D(\\text{\\textbf {1}}_{n}\\otimes \\Psi )\\rbrace \\subset {\\bf null}(L)$ implies that there exists a non-trivial balancing set $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ with division $D$ such that ${\\bf span}\\lbrace \\Psi \\rbrace \\subset {\\bf null}(\\mathcal {E}^{nb})$ .", "We will first show that actually, ${\\bf span}\\lbrace \\Psi \\rbrace ={\\bf null}(\\mathcal {E}^{nb})$ by raising the fact that $rank({\\bf null}(L))=rank(D({\\bf 1}_{n}\\otimes \\Psi ))=rank({\\bf 1}_{n}\\otimes \\Psi )=rank({\\bf 1}_{n})\\cdot rank(\\Psi )=rank(\\Psi )$ .", "If there is $\\psi ^{}\\in \\mathbb {R}^{d}$ for which $\\psi ^{}\\in {\\bf null}(\\mathcal {E}^{nb})$ but $\\psi ^{}\\notin {\\bf span}\\lbrace \\Psi \\rbrace $ , proposition 1) → 2) of Theorem REF states that $D({\\bf 1}_{n}\\otimes \\psi ^{})\\in {\\bf null}(L)$ , then let $\\Psi ^{\\prime }=[\\psi _{1},...,\\psi _{r},\\psi ^{}]$ , there is ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes \\Psi ^{\\prime })\\rbrace \\subset {\\bf null}(L)$ and ${\\bf null}(L)$ is raised by rank one.", "Therefore there exists $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ with division $D$ such that ${\\bf null}(L)=D(\\text{\\textbf {1}}_{n}\\otimes {\\bf null}(\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})))$ .", "Now suppose there exists another partition $(\\mathcal {V}_{1}^{^{\\prime }},\\mathcal {V}_{2}^{^{\\prime }})$ with a corresponding non-trivial balancing set $\\mathcal {E}^{nb}(\\mathcal {V}_{1}^{^{\\prime }},\\mathcal {V}_{2}^{^{\\prime }})$ , then by Corollary REF , $x^{}\\in D^{^{\\prime }}(\\text{\\textbf {1}}_{n}\\otimes \\text{{\\bf null}}(\\mathcal {E}^{nb}(\\mathcal {V}_{1}^{^{\\prime }},\\mathcal {V}_{2}^{^{\\prime }})))\\subset \\text{{\\bf null}}(L),$ while $D\\ne D^{^{\\prime }}$ and $D\\ne -D^{^{\\prime }}$ .", "Meanwhile Lemma REF states that since $\\bar{x}$ and $x^{}$ are composed of distinct gauge transformations $D$ and $D^{\\prime }$ , their linear combination does not yield any vector of the form $D({\\bf 1}_{n}\\otimes v)$ for $v\\in \\mathbb {R}^{d}$ , which implies $k_{1}\\bar{x}+k_{2}x^{}\\notin {\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes \\Xi )\\rbrace $ .", "Therefore one has $k_{1}\\bar{x}+k_{2}x^{}\\notin \\text{{\\bf null}}(L),$ for $k_{1},k_{2}\\ne 0$ , which is a contradiction.", "Thus $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ is the only non-trivial balancing set in the network and satisfies ${\\bf null}(\\mathcal {E}^{nb})={\\bf span}\\lbrace \\Xi \\rbrace $ , then the agents converge to the non-trivial intersecting null space of the unique NBS and $\\bar{x}_{i}\\in {\\bf null}(\\mathcal {E}^{nb})$ .", "A non-trivial balancing set is defined by a partition of nodes that aims at achieving structural balance on itself; Theorem REF states that when bipartite consensus is admitted on a matrix-weighted network, one is bound to find a non-trivial balancing set, a set of edges, as a third party that prevents the structural balance from happening, and is with a non-trivial intersecting null space.", "For any other grouping of agents, one would find their corresponding balancing set to have null spaces that intersect only trivially.", "Looking back on the definition of the non-trivial balancing set, we see that when the NBS is somehow unique, the bipartition of the agents' convergence states is mirrored in the particular grouping $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ of this NBS (which is also encoded in $D$ ).", "Even more noteworthy is the intersecting null space ${\\bf null}(\\mathcal {E}^{nb})$ that directly contributes to the Laplacian null space, as is indicated by ${\\bf null}(L)=D(\\text{\\textbf {1}}_{n}\\otimes \\text{{\\bf null}}(\\mathcal {E}^{nb}))$ , which means the agents converge to a linear combination of the vectors that span ${\\bf null}(\\mathcal {E}^{nb})$ .", "Based on this impression, the definition of the NBS is rather a rephrasing of those vectors of the crucial form $D(\\text{\\textbf {1}}_{n}\\otimes \\xi )$ whose role is immediately twofold: to split the agents into groups and to grant the convergence state of the network.", "Networks with A Positive-negative Spanning Tree In this subsection, we examine the matrix-weighted network with a positive-negative spanning tree.", "The following theorem is derived with respect to the non-trivial balancing set.", "Theorem 3 For a matrix-weighted network $\\mathcal {G}$ with a positive-negative spanning tree, under protocol (REF ), we have: 1) bipartite consensus is admitted if and only if it has a unique non-trivial balancing set; 2) trivial consensus is admitted when no non-trivial balancing set is present in the graph.", "Proof First we provide the proof of part 1).", "(Sufficiency) When the network has a unique non-trivial balancing set $\\mathcal {E}^{nb}$ with division $D$ , Corollary REF suggests that $D(\\text{\\textbf {1}}_{n}\\otimes \\text{{\\bf null}}(\\mathcal {E}^{nb}))\\subset \\text{{\\bf null}}(L)$ .", "We proceed to show that in fact, in the presence of the positive-negative spanning tree, $D({\\bf 1}_{n}\\otimes \\text{{\\bf null}}(\\mathcal {E}^{nb}))$ span the whole Laplacian null space.", "We know from the previous proof that to derive $\\text{{\\bf null}}(L)$ is to solve for $x$ that satisfies a series of equations $A_{ij}(x_{i}-\\text{{\\bf sgn}}(A_{ij})x_{j})={\\bf 0},\\forall (i,j)\\in \\mathcal {E}.$ Because for any two nodes there is a path with edges whose weight matrices are all positive (negative) definite, solve equation REF along the path, we can only derive $x_{p}=x_{q}$ or $x_{p}=-x_{q}$ for any pair of nodes $p,q\\in \\mathcal {V}$ .", "Thus any solution in the Laplacian null space could be represented as $D^{\\prime }({\\bf 1}_{n}\\otimes w)$ for some $w\\in \\mathbb {R}^{d}$ and a gauge matrix $D^{\\prime }$ .", "Suppose there exists $x^{}=D^{\\prime }(1_{n}\\otimes w)\\in \\text{{\\bf null}}(L)$ with $D^{\\prime }\\ne D$ and $D^{\\prime }\\ne -D$ , then according to Theorem REF , there exists a non-trivial balancing set $\\mathcal {E}^{nb}$ with division $D^{\\prime }$ in the network which contradicts our premise that $\\mathcal {E}^{nb}$ is unique.", "Now suppose there is $x^{}=D({\\bf 1}_{n}\\otimes w)\\in \\text{{\\bf null}}(L)$ with $w\\notin \\text{{\\bf null}}(\\mathcal {E}^{nb})$ , then with equation (REF ) we have $\\text{{\\bf blkdiag}}\\lbrace A_{k}\\rbrace HD(\\text{\\textbf {1}}_{n}\\otimes w)\\ne {\\bf 0}$ , hence $D(\\text{\\textbf {1}}_{n}\\otimes w)\\notin \\text{{\\bf null}}(L)$ which is clearly a contradiction.", "Therefore the Laplacian null space is spanned by $D(\\text{\\textbf {1}}_{n}\\otimes \\text{{\\bf null}}(\\mathcal {E}^{nb}))$ , and bipartite consensus is admitted.", "(Necessity) This is readily verified with Theorem REF .", "Now we proceed to prove part 2).", "We have established that in the presence of a positive-negative spanning tree, for any $p,q\\in \\mathcal {V}$ there is $x_{p}=\\pm x_{q}$ by solving (REF ) along the positive-negative definite path that connects them.", "As the tree contains no circle, the relative positivity and negativity of $x_{p}$ and $x_{q}$ naturally gives a partition $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ of the node set which defines a balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ .", "Note that the tree itself is structurally balanced with respect to this division, thus the edges of the tree are not included in $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ ; in other words, edges in $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ either has $A_{ij}\\prec 0(A_{ij}\\preceq 0)$ and connects within $\\mathcal {V}_{1}$ or $\\mathcal {V}_{2}$ (where the solution has been $x_{p}=x_{q}$ ), or has $A_{ij}\\succ 0(A_{ij}\\succeq 0)$ and connects between $\\mathcal {V}_{1}$ and $\\mathcal {V}_{2}$ (where the solution has been $x_{p}=-x_{q}$ ).", "When no non-trivial balancing set is found in the network, it means there exist two edges $\\mathcal {W}(e_{1})=\\mathcal {W}((i_{k_{1}},i_{k_{1}+1}))=A_{1}$ and $\\mathcal {W}(e_{2})=\\mathcal {W}((i_{k_{2}},i_{k_{2}+1}))=A_{2}$ in the balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ that have ${\\bf null}(A_{1})\\cap {\\bf null}(A_{2})=\\lbrace {\\bf 0}\\rbrace $ .", "Both edges satisfy $\\begin{array}{c}x_{k_{1}}-\\text{{\\bf sgn}}(A_{1})x_{k_{1}+1}=2x_{k_{1}}\\in \\text{{\\bf null}}(A_{1})\\\\x_{k_{2}}-\\text{{\\bf sgn}}(A_{2})x_{k_{2}+1}=2x_{k_{2}}\\in \\text{{\\bf null}}(A_{2})\\end{array},$ and since ${\\bf null}(A_{1})\\cap {\\bf null}(A_{2})=\\lbrace {\\bf 0}\\rbrace $ , the fact that $x_{k_{1}}=\\pm x_{k_{2}}$ gives $x_{k_{1}}=x_{k_{2}}={\\bf 0}$ , therefore $x_{i}={\\bf 0}$ for $i\\in \\mathcal {V}$ and a trivial consensus is admitted on the network.", "One fact suggested by Graph Theory is that a network may be spanned by trees with different choices of edges; one could consider when there is another positive-negative spanning tree that gives a distinct partition by solving eqn.", "(REF ).", "Since no non-trivial balancing set exists in the network, i.e., every balancing set is trivial for all possible partitions of $\\mathcal {V}$ , the above reasoning applies for all positive-negative spanning trees and the conclusion stands.", "In Theorem REF , the uniqueness of the non-trivial balancing set is proposed as a necessary condition for the bipartite consensus on matrix-weighted networks with general structural features.", "Therefore in Theorem REF , we confine ourselves to matrix-weighted networks with positive-negative spanning trees and have found the uniqueness of the NBS to be both necessary and sufficient, with the assistance of the intrinsic structural balance of the positive-negative spanning tree in the sufficient part.", "We have also established that to have at least one NBS is quite necessary for such networks to admit any steady-state other than the trivial consensus.", "For the derivation of the theorems so far, we mention that though some of the discussions in the proofs have touched on the notion of structural balance, the idea itself is not engaged in the formulation of the theorems, where the steady-state behaviour is directly associated with the existence (or non-existence) of the NBS.", "It is safe to say the non-trivial balancing set has taken the place of the structural balance as a proper indication of the system behaviour, and the graph-theoretic correspondence is partly rebuilt.", "Remark 5 Considering the necessary and sufficient condition for the bipartite consensus derived on the scalar-weighted network ([1]), which is the structural balance property of the network, we are aware that this is well incorporated into the framework of Theorem REF , since all weights that are scalar are the $1\\times 1$ positive/negative definite matrix weights, and the positive-negative spanning tree naturally exists.", "A Counter Example and A Sufficient Condition It is only natural, at this point, to ask if there is any possibility for the uniqueness of the NBS to be also conveniently sufficient even in the absence of a positive-negative spanning tree.", "However, we have come to a negative conclusion on this by raising the following counter-example.", "Figure: The matrix-weighted network 𝒢 counter \\mathcal {G}_{counter} for Example 3.The red solid (resp., dashed) lines denote edges weighted by positivedefinite (resp., semi-definite) matrices; the blue solid (resp., dashed)lines denote edges weighted by negative definite (resp., semi-definite)matrices.Example 3 The matrix-weighted network $\\mathcal {G}_{counter}$ is a 7-node structurally imbalanced network with the unique non-trivial balancing set being $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})=\\lbrace e_{23},e_{25},e_{46}\\rbrace $ , given by the illustration in Figure REF with the following weight matrix arrangements: $A_{23} & =\\begin{bmatrix}-2 & 2 & 0\\\\2 & -2 & 0\\\\0 & 0 & 0\\end{bmatrix}\\preceq 0, & v_{a}=\\begin{bmatrix}1\\\\1\\\\0\\end{bmatrix}, & v_{b}=\\begin{bmatrix}0\\\\0\\\\1\\end{bmatrix},\\\\A_{14} & =\\begin{bmatrix}1 & 1 & 0\\\\1 & 1 & 0\\\\0 & 0 & 0\\end{bmatrix}\\succeq 0, & v_{c}=\\begin{bmatrix}1\\\\-1\\\\0\\end{bmatrix}, & v_{b}=\\begin{bmatrix}0\\\\0\\\\1\\end{bmatrix},\\\\A_{47} & =\\begin{bmatrix}2 & -1 & 2\\\\-1 & 2 & -1\\\\2 & -1 & 2\\end{bmatrix}\\succeq 0, & v_{d}=\\begin{bmatrix}-1\\\\0\\\\1\\end{bmatrix},\\\\A_{12} & =\\begin{bmatrix}2 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{bmatrix}\\succ 0,$ and $A_{23}=A_{25}=A_{46},A_{12}=A_{13}=A_{45}=A_{56}=A_{17}.$ We have written down the vectors that span the null spaces of the semi-definite weight matrices.", "It is seen that the network $\\mathcal {G}_{counter}$ consists of four independent circles, three of which are negative and one is positive.", "The non-trivial balancing set must enclose edges that eliminate the negative circles simultaneously without generating any other one.", "The existence of the positive circle $\\lbrace 1,4,7\\rbrace $ has refrained the NBS from including $e_{14}$ as a result, despite that $e_{23},e_{14},e_{46}$ share the same eigenvector $v_{b}$ for the zero eigenvalue.", "$\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})=\\lbrace e_{23},e_{25},e_{46}\\rbrace $ is then unique as a non-trivial balancing set; however, we could see from Figure REF that the numerical solution suggests the network yields a non-trivial consensus solution, rather than a bipartite consensus solution, for the structurally imbalanced $\\mathcal {G}_{counter}$ .", "Therefore the uniqueness of the NBS alone is not a sufficient condition in any strict sense for general matrix-weighted networks.", "Figure: Numerical solutions for Example 3.", "The bipartite consensus is notadmitted despite the existence of a unique NBS.We would like to close the main study of this work with a sufficient condition on the bipartite consensus, albeit somewhat trivial, under which a node outside $\\mathcal {G}(\\mathcal {T})$ can merge with it through semi-definite paths.", "For now we denote the network with a positive-negative spanning tree and a non-trivial balancing set as $\\mathcal {G}^{nb}(\\mathcal {T})$ .", "According to Theorem REF , $\\mathcal {G}^{nb}(\\mathcal {T})$ admits the bipartite consensus, so that part of the agents converge to $\\zeta \\in \\mathbb {R}^{d}$ , of which we assume the signs to be $\\text{{\\bf sgn}}(x_{i})=1$ , while other agents would converge to $-\\zeta $ and their signs are written as $\\text{{\\bf sgn}}(x_{i})=-1$ .", "A node can merge with $\\mathcal {G}^{nb}(\\mathcal {T})$ if the expanded network obtains bipartite consensus altogether.", "Theorem 4 Consider a matrix-weighted network $\\mathcal {G}^{nb}(\\mathcal {T})$ with a positive-negative spanning tree and a non-trivial balancing set, which naturally has all its agents converging either to $\\zeta \\in \\mathbb {R}^{d}$ or $-\\zeta \\in \\mathbb {R}^{d}$ under (REF ).", "Suppose a vertex $i_{r}\\notin \\mathcal {G}^{nb}(\\mathcal {T})$ has $m$ paths $\\mathcal {P}_{k}=\\lbrace (i_{r},i_{r+1}^{k}),...,(i_{\|\\mathcal {P}_{k}\|}^{k},i_{\|\\mathcal {P}_{k}\|+1}^{k})\\rbrace ,k\\in \\underline{m}$ to reach $\\mathcal {G}^{nb}(\\mathcal {T})$ , each path has only its last vertex in $\\mathcal {G}^{nb}(\\mathcal {T})$ , which is $i_{\|\\mathcal {P}_{k}\|+1}^{k}\\in \\mathcal {G}^{nb}(\\mathcal {T}),k\\in \\underline{m}$ .", "Then $i_{r}$ merge with $\\mathcal {G}^{nb}(\\mathcal {T})$ if for any $k_{1},k_{2}\\in \\underline{m}$ , there is $\\text{{\\bf sgn}}(\\mathcal {P}_{k_{1}})\\text{{\\bf sgn}}(x_{\|\\mathcal {P}_{k_{1}}\|+1}^{k_{1}})=\\text{{\\bf sgn}}(\\mathcal {P}_{k_{2}})\\text{{\\bf sgn}}(x_{\|\\mathcal {P}_{k_{2}}\|+1}^{k_{2}})$ , and $\\bigcap _{k=1}^{m}\\text{{\\bf null}}(\\mathcal {P}_{k})=0$ .", "Proof Suppose $\\mathcal {P}_{1},\\mathcal {P}_{2}\\in \\lbrace \\mathcal {P}_{k}\\rbrace $ , $i_{\|\\mathcal {P}_{1}\|+1}^{1},i_{\|\\mathcal {P}_{2}\|+1}^{2}\\in \\mathcal {G}^{nb}(\\mathcal {T})$ , $x_{\|\\mathcal {P}_{1}\|+1}^{1},x_{\|\\mathcal {P}_{2}\|+1}^{2}$ denote their final states.", "Then there is $\\begin{array}{c}x_{r}-\\text{{\\bf sgn}}(\\mathcal {P}_{1})x_{\|\\mathcal {P}_{1}\|+1}^{1}\\in \\text{{\\bf null}}(\\mathcal {P}_{1}),\\\\x_{r}-\\text{{\\bf sgn}}(\\mathcal {P}_{2})x_{\|\\mathcal {P}_{2}\|+1}^{2}\\in \\text{{\\bf null}}(\\mathcal {P}_{2}),\\end{array}$ since $\\text{{\\bf sgn}}(\\mathcal {P}_{1})\\text{{\\bf sgn}}(x_{\|\\mathcal {P}_{1}\|+1}^{1})=\\text{{\\bf sgn}}(\\mathcal {P}_{2})\\text{{\\bf sgn}}(x_{\|\\mathcal {P}_{2}\|+1}^{2})$ , the left-hand sides are both about $x_{r}-\\text{{\\bf sgn}}(\\mathcal {P}_{1})\\text{{\\bf sgn}}(x_{\|\\mathcal {P}_{1}\|+1}^{1})\\zeta $ .", "With $\\text{{\\bf null}}(\\mathcal {P}_{1})\\bigcap \\text{{\\bf null}}(\\mathcal {P}_{2})=\\lbrace {\\bf 0}\\rbrace $ , $x_{r}$ has a unique solution $\\text{{\\bf sgn}}(\\mathcal {P}_{1})x_{\|\\mathcal {P}_{1}\|+1}^{1}$ thus is merged with $\\mathcal {G}^{nb}(\\mathcal {T})$ .", "Theorem REF extends our study of the spanning-tree case in Theorem REF to when there exist positive/negative semi-definite paths between agents, the conclusion being a sufficient condition.", "We also have the inference that on a general matrix-weighted network, if the subgraphs spanned by positive/negative-definite trees do not have their separate non-trivial balancing sets in the first place, then the network as a whole is incapable of achieving bipartite consensus.", "Simulation Example This section provides numerical examples of the theorems we have derived, based on the network constructed in Figure REF .", "Now we can see that $\\mathcal {G}_{1}$ is structurally imbalanced with a unique non-trivial balancing set $\\lbrace e_{23}\\rbrace $ , which yields a structurally balanced node partition $\\lbrace 1,5\\rbrace ,\\lbrace 2,3,4\\rbrace $ .", "Under dynamics (REF ) the agents admit bipartite consensus as in Figure REF , and the final states are determined by the intersecting null space of the NBS ${\\bf span}\\lbrace \\begin{bmatrix}1 & 1 & 0\\end{bmatrix}^{T}\\rbrace $ .", "Now suppose the edge weight $A_{34}$ is also semi-definite, and set $A_{34}=\\begin{bmatrix}1 & 0 & 1\\\\0 & 3 & 0\\\\1 & 0 & 1\\end{bmatrix}$ .", "Then $\\mathcal {G}_{1}$ has two non-trivial balancing sets that give different node partitions.", "While $\\lbrace e_{23}\\rbrace $ still partition the agents into $\\lbrace 1,5\\rbrace ,\\lbrace 2,3,4\\rbrace $ , $\\lbrace e_{34}\\rbrace $ produces partition $\\lbrace 1,5,3\\rbrace ,\\lbrace 2,4\\rbrace $ .", "Figure REF shows that bipartite consensus is not achieved under this circumstance.", "Figure: Agents do not admit bipartite consensus after the alteration of A 34 A_{34},due to the non-uniqueness of the non-trivial balancing sets.One could easily turn Figure REF into a graph without any non-trivial balancing set by setting $A_{23}$ negative-definite as $A_{23}=A_{12}$ .", "In this case $\\mathcal {G}_{1}$ has a positive-negative spanning tree, and as expected, the agents have only admitted a trivial consensus since there is no NBS in the graph, refer to Figure REF .", "Figure: Agents admit trivial consensus when 𝒢 1 \\mathcal {G}_{1} has no non-trivialbalancing set.", "Concluding Remarks In this paper, we have established the significance of the non-trivial balancing set to the bipartite consensus of matrix-weighted networks.", "It is shown that the uniqueness of such a set is a necessary condition in admitting the bipartite consensus.", "Moreover, if bipartite consensus is indeed achieved, the final states of the agents are determined by none other than the intersecting null space of the non-trivial balancing set.", "The uniqueness of the NBS is specifically studied on networks with positive-negative spanning trees, which turns out to be both necessary and sufficient for the bipartite consensus.", "Based on this conclusion, we have given the condition to extend the tree with semi-definite matrix-weighted paths while preserving the bipartite consensus on the resulting network.", "However, we are aware that this condition is formulated in an algorithmic fashion that does not involve much structural attribute of the network; for future research, we would expect the establishment of a sufficient condition for the bipartite consensus with the concept of the NBS that is more structure-based and applicable for specific control problems.", "Appendix 1.", "Proof for Lemma REF .", "Proof If $\\mathcal {G}$ is connected and $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ structurally balanced, construct a gauge transformation matrix $D$ such that ${\\bf blk}_{ii}(D)=I_{d}$ for $i\\in \\mathcal {V}_{1}$ and ${\\bf blk}_{ii}(D)=-I_{d}$ for $i\\in \\mathcal {V}_{2}$ , let $x\\in {\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes I_{d})\\rbrace $ , then $A_{ij}(x_{i}-{\\bf sgn}(A_{ij})x_{j})={\\bf 0}$ holds for all $(i,j)\\in \\mathcal {E}$ , because when $x_{i}=x_{j}$ , there is $i,j\\in \\mathcal {V}_{1}$ or $i,j\\in \\mathcal {V}_{2}$ , and ${\\bf sgn}(A_{ij})>0$ (${\\bf sgn}(A_{ij})\\ge 0$ ); when $x_{i}=-x_{j}$ , there is $i\\in \\mathcal {V}_{1}$ , $j\\in \\mathcal {V}_{2}$ or $i\\in \\mathcal {V}_{2}$ , $j\\in \\mathcal {V}_{1}$ , and ${\\bf sgn}(A_{ij})<0$ (${\\bf sgn}(A_{ij})\\le 0$ ).", "Therefore we have ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes I_{d})\\rbrace \\subset \\text{{\\bf null}}(L(\\mathcal {G}))$ .", "If $\\mathcal {G}$ is disconnected, note that $\\mathcal {G}$ is structurally balanced if and only if all its components are structurally balanced.", "Denote the components of $\\mathcal {G}$ as $\\mathcal {G}_{i}=(\\mathcal {V}_{i},\\mathcal {E}_{i},\\mathcal {A}_{i})$ where $i\\in \\underline{q}$ and $\|\\mathcal {V}_{i}\|=n_{i}$ .", "Let $L^{i}$ denote the matrix-valued Laplacian of $\\mathcal {G}_{i}$ for all $i\\in \\underline{q}$ .", "Then $L(\\mathcal {G})={\\bf blkdiag}\\lbrace L^{i}\\rbrace .$ Again there exist $D^{i}\\in \\mathbb {R}^{n_{i}d\\times n_{i}d}$ such that ${\\bf span}\\lbrace L^{i}D^{i}({\\bf 1}_{n_{i}}\\otimes I_{d})\\rbrace =\\lbrace {\\bf 0}\\rbrace $ for all $i\\in \\underline{q}$ .", "Therefore, one can choose $D={\\bf blkdiag}\\lbrace D^{i}\\rbrace $ , then there is ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes I_{d})\\rbrace \\subset \\text{{\\bf null}}(L(\\mathcal {G}))$ which completes the proof.", "Lemma 5 For a set of linearly independent vectors $v_{1},...,v_{r}\\in \\mathbb {R}^{d}$ , $2\\le r\\le d$ , with $\\forall k_{i}\\ne 0,i\\in \\underline{r}$ , the linear combination $x=k_{1}D_{1}(\\text{\\textbf {1}}_{n}\\otimes v_{1})+...+k_{r}D_{r}(\\text{\\textbf {1}}_{n}\\otimes v_{r})\\ne D(\\text{\\textbf {1}}_{n}\\otimes v)$ where $v\\in \\mathbb {R}^{d}$ and $D$ is a gauge transformation, if the sign patterns of the gauge transformations $D_{1},D_{2},...,D_{r}$ are distinct from each other, that is, there is no $D_{p},D_{q}$ with $D_{p}=D_{q}$ or $D_{p}=-D_{q},p,q\\in \\underline{r}$ .", "Proof Write $x$ in its block form as $x=\\text{{\\bf blk}}\\lbrace x_{1}^{T}x_{2}^{T}...x_{n}^{T}\\rbrace ^{T},x_{k}\\in \\mathbb {R}^{d}$ .", "Suppose we use $D_{1}(\\text{\\textbf {1}}_{n}\\otimes v_{1}),D_{2}(\\text{\\textbf {1}}_{n}\\otimes v_{2}),...,D_{r}(\\text{\\textbf {1}}_{n}\\otimes v_{r})$ for the linear combination, then $x={\\displaystyle \\sum _{j=1}^{r}}k_{j}D_{j}(\\text{\\textbf {1}}_{n}\\otimes v_{j})$ The blocks of $x$ are written as $\\begin{array}{c}x_{1}=z_{11}v_{1}+\\cdots +z_{1r}v_{r},\\\\\\vdots \\\\x_{l}=z_{l1}v_{1}+\\cdots +z_{lr}v_{r},\\\\\\vdots \\\\x_{t}=z_{t1}v_{1}+\\cdots +z_{tr}v_{r},\\\\\\vdots \\\\x_{n}=z_{n1}v_{1}+\\cdots +z_{nr}v_{r},\\end{array}$ where $\|z_{1j}\|=\|z_{2j}\|=...=\|z_{nj}\|=\|k_{j}\|$ for $j\\in \\underline{r}$ .", "The sign pattern of a gauge transformation is the sequence of signs of the diagonal blocks, $\\lbrace \\begin{array}{ccc}+1 & +1 & -1\\end{array}\\rbrace $ for $\\text{{\\bf blkdiag}}\\lbrace I_{d},I_{d},-I_{d}\\rbrace $ for instance.", "We use ${\\bf sgn}(D_{j}^{i})$ to denote the sign of the $i$ th diagonal block of gauge transformation $D_{j}$ , which can be either $+1$ or $-1$ .", "Then $z_{ij}={\\bf sgn}(D_{j}^{i})k_{j}$ .", "Note that the gauge transformations $D_{1}$ and $D_{r}$ are of different sign patterns, therefore there exist two blocks of $x$ , say, $x_{l}$ and $x_{t}$ , so that ${\\bf sgn}(D_{1}^{l})={\\bf sgn}(D_{r}^{l})$ and ${\\bf sgn}(D_{1}^{t})=-{\\bf sgn}(D_{r}^{t}).$ Suppose $x=D(1_{n}\\otimes v)$ , then we should have $x_{l}=\\pm x_{t}$ .", "If $x_{l}=x_{t}$ , then (a) suppose ${\\bf sgn}(D_{1}^{l})={\\bf sgn}(D_{1}^{t})$ , as a consequence of eqn.", "(REF ) and (REF ), there is ${\\bf sgn}(D_{r}^{l})=-{\\bf sgn}(D_{r}^{t})$ , i.e., $z_{l1}=z_{t1}$ and $z_{lr}=-z_{tr}$ .", "So when we equate $x_{l}$ and $x_{t}$ , there is $(z_{l2}-z_{t2})v_{2}+...+(z_{l,r-1}-z_{t,r-1})v_{r-1}+2z_{lr}v_{r}=0,$ then $v_{2},...,v_{r}$ becomes linearly dependent since there is at least $z_{lr}\\ne 0$ , thus we have derived a contradiction; (b) suppose ${\\bf sgn}(D_{1}^{l})=-{\\bf sgn}(D_{1}^{t})$ , then there is ${\\bf sgn}(D_{r}^{l})={\\bf sgn}(D_{r}^{t})$ , i.e., $z_{l1}=-z_{t1}$ and $z_{lr}=z_{tr}$ , so when we equate $x_{l}$ and $x_{t}$ , we have $2z_{l1}v_{1}+(z_{l2}-z_{t2})v_{2}+...+(z_{l,r-1}-z_{t,r-1})v_{r-1}=0,$ which contradicts the fact that $v_{1},...,v_{r-1}$ are linearly independent.", "For $x_{l}=-x_{t}$ , the contradictions can be derived similarly by discussing (a) ${\\bf sgn}(D_{1}^{l})={\\bf sgn}(D_{1}^{t})$ and (b) ${\\bf sgn}(D_{1}^{l})=-{\\bf sgn}(D_{1}^{t})$ .", "3.", "Proof for Lemma REF .", "Proof The case of $v_{1}$ and $v_{2}$ being linearly independent has been proved as the case of $r=2$ in Lemma REF .", "When $v_{2}=kv_{1}$ , suppose $x=\\alpha D_{1}(\\text{\\textbf {1}}_{n}\\otimes v_{1})+\\beta D_{2}(\\text{\\textbf {1}}_{n}\\otimes v_{2})=D(\\text{\\textbf {1}}_{n}\\otimes v),\\alpha \\ne 0,\\beta \\ne 0$ , then the blocks of $x$ are written as $x_{i}=({\\bf sgn}(D_{1}^{i})\\alpha +{\\bf sgn}(D_{2}^{i})\\beta )v_{1},i=1,...,n$ .", "Because we can find $x_{p}$ and $x_{q}$ with ${\\bf sgn}(D_{1}^{p})={\\bf sgn}(D_{2}^{p})$ and ${\\bf sgn}(D_{1}^{q})=-{\\bf sgn}(D_{2}^{q})$ , there is $x_{p}={\\bf sgn}(D_{1}^{p})(\\alpha +\\beta )v_{1},x_{q}={\\bf sgn}(D_{1}^{q})(\\alpha -\\beta )v_{1}$ .", "Let $x_{p}=x_{q}$ or $x_{p}=-x_{q}$ we can always derive $\\alpha =0$ or $\\beta =0$ , thus is a contradiction." ], [ "Problem Formulation and Motivation", "Consider a multi-agent network consisted of $n\\in \\mathbb {N}$ agents.", "The states of each agent $i\\in \\mathcal {V}$ is denoted by $x_{i}(t)=\\mathbb {R}^{d}$ where $d\\in \\mathbb {N}$ .", "The interaction protocol reads $\\dot{x}_{i}(t)=-\\sum _{j\\in \\mathcal {N}_{i}}\|A_{ij}\|(x_{i}(t)-\\text{{\\bf sgn}}(A_{ij})x_{j}(t)),i\\in \\mathcal {V},$ where $A_{ij}\\in \\mathbb {R}^{d\\times d}$ denotes the weight matrix on edge $(i,j)$ .", "The collective dynamics of the multi-agent network (REF ) can be characterized by $\\dot{x}(t)=-Lx(t),$ where $x(t)=[x_{1}^{T}(t),x_{2}^{T}(t),\\ldots ,x_{n}^{T}(t)]^{T}\\in \\mathbb {R}^{dn}$ and $L$ is the matrix-valued graph Laplacian.", "Definition 3 (Bipartite Consensus) The multi-agent network (REF ) is said to admit a bipartite consensus solution if there exists a solution $x$ such that $\\lim {}_{t\\rightarrow \\infty }x_{i}(t)=\\lim {}_{t\\rightarrow \\infty }x_{j}(t)\\ne 0$ or $\\lim _{t\\rightarrow \\infty }x_{i}(t)=-\\lim _{t\\rightarrow \\infty }x_{j}(t)\\ne {\\bf 0}$ for any $\\lbrace i,j\\rbrace \\subset \\mathcal {V}$ .", "When $\\lim {}_{t\\rightarrow \\infty }x_{i}(t)=\\lim {}_{t\\rightarrow \\infty }x_{j}(t)={\\bf 0}$ for all $\\lbrace i,j\\rbrace \\subset \\mathcal {V}$ , the network admits a trivial consensus.", "We employ the following example to motivate our work in this paper.", "Example 1 Consider the network $\\mathcal {G}_{1}$ in Figure REF which is structurally imbalanced with one negative circle.", "One may obtain a structurally balanced network from it by negating the sign of $e_{23}$ or alternatively, by negating the sign of $e_{34}$ .", "Figure: A structurally imbalanced matrix-weighted network 𝒢 1 \\mathcal {G}_{1}.The red solid (resp., dashed) lines denote edges weighted by positivedefinite (resp., semi-definite) matrices; the blue solid (resp., dashed)lines denote edges weighted by negative definite (resp., semi-definite)matrices.The edges are endowed with matrix-valued weights $A_{23} & =\\begin{bmatrix}-2 & 2 & 0\\\\2 & -2 & 0\\\\0 & 0 & -1\\end{bmatrix}\\preceq 0,\\\\A_{24} & =\\begin{bmatrix}1 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 1\\end{bmatrix}\\succeq 0,\\\\A_{12} & =\\begin{bmatrix}-2 & 0 & 0\\\\0 & -1 & 0\\\\0 & 0 & -1\\end{bmatrix}\\prec 0,$ and $A_{15}=-A_{23},A_{45}=A_{12},A_{34}=-A_{12}$ .", "Note that ${\\bf null}(A_{23})={\\bf null}(A_{15})={\\bf span}\\lbrace [\\begin{array}{ccc}1 & 1 & 0\\end{array}]^{T}\\rbrace $ and ${\\bf null}(A_{24})={\\bf span}\\lbrace [\\begin{array}{ccc}0 & 1 & 0\\end{array}]^{T}\\rbrace $ .", "The null space of the weight matrices of the remaining edges are trivially spanned by the zero vector.", "Under the above selection of edge weights, we examine the evolution of multi-agent system (REF ) on $\\mathcal {G}_{1}$ , yielding the state trajectories of each agent shown in Figure REF .", "Despite the fact that $\\mathcal {G}_{1}$ is structurally imbalanced, a bipartition of agents emerges by their steady-states, namely the agents of $\\mathcal {V}_{a}=\\lbrace 2,3,4\\rbrace $ converge to $[\\begin{array}{ccc}1.6525 & 1.6525 & 0\\end{array}]^{T}$ , others of $\\mathcal {V}_{b}=\\lbrace 1,5\\rbrace $ converge to $[\\begin{array}{ccc}-1.6525 & -1.6525 & 0\\end{array}]^{T}$ , and both of the steady states are spanned by $[\\begin{array}{ccc}1 & 1 & 0\\end{array}]^{T}$ .", "What we have noticed is that $[\\begin{array}{ccc}1 & 1 & 0\\end{array}]^{T}$ happens to span the null space of $A_{23}$ ; interestingly, by negating the sign of $A_{23}$ , the resulting network becomes structurally balanced and the structurally balanced partition of nodes is precisely $\\mathcal {V}_{a}=\\lbrace 2,3,4\\rbrace $ and $\\mathcal {V}_{b}=\\lbrace 1,5\\rbrace $ .", "Figure: State trajectories of multi-agent dynamics ()on 𝒢 1 \\mathcal {G}_{1}.The edge $e_{23}$ is, therefore, deemed highly correlated with the bipartite consensus solution of the network in Figure REF .", "A comparison with edges $e_{24}$ and $e_{34}$ elaborates on how $e_{23}$ is being essential in realizing the bipartite consensus.", "First, we notice that both $e_{23}$ and $e_{24}$ are of non-trivial null space, yet the dynamics did not converge to ${\\bf null}(A_{24})={\\bf span}\\lbrace [\\begin{array}{ccc}0 & 1 & 0\\end{array}]^{T}\\rbrace $ , presumably because the negation of $e_{24}$ does not yield any structurally balanced partition of the graph, which means $e_{24}$ is structurally unimportant.", "As for $e_{23}$ and $e_{34}$ , both being structurally critical in the above sense (structurally balanced partition $\\mathcal {V}_{a}=\\lbrace 1,3,5\\rbrace ,\\mathcal {V}_{b}=\\lbrace 2,4\\rbrace $ for $e_{34}$ ), we notice that the bipartition of the convergence has abided by the grouping of $e_{23}$ instead of that of $e_{34}$ , presumably because ${\\bf null}(A_{34})$ is merely ${\\bf span}\\lbrace {\\bf 0}\\rbrace $ and algebraically trivial.", "The edge $e_{23}$ is thus intuitively significant since it is the only edge in the graph that is non-trivial in both senses; it is where the structural and algebraic properties of the network intersect.", "Motivated by the above observations, we make the assumption that the bipartite consensus of the matrix-weighted network indicates the existence of a set of edges with non-trivial intersecting null spaces and, by their negation, restore the potential structural balance of the network.", "For the rest of this paper, we attempt to make a valid definition of these edges, and to examine the role they ought to play in the bipartite consensus of multi-agent systems on matrix-weighted networks." ], [ "Balancing Set of Matrix-weighted Networks", "A prominent structural feature of the edge we studied in Example REF concerns its negation of the sign, by which the structurally imbalanced network is rendered structurally balanced.", "This method was studied in [9], [7] which suggested that given any structurally imbalanced network, one can always transform it into a structurally balanced one with any preassigned node partition by negating the signs of the relevant edges.", "We refer to this approach in the literature and introduce the following concept to embed it in the context of the matrix-weighted multi-agent networks.", "Definition 4 (Balancing set) Let $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ denote a bipartition of node set $\\mathcal {V}$ in a network $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathcal {A})$ .", "Define the $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ as a set of edges such that a $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ structurally balanced network can be obtained if the sign of each edge in $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ is negated.", "Example 2 In Figure REF we have constructed a matrix-weighted network $\\mathcal {G}$ that is structurally imbalanced.", "Given a node partition $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ of $\\mathcal {G}$ , its balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ is consisted of the negative connections within $\\mathcal {V}_{1}$ and $\\mathcal {V}_{2}$ and the positive connections between them (edges colored in red).", "By negating the signs of the edges in $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ , we derive a graph $\\mathcal {G}^{\\prime }$ that is structurally balanced between $\\mathcal {V}_{1}$ and $\\mathcal {V}_{2}$ .", "Remark 1 The $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ is empty if and only if the network is $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ structurally balanced.", "For instance, the balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ of the structurally balanced $\\mathcal {G}^{\\prime }$ in Figure REF is empty since there is no edge to be negated.", "Figure: The negation operation in Example .Note that the matrix-valued weight plays a role in shaping the null space of the matrix-valued graph Laplacian, thus more constraints on the balancing set are needed to complete the definition.", "We proceed to quantitatively characterize the contribution of matrix-valued weights to the null space of the matrix-valued graph Laplacian.", "Definition 5 A set of matrices $A_{i}\\in \\mathbb {R}^{n\\times n},i\\in \\underline{l},$ are said to have non-trivial intersection of null spaces, or, to have non-trivial intersecting null space if $\\bigcap _{i=1}^{l}{\\bf null}(A_{i})\\ne \\left\\lbrace {\\bf 0}\\right\\rbrace .$ Definition 6 (Non-trivial balancing set) A $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ of a matrix-weighted network $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathcal {A})$ is a non-trivial $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ balancing set (NBS), denoted by $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ , if the weight matrices associated with edges in $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ have non-trivial intersection of null spaces.", "Remark 2 The non-trivial balancing set $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})=\\emptyset $ if and only if the corresponding balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})=\\emptyset $ .", "In this case, define $\\mathcal {W}(\\mathcal {E}^{nb})={\\bf 0}$ where ${\\bf 0}$ is the $d\\times d$ zero matrix.", "A matrix-weighted network $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathcal {A})$ has a unique non-trivial balancing set if there is only one bipartition $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ of $\\mathcal {V}$ such that the corresponding $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ -balancing set meets $\\bigcap _{e\\in \\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})}{\\bf null}(\\mathcal {W}(e))\\ne \\left\\lbrace {\\bf 0}\\right\\rbrace .$ In this case, we shall denote $\\text{{\\bf null}}(\\mathcal {E}^{nb})=\\bigcap _{e\\in \\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})}{\\bf null}(\\mathcal {W}(e))$ for brevity.", "Remark 3 It is noteworthy that to have only an empty non-trivial balancing set does not suggest there is no NBS in the graph; a graph without NBS is a structurally imbalanced graph for which any partition of nodes has a balancing set whose weight matrices share a trivial intersecting null space." ], [ "General Networks", "In this section, we set to examine the validity of the concept termed as the non-trivial balancing set through its correlation with the network steady-state, bearing in mind the question of to what extent is the non-trivial balancing set a satisfactory interpretation of the numerical solutions.", "We shall recall some facts about the algebraic structure of the Laplacian null space when bipartite consensus is achieved on matrix-weighted networks.", "Also, technical preparations Lemma REF and Lemma REF are presented for the proof of Theorem REF and Theorem REF , one is referred to the Appendix for their proofs.", "Lemma 2 [17] The matrix-weighted multi-agent network (REF ) achieves bipartite consensus if and only if there exists a gauge transformation $D$ such that $\\text{{\\bf null}}(L)=\\mathcal {S}=\\text{{\\bf span}}\\lbrace D(\\text{\\textbf {1}}_{n}\\otimes \\Psi )\\rbrace $ , where $D$ is a gauge transformation, $\\Psi =[\\psi _{1},\\psi _{2},...,\\psi _{s}]$ , and $\\psi _{i}$ , $i\\in \\underline{s}$ , $s\\le d$ , are orthogonal basis vectors in $\\mathbb {R}^{d}$ .", "Remark 4 The space $\\mathcal {S}$ in Lemma REF is defined as the bipartite consensus subspace in [17].", "And in [12], when $s=d$ , $\\mathcal {S}$ is proved to be the bipartite consensus subspace of a structurally balanced matrix-weighted network.", "Lemma 3 Let $\\mathcal {G}=(\\mathcal {V},\\mathcal {E},\\mathcal {A})$ be a matrix-weighted network with $n$ agents, each of dimension $d$ .", "If $\\mathcal {G}$ is structurally balanced with partition $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ , then there exists a gauge transformation $D\\in \\mathbb {R}^{nd\\times nd}$ such that ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes I_{d})\\rbrace \\subset \\text{{\\bf null}}(L(\\mathcal {G}))$ , and ${\\bf blk}_{ii}(D)=I_{d}$ for $i\\in \\mathcal {V}_{1}$ and ${\\bf blk}_{ii}(D)=-I_{d}$ for $i\\in \\mathcal {V}_{2}$ , where $L(\\mathcal {G})$ is the network Laplacian.", "Lemma 4 For any nonzero $v_{1},v_{2},v\\in \\mathbb {R}^{d}$ , the linear combination of $D_{1}(\\text{\\textbf {1}}_{n}\\otimes v_{1})$ and $D_{2}(\\text{\\textbf {1}}_{n}\\otimes v_{2})$ do not yield $D(\\text{\\textbf {1}}_{n}\\otimes v)$ whether or not $v_{1}$ and $v_{2}$ are linearly independent, where $D$ is a gauge transformation, $D_{1},D_{2}$ are gauge transformations with $D_{1}\\ne D_{2}$ and $D_{1}\\ne -D_{2}$ .", "In the Preliminary section, the gauge transformation is introduced as a block matrix $D={\\bf blkdiag}\\lbrace I_{d},-I_{d}\\rbrace $ for which the identity blocks $I_{d},-I_{d}$ are sequenced on the diagonal with a certain pattern of signs.", "Since the gauge matrix $D$ has $N$ $Nd$ -by-$d$ column blocks, the non-zero diagonal blocks could be considered to have a one-to-one correspondence with the well-indexed $N$ agents, reflecting either the relative positivity/negativity of their steady states, or their bipartition by structure.", "To exert this correspondence, consider a matrix-weighted network $\\mathcal {G}$ for which a partition $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ of the node set $\\mathcal {V}$ defines a balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ .", "We map this partition onto a gauge matrix $D$ such that ${\\bf blk}_{ii}(D)=I_{d}$ for $i\\in \\mathcal {V}_{1}$ and ${\\bf blk}_{ii}(D)=-I_{d}$ for $i\\in \\mathcal {V}_{2}$ , suppose the nodes are properly indexed.", "The node partition is then fully described by the pattern of the signs of the diagonal blocks, and we phrase $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ as a balancing set with division $D$ or of $D$ -division to notify how the nodes are actually partitioned given the edge set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ .", "We are ready to establish how the non-trivial balancing set is related to the null space of the matrix-valued Laplacians.", "Theorem 1 For a matrix-weighted network $\\mathcal {G}$ , the following properties are equivalent: 1) there exists a non-trivial balancing set $\\mathcal {E}^{nb}$ in $\\mathcal {G}$ with division $D$ , such that ${\\bf span}\\lbrace \\Xi \\rbrace \\subset {\\bf null}(\\mathcal {E}^{nb})$ , 2) ${\\bf span}\\lbrace D(\\text{\\textbf {1}}_{n}\\otimes \\Xi )\\rbrace \\subset \\text{{\\bf null}}(L(\\mathcal {G}))$ , where $D$ is a gauge transformation and $\\Xi =[\\xi _{1},...,\\xi _{r}]$ where $\\xi _{i}\\in \\mathbb {R}^{d},i\\in \\underline{r},0<r\\leqslant d$ are linearly independent.", "Proof For any matrix-weighted network the graph Laplacian can be expressed as $L=H^{T}\\text{{\\bf blkdiag}}\\lbrace \|A_{k}\|\\rbrace H,$ where $H$ is the signed incidence matrix and the blocks in $\\text{{\\bf blkdiag}}\\lbrace \|A_{k}\|\\rbrace $ are ordered the same as their appearances in $H$ .", "Thus, $Lx={\\bf 0}$ if and only if $\\text{{\\bf blkdiag}}\\lbrace \|A_{k}\|\\rbrace ^{\\frac{1}{2}}Hx={\\bf 0}$ , namely, $\|A_{ij}\|^{\\frac{1}{2}}(x_{i}-\\text{{\\bf sgn}}(A_{ij})x_{j})={\\bf 0},\\forall (i,j)\\in \\mathcal {E}.$ Note that $(x_{i}-\\text{{\\bf sgn}}(A_{ij})x_{j})^{T}\|A_{ij}\|(x_{i}-\\text{{\\bf sgn}}(A_{ij})x_{j})\\\\=\|\|\|A_{ij}\|^{\\frac{1}{2}}(x_{i}-\\text{{\\bf sgn}}(A_{ij})x_{j})\|\|^{2} & = & {\\bf 0},$ then $\|A_{ij}\|^{\\frac{1}{2}}(x_{i}-\\text{{\\bf sgn}}(A_{ij})x_{j})={\\bf 0}$ if and only if $A_{ij}(x_{i}-\\text{{\\bf sgn}}(A_{ij})x_{j})={\\bf 0},$ which implies $Hx\\in \\text{{\\bf null}}(\\text{{\\bf blkdiag}}\\lbrace A_{k}\\rbrace )$ where the weight matrices $A_{ij}$ are relabelled as $A_{k},k\\in \\underline{\|\\mathcal {E}\|}$ .", "1) → 2): Consider when the network has a non-trivial balancing set $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ with division $D$ .", "Without loss of generality, we block the signed incidence matrix $H$ as $H=[H_{1}^{T}H_{2}^{T}]^{T}$ where $H_{2}$ corresponds to edges in $\\mathcal {E}^{nb}$ whose weights have intersecting null space that is non-trivial.", "We know from Definition REF that the edges in $H_{1}$ constructs a structurally balanced subgraph, and since $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ is with division $D$ , the blocks of $D$ are assigned as ${\\bf blk}_{ii}(D)=I_{d}$ for $i\\in \\mathcal {V}_{1}$ and ${\\bf blk}_{ii}(D)=-I_{d}$ for $i\\in \\mathcal {V}_{2}$ .", "From Lemma REF the gauge transformation $D$ satisfies ${\\color {black}{\\color {black}{\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes I_{d})\\rbrace \\subset {\\bf null}(H_{1}^{T}H_{1})}}$ because $H_{1}^{T}H_{1}$ is the Laplacian matrix of a structurally balanced network with the same topology as the $H_{1}$ subgraph except the absolute weights are all identity matrices.", "It is then derived that ${\\bf span}\\lbrace H_{1}D(\\text{\\textbf {1}}_{n}\\otimes I_{d})\\rbrace =\\lbrace {\\bf 0}\\rbrace $ .", "Now consider $H_{2}D(\\text{\\textbf {1}}_{n}\\otimes \\xi _{p})$ where $\\xi _{p}\\in {\\bf span}\\lbrace \\xi _{1},...,\\xi _{r}\\rbrace & \\subset & \\text{{\\bf null}}(\\mathcal {E}^{nb})\\\\& = & \\bigcap _{A_{j}\\in \\mathcal {W}(\\mathcal {E}^{nb})}\\text{{\\bf null}}(A_{j}),p\\in \\underline{r}.$ For any row block $H_{2}^{j}\\in \\mathbb {R}^{d\\times nd}$ of $H_{2}$ , it is composed of either $\\pm I_{d}$ or 0, and the corresponding weight matrix $A_{j}$ breaks the structural balance of the $H_{1}$ subgraph.", "That means, if $H_{2}^{j}$ has two $I_{d}$ matrices, the corresponding weight matrix is negative (semi-)definite, and the $H_{1}$ subgraph puts the connected vertices in the same partition (e.g., both are in $\\mathcal {V}_{1}$ ), therefore $H_{2}^{j}D$ has two $I_{d}$ matrices or two $-I_{d}$ matrices.", "If $H_{2}^{j}$ has $I_{d}$ and $-I_{d}$ , the corresponding weight matrix is positive (semi-)definite and the $H_{1}$ subgraph puts the connected vertices in different partitions (one in $\\mathcal {V}_{1}$ and the other in $\\mathcal {V}_{2}$ ), then $H_{2}^{j}D$ has two $I_{d}$ matrices or two $-I_{d}$ matrices.", "Thus $H_{2}D({\\bf 1}_{n}\\otimes \\xi _{p})=2[\\pm \\xi _{p}^{T},...,\\pm \\xi _{p}^{T}]^{T}\\in \\mathbb {R}^{\|\\mathcal {E}^{nb}\|d\\times 1}$ and since $\\text{{\\bf span}}\\lbrace {\\bf 1}_{n}\\otimes \\xi _{p}\\rbrace \\subset \\text{{\\bf span}}\\lbrace \\text{\\textbf {1}}_{n}\\otimes I_{d}\\rbrace $ , $HD(\\text{\\textbf {1}}_{n}\\otimes \\xi _{p}) & =\\left[\\begin{array}{c}H_{1}D({\\bf 1}_{n}\\otimes \\xi _{p})\\\\H_{2}D({\\bf 1}_{n}\\otimes \\xi _{p})\\end{array}\\right]\\nonumber \\\\& =\\left[\\begin{array}{c}{\\bf 0}_{(\|\\mathcal {E}\|-\|\\mathcal {E}^{nb}\|)d}\\\\\\pm 2\\xi _{p}\\\\\\vdots \\\\\\pm 2\\xi _{p}\\end{array}\\right].$ Block the matrix $A$ as $A={\\bf blkdiag}\\lbrace \\mathcal {A}_{1},\\mathcal {A}_{2}\\rbrace $ , where matrix $\\mathcal {A}_{1}$ has the weight matrices of the $H_{1}$ subgraph as its diagonal blocks, and matrix $\\mathcal {A}_{2}$ , the $H_{2}$ subgraph.", "Then we have $\\text{{\\bf blkdiag}}\\lbrace \\mathcal {A}_{1},\\mathcal {A}_{2}\\rbrace HD(\\text{\\textbf {1}}_{n}\\otimes \\xi _{p})={\\bf 0}_{\|\\mathcal {E}\|d}.$ Because the $H_{2}$ subgraph contains edges in $\\mathcal {E}^{nb}$ and $\\xi _{p}\\in \\bigcap _{A_{j}\\in \\mathcal {W}(\\mathcal {E}^{nb})}\\text{{\\bf null}}(A_{j})$ , therefore $\\mathcal {A}_{2}H_{2}D({\\bf 1}_{n}\\otimes \\xi _{p})={\\bf 0}$ holds, for $p\\in \\underline{r}$ .", "Hence we have proved that ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes \\Xi )\\rbrace \\subset {\\bf null}(L)$ .", "2) → 1): Consider when the Laplacian has $D(\\text{\\textbf {1}}_{n}\\otimes \\xi _{p})\\subset \\text{{\\bf null}}(L)$ for $p\\in \\underline{r}$ , which means to assign the nodes with $+\\xi _{p}$ or $-\\xi _{p}$ according to the sign pattern of $D$ satisfies $A_{ij}(x_{i}-{\\bf sgn}(A_{ij})x_{j})={\\bf 0},\\forall (i,j)\\in \\mathcal {E}$ .", "The sign pattern of $D$ corresponds to a partition $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ which defines a balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ on the graph.", "For any edge $e_{lm}\\in \\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ , it either a) connects within $\\mathcal {V}_{1}$ or $\\mathcal {V}_{2}$ and has $A_{lm}\\prec 0(A_{lm}\\preceq 0)$ , or b) connects between $\\mathcal {V}_{1}$ and $\\mathcal {V}_{2}$ and has $A_{lm}\\succ 0(A_{lm}\\succeq 0)$ .", "Note that those in $\\mathcal {V}_{1}$ are assigned with $+\\xi _{p}$ by $D(\\text{\\textbf {1}}_{n}\\otimes \\xi _{p})$ while those in $\\mathcal {V}_{2}$ are assigned with $-\\xi _{p}$ .", "Therefore for a), eqn.", "(REF ) gives $A_{lm}(\\xi _{p}+\\xi _{p})={\\bf 0}$ and $\\xi _{p}\\in {\\bf null}(A_{lm})$ ; for b), eqn.", "(REF ) gives $A_{lm}(\\xi _{p}-(-\\xi _{p}))={\\bf 0}$ and $\\xi _{p}\\in {\\bf null}(A_{lm})$ .", "We now have for all $e_{lm}\\in \\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ , there is $\\xi _{p}\\in {\\bf null}(A_{lm}),p\\in \\underline{r}$ , thus the balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ with division $D$ has non-trivial intersecting null space, and ${\\bf span}\\lbrace \\Xi \\rbrace \\subset {\\bf null}(\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2}))$ .", "Corollary 1 For a non-trivial balancing set $\\mathcal {E}^{nb}$ in $\\mathcal {G}$ with division $D$ , it holds that $D({\\bf 1}_{n}\\otimes {\\bf null}(\\mathcal {E}^{nb}))\\subset {\\bf null}(L)$ .", "Proof This is a direct inference from the proposition 1) → 2) in Theorem REF by considering the bases that span ${\\bf null}(\\mathcal {E}^{nb})$ as $\\Xi $ .", "Theorem REF has illustrated how the existence of a non-trivial balancing set in the network interchanges with a set of vectors ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes \\Xi )\\rbrace $ in the Laplacian null space.", "Particularly, proposition 2) → 1) shows that as long as ${\\bf null}(L)$ includes vectors of the form ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes \\Xi )\\rbrace $ , they indicate the existence of a non-trivial balancing set $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ in $\\mathcal {G}$ whose partition follows the sign pattern of $D$ , and the columns of $\\Xi $ are included in the non-trivially intersecting null space ${\\bf null}(\\mathcal {E}^{nb})$ .", "The correlation plays a central role in establishing the fact that when bipartite consensus is admitted, there is at least one NBS in the matrix-weighted network.", "We are now able to derive a necessary condition on the bipartite consensus for matrix-weighted networks in general.", "Theorem 2 If the multi-agent system (REF ) admits a bipartite consensus solution with a steady state $\\bar{x}\\ne {\\bf 0}$ , then there exists a unique non-trivial balancing set $\\mathcal {E}^{nb}$ in $\\mathcal {G}$ such that $\\bar{x}_{i}\\in {\\bf null}(\\mathcal {E}^{nb})$ for all $i\\in \\mathcal {V}$ .", "Proof If bipartite consensus is achieved on the matrix-weighted network $\\mathcal {G}$ , then by Lemma REF there is ${\\bf null}(L)=\\text{{\\bf span}}\\lbrace D(\\text{\\textbf {1}}_{n}\\otimes \\Psi )\\rbrace $ where $\\psi _{1},...,\\psi _{r}$ are orthogonal vectors, and for the steady state we have $\\bar{x}\\in {\\bf span}\\lbrace D(\\text{\\textbf {1}}_{n}\\otimes \\Psi )\\rbrace $ , i.e., $\\bar{x}_{i}\\in {\\bf span}\\lbrace \\Psi \\rbrace $ .", "According to Theorem REF , ${\\bf span}\\lbrace D(\\text{\\textbf {1}}_{n}\\otimes \\Psi )\\rbrace \\subset {\\bf null}(L)$ implies that there exists a non-trivial balancing set $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ with division $D$ such that ${\\bf span}\\lbrace \\Psi \\rbrace \\subset {\\bf null}(\\mathcal {E}^{nb})$ .", "We will first show that actually, ${\\bf span}\\lbrace \\Psi \\rbrace ={\\bf null}(\\mathcal {E}^{nb})$ by raising the fact that $rank({\\bf null}(L))=rank(D({\\bf 1}_{n}\\otimes \\Psi ))=rank({\\bf 1}_{n}\\otimes \\Psi )=rank({\\bf 1}_{n})\\cdot rank(\\Psi )=rank(\\Psi )$ .", "If there is $\\psi ^{}\\in \\mathbb {R}^{d}$ for which $\\psi ^{}\\in {\\bf null}(\\mathcal {E}^{nb})$ but $\\psi ^{}\\notin {\\bf span}\\lbrace \\Psi \\rbrace $ , proposition 1) → 2) of Theorem REF states that $D({\\bf 1}_{n}\\otimes \\psi ^{})\\in {\\bf null}(L)$ , then let $\\Psi ^{\\prime }=[\\psi _{1},...,\\psi _{r},\\psi ^{}]$ , there is ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes \\Psi ^{\\prime })\\rbrace \\subset {\\bf null}(L)$ and ${\\bf null}(L)$ is raised by rank one.", "Therefore there exists $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ with division $D$ such that ${\\bf null}(L)=D(\\text{\\textbf {1}}_{n}\\otimes {\\bf null}(\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})))$ .", "Now suppose there exists another partition $(\\mathcal {V}_{1}^{^{\\prime }},\\mathcal {V}_{2}^{^{\\prime }})$ with a corresponding non-trivial balancing set $\\mathcal {E}^{nb}(\\mathcal {V}_{1}^{^{\\prime }},\\mathcal {V}_{2}^{^{\\prime }})$ , then by Corollary REF , $x^{}\\in D^{^{\\prime }}(\\text{\\textbf {1}}_{n}\\otimes \\text{{\\bf null}}(\\mathcal {E}^{nb}(\\mathcal {V}_{1}^{^{\\prime }},\\mathcal {V}_{2}^{^{\\prime }})))\\subset \\text{{\\bf null}}(L),$ while $D\\ne D^{^{\\prime }}$ and $D\\ne -D^{^{\\prime }}$ .", "Meanwhile Lemma REF states that since $\\bar{x}$ and $x^{}$ are composed of distinct gauge transformations $D$ and $D^{\\prime }$ , their linear combination does not yield any vector of the form $D({\\bf 1}_{n}\\otimes v)$ for $v\\in \\mathbb {R}^{d}$ , which implies $k_{1}\\bar{x}+k_{2}x^{}\\notin {\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes \\Xi )\\rbrace $ .", "Therefore one has $k_{1}\\bar{x}+k_{2}x^{}\\notin \\text{{\\bf null}}(L),$ for $k_{1},k_{2}\\ne 0$ , which is a contradiction.", "Thus $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ is the only non-trivial balancing set in the network and satisfies ${\\bf null}(\\mathcal {E}^{nb})={\\bf span}\\lbrace \\Xi \\rbrace $ , then the agents converge to the non-trivial intersecting null space of the unique NBS and $\\bar{x}_{i}\\in {\\bf null}(\\mathcal {E}^{nb})$ .", "A non-trivial balancing set is defined by a partition of nodes that aims at achieving structural balance on itself; Theorem REF states that when bipartite consensus is admitted on a matrix-weighted network, one is bound to find a non-trivial balancing set, a set of edges, as a third party that prevents the structural balance from happening, and is with a non-trivial intersecting null space.", "For any other grouping of agents, one would find their corresponding balancing set to have null spaces that intersect only trivially.", "Looking back on the definition of the non-trivial balancing set, we see that when the NBS is somehow unique, the bipartition of the agents' convergence states is mirrored in the particular grouping $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ of this NBS (which is also encoded in $D$ ).", "Even more noteworthy is the intersecting null space ${\\bf null}(\\mathcal {E}^{nb})$ that directly contributes to the Laplacian null space, as is indicated by ${\\bf null}(L)=D(\\text{\\textbf {1}}_{n}\\otimes \\text{{\\bf null}}(\\mathcal {E}^{nb}))$ , which means the agents converge to a linear combination of the vectors that span ${\\bf null}(\\mathcal {E}^{nb})$ .", "Based on this impression, the definition of the NBS is rather a rephrasing of those vectors of the crucial form $D(\\text{\\textbf {1}}_{n}\\otimes \\xi )$ whose role is immediately twofold: to split the agents into groups and to grant the convergence state of the network.", "Networks with A Positive-negative Spanning Tree In this subsection, we examine the matrix-weighted network with a positive-negative spanning tree.", "The following theorem is derived with respect to the non-trivial balancing set.", "Theorem 3 For a matrix-weighted network $\\mathcal {G}$ with a positive-negative spanning tree, under protocol (REF ), we have: 1) bipartite consensus is admitted if and only if it has a unique non-trivial balancing set; 2) trivial consensus is admitted when no non-trivial balancing set is present in the graph.", "Proof First we provide the proof of part 1).", "(Sufficiency) When the network has a unique non-trivial balancing set $\\mathcal {E}^{nb}$ with division $D$ , Corollary REF suggests that $D(\\text{\\textbf {1}}_{n}\\otimes \\text{{\\bf null}}(\\mathcal {E}^{nb}))\\subset \\text{{\\bf null}}(L)$ .", "We proceed to show that in fact, in the presence of the positive-negative spanning tree, $D({\\bf 1}_{n}\\otimes \\text{{\\bf null}}(\\mathcal {E}^{nb}))$ span the whole Laplacian null space.", "We know from the previous proof that to derive $\\text{{\\bf null}}(L)$ is to solve for $x$ that satisfies a series of equations $A_{ij}(x_{i}-\\text{{\\bf sgn}}(A_{ij})x_{j})={\\bf 0},\\forall (i,j)\\in \\mathcal {E}.$ Because for any two nodes there is a path with edges whose weight matrices are all positive (negative) definite, solve equation REF along the path, we can only derive $x_{p}=x_{q}$ or $x_{p}=-x_{q}$ for any pair of nodes $p,q\\in \\mathcal {V}$ .", "Thus any solution in the Laplacian null space could be represented as $D^{\\prime }({\\bf 1}_{n}\\otimes w)$ for some $w\\in \\mathbb {R}^{d}$ and a gauge matrix $D^{\\prime }$ .", "Suppose there exists $x^{}=D^{\\prime }(1_{n}\\otimes w)\\in \\text{{\\bf null}}(L)$ with $D^{\\prime }\\ne D$ and $D^{\\prime }\\ne -D$ , then according to Theorem REF , there exists a non-trivial balancing set $\\mathcal {E}^{nb}$ with division $D^{\\prime }$ in the network which contradicts our premise that $\\mathcal {E}^{nb}$ is unique.", "Now suppose there is $x^{}=D({\\bf 1}_{n}\\otimes w)\\in \\text{{\\bf null}}(L)$ with $w\\notin \\text{{\\bf null}}(\\mathcal {E}^{nb})$ , then with equation (REF ) we have $\\text{{\\bf blkdiag}}\\lbrace A_{k}\\rbrace HD(\\text{\\textbf {1}}_{n}\\otimes w)\\ne {\\bf 0}$ , hence $D(\\text{\\textbf {1}}_{n}\\otimes w)\\notin \\text{{\\bf null}}(L)$ which is clearly a contradiction.", "Therefore the Laplacian null space is spanned by $D(\\text{\\textbf {1}}_{n}\\otimes \\text{{\\bf null}}(\\mathcal {E}^{nb}))$ , and bipartite consensus is admitted.", "(Necessity) This is readily verified with Theorem REF .", "Now we proceed to prove part 2).", "We have established that in the presence of a positive-negative spanning tree, for any $p,q\\in \\mathcal {V}$ there is $x_{p}=\\pm x_{q}$ by solving (REF ) along the positive-negative definite path that connects them.", "As the tree contains no circle, the relative positivity and negativity of $x_{p}$ and $x_{q}$ naturally gives a partition $(\\mathcal {V}_{1},\\mathcal {V}_{2})$ of the node set which defines a balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ .", "Note that the tree itself is structurally balanced with respect to this division, thus the edges of the tree are not included in $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ ; in other words, edges in $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ either has $A_{ij}\\prec 0(A_{ij}\\preceq 0)$ and connects within $\\mathcal {V}_{1}$ or $\\mathcal {V}_{2}$ (where the solution has been $x_{p}=x_{q}$ ), or has $A_{ij}\\succ 0(A_{ij}\\succeq 0)$ and connects between $\\mathcal {V}_{1}$ and $\\mathcal {V}_{2}$ (where the solution has been $x_{p}=-x_{q}$ ).", "When no non-trivial balancing set is found in the network, it means there exist two edges $\\mathcal {W}(e_{1})=\\mathcal {W}((i_{k_{1}},i_{k_{1}+1}))=A_{1}$ and $\\mathcal {W}(e_{2})=\\mathcal {W}((i_{k_{2}},i_{k_{2}+1}))=A_{2}$ in the balancing set $\\mathcal {E}^{b}(\\mathcal {V}_{1},\\mathcal {V}_{2})$ that have ${\\bf null}(A_{1})\\cap {\\bf null}(A_{2})=\\lbrace {\\bf 0}\\rbrace $ .", "Both edges satisfy $\\begin{array}{c}x_{k_{1}}-\\text{{\\bf sgn}}(A_{1})x_{k_{1}+1}=2x_{k_{1}}\\in \\text{{\\bf null}}(A_{1})\\\\x_{k_{2}}-\\text{{\\bf sgn}}(A_{2})x_{k_{2}+1}=2x_{k_{2}}\\in \\text{{\\bf null}}(A_{2})\\end{array},$ and since ${\\bf null}(A_{1})\\cap {\\bf null}(A_{2})=\\lbrace {\\bf 0}\\rbrace $ , the fact that $x_{k_{1}}=\\pm x_{k_{2}}$ gives $x_{k_{1}}=x_{k_{2}}={\\bf 0}$ , therefore $x_{i}={\\bf 0}$ for $i\\in \\mathcal {V}$ and a trivial consensus is admitted on the network.", "One fact suggested by Graph Theory is that a network may be spanned by trees with different choices of edges; one could consider when there is another positive-negative spanning tree that gives a distinct partition by solving eqn.", "(REF ).", "Since no non-trivial balancing set exists in the network, i.e., every balancing set is trivial for all possible partitions of $\\mathcal {V}$ , the above reasoning applies for all positive-negative spanning trees and the conclusion stands.", "In Theorem REF , the uniqueness of the non-trivial balancing set is proposed as a necessary condition for the bipartite consensus on matrix-weighted networks with general structural features.", "Therefore in Theorem REF , we confine ourselves to matrix-weighted networks with positive-negative spanning trees and have found the uniqueness of the NBS to be both necessary and sufficient, with the assistance of the intrinsic structural balance of the positive-negative spanning tree in the sufficient part.", "We have also established that to have at least one NBS is quite necessary for such networks to admit any steady-state other than the trivial consensus.", "For the derivation of the theorems so far, we mention that though some of the discussions in the proofs have touched on the notion of structural balance, the idea itself is not engaged in the formulation of the theorems, where the steady-state behaviour is directly associated with the existence (or non-existence) of the NBS.", "It is safe to say the non-trivial balancing set has taken the place of the structural balance as a proper indication of the system behaviour, and the graph-theoretic correspondence is partly rebuilt.", "Remark 5 Considering the necessary and sufficient condition for the bipartite consensus derived on the scalar-weighted network ([1]), which is the structural balance property of the network, we are aware that this is well incorporated into the framework of Theorem REF , since all weights that are scalar are the $1\\times 1$ positive/negative definite matrix weights, and the positive-negative spanning tree naturally exists.", "A Counter Example and A Sufficient Condition It is only natural, at this point, to ask if there is any possibility for the uniqueness of the NBS to be also conveniently sufficient even in the absence of a positive-negative spanning tree.", "However, we have come to a negative conclusion on this by raising the following counter-example.", "Figure: The matrix-weighted network 𝒢 counter \\mathcal {G}_{counter} for Example 3.The red solid (resp., dashed) lines denote edges weighted by positivedefinite (resp., semi-definite) matrices; the blue solid (resp., dashed)lines denote edges weighted by negative definite (resp., semi-definite)matrices.Example 3 The matrix-weighted network $\\mathcal {G}_{counter}$ is a 7-node structurally imbalanced network with the unique non-trivial balancing set being $\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})=\\lbrace e_{23},e_{25},e_{46}\\rbrace $ , given by the illustration in Figure REF with the following weight matrix arrangements: $A_{23} & =\\begin{bmatrix}-2 & 2 & 0\\\\2 & -2 & 0\\\\0 & 0 & 0\\end{bmatrix}\\preceq 0, & v_{a}=\\begin{bmatrix}1\\\\1\\\\0\\end{bmatrix}, & v_{b}=\\begin{bmatrix}0\\\\0\\\\1\\end{bmatrix},\\\\A_{14} & =\\begin{bmatrix}1 & 1 & 0\\\\1 & 1 & 0\\\\0 & 0 & 0\\end{bmatrix}\\succeq 0, & v_{c}=\\begin{bmatrix}1\\\\-1\\\\0\\end{bmatrix}, & v_{b}=\\begin{bmatrix}0\\\\0\\\\1\\end{bmatrix},\\\\A_{47} & =\\begin{bmatrix}2 & -1 & 2\\\\-1 & 2 & -1\\\\2 & -1 & 2\\end{bmatrix}\\succeq 0, & v_{d}=\\begin{bmatrix}-1\\\\0\\\\1\\end{bmatrix},\\\\A_{12} & =\\begin{bmatrix}2 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{bmatrix}\\succ 0,$ and $A_{23}=A_{25}=A_{46},A_{12}=A_{13}=A_{45}=A_{56}=A_{17}.$ We have written down the vectors that span the null spaces of the semi-definite weight matrices.", "It is seen that the network $\\mathcal {G}_{counter}$ consists of four independent circles, three of which are negative and one is positive.", "The non-trivial balancing set must enclose edges that eliminate the negative circles simultaneously without generating any other one.", "The existence of the positive circle $\\lbrace 1,4,7\\rbrace $ has refrained the NBS from including $e_{14}$ as a result, despite that $e_{23},e_{14},e_{46}$ share the same eigenvector $v_{b}$ for the zero eigenvalue.", "$\\mathcal {E}^{nb}(\\mathcal {V}_{1},\\mathcal {V}_{2})=\\lbrace e_{23},e_{25},e_{46}\\rbrace $ is then unique as a non-trivial balancing set; however, we could see from Figure REF that the numerical solution suggests the network yields a non-trivial consensus solution, rather than a bipartite consensus solution, for the structurally imbalanced $\\mathcal {G}_{counter}$ .", "Therefore the uniqueness of the NBS alone is not a sufficient condition in any strict sense for general matrix-weighted networks.", "Figure: Numerical solutions for Example 3.", "The bipartite consensus is notadmitted despite the existence of a unique NBS.We would like to close the main study of this work with a sufficient condition on the bipartite consensus, albeit somewhat trivial, under which a node outside $\\mathcal {G}(\\mathcal {T})$ can merge with it through semi-definite paths.", "For now we denote the network with a positive-negative spanning tree and a non-trivial balancing set as $\\mathcal {G}^{nb}(\\mathcal {T})$ .", "According to Theorem REF , $\\mathcal {G}^{nb}(\\mathcal {T})$ admits the bipartite consensus, so that part of the agents converge to $\\zeta \\in \\mathbb {R}^{d}$ , of which we assume the signs to be $\\text{{\\bf sgn}}(x_{i})=1$ , while other agents would converge to $-\\zeta $ and their signs are written as $\\text{{\\bf sgn}}(x_{i})=-1$ .", "A node can merge with $\\mathcal {G}^{nb}(\\mathcal {T})$ if the expanded network obtains bipartite consensus altogether.", "Theorem 4 Consider a matrix-weighted network $\\mathcal {G}^{nb}(\\mathcal {T})$ with a positive-negative spanning tree and a non-trivial balancing set, which naturally has all its agents converging either to $\\zeta \\in \\mathbb {R}^{d}$ or $-\\zeta \\in \\mathbb {R}^{d}$ under (REF ).", "Suppose a vertex $i_{r}\\notin \\mathcal {G}^{nb}(\\mathcal {T})$ has $m$ paths $\\mathcal {P}_{k}=\\lbrace (i_{r},i_{r+1}^{k}),...,(i_{\|\\mathcal {P}_{k}\|}^{k},i_{\|\\mathcal {P}_{k}\|+1}^{k})\\rbrace ,k\\in \\underline{m}$ to reach $\\mathcal {G}^{nb}(\\mathcal {T})$ , each path has only its last vertex in $\\mathcal {G}^{nb}(\\mathcal {T})$ , which is $i_{\|\\mathcal {P}_{k}\|+1}^{k}\\in \\mathcal {G}^{nb}(\\mathcal {T}),k\\in \\underline{m}$ .", "Then $i_{r}$ merge with $\\mathcal {G}^{nb}(\\mathcal {T})$ if for any $k_{1},k_{2}\\in \\underline{m}$ , there is $\\text{{\\bf sgn}}(\\mathcal {P}_{k_{1}})\\text{{\\bf sgn}}(x_{\|\\mathcal {P}_{k_{1}}\|+1}^{k_{1}})=\\text{{\\bf sgn}}(\\mathcal {P}_{k_{2}})\\text{{\\bf sgn}}(x_{\|\\mathcal {P}_{k_{2}}\|+1}^{k_{2}})$ , and $\\bigcap _{k=1}^{m}\\text{{\\bf null}}(\\mathcal {P}_{k})=0$ .", "Proof Suppose $\\mathcal {P}_{1},\\mathcal {P}_{2}\\in \\lbrace \\mathcal {P}_{k}\\rbrace $ , $i_{\|\\mathcal {P}_{1}\|+1}^{1},i_{\|\\mathcal {P}_{2}\|+1}^{2}\\in \\mathcal {G}^{nb}(\\mathcal {T})$ , $x_{\|\\mathcal {P}_{1}\|+1}^{1},x_{\|\\mathcal {P}_{2}\|+1}^{2}$ denote their final states.", "Then there is $\\begin{array}{c}x_{r}-\\text{{\\bf sgn}}(\\mathcal {P}_{1})x_{\|\\mathcal {P}_{1}\|+1}^{1}\\in \\text{{\\bf null}}(\\mathcal {P}_{1}),\\\\x_{r}-\\text{{\\bf sgn}}(\\mathcal {P}_{2})x_{\|\\mathcal {P}_{2}\|+1}^{2}\\in \\text{{\\bf null}}(\\mathcal {P}_{2}),\\end{array}$ since $\\text{{\\bf sgn}}(\\mathcal {P}_{1})\\text{{\\bf sgn}}(x_{\|\\mathcal {P}_{1}\|+1}^{1})=\\text{{\\bf sgn}}(\\mathcal {P}_{2})\\text{{\\bf sgn}}(x_{\|\\mathcal {P}_{2}\|+1}^{2})$ , the left-hand sides are both about $x_{r}-\\text{{\\bf sgn}}(\\mathcal {P}_{1})\\text{{\\bf sgn}}(x_{\|\\mathcal {P}_{1}\|+1}^{1})\\zeta $ .", "With $\\text{{\\bf null}}(\\mathcal {P}_{1})\\bigcap \\text{{\\bf null}}(\\mathcal {P}_{2})=\\lbrace {\\bf 0}\\rbrace $ , $x_{r}$ has a unique solution $\\text{{\\bf sgn}}(\\mathcal {P}_{1})x_{\|\\mathcal {P}_{1}\|+1}^{1}$ thus is merged with $\\mathcal {G}^{nb}(\\mathcal {T})$ .", "Theorem REF extends our study of the spanning-tree case in Theorem REF to when there exist positive/negative semi-definite paths between agents, the conclusion being a sufficient condition.", "We also have the inference that on a general matrix-weighted network, if the subgraphs spanned by positive/negative-definite trees do not have their separate non-trivial balancing sets in the first place, then the network as a whole is incapable of achieving bipartite consensus.", "Simulation Example This section provides numerical examples of the theorems we have derived, based on the network constructed in Figure REF .", "Now we can see that $\\mathcal {G}_{1}$ is structurally imbalanced with a unique non-trivial balancing set $\\lbrace e_{23}\\rbrace $ , which yields a structurally balanced node partition $\\lbrace 1,5\\rbrace ,\\lbrace 2,3,4\\rbrace $ .", "Under dynamics (REF ) the agents admit bipartite consensus as in Figure REF , and the final states are determined by the intersecting null space of the NBS ${\\bf span}\\lbrace \\begin{bmatrix}1 & 1 & 0\\end{bmatrix}^{T}\\rbrace $ .", "Now suppose the edge weight $A_{34}$ is also semi-definite, and set $A_{34}=\\begin{bmatrix}1 & 0 & 1\\\\0 & 3 & 0\\\\1 & 0 & 1\\end{bmatrix}$ .", "Then $\\mathcal {G}_{1}$ has two non-trivial balancing sets that give different node partitions.", "While $\\lbrace e_{23}\\rbrace $ still partition the agents into $\\lbrace 1,5\\rbrace ,\\lbrace 2,3,4\\rbrace $ , $\\lbrace e_{34}\\rbrace $ produces partition $\\lbrace 1,5,3\\rbrace ,\\lbrace 2,4\\rbrace $ .", "Figure REF shows that bipartite consensus is not achieved under this circumstance.", "Figure: Agents do not admit bipartite consensus after the alteration of A 34 A_{34},due to the non-uniqueness of the non-trivial balancing sets.One could easily turn Figure REF into a graph without any non-trivial balancing set by setting $A_{23}$ negative-definite as $A_{23}=A_{12}$ .", "In this case $\\mathcal {G}_{1}$ has a positive-negative spanning tree, and as expected, the agents have only admitted a trivial consensus since there is no NBS in the graph, refer to Figure REF .", "Figure: Agents admit trivial consensus when 𝒢 1 \\mathcal {G}_{1} has no non-trivialbalancing set.", "Concluding Remarks In this paper, we have established the significance of the non-trivial balancing set to the bipartite consensus of matrix-weighted networks.", "It is shown that the uniqueness of such a set is a necessary condition in admitting the bipartite consensus.", "Moreover, if bipartite consensus is indeed achieved, the final states of the agents are determined by none other than the intersecting null space of the non-trivial balancing set.", "The uniqueness of the NBS is specifically studied on networks with positive-negative spanning trees, which turns out to be both necessary and sufficient for the bipartite consensus.", "Based on this conclusion, we have given the condition to extend the tree with semi-definite matrix-weighted paths while preserving the bipartite consensus on the resulting network.", "However, we are aware that this condition is formulated in an algorithmic fashion that does not involve much structural attribute of the network; for future research, we would expect the establishment of a sufficient condition for the bipartite consensus with the concept of the NBS that is more structure-based and applicable for specific control problems.", "Appendix 1.", "Proof for Lemma REF .", "Proof If $\\mathcal {G}$ is connected and $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ structurally balanced, construct a gauge transformation matrix $D$ such that ${\\bf blk}_{ii}(D)=I_{d}$ for $i\\in \\mathcal {V}_{1}$ and ${\\bf blk}_{ii}(D)=-I_{d}$ for $i\\in \\mathcal {V}_{2}$ , let $x\\in {\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes I_{d})\\rbrace $ , then $A_{ij}(x_{i}-{\\bf sgn}(A_{ij})x_{j})={\\bf 0}$ holds for all $(i,j)\\in \\mathcal {E}$ , because when $x_{i}=x_{j}$ , there is $i,j\\in \\mathcal {V}_{1}$ or $i,j\\in \\mathcal {V}_{2}$ , and ${\\bf sgn}(A_{ij})>0$ (${\\bf sgn}(A_{ij})\\ge 0$ ); when $x_{i}=-x_{j}$ , there is $i\\in \\mathcal {V}_{1}$ , $j\\in \\mathcal {V}_{2}$ or $i\\in \\mathcal {V}_{2}$ , $j\\in \\mathcal {V}_{1}$ , and ${\\bf sgn}(A_{ij})<0$ (${\\bf sgn}(A_{ij})\\le 0$ ).", "Therefore we have ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes I_{d})\\rbrace \\subset \\text{{\\bf null}}(L(\\mathcal {G}))$ .", "If $\\mathcal {G}$ is disconnected, note that $\\mathcal {G}$ is structurally balanced if and only if all its components are structurally balanced.", "Denote the components of $\\mathcal {G}$ as $\\mathcal {G}_{i}=(\\mathcal {V}_{i},\\mathcal {E}_{i},\\mathcal {A}_{i})$ where $i\\in \\underline{q}$ and $\|\\mathcal {V}_{i}\|=n_{i}$ .", "Let $L^{i}$ denote the matrix-valued Laplacian of $\\mathcal {G}_{i}$ for all $i\\in \\underline{q}$ .", "Then $L(\\mathcal {G})={\\bf blkdiag}\\lbrace L^{i}\\rbrace .$ Again there exist $D^{i}\\in \\mathbb {R}^{n_{i}d\\times n_{i}d}$ such that ${\\bf span}\\lbrace L^{i}D^{i}({\\bf 1}_{n_{i}}\\otimes I_{d})\\rbrace =\\lbrace {\\bf 0}\\rbrace $ for all $i\\in \\underline{q}$ .", "Therefore, one can choose $D={\\bf blkdiag}\\lbrace D^{i}\\rbrace $ , then there is ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes I_{d})\\rbrace \\subset \\text{{\\bf null}}(L(\\mathcal {G}))$ which completes the proof.", "Lemma 5 For a set of linearly independent vectors $v_{1},...,v_{r}\\in \\mathbb {R}^{d}$ , $2\\le r\\le d$ , with $\\forall k_{i}\\ne 0,i\\in \\underline{r}$ , the linear combination $x=k_{1}D_{1}(\\text{\\textbf {1}}_{n}\\otimes v_{1})+...+k_{r}D_{r}(\\text{\\textbf {1}}_{n}\\otimes v_{r})\\ne D(\\text{\\textbf {1}}_{n}\\otimes v)$ where $v\\in \\mathbb {R}^{d}$ and $D$ is a gauge transformation, if the sign patterns of the gauge transformations $D_{1},D_{2},...,D_{r}$ are distinct from each other, that is, there is no $D_{p},D_{q}$ with $D_{p}=D_{q}$ or $D_{p}=-D_{q},p,q\\in \\underline{r}$ .", "Proof Write $x$ in its block form as $x=\\text{{\\bf blk}}\\lbrace x_{1}^{T}x_{2}^{T}...x_{n}^{T}\\rbrace ^{T},x_{k}\\in \\mathbb {R}^{d}$ .", "Suppose we use $D_{1}(\\text{\\textbf {1}}_{n}\\otimes v_{1}),D_{2}(\\text{\\textbf {1}}_{n}\\otimes v_{2}),...,D_{r}(\\text{\\textbf {1}}_{n}\\otimes v_{r})$ for the linear combination, then $x={\\displaystyle \\sum _{j=1}^{r}}k_{j}D_{j}(\\text{\\textbf {1}}_{n}\\otimes v_{j})$ The blocks of $x$ are written as $\\begin{array}{c}x_{1}=z_{11}v_{1}+\\cdots +z_{1r}v_{r},\\\\\\vdots \\\\x_{l}=z_{l1}v_{1}+\\cdots +z_{lr}v_{r},\\\\\\vdots \\\\x_{t}=z_{t1}v_{1}+\\cdots +z_{tr}v_{r},\\\\\\vdots \\\\x_{n}=z_{n1}v_{1}+\\cdots +z_{nr}v_{r},\\end{array}$ where $\|z_{1j}\|=\|z_{2j}\|=...=\|z_{nj}\|=\|k_{j}\|$ for $j\\in \\underline{r}$ .", "The sign pattern of a gauge transformation is the sequence of signs of the diagonal blocks, $\\lbrace \\begin{array}{ccc}+1 & +1 & -1\\end{array}\\rbrace $ for $\\text{{\\bf blkdiag}}\\lbrace I_{d},I_{d},-I_{d}\\rbrace $ for instance.", "We use ${\\bf sgn}(D_{j}^{i})$ to denote the sign of the $i$ th diagonal block of gauge transformation $D_{j}$ , which can be either $+1$ or $-1$ .", "Then $z_{ij}={\\bf sgn}(D_{j}^{i})k_{j}$ .", "Note that the gauge transformations $D_{1}$ and $D_{r}$ are of different sign patterns, therefore there exist two blocks of $x$ , say, $x_{l}$ and $x_{t}$ , so that ${\\bf sgn}(D_{1}^{l})={\\bf sgn}(D_{r}^{l})$ and ${\\bf sgn}(D_{1}^{t})=-{\\bf sgn}(D_{r}^{t}).$ Suppose $x=D(1_{n}\\otimes v)$ , then we should have $x_{l}=\\pm x_{t}$ .", "If $x_{l}=x_{t}$ , then (a) suppose ${\\bf sgn}(D_{1}^{l})={\\bf sgn}(D_{1}^{t})$ , as a consequence of eqn.", "(REF ) and (REF ), there is ${\\bf sgn}(D_{r}^{l})=-{\\bf sgn}(D_{r}^{t})$ , i.e., $z_{l1}=z_{t1}$ and $z_{lr}=-z_{tr}$ .", "So when we equate $x_{l}$ and $x_{t}$ , there is $(z_{l2}-z_{t2})v_{2}+...+(z_{l,r-1}-z_{t,r-1})v_{r-1}+2z_{lr}v_{r}=0,$ then $v_{2},...,v_{r}$ becomes linearly dependent since there is at least $z_{lr}\\ne 0$ , thus we have derived a contradiction; (b) suppose ${\\bf sgn}(D_{1}^{l})=-{\\bf sgn}(D_{1}^{t})$ , then there is ${\\bf sgn}(D_{r}^{l})={\\bf sgn}(D_{r}^{t})$ , i.e., $z_{l1}=-z_{t1}$ and $z_{lr}=z_{tr}$ , so when we equate $x_{l}$ and $x_{t}$ , we have $2z_{l1}v_{1}+(z_{l2}-z_{t2})v_{2}+...+(z_{l,r-1}-z_{t,r-1})v_{r-1}=0,$ which contradicts the fact that $v_{1},...,v_{r-1}$ are linearly independent.", "For $x_{l}=-x_{t}$ , the contradictions can be derived similarly by discussing (a) ${\\bf sgn}(D_{1}^{l})={\\bf sgn}(D_{1}^{t})$ and (b) ${\\bf sgn}(D_{1}^{l})=-{\\bf sgn}(D_{1}^{t})$ .", "3.", "Proof for Lemma REF .", "Proof The case of $v_{1}$ and $v_{2}$ being linearly independent has been proved as the case of $r=2$ in Lemma REF .", "When $v_{2}=kv_{1}$ , suppose $x=\\alpha D_{1}(\\text{\\textbf {1}}_{n}\\otimes v_{1})+\\beta D_{2}(\\text{\\textbf {1}}_{n}\\otimes v_{2})=D(\\text{\\textbf {1}}_{n}\\otimes v),\\alpha \\ne 0,\\beta \\ne 0$ , then the blocks of $x$ are written as $x_{i}=({\\bf sgn}(D_{1}^{i})\\alpha +{\\bf sgn}(D_{2}^{i})\\beta )v_{1},i=1,...,n$ .", "Because we can find $x_{p}$ and $x_{q}$ with ${\\bf sgn}(D_{1}^{p})={\\bf sgn}(D_{2}^{p})$ and ${\\bf sgn}(D_{1}^{q})=-{\\bf sgn}(D_{2}^{q})$ , there is $x_{p}={\\bf sgn}(D_{1}^{p})(\\alpha +\\beta )v_{1},x_{q}={\\bf sgn}(D_{1}^{q})(\\alpha -\\beta )v_{1}$ .", "Let $x_{p}=x_{q}$ or $x_{p}=-x_{q}$ we can always derive $\\alpha =0$ or $\\beta =0$ , thus is a contradiction." ], [ "Simulation Example", "This section provides numerical examples of the theorems we have derived, based on the network constructed in Figure REF .", "Now we can see that $\\mathcal {G}_{1}$ is structurally imbalanced with a unique non-trivial balancing set $\\lbrace e_{23}\\rbrace $ , which yields a structurally balanced node partition $\\lbrace 1,5\\rbrace ,\\lbrace 2,3,4\\rbrace $ .", "Under dynamics (REF ) the agents admit bipartite consensus as in Figure REF , and the final states are determined by the intersecting null space of the NBS ${\\bf span}\\lbrace \\begin{bmatrix}1 & 1 & 0\\end{bmatrix}^{T}\\rbrace $ .", "Now suppose the edge weight $A_{34}$ is also semi-definite, and set $A_{34}=\\begin{bmatrix}1 & 0 & 1\\\\0 & 3 & 0\\\\1 & 0 & 1\\end{bmatrix}$ .", "Then $\\mathcal {G}_{1}$ has two non-trivial balancing sets that give different node partitions.", "While $\\lbrace e_{23}\\rbrace $ still partition the agents into $\\lbrace 1,5\\rbrace ,\\lbrace 2,3,4\\rbrace $ , $\\lbrace e_{34}\\rbrace $ produces partition $\\lbrace 1,5,3\\rbrace ,\\lbrace 2,4\\rbrace $ .", "Figure REF shows that bipartite consensus is not achieved under this circumstance.", "Figure: Agents do not admit bipartite consensus after the alteration of A 34 A_{34},due to the non-uniqueness of the non-trivial balancing sets.One could easily turn Figure REF into a graph without any non-trivial balancing set by setting $A_{23}$ negative-definite as $A_{23}=A_{12}$ .", "In this case $\\mathcal {G}_{1}$ has a positive-negative spanning tree, and as expected, the agents have only admitted a trivial consensus since there is no NBS in the graph, refer to Figure REF .", "Figure: Agents admit trivial consensus when 𝒢 1 \\mathcal {G}_{1} has no non-trivialbalancing set." ], [ "Concluding Remarks", "In this paper, we have established the significance of the non-trivial balancing set to the bipartite consensus of matrix-weighted networks.", "It is shown that the uniqueness of such a set is a necessary condition in admitting the bipartite consensus.", "Moreover, if bipartite consensus is indeed achieved, the final states of the agents are determined by none other than the intersecting null space of the non-trivial balancing set.", "The uniqueness of the NBS is specifically studied on networks with positive-negative spanning trees, which turns out to be both necessary and sufficient for the bipartite consensus.", "Based on this conclusion, we have given the condition to extend the tree with semi-definite matrix-weighted paths while preserving the bipartite consensus on the resulting network.", "However, we are aware that this condition is formulated in an algorithmic fashion that does not involve much structural attribute of the network; for future research, we would expect the establishment of a sufficient condition for the bipartite consensus with the concept of the NBS that is more structure-based and applicable for specific control problems." ], [ "Appendix", "1.", "Proof for Lemma REF .", "Proof If $\\mathcal {G}$ is connected and $(\\mathcal {V}_{1},\\mathcal {V}_{2})-$ structurally balanced, construct a gauge transformation matrix $D$ such that ${\\bf blk}_{ii}(D)=I_{d}$ for $i\\in \\mathcal {V}_{1}$ and ${\\bf blk}_{ii}(D)=-I_{d}$ for $i\\in \\mathcal {V}_{2}$ , let $x\\in {\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes I_{d})\\rbrace $ , then $A_{ij}(x_{i}-{\\bf sgn}(A_{ij})x_{j})={\\bf 0}$ holds for all $(i,j)\\in \\mathcal {E}$ , because when $x_{i}=x_{j}$ , there is $i,j\\in \\mathcal {V}_{1}$ or $i,j\\in \\mathcal {V}_{2}$ , and ${\\bf sgn}(A_{ij})>0$ (${\\bf sgn}(A_{ij})\\ge 0$ ); when $x_{i}=-x_{j}$ , there is $i\\in \\mathcal {V}_{1}$ , $j\\in \\mathcal {V}_{2}$ or $i\\in \\mathcal {V}_{2}$ , $j\\in \\mathcal {V}_{1}$ , and ${\\bf sgn}(A_{ij})<0$ (${\\bf sgn}(A_{ij})\\le 0$ ).", "Therefore we have ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes I_{d})\\rbrace \\subset \\text{{\\bf null}}(L(\\mathcal {G}))$ .", "If $\\mathcal {G}$ is disconnected, note that $\\mathcal {G}$ is structurally balanced if and only if all its components are structurally balanced.", "Denote the components of $\\mathcal {G}$ as $\\mathcal {G}_{i}=(\\mathcal {V}_{i},\\mathcal {E}_{i},\\mathcal {A}_{i})$ where $i\\in \\underline{q}$ and $\|\\mathcal {V}_{i}\|=n_{i}$ .", "Let $L^{i}$ denote the matrix-valued Laplacian of $\\mathcal {G}_{i}$ for all $i\\in \\underline{q}$ .", "Then $L(\\mathcal {G})={\\bf blkdiag}\\lbrace L^{i}\\rbrace .$ Again there exist $D^{i}\\in \\mathbb {R}^{n_{i}d\\times n_{i}d}$ such that ${\\bf span}\\lbrace L^{i}D^{i}({\\bf 1}_{n_{i}}\\otimes I_{d})\\rbrace =\\lbrace {\\bf 0}\\rbrace $ for all $i\\in \\underline{q}$ .", "Therefore, one can choose $D={\\bf blkdiag}\\lbrace D^{i}\\rbrace $ , then there is ${\\bf span}\\lbrace D({\\bf 1}_{n}\\otimes I_{d})\\rbrace \\subset \\text{{\\bf null}}(L(\\mathcal {G}))$ which completes the proof.", "Lemma 5 For a set of linearly independent vectors $v_{1},...,v_{r}\\in \\mathbb {R}^{d}$ , $2\\le r\\le d$ , with $\\forall k_{i}\\ne 0,i\\in \\underline{r}$ , the linear combination $x=k_{1}D_{1}(\\text{\\textbf {1}}_{n}\\otimes v_{1})+...+k_{r}D_{r}(\\text{\\textbf {1}}_{n}\\otimes v_{r})\\ne D(\\text{\\textbf {1}}_{n}\\otimes v)$ where $v\\in \\mathbb {R}^{d}$ and $D$ is a gauge transformation, if the sign patterns of the gauge transformations $D_{1},D_{2},...,D_{r}$ are distinct from each other, that is, there is no $D_{p},D_{q}$ with $D_{p}=D_{q}$ or $D_{p}=-D_{q},p,q\\in \\underline{r}$ .", "Proof Write $x$ in its block form as $x=\\text{{\\bf blk}}\\lbrace x_{1}^{T}x_{2}^{T}...x_{n}^{T}\\rbrace ^{T},x_{k}\\in \\mathbb {R}^{d}$ .", "Suppose we use $D_{1}(\\text{\\textbf {1}}_{n}\\otimes v_{1}),D_{2}(\\text{\\textbf {1}}_{n}\\otimes v_{2}),...,D_{r}(\\text{\\textbf {1}}_{n}\\otimes v_{r})$ for the linear combination, then $x={\\displaystyle \\sum _{j=1}^{r}}k_{j}D_{j}(\\text{\\textbf {1}}_{n}\\otimes v_{j})$ The blocks of $x$ are written as $\\begin{array}{c}x_{1}=z_{11}v_{1}+\\cdots +z_{1r}v_{r},\\\\\\vdots \\\\x_{l}=z_{l1}v_{1}+\\cdots +z_{lr}v_{r},\\\\\\vdots \\\\x_{t}=z_{t1}v_{1}+\\cdots +z_{tr}v_{r},\\\\\\vdots \\\\x_{n}=z_{n1}v_{1}+\\cdots +z_{nr}v_{r},\\end{array}$ where $\|z_{1j}\|=\|z_{2j}\|=...=\|z_{nj}\|=\|k_{j}\|$ for $j\\in \\underline{r}$ .", "The sign pattern of a gauge transformation is the sequence of signs of the diagonal blocks, $\\lbrace \\begin{array}{ccc}+1 & +1 & -1\\end{array}\\rbrace $ for $\\text{{\\bf blkdiag}}\\lbrace I_{d},I_{d},-I_{d}\\rbrace $ for instance.", "We use ${\\bf sgn}(D_{j}^{i})$ to denote the sign of the $i$ th diagonal block of gauge transformation $D_{j}$ , which can be either $+1$ or $-1$ .", "Then $z_{ij}={\\bf sgn}(D_{j}^{i})k_{j}$ .", "Note that the gauge transformations $D_{1}$ and $D_{r}$ are of different sign patterns, therefore there exist two blocks of $x$ , say, $x_{l}$ and $x_{t}$ , so that ${\\bf sgn}(D_{1}^{l})={\\bf sgn}(D_{r}^{l})$ and ${\\bf sgn}(D_{1}^{t})=-{\\bf sgn}(D_{r}^{t}).$ Suppose $x=D(1_{n}\\otimes v)$ , then we should have $x_{l}=\\pm x_{t}$ .", "If $x_{l}=x_{t}$ , then (a) suppose ${\\bf sgn}(D_{1}^{l})={\\bf sgn}(D_{1}^{t})$ , as a consequence of eqn.", "(REF ) and (REF ), there is ${\\bf sgn}(D_{r}^{l})=-{\\bf sgn}(D_{r}^{t})$ , i.e., $z_{l1}=z_{t1}$ and $z_{lr}=-z_{tr}$ .", "So when we equate $x_{l}$ and $x_{t}$ , there is $(z_{l2}-z_{t2})v_{2}+...+(z_{l,r-1}-z_{t,r-1})v_{r-1}+2z_{lr}v_{r}=0,$ then $v_{2},...,v_{r}$ becomes linearly dependent since there is at least $z_{lr}\\ne 0$ , thus we have derived a contradiction; (b) suppose ${\\bf sgn}(D_{1}^{l})=-{\\bf sgn}(D_{1}^{t})$ , then there is ${\\bf sgn}(D_{r}^{l})={\\bf sgn}(D_{r}^{t})$ , i.e., $z_{l1}=-z_{t1}$ and $z_{lr}=z_{tr}$ , so when we equate $x_{l}$ and $x_{t}$ , we have $2z_{l1}v_{1}+(z_{l2}-z_{t2})v_{2}+...+(z_{l,r-1}-z_{t,r-1})v_{r-1}=0,$ which contradicts the fact that $v_{1},...,v_{r-1}$ are linearly independent.", "For $x_{l}=-x_{t}$ , the contradictions can be derived similarly by discussing (a) ${\\bf sgn}(D_{1}^{l})={\\bf sgn}(D_{1}^{t})$ and (b) ${\\bf sgn}(D_{1}^{l})=-{\\bf sgn}(D_{1}^{t})$ .", "3.", "Proof for Lemma REF .", "Proof The case of $v_{1}$ and $v_{2}$ being linearly independent has been proved as the case of $r=2$ in Lemma REF .", "When $v_{2}=kv_{1}$ , suppose $x=\\alpha D_{1}(\\text{\\textbf {1}}_{n}\\otimes v_{1})+\\beta D_{2}(\\text{\\textbf {1}}_{n}\\otimes v_{2})=D(\\text{\\textbf {1}}_{n}\\otimes v),\\alpha \\ne 0,\\beta \\ne 0$ , then the blocks of $x$ are written as $x_{i}=({\\bf sgn}(D_{1}^{i})\\alpha +{\\bf sgn}(D_{2}^{i})\\beta )v_{1},i=1,...,n$ .", "Because we can find $x_{p}$ and $x_{q}$ with ${\\bf sgn}(D_{1}^{p})={\\bf sgn}(D_{2}^{p})$ and ${\\bf sgn}(D_{1}^{q})=-{\\bf sgn}(D_{2}^{q})$ , there is $x_{p}={\\bf sgn}(D_{1}^{p})(\\alpha +\\beta )v_{1},x_{q}={\\bf sgn}(D_{1}^{q})(\\alpha -\\beta )v_{1}$ .", "Let $x_{p}=x_{q}$ or $x_{p}=-x_{q}$ we can always derive $\\alpha =0$ or $\\beta =0$ , thus is a contradiction." ] ]	2011.14105
[ [ "Scalable Deep-Learning-Accelerated Topology Optimization for Additively\n Manufactured Materials" ], [ "Abstract Topology optimization (TO) is a popular and powerful computational approach for designing novel structures, materials, and devices.", "Two computational challenges have limited the applicability of TO to a variety of industrial applications.", "First, a TO problem often involves a large number of design variables to guarantee sufficient expressive power.", "Second, many TO problems require a large number of expensive physical model simulations, and those simulations cannot be parallelized.", "To address these issues, we propose a general scalable deep-learning (DL) based TO framework, referred to as SDL-TO, which utilizes parallel schemes in high performance computing (HPC) to accelerate the TO process for designing additively manufactured (AM) materials.", "Unlike the existing studies of DL for TO, our framework accelerates TO by learning the iterative history data and simultaneously training on the mapping between the given design and its gradient.", "The surrogate gradient is learned by utilizing parallel computing on multiple CPUs incorporated with a distributed DL training on multiple GPUs.", "The learned TO gradient enables a fast online update scheme instead of an expensive update based on the physical simulator or solver.", "Using a local sampling strategy, we achieve to reduce the intrinsic high dimensionality of the design space and improve the training accuracy and the scalability of the SDL-TO framework.", "The method is demonstrated by benchmark examples and AM materials design for heat conduction.", "The proposed SDL-TO framework shows competitive performance compared to the baseline methods but significantly reduces the computational cost by a speed up of around 8.6x over the standard TO implementation." ], [ "Introduction", "Topology optimization (TO) has been extensively applied to solving complex engineering design problems with a wide range of industrial applications, e.g.", "aerospace, building, and automotive.", "Unfortunately, TO has a major challenge in computational expense because it is an iterative and sequential procedure, which is not straightforward to be palatalized.", "A typical large-scale TO problem may involve hundreds or even thousands of design iterations, and for each iteration, the physical response needs to be solved to compute the gradient information through sensitivity analysis.", "To handle large-scale TO problem, which involves millions of design variables [1], [3], [15], the associated computational cost is highly prohibitive and will be larger and larger in the future since the problem are becoming more complex with more details [1].", "In general, there are several types of methods to improve the computational efficiency, including algorithm improvements [4], [20], e.g., advanced iterative solvers [21], [16], parallel computing [1], [2] including GPU computing [22], [44], [39], and deep learning techniques [32], [34], e.g., reparameterization [18], [17].", "Another typical strategy to improve the efficiency of TO simulations is to build surrogate modeling, which has been developed in a wide variety of contexts and disciplines, see [6], [9], [10], [12] and the reference therein.", "Surrogate modeling practice seeks to approximate the response of an original function (the loss function in this work), which is typically computationally expensive, by a cheaper-to-run surrogate.", "The loss function can then be evaluated by evaluating the surrogate directly without running expensive forward model executions.", "Compared to conventional TO algorithms, this approach is advantageous that it significantly reduces the number of forward model executions at a desired accuracy and allows evaluating the loss function in parallel.", "Several methods can be employed to construct the surrogate systems, including least-squares projection [26], [25], [11], [24], sparse interpolation [29], [28], [14], and compressive sensing [13], [23], [40], [42], [33], [31], etc.", "However, unlike the standard surrogate modeling task, when TO does not require an accurate surrogate model in the entire parameter space, because an optimizer only goes down a single path from the initial state to the final state to find the optimum.", "Thus, it is wasteful to build a global surrogate to the loss function.", "On the other hand, as TO is often a high-dimensional optimization problem, it is also challenging to build a global surrogate with sufficient accuracy even with the state-of-the-art neural network models.", "In this work, we first propose a general scalable framework integrating deep learning and parallel computing to accelerate the TO process for designing additively manufactured materials.", "The key idea is to dig into the iterative history data and use deep neural networks (DNN) to learn a surrogate gradient given a specific design.", "The learning step is achieved by utilizing parallel computing on multiple CPUs and a distributed DNN training on multiple GPUs, incorporated with a novel local sampling strategy, which aims to reduce the dimensionality to capture a low-dimensional search space.", "The proposed SDL-TO framework is demonstrated by several benchmark examples and compared with the start-of-the-art baseline methods.", "In the large-scale AM materials design example, SDL-TO significantly saves the computational cost and achieves a speed up of around 8.6x over the standard TO implementation." ], [ "SDL-TO: Scalable Deep Learning Accelerated Topology Optimization", "In this paper, we propose a scalable deep learning framework to accelerate TO, but we emphasize that it can be generally applied to a wide range of computational design optimization problems, specifically large-scale, high-dimensional, computationally intensive problems in computational sciences and engineering (CSE).", "The core idea is using an accurately learned gradient by DNN instead of the true gradient that is iteratively evaluated at each step to accelerate the optimization process.", "This is achieved by utilizing a local sampling strategy incorporating a scalable implementation on multiple CPUs and GPUs.", "Figure REF gives an overview of this process.", "Note that the proposed framework is universal in the sense that it can work with any gradient-based optimization based on computationally intensive physical models or simulators.", "Figure: SDL-TO framework to accelerating TO by learning the iterative history data and mapping between given design and its gradient.", "Using local sampling strategy with distributed computing and training on multiple CPUs and GPUs, the learned surrogate gradient enables an online update instead of expensive computing using a high-fidelity simulator or solver.Topology optimization formulation TO is a mathematical method that aims to optimize materials layout defined on a design domain with given boundary conditions, loads and volume constraints to minimize structural compliance $C$ (or equivalently the least strain energy).", "In this work, we use the modified Solid Isotropic Material with Penalization (SIMP) approach [36] with density-based approach to TO, where each element $e$ is assigned a density $x_e$ that determines its Young's modulus: $E_e(x_e) = E_{\\min } + x_e^p(E_0-E_{\\min }),\\quad x_e \\in [0,1]$ where $E_0$ is the stiffness of the material, $E_{\\min }$ is a very small stiffness assigned to void regions to prevent the stiffness matrix becoming singular.", "The modified SIMP approach differs from the classical SIMP approach [7], where elements with zero stiffness are avoided by using a small value.", "The modified mathematical formulation of the TO problem is $\\min _{{\\mathbf {x}}}: C(\\mathbf {x}) = \\mathbf {U}^T \\mathbf {K} \\mathbf {U} = \\sum _{e=1}^N E_e(x_e) \\mathbf {u}_e^T \\mathbf {k}_0 \\mathbf {u}_e, \\quad {\\rm s.t.}", "\\quad \\mathbf {KU=F}, \\ V(\\mathbf {x})= V_0, \\ 0 \\le \\mathbf {x} \\le 1$ where $\\mathbf {x}$ is the vector of design variables, $C$ is the structural compliance, $\\mathbf {K}$ is the global stiffness matrix, $\\mathbf {U}$ and $\\mathbf {F}$ are the global displacement and force vectors respectively, $\\mathbf {u}_e$ and $\\mathbf {k}_e$ are the element displacement vector and stiffness matrix respectively, $N$ is the number of elements used to discretize the design domain $\\Omega $ , $V(\\mathbf {x})$ and $V_0$ are the material volume and design domain volume respectively, $\\zeta $ is the prescribed volume fraction, and $p$ is the penalization power coefficient (typically $p=3$ ).", "TO baseline method The computationally limiting step in standard TO is the gradient estimation using an adjoint method [8], which typically needs to solve $\\mathbf {U=K^{-1}F}$ using the finite element method (FEM).", "The gradient is inserted into the Method of Moving Asymptotes (MMA) [38] algorithm that is the state-of-the-art optimizer, which has been extensively demonstrated to be versatile and well suited for a wide range TO problems.", "We implemented this algorithm using NumPy and compare our method with the standard TO process using MMA in a single CPU.", "Note that our scalable framework aims to accelerate the iterative optimization process and the comparison is based on the same solver and TO algorithm.", "The improvement of the TO algorithm self, the linear solver, or the physical simulator using parallel computing is out of the scope of this study.", "Online update with a local sampling strategy To large-scale TO problems, typically involving high resolution, the design space is so huge that it is very challenging to directly map the given design and its gradient using random samples.", "It is a non-trivial task to use a global surrogate model to cover the entire high-dimensional space in this case.", "To address this issue, we propose a local sampling strategy that allows us to capture the optimal low-dimensional gradient direction by generating random samples around the current design with a few history steps before.", "Since the standard TO algorithm uses gradient descent for a local search, the adjacent design does not show significant changes except for the initial stage.", "Therefore, we propose a local sampling strategy using local samples that are drawn from a Gaussian distribution $\\tilde{ \\mathbf {x}}_{rs}^{j+1} \\sim \\mathcal {N}(\\mathbf {x}_{\\textup {min}}^{j+1} +(\\mathbf {x}_{\\textup {max}}^{j+1}-\\mathbf {x}_{\\textup {min}}^{j+1})/2 , \\mathbf {\\sigma }) $ where $\\sigma $ is the standard deviation which is used to control the radius of the local search domain, and the upper and lower bound are defined by $\\tilde{ \\mathbf {x}}_{\\textup {max}}^{j+1}=\\max \\left\\lbrace \\tilde{ \\mathbf {x}}^{(j)},\\tilde{ \\mathbf {x}}^{(j-1)}, \\cdots , \\tilde{ \\mathbf {x}}^{(j-w)}\\right\\rbrace , \\quad \\tilde{ \\mathbf {x}}_{\\textup {min}}^{j+1}=\\min \\left\\lbrace \\tilde{ \\mathbf {x}}^{(j)},\\tilde{ \\mathbf {x}}^{(j-1)}, \\cdots , \\tilde{ \\mathbf {x}}^{(j-w)}\\right\\rbrace $ where $w$ is a lookback window size, which means how many history steps we will look back from the current state $\\tilde{ \\mathbf {x}}^{j+1}$ .", "As shown in Figure REF , we have a number of designs and their corresponding gradients from these local samples.", "A deep neural network is trained to map the design and its gradient and we therefore learn a accurate gradient for any specific design within this local search domain.", "Once we have the learned gradient in hand, we may continue gradient-based optimization without any expensive simulations, e.g., FEM, but a critical question is bought up: how many steps can we move forward to the optimal design?", "This is determined by defining a threshold $\\lambda ^$ of the cosine distance $d^c$ between current design $\\tilde{ \\mathbf {x}}^{j+1}$ and future design $\\tilde{ \\mathbf {x}}^{j+m}$ .", "If $d^c>\\lambda ^$ , the learning step consisting of simulation and training will be called, otherwise, the learned surrogate gradient will be used for online next step update of the design.", "Figure: Scalable implementation of the SDL-TO framework on multiple CPUs and GPUsScalable implementation of the SDL-TO framework A core contribution of the SDL-TO framework is to utilize high performance computing (HPC) sources (multiple CPUs and GPUs) to accelerate the sequentially iterative TO process in a parallel scheme.", "Figure REF shows the schema of our scalable implementation.", "Compared with the sequential TO iterative process $\\mathbf {x}_i, \\mathbf {x}_{i+1}, ..., \\mathbf {x}_{n}$ with high fidelity FEM simulation $h_s$ at each step, we accelerate optimization with an online update using the learned gradient $s_c$ incorporating with a local sampling strategy $\\ell _s$ , where local samples of $h_s$ are generated from a Gaussian distribution in Eq.", "(REF ) and sent to multiple CPUs for parallel simulations.", "Then we collect all $h_s$ data together for the distributed deep learning training on multiple GPUs to obtain a fast and accurate mapping between the current design and its gradient.", "In such a way, we only need to call a very small number of learning steps, but use the learned gradient $s_c$ for a fast design update.", "For example, we call the learning step at $\\mathbf {x}_{j+1}$ and then we can use the learned gradient for multiple later updating steps $\\mathbf {x}_{j+2}...,\\mathbf {x}_{k+1} (k>j+1)$ .", "Ideally, the total computational cost of $s_c$ equals $h_s$ but it needs to be fine-tuned and optimized due to the communication cost and a requirement of large enough computing resources.", "In this work, we use mpi4py for implementing the parallel simulation on multiple GPUs and Horovod [35] for the distributed training on multiple GPUs.", "All the parallel scheme is conducted and tested on the ORNL OLCF Summit supercomputer, where each Summit node consists of 2 IBM Power9 CPUs, 6 NVIDIA V100 GPUs, and NVLink for CPU-CPU and CPU-GPU communications." ], [ "Examples", "Benchmark bridge structure We first demonstrate our proposed framework on a benchmark bridge design problem.", "As shown in Figure REF , there are 80% void materials with $E_1=0.001$ and 20% solid materials with $E_2=1.0$ so that our goal is to optimize the layout of the materials to achieve minimum structural compliance of the support structure given a unit uniform load on the top.", "The SDL-TO framework shows the competitive performance of the objective loss values on two resolutions: 480$\\times $ 160 pixels (mesh grids) and 960$\\times $ 320 pixels (mesh grids).", "When the resolution is higher, our method outperforms the sequential TO (Seq-TO) method and shows smaller structures that benefit to improve the overall structural stiffness (the smaller objective (Obj) value, the better performance).", "Figure: Comparing baselines on the bridge structure design example on two mesh grids (480×\\times 160 pixels and 960×\\times 320 pixels).", "We found that the SDL-TO shows competitive performance on both the objective loss and the final design compared to the sequential TO that uses the MMA optimizer.The performance can be also demonstrated by a quantitative comparison of SDL-TO and Seq-TO in terms of iteration history and time-to-solution metric, as shown in Figure REF .", "The objective (loss) values are very close in terms of the iteration process.", "To illustrate the small differences, Figure REF (b) and (c) display the zoom-in results of two resolutions in the 75-200 iteration steps.", "Figure REF (d) shows the scalability performance in terms of time-to-solution.", "The Seq-TO is compared with multiple numbers of nodes including 16, 32, 64, and 128.", "It is interesting to note that the SDL-TO framework shows stronger scalability in the high-resolution case than the low-resolution case.", "This is probably due to a higher ratio of the communication overhead in the low-resolution case.", "Figure: Quantitative comparison of the SDL-TO and the Seq-TO on objective loss and time-to-solution.", "(a) Iterative history of two mesh grids, zoom-in results of (b) 480×\\times 160 pixels and (c) 960×\\times 320 pixels, and (d) time-to-solution performance as the number of nodes increases.Two materials heat conduction This is a large-scale TO problem that aims to address a challenging problem in additive manufacturing (AM): how to effectively distribute heating by optimizing the layout of two materials consisting of 40% good conductor with $k_1=1.0$ and 60% bad conductor with $k_2=0.001$ , as shown in Figure REF .", "Considering the AM constraints in practice, we use a very high resolution, 1024$\\times $ 1024 mesh grids in this case for a large-scale demonstration.", "Thus, the conventional sequential TO process is more computationally intensive than the bridge design example discussed above.", "Here, we assume to use a total of 200 iterations for the optimization design and compare the performance of the final design and time-to-solution between the Seq-TO method and SDL-TO method.", "Figure: Two materials heat conduction design problem.", "Compared with the Seq-TO algorithm that uses 200 iterations, the SDL-TO demonstrates a very close performance but only using 12 gradient learning steps.", "The other updates are achieved by using the learned surrogate gradient.By comparison of the different iterative designs, specifically on iteration 10, 20, 30, and 60, as shown in Figure REF , we can demonstrate that the SDL-TO framework using surrogate gradients learned by deep learning presents competitive performance on the final objective.", "For the iterative objective values shown in Figure REF , there are also almost no differences between the sequential TO methods and the proposed scalable TO approach.", "However, the SDL-TO framework only uses 12 gradient learning steps and then uses the surrogate gradient for a fast design update.", "In terms of the time-to-solution performance, the SDL-TO method outperforms the sequential TO method.", "As shown in Figure REF , the SDL-TO framework shows a speed up of 1.08x, 2.23x, 4.58x, and 8.60x using 16, 32, 64, and 128 nodes respectively.", "Figure: Iteration history and scalablility performance.", "The SDL-TO framework demonstrates a speed up of 8.6x accelerating over the standard TO implementation." ], [ "Related work", "Parameterization TO Hoyer et al.", "[17] proposed a neural reparameterization that improves topology optimization.", "They considered the use of implicit bias over functions by neural networks to improve the parameterization of TO and optimize the parameters of the neural networks instead of directly optimizing the densities on the mesh grid.", "This method can potentially achieve a better design with smaller objective loss values but it did not show how to accelerate the TO process within the proposed framework.", "Kallioras et al.", "[18] proposed a new representation using deep belief neural (DBN) coupled with the SIMP approach to accelerate the TO process.", "This method reduces the number of iterations but the overall time-to-solution is not ideal due to the relatively large expense of the DBN training and data generation.", "Neural networks and TO Several recent work focus on the use of convolutional neural network (CNN) [5], [45], [37], [41], [19] or generative adversarial network (GAN) [27], [43], [30] to replace the standard TO methods.", "Most of them start from creating a dataset of structures/materials via the standard TO and then train a machine learning model based on the dataset.", "These methods can only reproduce their training data that needs a large computational cost in advance.", "The trained model is strongly limited by the training dataset and is difficult to handle a general problem.", "In contrast, our method pursues a general framework to accelerate the TO process by using the iterative history data rather than the end of the final design." ], [ "Conclusion", "In this work, we develop a scalable framework that combined deep learning and parallel computing to accelerate the TO process for designing large-scale additively manufactured materials.", "The novel contribution is to deeply understand the iterative history data and utilize deep neural networks (DNN) to learn an accurate and fast surrogate gradient instead of the true gradient that is often time-consuming.", "The learning process is achieved by utilizing parallel computing on multiple CPUs and a distributed DNN training on multiple GPUs, incorporated with a local sampling strategy that enables to capture of the low-dimensional search space.", "The proposed SDL-TO framework is demonstrated by benchmark examples and compared with the standard sequential TO baseline methods.", "In the large-scale AM materials design example, SDL-TO shows competitive performance but significantly reduces the computational cost by achieving a speed up of 8.6x over the standard TO implementation." ], [ "Acknowledgments", "This work was supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research (ASCR), Applied Mathematics program under contract ERKJ352, ERKJ369; and by the Artificial Intelligence Initiative at the Oak Ridge National Laboratory (ORNL).", "This work used resources of the Oak Ridge Leadership Computing Facility, which is supported by the Office of Science of the U.S. Department of Energy under Contract No.", "DE-AC05-00OR22725." ] ]	2011.14177
[ [ "Wetting behaviour of a three-phase system in contact with a surface" ], [ "Abstract We extend the Cahn-Landau-de Gennes mean field theory of binary mixtures to understand the wetting thermodynamics of a three phase system, that is in contact with an external surface which prefers one of the phases.", "We model the system using a phenomenological free energy having three minima corresponding to low, intermediate and high density phases.", "By systematically varying the \\textit{(i)} depth of the central minimum, \\textit{(ii)} the surface interaction parameters, we explore the phase behavior, and wetting characteristics of the system across the triple point corresponding to three phase coexistence.", "We observe a non-monotonic dependence of the surface tension across the triple point that is associated with a complete to partial wetting transition.", "The methodology is then applied to study the wetting behaviour of a polymer-liquid crystal mixture in contact with a surface using a renormalised free energy.", "Our work provides a way to interrogate phase behavior and wetting transitions of biopolymers in cellular environments." ], [ "Introduction", "Wetting phenomena is ubiquitous in nature and arises in a variety of condensed matter systems ranging from classical fluids, to superconductors and Bose-Einstein condensates [1], [2], [3], [4].", "The most common example is a system having two bulk thermodynamic phases $\\phi ^{b}_{\\alpha }(T)$ , and $\\phi ^{b}_{\\beta }(T)$ , in contact with a surface that prefers one of them.", "For such systems, the wetting behavior can be understood using two equivalent formulations: i.e.", "in terms of the (i) contact angle $\\theta $ describing the geometric profile of a sessile drop of two coexisting bulk phases at a temperature $T < T_{c}$ , where $T_{c}$ corresponds to the bulk critical temparture, at the surface of a third phase [1], [3], [5], and (ii) profile $\\phi (z)$ , where $\\phi $ corresponds to the concentration of $\\alpha /\\beta $ phase as a function of the distance $z$ from the surface of the third phase, which happens to be a spectator [6].", "In terms of the contact angle $\\theta \\rightarrow 0$ , signals a transition from a partial to complete wetting, while in Cahn's approach, one has a macroscopic layer of one phase, $\\phi _{\\alpha }(T)$ in this case, residing at the surface, completely excluding the phase denoted by $\\phi ^{b}_{\\beta }(T)$ .", "A surface composition, $\\phi _{s}$ , intermediate between the densities of the two coexisting bulk phases, $\\phi ^{b}_{\\alpha }(T)$ and $\\phi ^{b}_{\\beta }(T)$ and decays smoothly to the bulk value of $\\phi ^{b}_{\\beta }(T)$ is a characteristic of partial wetting.", "This transition from complete to partial wetting (also known as the interface unbinding transition) can be effected by lowering the temperature.", "Cahn showed that as one approaches the bulk critical point from below, the interfacial energy between the two phases goes to zero faster than the difference between their individual surface energies with the spectator phase.", "This thus necessitates a partial to complete wetting transition [6] that has been well studied for small molecule mixtures.", "The Cahn argument also applies to polymeric mixtures [7], [12], [9], [10], [11], [8], [5], however, there are two important differences.", "While for small molecule mixtures, the wetting transition occurs close to the bulk critical point, for polymer solutions, due to the low value of the interfacial tension between the immiscible phases, the transition occurs far from the bulk criticality [12].", "Secondly, unlike small molecule mixtures, one can study the wetting transition for polymers as a function of the molecular weights of the individual components.", "The complete to partial wetting transition is associated with lateral migration of material that results in interfaces being perpendicular to the confining wall [13].", "For a symmetric mixture of small molecules confined between asymmetric walls, (i.e.", "where one wall preferentially attracts a phase) Parry and Evans [14] determined the concentration profile as a function of the confinement width and the temperature using mean field theory.", "This formulation was extended to polymeric fluids under symmetric [15] and asymmetric confinements [16], [17].", "In all these situations, the bulk thermodynamics of the system is described by a mean field free energy with two minima corresponding to the stable phases of the system and a square gradient term which accounts for the free energy cost associated with spatial variations .", "The surfaces prefer one of the phases and is modelled by a surface free energy that depends on the local density at the wall.", "The problem of minimisation of the coupled bulk and the surface energies to obtain the concentration profiles can be mapped to a geometrical problem of Hamiltonian flow in phase space [18].", "Pandit and Wortis were the first to advocate the use of such phase portraits as a way of visualising the solutions of the wetting profiles obeying appropriate boundary conditions [18].", "While the above discussion describes wetting in binary mixtures of simple or polymeric fluids, whose bulk thermodynamics is dictated by a free energy with two stable minimum at temperatures below a bulk critical temperature, there are many important physical situations where additional minima corresponding to locally stable phases may appear.", "Nematic ordering can induce additional local minimas in the free-energy landscapes as the anisotropic interactions are known to play an important role in the problem of polymer crystallisation [19], [20].", "Additionally, residual elastic interactions in the matrix arising from the presence of cross-links are known to severely modify free-energy landscapes of bulk mixtures and thus affect surface migration and wetting behaviour [21], [22].", "Other common examples are three phase systems e.g.", "in polymer nematic mixtures [23], [24], ternary amphiphiles [25], [26], [27], polymer-colloid mixtures [28], [29] or metallic alloys [30], [31].", "There has been a lot of recent interest in understanding the wetting thermodynamics in ternary mixtures [32], [33], [34], [35].", "Depending on temperature and interaction parameters several possibilities exist, e.g.", "one phase wets or spreads at the interface of the other two, or the three phases may meet along a line of common contact with three non-zero contact angles.", "The transition between these two states is an equilibrium, three-phase wetting transition and they appear in several varieties ranging from first to infinite order transitions [33].", "In this paper we present a consistent mean-field treatment of the thermodynamics of wetting for a two-component, three-phase system, which is in contact with an external surface, that acts as a spectator.", "The free energy of such a class of systems is modelled by two order parameters (i) one distinguishing between the ordered and disordered phases and (ii) one that distinguishes between two disordered phases differing in density.", "We follow the Hamiltonian phase portrait method to understand wetting for such a model, using a renormalised free energy, obtained by integrating out the order parameter that distinguishes between the the high density disordered and ordered phases.", "The renormalised free energy is thus expressed in terms of a single order parameter corresponding to the relative density of the phases.", "We demonstrate that the stable solution for the surface fraction identified from the multiple solutions which appear in the Cahn construction, corresponds to the one which minimises the total surface free energy.", "We systematically vary the stability of the intermediate phase and the values of the surface interaction parameters and demonstrate the change in the nature of surface wetting transition as a result.", "Finally, we apply this scheme to study the wetting phase diagram of a model polymer dispersed liquid crystal [23], [24] described by a free energy which accounts for both phase separation between low and high density polymer phases and the nematic ordering of the liquid crystalline component.", "In the next section we present the basic framework of the wetting calculations, which is followed by a section on application of this method on the wetting transition in a simple binary polymer mixture.", "This is followed be a section on the wetting thermodynamics in the three-phase systems and in the final section we apply this formalism on a model polymer-nematic mixture." ], [ "Wetting of a binary fluid in a semi-infinite geometry", "The basic aim of the wetting calculation is to minimise the total surface free-energy functional $\\Delta G_{surf}(\\phi _{s})$ : $\\Delta G_{surf}(\\phi _{s}) = \\Phi (\\phi _{s}) + \\int _{0}^{\\infty } \\left[ k(\\phi ) (\\frac{d\\phi }{dz})^2 + \\Delta f^{\\prime }(\\phi ) \\right] dz, $ where $\\Delta f^{\\prime }(\\phi )$ is the bulk free energy contribution (after the common-tangent construction, see below), $k(\\phi ) (\\frac{d\\phi }{dz})^2$ accounts the free energy cost arising from spatial gradients of the order parameter $\\phi $ , with $k(\\phi ) = \\frac{1}{36\\phi (1-\\phi )}$ , and $\\Phi (\\phi _{s})$ [8] accounts for the surface free-energy of the external surface located at $z=0$ .", "The bulk free-energy, $\\Delta f^{\\prime }(\\phi )$ , has a form that typically exhibits a single minimum at high temperatures, while it develops two distinct minima at lower temperatures, corresponding to two bulk thermodynamic phases.", "The thermodynamic equilibrium corresponding to the same chemical potential and osmotic pressure among the two coexisting thermodynamic phases is ensured by a common-tangent construction, $\\frac{\\partial f}{\\partial \\phi } \\bigg {\\vert }_{\\phi _{A}} = \\frac{f(\\phi _{B}) - f(\\phi _{A})}{\\phi _{B} - \\phi _{A}} \\nonumber \\\\\\frac{\\partial f}{\\partial \\phi } \\bigg {\\vert }_{\\phi _{B}} = \\frac{f(\\phi _{B}) - f(\\phi _{A})}{\\phi _{B} - \\phi _{A}}, $ where $\\phi _{A}$ and $\\phi _{B}$ are the two unknowns, which we identify as $\\phi _{\\alpha }(T)$ and $\\phi _{\\beta }(T)$ , with the convention, $\\phi _{\\alpha }(T)$ $\\le $ $\\phi _{\\beta }(T)$ .", "The free energy after the common tangent construction, $\\Delta f^{\\prime }(\\phi ) = f(\\phi ) - f(\\phi _{\\alpha }) - (\\phi - \\phi _{\\alpha }) \\frac{\\partial f}{\\partial \\phi } \\bigg {\\vert }_{\\phi _{\\alpha }}, $ enters the subsequent wetting calculations (see Fig.", "REF (a)).", "The minimisation of the total free energy $\\Delta G_{surf}(\\phi _s)$ (see Eq.", "REF ) is done in two steps.", "First, the bulk contribution is minimised as a function of $\\phi $ with the appropriate boundary conditions, i.e.", "the local density at the external surface should be $\\phi _s$ .", "The functional form that minimises the bulk contribution expressed in terms of $\\phi _s$ is then substituted back in Eq.", "REF .", "As a result, $\\Delta G_{surf}(\\phi _s)$ , the right hand side of Eq.", "REF becomes a function of the yet undetermined surface fraction, $\\phi _{s}$ .", "This function is again minimised with respect to $\\phi _s$ to obtain the surface fraction which then allows one to obtain the wetting profile.", "We use this framework to study wetting transition in a variety of systems.", "The equilibrium profiles, $\\phi (z)$ which minimise the Lagrangian density, $L(\\phi ,\\dot{\\phi })$ (the integrand of Eq.", "REF ) obey the Euler-Lagrange equations $\\frac{\\partial \\Delta f^{\\prime }}{\\partial \\phi } = 2k(\\phi ) \\ddot{\\phi }+ \\frac{\\partial k}{\\partial \\phi }(\\phi ) \\dot{\\phi }^2, $ where $\\dot{\\phi }$ = $\\frac{d \\phi }{dz}$ and $\\ddot{\\phi }$ = $\\frac{d^{2} \\phi }{dz^{2}}$ .", "The Hamiltonian can be obtained from the Lagrangian via a Legendre transformation given by $H(p,q) = p\\dot{q} - L(q,\\dot{q}) = \\frac{p^2}{4k(q)} - \\Delta f^{\\prime }(q), $ where the coordinate $q$ is $\\phi $ and the conjugate momentum, $p$ given by $p = \\frac{\\partial L}{\\partial \\dot{q}} = 2k(q)\\dot{q}.", "$ Since the Hamiltonian does not explicitly depend on $z$ , it is a conserved quantity, which leads to the following equation $k(\\phi ) \\dot{\\phi }^2 - \\Delta f^{\\prime }(\\phi ) = A, $ where the constant of integration $A = 0$ , since in the bulk, both $\\Delta f^{\\prime }(\\phi )$ and $\\dot{\\phi }$ are zero.", "Thus the minimal solution is given by $k(\\phi ) \\dot{\\phi }^2 = \\Delta f^{\\prime }(\\phi ), $ which implies the profile is given by, $\\frac{d \\phi }{dz} = \\sqrt{\\frac{\\Delta f^{\\prime }(\\phi )}{k(\\phi )}}.", "$ We take the positive sign of the root of Eq.", "REF if $\\phi < \\phi _{\\infty }$ , as is the case for all calculations outlined in this paper.", "Substituting this solution into Eq.", "REF , allows us to change the integration variable from the spatial coordinate $z$ to the density $\\phi $ .", "As a result, we can rewrite Eq.", "REF as, $\\Delta G_{surf}(\\phi _{s}) = \\Phi (\\phi _{s}) + \\int _{\\phi _{s}}^{\\phi _{\\infty }} 2 \\sqrt{\\Delta f^{\\prime }(\\phi ) k(\\phi )} d\\phi .", "$ In this work, we discuss a situation where the low density phase is preferred by the surface, i.e., $\\phi _{s} < \\phi _{\\infty }$ and we take the positive sign of the above square root.", "For $\\phi _{s}$ $>$ $\\phi _{\\infty }$ only $\\Phi (\\phi _{s})$ contributes to $\\Delta G_{surf}(\\phi _{s})$ .", "In the final stage of the minimisation scheme, we minimise $\\Delta G_{surf} (\\phi _{s})$ , given by Eq.", "REF with respect to $\\phi _{s}$ to obtain the undetermined surface fraction.", "The surface free-energy used in this work is of the following form : $\\Phi (\\phi _{s}) = h \\phi _{s} + \\frac{1}{2}g \\phi _{s}^2$ , with $h < 0$ and $g > 0$ .", "This choice makes the surface prefer a phase with $\\phi _{s} = -h/g$ .", "There are two ways to perform the final minimisation, either by numerically computing $\\Delta G_{surf} (\\phi _s)$ for various values of $\\phi _s$ and then finding its minima, or employing a Cahn-construction [6] by equating the first derivative of Eq.", "REF with respect to $\\phi _s$ to zero, yielding $\\frac{d \\Phi (\\phi _s)}{d \\phi _s} = 2 \\sqrt{\\Delta f^{\\prime }(\\phi _s) k(\\phi _s)} = \\Psi (\\phi _s).", "$ The surface fraction $\\phi _s$ , is then obtained from the intersection of the left and right hand side expressions of Eq.", "REF , numerically, which can result in multiple solutions but the stable roots are found by comparison of areas.", "As discussed below both these procedures yield the same value of surface fraction $\\phi _s$ .", "The profile is obtained by integrating Eq.", "REF , which yields $z = \\int _{\\phi _{s}}^{\\phi (z)} \\sqrt{\\frac{k(\\phi )}{\\Delta f^{\\prime }(\\phi )}} d \\phi .", "$ The boundary condition is obtained by substituting $z = 0$ in Eq.", "REF and REF and taking their ratio, which finally yields, $2k(\\phi _{s}) \\frac{d \\phi }{dz} \\bigg {\\vert }_{z = 0} = \\frac{d \\Phi (\\phi _{s})}{d \\phi _{s}}.$" ], [ "Wetting behaviour of binary polymer mixtures", "As a simple example, we consider the complete to partial wetting transition, as the temperature $T$ is deceased, in a binary mixture consisting of long ($N_{A}$ = 100) and short ($N_{B}$ = 50) polymers, in presence of an external surface at $z = 0$ , which prefers the short chain polymers (oligomers).", "The bulk thermodynamics is governed by a simple Flory-Huggins free energy [36] of the form, $f(\\phi ) = \\frac{\\phi }{N_{A}} \\ln \\phi + \\frac{(1 - \\phi )}{N_{B}} \\ln (1 - \\phi ) + \\chi \\phi (1 - \\phi ), $ where $\\phi $ is the composition of polymers and $(1 - \\phi )$ is the composition of the oligomers.", "The surface at $z = 0$ prefers the low $\\phi $ component with the bare surface energy of the form $\\Phi (\\phi _{s}) = h \\phi _{s} + \\frac{1}{2}g\\phi _{s}^{2},$ where $h < 0$ and $g > 0$ are the surface parameters.", "We choose $h = -0.00026$ and $g = 0.006$ .", "This implies that $(1 - \\phi )$ , i.e.", "the oligomer composition, is supposed to be high near this surface.", "The bulk phase of the polymer mixture becomes unstable when $\\chi $ is increased beyond the spinodal value $\\chi _{s}(\\phi _{0})$ , where $\\phi _{0}$ is the composition of the initially uniform mixture.", "The value of the Flory-Huggins $\\chi $ parameter at the spinodal is given by, $\\chi _{s}(\\phi _{0}) = \\frac{1}{2} \\left[ \\frac{1}{N_{A} \\phi _{0}} + \\frac{1}{N_{B} (1 - \\phi _{0})} \\right], $ Figure REF shows the transition from complete to partial wetting in a binary polymer mixture, in contact with an external surface, as the immiscibility parameter $\\chi $ is systematically increased (or the temperature of the system is decreased, since $\\chi \\propto 1/T$ ).", "Panel (b) shows the Cahn construction for the Flory-Huggins free energy for $\\chi = 1.01, \\text{(black)}, 1.05, \\text{(red) and} 1.08 \\chi _{s}(\\phi _{0}) \\text{(blue)}$ respectively, where $\\chi _{s}(\\phi _0)$ corresponds to the value of the immiscibility parameter at the spinodal (see Eq.", "REF ).", "As shown the Cahn construction yields multiple solutions, and the surface fraction, $\\phi _{s}$ , is chosen for which $\\Delta G_{surf}(\\phi _{s})$ is minimum (see panel (d)).", "This procedure is consistent with the area rule used for choosing the stable solution [3].", "The complete to partial wetting transition as the temperature is decreased is also evident from the change in the nature of the segregation profiles shown in Fig.", "REF (c).", "At higher temperatures, i.e.", "for $\\chi = 1.01 \\; \\text{and} \\; 1.05 \\chi _{s}(\\phi _{0})$ , the low $\\phi $ bulk phase (i.e.", "oligomers) wets the external surface ($\\phi _{s} < \\phi _{\\alpha }$ ) and completely expels the high density phase corresponding to polymers (see schematic in Fig.", "REF (c)).", "When the temperature is decreased, i.e., for $\\chi = 1.08 \\chi _{s}(\\phi _{0})$ , a partially wetting profile, corresponding to $\\phi _{\\alpha } < \\phi _{s} < \\phi _{\\beta }$ is observed at the surface.", "Figure: The schematic double minimum free energy with common tangent in panel (a) (see Eq. ).", "The Cahn construction associated with the wetting calculation for the binary polymer mixtures with surface energy parameters h=-0.00026h=-0.00026 and g=0.006g = 0.006 is shown in panel (b).", "The concentration profiles for long polymers shows a complete to partial wetting transition in panel (c).", "The effective surface free energy, obtained after minimising the bulk thermodynamics of the system as a function of the surface fraction is shown in panel (d)." ], [ "Wetting in a three-phase system", "While the bulk thermodynamics of binary polymeric mixtures always involves a free-energy with two local minima occurring at bulk densities, $\\phi _{\\alpha }(T)$ and $\\phi _{\\beta }(T)$ , complex mixtures with additional ordering fields, e.g.", "ternary amphiphiles [25], [27], mixtures of nematics and polymers [23], [24] (we would be specifically discussing wetting in these systems later in this manuscript), can have free energies with additional metastable minima.", "The study of the influence of an ordering field on wetting transitions is very interesting with several technological applications in electro-optical devices [38], [39] and high modulus fibres [40].", "In this section, we extend the square-gradient mean field theory of wetting of a binary mixture to a three-phase system.", "In particular, we discuss the role of metastability on the wetting thermodynamics by studying a phenomenological form of free energy with three distinct local minima, whose location and relative depths can be varied.", "Since we do not have an explicit temperature dependent free energy, we study the wetting transitions as (i) a function of the stability of the central minimum and (ii) by varying the surface parameters, $h$ and $g$ , which parametrizes $\\Phi (\\phi _{s})$ , the interactions of the external wall with the system.", "We focus on the Cahn-construction for a ternary system and provide a criterion that dictates whether the wetting transitions are first order or continuous in nature.", "The three-phase free-energy that we consider has a piece-wise parabolic form, $f(\\phi ) = min \\left[ f_{\\alpha }(\\phi ), f_{\\beta }(\\phi ), f_{\\gamma }(\\phi ) \\right], $ where the $min$ function chooses the minimum of three individual functions given by, $f_{\\alpha }(\\phi ) = a_{\\alpha }(\\phi - \\phi _{\\alpha })^{2} + f_{\\alpha }^{0} \\nonumber \\\\f_{\\beta }(\\phi ) = a_{\\beta }(\\phi - \\phi _{\\beta })^{2} + f_{\\beta }^{0} \\nonumber \\\\f_{\\gamma }(\\phi ) = a_{\\gamma }(\\phi - \\phi _{\\gamma })^{2} + f_{\\gamma }^{0}, $ with the following set of parameters : $\\phi _{\\alpha } = 0.1$ , $\\phi _{\\beta } = 0.5$ , and $\\phi _{\\gamma } = 0.9$ , $a_{\\alpha } = a_{\\beta } = a_{\\gamma } = 500$ and the relative heights of the three minima are set by $f_{\\alpha }^{0} = 1$ , $f_{\\beta }^{0} = 3.5$ , and $f_{\\gamma }^{0} = 5$ respectively.", "We study the effects of the bulk thermodynamics on the wetting behaviour by systematically varying the free energy parameters corresponding to intermediate values of $\\phi $ i.e.", "$f^{0}_{\\beta }$ .", "As a result the depth of the central minimum $h_{\\beta }$ , (see Figure REF ) is varied systematically by changing $f^{0}_{\\beta }$ , such that $-15 \\le f_{\\beta }^{0} \\le 10$ .", "The bare surface energy parameters are held fixed at $h = - 0.3 \\mu _{bulk}$ and $g = -12 h$ , where $\\mu _{bulk}$ corresponds to the slope of the red line in Figure REF (a).", "Next, we study the wetting transition as a function of the surface parameters, i.e.", "$h$ and $g$ , close to the triple point (see red curve in Fig.", "REF (a)).", "Figure: The triple-minimum free energy used for the calculation.", "The low (red), intermediate (blue), and high density (yellow) phases correspond to densities φ α \\phi _{\\alpha } = 0.1, φ β \\phi _{\\beta } = 0.5 and φ γ \\phi _{\\gamma } = 0.9 respectively.", "The variable h β h_{\\beta } indicates the height of the barrier between the two thermodynamically stable phases between which the system splits.The bulk phase diagram of the three-phase free energy as a function $h_{\\beta }$ is shown in Figure REF , where each region is designated by a colour of the phase/s that are stable in that region.", "For $h_{\\beta } > 0$ , the bulk free energy of a system, initially prepared with a uniform order parameter $\\phi _{0}$ , between $\\phi _{\\alpha }$ and $\\phi _{\\gamma }$ , is minimised by splitting between these two minima in a manner which preserves the initial order parameter value of $\\phi _{0}$ .", "Thus the common tangent for the subsequent wetting calculation is drawn between the the minimum at $\\phi _{\\alpha }$ and $\\phi _{\\gamma }$ and the $\\Delta f^{\\prime } (\\phi )$ for the subsequent wetting calculation should be constructed by subtracting off this common tangent from $f(\\phi )$ .", "Upon systematically decreasing $h_{\\beta }$ a situation arises when the minima of all three parabolic free energies lie on a common tangent (Fig.", "REF (a)).", "This is the triple point when the three phases coexist simultaneously.", "For $h_{\\beta } < 0$ , i.e.", "the $\\beta $ minimum corresponds to the most stable phase.", "If the initial composition is such that $\\phi _{0} < \\phi _{\\alpha }$ , a single phase with composition $\\phi _{\\alpha }$ is chosen.", "When $\\phi _{0}$ lies between the $\\alpha $ and the $\\beta $ minima, the bulk free energy is minimised by the system splitting between these two phases with the corresponding fractions following the lever rule [36] and the $\\Delta f^{\\prime } (\\phi )$ for the wetting calculation has been constructed by subtracting off this common tangent from $f(\\phi )$ .", "In this regime, the $\\gamma $ component of the free energy does not enter the wetting calculations, as the $\\alpha $ and the $\\beta $ minimum have the lowest free energies according to our chosen parameters and hence the common tangent for the wetting calculation is drawn between these two states.", "The order parameter value, $\\phi _{\\infty }$ , deep in the bulk is a value close to $\\phi _{\\beta }$ .", "For higher values of the initial composition, $\\phi _{0}$ , $\\beta $ phase becomes the most stable phase.", "Upon increasing $\\phi _{0}$ further, the bulk free-energy would be minimised when the system splits between the $\\beta $ and the $\\gamma $ minimum and in this situation the order parameter value deep inside the bulk, $\\phi _{\\infty }$ , would be close to $\\phi _{\\gamma }$ .", "Figure: The phase diagram for the three minimum free energy as a function of the stability of the phase β\\beta .Figure: The wetting thermodynamics as function of h β h_{\\beta }, when it is positive and at the triple point.", "Panel (a) shows the free energies, panel (b) shows the Cahn constructions, panel (c) shows the dependence of the minimised surface free energy on h β h_{\\beta } and panel (d) shows the order parameter profiles.Figure REF shows the wetting thermodynamics as function of $h_{\\beta } > 0$ and at the triple point, where the three phases coexist.", "Panel (a) shows the free energies corresponding to $h_{\\beta } = 7$ (black) and $h_{\\beta } = 0.001$ (red) respectively.", "We assume that the initial composition, $\\phi _0$ lies between $\\phi _{\\alpha }$ and $\\phi _{\\gamma }$ .", "Thus the bulk free-energy is minimised by the system splitting appropriately between $\\phi _{\\alpha }$ and $\\phi _{\\gamma }$ .", "We therefore draw a common tangent between these two minima, and the free energy, $\\Delta f^{\\prime }(\\phi )$ , which enters the wetting calculation is obtained by subtracting this common tangent from the free energy $f(\\phi )$ (see Eq.", "REF ).", "Panel (b) of Figure REF shows the corresponding Cahn constructions $h_{\\beta } = 0.001, 7$ respectively.", "The derivative of the surface free energy, $d \\Phi (\\phi _{s})/d \\phi _{s}$ , (blue line in panel (b)) intersects the curve $2 \\sqrt{\\Delta f^{\\prime }(\\phi _s) k(\\phi _s)}$ (RHS of Eq.", "REF ), only at one point, which yields the surface fraction, $\\phi _{s} < 0.1$ .", "The equilibrium value of the high-density phase, corresponds to the material concentration deep in the bulk, $\\phi _{\\infty } \\approx 0.9$ .", "Thus, these parameters, set the lower and upper limits of integration for the expressions appearing in Eq.", "REF and REF .", "Panel (c) shows the monotonically decreasing minimised surface free energy (the minimum of $\\Delta G_{surf}(\\phi _{s})$ ), or the surface tension, as a function of $h_{\\beta }$ and panel (d) shows the order parameter profiles.", "From Eq.", "REF it is clear that the surface tension has two contributions, one arising from the bare surface energy and the second from the area under the curve, $2 \\sqrt{\\Delta f^{\\prime }(\\phi _s) k(\\phi _s)}$ .", "In this case the surface fraction, $\\phi _{s}$ is independent of the variation in $h_{\\beta }$ , thus, while the bare surface energy remains unchanged the area under the curve, $2 \\sqrt{\\Delta f^{\\prime }(\\phi _s) k(\\phi _s)}$ , monotonically decreases with $h_{\\beta }$ .", "This leads to the monotonic decrease in surface tension with $h_{\\beta }$ .", "Similar behaviour has also been observed in calculations of surface tension in bulk systems with multiple minimas in the free energy landscape [37].", "It is clear from panel (d) that away from the triple point, when $h_{\\beta }$ is positive and high, the order-parameter profile starts from $\\phi _{s} <$ 0.1 ($\\alpha $ phase) and finally tends to its value of $\\phi _{\\infty }$ $\\sim $ 0.9 ($\\gamma $ phase) and the effect of the meta-stable $\\beta $ phase is negligible.", "Close to the triple point (see panel (d) of Figure REF ) there is a split interface with the surface wet by the $\\alpha $ phase thereby completely excluding the $\\beta $ and the $\\gamma $ phases from the surface.", "The $\\alpha $ phase is then wet by the $\\beta $ , which in turn is wet by the $\\gamma $ phase as one moves from the surface to the bulk.", "Schematic order parameter configurations for these two situations are shown in the insets in panel (d) of Figure REF .", "Figure REF summarises the thermodynamics of wetting as a function of $h_{\\beta }$ when it is negative and the initial composition, $\\phi _{0}$ , of the system is bracketed by $\\phi _{\\alpha }$ and $\\phi _{\\beta }$ (see Figure REF and the composition $\\phi _{0}^{1}$ marked in Figure REF (b)).", "Panel (a) of Figure REF shows the free-energies at two representative values of h$_{\\beta }$ and the common tangents constructed between the free-energy minimum corresponding to $\\phi _{\\alpha }$ and $\\phi _{\\beta }$ .", "Thus the relevant free energy $\\Delta f^{\\prime }(\\phi )$ , which enters the wetting calculation is obtained by subtracting this common tangent from the free energy $f(\\phi )$ shown in panel (a) of Figure REF .", "As a result, the value of the order parameter deep inside the bulk would be $\\sim $ $\\phi _{\\beta }$ = 0.5.", "As $h_{\\beta }$ becomes increasingly negative the value of $\\phi $ , in the vicinity of $\\phi _{\\alpha }$ , at which the common tangent between the $\\alpha $ and the $\\beta $ minimum intersects the free energy $f(\\phi )$ , decreases.", "This leads to an interesting behaviour in the wetting phenomena.", "Panel (b) shows the Cahn construction for determining the surface fraction.", "The location where the line corresponding to $\\frac{d \\Phi (\\phi _s)}{d \\phi _s}$ (blue line in panel (b)) becomes positive occurs at $\\phi _{s} = -h/g$ .", "For small absolute values of $h_{\\beta }$ , the value of $\\phi _{s}$ at which $2 \\sqrt{\\Delta f^{\\prime }(\\phi _s) k(\\phi _s)}$ becomes zero (or $\\Delta f^{\\prime }(\\phi _s)$ becomes zero) is greater than $\\phi _{s} = -h/g$ .", "This signifies a complete wetting of the surface by the $\\alpha $ phase as shown in the order parameter profile, black line in panel (c).", "As $h_{\\beta }$ becomes increasingly negative, a situation arises when the value of $\\phi _{s}$ at which $\\Delta f^{\\prime }(\\phi _s)$ becomes zero, is less than $\\phi _{s} = -h/g$ and this leads to a transition from complete to partial wetting and the red line in panel (c) yields a profile where the surface is partially wet by both the $\\alpha $ and the $\\beta $ phases.", "This transition from complete to partial wetting results in a non-monotonic dependence of the surface tension or the minimised surface free energy, $\\Delta G_{surf}(\\phi _{s})$ , shown in panel (d) of Figure REF .", "The value of $h_{\\beta }$ at which the non-monotonic behaviour in $\\Delta G_{surf}(\\phi _{s})$ arises, is that value where a transition from complete to partial wetting, of the surface by the $\\alpha $ phase, occurs.", "This is shown in in the inset of panel (d), which shows the dependence of the surface fraction, $\\phi _{s}$ on $h_{\\beta }$ .", "This dependence of the surface tension is unlike what had been observed in the situation when $h_{\\beta }$ was positive.", "Figure: The wetting thermodynamics as function of h β h_{\\beta }, when it is negative and when the α\\alpha and the β\\beta phases coexist.", "Panel (a) shows the free energies, panel (b) shows the Cahn constructions, panel (c) shows the segregation profiles andpanel (d) shows the dependence of the minimised surface free energy on h β h_{\\beta }.", "The inset to panel (d) shows the dependence of thesurface fraction φ s \\phi _{s} on h β h_{\\beta }, which signifies a transition from complete to partial wetting as one decreases h β h_{\\beta }.For negative $h_{\\beta }$ , and $\\phi _{0} \\sim \\phi _{\\beta }$ , the $\\beta $ minimum is the only stable state available, which minimises the free energy of the system.", "In this situation the reconstructed free-energy for the wetting calculation is obtained by drawing a horizontal tangent to the full free-energy at $\\phi _{\\beta }$ and subtracting this line from $f(\\phi )$ .", "The summary of the wetting calculation in this regime is presented in Figure REF , where panel (a) shows the free-energies and the horizontal tangent for two chosen values of $h_{\\beta }$ .", "Panel (b) of Figure REF shows the Cahn plots for obtaining the surface fraction and in these situations there is only one intersection between the red and black bulk contributions of $2 \\sqrt{\\Delta f^{\\prime }(\\phi _s) k(\\phi _s)}$ and the surface contribution arising from the $\\frac{d \\Phi (\\phi _s)}{d \\phi _s}$ term is shown in blue.", "With decreasing $h_{\\beta }$ , the value of surface fraction $\\phi _{s}$ systematically increases (see the Cahn plots in panel (b) of Figure REF ).", "Thus, in this situation, the two terms contributing to the surface tension in Eq.", "REF has opposite dependence with decreasing $h_{\\beta }$ .", "While the bare surface energy increases with $\\phi _{s}$ the area under $2 \\sqrt{\\Delta f^{\\prime }(\\phi _s) k(\\phi _s)}$ decreases, with the bare surface energy contributing more and this leads to the initial increase in the surface tension with decreasing $h_{\\beta }$ (see panel (c)).", "Once $h_{\\beta }$ falls below $h_{\\beta } \\lesssim -8$ , the surface line in panel (b) moves from the parabola corresponding to $\\alpha $ minimum to the one corresponding to the $\\beta $ minimum.", "After this point the surface fraction remains invariant upon further decrease of $h_{\\beta }$ and as a result the surface tension in panel (c) also shows a plateau.", "Panel (d) of Figure REF shows the segregation profiles for two values of $h_{\\beta }$ and in both these situations one observes partial wetting and the inset shows a two-dimensional, schematic representation of the order parameter profile.", "Figure: The wetting thermodynamics as function of h β h_{\\beta }, when it is negative and the β\\beta phase is the most stable one.", "Panel (a) shows the free energies, panel (b) shows the Cahn constructions, panel (c) shows the dependence of the minimised surface free energy on h β h_{\\beta } and panel (d) shows the segregation profiles.If the initial composition, $\\phi _{0}$ , is bracketed by $\\phi _{\\beta }$ and $\\phi _{\\gamma }$ , there are two possibilities for minimising the bulk free energy, either (a) the $\\phi _{0}$ is divided between the $\\beta $ and the $\\gamma $ minimum by order-parameter conservation and the minimum free energy is given by $F_{A}$ or (b) the system tries to minimize its free-energy by splitting into the three minimas conserving the order parameter and the minimum free energy is given by $F_{B}$ .", "This second possibility arises as $f(\\phi _{\\alpha }) < f(\\phi _{\\gamma })$ .", "We prove below that $F_{A}$ is always less than $F_{B}$ , which means that an initial uniform composition, $\\phi _{0}$ , which is between $\\phi _{\\beta }$ and $\\phi _{\\gamma }$ , will always be split into order-parameter values obtained by drawing a common tangent between the $\\beta $ and the $\\gamma $ minimum.", "The free-energy $F_{A}$ is given by $F_{A} = \\frac{\\phi _{\\gamma } - \\phi _{0}}{\\phi _{\\gamma } - \\phi _{\\beta }} f(\\phi _{\\beta }) + \\frac{\\phi _{0} - \\phi _{\\beta }}{\\phi _{\\gamma } - \\phi _{\\beta }} f(\\phi _{\\gamma }).$ We explore the possibility of lowering the free energy of the system by splitting it into three minima.", "Assuming the mass conservation constraint, $\\phi _{0} = f_{\\alpha } \\phi _{\\alpha } + f_{\\beta } \\phi _{\\beta } + (1 - f_{\\alpha } - f_{\\beta }) \\phi _{\\gamma }.$ The above equation allows us to express the fractions $f_{\\alpha }$ and $f_{\\gamma }$ in terms of the fraction $f_{\\beta }$ , $f_{\\alpha } = \\frac{(\\phi _{\\gamma } - \\phi _{0}) - f_{\\beta }(\\phi _{\\gamma } - \\phi _{\\beta })}{(\\phi _{\\gamma } - \\phi _{\\alpha })} \\nonumber \\\\f_{\\gamma } = \\frac{(\\phi _{0} - \\phi _{\\alpha }) - f_{\\beta }(\\phi _{\\beta } - \\phi _{\\alpha })}{(\\phi _{\\gamma } - \\phi _{\\alpha })}.$ From the above fractions one can write the free energy, where the initial order parameter has been partitioned into the three free energy minimum, in the following form, $F_{B} &=& \\left[ \\frac{(\\phi _{\\gamma } - \\phi _{0}) - f_{\\beta }(\\phi _{\\gamma } - \\phi _{\\beta })}{(\\phi _{\\gamma } - \\phi _{\\alpha })} \\right]f(\\phi _{\\alpha }) + \\nonumber \\\\ &f_{\\beta } f(\\phi _{\\beta })& + \\left[ \\frac{(\\phi _{0} - \\phi _{\\alpha }) - f_{\\beta }(\\phi _{\\beta } - \\phi _{\\alpha })}{(\\phi _{\\gamma } - \\phi _{\\alpha })} \\right]f(\\phi _{\\gamma })$ It is evident from the above expressions that $f_{\\gamma } > f_{\\alpha }$ , owing to the choice of parameters for our model free-energy, and both of them linearly decrease as one increases $f_{\\beta }$ , due to the constraint that their sum should be equal to unity.", "Thus upon systematically increasing $f_{\\beta }$ , $f_{\\alpha }$ reaches zero first and this occurs when $f_{\\beta } = \\frac{\\phi _{\\gamma } - \\phi _{0}}{\\phi _{\\gamma } - \\phi _{\\beta }}$ and $f_{\\gamma } = \\frac{\\phi _{0} - \\phi _{\\beta }}{\\phi _{\\gamma } - \\phi _{\\beta }}$ .", "At this point the free energy of the system is $F_{A}$ and thus this proves that $F_{B}$ cannot be lower than $F_{A}$ , implying that when $\\phi _{\\beta } < \\phi _{0} < \\phi _{\\alpha }$ , the lowest free energy would be obtained by splitting between $\\beta $ and the $\\gamma $ minimum.", "This thus implies that the relevant common tangent must be between the free-energy minimum at $\\phi _{\\beta }$ and $\\phi _{\\gamma }$ and the $\\Delta f^{\\prime } (\\phi )$ should be constructed by subtracting off this common tangent from $f(\\phi )$ .", "Figure: The wetting thermodynamics as function of h β h_{\\beta }, when it is negative and when the β\\beta and the γ\\gamma phases coexist.", "Panel (a) shows the free energies, panel (b) shows the Cahn constructions, panel (c) shows the dependence of the minimised surface free energy on h β h_{\\beta } and panel (d) shows the segregation profiles.Figure REF summarises the wetting thermodynamics for negative h$_{\\beta }$ , when the initial composition $\\phi _{0}$ , is split between the $\\phi _{\\beta }$ and $\\phi _{\\gamma }$ minimum (the composition $\\phi _{0}^{2}$ in Figure REF (b)).", "Panel (a) of Figure REF shows the free-energies and the common tangents, panel (b) shows the Cahn plots yielding the surface fraction, $\\phi _{s}$ .", "Panel (c) shows the variation of the minimised surface free energy as a function of the decreasing $h_{\\beta }$ and panel (d) shows the segregation profiles for two values of $h_{\\beta }$ .", "The inset to panel (d) shows a schematic, two-dimensional order parameter profile, which signifies that the surface is partially wetted by both $\\alpha $ and $\\beta $ phases.", "In this situation, the minimised surface free energy, $\\Delta G_{surf}(\\phi _{s})$ , increases with decreasing $h_{\\beta }$ .", "This can be physically understood from the fact that the bare surface free energy is minimum for $\\phi _{s} \\sim $ 0.083 and it increases for higher values of $\\phi _{s}$ .", "With decreasing $h_{\\beta }$ , the value of $\\phi _{s}$ increases, thus it leads to a monotonic increase of the total surface free energy.", "Figure: The wetting thermodynamics as function of the hh and gg parameters, close to the triple point, where all the three phases coexist.", "Panel (a) shows the free energies (black line) and the common tangent in red, panel (b) shows the Cahn construction, when h=-0.2μ bulk h = -0.2 \\mu _{bulk} and corresponding to the smallest and the largest g values considered.", "Panel (c) shows the surface fractions as a function of parameter g, for h=-0.2μ bulk h = -0.2 \\mu _{bulk} (black line) and h=-1.2μ bulk h = -1.2 \\mu _{bulk} (red line).", "Panel (d) shows the order parameter profiles for three values of g corresponding to h=-0.2μ bulk h = -0.2 \\mu _{bulk}.In the final set of calculations with the model three-minimum free energy we compute the wetting phase diagram when the system is close to the triple point (where all three phases coexist) and vary the parameters, $h$ and $g$ , which parametrises bare surface free-energy, $\\Phi (\\phi _{s})$ .", "Panel (a) of Figure REF shows the triple-minimum free energy close to triple point and a common tangent showing the coexistence of all the three phases.", "In these calculations the value of the parameter $g$ is varied systematically form $g_{min} = -2h$ to $g_{max} = -20h$ .", "The value of $h$ is again varied between $h_{min} = -0.2 \\mu _{bulk}$ to $h_{max} = -1.2 \\mu _{bulk}$ , where $\\mu _{bulk}$ is the slope of the common tangent in panel (a).", "The corresponding Cahn plots for the lines $\\frac{d \\Phi (\\phi _s)}{d \\phi _s}$ , with the smallest and largest slopes are shown in panel (b), where $h = -0.2 \\mu _{bulk}$ .", "The surface lines correspond to $\\frac{d \\Phi (\\phi _s)}{d \\phi _s} = h + g \\phi _{s}$ , thus h is the intercept of the surface line and $g$ is its slope.", "In panel (c) of Figure REF we observe that at low absolute value of the parameter $h$ , we observe two first order transitions (black line), for the surface fraction as a function of the parameter $g$ , of which the first transition occurring at a value of $-g/h \\sim 5$ is between two partially wet states, whereas the transition occurring at $-g/h \\sim 13$ is a transition between partial to complete wetting states.", "Upon increasing the absolute value of $h$ (red line), the first order transition at occurring at higher value of $g$ , transforms to a continuous transition and also the jump in the surface fraction, $\\phi _{s}$ , occurring at low $g/h$ , also decreases.", "The first order transitions occur when the line corresponding to the derivative of the surface free-energy, $\\frac{d \\Phi (\\phi _s)}{d \\phi _s}$ , cuts the curve $2 \\sqrt{\\Delta f^{\\prime }(\\phi _s) k(\\phi _s)}$ simultaneously at three values of $\\phi _{s}$ and this only happens when the slope of the $\\frac{d \\Phi (\\phi _s)}{d \\phi _s}$ line is small as in panel (b).", "When the magnitude of $h$ increases, the $\\frac{d \\Phi (\\phi _s)}{d \\phi _s}$ line never cuts the curve described by $2 \\sqrt{\\Delta f^{\\prime }(\\phi _s) k(\\phi _s)}$ simultaneously at three points and transitions tuned by varying parameter $g$ become continuous in nature [4].", "Panel (d) of Figure REF shows the order parameter profiles for the three values of g, when h is set to -0.2 $\\mu _{bulk}$ .", "At the highest absolute value $g$ (blue line) we observe a complete wetting of the surface by the $\\alpha $ phase.", "As the system is close to the triple point and as the common tangent simultaneously passes through all the three minima, the $\\alpha $ phase at the surface is wet by the $\\beta $ phase and finally the $\\gamma $ phase emerges deep in the bulk.", "for lower values of the parameter $g \\sim -12 h$ (red line), one observes the $\\beta $ phase at the surface, which then leads to the $\\gamma $ phase in the bulk." ], [ "Wetting of Polymer Dispersed Liquid Crystal mixtures:", "We apply the methodology developed for studying the wetting thermodynamics of a generic three-phase system to the case of polymer dispersed liquid crystals.", "Here, we use as an example a model of PDLC previously studied by Matsuyama et al.", "[23], [24] for describing the bulk thermodynamics of a mixture of polymers and nematogens.", "A Flory-Huggins type free energy of the mixture, depending on two order parameters, is given by the free energy, $f(\\phi ,S) = f_{iso}(\\phi ) + \\nu \\phi ^{2} f_{nem}(S),$ where $\\phi $ is the composition of the nematic component, $(1 - \\phi )$ is the composition of the polymer.", "The term $f_{iso}(\\phi )$ is the Flory-Huggins like isotropic part of the free-energy, given by, $f_{iso}(\\phi ) = \\frac{1-\\phi }{n_{P}} \\ln (1 - \\phi ) + \\frac{\\phi }{n_{l}} \\ln \\phi + \\chi \\phi (1 - \\phi ),$ where $n_{P}$ is the length of the polymer, $n_{l}$ is the length of the nematogens and $\\chi $ is the Flory-Huggins parameter controlling the thermodynamics of mixing.", "The nematic part of the free energy is given by $f_{nem}(S)$ , with $S$ denoting the nematic order parameter, is given by, $f_{nem}(S) = \\frac{1}{2} \\left[(1 - \\frac{\\eta }{3}) S^{2} - \\frac{\\eta }{9} S^{3} + \\frac{\\eta }{6} S^{4} \\right].$ In Eq.", "REF $\\eta $ is a factor dependent of the local nematic density $\\phi $ , which couples the polymeric and the liquid crystalline degrees of freedom and is given by $\\eta $ = $n_{l} \\nu \\phi $ .", "The parameter $\\nu $ controls the isotropic to nematic transition and is given by $\\nu = \\frac{2.7}{n_{l}} \\left( \\frac{T_{NI}}{T} \\right).$ As a result of this, $\\eta $ is given by, $\\eta = 2.7 \\left( \\frac{T_{NI}}{T} \\right) \\phi $ Similarly, $\\chi $ , the parameter controlling the phase separation is given by, $\\chi = \\frac{1}{n_{P}} \\left( \\frac{T_{c}}{T} \\right)$ Thermodynamic equilibrium necessitates the minimisation of the total free energy, which we achieve in two steps : first we minimise the nematic part of the free energy and obtain a value of the nematic order parameter $S$ (which is a function of $\\eta $ , which inturn is a function of $\\phi $ ).", "This order parameter $S$ is then substituted back into the free energy, which now becomes a renormalised function of $\\phi $ .", "Upon minimising $f_{nem}(S)$ , we get the following equation for the non-zero roots, $\\frac{2 \\eta }{3} S^{2} - \\frac{\\eta }{3}S + (1 - \\frac{\\eta }{3}) = 0$ This equation has two roots, of which the positive (below $T_{NI}$ only the positive root contributes) is given by $S_{+} = \\frac{\\eta /3 + \\sqrt{\\eta ^{2} - 8\\eta /3}}{(4 \\eta /3)}$ The order parameter $S_{+}$ is now substituted back into the expression of $f(\\phi , S)$ resulting in a renormalized free energy which is a function of $\\phi $ alone.", "The thermodynamics of this model is derived form this modified free energy.", "We study a system for which $n_{p}$ = 20, $n_{l}$ = 2 and $\\nu /\\chi $ = 3.1 and we are close to the triple point of the system at $\\tau $ = 0.969, where the two isotropic phases, $I_{1}$ and $I_{2}$ and the nematic phase $N$ are in coexistence.", "The bulk free energy or the free-energy difference of the system with respect to an initially homogeneous state, which enters the wetting calculation is given by, $\\Delta f(\\phi ,S) = f(\\phi ,S) - f(\\phi _{0},0) - (\\phi - \\phi _{0})(\\partial f/\\partial \\phi )_{\\phi = \\phi _{0}}$ where $\\phi _{0}$ refers to the order-parameter of the initially homogeneous system and its value is taken as 0.6 in the subsequent calculations.", "It is also assumed that the surface prefers the polymeric component characterised by low value of the order parameter $\\phi $ .", "This free energy is shown in panel (a) of Figure REF , which has three minima around $\\phi $ $\\sim $ 0.6 (isotropic), 0.88 (isotropic) and 0.99 (nematic).", "The parameters describing the surface interaction energy, $\\Phi (\\phi _s) = h\\phi _{s} + \\frac{1}{2}g\\phi _{s}^2$ , are the following : $g$ is varied between $-2h$ and $-100h$ , where $h$ is varied between -2$\\mu $ and -8$\\mu $ , where $\\mu $ is the slope of the common tangent between the minima at $\\phi = 0.6$ and the one at $\\phi = 0.88$ , in panel (a) of Figure REF .", "We observe qualitatively similar features in wetting behaviour to our previously discussed model three-minimum free energy.", "Panel (b) shows the Cahn construction for the surface lines shown for the minimum and maximum $g$ corresponding to $h = -2 \\mu $ .", "We observe in panel (c) that at low absolute value of the parameter $h$ , the surface fraction undergoes a first order transitions (black line), as a function of the parameter $g$ , while at higher absolute values of parameter h one observes continuous transition in the surface fraction (red line).", "Panel (d) shows the profile of the order parameter corresponding to the surface line shown in blue in panel (b).", "Figure: The renormalised free-energy is shown in panel (a) (after the minimisation has been performed on the nematic part of the free-energy) as a function of the nematic volume fraction φ\\phi , showing the low density isotropic phases I 1 I_{1}, the high density isotropic phase I 2 I_{2} and the nematic phase NN.", "Panel (b) shows the Cahn construction, with the surface lines shown for the minimum and maximum gg corresponding to h=-2μh = -2 \\mu .", "Panel (c) shows the variation of the surface fraction as a function of the parameter gg (h=-2μh = -2\\mu is shown in black, while h=-20μh = -20 \\mu is shown in red).", "Panel (d) shows the profile of the order parameter corresponding to the surface line shown in blue in panel (b).Conclusions: We discuss a mean-field theory for the thermodynamics of wetting in complex mixtures, where there are three minimum in the bulk free-energy landscape when exposed to a surface, whic prefers one of the components.", "Such a free-energy landscape can arise in a variety of complex mixtures like polymer nematic mixtures, ternary amphiphiles, polymer-colloid mixtures or metallic alloys.", "Interaction with the external surface are accounted via local potentials.", "We apply the Cahn-Landau-De Gennes mean field theory to understand the wetting thermodynamics of such a system as we sytematically vary the height of the central minimum and we find that the surface tension decreases monotonically with the height of this minimum, when it is unstable.", "As the central minimum becomes stable the phase diagram bifurcates and we observe a non-monotonic dependence of the surface tension on the stability of the central minimum, in one of the branches, which is associated with a complete to partial wetting transition.", "In the other branch we observe a monotonic increase in surface tension with an increasing stability of the central minimum.", "Close to the triple point, the wetting phase diagram computed by varying the bare surface energy parameters, $h$ and $g$ , yields two first order transitions in the surface fraction as a function $g$ , for low values of the parameter $h$ .", "Upon increasing the absolute values of $h$ , we observe that the first order transition in surface fractions give way to continuous transitions.", "A geometric understanding of these phenomena is discussed.", "Finally we present the wetting calculations for a polymer-nematic mixture, whose free energy actually has a three-minimum structure and show that the qualitative results obtained for our generic three-minimum free energy also holds for the polymer-nematic mixture.", "We hope that our theoretical work will prompt experimental studies in understanding wetting and phase behavior of biopolymer solutions in cellular environments.", "Our results are also applicable to polymer dispersed liquid crystals (PDLC) which are an important class of materials with applications ranging from novel bulk phenomena in electro-optic devices [41] to very rich and unique surface phenomena like tunable surface roughness [42] and electric-field driven meso-patterning on soft surfaces [43], [44].", "Acknowledgements: BC and BM acknowledge funding support from EPSRC via grant EP/P07864/1, and P $\\&$ G, Akzo-Nobel, and Mondelez Intl.", "Plc.", "We would also like to acknowledge S. Biswas for help in preparing the figures." ] ]	2011.14202
[ [ "HOQST: Hamiltonian Open Quantum System Toolkit" ], [ "Abstract We present an open-source software package called \"Hamiltonian Open Quantum System Toolkit\" (HOQST), a collection of tools for the investigation of open quantum system dynamics in Hamiltonian quantum computing, including both quantum annealing and the gate-model of quantum computing.", "It features the key master equations (MEs) used in the field, suitable for describing the reduced system dynamics of an arbitrary time-dependent Hamiltonian with either weak or strong coupling to infinite-dimensional quantum baths.", "This includes the Redfield ME, the polaron-transformed Redfield ME, the adiabatic ME, the coarse-grained ME, and the universal Lindblad ME.", "HOQST also includes the stochastic Schrodinger equation with spin-fluctuators.", "We present an overview of the theories behind the various MEs and provide examples to illustrate typical workflows in HOQST.", "We present an example that shows that HOQST can provide order of magnitude speedups compared to QuTiP, for problems with time-dependent Hamiltonians.", "The package is ready to be deployed on high-performance computing (HPC) clusters and is aimed at providing reliable open-system analysis tools for noisy intermediate-scale quantum (NISQ) devices.", "The HOQST Github repository (https://github.com/USCqserver/OpenQuantumTools.jl) provides the starting point for users.", "Detailed information can be found in the README file." ], [ "Introduction", "The theory of open quantum system has been an important subfield of quantum physics during the past decades with a rich collection of well established method [11], [12], [13], [14], [15], [16].", "Since perfect isolation of quantum systems is impossible, any quantum mechanical system must be treated as an open system in practice.", "The theory of open quantum system thus plays a major role in various applications of quantum physics, e.g., quantum optics [17], [18], [19], quantum control [20], and quantum computing (QC) [21].", "It becomes even more relevant in the context of Hamiltonian quantum computing (HQC), broadly defined as `analog' QC performed via continuously and smoothly driven Hamiltonians, as opposed to `discrete' gate-model QC, where Hamiltonians are driven discontinuously.", "Well-known example of HQC include adiabatic quantum computing (AQC) [22] and quantum annealing (QA) [23] (see, e.g., Refs.", "[24], [25] for reviews), as well as holonomic QC [26], [27].", "For example, in QA the Hamiltonian needs to move continuously from the the initial driver Hamiltonian to the final problem Hamiltonian, and therefore, unlike idealized gate-model quantum computers whose description often involves effective noise channels (completely positive maps), practical quantum annealers are better described by noise models derived directly from first principles [28], [29], [30], [31], [32].", "As the entire field of QC is now in the noisy intermediate-scale quantum (NISQ) era , an efficient and evolving framework of open quantum system simulation is essential for advancing our understanding of noise in quantum devices, as well for helping in the search for more effective error suppression and correction techniques .", "Moreover, the distinction between analog and discrete models of QC is to some degree arbitrary, since in reality even gate-model QC involves continuous driving due to the finite bandwidth of signal generators and controllers.", "We thus view the gate-model of QC as part of HQC for the purposes of this work.", "At present, there is an increasing number of software tools being developed for open system simulations.", "An important example of open-source software in this area is QuTiP , .", "It is one of the first packages in the field to adopt the modern software engineering paradigm, and is actively maintained on Github since its release, with new features and enhancements being continuously added.", "However, since QuTiP is designed to be as general as possible, it lacks several tools and the computational performance required to address the new challenges we are now facing in the field of HQC.", "Inspired by this unique challenge, and by the success of QuTiP, we present here a complementary and alternative open system simulation framework, which we call “Hamiltonian Open Quantum System Toolkit\" (HOQST).", "As the name suggests, the goal of HOQST is not to cover the entire field of open quantum system simulation but to focus on Hamiltonian QC, while retaining the flexibility to simulate systems subject to arbitrary time-dependent Hamiltonians.", "This focus gives us the ability to adopt domain-specific design choices and optimizations.", "The resulting implementation distinguishes itself from other available software by offering the following advantages: HOQST is written in the Julia programming language , which is designed for high performance computing.", "HOQST is built upon the excellent ordinary differential equations (ODE) package DifferentialEquations.jl [10]; thus HOQST also benefits from progress in the field of ODE solvers.", "Focusing solely on Hamiltonian QC, HOQST features several newly developed open quantum system tools such as recently published master equations.", "HOQST includes tools that work beyond the weak coupling limit.", "HOQST provides a naive interface for HPC clusters.", "HOQST provides native GPU support with the CUDA.jl package.", "HOQST is developed following the Julia design philosophy .", "We would like it to be as user-friendly as possible without compromising performance.", "Although there is room for optimization, the first release of HOQST features reliable and efficient implementations of several key master equations (MEs) adopted in the HQC field, together with a highly modularized framework suitable for future development.", "Since the HOQST project started as an attempt to build a tool to simulate quantum annealing, it displays a certain bias towards QA.", "However, we reemphasize that it is broadly applicable to open quantum systems evolving subject to any time-dependent Hamiltonian.", "This paper is organized as follows.", "In Sec.", "we briefly review the open system models and corresponding master equations supported by this package.", "In Sec.", "we introduce several numerical techniques to speed up computations.", "In Sec.", "we introduce the capabilities of HOQST and provide a comparison with other quantum simulators.", "In Sec.", "we provide examples to illustrate typical workflows of HOQST.", "We conclude in Sec.", ", and provide various additional technical details in the Appendix." ], [ "Methods", "Throughout this work we consider a quantum mechanical system $\\mathrm {S}$ coupled to a bath $\\mathrm {B}$ .", "The total Hamiltonian is assumed to have the following form $H = H_{\\mathrm {S}} + H_{\\mathrm {I}} + H_{\\mathrm {B}} \\ ,$ where $H_{\\mathrm {S}}$ and $H_{\\mathrm {B}}$ denote, respectively, the free system and bath Hamiltonians.", "$H_{\\mathrm {I}}$ is the system-bath interaction, which is often written as $H_\\mathrm {I} = \\sum _\\alpha g_\\alpha A_\\alpha \\otimes B_\\alpha \\ ,$ where $A_\\alpha $ and $B_\\alpha $ are dimensionless Hermitian operators acting on the system and the bath, respectively, and exclude both $I_{\\mathrm {S}}$ and $I_{\\mathrm {B}}$ (the identity operators on the system and the bath, respectively).", "The parameters $g_\\alpha $ are sometimes absorbed into $B_\\alpha $ but are kept explicit in HOQST, are have units of energy.", "In addition, we assume for simplicity the factorized initial condition $\\rho (0) = \\rho _{\\mathrm {S}}(0) \\otimes \\rho _{\\mathrm {B}}$ for the joint system-bath state at the initial time $t=0$ , where $\\rho _\\mathrm {B}$ is a Gibbs state at inverse temperature $\\beta $ $\\rho _\\mathrm {B} = \\frac{e^{-\\beta H_{\\mathrm {B}}}}{[e^{-\\beta H_{\\mathrm {B}}}]} \\ ,$ though we note that factorization is not necessary for a valid description of open system dynamics , .", "We work in units such that $k_B=1$ , so that $\\beta $ has units of inverse energy, or time (since we also set $\\hbar =1$ ).", "Before proceeding, we transform the original Hamiltonian Eq.", "(REF ) into a rotating frame defined by $U(t)$ , i.e.", "$\\tilde{H}(t)=U^\\dagger (t)HU(t) \\ .$ The specific form of the unitary operator $U(t)$ will lead to different master equations and will be discussed in detail in subsequent sections.", "The goals of the rotation are to remove the pure-bath Hamiltonian $H_{\\mathrm {B}}$ and to identify terms that remains small in different system bath coupling regimes.", "As long as $U(t)$ acts non-trivially on the bath, we may assume without loss of generality we that the rotating frame Hamiltonian $\\tilde{H}$ has the following form: $\\tilde{H} = \\tilde{H}_{\\mathrm {S}} + \\tilde{H}_{\\mathrm {I}} \\ ,$ where $\\tilde{H}_{\\mathrm {S}}$ acts only the system and $\\tilde{H}_{\\mathrm {I}}$ acts jointly on the system and the bath.", "The Liouville von Neumann equation in this rotating frame is $\\frac{\\partial }{\\partial t} \\tilde{\\rho }(t) = -i[\\tilde{H}(t),\\rho (t)] \\equiv \\tilde{\\mathcal {L}}(t)\\rho (t) \\ ,$ where $\\tilde{\\mathcal {L}}(t)$ denotes the Liouvillian superoperator.", "We work in units of $\\hbar =1$ throughout.", "Once again we can always write $\\tilde{H}_\\mathrm {I} = \\sum _\\alpha g_\\alpha \\tilde{A}_\\alpha \\otimes \\tilde{B}_\\alpha $ where $\\tilde{A}_\\alpha $ and $\\tilde{B}_\\alpha $ are, respectively, system and bath operators (excluding identity).", "However, it is important to note that $\\tilde{A}_\\alpha $ and $\\tilde{B}_\\alpha $ do not necessarily correspond to $U^\\dagger (t)A_\\alpha U(t)$ and $U^\\dagger (t)B_\\alpha U(t)$ in Eq.", "(REF ) because $U(t)$ may not preserve the tensor product structure (i.e., we allow for $U^\\dagger A\\otimes B U \\ne U^\\dagger A U \\otimes U^\\dagger B U$ ).", "Let ${\\tilde{X}} = \\left(\\tilde{X} \\tilde{\\rho }_B \\right)$ denote the expectation value of any rotating frame bath operator $\\tilde{X}$ with respect to $\\tilde{\\rho }_B$ .", "Then the two-point correlation function is $C_{\\alpha \\beta }(t_1, t_2) = g_\\alpha g_\\beta {\\tilde{B}_\\alpha (t_1) \\tilde{B}_\\beta (t_2)} \\ .$ If the correlation function is time-translation-invariant $C_{\\alpha \\beta }(t_1, t_2) = C_{\\alpha \\beta }(t_1-t_2, 0) \\equiv C_{\\alpha \\beta }(\\tau )\\ ,\\quad \\tau \\equiv t_1-t_2\\ ,$ then the noise spectrum of the bath can be properly defined by taking the Fourier transform $\\gamma _{\\alpha \\beta }(\\omega ) = \\int _{-\\infty }^{\\infty } C_{\\alpha \\beta }(\\tau ) e^{i\\omega \\tau }{\\omega } \\ .$ The widely used Ohmic bath case is $\\gamma ^{\\text{Ohmic}}_{\\alpha \\beta }(\\omega ) = 2\\pi \\eta g_\\alpha g_\\beta \\frac{\\omega e^{-{\\omega }/\\omega _c}}{1-e^{-\\beta \\omega }} .$ HOQST provides a built-in Ohmic bath object.", "It also provides flexible APIs to define a CustomBath object by specifying either the correlation function or the noise spectrum.", "However, to avoid the added computational cost of the Fourier transform and the numerical instability associated with large frequencies, HOQST does not convert these quantities into each other, so the user is responsible for supplying the correct function depending on the solver used." ], [ "Timescales", "Following [4], we define the two timescales $\\frac{1}{\\tau _{SB}} = \\int _0^{\\infty }\|C (\\tau )\|d\\tau \\ , \\quad \\tau _B &= \\frac{\\int _0^{t_f}\\tau \|C(\\tau )\|d\\tau }{\\int _0^{\\infty }\|C(\\tau )\|d\\tau }$ Here $t_f$ is the total evolution time, used as a cutoff which can often be taken as $\\infty $ .", "The quantity $\\tau _{SB}$ is the fastest system decoherence timescale, or timescale over which the system density matrix $\\rho _S$ changes due to the coupling to the bath, in the interaction picture.", "The quantity $\\tau _B$ is the characteristic timescale of the decay of $C(\\tau )$ .", "Note that the expression for $\\tau _B$ becomes an identity if we choose $\|C(\\tau )\| \\propto e^{-\\tau /\\tau _B}$ and take the limit $t_f\\rightarrow \\infty $ .", "As shown in Ref.", "[4], $\\tau _{SB}$ and $\\tau _B$ are the only two parameters relevant for determining the range of applicability of the various master equations discussed here (with the ULE case being no exception, but discussed separately in Ref. [5]).", "For convenience we collect the corresponding error bounds here, before discussing the various MEs.", "Namely, the error bound of the Redfield master equation is $\\Vert \\rho _{\\text{true}}(t) -\\rho _R(t) \\Vert _1 \\le O\\left(\\frac{\\tau _B}{\\tau _{SB}} e^{12t/\\tau _{SB}}\\right)\\text{ln}\\left(\\frac{\\tau _{SB}}{\\tau _B} \\right)\\ ,$ where $\\rho _{\\text{true}}(t)$ denotes the true (approximation-free) state.", "The error bound of the Davies-Lindblad master equation is $\\Vert \\rho _{\\text{true}}(t) -\\rho _D(t) \\Vert _1 \\le O\\left( \\left(\\frac{\\tau _B}{\\tau _{SB}} +\\frac{1}{\\sqrt{\\tau _{SB}\\delta E}}\\right)e^{12t/\\tau _{SB}}\\right)\\ ,$ where $\\delta E = $ min$_{i\\ne j}\|E_i-E_j\|$ is the level spacing, with $E_i$ the eigenenergies of the system Hamiltonian $H_{\\mathrm {S}}$ .", "This original version of this ME does not directly allow for time-dependent driving, and we shall consider an adiabatic variant that does, the AME [3].", "The same error bound should apply in this case, since the difference is only in that the Lindblad operators are rotated in the AME case with the (adiabatically) changing eigenstates of $H_{\\mathrm {S}}$ , and the norms used to arrive at Eq.", "(REF ) are invariant under this unitary transformation.", "The error bound of the coarse-grained master equation is $\\Vert \\rho _{\\text{true}}(t) -\\rho _{C}(t) \\Vert _1 \\le O\\left(\\sqrt{\\frac{\\tau _B}{\\tau _{SB}}}e^{6t/\\tau _{SB}}\\right) \\ .$ The error of the PTRE master equation can be separated into two parts.", "The first part comes from truncating the expansion to 2nd order.", "It can be bounded using the same expression as in Eq.", "(REF ), with timescales defined by the polaron frame correlation $K(\\tau )$ : $\\frac{1}{\\tau _{SB}} = \\Delta ^2_m\\int _0^{\\infty }\|K (\\tau )\|d\\tau \\ , \\quad \\tau _B &= \\frac{\\int _0^{t_f}\\tau \|K(\\tau )\|d\\tau }{\\int _0^{\\infty }\|K(\\tau )\|d\\tau } \\ .$ A detailed discussion of the above quantities is presented in and , and as far as we know the error bounds we derive here for the PTRE are new.", "We mention here that if the system-bath coupling strength $g_\\alpha $ in Eq.", "(REF ) is sent to infinity, both $1/\\tau _{SB}$ and $\\tau _B/\\tau _{SB}$ go to 0.", "Thus the PTRE works in the strong coupling regime.", "The second part of the error is caused by ignoring the 1st and 2nd order inhomogeneous terms, which themselves are due to the polaron transformation breaking the factorized initial condition.", "We do not have a bound on this error yet, but numerical studies suggest it is small when $\\rho _\\mathrm {S}(0)$ is diagonal .", "These bounds assume a Gaussian bath.", "For a non-Gaussian bath extra timescales relating to higher-order correlation functions generally appear, and the error bounds will contain additional terms." ], [ "Cumulant expansion", "The cumulant expansion is a technique originally designed for the perturbation expansion of stochastic differential equations , .", "This technique can be generalized to the open quantum system setting and allows a systematic description of the reduced system dynamics [13], .", "By applying this technique in different rotating frames, master equations with different ranges of applicability can be derived [13], [32], [2].", "Defining the projection operator $\\mathcal {P}$ $\\mathcal {P}\\rho = \\mathrm {tr}_\\mathrm {B} \\lbrace \\rho \\rbrace \\otimes \\rho _\\mathrm {B} \\equiv \\rho _\\mathrm {S}\\otimes \\rho _\\mathrm {B} \\ ,$ the formal cumulant expansion of Eq.", "(REF ) is $\\frac{\\partial }{\\partial t} \\mathcal {P}\\tilde{\\rho }(t) = \\sum _n \\mathcal {K}_n(t) \\mathcal {P}\\tilde{\\rho }(t)\\ ,$ where the nth order generator $\\mathcal {K}_n(t)$ is $\\mathcal {K}_n(t) = \\int _0^t {t_1} \\int _0^{t_1}{t_2} \\cdots \\int _0^{t_{n-2}}{t_{n-1}}{\\tilde{\\mathcal {L}}(t)\\tilde{\\mathcal {L}}(t_1)\\cdots \\tilde{\\mathcal {L}}(t_{n-1})}_{\\mathrm {oc}}\\ ,$ and the quantities ${\\tilde{\\mathcal {L}}(t)\\tilde{\\mathcal {L}}(t_1)\\cdots \\tilde{\\mathcal {L}}(t_{n-1})}_{\\mathrm {oc}}$ are known as ordered cumulants [13], .", "In HOQST, we consider only the first and second order generators, which are given by: ${\\tilde{\\mathcal {L}}(t)}_{oc} = \\mathcal {P}\\tilde{\\mathcal {L}}(t)\\mathcal {P} \\ , \\quad {\\tilde{\\mathcal {L}}(t)\\tilde{\\mathcal {L}}(t_1)}_{oc} = \\mathcal {P}\\tilde{\\mathcal {L}}(t)\\tilde{\\mathcal {L}}(t_1)\\mathcal {P} \\ .$ Incorporating higher order cumulants generally leads to more accurate results [13]." ], [ "Redfield equation", "The oldest and one of the most well-known MEs in this category is the Redfield equation [1] (also known as TCL2 [13], where TCL$n$ stands for time-convolutionless at level $n$ , arising from an expansion up to an including $\\mathcal {K}_n(t)$ ), which directly follows from Eq.", "(REF ) after choosing the rotation $U(t)$ to be $U(t) = U_\\mathrm {S}(t)\\otimes U_\\mathrm {B}(t) \\ , \\quad U_\\mathrm {S}(t)=T_+ \\exp {-i \\int _0^t H_\\mathrm {S}(\\tau ){\\tau }} \\ , \\quad U_\\mathrm {B}(t) = \\exp {-iH_\\mathrm {B}t} \\ ,$ where $T_+$ denotes the forward time-ordering operator.", "After rotating back to the Schrödinger picture, one of the most common forms of the Redfield equation is $\\dot{\\rho }_{\\mathrm {S}}(t) = -i{H_\\mathrm {S}(t)}{\\rho _\\mathrm {S}(t)} - \\sum _\\alpha {A_\\alpha (t)}{\\Lambda _\\alpha (t)\\rho _\\mathrm {S}(t)} + h.c.$ where $\\Lambda _\\alpha (t) = \\sum _\\beta \\int _0^{t} C_{\\alpha \\beta }(t-\\tau )U_\\mathrm {S}(t, \\tau )A_\\beta (\\tau )U_\\mathrm {S}^\\dagger (t, \\tau ) {\\tau } \\ .$ This is the form used in HOQST.", "The error bound for the Redfield ME is given in Eq.", "(REF ).", "We note that the current release of HOQST supports correlated baths for the Redfield and adiabatic master equation solvers.", "However, for simplicity, we henceforth focus on uncorrelated baths where $C_{\\alpha \\beta }(t)=\\delta _{\\alpha \\beta }C_\\alpha (t)$ .", "The most significant drawback of the Redfield equation is the fact that it does not generate a completely-positive evolution, and in particular can result in unphysical negative states (density matrices with negative eigenvalues).", "Though formal fixes for this problem have been proposed , , to address the issue in HOQST an optional positivity check routine is implemented at the code level, which can stop the solver if the density matrix become negative.", "In addition, three variants of Redfield equation that guarantee positivity, namely the adiabatic master equation (AME) [3], the coarse-grained master equation (CGME) [4], , and the universal Lindblad equation (ULE) [5], are included in HOQST.", "We detail these MEs next." ], [ "Coarse-grained master equation (CGME)", "The CGME can be obtained from Eq.", "(REF ) by first time-averaging the Redfield part, i.e., shifting $t \\mapsto t+ t_1$ and applying $\\frac{1}{T_a}\\int _{T_a/2}^{T_a/2}{t_1}$ .", "One then neglects a part of the integral to regain complete positivity.", "The result is [4]: $\\dot{\\rho } = -i{H_\\mathrm {S}+H_{\\mathrm {LS}}}{\\rho } + \\sum _\\alpha \\frac{1}{T_a}\\int _{T_a/2}^{-T_a/2}{t_1}\\int _{T_a/2}^{-T_a/2}{t_2} C_\\alpha (t_2-t_1)\\Big [A_\\alpha (t+t_1)\\rho _\\mathrm {S}A^\\dagger _\\alpha (t+t_1) -\\frac{1}{2}{A_\\alpha (t+t_2) A_\\alpha (t+t_1)}{\\rho _\\mathrm {S}}\\Big ] \\ ,$ where $A_\\alpha (t+t_1) = U^\\dagger (t+t_1, t) A_\\alpha (t) U(t+t_1, t)$ and the Lamb shift is given by $H_{\\mathrm {LS}} = \\frac{i}{2T_a}\\int _{T_a/2}^{-T_a/2}{t_1}\\int _{T_a/2}^{-T_a/2}{t_2} \\operatorname{sgn}(t_1-t_2) C_\\alpha (t_2-t_1) A(t+t_2)A(t+t_1) \\ .$ The quantity $T_a$ is the coarse-graining time, a phenomenological parameter that can be manually specified or automatically chosen based on the bath correlation function .", "HOQST uses a multidimensional `h-adaptive' algorithm based on to perform the 2-dimensional integration.", "The error bound for the CGME is given in Eq.", "(REF )." ], [ "Universal Lindblad equation (ULE)", "The ULE [5] is a Lindblad-form master equation that shares the same error bound as the Redfield equation, i.e., Eq.", "(REF ).", "The formal form of the ULE is identical to the Lindblad equation: $\\dot{\\rho }_{\\mathrm {S}}(t) = -i{H_\\mathrm {S}(t)+H_{\\mathrm {LS}}(t)}{\\rho _\\mathrm {S}(t)} + \\sum _\\alpha \\Bigg [ L_\\alpha (t)\\rho L_\\alpha ^\\dagger (t)-\\frac{1}{2}{L_\\alpha ^\\dagger (t)L_\\alpha (t)}{\\rho } \\Bigg ]\\ ,$ where the time-dependent Lindblad operators are $L_\\alpha (t) = \\int _{-\\infty }^{\\infty } g_\\alpha (t-\\tau )U_\\mathrm {S}(t,\\tau )A_\\alpha (\\tau )U^\\dagger (t,\\tau ) {\\tau } \\ ,$ and the Lamb shift is $H_\\mathrm {LS}(t) = \\sum _\\alpha \\frac{1}{2i}\\int _{-\\infty }^{\\infty }{s}{s^{\\prime }}U(t, s)A_\\alpha (s) g_\\alpha (s-t)U(s,s^{\\prime })g_\\alpha (t-s^{\\prime })A_\\alpha (s^{\\prime })U^\\dagger (t, s^{\\prime }) \\operatorname{sgn}(s-s^{\\prime }) \\ .$ In the above expression, $g_\\alpha (t)$ is called the jump correlation and is the inverse Fourier transform of the square root of the noise spectrum $g_\\alpha (t) = \\frac{1}{2\\pi }\\int _{-\\infty }^{\\infty }\\sqrt{\\gamma _\\alpha (\\omega )}e^{-i \\omega t} {\\omega } \\ .$ The integration limits of Eqs.", "(REF ) and (REF ) are problematic in practice because the unitary $U_\\mathrm {S}(t)$ does not go beyond $[0, t_f]$ .", "In numerical implementation, we replace the integral limit with $\\int _0^{t_f}$ .", "This is a good approximation when $g(t)$ decays much faster than $t_f$ .", "This is the form of ULE used in HOQST." ], [ "Adiabatic master equation (AME)", "To derive the AME, we replace $U_\\mathrm {S}(t-\\tau )$ in Eq.", "(REF ) with the `ideal adiabatic evolution' and apply the standard Markov assumption and the rotating wave approximation (RWA).", "The resulting equation is $\\dot{\\rho }_\\mathrm {S}(t) = -i{H_\\mathrm {S}(t)+H_{\\mathrm {LS}}(t)}{\\rho _\\mathrm {S}(t)} + \\sum _{\\alpha \\beta }\\sum _\\omega \\gamma _{\\alpha \\beta }(\\omega ){L_{\\omega ,\\beta }(t)\\rho _\\mathrm {S}(t)L^\\dagger _{\\omega ,\\alpha }(t)-\\frac{1}{2}{L^\\dagger _{\\omega ,\\alpha }(t)L_{\\omega ,\\beta }(t)}{\\rho _\\mathrm {S}(t)}} \\ .$ The AME is in Davies form and the Lindblad operators are defined by $L_{\\omega ,\\alpha }(t) = \\sum _{\\varepsilon _b -\\varepsilon _a = \\omega } {\\psi _a}{A_\\alpha }{\\psi _b}{\\psi _a}{\\psi _b}\\ ,$ where $\\varepsilon _{a}$ is the instantaneous energy of the $a$ 'th level of the system Hamiltonian, i.e., $H_\\mathrm {S}(t){\\psi _a(t)} =\\varepsilon _{a}(t){\\psi _a(t)}$ .", "Finally, the Lamb shift term is $H_{\\mathrm {LS}}(t) = \\sum _{\\alpha \\beta }\\sum _{\\omega } L^\\dagger _{\\omega ,\\alpha }(t) L_{\\omega ,\\beta }(t)S_{\\alpha \\beta }(\\omega ) \\ ,$ where $S_{\\alpha \\beta }(\\omega ) = \\frac{1}{2\\pi }\\int _{-\\infty }^{+\\infty } \\gamma _{\\alpha \\beta }(\\omega ^{\\prime })\\mathcal {P}(\\frac{1}{\\omega -\\omega ^{\\prime }}){\\omega ^{\\prime }} ,$ with $\\mathcal {P}$ denoting the Cauchy principal value.", "The error bound is given in Eq.", "(REF ).", "If the RWA is not applied, the resulting equation is called the one-sided AME: $\\dot{\\rho }_\\mathrm {S}(t) = -i{H_\\mathrm {S}(t)}{\\rho _\\mathrm {S}(t)} + \\sum _{\\alpha \\beta }\\sum _\\omega \\Gamma _{\\alpha \\beta }(\\omega ){L_{\\omega ,\\beta }(t)\\rho _\\mathrm {S}(t)}{A_\\alpha } + h.c. \\ ,$ where $\\Gamma _{\\alpha \\beta }(\\omega ) = \\int _0^{\\infty } C_{\\alpha \\beta }(t)e^{i\\omega t} {t} = \\frac{1}{2} \\gamma _{\\alpha \\beta }(\\omega )+\\mathrm {i} S_{\\alpha \\beta }(\\omega ) \\ .$ These two forms of the AME behave differently when the energy gaps are small because the RWA breaks down in such regions [4].", "More importantly, like the Redfield equation, the one-sided AME does not generate a completely-positive evolution.", "The code-level positivity check routine works with this version of the AME as well." ], [ "Classical $1/f$ noise", "HOQST includes the ability to model $1/f$ noise, which is an important and dominant source of decoherence in most solid-state quantum NISQ platforms [6], in particular those based on superconducting qubits , , , .", "Fully quantum treatments of $1/f$ noise have been proposed , , including for quantum annealing [32].", "In HOQST we adopt the simpler approach of modeling $1/f$ noise as classical stochastic noise generated by a summation of telegraph processes, which has proved to be a good approximation to the fully quantum version [6].", "Specifically, we provide a quantum-trajectory simulation of the following stochastic Schrödinger equation ${\\dot{\\Phi }} = -i \\Big (H_\\mathrm {S}+\\sum _{\\alpha } \\delta _\\alpha (t) A_\\alpha \\Big ) {\\Phi } \\ ,$ where each $\\delta _\\alpha (t)$ is a sum of telegraph processes $\\delta _\\alpha (t) = \\sum _{i=1}^N T_i(t) \\ ,$ where $T_i(t)$ switches randomly between $\\pm b_i$ with rate $\\gamma _i$ .", "As shown in [6], in the limit of $N\\rightarrow \\infty $ and $b_i\\rightarrow \\bar{b}$ , if the $\\gamma _i$ 's are log-uniformly distributed in the interval $[\\gamma _{\\min },\\gamma _{\\max }]$ (with $\\gamma _{\\max } \\gg \\gamma _{\\min }$ ), the noise spectrum of $\\delta _\\alpha (t)$ approaches a $1/f$ spectrum within the same interval.", "Empirically, we find that a good approximation can be achieved with relatively small $N$ ." ], [ "Hybrid model", "The most significant drawback of a purely classical noise model is that if its steady state is unique then it is the maximally mixed state.", "To see this, we first realize that each trajectory of Eq.", "(REF ) generates a unitary acting on the space $\\mathcal {S}(\\mathcal {H}_S)$ of density matrices $\\mathcal {U}_k(t)\\rho _\\mathrm {S} = U_k(t) \\rho _\\mathrm {S} U_k^\\dagger (t) \\ .$ Averaging over the trajectories over a distribution $p(k)$ creates a unital (identity preserving) map from $\\mathcal {S}(\\mathcal {H}_S)$ into itself $\\bar{\\mathcal {U}}(t) \\rho _\\mathrm {S}(0) = \\int p(k) \\mathcal {U}_k(t)\\rho _\\mathrm {S} {k}\\ .$ If the steady state $\\rho _\\infty $ is unique then we can define it as $\\lim _{t\\rightarrow \\infty }\\bar{\\mathcal {U}}(t) \\rho _\\mathrm {S}(0) = \\rho _\\infty \\quad \\forall \\rho _\\mathrm {S}(0) \\ .$ By unitality it would then follow that $\\rho _\\infty = {I}$ , since we can choose $\\rho _\\mathrm {S}(0) = I$ .", "However, this is not what is observed in real devices, e.g., in experiments with superconducting flux [30] or transmon qubits.", "To account for this, HOQST includes a hybrid classical-quantum noise model: $\\dot{\\rho } = -i {H_\\mathrm {S}+\\sum _{\\alpha } \\delta _\\alpha (t) A_\\alpha }{\\rho } +\\mathcal {L}(\\rho ) \\ ,$ where $\\delta _\\alpha (t)$ is the same random process as in Eq.", "(REF ), and $\\mathcal {L}$ is the superoperator generated by the cumulant expansion (REF ).", "At present, HOQST supports the combination of $1/f$ noise with both the Redfield and adiabatic master equations." ], [ "Polaron transform", "If the bath operators in Eq.", "(REF ) are bosonic $B_\\alpha = \\sum _{k} \\lambda _{k} (b^\\dagger _{\\alpha , k} + b_{\\alpha , k}) \\ , \\qquad H_{\\mathrm {B}}= \\sum _{\\alpha , k} \\omega _{\\alpha , k} b^\\dagger _{\\alpha ,k}b_{\\alpha , k} \\ ,$ we can choose the joint system-bath unitary $U(t)$ in Eq.", "(REF ) as [2] $U_p(t) = \\exp { - i \\sum _{\\alpha , k} A_\\alpha ^d \\frac{g_{\\alpha }\\lambda _{k}}{i\\omega _{\\alpha , k}}(b^\\dagger _{\\alpha , k} - b_{\\alpha , k})} U_\\mathrm {B}(t) \\ ,$ where $A^{d}_\\alpha $ is the diagonal component of $A_\\alpha $ in the interaction Hamiltonian (REF ) and $U_\\mathrm {B}(t)$ is given in Eq.", "(REF )We use $\\lambda $ instead of $g$ in $B_\\alpha $ to distinguish it from the expansion parameter in Eq.", "(REF )..", "The corresponding second order ME (REF ) is known as the polaron-transformed Redfield equation (PTRE) [2] or the noninteracting-blip approximation (NIBA) , [15].", "The PTRE has a different range of applicability than the previous MEs we have discussed.", "Whereas the latter apply under weak-coupling conditions, the transformation defined in Eq.", "(REF ) leads to a complementary range of applicability under strong-coupling.", "This particular form of Eq.", "(REF ) does not preserve the factored initial state, so that inhomogeneous terms are present after the transformation.", "However, if $\\rho _\\mathrm {S}(0)$ is diagonal then numerical studies of the effects of the inhomogeneous terms suggest that they can be ignored .", "In addition, the PTRE can be extended beyond the spin-boson model by choosing a different form of the joint system-bath unitary (REF ), as , [32]Note that in references , [32], different lower integration limits, 0 or $-\\infty $ , are chosen for the rotation (REF ).", "Similar results are derived despite this technical difference.", "$U_p(t) = U_\\mathrm {B}(t) T_+\\exp {-i\\sum _\\alpha A^{d}_\\alpha g_{\\alpha } \\int _{0}^t B_\\alpha (\\tau ){\\tau }} \\ .$ The two transformations in Eq.", "(REF ) and (REF ) lead to MEs with identical structure but slightly different expressions (see for details).", "Whether those differences make any physical significance is an interesting topic for further study.", "In this paper we choose to work with Eq.", "(REF ).", "Because the general form of the PTRE is unwieldy, we present its form for a standard quantum annealing model $H(t) &= H_{\\mathrm {S}}(t) + H_{\\mathrm {I}}+ H_{\\mathrm {B}}\\ , \\quad H_{\\mathrm {I}}= \\sum _i g_i \\sigma _i^z\\otimes B_i \\\\H_{\\mathrm {S}}(t) & =a(t)H_\\mathrm {driver} + b(t)H_\\mathrm {prob}\\ ,$ where $a(t)$ and $b(t)$ are the annealing schedules, and $H_\\mathrm {driver}$ and $H_\\mathrm {prob}$ are the standard driver and problem Hamiltonians, respectively: $H_\\mathrm {driver} = -\\sum _i \\sigma ^x_i\\ , \\quad H_\\mathrm {prob} = \\sum _i h_i \\sigma _i^z + \\sum _{i<j} J_{ij} \\sigma _i^z \\sigma _j^z \\ ,$ where the Pauli matrix $\\sigma ^x$ acting on qubit $i$ is denoted by $\\sigma ^x_{i}$ , etc.", "The transformed Hamiltonian is $\\tilde{H}(t) = a(t)\\bigg [\\sum _i \\sigma ^+_i \\otimes \\xi _i^+ (t) + \\sigma ^-_i\\otimes \\xi _i^- (t)\\bigg ] +b(t)H_\\mathrm {prob} \\ ,$ where $\\xi _i^{\\pm }(t) =U^\\dagger _{\\mathrm {B}}(t) \\exp {\\pm \\sum _k \\frac{2g_i\\lambda _k}{\\omega _k}(b_{i,k}^\\dagger - b_{i,k})} U_{\\mathrm {B}}(t)\\ .$ The Redfield equation corresponding to Eq.", "(REF ) is ${}{t}\\tilde{\\rho }_{\\mathrm {S}}(t) = -i{\\tilde{H}_\\mathrm {S}(t)+a(t)\\sum _i\\kappa _i\\sigma _x}{\\tilde{\\rho }_\\mathrm {S}(t)} - \\sum _{i,\\alpha } {\\sigma _i^\\alpha }{\\Lambda _i^\\alpha (t)\\tilde{\\rho }_\\mathrm {S}(t)} + h.c. \\ ,$ where $\\Lambda _i^\\alpha (t) &= a(t)\\sum _{\\beta } \\int _0^{t} a(\\tau )K_i^{\\alpha \\beta }(t, \\tau )\\tilde{U}_\\mathrm {S}(t, \\tau )\\sigma _i^\\beta \\tilde{U}_\\mathrm {S}^\\dagger (t, \\tau ) {\\tau } \\\\K_i^{\\alpha \\beta }(t, \\tau ) &={\\xi _i^\\alpha (t) \\xi _i^\\beta (\\tau )}\\\\\\kappa _i &= {\\xi _i^{\\pm }(t)}$ and $\\tilde{U}_\\mathrm {S}(t, \\tau ) = T_+\\exp {-i\\int _\\tau ^t \\tilde{H}_\\mathrm {S}(\\tau ^{\\prime }) {\\tau ^{\\prime }}} \\ , \\quad \\tilde{H}_{\\mathrm {S}}(t) = b(t)H_\\mathrm {prob} \\ .$ Here $K_i^{\\alpha \\beta }(t, \\tau )$ is the two-point correlation function in the polaron frame [akin to the correlation function defined in Eq.", "(REF )], and $\\kappa _i$ corresponds to the first order cumulant generator in Eq.", "(REF ) and is also known as the reorganization energy; it contributes a Lamb-shift-like term in Eq.", "(REF ).", "It is also worth mentioning that the polaron transformation (REF ) can be done partially, which means that in Eqs.", "(REF ) and (REF ), $\\alpha $ can be summed over a subset of system-bath coupling terms.", "To solve this form of the PTRE in HOQST, the user can define a new correlation function $C_i^{\\alpha \\beta }(t,\\tau ) = a(t)a(\\tau )K_{\\alpha \\beta }(t, \\tau )$ and use the Redfield solver.", "An alternative approach is to make the Markov approximation in Eq.", "(REF ) $\\int _0^t a(\\tau ) \\cdots {\\tau } \\rightarrow a(t)\\int _0^\\infty \\cdots {\\tau }$ and write Eq.", "(REF ) in Davies form (see for more details).", "This leads to the same expression as the AME [Eq.", "(REF )], but with different Lindblad operators: $L_i^{\\omega ,\\alpha }(t) = a(t)\\sum _{\\varepsilon _b -\\varepsilon _a = \\omega } {\\psi _a}{\\sigma _i^\\alpha }{\\psi _b}{\\psi _a}{\\psi _b} \\ ,$ where now ${\\psi _a}$ is the energy eigenstate of the Hamiltonian $\\tilde{H}_\\mathrm {S}(t)$ , and the noise spectrum is $\\gamma _i^{\\alpha \\beta }(\\omega ) = \\int _{-\\infty }^{\\infty }K_i^{\\alpha \\beta }(t)e^{i \\omega t}{t} \\ .$ Then the AME solver can be used to solve this Lindblad-form PTRE." ], [ "Redfield backward integration", "To solve the Redfield or Redfield-like master equation (REF ), one needs to integrate the unitary $U_{\\mathrm {S}}$ backward in time at each ODE step.", "Such integrations are computationally expensive for long evolution times and become the bottleneck of the solver.", "To improve the efficiency of the solver, we introduce an additional parameter $T_a$ as the lower integration limit: $\\Lambda _\\alpha (t) = \\sum _\\beta \\int _{T_a}^{t} C_{\\alpha \\beta }(t-\\tau )U_\\mathrm {S}(t, \\tau )A_\\beta (\\tau )U_\\mathrm {S}^\\dagger (t, \\tau ) {\\tau } \\ .$ To justify this, note first that ${\\int _0^{T_a} C_{\\alpha \\beta }(t-\\tau )U_\\mathrm {S}(t, \\tau )A_\\beta (\\tau )U_\\mathrm {S}^\\dagger (t, \\tau ) {\\tau }} \\le \\int _{t-T_a}^{t} {C(\\tau ^{\\prime })} {\\tau ^{\\prime }} \\ ,$ where ${\\cdot }$ is any unitarily invariant norm.", "To obtain this inequality, we perform a change of variable $t-\\tau \\rightarrow \\tau ^{\\prime }$ and make use of the fact that the operator $A_\\beta (t)$ can always be normalized by absorbing a constant factor into the corresponding bath operator $B_\\beta $ .", "Second, note that in most applications the bath correlation function $C(\\tau ^{\\prime })$ decays fast compared with the total evolution time.", "As a result, the r.h.s.", "of Eq.", "(REF ) is small for sufficiently large $t$ .", "The neglected part, i.e., the integral over $[0,T_a]$ , can thus be safely ignored so long as the r.h.s.", "of Eq.", "(REF ) is below the error tolerance of the numerical integration algorithm.", "The same technique can also be applied to the ULE.", "The integration limits in Eqs.", "(REF ) and (REF ) can be localized around $t$ , i.e.", "replaced by $\\int _{t-T_a}^{t+T_a}$ .", "However, choosing an appropriate $T_a$ is a process of trial and error.", "The user needs to determine its value in a case-by-case manner." ], [ "Precomputing the Lamb shift", "HOQST adopts numerical techniques proposed in to calculate Cauchy principle value in the Lamb shift (REF ).", "Instead of evaluating it at each ODE step, to speed up the computations all the ME solvers support precomputing the Lamb shift on a predefined grid and use interpolation to fill up the values between the grid points." ], [ "Adiabatic frame", "For a typical annealing process, the total annealing time is usually much larger than the inverse energy scale of the problem $t_f \\gg \\frac{1}{\\min _{s\\in {0,1}}\\big [\\max (A(s), B(s))\\big ]} \\ .$ Informally, the frequency of the oscillation between the real and imaginary part of the off-diagonal elements of $\\rho _\\mathrm {S}$ in the neighborhood of $s$ is positively proportional to both $A(s)$ and $B(s)$ .", "As a consequence, directly solving the dynamics in the Schrödinger picture is challenging because the algorithm needs to deal with the fast oscillations induced by the Hamiltonian, thus impacting the step size.", "HOQST includes an optional pre-processing step to rotate the Hamiltonian into the adiabatic frame .", "If the evolution is in the adiabatic limit, the off-diagonal elements of the density matrix in this frame should approximately vanish.", "The fast oscillation is absent and a large step size can be taken by the ODE solver.", "This technique provides advantages if the user wishes to repeatedly solve the same problem with different parameters.", "We briefly summarize the adiabatic frame transformation, following Ref. .", "For general multi-qubit annealing, the system Hamiltonian given in Eqs.", "(REF ) and (REF ) can be formally diagonalized and written as $H_\\mathrm {S}(s) = \\sum _n E_n{s}{n} \\ ,$ where ${{n{s}}}$ is the instantaneous energy eigenbasis [eigenvectors of $H_\\mathrm {S}(s)$ ], and $s=t/t_f$ is the dimensionless instantaneous time.", "We assume that $H_\\mathrm {S}(s)$ is real for all $s$ .", "The system density matrix can be written in the instantaneous energy eigenbasis: $\\rho {s} = \\sum _{nm}\\rho _{nm}{n}{m} \\ .$ We call the associated matrix $\\tilde{\\rho } = {\\rho _{nm}}$ the density matrix in the adiabatic frame.", "It can shown that $\\dot{\\rho }_{nm} =-it_f{E_n-E_m}\\rho _{nm} -i{-i\\sum _{n^{\\prime }\\ne n}{n}{\\dot{n^{\\prime }}}\\rho _{n^{\\prime }m} +i \\sum _{m^{\\prime }\\ne m}\\rho _{nm^{\\prime }}{m^{\\prime }}{\\dot{m}}}\\ ,$ where the dot denotes differentiation with respect to $s$ .", "I.e., $\\tilde{\\rho }$ obeys the von Neumann equation $\\dot{\\tilde{\\rho }} = -i{\\tilde{H}}{\\tilde{\\rho }}$ , with the effective Hamiltonian $\\tilde{H} =\\begin{pmatrix}t_fE_0 & -i {0}{\\dot{1}} & \\dots \\\\i{0}{\\dot{1}} & t_fE_1 & \\dots \\\\\\vdots & & \\ddots \\end{pmatrix} \\ .$" ], [ "Quantum trajectories method", "HOQST implements a quantum-trajectory solver for the AME based on Ref. [31].", "Using the native distributed memory parallel computing interface of both Julia and DifferentialEquations.jl, the quantum-trajectory simulations can take advantage of HPC clusters with minimum changes in the code.", "In addition, classical $1/f$ noise can be infused into the AME trajectory solver to generate the hybrid dynamics described in Eq.", "(REF ).", "HOQST provides two sets of APIs for direct and trajectory solvers.", "The APIs for direct solvers, which are listed in Table REF , adopt the naming convention solve_name, where name is the type of equation to be solved.", "Table: The common solver APIs.", "The PTRE can be solved using solve_redfield and solve_ame by defining the corresponding polaron frame Hamiltonian and bath.The APIs for the quantum-trajectory solver is built around a single function build_ensembles.", "Calling this function builds an EnsembleProblem object that can be distributed to remote workers for parallel simulation .", "An example of a complete workflow is given in Sec. .", "Recent developments in the field of QC have led to an explosion of quantum software platforms, such as , , , .", "A review of various major platforms is given in .", "Because some of these platforms include the capability to simulate noisy quantum circuits, we briefly compare their respective noise models and solver types in Table REF .", "Furthermore, for packages that support arbitrary time-dependent Hamiltonian and rely on MEs as solvers, we list their compatible MEs in Table REF .", "Table: Comparison chart for noise models and solver typesTable: Comparison chart for ME support" ], [ "Examples", "HOQST has a large collection of tutorials located at a dedicated Github repo .", "Tables REF and REF list the introductory-level and advanced tutorials, respectively.", "Table: List of introductory-level tutorialsTable: List of advanced tutorialsAs an illustrative yet non-trivial example, we next discuss the simulation of a three-qubit quantum annealing entanglement witness experiment , .", "This example will serve to illustrate the typical workflow of HOQST, which includes: Define the system Hamiltonian and initial state.", "Define the coupling operators and bath objects.", "(Optional) Combine coupling and bath objects into InteractionSet if there are multiple system-bath couplings.", "Combine these objects into an AnnealingIt is called Annealing for historical reasons.", "Evolution is also supported.", "object.", "(Optional) Construct EnsembleProblem from Annealing object.", "Call the solver.", "Handle the solution." ], [ "Entanglement witness experiment modeling", "The entanglement witness experiment was proposed to provide evidence of entanglement in a D-Wave quantum annealing device .", "An open system analysis of these experiments was performed using the AME in Ref. .", "The crux of the experiment is actually a form of tunneling spectroscopy , where the goal is to find the energy gaps of the Hamiltonian $-a(s) \\sum _{i} \\sigma _{x}^{i}+b(s) H_{\\mathrm {Ising}}$ .", "This is done by observing the location of a peak in the tunneling rate as measured using a probe qubit.", "The Hamiltonian of the 3-qubit-version of the experiment is $H_\\mathrm {S}(\\tau )=-a(s(\\tau )) \\sum _{i=1}^{2} \\sigma ^{x}_{i}-a(s_p(\\tau )) \\sigma ^{x}_{p}+b(s(\\tau )) H_{\\mathrm {Ising}} \\ ,$ where $a(s)$ and $b(s)$ are the annealing schedules, and $s(\\tau )$ and $s_p(\\tau )$ are functions of the dimensionless time $\\tau = t / t_f$ .", "The Hamiltonian consists of two system qubits coupled to an ancilla system qubit, as shown in Fig.", "REF .", "The aforementioned location of the tunneling rate peak can be controlled by varying $h_P$ , and this information can be used to extract the energy gaps as a function of $s$ .", "We refer interested readers to Refs.", ", for more information.", "Figure: The Ising Hamiltonian H Ising H_{\\text{Ising}} in Eq.", "() of the 3-qubit entanglement witness experiment.", "The goal is to demonstrate entanglement between qubits 1 and 2, by probing the ancilla qubit (denoted p).", "In our simulations, J 1P J_{1P} and J S J_S are fixed at J 1P =-1.8J_{1P} = -1.8, J S =-2.5J_S=-2.5.The annealing schedules and annealing parameters are illustrated in Fig.", "REF .", "The particular chip used in the experiments is described in Ref. .", "To extract the tunneling rate, we first perform the simulation with the initial `all-one' state ${\\psi (0)}={1}^{\\otimes 3}$ for different $h_p$ and $t_2 = \\tau _2 t_f$ values.", "The population of the all-one state at the end of anneal is then obtained as a function of $h_p$ and $\\tau _2$ : $P_{{\\mathbf {1}}}(h_p, t_2) = {{\\psi (t_f)}{\\psi (0)}}^2 $ .", "Lastly, we fit $P_{{\\mathbf {1}}}(h_p, t_2)$ to the function $a e^{b t_2} + c e^{d t_2}$ , from which the rate $\\Gamma $ can be estimated $\\Gamma (h_p) = -{P_{{\\mathbf {1}}}}{t_2}\\bigg \\vert _{t_2=0} = -ab - cd \\ .$ Figure: (a) Annealing schedules.", "(b) Annealing parameters as functions of dimensionless time.", "There are three stages in the experiment: (1) first evolve s(τ)s(\\tau ) from 1 to s * s^{} for τ∈[0,τ 1 ]\\tau \\in [0,\\tau _1] and then evolve s p (τ)s_p(\\tau ) from 1 to s p s_p^* for τ∈[τ 1 ,2τ 1 ]\\tau \\in [\\tau _1,2\\tau _1]; (2) pause for a time τ 2 \\tau _2; (3) reverse the first stage.", "In our simulation, we choose s * =0.339s^=0.339, τ 1 t f =10μs\\tau _1 * t_f = 10\\mu s following and s p * =0.612s_p^=0.612 such that 2A(s p )≈1 MHz 2A(s_p^)\\approx 1\\mathrm {MHz} .", "The value of τ 2 \\tau _2 is varied to obtain the tunneling rate.For the open system model, we follow the strategy used in Ref. .", "We assume the qubits are coupled to independent baths $H(t) = H_\\mathrm {S}(t) + \\sum _{i=1}^{2}g_i\\sigma ^z_i\\otimes B_i + g_p\\sigma _z^p\\otimes B_p + H_{\\mathrm {B}}\\ ,$ but the bath coupling to the probe qubit $g_p$ is much stronger than the coupling to the two system qubits $g_1=g_2=g_s$ .", "In addition, we assume the bath is Ohmic [Eq.", "(REF )] with coupling strength $\\eta g_s^2 / \\hbar ^2 = 1.2732 \\times 10^{-4}$ , cut-off frequency $f_c = 4\\mathrm {GHz}$ , and temperature $T = 12.5\\mathrm {mK}$ .", "We run the simulation with different models of $B_p$ : Ohmic bath with interaction strength $g_p = 10 g_s$ : Sec.", "REF .", "Hybrid Ohmic bath whose coupling strength to the Ohmic component is $g_p = 10 g_s$ and varying macroscopic resonant tunneling (MRT) width: Sec.", "REF .", "The following pseudo-code block demonstrates the workings of an AME simulation in HOQST, following the workflow above.", "The exact codes are hosted in the tutorial repo .", "jp = -1.8; js = -2.5; hp = 0.5; the natural unit of time in HOQST is (ns) τ1 = 10000; τ2 = 5000; tf = 4τ1 + τ2; construct each term in the Hamiltonian Hlist = [-σx ⊗ σi ⊗ σi, -σi ⊗ σx ⊗ σi - σi ⊗ σi ⊗ σx, localfieldterm([hp, -jp, 0], [1, 2, 3], 3) + twolocalterm([jp, js], [[1, 2], [2, 3]], 3) ] we assume here s and sp are already defined according to figure 2 flist = [(t)->A(sp(t)), (t)->A(s(t)), (t)->B(s(t))] define the system Hamiltonian H = DenseHamiltonian(flist, Hlist) define the initial state u0 = PauliVec[3][2] ⊗ PauliVec[3][2] ⊗ PauliVec[3][2] define the system-bath coupling couplingohmic = ConstantCouplings([\"10ZII\", \"IZI\", \"IIZ\"], unit = ħ) bathohmic = Ohmic(1.2732e-4, 4, 12.5) combine the components into an Annealing object annealing = Annealing(H, u0, coupling = coupling, bath = bath) call the solver sol = solveame(annealing, tf, alg = Tsit5(), ωhint = range(-15,15,length=200), ddiscontinuities = [τ1, 2τ1, 2τ1+τ2, 3*τ1+τ2], saveat = range(0,tf,length=100), maxiters = 1e8, reltol = 1e-5 ) The solution sol is returned as an ODESolution type, the details of which are listed in Ref.", "." ], [ "AME with an Ohmic bath", "Simulations using two different flavors of the AME, the one-sided AME (REF ) and the Lindblad form AME (REF ), are shown in Fig.", "REF .", "The results demonstrate that these two AME flavors only differ significantly near the small gap region." ], [ "PTRE", "To use the PTRE for the entanglement witness problem, we perform the polaron transformation (REF ) only on the probe qubit and leave the system qubits unchanged.", "After following through the same procedure as detailed in Sec.", "REF and , the following ME can be derived: ${}{t}\\tilde{\\rho }_{\\mathrm {S}}(t) = -i{\\tilde{H}_\\mathrm {S}(t)+\\tilde{H}_\\mathrm {LS}(t)}{\\tilde{\\rho }_\\mathrm {S}(t)} + \\mathcal {L}_A\\Big [\\tilde{\\rho }_\\mathrm {S}(t)\\Big ] + \\mathcal {L}_P\\Big [\\tilde{\\rho }_\\mathrm {S}(t)\\Big ] \\ ,$ where $\\tilde{H}_\\mathrm {S}(t) = -a(t) \\sum _{i=1}^{2} \\sigma ^{x}_{i}+b(t) H_{\\mathrm {Ising}} \\ ,$ and the Liouville operators $\\mathcal {L}_A$ and $\\mathcal {L}_P$ corresponds to the AME part and PTRE part of this equation, respectively: $\\mathcal {L}_A(\\rho ) &= \\sum _{i=1}^2\\sum _\\omega \\gamma (\\omega ){L_{\\omega ,i}(t)\\rho L^\\dagger _{\\omega ,i}(t)-\\frac{1}{2}{L^\\dagger _{\\omega ,i}(t)L_{\\omega ,i}(t)}{\\rho }}\\\\\\mathcal {L}_P(\\rho ) &= \\sum _{\\alpha \\in {+,-}}\\sum _\\omega \\gamma _P(\\omega ){L_p^{\\omega ,\\alpha }(t)\\rho L_p^{\\omega ,\\alpha \\dagger }(t)-\\frac{1}{2}{L_p^{\\omega ,\\alpha \\dagger }(t)L_p^{\\omega ,\\alpha }(t)}{\\rho }} \\ ,$ where the Lindblad operators are defined in Eq.", "(REF ) and Eq.", "(REF ) respectively.", "The function $\\gamma (\\omega )$ is the standard Ohmic spectrum and $\\gamma _P(\\omega )$ is the polaron frame spectrum with a hybrid Ohmic form , [32] discussed in .", "We provide the explicit form of $\\gamma _P(\\omega )$ here $\\gamma _P(\\omega ) = \\int K(t) e^{i\\omega t} {t} = \\int \\frac{{x}}{2\\pi } G_L(\\omega -x)G_H(x) {x} \\ ,$ where $G_L(\\omega ) = \\sqrt{\\frac{\\pi }{2W^2}}\\exp \\Bigg [-\\frac{(\\omega -4\\varepsilon _L)^2}{8W^2}\\Bigg ] \\ ,$ and $G_H(\\omega ) = \\frac{4\\gamma (\\omega )}{\\omega ^2 + 4\\gamma (0)^2} \\ .$ $G_L(\\omega )$ is the contribution of low frequency component, characterized by the MRT width $W$ .", "Because $W$ and $\\varepsilon _L$ are connected through the fluctuation-dissipation theorem we have $W^2 = 2\\varepsilon _L T$ ; thus, hybridizing low frequency noise with an Ohmic bath introduces one additional parameter." ], [ "Results", "The tunneling rates obtained via different ME simulations are compared with the experimental results in Fig.", "REF .", "In comparison with the AME, the PTRE exhibits a larger Gaussian line-width broadening and closer agreement with the experimental data.", "There are two additional observations, which will be the subject of future studies: If we increase $W$ while fixing $T$ , the curve is stretched to the right.", "This the result of the fluctuation-dissipation theorem, where $\\varepsilon _L$ scales quadratically with $W$ .", "Such a shift can be compensated by increasing the temperature together with $W$ [see the black dashed curve in Fig.", "(REF )].", "To better match the experimental curve, we need a higher $T$ than the reported value in Ref. .", "There is still a mismatch between the theoretical and experimental amplitudes of the tunneling rate curves, whose physical origin is currently unclear.", "Figure: Tunneling rates obtained via different MEs compared with the experimental results (extracted from using WebPlotDigitizer ).", "The bath coupled to the system qubits is Ohmic with parameters: ηg s 2 /ℏ 2 =1.2732×10 -4 \\eta g_s^2 / \\hbar ^2 = 1.2732 \\times 10^{-4}, f c =4 GHz f_c = 4\\mathrm {GHz} and T=12.5 mK T = 12.5\\mathrm {mK}.", "The corresponding models and parameters of the bath coupled to the probe qubit used for different simulations are (1) AME: Ohmic with g p =10g s g_p=10g_s.", "(2) PTRE: hybrid-Ohmic with g p =10g s g_p=10g_s and various WW, TT values.", "The cut-off frequency f c f_c is the same across different models." ], [ "Conclusions", "We presented a software package called Hamiltonian Open Quantum System Toolkit (HOQST).", "It is user-friendly and written in Julia.", "It supports various master equations with a wide joint range of applicability, as well as stochastic Hamiltonians to model $1/f$ noise.", "We briefly reviewed the theories behind these master equations and illustrated how to use HOQST to simulate the open system dynamics in a 3-qubit entanglement witness experiment.", "We also derived new error bounds for the polaron-transformed Redfield equation.", "We expect HOQST to be useful for researchers working in the field of open quantum systems, dealing with systems governed by time-dependent Hamiltonians.", "HOQST provides both basic and advanced numerical simulation tools in this area, which can be applied to simulate superconducting qubits of all types, trapped ions, NV centers, silicon quantum dot qubits, etc.", "Future releases of HOQST will expand both the suite of open system models and range of quantum control and computation models it supports." ], [ "Acknowledgments", "The authors are grateful to Jenia Mozgunov, Tameem Albash, Ka Wa Yip and Vinay Tripathi for useful discussions and feedback.", "The authors also thanks Grace Chen for the HOQST logo design.", "This research is based upon work (partially) supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) and the Defense Advanced Research Projects Agency (DARPA), via the U.S. Army Research Office contract W911NF-17-C-0050.", "The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, DARPA, ARO, or the U.S. Government.", "The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.", "The authors acknowledge the Center for Advanced Research Computing (CARC) at the University of Southern California for providing computing resources that have contributed to the research results reported within this publication.", "URL: https://carc.usc.edu." ], [ "Two different formulations", "We illustrate two different formulations of the polaron transformation via the simplest Hamiltonian $H &= H_{\\mathrm {S}}\\otimes I_B + H_{\\mathrm {SB}} + I_S\\otimes H_{\\mathrm {B}}\\\\&\\equiv (\\varepsilon \\sigma _z + \\Delta \\sigma _x)\\otimes I_B + g\\sigma _z\\otimes \\sum _k \\lambda _k(b_k^\\dagger + b_k) + I_S\\otimes \\left(\\sum _k \\omega _k b_k^\\dagger b_k\\right) \\ .$ The standard polaron transformation rotates this Hamiltonian via the unitary [2] $U_p = \\exp {- i\\sigma _z \\sum _k \\frac{g\\lambda _k}{i\\omega _k}(b^\\dagger _k - b_k)}$ which leads to the interaction picture Hamiltonian $\\tilde{H} = \\varepsilon \\sigma _z + \\Delta \\xi _+\\sigma _+ + \\Delta \\xi _-\\sigma _- + H_{\\mathrm {B}}\\ ,$ where $\\xi _\\pm = \\exp {\\pm 2i \\sum _k \\frac{g\\lambda _k}{i\\omega _k}(b^\\dagger _k - b_k)} \\ .$ The only quantities that matter in the 2nd order TCL formalism (REF ) are the average and two-point correlation function of $\\xi _\\pm (t)$ given by ${\\xi _{\\pm }(t)} &= _\\mathrm {B}\\lbrace U^\\dagger _\\mathrm {B}(t) \\xi _\\pm U_\\mathrm {B}(t)\\rho _\\mathrm {B}\\rbrace = _\\mathrm {B}\\lbrace \\xi _\\pm \\rho _\\mathrm {B}\\rbrace \\\\{\\xi _\\alpha (t)\\xi _\\beta (0)} &=_\\mathrm {B}\\lbrace U^\\dagger _\\mathrm {B}(t) \\xi _\\alpha U_\\mathrm {B}(t) \\xi _\\beta \\rho _\\mathrm {B}\\rbrace \\ ,$ where $\\alpha , \\beta \\in \\lbrace +, -\\rbrace $ and $U_\\mathrm {B}(t) = \\exp {-iH_{\\mathrm {B}}t}$ .", "To arrive at the second equality in Eq.", "(REF ) we used $[\\rho _\\mathrm {B},U_\\mathrm {B}]=0$ , which is true for $\\rho _\\mathrm {B}$ in a Gibbs state, as in Eq.", "(REF ).", "An alternative approach [32] is to first rotate the Hamiltonian (REF ) w.r.t.", "$H_{\\mathrm {B}}$ and then rotate it again by $U_d(t) = T_+\\exp {-i \\sigma _z \\int _{0}^t B(\\tau ){\\tau }} \\ ,$ where $B(\\tau ) = U^\\dagger _\\mathrm {B}(\\tau ) g\\sum _k \\lambda _k(b_k^\\dagger + b_k) U_\\mathrm {B}(\\tau ) \\ .$ It is worth noting that in this formalism the bath does not need to be bosonic.", "Let us define $\\Xi _\\pm (t) = T_+\\exp {\\pm i \\int _{0}^t B(\\tau ) {\\tau }} \\ .$ Then the effective Hamiltonian in this two-fold interaction picture is $\\tilde{H} = \\varepsilon \\sigma _z + \\Delta \\zeta _+(t)\\sigma _+ + \\Delta \\zeta _-(t)\\sigma _- \\ ,$ where $\\zeta _\\pm (t) = \\Xi ^\\dagger _\\mp (t) \\Xi _\\pm (t) \\ .$ Again, the 2nd order ME only depends on ${\\zeta _{\\pm }(t)} &= _\\mathrm {B}\\lbrace \\Xi ^\\dagger _\\mp (t) \\Xi _\\pm (t)\\rho _\\mathrm {B}\\rbrace \\\\{\\zeta _\\alpha (t_1)\\zeta _\\beta (t_2)} &=_\\mathrm {B}\\lbrace \\Xi ^\\dagger _{\\bar{\\alpha }}(t_1) \\Xi _\\alpha (t_1)\\Xi ^\\dagger _{\\bar{\\beta }}(t_2) \\Xi _\\beta (t_2)\\rho _\\mathrm {B}\\rbrace \\ ,$ where $\\bar{\\alpha }$ means the opposite symbol of $\\alpha $ in $\\lbrace +,-\\rbrace $ .", "Appendices D.4 and D.5 in [32] give detailed calculations for Eq.", "(REF ) and ()." ], [ "Correlation function in polaron frame", "We now discuss Eqs.", "eq:polaronk1, eq:polaronk2,eq:generalpolaronk1, eq:generalpolaronk2 for the spin-boson model.", "We first introduce the bath spectral function $J(\\omega ) = \\sum _k \\lambda ^2_k \\delta (\\omega -\\omega _k) \\ ,$ which is usually taken into the continuous limit.", "The standard result , for Eqs.", "(REF ) and () are $\\kappa = {\\xi _{\\pm }(t)} = \\exp {-2g^2\\sum _k\\frac{\\lambda _k^2}{\\omega _k^2}\\coth (\\beta \\omega _k/2)} = \\exp {-2g^2\\int _0^\\infty \\frac{J(\\omega )}{\\omega ^2}\\coth (\\beta \\omega /2){\\omega }} \\ ,$ and ${\\xi _\\alpha (t)\\xi _\\alpha (0)} &= \\kappa ^2(e^{-\\phi (t)}-1) \\\\{\\xi _\\alpha (t)\\xi _\\beta (0)}&=\\kappa ^2(e^{\\phi (t)}+1) \\ ,$ where $\\phi (t)=4g^2\\int _0^\\infty {\\omega }\\frac{J(\\omega )}{\\omega ^2}\\Big (\\coth (\\frac{\\beta \\omega }{2})\\cos (\\omega t)-i\\sin (\\omega t)\\Big ) \\ .$ For spectral functions which lead to $\\kappa \\rightarrow 0$ and $\\phi (t) \\rightarrow \\infty $ (e.g., an Ohmic bath), $\\kappa ^2 e^{\\phi (t)}$ converges to a finite number.", "Thus the standard result for an Ohmic bath is ${\\xi _+(t)\\xi _+(0)} &= {\\xi _-(t)\\xi _-(0)} = 0\\\\{\\xi _+(t)\\xi _-(0)} &= {\\xi _-(t)\\xi _+(0)} = \\exp (-4Q_2(t)-4iQ_1(t)) \\ ,$ where $Q_2(t)$ and $Q_1(t)$ are $Q_1(t) &\\equiv g^2\\int _0^\\infty \\frac{J(\\omega )}{\\omega ^2} \\sin (\\omega t) {\\omega } \\\\Q_2(t) &\\equiv g^2\\int _0^\\infty \\frac{J(\\omega )(1-\\cos (\\omega t))}{\\omega ^2}\\coth (\\beta \\hbar \\omega /2) {\\omega } \\ .$ By substituting the Ohmic spectral function $J(\\omega ) = \\eta \\omega e^{\\omega /\\omega _c}$ , the explicit form of $Q_1$ and $Q_2$ can be calculated.", "Here we take another approach by noticing that $-Q_2(t)-iQ_1(t) = \\int _0^t\\int _{-\\infty }^{0} C(t_1, t_2) {t_1} {t_2}\\ ,$ where $C(t_1, t_2)$ is the bath correlation function defined in Eq.", "(REF ) that can also be expressed in terms of the spectral function $C(t_1, t_2) = g^2\\int _0^{\\infty }{\\omega } J(\\omega )\\coth (\\beta \\omega /2)\\cos (\\omega (t_1-t_2)) - iJ(\\omega )\\sin (\\omega (t_1-t_2)) \\ .$ Thus Eq.", "() can be rewritten as ${\\xi _+(t)\\xi _-(0)} = {\\xi _-(t)\\xi _+(0)} = \\exp {4\\int _{-\\infty }^{\\infty } \\gamma (\\omega ) \\frac{e^{-i\\omega t}-1}{\\omega ^2}{\\omega }} \\ ,$ where $\\gamma (\\omega )$ is the spectral density defined in Eq.", "(REF ).", "For Eq.", "(REF ) and (), we follow Ref.", "[32] and write down the the result here for comparison.", "The 1st order average is ${\\zeta _{\\pm }(t)} = \\exp {-4Q_2(t)-4iQ_1(t)} \\ ,$ which is supposed to decrease exponentially fast for $t>0$ .", "The two-point correlation functions are ${\\zeta _+(t)\\zeta _+(0)} &= {\\zeta _-(t)\\zeta _-(0)} = 0\\\\{\\zeta _+(t_1)\\zeta _-(t_2)} &= {\\zeta _-(t_1)\\zeta _+(t_2)} = \\exp \\bigg \\lbrace -4Q_2(t)-4iQ_1(t) +4i\\big [C(t_1)-C(t_2)\\big ]\\bigg \\rbrace \\ .$ The difference between Eqs.", "eq:polaroncorrelationohmic and eq:generalpolaroncorrelationohmic are easily noticeable.", "The physical origins and significance of this difference is a subject for future studies." ], [ "Error analysis", "We present an error analysis based on the standard form of the polaron transformation (REF ).", "Our strategy is to divide the error into separate parts and use Lemma 1 in Ref.", "[4] to estimate the total error of the master equation.", "Besides those already presented in [4], there are two additional error terms in the PTRE, the physical origin of which are the inhomogeneous terms resulting from the breakdown of the factorized initial condition by $U_p$ : $U_p^\\dagger \\rho _\\mathrm {S}(0)\\otimes \\rho _\\mathrm {B} U_p \\ne \\tilde{\\rho }_\\mathrm {S}(0) \\otimes \\tilde{\\rho }_\\mathrm {B} \\ .$ In addition to the projector defined in Eq.", "(REF ), we define its complementary $\\mathcal {Q} \\equiv I -\\mathcal {P}$ .", "The standard projection operator technique [13] leads to $\\frac{\\partial }{\\partial t} \\mathcal {P}\\tilde{\\rho }(t) = \\sum _n \\mathcal {K}_n(t) \\mathcal {P}\\tilde{\\rho }(t) + \\sum _n \\mathcal {I}_n(t)\\mathcal {Q}\\tilde{\\rho }(0)\\ ,$ where the rightmost expression is the contribution of the inhomogeneous terms.", "Ref.", "calculates $\\mathcal {I}_n(t)$ up to $n=4$ and shows they are identical to those for $\\mathcal {K}_n(t)$ except that the final $\\mathcal {P}$ in each term is replaced by a $\\mathcal {Q}$ .", "Here we conjecture that this is also true for $n>4$ , and then use the same error bound as in Ref.", "[4] (called the Born approximation error there), to bound the error of truncating to the order of $\\mathcal {I}_2$ : $\\mathcal {E}^{(\\infty )}_I = {\\sum _{n=1}^{\\infty } \\mathcal {I}_n(t)\\mathcal {Q}\\tilde{\\rho }(0) - \\sum _{n=1}^{2} \\mathcal {I}_n(t)\\mathcal {Q}\\tilde{\\rho }(0)}_1 \\ .$ To simplify the notation, we define $\\rho _{\\text{true}}(t)$ as the solution of Eq.", "(REF ) when $n \\rightarrow \\infty $ .", "The PTRE error bound can be written as the sum of three terms: $\\Vert \\rho _{\\text{true}}(t) -\\rho _{\\text{PTRE}}(t) \\Vert _1 \\le \\mathcal {E}_{BM} + \\mathcal {E}_{I}^{(\\infty )} + \\mathcal {E}_{I}^{(2)} \\ ,$ where $\\mathcal {E}_{BM}$ is the error bound presented in Ref.", "[4]:The error bound in Ref.", "[4] includes the error due to changing $\\int ^{t} \\rightarrow \\int ^{\\infty }$ in the Redfield equation.", "We employ Eq.", "(REF ) despite not making this approximation, because the upper bound remains valid without it.", "$\\mathcal {E}_{BM} \\le O\\left(e^{\\frac{12 t}{\\tau _{S B}}} \\frac{\\tau _{B}}{\\tau _{S B}}\\right) \\ln \\frac{\\tau _{S B}}{\\tau _{B}} \\ ,$ $\\mathcal {E}_I^{(\\infty )}$ is given in Eq.", "(REF ) and $\\mathcal {E}^{(2)}_I={\\sum _{n=1}^{2} \\mathcal {I}_n(t)\\mathcal {Q}\\tilde{\\rho }(0)}_1$ .", "Here we argue that $\\mathcal {E}_I^{(\\infty )}$ can be bounded by the the same expression given in (REF ).", "Because $\\mathcal {Q}\\tilde{\\rho }(0) = \\tilde{\\rho }(0) - \\mathcal {P}\\tilde{\\rho }(0)$ , the error (REF ) can be bounded by the triangle inequality $\\mathcal {E}^{(\\infty )}_I \\le {\\sum _{n=1}^{\\infty } \\mathcal {I}_n(t)\\tilde{\\rho }(0) - \\sum _{n=1}^{2} \\mathcal {I}_n(t)\\tilde{\\rho }(0)}_1 + {\\sum _{n=1}^{\\infty } \\mathcal {I}_n(t)\\mathcal {P}\\tilde{\\rho }(0) - \\sum _{n=1}^{2} \\mathcal {I}_n(t)\\mathcal {P}\\tilde{\\rho }(0)}_1 \\ ,$ where the second term automatically satisfies the bound given in Eq.", "(REF ).", "To bound the first term, we first write out $\\tilde{\\rho }(0)$ explicitly for the case of a single system qubit and $H_{\\mathrm {I}}= g\\sigma ^z\\otimes \\sum _{k} \\lambda _{k} (b^\\dagger _{k} + b_{k})$ : $\\tilde{\\rho }(0) &= U_p^\\dagger \\rho _\\mathrm {S}(0)\\otimes \\rho _\\mathrm {B} U_p \\\\&={e^{iD}{0}+e^{-iD}{1}} \\rho _\\mathrm {S}(0) \\otimes \\rho _\\mathrm {B} {e^{-iD}{0}+e^{iD}{1}} \\\\&=\\rho _\\mathrm {S}^{00} e^{iD}\\rho _\\mathrm {B}e^{-iD} + \\rho _\\mathrm {S}^{11}e^{-iD}\\rho _\\mathrm {B}e^{iD} + \\rho _\\mathrm {S}^{10}e^{-iD}\\rho _\\mathrm {B}e^{-iD} + \\rho _\\mathrm {S}^{01}e^{iD}\\rho _\\mathrm {B}e^{iD}\\ ,$ where $U_p$ was defined in Eq.", "(REF ), and $D = \\sum _k \\frac{g\\lambda _k}{i\\omega _k}(b^\\dagger _k - b_k) \\ , \\qquad \\rho _\\mathrm {S}^{ij} = {i}{\\rho _\\mathrm {S}(0)}{j}{i}{j}\\ .$ Using our conjecture, the formal expansion of $\\mathcal {I}_n(t)$ is the same as $\\mathcal {K}_n(t)$ with the average operation in the rightmost term being replaced by ${\\cdot }_{mn} = _\\mathrm {B}{\\cdot \\ \\rho ^{mn}_\\mathrm {B}} \\ ,$ where $\\rho _\\mathrm {B}^{mn} = e^{n iD}\\rho _\\mathrm {B}e^{m iD}$ and $m,n \\in {-1,+1}$ .", "To illustrate this point, let us consider one term in the 4th order expansion $\\mathcal {I}_4(t)\\tilde{\\rho }(0)$ : $\\int _0^{t}\\int _0^{t_1}\\int _0^{t_2}\\operatorname{Tr}_{\\mathrm {B}}\\left[\\tilde{\\mathcal {L}}(t) \\tilde{\\mathcal {L}}\\left(t_{3}\\right) \\rho _{\\mathrm {B}}\\right] \\operatorname{Tr}_{\\mathrm {B}}\\left[\\tilde{\\mathcal {L}}\\left(t_{1}\\right) \\tilde{\\mathcal {L}}\\left(t_{2}\\right) \\tilde{\\rho }(0)\\right] {t_1}{t_2}{t_3}\\ .$ After substituting Eq.", "(REF ) into the above expression and expanding the commutators, one term in the summation is $\\mathcal {F}_{++} = \\Delta ^4\\int _0^{t}\\int _0^{t_1}\\int _0^{t_2} {t_1}{t_2}{t_3} {\\xi _+(t)\\xi _-(t_3)} {\\xi _+(t_1)\\xi _-(t_2)}_{++} \\sigma ^+ \\sigma ^-\\sigma ^+ \\sigma ^- \\rho ^{01}_\\mathrm {S} \\ ,$ where we use $+, -$ in the subscript instead of $+1, -1$ to simplify the notation.", "Thus, the next step is to calculate the average and two point correlation function under the new “average” operator (REF ): ${\\xi _\\pm (t)}_{mn} &= _\\mathrm {B}\\lbrace e^{n i D}U^\\dagger _\\mathrm {B}(t) \\xi _\\pm U_\\mathrm {B}(t) e^{m i D}\\rho _\\mathrm {B}\\rbrace \\\\{\\xi _\\alpha (t_1)\\xi _\\beta (t_2)}_{mn} &=_\\mathrm {B}\\lbrace e^{n i D}U^\\dagger _\\mathrm {B}(t_1) \\xi _\\alpha U_\\mathrm {B}(t_1) U^\\dagger _\\mathrm {B}(t_2) \\xi _\\beta U_\\mathrm {B}(t_2) e^{m i D}\\rho _\\mathrm {B}\\rbrace \\ .$ Eq.", "() can be explicitly carried out using the following identities: $\\xi _{\\pm }(t) \\equiv U^\\dagger _\\mathrm {B}(t) \\xi _\\pm U_\\mathrm {B}(t) &= \\exp {\\pm 2i U^\\dagger _\\mathrm {B}(t)\\sum _k\\frac{g\\lambda _k}{i\\omega _k}{b^\\dagger _k - b_k} U_\\mathrm {B}(t)} \\\\&= \\exp {\\pm 2i \\sum _k \\frac{g\\lambda _k}{i\\omega _k}{b^\\dagger _k e^{i\\omega _k t} - b_k e^{-i\\omega _k t}}} \\\\{b^\\dagger _k e^{i\\omega _k t} - b_k e^{-i\\omega _k t}}{b^\\dagger _{k^{\\prime }} - b_{k^{\\prime }}} &= 2i\\sin (\\omega _k t)\\delta _{kk^{\\prime }} \\ ,$ and the Baker-Campbell-Hausdorff (BCH) formula $e^X e^Y = e^{X+Y+\\frac{1}{2}{X}{Y}+\\dots } \\ .$ After some algebraic manipulations, we get ${\\xi _\\pm (t)}_{mn} = e^{\\pm 4in \\sum _k \\frac{g^2\\lambda ^2_k}{\\omega _k^2}\\sin (\\omega _kt)} {\\xi _\\pm (t)e^{i(n+m)D}} = {\\left\\lbrace \\begin{array}{ll}f^\\pm _n(t){\\xi _\\pm (t)} & m \\ne n\\\\f^\\pm _n(t){\\xi _\\pm (t)\\xi _l(0)} & m=n, \\ l = \\operatorname{sgn}(n)\\end{array}\\right.}", "\\ ,$ where $f^\\alpha _n(t) = \\exp {4i\\alpha n \\sum _k \\frac{g^2\\lambda ^2_k}{\\omega _k^2}\\sin (\\omega _kt)}$ .", "For simplicity, we consider only the Ohmic bath case, for which ${\\xi _\\pm (t)}=0$ and ${\\xi _\\pm (t)\\xi _l(0)}$ decay exponentially for $t>0$ .", "As a result, we can ignore the first order term ${\\xi _\\pm (t)}_{mn}\\approx 0$ .", "The two-point correlation function is ${\\xi _\\alpha (t_1)\\xi _\\beta (t_2)}_{mn}=f^\\alpha _n(t_1)f^\\beta _n(t_2){\\xi _\\alpha (t_1)\\xi _\\beta (t_2)e^{i(n+m)D}} = {\\left\\lbrace \\begin{array}{ll}f^\\alpha _n(t_1)f^\\beta _n(t_2){\\xi _\\alpha (t_1)\\xi _\\beta (t_2)} & m \\ne n\\\\f^\\alpha _n(t_1)f^\\beta _n(t_2){\\xi _\\alpha (t_1)\\xi _\\beta (t_2)\\xi _l(0)} & m=n, \\ l = \\operatorname{sgn}(n)\\end{array}\\right.}", "\\ .$ where the second line equals 0 because ${\\xi _\\pm (t)}=0$ and the noise is Gaussian.", "The above result can be generalized to the multi-point correlation function ${\\xi _{\\alpha _1}(t_1)\\xi _{\\alpha _2}(t_2)\\cdots \\xi _{\\alpha _k}(t_k)}_{mn} = {\\left\\lbrace \\begin{array}{ll}f^{\\alpha _1}_n(t_1)f^{\\alpha _2}_n(t_2)\\cdots f^{\\alpha _k}_n(t_k){\\xi _{\\alpha _1}(t_1)\\xi _{\\alpha _2}(t_2)\\cdots \\xi _{\\alpha _k}(t_k)} & m \\ne n\\\\f^{\\alpha _1}_n(t_1)f^{\\alpha _2}_n(t_2)\\cdots f^{\\alpha _k}_n(t_k){\\xi _{\\alpha _1}(t_1)\\xi _{\\alpha _2}(t_2)\\cdots \\xi _{\\alpha _k}(t_k)\\xi _l(0)} & m=n, \\ l = \\operatorname{sgn}(n)\\end{array}\\right.}", "\\ .$ For the $m \\ne n$ case, the correlation function under ${\\cdot }_{mn}$ is the same as the correlation function under ${\\cdot }$ up to an additional phase factor.", "For the $m = n$ case, the right hand side of Eq.", "(REF ) can be decomposed into two-point correlation functions by Wick’s theorem.", "After the decomposition, each term would appear inside the integral as $\\int _0^{t_1}{t_2}\\int _0^{t_2}{t_3}\\cdots \\int _0^{t_{k-1}}{t_k} {\\xi _{\\alpha _1}(t_{\\alpha _1})\\xi _{\\alpha _2}(t_{\\alpha _2})} \\cdots {\\xi _{\\alpha _s}(t_s)\\xi _l(0)} \\ .$ Again, because the particular pair ${\\xi _{\\alpha _s}(t_s)\\xi _l(0)}$ decreases exponentially for $t_s > 0$ and thus has approximately zero overlap with other two point correlation functions under the integral, we ignore all these terms.", "Finally, we can sum up every term over the indices $m$ and $n$ .", "For example, for terms that have the same form as Eq.", "(REF ), the summation is $\\mathcal {F}_{++} + \\mathcal {F}_{+-} + \\mathcal {F}_{-+} + \\mathcal {F}_{--} = \\Delta ^4\\int _0^{t}\\int _0^{t_1}\\int _0^{t_2} {t_1}{t_2}{t_3} {\\xi _+(t)\\xi _-(t_3)} {\\xi _+(t_1)\\xi _-(t_2)} \\sigma ^+ \\sigma ^-\\sigma ^+ \\sigma ^- \\tilde{o} \\ ,$ where $\\tilde{o} = \\rho _\\mathrm {S}^{00}f^+_{-}(t_1)f^-_{-}(t_2) + \\rho _\\mathrm {S}^{11}f^+_{+}(t_1)f^-_{+}(t_2)$ .", "Eq.", "(REF ) can be bounded by ${\\mathcal {F}_{++} + \\mathcal {F}_{+-} + \\mathcal {F}_{-+} + \\mathcal {F}_{--}}_1 \\le \\Delta ^4\\int _0^{t}\\int _0^{t_1}\\int _0^{t_2} {t_1}{t_2}{t_3}{{\\xi _+(t)\\xi _-(t_3)}}{{\\xi _+(t_1)\\xi _-(t_2)}} \\ ,$ where we make use of the fact $\|{\\tilde{o}}\| \\le \|\\rho _\\mathrm {S}^{00}\| + \|\\rho _\\mathrm {S}^{11}\|= 1$ .", "Formally applying the same technique to every term after expanding $\\mathcal {I}_n(t)\\tilde{\\rho }(0)$ , allows us to show that the first part of Eq.", "(REF ) also satisfies the error bound given in Eq.", "(REF ).In this case we do not have the Markov approximation error and the error due to changing the integration limit.", "But the bound can still be relaxed to the form of Eq.", "(REF ).", "In conclusion, the error due to truncation of the inhomogeneous parts to second order has the same scaling as the error due to truncation of the homogeneous parts [Eq.", "(REF )].", "As a result, the total truncation error can be combined using a single big-$O$ notation: $\\Vert \\rho _{\\text{true}}(t) -\\rho _{\\text{PTRE}}(t) \\Vert _1 \\le O\\left(\\frac{\\tau _B}{\\tau _{SB}} e^{12t/\\tau _{SB}}\\right)\\text{ln}\\left(\\frac{\\tau _{SB}}{\\tau _B} \\right)\\ + \\mathcal {E}_I^{(2)},$ We discuss the timescales in the polaron frame in .", "Before proceeding, we remark that: The same analysis also applies to the multi-qubit PTRE described in Eq.", "(REF ) by replacing $\\rho ^{ij}_\\mathrm {S}$ with $\\rho _k^{ij}$ $\\rho ^{ij}_k = Z_k \\rho _\\mathrm {S}(0) Z_k \\ ,$ where $k$ is the index for different qubits.", "The analysis is also applicable for time-dependent $\\Delta (t)$ .", "In this case, the corresponding two-point correlation function is defined as $C^{\\alpha \\beta }(t_1, t_2) = \\Delta (t_1)\\Delta (t_2){\\xi _\\alpha (t_1)\\xi _\\beta (t_2)}$ , which can be upper bounded by ${C^{\\alpha \\beta }(t_1, t_2)} \\le \\Delta ^2_m{{\\xi _\\alpha (t_1)\\xi _\\beta (t_2)}} \\ ,$ where $\\Delta _m = \\max _{t\\in [0, t_f]} \\Delta (t)$ .", "This introduces a constant $\\Delta _m$ in the definition of the timescales.", "The error bound we analyzed in this section does not include the error due to ignoring the first and second order inhomogeneous terms $\\mathcal {E}_I^{(2)}$ (which is usually the case in the standard PTRE treatment).", "Numerical studies show that they can be ignored if $\\rho _\\mathrm {S}(0)$ is diagonal .", "More rigorous error bounding is an interesting topic for future study." ], [ "Time scales", "The time scales given in Eq.", "(REF ) can be computed directly for the polaron frame two-point correlation function ().", "Because the error bound in Eq.", "(REF ) scales with $1/\\tau _{SB}$ and $\\tau _B/\\tau _{SB}$ , we explicitly write out those two quantities $\\frac{1}{\\tau _{SB}} &= \\Delta _m^2\\int _0^\\infty {K(\\tau )} {\\tau } =\\int _0^\\infty \\exp {-4 g^2\\int _0^\\infty \\frac{J(\\omega )(1-\\cos (\\omega \\tau ))}{\\omega ^2}\\coth (\\beta \\hbar \\omega /2) {\\omega }} {\\tau }\\\\\\frac{\\tau _B}{\\tau _{SB}} &= \\Delta _m^2\\int _0^{t_f} \\tau {K(\\tau )} {\\tau } = \\int _0^\\infty \\tau \\exp {-4 g^2\\int _0^\\infty \\frac{J(\\omega )(1-\\cos (\\omega \\tau ))}{\\omega ^2}\\coth (\\beta \\hbar \\omega /2) {\\omega }} {\\tau } \\ ,$ where $\\Delta _m = \\max _{t\\in [0, t_f]} \\Delta (t)$ .", "We can see that, when the system-bath coupling strength is sent to infinity, both of these quantities go to zero: $1/\\tau _{SB}\\rightarrow 0$ and $\\tau _B/\\tau _{SB} \\rightarrow 0$ as $g\\rightarrow \\infty $ .", "Thus, the PTRE works in the strong coupling regime." ], [ "Lindblad form of the PTRE via the adiabatic approximation", "In this section, we derive an adiabatic form of the PTRE from Eq.", "(REF ): ${}{t}\\tilde{\\rho }_{\\mathrm {S}}(t) = -i{\\tilde{H}_\\mathrm {S}(t)+a(t)\\sum _i\\kappa _i\\sigma _x}{\\tilde{\\rho }_\\mathrm {S}(t)} - \\sum _{i} {\\sigma _i^\\alpha }{\\Lambda _i^\\alpha (t)\\tilde{\\rho }_\\mathrm {S}(t)} + \\text{h.c.} \\ ,$ where $\\Lambda _i^\\alpha (t) &= a(t)\\sum _{\\beta } \\int _0^{t} a(\\tau )K_i^{\\alpha \\beta }(t - \\tau )\\tilde{U}_\\mathrm {S}(t, \\tau )\\sigma _i^\\beta \\tilde{U}_\\mathrm {S}^\\dagger (t, \\tau ) {\\tau } \\\\K_i^{\\alpha \\beta }(t - \\tau ) & ={\\xi _i^\\alpha (t-\\tau ) \\xi _i^\\beta (0)}\\\\\\kappa _i &= {\\xi _i^{\\pm }(t)}\\ ,$ and $\\tilde{U}_\\mathrm {S}(t, \\tau ) = T_+\\exp {-i\\int _\\tau ^t \\tilde{H}_\\mathrm {S}(\\tau ^{\\prime }) {\\tau ^{\\prime }}} \\ .$ Three approximations need to be applied to obtain the final result:" ], [ "Markov approximation", "The first step is to move the function $a(t)$ in Eq.", "(REF ) outside the integral $\\int _0^t a(\\tau ) \\cdots {\\tau } \\rightarrow a(t)\\int _0^t \\cdots {\\tau } \\ ,$ then perform a change of variable $t-\\tau \\rightarrow \\tau $ .", "Eq.", "(REF ) becomes: $\\Lambda _i^\\alpha (t) = a^2(t)\\sum _{\\beta } \\int _0^{t} K_i^{\\alpha \\beta }(\\tau )\\tilde{U}_\\mathrm {S}(t, t-\\tau )\\sigma _i^\\beta \\tilde{U}_\\mathrm {S}^\\dagger (t, t-\\tau ) {\\tau } \\ .$ Finally, the integration limit is taken to infinity: $\\int _0^{t} \\rightarrow \\int _0^\\infty $ .", "A detailed description of the adiabatic approximation is given in Ref. [3].", "The core idea is to rewrite the unitary in Eq.", "(REF ) as $\\tilde{U}_{\\mathrm {S}}(t, t-\\tau )=\\tilde{U}_\\mathrm {S}(t,0)\\tilde{U}^\\dagger _{\\mathrm {S}}(t-\\tau , 0) \\ ,$ and then replace $\\tilde{U}_\\mathrm {S}$ with an appropriate adiabatic evolution $\\tilde{U}_\\mathrm {S}(t-\\tau ,0) &\\approx e^{i\\tau \\tilde{H}_\\mathrm {S}(t)} U^{\\mathrm {ad}}_\\mathrm {S}(t, 0) \\\\\\tilde{U}_\\mathrm {S}(t,0) &\\approx U^{\\mathrm {ad}}_\\mathrm {S}(t, 0) \\ ,$ where $U^{\\mathrm {ad}}_\\mathrm {S}(t, 0)$ is the ideal adiabatic evolution $U_{\\mathrm {S}}^{\\mathrm {ad}}\\left(t, t^{\\prime }\\right)=\\sum _{a}\\left\|\\varepsilon _{a}(t)\\right\\rangle \\left\\langle \\varepsilon _{a}\\left(t^{\\prime }\\right)\\right\| \\mathrm {e}^{-\\mathrm {i} \\mu _{a}\\left(t, t^{\\prime }\\right)} \\ ,$ with a phase $\\mu _{a}\\left(t, t^{\\prime }\\right)=\\int _{t^{\\prime }}^{t} \\mathrm {d} \\tau \\left[\\varepsilon _{a}(\\tau )-\\phi _{a}(\\tau )\\right] \\ .$ In the above expressions, ${\\varepsilon _{a}(t)}$ is the instantaneous eigenstate of $\\tilde{H}_\\mathrm {S}(t)$ and $\\phi _{a}(t)={i}\\left\\langle \\varepsilon _{a}(t) \\mid \\dot{\\varepsilon }_{a}(t)\\right\\rangle $ is the Berry connection.", "After this procedure, Eq.", "(REF ) becomes: $\\Lambda _{i}^\\alpha (t) = a^2(t)\\sum _{\\beta ,a,b} \\Gamma _i^{\\alpha \\beta }(\\omega _{ba})L_{i\\beta }^{ab}(t) \\ ,$ where $\\omega _{ba} = \\varepsilon _b - \\varepsilon _a$ , and $L_{i\\beta }^{ab}(t) = {\\varepsilon _a(t)}{\\sigma _i^\\beta }{\\varepsilon _b(t)} {\\varepsilon _a(t)}{\\varepsilon _b(t)} \\ .$ $\\Gamma _i^{\\alpha \\beta }$ is the one-sided Fourier transform of the correlation function $\\Gamma _i^{\\alpha \\beta }(\\omega ) = \\int _0^{\\infty }{t} K_i^{\\alpha \\beta }(t) e^{i\\omega t} \\ .$ Substituting Eq.", "(REF ) into Eq.", "(REF ), we have the one-sided adiabatic PTRE.", "To perform the RWA, we first need to move the one-sided adiabatic PTRE into the interaction picture with respect to $\\tilde{H}_\\mathrm {S}(t)$ via the approximated unitary in Eq.", "(REF ).", "The non-Hamiltonian part of Eq.", "(REF ) is transformed into $&- \\tilde{U}^\\dagger _\\mathrm {S}(t)\\sum _{i} {\\sigma _i^\\alpha }{\\Lambda _i^\\alpha (t)\\tilde{\\rho }_\\mathrm {S}(t)}\\tilde{U}_\\mathrm {S}(t) + h.c. \\approx \\\\ &\\quad \\sum _{a,b,a^{\\prime },b^{\\prime }} e^{-i{\\mu _{ba}(t,0)+\\mu _{b^{\\prime }a^{\\prime }}(t,0)}}a^2(t)\\sum _i\\Gamma _i^{\\alpha \\beta }(\\omega ) \\bigg \\lbrace {\\varepsilon _a(t)}{\\sigma _i^\\beta }{\\varepsilon _b(t)} \\Pi _{ab}(0) \\accentset{\\approx }{\\rho }_\\mathrm {S}(t)\\Pi _{a^{\\prime }b^{\\prime }}(0){\\varepsilon _{a^{\\prime }}(t)}{\\sigma _i^\\alpha }{\\varepsilon _{b^{\\prime }}(t)} \\\\&\\quad - {\\varepsilon _{a^{\\prime }}(t)}{\\sigma _i^\\alpha }{\\varepsilon _{b^{\\prime }}(t)} {\\varepsilon _a(t)}{\\sigma _i^\\beta }{\\varepsilon _b(t)} \\Pi _{ab}(0)\\Pi _{a^{\\prime }b^{\\prime }}(0) \\accentset{\\approx }{\\rho }_\\mathrm {S}(t)\\bigg \\rbrace + \\text{h.c.} \\ ,$ where $\\Pi _{ab}(0)$ is a shorthand notation for ${\\varepsilon _a(0)}{\\varepsilon _b(0)}$ .", "The RWA amounts to keeping only terms where either $a^{\\prime }=b$ , $b^{\\prime }=a$ or $a=b$ , $a^{\\prime }=b^{\\prime }$ .", "After the fast-oscillating terms are ignored, we move back to the polaron frame and make use of Eq.", "(REF ) to derive the Lindblad-form PTRE: $\\dot{\\tilde{\\rho }}_\\mathrm {S}(t) = -i{\\tilde{H}_\\mathrm {S}(t)+\\tilde{H}_{\\mathrm {LS}}(t)+a(t)\\sum _i\\kappa _i\\sigma _x}{\\tilde{\\rho }_\\mathrm {S}(t)} + \\sum _{i,\\alpha ,\\beta }\\sum _\\omega \\gamma _i^{\\alpha \\beta }(\\omega ){L_{i}^{\\omega ,\\beta }(t)\\tilde{\\rho }_\\mathrm {S}(t)L^{\\omega ,\\bar{\\alpha }\\dagger }_{i}(t)-\\frac{1}{2}{L^{\\omega ,\\bar{\\alpha }\\dagger }_{i}(t)L_{i}^{\\omega ,\\beta }(t)}{\\tilde{\\rho }_\\mathrm {S}(t)}} \\ ,$ where the new Lindblad operators are given by $L_{i}^{\\omega ,\\alpha }(t) = a(t)\\sum _{\\varepsilon _b - \\varepsilon _a = \\omega }{\\varepsilon _a(t)}{\\sigma ^\\alpha _i}{\\varepsilon _b(t)}{\\varepsilon _a(t)}{\\varepsilon _b(t)} \\ .$ The notation $\\bar{\\alpha }$ again means the opposite symbol of $\\alpha $ in ${+, -}$ .", "Since $\\sigma _i^{\\pm }$ are Hermitian conjugates of each other: $L_i^{-\\omega , \\alpha }(t) = {L_i^{\\omega .", "\\bar{\\alpha }}(t)}^\\dagger \\ .$ Finally, the Lamb shift term is $\\tilde{H}_{\\mathrm {LS}}(t) = \\sum _{i,\\alpha ,\\beta }\\sum _{\\omega } L_i^{\\omega ,\\bar{\\alpha }\\dagger }L_i^{\\omega ,\\beta }(t)S_i^{\\alpha \\beta }(\\omega ) \\ .$ Because the techniques used here are the same as those in [3], [4], the error bounds in these references are still valid." ], [ "Hybrid noise", "The two point correlation function in the polaron frame [Eq.", "(REF )] can be extended to a hybrid noise model .", "The core assumption here is that the bath spectral density can be separated into low frequency and high frequency parts: $\\gamma (\\omega ) = \\gamma _L(\\omega ) + \\gamma _H(\\omega ) \\ .$ The integral inside the exponent of Eq.", "(REF ) can also be separated into $f_L(t) = \\int _{-\\infty }^{\\infty } {\\omega } \\gamma _L(\\omega ) \\frac{e^{-i\\omega t}-1}{\\omega ^2} \\ ,\\quad f_H(t) = \\int _{-\\infty }^{\\infty } {\\omega } \\gamma _H(\\omega ) \\frac{e^{-i\\omega t}-1}{\\omega ^2} \\ .$ If $\\gamma _L(\\omega )$ is concentrated on low frequencies, a 2nd order Taylor expansion of $e^{-i\\omega t}$ is justified.", "As a result, the first term in Eq.", "(REF ) can be written as $f_L(t) = -\\frac{1}{2}W^2t^2-i\\varepsilon _L t$ where $W^2 = \\int \\gamma _L(\\omega ) {\\omega } \\quad \\varepsilon _L = \\mathcal {P} \\int \\frac{\\gamma _L(\\omega )}{\\omega } {\\omega } \\ ,$ and $\\mathcal {P}$ stands for the Cauchy principal value.", "$W$ and $\\varepsilon _L$ , usually known as the MRT linewidth and reorganization energy, respectively, are experimentally measurable quantities that are connected through the fluctuation-dissipation theorem : $W^2 = 2\\varepsilon _L T$ .", "Finally, the two point correlation function under this hybrid noise model is $K(t) = {\\xi _+(t)\\xi _-(0)} = {\\xi _-(t)\\xi _+(0)} = e^{-4i\\varepsilon _L t - 2W^2t^2} \\exp {4\\int {\\omega } \\gamma _H(\\omega ) \\frac{e^{-i\\omega t}-1}{\\omega ^2}} \\ .$ The corresponding spectral density of $K(t)$ is $\\gamma _K(\\omega ) = \\int K(t) e^{i\\omega t} {t} = \\int \\frac{{x}}{2\\pi } G_L(\\omega -x)G_H(x) {x} \\ ,$ where $G_L(\\omega )$ and $G_H(\\omega )$ are the Fourier transforms of $\\exp {4f_L(t)}$ and $\\exp {4f_H(t)}$ , respectively.", "The low frequency component is Gaussian: $G_L(\\omega ) = \\int {\\omega } e^{-4i\\varepsilon _L t - 2W^2t^2} e^{i\\omega t}= \\sqrt{\\frac{\\pi }{2W^2}}\\exp \\Bigg [-\\frac{(\\omega -4\\varepsilon _L)^2}{8W^2}\\Bigg ] \\ ,$ while the high frequency part can be approximated as a single Lorentzian [32]: $G_H(\\omega ) = \\frac{4\\gamma _H(\\omega )}{\\omega ^2 + 4\\gamma _H(0)^2} \\ .$ Finally, it should be mentioned that the fluctuation-dissipation theorem guarantees that the Kubo-Martin-Schwinger (KMS) condition be satisfied for $G_L(\\omega )$ .", "So if $\\gamma _H(\\omega )$ satisfies KMS condition, the spectral density in the polaron frame also satisfies the KMS condition $\\gamma _K({\\omega }) = e^{\\beta {\\omega }}\\gamma _K(-{\\omega }) \\ .$" ] ]	2011.14046
[ [ "On Classifying the Effects of Policy Announcements on Volatility" ], [ "Abstract The financial turmoil surrounding the Great Recession called for unprecedented intervention by Central Banks: unconventional policies affected various areas in the economy, including stock market volatility.", "In order to evaluate such effects, by including Markov Switching dynamics within a recent Multiplicative Error Model, we propose a model--based classification of the dates of a Central Bank's announcements to distinguish the cases where the announcement implies an increase or a decrease in volatility, or no effect.", "In detail, we propose two smoothed probability--based classification methods, obtained as a by--product of the model estimation, which provide very similar results to those coming from a classical k--means clustering procedure.", "The application on four Eurozone market volatility series shows a successful classification of 144 European Central Bank announcements." ], [ "Introduction", "Since the onset of the Great Recession, many Central Banks resorted to unconventional monetary policy in order to mitigate the consequences that the crisis had on the real economy and financial markets as well.", "All the related measures were introduced by means of monetary policy announcements; recent literature focused on the consequences for the real economy (e.g.", "[1], [2]) and for the financial markets, in particular their volatility [3], [4], [5], all viewing the effect of the announcements as a constant factor in the models.", "The actual strength of an announcement, however, turns out to be a consequence of the conditions in which the measure is adopted, its wording, how it constitutes a surprise relative to the consensus, how divergent expectations are, and so on.", "The impact on financial markets, in particular on their volatility, is a consequence of the immediate adjustment of the asset prices to the new information, and the formation of new equilibria after the announcement.", "Within the class of Multiplicative Error Models (MEMs) by [6], the recent work by [7] is a first attempt at measuring the unconventional policy effects as an unobservable component of volatility, distinguishing the effects due to policy implementation (as represented by a balance sheet–based continuous proxy variable) from those on the announcement day (related to a dummy variable).", "In what follows, we chose to modify their univariate Asymmetric Composite Model (ACM – cf.", "also the previous contribution by [8]): here, rather than resorting to a dummy variable, we take the view that effects of an announcement could be measured in terms of changes in volatility level across two alternative nonobservable regimes; to this end, we specify volatility dynamics as Markov Switching (MS), without revealing the model when the announcement took place.", "As a by–product, we suggest a classification rule for the underlying announcements, according to whether they have a notable impact on volatility via a change in regime or they induced a permanence in the same regime.", "Such a classification is possible only thanks to the MS approach and would not be reproduceable for other volatility models (notably, the plain Asymmetric (A)MEM by [9], or the HAR by [10]).", "In spite of it not being a customary classification method based on distance measures, our approach is still within the large literature concerning model–based clustering (for a recent update on the state of the art, cf.", "[11] and convenient reviews of these methods are in [12]).", "Interestingly, next to the whole and subsequence areas of analysis, [13] mention also the time-point analysis strain of research, where we place our approach, aimed at detecting both expected and unusual patterns through the identification of dynamic changes in time series features.", "By the same token, our proposal is clearly different from the model–based techniques aimed at developing clustering in volatility of financial markets, such as [14], [15], [16], [17], where the classification involves the full time series and not individual observations.", "The same considerations can be made by comparing our approach with [18], where an ACM model similar to the one proposed in this paper is adopted, but, again, with the aim to classify the similarities between entire financial time series.", "Based on the estimation of our model on time series of realized volatility (for four Eurozone stock indices, i.e., CAC40, DAX30, FTSEMIB and IBEX35), we consider 144 announcements and we classify how such announcements had an effect on the level of each market volatility: the groups we get are named Plank (a neutral effect), Squat (a decrease in volatility) and Jump (an increase in volatility).", "Built as a simple processing of the smoothed MS probabilities of being in either a low or a high volatility regime, our classification techniques are simple to implement and deliver results very similar to a benchmark k–means clustering approach.", "The paper is structured as follows.", "We detail our MS time-point analysis approach to classification in Section , within which the model is presented in Subsection REF and the proposed classification procedures in Subsection REF .", "The empirical application is contained in Section , where we discuss data features and the framework of events occurring in our sample period; estimation results are discussed in Subsection REF and the corresponding classifications in Subsection REF .", "Finally, Section contains some concluding remarks.", "Volatility modelling exploits the availability of high frequency data, favouring a decoupling of measurement and modelling with respect to the more traditional approach based on GARCH models [19], [20].", "The so–called Realized Volatility (RV) is recognized to have better properties in measurement than the outcome of the estimated GARCH–based conditional variance of returns [21].", "As per forecasting, several conditional models are available for RV; one of them is the MEM, proposed by [6], which takes volatility dynamics in terms of the product of two positive time-varying factors, one representing its conditional mean and the other a positive–valued disturbance.", "Several improvements allow to capture stylized facts and to accommodate specific cases; in particular the [9] specification introduces the asymmetric and predetermined variable effects.", "In what follows, with an eye to capturing policy effects, we extend the MS–AMEM by [22] to additively accomodate policy–induced effects (component $\\xi _{t,s_t}$ ) next to the volatility dynamics (the base volatility component $\\varsigma _{t}$ ): $\\begin{array}{l}RV_{t}=\\mu _{t,s_t}\\epsilon _{t}, \\, \\epsilon _{t}\|\\mathbf {I}_{t-1}\\sim Gamma(\\vartheta _{s_t} ,\\frac{1}{\\vartheta _{s_t}})\\\\\\mu _{t,s_t}=\\varsigma _{t}+\\xi _{t,s_t}\\\\\\varsigma _{t}=\\omega +\\alpha RV_{t-1}+\\beta \\varsigma _{t-1}+\\gamma D_{t-1}RV_{t-1}\\\\\\xi _{t,s_t}=\\varphi _0+\\varphi _{1} s_t+ \\delta (E\\left(x_{t}\|\\mathbf {I}_{t-1}\\right)-\\bar{x})+ \\psi \\xi _{t-1,s_{t-1}}\\end{array} $ As with any other MEM, the realized volatility at time $t$ , $RV_t$ , is seen as the product of a conditional (on the past information set $\\mathbf {I}_{t-1}$ ) expectation term $\\mu _t$ times a unit mean error term $\\epsilon _t$ following a Gamma distribution.In suggesting the Asymmetric MEM, [9] justify the adoption of a Gamma distribution in view of its flexibility and the fact that it nests other notable distributions (such as the exponential and the Chi-square).", "In a more general framework for MEMs, [23] show that the first order conditions for a Gamma–based likelihood function coincide with the objective function of a semiparametric GMM based approach.", "In our approach, in line with [7], the expected conditional volatility is decomposed as the sum of $\\varsigma _t$ , evolving as a GARCH–like process (with asymmetric effects tied to the negative sign of past returns captured by a dummy variable, $D_{t-1}$ ), and a policy–specific and regime–dependent term $\\xi _{t,s_t}$ , which follows an AR(1) model.", "The driving variable for this dynamics is $x_t$ – a proxy for unconventional policy measures, entering the model as the deviation of its conditional expectation from a long term mean $\\bar{x}$ – which accounts for the Central Banks' balance sheet composition.", "In the original ACM model by [7]Actually, this model is derived from the general framework in [8], where the mean of the conditional volatility is the sum of two unobservable components: the particular ACM specification proposed by [24] to model spillover effects in financial markets is adapted to representing the base volatility and the unconventional policy effect respectively.", "the last equation is: $\\begin{array}{c}\\xi _{t}=\\delta (E\\left(x_{t}\|\\mathbf {I}_{t-1}\\right)-\\bar{x})+ \\varphi (\\Lambda _{t}-\\bar{\\Lambda })+ \\psi \\xi _{t-1},\\end{array}$ where, the $\\Lambda _t$ term is a dummy variable representing the effect of the announcements and is taken as deviation from its long–term mean, also here; the days of the announcements are not random variables, as they are put in the calendar in advance by the ECB.", "Contrary to the approach by [7], we are not considering the announcements explicitly in the model via dummy variables; what constitutes the novelty in the present econometric context is that we detect their presence when a change in regime is attributable to a market volatility reaction to the announcement.", "More in detail, we extend the ACM model by [7] in making the policy–specific component $\\xi _t$ become $\\xi _{t,s_t}$ , i.e., we consider a dichotomic discrete latent variable $s_t=0,1$ , representing the regime at time $t$ .", "When $s_t=0$ , the time series is in a low volatility regime with intercept $\\varphi _0$ , and increases by the term $\\varphi _1\\ge 0$ in the high volatility regime ($s_t=1$ ).", "In practice, though, as the regime is not observable, such an intercept turns out to be time–varying, as discussed below.", "The dynamics of the state variable $s_t$ is driven by a first–order Markov chain, that is: $Pr(s_{t}=j\|s_{t-1}=i,s_{t-2}\\ldots )=Pr(s_{t}=j\|s_{t-1}=i)=p_{ij}.$ Building also on the evidence and the discussion presented in [22], we obtain the important result of expanding the flexibility of the model, since the distribution of the error term now follows a mixture of two Gamma densities.", "Positiveness ($\\omega >0$ , $\\alpha , \\beta , \\gamma \\ge 0$ ) and stationarity ($\\alpha +\\beta +\\frac{\\gamma }{2}<1$ and $\\left\|\\psi \\right\|<1$ ) conditions established in the case of the ACM are regime–independent and hence are valid for the MS–ACM as well.", "The Markov process assumption implies the dependence only on the current state and implicitly assumes that the sojourn distribution is geometric.", "In principle, we can assume a more general dependence on the duration of the state or on $p$ lagged states, adopting semi–Markov processes and transition distribution models.", "In the present framework, however, we rely on customary assumption of market efficiency, based on the tenet that information is instantly and completely incorporated into the current price.", "This theory is supported by the empirical evidence with daily data and is consistent with the econometric literature of MS models in the GARCH framework (see, for example, [25], [26], [27]).", "The likelihood function of the MS–ACM is obtained by means of the so called Hamilton filter and smoother, as described in [28] (Ch.", "22), adopting the approximated solution proposed by [29] to solve the path dependence issue, a computational problem due to the dependence of $\\mu _t$ on all past values of $s_t$ .", "In fact, at the end of the recursive Hamilton filter, we would have to keep track of all possible paths obtained by all the combinations of regimes from the first time until the last one, with $2^T$ possible different scenarios.", "To by–pass this problem, [29] proposes to collapse the 4 possible values of $\\mu _t$ at time $t$ , obtained at the end of each step of the Hamilton filter, into 2 values, averaging and weighing them with the corresponding conditional probabilities $Pr[s_{t-1}=i,s_{t}=j\|\\mathbf {I}_t]$ (obtained in the same Hamilton filter step).", "This approximation is the most popular solution, albeit not exact, causing a bias in the likelihood function; in practice, though, the collapsing procedure does not involve substantial errors in the final estimation: as shown by [22] through simulation experiments, the approximated likelihood is satisfactory when the number of observations is above 2000, an attainable goal with daily financial time series.", "As a recent contribution, [30] have generalized and improved the collapsing procedure for MS–GARCH models, implementing optimal particle filters to explicit the likelihood; the particle selection step is performed deterministically and not stochastically, favoring the implementation and reducing the computational effort.", "The authors compare their approximate simple solution with the simulation–based approach of [31], showing a small bias in practical situations.", "Moreover, this approach avoids the typical problem of the particle filters, which provide likelihood estimates that are discontinuous functions of the parameters.", "A recent simulated particle filter approach was proposed by [32], as a modification to the classical setup, obtaining estimated likelihood functions amenable to numerical optimization and able to work also in presence of missing data.", "The comparison of these different estimation procedures in the MS–AMEM framework is interesting, but is beyond the scope of the present work.", "The so–called smoothed probabilities $P[s_t\|\\mathbf {I}_T]$ provided by the Hamilton smoother are used to make inference on the regime conditional on the full information available, $\\mathbf {I}_T$ ; a rule of thumb consists of assigning the observation at time $t$ to regime 1 if $\\hat{p}_t\\equiv P[s_t=1\|\\mathbf {I}_T]>0.5$ , otherwise to regime 0.", "In our context, the smoothed probabilities are further used to estimate the intercept in (REF ), as a weighted average of $\\hat{\\varphi }_0$ and $\\hat{\\varphi }_1$ : $\\hat{\\varphi }_t=\\hat{\\varphi }_0 (1-\\hat{p}_t)+(\\hat{\\varphi }_0+\\hat{\\varphi }_1) \\hat{p}_t$ where a hat indicates the QML estimate of the parameter.", "The interesting feature of our model is that the intercept of $\\xi _{t,s_t}$ , the coefficient $\\hat{\\varphi }_t$ , is time–varying, providing the result that, differently from Eq.", "(REF ), a change in regime will induce a change in the level of the series with varying intensity, without feeding the model the dates of the announcements.", "This is in contrast with a model relying on a dummy variable for announcements which would force the effect on volatility dynamics to be constant.", "When we overlap the dates of the announcements to the estimated results, we can thus monitor a possible change in the volatility level and its amplitude at each date.", "As a consequence, we propose two different methods to classify the policy announcements based on the changes of $\\hat{\\varphi }_t$ when $t$ is an announcement day, relative to the day before." ], [ "Classification of announcements", "In formal terms, the $N$ dates of announcements (i.e., when $\\Lambda _t=1$ ) are selected within the overall time series and, for those, the values $\\hat{\\varphi }_t-\\hat{\\varphi }_{t-1}$ are calculated.", "Notice that, from (REF ): $\\Delta \\hat{\\varphi }_\\tau \\equiv \\hat{\\varphi }_\\tau -\\hat{\\varphi }_{\\tau -1}=\\hat{\\varphi }_1(\\hat{p}_\\tau -\\hat{p}_{\\tau -1})\\equiv \\hat{\\varphi }_1\\Delta \\hat{p}_\\tau \\quad \\forall ~ \\tau =t: \\Lambda _t=1,$ that is, such announcement effects on the volatility level can be evaluated through the variations in the smoothed probabilities directly.", "Thus, a first form of classification of the $N$ announcement is to apply a clustering algorithm to obtain groups with similar $\\Delta \\hat{p}_t$ .", "However, by the approach detailed in Eq.", "(REF ), one can notice that the announcement effect (a large movement in the intercept) will be estimated to be substantial, the larger the movement between the probability of being in regime 1 at time $t-1$ and the corresponding one at time $t$ .", "An alternative to a clustering–based method can therefore be suggested, exploiting the customary mapping of smoothed probabilities into a regime classification based on the threshold $\\hat{p}_t=0.5$ .", "A first smoothed probability–based classification (dubbed SP-level) can be obtained directly from the position of the probabilities $\\hat{p}_t$ , respectively, $\\hat{p}_{t-1}$ , relative to the threshold value $0.5$ .", "In this case we suggest 4 groups, with an immediate interpretation, in the following way: No effect and low volatility – (low) Plank, if $\\hat{p}_t\\le 0.5$ and $\\hat{p}_{t-1}\\le 0.5$ ; No effect and high volatility – (high) Plank, if $\\hat{p}_t\\ge 0.5$ and $\\hat{p}_{t-1}\\ge 0.5$ ; Decrease in volatility – Squat, if $\\hat{p}_t\\le 0.5$ and $\\hat{p}_{t-1}\\ge 0.5$ ; Increase in volatility – Jump, if $\\hat{p}_t\\ge 0.5$ and $\\hat{p}_{t-1}\\le 0.5$ .", "A second smoothed probability–based classification (called SP–diff, since it is based on $\\Delta \\hat{p}_t$ ) gives three groups as follows: No effect – Plank, if $-0.5 \\le \\Delta \\hat{p}_t \\le 0.5$ ; this group will contain cases with a moderate effect, with or without regime change (i.e.", "irrespective of whether the threshold is crossed), since subsequent $\\hat{p}_t$ 's are close to one another; Decrease in volatility – Squat, if $\\Delta \\hat{p}_t < -0.5$ ; in this case the volatility at time $t$ is attributed to regime 0 whereas the volatility at time $t-1$ to regime 1, with a sharper change in the value of the probability by more than 50%; Increase in volatility – Jump, if $\\Delta \\hat{p}_t>0.5$ ; in this case the volatility at time $t$ is attributed to regime 1, whereas the volatility at time $t-1$ to regime 0, and the change in the value of the probability is more than 50%.", "The diff–groups here are different from the level–groups, even if they have the same label, since in this second approach we would classify as a Plank a change in regime between $t-1$ and $t$ , not accompanied by a relevant change in $\\hat{p}_t$ 's.", "By both classifications, however, Squats and Jumps come when the MS model produces a sharp mapping into regimes, and therefore should give similar results.", "If well designed, the three classifications should provide similar results, and evidence will be provided below; both smoothed probability–based classifications have the advantage of being immediately applicable each time an announcement is provided.Notice that our decoding is local, given that the final clustering depends on the smoothed probabilities derived from the Hamilton filter and Kim smoother.", "To evaluate the degree of uncertainty attributable to these classification methods based on the smoothed probabilities, we propose a new index.", "We note that the sharpest assignment of the volatility to state 0, respectively, 1, is obtained when the smoothed probability is exactly equal to 0 or 1.", "In reference to our distinction between Plank (P), Squat (S) and Jump (J), such a case implies that the difference of the contiguous probabilities would then be equal to 0 in the case P, (-1) in the case S and (+1) in the case J.Note that the SP–level and SP–diff coincide, in such a case.", "As a consequence, we suggest the following index: $U=\\frac{2}{N} \\sum _{\\tau =1}^{N} \\left[\\left\|\\Delta ^{(P)}\\hat{p}_t\\right\|+\\left\|\\Delta ^{(S)}\\hat{p}_t+1\\right\|+\\left\|\\Delta ^{(J)}\\hat{p}_t-1\\right\|\\right],$ which varies between 0 (sharpest classification) and 1 (case of maximum uncertainty).Since the differences in the group P assume values between -0.5 and 0.5, in group S between -1 and -0.5, in group J between 0.5 and 1, we multiply the average in Eq.", "(REF ) by 2 to map the result into $[0,1]$ ." ], [ "An Empirical Application", "We consider the annualized realized kernel volatility of four Eurozone stock indices (CAC40, DAX30, FTSEMIB and IBEX35) as provided by the Oxford Man InstituteData are available at https://realized.oxford-man.ox.ac.uk/data/download for the period from June 1, 2009 to December 31, 2019 (daily data, 2685 observations).", "Their profiles, shown in Figure REF , behave similarly between one another, exhibiting bursts of market activity occurring in correspondence with some events of relevance: two remarkable peaks are recognizable in the first part of the sample, coinciding with the flash crash on May 6, 2010 and with the Black Monday on August 8, 2011 (depicted as blue–dashed vertical lines in Figure REF ).", "Similarly, some volatility spikes correspond to monetary policy announcements (red–dashed line); this is the case, for example, of the so–called conventional monetary policy decisions concerning the interest rates (on August 2, 2012; November 7, 2013; December 4, 2014; December 3, 2015) and the unconventional policy decision on March 10, 2016, when the Corporate and Public Sector Purchases Programmes (CSPP and PSPP, respectively) were included in the Expanded Asset Purchases Programme (EAPP).", "In addition, volatility clustering emerges quite clearly throughout, with a long period of low volatility starting in July 2012, whereas short–lived periods of high volatility are observed at the beginning of the sample, corresponding to the Greek sovereign debt crisis in May 2010 and in the mid of 2011 when the sovereign debt crisis exploded spilling over beyond the Eurozone.", "We proxy for the implementation effects, the term $E\\left(x_{t}\|\\mathbf {I}_{t-1}\\right)$ in Eq.", "(REF ), via the ratio of the amount of securities held for unconventional policy purposes to that employed for conventional policy measuresData are obtained from the ECB website and Datastream..", "The conditional expectation of $x_t$ is estimated through the ARIMA(4,1,1) model, according to a preliminary order identification procedure.", "Finally, the list of monetary policy announcements consists of $N=144$ events, constructed starting from the ECB press releasesAvailable at https://www.ecb.europa.eu/press/pr/activities/mopo/html/index.en.html, each setting $\\Lambda _t=1$ .", "Figure: Realized Volatility series and relevant events (red and blue dashed-lines) for financial markets.", "Sample Period: June 1, 2009 to December 31, 2019In what follows, we chose to set the number of states to 2, after a preliminary investigation among the alternatives.", "As it is well known, in the MS world competing models cannot be compared with classical tests for the problem of nuisance parameters present only under the alternative hypothesis ([33]).", "This notwithstanding, we have estimated 3-state MS models for each market and compared them with the 2-state and the no MS cases, relying on the information criteria as a tool for model selection ([34]); across all series, the 2 state case is preferred in terms of BIC.", "This result is further confirmed by the results of the 3–state case estimate, where the switching coefficient $ \\varphi $ of the third state is not significant.Results are available among the supplemental material." ], [ "Estimation results", "Estimation results of our MS–ACM are shown in Table REF for the volatilities of each financial index.Initializing ML estimation from 11 sets across a rather wide range of random starting values for each model, we converge to the same coefficient estimates, up to the third digit.", "On average, the computational time of the full estimation procedure, using a GAUSS code running on an I5–7th gen 2.5 Ghz processor, is approximately 2 minutes.", "They support our choice to assume the constant of the $\\xi _{t,st}$ component as switching: in particular, the constant in the low volatility regime is equal to zero across indices (a regime without monetary policy announcements), whereas it increases remarkably in the high volatility regime.", "As a consequence, we removed $\\varphi _0$ from all models, and, for the same reasons, the AR coefficient $\\psi $ .", "The unconventional policy proxy significantly enters the model with a negative sign, as it is expected to reduce the volatility level, with the strongest impact observed for the IBEX35 (-1.09) and the weakest occurring for the DAX30 (-0.44).", "As per the probability coefficients, the probability of remaining in the low regime is higher than that of the high volatility regime, leading to an average duration of 1 day (calculated as $\\frac{1}{1-p_{ii}}, i=0,1$ ) in the high regime for all markets, while the duration in the low regime ranges between 14 days (FTSEMIB) and 53 days (DAX30).", "Some model diagnostics reinforces the quality of our approach: in Table REF we show a few tests on the residuals.", "The Ljung–Box statistics, calculated for lags 1, 5, 10, demonstrate that no substantial autocorrelation remains in the residuals, which also show sample mean equal to 1, in line with the hypothesis about the center of the Gamma distribution.", "Additionally, a Kolmogorov-Smirnov test fails to reject the null hypothesis (even at a 10% significance level) that the distribution is a mixture of two Gamma densities weighted by the corresponding ergodic probabilities (derived from the estimated Markov Chain).", "Furthermore, in order to compare the goodness of fit across alternative models, we built an in-sample Model Confidence Set ([35]); the competing models, next to our MS–ACM (Eq.", "REF ), are the classical AMEM ([9]), the AMEMX (an AMEM with $x_{t}$ added as a regressor), and the ACM (Eq.", "REF ).", "The procedure always selects our MS model as the best model and none of the alternatives falls into the best set.", "Finally, as a further check, we calculated the cross correlations at lag 1 between the residuals across markets; the results show very low residual lagged cross correlations (all below $0.1$ and most below $0.05$ ), pointing to the fact of no evidence of dynamic interdependence.", "Evidently, this issue should be addressed within a multivariate analysis, but this goes beyond the scope of this work.", "In Figure REF , we reproduce the time–varying constant $\\hat{\\varphi }_t$ , superimposing the dates at which an announcement occurred (an information not provided to the model).", "In line with an immediate absorption of the news, the series jump in correspondence of monetary policy announcements, more frequently so for the FTSEMIB and the IBEX35, than for the CAC40 and the DAX30.", "In order to add readability to the results, we consider three meaningful dates, in detail: August 4, 2011, when the ECB gave additional details on a Longer Term Refinancing Operation (LTRO); August 2, 2012, when the ECB communicated to the market that there would have been no changes of interest rates, and December 3, 2015, when, conversely, only the interest rate on deposit facility was decreased by 10bps.", "Note that these announcements depressed returns in all cases, with the worst loss happening at the unconventional policy announcement (August 4, 2011), between 4.41% (DAX30) and 6.9% (FTSEMIB).", "Interestingly, in the case of the two conventional policy announcements, the estimated probability to be in the high regime is near 1; on the day of the unconventional policy announcement (August 4, 2011), the process appears to be in the low regime, an empirical support to the idea that unconventional policies were successful in reducing stock market volatility.This fact is also confirmed by the negative sign of the $\\delta $ coefficient in Table REF .", "Finally, we observe a reduction in $\\hat{\\varphi }_t$ after the announcements, as evidence of the short estimated permanence in the high volatility regime.", "As a general remark, we find support for our motivation to address the very stylized facts of RV within the MEM class: persistence, captured by an ARMA representation of the dynamics; asymmetric response of volatility to the sign of past returns, captured by a significant asymmetric term; lack of serial correlation in the residuals, ascertained by the usual Ljung–Box test statistics; the addition of MS features advanced here allows us to improve the goodness of fit, as discussed before, and a different consideration of quiet and turbulent periods.", "Table: Estimation results (robust sandwich form of the standard errors in parentheses) of four MS–ACMs relative to different Eurozone volatility series.", "Sample period: June 1, 2009 - December 31, 2019.Table: Residual diagnostics of four MS-ACMs relative to different Eurozone volatility series: p-values of Ljung–Box statistics for different lags, mean of residuals (standard deviation in parentheses) and Kolmogorov-Smirnov statistic.Figure: ϕ ^ t \\hat{\\varphi }_t as a weighted average resulting from the MS–ACM (blue line) and monetary policy announcements (red dots).", "The period depicted spans July 1, 2011 to December 31, 2015.", "Black dots correspond to three meaningful announcements (details in the text): August 4, 2011, August 2, 2012, and December 3, 2015." ], [ "Classification Results", "The estimated smoothed probabilities are used to classify the 144 dates of announcements into the three categories: Plank, Squat, and Jump.", "We calculated the variable $\\Delta \\hat{p}_t$ (see Eq.", "(REF )) and apply a classical k–means algorithm, minimizing the sum of squares from points to the assigned cluster centers (the average of $\\Delta \\hat{p}_t$ within each group), to form three groups.", "In the first column of Table REF we show the number of dates belonging to each group for each market.", "Most of the announcements do not cause a change in volatility (Plank); however, a remarkable percentage of the announcements cause a switching from the low to the high volatility for FTSEMIB and IBEX35, that is 15.3% and 8.3%, respectively, whereas it is less than 5% for CAC40 and DAX30.", "Some important announcements belonging to this category refer both to conventional policies (for example, the decrease of interest rates established on December 3, 2015, when they became negative) and unconventional policies (e.g.", "the announcement on March 10, 2016, when the amount of securities purchased within the implementation of the EAPP passes from 60 to 80 billion per month; the announcement on September 12, 2019, when the ECB decided to run the EAPP as long as necessary).", "Finally, a smaller percentage of the announcements caused a switching from the high to the low volatility regimes: some examples are represented by the details on the Covered Bond Purchases Program (CBPP) released on June 4, 2009 (for the CAC40), the Security Market Program (SMP) on May 10, 2010 (for the FTSEMIB) and the announcement on June 8, 2017 concerning details on the EAPP (for the IBEX35).", "Our analysis confirms the results in [7] where the dummy representing the announcements has a positive sign and is more significant for the FTSEMIB and the IBEX35.", "The announcement effect seems to be more pronounced for the volatility of the Italian and Spanish markets than for the French and German ones, in line with the more stable performance of the latter during this turbulent period.", "In the last two columns of Table REF we show the clustering obtained with the two smoothed probability–based approaches (we merged the two Plank cases in the SP–level classification, in order to make the results comparable with the other classifications).", "The outcome seems quite similar across methods, with a larger number of cases identified as Plank announcements relative to what the k–means clustering delivers.", "Also, considering the SP–level approach, we can notice that almost all Plank cases are identified when the regime was one of low volatility at time $t-1$ .", "The centers of each group are around 0 for the Plank group across classification methods.", "Some differences are present in the centers of the other two categories, in particular between the k–means method and the two smoothed probability–based approaches; FTSEMIB seems to be the one market with less sharp classifications when derived from the three methods.", "In Table REF we report the values of $U$ for each classification method and each series.", "In general, SP-diff presents the lowest $U$ values, but also SP-level has a very similar performance.", "A formal evaluation of the differences in the classifications obtained with the three alternative methods can be conducted by means of the adjusted Rand index [37], [38].", "Such a method is generally used to compare the groups obtained by a certain algorithm with respect to a benchmark clustering; in our case, we use it to verify the similarity of clustering methods by taking possible pairs in turn.", "The Rand index ranges in [0, 1], and takes on value 1 when the two methods provide the same clustering, and value 0 in the case of maximum difference between them.", "In Table REF we show the values of this index across volatility series; they are always larger than 0.9 (with the slight exception of the comparison between k–means and SP–diff for the FTSEMIB, which is 0.85), with value 1 in the case of the two smoothed probability–based methods for the CAC40 and the DAX30.", "Given that the three alternative methods provide very similar results, the smoothed probability–based solutions receive a good support as an immediate by–product of the model estimation, not requiring further statistical clustering algorithms.", "Furthermore, to verify if the announcements are classified in a similar way, this time across the four volatility series, we calculate the Rand index for the same clustering method, but for different markets (Table REF ); once again, we get a strong agreement in the classification results, with Rand indices always larger than 0.74, with a clear similarity between the CAC40 and the DAX30.", "Moreover, it seems that the SP–diff method provides more similar patterns among markets relative to the other two methods.", "By comparison, the k–means provides lower values, but still high in terms of similarity.", "We can conclude that the smoothed probability–based classifications confirm their good performance in providing reliable results, also in view of their very practical derivation, using the more formal k–means statistical clustering as a benchmark classification.", "Table: Classification of announcements for the Eurozone volatility series using three alternative algorithms.", "The numbers in parentheses are the centers of the corresponding group.", "In the group Plank of the SP–level classification, the number in square brackets represents the cases with high volatility both at tt and t-1t-1 (High–Plank).Table: Evaluation of uncertainty by UU index for three alternative classification methods of four Eurozone volatility series.Table: Adjusted Rand index between classification methods by volatility series.Table: Adjusted Rand index between pairs of volatility series by classification method." ], [ "Concluding Remarks", "In this paper we derive a novel Markov Switching Multiplicative Error Model to include a component related to monetary policy actions: such a model extends the recent MEM–class contribution by [7] which accommodates an additive component related to volatility dynamics induced by policy measures.", "As a relevant by–product, we advance a simple–to–obtain suggestion on how to map the information on estimated volatility regimes to classify announcements of a Central Bank in terms of their impact on volatility levels.", "Recent econometric literature is producing great efforts in evaluating these transmission mechanisms in real and financial economies, but generally they assume different announcements as producing the same effect.", "In the model we propose within a more realistic scenario, the announcements are allowed to have different importance, and no prior classification is imposed.", "With MS features, our model has the merit of extracting from volatility an unobservable signal attributable to the unconventional policy effects, with jumps in its intercept as a consequence of a policy announcement: the estimated parameters allow us to derive a procedure to map the variations in the switching intercept into groups interpretable as policy effects on volatility.", "We propose two smoothed probability–based classifications, one based on the thumb rule of the classification derived from the mode of the regime (SP–level method), and another based on the differences of the same smoothed probabilities (we call it SP–diff).", "The smoothed probabilities are a by–product of the Maximum Likelihood estimation, so they allow for an immediate classification of the announcements.", "The application on four European volatility series in the empirical application shows how the smoothed probability–based classifications provide a very similar clustering with respect to a statistical clustering algorithm (k–means).", "Such a classification method based on the smoothed probabilities is extendable to all cases when a MS model is a suitable representation of the data.", "Given the time series framework, the procedure could be extended to be adapted to the real–time classification of the announcements, when new observations are available, and use the framework in a forecasting context, not examined here.", "Given that the classification need to be reapplied to each new announcement, it remains to be seen how robust the previous classification is to the inclusion of new data or, for that matter, to outliers (see, for example, [39]).", "Moreover, it could be interesting to verify whether such a classification method is robust with respect to alternative time–varying models, such as the smooth-transition MEM proposed in [22], and alternative clustering methods.", "An interesting extensionWe thank an anonymous referee to have suggested this possibility.", "is to model the duration in a certain regimes as a function of the announcements.", "One way to investigate this concern is to adopt a time–varying transition probability (TVTP) matrix, where the probability to stay in the same regime at time $t$ depends on the state at $t-1$ and the dummy $\\Lambda _t=1$ when there is an announcement at time $t$ .", "As a consequence, we are able to estimate the duration of a certain state in the case of no announcement and in the case of announcement.", "We did estimate such a TVTP MS–ACM and we indeed confirm that both probabilities, to stay in regime 0 and in regime 1, change in correspondence of the announcements, in the direction of a switch from regime 0 to 1 or the permanence in regime 1 (the coefficient for $p_{11}$ is, however, not significant, in view perhaps of the short duration of regime 1).", "This model outperforms the standard model in terms of Likelihood Ratio test, but, when used for classification purposes (the main goal in this context), it introduces less sharp results relative to the MS–ACM case, as it provides smoothed probabilities often close to 0.5.", "To save space, these results are available in the supplemental material section." ] ]	2011.14094
[ [ "Ideal Triangulation and Disk Unfolding of a Singular Flat Surface" ], [ "Abstract An ideal triangulation of a singular flat surface is a geodesic triangulation such that its vertex set is equal to the set of singular points of the surface.", "Using the fact that each pair of points in a surface has a finite number of geodesics having length $\\leq L$ connecting them, where $L$ is any positive number, we prove that each singular flat surface has an ideal triangulation provided that the surface has singular points when it has no boundary components, or each of its boundary components has a singular point.", "Also, we prove that such a surface contains a finite number of geodesics which connect its singular points so that when we cut the surface through these arcs we get a flat disk with a non-singular interior." ], [ "Introduction", "Let $S_{{g},{n}}$ be a closed orientable surface of genus ${g}\\ge 0$ with ${n}\\ge 0$ punctured points.", "It is well known that $S_{{g},{n}}$ admits a complete finite-area Riemannian metric of constant curvature equal to -1 when the Euler characteristic $\\chi (S_{{g},{n}})$ of $S_{{g},{n}}$ is negative.", "Indeed, in this case there are many Riemannian metrics of constant curvature equal -1 on $S_{{g},{n}}$ .", "The space of equivalence classes of this metrics is called Teichmüller space of $S_{{g},{n}}$ and denoted by $S_{{g},{n}})$ .", "Also, the surface $S_{1,0}$ admits a Riemannian metric of the constant curvature equal to 0.", "It is known that $S_{{g},{n}})$ is homeomorphic to $\\mathbb {R}^{6g-6+2n}$ when $\\chi (S_{{g},{n}})<0$ .", "There are several ways to see this.", "For example, one can form a pants decomposition of $S_{{g},{n}}$ and consider the length parameters and the twist parameters.", "The coordinates obtained this way are called the Fenchel-Nielsen coordinates.", "See [10] or [3] for a proof of this fact.", "If the surface is not compact, that is ${n}>0$ , and $\\chi (S_{{g},{n}})<0$ , then there is another way to parametrize the Teichmüller space.", "The main ingredient of this parametrization is an ideal triangulation of a given hyperbolic structure.", "An ideal triangulation of a hyperbolic surface is a triangulation such that each edge is a geodesic and the vertex set of the triangulation is the set of punctures of the surface.", "Any hyperbolic structure on such a surface is formed by an ideal triangulation and one may parametrize the Teichmüller space by using ideal triangulations.", "See [7] for details.", "Let $\\text{Diff}^+(S_{{g},{n}})$ be the group of all orientation preserving diffeomorphisms of $S_{{g},{n}}$ and $\\text{Diff}_0(S_{{g},{n}})$ be the subgroup of diffeomorphisms which are isotopic to the identity.", "$\\text{Diff}^+(S_{{g},{n}})/\\text{Diff}_0(S_{{g},{n}})$ is called the mapping class group and denoted by ${Mod}(S_{{g},{n}})$ .", "If one endowes Teichmüller space with the Teichmüller's metric, then the mapping class group acts on it by isometries.", "Also, this action is properly discontinuous.", "The proof of this fact is non-trivial and the following lemma is one of the main ingredients of the proof.", "Before stating the lemma, let us recall what the raw length spectrum of a hyperbolic metric is.", "Let $X$ be a complete hyperbolic surface which has finite area.", "Let $c$ be the isotopy class of a simple closed curve which is not homotopic to a point or a puncture.", "We call such a simple closed curve essential.", "Then it is a well-known fact that $c$ contains a unique geodesic.", "Let $l_X(c)$ be the length of this simple closed geodesic.", "Let us define the raw length spectrum of $X$ as the set ${rls}(X)=\\lbrace l_X(c) \\rbrace \\subset \\mathbb {R}_+,$ where $c$ ranges over all isotopy classes of simple closed curves which are essential.", "Lemma 1 ([3]) Let $X$ be a hyperbolic srface which has finite area.", "Then the set ${rls}$ is a closed, discrete subset of $\\mathbb {R}$ .", "Furthermore, for each $L \\in \\mathbb {R}$ the set $\\lbrace c: c \\ \\text{is an isotopy class of an essential simple curve in }\\ X \\ \\text{with} \\ l_X(c)\\le L \\rbrace $ is finite.", "Before stating the main results of this paper, let us recall what a singular flat surface is.", "Roughly speaking, a singular flat surface is a surface which is obtained by gluing Euclidean triangles along their edges appropriately.", "It is known that any compact surface with genus greater than or equal to one admits a singular flat metric.", "A geodesic arc on a singular flat surface is a continuous map from a closed interval to the surface which is locally length minimizing.", "It is well-known that the space of singular flat surfaces having same topology and prescribed angle data can be identified with the Teichmüller space of a punctured surface.", "See [12].", "Therefore, singular flat surfaces form a nice family to study Teichmüller theory.", "In [5], the authors proved a result about singular flat surfaces which is similar to Lemma REF .", "The proof is analytic and based on Arzelo-Ascoli Theorem.", "The authors use this result to obtain a generalized flip algorithm to construct Delaunay triangulation and Voronoi diagram on given singular flat surface.", "Although this result was proven for only surfaces without boundary, the reasoning of the authors is valid for the singular flat surfaces with a boundary component.", "Here is the that we mentioned above: Proposition 1 ([5]) Let $S$ be a compact singular flat surface.", "For any pair of points $p,q \\in S$ and for any $L\\ge 0$ , the number of geodesics arcs of length $\\le L$ joining $p$ and $q$ is finite.", "We now state the main result of the present paper.", "It is well known that a compact singular flat surface can be triangulated with Euclidean triangles.", "A proof of this fact may be found in [11].", "Indeed one can cut any singular flat surface through piecewise geodesic arcs to obtain a disk with no singular points in its interior, where an interior point is called singular if it has an angle not equal to $2\\pi $ .", "See Section .", "Then one can show that such a disk can be triangulated and this triangulation gives us a triangulation of the surface that we started with.", "However, there are several problems with such a triangulation.", "First of all, there may be a lot of triangles required and we may not keep track of the number of triangles involved.", "Also, it does not give us any clue to construct a singular flat surface from a disk or a family of triangles.", "If the surface has genus 0 and the angle at each singular point is less than $2\\pi $ , then it can be triangulated in a nice way.", "Indeed Thurston [9] proved that such a surface can be triangulated in a way that the vertex set of the triangulation coincides with the set of singular points of the surface.", "He proved this result by considering Voronoi diagrams in flat surfaces.", "He used this result to parametrize the space of equivalence classes of singular flat spheres with prescribed angle data.", "We give an example of such a triangulation in Figure REF .", "Glue the edges of the triangle with the same label to get a flat sphere with four singular points having angle $\\pi $ .", "Then the given triangulation of the triangle induces a triangulation of the tetrahedron.", "Figure: A triangulation of the flat sphere with four singular points.In this paper, we generalize Thurston's result to arbitrary flat surfaces with or without boundary.", "More precisely, we prove that any singular flat surface without a boundary component has a triangulation such that the set of singular points coincides with the vertex set of the triangulation provided that the surface has a singular point.", "If the surface has a boundary component, we show that the surface has such a triangulation if each of its boundary components has a singular point.", "Note that we do not use the result of Thurston in the proof.", "Instead, we use Proposition REF to reduce the problem to the case of a disk by an inductive reasoning.", "Also, we consider non-orientable surfaces with or without boundary.", "The second main result of the present paper may be considered as a weak generalization of the Alexandrov Unfolding Process.", "See [1] and [4].", "Alexandrov Unfolding Theorem states that every flat sphere having angle less than $2\\pi $ at each singular point can be cut through some geodesic connecting some of its vertices so that the resulted shape is a disk and this disk can be embedded into Euclidean plane.", "It implies that any flat sphere with appropriate angle data can be obtained from a planer polygon of a suitable type.", "We give an example of such a polygon in Figure REF .", "Glue the edges of the polygon having the same label to obtain a flat sphere with 5 singular points.", "We show that any singular flat surface can be obtained from a flat disk with a non-singular interior with a relatively small number of edges.", "More precisely, we prove that any singular flat surface can be cut through a finite number of geodesics which may intersect only at their end points so that the resulting flat surface is a disk provided that the initial singular flat surface has enough singular points.", "Our result is weak generalization of Alexandrov Unfolding Theorem since it says nothing about embedding of the resulting disk into Euclidean plane.", "Figure: A flat sphere from a planer polygon." ], [ "Singular Flat Surface", "Let $S$ be a compact surface perhaps with boundary.", "A singular flat metric on $S$ is a singular Riemannian metric so that each interior point of the surface has a neighborhood that is isometric to that of a neighborhood of the apex of a cone, and each boundary point has a neighborhood isometric to that of a neighborhood of the apex of a cut of a cone.", "See Figure REF .", "A surface with a singular flat metric is called singular flat surface.", "In particular, there is a well defined notion of angle for each point in a singular flat surface.", "Let us denote the angle at $p \\in S$ by $\\theta _p$ .", "An interior point $p$ of $S$ is called singular if $\\theta _p\\ne 2\\pi $ .", "Otherwise, it is called non-singular.", "A boundary point $p$ is called singular if $\\theta _p\\ne \\pi $ .", "Otherwise, it is called non-singular.", "The values $2\\pi -\\theta _p$ and $ \\pi -\\theta _p$ are called the curvature of $S$ at $p$ if $p$ is an interior point or a boundary point, respectively.", "Thus a point is non-singular if and only if its curvature is 0.", "It is known that a compact singular flat surface can be triangulated.", "That is, it can be obtained by gluing Euclidean triangles along their edges appropriately.", "See [11] for a proof.", "Note that the triangulation given in [11] is not necessarily ideal.", "That is, it may not be true that the set of vertices of the triangulation coincides with the set of singular points of the surface.", "In [8], it was proven that any complete singular flat metric in a possibly non-compact surface can be triangulated by finitely many isometry types of Euclidean triangles.", "Figure: A cone and a cut of a coneThe following formula is called Gauss-Bonnet formula.", "It's proof may be found in [11].", "Proposition 2 (Gauss-Bonnet Formula) Let $S$ be a compact orientable singular flat surface.", "Then $\\sum _{p \\in S}\\kappa _p= 2\\pi \\chi (S).$ A singular flat surface comes with additional structures.", "First of all, each path in $S$ has a length.", "This makes a singular Euclidean surface a length space.", "That is, we can define a metric on $S$ by considering the infimum of the lengths of the curves that join two points on the surface.", "When $S$ is compact, it follows from the general properties of length spaces that for any two points there is a length minimizing path between them and each homotopy class of paths joining two points has a length minimizing representative.", "Also, there is a length minimizing geodesic in any free homotopy class of loops.", "See [2] Note that by a geodesic, we mean a locally length minimizing path." ], [ "Labeled Singular Flat Surfaces and Their Raw Length Spectrums ", "In this section we define labeled singular flat surfaces and their raw length spectrums.", "First, we introduce the notion of the labeled singular flat surface which is essentially a singular flat surface together with a finite set of points which we call the label set.", "This notion is necessary to prove the existence of a triangulation with desired properties.", "Proof of this fact is given by induction on the number of singular points and genus of a given surface, and it is obtained by cutting the surface through arcs.", "When such a cutting operation is performed on a surface, one can get non-singular points from the singular ones.", "Thus, to keep track of these points, we introduce labeled singular flat surfaces.", "Definition 1 A labeled singular flat surface is a compact singular flat surface $S$ together with a finite set of point $P\\subset S$ so that $P$ is non-empty, each singular point of $S$ is in $P$ , each boundary component of $S$ has a non-empty intersection with $P$ .", "Definition 2 An arc on a labeled flat surface $(S,P)$ is a geodesic which joins two points of $P$ and whose interior is a subset of $S- P$ .", "Note that we assume that constant paths are not arcs.", "Each arc has a constant speed parametrization and we regard two different constant speed parametrization of the same arc equivalent.", "Definition 3 Let $(S, P)$ be a labeled singular flat surface.", "The raw length spectrum of $(S,P)$ is the set of lengths of arcs on $S$ , that is, ${rls}(S,P)=\\lbrace l_S(c): c \\ \\text{is an arc on} \\ S \\rbrace \\subset \\mathbb {R},$ where $l_S(c)$ length of the arc $c$ .", "Corollary 1 ${rls}(S,P)$ is a discrete and closed subset of $\\mathbb {R}$ .", "Indeed, for any $L \\ge 0$ the set $\\lbrace c: c \\ \\text{is an arc such that}\\ l_S(c)\\le L \\rbrace $ is finite.", "These statements follow immediately from Proposition REF .", "Note that when $L$ is sufficiently large ${rls}(S,P)$ contains a positive number.", "Definition 4 Let $(S,p)$ be a labeled flat singular surface.", "A multiple arc is a path which is a finite union of connected arcs.", "We define the multiple raw length spectrum as ${mrls}=\\lbrace l_S(c): c \\ \\text{is a multiple arc on} \\ S\\rbrace ,$ where $l_S(c)$ is the length of $c$ with respect to singular flat metric on $S$ .", "The following corollary is immediate.", "Corollary 2 ${mrls}(S,P)$ is a discrete and closed subset of $\\mathbb {R}$ .", "Indeed, for any $L \\ge 0$ the set $\\lbrace c: c \\ \\text{is an multiple arc such that}\\ l_S(c)\\le L \\rbrace $ Consider the set of geodesics which start and end at same point of $P$ .", "Call this set $Loop(S,P)$ .", "Note that we do not include constant loops to this set.", "Since each such a geodesic a multiple arc, the following corollary is immediate.", "Corollary 3 $Loop(S,P)$ is discrete and and closed.", "Indeed, for any $L\\ge 0$ , the set $\\lbrace c \\in Loop(S,P): \\ \\text{length of c is less than or equal to} \\ L\\rbrace $ is finite.", "A singular flat surface which has a trivial holonomy group is called a translation surface.", "A geodesic segment which has non-singular interior and connects two singular points of the surface is called a saddle connection.", "It is known that the set of holonomy vectors of saddle connections of translation surface is a discrete subset of $\\mathbb {R}^2$ .", "See [6] for a proof." ], [ "Ideal Triangulation", "In this section, we show that each flat surface with enough singular points have a triangulation whose set of vertices coincides with the set of singular points of the surface.", "We will make what it means that a surface has enough singular points clear.", "This amounts to discard some surfaces which have obviously do not have such a triangulation.", "For example, flat tori can not have such triangulations because they do not have singular points.", "Also, a singular flat surface with a non-singular boundary component can not have such a triangulation.", "Indeed, we show that any singular flat surface with boundary having a singular point in each of its boundary component have such a nice triangulation.", "Furthermore, if the surface does not have boundary components, then it can be triangulated in this manner when it contains singular points.", "The proof of existence of such a triangulation is based on induction on the number of boundary components, genus and the number of singular points of the surface.", "We will cut a given singular flat surface several times through geodesics that join its singular points to apply the induction hypothesis.", "It may happen that, if one cuts a singular flat surface through a geodesic starting at a singular point, then one gets non-singular points.", "This will be the point where the notion of labeled singular flat surface will help us.", "Now we define explicitly what we mean by a nice triangulation.", "Definition Let $S$ be a labeled singular flat surface with the label set $P$ .", "An ideal triangulation of $S$ is a triangulation of $S$ such that each edge of the triangulation is a simple arc, the vertex set of the triangulation is $P$ , the interior of each edge does not contain any point of $P$ .", "Note that we say that a path $c:[a,b]\\rightarrow S$ is simple if $t<t^{\\prime }\\in [a,b]$ with $c(t)=c(t^{\\prime })$ implies that $t=a$ and $t=b$ .", "We will prove that any labeled singular flat surface has an ideal triangulation.", "Note that we do not discard non-orientable surfaces and surfaces with boundary.", "Some triangles of an ideal triangulation may have two edges and two vertices.", "For example, one can glue the edges of an isosceles triangle which have equal lengths to get such a triangulation on a singular flat surface." ], [ "Orientable surfaces", "Now we start to prove existence of an ideal triangulation.", "We start with the simplest case; the case of a flat disk.", "Recall that a flat disk is a labeled singular flat surface which is homeomorphic to a closed disk and has a non-singular interior.", "Note that polygons are examples of flat disks, but it is not true that any flat disk can be embedded into the Euclidean plane.", "Definition 5 A labeled flat disk is a singular flat disk which is homeomorphic to a closed disk and does not have any labeled points on its interior.", "Lemma 2 Each labeled flat disk has an ideal triangulation.", "Any flat disk has at least three singular boundary points.", "Therefore, if the label set contains three points, then the disk is indeed a triangle, hence it has an obvious ideal triangulation.", "We do induction on the number of labeled points.", "Assume that a flat disk has a label set which contains more than three points.", "Take a point $v$ which is on the boundary and singular.", "For each other labeled point, consider a length minimizing path joining $v$ and this point.", "It follows that there exists a labeled point $w$ such that the path joining $v$ and $w$ is not completely contained on the boundary.", "Note that this is not true if the label set has only three points.", "Since the path is length minimizing, it follows that it has a subpath such that the end points of this subpath are (not necessarily the same) labeled points and its interior is contained in the interior of the disk.", "If we cut the disk through this path, then we get two flat disks so that each of them has less labeled points than the original disk.", "By the induction hypothesis, these disks have ideal triangulations, and it is clear that we can glue this disks back to get an ideal triangulation of the original disk.", "Now we consider singular flat surfaces which are homeomorphic to a closed disk possibly with singular interiors.", "Lemma 3 A labeled flat surface which is homeomorphic to a closed disk has an ideal triangulation.", "If the disk has no labeled interior points, in Lemma REF we proved that it has an ideal triangulation.", "Now we do induction on the number of labeled interior points.", "Since the base case is done, we assume that the disk has labeled interior points.", "Take a labeled boundary point of the disk and call it $v$ .", "Also take a labeled interior point $w$ .", "There is a (not necessarily unique) length minimizing path between these two points.", "This path has a subpath which connects a labeled boundary point and a labeled interior point so that interior of it has an empty intersection with $P$ .", "Note that these new points may be different than $v$ and $w$ .", "Cut the surface through this subpath and form a new labeled singular flat surface by adding the two new boundary points to the label set.", "This new labeled singular flat surface has less labeled interior points than the one that we started with.", "So it has an ideal triangulation by the induction hypothesis.", "This triangulation induces an ideal triangulation of the surface that we started with.", "Now we consider the case of the sphere.", "Note that Thurston [9] also considered this case and obtained an ideal triangulation for each singular flat sphere having non-negative curvature data.", "His method depends on existence of Veronoi regions on a given flat surface.", "Our approach is quite different than his approach.", "Lemma 4 Any labeled singular flat surface which is homeomorphic to a sphere has an ideal triangulation.", "Note that Gauss-Bonnet formula implies that any singular flat metric on the sphere has at least three singular points.", "Take two labeled points in such a surface and consider a length minimizing path joining them.", "Then this path has a subpath which joins two labeled points and has an interior having non-empty intersection with the label set.", "Cutting the sphere through this path, we obtain a labeled singular flat surface which is homeomorphic to a closed disk.", "Lemma REF implies that this disk has an ideal triangulation and it is clear that this triangulation induces an ideal triangulation of the sphere.", "If $S$ is a surface, we denote the surface obtained by removing interiors of $b$ disjoint closed disks from the interior of $S$ by $S[b]$ .", "Now we consider a singular flat metric on a surface of genus zero with several boundary components.", "An example of such a flat surface may be found in Figure REF .", "Figure: A singular flat surface which is homeomorphic to S 2 [3]S^2[3].Lemma 5 If a labeled singular flat surface is homeomorphic to $S^2[b]$ , $b\\ge 0$ , then it admits an ideal triangulation.", "The case $b\\le 1$ was done in Lemma REF and Lemma REF .", "We do induction on $b$ .", "Assume that $b>1$ .", "Take two distinct boundary components.", "Let $w$ and $w^{\\prime }$ be labeled points in these boundary components so that they do not lie in the same component.", "Consider a length minimizing geodesic which connects $w$ and $w^{\\prime }$ .", "Note that if this geodesic makes a non-zero angle between any boundary component at any point distinct than $w$ and $w^{\\prime }$ , then this point is a singular point.", "Furthermore, since the geodesic is length minimizing, it does not intersect itself.", "Now consider the intersection of $P$ with this geodesic.", "If we cut the surface through this geodesic, then we get a singular flat surface which is homeomorphic to $S^2[b^{\\prime }]$ for some $b^{\\prime }<b$ .", "Each point on the intersection set of the geodesic and $P$ gives us two new points on the boundary of the resulted surface, we add these points to $P$ to form a new label set.", "This new surface together with this new label set has an ideal triangulation and this triangulation induces an ideal triangulation of the original labeled flat surface.", "Lemma 6 Let $(S,P)$ be a labeled flat surface whose genus is greater than or equal to 1 and which has at most one boundary component.", "Let ${S}_P$ be the subset of $Loop(S,P)$ which consists of the curves that are not homotopic to a point and loops winding about the boundary component several times.", "Then the shortest element of ${S}_P$ is a simple geodesic.", "Since the genus of the surface is positive, ${S}_P$ is not empty.", "Note that such an element is not unique and its existence follows by Corollary REF .", "Firstly, assume that the surface has no boundary components.", "Assume that this geodesic is not simple.", "Consider the parametric representation $\\alpha : [0,1] \\rightarrow S$ of this geodesic such that $\\alpha (0)=\\alpha (1)=x \\in P$ .", "Let $b\\in (0,1)$ be the smallest value so that there exists $b^{\\prime } >b $ with $\\alpha (b)=\\alpha (b^{\\prime })$ .", "Call this point $\\alpha (b)=\\alpha (b^{\\prime })=y$ .", "Then the loop which is obtained by restricting $\\alpha $ to the interval $[b,b^{\\prime }]$ is not homotopic to a point since the path $\\alpha $ is length minimizing.", "Also the point $y$ can not be in $P$ .", "Indeed, if $y$ is in $P$ , then the curve obtaining by restricting $\\alpha $ to $[b,b^{\\prime }]$ is in ${S}_P$ and has length less than the length of $\\alpha $ .", "In particular, $y$ is a non-singular point.", "Now consider the path obtaining by the juxtaposition of the curves $\\alpha \\vert _{[0,b]}$ and $\\alpha \\vert _{[b^{\\prime },1]}$ .", "If this loop is homotopic to a point, then there is a piecewise geodesic path based at $x$ which has length less than the length of $\\alpha $ and which is not homotopically trivial.", "See Figure REF But this means that there is a loop based at $x$ whose length is less then the length of $\\alpha $ , which is impossible by the choice of $\\alpha $ .", "Now assume that the surface has one boundary component.", "Assume that the geodesic is not simple.", "Consider the parametric representation $\\alpha : [0,1] \\rightarrow S$ of this geodesic such that $\\alpha (0)=\\alpha (1)=x \\in P$ .", "As before, let $b\\in (0,1)$ be the smallest value so that there exists $b^{\\prime } >b $ with $\\alpha (b)=\\alpha (b^{\\prime })$ .", "Call this point $\\alpha (b)=\\alpha (b^{\\prime })=y$ .", "The curve obtaining by restricting $\\alpha $ to the interval $[b,b^{\\prime }]$ can not be homotopic to a point since $\\alpha $ is length minimizing.", "Assume that it is homotopic to a curve winding about the boundary component several times.", "Then the curve obtaining by juxtaposing the curves $\\alpha \\vert _{[0,b]}$ and $\\alpha \\vert _{[b^{\\prime },1]}$ can not be homotopic to a point or the boundary component.", "But this curve has length than the length of $\\alpha $ , which is a contradiction.", "Therefore the curve obtained by restricting $\\alpha $ to $[b,b^{\\prime }]$ is not homotopic to a point or the boundary.", "As above it follows that $y$ is not in $P$ .", "In particular, $y$ is not a singular point.", "Now let $\\alpha ^{\\prime }$ be the curve joining $y$ and $x$ and which traces the same set with $\\alpha \\vert _{[0,b]}$ but in opposite direction.", "Assume that $\\alpha ^{\\prime }$ has length less than or equal to $\\alpha \\vert _{[b^{\\prime },1]}$ .", "Then the curve obtained by juxtaposing the curves $\\alpha \\vert _{[0,b^{\\prime }]}$ and $\\alpha ^{\\prime }$ is not homotopic to the boundary or a single point.", "But it has length less than or equal to the length of $\\alpha $ .", "Also, since $y$ is non-singular, we can modify this curve near $y$ to get a curve in same homotopy class but which has strictly smaller length.", "This contradicts with the choice of $\\alpha $ .", "If the length of $\\alpha \\vert _{[b^{\\prime },1]}$ is less than the length of $\\alpha ^{\\prime }$ , we can proceed in a similar way to get a contradiction.", "Figure: The polygonal loop α\\alpha is described in a torus.", "Consider the quadrilateral formed by red line segments.", "If we modify α\\alpha by replacing its diagonals by red segments, then we get a path whose length is strictly less than the length of α\\alpha .Note that there is a notion of genus for non-orientable surfaces.", "Lemma REF does not discard the case of non-orientable surfaces since in the proof we made no assumption about orientability of $S$ .", "Proposition 3 Let $(S,P)$ be an orientable labeled singular surface.", "Then $S$ has an ideal triangulation.", "Assume that $S$ has more than one boundary components.", "In that case, as in the proof Lemma REF , we may take two points in two different boundary components and cut the surface through a length minimizing geodesic between these points.", "Each point in the intersection of $P$ and this geodesic gives us two points in the new surface.", "By including these points to the label set we can get a new label set.", "Note that after this operation the number of the boundary components decreases.", "So we may repeat the process until we get one boundary component.", "Therefore we assume that $S$ has at most one boundary component.", "Now we do induction on the genus of $S$ .", "If $S$ has genus zero, then it has an ideal triangulation by Lemma REF .", "Assume that genus of $S$ is positive.", "Lemma REF implies that there exists a point $x \\in P$ and a simple geodesic loop based at $x$ which is not homotopic to a point or a boundary component.", "Cut the surface through this geodesic loop.", "Then there are two possible cases.", "This geodesic is separating.", "In this case, we obtain two surfaces so that each of them having smaller genus than the original surface.", "We can form a labeled set for each of these surfaces accordingly.", "By induction hypothesis, these surfaces admit ideal triangulations.", "It is clear that these triangulations induce an ideal triangulation of the original surface.", "The geodesic is non-separating.", "In this case we obtain a surface which has genus less than the genus of the original surface.", "Note that each point on the geodesic induces two points on the boundary of new surface.", "By adding these points to the labeled set, we may form a new labeled set.", "By induction hypothesis, this surface has an ideal triangulation and this ideal triangulation induces an ideal triangulation of the original surface.", "The following corollaries are immediate.", "Corollary 4 Let $S$ be a compact orientable surface with boundary together with a singular flat metric.", "If each boundary component of $S$ has a singular point then $S$ has an ideal triangulation with respect to its set of singular points.", "Corollary 5 Let $S$ be a closed orientable surface without boundary and with a singular flat metric.", "If the set of singular points of $S$ is not empty, then $S$ admits an ideal triangulation with respect to its set of singular points." ], [ "Non-orientable Surfaces", "Our next objective is to show that any non-orientable labeled flat surface has an ideal triangulation.", "Remark A non-singular Möbius strip with a label set $P$ having cardinality $n\\ge 1$ has an ideal triangulation.", "In Figure REF , you can find an example of such a triangulation when the cardinality of $P$ is 1.", "Figure: An ideal triangulation of the Möbius band with one labeled point.Lemma 7 A labeled singular flat surface which is homeomorphic to the Möbius band admits an ideal triangulation.", "Take a labeled boundary point of the surface and consider a length minimizing geodesic which is in the homotopy class of the loop shown in Figure REF .", "Note that since the geodesic is length minimizing, it does not have self-intersection, that is, it is simple.", "After this operation we get a disk with a singular flat metric on it.", "We can form a new label set and get an ideal triangulation of this disk.", "This triangulation induces an ideal triangulation of the Möbius band.", "Theorem 1 Any compact labeled singular flat surface admits an ideal triangulation.", "If the surface is orientable, then the statement was proven in Proposition REF .", "Assume that the surface is non-orientable and it is not homeomorphic to the Möbius band.", "We do the following operation on the surface: Reduce the number of boundary components: if the surface has more than one boundary component, we reduce the number of boundary components to one by cutting the surface through arcs joining its two boundary components repeatedly.", "If the surface has less than or equal to one boundary component, then we cut the surface through the loop whose existence is given by Lemma REF .", "After this process, we get one or two singular flat surface.", "If any of these singular flat surfaces is orientable or homeomorphic to a Möbius band, then we apply no more cutting operation to it.", "Otherwise, we apply these operations again and arrange a new set of labeled points for it.", "Since the surface is of finite type, after finitely many run of the process we will get a finite number of singular flat surfaces each of which is either homeomorphic to Möbius band or orientable.", "By Lemma REF and Proposition REF these surfaces admit ideal triangulations.", "These triangulations give an ideal triangulation of the initial surface." ], [ "Disk Unfolding", "Consider a singular flat surface which is homeomorphic to the sphere $S^2$ such that curvature at each singular point is positive.", "Assume that it has $n$ singular points and the curvature at each of these points is positive.", "Pick a singular point and call it the base point.", "Consider a length minimizing geodesic for each other singular point which connects the base point and this singular point.", "Then these geodesics do not have self-intersection and they do not intersect with each other except at the base point.", "The surface we obtain after cutting the sphere through these geodesics is a flat disk with a non-singular interior.", "Alexandrov Unfolding Theorem states [1] that this flat disk can disk can be embedded isometrically into Euclidean plane.", "Shortly, it states that any singular flat surface can be obtained from a planer polygon.", "In Section , we obtained any singular flat surface from a flat disk.", "But, in the construction, we used an arbitrary number of geodesic segments.", "Actually, we cut the surface through geodesics which are not necessarily arcs.", "In this section, we show that we can do this unfolding by using only arcs.", "Definition 6 Let $(S,P)$ be a labeled singular flat surface.", "We say that $(S,P)$ admits a disk unfolding if there exists a finite set of simple arcs of $S$ which can intersect only at their end points so that when we cut the surface through these arcs we get a flat disk with a non-singular interior.", "Lemma 8 Any singular flat surface homeomorphic to $S^2[b]$ admits a disk unfolding.", "The proof of this statement is similar to the proof of Lemma REF .", "Therefore we omit the proof.", "Proposition 4 Any orientable labeled flat surface admits a disk unfolding.", "We do induction on the genus.", "The case $g=0$ is explained in Lemma REF .", "Assume that the surface has genus greater than zero.", "If the surface has more than one boundary componet, then we can cut the surface through length minimizing geodesics, which are formed indeed finite unions of arcs, to reduce the number of the boundary components.", "Therefore we may assume that the surface has at most one boundary component.", "Lemma REF implies that there exists a simple multiple arc which is not homotopic to the boundary component and a point.", "Now there are two cases to consider.", "The arc is non-separating.", "If we cut the surface through this arc.", "We get a surface with smaller genus.", "By induction hypothesis, this surface admits a disk unfolding and we are done.", "The arc is separating.", "In this case, we obtain two surfaces such that each of them has smaller genus.", "Each of these surfaces admits disk unfolding.", "Glue these disks through the corresponding edges, the edges which are induced by the simple arc, to get a disk together with a flat metric.", "Now this disk admits a disk unfolding and we are done.", "Lemma 9 Any labeled singular flat surface which is homeomorphic to the Möbius band admits a disk unfolding.", "The proof Lemma REF shows that we can cut this surface through a loop based at its boundary so that the resulting surface is a disk.", "Then the result follows from Proposition REF .", "Theorem 2 Any labeled singular flat surface admits a disk unfolding.", "If the surface is orientable, we proved the statement in Proposition REF .", "Assume that the surface is not orientable.", "As in the proof of Theorem REF , we can cut the surface through the simple arcs to get a finite number of orientable surfaces and Möbius band.", "By Proposition REF and Lemma REF each of these surfaces admits a disk unfolding.", "By gluing these unfoldings accordingly, we may get a disk unfolding." ] ]	2011.14138
[ [ "Positive enumerable functors" ], [ "Abstract We study reductions well suited to compare structures and classes of structures with respect to properties based on enumeration reducibility.", "We introduce the notion of a positive enumerable functor and study the relationship with established reductions based on functors and alternative definitions." ], [ "Introduction", "In this article we study notions of reductions that let us compare classes of structures with respect to their computability theoretic properties.", "Computability theoretic reductions between classes of structures can be formalized using effective versions of the category theoretic notion of a functor.", "While computable functors have already been used in the 80's by Goncharov , the formal investigation of this notion was only started recently after R. Miller, Poonen, Schoutens, and Shlapentokh explicitly used a computable functor to obtain a reduction from the class of graphs to the class of fields.", "Their result shows that fields are universal with respect to many properties studied in computable structure theory.", "In the third author studied effective versions of functors based on enumeration reducibility and their relation to notions of interpretability.", "There, it was shown that the existence of a computable functor implies the existence of an enumerable functor effectively isomorphic to it.", "In that article there also appeared an unfortunately incorrect claim that enumerable functors are equivalent to a variation of effective interpretability, a notion equivalent to computable functors .", "Indeed, it was later shown in Rossegger's thesis , that the existence of a computable functor implies the existence of an enumerable functor and thus enumerable functors are equivalent to the original notion.", "Hence, enumerable functors are equivalent to the original version of effective interpretability.", "In this paper we provide a simple proof of the latter result.", "It is not very surprising that enumerable and computable functors are equivalent, as the enumeration operators witnessing the effectiveness of an enumerable functor are given access to the atomic diagrams of structures, which are total sets.", "The main objective of this article is the study of positive enumerable functors, an effectivization of functors that grants the involved enumeration operators access to the positive diagrams of structures instead of their atomic diagrams.", "While computable functors are well suited to compare structures with respect to properties related to relative computability and the Turing degrees, positive enumerable functors provide the right framework to compare structures with respect to their enumerations and properties related to the enumeration degrees.", "The paper is organized as follows.", "In sec:defandfirstres, we first show that computable functors and enumerable functors are equivalent, and then begin the study of positive enumerable functors and reductions based on them.", "We show that reductions by positive enumerable bi-transformations preserve enumeration degree spectra, a generalization of degree spectra considering all enumerations of a structure introduced by Soskov .", "We then exhibit an example consisting of two structures which are computably bi-transformable but whose enumeration degree spectra are different.", "This implies that positive enumerable functors and computable functors are independent notions.", "Towards the end of the section we compare different possible definitions of positive enumerable functors and extend our results to reductions between arbitrary classes of structures." ], [ "Computable and enumerable functors", "In this article we assume that our structures are in a relational language $(R_i)_{i\\in \\omega }$ where each $R_i$ has arity $a_i$ and the map $i\\mapsto a_i$ is computable.", "We furthermore only consider countable structures with universe $\\omega $ .", "We view classes of structures as categories where the objects are structures in a given language $\\mathcal {L}$ and the morphisms are isomorphisms between them.", "Recall that a functor $F:\\mathfrak {C}\\rightarrow \\mathfrak {D}$ maps structures from $\\mathfrak {C}$ to structures in $\\mathfrak {D}$ and maps isomorphisms $f:\\mathcal {A}\\rightarrow \\mathcal {B}$ to $F(f):F(\\mathcal {A})\\rightarrow F(\\mathcal {B})$ preserving composition and identity.", "The smallest classes we consider are isomorphism classes of a single structure $\\mathcal {A}$ , $ Iso(\\mathcal {A})=\\lbrace \\mathcal {B}: \\mathcal {B}\\cong \\mathcal {A}\\rbrace .$ We will often talk about a functor from $\\mathcal {A}$ to $\\mathcal {B}$ , $F:\\mathcal {A}\\rightarrow \\mathcal {B}$ when we mean a functor $F:Iso(\\mathcal {A})\\rightarrow Iso(\\mathcal {B})$ .", "Depending on the properties that we want our functor to preserve we may use different effectivizations, but they will all be of the following form.", "Generally, an effectivization of a functor $F:\\mathfrak {C}\\rightarrow \\mathfrak {D}$ will consist of a pair of operators $(\\Phi ,\\Phi _)$ and a suitable coding $C$ such that for all $\\mathcal {A}\\in \\mathfrak {C}$ , $\\Phi (C(\\mathcal {A}))=C(F(\\mathcal {A}))$ , for all $\\mathcal {A},\\mathcal {B}\\in \\mathfrak {C}$ and $f\\in Hom(\\mathcal {A},\\mathcal {B})$ , $\\Phi (C(\\mathcal {A}),C(f),C(\\mathcal {B}))= C(F(f))$ .", "In this article the operators will either be enumeration or Turing operators.", "If the coding is clear from context we will omit the coding function, i.e., we write $\\Phi (\\mathcal {A})$ instead of $\\Phi (C(\\mathcal {A}))$ .", "The most common coding in computable structure theory is the following.", "Let $\\mathcal {A}$ be a structure in relational language ${(R_i)}_{i\\in \\omega }$ .", "Then the atomic diagram $D(\\mathcal {A})$ of $\\mathcal {A}$ is the set $ \\bigoplus _{i\\in \\omega } R_i^{\\mathcal {A}}\\oplus \\bigoplus _{i\\in \\omega }\\lnot {R_i^{\\mathcal {A}}}.$ In the literature one can often find different definitions of the atomic diagram.", "It is easy to show that all of these notions are Turing and enumeration equivalent.", "The reason why we chose this definition is that it is conceptually easier to define the positive diagram and deal with enumerations of structures like this.", "We are now ready to define various effectivizations of functors.", "[,] A functor $F:\\mathfrak {C}\\rightarrow \\mathfrak {D}$ is computable if there is a pair of Turing operators $(\\Phi ,\\Phi _)$ such that for all $\\mathcal {A},\\mathcal {B}\\in \\mathfrak {C}$ $\\Phi ^{D(\\mathcal {A})}=D(F(\\mathcal {A}))$ , for all $f\\in Hom(\\mathcal {A},\\mathcal {B})$ , $\\Phi ^{D(\\mathcal {A})\\oplus Graph(f)\\oplus D(\\mathcal {B})}=F(f)$ .", "A functor $F:\\mathfrak {C}\\rightarrow \\mathfrak {D}$ is enumerable if there is a pair $(\\Psi ,\\Psi _)$ where $\\Psi $ and $\\Psi _$ are enumeration operators such that for all $\\mathcal {A},\\mathcal {B}\\in \\mathfrak {C}$ $\\Psi ^{D(\\mathcal {A})}=D(F(\\mathcal {A}))$ , for all $f\\in hom(\\mathcal {A},\\mathcal {B})$ , $\\Psi _^{D(\\mathcal {A})\\oplus Graph(f)\\oplus D(\\mathcal {B})}=Graph(F(f))$ .", "In enumerable functors were defined differently, using a Turing operator instead of an enumeration operator for the homomorphisms.", "The definition was as follows.", "[] A functor $F:\\mathfrak {C}\\rightarrow \\mathfrak {D}$ is $\\star $ -enumerable if there is a pair $(\\Psi ,\\Phi _)$ where $\\Psi $ is an enumeration operator and $\\Phi _$ is a Turing operator such that for all $\\mathcal {A},\\mathcal {B}\\in \\mathfrak {C}$ $\\Psi ^{D(\\mathcal {A})}=D(F(\\mathcal {A}))$ , for all $f\\in hom(\\mathcal {A},\\mathcal {B})$ , $\\Phi _^{D(\\mathcal {A})\\oplus Graph(f)\\oplus D(\\mathcal {B})}=Graph(F(f))$ .", "It turns out that the two definitions are equivalent and we will thus stick with def:enumerablefunctor which seems to be more natural.", "A functor $F:\\mathcal {A}\\rightarrow \\mathcal {B}$ is enumerable if and only if it is $\\star $ -enumerable.", "Say we have an enumerable functor given by $(\\Psi ,\\Psi _)$ and an isomorphism $f:\\tilde{\\mathcal {A}}\\rightarrow \\hat{\\mathcal {A}}$ for $\\tilde{\\mathcal {A}}\\cong \\hat{\\mathcal {A}}\\cong \\mathcal {A}$ .", "We can compute the isomorphism $F(f)$ by enumerating $Graph(F(f))$ using $\\Psi _^{\\tilde{\\mathcal {A}}\\oplus f\\oplus \\hat{\\mathcal {A}}}$ .", "For every $x$ we are guaranteed to enumerate $(x,y)\\in Graph(F(f))$ for some $y$ as the domain of $\\mathcal {A}$ is $\\omega $ .", "This is uniform in $\\tilde{\\mathcal {A}}$ , $f$ and $\\hat{\\mathcal {A}}$ .", "Thus there is a Turing operator $\\Phi _$ such that $(\\Psi ,\\Phi _)$ witnesses that $F$ is $\\star $ -enumerable.", "Now, say $F$ is $\\star $ -enumerable as witnessed by $(\\Psi ,\\Phi _)$ .", "For every $\\sigma ,x,y$ with $\\Phi ^\\sigma _(x)\\downarrow =y$ such that $\\sigma $ can be split into $\\sigma _0\\oplus \\sigma _1\\oplus \\sigma _2$ where $\\sigma _0$ , $\\sigma _2$ are partial characteristic functions of finite structures in a finite sublanguage $L$ of the language of $\\mathcal {A}$ and $\\sigma _1$ is the partial graph of a function, consider the set $X^{x,y}_\\sigma =\\lbrace &(B\\oplus Graph(\\tau )\\oplus C,\\langle x,y\\rangle ): B, C\\text{ are atomic diagrams of finite }\\\\& L\\text{-structures}, B\\text{ compatible with }\\sigma _0,\\,C\\text{ compatible with }\\sigma _2,\\\\& \\sigma _1(u,v)=1\\rightarrow \\tau (u)=v, \\text{ and } \\sigma _1(u,v)=0\\rightarrow \\tau (u)=z\\\\& \\text{where } z\\notin range(\\sigma _1)\\rbrace .$ We can now define our enumeration operator as $\\Psi _\\star =\\bigcup _{x,y,\\sigma :\\Phi _\\star ^\\sigma (x)\\downarrow =y} X^{x,y}_\\sigma $ .", "Given an enumeration of $\\Phi _$ we can produce an enumeration of $\\Psi _$ , so $\\Psi _$ is c.e.", "It remains to show that $\\Psi _^{\\hat{\\mathcal {A}}\\oplus f\\oplus \\tilde{\\mathcal {A}}}=\\Phi _^{\\hat{\\mathcal {A}}\\oplus f\\oplus \\tilde{\\mathcal {A}}}$ .", "Say $\\Phi _^{\\tilde{\\mathcal {A}}\\oplus f\\oplus \\hat{\\mathcal {A}}}(x)=y$ .", "Then there is $\\sigma \\preceq \\tilde{\\mathcal {A}}\\oplus f\\oplus \\hat{\\mathcal {A}}$ such that $(\\sigma ,x,y)\\in \\Phi _$ and thus by the construction of $X_{\\sigma }$ there is $B\\subseteq D(\\tilde{\\mathcal {A}})$ , $C\\subseteq D(\\hat{\\mathcal {A}})$ and $Graph(\\tau )\\subseteq Graph(f)$ such that $(B\\oplus Graph(\\tau )\\oplus C, \\langle x,y\\rangle )\\in X_\\sigma $ .", "Thus $\\langle x,y\\rangle \\in \\Psi _^{\\tilde{\\mathcal {A}}\\oplus f\\oplus \\hat{\\mathcal {A}}}$ .", "On the other hand say $\\langle x,y\\rangle \\in \\Psi _^{\\tilde{\\mathcal {A}}\\oplus f\\oplus \\hat{\\mathcal {A}}}$ .", "Then, there is $(B\\oplus Graph(\\tau )\\oplus C,\\langle x,y\\rangle )\\in \\Psi _$ with $B\\subseteq \\tilde{\\mathcal {A}}$ , $Graph(\\tau )\\subseteq Graph(f)$ and $C\\subseteq \\hat{\\mathcal {A}}$ .", "Furthermore, there is $\\sigma \\preceq \\chi _{B\\oplus Graph(\\tau )\\oplus C}$ such that $(\\sigma ,x,y)\\in \\Phi _$ .", "Thus $\\Psi _^{\\tilde{\\mathcal {A}}\\oplus f\\oplus \\hat{\\mathcal {A}}}=Graph(F(f))$ for any $\\hat{\\mathcal {A}}\\cong \\tilde{\\mathcal {A}}\\cong \\mathcal {A}$ and $f:\\tilde{\\mathcal {A}}\\cong \\hat{\\mathcal {A}}$ and hence $F$ is enumerable.", "$\\Box $ In it was shown that the existence of an enumerable functor implies the existence of a computable functor and in the converse was shown.", "We give a simple proof of the latter.", "If $F:\\mathcal {A}\\rightarrow \\mathcal {B}$ is a computable functor, then it is enumerable.", "Given a computable functor $F$ we will show that $F$ is $\\star $ -enumerable.", "That $F$ is then also enumerable follows from prop:starenumiffenum.", "Let $D(L_\\mathcal {A})$ be the collection of finite atomic diagrams in the language of $\\mathcal {A}$ .", "To every $p\\in D(L_\\mathcal {A})$ we associate a finite string $\\alpha _p$ in the alphabet $\\lbrace 0,1,\\uparrow \\rbrace $ so that if $p$ specifies that $R_i$ holds on elements coded by $u$ , then we set that $\\lnot R_i$ does not hold on these elements.", "More formally, $\\alpha _p(x)=1$ if $x\\in p$ , $\\alpha _p(x)=0$ if $x=2\\langle i, u\\rangle $ and $2\\langle i,u\\rangle +1\\in p$ or $x=2\\langle i,u\\rangle +1$ and $2\\langle i, u\\rangle \\in p$ , and $\\alpha _p(x)=\\uparrow $ if $x$ is less than the largest element of $p$ and none of the other cases fits.", "We also associate a string $\\tilde{\\alpha }_p\\in 2^{\|\\alpha _p\|}$ with $p$ where $\\tilde{\\alpha }_p(x)=1$ if and only if $\\alpha _p(x)=1$ and $\\tilde{\\alpha }_p(x)=0$ if and only if $\\alpha _p(x)=0$ or $\\alpha _p(x)\\uparrow $ .", "Let the computability of $F$ be witnessed by $(\\Phi ,\\Phi _)$ .", "We build the enumeration operator $\\Psi $ as follows.", "For every $p\\in D(L_\\mathcal {A})$ and every $x$ if $\\Phi ^{\\tilde{\\alpha }_p}(x)\\downarrow =1$ and every call to the oracle during the computation is on an element $z$ such that $\\alpha _p(z)\\ne \\uparrow $ , then enumerate $(p,x)$ into $\\Psi $ .", "This finishes the construction of $\\Psi $ .", "Now, let $\\hat{\\mathcal {A}}\\cong \\mathcal {A}$ .", "We have that $x\\in \\Psi ^{\\hat{\\mathcal {A}}}(x)$ if and only if there exists $p\\in D(L_\\mathcal {A})$ such that $p \\subseteq D(\\hat{\\mathcal {A}})$ and $(p,x)\\in \\Psi $ .", "We further have that $(p,x)\\in \\Psi $ if and only if $\\Phi ^{\\tilde{\\alpha }_p}(x)\\downarrow =1$ if and only if $\\Phi ^{\\hat{\\mathcal {A}}}(x)=1$ .", "Thus $F$ is enumerable using $(\\Psi ,\\Phi _)$ .", "$\\Box $ Combining thm:compimpenum with the results from we obtain that enumerable functors and computable functors defined using the atomic diagram of a structure as input are equivalent notions.", "This is not surprising.", "After all, the atomic diagram of a structure always has total enumeration degree and there is a canonical isomorphism between the total enumeration degrees and the Turing degrees.", "In order to make this equivalence precise we need another definition.", "[] A functor $F: \\mathfrak {C} \\rightarrow \\mathfrak {D}$ is effectively isomorphic to a functor $G: \\mathfrak {C} \\rightarrow \\mathfrak {D}$ if there is a Turing functional $\\Lambda $ such that for any $\\mathcal {A} \\in \\mathfrak {C}$ , $\\Lambda ^\\mathcal {A} : F(\\mathcal {A}) \\rightarrow G(\\mathcal {A})$ is an isomorphism.", "Moreover, for any morphism $h\\in Hom(\\mathcal {A}, \\mathcal {B})$ in $\\mathfrak {C}$ , $\\Lambda ^\\mathcal {B} \\circ F(h) = G(h) \\circ \\Lambda ^\\mathcal {A}$ .", "That is, the diagram below commutes.", "FA) at (0,2) $F(\\mathcal {A})$ ; FB) at (0,0) $F(\\mathcal {B})$ ; GA) at (3,2) $G(\\mathcal {A})$ ; GB) at (3,0) $G(\\mathcal {B})$ ; [->] (FA) – node[above] $\\Lambda ^\\mathcal {A}$ (GA); [->] (FB) – node[above] $\\Lambda ^\\mathcal {B}$ (GB); [->] (FA) – node[left] $F(h)$ (FB); [->] (GA) – node[right] $G(h)$ (GB); The following is an immediate corollary of thm:compimpenum and .", "Let $F:\\mathcal {A}\\rightarrow \\mathcal {B}$ be a functor.", "Then $F$ is computable if and only if there is an enumerable functor $G:\\mathcal {A}\\rightarrow \\mathcal {B}$ effectively isomorphic to $F$ .", "[] Suppose $F: \\mathfrak {C} \\rightarrow \\mathfrak {D}$ , $G: \\mathfrak {D} \\rightarrow \\mathfrak {C}$ are functors such that $G \\circ F$ is effectively isomorphic to $Id_\\mathfrak {C}$ via the Turing functional $\\Lambda _\\mathfrak {C}$ and $F \\circ G$ is effectively isomorphic to $Id_\\mathfrak {D}$ via the Turing functional $\\Lambda _\\mathfrak {D}$ .", "If furthermore, for any $\\mathcal {A} \\in \\mathfrak {C}$ and $\\mathcal {B} \\in \\mathfrak {D}$ , $\\Lambda _\\mathfrak {D}^{F(\\mathcal {A})}= F(\\Lambda _\\mathfrak {C}^{\\mathcal {A}}) :F(\\mathcal {A}) \\rightarrow F(G(F(\\mathcal {A})))$ and $\\Lambda _\\mathfrak {C}^{G(\\mathcal {B})}=G(\\Lambda _\\mathfrak {D}^{\\mathcal {B}}) : G(\\mathcal {B}) \\rightarrow G(F(G(\\mathcal {B})))$ , then $F$ and $G$ are said to be pseudo inverses.", "[] Two structures $\\mathcal {A}$ and $\\mathcal {B}$ are computably bi-transformable if there are computable pseudo-inverse functors $F:\\mathcal {A}\\rightarrow \\mathcal {B}$ and $G:\\mathcal {B}\\rightarrow \\mathcal {A}$ .", "If the functors in def:compbitrans are enumerable instead of computable then we say that $\\mathcal {A}$ and $\\mathcal {B}$ are enumerably bi-transformable.", "As an immediate corollary of thm:enumiffcomp we obtain the following.", "Two structures $\\mathcal {A}$ and $\\mathcal {B}$ are enumerably bi-transformable if and only if they are computably bi-transformable." ], [ "Effectivizations using positive diagrams", "We now turn our attention to the setting where we only have positive information about the structures.", "We follow Soskov in our definitions.", "See also the survey paper by Soskova and Soskova on computable structure theory and enumeration degrees.", "Let $\\mathcal {A}$ be a structure in relational language ${(R_i)}_{i\\in \\omega }$ .", "The positive diagram of $\\mathcal {A}$, denoted by $P(\\mathcal {A})$ , is the set $ =\\oplus \\ne \\oplus \\bigoplus _{i\\in \\omega } R_i^{\\mathcal {A}}.$ We are interested in the degrees of enumerations of $P(\\mathcal {A})$ .", "To be more precise let $f$ be an enumeration of $\\omega $ and for $X\\subseteq \\omega ^n$ let $ f^{-1}(X)=\\lbrace \\langle x_1,\\dots ,x_n\\rangle : ( f(x_1),\\dots ,f(x_n))\\in X\\rbrace .$ Given $\\mathcal {A}$ let $f^{-1}(\\mathcal {A})=f^{-1}(=)\\oplus f^{-1}(\\ne )\\oplus f^{-1}(R_0^{\\mathcal {A}})\\oplus \\dots $ .", "Notice that if $f=id$ , then $f^{-1}$ is just the positive diagram of $\\mathcal {A}$ .", "The enumeration degree spectrum of $\\mathcal {A}$ is the set $ eSp(\\mathcal {A})=\\lbrace d_e(f^{-1}(\\mathcal {A})): f \\text{ is an enumeration of } \\omega \\rbrace .$ If $\\mathbf {a}$ is the least element of $eSp(\\mathcal {A})$ , then $\\mathbf {a}$ is called the enumeration degree of $\\mathcal {A}$ .", "In order to obtain a notion of reduction that preserves enumeration spectra we need an effectivization of functors where we use positive diagrams of structures as coding.", "It is clear that for computable functors this makes no difference as $P(\\mathcal {A})\\equiv _T D(\\mathcal {A})$ .", "For enumerable functors it does make a difference.", "We also need to replace the Turing operators in the definition of pseudo inverses with enumeration operators.", "The new notions are as follows.", "A functor $F:\\mathfrak {C}\\rightarrow \\mathfrak {D}$ is positive enumerable if there is a pair $(\\Psi ,\\Psi _)$ where $\\Psi $ and $\\Psi _$ are enumeration operators such that for all $\\mathcal {A},\\mathcal {B}\\in \\mathfrak {C}$ $\\Psi ^{P(\\mathcal {A})}=P(F(\\mathcal {A}))$ , for all $f\\in hom(\\mathcal {A},\\mathcal {B})$ , $\\Psi _^{P(\\mathcal {A})\\oplus Graph(f)\\oplus P(\\mathcal {B})}=Graph(F(f))$ .", "A functor $F: \\mathfrak {C} \\rightarrow \\mathfrak {D}$ is enumeration isomorphic to a functor $G: \\mathfrak {C} \\rightarrow \\mathfrak {D}$ if there is an enumeration operator $\\Lambda $ such that for any $\\mathcal {A} \\in \\mathfrak {C}$ , $\\Lambda ^{P(\\mathcal {A})} : F(\\mathcal {A}) \\rightarrow G(\\mathcal {A})$ is an isomorphism.", "Moreover, for any morphism $h\\in Hom(\\mathcal {A}, \\mathcal {B})$ in $\\mathfrak {C}$ , $\\Lambda ^{P(\\mathcal {B})} \\circ F(h) = G(h) \\circ \\Lambda ^{P(\\mathcal {A})}$ .", "Suppose $F: \\mathfrak {C} \\rightarrow \\mathfrak {D}$ , $G: \\mathfrak {D} \\rightarrow \\mathfrak {C}$ are functors such that $G \\circ F$ is enumeration isomorphic to $Id_\\mathfrak {C}$ via the enumeration operator $\\Lambda _\\mathfrak {C}$ and $F \\circ G$ is enumeration isomorphic to $Id_\\mathfrak {D}$ via the enumeration operator $\\Lambda _\\mathfrak {D}$ .", "If, furthermore, for any $\\mathcal {A} \\in \\mathfrak {C}$ and $\\mathcal {B} \\in \\mathfrak {D}$ , $\\Lambda _\\mathfrak {D}^{P(F(\\mathcal {A}))}=F(\\Lambda _\\mathfrak {C}^{P(\\mathcal {A})}) :F(\\mathcal {A}) \\rightarrow F(G(F(\\mathcal {A})))$ and $\\Lambda _\\mathfrak {C}^{P(G(\\mathcal {B}))}=G(\\Lambda _\\mathfrak {D}^{P(\\mathcal {B})}) : G(\\mathcal {B}) \\rightarrow G(F(G(\\mathcal {B})))$ , then $F$ and $G$ are said to be enumeration pseudo inverses.", "Two structures $\\mathcal {A}$ and $\\mathcal {B}$ are positive enumerably bi-transform-able if there are positive enumerable enumeration pseudo-inverse functors $F:\\mathcal {A}\\rightarrow \\mathcal {B}$ and $G:\\mathcal {B}\\rightarrow \\mathcal {A}$ .", "Let $\\mathcal {A}$ and $\\mathcal {B}$ be positive enumerably bi-transformable.", "Then $eSp(\\mathcal {A})=eSp(\\mathcal {B})$ .", "Say $\\mathcal {A}$ and $\\mathcal {B}$ are positive enumerably bi-transformable by $F:\\mathcal {A}\\rightarrow \\mathcal {B}$ and $G:\\mathcal {B}\\rightarrow \\mathcal {A}$ .", "Let $f$ be an arbitrary enumeration of $\\omega $ , then, viewing $f^{-1}(\\mathcal {A})/f^{-1}(=)$ as a structure on $\\omega $ by pulling back a canonical enumeration of the least elements in its $=$ -equivalence classes, we have that there is $\\hat{\\mathcal {A}}\\cong \\mathcal {A}$ such that $P(\\hat{\\mathcal {A}})=f^{-1}(\\mathcal {A})/f^{-1}(=)$ and $P(\\hat{\\mathcal {A}})\\le _ef^{-1}(\\mathcal {A})$ .", "As $F$ is positive enumerable we have that $f^{-1}(\\mathcal {A})\\ge _eP(F(\\hat{\\mathcal {A}}))$ .", "Furthermore, we shall see that $f^{-1}(F(\\hat{\\mathcal {A}}))\\le _e f^{-1}(\\mathcal {A})$ and that $f^{-1}(\\mathcal {A})/f^{-1}(=)=P(F(\\hat{\\mathcal {A}}))$ .", "Given an enumeration of $f^{-1}(\\mathcal {A})$ and an enumeration of $P(F(\\hat{\\mathcal {A}}))$ , we may first order the equivalence classes of $f^{-1}(=)$ by their least elements and then, if $R_i(a_1,\\dots ,a_n)\\in P(F(\\hat{\\mathcal {A}}))$ we enumerate $R_i(b_1,\\dots ,b_n)$ for all $b_1,\\dots ,b_n\\in \\omega $ such that $b_j$ is in the $a_j$th equivalence class of $f^{-1}(=)$ .", "It is not hard to see that this gives an enumeration of a set $X$ such that $f^{-1}(=)\\oplus f^{-1}(\\ne )\\oplus X=f^{-1}(F(\\hat{\\mathcal {A}}))$ , that $f^{-1}(F(\\hat{\\mathcal {A}}))/f^{-1}(=)=P(F(\\hat{\\mathcal {A}}))$ , and since by construction $f^{-1}(F(\\hat{\\mathcal {A}}))\\le _e P(F(\\hat{\\mathcal {A}}))\\oplus f^{-1}(\\mathcal {A})$ we have $f^{-1}(F(\\hat{\\mathcal {A}}))\\le _e f^{-1}(\\mathcal {A})$ .", "We can apply the same argument with $G$ in place of $F$ and $F(\\hat{\\mathcal {A}})$ in place of $\\mathcal {A}$ to get that $f^{-1}(G(F(\\hat{\\mathcal {A}})))/f^{-1}(=)=P(G(F(\\hat{\\mathcal {A}})))$ and $ f^{-1}(G(F(\\hat{\\mathcal {A}})))\\le _e f^{-1}(F(\\hat{\\mathcal {A}}))\\le _e f^{-1}(\\mathcal {A}).$ At last, recall that, as $\\mathcal {A}$ and $\\mathcal {B}$ are positive enumerably bi-transformable, there is an enumeration operator $\\Psi $ such that $\\Psi ^{P(G(F(\\hat{\\mathcal {A}})))}$ is the enumeration of the graph of an isomorphism $i:G(F(\\hat{\\mathcal {A}}))\\cong \\hat{\\mathcal {A}}$ .", "But then $(f\\circ i)^{-1}(G(F(\\hat{\\mathcal {A}})))=f^{-1}(\\mathcal {A})$ and $ f^{-1}(\\mathcal {A})\\le _e f^{-1}(G(F(\\hat{\\mathcal {A}})))\\le _e f^{-1}(F(\\hat{\\mathcal {A}}))\\le _e f^{-1}(\\mathcal {A}).$ This shows that $eSp(\\mathcal {A})\\subseteq eSp(\\mathcal {B})$ .", "The proof that $eSp(\\mathcal {B})\\subseteq eSp(\\mathcal {A})$ is analogous.", "$\\Box $ There are computably bi-transformable structures $\\mathcal {A}$ and $\\mathcal {B}$ such that $eSp(\\mathcal {A})\\ne eSp(\\mathcal {B})$ .", "In particular, $\\mathcal {A}$ and $\\mathcal {B}$ are not positive enumerably bi-transformable.", "Let $\\mathcal {A}=(\\omega ,\\underline{0},s, K)$ where $s$ is the successor relation on $\\omega $ , $\\underline{0}$ the first element, and $K$ the membership relation of the halting set.", "Assume $\\mathcal {B}=(\\omega ,\\underline{0},s,\\overline{K})$ is defined as $\\mathcal {A}$ except that $\\overline{K}(x)$ if and only if $\\lnot K(x)$ .", "There is a computable functor $F:\\mathcal {A}\\rightarrow \\mathcal {B}$ taking $\\hat{\\mathcal {A}}=(\\omega , \\underline{0}^{\\hat{\\mathcal {A}}},s^{\\hat{\\mathcal {A}}}, K^{\\hat{\\mathcal {A}}})\\cong \\mathcal {A}$ to $F(\\hat{\\mathcal {A}})=(\\omega ,\\underline{0}^{\\hat{\\mathcal {A}}},s^{\\hat{\\mathcal {A}}}, \\lnot K^{\\hat{\\mathcal {A}}})$ and acting as the identity on isomorphisms.", "Furthermore, $F$ has a computable inverse and thus $\\mathcal {A}$ is computably bi-transformable to $\\mathcal {B}$ .", "However, $\\mathcal {A}$ has enumeration degree $\\mathbf {0}^{\\prime }_e$ and $\\mathcal {B}$ has enumeration degree $\\overline{\\mathbf {0}^{\\prime }}_e$ .", "Thus there cannot be a positive enumerable functor from $\\mathcal {B}$ to $\\mathcal {A}$ .", "$\\Box $ The following shows that computable functors and positive enumerable functors are independent.", "There are structures $\\mathcal {A}$ and $\\mathcal {B}$ such that $\\mathcal {A}$ is positive enumerably bi-transformable with $\\mathcal {B}$ but $\\mathcal {A}$ is not computably bi-transformable with $\\mathcal {B}$ .", "Let $\\mathcal {A}$ be as in prop:compfunctorsnoeSp, i.e., $\\mathcal {A}=(\\omega ,\\underline{0},s, K)$ and $\\mathcal {B}=(\\omega , \\underline{0},s)$ .", "Then it is not hard to see that $\\mathcal {A}$ is positive enumerably bi-transformable with $\\mathcal {B}$ .", "However, there can not be a computable functor from $\\mathcal {B}$ to $\\mathcal {A}$ as $\\mathcal {B}$ has Turing degree $\\mathbf {0}$ and $\\mathcal {A}$ has Turing degree $\\mathbf {0}^{\\prime }$ .", "$\\Box $ We have seen in prop:starenumiffenum that $\\star $ -enumerable functors and enumerable functors are equivalent.", "Positive enumerable functors also admit a different definition.", "A functor $F:\\mathfrak {C}\\rightarrow \\mathfrak {D}$ is positive $\\star $ -enumerable if there is a pair $(\\Psi ,\\Phi _)$ where $\\Psi $ is an enumeration operator and $\\Phi _$ is a Turing operator such that for all $\\mathcal {A},\\mathcal {B}\\in \\mathfrak {C}$ $\\Psi ^{P(\\mathcal {A})}=P(F(\\mathcal {A}))$ , for all $f\\in hom(\\mathcal {A},\\mathcal {B})$ , $\\Phi _^{P(\\mathcal {A})\\oplus Graph(f)\\oplus P(\\mathcal {B})}=Graph(F(f))$ .", "Every positive enumerable functor is positive $\\star $ -enumerable.", "Let $F:\\mathcal {A}\\rightarrow \\mathcal {B}$ be given by $(\\Psi ,\\Psi _)$ and let $f:\\tilde{\\mathcal {A}}\\cong \\hat{\\mathcal {A}}$ for $\\tilde{\\mathcal {A}}\\cong \\hat{\\mathcal {A}}\\cong \\mathcal {A}$ .", "Now we can define a procedure computing $F(f)$ as follows.", "Given $x$ , and $\\tilde{\\mathcal {A}}\\oplus f\\oplus \\hat{\\mathcal {A}}$ enumerate $\\Psi _^{\\tilde{\\mathcal {A}}\\oplus f\\oplus \\hat{\\mathcal {A}}}$ until $\\langle x,y\\rangle \\searrow \\Psi _^{\\tilde{\\mathcal {A}}\\oplus f\\oplus \\hat{\\mathcal {A}}}$ for some $y$ .", "This is uniform in $\\tilde{\\mathcal {A}}\\oplus f\\oplus \\hat{\\mathcal {A}}$ and thus there exists a Turing operator $\\Phi _$ with this behaviour.", "The pair $(\\Psi ,\\Phi _)$ then witnesses that $F$ is $\\star $ -enumerable.", "$\\Box $ There is positive $\\star $ -enumerable functor that is not enumeration isomorphic to any positive enumerable functor.", "We will build two structures $\\mathcal {A}$ and $\\mathcal {B}$ such that there is a positive $\\star $ -enumerable functor $F:\\mathcal {A}\\rightarrow \\mathcal {B}$ that is not positive enumerable.", "The structure $\\mathcal {A}$ is a graph constructed as follows.", "It has a vertex $a$ with a loop connected to $a$ and a cycle of size $n$ for every natural number $n$ .", "If $n\\in \\emptyset ^{\\prime }$ then there is an edge between $a$ and one element of the $n$ cycle, otherwise there is no such edge.", "Clearly, $deg_T(P(\\mathcal {A}))=\\mathbf {0}^{\\prime }$ and $P(\\mathcal {A})\\lnot \\ge _e \\overline{\\emptyset }^{\\prime }$ .", "The structure $\\mathcal {B}$ is a typical graph that witnesses that there is a structure with degree of categoricity $\\mathbf {0}^{\\prime }$ (that is, $\\mathbf {0}^{\\prime }$ is the least degree computing an isomorphism between any two computable copies of $\\mathcal {B}$ ).", "Let us recap how we build two copies of $\\mathcal {B}$ , $\\mathcal {B}_1$ and $\\mathcal {B}_2$ such that $\\mathbf {0}^{\\prime }$ is the least degree computing isomorphism between $\\mathcal {B}_1$ and $\\mathcal {B}_2$ .", "Both graphs consist of an infinite ray with a loop at its first element.", "Let $v_i$ be the $i^{th}$ element in the ray in $\\mathcal {B}_1$ and $\\hat{v}_i$ be the $i^{th}$ element in the ray in $\\mathcal {B}_2$ .", "Now for every $v_i$ there are two elements $a_i$ and $b_i$ with $v_iEa_i$ and $v_iEb_i$ .", "Likewise for every $\\hat{v}_i$ there are two elements $\\hat{a}_i$ and $\\hat{b}_i$ with $\\hat{v}_iE\\hat{a}_i$ and $\\hat{v}_iE \\hat{b}_i$ .", "Furthermore there are additional vertices $s_i$ , $\\hat{s}_i$ with $a_iE s_i$ and $\\hat{a}_iE \\hat{s}_i$ .", "Take an enumeration of $\\emptyset ^{\\prime }$ .", "If $i\\searrow \\emptyset ^{\\prime }$ , then add vertices $b_iE\\cdot E\\cdot $ and $\\hat{s}_i E \\cdot $ , $\\hat{b}_i E \\cdot $ .", "This finishes the construction of $\\mathcal {B}$ .", "It is not hard to see that there is a unique isomorphism $f:\\mathcal {B}_1\\rightarrow \\mathcal {B}_2$ and that $deg(f)=\\mathbf {0}^{\\prime }$ and $Graph(f)\\ge _e \\overline{\\emptyset }^{\\prime }$ .", "We now construct the functor $F$ .", "Given an enumeration of $P(\\hat{\\mathcal {A}})$ for $\\hat{\\mathcal {A}}\\cong \\mathcal {A}$ we wait until we see the cycle containing 0 (any natural number would work).", "If it is of even length, or 0 is the special vertex $a$ , we let $F(\\hat{\\mathcal {A}})=\\mathcal {B}_1$ and if it is of odd length we let $F(\\hat{\\mathcal {A}})=\\mathcal {B}_2$ .", "Clearly given any enumeration of a copy of $\\mathcal {A}$ this procedure produces an enumeration of a copy of $\\mathcal {B}$ .", "As $\\mathcal {B}$ is rigid we just let $F(f:\\hat{\\mathcal {A}}\\rightarrow \\tilde{\\mathcal {A}})=g:F(\\hat{\\mathcal {A}})\\rightarrow F(\\tilde{\\mathcal {A}})$ where $g$ is the unique isomorphism between $F(\\hat{\\mathcal {A}})$ and $F(\\tilde{\\mathcal {A}})$ .", "Note that there is a Turing operator $\\Theta $ such that $\\Theta ^{P(\\hat{\\mathcal {A}})}=\\emptyset ^{\\prime }$ for any $\\hat{\\mathcal {A}}\\cong \\mathcal {A}$ and that the isomorphism between $F(\\hat{\\mathcal {A}})$ and $F(\\tilde{\\mathcal {A}})$ can be computed uniformly from $P(F(\\hat{\\mathcal {A}}))\\oplus P(F(\\tilde{\\mathcal {A}}))\\oplus \\emptyset ^{\\prime }$ .", "Thus, there is an operator $\\Phi _$ witnessing that $F$ is positive $\\star $ -enumerable.", "To see that $F$ is not positive enumerable consider two copies $\\hat{\\mathcal {A}}$ and $\\tilde{\\mathcal {A}}$ of $\\mathcal {A}$ with $deg_e(P(\\hat{\\mathcal {A}}))=deg_e(P(\\tilde{\\mathcal {A}}))=\\mathbf {0}^{\\prime }_e$ such that 0 is part of an even cycle in $\\hat{\\mathcal {A}}$ and part of an odd cycle in $\\tilde{\\mathcal {A}}$ .", "Notice that there is $f:\\hat{\\mathcal {A}}\\rightarrow \\tilde{\\mathcal {A}}$ such that $P(\\hat{\\mathcal {A}})\\oplus P(\\tilde{\\mathcal {A}})\\ge _e P(\\hat{\\mathcal {A}})\\oplus Graph(f:\\hat{\\mathcal {A}}\\rightarrow \\tilde{\\mathcal {A}})\\oplus P(\\tilde{\\mathcal {A}})$ , and also that $P(\\hat{\\mathcal {A}})\\oplus P(\\tilde{\\mathcal {A}})\\lnot \\ge _e \\overline{\\emptyset }^{\\prime }$ .", "But $Graph(g:F(\\hat{\\mathcal {A}})\\rightarrow F(\\tilde{\\mathcal {A}}))\\ge _e\\overline{\\emptyset }^{\\prime }$ as $F(\\hat{\\mathcal {A}})=\\mathcal {B}_1$ and $F(\\tilde{\\mathcal {A}})=\\mathcal {B}_2$ .", "Thus there can not be an enumeration operator witnessing that $F$ is positive enumerable.", "Assume $F$ was enumeration isomorphic to a positive enumerable functor $G$ and that this isomorphism is witnessed by $\\Lambda $ .", "Then, taking $\\hat{\\mathcal {A}}$ , $\\tilde{\\mathcal {A}}$ and $f:\\hat{\\mathcal {A}}\\rightarrow \\tilde{\\mathcal {A}}$ as in the above paragraph we have that $P(\\hat{\\mathcal {A}})\\oplus P(\\tilde{\\mathcal {A}})\\ge _e Graph(G(f))$ .", "But then $P(\\hat{\\mathcal {A}})\\oplus P(\\tilde{\\mathcal {A}})\\ge _e Graph(\\Lambda ^{P(\\hat{\\mathcal {A}})}\\circ G(f)\\circ {\\Lambda ^{P(\\tilde{\\mathcal {A}})}}^{-1})=Graph(F(f))\\ge _e \\overline{\\emptyset }^{\\prime }.$ This is a contradiction since $deg_e(P(\\hat{\\mathcal {A}})\\oplus P(\\tilde{\\mathcal {A}}))=\\mathbf {0}^{\\prime }_e$ .", "$\\Box $" ], [ "Reductions between arbitrary classes", "So far we have seen how we can compare structures with respect to computability theoretic properties.", "Our notions can be naturally extended to allow the comparison of arbitrary classes of structures.", "[] Let $\\mathfrak {C}$ and $\\mathfrak {D}$ be classes of structures.", "The class $\\mathfrak {C}$ is uniformly computably transformably reducible, short u.c.t.", "reducible, to $\\mathfrak {D}$ if there are a subclass $\\mathfrak {D}^{\\prime }\\subseteq \\mathfrak {D}$ and computable functors $F:\\mathfrak {C}\\rightarrow \\mathfrak {D}^{\\prime }\\subseteq \\mathfrak {D}$ and $G:\\mathfrak {D}^{\\prime }\\rightarrow \\mathfrak {C}$ such that $F$ and $G$ are pseudo-inverses.", "Let $\\mathfrak {C}$ and $\\mathfrak {D}$ be classes of structures.", "The class $\\mathfrak {C}$ is uniformly (positive) enumerably transformably reducible, short u.e.t., (u.p.e.t.)", "reducible, to $\\mathfrak {D}$ if there is a subclass $\\mathfrak {D}^{\\prime }\\subseteq \\mathfrak {D}$ and (positive) enumerable functors $F:\\mathfrak {C}\\rightarrow \\mathfrak {D}^{\\prime }\\subseteq \\mathfrak {D}$ and $G:\\mathfrak {D}^{\\prime }\\rightarrow \\mathfrak {C}$ such that $F$ and $G$ are pseudo-inverses.", "prop:compfunctorsnoeSp,prop:posenumnotcomp show that u.p.e.t.", "and u.c.t reductions are independent notions.", "There are classes of structures $\\mathfrak {C}_1,\\mathfrak {C}_2$ and $\\mathfrak {D}_1,\\mathfrak {D}_2$ such that $\\mathfrak {C}_1$ is u.c.t.", "reducible to $\\mathfrak {D}_1$ but $\\mathfrak {C}_1$ is not u.p.e.t.", "reducible to $\\mathfrak {D}_1$ .", "$\\mathfrak {C}_2$ is u.p.e.t.", "reducible to $\\mathfrak {D}_2$ but $\\mathfrak {C}_2$ is not u.c.t.", "reducible to $\\mathfrak {D}_2$ .", "Similar to cor:bitransenumiffcomp we obtain the equivalence of u.e.t.", "and u.c.t reductions.", "Let $\\mathfrak {C}$ and $\\mathfrak {D}$ be arbitrary classes of countable structures.", "Then $\\mathfrak {C}$ is u.e.t.", "reducible to $\\mathfrak {D}$ if and only if it is u.c.t.", "reducible to $\\mathfrak {D}$ ." ] ]	2011.14160
[ [ "Class-agnostic Object Detection" ], [ "Abstract Object detection models perform well at localizing and classifying objects that they are shown during training.", "However, due to the difficulty and cost associated with creating and annotating detection datasets, trained models detect a limited number of object types with unknown objects treated as background content.", "This hinders the adoption of conventional detectors in real-world applications like large-scale object matching, visual grounding, visual relation prediction, obstacle detection (where it is more important to determine the presence and location of objects than to find specific types), etc.", "We propose class-agnostic object detection as a new problem that focuses on detecting objects irrespective of their object-classes.", "Specifically, the goal is to predict bounding boxes for all objects in an image but not their object-classes.", "The predicted boxes can then be consumed by another system to perform application-specific classification, retrieval, etc.", "We propose training and evaluation protocols for benchmarking class-agnostic detectors to advance future research in this domain.", "Finally, we propose (1) baseline methods and (2) a new adversarial learning framework for class-agnostic detection that forces the model to exclude class-specific information from features used for predictions.", "Experimental results show that adversarial learning improves class-agnostic detection efficacy." ], [ "Introduction", "Human visual scene understanding relies on the ability to detect and recognize objects in one's field of view.", "This naturally carries over to machine scene understanding through computer vision techniques.", "Hence, the field of object detection has garnered tremendous research interest [54] and large advances have been made in improving detection systems, especially since the adoption of deep learning.", "The object detection task is formulated as the prediction of bounding boxes and classes for objects in a given image.", "This requires densely labeled data that contains annotations for all objects in training images.", "However, creating such datasets is extremely challenging and expensive.", "Therefore, conventional object detection focuses instead on the reduced task of locating and recognizing “objects of interest” corresponding to limited types of objects that are labeled in the training data, with objects of unknown types treated as background content.", "Notably, detection of unknown object-types is explicitly penalized in the widely adopted mean average precision metric used for benchmarking.", "This hinders the adoption of trained detectors in real-world applications due to the added cost of retraining them for application-specific object types.", "Furthermore, such detectors cannot be used in applications like obstacle detection, where it is more important to determine the location of all objects present in the scene than to find specific kinds of objects.", "In order to address the aforementioned limitations of conventional class-aware detection, we propose class-agnostic object detection as a new research task with the goal of predicting bounding boxes for all objects present in an image irrespective of their object-types.", "Intuitively, this task additionally seeks to detect objects of unknown types, which are not present or annotated in the training data.", "This challenging yet high-reward goal of generalization to unseen object types would benefit downstream applications (e.g., application-specific object classification, object retrieval from large databases, etc.)", "that can consume such class-agnostic detections.", "Besides the problem formulation, we propose training and evaluation protocols for class-agnostic detection with generalization and downstream utility as primary goals.", "Generalization is evaluated on PASCAL VOC [9], MS COCO [28], and Open Images [24] in the form of recall of unseen classes.", "Specifically, models trained on VOC are evaluated on unseen classes of COCO and those trained on COCO are tested on non-overlapping classes of Open Images.", "Furthermore, the VOC dataset is split into seen and unseen classes in order to measure recall of seen and unseen classes within the same dataset.", "Utility of trained detectors is evaluated as the accuracy of pretrained ImageNet [40] classifiers on ObjectNet [2] images cropped using the bounding boxes predicted by the detectors.", "We present a few baseline methods for class-agnostic object detection – region proposals of two-stage detectors, pretrained class-aware models used as-is or finetuned for binary object versus not classification, and detection models trained from scratch for the said binary classification.", "Finally, we propose a new adversarial training framework that forces the detection model to exclude class-specific information from the features used for making predictions.", "Experimental results show that our adversarial framework improves class-agnostic detection efficacy for both Faster R-CNN [39] (two-stage) and SSD [29] (one-stage).", "The major contributions of this paper are: a novel class-agnostic object detection problem formulation as a new research direction, training and evaluation protocols for benchmarking and advancing research, a new adversarial learning framework for class-agnostic detection that penalizes the model if object-type is encoded in embeddings used for predictions." ], [ "Related Work", "Zou et al.", "[54] provide a comprehensive survey of advances in object detection in the past 20 years.", "Some conventional detection works [46], [50], [36], [20], [25] describe the task as “generic” object detection but they use it to signify their focus on common objects of known types, which is distinct from our work on detecting objects of all types.", "Two-stage object detectors like Faster R-CNN [39] employ a region proposal module that identifies a set of regions in a given image that could contain objects, which are then used by a downstream detector module to localize and identify objects.", "A number of works [12], [33], [5], [23], [43], [35], [7], [52] have been proposed recently that aim to improve the quality of such object proposals and reduce their number in order to speed up the final detection task.", "A majority of proposal methods are trained end-to-end with the detector module, which biases them to objects of known types.", "Methods also exist for generating [32], [6], [53], [44] or filtering [38], [31], [30] object proposals based on edge-related features and objectness metrics.", "However, as defined, these proposals do not correspond directly to final detections and must be fed to a detector module to infer final detections.", "In contrast, the proposed task of class-agnostic detection is aimed at predicting final bounding boxes for all objects in an image.", "Two-stage methods for class-agnostic detection could employ these works to generate intermediate object proposals.", "Some works [14], [49] focus on generating pixel-level objectness scores in order to segment object regions out of background content.", "These works produce objectness heatmaps which could sometimes be used to detect individual objects if they do not overlap and are strikingly distinct.", "A few works [13], [8] have shown the efficacy of convolutional neural networks like AlexNet [21] for localizing objects irrespective of their classes.", "In this work, we take this idea further and formally define the task of class-agnostic object detection along with training and evaluation protocols for benchmarking current and future research.", "Kuen et al.", "[22] show how weights from object recognition networks can be transferred to detection models to scale-up the number of object-types for detection.", "This line of research is also related to few-shot [34], [45], [19], [10], [51], [48], [47] and zero-shot [1], [26], [37] object detection, which are targeted towards detecting objects of novel types given a few reference samples or descriptions, respectively.", "Figure: Adversarial framework for class-agnostic object detection for both one- and two-stage models.", "The upstream module with the backbone (e.g., VGG16, ResNet-50, etc.", "), Region Proposal Network (two-stage) and ROI feature extractor (two-stage) remain unchanged, along with the box regressor.", "The conventional object-type classification is replaced with a binary object-or-not classifier, and a new adversarial object-type discriminator is attached during training.", "The bottom part of the figure shows the training procedure – iterating between five discriminator updates for each update to the other modules." ], [ "Class-agnostic Object Detection", "In this section, we describe the proposed task of class-agnostic object detection and contrast it with conventional class-aware object detection.", "We then discuss a few intuitive baseline models for the proposed task." ], [ "From Class-aware to Class-agnostic Detection", "The conventional formulation of object detection can be described as the task of locating and classifying objects in a given image.", "The goal of models developed to solve this task is to predict bounding box coordinates and class labels for objects present in the image.", "Given this formulation, a plethora of models have been proposed that train on images with object annotations.", "However, due to the difficulty and cost associated with collecting and labeling large datasets, annotations are typically collected for a limited number of object categories.", "This has inadvertently reduced the original formulation to the task of detecting “objects of interest”, corresponding to object types that are annotated in training datasets.", "Although reduced in form, the conventional formulation remains a challenging problem, with direct application in use-cases with a fixed set of known object types.", "In this work, we propose a sibling task of class-agnostic object detection that aims to detect all objects in a given image irrespective of their object-types.", "More specifically, the goal of this task is to predict bounding boxes for all objects present in an image but not their category.", "Furthermore, given that most available training datasets do not contain dense annotations for all kinds of objects, an additional implicit goal for models developed for this task is to generalize to objects of unknown types, i.e., those for which annotations are not available in the training dataset.", "This is in direct contrast to conventional class-aware object detection, which treats unknown objects as background content.", "As compared to conventional detectors, class-agnostic models can be more easily adopted in complex real-life applications like object retrieval from a large and diverse database, recognition of application-specific object-types through training a downstream object recognition model (instead of retraining the full detection network), etc." ], [ "Baseline Models", "We identify three intuitive and straightforward baseline models for solving the class-agnostic detection task, selected due to their ease of implementation and natural curiosity about their performance.", "The baseline models are: region proposal network of a two-stage detector, class-aware detector trained for object-type classification and bounding box regression, pretrained class-aware model finetuned end-to-end for object-or-not binary classification instead of object-type, along with bounding box regression, detection model trained from scratch for object-or-not binary classification and bounding box regression." ], [ "General Framework", "Conventional class-aware detection focuses on detecting “objects of interest”, which inherently requires models to be able to distinguish between types of objects in a closed known set.", "Intuitively, models achieve this by encoding features that are discriminative of object-types.", "However, for class-agnostic detection and for models to be able to detect previously unseen types of objects, detectors should encode features that more effectively distinguish objects from background content and individual objects from other objects in the image, without discriminating between object-types.", "Naïvely training conventional object detectors for the binary classification task of object-or-not along with bounding box regression is not sufficient to ensure that models focus on class-agnostic features and more importantly, ignore type-distinguishing features so that they can better generalize to unseen object-types.", "In order to overcome this problem, we propose to train class-agnostic object detectors in an adversarial fashion such that models are penalized for encoding features that contain object-type information.", "We begin with observing a popular two-part pattern in the model design of both one-stage and two-stage conventional object detectors.", "The first upstream part of a detection model learns a set of convolutional features from entire images (one-stage) or regions of interest (two-stage).", "The second downstream part consumes these features and passes them through classification and regression branches for object-type and bounding box prediction, respectively.", "This two-part setup allows for external control on the information output by the first part and consumed by the second.", "We propose to augment class-agnostic detectors with adversarial discriminator branches that attempt to classify object-types (annotated in the training data) from the features output by the upstream part of detection networks, and penalize the model training if they are successful [16], [17], [18], [15].", "The models are trained in an alternating manner such that the discriminators are frozen when the rest of the model is updated and vice versa.", "While updating the discriminators, we use the standard categorical cross-entropy loss with object-types used as prediction targets.", "On the other hand, while training the rest of the model, we minimize (a) the cross-entropy loss for object-or-not classification, (b) smooth L1 loss for bounding box regression, and (c) the negative entropy of discriminator predictions.", "This entropy maximization forces the upstream part of detection models to exclude object-type information from the features it outputs.", "The discriminator is updated five times for every update to the rest of the model and the negative entropy is weighted with a multiplier $\\alpha $ (tuned on $\\lbrace 0.1, 1\\rbrace $ ) in the overall objective.", "fig:model summarizes the complete framework.", "During test-time inference, the discriminators are detached from the model, giving back the original network with the standard layers and parameter-count.", "Thus, our framework does not cause performance delays." ], [ "Model Instantiation", "We demonstrate the applicability of the proposed adversarial framework to both Faster R-CNN (FRCNN), a two-stage detector, and SSD, a one-stage detector.", "We use the publicly available MMDetection [4] framework for implementing the models and running experiments.", "We train two versions of the adversarial models – one is trained from scratch and the other is finetuned from a pretrained baseline class-aware model.", "Further details are as follows.", "Faster R-CNN.", "The FRCNN model first generates regions of interest in the input image, which is followed by extracting features from and making object-type and bounding box predictions for each region.", "We create a class-agnostic adversarial version of FRCNN by (1) replacing the multi-class object-type classification layer with a binary object-or-not layer, and (2) attaching an adversarial discriminator on top of the feature extraction layer that provides inputs to the classification and regression heads.", "Thus, during training, this feature layer serves three prediction heads instead of the standard two.", "In our experiments, we use the standard FRCNN model available in MMDetection, which includes a ResNet-50 backbone and a Feature Pyramid Network [27].", "SSD.", "An SSD model utilizes features from several layers of its backbone network to detect objects of different scales, corresponding to the depth-levels of the backbone layers.", "Specifically, SSD models contain classification and regression layers for making predictions at each depth-level.", "In order to create a class-agnostic adversarial version of the SSD model, we (1) replace each object-type classification layer with a binary object-or-not layer, and (2) attach an adversarial discriminator at each depth-level where predictions are made.", "Thus, during training, each prediction level in the resulting model has three prediction heads instead of the conventional two.", "We use the standard SSD-300 with VGG-16 pretrained on ImageNet as the backbone.", "Figure: Generalization results for FRCNN models trained on the seen VOC dataset.", "The top row shows macro-level AR@kk for seen and unseen classes in VOC and their harmonic mean (AR-HM).", "FRCNN-agnostic-adv performs the best overall.", "The second row shows micro-level results for the easy, medium, and hard unseen classes.", "FRCNN-agnostic-adv performs the best on the hard and easy classes with recall drop for the medium class.", "The last row provides results of evaluation on the COCO data of 60 unseen classes.", "FRCNN-agnostic-adv achieves the best AR@kk for objects of all sizes." ], [ "Training and Evaluation Protocols", "We propose two kinds of experiments for evaluating class-agnostic detection, geared towards measuring (1) generalization to unseen object types, and (2) the downstream utility of trained models.", "While there are several ways to design experiments for (1) and (2), we propose first steps with potential for refinement in future work." ], [ "Generalization to Unseen Object-types.", "Class-agnostic detectors should, by definition, not be limited to object types seen during training.", "Hence, it is important to evaluate their efficacy at identifying unseen types of objects.", "We measure this performance as average recall (AR; also known as detection rate [53]) at various number ($k \\in \\lbrace 3, 5, 10, 20, 30, 100, 300, 1000\\rbrace $ ) of allowed bounding box predictions, i.e., AR@$k$ ($\\text{IoU} \\ge 0.5$ ).", "The goal here is to achieve high AR@$k$ at all $k$ levels.", "We design two sets of generalization experiments based on this setup.", "Experiment I.", "We split the VOC dataset into 17 seen classes and three unseen classes.", "Seen classes in the VOC 07+12 training set are used for learning the models while both seen and unseen classes in the VOC 07 validation set are used for evaluation as AR@$k$ of seen and unseen classes.", "The harmonic mean of seen and unseen AR@$k$ , for each $k$ , is reported as a measure of overall performance.", "In order to select the three unseen classes, we compute the confusion matrixhttps://github.com/kaanakan/object_detection_confusion_matrix of a 300-layer SSD model trained on the standard 20-class VOC 07+12 dataset, and use F1 scores to determine easy, medium, and hard classes.", "Specifically, the class with the highest F1 is the hardest for generalization as it is the most distinguishable from the other classes, making it the most difficult to generalize to if it were not seen during training.", "Similarly, the class with the lowest F1 is picked as the easy class.", "The medium class is the one with the median F1.", "Thus, the unseen set comprises cow (easy), boat (medium) and tvmonitor (hard) classes, while the rest are considered seen.", "In addition to the macro AR@$k$ scores, micro AR@$k$ are reported for each unseen class.", "Besides the evaluation on VOC, the (VOC) models are tested for generalization on the 60 non-VOC classes in the COCO 2017 validation set using the same AR@$k$ metric.", "AR@1000 is further reported for small, medium, and large sized objects belonging to the 60 classes.", "Table: Generalization results for experiments I and II.", "AR@1000-Unseen shows results for unseen VOC classes, AR@1000-COCO for unseen COCO classes, and AR@1000-OI for unseen Open Images classes.", "“Ovr”, “Med”, “Sml”, and “Lrg” stand for overall, medium, small, and large, respectively.", "“-aw-” and “-ag-” in the model name indicate whether the model is class-aware or -agnostic, “-ft-” tells whether the model was finetuned from the class-aware baseline, and “-ad” represents models trained adversarially.", "FRCNN-aw-prop refers to the evaluation of the FRCNN-aw proposals from its first stage Region Proposal Network.", "Class-agnostic models generalize better than class-aware models with those trained adversarially from scratch performing the best overall.Experiment II.", "In this experiment, models are trained on the COCO 2017 training set and evaluated on non-overlapping classes in the Open Images V6 test set.", "We first normalize the class names in both COCO and Open Images to find exact matches between classes in the two datasets.", "We identify a few classes that have slightly different names in the two datasets and normalize them manually.", "We then build a semantic tree using the publicly available Open Images class hierarchy and exclude all classes that either match exactly with a COCO class or have a COCO class as a predecessor or successor.", "The remaining classes are used for testing.", "AR@$k$ is used as the evaluation metric." ], [ "Downstream Utility", "A major motivation for research in the domain of class-agnostic object detection is the potential for widespread downstream utilization of class agnostic bounding boxes for extracting objects in images and using them for various visual reasoning tasks.", "In this work, we propose evaluation of downtream utility in terms of object recognition on images in the ObjectNet [2] dataset.", "Detection models are used to predict $M$ bounding boxes for each image, which are then use to crop the image into $M$ versions.", "Pretrainedhttps://pytorch.org/docs/stable/torchvision/models.html ImageNet classifiers are then used to predict the object class from the cropped images.", "Here we use ResNet-152 [11], MobileNet-v2 [41], and Inception-v3 [42] models.", "Two metrics are used for evaluation – Accuracy@$M$ for $M \\in \\lbrace 1, 5, 10\\rbrace $ , and Best-overlap (BO) accuracy.", "For Accuracy@$M$ , the classifier's prediction on at least one of the $M$ crops needs to be correct for the image to be considered as successfully classified.", "BO-accuracy is calculated using bounding boxes that have the highest intersection over union (IoU) with the ground-truth [3] bounding boxes.", "Figure: Generalization results for SSD models trained on the seen VOC dataset.", "The top row shows macro-level AR@kk for seen and unseen classes in VOC as well as their harmonic mean (AR-HM).", "SSD-agnostic-adv performs the best on AR-Unseen and AR-HM, with a drop in AR-Seen, but the models that outperform SSD-agnostic-adv on AR-Seen do significantly worse on AR-Unseen and AR-HM.", "The second row shows micro-level results for the easy, medium, and hard unseen classes.", "SSD-agnostic-adv performs the best in all categories.", "The last row provides results of evaluation on the COCO data of 60 unseen classes.", "SSD-agnostic-adv achieves the best AR@kk with a slight reduction for small-sized objects." ], [ "Generalization to Unseen Object-types", "Experiment I.", "We train the baseline models and the adversarial models on the VOC seen set and evaluate them on both the VOC set and the COCO unseen classes.", "Results of this experiment with FRCNN as the base model are presented in fig:frcnnvoc.", "Class-agnostic models outperform standard class-aware models in overall performance, especially for unseen VOC classes.", "Breakdown of recall for unseen classes reveals that the margin in recall becomes larger as the difficulty of generalizability is increased, with the largest performance gains on the hard unseen class.", "Results on the COCO unseen classes show that the adversarial model performs the best overall and for objects of all sizes.", "fig:ssdvoc summarizes the results with SSD as the base model.", "The results show that the class-agnostic SSD models largely outperform the conventional class-aware SSD model.", "Furthermore, the proposed adversarial model performs the best overall with some reduction in average recall for the VOC seen set.", "Breakdown of recall for the easy, medium, and hard unseen classes shows that the adversarial model performs the best across the board.", "Finally, results on COCO show consistent improvements from adversarial learning, especially for medium and large sized objects.", "Hence, class-agnostic models perform better than class-aware models in generalizing to objects of unseen types and the proposed adversarial model performs the best overall.", "Experiment II.", "In this experiment, the baseline and the adversarial models are trained on the COCO dataset and evaluated on the non-overlapping classes of the Open Images, as outlined in subsec:gen. fig:ssdfrcnncoco summarizes the results for both FRCNN and SSD models.", "In both cases, the class-agnostic models generalize better than the class-aware models overall, with the proposed class-agnostic models trained from scratch achieving the best AR scores.", "tab:generalization presents AR@1000 generalization scores for experiments I and II, showing that the adversarially trained (from scratch) models perform the best in both settings.", "We find that the recall of models that are finetuned from the pretrained class-aware baselines are worse than those that are trained from scratch.", "We attribute this to the difficulty of unlearning discriminative features for object-type classification and realigning to learn type-agnostic features, which prevents the finetuned models from achieving the same performance as those that are trained from scratch.", "Figure: Generalization results of (a) FRCNN and (b) SSD models trained on the COCO dataset and evaluated on the non-overlapping classes in the Open Images dataset.", "Results show that class-agnostic models generalize better than the class-aware models, with the SSD-agnostic-adv and FRCNN-agnostic-adv models achieving the best recall.Table: Downstream ObjectNet recognition results.", "Images are cropped using boxes predicted by the detectors and pretrained ImageNet models are used for classification from the cropped images.", "Averages of accuracies across classifiers are also reported.", "BO-acc is the accuracy when the predicted box with the highest IoU with the ground-truth box is used.", "Accuracies are also presented for top MM proposals as Acc@MM.", "Results for uncropped images and those cropped using ground-truth boxes are provided for reference.", "“-aw-” and “-ag-” in the model name indicate whether the model is class-aware or -agnostic; “-ad” represents models trained adversarially.", "Results of other baseline detectors are provided in the supplementary material.", "Adversarially trained class-agnostic models achieve the best results." ], [ "Downstream Object Recognition", "We evaluate the object recognition performance for images cropped using bounding boxes predicted by the class-aware and class-agnostic models trained for the generalization experiments in subsec:gen. tab:objectnet summarizes the results of these experiments.", "Results show that the adversarially trained class-agnostic models perform better than the baselines in general on both Accuracy@$M$ and Best-overlap accuracy (as described in subsec:downstream), irrespective of the ImageNet classifier used.", "These results correlate with the generalization results presented in subsec:resultsgen, indicating that detection models that generalize better to unseen object-types have higher downstream utility." ], [ "Conclusion and Future Work", "Conventional class-aware objection detection models perform well at detecting “objects of interest” that belong to known objects types seen during training.", "However, these models do not generalize well to unseen object-types, limiting their incorporation in real-world applications.", "In this work, we have formulated a novel task of class-agnostic object detection, where the goal is to detect objects of known and unknown types irrespective of their category.", "Furthermore, we have presented training and evaluation protocols for benchmarking models and advancing future research in this direction.", "These include two sets of experiments – (1) one for generalization to new object-types and (2) another for downstream utility in terms of object recognition.", "Finally, we have presented a few intuitive baselines and proposed a new adversarially trained model that penalizes the objective if the learned representations encode object-type.", "Results show that the proposed adversarial class-agnostic model outperforms the baselines on both generalization and downstream-utility experiments, for both one-stage (SSD) and two-stage (Faster R-CNN) detectors.", "This work serves to establish a new direction of research in object detection.", "As such, the possibilities for future research are endless, including but not limited to — (1) further refinement of the problem formulation, (2) improved as well as additional training and evaluation protocols, and (3) novel methods for class-agnostic object detection." ] ]	2011.14204
[ [ "Crossing Symmetry for Long Multiplets in 4D $\\mathcal{N}=1$ SCFTs" ], [ "Abstract In this work we construct the crossing symmetry equations for mixed correlators of two long and two BPS operators in 4D $\\mathcal{N}=1$ SCFTs.", "The analysis presented here illustrates how our general group theoretic approach to long superblocks and tensor structures of superconformal algebras can be applied to give explicit ready-to-use expressions.", "In the case at hand, we obtain a system of four crossing symmetry equations for the relevant OPE coefficients.", "One of these four equations coincides with the equation found and analysed by Li, Meltzer and Stergiou by restricting to the superprimary component of the long multiplets.", "The other three equations are new and they provide powerful additional constraints on the same OPE data." ], [ "Introduction", "After its revival in [1], the conformal bootstrap has proved to be a powerful approach to conformal field theories (CFTs) which offers insights, particularly about their landscape, that is currently unattainable by other methods.", "Within the bootstrap approach, CFTs are found as solutions to an infinite set of crossing symmetry equations.", "In order to write down these equations, one needs to have a good control over two kinematical ingredients - conformal partial waves and tensor structures.", "The former are a natural basis for the expansion of correlation functions in terms of conformal invariants while the latter capture the nontrivial transformation behaviour of correlation functions under rotations.", "In their pioneering work [2], [3], Dolan and Osborn realised that four-point conformal blocks can be characterised as solutions of Casimir differential equations, and used this fact to find series expansions for them.", "While these results applied to blocks for correlation functions of scalars in even number of spacetime dimensions, they have since been extended to spinning operators and odd dimensions.", "In most cases, explicit general formulas for the blocks, e.g.", "in terms of special functions, do not exist, but any given conformal block can be computed in an algorithmic way by starting from the scalar ones and using weight-shifting operators.", "For this and other techniques of finding spinning conformal blocks, see [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15].", "Given that most conformal field theories we know about possess some amount of supersymmetry, in particular in $d=4$ dimensions, it is certainly of interest to adapt and apply the bootstrap programme to superconformal field theories (SCFTs) and clearly a lot of research has explored this important direction already.", "The superconformal bootstrap programme meets some obvious challenges, namely the fact that even the smallest supermultiplets include several primaries for the bosonic conformal symmetry, certainly also including fields with spin.", "Hence, studying SCFTs as conformal field theories without taking the additional constraining power of supersymmetry into account seems a rather daunting task.", "On the other hand, the kinematical aspects and in particular the partial waves for fields in generic representations of the superconformal algebra are very poorly developed.", "So far, the strategy has been largely to either restrict the bootstrap analysis to four-point function of chiral (half-BPS) operators or to four-point functions of the superprimary component.", "For these two cases, it has been possible to construct the relevant blocks, see e.g.", "[16], [17], [18], [19], [24], [26], [20], [23], [21], [22], [25], [27], [28] for more recent work and references to earlier literature.", "On the other hand, it was demonstrated in [29] that superconformal bootstrap imposes significantly more powerful constraints when applied to correlators involving long or at least semi-short operators.", "This should not come as a surprise since a long multiplet bootstrap combines the usual power of a mixed correlator bootstrap with the full constraints of supersymmetry that keep the number of independent dynamical variables small.", "In [29], the long multiplet bootstrap was only explored for $\\mathcal {N} = (2,0)$ SCFTs in $d=2$ dimensions since for this case it was possible to construct all the relevant blocks with rather traditional techniques, see also [30] for another instance of the long multiplet bootstrap 2-dimensional SCFTs.", "For promising applications to higher dimensions, our ignorance about generic superconformal blocks has been the main bottleneck.", "In fact, with the exception of [31] there had been very few attempts even to advance beyond the present status.", "In [32], [33] we launched a programme to resolve this issue of superconformal partial wave decompositions for long operators, at least for superconformal algebras of type I, for which the (internal) R-symmetry group $U$ contains an abelian factor $U(1)$ .", "In particular, all superconformal algebras in $d=4$ are of type I.", "As usual, the analysis consists of two parts.", "Starting with some correlator of fields that are inserted at points $x_i$ one first needs to find an appropriate set of four-point tensor structures that reduce the correlator to functions of superconformal cross ratios.", "Once these tensor structures are found, one derives and solves Casimir differential equations for a basis of superconformal blocks.", "We addressed the latter step in [33] after we had dealt with the former in [32].", "Both parts of the construction are heavily based on tools from group theory and harmonic analysis, following an earlier group theoretic approach to spinning conformal blocks that was developed in [34], [35], [36].", "With the choice of tensor structures and coordinates this approach provides, the Casimir differential equations for the blocks assume a universal form of a matrix Schrödinger problem of Calogero-Sutherland type.", "For this reason we shall refer to the group theoretic coordinates as Calogero-Sutherland coordinates and to the group theoretic choice of tensor structures as the Calogero-Sutherland gauge.", "When applied to superconformal algebras of type I, it turns out that the Casimir differential equations in the Calogero-Sutherland gauge take a particularly simple form - they can be written as a nilpotent perturbation of a set of Casimir equations for spinning fields of the (bosonic) conformal algebra.", "This implies that one can construct the superblocks through a finite order perturbation theory from spinning bosonic ones.", "While the general framework developed in [32] is fully algorithmic, it remained a bit formal.", "It is the main goal of the present work to illustrate how to apply this formalism in a concrete example relevant to 4-dimensional SCFTs.", "Our presentation will focus on the derivation of the crossing symmetry equations, i.e.", "the compatibility of the $s-$ and $t-$ channel superconformal partial wave expansions.", "In these equations, the relevant tensor structures only enter through the quotient of the $s-$ and $t-$ channel quantities.", "This quotient, which we dubbed the crossing factor in [33], is a matrix of conformal invariants, i.e.", "it only depends on cross ratios.", "Therefore, the crossing matrix is a little easier to compute than the individual tensor structures, see below.", "The example we will discuss here is relevant for the study of 4-dimensional $\\mathcal {N}=1$ superconformal field theories and concerns correlations functions $G_4(x_i) = \\langle \\bar{\\varphi }_1(x_1) \\mathcal {R}(x_2) \\varphi _3(x_3) \\mathcal {R}(x_4) \\rangle ,$ of a chiral field $\\varphi _3$ , an anti-chiral $\\bar{\\varphi }_1$ and two identical long multiplets $\\mathcal {R}$ with real scalar superprimary component $R$ .", "For this example we shall derive the full set of long multiplet crossing equations.", "Once all the algebraic dust of [32] and [33] has settled, the equations take a fully explicit form that is ready-to-use.", "Crossing symmetry for four-point functions of the corresponding superprimary fields has been studied before, most notably in [37].", "In spelling out our result and comparing it to the equation in [37] we will adopt notations of this earlier work, in particular for the dynamical operator product coefficients in the crossing equations.", "To understand the main features of the crossing equations let us briefly recall that the operator product of $\\bar{\\varphi }$ and $\\mathcal {R}$ contains four different families of supermultiplets with superprimaries $\\mathcal {O}$ labelled by their weights, spins and $U(1)$ internal charges.", "It is important to stress that 3-point functions of the superprimary field $\\mathcal {O}$ with $\\bar{\\varphi }$ and $R$ can vanish and that a non-trivial three-point function requires to pass from $\\mathcal {O}$ to one of its superdescendants.", "In the case at hand, this comment applies to three of the four families of operators we mentioned above.", "Following [37], we shall denote the associated three-point couplings by $ c_{\\bar{\\varphi }R (\\bar{Q}^2\\mathcal {O})_l}\\ , \\quad \\hat{c}_{\\bar{\\varphi }R (\\bar{Q}\\mathcal {O})_l}\\ , \\quad \\check{c}_{\\bar{\\varphi }R(\\bar{Q}\\mathcal {O})_l} \\ , \\quad \\bar{c}_{\\bar{\\varphi }R\\mathcal {O}_l} \\ .$ This notation displays the relevant superdescendant explicitly.", "Let us point out that the normalisation of any superdescendant is determined through supersymmetry by the (canonical) normalisation of the superprimary.", "With these notations being set up, the crossing equations we are about to derive assume the following form $ \\bar{\\sum }\|c_{\\bar{\\varphi }R (\\bar{Q}^2\\mathcal {O})_l}\|^2\\ \\mathcal {F}_1 +\\hat{\\sum }\|\\hat{c}_{\\bar{\\varphi }R (\\bar{Q}\\mathcal {O})_l}\|^2\\, \\hat{\\gamma }\\ \\mathcal {F}_2+ \\check{\\sum }\|\\check{c}_{\\bar{\\varphi }R(\\bar{Q}\\mathcal {O})_l}\|^2 \\, \\check{\\gamma }\\ \\mathcal {F}_3+ \\sum \|\\bar{c}_{\\bar{\\varphi }R\\mathcal {O}_l}\|^2 \\, \\bar{\\gamma }\\ \\mathcal {F}_4 = 0\\ .$ The four different terms in this equation correspond four independent four-point tensor structures.", "Each of the summations runs over intermediate supermultiplets.", "The last summation $\\bar{\\Sigma }$ , for example, includes all superprimaries $\\mathcal {O}^{\\Delta }_l$ of any spin $l$ and weight $\\Delta $ such that the three-point function with the fields $\\bar{\\varphi }$ and $R$ does not vanish etc.", "The factors $\\gamma $ that appear in three of the terms in eq.", "(REF ) are some explicitly known rational functions of the weight $\\Delta $ and the spin $l$ that are needed to match our conventions for the normalisation of superconformal blocks with the ones used in [37], see below.", "So the main kinematic ingredient in eq.", "(REF ) are the objects $\\mathcal {F}_j$ which take the form $ \\mathcal {F}_j = G_j(\\alpha _i^{-1}) - M_{st}(\\alpha _i) G_j(\\alpha _i),$ of a weighted sum of $s-$ and $t-$ channel superconformal blocks $G_j$ , as usual.", "The arguments $\\alpha _i$ are functions of the two cross ratios defined in eq.", "(REF ) and the weight factor $M_{st}(\\alpha _i)$ is the crossing factor we mentioned before, i.e.", "the ratio of $s-$ and $t-$ channel tensor structures.", "Superconformal blocks $G_j$ possess four independent components $G_{jk}$ , $k=1, \\dots ,4$ .", "We construct them explicitly in terms of well-known (spinning) bosonic blocks in eqs.", "(REF )-().", "All these partial waves can be expressed in terms of Gauss' hypergeometric function through formulas given in the appendix E. The crossing factor $M_{st}$ is a $4 \\times 4$ matrix spelled out in eq.", "(REF ).", "Together, these formulas allow to evaluate the four components of $\\mathcal {F}_j$ for any value of the cross ratios.", "In conclusion, our equation (REF ) provides a set of four independent crossing equations in which all the input from kinematics is explicitly known.", "In comparison, the work of Li, Meltzer and Stergiou contains only a single such crossing equation for the correlator (REF ) that arises from the superprimary component $R$ of the long multiplet $\\mathcal {R}$ , see eq.", "$(4.9)$ of [37].", "As one can check by explicit comparison of the superblocks, this equation coincides with the first of the four components in our crossing equation (REF ).", "In fact, the partial waves of [37] can be obtain from the superconformal blocks that we construct in eqs.", "(REF )-() as wavefunctions of a matrix Calogero-Sutherland problem, by restricting to their first components.", "More precisely $F = \\Lambda \\mathcal {F}_{11}\\ , \\quad \\hat{\\mathcal {F}} = \\hat{\\gamma }\\Lambda \\mathcal {F}_{21} \\ , \\quad \\check{\\mathcal {F}} = \\check{\\gamma }\\Lambda \\mathcal {F}_{31} \\ , \\quad \\bar{\\mathcal {F}} = \\bar{\\gamma }\\Lambda \\mathcal {F}_{41} \\ .$ Explicit formulas for the coefficients $\\hat{\\gamma }, \\check{\\gamma }$ and $\\gamma $ are given in eq.", "(REF ) and the factor $\\Lambda $ is a simple function of the cross ratios that maps blocks from Calogero-Sutherland gauge to the more standard gauge used in [37], see [38].", "Since this factor is the same for all four blocks, $\\Lambda $ does not appear in the crossing equation.", "Hence our result (REF ) is fully compatible with the work of Li et al.", "and extends it, as we obtain three more equations that the same dynamical coefficients have to satisfy.", "In this sense, the crossing symmetry in the long multiplet bootstrap is significantly more constraining that its restriction to superprimaries.", "Let us now briefly describe the outline of this paper.", "In section 2 we adapt the formalism developed in [33] to the computation of the crossing factor $M_{st}(\\alpha _i)$ that appears in eq.", "(REF ).", "The section includes a brief review of the main steps that need to be performed for the group theoretic construction of crossing factors.", "Then these steps are carried out for the example at hand.", "For the initial steps we keep the explicit calculations a bit more general so that many of the formulas we present still apply to superconformal algebras of the form $\\mathfrak {sl}(2m\|\\mathcal {N})$ .", "But in the final computation of $M_{st}$ we then focus on the crossing factor for the correlator (REF ) in 4-dimensional $\\mathcal {N}=1$ SCFTs.", "Section 3 is devoted to the computation of the superconformal blocks $G_j,j=1, \\dots , 4$ .", "This is achieved by adapting the framework of [32] to the correlator (REF ).", "In the first subsection, we review the form of the Laplacian on type I supergroups and discuss its rather astonishing features.", "Then we reduce the Laplacian on the supergroup $G=SL(4\|1)$ to a subspace of so-called $K$ -spherical functions on $G$ that is determined by the choice of fields that appear in the 4-point function (REF ).", "After this reduction we obtain a $4 \\times 4$ matrix Schrödinger operator whose eigenfunctions are superconformal blocks in Calogero-Sutherland gauge.", "These eigenfunctions are found explicitly in the third subsection before all our results are put together in the final subsection to derive the crossing equation (REF ).", "Our paper ends with an outlook to further applications of methods from [32], [33].", "A number of appendices contain some Lie-theoretic background and the set of conventions used throughout the main text." ], [ "Tensor Structures and Crossing Factor", "In this section we compute the crossing factor $M$ for the correlation function (REF ).", "The first subsection contains a brief review that of the general procedure that was developed in [33].", "Here we shall present it in the form of a algorithm that runs through three well defined steps of group theoretic calculations.", "These are then executed in the remaining subsections.", "Part of the necessary formulas are derived for the larger family of superalgebras $\\mathfrak {sl}(2m\|\\mathcal {N})$ and hence apply beyond the superconformal algebra of $\\mathcal {N}=1$ SCFTs in $d=4$ dimensions." ], [ "Review of the group theoretic approach", "Within the group theoretic approach, in which the Casimir equations for conformal blocks take the form of Calogero-Sutherland eigenvalue equations, the derivation of the crossing symmetry equations relies on two decompositions of the (super)conformal group, the Bruhat decomposition and the Cartan decomposition.", "The Bruhat factorisation is intimately related to conformal Ward identities and can be used to extend conformal fields to functions on the group.", "On the other hand, the Cartan factorisation provides a definition of Calogero-Sutherland (radial) coordinates.", "The relation between the two decompositions for the bosonic conformal group $SO(d+1,1)$ is well-known and this allowed us to compute the crossing factors of spinning bosonic fields in [33].", "Our goal is to adapt this program to superconformal groups of type I.", "Given some superconformal group $G$ , one may think of superspace $M$ as the subgroup that is generated translations and supertranslations.", "Let $\\lbrace X_a\\rbrace $ be the associated generators in the Lie superalgebra and denote by $x = \\lbrace x^a\\rbrace $ the corresponding superspace coordinates.", "The supergroup element $m(x) \\subset G$ is then given by $m(x) = e^{x^a X_a}\\ .$ We shall use this map to identify the superspace $M$ , i.e.", "the graded algebra generated by the superspace coordinates with the associated subgroup $M \\subset G$ .", "Given any pair of superspace coordinates $x_i, x_j$ one can define coordinates $x_{ij}$ in the graded tensor product $M_i\\otimes M_j$ of the individual superspaces, $m(x_{ij}) = m(x_j)^{-1} m(x_i)\\ .", "$ Any superconformal group contains the so-called Weyl inversion.", "This is an element of the underlying bosonic Lie group $G_{(0)}$ that is defined by $w = e^{\\pi \\frac{K_d-P_d}{2}},$ in terms of the generators $P_d$ of translations and $K_d$ of special conformal transformations.", "In bosonic theories, $w$ is the composition of the conformal inversion with the reflection in the hyperplane orthogonal to the unit vector $e_d$ .", "We use the Weyl inversion to define a second family of supergroup elements $n(x)$ by $w^{-1} m(x^a) w = n(x^a)\\ .", "$ Note that the elements $n(x)$ can be written by exponentiating a linear combination of the generators of (super)special conformal transformations.", "With this basic notation set up, we can now state the first step of our construction.", "In mathematical terms it instructs us to compute the Bruhat decomposition of $w m(x)$ , $w m(x) = m(y(x)) n(z(x)) k(t(x))\\ .", "$ The Bruhat decomposition is used to write an arbitrary supergroup element $g$ as a product of an element $m(y) \\in M$ , and element $n(z) \\in N = w^{-1} M w$ and a third element $k$ that lies in the (bosonic) subgroup $K$ that is generated by rotations, dilations and internal (R-symmetry) transformations.", "In other words, it is associated to the decomposition $\\mathfrak {g} = \\mathfrak {m} \\oplus \\mathfrak {n} \\oplus \\mathfrak {k}, \\nonumber $ of Lie superalgebra $\\mathfrak {g}$ of $G$ into (super)translation, a (super)special conformal transformation and elements that commute with with the generator of dilations.", "By performing the the Bruhat decomposition (REF ) for the special elements $g = w m(x)$ we determine three sets of functions $y(x)$ , $z(x)$ and $t(x)$ on superspace.", "We shall often simply write $y,z,t$ or $y_{ij}, z_{ij}, t_{ij}$ without specific reference to the dependence on $x$ , $x_{ij}$ etc.", "Let us also note that $y$ simply encodes how the Weyl inversion acts on superspace.", "Once we have determined the functions $y,z,t$ by performing the Bruhat decomposition (REF ) of $wm$ , we can move to the second step of our algorithm.", "It involves another factorization of (super)group elements $g$ , namely the the Cartan decomposition.", "In fact, we will make use of two closely related decompositions which we will refer to as unprimed and primed Cartan decompositions.", "In these, supergroup elements are written in factorised form as $ g = k_l \\eta ^{\\prime }_l a \\eta ^{\\prime }_r k_r = \\eta _l k_l a k_r \\eta _r\\ .$ Here $k_l, k_r\\in K$ and $a$ lies in the two-dimensional abelian $A$ group that is generated by $\\lbrace P_1+K_1,P_2-K_2\\rbrace $ .", "We shall parametrise $A$ by local coordinates $(u_1,u_2)$ according to $a(u_1,u_2) = e^{\\frac{u_1+u_2}{4}(P_1+K_1) -i \\frac{u_1-u_2}{4}(P_2 - K_2)}\\ .$ The factors $\\eta _{l,r}$ and $\\eta ^{\\prime }_{l,r}$ are associated with fermionic generators.", "More specifically, $\\eta _l,\\eta ^{\\prime }_l$ are obtained by exponentiation of generators of negative $R$ -charge and $\\eta _r,\\eta ^{\\prime }_r$ from generators with positive charge.", "That factors $k_{l,r}$ and $a$ are the same in the two versions of the Cartan decomposition follows from the Baker-Campbell-Hausdorff formula.", "There is a freedom in the choice of $k_{l,r}$ coming from the fact that $A$ has a non-trivial stabiliser group $B$ in $K$ , but the crossing equations do not depend on this choice, see [33].", "Having introduced the Cartan decomposition, we can now state the second step in our construction explicitly.", "To this end, let us consider four points $x_i, i=1, \\dots ,4,$ in superspace and assume that we have calculated the functions $y_{ij},\\ z_{ij}$ and $t_{ij}$ from the Bruhat decomposition of $wm(x_{ij})$ .", "We use this data to construct the following family of supergroup elements $g(x_i) = n(y_{21})^{-1} m(x_{31}) n(y_{43})\\ ,$ that depends on the superspace coordinates $x_i$ of our four external points.", "The main challenge of step 2 is to perform the Cartan decomposition of $g(x_i)$ .", "We can do that once and for all but then need to apply these decompositions to different `channels'.", "These channels can be labelled by permutations $\\sigma $ of the four points with the $s$ -channel associated to the trivial permutation $\\sigma =1$ and the $t$ -channel to the permutation $\\sigma =(24)$ that exchanges point $x_2$ and $x_4$ .", "The corresponding Cartan decompositions read $g_\\sigma (x_i) = g(x_{\\sigma (i)}) = k_{\\sigma ,l}(x_i) \\eta ^{\\prime }_{\\sigma ,l}(x_i)a_\\sigma (x_i)\\eta ^{\\prime }_{\\sigma ,r}(x_i) k_{\\sigma ,r}(x_i) \\ .$ Finding explicit formulas for the various factors requires some work even for bosonic theories [36].", "But depending on the precise setup, some shortcuts are possible, see below.", "Having constructed the Cartan decompositions (REF ) we now turn to the third and final step, namely the construction of the crossing factor for the transformation from the $s-$ to the $t-$ channel.", "By definition, the crossing factor is a matrix that acts on the finite dimensional space of polarizations of the external fields.", "It may be regarded as a quotient of $s-$ and $t-$ channel tensor structures.", "Let us suppose that the superprimaries of our four supermultiplets transform in representations $\\rho _i$ of the bosonic subgroup $K$ .", "The choice of representations $\\rho _i$ amounts to the choice of conformal weights $\\Delta _i$ , spins $\\lambda _i$ and $R$ -charges $r_i$ .", "The associated carrier spaces $V_i$ are spanned by the polarization vectors of the superprimaries.", "For scalar superprimaries in theories with abelian $R$ -symmetry, for example, the representations $\\rho _i$ are 1-dimensional.", "Given any choice $\\rho _i$ of representations, we define the super-crossing factor as $\\mathcal {M}_{st} = \\mathcal {P}_t \\Big ( \\rho _1(\\kappa _1)\\otimes \\rho _2(\\kappa _2)\\otimes \\rho _3(\\kappa _3)\\otimes \\rho _4(\\kappa _4) \\Big )\\mathcal {P}_s\\ .$ In this formula $\\mathcal {P}_s$ and $\\mathcal {P}_t$ denote projectors to the subspace of polarizations that are invariant under the action of the subgroup $B$ that stabilises four points, see [33] for details.", "More importantly, the arguments $\\kappa _i \\in K \\subset G$ are constructed as $& \\kappa _1 = k_{t,l}^{-1}k_{s,l}, \\quad \\kappa _2 = k^{w}_{t,r} k(t_{23})k(t_{21})^{-1} k^{w}_{s,l}, \\\\[2mm]& \\kappa _3 = k_{t,r}k_{s,r}^{-1}, \\quad \\kappa _4 = (k^{w}_{t,l})^{-1} k(t_{41})k(t_{43})^{-1} (k^{w}_{s,r})^{-1}, $ from factors that have been determined in previous steps and $k^w = w k w^{-1}$ .", "By construction $\\mathcal {M}$ is a family of matrices that acts in the tensor product $\\otimes _{i=1}^4 V_i$ of the polarization spaces.", "The matrix elements are functions of the four sets of superspace coordinates $x_i$ .", "As one might suspect and we proved in [33], the crossing factor $\\mathcal {M}_{st}$ is a superconformal invariant, i.e.", "it depends on the external points $x_i$ only through their cross ratios.", "But as it stands, these still include fermionic (nilpotent) cross ratios in addition to the pair of bosonic cross ratios.", "To capture the contribution of all the components of the supermultiplets to the crossing equations one finally has to perform a Taylor expansion in the fermionic cross ratios to construct the crossing factor $M_{st}$ that now acts on the space of $B$ -invariant polarisations of the supermultiplet, not just its superprimaries.", "Let us summarize once again the key steps of our construction.", "In order to derive the crossing factor, one needs to determine two factorisations (REF ) and (REF ).", "The results are then substituted in eqs.", "(REF ), () and (REF ) to find the super-crossing factor.", "A final expansion in fermionic invariants then gives the crossing factor $M$ that appears in the crossing equations.", "In concrete implementations one can often shortcut the full Cartan decomposition.", "Using conformal invariance of the crossing factor it is usually possible to make a particular gauge choice for the set of $x_i$ that does not effect the values of cross ratios but significantly simplifies the Cartan decomposition of $g_\\sigma (x_i)$ .", "We can now detail the content of the remaining subsections.", "In section 2.2 we obtain the Bruhat decomposition (REF ) for two infinite families of type I superconformal groups, $SL(2\|\\mathcal {N})$ and $SL(4\|\\mathcal {N})$ .", "Next in section 2.3 we find the Cartan factors (REF ) associated with the subgroup $A$ , dilations and $U(1)_R$ -symmetries.", "These factors are sufficient to write down the super-crossing factor if the fields in the correlation function are scalars, which is also done at the end of subsection 2.3.", "Though explicit, the resulting expression for the crossing factor is still somewhat complicated.", "In the last subsection, we specialise the discussion to the case of two long and two short operators in an $\\mathcal {N}=1$ SCFT, for which the formulas simplify significantly.", "Finally, we will perform the remaining Taylor expansion in the fermionic invariants and write the crossing factor $M_{st}$ , thereby completing the task we had set out for this section." ], [ "Bruhat decomposition for $\\mathfrak {sl}(2m\|\\mathcal {N})$", "Let $\\mathfrak {g}$ be a simple complex superconformal algebra of type I.", "These include a few isolated Lie superalgebras, namely $\\mathfrak {osp}(2\|4),\\mathfrak {psl}(2\|2)$ and $\\mathfrak {psl}(4\|4)$ , as well as two infinite families $\\mathfrak {sl}(2\|\\mathcal {N})$ and $\\mathfrak {sl}(4\|\\mathcal {N})$ .", "We shall focus on these infinite families, i.e.", "we set $\\mathfrak {g} = \\mathfrak {sl}(2m\|\\mathcal {N})$ .", "For this choice of $\\mathfrak {g}$ one finds $\\mathfrak {g}_{(0)} = \\mathfrak {sl}(2m) \\oplus \\mathfrak {sl}(\\mathcal {N}) \\oplus \\mathfrak {u}(1),\\quad \\mathfrak {g}_{+} = (2m,\\overline{\\mathcal {N}},1),\\quad \\mathfrak {g}_- = (\\overline{2m},\\mathcal {N},-1)\\ .$ The summand $\\mathfrak {sl}(2m)$ is the bosonic conformal Lie algebra with a basis $\\lbrace D,P_{\\dot{\\alpha }}^{\\ \\beta }, K_{\\alpha }^{\\ \\dot{\\beta }}, M_{\\alpha }^{\\ \\beta },M_{\\dot{\\alpha }}^{\\ \\dot{\\beta }}\\rbrace $ .", "The other two summands in $\\mathfrak {g}_{(0)}$ form the internal symmetry algebra and their basis elements are denoted by $R_{I}^{\\ J}$ and $R$ , respectively.", "Here, the indices $I,J = 1,...,\\mathcal {N}$ are that of the fundamental representation of $\\mathfrak {sl}(\\mathcal {N})$ .", "Indices $\\alpha ,\\dot{\\alpha }=1,...,m$ are that of the fundamental and the anti-fundamental representation of the rotation Lie algebra inside $\\mathfrak {sl}(2m)$ .", "Of course, in the case $m=1$ , there are no rotations and these indices run over a one element set.", "Finally, the spaces $\\mathfrak {q}_\\pm $ and $\\mathfrak {s}_\\pm $ have dimension $m\\mathcal {N}$ .", "The representations of $\\mathfrak {k}$ which they carry are indicated by the type of indices of their generators $\\mathfrak {q}_+ = \\text{span}\\lbrace Q_{\\dot{\\alpha }}^{\\ J}\\rbrace ,\\quad \\mathfrak {q}_- =\\text{span}\\lbrace Q_I^{\\ \\beta }\\rbrace ,\\quad \\mathfrak {s}_+ = \\text{span}\\lbrace S_{\\alpha }^{\\ J}\\rbrace ,\\quad \\mathfrak {s}_- = \\text{span}\\lbrace S_I^{\\ \\dot{\\beta }}\\rbrace \\ .$ The dual basis of $\\mathfrak {g}_{(1)}^\\ast $ to this one will be denoted by $\\lbrace q^{\\dot{\\alpha }}_{\\ J},q^I_{\\ \\beta },s^{\\alpha }_{\\ J},s^I_{\\ \\dot{\\beta }}\\rbrace $ .", "Spaces $\\mathfrak {g}_\\pm $ carry representations of $\\mathfrak {g}_{(0)}$ which are dual to each other.", "Explicitly, the dual bases are $(S_\\alpha ^{\\ I})^* = Q_I^{\\ \\alpha },\\quad (Q_{\\dot{\\alpha }}^{\\ I})^* = S_I^{\\ \\dot{\\alpha }}.$ The Lie superalgebra $\\mathfrak {g}$ has a fundamental $2m+\\mathcal {N}$ -dimensional representation.", "We will denote by $E_i^{\\ j}$ the matrix with 1 at position $(i,j)$ and zeros elsewhere, $i,j=1,...,2m+\\mathcal {N}$ .", "Such indices are split in three pieces $\\dot{\\alpha },\\alpha ,I$ , that is, we write $A = A^i_{\\ j} E_{i}^{\\ j} = \\begin{pmatrix}A^{\\dot{\\alpha }}_{\\ \\dot{\\beta }} & A^{\\dot{\\alpha }}_{\\ \\beta } & A^{\\dot{\\alpha }}_{\\ J}\\\\A^{\\alpha }_{\\ \\dot{\\beta }} & A^{\\alpha }_{\\ \\beta } & A^{\\alpha }_{\\ J}\\\\(-1)^{\|A^I_{\\ \\dot{\\beta }}\|}A^I_{\\ \\dot{\\beta }} & (-1)^{\|A^I_{\\ \\beta }\|}A^I_{\\ \\beta } & (-1)^{\|A^I_{\\ J}\|}A^I_{\\ J}\\end{pmatrix}.$ We can choose the generators so that the $\\mathfrak {sl}(2m)$ and $\\mathfrak {sl}(\\mathcal {N})$ algebras sit in the top left and bottom right corners, respectively, while the subspaces $\\mathfrak {g}_\\pm $ occupy the top right and bottom left corners.", "Schematically $\\begin{pmatrix}\\mathfrak {k}\\cap \\mathfrak {sl}(2m) & \\mathfrak {g}_1 & \\mathfrak {q}_+ \\\\\\mathfrak {g}_{-1} & \\mathfrak {k}\\cap \\mathfrak {sl}(2m) & \\mathfrak {s}_+ \\\\\\mathfrak {s}_- & \\mathfrak {q}_- & \\mathfrak {sl}(\\mathcal {N})\\end{pmatrix}\\ .$ For the precise definition of the fundamental representation, see the appendix B.The early works [39], [40], [41] use the same representation both of the Lie superalgebra and the supergroup and have inspired some of our calculations.", "In the remainder of this subsection, we will derive expression for $y(x),z(x),t(x)$ appearing in eq.", "(REF ) .", "In order to do this, we spell out the supermatrices representing various factors in this equation.", "The Weyl inversion and its inverse take the form $w = \\begin{pmatrix}0 & -w^{\\dot{\\alpha }}_{\\ \\beta } & 0 \\\\-w^{\\alpha }_{\\ \\dot{\\beta }} & 0 & 0 \\\\0 & 0 & \\delta ^I_{\\ J}\\end{pmatrix},\\quad w^{-1} = \\begin{pmatrix}0 & w^{\\dot{\\alpha }}_{\\ \\beta } & 0 \\\\w^{\\alpha }_{\\ \\dot{\\beta }} & 0 & 0 \\\\0 & 0 & \\delta ^I_{\\ J}\\end{pmatrix},$ where $-w^{\\dot{\\alpha }}_{\\ \\beta } = w^{\\alpha }_{\\ \\dot{\\beta }} = \\sigma _2 $ for $m=2$ and $w^{\\dot{\\alpha }}_{\\ \\beta } = - w^{\\alpha }_{\\ \\dot{\\beta }}=1$ for $m=1$ .", "The superspace $\\mathcal {M} = G/P$ is generated by variables $x^{\\dot{\\alpha }}_{\\ \\beta }, \\theta ^{\\dot{\\alpha }}_{\\ J}, \\bar{\\theta }^{I}_{\\ \\beta }$ , obeying the usual (anti)commutation relations.", "We see that $m(x) = e^{x^{\\dot{\\alpha }}_{\\ \\beta }P_{\\dot{\\alpha }}^{\\ \\beta } + \\theta ^{\\dot{\\alpha }}_{\\ J}Q_{\\dot{\\alpha }}^{\\ J} + \\bar{\\theta }^{I}_{\\ \\beta } Q_I^{\\ \\beta }}=\\begin{pmatrix}\\delta ^{\\dot{\\alpha }}_{\\ \\dot{\\beta }} & X^{\\dot{\\alpha }}_{\\ \\beta } & \\theta ^{\\dot{\\alpha }}_{\\ J}\\\\0 & \\delta ^{\\alpha }_{\\ \\beta } & 0\\\\0 & -\\bar{\\theta }^{I}_{\\ \\beta } & \\delta ^{I}_{\\ J}\\end{pmatrix},\\ \\ \\text{with} \\ \\ X^{\\dot{\\alpha }}_{\\ \\beta } = x^{\\dot{\\alpha }}_{\\ \\beta }-\\frac{1}{2}\\theta ^{\\dot{\\alpha }}_{\\ K} \\bar{\\theta }^{K}_{\\ \\beta }\\ .$ Using $w^{\\dot{\\alpha }}_{\\ \\delta } w^{\\delta }_{\\ \\dot{\\beta }} = -\\delta ^{\\dot{\\alpha }}_{\\ \\dot{\\beta }}$ and $w^{\\alpha }_{\\ \\dot{\\delta }} w^{\\dot{\\delta }}_{\\ \\beta } = -\\delta ^{\\alpha }_{\\ \\beta }$ we get for elements $n(x)$ $n(x) = w^{-1} m(x) w = \\begin{pmatrix}\\delta ^{\\dot{\\alpha }}_{\\ \\dot{\\beta }} & 0 & 0 \\\\-w^\\alpha _{\\ \\dot{\\gamma }}X^{\\dot{\\gamma }}_{\\ \\delta }w^{\\delta }_{\\ \\dot{\\beta }} & \\delta ^{\\alpha }_{\\ \\beta } & w^{\\alpha }_{\\ \\dot{\\gamma }}\\theta ^{\\dot{\\gamma }}_{\\ J}\\\\\\bar{\\theta }^I_{\\ \\delta } w^{\\delta }_{\\ \\dot{\\beta }} & 0 & \\delta ^{I}_{\\ J}\\end{pmatrix}.$ Finally, elements of the subgroup $K$ assume the form $k(t) = \\begin{pmatrix}e^{\\frac{\\mathcal {N}\\kappa }{\\mathcal {N}-2m}+\\frac{1}{2}\\lambda }(r_1)^{\\dot{\\alpha }}_{\\ \\dot{\\beta }} & 0 & 0 \\\\0 & e^{\\frac{\\mathcal {N}\\kappa }{\\mathcal {N}-2m}-\\frac{1}{2}\\lambda }(r_2)^{\\alpha }_{\\ \\beta } & 0 \\\\0 & 0 & e^{\\frac{2m\\kappa }{\\mathcal {N}-2m}}U^{I}_{\\ J}\\end{pmatrix} \\equiv \\text{diag}(k_1,k_2,k_3)\\ .$ Matrices $r_{1,2}$ are purely rotational.", "That is, they belong to $SL(2,\\mathbb {C})$ for $m=2$ and are equal to 1 if $m=1$ .", "In the following we will suppress indices where no confusion can arise.", "They can be put back at any point by looking at what type of indices a certain object carries and contracting over the appropriate number and type of dummy indices.", "We shall agree to write $J=w^{\\alpha }_{\\ \\dot{\\beta }}$ , then $-J^{-1}=w^{\\dot{\\alpha }}_{\\ \\beta }$ .", "With these conventions, the above expressions can be rewritten as $w= \\begin{pmatrix}0 & J^{-1} & 0 \\\\-J & 0 & 0\\\\0 & 0 & 1\\end{pmatrix},\\ w^{-1} = \\begin{pmatrix}0 & -J^{-1} & 0 \\\\J & 0 & 0\\\\0 & 0 & 1\\end{pmatrix},\\ m(x) = \\begin{pmatrix}1 & X & \\theta \\\\0 & 1 & 0\\\\0 & -\\bar{\\theta }& 1\\end{pmatrix},\\ n(x) = \\begin{pmatrix}1 & 0 & 0 \\\\-JXJ & 1 & J\\theta \\\\\\bar{\\theta }J & 0 & 1\\end{pmatrix}\\ .$ Therefore, the equation $w m(x) = m(y)n(z)k(t)$ reads $\\begin{pmatrix}0 & J^{-1} & 0 \\\\-J & -J X & -J\\theta \\\\0 & -\\bar{\\theta }& 1\\end{pmatrix} = \\begin{pmatrix}(1 - YJZJ + \\eta \\bar{\\zeta }J)k_1 & Y k_2 & (YJ\\zeta + \\eta ) k_3\\\\-JZJ k_1 & k_2 & J\\zeta k_3\\\\(\\bar{\\eta }JZJ + \\bar{\\zeta }J)k_1 & -\\bar{\\eta }k_2 & (1-\\bar{\\eta }J\\zeta ) k_3\\end{pmatrix}\\ .$ Here, the notation is $y = (y^{\\dot{\\alpha }}_{\\ \\beta },\\eta ^{\\dot{\\alpha }}_{\\ J},\\bar{\\eta }^I_{\\ \\beta })$ , $z = (z^{\\dot{\\alpha }}_{\\ \\beta },\\zeta ^{\\dot{\\alpha }}_{\\ J},\\bar{\\zeta }^I_{\\ \\beta })$ and $Y,Z$ are introduced analogously to $X$ .", "To write down the solution for $y,z$ and $t$ we introduce $T = 1 + \\bar{\\theta }X^{-1}\\theta ,\\quad \\Lambda = 1+X^{-1}\\theta \\bar{\\theta }\\ .$ Then one observes that $\\Lambda ^{-1} = 1-X^{-1}\\theta T^{-1}\\bar{\\theta }$ .", "Using this, the solution to the above system is found $&(Y,\\eta ,\\bar{\\eta }) = (- (JXJ)^{-1},-(XJ)^{-1}\\theta T^{-1},-\\bar{\\theta }(JX)^{-1}),\\\\&(Z,\\zeta ,\\bar{\\zeta }) = (- X - \\theta \\bar{\\theta },-\\theta T^{-1},-\\bar{\\theta }\\Lambda ),\\\\&(k_1,k_2,k_3) = (-((X+\\theta \\bar{\\theta })J)^{-1},-JX,T)\\ .$ In particular $z^{\\dot{\\alpha }}_{\\ \\beta } = -x^{\\dot{\\alpha }}_{\\ \\beta }$ , as in the bosonic theory.", "This completes our analysis of the equation (REF )." ], [ "Cartan decomposition and crossing factor", "Having determined the decomposition (REF ), we can turn to the crossing equations.", "We shall consider $s$ - and $t$ -channels and use superconformal transformations to set $x_1 = (a e_1 + b e_2,\\theta _1,\\bar{\\theta }_1),\\quad x_2 = (0,0,0),\\quad x_3 = (e_1,\\theta _3,\\bar{\\theta }_3),\\quad x_4 = (\\infty e_1,0,0)\\ .$ To write the crossing symmetry equations, one should consider the primed Cartan decomposition of $G$ .", "We start from its unprimed cousin $g = e^{q^I_{\\ \\beta } Q_I^{\\ \\beta } + s^I_{\\ \\dot{\\beta }} S_{I}^{\\ \\dot{\\beta }}} \\, k_l a(u_1,u_2) k_r\\, e^{q^{\\dot{\\alpha }}_{\\ J}Q_{\\dot{\\alpha }}^{\\ J} + s^{\\alpha }_{\\ J} S_{\\alpha }^{\\ J}}\\ .$ In the fundamental representation this reads $g = \\begin{pmatrix}\\delta ^{\\dot{\\alpha }}_{\\ \\dot{\\gamma }} & 0 & 0\\\\0 & \\delta ^{\\alpha }_{\\ \\gamma } & 0\\\\-s^I_{\\ \\dot{\\gamma }} & -q^I_{\\ \\gamma } & \\delta ^{I}_{\\ K}\\end{pmatrix} \\begin{pmatrix}e^{\\frac{\\mathcal {N}\\kappa }{\\mathcal {N}-2m}} (g_b)^{\\dot{\\gamma }}_{\\ \\dot{\\delta }} & e^{\\frac{\\mathcal {N}\\kappa }{\\mathcal {N}-2m}} (g_b)^{\\dot{\\gamma }}_{\\ \\delta } & 0 \\\\e^{\\frac{\\mathcal {N}\\kappa }{\\mathcal {N}-2m}} (g_b)^{\\gamma }_{\\ \\dot{\\delta }} & e^{\\frac{\\mathcal {N}\\kappa }{\\mathcal {N}-2m}} (g_b)^{\\gamma }_{\\ \\delta } & 0\\\\0 & 0 & e^{\\frac{2m\\kappa }{\\mathcal {N}-2m}} U^K_{\\ L}\\end{pmatrix} \\begin{pmatrix}\\delta ^{\\dot{\\delta }}_{\\ \\dot{\\beta }} & 0 & q^{\\dot{\\delta }}_{\\ J} \\\\0 & \\delta ^{\\delta }_{\\ \\beta } & s^{\\delta }_{\\ J} \\\\0 & 0 & \\delta ^{L}_{\\ J}\\end{pmatrix},$ where $g_b = k_l^b a k_r^b$ is an element of the bosonic conformal group.", "We see that the top left $2m\\times 2m$ corner is simply a scalar multiple of $g_b$ .", "When written without indices, generators of $\\mathfrak {g}_-^\\ast $ will carry a bar, to be distinguished from generators of $\\mathfrak {g}_+^\\ast $ .", "With this convention, the above Cartan decomposition reads $g = e^{\\frac{\\mathcal {N}\\kappa }{\\mathcal {N}-2m}}\\begin{pmatrix}A & B & Aq+Bs\\\\C & D & Cq+Ds\\\\-\\bar{s} A -\\bar{q} C & -\\bar{s} B - \\bar{q} D & e^{-\\kappa } U-(\\bar{s} A +\\bar{q} C)q -(\\bar{s} B + \\bar{q} D)s\\end{pmatrix}\\ .$ Here $A,B,C,D$ are $m\\times m$ blocks of $g_b$ and how to extract Cartan coordinates from them was explained in [36].", "The elements that we want to decompose are $g_s(x_i) = n(y_{21})^{-1} m(x_{31}) n(y_{43}),\\quad g_t(x_i) = n(y_{41})^{-1} m(x_{31}) n(y_{23})\\ .$ As can be seen from the solutions of eq.", "(REF ) found in the previous subsection, when $x$ is sent to $(\\infty e_1,\\theta ,\\bar{\\theta })$ then $y(x)=0$ and consequently $n(y(x))=1$ .", "Therefore, in the special configuration that we chose, one has $g_s(x_i) = n(y_{21})^{-1} m(x_{31}),\\quad g_t(x_i) = m(x_{31}) n(y_{23})\\ .$ Thus we are led to consider the decomposition of elements that take the general form $n(y)m(x)$ and $m(x)n(y^{\\prime })$ .", "We treat these in turn.", "In the notation of the previous subsection $n(y)m(x) = \\begin{pmatrix}1 & X & \\theta \\\\- JYJ & 1 - JYJX - J\\eta \\bar{\\theta }& J\\eta - JYJ\\theta \\\\\\bar{\\eta }J & \\bar{\\eta }JX - \\bar{\\theta }& 1 + \\bar{\\eta }J \\theta \\end{pmatrix}\\ .$ One immediately finds $s_s = (1 - J \\eta \\bar{\\theta })^{-1} J \\eta ,\\ q_s = \\theta - X s_s,\\ \\bar{q}_s = \\bar{\\theta }(1-J\\eta \\bar{\\theta })^{-1},\\ \\bar{s}_s = (\\bar{q}_s J Y - \\bar{\\eta })J,\\ e^{\\frac{2m\\kappa _s}{\\mathcal {N}-2m}}U_s = 1 + \\bar{\\theta }s_s\\ .$ The last expression can be simplified by substituting for $s$ and performing the following manipulation $1 + \\bar{\\theta }s_s = 1+ \\bar{\\theta }(1 - J \\eta \\bar{\\theta })^{-1} J \\eta = 1 + \\bar{\\theta }\\Big (\\sum _{n=0}^{\\infty } (J \\eta \\bar{\\theta })^n\\Big )J\\eta = \\sum _{n=0}^{\\infty }(\\bar{\\theta }J\\eta )^n = (1-\\bar{\\theta }J\\eta )^{-1}\\ .", "\\nonumber $ Therefore, taking the determinant of the last equation in (REF ) gives $e^{\\frac{2m\\mathcal {N}\\kappa _s}{\\mathcal {N}-2m}} = \\text{det}(1 - \\bar{\\theta }J\\eta )^{-1}\\ .$ Next, by looking at determinants of top left four $m\\times m$ blocks we obtain the coordinates associated with dilations $e^{2(\\lambda _{s,l} + \\lambda _{s,r})} = -\\text{det}(J^{-1}-YJX-\\eta \\bar{\\theta })^{-1},\\quad e^{2(\\lambda _{s,l} - \\lambda _{s,r})} = \\text{det}X\\text{det}Y^{-1},$ as well as the coordinates $(u_1,u_2)$ of the abelian torus $& \\sinh ^2\\frac{u^s_1}{2}\\sinh ^2\\frac{u^s_2}{2} = \\text{det}X\\text{det}Y\\text{det}(1 - \\bar{\\theta }J\\eta ),\\\\& \\cosh ^2\\frac{u^s_1}{2} \\cosh ^2\\frac{u^s_2}{2} = -\\text{det}(J^{-1}-YJX-\\eta \\bar{\\theta })\\text{det}(1 - \\bar{\\theta }J\\eta )\\ .$ We have already put a label $s$ on the coordinates, as they are indeed the $s$ -channel coordinates for appropriate choices of $x$ and $y$ as in eq.", "(REF ).", "For the other channel, we decompose $m(x)n(y^{\\prime }) = \\begin{pmatrix}1 - XJY^{\\prime }J +\\theta \\bar{\\eta }^{\\prime } J & X & XJ\\eta ^{\\prime }+\\theta \\\\-JY^{\\prime }J & 1 & J\\eta ^{\\prime }\\\\\\bar{\\theta }JY^{\\prime }J+\\bar{\\eta }^{\\prime } J & -\\bar{\\theta }& 1-\\bar{\\theta }J \\eta ^{\\prime }\\end{pmatrix}\\ .$ Following similar steps as above, we find $q_t = (1+\\theta \\bar{\\eta }^{\\prime } J)^{-1}\\theta ,\\ s_t=J(\\eta ^{\\prime }+Y^{\\prime }J q_t),\\ \\bar{s}_t = -\\bar{\\eta }^{\\prime } J (1+\\theta \\bar{\\eta }^{\\prime } J)^{-1},\\ \\bar{q}_t = \\bar{\\theta }- \\bar{s}_t X,\\ e^{\\frac{2m\\kappa _t}{\\mathcal {N}-2m}}U_t = 1 - \\bar{\\eta }J q_t,$ and therefore $e^{\\frac{2m\\kappa _t}{\\mathcal {N}-2m}} = \\text{det}(1 + \\bar{\\eta }^{\\prime } J\\theta )^{-1}\\ .$ Dilation coordinates are now $e^{2(\\lambda _{t,l} + \\lambda _{t,r})} = -\\text{det}(J^{-1}-XJY^{\\prime }+\\theta \\bar{\\eta }^{\\prime }),\\quad e^{2(\\lambda _{t,l} - \\lambda _{t,r})} = \\text{det}X\\text{det}Y^{\\prime -1}\\ .$ Finally the coordinates on the abelian torus read $& \\sinh ^2\\frac{u^t_1}{2}\\sinh ^2\\frac{u^t_2}{2} = \\text{det}X\\text{det}Y^{\\prime } \\text{det}(1+\\bar{\\eta }^{\\prime } J\\theta ),\\\\& \\cosh ^2\\frac{u^t_1}{2} \\cosh ^2\\frac{u^t_2}{2} = -\\text{det}(J^{-1}-XJY^{\\prime }+\\theta \\bar{\\eta }^{\\prime })\\text{det}(1+\\bar{\\eta }^{\\prime } J\\theta )\\ .$ Expressions written so far are sufficient to determine the crossing factor for fields which transform trivially under rotations and $SU(\\mathcal {N})$ internal symmetries.", "For applications that we have in mind in this work these conditions are satisfied.", "Assuming that a field transforms trivially both under spatial rotations and $SU(\\mathcal {N})$ internal symmetries, it is associated with a 1-dimensional representation $\\rho _{\\Delta ,r}$ of $K$ .", "Here $\\Delta $ is the conformal weight and $r$ the $U(1)_R$ -charge of the field.", "Our parametrisation of $K$ is such that $\\rho _{\\Delta ,r}(e^{\\lambda D + \\kappa R}e^{r^\\alpha _{\\ \\beta } M_{\\alpha }^{\\ \\beta } + r^{\\dot{\\alpha }}_{\\ \\dot{\\beta }} M_{\\dot{\\alpha }}^{\\ \\dot{\\beta }} + u^I_{\\ J} R_I^{\\ J}}) = e^{-\\Delta \\lambda + r\\kappa }.$ Therefore, the tensor factors appearing in $\\mathcal {M}_{st}$ are $& \\rho _3(\\kappa _3) = e^{\\Delta _3(\\lambda _{s,r}-\\lambda _{t,r})},\\ \\ \\rho _4(\\kappa _4) = e^{-\\Delta _4(\\lambda _{t,l}+\\lambda _{s,r})-r_4\\kappa _t},\\\\[2mm]& \\rho _1(\\kappa _1) = e^{\\Delta _1(\\lambda _{t,l}-\\lambda _{s,l})+r_1(\\kappa _s-\\kappa _t)},\\ \\ \\rho _2(\\kappa _2) = e^{\\Delta _2(\\lambda _{t,r}+\\lambda _{s,l})+r_2\\kappa _s}\\rho _2(k(t_{23})k(t_{21})^{-1})\\ .$ In the last expression we have used that the middle two factors in $\\kappa _4$ cancel out in our gauge.", "All the coordinates appearing on right hand sides of previous equations have been spelled out and one simply substitutes for them to find the product." ], [ "Application to 4-dimensional $\\mathcal {N}=1$ SCFTs", "Let us apply the results from previous two subsections to the complexified $\\mathcal {N}=1$ superconformal algebra in $d=4$ dimensions, $\\mathfrak {g} = \\mathfrak {sl}(4\|1)$ .", "We use the same notation as above, only $\\mathfrak {sl}(\\mathcal {N})$ indices become redundant, as this summand disappears for $\\mathcal {N}=1$ .", "The correlation function we want to consider is that of two long multiplets $\\mathcal {O}$ , along with one anti-chiral field $\\bar{\\varphi }_1$ and one chiral $ \\varphi _3$ , see eq.", "(REF ).", "The fields have conformal weights $\\Delta _i$ and $R$ -charges $r_i$ , and we assume that $\\sum r_i = 0$ .", "Therefore, we can write $r = r_1 + r_2 = - r_3 - r_4$ .", "Chirality conditions further imply $\\Delta _1 = - \\frac{3}{2} r_1, \\quad \\Delta _3 = \\frac{3}{2} r_3\\ .$ The general solution for $y(x)$ specialises in the case $m=2,\\ \\mathcal {N}=1$ to $y = - \\frac{1+\\frac{\\Omega }{4\\text{det}x}}{\\text{det}x}x^{t},\\ \\eta = \\frac{-i}{\\text{det}x}\\begin{pmatrix}x^{\\dot{2}}_{\\ 1}\\theta ^{\\dot{1}} - x^{\\dot{1}}_{\\ 1}\\theta ^{\\dot{2}} + \\frac{1}{2} \\bar{\\theta }_1 \\theta ^{\\dot{1}}\\theta ^{\\dot{2}}\\\\x^{\\dot{2}}_{\\ 2}\\theta ^{\\dot{1}} - x^{\\dot{1}}_{\\ 2}\\theta ^{\\dot{2}} + \\frac{1}{2} \\bar{\\theta }_2 \\theta ^{\\dot{1}}\\theta ^{\\dot{2}}\\end{pmatrix},\\ \\bar{\\eta }^T = \\frac{-i}{\\text{det}x}\\begin{pmatrix}x^{\\dot{1}}_{\\ 1}\\bar{\\theta }_2 - x^{\\dot{1}}_{\\ 2}\\bar{\\theta }_1 - \\frac{1}{2} \\theta ^{\\dot{1}} \\bar{\\theta }_1\\bar{\\theta }_2\\\\x^{\\dot{2}}_{\\ 1}\\bar{\\theta }_2 - x^{\\dot{2}}_{\\ 2}\\bar{\\theta }_1 - \\frac{1}{2} \\theta ^{\\dot{2}} \\bar{\\theta }_1\\bar{\\theta }_2\\end{pmatrix}\\ .\\nonumber $ In these formulas, $x$ and $y$ denote $2\\times 2$ matrices of bosonic coordinates of super-points $x$ and $y$ .", "This is a slight abuse of notation, but in any equation the meaning of symbols $x,y$ is clear from the context.", "By $\\Omega $ we denote the element $\\theta ^{\\dot{1}}\\theta ^{\\dot{2}}\\bar{\\theta }_1\\bar{\\theta }_2$ .", "The covariant derivatives, realising the right-regular action of $\\mathfrak {m}$ , read in our coordinates $D_{\\dot{\\alpha }}^{\\ I} = \\partial _{\\dot{\\alpha }}^{\\ I} + \\frac{1}{2}\\bar{\\theta }^{I}_{\\ \\beta }\\partial _{\\dot{\\alpha }}^{\\ \\beta },\\quad \\bar{D}_{I}^{\\ \\alpha } = -\\partial _{I}^{\\ \\alpha }-\\frac{1}{2}\\theta ^{\\dot{\\beta }}_{\\ I}\\partial _{\\dot{\\beta }}^{\\ \\alpha }\\ .$ One can verify that they anti-commute with the right-invariant vector fields written in the appendix B.", "Let us introduce the corresponding chiral and anti-chiral coordinates $x^{\\prime \\dot{\\alpha }}_{\\ \\beta } = x^{\\dot{\\alpha }}_{\\ \\beta } + \\frac{1}{2}\\theta ^{\\dot{\\alpha }}_{\\ I}\\bar{\\theta }^{I}_{\\ \\beta },\\quad x^{\\prime \\prime \\dot{\\alpha }}_{\\ \\beta } = x^{\\dot{\\alpha }}_{\\ \\beta } - \\frac{1}{2}\\theta ^{\\dot{\\alpha }}_{\\ I}\\bar{\\theta }^{I}_{\\ \\beta }\\ .$ We further set $\\theta ^{\\prime } = \\theta ^{\\prime \\prime } = \\theta $ and $\\bar{\\theta }^{\\prime } = \\bar{\\theta }^{\\prime \\prime } = \\bar{\\theta }$ .", "Then the following equalities hold $D x^{\\prime \\prime } = D\\bar{\\theta }^{\\prime \\prime } = 0, \\quad \\bar{D} x^{\\prime } = \\bar{D} \\theta ^{\\prime } = 0\\ .$ The chirality conditions satisfied by the fields allow us to set $\\theta _1$ and $\\bar{\\theta }_3$ to zero.", "Let us write $\\alpha = a + ib$ , $\\alpha ^\\ast = a - ib$ and fix the insertion points to positions as explained in the previous subsection.", "Further, we write $y = - y_{21}$ and $y^{\\prime }=y_{23}$ .", "Then a computation gives $& y = \\Big ( \\begin{pmatrix}-1/\\alpha ^\\ast & 0\\\\0 & 1/\\alpha \\end{pmatrix},\\ \\begin{pmatrix}0\\\\0\\end{pmatrix},\\ i\\begin{pmatrix}(\\bar{\\theta }_1)_2/\\alpha ^\\ast \\\\(\\bar{\\theta }_1)_1/\\alpha \\end{pmatrix}^t\\Big ),\\ y^{\\prime } = \\Big ( \\begin{pmatrix}1 & 0\\\\0 & -1\\end{pmatrix},\\ i\\begin{pmatrix}(\\theta _3)^{\\dot{2}}\\\\(\\theta _3)^{\\dot{1}}\\end{pmatrix},\\ \\begin{pmatrix}0\\\\0\\end{pmatrix}^t\\Big )\\ .$ Next, the factor $m(x_{31})$ is found $m(x_{31}) = \\begin{pmatrix}1 & X & \\theta _3\\\\0 & 1 & 0\\\\0 & \\bar{\\theta }_1 & 1\\end{pmatrix}, \\ \\text{with}\\ X = X_3 - X_1 = \\begin{pmatrix}-1 + \\alpha & 0\\\\0 & 1 - \\alpha ^\\ast \\end{pmatrix}\\ .$ We are now ready to consider the Cartan decomposition of $n(y)m(x)$ and $m(x)n(y^{\\prime })$ .", "Using the formulas of the previous subsection, the fermionic coordinates and dilation coordinates are $& q_s = q_t = \\theta _3,\\quad \\bar{q}_s = \\bar{q}_t = -\\bar{\\theta }_1,\\quad s_s = \\bar{s}_s = s_t = \\bar{s}_t = 0,\\\\& e^{4\\lambda _{s,l}} = \\alpha ^2(\\alpha ^\\ast )^2(1-\\alpha )(1-\\alpha ^\\ast ),\\quad e^{4\\lambda _{s,r}} = \\frac{1}{(1-\\alpha )(1-\\alpha ^\\ast )},\\\\& e^{4\\lambda _{t,l}} = \\alpha \\alpha ^\\ast (1-\\alpha )(1-\\alpha ^\\ast ),\\quad e^{4\\lambda _{t,r}} = \\frac{\\alpha \\alpha ^\\ast }{(1-\\alpha )(1-\\alpha ^\\ast )}\\ .$ The other factors that appear in $k_{s/t,l/r}$ , which are products of rotations and $R$ -symmetry transformations, assume the following diagonal form $r_{s/t,l} = \\begin{pmatrix}L_{s/t} & 0 & 0\\\\0 & L_{s/t}^{-1} & 0\\\\0 & 0 & 1\\end{pmatrix},\\ r_{s/t,r}=\\begin{pmatrix}R_{s/t} & 0 & 0\\\\0 & R_{s/t}^{-1} & 0\\\\0 & 0 & 1\\end{pmatrix}, \\ \\ \\text{with} \\ \\ L_{s,t} = \\begin{pmatrix}l_{s,t} & 0\\\\0 & l_{s,t}^{-1}\\end{pmatrix},\\ \\ R_{s,t} = \\begin{pmatrix}r_{s,t} & 0\\\\0 & r_{s,t}^{-1}\\end{pmatrix},$ and $l_{s,t},\\ r_{s,t}$ are in turn given by $l_s = \\Big (\\frac{\\alpha ^2(1-\\alpha )}{(\\alpha ^\\ast )^2(1-\\alpha ^\\ast )}\\Big )^{1/8},\\ r_s = \\Big (\\frac{1-\\alpha ^\\ast }{1-\\alpha }\\Big )^{1/8},\\ l_t = \\sqrt{-i}\\Big (\\frac{\\alpha (1-\\alpha )}{\\alpha ^\\ast (1-\\alpha ^\\ast )}\\Big )^{1/8},\\ r_t = \\frac{1}{\\sqrt{-i}}\\Big (\\frac{\\alpha (1-\\alpha ^\\ast )}{\\alpha ^\\ast (1-\\alpha )}\\Big )^{1/8}\\ .", "\\nonumber $ Finally, the coordinates on the torus are $\\cosh ^2\\frac{u_1^s}{2} = \\frac{1}{\\alpha },\\quad \\cosh ^2\\frac{u_2^s}{2} = \\frac{1}{\\alpha ^\\ast },\\quad \\cosh ^2\\frac{u_1^t}{2} = \\alpha ,\\quad \\cosh ^2\\frac{u_2^t}{2} = \\alpha ^\\ast \\ .$ This completes the determination of Cartan coordinates of the elements $g_s$ and $g_t$ .", "To find the matrix $\\mathcal {M}_{st}$ , it is still required to determine $k(t_{21})$ and $k(t_{23})$ .", "These are $k(t_{21}) = (\\alpha \\alpha ^\\ast )^{-D},\\quad k(t_{23}) = 1\\ .$ This allows for the computation of factors appearing in $\\mathcal {M}_{st}$ .", "The computation gives $\\rho _i(\\kappa _i) = (\\alpha \\alpha ^\\ast )^{-\\frac{\\Delta _i}{4}}\\ .$ To derive the crossing equations, there is one remaining step, namely to perform the expansion in nilpotent invariants in both channels.", "In order to do this, we need to switch to the primed Cartan coordinates by moving the exponentials containing fermionic variables past the elements of the left and right $K$ -subgroups.", "We have in both channels that $s^{\\prime } = \\bar{s}^{\\prime } = 0$ and $\\bar{q}_s^{\\prime } = -\\bar{\\theta }_1 L_s^{-1} e^{-\\frac{1}{2}\\lambda _{s,l}},\\quad q_s^{\\prime } = R_s e^{\\frac{1}{2}\\lambda _{s,r}}\\theta _3,\\quad \\bar{q}^{\\prime }_t = -\\bar{\\theta }_1 L_t^{-1} e^{-\\frac{1}{2}\\lambda _{t,l}},\\quad q_t^{\\prime } = R_t e^{\\frac{1}{2}\\lambda _{t,r}}\\theta _3\\ .$ Recall that $B$ is the commutant in $G_{(0)}$ of the 2-dimensional abelian group $A$ .", "In the case at hand, $B=SO(2)\\times SO(2)$ and Lie algebras of $A$ and $B$ are $\\mathfrak {a} = \\text{span}\\lbrace P_1 + K_1,\\ P_2 - K_2\\rbrace ,\\quad \\mathfrak {b} = \\text{span}\\lbrace R,M_1^{\\ 1} + M_{\\dot{1}}^{\\ \\dot{1}}\\rbrace \\ .$ Irreducible finite-dimensional representations of $\\mathfrak {k}$ are labelled by two spins, a conformal weight and an $R$ -charge, $(j_1,j_2)^{\\Delta }_r$ .", "In such notation, the four modules $\\mathfrak {q}_\\pm ,\\mathfrak {s}_\\pm $ are $\\mathfrak {q}_+ = (0,1/2)^{1/2}_1,\\quad \\mathfrak {q}_- = (1/2,0)^{1/2}_{-1},\\quad \\mathfrak {s}_+ = (1/2,0)^{-1/2}_1,\\quad \\mathfrak {s}_- = (0,1/2)^{-1/2}_{-1}\\ .$ According to our general theory, [32], blocks for the correlation function (REF ) are functions on the double coset with values in the space $\\left( V_{(12)} \\otimes \\Lambda \\mathfrak {q}_- \\otimes V_{(34)} \\otimes \\Lambda \\mathfrak {q}_+ \\right)^{\\mathfrak {b}} = \\left( \\Lambda \\mathfrak {q} \\right)^\\mathfrak {b}\\ .", "$ Under the action of $\\mathfrak {k}$ the 16-dimensional exterior algebra inside the brackets transforms as $\\Lambda \\mathfrak {q} \\cong \\mathbb {1}^0_0 \\oplus \\mathbb {1}^1_2 \\oplus \\mathbb {1}^{1}_{-2} \\oplus \\mathbb {1}^2_0 \\oplus (1/2,0)^{1/2}_{-1} \\oplus (0,1/2)^{1/2}_1 \\oplus (1/2,1/2)^1_0\\oplus (1/2,0)^{3/2}_{1} \\oplus (0,1/2)^{3/2}_{-1}\\ .", "\\nonumber $ We have written $\\mathbb {1}$ for the trivial representation of $SU(2)\\times SU(2)$ .", "Two of the singlets are $\\mathfrak {b}$ -invariant and the 4-dimensional representation $(1/2,1/2)$ contains a 2-dimensional invariant subspace.", "Hence, the space of invariants is 4-dimensional and spanned by $W_{\\bar{\\varphi }\\mathcal {O}\\varphi \\mathcal {O}} = (\\Lambda \\mathfrak {q})^B = \\text{span}\\lbrace 1,Q_{\\dot{1}}Q^1,Q_{\\dot{2}}Q^2, Q_{\\dot{1}}Q^1 Q_{\\dot{2}}Q^2\\rbrace \\ .$ Indeed, from the bracket relations given in the appendix, one checks that these combinations of generators commute, in the universal enveloping algebra $U(\\mathfrak {g})$ , with $M_1^{\\ 1} + M_{\\dot{1}}^{\\ \\dot{1}}$ and $R$ .", "In the two channels, the invariant combinations read $& (\\bar{q}^{\\prime }_t)_1 (q^{\\prime }_t)^{\\dot{1}} = -i (1-\\alpha )^{-1/2} (\\bar{\\theta }_1)_1 (\\theta _3)^{\\dot{1}},\\ (\\bar{q}^{\\prime }_t)_2 (q^{\\prime }_s)^{\\dot{2}} = i (1-\\alpha ^\\ast )^{-1/2} (\\bar{\\theta }_1)_2 (\\theta _3)^{\\dot{2}},\\\\& (\\bar{q}^{\\prime }_s)_1 (q^{\\prime }_s)^{\\dot{1}} = - \\alpha ^{-1/2} (1-\\alpha )^{-1/2} (\\bar{\\theta }_1)_1 (\\theta _3)^{\\dot{1}},\\ (\\bar{q}^{\\prime }_s)_2 (q^{\\prime }_s)^{\\dot{2}} = -(\\alpha ^\\ast )^{-1/2} (1-\\alpha ^\\ast )^{-1/2} (\\bar{\\theta }_1)_2 (\\theta _3)^{\\dot{2}}\\ .$ Putting everything together, the crossing factor between $s$ - and $t$ -channels reads $ M_{st} = (\\alpha \\alpha ^\\ast )^{\\frac{7}{4}-\\frac{1}{4} \\sum \\Delta _i}\\begin{pmatrix}1 & 0 & 0 & 0\\\\0 & i\\sqrt{\\alpha } & 0 & 0\\\\0 & 0 & -i\\sqrt{\\alpha ^\\ast } & 0\\\\0 & 0 & 0 & \\sqrt{\\alpha \\alpha ^\\ast }\\end{pmatrix}^{-1}\\ .$ The factor $(\\alpha \\alpha ^\\ast )^{7/4}$ is the ratio of Haar measure densities in the two channels.", "We may observe that variables $\\alpha ,\\alpha ^\\ast $ are related to the usual variables $z,\\bar{z}$ by $ \\alpha = \\frac{z}{z-1}, \\quad \\alpha ^\\ast = \\frac{\\bar{z}}{\\bar{z} - 1}\\ .$ Thus the top left entry of the crossing matrix is the one that we would get in the bosonic theory, [33], as expected." ], [ "Superconformal Blocks for 4-dimensional $\\mathcal {N}=1$ SCFTs", "In this section we will compute the superconformal blocks for the correlator (REF ), which will allow us to write the associated crossing equation.", "According to the general strategy of the group theoretical approach, the relevant Casimir operators for conformal blocks descend from the Laplace operator on the (super)conformal group.", "Hence, the first subsection is devoted to the construction of the Laplacian in the (unprimed) Cartan coordinates on a supergroup of type I which were introduced in section 2.3.", "To obtain the Casimir operators we need to restrict the full Laplacian from the set of all (vector valued) functions on $G$ to the subspace of so-called $K$ -spherical functions, see below.", "The reduction of the Laplacian to the relevant $K$ -spherical functions on $SL(4\|1)$ is performed in the second subsection.", "In our group theoretical approach, the reduced (Casimir) operators take the form of a matrix Schrödinger operator $H$ .", "Its eigenfunctions are constructed in the third subsection in terms of ordinary (Gauss-type) hypergeometric functions.", "The precise relation of these eigenfunctions with superconformal blocks for the various operators that contribute to the correlation function (REF ) is detailed in the final subsection before we derive the crossing symmetry equations we stated in the introduction." ], [ "The Laplacian in Cartan coordinates", "In our harmonic analysis approach to superconformal correlations, four-point functions are represented as so-called $K$ -spherical functions on the superconformal group.", "These are vector-valued functions covariant with respect to both left and right regular action of the subgroup $K\\subset G$ , with covariance laws that are determined by representations that label the four fields of the correlator, see [33].", "When we perform the map from correlation functions to $K$ -spherical functions, the superconformal Casimir operator that is conventionally used to characterize the superconformal blocks is carried to the Laplace-Beltrami operator on $G$ .", "Thus the question of computing partial waves becomes one in harmonic analysis.", "The Laplacian on a Lie group or a supergroup can be constructed as the quadratic Casimir made out of right-invariant vector fields (or left-invariant, the two prescriptions give the same operator).", "In any given coordinate system it is a second order differential operator that is typically very complicated.", "However, since it commutes with the left and right regular actions, the Laplacian acts within the space of $K$ -spherical functions and, since the work of Berezin [42], is known to reduce to a simple operator on this space.", "These classical results apply to ordinary Lie groups, but as we have shown in [32], admit a very satisfactory extension to include supersymmetry of type I.", "The simplicity of the Laplacian now follows from its expression in the unprimed Cartan coordinates that we now review.", "Let $\\lbrace X^A\\rbrace $ be a basis for a type I Lie superalgebra $\\mathfrak {g}$ .", "We write $\\lbrace X^A\\rbrace = \\lbrace X^a,X^\\mu ,X_\\mu \\rbrace $ with $\\mathfrak {g}_{(0)} = \\text{span}\\lbrace X^a\\rbrace , \\quad \\mathfrak {g}_+ = \\text{span}\\lbrace X^\\mu \\rbrace ,\\quad \\mathfrak {g}_- = \\text{span}\\lbrace X_\\mu \\rbrace \\ .$ Spaces $\\mathfrak {g}_+$ and $\\mathfrak {g}_-$ carry representations of the even subgroup $G_{(0)}$ (under the adjoint action) that are dual to each other $g_{(0)} X^\\mu g_{(0)}^{-1} = \\pi (g_{(0)})^\\mu _{\\ \\nu } X^\\nu , \\quad g_{(0)} X_\\nu g_{(0)}^{-1} = \\pi (g_{(0)}^{-1})^\\mu _{\\ \\nu } X_\\mu \\ .$ Let us denote by $K^{ab}$ the Killing form of the even subalgebra $\\mathfrak {g}_{(0)}$ .", "It can be used to construct the quadratic Casimir element $C_2 = K_{ab} X^a X^b - X^\\mu X_\\mu + X_\\mu X^\\mu \\ .$ In the present notation, the (unprimed) Cartan decomposition of $G$ may be written as $g = e^{x^\\mu X_\\mu } g_{(0)} e^{x_\\nu X^\\nu }\\ .$ The middle factor $g_{(0)}$ is an element of the underlying Lie group $G_{(0)} =G_{bos}\\times U$ .", "We use Cartan coordinates for $G_{bos}$ as in [36] and some arbitrary coordinates for the internal symmetry group $U$ .", "Now a short computation of right-invariant vector fields leads to the following expression for the Laplacian, see e.g.", "[43], $\\mathcal {R}_{C_2} = \\mathcal {R}_{C_2}^{(0)} -2\\pi (g_{(0)}^{-1})^\\mu _{\\ \\nu }\\partial _{x_\\nu }\\partial _{x^\\mu } - K_{ab}\\pi (X^a)^\\mu _{\\ \\mu }\\mathcal {R}^{(0)}_{X^b}\\ .$ This is a remarkable formula.", "The first term on the right-hand side is the Laplacian of the underlying Lie group $G_{(0)}$ .", "The second term involves second order derivatives with respect to fermionic coordinates only, with coefficients depending on bosonic coordinates.", "Hence, this term is nilpotent.", "In the last term, the first order differential operators $\\mathcal {R}^{(0)}_{X^b}$ denote right-invariant vector fields on $G_{(0)}$ .", "These are multiplied with the trace of the bosonic generators $X^a$ in the representation on fermionic generators.", "In what follows we will expand scalar functions on the supergroup $G$ in the fermionic coordinates, thereby writing them as vector-valued functions on $G_{(0)}$ .", "In the process, the Laplacian on the supergroup turns into a matrix-valued operator which, and after restriction to the appropriate subspace of $K$ -spherical functions, can be reduced to the two-dimensional space $A$ that is parametrised by $u_1$ and $u_2$ .", "Once this is done, the bosonic Laplacian becomes a matrix-valued Calogero-Sutherland Hamiltonian whose eigenfunctions are spinning bosonic conformal blocks, [34], [35].", "Parameters of the Hamiltonian depend both on the quantum numbers of fields in the correlator and on the representation of $K$ carried by $\\mathfrak {g}_+$ , see [32].", "This follows from the fact that the unprimed fermionic Cartan coordinates transform non-trivially under the regular action of $K$ .", "As can be readily seen from eq.", "(REF ), the second term on the right-hand side reduces to a nilpotent, upper triangular matrix of functions in the two variables $u_1$ and $u_2$ .", "The third term can at most have one non-zero contribution, which arises from the $U(1)_R$ -generator, the only one that is not traceless in the representation $\\pi $ .", "Since this generator is also used in the formulation of $K$ -covariance laws, the whole term reduces to a matrix of constants.", "These comments may become more transparent in the example of $SL(4\|1)$ that we will now treat in detail." ], [ "Casimir equations for 4-dimensional $\\mathcal {N}=1$ SCFTs", "Let us now apply the formula (REF ) for the Laplacian to derive the Casimir equations for the correlation function (REF ).", "We will make use of both primed and unprimed Cartan coordinates on the supergroup $SL(4\|1)$ .", "The unprimed ones read $g = e^{q_\\alpha Q^\\alpha + s_{\\dot{\\alpha }} S^{\\dot{\\alpha }}} e^{\\kappa R} e^{\\lambda _l D} r_le^{\\frac{u_1+u_2}{4}(P_1+K_1) - i\\frac{u_1-u_2}{4}(P_2-K_2)} r_re^{\\lambda _r D}e^{q^{\\dot{\\alpha }}Q_{\\dot{\\alpha }} + s^\\alpha S_{\\alpha }},$ with $r_l = e^{\\varphi ^l_1 X_1} e^{\\theta ^l_1 Z_1} e^{\\psi ^l_1 X_1} e^{\\varphi ^l_2 X_2} e^{\\theta ^l_2 Z_2}e^{\\psi ^l_2 X_2}\\ ,\\quad r_r = e^{\\varphi ^r_1 X_1} e^{\\theta ^r_1 Z_1} e^{\\psi ^r_1 X_1}e^{\\varphi ^r_2 X_2} e^{\\theta ^r_2 Z_2} e^{\\psi ^r_2 X_2},$ and $\\psi _2^l = -\\psi _1^l$ .", "These coordinates are useful because they make the Laplacian take a particularly simple form.", "On the other hand, the primed Cartan coordinates $g = e^{\\kappa R} e^{\\lambda _l D} r_l e^{q^{\\prime }_\\alpha Q^\\alpha + s^{\\prime }_{\\dot{\\alpha }} S^{\\dot{\\alpha }}}e^{\\frac{u_1+u_2}{4}(P_1+K_1) - i\\frac{u_1-u_2}{4}(P_2-K_2)} e^{q^{\\prime \\dot{\\alpha }}Q_{\\dot{\\alpha }}+ s^{\\prime \\alpha }S_{\\alpha }} r_r e^{\\lambda _r D},$ are well suited to formulate the restriction to $K$ -spherical functions, i.e.", "for the formulation of the left and right $K$ -covariance laws.", "The explicit relation between coordinate systems (REF ) and (REF ) is written in the appendix C. For the four-point function (REF ), the associated covariant functions $f$ on the supergroup satisfy covariance properties $(\\partial _{\\lambda ^{\\prime }_l} - 2a) f = (\\partial _{\\lambda ^{\\prime }_r} - 2b) f = (\\partial _{\\kappa ^{\\prime }} - r) f= \\partial _{\\varphi _1^{^{\\prime }l}} f = ... = \\partial _{\\psi _2^{^{\\prime }r}} f = 0\\ .$ The parameters $a$ and $b$ are related to conformal weights of the fields in the correlation function by $2a = \\Delta _2 - \\Delta _1$ and $2b = \\Delta _3 - \\Delta _4$ .", "As we explained above, $\\lambda _l^{\\prime } =\\lambda _l$ and similarly for all other variables that appear in eq.", "(REF ), but the partial derivatives of course depend on the full system of coordinates and $\\partial _{\\lambda ^{\\prime }_l}\\ne \\partial _{\\lambda _l}$ etc.", "Due to the shortening of operators at positions 1 and 3, there are further differential equations that $f$ obeys, namely $\\partial _{s_{\\dot{\\alpha }}} f = \\partial _{s^\\alpha } f = 0\\ .$ Having described covariance properties of $f$ , we can perform the reduction of the Laplacian.", "From now on, we focus on one channel, say the $s$ -channel, the discussion for the other one being entirely analogous.", "Upon expansion in the Grassmann coordinates, the function $f$ is regarded as a vector-valued function on the underlying Lie group $f : G_{(0)} \\xrightarrow{} \\Lambda \\mathfrak {q}\\ .$ On the two-dimensional abelian group $A$ generated by $\\lbrace P_1+K_1,P_2-K_2\\rbrace $ , the function $f$ restricts to $\\omega ^{1/2} G$ with $G = G^{(1)}(u_i) + G^{(2)}(u_i) q_1 q^{\\dot{1}} + G^{(3)}(u_i) q_2 q^{\\dot{2}} + G^{(4)}(u_i) q_1 q^{\\dot{1}}q_2 q^{\\dot{2}}\\ .$ The conventional factor $\\omega ^{1/2}$ is defined in [36].", "Other components of $G$ vanish due to requirements of $B$ -invariance.", "After the reduction, the Laplacian restricts to the operator $H = H_0 + A,$ that acts on $G$ according to the general theory from [32].", "Let us first apply this theory to find the unperturbed part $H_0$ .", "This is the $4\\times 4$ matrix of differential operators $H_0 = -\\begin{pmatrix}H_{\\textit {sc}}^{a,b} + \\frac{3}{16} r^2 + \\frac{3}{4} r & 0 & 0 \\\\0 & H^{a+\\frac{1}{4},b-\\frac{1}{4}}_{\\frac{1}{2}}+\\frac{3}{16} (r-1)^2 + \\frac{3}{4} (r-1) & 0 \\\\0 & 0 & H_{\\textit {sc}}^{a+\\frac{1}{2},b-\\frac{1}{2}} + \\frac{3}{16} (r-2)^2 + \\frac{3}{4} (r-2)\\end{pmatrix}\\ .$ Here the operators on the diagonal are the scalar and the seed Calogero-Sutherland Hamiltonians.", "Explicitly $ H^{a,b}_{\\textit {sc}}=-\\frac{1}{2}\\partial ^2_{u_1} - \\frac{1}{2}\\partial ^2_{u_2} + \\frac{1}{2}\\left(\\frac{(a+b)^2-\\frac{1}{4}}{\\sinh ^2 u_1}-\\frac{a b}{\\sinh ^2\\frac{u_1}{2}} +\\frac{(a+b)^2-\\frac{1}{4}}{\\sinh ^2 u_2}-\\frac{a b}{\\sinh ^2\\frac{u_2}{2}}\\right)+\\frac{5}{4},$ and $H^{a,b}_{\\frac{1}{2}}=\\begin{pmatrix}H^{a,b}_{sc} -\\frac{1}{16} & 0 \\\\0 & H^{a,b}_{sc}-\\frac{1}{16}\\end{pmatrix}+\\\\[2mm]\\frac{1}{32}\\omega \\begin{pmatrix}\\frac{1}{\\sinh ^2\\frac{u_1}{2}}+\\frac{1}{\\sinh ^2\\frac{u_2}{2}}+\\frac{4}{\\sinh ^2\\frac{u_1-u_2}{4}}-\\frac{4}{\\cosh ^2\\frac{u_1+u_2}{4}} & 4(b-a)\\left(\\frac{1}{\\sinh ^2\\frac{u_1}{2}}-\\frac{1}{\\sinh ^2\\frac{u_2}{2}}\\right) \\\\ \\nonumber 4(b-a) \\left( \\frac{1}{\\sinh ^2\\frac{u_1}{2}}-\\frac{1}{\\sinh ^2\\frac{u_2}{2}}\\right) & \\frac{1}{\\sinh ^2\\frac{u_1}{2}}+\\frac{1}{\\sinh ^2\\frac{u_2}{2}}+\\frac{4}{\\sinh ^2\\frac{u_1+u_2}{4}}-\\frac{4}{\\cosh ^2\\frac{u_1-u_2}{4}}\\end{pmatrix}\\omega ^{-1},$ with $\\omega = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}1 & -1\\\\1 & 1\\end{pmatrix}\\ .$ Before we spell out the form of the nilpotent term $A$ , let us make a few comments on the derivation of $H_0$ .", "We focus on the most non-trivial part of $H_0$ , namely the seed Hamiltonian.", "Discussion of the two scalar Hamiltonians then follows by similar arguments.", "The covariant function $f$ contains among its 16 components the 4-component function $f_{-+} : G_{(0)} \\xrightarrow{} \\mathfrak {q}_- \\otimes \\mathfrak {q}_+, \\quad f_{-+} =f^{\\alpha }_{\\ \\dot{\\alpha }} q_\\alpha q^{\\dot{\\alpha }}\\ .$ We can express $f_{-+}$ in the primed Cartan coordinates as $f_{-+} = e^{-\\kappa +\\frac{1}{2}\\lambda _l-\\frac{1}{2}\\lambda _r} f^{\\alpha }_{\\ \\dot{\\alpha }} \\mathcal {L}_{\\alpha }^{\\ \\beta }\\mathcal {R}^{\\dot{\\alpha }}_{\\ \\dot{\\beta }} q^{\\prime }_\\beta q^{\\prime \\dot{\\beta }}\\ .$ The last equation defines $SU(2)$ matrices $\\mathcal {L},\\mathcal {R}$ .", "Their explicit form is easily written down using the formulas of the appendix C. In the primed Cartan coordinates, the dilation covariance laws simply read $f_i(e^{\\kappa R + \\lambda _l D}g_{(0)}e^{\\lambda _r D}) = e^{r\\kappa + 2a \\lambda _l + 2b\\lambda _r} f_i(g_{(0)})\\ .$ We have performed the expansion in fermionic primed Cartan coordinates.", "The index $i$ denotes any component of $f^{\\alpha }_{\\ \\dot{\\alpha }}$ .", "Thus, from eq.", "(REF ) we see that $a$ and $b$ receive shifts by $\\pm 1/4$ , respectively.", "The two constant terms involving the $r$ -charge in the upper left matrix element of $H_0$ come from $-R^2/4$ in the quadratic Casimir and from the third term in the Laplacian that involves the trace of $R$ in the representation $\\mathfrak {g}_+$ , respectively.", "Again, in the seed Hamiltonian, $r$ is shifted by $-1$ due to eq.", "(REF ).", "Next, consider the covariance law with respect to rotations.", "Similarly as in [35], these lead us to the seed Hamiltonian $H_{1/2}$ (matrices $\\mathcal {L},\\mathcal {R}$ here are different from those used in [35] but so is the projector $\\mathcal {P}$ , and they lead to the same Hamiltonian.", "The two calculations are related by a change of basis).", "Having described the unperturbed Hamiltonian, let us turn to the perturbation $A$ , which reads $A = -2\\begin{pmatrix}0& a^1_{\\ 3} & a^2_{\\ 4} & 0 \\\\0 & 0 & 0 & a^2_{\\ 4} \\\\0 & 0 & 0 & a^1_{\\ 3}\\\\0 & 0 & 0 & 0\\end{pmatrix} = -2\\begin{pmatrix}0& \\sinh \\frac{u_1}{2} & -\\sinh \\frac{u_2}{2} & 0 \\\\0 & 0 & 0 & -\\sinh \\frac{u_2}{2} \\\\0 & 0 & 0 & \\sinh \\frac{u_1}{2}\\\\0 & 0 & 0 & 0\\end{pmatrix}.$ To derive this, notice that in the representation $\\mathfrak {g}_+$ of $G_{(0)}$ , with the basis $\\lbrace X^\\mu \\rbrace =\\lbrace Q_{\\dot{1}},Q_{\\dot{2}},S_1,S_2\\rbrace $ (and with this order of basis vectors) the element $a(u_1,u_2)$ is given by $\\pi \\left(a(u_1,u_2)^{-1}\\right)^\\mu _{\\ \\nu } = \\begin{pmatrix}\\cosh \\frac{u_1}{2}& 0 & \\sinh \\frac{u_1}{2} & 0\\\\0 & \\cosh \\frac{u_2}{2} & 0 & -\\sinh \\frac{u_2}{2}\\\\\\sinh \\frac{u_1}{2} & 0 & \\cosh \\frac{u_1}{2} & 0\\\\0 & -\\sinh \\frac{u_2}{2} & 0 & \\cosh \\frac{u_2}{2}\\end{pmatrix} \\equiv (a^{\\mu }_{\\ \\nu })\\ .$ Basis of $\\mathfrak {g}_-$ that is dual to the above basis of $\\mathfrak {g}_+$ is $\\lbrace S^{\\dot{1}},S^{\\dot{2}},Q^1,Q^2\\rbrace $ .", "Now, an inspection of the second term in eq.", "(REF ) leads to eq.", "(REF ).", "To summarise, the Laplacian eigenvalue equation on $SL(4\|1)$ reduces on the space of $K$ -spherical functions to the eigenvalue problem of the operator $H$ written in eq.", "(REF ).", "The two ingredients of this operator are a $4\\times 4$ matrix of differential operators $H_0$ and a nilpotent matrix of functions $A$ in the two cross rations, see eqs.", "(REF ) and (REF ).", "We now turn to eigenfunctions of $H$ , that is, to the construction of the superconformal blocks for the correlator (REF )." ], [ "Construction of the superconformal blocks", "Having set up the Casimir equation for our superblocks in Calogero-Sutherland gauge we now come to the main task of this section, namely to solve these equations and thereby to construct the associated superconformal blocks.", "Our formulas will be entirely explicit in that we construct the superblocks as a finite linear combination of a set of known special functions.", "These functions appear already as (bosonic) conformal blocks of the component fields in the supermultiplets or, put differently, in the solution of the eigenvalue problem for the unperturbed Hamiltonian $H_0$ .", "To set the stage, let us introduce the eigenfunctions for the two different differential operators that appear along the diagonal of $H_0$ , see eqs.", "(REF ) and (REF ), $H^{a,b}_{\\textit {sc}}\\, \\phi ^{a,b}_{\\Delta ,l} & = C_{\\Delta ,l}^{\\textit {sc}}\\, \\phi ^{a,b}_{\\Delta ,l},\\ \\ \\ \\quad \\quad \\quad C_{\\Delta ,l}^{\\textit {sc}}=-\\frac{1}{4}\\Delta (\\Delta -4)-\\frac{1}{4}l(l+2),\\\\H^{a,b}_{\\frac{1}{2}}\\, \\Psi ^{(a,b)}_{\\pm ,\\Delta ,l} & = C^{\\textit {seed}}_{\\Delta ,l}\\, \\Psi ^{(a,b)}_{\\pm ,\\Delta ,l},\\ \\ \\ \\quad C^{\\textit {seed}}_{\\Delta ,l} = -\\frac{1}{4}\\Delta (\\Delta -4)-\\frac{1}{4} l(l+3)-\\frac{3}{8}\\ .", "$ The eigenfunctions $\\phi ^{a,b}_{\\Delta ,l}$ are related to the usual scalar conformal blocks through a simple transformation, see [11].", "The precise relation of the eigenfunctions $\\Psi ^{a,b}_{\\pm ,\\Delta ,l}$ with the simplest spinning seed conformal blocks of [9] was found in [35].", "Recall that the eigenvalue equation for $\\Psi $ is a matrix equation and hence $\\Psi $ has two components.", "The subscript $\\pm $ on $\\Psi $ labels two linearly independent solutions with the same eigenvalue.", "Explicit formulas for these functions can be found in the appendix E. They can all be written in terms of Gauss' hypergeometric function as finite sums of products thereof.", "These concrete formulas are needed e.g.", "for numerical evaluation of the crossing symmetry constraints we spell out below.", "But for our derivation of these constraints we only need a few very basic formulas that tell us how they behave under multiplication with the matrix elements of the nilpotent potential $A$ that was given in eq.", "(REF ), $ &2\\begin{pmatrix}-\\sinh \\frac{u_1}{2}\\ , \\ & \\sinh \\frac{u_2}{2}\\end{pmatrix}\\Psi ^{a+\\frac{1}{4},b-\\frac{1}{4}}_{+,\\Delta +1,l} = \\gamma ^{+1}_{\\Delta ,l}\\,\\phi ^{a,b}_{\\Delta +\\frac{1}{2},l}+ \\gamma ^{+2}_{\\Delta ,l} \\, \\phi ^{a,b}_{\\Delta +\\frac{3}{2},l+1},\\\\[4mm]&2\\begin{pmatrix}-\\sinh \\frac{u_1}{2}\\ , \\ & \\sinh \\frac{u_2}{2}\\end{pmatrix} \\Psi ^{a+\\frac{1}{4},b-\\frac{1}{4}}_{-,\\Delta +1,l} = \\gamma ^{-1}_{\\Delta ,l} \\,\\phi ^{a,b}_{\\Delta +\\frac{1}{2},l+1} + \\gamma ^{-2}_{\\Delta ,l}\\, \\phi ^{a,b}_{\\Delta +\\frac{3}{2},l},\\\\[2mm]& 2 \\begin{pmatrix}\\sinh \\frac{u_2}{2}\\\\-\\sinh \\frac{u_1}{2}\\end{pmatrix} \\phi ^{a+\\frac{1}{2},b-\\frac{1}{2}}_{\\Delta +1,l} = \\gamma ^{1+}_{\\Delta ,l}\\,\\Psi ^{a+\\frac{1}{4},b-\\frac{1}{4}}_{+,\\Delta +\\frac{1}{2},l} + \\gamma ^{2+}_{\\Delta ,l}\\,\\Psi ^{a+\\frac{1}{4},b-\\frac{1}{4}}_{+,\\Delta +\\frac{3}{2},l-1}+ \\gamma ^{1-}_{\\Delta ,l}\\,{\\Psi }^{a+\\frac{1}{4},b-\\frac{1}{4}}_{-,\\Delta +\\frac{1}{2},l-1} + \\gamma ^{2-}_{\\Delta ,l}\\,{\\Psi }^{a+\\frac{1}{4},b-\\frac{1}{4}}_{-,\\Delta +\\frac{3}{2},l}\\ .$ Here $& \\gamma ^{+1}_{\\Delta ,l} = \\frac{\\sqrt{2}}{i(-1)^{a+b}}\\frac{l+2}{l+1}, \\quad \\gamma ^{+2}_{\\Delta ,l} = \\frac{i(-1)^{-a-b}}{2\\sqrt{2}} \\frac{(4a-2l-2\\Delta -1)(2\\Delta -3)(4b+2l+2\\Delta +1)}{(2\\Delta -1)(2\\Delta +2l+1)(2\\Delta +2l+3)},\\\\[2mm]& \\gamma ^{-1}_{\\Delta ,l} = \\frac{2\\sqrt{2}}{i(-1)^{a+b}}, \\quad \\gamma ^{-2}_{\\Delta ,l} =\\frac{(-1)^{-a-b}}{i\\sqrt{2}}\\frac{(l+2)(4a+2l-2\\Delta +5)(2\\Delta -3)(-4b+2l-2\\Delta +5)}{(l+1)(2l-2\\Delta +3)(2l-2\\Delta +5)(2\\Delta -1)},\\\\[2mm]& \\gamma ^{1+}_{\\Delta ,l} = i (-1)^{a+b} 2\\sqrt{2}, \\quad \\gamma ^{2+}_{\\Delta ,l}=\\frac{i(-1)^{a+b}}{\\sqrt{2}} \\frac{ \\Delta l (2a+l-\\Delta +2)(-2b+l-\\Delta +2)}{(\\Delta -1)(l+1)(l-\\Delta +1)(l-\\Delta +2)},\\\\[2mm]& \\gamma ^{1-}_{\\Delta ,l} = i(-1)^{a+b}\\sqrt{2}\\frac{l}{l+1},\\ \\gamma ^{2-}_{\\Delta ,l}=\\frac{(-1)^{a+b}}{2\\sqrt{2}i}\\frac{\\Delta (2a-l-\\Delta )(2b+l+\\Delta )}{(\\Delta -1)(l+\\Delta )(l+\\Delta +1)}\\ .", "$ As far as we are aware, these identities involving scalar and seed conformal blocks are new, although their form has a clear representation-theoretic origin.", "Rather than computing the coefficients $\\gamma $ in terms of $SO(6)$ Clebsch-Gordan coefficients, they are most easily obtained by picking any point $(u_1,u_2)$ and Taylor-expanding both sides of the above equations.", "This produces a system of linear equations and it is clear that going to high enough order gives sufficiently many equations to fix the eight coefficients $\\gamma $ .", "A natural choice for the expansion point is to take $u_1\\gg u_2\\rightarrow \\infty $ , which corresponds to the OPE limit for the scalar and seed blocks.", "It is easy to promote the functions $\\Psi $ and $\\phi $ to eigenfunctions of $H_0$ .", "To this end let us denote the standard basis of $\\mathbb {C}^4$ by $\\lbrace e_1,...,e_4\\rbrace $ .", "The eigenfunctions of $H_0$ are therefore of the form $G_1^0 = \\phi ^{a,b}_{\\Delta +1,l}e_1,\\quad G_2^0 = \\Psi ^{a+\\frac{1}{4},b-\\frac{1}{4}}_{+,\\Delta +1,l},\\quad G_3^0 = \\Psi ^{a+\\frac{1}{4},b-\\frac{1}{4}}_{-,\\Delta +1,l},\\quad G_4^0 = \\phi ^{a+\\frac{1}{2},b-\\frac{1}{2}}_{\\Delta +1,l}e_4,$ where it is understood that the two non-zero components of $G_2$ and $G_3$ are in the space that is spanned by $e_2$ and $e_3$ .", "We shall work with this notation throughout the remaining part of this section whenever we write $\\Psi _\\pm $ .", "The solution of the eigenvalue problem for $H$ , which can be obtained by a finite perturbation of $G^0_i$ , will be denoted by $G_i$ , as before.", "With the help of the explicit expressions for $G^0_i$ , the formula (REF ) for the nilpotent potential and our auxiliary formulas (REF )-() one findsThe tensor product of any given finite dimensional $SO(d,2)$ representation $T_\\nu $ labelled by a Young tableau $\\nu $ and the induced representation $\\pi _{\\Delta ,\\mu }$ can be decomposed into a finite sum of induced representations (e.g.", "see [10]) as $T_\\nu \\otimes \\pi _{\\Delta ,\\mu }=\\bigoplus \\limits _{i=-j}^j \\bigoplus _{\\lambda \\in \\nu ^i\\otimes \\mu }\\pi _{\\Delta +i,\\lambda } $ where indices $(i,\\nu ^i)$ are defined through the decomposition of the $SO(d+2)$ representation $\\nu $ with respect to its $SO(2)\\times SO(d)$ subgroup and enumerate a (semi) integer $SO(2)$ conformal weight $i$ along with an $SO(d)$ Young tableau $\\nu ^i$ .", "Bosonic conformal blocks are particular matrix elements of some representation $\\pi _{\\Delta ,\\mu }$ of $G_{(0)}$ , [34].", "In the course of perturbation theory they are multiplied by matrix elements of the fundamental representation $\\pi =\\pi _f$ of $G_{(0)}$ .", "Therefore, the bosonic blocks that appear in the $n$ -th order of the perturbation theory are matrix elements of $\\pi _{\\Delta ,\\mu }\\otimes \\pi _f^{\\otimes n}$ .", "This fixes the functional form of our solutions.", "$& G_1 = G^0_1 = \\phi ^{a,b}_{\\Delta +1,l}e_1, \\\\[2mm]& G_2 = \\Psi ^{a+\\frac{1}{4},b-\\frac{1}{4}}_{+,\\Delta +1,l}+\\left(c^{+1}_{\\Delta ,l}\\, \\phi ^{a,b}_{\\Delta +\\frac{1}{2},l}+c^{+2}_{\\Delta ,l}\\, \\phi ^{a,b}_{\\Delta +\\frac{3}{2},l+1}\\right)e_1, \\\\[2mm]& G_3 = \\Psi ^{a+\\frac{1}{4},b-\\frac{1}{4}}_{-,\\Delta +1,l}+ \\left(c^{-1}_{\\Delta ,l}\\,\\phi ^{a,b}_{\\Delta +\\frac{1}{2},l+1}+ c^{-2}_{\\Delta ,l}\\, \\phi ^{a,b}_{\\Delta +\\frac{3}{2},l}\\right)e_1, \\\\[2mm]& G_4 = \\phi ^{a+\\frac{1}{2},b-\\frac{1}{2}}_{\\Delta +1,l}e_4 + c^{1+}_{\\Delta ,l}\\, \\Psi ^{a+\\frac{1}{4},b-\\frac{1}{4}}_{+,\\Delta +\\frac{1}{2},l}+ c^{2+}_{\\Delta ,l}\\, \\Psi ^{a+\\frac{1}{4},b-\\frac{1}{4}}_{+,\\Delta +\\frac{3}{2},l-1} + c^{1-}_{\\Delta ,l}\\,\\Psi ^{a+\\frac{1}{4},b-\\frac{1}{4}}_{-,\\Delta +\\frac{1}{2},l-1} + c^{2-}_{\\Delta ,l}\\,{\\Psi }^{a+\\frac{1}{4},b-\\frac{1}{4}}_{-,\\Delta +\\frac{3}{2},l} + \\nonumber \\\\[2mm]& \\hspace*{56.9055pt} +\\left(k^{00}_{\\Delta ,l}\\, \\phi ^{a,b}_{\\Delta ,l} +k^{01}_{\\Delta ,l}\\, \\phi ^{a,b}_{\\Delta +1,l+1} +k^{10}_{\\Delta ,l}\\, \\phi ^{a,b}_{\\Delta +1,l-1}+k^{11}_{\\Delta ,l}\\, \\phi ^{a,b}_{\\Delta +2,l} \\right)e_1\\ .", "$ The construction of these four solutions requires increasing orders of perturbation theory.", "The solution $G_1$ is obviously obtained in zeroth order and equal to the scalar bosonic block $G_1^0$ .", "The second and the third solution $G_2$ and $G_3$ are obtained at first order while the last solution $G_4$ required to go to second order.", "In order to complete the solution, we just need to spell out the various coefficients.", "Those that arose from first order perturbation theory are directly related to the coefficients $\\gamma $ we introduced in eqs.", "(REF )-() as, $c^{\\pm i} = \\frac{\\gamma ^{\\pm i}}{\\frac{3}{16}(2r+3) + C^{\\textit {sc}}_{\\pm i} - C^{\\textit {seed}}_{\\Delta +1,l}}\\ \\quad , \\quad \\quad c^{i \\pm } = \\frac{\\gamma ^{i \\pm }}{\\frac{3}{16}(2r+1)+C^{\\textit {seed}}_{i\\pm }-C^{\\textit {sc}}_{\\Delta +1,l}}$ where $\\begin{pmatrix} C^\\sigma _{+1} & C^\\sigma _{+2} \\\\ C^\\sigma _{-1} & C^\\sigma _{-2} \\end{pmatrix} =\\begin{pmatrix} C^\\sigma _{\\Delta + \\frac{1}{2},l} & C^\\sigma _{\\Delta + \\frac{3}{2},l+1} \\\\C^\\sigma _{\\Delta + \\frac{1}{2}, l+1} & C^\\sigma _{\\Delta +\\frac{3}{2}, l} \\end{pmatrix}\\ ,\\quad \\begin{pmatrix} C^\\sigma _{1+} & C^\\sigma _{1-} \\\\ C^\\sigma _{2+} & C^\\sigma _{2-} \\end{pmatrix} =\\begin{pmatrix} C^\\sigma _{\\Delta + \\frac{1}{2},l} & C^\\sigma _{\\Delta + \\frac{1}{2}, l-1} \\\\C^\\sigma _{\\Delta + \\frac{3}{2}, l-1} & C^\\sigma _{\\Delta +\\frac{3}{2}, l} \\end{pmatrix}\\ ,$ for $\\sigma = \\textit {sc},\\textit {seed}$ .", "The second order coefficients in the second line of $G_4$ involve products of the coefficients $\\gamma $ and they read as follows, $ & k^{00}_{\\Delta ,l} = \\frac{\\gamma ^{1+}_{\\Delta ,l}\\gamma ^{+1}_{\\Delta -\\frac{1}{2},l}+\\gamma ^{1-}_{\\Delta ,l}\\gamma ^{-1}_{\\Delta -\\frac{1}{2},l-1}}{\\frac{3}{4}(r+1)-C^{\\textit {sc}}_{\\Delta +1,l}+C^{\\textit {sc}}_{\\Delta ,l}}\\ ,\\quad \\quad \\quad k^{01}_{\\Delta ,l} = \\frac{\\gamma ^{1+}_{\\Delta ,l}\\gamma ^{+2}_{\\Delta -\\frac{1}{2},l} + \\gamma ^{2-}_{\\Delta ,l}\\gamma ^{-1}_{\\Delta +\\frac{1}{2},l}}{\\frac{3}{4}(r+1)-C^{\\textit {sc}}_{\\Delta +1,l}+C^{\\textit {sc}}_{\\Delta +1,l+1}},\\\\[2mm]& k^{10}_{\\Delta ,l} = \\frac{\\gamma ^{2+}_{\\Delta ,l}\\gamma ^{+1}_{\\Delta +\\frac{1}{2},l-1} + \\gamma ^{1-}_{\\Delta ,l}\\gamma ^{-2}_{\\Delta -\\frac{1}{2},l-1}}{\\frac{3}{4}(r+1)-C^{\\textit {sc}}_{\\Delta +1,l}+C^{\\textit {sc}}_{\\Delta +1,l-1}}\\ ,\\quad \\quad k^{11}_{\\Delta ,l} = \\frac{\\gamma ^{2+}_{\\Delta ,l}\\gamma ^{+2}_{\\Delta +\\frac{1}{2},l-1} + \\gamma ^{2-}_{\\Delta ,l}\\gamma ^{-2}_{\\Delta +\\frac{1}{2},l}}{\\frac{3}{4}(r+1)-C^{\\textit {sc}}_{\\Delta +1,l}+C^{\\textit {sc}}_{\\Delta +2,l}}\\ .", "$ Clearly, Hamiltonians $H_0$ and $H$ have the same spectrum - the eigenvalue of $G_i$ equals that of $G^0_i$ .", "The dependence of these eigenvalues on the weight $\\Delta $ and spin $l$ of the intermediate field can be read off from (REF )-(), $& C_2 = C_3 = \\frac{1}{4} \\Delta (\\Delta -2) + \\frac{1}{4} l(l+3) - \\frac{3}{16}(r+1)^2 +\\frac{3}{8},\\\\[2mm]& C_1 = \\frac{1}{4} \\Delta (\\Delta -2) + \\frac{1}{4} l(l+2) -\\frac{3}{16}(r+2)^2,\\ C_4 =\\frac{1}{4} \\Delta (\\Delta -2) + \\frac{1}{4} l(l+2) -\\frac{3}{16}r^2 \\ .$ This completes our discussion of the solution to the eigenvalue problem of $H$ and hence our construction of the superconformal blocks for the four-point functions under consideration.", "Let us stress again that all these results are completely explicit, with explicitly known coefficients $c,k$ in eqs.", "(REF ), (REF ), () and explicitly known functions $\\phi $ and $\\Psi _\\pm $ , see the appendix E." ], [ "Assembling the pieces: The crossing equations", "In this last subsection, we will derive the crossing symmetry equations for the four-point function (REF ) with identical long operators $\\mathcal {O}_2 = \\mathcal {O}_4 \\equiv \\mathcal {R}$ of vanishing $R$ -charge.", "In order to achieve this, we first need to associate eigenfunctions of $H$ found in the previous subsection with superconformal blocks representing the propagation of superconformal representations in the operator products $\\bar{\\varphi }\\times R$ and $\\varphi \\times R$ .", "There are four possible kinds of representations appearing in these operator products.", "Their quantum numbers are, [37] $\\rho _1 = (j,j)^\\Delta _{-r-2}, \\quad \\rho _2 = (j,j+1/2)^\\Delta _{-r-1}, \\quad \\rho _3 = (j+1/2,j)^\\Delta _{-r-1}, \\quad \\rho _4 = (j,j)^\\Delta _{-r}\\ .$ We will write $l = 2j$ .", "One observes that the eigenvalues $C_i$ written in (REF )-() coincide with the values of the quadratic Casimir in representations $\\rho _i$ (see the appendix D).", "Therefore, we can identify the propagation of the operator labelled by $\\rho _i$ with the superconformal block $G_i$ from above.", "There still remains the question of normalisation of these blocks.", "To settle it, let us focus on the $e_1$ -component of the four solutions.", "From the explicit formulas of the previous section, one can directly verify that $& \\frac{c^{+2}_{\\Delta ,l}}{c^{+1}_{\\Delta ,l}} = -\\frac{\\hat{c}_2}{\\hat{c}_1},\\quad \\frac{c^{-2}_{\\Delta ,l}}{c^{-1}_{\\Delta ,l}} = -\\frac{\\check{c}_2}{\\check{c}_1},\\quad \\frac{k^{01}_{\\Delta ,l}}{k^{00}_{\\Delta ,l}} = -\\bar{c}_1,\\quad \\frac{k^{10}_{\\Delta ,l}}{k^{00}_{\\Delta ,l}} = -\\bar{c}_2,\\quad \\frac{k^{11}_{\\Delta ,l}}{k^{00}_{\\Delta ,l}} = \\bar{c}_1 \\bar{c}_2\\ .$ Here, the coefficients $\\hat{c}_i, \\check{c}_i, \\bar{c}_i$ are taken from [37], with the substitution $l\\xrightarrow{}l+1$ in coefficients $\\check{c}_i$ in order to synchronise conventions.", "We will only need explicit formulas for two of these coefficients below, $ \\hat{c}_1 = \\frac{l+2}{(l+1)(2(\\Delta - l - \\Delta _\\varphi ) -3)} \\ , \\quad \\check{c}_1 = \\frac{1}{2(\\Delta + l -\\Delta _\\varphi ) + 3} \\ .$ After stripping of a conventional prefactor, the four-point function in [37] decomposes over superconformal blocks as $g(z_i) = \\sum \|c_{\\bar{\\varphi }R(\\bar{Q}^2\\mathcal {O}_l})\|^2 \\mathcal {G}(z_i) +\\sum \|c_{\\bar{\\varphi }R(\\bar{Q}\\mathcal {O})_l}\|^2 \\mathcal {\\hat{G}}(z_i) +\\sum \|c_{\\bar{\\varphi }R(\\bar{Q}\\mathcal {O})_l}\|^2 \\mathcal {\\check{G}}(z_i) +\\sum \|\\bar{c}_{\\bar{\\varphi }R\\mathcal {O}_l}\|^2 \\mathcal {\\bar{G}}(z_i)\\ .$ Here $\\mathcal {G},\\mathcal {\\hat{G}}, ...$ are sums of Dolan-Osborn scalar conformal blocks with appropriate coefficients, see eqs.", "(2.32),(2.25),(2.29),(2.19) of [37] for details.", "The Calogero-Sutherland scalar blocks are related to Dolan-Osborn ones by $\\phi ^{a,b}_{\\Delta ,l} = \\Lambda ^{a,b}(z_i)g^{a,b}_{\\Delta ,l}$ , where $\\Lambda ^{a,b}$ is an explicit function of cross ratios whose precise expression we will not need, see [38].", "The relation of the zero component of our function $G$ to the function $g$ defined in eq.", "(REF ) mimics that between the blocks, $G^0 =\\Lambda ^{a,b}(z_i) g$ .", "Therefore, the function $G$ on the abelian group $A$ that encodes the correlation function (REF ) in our approach decomposes over the superconformal blocks $G_j$ , $j=1,...,4$ , we determined on the previous subsection with coefficients $G(\\alpha _i) &= \\sum \|c_{\\bar{\\varphi }R(\\bar{Q}^2\\mathcal {O})_l}\|^2 G_1(\\alpha _i)+ \\sum \\frac{\\hat{c}_1}{c^{+1}_{\\Delta ,l}}\|c_{\\bar{\\varphi }R(\\bar{Q}\\mathcal {O})_l}\|^2 G_2(\\alpha _i)\\\\[2mm]& + \\sum \\frac{\\check{c}_1}{c^{-1}_{\\Delta ,l}} \|c_{\\bar{\\varphi }R(\\bar{Q}\\mathcal {O})_l}\|^2 G_3(\\alpha _i) +\\sum \\frac{1}{k^{00}_{\\Delta ,l}} \|\\bar{c}_{\\bar{\\varphi }R\\mathcal {O}_l}\|^2 G_4(\\alpha _i)\\ .$ We write interchangeably $\\alpha =\\alpha _1$ and $\\alpha ^\\ast =\\alpha _2$ .", "The relation of $\\alpha _i$ with the cross ratios was stated in eq.", "(REF ).", "Combining the block decomposition (REF ) of the correlation function $G$ with the crossing symmetry relation between $s-$ and $t-$ channel, $G(1/\\alpha ,1/\\alpha ^\\ast ) = M_{st}(\\alpha ,\\alpha ^\\ast ) G(\\alpha ,\\alpha ^\\ast )\\ $ with $M_{st}$ as defined in eq.", "(REF ) we have thereby derived the crossing relations we anticipated in eq.", "(REF ) of the introduction with coefficients given by $ \\hat{\\gamma } = \\frac{\\hat{c}_1}{c^{+1}_{\\Delta ,l}}\\ , \\quad \\check{\\gamma } =\\frac{\\check{c}_1}{c^{-1}_{\\Delta ,l}} \\, \\quad \\bar{\\gamma } = \\frac{1}{k^{00}_{\\Delta ,l}} \\ .$ with the coefficients $\\hat{c}_1$ and $\\check{c}_1$ defined in eq.", "(REF ), as well as the coefficients $c^{\\pm 1}_{\\Delta ,l}$ and $k^{00}_{\\Delta ,l}$ we introduced in eqs.", "(REF ) and (REF ), respectively.", "All these expressions are rational functions of the conformal weight $\\Delta $ and the spin $l$ .", "Thereby we have completed our derivation of the crossing equation (REF )." ], [ "Conclusions", "In this work we constructed superconformal blocks and crossing symmetry equations for the four-point function of two long, one chiral and one anti-chiral scalar operator in 4-dimensional $\\mathcal {N}=1$ SCFTs.", "The two ingredients that went into the derivation, namely the construction of superblocks $G_j$ and of the crossing factor $M_{st}$ applied general techniques developed in [32] and [33].", "This is indeed the first time a full set of long multiplet bootstrap equations has been written down in dimension higher than two.", "In the course of solving the Casimir differential equations for the relevant superconformal blocks, we also derived certain relations between scalar and seed bosonic conformal blocks which seem to be new.", "As should be clear from the previous sections, our methods are by no means restricted to the above class of correlators.", "They are completely algorithmic and can be applied to a wide class of correlation functions involving long multiplets in SCFTs with superconformal symmetry of type I.", "Let us mention some directions that can be pursued in the future.", "The approach to superconformal partial waves we developed in [32] works particular well for long multiplets.", "While some shortening conditions may be implemented before even writing down the Casimir differential equations, others can only be implemented once the solutions to these equations have been constructed.", "Which case occurs depends on how compatible the specific set of shortening conditions is with our choice or Cartan coordinates, see [32] for details.", "The four-point functions (REF ) we addressed here involve two BPS operators.", "Their shortening conditions belong to the first class that are easy to implement before writing down the equations.", "While the Casimir equations for the correlation function of four long multiplets $\\mathcal {R}$ turn out to possess 36 components, the shortening conditions of the chiral and anti-chiral field reduces the operators to a $4 \\times 4$ matrix system.", "These statements are particular to $\\mathfrak {sl}(4\|1)$ , but the same simplifications occur for any superconformal algebra of type I.", "Also, the simplifications in the crossing factor that we saw in section 2.4 have certain universality and appear for other algebras of the infinite family $\\mathfrak {sl}(m\|\\mathcal {N})$ .", "Therefore, correlators of the type $(\\ref {4-point-function})$ in a large class of SCFTs are very well-suited to the Calogero-Sutherland gauge.", "In order to perform a numerical long multiplet bootstrap analysis for $\\mathcal {N}=1$ SCFTs in $d=4$ , one would need to include the four-point function of the superfield $\\mathcal {R}$ in order to exploit positivity.", "This four-point function can be addressed with the same techniques as used in this work.", "Writing down the Casimir equations is actually not that difficult.", "Finding solutions meets two technical challenges.", "On the one hand, the unperturbed Schrödinger problem for $H_0$ now involves a larger set of 4-dimensional spinning blocks, including both higher seed and non-seed blocks.", "Of course, all of these are known in the literature and can be mapped to the Calogero-Sutherland gauge using results of [36].", "On the other hand, the nilpotent perturbation theory has to be taken to a higher order before it truncates.", "This requires a vast extension of the relations (REF )-().", "While all this is in principle straightforward it eventually results in relatively large sums of bosonic blocks.", "The superprimary component of the associated crossing symmetry equation has been worked out in [37].", "So, the main task that remains is to construct the other 35 components of this system of equations.", "Given that one would expect significantly stronger constraints on the dynamics of $\\mathcal {N}=1$ SCFTs it may be well worth the effort to carry this out explicitly.", "Similar comments apply to the important question of bootstrapping the stress tensor multiplet in four-dimensional $\\mathcal {N}=2$ SCFTs.", "It would require significant technical effort, but our method does apply to this case as well.", "Let us note that, while the method follows a clear algorithm, not all of its steps have been automatised by computer programs yet.", "Even without this, we were able to treat the highly non-trivial correlator $(\\ref {4-point-function})$ because symmetries are exploited to such an extent that computations in the end become quite simple.", "Therefore, the idea that the fully automatised program will turn the difficult problems mentioned above into tractable or even routine ones holds some promise.", "In another direction, it would be interesting to study from the Calogero-Sutherland side the observed relations between three type I superconformal algebras in dimensions two, three and four, namely $\\mathfrak {sl}(2\|1)\\oplus \\mathfrak {sl}(2\|1)$ , $\\mathfrak {osp}(2\|4)$ and $\\mathfrak {sl}(4\|1)$ .", "These Lie superalgebras have been observed to possess similar properties which allow for them to be studied in a somewhat uniform manner, [46].", "We have already analysed $\\mathfrak {sl}(2\|1)$ in [32] and $\\mathfrak {sl}(4\|1)$ here.", "The remaining algebra, $\\mathfrak {osp}(2\|4)$ , which is the only type I superconformal algebra in three dimensions, is in many respect similar to $\\mathfrak {sl}(4\|1)$ - it has the same number of odd generators and admits the same kind of short representations.", "Its analysis would thus be analogous to the one presented here, with one difference: bosonic blocks in three dimensions are more complicated than in four and cannot be written as finite sums of Gauss' hypergeometric functions.", "It would be interesting to compute the Calogero-Sutherland model associated to $\\mathfrak {osp}(2\|4)$ and compare it to the ones found here and in [32].", "As a final comment, the harmonic analysis approach can be extended to defect SCFTs.", "The corresponding bosonic theory of defect conformal blocks and tensor structures has been constructed in [47].", "Similarly to the non-defect case, the theory admits a natural generalisation to include supersymmetry.", "Since both the theory of [32], [33] and that of [47] are highly structured and have various advantages over the conventional approaches, there is reason to believe that the same will be true for the combination of the two.", "Defects in superconformal theories have a rich structure and diverse applications, so exploring this direction further certainly seems worthwhile.", "Acknowledgements: We thank Aleix Gimenez-Grau, Misha Isachenkov, Denis Karateev, Madalena Lemos, Pedro Liendo, Junchen Rong, Andreas Stergiou and Philine van Vliet for comments and fruitful discussion.", "The work of ES is supported by Ministry of Science and Higher Education of the Russian Federation, agreement № 075-15-2019-1620 and by ERC grant 648630 IQFT.", "VS and IB acknowledge support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2121 ,,Quantum Universe” – 390833306." ], [ "Superconformal Algebras of Type I", "In this appendix we define what is meant by a superconformal algebra and introduce types I and II.", "While some of the discussion of the main text applies equally well to both types, the construction of Casimir equations relies on the algebra being of type I.", "Let $\\mathfrak {g} = \\mathfrak {g}_{(0)} \\oplus \\mathfrak {g}_{(1)}$ be a finite-dimensional Lie superalgebra.", "We say that $\\mathfrak {g}$ is a superconformal algebra if its even part $\\mathfrak {g}_{(0)}$ contains the conformal Lie algebra $\\mathfrak {so}(d+1,1)$ as a direct summand and the odd part $\\mathfrak {g}_{(1)}$ decomposes as a direct sum of spinor representations of $\\mathfrak {so}(d)\\subset \\mathfrak {so}(d+1,1)$ under the adjoint action.", "If this is the case, we denote the dilation generator of the bosonic conformal Lie algebra by $D$ .", "Eigenvalues with respect to $\\text{ad}_D$ give a decomposition of $\\mathfrak {g}$ into the sum of eigenspaces $\\mathfrak {g} = \\mathfrak {g}_{-1}\\oplus \\mathfrak {g}_{-1/2} \\oplus \\mathfrak {g}_0\\oplus \\mathfrak {g}_{1/2}\\oplus \\mathfrak {g}_{1} = \\mathfrak {g}_{-1} \\oplus \\mathfrak {s} \\oplus \\mathfrak {k} \\oplus \\mathfrak {q}\\oplus \\mathfrak {g}_{1}\\ .$ The even part of $\\mathfrak {g}$ is composed of $\\mathfrak {g}_{\\pm 1}$ and $\\mathfrak {k}$ where $\\mathfrak {g}_{-1}={\\mathfrak {n}}$ contains the generators $K_\\mu $ of special conformal transformations while $\\mathfrak {g}_{1} = \\mathfrak {n}$ is spanned by translations $P_\\mu $ .", "Dilations, rotations and internal symmetries make up $ \\mathfrak {k} = \\mathfrak {so}(1,1) \\oplus \\mathfrak {so}(d) \\oplus \\mathfrak {u} \\ .", "$ Generators of $\\mathfrak {g}_{\\pm 1/2}$ , are supertranslations $Q_\\alpha $ and super special conformal transformations $S_\\alpha $ .", "We shall also denote these summands as $\\mathfrak {s}= \\mathfrak {g}_{-1/2}$ and $\\mathfrak {q} = \\mathfrak {g}_{1/2}$ .", "All elements of non-positive degree make up a subalgebra $\\mathfrak {p}$ of $\\mathfrak {g}$ that will be referred to as the parabolic subalgebra $\\mathfrak {p} = \\mathfrak {g}_{-1} \\oplus \\mathfrak {g}_{-1/2} \\oplus \\mathfrak {g}_0\\ .$ There is a unique (connected) corresponding subgroup $P\\subset G$ such that $\\mathfrak {p} = \\mathit {Lie}(P)$ .", "The superspace can be identified with the supergroup of translations and supertranslations.", "It is defined as the homogeneous space $M = G/P$ .", "The above structure is present in any superconformal algebra.", "In this work, we shall mainly consider those $\\mathfrak {g}$ which satisfy an additional condition of being of type I.", "This means that the odd subspace decomposes as a direct sum of two irreducible representations of $\\mathfrak {g}_{(0)}$ under the adjoint action $\\mathfrak {g}_{(1)} = \\mathfrak {g}_+ \\oplus \\mathfrak {g}_-\\ .$ The two modules $\\mathfrak {g}_\\pm $ are then necessarily dual to each other and further satisfy $\\lbrace \\mathfrak {g}_\\pm , \\mathfrak {g}_\\pm \\rbrace = 0 \\ .$ In addition, the bosonic algebra assumes the form $\\mathfrak {g}_{(0)} = [\\mathfrak {g}_{(0)},\\mathfrak {g}_{(0)}] \\oplus \\mathfrak {u}(1)\\ .$ The $\\mathfrak {u}(1)$ summand is a part of the internal symmetry algebra.", "Its generators will be denoted by $R$ .", "All elements in $\\mathfrak {g}_+$ possess the same $R$ -charge.", "The same is true for the elements of $\\mathfrak {g}_-$ , but the $R$ -charge of these elements has the opposite value.", "Elements in the even subalgebra $\\mathfrak {g}_{(0)}$ , on the other hand, commute with $R$ .", "Let us denote the intersections of the subspaces $\\mathfrak {q}$ and $\\mathfrak {s}$ with $\\mathfrak {g}_\\pm $ by $ \\mathfrak {q}_\\pm = \\mathfrak {q}\\cap \\mathfrak {g}_{\\pm } \\quad ,\\ \\mathfrak {s}_\\pm =\\mathfrak {s}\\cap \\mathfrak {g}_\\pm \\ .$ The subspaces $\\mathfrak {q}_\\pm $ and $\\mathfrak {s}_\\pm $ do not carry a representation of $\\mathfrak {g}_{(0)}$ , but they do carry a representation of $\\mathfrak {k}$ .", "This also means that in type I superconformal algebras, the action of $\\mathfrak {k}$ on super-translations decomposes into two or more irreducible representations.", "It turns out that $\\text{dim}(\\mathfrak {q}_\\pm ) = \\text{dim}(\\mathfrak {s}_\\pm ) = \\text{dim}(\\mathfrak {g}_{(1)})/4\\ .$ The full list of type I superconformal algebras, which follows directly from Kac's classification [44], is $\\mathfrak {sl}(2\|\\mathcal {N}),\\ \\mathfrak {sl}(2\|\\mathcal {N}_1)\\oplus \\mathfrak {sl}(2\|\\mathcal {N}_2)\\ \\mathfrak {psl}(2\|2),\\ \\mathfrak {osp}(2\|4),\\ \\mathfrak {sl}(4\|\\mathcal {N}),\\ \\mathfrak {psl}(4\|4)\\ .$ The presented list is that of complexified Lie superalgebras - for different spacetime signatures one considers their various real forms." ], [ "Conventions for $\\mathfrak {sl}(2m\|\\mathcal {N})$", "In this appendix we collect our conventions for the class of Lie superalgebras $\\mathfrak {g} = \\mathfrak {sl}(2m\|\\mathcal {N})$ .", "Before doing that, let us state conventions regarding spinors that are used throughout the main text.", "Greek indices from the middle of the alphabet $\\mu , \\nu ...=1,...,4$ are Euclidean spacetime indices and are raised and lowered using the flat Euclidean metric $g_{\\mu \\nu } = \\delta _{\\mu \\nu }$ .", "The undotted and dotted Greek indices from the beginning of the alphabet $\\alpha ,\\dot{\\alpha }... = 1,2$ are labelling vectors in the $(1/2,0)$ and $(0,1/2)$ representations of $\\mathfrak {so}(4)$ , respectively.", "They are raised and lowered using the Levi-Civita symbol $\\psi _\\alpha = \\varepsilon _{\\alpha \\beta }\\psi ^\\beta ,\\quad \\varepsilon _{\\alpha \\beta } = \\begin{pmatrix}0 & -1\\\\1 & 0\\end{pmatrix}\\ .$ The same rule holds for dotted indices.", "To convert a vector index into a pair of a fundamental and and an anti-fundamental index, we make use of the matrices $(\\gamma _\\mu )^{\\dot{\\alpha }}_{\\ \\alpha } = (-\\sigma _3,-i I_2,\\sigma _1,-\\sigma _2) \\quad \\text{i.e.}", "\\quad x^{\\dot{\\alpha }}_{\\ \\alpha } = \\begin{pmatrix}-x_1 - i x_2 & x_3+i x_4\\\\x_3 - i x_4 & x_1-i x_2\\end{pmatrix}\\ .$ We now give the bracket relations in $\\mathfrak {g}$ .", "The even subalgebra $\\mathfrak {g}_{(0)}$ has brackets $&[D, P_{\\dot{\\alpha }}^{\\ \\beta }] = P_{\\dot{\\alpha }}^{\\ \\beta }\\ , \\ [D, K_{\\alpha }^{\\ \\dot{\\beta }} ]=- K_{\\alpha }^{\\ \\dot{\\beta }}, \\\\[2mm]&[M_{\\alpha }^{\\ \\beta },P_{\\dot{\\gamma }}^{\\ \\delta }] = \\frac{1}{2} \\delta _{\\alpha }^{\\ \\beta } P_{\\dot{\\gamma }}^{\\ \\delta } - \\delta _{\\alpha }^{\\ \\delta }P_{\\dot{\\gamma }}^{\\ \\beta },\\ [M_{\\dot{\\alpha }}^{\\ \\dot{\\beta }},P_{\\dot{\\gamma }}^{\\ \\delta }] = -\\frac{1}{2} \\delta _{\\dot{\\alpha }}^{\\ \\dot{\\beta }} P_{\\dot{\\gamma }}^{\\ \\delta } + \\delta _{\\dot{\\gamma }}^{\\ \\dot{\\beta }}P_{\\dot{\\alpha }}^{\\ \\delta }, \\\\[2mm]&[M_{\\alpha }^{\\ \\beta },K_{\\gamma }^{\\ \\dot{\\delta }}] = -\\frac{1}{2} \\delta _{\\alpha }^{\\ \\beta } K_{\\gamma }^{\\ \\dot{\\delta }} + \\delta _{\\gamma }^{\\ \\beta }K_{\\alpha }^{\\ \\dot{\\delta }},\\ [M_{\\dot{\\alpha }}^{\\ \\dot{\\beta }},K_{\\gamma }^{\\ \\dot{\\delta }}] = \\frac{1}{2} \\delta _{\\dot{\\alpha }}^{\\ \\dot{\\beta }} K_{\\gamma }^{\\ \\dot{\\delta }} - \\delta _{\\dot{\\alpha }}^{\\ \\dot{\\delta }}K_{\\gamma }^{\\ \\dot{\\beta }}, \\\\[2mm]& [M_{\\dot{\\alpha }}^{\\ \\dot{\\beta }}, M_{\\dot{\\gamma }}^{\\ \\dot{\\delta }}] = \\delta _{\\dot{\\gamma }}^{\\ \\dot{\\beta }} M_{\\dot{\\alpha }}^{\\ \\dot{\\delta }} - \\delta _{\\dot{\\alpha }}^{\\ \\dot{\\delta }} M_{\\dot{\\gamma }}^{\\ \\dot{\\beta }},\\ [M_{\\alpha }^{\\ \\beta }, M_{\\gamma }^{\\ \\delta }] = \\delta _{\\gamma }^{\\ \\beta } M_{\\alpha }^{\\ \\delta } - \\delta _{\\alpha }^{\\ \\delta } M_{\\gamma }^{\\ \\beta }, \\\\[2mm]& [K_{\\alpha }^{\\ \\dot{\\beta }},P_{\\dot{\\gamma }}^{\\ \\delta }] = \\delta _{\\dot{\\gamma }}^{\\ \\dot{\\beta }}M_{\\alpha }^{\\ \\delta } - \\delta _{\\alpha }^{\\ \\delta }M_{\\dot{\\gamma }}^{\\ \\dot{\\beta }} - 2 \\delta _{\\dot{\\gamma }}^{\\ \\dot{\\beta }}\\delta _{\\alpha }^{\\ \\delta } D\\ .$ Next, the brackets between even and odd generators read $& [R,Q_{\\dot{\\alpha }}^{\\ J}] = Q_{\\dot{\\alpha }}^{\\ J},\\ [R,Q_{I}^{\\ \\beta }] = -Q_{I}^{\\ \\beta },\\ [R,S_{\\alpha }^{\\ J}] = S_{\\alpha }^{\\ J},\\ [R,S_{I}^{\\ \\dot{\\beta }}] = -S_{I}^{\\ \\dot{\\beta }}, \\\\[2mm]& [D,Q_{\\dot{\\alpha }}^{\\ J}] = \\frac{1}{2} Q_{\\dot{\\alpha }}^{\\ J},\\ [D,Q_{I}^{\\ \\beta }] = \\frac{1}{2} Q_{I}^{\\ \\beta },\\ [D,S_{\\alpha }^{\\ J}] = -\\frac{1}{2} S_{\\alpha }^{\\ J},\\ [D,S_{I}^{\\ \\dot{\\beta }}] = -\\frac{1}{2} S_{I}^{\\ \\dot{\\beta }},\\\\[2mm]& [M_{\\alpha }^{\\ \\beta },Q_{K}^{\\ \\delta }] = \\frac{1}{2} \\delta _{\\alpha }^{\\ \\beta } Q_{K}^{\\ \\delta } - \\delta _{\\alpha }^{\\ \\delta }Q_{K}^{\\ \\beta },\\ [M_{\\dot{\\alpha }}^{\\ \\dot{\\beta }},Q_{\\dot{\\gamma }}^{\\ L}] = -\\frac{1}{2} \\delta _{\\dot{\\alpha }}^{\\ \\dot{\\beta }} Q_{\\dot{\\gamma }}^{\\ L} + \\delta _{\\dot{\\gamma }}^{\\ \\dot{\\beta }}Q_{\\dot{\\alpha }}^{\\ L}, \\\\[2mm]& [M_{\\alpha }^{\\ \\beta },S_{\\gamma }^{\\ L}] = -\\frac{1}{2} \\delta _{\\alpha }^{\\ \\beta } S_{\\gamma }^{\\ L} + \\delta _{\\gamma }^{\\ \\beta }S_{\\alpha }^{\\ L},\\ [M_{\\dot{\\alpha }}^{\\ \\dot{\\beta }},S_{K}^{\\ \\dot{\\delta }}] = \\frac{1}{2} \\delta _{\\dot{\\alpha }}^{\\ \\dot{\\beta }} S_{K}^{\\ \\dot{\\delta }} - \\delta _{\\dot{\\alpha }}^{\\ \\dot{\\delta }}S_{K}^{\\ \\dot{\\beta }}, \\\\[2mm]& [P_{\\dot{\\alpha }}^{\\ \\beta },S_{\\gamma }^{\\ L}] = \\delta _{\\gamma }^{\\ \\beta }Q_{\\dot{\\alpha }}^{\\ L},\\ [P_{\\dot{\\alpha }}^{\\ \\beta },S_{K}^{\\ \\dot{\\alpha }}] = -\\delta _{\\dot{\\alpha }}^{\\ \\dot{\\delta }}Q_{K}^{\\ \\beta },\\ [K_{\\alpha }^{\\ \\dot{\\beta }},Q_{\\dot{\\gamma }}^{\\ L}] = \\delta _{\\dot{\\gamma }}^{\\ \\dot{\\beta }}S_{\\alpha }^{\\ L},\\ [K_{\\alpha }^{\\ \\dot{\\beta }},Q_{K}^{\\ \\delta }] = -\\delta _{\\alpha }^{\\ \\delta }S_{K}^{\\ \\dot{\\beta }}\\ .$ Finally, we give the brackets between odd generators $& \\lbrace Q_{\\dot{\\alpha }}^{\\ J},Q_{I}^{\\ \\beta }\\rbrace = \\delta _{I}^{\\ J} P_{\\dot{\\alpha }}^{\\ \\beta },\\ \\lbrace S_{\\alpha }^{\\ J},S_{I}^{\\ \\dot{\\beta }}\\rbrace = \\delta _{I}^{\\ J} K_{\\alpha }^{\\ \\dot{\\beta }}, \\\\[2mm]& \\lbrace Q_{\\dot{\\alpha }}^{\\ J}, S_{I}^{\\ \\dot{\\beta }}\\rbrace = \\delta _{I}^{\\ J}M_{\\dot{\\alpha }}^{\\ \\dot{\\beta }} + \\delta _{\\dot{\\alpha }}^{\\ \\dot{\\beta }} R_{I}^{\\ J} + \\delta _{I}^{\\ J}\\delta _{\\dot{\\alpha }}^{\\ \\dot{\\beta }}(aD+bR),\\\\[2mm]& \\lbrace Q_{I}^{\\ \\beta }, S_{\\alpha }^{\\ J}\\rbrace = \\delta _{I}^{\\ J}M_{\\alpha }^{\\ \\beta } + \\delta _{\\alpha }^{\\ \\beta } R_{I}^{\\ J} + \\delta _{I}^{\\ J}\\delta _{\\alpha }^{\\ \\beta }(cD+dR)\\ .$ Throughout the text, we use the fundamental, $2m+\\mathcal {N}$ -dimensional representation of $\\mathfrak {g}$ .", "In this representation, the generators are given by $D =\\frac{1}{2}\\text{diag}(I_m,-I_m,0),\\quad R = \\frac{1}{\\mathcal {N}-2m} \\text{diag}(\\mathcal {N}I_m,\\mathcal {N}I_m,2m I_{\\mathcal {N}})\\ .$ $& M_{\\dot{1}}^{\\ \\dot{2}} = E_{\\dot{1}}^{\\ \\dot{2}},\\quad M_{\\dot{2}}^{\\ \\dot{1}} = E_{\\dot{2}}^{\\ \\dot{1}},\\quad M_{\\dot{1}}^{\\ \\dot{1}} = - M_{\\dot{2}}^{\\ \\dot{2}} = \\frac{1}{2} (E_{\\dot{1}}^{\\ \\dot{1}} - E_{\\dot{2}}^{\\ \\dot{2}}),\\\\& M_{1}^{\\ 2} = E_{1}^{\\ 2},\\quad M_{2}^{\\ 1} = E_{2}^{\\ 1},\\quad M_{1}^{\\ 1} = - M_{2}^{\\ 2} = \\frac{1}{2} (E_{1}^{\\ 1} - E_{2}^{\\ 2})\\ .$ $P_{\\dot{\\alpha }}^{\\ \\beta } = E_{\\dot{\\alpha }}^{\\ \\beta },\\quad Q_{\\dot{\\alpha }}^{\\ J} = E_{\\dot{\\alpha }}^{\\ J},\\quad Q_I^{\\ \\beta } = E_I^{\\ \\beta },\\quad K_{\\alpha }^{\\ \\dot{\\beta }} = E_{\\alpha }^{\\ \\dot{\\beta }},\\quad S_{\\alpha }^{\\ J} = E_{\\alpha }^{\\ J},\\quad S_{I}^{\\ \\dot{\\beta }} = E_I^{\\ \\dot{\\beta }}\\ .$ $R_I^{\\ J} = E_I^{\\ J}, \\quad R_I^{\\ I} = \\frac{1}{2} (E_I^{\\ I} - E_{I+1}^{\\ I+1})\\ .$ The standard differential operators representing the action of translation and supertranslation generators on the superspace are $p_{\\dot{\\alpha }}^{\\ \\alpha } = \\partial _{\\dot{\\alpha }}^{\\ \\alpha },\\quad q_{\\dot{\\alpha }}^{\\ I} = \\partial _{\\dot{\\alpha }}^{\\ I} - \\frac{1}{2}\\bar{\\theta }^I_{\\ \\beta }\\partial _{\\dot{\\alpha }}^{\\ \\beta },\\quad q_I^{\\ \\alpha } = \\partial _{I}^{\\ \\alpha }-\\frac{1}{2}\\theta ^{\\dot{\\beta }}_{\\ I}\\partial _{\\dot{\\beta }}^{\\ \\alpha }\\ .$" ], [ "Killing form", "As the standard basis of $\\mathfrak {g}_{(0)}$ we take the generators in the order $\\lbrace X^a\\rbrace = \\lbrace R,D,M_{\\dot{1}}^{\\ \\dot{1}},M_{\\dot{1}}^{\\ \\dot{2}},M_{\\dot{2}}^{\\ \\dot{1}},M_1^{\\ 1},M_1^{\\ 2},M_2^{\\ 1},P_{\\dot{1}}^{\\ 1},P_{\\dot{1}}^{\\ 2},P_{\\dot{2}}^{\\ 1},P_{\\dot{2}}^{\\ 2},K_1^{\\ \\dot{1}},K_1^{\\ \\dot{2}},K_2^{\\ \\dot{1}},K_2^{\\ \\dot{2}}\\rbrace \\ .$ The Killing form is given in terms of the supertrace in the fundamental representation $K^{ab} = \\frac{1}{4}\\text{str}(X^a X^b)\\ .$ In particular, this fixes the the normalisation of the quadratic Casimir as $C_2 = -\\frac{3}{16} R^2 + \\frac{1}{4} D^2 + ...\\ .$" ], [ "Cartan coordinates", "Here we write the relation between primed and unprimed Cartan coordinates.", "The primed bosonic coordinates are equal to the unprimed, so we only need to consider the fermionic ones.", "Let $L$ be the matrix $L(\\varphi ,\\theta ,\\psi ) = \\begin{pmatrix}e^{\\frac{i}{2}(\\varphi + \\psi )}\\cos \\frac{\\theta }{2} & - i e^{\\frac{i}{2}(\\psi - \\varphi )}\\sin \\frac{\\theta }{2}\\\\- i e^{\\frac{i}{2}(\\varphi - \\psi )}\\sin \\frac{\\theta }{2} & e^{-\\frac{i}{2}(\\varphi + \\psi )}\\cos \\frac{\\theta }{2} \\end{pmatrix}\\ .$ Then the relation between two sets of coordinates reads $& \\begin{pmatrix}q_1^{\\prime }\\\\q_2^{\\prime }\\end{pmatrix} = e^{\\kappa - \\frac{1}{2}\\lambda _l}L(\\varphi ^l_2,\\theta ^l_2,-\\psi ^l_1)\\begin{pmatrix}q_1\\\\q_2\\end{pmatrix},\\quad \\begin{pmatrix}q^{\\prime \\dot{1}}\\\\q^{\\prime \\dot{2}}\\end{pmatrix} = e^{\\frac{1}{2}\\lambda _r}L(\\psi _1^r,-\\theta _1^r,\\varphi _1^r)\\begin{pmatrix}q^{\\dot{1}}\\\\q^{\\dot{2}}\\end{pmatrix},\\\\& \\begin{pmatrix}s^{\\prime }_{\\dot{1}}\\\\s^{\\prime }_{\\dot{2}}\\end{pmatrix} = e^{\\kappa + \\frac{1}{2}\\lambda _l}L(\\varphi ^l_1,-\\theta ^l_1,\\psi ^l_1)\\begin{pmatrix}s_{\\dot{1}}\\\\s_{\\dot{2}}\\end{pmatrix},\\quad \\begin{pmatrix}s^{\\prime 1}\\\\s^{\\prime 2}\\end{pmatrix} = e^{-\\frac{1}{2}\\lambda _r}L(\\psi _2^r,\\theta _2^r,\\varphi _2^r)\\begin{pmatrix}s^1\\\\s^2\\end{pmatrix}\\ .$ From these equations, one gets relations between partial derivatives.", "Among others, we have $\\partial _{\\kappa ^{\\prime }} = \\partial _\\kappa - q_\\alpha \\partial _{q_\\alpha }, \\quad \\partial _{\\lambda _l^{\\prime }} = \\partial _{\\lambda _l} + \\frac{1}{2} q_\\alpha \\partial _{q_\\alpha }, \\quad \\partial _{\\lambda _r^{\\prime }} = \\partial _{\\lambda _r} - \\frac{1}{2} q^{\\dot{\\alpha }}\\partial _{q^{\\dot{\\alpha }}},\\\\$ from which follow covariance conditions in the unprimed coordinates $r = \\partial _\\kappa -\\begin{pmatrix}0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & 2\\end{pmatrix}, \\quad 2a = \\partial _{\\lambda _l} + \\frac{1}{2} \\begin{pmatrix}0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & 2\\end{pmatrix}, \\quad 2b = \\partial _{\\lambda _r} - \\frac{1}{2} \\begin{pmatrix}0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0\\\\0 & 0 & 1 & 0\\\\0 & 0 & 0 & 2\\end{pmatrix}\\ .$ Similarly $& 0 = \\partial _{\\varphi ^r_1} + \\frac{i}{2}\\cos \\theta _1^r\\left(q^{\\dot{2}}\\partial _{q^{\\dot{2}}}-q^{\\dot{1}}\\partial _{q^{\\dot{1}}}\\right) + \\frac{1}{2}\\sin \\theta ^r_1\\left(e^{-i\\psi _1^r} q^{\\dot{2}}\\partial _{q^{\\dot{1}}} - e^{i\\psi _1^r}q^{\\dot{1}}\\partial _{q^{\\dot{2}}}\\right),\\\\& 0 = \\partial _{\\psi _1^r} - \\frac{i}{2}q^{\\dot{1}}\\partial _{q^{\\dot{1}}} + \\frac{i}{2}q^{\\dot{2}}\\partial _{q^{\\dot{2}}}, \\quad 0 = \\partial _{\\theta ^r_1} - \\frac{i}{2}e^{-i\\psi ^r_1} q^{\\dot{2}}\\partial _{q^{\\dot{1}}} - \\frac{i}{2}e^{i\\psi ^r_1}q^{\\dot{1}}\\partial _{q^{\\dot{2}}},\\\\& 0 = \\partial _{\\varphi ^l_2} - \\frac{i}{2} q_1 \\partial _{q_1} + \\frac{i}{2} q_2 \\partial _{q_2}, \\quad 0 = \\partial _{\\theta ^l_2} + \\frac{i}{2} e^{-i\\varphi ^l_2}q_2\\partial _{q_1} + \\frac{i}{2} e^{i\\varphi ^l_2} q_1\\partial _{q_2},\\\\& 0 = \\partial _{\\psi ^l_1} + \\frac{i}{2}\\cos \\theta ^l_2 \\left(q_1\\partial _{q_1} - q_2\\partial _{q_2}\\right) + \\frac{1}{2}\\sin \\theta ^l_2\\left(e^{-i\\varphi ^l_2} q_2\\partial _{q_1} - e^{i\\varphi ^l_2} q_1\\partial _{q_2}\\right)\\ .$" ], [ "Representations of $\\mathfrak {sl}(4\|1)$ and {{formula:b729d579-843e-412a-b120-a497544049ae}}", "Parabolic Verma modules of $\\mathfrak {g} = \\mathfrak {sl}(4\|1)$ are labelled by quantum numbers $(j_1,j_2,\\Delta ,r)$ .", "Two half-integer spins $(j_1,j_2)$ specify a finite dimensional representation of the rotation Lie algebra $\\mathfrak {so}(4) = \\mathfrak {su}(2)\\oplus \\mathfrak {su}(2)$ , while $\\Delta $ and $r$ are weights of the generators $D$ and $R$ , respectively.", "The value of the quadratic Casimir in the representation $(j_1,j_2,\\Delta ,r)$ is $C_2(j_1,j_2,\\Delta ,r) = \\frac{1}{2} j_1 (j_1 + 1) + \\frac{1}{2} j_2 (j_2 + 1) + \\frac{1}{4} \\Delta (\\Delta - 2) - \\frac{3}{16} r^2\\ .$ Sometimes, the labels $\\Delta ,r$ are traded for $q,\\bar{q}$ which are defined by $\\Delta = q + \\bar{q}, \\quad r = \\frac{2}{3} (q - \\bar{q})\\ .$ Symmetric traceless representations of $\\mathfrak {so}(4)$ have $j_1 = j_2 = l/2$ and the quadratic Casimir is then $C_2(l,\\Delta ,r) = \\frac{1}{4} l(l+2) + \\frac{1}{4}\\Delta (\\Delta - 2) - \\frac{3}{16} r^2\\ .$ The conformal algebra in four dimensions, $\\mathfrak {g}^b = \\mathfrak {so}(1,5)$ has highest weight representations labelled by $(j_1,j_2,\\Delta )$ .", "The value of the quadratic Casimir in these representations is $C_2(\\Delta ,j_1,j_2) = \\frac{1}{2} j_1 (j_1 + 1) + \\frac{1}{2} j_2 (j_2 + 1) + \\frac{1}{4}\\Delta (\\Delta - 4)\\ .$ Without loss of generality we assume $j_1\\le j_2$ and use alternative notation $l = 2j_1, p = 2(j_2 - j_1)$ .", "Then the Casimir can be rewritten as $C_2(\\Delta ,l,p) = \\frac{1}{4} l^2 + \\frac{1}{8}(p+2)(p+2l) + \\frac{1}{4}\\Delta (\\Delta -4)\\ .$ In particular, in the main text we are interested in the cases $p=0$ and $p=1$ , which give $C_2(\\Delta ,l,0) = \\frac{1}{4} l(l+2) + \\frac{1}{4} \\Delta (\\Delta -4), \\quad C_2(\\Delta ,l,1) = \\frac{1}{4} l(l+3) + \\frac{1}{4}\\Delta (\\Delta -4) + \\frac{3}{8}\\ .$" ], [ "Bosonic conformal blocks", "In this appendix, we write the scalar and seed blocks that are used throughout the main text.", "We define $\\mathcal {F}^{-(a,b;c)}_{\\rho _1,\\rho _2}(z,\\bar{z}) = z^{\\rho _1}\\ _2F_1(a+\\rho _1,b+\\rho _1;c+2\\rho _1;z)\\ \\bar{z}^{\\rho _2}\\ _2F_1(a+\\rho _2,b+\\rho _2;c+2\\rho _2;\\bar{z}) - (z \\leftrightarrow \\bar{z})\\ .$ Scalar blocks in the Calogero-Sutherland gauge read $\\phi ^{a,b}_{\\Delta ,l}(z,\\bar{z}) = \\frac{((z-1)(\\bar{z}-1))^{\\frac{a+b}{2}+\\frac{1}{4}}}{(z \\bar{z})^\\frac{1}{2}} \\mathcal {F}^{-(a,b;0)}_{\\frac{\\Delta +l}{2},\\frac{\\Delta -l-2}{2}}(z,\\bar{z})\\ .$ Let $G = (G_0^{(1)},G_1^{(1)})^T$ and $\\bar{G} = (\\bar{G}_0^{(1)},\\bar{G}_1^{(1)})$ be the seed and conjugate seed conformal blocks from [9] and set $\\Psi ^{a,b}_{\\Delta ,l} = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}1 & -1\\\\1 & 1\\end{pmatrix} S^{-1} G,\\quad \\bar{\\Psi }^{a,b}_{\\Delta ,l} = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}1 & -1\\\\1 & 1\\end{pmatrix} S^{-1} \\bar{G}\\ .$ The matrix $S$ is defined in the appendix A of [35], where in the notation of that paper $x = u_1$ and $y = u_2$ ." ] ]	2011.14116
[ [ "Risk-Monotonicity in Statistical Learning" ], [ "Abstract Acquisition of data is a difficult task in many applications of machine learning, and it is only natural that one hopes and expects the population risk to decrease (better performance) monotonically with increasing data points.", "It turns out, somewhat surprisingly, that this is not the case even for the most standard algorithms that minimize the empirical risk.", "Non-monotonic behavior of the risk and instability in training have manifested and appeared in the popular deep learning paradigm under the description of double descent.", "These problems highlight the current lack of understanding of learning algorithms and generalization.", "It is, therefore, crucial to pursue this concern and provide a characterization of such behavior.", "In this paper, we derive the first consistent and risk-monotonic (in high probability) algorithms for a general statistical learning setting under weak assumptions, consequently answering some questions posed by Viering et al.", "2019 on how to avoid non-monotonic behavior of risk curves.", "We further show that risk monotonicity need not necessarily come at the price of worse excess risk rates.", "To achieve this, we derive new empirical Bernstein-like concentration inequalities of independent interest that hold for certain non-i.i.d.~processes such as Martingale Difference Sequences." ], [ "Acknowledgments", "Acquisition of data is a difficult task in many applications of machine learning, and it is only natural that one hopes and expects the populating risk to decrease (better performance) monotonically with increasing data points.", "It turns out, somewhat surprisingly, that this is not the case even for the most standard algorithms such as empirical risk minimization.", "Non-monotonic behaviour of the risk and instability in training have manifested and appeared in the popular deep learning paradigm under the description of double descent.", "These problems highlight bewilderment in our understanding of learning algorithms and generalization.", "It is, therefore, crucial to pursue this concern and provide a characterization of such behaviour.", "In this paper, we derive the first consistent and risk-monotonic algorithms for a general statistical learning setting under weak assumptions, consequently resolving an open problem [48] on how to avoid non-monotonic behaviour of risk curves.", "Our work makes a significant contribution to the topic of risk-monotonicity, which may be key in resolving empirical phenomena such as double descent." ], [ "Introduction", "Guarantees on the performance of machine learning algorithms are desirable, especially given the widespread deployment.", "A traditional performance guarantee often takes the form of a generalization bound, where the expected risk associated with hypotheses returned by an algorithm is bounded in terms of the corresponding empirical risk plus an additive error which typically converges to zero as the sample size increases.", "However, interpreting such bounds is not always straight forward and can be somewhat ambiguous.", "In particular, given that the error term in these bounds goes to zero, it is tempting to conclude that more data would monotonically decrease the expected risk of an algorithm such as the Empirical Risk Minimizer (ERM).", "However, this is not always the case; for example, [32] showed that increasing the sample size by one, can sometimes make the test performance worse in expectation for commonly used algorithms such as ERM in popular settings including linear regression.", "This type of non-monotonic behaviour is still poorly understood and indeed not a desirable feature of an algorithm since it is expensive to acquire more data in many applications.", "Non-monotonic behaviour of risk curves [43]—the curve of the expected risk as a function of the sample size—has been observed in many previous works [21], [42], [44], [23] (see also [32] for a nice account of the literature).", "At least two phenomena have been identified as being the cause behind such behaviour.", "The first one, coined peaking [29], [22], or double descent according to more recent literature [6], [45], [7], [18], [16], [37], [39], [40], [17], [14], [13], [15], [41], is the phenomenon where the risk curve peaks at a certain sample size $n$ .", "This sample size typically represents the cross-over point from an over-parameterized to under-parameterized model.", "For example, when the number of data points is less than the number of parameters of a model (over-parameterized model), such as Neural Networks, the expected risk can typically increase until the point where the number of data points exceeds the number of parameters (under-parameterized model).", "The second phenomenon is known as dipping [31], [30], where the risk curve reaches a minimum at a certain sample size $n$ and increases after that—never reaching the minimum again even for very large $n$ .", "This phenomenon typically happens when the algorithm is trained on a surrogate loss that differs from the one used to evaluate the risk [9].", "It is becoming more apparent that the two phenomena just mentioned (double descent and dipping) do not fully characterize when non-monotonic risk behaviour occurs.", "[32] showed that non-monotonic risk behaviour could happen outside these settings and formally prove that the risk curve of ERM is non-monotonic in linear regression with prevalent losses.", "The most striking aspect of their findings is that the risk curves in some of the cases they study can display a perpetual “oscillating” behaviour; there is no sample size beyond which the risk curve becomes monotone.", "In such cases, the non-monotonicity of the risk cannot be attributed to the peaking/double descent phenomenon.", "Moreover, they rule out the dipping phenomenon by studying the ERM on the actual loss (not a surrogate loss).", "The findings of [32] stress our current lack of understanding of generalization.", "This was echoed more particularly by [48], who posed the following question as part of a COLT open problem: $\\text{\\emph {How can we provably avoid non-monotonic behaviour?", "}}$" ], [ "Contributions.", "In this work, we answer the above question by presenting algorithms that are both consistent and risk-monotonic under weak assumptions.", "We study the guarantees of our algorithms in the standard statistical learning setting with bounded losses.", "For the first variant of our algorithm, we require a finite Rademacher complexity of the loss composed with the hypothesis class.", "Under this condition, we show that the algorithm is risk-monotonic and provide the rate of convergence of its excess-risk, which matches (up to log-factors) the optimal rate [20], [46], [10], [50] one would get without further assumptions on the loss function or the data-generating distributionAdditional assumptions may include mixability/exp-concavity of the loss [47], [36], and/or the Bernstein/Tsybakov condition on the data-generating distribution [11], [3].. We present a second variant of the algorithm that is also risk-monotonic and achieves faster excess-risk rates under the Bernstein condition [4].", "Our results exceed the expectations expressed by [48] since our algorithms have monotonic risk curves for all sample sizes $n\\ge 1$ as opposed to only for large enough $n$ .", "Our results also refute the conjecture by [19] who predicted that consistent and risk-monotonic predictors in binary classification (so-called smart rules) do not exist." ], [ "Approach Overview.", "Given $n$ samples, the key idea behind our approach is to iteratively generate a sequence of hypotheses $\\hat{h}_1, \\hat{h}_2, \\dots $ leading up to $\\hat{h}_n$ , where we only allow consecutive hypotheses, say $\\hat{h}_{k-1}$ and $\\hat{h}_{k}$ to differ if we can guarantee (with high enough confidence) that the risk associated with $\\hat{h}_k$ is lower than that of $\\hat{h}_{k-1}$ .", "To test for this, we compare the empirical losses of $\\hat{h}_{k-1}$ and $\\hat{h}_{k}$ , taking into account the potential gap between empirical and population expectations.", "Further, we provide a way of estimating this gap using the observed samples." ], [ "Related Works.", "Much work has already been done in efforts to mitigate the non-monotonic behaviour of risk curves [49], [41], [32].", "For example, in the supervised learning setting with the zero-one loss, [8] introduced the “memorize” algorithm that predicts the majority label on any test instance $x$ that was observed during training; otherwise, a default label is predicted.", "[8] showed that this algorithm is risk-monotonic.", "However, it is unclear how their result could generalize beyond the particular setting they considered.", "Risk-monotonic algorithms are also known for the case where the model is correctly specified (see [32] for an overview); in this paper, we do not make such an assumption.", "Closer to our work is that of [49] who, like us, also used the idea of only updating the current predictor for sample size $n$ if it has a lower risk than the predictor for sample size $n-1$ .", "They determine whether this is the case by performing statistical tests on a validation set (or through cross-validation).", "They introduce algorithm wrappers that ensure that the risk curves of the final algorithms are monotonic with high probability.", "However, their results are specialized to the 0-1 loss, and they do not answer the question by [48] on the existence of learners that guarantee a monotonic risk in expectation." ], [ "Outline.", "In Section , we introduce the setting, notation, and relevant definitions.", "In Section , we present our algorithms and state their guarantees.", "Section is dedicated to the proofs of monotonicity and consistency of one of our algorithms.", "All remaining proofs are differed to Appendices and .", "We conclude with a discussion in Section ." ], [ "Preliminaries", "In this section, we present our notation and the relevant definitions needed for the rest of the paper." ], [ "Setting and Notation.", "Let $\\mathcal {Z}$ [resp.", "$\\mathcal {H}$ ] be an arbitrary feature [resp.", "hypothesis] space, and let $\\ell : \\mathcal {H}\\times \\mathcal {Z}\\rightarrow [0,1]$ be a bounded loss function.", "We denote by ${P}(\\mathcal {Z})$ the set of probability measures over $\\mathcal {Z}$ , and by ${F}(\\mathcal {Z}, \\mathbb {R})$ the set of all bounded measurable functions from $\\mathcal {Z}$ to $\\mathbb {R}$ .", "Data is represented by a random variable $Z\\in \\mathcal {Z}$ which we assume to be distributed according to an unknown distribution $P\\in {P}(\\mathcal {Z})$ .", "A learning problem is a tuple $(P, \\ell , \\mathcal {H})$ .", "We assume throughout that $Z_1, Z_2 , \\dots $ are i.i.d.", "copies of $Z$ , and we denote by $\\widehat{P}_n$ the empirical distribution defined by $\\widehat{P}_n(\\cdot ) \\frac{1}{n}\\sum _{i=1}^n _{Z_i}(\\cdot ),\\quad n\\in \\mathbb {N}, \\nonumber $ where $ _{z}(\\cdot )$ represents the Dirac distribution at $z\\in \\mathcal {Z}$ .", "We denote the empirical and population risks by, respectively, $\\widehat{L}_n(h)\\mathbf {E}_{\\widehat{P}_n(Z)}[\\ell (h, Z)] \\quad \\text{and} \\quad L(h)\\mathbf {E}_{P(Z)}[\\ell (h, Z)],\\nonumber $ for $n\\in \\mathbb {N}$ and $h\\in \\mathcal {H}$ .", "To simplify notation, we write $Z_{1:n} (Z_1, \\dots , Z_n)$ , for $n\\in \\mathbb {N}$ , and we let $P^n P \\times \\dots \\times P$ be the product distribution over $\\mathcal {Z}^n$ .", "In what follows, it will be useful to define the following function class: $\\ell \\circ \\mathcal {H}\\lbrace z\\mapsto \\ell (h,z): h\\in \\mathcal {H}\\rbrace .", "\\nonumber $ We adopt standard non-asymptotic big-oh notation; for functions $f,g: \\mathbb {N}\\rightarrow \\mathbb {R}_{\\ge 0}$ , we write $f \\le O(g)$ if there exists some universal constant $C>0$ such that $f(n) \\le C g(n)$ , for all $n\\in \\mathbb {N}$ .", "We also write $f\\le \\widetilde{O}(g)$ , if there exists a $C>0$ such that $f(n)$ is less than $C g(n)$ up to a multiplicative poly-log-factor in $n$ , for all $n\\in \\mathbb {N}$ .", "We now present a series of standard definitions we require: Definition 1 (Rademacher Complexity) Let $Z_1,\\dots ,Z_n$ be i.i.d.", "random variables in some set $\\mathcal {Z}$ .", "The Rademacher complexity of a function class $\\mathcal {F}\\subseteq {F}(\\mathcal {Z}, \\mathbb {R})$ is defined by $\\mathfrak {R}_n(\\mathcal {F}) & \\mathbf {E}\\left[\\sup _{f\\in \\mathcal {F}} \\frac{1}{n} \\sum _{i=1}^n \\sigma _i f(Z_i)\\right],\\nonumber $ where $\\sigma _1, \\dots , \\sigma _n$ are i.i.d.", "Rademacher random variables (i.e.", "$\\mathbf {P}[\\sigma _i=\\pm 1]=1/2$ , $i\\in [n]$).", "Definition 2 (Consistency) An algorithm that for each sample size $n\\in \\mathbb {N}$ and any distribution $P\\in {P}(\\mathcal {Z})$ outputs a hypothesis $\\hat{h}_n \\in \\mathcal {H}$ based on i.i.d.", "samples $Z_1,\\dots , Z_n \\sim P$ , is consistent if the risk $\\mathbf {E}_{P(Z)}[\\ell (\\hat{h}_n, Z)]$ converges to $\\mathcal {H}$ -optimal risk $\\inf _{h \\in \\mathcal {H}} \\left\\lbrace \\mathbf {E}_{P(Z)}[ \\ell (h, Z)]\\right\\rbrace $ in probability as $n\\rightarrow \\infty $ .", "Definition 3 (Risk-Monotonicity) An algorithm that for each sample size $n\\in \\mathbb {N}$ and any distribution $P\\in {P}(\\mathcal {Z})$ outputs a hypothesis $\\hat{h}_n \\in \\mathcal {H}$ based on i.i.d.", "samples $Z_1,\\dots , Z_n \\sim P$ , is risk-monotonic if $\\mathbf {E}_{P^n(Z_{1:n})}[ L(\\hat{h}_n)]] \\le \\mathbf {E}_{P^{n-1}(Z_{1:n-1})}[L(\\hat{h}_{n-1})], \\quad \\text{for all } n \\in \\mathbb {N}.$ The notion of monotonicity we just defined corresponds to the strongest notion of monotonicity considered by [32], [48], which they refer to as global $\\mathcal {Z}$ -monotonicity." ], [ "Risk-Monotonic Algorithms", "In this section, we present risk-monotonic algorithms for a bounded loss $\\ell $ .", "Our first algorithm requires the Rademacher complexity of the function class $\\ell \\circ \\mathcal {H}$ be finite, while for the second algorithm we assume that the hypothesis class $\\mathcal {H}$ is finite and show that risk-monotonicity is possible while achieving fast excess-risk rates under the Bernstein condition; in subsection REF , we discuss how the finiteness-of-$\\mathcal {H}$ assumption may be removed (we make the assumption to simplify the exposition).", "The two algorithms we present in this section are instantiations of Algorithm with different choices of input sequence $(\\delta _k)$ .", "Naturally, both algorithms share the same underlying idea; given a sample size $n$ and the task of generating a hypothesis $\\hat{h}_n = \\hat{h}_n(Z_{1:n})$ , the approach we take is to generate a sequence of hypotheses $\\hat{h}_1=\\hat{h}_1(Z_1), \\hat{h}_2=\\hat{h}_2(Z_{1:2}), \\dots ,$ leading up to $\\hat{h}_n$ , where we only allow two consecutive hypotheses, say $\\hat{h}_{k-1}$ and $\\hat{h}_{k}$ , to differ if we can guarantee that the risk associated with $\\hat{h}_{k}$ is smaller than that of $\\hat{h}_{k-1}$ .", "Doing this ensures that the hypotheses $(\\hat{h}_k)$ have non-increasing risk as a function of $k$ .", "To test whether a hypothesis $\\tilde{h}$ has a smaller risk than $\\hat{h}$ given $n$ sample, we use the fact that $\\left\|\\widehat{L}_n(\\tilde{h}) - L(\\tilde{h})\\right\|\\le \\epsilon _n \\quad \\text{and} \\quad \\left\|\\widehat{L}_n(\\hat{h}) - L(\\hat{h})\\right\| \\le \\epsilon _n, $ with high probability, for some error $\\epsilon _n$ that can be obtained through a concentration argument of the empirical risk.", "Using [eq:fact](eq:fact), we can be sure (with high probability) that the population risk of $\\tilde{h}$ is less than that of $\\hat{h}$ if $\\widehat{L}_n(\\tilde{h}) -\\widehat{L}_n(\\hat{h})\\le - 2\\epsilon _n$ .", "Therefore, we will essentially set $\\delta _n = 2 \\epsilon _n$ , for all $n\\in \\mathbb {N}$ , in Algorithm .", "We will apply two different concentration arguments to the empirical risk, leading to two different “gap” sequences $(\\delta _k)$ , and thus two different algorithm variants.", "The first algorithm relies on a uniform convergence argument (which is why we require a finite Rademacher complexity of $\\ell \\circ \\mathcal {H}$ ).", "We show that this algorithm is risk-monotonic with an excess-risk rate matching the optimal rate one would get without imposing further constraints on the loss or the data-generating distribution.", "The second (risk-monotonic) variant of our algorithm relies on the empirical Bernstein inequality [34].", "The resulting expression of the gap sequence $(\\delta _k)$ allows us to show a fast convergence rate of the excess-risk under the Bernstein condition.", "We now present the first variant of our algorithm: [btp] Greedy Empirical Risk Minimization ($\\mathsf {GERM}$ ).", "[1] Samples $Z_1,\\dots , Z_n$ .", "An arbitrary initial hypothesis $\\hat{h}_0 \\in \\mathcal {H}$ .", "A sequence $(\\delta _k)$ .", "$k=1,\\dots ,n$ Set $\\displaystyle \\tilde{h}_k \\in \\operatornamewithlimits{arg\\,min}_{h\\in \\mathcal {H}} \\sum _{i=1}^k \\ell (h,Z_i)$ .", "blue!70!black// ERM computation $\\displaystyle \\frac{1}{k}\\sum _{i=1}^k\\ell (\\tilde{h}_k,Z_i)- \\frac{1}{k}\\sum _{i=1}^k\\ell (\\hat{h}_{k-1},Z_i)\\le - \\delta _k$ Set $\\hat{h}_{k} = \\tilde{h}_{k}$ .", "Set $\\hat{h}_k = \\hat{h}_{k-1}$ .", "Return $\\hat{h}_n$ ." ], [ "Greedy Empirical Risk Minimization via Uniform Convergence", "For the variant of Algorithm we consider in this subsection, we require the following assumption on the Rademacher complexity of the function class $\\ell \\circ \\mathcal {H}$ : Assumption 1 The function class $\\ell \\circ \\mathcal {H}$ satisfies $\\mathfrak {R}_k(\\ell \\circ \\mathcal {H}) <+\\infty $ , for all $k\\in \\mathbb {N}$ .", "Furthermore, for our algorithm to be consistent, we will also need that $\\mathfrak {R}_k(\\ell \\circ \\mathcal {H}) \\rightarrow 0$ as $k\\rightarrow \\infty $ .", "We will instantiate Algorithm with the sequence $(\\delta _k)$ given by: $\\delta _k 4 \\bar{\\mathfrak {R}}_k + \\sqrt{{2 \\ln (2k)}/{k}}+{2}/{k}, \\ \\ \\text{for all } k\\in \\mathbb {N}, $ where $(\\bar{\\mathfrak {R}}_k)$ are high-probability upper bounds on the Rademacher complexities $(\\mathfrak {R}_k(\\ell \\circ \\mathcal {H}))$ ; in particular, we require $(\\bar{\\mathfrak {R}}_k)$ to satisfy $\\mathbf {P}\\left[\\bar{\\mathfrak {R}}_k \\ge \\mathfrak {R}_k(\\ell \\circ \\mathcal {H}) \\right] \\ge 1 - 1/k, \\ \\text{ for all } k \\in \\mathbb {N}.", "$ A candidate sequence $(\\bar{\\mathfrak {R}}_k)$ that satisfies [eq:rad](eq:rad), and which can be evaluated using empirical samples, is given by $\\bar{\\mathfrak {R}}_k = 0\\vee \\sup _{h \\in \\mathcal {H}} \\left\\lbrace \\frac{1}{k}\\sum _{i=1}^k \\sigma _i \\cdot \\ell (h, Z_i) + \\sqrt{\\frac{2 \\ln (2 k)}{k}}\\right\\rbrace , \\quad \\text{for } k \\in \\mathbb {N}, $ where $\\sigma _1, \\dots , \\sigma _k$ are i.i.d.", "Rademacher random variables.", "The fact that the choice of $(\\bar{\\mathfrak {R}}_k)$ in [eq:emp](eq:emp) satisfies [eq:rad](eq:rad) follows directly from the following proposition whose proof (see Appendix ) is based on a standard application of McDiarmid's inequality: Proposition 1 For i.i.d.", "random variables $Z_1, \\dots , Z_n \\in \\mathcal {Z}$ and i.i.d.", "Rademacher variables $\\sigma _1, \\dots , \\sigma _n$ , we have, for all $\\delta \\in (0,1)$ , $\\mathbf {P}\\left[ \\left\|\\mathfrak {R}_n(\\ell \\circ \\mathcal {H}) - \\sup _{h \\in \\mathcal {H}} \\frac{1}{n}\\sum _{i=1}^n \\sigma _i \\cdot \\ell (h, Z_i) \\right\| \\le \\sqrt{\\frac{2 \\ln (2/\\delta )}{n}} \\right]\\ge 1 - 1/\\delta .\\nonumber $ We now present the guarantees of $\\mathsf {GERM}$ (the proofs are postponed to Section ).", "We begin with risk-monotonicity: Theorem 2 Algorithm with $(\\delta _k)$ as in [eq:deltak](eq:deltak) and $(\\bar{\\mathfrak {R}}_k)$ satisfying [eq:rad](eq:rad) is risk-monotonic according to Definition REF .", "As discussed above, a candidate choice for $\\bar{\\mathfrak {R}}_k$ that satisfies the condition of Theorem REF is given in [eq:emp](eq:emp).", "We remark, however, that $(\\bar{\\mathfrak {R}}_k)$ as selected in [eq:emp](eq:emp) requires the optimization of an empirical objective over $\\mathcal {H}$ , and may be replaced by a more practical choice, given [eq:rad](eq:rad) is satisfied.", "For example, in some settings, a deterministic upper bound on $\\mathfrak {R}_k(\\ell \\circ \\mathcal {H})$ trivially satisfies [eq:rad](eq:rad) and if available, is another viable choice for $\\bar{\\mathfrak {R}}_k$ .", "We now show that our algorithm is also consistent under appropriate conditions: Theorem 3 Let $\\hat{h}_n$ be the output of Algorithm with $(\\delta _k)$ as in [eq:deltak](eq:deltak) and $(\\bar{\\mathfrak {R}}_k)$ satisfying [eq:rad](eq:rad).", "Then, for all $n$ , with probability at least $1-2/n$ , $\\mathbf {E}_{P(Z)}[\\ell (\\hat{h}_n, Z)] \\le \\inf _{h \\in \\mathcal {H}} \\left\\lbrace \\mathbf {E}_{P(Z)}[ \\ell (h, Z)]\\right\\rbrace +12\\bar{\\mathfrak {R}}_n + 3\\sqrt{\\frac{2\\ln (2n)}{n}}+ \\frac{2}{n}.", "$ Theorem REF implies that Algorithm , with the choices of $(\\delta _k)$ and $(\\bar{\\mathfrak {R}}_k)$ as in the theorem statement, is consistent whenever the sequence $(\\bar{\\mathfrak {R}}_k)$ satisfies $\\lim _{k\\rightarrow \\infty }\\bar{\\mathfrak {R}}_k= 0$ in probability.", "We also note that if $\\bar{\\mathfrak {R}}_k \\le O(\\mathfrak {R}_k(\\ell \\circ \\mathcal {H}) + (\\ln (k)/k)^{1/2})$ with high probability (this is the case for $(\\bar{\\mathfrak {R}}_k)$ as in [eq:emp](eq:emp) by Proposition REF ), then the rate achieved in [eq:rate1](eq:rate1) matches (up to log-factors) the optimal excess-risk rateTechnically, the excess-risk lower bound presented in e.g.", "[10] is expressed in terms of the VC-dimension.", "Bounding the Rademacher complexity in terms of the VC-dimension using Sauer's Lemma, see e.g.", "[11], establishes the optimality of the excess-risk rate we present.", "when no additional assumptions are made about the loss or the data-generating distribution [20], [46], [10], [50].", "We formalize this in the next corollary by presenting the excess-risk rate of Algorithm with the particular choice of sequence $(\\bar{\\mathfrak {R}}_k)$ in [eq:emp](eq:emp): Corollary 4 Algorithm with $(\\delta _k)$ as in [eq:deltak](eq:deltak) and $(\\bar{\\mathfrak {R}}_k)$ as in [eq:emp](eq:emp) is risk-monotonic, and its output $\\hat{h}_n$ satisfies, for all $n$ , with probability at least $1-2/n$ , $\\mathbf {E}_{P(Z)}[\\ell (\\hat{h}_n, Z)] \\le \\inf _{h \\in \\mathcal {H}} \\left\\lbrace \\mathbf {E}_{P(Z)}[ \\ell (h, Z)]\\right\\rbrace +12\\mathfrak {R}_n(\\ell \\circ \\mathcal {H}) + 4\\sqrt{\\frac{2\\ln (2n)}{n}}+ \\frac{2}{n}.", "\\nonumber $ The corollary follows directly from Theorems REF and REF , and Proposition REF .", "Remark 1 The risk-monotonicity of Algorithm in both variants we consider in this subsection and the next is ensured by the case distinction in Line .", "Therefore, the ERM hypothesis $\\tilde{h}_k$ in Line of the algorithm may be replaced by any other consistent hypothesis.", "The resulting algorithm will still be risk-monotonic and consistent.", "The corresponding proofs are easily extended to this case.", "Next, we present the second variant of our algorithm that relies on the empirical Bernstein inequality instead of uniform convergence to select the sequence of hypotheses $(\\hat{h}_k)$ .", "Doing this allows for a better convergence rate of the excess-risk under the Bernstein condition." ], [ "Greedy Empirical Risk Minimization via Empirical Bernstein", "In this subsection, we assume that the hypothesis set $\|\\mathcal {H}\|$ is finite and instantiate Algorithm with the sequence $(\\delta _k)$ defined by $\\delta _k \\sqrt{\\frac{2 \\sum _{i=1}^k (\\ell (\\tilde{h}_k,Z_i)- \\ell (\\hat{h}_{k-1},Z_i))^2 \\ln (2k \|\\mathcal {H}\|^2)}{(k-1)^2}} +\\frac{5\\ln (2k\|\\mathcal {H}\|^2)}{k-1} + \\frac{2}{k}, \\quad k \\in \\mathbb {N}, $ where $(\\tilde{h}_k)$ and $(\\hat{h}_k)$ are as in Algorithm .", "One way to dispose of the finiteness assumption of the hypothesis class $\\mathcal {H}$ is to consider a randomized hypothesis, instead of the ERM in Line of Algorithm , and apply existing PAC-Bayesian bounds to determine the appropriate gap sequence $(\\delta _k)$ (the PAC-Bayesian bounds presented in e.g.", "[38] lead to a gap sequence that is compatible with the analysis we carry for the results of this subsection).", "As noted in Remark REF , swapping the ERM for another predictor (randomized or not) need not affect risk-monotonicityIn the case of randomized predictors, the definition of risk-monotonicity in [eq:monotone](eq:monotone) and Line of Algorithm need be modified to incorporate expectations over the randomness of the predictors..", "However, to keep the exposition simple, and to avoid obscuring the main point—which is how to achieve fast excess-risk rates under the Bernstein condition while maintaining risk-monotonicity—we decided to focus on the case where the class $\\mathcal {H}$ is finite.", "The choice of sequence $(\\delta _k)$ in [eq:deltak2](eq:deltak2) will allow us to show a fast rate of convergence of the excess-risk (faster than the one obtained in Corollary REF ) under the following Bernstein condition: Definition 4 [Bernstein Condition] The learning problem $(P, \\ell , \\mathcal {H})$ satisfies the $(\\beta ,B)$ -Bernstein condition, for $\\beta \\in [0,1]$ and $B>0$ , if for all $h\\in \\mathcal {H}$ , $\\nonumber \\mathbf {E}_{P(Z)}\\left[\\left(\\ell (h,Z)- \\ell (h_\\star , Z)\\right)^2\\right] \\le B \\cdot \\mathbf {E}_{P(Z)}\\left[\\ell (h, Z)- \\ell (h_\\star , Z)\\right]^{\\beta },$ where $h_* \\in \\arg \\inf _{h\\in \\mathcal {H}} \\mathbf {E}_{P(Z)}[\\ell (h, Z)]$ is a risk minimizer within the closer of $\\mathcal {H}$ .", "The Bernstein condition [2], [5], [4], [24], [28] essentially characterizes the easiness of the learning problem.", "In particular, it implies that the variance of the excess-loss random variable $\\ell (h,Z)- \\ell (h_\\star , Z)$ vanishes when the risk associate with the hypothesis $h\\in \\mathcal {H}$ gets closer to the $\\mathcal {H}$ -optimal risk $L(h_\\star )$ .", "For bounded loss functions, the Bernstein condition with $\\beta =0$ always holds.", "The Bernstein condition with $\\beta =1$ —also known as the Massart noise condition [33]—corresponds to the easiest learning setting.", "It holds, for example, when $\\mathcal {H}$ is convex and $h \\mapsto \\ell (h,z)$ is exp-concave, for all $z\\in \\mathcal {Z}$ [24], [36] (for more examples of learning settings where a Bernstein condition holds see [28]).", "The case where $\\beta \\in (0,1)$ interpolates naturally between these two extreme cases, where intermediate excess-risk rates are achievable.", "We start by the statement of risk-monotonicity (the proof is postponed to Appendix ): Theorem 5 Algorithm with $(\\delta _k)$ as in [eq:deltak2](eq:deltak2) is risk-monotonic according to Definition REF .", "The proof of the theorem is the same as that of Theorem REF except for minor changes.", "Note that the algorithm does not require any Bernstein condition to ensure risk-monotonicity.", "We only use the condition in the next theorem to show intermediate excess-risk rates: Theorem 6 Let $B>0$ and $\\beta \\in [0,1]$ , and suppose that the $(\\beta , B)$ -Bernstein condition holds.", "Then, the output $\\hat{h}_n$ of Algorithm with $(\\delta _k)$ as in [eq:deltak2](eq:deltak2) satisfies, for all $n$ , with probability at least $1-4/n$ , $\\mathbf {E}_{P(Z)}[\\ell (\\hat{h}_n, Z)] \\le \\inf _{h \\in \\mathcal {H}} \\left\\lbrace \\mathbf {E}_{P(Z)}[ \\ell (h, Z)]\\right\\rbrace + O\\left(\\left( \\frac{\\ln (n\|\\mathcal {H}\|)}{n} \\right)^{\\frac{1}{2-\\beta }} +\\frac{\\ln (n\|\\mathcal {H}\|)}{n} \\right).", "\\nonumber $ We note that the excess-risk rate achieved in Theorem REF interpolates nicely between the fast $\\widetilde{O}( {(\\ln \|\\mathcal {H}\|)}/{n})$ rate achieved when $\\beta =1$ (the easiest learning setting), and the worst-case rate $\\widetilde{O}(\\sqrt{( \\ln \|\\mathcal {H}\|)/n})$ achieved when $\\beta =0$ (this always holds for a bounded loss).", "Crucially, the bound in Theorem REF is, in general, the best one can hope for up to log-factors (see e.g. [28]).", "In the next section, we present the proofs of Theorems REF and REF (those of Theorems REF and REF are postponed to Appendix )." ], [ "Proofs of Theorems ", "In the proofs of Theorems REF and REF , we need the following standard uniform convergence result: Theorem 7 ([11]) Let $n\\in \\mathbb {N}$ and $Z,Z_1,\\dots , Z_n$ be i.i.d.", "random variables such that $Z\\sim P$ .", "Further, let $\\mathcal {F}\\subseteq {F}(\\mathcal {Z},[0,1])$ be a class of functions bounded between 0 and 1.", "Then, for any $\\delta \\in (0,1)$ , with probability at least $1-\\delta $ , $\\sup _{f \\in \\mathcal {F}} \\left\|\\mathbf {E}_{P(Z)}[f(Z)] -\\mathbf {E}_{\\widehat{P}_n(Z)}[f(Z)]\\right\| \\le 2 \\mathfrak {R}_n(\\mathcal {F}) + \\sqrt{\\frac{\\ln (2/\\delta )}{2n}}.\\nonumber $ .", "From this result and a union bound, we arrive at the following useful corollary: Corollary 8 Let $n\\in \\mathbb {N}$ and $Z,Z_1,\\dots , Z_n$ be i.i.d.", "random variables such that $Z\\sim P$ .", "Further, let $\\bar{\\mathfrak {R}}_n>0$ be such that $\\mathbf {P}[\\bar{\\mathfrak {R}}_n \\ge \\mathfrak {R}_n(\\ell \\circ \\mathcal {H})]\\ge 1 - 1/n$ , and define the event $\\mathcal {E}_n \\left\\lbrace \\left\|L(h) -\\widehat{L}_n(h)\\right\| \\le \\epsilon _n, \\text{ for all } h\\in \\mathcal {H}\\right\\rbrace , \\quad \\text{where}\\quad \\epsilon _n 2 \\bar{\\mathfrak {R}}_n + \\sqrt{\\frac{\\ln (2n)}{2n}}.", "\\nonumber $ Then, $\\mathbf {P}[\\mathcal {E}_n] \\ge 1 - 2/n$ .", "[Proof of Theorem REF ] First, note that by linearity of the expectation it suffices to show that $\\mathbf {E}_{P^n(Z_{1:n})}\\left[ L(\\hat{h}_n) - L(\\hat{h}_{n-1}) \\right]\\le 0.\\nonumber $ Moving forward, we define $\\Delta _n L(\\hat{h}_n) - L(\\hat{h}_{n-1})$ .", "Let $\\epsilon _n$ and $\\mathcal {E}_n$ be as in Corollary REF with $\\bar{\\mathfrak {R}}_n$ being the high probability upper bound on $\\mathfrak {R}_n(\\ell \\circ \\mathcal {H})$ that Algorithm has access to ($\\bar{\\mathfrak {R}}_n$ satisfies [eq:rad](eq:rad)).", "By Corollary REF , we have $\\mathbf {P}[\\mathcal {E}_n]\\ge 1-2/n$ .", "Now, by the law of the total expectation, we have $\\mathbf {E}[\\Delta _n] &= \\mathbf {P}[\\lbrace \\hat{h}_{n} \\equiv \\hat{h}_{n-1}\\rbrace ] \\cdot \\mathbf {E}[\\Delta _n \\mid \\lbrace \\hat{h}_{n} \\equiv \\hat{h}_{n-1}\\rbrace ]\\nonumber \\\\ &\\quad +\\mathbf {P}[\\lbrace \\hat{h}_{n} \\lnot \\equiv \\hat{h}_{n-1}\\rbrace ] \\cdot \\mathbf {E}[\\Delta _n \\mid \\lbrace \\hat{h}_{n} \\lnot \\equiv \\hat{h}_{n-1}\\rbrace ] , \\nonumber \\\\ & \\le \\mathbf {E}[\\Delta _n \\mid \\lbrace \\hat{h}_{n} \\lnot \\equiv \\hat{h}_{n-1}\\rbrace ], $ where the last inequality follows by the fact that if $\\hat{h}_n \\equiv \\hat{h}_{n-1}$ , then $\\Delta _n =0$ .", "We note that the expectation in [eq:one](eq:one) is with respect to $P^n(Z_{1:n})$ .", "From [eq:one](eq:one), it follows that $\\mathbf {E}[\\Delta _n] & \\le \\mathbf {P}[\\mathcal {E}_n]\\cdot \\mathbf {E}[\\Delta _n \\mid \\lbrace \\hat{h}_{n} \\lnot \\equiv \\hat{h}_{n-1}\\rbrace \\cap \\mathcal {E}_n] + \\mathbf {P}[\\mathcal {E}_n^{\\mathrm {c}}] \\cdot \\mathbf {E}[\\Delta _n \\mid \\lbrace \\hat{h}_{n} \\lnot \\equiv \\hat{h}_{n-1}\\rbrace \\cap \\mathcal {E}_n^{\\mathrm {c}}], \\nonumber \\\\& \\le \\mathbf {E}[\\Delta _n \\mid \\lbrace \\hat{h}_{n} \\lnot \\equiv \\hat{h}_{n-1}\\rbrace \\cap \\mathcal {E}_n] + 2/n,$ where the last inequality follows by the fact that the loss $\\ell $ takes values in $[0,1]$ and that $\\mathbf {P}[\\mathcal {E}_n^{\\mathrm {c}}] \\le 2/n$ .", "Now, if $\\hat{h}_n \\lnot \\equiv \\hat{h}_{n-1}$ , then by Line of Algorithm , we have $\\widehat{L}_n(\\tilde{h}_n) \\le \\widehat{L}_n(\\hat{h}_{n-1}) -\\delta _n,$ where we recall that $\\delta _n$ is as in [eq:deltak](eq:deltak) (with $k=n$ ).", "Under the event $\\mathcal {E}_n$ , we have $L(\\tilde{h}_n) -\\epsilon _n \\le \\widehat{L}_n(\\tilde{h}_n) \\quad \\text{and}\\quad \\widehat{L}_n(\\hat{h}_{n-1}) \\le L(\\hat{h}_{n-1})+ \\epsilon _n.", "\\nonumber $ This, in combination with [eq:sup](eq:sup), implies that under the event $\\mathcal {E}_n \\cap \\lbrace \\hat{h}_n \\lnot \\equiv \\hat{h}_{n-1}\\rbrace $ , $\\Delta _n +2/n &= L(\\tilde{h}_n)- L(\\hat{h}_{n-1})+2/n, \\nonumber \\\\ &\\le 2/n-\\delta _n +2\\epsilon _n, \\nonumber \\\\& = 0,\\nonumber $ where in the last equality we substituted the expression of $\\delta _n$ in [eq:deltak](eq:deltak) (with $k=n$ ).", "As a result, we have $\\mathbf {E}_{P^n(Z_{1:n})}[ \\Delta _n +2/n \\mid \\lbrace \\hat{h}_{n} \\lnot \\equiv \\hat{h}_{n-1}\\rbrace \\cap \\mathcal {E}_n]\\le 0.$ Combining [eq:two](eq:two) and [eq:last](eq:last) yields the desired result.", "[Proof of Theorem REF ] Let $\\epsilon _n$ and $\\mathcal {E}_n$ be as in Corollary REF with $\\bar{\\mathfrak {R}}_n$ being the high probability upper bound on $\\mathfrak {R}_n(\\ell \\circ \\mathcal {H})$ that Algorithm has access to ($\\bar{\\mathfrak {R}}_n$ satisfies [eq:rad](eq:rad)).", "By Corollary REF , we have $\\mathbf {P}[\\mathcal {E}_n]\\ge 1 -2/n$ .", "For the rest of this proof, we will condition on the event $\\mathcal {E}_n$ .", "By definition of $\\tilde{h}_n$ in Algorithm and Corollary REF , we have $L(\\tilde{h}_{n}) &\\le \\widehat{L}_n(\\tilde{h}_n)+\\epsilon _n , & (\\text{under the event $\\mathcal {E}_n$}) \\nonumber \\\\& =\\inf _{h \\in \\mathcal {H}} \\widehat{L}_n(h)+\\epsilon _n, & (\\text{$\\tilde{h}_n$ is the ERM}) \\nonumber \\\\&\\le \\inf _{h \\in \\mathcal {H}} L(h)+2\\epsilon _n.", "& (\\text{under the event $\\mathcal {E}_n$}) $ We now consider two cases pertaining to the condition in Line of Algorithm :" ], [ "Case 1.", "Suppose that the condition in Line of Algorithm is satisfied for $k=n$ .", "In this case, $L(\\hat{h}_n)] & = L(\\tilde{h}_n), & \\nonumber \\\\& \\le \\inf _{h \\in \\mathcal {H}} L(h) + 2 \\epsilon _n .", "& (\\text{by } [eq:supstuss]{(\\ref {eq:supstuss})}) $" ], [ "Case 2.", "Suppose the condition in Line does not hold for $k=n$ .", "This means $\\hat{h}_{n}\\equiv \\hat{h}_{n-1}$ , and so $L(\\tilde{h}_{n}) &\\ge \\widehat{L}_n(\\tilde{h}_n) -\\epsilon _n,& (\\text{under the event $\\mathcal {E}_n$}) \\nonumber \\\\ & > \\widehat{L}_n(\\hat{h}_n) -\\delta _n -\\epsilon _n, & (\\text{condition in Line \\ref {line:4} is false}) \\nonumber \\\\ & \\ge L(\\hat{h}_n) -\\delta _n -2\\epsilon _n.", "& (\\text{under the event $\\mathcal {E}_n$}) $ Thus, by combining [eq:supstuss](eq:supstuss) and [eq:case1](eq:case1), we get $L(\\hat{h}_{n}) \\le \\inf _{h \\in \\mathcal {H}} L(h)+ 4 \\epsilon _n + \\delta _n.$ From [eq:postsup](eq:postsup) and [eq:case2](eq:case2), we conclude that with probability $\\mathbf {P}[\\mathcal {E}_n]\\ge 1 -2/n$ : $\\mathbf {E}_{P(Z)}[\\ell (\\hat{h}_{n}, Z)] \\le \\inf _{h \\in \\mathcal {H}} \\left\\lbrace \\mathbf {E}_{P(Z)}[ \\ell (h, Z)]\\right\\rbrace + 4 \\epsilon _n + \\delta _n,\\nonumber $ which implies the desired result." ], [ "Discussion and Future Work", "In this paper, we derived the first consistent and risk-monotonic algorithms for a general statistical learning setting with bounded losses." ], [ "Computational Considerations.", "From a computational perspective, the main setback of Algorithm is that returning the final hypothesis $\\hat{h}_n$ requires performing $n$ -ERM computations to evaluate the intermediate hypotheses $(\\tilde{h}_k)_{k\\in [n]}$ .", "Nevertheless, in practice, ERM solutions for sample sizes $k$ and $k+1$ may be close to each other, and this fact may be leveraged to efficiently generate the ERM sequence $(\\tilde{h}_k)$ .", "When the loss $\\ell $ is convex in the first argument, it is possible to efficiently generate the final predictor $\\hat{h}_n$ using tools from online convex optimization.", "However, in general it is unclear whether risk-monotonicity can be achieved without the (greedy) for-loop procedure of Algorithm .", "We note also that if one only wants a decreasing risk after some sample size $s\\in \\mathbb {N}$ then computing the hypotheses $(\\hat{h}_{k})_{k<s}$ is unnecessary.", "In this case, the for-loop in Algorithm need only start at $k=s$ ; the resulting hypotheses would satisfy the monotonicity condition in [eq:monotone](eq:monotone) for all $n\\ge s$ (the proofs of Theorems REF and REF can easily be modified to show this)." ], [ "Extensions.", "Some important questions remain open along the axes of assumptions.", "In particular, can we remove the boundedness condition on the loss while retaining risk-monotonicity?", "Lifting the boundedness assumption may be key in resolving another COLT open problem [25] regarding achievable risk rates of log-loss Bayesian predictorsWe remark that their setting is slightly different from ours, in that their predictors may be improper, i.e.", "not within the hypothesis class $\\mathcal {H}$ ..", "Finally, studying risk-monotonicity in high probability instead of in expectation would also be an interesting future research direction as in some particular settings, monotonicity of individual risk curves may be more desirable [49].", "Our results build foundations for these avenues, which are promising subjects for future work.", "We thank Peter Grünwald for his useful suggestions.", "This work was supported by the Australian Research Council and Data61." ], [ "Estimating the Rademacher Complexity", "In this appendix, we derive an estimator of the Rademacher complexity of a class $\\mathcal {F}$ consisting of functions taking values in the interval $[0,1]$ .", "We will be using McDiarmid's inequality: Theorem 9 ([35]) Let $c>0$ and $X_1, \\dots , X_n$ be independent random variables taking values in a set $A$ , and assume that $f : \\mathcal {A}^n \\rightarrow \\mathbb {R}$ satisfies, $\\sup _{x_1, \\dots , x_n,x_i^{\\prime } \\in \\mathcal {A}} \\left\| f(x_1, \\dots , x_n) - f(x_1, \\dots , x_{i-1}, x_i^{\\prime }, x_{i+1}, \\dots , x_n) \\right\| \\le c,\\quad \\text{for all $1\\le i\\le n$.}", "$ Then, for every $t >0$ , $\\mathbf {P}[\| f(X_1, \\dots , X_n) - \\mathbf {E}[f(X_1, \\dots , X_n)]\| \\ge t] \\le 2 \\exp (-2 t^2/(n c^2))$ .", "Lemma 10 For $\\mathcal {A}\\lbrace \\pm 1 \\rbrace \\times \\mathcal {Z}$ , the function $f : \\mathcal {A}^n \\rightarrow \\mathbb {R}$ defined by $f((\\sigma _1, z_1), \\dots , (\\sigma _n, z_n)) \\sup _{h \\in \\mathcal {H}} \\frac{1}{n} \\sum _{i=1}^n \\sigma _i \\cdot \\ell (h, z_i),\\nonumber $ satisfies [eq:mccond](eq:mccond) with $c = 2/n$ and $x_i(\\sigma _i, z_i)$ , $i \\in [n]$ .", "[Proof] For any $i\\in [n]$ and $x_1 =(\\sigma _1, z_1), \\dots , x_n=(\\sigma _n, z_n), x_i^{\\prime }=(\\sigma _i^{\\prime }, z_i^{\\prime })\\in \\mathcal {A}$ , we have $\\left\| f(x_1, \\dots , x_n) - f(x_1, \\dots , x_{i-1}, x_i^{\\prime }, x_{i+1}, \\dots , x_n) \\right\| & \\le \\sup _{h \\in \\mathcal {H}}\\frac{1}{n} \\left\| \\sigma _i \\cdot \\ell (h, z_i) - \\sigma ^{\\prime }_i \\cdot \\ell (h, z^{\\prime }_i) \\right\|,\\nonumber \\\\& \\le \\frac{2}{n}, \\nonumber $ where the last inequality follows by the fact that the loss $\\ell $ takes values in $[0,1]$ and $\\sigma _i \\in \\lbrace \\pm 1\\rbrace $ , for all $i\\in [n]$ .", "This completes the proof.", "[Proof of Proposition REF] By definition of the Rademacher inequality, we have $\\mathfrak {R}_n(\\ell \\circ \\mathcal {H}) & =\\mathbf {E}\\left[\\sup _{h \\in \\mathcal {H}} \\frac{1}{n}\\sum _{i=1}^n \\sigma _i \\cdot \\ell (h, Z_i)\\right].$ Thus, the desired inequality follow directly by McDiarmid's inequality (Theorem REF ) and Lemma REF ." ], [ "Proof of Theorems ", "Within this appendix, it will be convenient to adopt the ESI notation [28]: Definition 5 (Exponential Stochastic Inequality (ESI) notation) Let $\\eta >0$ , and $X$ , $Y$ be any two random variables with joint distribution $D$ .", "We define $X ^D_{\\eta } \\ Y \\ \\ \\iff \\ \\ X -Y^D_{\\eta } \\ 0 \\ \\ \\iff \\ \\ \\mathbf {E}_{D(X, Y)} \\left[e^{\\eta (X- Y)} \\right] \\le 1.\\nonumber $ In what follows, we denote by $X_h(Z)\\ell (h, Z)- \\ell (h_,Z)$ , $h\\in \\mathcal {H}$ , the excess-loss random variable, where $h_$ is the risk minimizer within $\\mathcal {H}$ .", "Let $ \\Phi _\\eta \\frac{1}{\\eta } \\ln \\mathbf {E}_{P(Z)}\\left[e^{-\\eta X_h(Z)}\\right]$ be the normalized cumulant generating function of $X_h(Z)$ .", "We note that since the loss $\\ell $ takes values in the interval $[0,1]$ , we have $X_{h}(z) \\in [-1,1],\\ \\ \\text{ for all }(h,z)\\in \\mathcal {H}\\times \\mathcal {Z}.$ We now present some existing results pertaining to the excess-loss random variable $X_h(Z)$ and its normalized cumulant generating function, which will be useful in the proof of Theorem REF : Lemma 11 ([28]) Let $h\\in \\mathcal {H}$ , $X_h$ , and $\\Phi _\\eta $ be as above.", "Then, for all $\\eta \\ge 0$ , $\\alpha _\\eta \\cdot X_h(Z)^2 - X_h(Z) _{\\eta }\\Phi _{2\\eta } + \\alpha _\\eta \\cdot \\Phi _{2\\eta }^2, \\quad \\text{where } \\alpha _\\eta \\frac{\\eta }{1+\\sqrt{1+4 \\eta ^2}}.", "\\nonumber $ Lemma 12 ([28]) If the $(\\beta ,B)$ -Bernstein condition holds for $(\\beta ,B) \\in [0,1]\\times \\mathbb {R}_{>0}$ , then for $\\Phi _\\eta $ as in [eq:cumulant](eq:cumulant), it holds that $\\Phi _\\eta \\le (B \\eta )^{\\frac{1}{1-\\beta }}, \\quad \\text{for all $\\eta \\in (0,1]$}.", "\\nonumber $ Lemma 13 ([12]) For $\\Phi _\\eta $ as in [eq:cumulant](eq:cumulant), it holds that $\\Phi _\\eta \\le \\frac{\\eta }{2}, \\quad \\text{for all $\\eta \\in \\mathbb {R}$}.\\nonumber $ Lemma 14 ([12]) The excess-loss random variable $X_h(Z)$ satisfies $X_h(Z) - \\mathbf {E}_{P(z)}[X_h(Z)] _{\\eta } \\eta \\cdot \\mathbf {E}_{P(Z)}[X_h(Z)^2], \\quad \\text{for all $\\eta \\in [0,1]$}.\\nonumber $ In the next subsection, we present a sequence of intermediate results needed in the proofs of Theorems REF and REF ." ], [ "Intermediate Results", "The proofs of all the results in this subsection are postponed to Subsection REF .", "We begin by a result pertaining to the concentration of the empirical risk using the empirical Bernstein inequality [34]: Lemma 15 Let $\\ell : \\mathcal {H}\\times \\mathcal {Z}\\rightarrow [0,1]$ and suppose that $\\mathcal {H}$ is a finite set.", "Further, let $Z,Z_1, \\dots , Z_n$ be i.i.d.", "random variables such that $Z\\sim P$ .", "Then, for all $\\delta \\in (0,1)$ , with probability at least $1-\\delta $ , $L(h) -L(h^{\\prime }) \\le \\widehat{L}_n(h) -\\widehat{L}_n(h^{\\prime }) + \\sqrt{\\frac{2 \\sum _{i=1}^n (\\ell (h,Z_i)- \\ell (h^{\\prime },Z_i))^2 \\ln (2\|\\mathcal {H}\|^2/\\delta )}{(n-1)^2}} +\\frac{5\\ln (2\|\\mathcal {H}\|^2/\\delta )}{n-1},$ for all $h,h^{\\prime } \\in \\mathcal {H}$ .", "The next lemma provides a way of bounding the square-root term in [eq:useful](eq:useful) under the Bernstein condition: Lemma 16 Let $c\\in \\lbrace 2,2^4\\rbrace $ , $\\beta \\in [0,1]$ , $B>0$ , $n\\in \\mathbb {N}$ , and suppose that the $(\\beta ,B)$ -Bernstein condition holds.", "Then, for any $\\delta \\in (0,1)$ , with probability at least $1-\\delta $ , $\\sqrt{\\frac{c \\sum _{i=1}^n (\\ell (h,Z_i)- \\ell (h_\\star ,Z_i))^2 \\ln (2\|\\mathcal {H}\|^2/\\delta )}{(n-1)^2}}\\le \\frac{L(h) -L(h_\\star )}{2}+ O\\left( \\max _{\\beta ^{\\prime } \\in \\lbrace 1,\\beta \\rbrace }\\left( \\frac{\\ln (\|\\mathcal {H}\|/\\delta )}{n} \\right)^{\\frac{1}{2-\\beta ^{\\prime }}} \\right) ,$ for all $h\\in \\mathcal {H}$ , where $h_\\star \\in \\mathrm {arg\\ inf}_{h\\in \\mathcal {H}} L(h)$ .", "Using the previous two lemmas, we derive the excess-risk rate of ERM under the Bernstein condition: Lemma 17 Let $\\beta \\in [0,1]$ and $B>0$ .", "Under the $(\\beta , B)$ -Bernstein condition, the ERM $\\tilde{h}_n \\in \\textrm {arg\\ inf}_{h \\in \\mathcal {H}} \\mathbf {E}_{\\widehat{P}_n(Z)}[\\ell (h,Z)]$ satisfies, with probability at least $1-2/n$ , $L(\\tilde{h}_n) - L(h_\\star ) \\le O\\left(\\left( \\frac{\\ln (n\|\\mathcal {H}\|)}{n} \\right)^{\\frac{1}{2-\\beta }} +\\frac{\\ln (n\|\\mathcal {H}\|)}{n} \\right).\\nonumber $" ], [ "Proofs of Theorems ", "In what follows, it will be useful to define the event $\\mathcal {E}_n \\left\\lbrace L(\\tilde{h}_n)-L(\\hat{h}_{n-1}) \\le \\widehat{L}_n(\\tilde{h}_n) -\\widehat{L}_n(\\hat{h}_{n-1}) +\\epsilon _n \\right\\rbrace ,\\ n \\in \\mathbb {N}, \\\\\\text{where} \\quad \\epsilon _n \\sqrt{\\frac{2 \\sum _{i=1}^n (\\ell (\\tilde{h}_n,Z_i)- \\ell (\\hat{h}_{n-1},Z_i))^2 \\ln (2n\|\\mathcal {H}\|^2)}{(n-1)^2}} +\\frac{5\\ln (2n\|\\mathcal {H}\|^2)}{n-1}, $ and $(\\tilde{h}_{k})$ and $(\\hat{h}_{k})$ are as in Algorithm with the choice of $(\\delta _k)$ in [eq:deltak2](eq:deltak2).", "We note that by Lemma REF , we have $\\mathbf {P}[\\mathcal {E}_n]\\ge 1-1/n\\ge 1 -2/n$ , for all $n$ .", "We begin by the proof of risk-monotonicity: [Proof of Theorem REF] Let $\\Delta _n L(\\hat{h}_{n}) - L(\\hat{h}_{n-1})$ .", "Using the definitions of $\\mathcal {E}_n, \\epsilon _n,$ and $\\delta _n$ as in [eq:event2](eq:event2), [eq:espilon](eq:espilon), and [eq:deltak2](eq:deltak2) (with $k=n$ ), respectively, and following exactly the same steps as in the proof of Theorem REF , we arrive at $\\mathbf {E}[\\Delta _n] \\le \\mathbf {E}[\\Delta _n \\mid \\lbrace \\hat{h}_{n} \\lnot \\equiv \\hat{h}_{n-1}\\rbrace \\cap \\mathcal {E}_n] + 2/n.$ Now, if $\\hat{h}_n \\lnot \\equiv \\hat{h}_{n-1}$ , then by Line of Algorithm , we have $\\widehat{L}_n(\\tilde{h}_n) \\le \\widehat{L}_n(\\hat{h}_{n-1}) -\\delta _n,$ Under the event $\\mathcal {E}_n$ , we have $L(\\tilde{h}_n)-L(\\hat{h}_{n-1}) \\le \\widehat{L}_n(\\tilde{h}_n) -\\widehat{L}_n(\\hat{h}_{n-1}) +\\epsilon _n.", "\\nonumber $ This, in combination with [eq:sup0](eq:sup0), implies that under the event $\\mathcal {E}_n \\cap \\lbrace \\hat{h}_n \\lnot \\equiv \\hat{h}_{n-1}\\rbrace $ , $\\Delta _n +2/n &= L(\\tilde{h}_n)- L(\\hat{h}_{n-1})+2/n, \\nonumber \\\\ &\\le 2/n-\\delta _n +\\epsilon _n, \\nonumber \\\\& = 0,\\nonumber $ where in the last equality we substituted the expression of $\\delta _n$ in [eq:deltak2](eq:deltak2) (with $k=n$ ).", "As a result, $\\mathbf {E}_{P^n(Z_{1:n})}[ \\Delta _n +2/n \\mid \\lbrace \\hat{h}_{n} \\lnot \\equiv \\hat{h}_{n-1}\\rbrace \\cap \\mathcal {E}_n]\\le 0.$ Combining [eq:two2](eq:two2) and [eq:last0](eq:last0) yields the desired result.", "[Proof of Theorem REF] Let $\\epsilon _n$ and $\\delta _n$ be as in [eq:espilon](eq:espilon) and [eq:deltak2](eq:deltak2) (with $k=n$ ), respectively.", "We consider two cases pertaining to the condition in Line of Algorithm :" ], [ "Case 1.", "Suppose that the condition in Line of Algorithm is satisfied for $k=n$ .", "In this case, we have, by Lemma REF , $L(\\hat{h}_n) -L(h_\\star )& = L(\\tilde{h}_n)-L(h_\\star ) \\nonumber \\\\& \\le O\\left(\\left( \\frac{\\ln (n\|\\mathcal {H}\|)}{n} \\right)^{\\frac{1}{2-\\beta }} +\\frac{\\ln (n\|\\mathcal {H}\|)}{n} \\right), $ with probability at least $1-2/n$ ." ], [ "Case 2.", "Now suppose the condition in Line does not hold for $k=n$ .", "This means that $\\hat{h}_{n}\\equiv \\hat{h}_{n-1}$ , and so $\\widehat{L}_n(\\hat{h}_{n}) - \\widehat{L}_n (\\tilde{h}_{n}) \\le \\delta _n.", "$ Using this and Lemma REF , we have, with probability at least $1-1/n$ , $L(\\hat{h}_n) & = L(\\tilde{h}_{n}) + (L(\\hat{h}_{n})- L(\\tilde{h}_{n})),& \\nonumber \\\\& \\le L(\\tilde{h}_{n}) + \\widehat{L}_n(\\hat{h}_{n}) - \\widehat{L}_n (\\tilde{h}_{n}) + \\epsilon _n, & (\\text{Lemma \\ref {lem:bernerror}}) \\nonumber \\\\& \\le L(\\tilde{h}_{n}) + \\delta _n + \\epsilon _n, & \\text{(by [eq:notsatisfied]{(\\ref {eq:notsatisfied})})}\\nonumber \\\\& = L(\\tilde{h}_{n}) + \\sqrt{\\frac{8 \\sum _{i=1}^n (\\ell (\\tilde{h}_n,Z_i)- \\ell (\\hat{h}_{n},Z_i))^2 \\ln (2n\|\\mathcal {H}\|^2)}{(n-1)^2}} +\\frac{10\\ln (2n\|\\mathcal {H}\|^2)}{n-1} +\\frac{2}{n}, & (\\hat{h}_n \\equiv \\hat{h}_{n-1}) \\nonumber \\\\& \\le L(\\tilde{h}_{n}) + \\sqrt{\\frac{16 \\sum _{i=1}^n (\\ell (\\tilde{h}_n,Z_i)- \\ell ( h_\\star ,Z_i))^2 \\ln (2n\|\\mathcal {H}\|^2)}{(n-1)^2}} +\\frac{10\\ln (2n\|\\mathcal {H}\|^2)}{n-1} +\\frac{2}{n} &\\nonumber \\\\& \\quad +\\sqrt{\\frac{16 \\sum _{i=1}^n (\\ell (\\hat{h}_n,Z_i)- \\ell ( h_\\star ,Z_i))^2 \\ln (2n\|\\mathcal {H}\|^2)}{(n-1)^2}}, &$ where to obtain the last inequality, we used the fact that $(a-c)^2 \\le 2(a-b)^2+ 2(b - c)^2$ and $\\sqrt{a+b} \\le \\sqrt{a}+\\sqrt{b}$ for all $a,b,c \\in \\mathbb {R}_{\\ge 0}$ .", "Now, by [eq:case1inter](eq:case1inter), Lemma REF , and a union bound, we obtain, with probability at least $1-2/n$ , $L(\\hat{h}_n)- L(h_\\star ) & \\le L(\\tilde{h}_{n}) -L(h_\\star )+ \\frac{L(\\tilde{h}_n) -L(h_\\star )}{2} + \\frac{L(\\hat{h}_n) -L(h_\\star )}{2} + \\frac{2}{n} & \\nonumber \\\\& \\quad + O\\left( \\max _{\\beta ^{\\prime } \\in \\lbrace 1,\\beta \\rbrace }\\left( \\frac{\\ln (n\|\\mathcal {H}\|)}{n} \\right)^{\\frac{1}{2-\\beta ^{\\prime }}} \\right)+ \\frac{10\\ln (2n\|\\mathcal {H}\|^2)}{n-1},\\nonumber &{which, after re-arranging, becomes}\\frac{L(\\hat{h}_n)- L(h_\\star )}{2} &\\le \\frac{3(L(\\tilde{h}_n) - L(h_\\star ))}{2} + O\\left(\\max _{\\beta ^{\\prime } \\in \\lbrace 1,\\beta \\rbrace }\\left( \\frac{\\ln (n\|\\mathcal {H}\|)}{n} \\right)^{\\frac{1}{2-\\beta ^{\\prime }}} \\right)+ \\frac{10\\ln (2n\|\\mathcal {H}\|^2)}{n-1}+\\frac{2}{n}.&$ Combining [eq:takeit](eq:takeit) with Lemma REF , and applying a union bound, we get, with probability at least $1-4/n$ , $L(\\hat{h}_n)- L(h_\\star ) & \\le O\\left(\\left( \\frac{\\ln (n\|\\mathcal {H}\|)}{n} \\right)^{\\frac{1}{2-\\beta }} +\\frac{\\ln (n\|\\mathcal {H}\|)}{n} \\right).", "$ The combination of [eq:postsup0](eq:postsup0) and [eq:case22](eq:case22) lead to the desired result." ], [ "Proofs of Intermediate Results", "[Proof of Lemma REF ] The proof follows by the empirical Bernstein inequality [34] with the function $f : (\\mathcal {H}\\times \\mathcal {H}) \\times \\mathcal {Z}\\rightarrow [0,1]$ defined by $f((h,h^{\\prime }),z)= \\left(\\ell (h,z)- \\ell (h^{\\prime },z)+1\\right)/2.\\nonumber $ In particular, [34] implies that, for any $\\delta \\in (0,1)$ , with probability at least $1-\\delta $ , $\\mathbf {E}_{P(Z)}[f((h,h^{\\prime }),Z)] \\le \\mathbf {E}_{\\widehat{P}_n(Z)}[f((h,h^{\\prime }),Z)] + \\sqrt{\\frac{2 \\widehat{V}_n\\cdot \\ln (2 \|\\mathcal {H}\|^2/\\delta )}{n-1}} +\\frac{7 \\ln (2 \|\\mathcal {H}\|^2/\\delta )}{3(n-1)}, $ for all $h,h^{\\prime }\\in \\mathcal {H}$ , where $\\widehat{V}_n$ is the sample variance: $\\widehat{V}_n &\\frac{1}{n(n-1)}\\sum _{1\\le i<j\\le n} (f((h,h^{\\prime }),Z_i) - f((h,h^{\\prime }),Z_j))^2, \\nonumber \\\\ &= \\frac{1}{4n(n-1)}\\sum _{1\\le i<j\\le n} (\\ell (h,Z_i) -\\ell (h^{\\prime },Z_i) - \\ell (h,Z_j) +\\ell (h^{\\prime },Z_j))^2, \\nonumber \\\\& = \\frac{n}{4(n-1)} \\mathbf {E}_{\\widehat{P}_n(Z)} \\left[ \\left( \\ell (h,Z)-\\ell (h^{\\prime },Z) - \\mathbf {E}_{\\widehat{P}_n(Z^{\\prime })} [ \\ell (h,Z^{\\prime })-\\ell (h^{\\prime },Z^{\\prime })] \\right)^2 \\right], \\nonumber \\\\& = \\frac{n}{4(n-1)} \\left( \\mathbf {E}_{\\widehat{P}_n(Z)} \\left[ \\left( \\ell (h,Z)-\\ell (h^{\\prime },Z) \\right)^2 \\right] - \\mathbf {E}_{\\widehat{P}_n(Z^{\\prime })} [ \\ell (h,Z^{\\prime })-\\ell (h^{\\prime },Z^{\\prime })]^2 \\right), \\nonumber \\\\& \\le \\frac{n}{4(n-1)} \\mathbf {E}_{\\widehat{P}_n(Z)} \\left[ \\left( \\ell (h,Z)-\\ell (h^{\\prime },Z) \\right)^2 \\right].", "$ Plugging [eq:newvar](eq:newvar) into [eq:empbern](eq:empbern) and multiplying the resulting inequality by 2, leads to the desired inequality.", "[Proof of Lemma REF] To simplify notation, we define $\\varepsilon _n \\sqrt{\\frac{2 \\sum _{i=1}^n (\\ell (\\tilde{h}_n,Z_i)- \\ell (h_\\star ,Z_i))^2 \\ln (2n\|\\mathcal {H}\|^2)}{(n-1)^2}}+\\frac{5\\ln (2n\|\\mathcal {H}\|^2)}{n-1}.\\nonumber $ By Lemma REF , we have, with probability at least $1-1/n$ , $L(\\tilde{h}_n) & = L(h_\\star ) + (L(\\tilde{h}_{n})- L(h_\\star )),& \\nonumber \\\\& \\le L(h_\\star ) + \\widehat{L}_n(\\tilde{h}_{n}) - \\widehat{L}_n (h_\\star ) + \\varepsilon _n, & (\\text{Lemma \\ref {lem:bernerror}}) \\nonumber \\\\& \\le L(h_\\star ) + \\varepsilon _n, & \\text{($\\tilde{h}_n$ is the ERM)}\\nonumber \\\\& = L(h_\\star ) + \\sqrt{\\frac{2 \\sum _{i=1}^n (\\ell (\\tilde{h}_n,Z_i)- \\ell ( h_\\star ,Z_i))^2 \\ln (2n\|\\mathcal {H}\|^2)}{(n-1)^2}} +\\frac{5\\ln (2n\|\\mathcal {H}\|^2)}{n-1}.", "&$ By applying Lemma REF to bound the middle term on the RHS of [eq:case0](eq:case0), we get with probability at least $1-2/n$ , $L(\\tilde{h}_n) & = L(h_\\star ) + \\frac{L(\\tilde{h}_n)- L(h_\\star )}{2} + O\\left( \\max _{\\beta ^{\\prime } \\in \\lbrace 1,\\beta \\rbrace }\\left( \\frac{\\ln (n\|\\mathcal {H}\|)}{n} \\right)^{\\frac{1}{2-\\beta ^{\\prime }}} \\right) + \\frac{5\\ln (2n\|\\mathcal {H}\|^2)}{n-1},\\nonumber \\\\& = L(h_\\star ) + \\frac{L(\\tilde{h}_n)- L(h_\\star )}{2} + O\\left( \\max _{\\beta ^{\\prime } \\in \\lbrace 1,\\beta \\rbrace }\\left( \\frac{\\ln (n\|\\mathcal {H}\|)}{n} \\right)^{\\frac{1}{2-\\beta ^{\\prime }}} \\right).", "$ After rearranging [eq:needarrange](eq:needarrange), we obtain the desired result.", "[Proof of Lemma REF] We use the fact that $\\sqrt{xy} \\le (\\nu x + y/\\nu )/2$ , for all $\\nu >0$ , and apply it to the LHS of [eq:defrn](eq:defrn) with $\\nu = \\frac{\\eta \\cdot (n-1)}{8 n},\\quad x = \\frac{1}{n-1}\\sum _{i=1}^n (\\ell (\\tilde{h}_n,Z_i)- \\ell (\\hat{h}_{n-1},Z_i))^2, \\quad \\text{and}\\quad y = \\frac{c \\ln (2\|\\mathcal {H}\|^2/\\delta )}{n-1},$ which leads to, for all $\\eta >0$ , $r_{n,\\delta }(h)& \\sqrt{\\frac{c \\sum _{i=1}^n (\\ell (h,Z_i)- \\ell (h_\\star ,Z_i))^2 \\ln (2\|\\mathcal {H}\|^2/\\delta )}{(n-1)^2}},\\nonumber \\\\ & \\le \\frac{\\eta }{16 n}\\sum _{i=1}^n (\\ell (h,Z_i)- \\ell (h_\\star ,Z_i))^2 + \\frac{4 n c \\ln (2\|\\mathcal {H}\|^2/\\delta )}{(n-1)^2\\eta }, \\nonumber \\\\& \\le \\frac{\\eta }{16 n}\\sum _{i=1}^n (\\ell (h,Z_i)- \\ell (h_\\star ,Z_i))^2 + \\frac{8c \\ln (2\|\\mathcal {H}\|^2/\\delta )}{(n-1)\\eta }, $ where the last inequality follows by the fact that $n \\le 2 (n-1)$ , for all $n>1$ .", "Let $C_\\beta \\left((1-\\beta )^{1-\\beta } \\beta ^\\beta \\right)^{\\frac{\\beta }{1-\\beta }} +3/2(2B)^{\\frac{1}{1-\\beta }}.$ By combining [eq:prebbern](eq:prebbern) and Lemma REF , we get, for any $\\delta \\in (0,1)$ and $\\eta \\in [0,1/2]$ , with probability at least $1-\\delta $ , $\\forall h \\in \\mathcal {H}, \\ \\ r_{n,\\delta }(h) &\\le (L(h) -L(h_\\star ))/2 + C_{\\beta } \\cdot \\eta ^{\\frac{1}{1-\\beta }}/4 +\\frac{ \\ln (\|\\mathcal {H}\|/\\delta )}{2n \\eta }+ \\frac{8c \\ln (2\|\\mathcal {H}\|^2/\\delta )}{(n-1)\\eta }, \\nonumber \\\\& \\le (L(h) -L(h_\\star ))/2 + C_{\\beta } \\cdot \\eta ^{\\frac{1}{1-\\beta }}/4 + \\frac{(16c +1/2) \\ln (2\|\\mathcal {H}\|^2/\\delta )}{n\\cdot \\eta } , $ where the last inequality follows by the fact that $n\\le 2(n-1)$ and $\|\\mathcal {H}\|\\ge 1$ .", "Now, minimizing the RHS of [eq:allh](eq:allh) over $\\eta \\in (0,1/2)$ and invoking Lemma REF , we get, for any $\\delta \\in (0,1)$ , with probability at least $1-\\delta $ , $r_{n,\\delta }(h) & \\le \\frac{L(h)-L(h_\\star )}{2} + \\frac{C_\\beta \\cdot (3 - 2\\beta )}{4(1 -\\beta )} \\left(\\frac{4(1-\\beta )(16 c +1/2) \\ln (2 \|\\mathcal {H}\|^2/\\delta )}{C_\\beta \\cdot n} \\right)^{\\frac{1}{2-\\beta }}\\nonumber \\\\ & \\quad + \\frac{2(16c +1/2) \\ln (2\|\\mathcal {H}\|^2/\\delta )}{n},\\nonumber \\\\& \\le \\frac{L(h)-L(h_\\star )}{2} + \\frac{C_\\beta ^{\\frac{1-\\beta }{2-\\beta }}\\cdot (3 - 2\\beta )}{4(1 -\\beta )} \\left(\\frac{4(1-\\beta )(16 c +1/2) \\ln (2 \|\\mathcal {H}\|^2/\\delta )}{ n} \\right)^{\\frac{1}{2-\\beta }}\\nonumber \\\\ & \\quad + \\frac{2(16c +1/2) \\ln (2\|\\mathcal {H}\|^2/\\delta )}{n}, $ for all $h\\in \\mathcal {H}$ .", "Combining [eq:new](eq:new) with the fact that $\\beta \\mapsto C_\\beta ^{\\frac{1-\\beta }{2-\\beta }}$ is bounded in $[0,1)$ , we get the desired result." ], [ "Additional Technical Lemmas", "In this subsection, we present additional technical lemmas we require.", "The following two Propositions about ESI are imported from [38]: Proposition 18 [From ESI to Probability] For $\\eta > 0$ , if $X _{\\eta } Y$ then for all $ \\delta \\in (0,1)$ , $X\\le Y + \\frac{\\ln (1/\\delta )}{ \\eta }$ , with probability at least $1-\\delta $ .", "Proposition 19 [ESI Transitivity and Chain Rule] (a) Let $Z_1, \\dots , Z_n$ be any random variables on $\\mathcal {Z}$ (not necessarily independent).", "If for some $(\\gamma _i)_{i \\in [n]} \\in (0,+\\infty )^n$ , $Z_{i} _{\\gamma _i} 0$ , for all $i\\in [n]$ , then $\\sum _{i=1}^n Z_{i} _{\\nu _n} 0, \\quad \\text{where $\\nu _n \\left(\\sum _{i=1}^n \\frac{1}{\\gamma _i}\\right)^{-1}$} \\text{\\ \\ \\emph {(}so if $\\forall i \\in [n], \\gamma _i = \\gamma >0$ then $\\nu _n = \\gamma /n$\\emph {)}}.", "\\nonumber $ (b) Suppose now that $Z_1, \\ldots , Z_n$ are i.i.d.", "and let $f:\\mathcal {Z}\\times \\bigcup _{i=1}^n \\mathcal {Z}^i \\rightarrow \\mathbb {R}$ be any real-valued function.", "If for some $\\eta >0$ , $f(Z_i; z_{<i}) _\\eta 0$ , for all $i \\in [n]$ and all $z_{<i} \\in \\mathcal {Z}^{i-1}$ , then $\\sum _{i=1}^n f(Z_i; Z_{<i}) _{\\eta } 0$ .", "The upcoming lemmas will help us bound the sequence $(\\delta _k)$ in [eq:deltak2](eq:deltak2) under the Bernstein condition.", "Lemma 20 Let $\\beta \\in [0,1]$ , $B>0$ , $n\\in \\mathbb {N}$ , and suppose that the $(\\beta ,B)$ -Bernstein condition holds.", "Then, for any $\\eta \\in [0,1/2]$ and $\\delta \\in (0,1)$ , with probability at least $1-\\delta $ , $\\frac{\\eta }{n} \\sum _{i=1}^n (\\ell (h, Z_i) - \\ell (h_\\star ,Z_i))^2\\le 8 (L(h) -L(h_\\star )) + 4 C_{\\beta } \\cdot \\eta ^{\\frac{1}{1-\\beta }} +\\frac{8 \\ln (\|\\mathcal {H}\|/\\delta )}{n \\eta }, \\forall h\\in \\mathcal {H},\\nonumber $ where $h_\\star \\in \\mathrm {arg\\ inf}_{h\\in \\mathcal {H}} L(h)$ and $C_\\beta \\left((1-\\beta )^{1-\\beta } \\beta ^\\beta \\right)^{\\frac{\\beta }{1-\\beta }} +3/2(2B)^{\\frac{1}{1-\\beta }}$ .", "[Proof] Let $\\delta \\in (0,1)$ and define $X_h(z)\\ell (h, z)- \\ell (h_*,z)$ , for $z\\in \\mathcal {Z}$ .", "Under the $(\\beta , B)$ -Bernstein condition, Lemmas REF -REF imply, for all $\\eta \\in [0,1]$ $\\alpha _\\eta \\cdot X_h(Z_i)^2 _{\\eta } X_h(Z_i)+ 3/2 \\left(2B \\eta \\right)^{\\frac{1}{1-\\beta }},\\quad \\text{where} \\ \\ \\alpha _\\eta \\frac{\\eta }{1+\\sqrt{1+4\\eta ^2}}.", "$ Now, due to the Bernstein inequality (Lemma REF ), we have for all $\\eta \\in [0,1]$ , $X_h(Z_i)& _{\\eta } \\mathbf {E}_{P(Z)}[X_h(Z)] + \\eta \\cdot \\mathbf {E}_{P(Z)}[X_h(Z)^2], \\nonumber \\\\& _{\\eta } \\mathbf {E}_{P(Z)}[X_h(Z)] + \\eta \\cdot \\mathbf {E}_{P(Z)}[X_h(Z)]^{\\beta }, \\quad \\text{(by the Bernstein condition)} \\nonumber \\\\&_{\\eta } 2 \\mathbf {E}_{Z \\sim P}[X_h(Z)]+ c_{\\beta }^{\\frac{\\beta }{1-\\beta }} \\cdot \\eta ^{\\frac{1}{1-\\beta }}, \\quad \\text{ where } c_{\\beta }(1-\\beta )^{1-\\beta }\\beta ^{\\beta }.", "$ The last inequality follows by the fact that $z^{\\beta } = c_{\\beta } \\cdot \\inf _{\\nu >0} \\lbrace z/\\nu + \\nu ^{\\frac{\\beta }{1-\\beta }} \\rbrace $ , for $z\\ge 0$ (in our case, we set $\\nu = c_{\\beta } \\eta $ to get to [eq:individual](eq:individual)).", "By chaining [eq:koolen](eq:koolen) with [eq:individual](eq:individual) using Proposition REF -(a), we get: $\\alpha _\\eta \\cdot X_h(Z_i)^2 &_{\\frac{\\eta }{2}}\\ 2 \\mathbf {E}_{P(Z)}[X_h(Z)] + c_{\\beta }^{\\frac{\\beta }{1-\\beta }} \\cdot \\eta ^{\\frac{1}{1-\\beta }} + 3/2 (2B\\eta )^{\\frac{1}{1-\\beta }}.", "\\nonumber \\\\& _{\\frac{\\eta }{2}}\\ 2 \\mathbf {E}_{P(Z)}[X_h(Z)] +C_\\beta \\cdot \\eta ^{\\frac{1}{1-\\beta }}, \\ \\text{with}\\ \\ C_\\beta c_{\\beta }^{\\frac{\\beta }{1-\\beta }} +3/2(2B)^{\\frac{1}{1-\\beta }}.", "$ Summing [eq:chained](eq:chained) for $i=1,\\dots ,n$ , using Proposition REF -(b), we get $\\alpha _\\eta \\sum _{i=1}^n X_h(Z_i)^2 &_{\\frac{ \\eta }{2}}\\ 2n\\mathbf {E}_{Z \\sim P}[X_h(Z)] + n C_\\beta \\cdot \\eta ^{\\frac{1}{1-\\beta }}.\\nonumber $ Moving to an in probability statement using Proposition REF and applying a union bound for $h \\in \\mathcal {H}$ , we get for all $\\eta \\in [0,1]$ , with probability at least $1-\\delta $ , $\\alpha _\\eta \\sum _{i=1}^n X_h(Z_i)^2 \\le 2n \\mathbf {E}_{Z \\sim P}[X_h(Z)] + n C_\\beta \\cdot \\eta ^{\\frac{1}{1-\\beta }} +\\frac{2\\ln (\|\\mathcal {H}\|/\\delta )}{\\eta }, \\quad \\text{for all }h \\in \\mathcal {H}.", "$ After dividing by $n$ and using the fact that $\\alpha _\\eta \\ge \\eta /4$ , for all $\\eta \\in [0,1/2]$ , [eq:lastone](eq:lastone) leads to the desired result.", "Lemma 21 For $A, B>0$ , we have $\\inf _{\\eta \\in (0,1/2)} \\left\\lbrace A \\eta ^{\\frac{1}{1-\\beta }} + B/\\eta \\right\\rbrace \\le \\frac{A (3-2 \\beta )}{1-\\beta } \\left(\\frac{(1-\\beta ) B}{A}\\right)^{\\frac{1}{2-\\beta }}+2 B.", "$ [Proof] The unconstrained minimizer of the LHS of [eq:obj](eq:obj) is given by $\\eta _\\star \\left(\\frac{(1-\\beta ) B}{A}\\right)^{\\frac{1-\\beta }{2-\\beta }}.", "\\nonumber $ If $\\eta _\\star \\le 1/2$ , then $\\inf _{\\eta \\in (0,1/2]} \\left\\lbrace A \\eta ^{\\frac{1}{1-\\beta }} + B/\\eta \\right\\rbrace \\le A \\eta _\\star ^{\\frac{1}{1-\\beta }} + B/\\eta _\\star = \\frac{A (2- \\beta )}{1-\\beta } \\left(\\frac{(1-\\beta ) B}{A}\\right)^{\\frac{1}{2-\\beta }}.", "$ Now if $\\eta _\\star > 1/2$ , we have $(1/2)^{\\frac{1}{1-\\beta }} < \\left(\\frac{(1-\\beta ) B}{A}\\right)^{\\frac{1}{2-\\beta }}$ , and so, we have $\\inf _{\\eta \\in (0,1/2]} \\left\\lbrace A \\eta ^{\\frac{1}{1-\\beta }} + B/\\eta \\right\\rbrace &\\le A (1/2)^{\\frac{1}{1-\\beta }} + 2 B, \\nonumber \\\\& \\le A \\left(\\frac{(1-\\beta ) B}{A}\\right)^{\\frac{1}{2-\\beta }} + 2 B.$ By combining [eq:case11](eq:case11) and [eq:case21](eq:case21) we get the desired result." ] ]	2011.14126
[ [ "Understanding the velocity distribution of the Galactic Bulge with\n APOGEE and Gaia" ], [ "Abstract We revisit the stellar velocity distribution in the Galactic bulge/bar region with APOGEE DR16 and {\\it Gaia} DR2, focusing in particular on the possible high-velocity (HV) peaks and their physical origin.", "We fit the velocity distributions with two different models, namely with Gauss-Hermite polynomial and Gaussian mixture model (GMM).", "The result of the fit using Gauss-Hermite polynomials reveals a positive correlation between the mean velocity ($\\bar{V}$) and the \"skewness\" ($h_{3}$) of the velocity distribution, possibly caused by the Galactic bar.", "The $n=2$ GMM fitting reveals a symmetric longitudinal trend of $\|\\mu_{2}\|$ and $\\sigma_{2}$ (the mean velocity and the standard deviation of the secondary component), which is inconsistent to the $x_{2}$ orbital family predictions.", "Cold secondary peaks could be seen at $\|l\|\\sim6^\\circ$.", "However, with the additional tangential information from {\\it Gaia}, we find that the HV stars in the bulge show similar patterns in the radial-tangential velocity distribution ($V_{\\rm R}-V_{\\rm T}$), regardless of the existence of a distinct cold HV peak.", "The observed $V_{\\rm R}-V_{\\rm T}$ (or $V_{\\rm GSR}-\\mu_{l}$) distributions are consistent with the predictions of a simple MW bar model.", "The chemical abundances and ages inferred from ASPCAP and CANNON suggest that the HV stars in the bulge/bar are generally as old as, if not older than, the other stars in the bulge/bar region." ], [ "introduction", "Classical bulges or pseudo-bulges [23], commonly found in the central regions of disk galaxies, encode essential information about the formation and evolution history of their host galaxies.", "Near-infrared observations revealed that the inner regions of the Milky Way (MW) host an asymmetric boxy-bulge [62].", "Its morphology suggests that it has a different origin from spherical classical bulges which are likely produced by mergers.", "[8] postulated that the MW bulge is a tilted bar with the near-end of the bar at positive longitudes.", "Later studies confirmed that the MW hosts a stellar bar viewed almost end-on [56], [40], [58], [28], [4], [41], [46], [50], [24], [60], [61], [37], [64].", "However, many of its structural, kinematical and chemical properties are still under debate [6], [49], [3].", "A dynamically cold ($\\sigma _{V}\\approx $ $30\\; {\\rm km}\\;{\\rm s}^{-1}$ ) high-$V_{\\rm GSR}$$V_{\\rm GSR}$ is the line-of-sigh velocity with respect to the Galactic Standard of Rest.", "$\\overline{V}_{GSR}$ and $\\sigma _{V}$ denote the mean and standard deviation of the high-$V_{\\rm GSR}$ peak.", "peak ($\\overline{V}_{GSR}\\approx $ $200\\; {\\rm km}\\;{\\rm s}^{-1}$ ) in the stellar velocity distribution of the Galactic bar/bulge was first reported by [38] using the Apache Point Observatory Galactic Evolution Experiment (APOGEE; [29]) commissioning data (DR10).", "The stars contributing to the high-$V_{\\rm GSR}$ peak are located at a heliocentric distance of $5-10 \\;{\\rm kpc}$ , indicating that they are not likely associated with the halo or the tidal tails of the Sagittarius dwarf galaxy.", "[38] concluded that they are possibly bulge/bar stars orbiting in the Galactic bar potential.", "[33] suggested that stars in bar-supported 2:1 resonant orbits (e.g.", "$x_{1}$ -orbits) could cause a high-$V_{\\rm GSR}$ peak.", "However, using a self-consistent $N$ -body simulation of the Galactic bar, [27] found only a shoulder-like structure instead of a distinct peak at high-$V_{\\rm GSR}$ .", "Using simulations which included star formation, [1] argued that young stars ($\\sim $ 2 Gyrs) on $x_{1}$ -orbits captured by the growing bar, may also produce a high-$V_{\\rm GSR}$ peak.", "The $x_{1}$ -orbits are the most important type of periodic orbits in almost all numerical models of the galactic bar.", "However, in models of highly elongated bars the propeller orbits, which are “distant relatives” in fact, of the $x_{1}$ orbital family play the dominant role [22].", "[31] argued that the propeller orbits can be used to explain the high-$V_{\\rm GSR}$ peak in three in-plane fields.", "In another scenario still, a kpc-scale nuclear disk supported by $x_{2}$ -orbits contributes to the formation of the high-$V_{\\rm GSR}$ peak [12], [13].", "$x_{2}$ -orbits are generally less spatially extended than $x_{1}$ -orbits and their major axis is perpendicular to the bar.", "To better distinguish between these scenarios for the high-$V_{\\rm GSR}$ (hereafter HV) peaks, in this work we study the bulge kinematics and chemical information based on the APOGEE DR16 and Gaia DR2 surveys.", "In our previous work [65], we have studied the velocity distributions in the MW bulge/bar using APOGEE DR13.", "We found 3 fields showing a HV peak at positive longitudes, i.e.", "fields ($l$ , $b$ ) $=$ $(6,0)$ , $(10,-2)$ and $(10,2)$ , in the Galactic bulge region.", "We also found that HV stars show similar chemical abundance ([M/H], [$\\alpha $ /M] and [C/N])These chemical abundances are derived from ASPCAP.", "[M/H] is the calibrated metallicity.", "[$\\alpha $ /M] is the calibrated $\\alpha $ -enhancement.", "[C/N]=[C/Fe]$-$ [N/Fe] is the relative carbon to nitrogen ratio.", "distributions compared to lower $V_{\\rm GSR}$ stars (the main component in the velocity distribution), indicating that these two components may have a similar age composition.", "Moreover, in contradiction with previous predictions [1], we found that both young and old stellar populations show HV features.", "APOGEE-1 observed only the positive longitude side of the MW bulge/bar.", "With the goal of better understanding the origin of the HV peaks, in this paper we revisit the stellar kinematics and chemistry of the Galactic bulge/bar using APOGEE DR16 which contains all observed data from APOGEE-1 and APOGEE-2, and covers both positive and negative longitude fields.", "The paper is organized as follows.", "The sample selection and the stellar parameters estimation are described in §.", "The kinematics of the bulge/bar stars is investigated in §, including the additional information from the radial velocity ($V_{\\rm R}$ ) and tangential velocity ($V_{\\rm T}$ ) in Galactocentric coordinates.", "In § we compare the chemical abundances and age distributions of different sub-samples.", "The results are discussed in § and summarized in §.", "APOGEE is part of the Sloan Digital Sky Survey IV [7], aiming to map the kinematic and chemical structures of the MW [29].", "It has built a database of high-resolution (R$\\sim $ 22,500), near-infrared (1.51 to 1.69 $\\mu $ m H-band) spectra for over $10^{5}$ giant stars.", "Stars in all the major components of the MW were sampled, including the “dust-hidden” parts.", "APOGEE-1 observed the MW bulge/bar mainly at $l > 0^{}$ , whilst APOGEE-2 survey covers the whole bulge/bar region.", "APOGEE-2 North (APOGEE-2N) and APOGEE-2 South (APOGEE-2S) are two complementary components of APOGEE-2.", "APOGEE-2N continues observations at Apache Point Observatory using the Sloan 2.5 m telescope [20] and original APOGEE spectrograph [63].", "Meanwhile APOGEE-2S observes the Southern Hemisphere using a cloned APOGEE spectrograph at the 2.5 m Irénée du Pont telescope at the Las Campanas Observatory (LCO) [9] .", "Gaia [17], [18] is a European space mission to create the most accurate three-dimensional map of the MW.", "The second Gaia data release (Gaia DR2) provides high-precision parallaxes and proper motions for 1.3 billion sources down to magnitude G$\\sim $ 21 mag.", "The proper motions of the cross-matched sampleGaia DR2 cross-matches with APOGEE DR16 have been included in the APOGEE DR16 catalog.", "From APOGEE DR16 we select our main sample (Sample A) by removing stars with low signal-to-noise ratio ($\\rm SNR < 50$ ) and high surface gravity (log$(g)> 3.8$ ) to avoid foreground dwarf contamination.", "To exclude globular clusters, binaries, and variable stars, we remove stars with large radial velocity variations during multiple visits, i.e., ${\\rm VSCATTER} > 1\\; {\\rm km}\\;{\\rm s}^{-1}$ [39], [14].", "In total, Sample A contains $\\sim $ 33000 stars within $-20<l<20$ and $-10<b<0$ , and we use it in §REF to investigate the velocity distributions.", "We build a 6D phase-space catalog with the addition of proper motions from Gaia and stellar distances from StarHorse [45].", "Using coordinates transformation we can compute the Galactocentric radius ($R$ ), azimuth angle ($\\phi $ ), vertical position ($Z$ ), radial velocity ($V_{\\rm R}$ ), tangential velocity ($V_{\\rm T}$ ) and vertical velocity ($V_{\\rm Z}$ ) in the Galactic cylindrical coordinate system.", "To study the bulge 3D kinematics we select a sub-sample (Sample B) from Sample A which contains both StarHorse distances and Gaia proper motion measurements.", "This sub-sample is used to study the proper motion distributions and the correlations between the radial and tangential velocities in §REF and §REF .", "After removing stars with proper motion uncertainties greater than 1.5 mas/yr and stars with $d_{\\rm err}/d > 30\\%$ , $\\sim $ 22000 stars are left in Sample B.", "In order to investigate the chemical abundances and the age distributions of the HV stars, we select another sub-sample (Sample C) from Sample A by requiring ASPCAPFLAG=0 for stars with reliable stellar parameters determined by the ASPCAP pipeline.", "Sample C contains $\\sim $ 22000 stars that are analyzed in § to investigate the chemical and age properties of stars at different velocities.", "Following [65], to calculate $V_{\\rm GSR}$ , $V_{\\rm R}$ and $V_{\\rm T}$ we adopt the solar position at $(X_{\\odot }, Y_{\\odot }, Z_{\\odot }) = (8.34, 0, 0.027) \\;{\\rm kpc}$ [47] and the peculiar motion $(U_{\\odot }, V_{\\odot }, W_{\\odot }) = (11.10, 12.24, 7.25) \\; {\\rm km}\\;{\\rm s}^{-1}$ [48], and the circular speed in the Local Standard of Rest of $239\\; {\\rm km}\\;{\\rm s}^{-1}$ [32].", "In the following analysis, we separate stars in each sample into four groups according to their $V_{\\rm GSR}$ and $R$ .", "For example, in the bulge fields with $l > 0^{}$ , stars are divided into the disk main component ($R \\ge 4 \\;{\\rm kpc}, V_{\\rm GSR} < 180 \\; {\\rm km}\\;{\\rm s}^{-1}$ ), the disk HV component ($R \\ge 4 \\;{\\rm kpc}, V_{\\rm GSR} \\ge 180 \\; {\\rm km}\\;{\\rm s}^{-1}$ ), the bulge main component ($R < 4 \\;{\\rm kpc}, V_{\\rm GSR} < 180 \\; {\\rm km}\\;{\\rm s}^{-1}$ ) and the bulge HV component ($R < 4 \\;{\\rm kpc}, V_{\\rm GSR} \\ge 180 \\; {\\rm km}\\;{\\rm s}^{-1}$ ).", "Accordingly, stars with $l < 0^{}$ can also be divided into these four components with $V_{\\rm GSR}$ separated at $-180 \\; {\\rm km}\\;{\\rm s}^{-1}$ .", "We also compare the observational results to a self-consistent $N$ -body barred galaxy model from [50] (hereafter S10).", "In this model one million particles initially distributed as a pure disk galaxy evolve in a rigid dark matter halo potential.", "A bar quickly forms and buckles in the vertical direction to form the inner boxy/peanut shaped bulge.", "The disk substructures become steady after $\\sim $ 2.4 Gyr.", "As shown in [50], this model reproduces the bulge kinematics and morphology remarkably well." ], [ "Stellar Age Estimation", "To estimate the stellar ages we employ the CANNON, a data driven method to compute stellar parameters and abundances from spectra [34], [35].", "Under the assumptions that the continuum-normalized flux is a polynomial of stellar labels and the spectral model is characterized by the coefficient vector of this polynomial, stellar labels are computed by maximizing the likelihood function.", "[35] used 5 stellar labels including effective tempreture $T_{\\rm eff}$ , surface gravity log$(g)$ , metallicity [Fe/H], $\\alpha $ -enhancement [$\\alpha $ /Fe] and mass.", "In our study we tried several sets of stellar labels and selected the set $\\lbrace $$T_{\\rm eff}$ , log$(g)$ , [M/H], [$\\alpha $ /M], [C/Fe], [N/Fe], mass, age$\\rbrace $ that provides the best age estimation.", "We trained a model using age and mass from APOKASC catalog v6.5.4 [44] and the corresponding spectra and stellar parameters ($T_{\\rm eff}$ , log$(g)$ , [M/H], [$\\alpha $ /M], [C/Fe], [N/Fe]) from APOGEE DR16.", "This sample includes $\\sim $ 6000 stars, with 70% for training and 30% for testing.", "Our test results are shown in the Appendix.", "The stellar parameters in the test sample are well coveredIn the bulge region, the log$(g)$ range of our sample (0$-$ 3.5) is wider than the training sample (1$-$ 3.5), which may introduce additional uncertainties to the estimated stellar parameters for stars with log$(g)$ less than 1.", "But it will not statistically change the age distributions of different subgroups.", "Similar age distributions could be obtained even if we excluded those stars with log$(g) <1$ ., confirming the robustness of our method.", "The typical uncertainty of the stellar age is $\\sim 0.3$ dex, similar to the results in [53] using the CANNON on the same training sample (APOKASC).", "The trained model was then applied to the whole APOGEE DR16 sample." ], [ "Velocity Distributions", "The spatial distribution of Sample A in the Galactic bulge/bar region is shown in Figure REF .", "We group the stars in $2\\times 2$ bins within $\|l\|<21$ and $\|b\|< 11$ (see dashed grid in Fig.", "REF ).", "Fig.", "REF shows the $V_{\\rm GSR}$ distributions inside the grid, where each sub-panel corresponds to a $2\\times 2$ bin.", "We show the histograms and the kernel density estimations (KDEs).", "Different choices of bin widths or kernel bandwidth affect the shape of the histograms/KDEs.", "To balance the sampling noise and at the same time to achieve a better resolution of the density estimation of the velocity distribution, we use the method described in [16] to get the optimal bin width and the method in [52] to get the optimal kernel bandwidth.", "The HV feature is noticeable in several fields (red histograms) near the mid-plane.", "In most of the bulge/bar fields the velocity distributions show a skewed-Gaussian profile consistent with previous studies, e.g., [27] and [65].", "There are two types of HV features, namely the distinctive HV peak and the HV shoulder.", "For the distinctive HV peak, a local minimum is required between the main component and the HV peak in the velocity distribution, as seen in the $(6,0)$ field.", "The HV shoulder is usually a skewed Gaussian profile extending towards HV values, e.g., $(8, 0)$In [27], the HV shoulder was not considered as a smaller cold HV peak enshrouded by the main low velocity peak.", "The `shoulder' corresponds to the skewed asymmetric shape of the distribution, which may be caused by different mechanisms compared to the high velocity peak.", "We follow the same definition in this paper.", "To find the mean velocity and velocity dispersion of the HV feature, we fit the velocity distributions with a Gaussian mixture model (GMM) provided by scikit-learn [42] which implements an expectation-maximization (EM) algorithmAn extra condition $\\eta _{n}>0.01$ is adopted.", "To avoid outliers in GMM fitting we exclude stars with $\\left\|V_{\\rm GSR}\\right\|>400\\; {\\rm km}\\;{\\rm s}^{-1}$ .", "scikit-learn gives random initial coefficient.", "For every fitting we run the procedure for several times to select the best model with the lowest BIC..", "Since a GMM with 2 components has been commonly adopted by previous studies [38], [64], we also initially set the number of Gaussians to $n=$ 2.", "A dynamically cold velocity component is defined to have a small velocity dispersion ($\\sigma <40\\; {\\rm km}\\;{\\rm s}^{-1}$ )The velocity dispersion $\\sigma $ of the cold HV peak reported in [38] is around 30$\\; {\\rm km}\\;{\\rm s}^{-1}$ .", "In this paper we adopt a slightly higher value (40$\\; {\\rm km}\\;{\\rm s}^{-1}$ ) to ensure the detection of possible cold HV peaks in the bulge region..", "The weight ($\\eta $ ), mean ($\\mu $ ), and standard deviation ($\\sigma $ ) for two Gaussian components are shown in Fig.", "REF .", "The main components (with $\\eta >0.5$ ) and the secondary component (with $\\eta < 0.5$ ) show a similar longitudinal trend for both $\\mu $ and $\\sigma $ .", "In most fields, the secondary component has a higher absolute mean velocity than the main component, therefore we dub it the HV component.", "In the $\\sigma _{2}$ distribution we notice a few cold peaks ( $\\sigma _{2}\\sim 40\\; {\\rm km}\\;{\\rm s}^{-1}$ , light blue) at $\|l\|\\sim 6$ .", "From Fig.", "REF we can see that the raw observational fields are not uniformly distributed in the bulge area.", "To reduce the binning effect we also try to group stars by their raw fields and apply the GMM, with the fitting results shown in Fig.", "REF ; fields where we find a cold HV component are colored in red inside the bulge region and in magenta outside the bulge regionField 010-07-C and some other fields around the Sagittarius Dwarf Spheroidal also show a cold HV peak.", "We choose not to include those fields in this study since they are probably not related to the Galactic bulge/bar structure [65].. Binning stars by raw fields, in the bulge region $(\|l\| < 10, \|b\| < 10)$ , with the threshold of $\\sigma <40\\; {\\rm km}\\;{\\rm s}^{-1}$ , only one raw field $(6,0)$ shows a cold peak.", "Several theoretical works [12], [1] predict cold HV features in the disk region delimited by $5<\|l\|<9$ and $\|b\|<1$ (see red boxes in Fig.", "REF ).", "However, only a few stars were observed in the red box at negative longitudes.", "We could only use the data in $(\\pm 5,0)$ and $(\\pm 10,0)$ to discriminate between different models.", "Since all previous models predict the most obvious HV features near the mid-plane ($b= 0^{}$ ), we show the longitudinal profile of the GMM parameters for the stellar velocity distributions near the mid-plane, in Fig.", "REF .", "Only stars within $\|b\|<1$ are included (the same as the $b=0$ bins in Fig.", "REF ).", "In Fig.", "REF a the number of Gaussians is fixed to $n=2$ .", "The components with $\\sigma < 40\\; {\\rm km}\\;{\\rm s}^{-1}$ are marked with a black circle, within $\|b\|<1$ and $\|l\|<15$ there are no cold HV peaksIn Fig.", "REF a the $\\sigma $ value of the secondary peak in bin $(6,0)$ is slightly greater than that in the raw $(6,0)$ field.", "Since the $2\\times 2$ bin covers a larger area than the raw observational $(6,0)$ field, the inclusion of nearby stars might make the secondary peak hotter.. As shown in Fig.", "REF , the main and the secondary components show comparable $\\sigma $ and similar longitudinal trends; in the bulge region ($\|l\|<10$ ) the velocity dispersions ($\\sigma _{1}$ and $\\sigma _{2}$ ) are larger than that of the disk dominated region ($\|l\|>10$ ).", "Fixing the number of Gaussians to $n=2$ gives the second simplest GMM but not the best GMM.", "Therefore we also try to free the number of Gaussians to fit the velocity distribution.", "The best model giving the lowest Bayesian Information Criterion (BIC) value is adopted.", "As shown in Fig.", "REF b, at each longitude, the number of points reflects the number of Gaussians $n$ used in the fitting, where $n$ provides the minimum BIC.", "In the disk region ($\|l\|>10$ ) we usually found $n = 1-3$ while in the bulge/bar region ($-10<l<10$ ) we usually found $n=3-4$ .", "This indicates that the velocity distributions in the MW central bulge are complicated, which could also be considered as an additional evidence that there might not be a distinct “cold HV structure” in the bulge region." ], [ "Gauss-Hermite Polynomials Modeling", "In the previous subsection we saw a longitudinal global trend of the velocity and velocity dispersion, suggesting the high-$V_{GSR}$ feature is possibly related to a global, large scale structure rather than a localized phenomenon.", "Gauss-Hermite polynomials are also commonly used to fit quasi-Gaussian profiles [5], with coefficients $h_{n}$ describe higher order deviations form a Gaussian.", "We adopt the fitting method in [59], [5].", "As it has already been demonstrated in simulations, the relation between the first and the third coefficients ($\\,\\overline{\\!", "{V}}$ and $h_{3}$ ) of a Gauss-Hermite polynomial, which describes the line-of-sight velocity distribution, can be used to characterize the morphology of a galaxy: at large inclination angles these two parameters are positively correlated in the bar region, and negatively correlated in the disk dominated region [10], [21], [26].", "The $\\,\\overline{\\!", "{V}}-h_{3}$ correlation is a useful tool to identify the bar structures in external disk galaxies with IFU observations.", "The correlation has also been seen in the MW bulge [64], [65], although the viewing point is inside the Galaxy at the solar position.", "For a barred galaxy viewed edge-on, the most significant $\\,\\overline{\\!", "{V}}-h_{3}$ correlation would show up in the fields close to mid-plane.", "Considering stars near the mid-plane $(-2<b<2)$ , the longitudinal profiles of the four best-fit Gauss-Hermite coefficients $\\,\\overline{\\!", "{V}}$ , $\\sigma $ , $h_{3}$ and $h_{4}$ are shown in Fig.", "REF in blue.", "Sample A is used, groupted by 4$$ $\\times $ 4$$ bins, within $32<l<-32$ and $-2<b<2$ .", "Blue circles in the figure mark the central longitude of each bin.", "For comparison, we computed the same values for the S10 model on the same ($l, b$ ) grid and the results are shown in red.", "The shaded region in the figure shows the statistical errors calculated by the bootstrapping method.", "Since there are $\\sim $ 500 $-$ 2500 stars in most bins, the statistical errors are small.", "In addition, the stellar position and velocity measurements have small errors, which would not introduce large $h_{3}$ uncertainties.", "However, some other factors could severely affect the $h_{3}$ measurement, e.g., the incompleteness of the observational data or absence of multiple stellar populations in the S10 model.", "Overall, the profiles of the Gauss-Hermite coefficients between the observation and simulation are consistent.", "In S10, $h_{3}$ reaches its local maximum at $l\\sim 11$ at positive longitudes and its local minimum at $l\\sim -9$ at negative longitudes.", "These two extrema are slightly asymmetric.", "This is expected since the turning point in the $h_{3}$ profile might be related to the bar length, and the far-side of the bar ends at smaller $\|l\|$ than the near-side.", "The kinematic profiles of APOGEE stars are therefore consistent with theoretical expectations.", "In the bulge/bar region, $h_{3}$ is positively correlated with $\\,\\overline{\\!", "{V}}$ .", "The turning points are also similar to the model; at negative longitudes (far-side of the bar), the turning point of the $h_{3}$ profile appears at slightly lower $\|l\|$ than the positive longitude (near side of the bar).", "The low values of $h_{3}$ at $l<-15$ are unexpected but they may be explained by the survey incompleteness at larger distances: in simulations all stars along the line-of-sight are included, while the survey is magnitude limited and may not reach the faint/more distant stars.", "Generally speaking, considering the variation of the shape of the velocity distribution, the $h_3$ and $\\,\\overline{\\!", "{V}}$ profiles and correlation are consistent with predictions of barred models.", "The HV feature is a natural consequence of a bar." ], [ "Limitations with $V_{\\rm GSR}$ Distributions", "Identifying a dynamically cold feature (in this case the HV component) may shed light on the orbital composition and the substructures of the Galactic bulge.", "Some earlier works related this cold component to certain resonant orbits [33], [1], [31], while others argued that this cold component may be a signature of a kpc-scale nuclear disk [12].", "However, according to our results, the criterion for identifying cold peaks is relatively subjective.", "The answer to this question heavily depends on the choice of the fitting methods for the velocity distribution, which may vary between different studies.", "To some extent, enlarging the sample size may not be enough to clarify whether there is indeed a peak in the velocity distribution.", "Other stellar properties are also needed.", "Therefore, we consider the additional information of the tangential velocities, chemical abundances and ages in the following sections, which could hopefully provide more constraints on the origin of the HV stars.", "Figure: V GSR -μ l V_{\\rm GSR}-\\mu _{l} distributions in some key fields near the mid-plane; (a)-(e) show the simulated V GSR -μ l V_{\\rm GSR}-\\mu _{l} distribution from S10, of particles in different distances.", "Row (f) shows observational V GSR -μ l V_{\\rm GSR}-\\mu _{l} distribution (Sample B is used here).", "The longitudes decrease from left to right from l=10l = 10 to -20-20.", "The S10 model distances decrease from top to bottom.", "In (a) all the particles in the S10 model along the line-of-sight are included.", "Legends are the same as Fig.", ".", "Without selecting stars on particular orbits, the S10 model could reproduce the observed V GSR -μ l V_{\\rm GSR}-\\mu _{l} distributions in most mid-plane fields.", "The agreement is particularly good if we consider particles in front of the Galactic Center (GC), where most of the APOGEE stars lie, with heliocentric distances between 5 and 7 kpc (see Fig.", "d).Figure: V R -V T V_{\\rm R}-V_{\\rm T} distributions of stars in Sample B in the same region as Fig.", ", binned by grid.", "Legends are the same as Fig.", ".", "V R -V T V_{\\rm R}-V_{\\rm T} distributions of the HV stars (red) show coherence in different latitudes." ], [ "$V_{\\rm GSR}-\\mu _{l}$ Distributions", "The propeller orbits, “distant relatives” of the $x_{1}$ -orbits, were recently suggested to contribute to the observed HV peaks [31].", "The propeller orbit family can match the HV peak both in radial velocity and proper motion at $l=4$ (see Fig.", "15 in [31]).", "Fig.", "REF shows the distribution of our Sample B in the $V_{\\rm GSR}-\\mu _{l}$ space.", "Each sub-panel of the figure corresponds to a 2$\\times 2$ field in the dashed grid of Fig.", "REF .", "It is apparent that HV stars (red points) at positive longitudes show a larger $\|\\mu _{l}\|$ ; while at negative longitudes these HV stars show a smaller/similar $\|\\mu _{l}\|$ .", "These trends are coherent in the whole bulge and are consistent with the trends observed by [31] for the three fields they studied, at $l = 8, 6$ and 4.", "The simulations, contrary to the observational data, are not affected by observational errors and it is therefore easy to group the particles by their distances.", "Thus, we use the S10 model to study the distance dependence of the $V_{\\rm GSR}-\\mu _{l}$ distributions by grouping stars according to their Galactic coordinates and distances, as shown in Fig.", "REF .", "As mentioned in §, the stars are divided into disk and bulge components, which are further separated into the HV and the main components.", "In Fig.", "REF these components are labeled with different colors (see caption).", "We compare the observations (Fig.", "REF f) to the S10 model (Fig.", "REF a-e) in the $V_{\\rm GSR}-\\mu _{l}$ space, where the S10 model particles are further split into different sub-groups according to their heliocentric distances (e.g.", "Fig.", "REF b contains particles with 9 < $d$ < 11 kpc).", "We focus on key raw observational fields in the mid-plane of the bulge region, with $\|b\|<1$ and $\|l\|>4$ which contain a distinct HV component.", "Although fields at $l = -13, -15$ and $-20$ are outside the bulge region, we also include them since they are newly observed by APOGEE-2.", "The particles in the S10 model are selected within the same $(l, b)$ range as the observations.", "Without selecting stars on particular orbits, the S10 model could reproduce the observed $V_{\\rm GSR}-\\mu _{l}$ distributions in most mid-plane fields.", "The agreement is particularly good if we consider particles in front of the Galactic Center (GC), where most of the APOGEE stars lie, with heliocentric distances between 5 and 7 kpc (see Fig.", "REF d).", "The agreement between data and observations is less good in the (4$$ , 0$$ ) field where the APOGEE stars appear to split into two clumps corresponding to the main and HV components as already reported by [31].", "However, the discrepancy between the S10 model and observations in this field could be caused by small number statistics as discussed in the following subsection, where we compare the $V_{\\rm R}-V_{\\rm T}$ distributions between the observations and the model.", "Our result seems to indicate a possible connection between the HV stars and the propeller orbit proposed in [31], especially in the raw $(4,0)$ field .", "However, in this field, the velocity distribution of the full APOGEE sample (Sample A) shows no distinct HV peak (see Fig.", "REF ), which is inconsistent with the prediction of the propeller orbit scenario.", "More observations with increased number of stars and new fields at negative longitudes are needed to confirm this interpretation for the HV peak.", "Figure: V R -V T V_{\\rm R}-V_{\\rm T} distributions of the same bulge fields in the simulation S10 (row a) and the observed data in Sample B (row f).", "Rows (b)-(e) are the same as row (a) but only contain stars with different heliocentric distances.", "The layout and legends are the same with Fig.", ".Figure: Comparison of the V GSR V_{\\rm GSR} distributions between Sample A (initial APOGEE DR16 sample, purple lines) and Sample B (APOGEE DR16 sample with Gaia proper motions and StarHorse distances, black lines).", "The number of stars in each field is also shown in the corresponding panel.", "In most of the fields the velocity distributions are similar, indicating that the cross-matching with Gaia would not change the velocity profiles much." ], [ "$V_{\\rm R}-V_{\\rm T}$ Distributions", "Using the distance information provided by StarHorse, we can convert the $Gaia$ DR2 proper motions to the velocity space, i.e.", "we compute the radial velocity $V_{\\rm R}$ and tangential velocity $V_{\\rm T}$ with respect to the Galactic Center.", "The $V_{\\rm R}-V_{\\rm T}$ distributions are shown in Fig.", "REF .", "In this figure the disk main component (green) shows a smaller velocity dispersion than the bulge/bar main component (blue), as the bulge is generally hotter than the disk.", "The distributions of HV stars (magenta and red) show longitudinal dependence: in the mid-plane ($b=0$ ), the HV stars generally have negative $V_{\\rm R}$ for $l > 0$ and positive $V_{\\rm R}$ for $l < 0$ .", "However, in fields above or below the mid-plane, HV stars follow a banana-like distribution in $V_{\\rm R}-V_{\\rm T}$ space in both positive and negative $V_{\\rm R}$ .", "Both peak-like and shoulder-like HV features could be seen in Fig.", "REF .", "If stars in the peak-like feature have different origins (e.g.", "due to different kinds of resonant orbits) compared to those in the shoulder-like feature, they may show distinct features in the $V_{\\rm R}-V_{\\rm T}$ space.", "However, the $V_{\\rm R}-V_{\\rm T}$ distributions of the HV stars in different bulge/bar fields are quite similar, which may imply that HV features share the same origin regardless of the shape of the distributions (i.e.", "peak-like or shoulder-like).", "Figure: Errors in distance, V R V_{\\rm R}, and V T V_{\\rm T}, as a function of distance for Sample B.", "Legends are the same as Fig.", ".", "Errors at positive longitudes are generally smaller than that at negative longitudes.Fig.", "REF is similar to Fig.", "REF but now showing the comparison between the observations and the S10 model in the $V_{\\rm R}-V_{\\rm T}$ space.", "Within the bulge/bar dominated region $\|l\|<10$ , the disk particles have smaller velocity dispersions along the radial and tangential directions than the bulge/bar component.", "The disk particles constitute a larger fraction of the sample in the fields at $l = -13, -15$ and $-20$ outside the bulge dominated region.", "In these fields, the disk and the bulge/bar components have similar velocity dispersions.", "Fig.", "REF a, the HV particles showing a banana-shaped distribution can be easily understood since only stars with a large $V_{\\rm R}$ or $V_{\\rm T}$ can contribute to the HV.", "As the distance varies, the $V_{\\rm R}$ changes sign because its definition is relative to Galactic Center; as $l$ varies, $V_{\\rm R}$ changes sign because of the symmetry of bar orbits.", "For fields at positive longitudes, the location of the HV stars in the $V_{\\rm R}-V_{\\rm T}$ plane (red points) gradually shifts from bottom left to bottom right with increasing distance, with a turning point at around $d=8\\;{\\rm kpc}$ .", "This could explain the observed different $V_{\\rm R}-V_{\\rm T}$ distribution of the HV stars in Fig.", "REF .", "In mid-plane fields, most HV stars appear only on one side (at positive or negative $V_{\\rm R}$ ).", "However, in off-plane fields the distribution of HV stars becomes roughly symmetric and banana-like and they can appear at both positive and negative $V_{\\rm R}$ .", "This is probably due to stronger dust extinction in the mid-plane, which obscures distant stars.", "Fig.", "REF f shows the APOGEE $V_{\\rm R}-V_{\\rm T}$ distributions in the same fields as the other rows.", "Similar to Fig.", "REF d, HV stars at positive longitudes generally show negative $V_{\\rm R}$ .", "This is consistent with our arguments that at positive longitudes most of the HV stars are within 8 kpc from the Sun, while at negative longitudes the distance distributions are broader and no clear trend can be observed.", "In Fig.", "REF and Fig.", "REF Sample B is used, as a sub-sample of Sample A.", "Sample B contains only stars with $Gaia$ DR2 proper motions and StarHorse distances, and acceptable errors on proper motion and distance.", "In Fig.", "REF we compare the $V_{\\rm GSR}$ distributions of Sample A (purple) and Sample B (black).", "In most fields, they are very similar, confirming that Sample B is not strongly affected by selection effects.", "Stars in Sample A could reach a StarHorse distance of 10 kpc.", "In most fields, the distance distribution of Sample B does not change much and can still reach 10 kpc, indicating that Sample B extends far enough to reach the bulge/bar also at negative longitudes.", "The $V_{\\rm R}-V_{\\rm T}$ distributions of the S10 model at $l = -15$ and $-20$ shown in Fig.", "REF a only contains a small fraction of bulge/bar stars (blue and red), with the majority as disk stars (green and magenta).", "The corresponding APOGEE fields in Fig.", "REF f also show that the disk is the dominant population at these longitudes.", "Therefore, the HV feature in these two fields might not be related to the bulge/bar.", "Sample A in these two fields shows a HV peak.", "After cross-matching with Gaia and removing stars with large distance uncertainty, the HV components in the velocity distributions become less noticeable as shown in the right two panels in Fig.", "REF .", "This means that after the cross-match more HV stars may have been removed from the sample compared to the main component.", "Therefore it is difficult to draw any further insights on the origin of the HV stars from the $V_{\\rm R}-V_{\\rm T}$ distributions." ], [ "Uncertainties", "When studying the $V_{\\rm R}-V_{\\rm T}$ distributions we exclude stars with large proper motion errors ($>$ 1.5 mas/yr) and large relative distance errors ($d_{\\rm err}/d>0.3$ ).", "The proper motion uncertainties and the cross-terms are provided by $Gaia$ DR2, the distance errors by StarHorse and the radial velocity errors by APOGEE-2.", "We compute the uncertainties for the Galactocentric velocities along the radial and tangential directions, propagating the covariance matrix from one coordinate system to another.", "For the same sub-sample in Fig.", "REF , we show the heliocentric distance errors from StarHorse and the newly derived $V_{\\rm R}$ and $V_{\\rm T}$ errors in Fig.", "REF , as a function of StarHorse distance.", "Within $\|l\|<5$ , the $V_{\\rm R}$ errors are small since they are mainly contributed by the $V_{\\rm los}$ errors which are very small.", "With increasing $\|l\|$ , the $V_{\\rm R}$ uncertainties become larger, reaching up to 30 $\\; {\\rm km}\\;{\\rm s}^{-1}$ .", "The $V_{\\rm T}$ errors show the opposite trend: for small $\|l\|$ , $V_{\\rm T,err}$ are large.", "From the analysis of the velocity errors we can conclude that $V_{\\rm R, err}$ is acceptable in the bulge region.", "Note that velocity errors at positive longitudes are smaller than at negative longitudes, which might also explain why clear trends of HV stars in the $V_{\\rm R}-V_{\\rm T}$ distributions could only be seen in the near side of the bar.", "In this section we study the chemical abundance distributions of bulge/bar stars at different velocities in fields along the mid-plane ($b=0$ ).", "As mentioned in $§$ we selected Sample C from the main Sample A with the ASPCAP stellar parameters available (ASPCAPFLAG=0).", "Stars are then grouped by their $V_{\\rm GSR}$ and Galactocentric radius $R$ and further divided into four components (i.e., disk main, bulge main, disk HV and bulge HV components) as described in $§$ .", "Similar to [65], we also use the chemical abundances ([M/H], [$\\alpha $ /M] and [C/N]) as age proxies.", "The metallicity and $\\alpha $ -abundance trace the age from the chemical evolution aspect, while [C/N] traces the age from the stellar evolution aspect [30].", "Massive stars (which are younger in general) would have a lower [C/N] after their first dredge-up phase during the stellar evolution process.", "This relation is valid for our sample as most stars have experienced their first dredge-up phase since their log$\\,(g)<3.5$ .", "A recent study by [51] argued that for metal-poor ([Fe/H]<$-$ 0.5), evolved (log$(g)<2$ ) red giants, [C/N] may not be a reliable age indicator, but our conclusion here should not be affected since such stars only contribute a small fraction to our sample ($\\sim $ 4%).", "To study the chemical properties of stars with different velocities, we select stars from Sample C. For the same fields in Fig.", "REF we plot the distributions of the three chemical parameters of the four components (see Fig.", "REF ).", "Our sample covers a large distance range (up to 10 kpc) from the Sun.", "However, bulge stars (with Galactocentric radius $R<4$ ) only contribute a small fraction in the sub-sample.", "There are more metal poor or $\\alpha $ -enhanced stars in the bulge/bar fields ($\|l\|<10$ ) compared to the disk fields.", "Velocity distributions in the raw field $(6,0)$ shows the coldest HV peak to others in the bulge/bar region.", "The chemical distribution of stars in this field is no different to that of the other bulge fields within $\|l\|<10$ (see Fig.", "REF ).", "HV stars at $(4,0)$ show quite different chemical abundance, but the number of stars in this field is small.", "A larger sample with more accurate kinematical and chemical information is needed to better understand the origin of HV stars in this field.", "Generally speaking, in the bulge region $(\|l\|<10)$ , the bulge/bar components (blue and red) have relatively low metallicity, high [$\\alpha $ /M], and high [C/N] compared to the disk components (green and magenta).", "We find no significant difference in the chemical abundance distributions of the bulge/bar main component and the HV component, indicating that they belong to a similar stellar population, consistent with [65].", "As mentioned in §2.2, we have estimated the stellar ages from the CANNON pipeline.", "Fig.", "REF d shows the direct age comparison between the four components.", "In most of these fields, stars in the disk components (green and magenta) are relatively younger than those in the bulge/bar components (blue and red).", "On the other hand, the age distributions of the two bulge/bar components (blue and red) are similar.", "This result is consistent with the conclusion of our previous work.", "In the bulge/bar region, there is no clear age difference between stars in the HV peak and the main component [65]." ], [ "Comparison with Previous Results", "The identification of a secondary peak in the velocity distribution is not straightforward.", "Different methods have been utilized in previous studies.", "For example, [38] used a double-Gaussian model to fit the velocity distributions to extract a secondary peak at high velocity with small velocity dispersion in several bulge fields.", "In our previous work, [65] measured the strength of the HV peak/shoulder by calculating the deviation of the velocity distribution from a Gaussian profile.", "Here, besides the Gauss-Hermite approach, we also fitted the data with a GMM with scikit-learn in §REF .", "In [27] the HV shoulder is not considered as a weaker Gaussian enshrouded by the main one but was described as a skewed asymmetric shape of the distribution.", "There might be different mechanisms between the HV peak and the HV shoulder, which may related to the formation and evolution history of the Galactic bulge.", "Comparing distributions in other dimensions, i.e., $V_{\\rm R}-V_{\\rm T}$ and chemical abundance distributions, we find no obvious difference between the fields showing a shoulder-like and a peak-like HV feature.", "Thus HV peaks and shoulders might not be essentially different.", "Instead, it is more important to confirm whether the HV feature is cold.", "If we chose $\\sigma <40\\; {\\rm km}\\;{\\rm s}^{-1}$ as a threshold for a cold peak inside the bulge/bar, then the secondary peaks in most regions would not be cold (see Fig.", "REF and Fig.", "REF ).", "[38] reported several fields showing a cold HV peak at positive longitudes ($4<l<14$ ), with a spatial distribution slightly asymmetric - with colder HV peaks at negative latitudes.", "At negative longitudes HV peaks have also been reported at $(-6, 0)$ [2] and $(-8.5, 0)$ [57].", "On the other hand, the HV features could be widely spread in the bulge/bar region.", "Although the two fields mentioned above are not covered by APOGEE DR16, the general longitudinal trend of the secondary peak is consistent with these previous studies.", "The longitudinal trend of the velocity distributions is also consistent with previous works and model predictions of bar kinematics.", "As shown in Fig.", "REF , at $5<\|l\|<20$ the secondary component in the GMM, i.e., the HV feature, shows similar mean values ($\\mu \\sim 200\\; {\\rm km}\\;{\\rm s}^{-1}$ ), which is consistent with [64].", "Fitting the profiles with Gauss-Hermite polynomial the correlation between its coefficients $\\,\\overline{\\!", "{V}}$ and $h_{3}$ is also consistent with the theoretical expectations of a bar structure [10], [21], [26].", "[65] found no chemical abundance difference between stars in the HV peak and the main component, which implied that their age composition is similar.", "By comparing the updated chemical abundances [M/H], [$\\alpha $ /M] and [C/N] for stars in more fields we still find no distinction between these two components.", "Further more, the age distribution in Fig.", "REF also supports the conclusion that these two component have a similar age composition.", "The present work provides significant improvements over our previous work.", "[65] used APOGEE DR13, which mainly covers the positive longitudes side of the bulge/bar.", "This work uses APOGEE DR16, which contains more fields at negative longitudes.", "The full bulge coverage is very important in discriminating between different model predictions.", "Moreover, we estimate the stellar ages using the CANNON pipeline which enables us to study the kinematic properties of stars in different age ranges.", "We also utilized the Gaia data and the StarHorse catalog to get robust distances and 6D phase-space information for stars in the bulge region.", "Therefore, we can compare the data with model predictions in new phase-space, e.g., the $V_{R}-V_{T}$ space." ], [ "Metallicity and Age bias", "As shown in Fig.", "REF , the bulge main component (blue) and the bulge HV component (red) have similar chemical abundances and age distributions.", "In most fields, the foreground disk components (green and magenta) are slightly more metal-rich than the bulge/bar components (blue and red), implying a global positive radial metallicity gradient from the inner region to the outer disk.", "Such positive metallicity gradient, different from the weak negative radial metallicity gradient revealed by GIBS and ARGOS [36], [66], was also reported in [15] using APOGEE DR13.", "The positive radial metallicity gradient might be due to the potential metallicity bias discussed in [19].", "In the spectroscopic analysis due to the cool edge of the model grid (ASPCAP, $T_{\\rm eff} \\sim 3500$ K), some metal-rich old stars were missing.", "Metal-rich stars in the MW disk are usually young so this selection effect makes bulge stars more metal-poor on average.", "For the same reason stars in the red giant branch are better represented at younger ages statistically [19], which is also the case of our Sample C shown in Fig.", "REF .", "However, the missing old metal-rich stars should still be included in Fig.", "REF with Sample B.", "Therefore, no such age bias should exist in the velocity distributions." ], [ "The Origin of the HV Component", "Several models were proposed to explain the HV peak.", "A simple conjecture of its origin is stars in the resonant orbits.", "The HV peaks might have a gaseous counterpart, as HV peaks could also be seen in the $(l,V_{\\rm los})$ distributions, which are usually interpreted as gas following $x_{1}$ -orbits [54], [25].", "As the most populous orbits in barred potentials [55], [33] showed that the $x_{1}$ orbital family is able to simultaneously generate peaks close to the observed value.", "This scenario is also supported by [1].", "With a barred model containing star formation, they further pointed out that at a given time, $\\sim 40-50$ percent of HV stars are in $x_{1}$ -like orbits and they are preferentially young, since new born stars are easily trapped by $x_{1}$ -orbits.", "Recently, [31] suggested that the propeller orbit, as a “distant relative” of the $x_{1}$ family, could also be responsible for the observed HV peaks in the MW bar, and the fraction of propeller orbits could provide constrains to MW modeling.", "In models of very elongated bars, propeller orbits are common and play a very dominant role [22].Connections and differences between the $x_{1}$ and the propeller families could be found in Fig.", "6 of [22].", "The $x_{2}$ -orbits belong to another important resonant orbital family in the bar region.", "[12] suggested that a kpc-scale nuclear disc (or ring), composed of stars on $x_{2}$ -orbits, is responsible for the HV peak.", "Table: GMM fitting (n=2) parametersTable: GMM fitting (n=3) parameters of the weakest componentWe could use the observational data to discriminate between these models, since the $x_{1}$ and the $x_{2}$ scenarios predict distinct secondary peaks at different longitudes with different $\\mu $ and $\\sigma $ .", "[1] predicted HV peaks at $l=3- 10, \|b\|<1$ .", "In their Fig.", "4, the velocity profiles at the same $\|l\|$ are almost symmetric, i.e., have the same $\|\\mu \|$ and $\\sigma $ .", "At positive longitudes, [11] predicted HV peaks at $l=8- 12$ , also in the mid-plane.", "As seen in their Fig.", "7, HV peaks at negative longitudes show smaller $\|\\mu \|$ and smaller $\\sigma $ .", "Although some key fields are not covered by APOGEE-2, we still have the $(\\pm 5, 0)$ and $(\\pm 10, 0)$ fields to discriminate between $x_{1}$ and $x_{2}$ -orbits scenarios.", "Tab.", "REF and Tab.", "REF show the GMM fitting results for $n=2$ and $n=3$ respectively.", "The velocity dispersions $\\sigma $ of the HV component ($\\sigma _{2,2} $ for $n=2$ and $\\sigma _{3,3}$ for $n=3$ ) are similar at the same $\|l\|$ .", "The peak velocities $\|\\mu _{2,2}\|$ and $\|\\mu _{3,3}\|$ at negative longitudes are slightly larger than those at positive longitudes.", "This symmetry could also be seen in Figures REF and REF .", "This observational result is inconsistent with the [12] model predictions that the HV peaks at negative longitudes should have smaller peak velocity and velocity dispersion.", "The propeller orbit, a “distant relative” of the $x_1$ orbit, has also been suggested to contribute to the observed HV peaks in the MW bar [31].", "They argued that this kind of orbits could explain the observed proper motions of stars in fields at $ l = 4, 6, 8$ in the mid-plane $(b=0)$ .", "To make a direct comparison with their results, we plot the same $V_{\\rm GSR}-\\mu _{l}$ distributions of stars in several key fields in the mid-plane.", "The S10 model is also included for comparison (see Fig.", "REF ).", "In this figure, S10 could match the observational data in most fields, except for $(4, 0)$ , where the distribution shows a large slope and is clearly divided into two clumps.", "In the same raw field we also found that the distances of HV stars are concentrated at $d=6.5\\;{\\rm kpc}$ .", "Interestingly, according to Fig.", "15 in [31], $l=4$ is close to the line-of-sight tangential point of the propeller orbit at a distance of $\\sim 7\\;{\\rm kpc}$ from the Sun, indicating a possible connection between the HV stars at (4$$ , 0$$ ) with the propeller orbit.", "As shown in Fig.", "REF , HV stars in different bins show a latitudinal similarity and a clear longitudinal trend in the $V_{\\rm R}-V_{\\rm T}$ distributions, regardless of the shapes of velocity profiles.", "The good agreement between the self-consistent barred model S10 and the APOGEE observations further supports the tight relation between the bar and the HV stars.", "The $(4,0)$ field seems to be an exception, but in this field the number of HV stars is still too small to be conclusive.", "Generally speaking, the HV feature has a tight relation with the bar but may not be necessarily related to some special resonant orbits, since in S10 all the particles in the simulation within a certain radius range ($5<d<7\\;{\\rm kpc}$ ) are included without selecting any particular type of orbits.", "In addition to kinematics, chemical abundances and age information can help to further constrain the origin of the HV peak.", "[1] suggested that young stars with age $<2\\;{\\rm Gyr}$ contribute to a HV peak in the velocity distribution.", "In this scenario the fraction of young stars is higher in the HV peak than the main component, since both young and old stars could contribute to the main component, but only young stars contribute to the HV peak.", "The nuclear disk scenario in [11] also implies that stars on the $x_2$ orbit are generally younger than the bulge/bar main component, since the nuclear disk forms later inside the bar structure.", "In this study, with more fields, better derived chemical abundances and estimated age from the CANNON, we still find no evidence for younger stars in the HV component than the main component, as shown in Fig.", "REF , consistent with [65].", "The raw field $(6, 0)$ shows the coldest secondary peak in $V_{\\rm GSR}$ distribution.", "However, the $V_{\\rm R}-V_{\\rm T}$ distribution, chemical abundance distribution and the age distribution of stars in this field show a similar trend to stars in the other fields.", "Thus it might not have a unique origin.", "We also checked the globular clusters distribution in the inner galaxy [43] and found no corresponding globular clusters near the $(6,0)$ field.", "Using the BIC to select the best number of Gaussians $n$ in the GMM, we find that $n = 3-5$ Gaussians are usually needed in the bulge/bar region, indicating that velocity distributions in the bulge/bar region are relatively complicated.", "More observations are needed in the future to verify the existence of the cold HV peaks in the bulge.", "The number of stars in each field should be increased to achieve better statistics.", "In addition, more spatial coverage at negative longitude is necessary.", "For example, $(4, 0)$ and $(-6,0)$ are the two key fields to distinguish between different scenarios.", "However, the sample size in the $(4, 0)$ field is relatively small, and the $(-6, 0)$ field even lacks observational coverage.", "With more targets observed and larger spatial coverage in the bulge region, predictions from different scenarios could be better verified." ], [ "Summary and Conclusion", "We revisit the stellar velocity distributions in the bulge/bar region with APOGEE DR16 and Gaia DR2 , focusing on the possible HV peak and its physical origin.", "Our sample covers the Galactic bulge in both the positive and negative longitudes, allowing for a comprehensive analysis of the Galactic bulge.", "We adopt StarHorse distances and Gaia proper motion to derive the radial velocity ($V_{\\rm R}$ ) and tangential velocity ($V_{\\rm T}$ ).", "Stellar ages were estimated with the CANNON.", "We fit the velocity distributions with two different models, namely the GMM and the Gauss-Hermite polynomial.", "In the bulge region, the Gauss-Hermite coefficients $h_{3}$ are positively correlated with the mean velocity $\\,\\overline{\\!", "{V}}$ , which may be due to the Galactic bar structure.", "At $\|l\|>10$ (the disk dominated region), $h_{3}$ is anti-correlated with $\\,\\overline{\\!", "{V}}$ , consistent with theoretical expectations.", "Fitting the $V_{\\rm GSR}$ profiles with a 2-component GMM we find that most of the secondary components in the bulge region are not dynamically cold (with a threashold of $\\sigma <40\\; {\\rm km}\\;{\\rm s}^{-1}$ ).", "Although some key fields within $-10< l < -5$ lack the coverage, we still find symmetric longitudinal trends in mean velocity ($\\mu $ ) and standard deviation ($\\sigma $ ) distributions of the secondary component, which imply a bar-like kinematic.", "In the bulge region, the coldest HV peak show up at $\|l\|\\sim 6$ .", "If we free the number of Gaussians in GMM fitting, $n=3-4$ are usually required, implying a complex velocity distribution in the MW bulge.", "With the additional tangential motion information from $Gaia$ , we find that the HV stars show similar patterns in the radial-tangential velocity distribution $(V_{\\rm R}-V_{\\rm T})$ , regardless of the existence of a distinct cold HV peak.", "Moreover, without specifically selecting certain kinds of resonant orbits, a simple MW bar model could reproduce well the observed $V_{\\rm R}-V_{\\rm T}$ (or $V_{\\rm GSR}-\\mu _{l}$ ) distributions.", "The chemical abundances and the age inferred from the ASPCAP and the CANNON implies that, the HV stars in the bulge/bar region $(R < 4\\;{\\rm kpc})$ are generally as old as the other stars in the bar region.", "Unfortunately, currently no single model can well explain all the observational results.", "The models mainly fall into two categories, i.e., the $x_{1}$ -orbit scenario and the $x_{2}$ -orbit scenario: (1) The $x_{1}$ -orbit scenario: [1] argued that the HV peak is contributed by young stars captured on $x_{1}$ -orbits by the bar potential, since the selection function of APOGEE might be more sensitive to young stars.", "In our study, we found that the APOGEE-2 HV stars do show bar-like kinematics.", "$x_1$ orbits are the back-bone of the bar, and therefore they likely contribute to the observed HV feature, but in observations we also found that these HV stars are not young (older than 8 Gyrs).", "[31] presented a new model for galactic bars, in which propeller orbits (a “distant relative” of $x_{1}$ orbits) plays a dominant role in the orbital structure.", "They suggested that the propeller orbits, which are not necessarily young, may be responsible for the observed HV peaks, since their proper motions match the observations well.", "Fig.", "REF shows that without selecting certain kinds of orbits, a barred model could also well reproduce the observed $V_{\\rm GSR}-\\mu _{l}$ distributions.", "This indicates that other orbital families, different from the propeller orbits, can also form similar HV features.", "Thus, the propeller orbits are sufficient but not necessary to explain the HV features.", "(2) The $x_2$ -orbit scenario: For the kpc-scale nuclear disk model composed by $x_2$ orbits [12], [11], there are three predictions: the HV peaks have (a) smaller $\|\\mu \|$ and (b) smaller $\|\\sigma \|$ at negative longitudes, in comparison to the corresponding positive longitudes with the same $\|l\|$ ; (c) the nuclear disk must form later than the stars in the main component, hinting for a possible age difference with the HV peak stars, which are younger than the main component.", "According to the observational data, the fields at $l = \\pm 10^\\circ $ are inconsistent with all these predictions.", "In addition, the longitudinal symmetric distributions of $\\mu , \\sigma $ , and $h_{3}$ of the secondary component shown in Figures REF -REF are inconsistent with the $x_{2}$ -orbit predictions.", "More observations and theoretical efforts are needed to fully understand the origin of the HV peaks/shoulders in the bulge/bar region.", "We thank the referee for the constructive and valuable comments that helped to improve this paper.", "The research presented here is partially supported by the National Key R&D Program of China under grant No.", "2018YFA0404501; by the National Natural Science Foundation of China under grant Nos.", "11773052, 11761131016, 11333003; by the “111” Project of the Ministry of Education under grant No.", "B20019, and by the MOE Key Lab for Particle Physics, Astrophysics and Cosmology.", "This work made use of the Gravity Supercomputer at the Department of Astronomy, Shanghai Jiao Tong University, and the facilities of the Center for High Performance Computing at Shanghai Astronomical Observatory.", "J.G.F-T is supported by FONDECYT No.", "3180210 and Becas Iberoamérica Investigador 2019, Banco Santander Chile.", "Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions.", "SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah.", "The SDSS web site is www.sdss.org.", "SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.", "This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/consortium).", "Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.", "Fig.", "REF shows the comparison between the test data from APOKASC and the predicted values from the CANNON.", "Figure: CANNON result.", "x: test data from APOKASC; y: predict values from the CANNON." ] ]	2011.14065
[ [ "Fair and Diverse Allocation of Scarce Resources" ], [ "Abstract We aim to design a fairness-aware allocation approach to maximize the geographical diversity and avoid unfairness in the sense of demographic disparity.", "During the development of this work, the COVID-19 pandemic is still spreading in the U.S. and other parts of the world on large scale.", "Many poor communities and minority groups are much more vulnerable than the rest.", "To provide sufficient vaccine and medical resources to all residents and effectively stop the further spreading of the pandemic, the average medical resources per capita of a community should be independent of the community's demographic features but only conditional on the exposure rate to the disease.", "In this article, we integrate different aspects of resource allocation and seek a synergistic intervention strategy that gives vulnerable populations with higher priority when distributing medical resources.", "This prevention-centered strategy is a trade-off between geographical coverage and social group fairness.", "The proposed principle can be applied to other scarce resources and social benefits allocation." ], [ "Introduction", "The COVID-19 pandemic has been widely spreading around the globe and caused hundreds of thousands of deaths, crashed the healthcare systems of many countries, and stalled almost all social and economical activities, which leads to an astronomical amount of financial loss.", "To battle the spreading COVID-19 pandemic, several leading countries and organizations have devoted a significant amount of resources to develop vaccines, new diagnostics, and anti-infective treatments for the novel coronavirus [17].", "More than 60 candidate vaccines are now in development worldwide, and several have entered early clinical trials in human volunteers, according to the World Health Organization [39].", "Once the development of the treatments is approved for public use, it is of paramount importance that these resources are quickly dispatched to all the communities in a country because one weak link in the defense against the virus would leave the neighboring communities even the whole state vulnerable and exposed to the spreading of the disease.", "However, the existing healthcare infrastructures in many states and cities are insufficient to provide universal supplies of vaccinations and treatments.", "Hence, targeting the high risk population and setting fair principles for prioritizing the allocation accordingly will save lives, curb the further spreading of the virus, and prevent sequential global pandemic outbreaks in the following years.", "In United States, due to the disparities [43] between different population subgroups in terms of health, financial stability, and accessibility to health care services, certain low-income and minority populated communities are particularly vulnerable to COVID-19.", "For example, among Chicago's minority groups, the Latinx and Black groups have been hit much harder by the COVID-19 pandemic than any other ethnic groups.", "By the latest report (September 15, 2020) from the Chicago Department of Public Health (CDPH), among the death tolls caused by COVID-19, 33.0% are from the Latinx group and 42.8% are from Black & non-Latinx group [49].", "These two ethnic groups also lead in the number of COVID-19 confirmed cases by a large margin compared to the rest ethnic groups.", "These ratios are disproportional to the percentages of these ethnic groups of the total population in Chicago, which mainly consists of 32.3% white, 28.7% Hispanic, and 30.9% African American, according to US Census Bureau[3].", "The development of vaccine and other effective treatment medicines against any novel infectious disease generally takes a long time due to the high uncertainties in all the stages in the process, including clinical trial.", "Even if the vaccines and treatments are successfully developed, they are only available in limited quantities initially due to manufacturing, logistic and financial constraints.", "Due to the huge gap between available medical resources and the entire population in need of them, especially during the early production stages, the allocation of such resources becomes a difficult and yet pressing issue.", "Moreover, due to demographic distribution and occupational factors of the residents in different geographical regions, resource allocation based on solely maximum geographical uniformity can lead to an unfair distribution system and propagate biases across population subgroups of the population.", "We design a fairness-aware allocation framework for vaccine and scarce treatment resources considering both Geographical Diversity and Social Group Fairness as the guiding principles for prevention-centered strategies.", "The social group fairness is based on the general fairness notion of Equality of Opportunity [31], [25].", "An allocation strategy is fair if the average amount of resources an individual receives only depends on the individual's exposure rate to the disease and is independent of the individual's demographic or social-economic background.", "We also consider an allocation strategy to be diverse if the geographical location does not affect the averaged resources an individual can receive.", "Based on such notions of fairness and diversity, we formally define and formulate them into inequality constraints.", "Our proposed research provides a solution to distribute the scarce medical resources in a fair manner to all communities and protect certain minorities and low-income groups that are more vulnerable to pandemic's impact.", "Not only can such a solution help stop the spreading pandemic more effectively, but also push for justice and fairness in healthcare decision making." ], [ "Related Work", "Resource allocation has been a classic and important problem in many domains such as economy (e.g,[30]), management science (e.g,[14]), emergency response (e.g,[28]), etc.", "More recent works have addressed the need for an integrated approach with different purposes such as material allocation in a production plan [8], supplier selection, and order allocation application[51].", "Healthcare resource allocation is one of the most challenging allocation problems.", "A large number works have been developed to identify effective strategies for it [7], [54], [60], [41], [27], [33], [12], [26].", "Recently, the fairness of algorithmic decision making has been the center of attention of many researchers, who are designing and developing algorithms for different purposes, such as machine learning, ranking, and social welfare [5], [4], [64], [23].", "Mitigating the bias of an outcome from a decision model, which is mainly caused by the inherent bias in the data and societal norm, will ensure that the outcome is not favorable or adversarial toward any specific subgroups [64], [4], [16], [24], [31], [62], [59].", "One of the critical problems where the fairness of the outcome matters significantly is scarce resource allocation.", "The notion of fairness in resource allocation has been introduced in [6] and later with a more precise definition in [10].", "Fairness was firstly adopted in bandwidth allocation problem for computer network systems [15], [18], [36], [35], [46], [47], [38], [34], [11].", "In these settings, the amount of resources requested can be modified by different users.", "Besides, service allocation mainly covers a group of users and is not necessarily one-to-one allocation.", "These settings differ from the resource allocation for scarce treatments which is a one-to-one allocation problem.", "Now fairness has been addressed in many resource and service allocation methods [58], [41], [40], [25], [37].", "The importance of ethical consideration in resource allocation and principal guidelines have been discussed in [61].", "Here we highlight a few and emphasize our difference from them.", "In [25], the authors formalize a general notion of fairness for allocation problems and investigate its algorithmic consequences when the decision-maker does not know the distribution of different subgroups (defined by creditworthy or criminal background) in the population.", "The distribution estimation is accomplished using censored feedback (individuals who received the resource, not the true number).", "In our work, we estimate the distribution of different social groups from the data using Bayes rule.", "Singh [53] considers a fair allocation of multiple resources to multiple users and have proposed a general optimization model to study the allocation.", "Our proposed method differs from [53] on two aspects.", "First, [53] assumes multi-resources and multi-type users, whereas we mainly focus on resource allocation problem across different regions and different population subgroups.", "Second, [53] aims to maximize the coverage, whereas we consider a fairness-diversity trade-off by minimizing the diversity and fairness gaps, simultaneously, across different regions.", "Donahue et al.", "[20] considers the problem of maximizing resource utilization when the demands for the resource are distributed across multiple groups and drawn from probability distributions.", "They require equal probabilities of receiving the resource across different groups to satisfy fairness and provide upper bounds on the price of fairness over different probability distributions.", "In our proposed model, we utilize a similar fairness requirement while requiring diversity (population) consideration across different regions.", "Thus, our work is an intersection between fairness and diversity.", "Furthermore, we deal with the one-to-one allocation instead of the coverage problem.", "One recent works related to COVID-19 resource allocation designed a vulnerability indicator for racial subgroups that can be used as guidelines for medical resource allocation [48].", "The proposed model cannot identify other vulnerable subgroups that are not geographically clustered, thus not able to form spatially concentrated communities.", "In our paper, we directly identify subgroups' vulnerability using exposure rates estimated from COVID-19 cases and death.", "We focus on the available data to estimate the necessary parameters and perform an empirical study using the proposed Algorithm REF ." ], [ "Summary of Contributions", "In this paper, we aim to design a fairness-aware allocation strategy that considers the trade-off between geographical diversity and social fairness (demographic disparity) in allocating resources.", "We provide a new aspect to the classic allocation problem while seeking a synergistic intervention strategy that prioritizes disadvantaged people in the distribution of scarce resources.", "The proposed approach is different from the existing works using the maximum utilization [13] or maximum coverage [53], [25] objective.", "The nature of the treatment allocation problem is not the same as the coverage problems, since the former is a one-to-one assignment problem, whereas in the latter the resource can cover more than one user such as police or doctor allocation[25], [20].", "In our work, we aim to study scarce resource allocation considering the trade-off between geographical diversity and social fairness.", "More specifically, we do not seek to share the vaccines equally among groups of populations.", "Instead, we aim to emphasize the protection of the vulnerable populations or the ones at high risk, depending on their exposure rate, in order to prevent death/spread of the disease.", "We focus on the available data to estimate the necessary parameters and perform an empirical study using the proposed Algorithm REF .", "However, our proposed model can be generalized to resource allocation in other scenarios such as disaster relief.", "In § , we first model the notions of geographical diversity and social group fairness for the allocation of scarce medical treatments and vaccines.", "We then formulate the allocation problem into an Integer Program (IP) problem, which incorporates the diversity and fairness as constraints, in addition to capacity constraint, i.e., the amount of available resources.", "Fairness and diversity constraints are bounded by user-defined hyperparameters, $\\epsilon _d$ , and $\\epsilon _f$ , which are the allowed diversity and fairness gaps, respectively.", "To obtain a feasible solution for the original IP problem efficiently, we relax the IP problem to a Linear Programming (LP) problem.", "Moreover, the fairness and diversity constraints are much more complicated than the capacity constraint, and to deal with them efficiently, we use the penalty method, which is a common practice to solve constrained optimization.", "The two constraints are combined into a single objective function using a trade-off hyperparameter $\\alpha $ .", "To guarantee that the converted problem is equivalent to the original feasibility problem we provide theoretical proofs and subsequently, a binary search algorithm is used to obtain a feasible range of $\\alpha $ .", "We evaluate the allocation scenario under different $(\\epsilon _d, \\epsilon _f)$ values and provide the corresponding feasible range for $\\alpha $ , accordingly.", "Different levels of the trade-off between diversity and fairness are presented.", "The proposed framework can be applied at different stages of the pandemic to estimate the exposure rate of population subgroups, and obtain a feasible allocation considering both population size and exposed population.", "In § , we evaluate the performance of the proposed model using COVID datasets in Chicago for vaccine and scarce treatment allocations.", "The results demonstrate the impact of incorporating fairness criteria in the allocation model compared to the diverse allocation and uniform allocation.", "The paper concludes in § ." ], [ " Traditional resource allocation treatments have been widely studied in the optimization literature [30], [14], [28] where the limited resource is being distributed to areas based on a single objective (e.g., function of total population).", "However, regions might have different needs or require higher priority in a medical resource allocation setting (e.g.vaccine).", "An allocation treatment that is solely based on geographical diversity, suggests an equalized distribution among the regions, which may not provide equity in the sense of social fairness.", "To assure the concept of “equity” as well as “equality”, we aim to model and incorporate fairness notion on demographic subgroups as another principal in the allocation decision process.", "Identifying each region's priority or risk level, is a critical task since it can save many lives and protect the vulnerable sensitive subgroups.", "We aim to design a fair and diverse resource allocation framework through modeling the Social Group Fairness and Geographical Diversity and modeling their trade-off in the optimization setting.", "The optimization model minimizes the fairness and diversity gaps across different subgroups of different regions while satisfying the capacity constraint to protect a vulnerable population.", "A trade-off analysis is performed to demonstrate the price of fairness incorporation as the fairness gap (the maximum allocation gap between demographic subgroups) with and without fairness consideration in the allocation.", "We propose a tuning approach to identify an optimal range for the trade-off hyperparameter($\\alpha $ ) in order to provide the best possible allocation solution under different scenarios." ], [ "We consider a centralized decision maker for allocating available $b$ units of vaccines to a set of centers denoted by $M$ .", "They include clinics, hospitals, pharmacies, etc.", "For convenience, we assume the entire area covered by the $M$ centers to be a city, but it can be a county, or other administrative district.", "Let $x_j$ be the decision variable denoting the amount of vaccines to be allocated to center $j\\in M$ .", "Let $z_{j}$ be the region, such as the list of zip-code areas, that are assigned to be covered by center $j$ .", "For simplification, we assume there is no overlap between the list of zip-code areas covered by different centers (even if they are close in distance), i.e.", "$z_l \\cap z_k=\\emptyset , \\forall l, k\\in M$ .", "We also assume the policy that residents can only receive the resources from the center that covers the region where they reside.", "Such policy is not uncommon in practice, especially in distribution of scarce resources.", "Next we introduce some key concepts and their notation.", "If we consider the geographical location of any individual reside to be a random variable, denoted by $Z$ , then $\\lbrace j\\in M, z_j\\rbrace $ are the possible values for $Z$ .", "Therefore, $P(Z=z_j)$ is the proportion of the population who reside in $z_j$ .", "Let $U_1,\\cdots ,U_p$ be discrete-valued sensitive variables corresponding to demographic and socioeconomic attributes, and $S_i$ for $i=1,\\ldots ,p$ be the set of possible levels for each of the sensitive variable.", "For instance, if $[U_1,U_2,U_3]$ represent three attributes, income, race, gender, respectively, then $S_1=\\lbrace \\text{low, medium, high}\\rbrace $ , $S_2=\\lbrace \\text{black, latinx, white, others}\\rbrace $ , and $S_3=\\lbrace \\text{female, male}\\rbrace $ .", "The combinations of all the levels of $U_1,\\ldots , U_p$ is a set denoted by $\\mathcal {G}$ and indexed by a set $I$ , i.e., $\\mathcal {G}=\\lbrace g_i, \\text{ for }i \\in I\\rbrace $ .", "In other words, $\\mathcal {G}=S_1\\times S_2\\times \\cdots \\times S_p$ .", "For any $g\\in \\mathcal {G}$ , it corresponds to a possible combination of levels of $U_1,\\ldots , U_p$ , such as $<low-income, black, female>$ , and there can be a group of population whose values of the sensitive variables $(U_1,\\ldots , U_p)$ are equal to $g$ .", "For short, we call it social group $g$ .", "Let $s_{i,j}$ be the population size of the social group $g_i$ who reside in region $z_j$ , where $i\\in I, j \\in M$ .", "Therefore, $s_{i,j}/\\sum _{r\\in I,k\\in M} s_{l,j}=P([U_1,\\ldots , U_p, Z]=[g_i,z_j])$ .", "Let $E$ be a binary random variable with $E=1$ representing the individual is exposed to the infectious disease and $E=0$ otherwise.", "So $P(E=1\|g_i)$ is the exposure rate of the social group $g_i$ and $P(E=1\|g_i, z_j)$ the exposure rate of the social group $g_i$ living the area of $z_j$ .", "It is intuitive to assume these exposure rates depend on the social groups and regions.", "We will discuss more on some reasonable assumptions on the exposure rates and how to estimate them later.", "A key concept in resource allocation is the amount of the resource per capita.", "It is a ratio between the quantity of available resource and the size of the population who are going to receive the resource.", "Denote $V$ as the amount of resource one individual receives.", "One important assumption we make here is that $V$ follows a discrete uniform distribution, and there are three parameters involved, the amount of resource $X$ , $\\mathcal {Y}$ the population who are to receive the $X$ amount of resource, and size of the population $Y=card(\\mathcal {Y})$ , the cardinality of $\\mathcal {Y}$ .", "Therefore, the mean value of $V$ is $\\mathbb {E}(V\|X, \\mathcal {Y}, Y)=\\frac{X}{Y}$ is the resources per capita, and it varies with respect to the three parameters.", "The focus of this article is on the following problem.", "Given a limited amount of resource $b$ , such as vaccines, how should the decision maker allocate the amount of resource $x_j$ to each center $j$ satisfying geographical diversity (quantified by $\\mathcal {D}(x)$ ), and social fairness (quantified by $\\mathcal {F}(x)$ )." ], [ "In this part, we define geographical diversity and social group fairness, which the latter is based on the equality of opportunity notion of fairness [24].", "To quantify the geographical diversity and social fairness, we introduce $\\mathcal {D}(x)$ and $\\mathcal {F}(x)$ in Equations (REF ) and (REF ).", "Definition 1 (Geographical Diversity) Geographical Diversity of allocation of limited resource to a set of centers $M$ is satisfied if $\\forall j \\in M$ , on average the resource per capita is invariant with respect to the location of the groups of the population, i.e., $\\mathbb {E}(V\|x_j, z_j, \\sum _{i\\in I}s_{i,j})$ does not vary with respect to the location $z_j$ .", "The notation $\\mathbb {E}(V\|x_j, z_j, \\sum _{i\\in I}s_{i,j})$ is simplified and we use $z_j$ to refer to the population who reside in region $z_j$ .", "Let $s_{i,j}$ be the population size of the social group $g_i$ and $\\mathbb {E}(V\|\\sum _{j\\in M} x_j, \\sum _{i\\in I,j\\in M}s_{i,j})$ denote the averaged resource per capita over the entire city under the consideration of the resource allocation plan, and the location is omitted since it is obvious.", "It is straightforward to formulate the geographical diversity $\\mathcal {D}_j(x)$ for $\\forall j \\in M$ as follows.", "$\\mathcal {D}_j(x) = {\\mathbb {E}(V\|x_j, z_j, \\sum _{i\\in I}s_{i,j})-\\mathbb {E}(V\|\\sum _{j\\in M}x_j, \\sum _{i\\in I,j\\in M}s_{i,j})}={\\frac{x_j}{\\sum \\limits _{i\\in I}s_{i,j}}-\\frac{\\sum \\limits _{j\\in M} x_j}{\\sum \\limits _{j\\in M} \\sum \\limits _{i\\in I} s_{i,j}}}.$ So if geographical diversity is strictly met, $\\mathcal {D}_j(x)=0$ for all $j$ .", "The geometrical diversity represents the conventional or minimal requirement for resource distribution, that the resources should be evenly distributed among all geographical regions across the entire city.", "Next, we introduce the concept of social fairness.", "In this definition, we emphasize the even distribution of resource among the endangered population, i.e., who are exposed to the infectious disease, disregarding the social groups.", "Definition 2 (Social Fairness) Social Fairness of allocation of limited resource to a set of centers $M$ is satisfied if $\\forall i \\in I $ , the averaged resource per capita is invariant with respect to the values of $U_1,\\cdots ,U_p$ of the group of population, i.e., $\\mathbb {E}(V\|E=1, g_i)$ does not vary with respect to the social group $g_i$ .", "The notation $\\mathbb {E}(V\|E=1, g_i)$ is simplified and we use $E=1$ and $g_i$ to denote the exposed individuals in social group $g_i$ .", "Directly translating this definition into formula, the fairness principle should be $\\mathcal {F}_i(x)={\\mathbb {E}(V\|E=1, g_i) - \\mathbb {E}(V\|E=1) }, \\quad \\forall i\\in I.$ However, the calculation of $\\mathbb {E}(V\|E=1, g_i)$ and $\\mathbb {E}(V\|E=1)$ are not so straightforward, and we derive them as follows.", "We first calculate the resource per capita for the exposed individuals who reside in $z_j$ , disregarding the social group, i.e., $\\mathbb {E}(V\|E=1, z_j)= \\frac{x_j}{\\sum \\limits _{l \\in I} s_{l,j}\\times P(E=1\|z_j, g_l)}, \\quad \\forall j \\in M$ in which the denominator is the amount of exposed individuals in $z_j$ .", "Then, E(V\|E=1, gi)=j M E(V\|E=1, zj)P(Z=zj\|E=1, gi) =jM E(V\|E=1, zj)P(E=1, [U1,..., Up]=gi, Z=zj)P(E=1, [U1,..., Up]=gi) =jM xjl I sl,jP(E=1\|zj, gl)si,jP(E=1\|gi,zj)/(i', j' si', j')kM si,kP(E=1\|gi, zk)/(i', j' si', j') =jM xjl I sl,jP(E=1\|zj, gl)si,jP(E=1\|gi,zj)kM si,kP(E=1\|gi, zk) In this article, we assume that the chance of exposure of an individual only depends on the individual's social attributes, and is independent of the geographical location, i.e., $P(E=1\|U_1,\\ldots , U_p, Z)=P(E=1\|U_1,\\ldots , U_p)$ .", "Equivalently, $\\forall i \\in I$ and $\\forall j \\in M$ , $P(E=1\|g_i, z_j)=P(E=1\|g_i)$ .", "This assumption is reasonable since in many U.S. cities as in Chicago, the geometrical locations of the residents are in fact highly correlated with the social and economical status of the residents.", "For example, in Figure REF , the percentage of the major ethnic groups, White, Hispanic, and African-American, are shown in three heat maps.", "It is very clear that the geometrical locations of the residents and the racial groups are heavily correlated.", "Of course, this assumption also makes the rest of the formulation much simpler.", "Under this assumption, we can simply obtain $\\mathbb {E}(V\|E=1, g_i)=\\sum \\limits _{j \\in M}\\frac{x_j}{\\sum \\limits _{l \\in I} s_{l,j}\\times P(E=1\|g_l)}\\frac{s_{i,j}}{\\sum \\limits _{k\\in M} s_{i,k}}.$ Figure: African-AmericanNext, to obtain $\\mathbb {E}(V\|E=1)$ , we need to integrate Equation (REF ) with respect to $g_i$ , i.e., E(V\|E=1)=i I E(V\|E=1, gi)P([U1,..., Up]=gi\|E=1) =i I E(V\|E=1, gi)P(E=1\|gi)P([U1,..., Up]=gi)P(E=1) =i I E(V\|E=1, gi)P(E=1\|gi)kM si,k/(i', j' si', j')rI, k'M sr,k' P(E=1\|zk', gr)/(i', j' si', j') =i I (j Mxjl I sl,jP(E=1\|gl)si,jkM si,k )P(E=1\|gi) kM si,krI, k'M sr,k' P(E=1\|gr) =i I j Mxjl I sl,jP(E=1\|gl)P(E=1\|gi)si,jrI, k'M sr,k' P(E=1\|gr).", "Based on the derivations, we can formulate the fairness for $\\forall i\\in I$ , Fi(x)=E(V\|E=1, gi) - E(V\|E=1) = * j Mxjl I sl,jP(E=1\|gl)si,jkM si,k- i I j Mxjl I sl,jP(E=1\|gl)si,jP(E=1\|gi)rI, k'M sr,k' P(E=1\|gr) .", "Similar to the definition of $\\mathcal {D}_j(x)$ , if the fairness is strictly met, $\\mathcal {F}_i(x)=0$ for all $i\\in I$ ." ], [ "As explained above, if the diversity and fairness requirements are strictly met, all $\\mathcal {D}_j(x)$ and $\\mathcal {F}_i(x)$ should be 0.", "However, such constraints are too restrictive and can be difficult or impossible to satisfy for all regions and social groups.", "Define $\\mathcal {D}(x)=\\max _{j\\in M}\\mathcal {D}_j(x)$ and $\\mathcal {F}(x)=\\max _{i\\in I}\\mathcal {F}_i(x)$ .", "$\\mathcal {D}(x)$ and $\\mathcal {F}(x)$ are auxiliary decision variables that signify the tight upper bounds on the diversity and fairness constraints, i.e., $\\mathcal {D}_j(x)\\le \\mathcal {D}(x)$ and $\\mathcal {F}_i(x)\\le \\mathcal {F}(x)$ for any $j \\in M$ and $i \\in I$ .", "Ideally, we want to find the a feasible solution $x$ such that both upper bounds are equal to zero.", "In addition, $x_j$ for $j=1,\\ldots , M$ should satisfy the capacity constraint, i.e., $\\sum _{j\\in M} x_j=b$ .", "Seeking to achieve the geographical diversity and social fairness simultaneously, one can formulate this problem as multi-objective (MO) minimization.", "P1: x (D(x), F(x)) Dj(x)D(x), j M Fi(x)F(x), i I jM xj= b xj 0, j M xjZ, j M. The integer constraint is because usually, resource such as vaccines are counted in integers and one individual only receives one vaccination.", "The integer constraints in $P1$ can be relaxed, particularly in practice when $b$ is large.", "A common solution to the multi-objective optimization problem is to use the weighted sum method, which leads to a simpler minimization problem $P2$ .", "After removing the absolute operation in the constraints, we obtain the relaxed linear programming (LP) problem.", "P2: x (1-) D(x) +F(x) s.t.", "D+j(x)D(x), j M D-j(x )D(x), j M F+i(x)F(x), i I F-i(x)F(x), i I jM xj= b xj 0, j M. where $\\mathcal {D}^+_j(x)$ refers to the positive side of the absolute value function, $\\mathcal {D}_j(x)$ , and $\\mathcal {D}^-_j(x)$ refers to the negative side of the absolute function.", "Similarly, for $\\mathcal {F}_j(x)$ , we have $\\mathcal {F}^+_j(x)$ and $\\mathcal {F}^-_j(x)$ .", "Here $\\alpha \\in [0,1]$ is a hyperparameter that controls the trade-off between the importance of fairness and diversity.", "If $\\alpha >0.5$ , the objective function focuses more on achieving the fairness, and it focuses more on the diversity if $\\alpha < 0.5$ .", "Figure: Rounding HeuristicIf we relax the integer constraints of $P1$ , based on the multi-objective optimization theory, it is easy to conclude that for any $x^$ in the Pareto front of $P1$ , there exists an $\\alpha ^\\in [0,1]$ such that $x^$ is an optimal solution of $P2$ .", "This is because the remaining constraints of $P1$ , including $\\mathcal {D}_j(x) \\le \\mathcal {D}(x) $ for all $j \\in M$ , $\\mathcal {F}_i(x)\\le \\mathcal {F}(x)$ for all $i\\in I$ , $\\sum _j x_j =b$ , and $x_j \\ge 0$ for all $j \\in M$ , form a convex polyhedron.", "Particularly, the constraints $\\mathcal {D}_j(x) \\le \\mathcal {D}(x)$ and $\\mathcal {F}_i(x)\\le \\mathcal {F}(x)$ are all linear in $x$ .", "Thus, we have the above conclusion.", "In practice, the simplex method can be used to find the optimal solution of $P2$ efficiently with common optimization libraries like IBM CPLEX and SciPy.", "Once the optimal solution of the LP-relaxation problem, $P2$ , is obtained for a given $\\alpha $ value, we need to round the solution into integers.", "But the rounded solution is not necessarily feasible for the minimization problem $P2$ .", "To ensure the feasibility of the rounded solution we employ a heuristic rounding approach in Algorithm REF , which is similar to the one in [55].", "The algorithm starts with rounding the LP relaxation solution to the nearest integer values.", "Next, if the capacity constraint $\\sum _{j \\in M} x_j=b$ is not satisfied, the algorithm reduces $x_j$ for the top populated region based on the total exceeded amount.", "If the resource constraint is under-satisfied, the algorithm increases the top exposed populated areas based on the total remaining resources.", "Note that in the Diverse-only ($\\alpha =0$ ) and Fairness-only ($\\alpha =1$ ) scenarios, the algorithm only considers the population and exposed population to sort $x_j$ 's, respectively." ], [ "Feasibility", "As we discussed in § , the ideal situation would be minimizing the upper bounds $\\mathbb {D}(x)$ and $\\mathcal {F}(x)$ to the full extent.", "An important observation to make here is that the ideal case with zero upper bounds both on the fairness and the diversity constraints is almost always impossible, due to the underlying biases in data and historical discrimination in society.", "Therefore, in practice, a small positive upper bound threshold is considered to be satisfied on fairness and diversity.", "For example, the fairness requirement can be thought of as the US Equal Employment Opportunity Commission’s ”four-fifths rule,” which requires that ”the selection rate for any race, sex, or ethnic group [must be at least] four-fifths (4/5) (or eighty percent) of the rate for the group with the highest rate”Uniform Guidelines on Employment Selection Procedures, 29 C.F.R.", "§1607.4(D) (2015).. We consider a similar requirement for diversity as well.", "The solution obtained under the MO model, $P2$ , is unable to guarantee to satisfy these requirements.", "For example, a solution with zero unfairness (but not satisfactory on the level of diversity) can be on the Pareto front solution of MO – hence might be the optimal output – even though it is not a valid solution.", "Subsequently, we introduce two control parameters $\\epsilon _d$ and $\\epsilon _f$ , which are the acceptable thresholds for the diversity and fairness requirements, i.e., $\\mathcal {D}(x)\\le \\epsilon _d$ and $\\mathcal {F}(x)\\le \\epsilon _f$ .", "The violation threshold parameters $\\epsilon _{\\mathcal {D}}, \\epsilon _{\\mathcal {F}} \\in [0,1]$ , are user-defined values that determine how diverse and fair the allocation should be.", "The value $\\epsilon _{\\mathcal {F}} = 0$ corresponds to a fully fair allocation, whereas $\\epsilon _{\\mathcal {F}} = 1$ corresponds to a completely fairness-ignorant allocation that solely considers the diversity.", "These violations must be relatively small by which resources allocated to a particular region and particular group nearly achieve the required level of diversity and fairness, respectively.", "Note that as quantitative metric, we say that allocation is considered as fair if $\\mathcal {F}(x)\\le \\epsilon _f$ , and allocation is considered as diverse if $\\mathcal {D}(x) \\le \\epsilon _d$ .", "Depending on the constraints, the weights, and the available scarce resources, no feasible solution may exist for the optimization problem.", "In this case, the decision-maker will have no choice but to relax the constraints.", "In § we will discuss the allocation under different scenario.", "Fortunately, as we shall prove in Theorem REF , if the problem has a feasible solution given the fairness and diversity constraints, there must exist an $\\alpha $ value under which the optimum solution of $P2$ is feasible and vice versa.", "Proposition 1 Let be $X^{\\prime }=\\lbrace x\|\\mathcal {D}(x)<\\epsilon _d, \\mathcal {F}(x)<\\epsilon _f, \\sum _{j\\in M} x_j=b, x_j \\ge 0\\rbrace $ : If $X^{\\prime }\\ne \\emptyset $ : given $x^ \\in X^{\\prime }$ , $\\exists \\alpha ^* $ such that $x^$ is an optimal solution of $P2$ .", "If $\\lnot \\exists ~\\alpha ^ $ such that $x^$ is an optimal solution of $P2$ that satisfies both conditions $\\mathcal {D}(x)<\\epsilon _d$ and $ \\mathcal {F}(x)<\\epsilon _f$ , then $X^{\\prime } = \\emptyset $ .", "We now need to find the $\\alpha $ value for which an optimum of $P2$ is a feasible satisfying diversity and fairness requirements.", "To do so, a naive approach would be a brute-force search in $[0,1]$ with a small step sizes added to $\\alpha $ .", "However, this is not a practical approach since it needs to solve the optimization problem in each iteration and as the step size decreases the exhaustive search increases and the number of times that it needs to solve $P2$ increases, accordingly.", "Therefore, we first obtain the monotone properties of optimal upper bound $\\mathcal {D}(x)$ and $\\mathcal {F}(x)$ with respect to $\\alpha $ .", "This result is used to find proper $\\alpha $ value later in § REF .", "Proposition 2 Let $x_1^$ and $x_2^$ be optimum solutions of $P2$ given $\\alpha _1$ and $\\alpha _2$ , respectively.", "It can be shown that if $\\alpha _1\\le \\alpha _2$ , then $\\mathcal {F}(x_2^)\\le \\mathcal {F}(x_1^)$ and $\\mathcal {D}(x_1^)\\le \\mathcal {D}(x_2^)$ ." ], [ "Choosing Trade-off Parameter, $\\alpha $", "The parameter $\\alpha $ in $P2$ controls the trade-off between the diversity and fairness of the allocation decision.", "Using the monotone property of $\\mathcal {D}(x)$ and $\\mathcal {F}(x)$ to $\\alpha $ in Proposition 1, we now propose an approach to find a proper $\\alpha $ value that satisfies both fairness and diversity constraints for given $(\\epsilon _d,\\epsilon _f)$ .", "Using an example we show the high-level idea of the tuning algorithm under different scenarios.", "In Figure REF , the monotonically decreasing red curve corresponds to the fairness constraint and the monotonically increasing blue curve corresponds to the diversity constraint.", "The dashed horizontal lines corresponds to the thresholds allowed for diversity $\\mathcal {D}(x)$ and fairness $\\mathcal {F}(x)$ constraints, respectively.", "For $\\alpha < \\alpha _1$ we can see that $\\mathcal {F}(x)$ exceeds the allowed threshold of $\\epsilon _f$ .", "Based on Proposition 1, we should remove the range $[0, \\alpha _1)$ from consideration.", "For $\\alpha > \\alpha _3$ , $\\mathcal {D}(x)$ exceeds the allowed threshold of $\\epsilon _d$ , hence, we can remove the range $(\\alpha _3, 1]$ from consideration.", "For $\\alpha =\\alpha _2$ , both $\\mathcal {D}(x)\\le \\epsilon _d$ and $\\mathcal {F}(x)\\le \\epsilon _f$ are satisfied.", "The color bar below the plot in Figure REF shows different ranges of $\\alpha $ .", "The green segment corresponds to the range of $\\alpha $ values for which the optimum solution of $P2$ satisfies $\\mathcal {D}(x)\\le \\epsilon _d$ and $\\mathcal {F}(x)\\le \\epsilon _f$ .", "Consequently, we can prune the infeasible intervals of $\\alpha $ by evaluating the violation of diversity and fairness thresholds.", "Figure: Tuning α\\alpha Algorithm REF represents the proposed tuning algorithm.", "The algorithm starts from the middle of the $\\alpha $ interval, $\\alpha _m$ , and solves $P2$ with that.", "Then, it splits the $\\alpha $ region into half to further prune the region until a narrow feasible range of $\\alpha $ is obtained.", "In each iteration, if the diversity threshold is not satisfied by the solution obtained from $P2$ the algorithm prunes the interval larger than $\\alpha _m$ .", "Similarly, if the fairness threshold is not satisfied, the algorithm prunes the interval smaller than $\\alpha _m$ .", "If neither of the thresholds is satisfied with the solution obtained from $P2$ using $\\alpha _m$ , there does not exist a feasible solution for $(\\epsilon _d, \\epsilon _f)$ .", "Finally, when both constraints are satisfied the algorithm returns a valid solution for $P2$ , which can be rounded to obtain an integer solution for $P1$ ." ], [ "Price of Fairness", "We now define the Price of Fairness (PoF) as the difference of the fairness gap of the optimal solution of $P2$ that is obtained with and without any fairness constraints.", "As we described in § , Equation (REF ) is the fairness constraint that is defined for each social group.", "Solving $P2$ with and without the fairness constraints, we can obtain different allocation solutions and consequently different fairness and diversity gaps.", "This will allow us to compare fair-diverse allocation performance with the Diverse-only allocation to analyze the impact of fairness constraints, namely PoF.", "The PoF metric assists the decision-maker on the fairness scheme to be considered, mainly, $\\epsilon _d$ and $\\epsilon _f$ .", "Note that there is a trade-off between diversity and fairness gap in $P2$ , when the allocation scheme is more focused on diversity, smaller $\\epsilon _f$ , the control parameter $\\alpha $ become close to 0.", "Subsequently, $P2$ minimizes the diversity upper bound subject to only the capacity constraint.", "Comparing the solution of such a problem with the one that has more focus on fairness, $\\alpha $ close to 1, we expect to see a larger fairness gap and smaller diversity gap.", "In §, we evaluate PoF of the allocation solution obtained under different scenarios." ], [ "Case Study: Resource Allocation for COVID-19 Relief in US cities ", "In this section, we apply the proposed fair and diverse allocation strategy to the planning of distribution of medical resources for COVID-19 relief among some US cities (Chicago, New York, Baltimore).", "The city of Chicago is a segregated city with relatively high Hispanic and Black population[9].", "Several research studies have addressed a positive correlation among proportions of Black and Hispanic communities and COVID-19 cases [9], [2], [42], [63].", "More specifically, in [9], both positive and statistically significant associations between the proportions of Black and Hispanic population and per capita COVID-19 cases have been identified using data from six segregated cities in the US.", "In particular, New York City has the highest rate of confirmed cases, followed by Chicago and Baltimore.", "That is, we present our instance problems based on these cities.Even though our case study is limited to three cities, it includes the first (New York City) and third (Chicago) largest cities in the USA, and presents our approach for cities with higher vulnerable population (e.g., Baltimore).", "We rely on the US census for population and demographic distribution data.", "For the COVID-19 cases, death tolls, and hospitalization we separately collect the data that is provided by governmental departments of each considered cities.", "The main sources of each datasets are described in Section 3.1.", "In the subsequent section, we first perform a descriptive analysis of the data to motivate the necessity of the fair and diverse distribution of medical resources.", "Next, we evaluate the performance of our proposed Fair-Diverse allocation approach and identify a reasonable trade-off parameter $\\alpha $ based on the binary search Algorithm REF .", "Lastly, we will calculate the price of fairness (PoF) to highlight the role of fairness constraints in our optimization setting.", "All of the analytical results, optimization models, and algorithms were implemented in Python 3.7 using docplex and sklearn packages.", "The codess are available on github https://github.com/nnezam2/Fair-Resource-Allocation/blob/master/README.md." ], [ "Data Description", "Population Dataset: The uszipcodehttps://uszipcode.readthedocs.io/ Python package provides detailed geographic, demographic, socio-economic, real estate, and education information at the state, city, and even zip code level for different areas within the US.", "Based on the documentation, this package uses an up-to-date database by having a crawler running every week to collect different data points from multiple data sources.", "This dataset does not provide the intersectional population.", "We refer to this dataset as Pop.", "throughout the experiment section.", "City of Chicago COVID-19 Database [44] provides daily data on COVID-19 positive-tested cases, death tolls, hospitalizations, and other individuals' attributes (e.g., age, race, gender) to track the pandemic in this city.", "In our study, we primarily use the COVID-19 Daily Cases data to reveal the inequality among different population subgroups.", "We refer to this dataset as Chicago-COVID-Cases throughout the experiment section.", "The city of Chicago COVID-19 database also provides daily data on COVID-19 Cases, Tests, and Deaths by ZIP Code dataset REF .", "We refer to this dataset as Chicago-COVID-Zipcode throughout the experiment section.", "New York City (NYC) COVID-19 Data Repository https://github.com/nychealth/coronavirus-data consists of different COVID-19 related datasets including daily, weekly, monthly data, data on SARS-CoV-2 variants, the cumulative COVID-19 cases, etc.", "For the purpose of our analysis, we use COVID-19 cases and death totals by age, race, and gender since the start of the COVID-19 outbreak in NYC, February 29, 2020.", "We refer to this dataset as NYC-COVID-Zipcode throughout the experiment section.", "City of Baltimore COVID-19 data Dashboardhttps://coronavirus.baltimorecity.gov/ provides different statistics and visualization for COVID-19 data in Baltimore.", "The main source of this data is the Maryland Department of health[45], which is updated on a daily basis.", "This dataset includes COVID-19 the number of cases and death at the zip-code level, and total cases and death tolls by age, race, gender.", "We refer to this dataset as BLT-COVID-Zipcode throughout the paper." ], [ "Descriptive Analysis on COVID-19 risk factor among different population subgroups, and different regions (zip-codes) ", "Utilizing the Chicago-COVID-Cases dataset, we first identify the contribution of each demographic attribute to the total number of COVID-19 cases and deaths in each region of the Chicago area.", "This analysis reveals key insights into the actual importance and differentiation among demographic subgroups.", "Next, we perform a PCA analysis [1] using Pop.", "merged with the Chicago-COVID-Zipcode and produce a Biplot, as shown in Figure REF , to identify and compare the contribution of different attributes to the COVID-19 death and case numbers.", "Note that we used cross-validation to tune the number of components of the PCA method.", "To better visualize the impact of these attributes on different regions, we implement a $K$ -means clustering [32] method to partition our geographical areas (Zipcodes) based on the numbers of COVID-19 deaths and cases.", "We cluster them into three categories and label them as high, medium, and low impacted areas, accordingly.", "Our findings in this part show that areas with higher COVID-19 deaths and cases tend to have a higher population, higher elderly population (compared with the young one), more Black Or African American, Latinos population (compared with other races), and lower median income.", "Furthermore, we believe that the revealed negative correlation between income and COVID-19 cases should raise serious concerns, in future decision-making procedures.", "One justification for that is lower-income individuals are mostly daily-paid and cannot afford living expenses if they are self-quarantined or stop working.", "Consequently, the pandemic inevitably affects the areas harder with larger lower-income populations.", "However, we do not have COVID-19 data based on income level.", "Hence, we do not consider income in this study.", "In brief, we can observe notable differences in contribution to the COVID-19 positive cases and death tolls across specific demographic attributes (e.g, older age).", "This fact reveals the critical role of a Fair-Diverse model, which considers not only the overall population size but also the unrepresentative (exposed) population for scarce resource allocation problem.", "Figure: Exposure Rates of Population Subgroups-Chicago City" ], [ "Estimating the Distribution of Exposed Population", "Estimating the marginal distribution of the number of high-risk individuals in each demographic group, i.e., $P(E=1\|g_i)$ , requires the data of positive-tested (infected) cases and death tolls.", "In this article, we intend to propose a resource allocation plan for vaccines and treatments for the COVID-19 pandemic.", "However, we only have access to the count of individuals of each social group who were infected from COVID-19, namely $P(g_i\|E=1)$ .", "The probability $P(E=1\|g_i)$ can be calculated from the Bayes formula $P(E=1\|g_i)= \\frac{P(g_i\|E=1)P(E=1)}{P(g_i)}.$ We will use $P(E=1\|g_i)$ , in the §REF to form the fairness constraints of the optimization problem, Equation REF , and calculate the exposed population in each zip-code.", "Table REF presents the exposure rates for different population subgroups in Chicago City (The exposure rates for other cases (cities) are provided in Appendix REF , REF ." ], [ "Fair-Diverse Allocation", "To evaluate the proposed Fair-Diverse model, we construct Diverse-only, Fair-only, and equalized importance or alpha=0.5 models, and compare the allocation solutions as well as the resulted fairness and diversity gaps among them.", "In our terminology, Diverse-only corresponds to an allocation that is merely based on diversity constraint, Equation REF , and is not considering any other (e.g., fairness) measures.", "Similarly, Fair-only corresponds to a model solely based on fairness constraint.", "Furthermore, alpha=0.5 refers to a model that has equalized weights on Fairness and Diversity constraints in the optimization setting.", "As mentioned in § , we propose a diversity and fairness trade-off problem as in $P2$ .", "As long as the resource constraint is satisfied, $P2$ has an optimal solution given an $\\alpha $ value.", "However, the optimal solution obtained from $p2$ might not be feasible given the diversity and fairness requirements $\\epsilon _d$ and $\\epsilon _f$ as mentioned in § REF .", "That is we apply our proposed binary search algorithm to discover a range for $\\alpha $ that results in a feasible solution.", "To evaluate the performance of the Algorithm REF , we will consider three US cities and different sensitive attributes ( Race, Age, and Gender) and solve each instance problem separately.", "We will then show the allocation results for each problem using the four above-mentioned models and discuss $\\alpha $ ranges in detail.", "The total number of zip-code areas varies for each city with 177 regions in New York City, 58 in Chicago, and 36 in Baltimore.", "The resulted allocations of each problem is sorted by population size for the top 15 area (zip-codes).", "City of Chicago We consider $b=200000$ units of vaccines as the total available resources to be allocated for the city of Chicago.", "Using Pop., and Chicago-COVID-Zipcode datasets, we evaluate the allocation results of our proposed resource allocation framework at the zip-code level in the city of Chicago.", "The first instance problem considers Race as the sensitive attribute and the associated Fair-Diverse model captures inequalities across different racial subgroups.", "The results for the top 15 populated areas are reported in Table REF .", "The baseline values for $\\epsilon _{f}$ and $\\epsilon _{d}$ are derived from the alpha=0.5 model and are equal to 0.24 and 0.007.", "Note that we do not use the tuning algorithm for this alpha=0.5 model.", "For Fair-Diverse model $\\epsilon _{f}$ and $\\epsilon _{d}$ are both set to be 0.025 to decrease the fairness gap compared to the baseline value.", "The resulted tuned range for $\\alpha $ is between 0.54 and 0.86 in this case (midpoint=0.70).", "Looking at Table REF , the area associated with zip-code \"60639\" for instance, receives a lower number of vaccines using Diverse-only model but higher in both alpha=0.5 and Fair-Diverse model ($\\alpha =0.70$ ) due to having higher total exposed population.", "In contrast, the Fair-only closes the fairness gap to the full extent, ($\\epsilon _{f}=0$ ), and as a result obtains an extreme allocation solution in which only a few areas (zip-codes) receive vaccines.", "Undoubtedly, this could not be a desirable allocation solution under certain fairness and diversity requirements.", "Since the table represents the top 15 populated areas, we cannot observe all areas with positive allocation using Fair-only model.", "Table: Resource Allocation (Racial groups): Top 15 populated areas- ChicagoThe second instance problem considers Age as the sensitive attribute and the associated Fair-Diverse model captures inequalities across different age groups.", "The results for the top 15 populated areas are reported in Table REF .", "The baseline values for $\\epsilon _{f}$ and $\\epsilon _{d}$ are derived from the alpha=0.5 problem and are equal to 0.012 and 0.", "Next, we run the binary search algorithm to tune the $\\alpha $ value and find a feasible solution for Fair-Diverse model.", "In this case, $\\epsilon _{f}$ and $\\epsilon _{d}$ are both set to be 0.003.", "The resulted tuned range for $\\alpha $ is between 0.57 and 1 (midpoint=0.78).", "As mentioned previously, the Diverse-only model assigns vaccines to areas only based on the total population, and the Fair-only model obtains an extreme allocation solution in which only a few areas (zip-codes) receive vaccines.", "Therefore, none of these models are capable of delivering a fair and diverse allocation solution.", "Note that in this instance problem, alpha=0.5 model is not doing any better than the Diverse-only model since the weight on the fairness component is not adequate to change the results (diversity-gap is dominant).", "This result can further reveal the necessity of the proposed tuning algorithm.", "Table: Resource Allocation (Age groups): Top 15 populated areas-ChicagoFinally, a noteworthy instance occurs when we consider gender as the sensitive attribute and the Fair-Diverse model attempts to eliminate the unfairness between male and female subgroups.", "The results for the top 15 populated areas are reported in Table REF .", "The baseline values for $\\epsilon _{f}$ and $\\epsilon _{d}$ derived from the alpha=0.5 problem and are both close to zero ( 6.04e-05 and 0).", "It is worth mentioning that this is because of the similar gender population distribution across different regions.", "Consequently, the fairness and diversity requirements can be satisfied even with Diverse-only model.", "We can still run the binary search algorithm to tune the $\\alpha $ value and find a feasible solution for Fair-Diverse model by closing the fairness gap further.", "To do this, the $\\epsilon _{f}$ and $\\epsilon _{d}$ values in Fair-Diverse model should be set to 0 and 0.1.", "The resulted tuned range or $\\alpha $ is between 0.92 and 1 in this case (midpoint=0.96).", "Looking at Table REF and exposure rates (shown in Table REF ), we notice that, in this specific instance problem, the exposed population size is, in actuality, aligned with the total population size in different areas.", "In other words, highly populated areas tend to have higher exposed populations as well.", "Therefore, the resulted allocations from Diverse-only, alpha=0.5, and Fair-Diverse models are very close and even equal in some cases.", "Table: Resource Allocation (Gender groups): Top 15 populated areas-ChicagoFigures REF and REF present the results for the top 15 populated regions in the Chicago City for racial and age instance problems.", "Note that in both instance problems, the total population equals the summation of all associated groups (e.g Age groups) due to having unknown labels in the data.", "In Figure REF , for example, \"60614\" and \"60634\" regions receive less vaccines both under Fair-Diverse and alpha=0.5 models due to having higher exposed population, Table REF .", "The Diverse-only does not consider the exposed population, therefore, the associated allocations are higher for these regions.", "Besides, the allocation obtained from the tuned range of $\\alpha $ is significantly different from the allocation obtained with the alpha=0.5 model since the latter does not satisfy the fairness requirement $\\epsilon _f$ .", "Moving to another instance problem, Figure REF represents the allocation results for the age instance problem.", "Based on the plot, we can notice that the alpha=0.5 and Diverse-only allocation solutions overlap.", "This can be justified by the fact that the 50% emphasis on fairness is not sufficient to close the age subgroups disparities, and it requires a higher $\\alpha $ value as we obtained through the tuning algorithm.", "That being said, the tuned $\\alpha $ value, in this case, is 0.78, which is substantially higher than $0.5$ .", "For example, \"60614\" and \"60609\" regions receive less vaccines using Fair-Diverse model with $\\alpha $ tuning since they have relatively lower exposed population, Table REF .", "For observing more interactive visualization tools, please check our newly created web application on the Chicago City datasets using Rshineyhttps://nazanin.shinyapps.io/Fair_Resource_Allocation/.", "Figure: Resource Allocation (Age groups):Top 15 populated areas-ChicagoNewYork City (NYC) Utilizing Pop., and NYC-COVID-Zipcode datasets, we assess our proposed resource allocation framework at zip-code level for New York City.", "In this problem, we consider the total number of vaccines available $b$ to 500000 since NYC has higher population compared with Chicago.", "Similar to the Chicago case study, the first instance problem for NYC considers Race, and the second instance consider Age as the sensitive attributes.", "We ignore the description of Gender instance problem in this section (please see the appendix for the results).", "The first instance problem considers Race with the aim of capturing inequalities across different racial subgroups.", "The results for the top 15 populated areas in NYC are reported in Table REF .", "The $\\epsilon _{f}$ and $\\epsilon _{d}$ derived from the alpha=0.5 model, are 0.10 and 0.0007 respectively.", "For Fair-Diverse model $\\epsilon _{f}$ and $\\epsilon _{d}$ are both set to be 0.017 to decrease the fairness gap compared to the baseline value.", "The tuned $\\alpha $ range is between 0.66 and 0.69 (midpoint=0.67).", "Looking at Table REF , the area associated with zip-code \"10467\", receives a lower number of vaccines using Diverse-only model but higher numbers in both alpha=0.5 and Fair-Diverse model ($\\alpha =0.67$ ) due to having higher total exposed population and more risks.", "In contrast, the Fair-only closes the fairness gap to the full extent ($\\epsilon _{f}=0$ ), and as a result, obtains an extreme allocation solution in which only a few areas (zip-codes) receive vaccines.", "As shown in Table REF , none of the top 15 populated areas will receive vaccines under this extreme condition.", "As a result, this could not be an applicable allocation solution.", "The second instance problem considers Age to captures inequalities across different age groups.", "The results for the top 15 populated areas are reported in Table REF .", "The baseline values for $\\epsilon _{f}$ and $\\epsilon _{d}$ are derived from the alpha=0.5 problem and are equal to 0.008 and 0 while the $\\alpha $ value range is between zero and one.", "Next, we run the binary search algorithm to tune the $\\alpha $ value and find a feasible solution for Fair-Diverse model with $\\epsilon _{f}$ and $\\epsilon _{d}$ both set to 0.003.", "The resulted tuned range for $\\alpha $ is between 0.68 and 1 (midpoint=0.84).", "As mentioned previously, the Diverse-only model assigns vaccines to areas merely based on the total population, and the Fair-only model obtains an extreme allocation solution in which only a few areas (zip-codes) receive vaccines.", "Therefore, these models are not capable of delivering a fair and diverse allocation solution.", "Note that in this instance problem, alpha=0.5 model is not doing any better than the Diverse-only model since the weight on the fairness component is not adequate which further reveals the necessity of the proposed tuning approach.", "Figure: Resource Allocation (Age groups):Top 15 populated areas-NYCTable: Resource Allocation (Racial groups): Top 15 populated areas- NYCFigures REF and REF present the results for the top 15 populated regions in the New York City for racial and age instance problems.", "Note that in both instance problems, the total population equals the summation of all associated groups (e.g Age groups) due to having unknown labels in the data.", "In Figure REF , \"11236\" and \"11219\" zip-codes receive less vaccines both under Fair-Diverse and alpha=0.5 models due to having lower exposed population, Table REF .", "The Diverse-only does not consider the exposed population, therefore, the associated allocations are higher for these regions.", "Besides, the allocation obtained from the tuned range of $\\alpha $ is significantly different from the allocation obtained with the alpha=0.5 model since the latter does not satisfy the fairness requirement $\\epsilon _f$ .", "Moving to another instance problem, Figure REF represents the allocation results for the age instance problem.", "Based on the plot, we can notice that the alpha=0.5 and Diverse-only allocation solutions overlap.", "This can be justified by the fact that the 50% emphasis on fairness is not sufficient to close the age subgroups disparities, and it requires a higher $\\alpha $ value as we obtained through the tuning algorithm.", "That being said, the tuned $\\alpha $ value, in this case, is 0.84, which is substantially higher than $0.5$ .", "For example, \"11226\" and \"11211\" regions receive less vaccines using Fair-Diverse model comparing with Diverse-only or alpha=0.5 since they have relatively lower exposed population, Table REF .", "Table: Resource Allocation (Age groups): Top 15 populated areas- NYCBaltimore City Utilizing Pop., and BLT-COVID-Zipcode datasets, we empirically tested our proposed resource allocation framework at the zip-code level.", "In this problem, we modify the total number of vaccines available $b = 100000$ since Baltimore has a relatively lower population compared to Chicago and New York City.", "However, similar to other case studies, the first instance problem considers Race, and the second instance considers Age as the sensitive attributes and ignore the description of Gender instance problem (please see the appendix for the results).", "The Race instance problem captures the inequalities across different racial subgroups.", "The results for the top 15 populated areas in Baltimore are reported in Table REF .", "The $\\epsilon _{f}$ and $\\epsilon _{d}$ derived from the alpha=0.5 model, are 0.046 and 0.016 respectively.", "For Fair-Diverse model $\\epsilon _{f}$ and $\\epsilon _{d}$ are both set to be 0.025 to decrease the fairness gap compared to the baseline value.", "The tuned $\\alpha $ range is between 0.57 and 1 (midpoint=0.78).", "Looking at Table REF , the area associated with zip-code \"21230\", receives a higher number of vaccines using Diverse-only model but lower numbers in both alpha=0.5 and Fair-Diverse model ($\\alpha =0.78$ ) due to having lower total exposed population and less risk level.", "On the other hand, the Fair-only closes the fairness gap ($\\epsilon _{f}=0$ ), and obtains an extreme allocation solution in which only a few areas (zip-codes) receive vaccines.", "As shown in Table REF , four regions among the top 15populated areas receive vaccines under this extreme condition.", "As a result, this could not be an applicable allocation solution.", "The Age instance problem captures inequalities across different age groups.", "The results for the top 15 populated areas are reported in Table REF .", "The baseline values for $\\epsilon _{f}$ and $\\epsilon _{d}$ are derived from the alpha=0.5 problem and are equal to 0.014 and 0 while the $\\alpha $ value range is between zero and one.", "Next, we run the binary search algorithm to tune the $\\alpha $ value and find a feasible solution for Fair-Diverse model with $\\epsilon _{f}$ and $\\epsilon _{d}$ both set to 0.007.", "The resulted tuned range for $\\alpha $ is between 0.76 and 1 (midpoint=0.88).", "As mentioned previously, the Diverse-only model assigns vaccines to areas merely based on the total population, and the Fair-only model obtains an extreme allocation solution in which only four areas (zip-codes) receive vaccines.", "Therefore, these models are not capable of delivering a fair and diverse allocation solution.", "Note that in this instance problem, alpha=0.5 model is not doing any better than the Diverse-only model since the weight on the fairness component is not adequate which further reveals the necessity of the proposed tuning approach.", "Figure: Resource Allocation (Age groups):top 15 populated areas-BaltimoreTable: Resource Allocation (Racial groups): Top 15 populated areas- BaltimoreFigures REF and REF show the results for the top 15 populated regions in the Baltimore City for racial and age instance problems.", "Note that in both instance problems, the total population equals the summation of all associated groups (e.g Age groups) due to having unknown labels in the data.", "In Figure REF , and zip-codes receive less vaccines both under Fair-Diverse and alpha=0.5 models due to having lower exposed population, Table REF .", "The Diverse-only does not consider the exposed population, therefore, the associated allocations are higher for these regions.", "Besides, the allocation obtained from the tuned range of $\\alpha $ is significantly different from the allocation obtained with the alpha=0.5 model since the latter does not satisfy the fairness requirement $\\epsilon _f$ .", "Moving to the next instance problem, Figure REF represents the allocation results for the age instance problem.", "Based on the plot, we can notice that the alpha=0.5 and Diverse-only allocation solutions overlap.", "This can be justified by the fact that the 50% emphasis on fairness is not sufficient to close the age subgroups disparities, and it requires a higher $\\alpha $ value as we obtained through the tuning algorithm.", "That being said, the tuned $\\alpha $ value, in this case, is 0.88, which is substantially higher than $0.5$ .", "For example, \"21217\" and \"21223\" regions receive less vaccines using Fair-Diverse model comparing with Diverse-only or alpha=0.5 since they have relatively lower exposed population, Table REF .", "Table: Resource Allocation (Age groups): Top 15 populated areas- BaltimoreAs we discussed in § REF , the allocation solution that is obtained from $P2$ does not necessarily satisfy the fairness and diversity requirements (under any $\\alpha $ value).", "To demonstrate the performance of the tuning algorithm, which always returns a range for $\\alpha $ under which the optimal solution of $P2$ is feasible, we now study the impact of $\\epsilon _f$ and $\\epsilon _d$ on the optimal solution under different models using the city of Chicago case study.", "The plots in REF are based on the racial instance problem.", "These figures reveal that under any fairness and diversity requirements (given $\\epsilon _f$ and $\\epsilon _d$ values) the $\\alpha $ tuning algorithm returns a feasible solution for $\\emph {Fair-Diverse}$ model.", "Note that, this is not the case for other models (Fair-only, Diverse-only and alpha=0.5) as it can be observed from the Figures REF and REF .", "In other words, the optimal solutions obtained from the Diverse-only and alpha=0.5 models does not satisfy the fairness requirement ($\\epsilon _{f} \\le 0.03$ and $\\epsilon _{f} \\le 0.2$ ), and the solution obtained from the Fair-only model fails to satisfy the diversity requirement ($\\epsilon _{d} \\le 0.3$ ) in Figures REF and REF .", "However, if we relax the fairness requirement to $\\epsilon _f \\le 0.3$ , Figure REF , the alpha=0.5 model can indeed achieve a feasible solution.", "It is worth mentioning that the diversity requirement is easier to achieve compared to the fairness requirement due to the larger inherent disparities in exposed population.", "Figure: Impact of the ϵ f \\epsilon _f and ϵ d \\epsilon _d on the diversity-fairness trade-off" ], [ "Price of Fairness", "In this section, we will compare the fairness and diversity gaps under different models and population subgroups to discuss the price of fairness (PoF).", "The results are obtained based on the aforementioned racial group instance problem for the city of Chicago.", "We present the results under tuned $\\alpha $ value (0.71) in this part.", "Firstly, Figure REF reveals that the Fair-Diverse model reduces fairness and diversity gaps more compared to Diverse-only and Fair-only models.", "Although the Diverse-only model eliminates the diversity gap, it fails to decrease the fairness gap.", "Similarly, the Fair-only model eliminates the fairness gap but it fails to decrease the diversity gap.", "Moreover, Figure REF shows some considerable reduction in the gaps across different population subgroups (racial groups) using the Fair-Diverse model.", "The comparison between the results with the Diverse-only and the uniform allocation solution reveals the necessity of the Fair-Diverse model in closing the gaps, and therefore, reducing the disparities across different population subgroups.", "Lastly, Figure REF illustrates the trade-off between fairness and diversity gaps considering different $\\alpha $ values.", "Note that the $\\alpha $ values represent the midpoint of the feasible range.", "We can observe that increasing the $\\alpha $ value, which is the weight on the fairness component in $P2$ , decreases the fairness gap as expected.", "However, the diversity gap will increase due to the fairness-diversity trade-off.", "Finally, as discussed in § REF , the Price of Fairness (PoF) can be evaluated using the difference of the fairness gap from the optimal solution of P2 that is obtained with and without any fairness constraints.", "POF could be defined as the fraction of the fairness gap that is obtained from the (Diverse-only) allocation to the allocation solution based on the Fair-Diverse model.", "If an allocation solution decreases the fairness gap more than the Fair-Diverse model, the POF is less than one.", "Otherwise, PoF is > 1 and we will need to find a balance between the fairness and diversity objectives to decide on which allocation to choose.", "In the case of the Racial instance problem with $\\alpha =0.71$ , the PoF equals $\\frac{0.4419}{0.025}=17.67$ , which is significantly larger than 1.", "Figure: Fairness and Diversity gaps based on different α\\alpha values" ], [ "Discussion", "In this section, we discuss the scope and limitation of the proposed fair-diverse resource allocation method, which is introduced in §.", "Through the three case studies in §REF , we can see that the proposed method is applicable to the early stage of medical resource allocation when the resource is still considered to be scarce.", "As long as the required data are available and accurate, the proposed allocation scheme can be applied.", "We want to point out to some limitations of that the proposed approach can be further improved upon.", "First, the exposure rate estimation can be improved.", "Recall that in § the COVID-19 exposure rate for each intersectional group is denoted by $P(e\|g_{i})$ .", "However, the primary challenge while estimating COVID-19 exposure rates for intersectional subgroups (e.g., <black,female,age30-39>), is the lack of intersectional population data in each Zipcode, denotes by $S_{ij}$ in §.", "Given the availability of the required data format, one could use a Poisson regression model to obtain the intersectional exposure rates.", "Second, the proposed method does not consider the profession of residents in prioritizing the sub-groups.", "In the U.S., during the early vaccination process, priority was given to the sub-groups based on age, underlying health conditions, and professions.", "Front-line workers such as health care workers, grocery employees, K-12 educators, etc., whose works are essential to normal social functions were among the first one or two batches of vaccine receivers.", "In our work, we have estimated the COVID-19 exposure rate from the daily infected cases.", "It indirectly considers the residents' profession since front-line workers are expected to have a larger chance to be exposed to the virus.", "But we are not able to directly include the profession of resident as a characteristic to define the sub-group of populations.", "It is mainly due to the lack of data that has the job descriptions for residents (or the percentage) in each Zipcode area.", "We plan to work with city officials or non-profit organizations that have more detailed population data and further improve our proposed method.", "Furthermore, estimating the intersectional population size could not be an appropriate implementation approach.", "Regarding the aforementioned data limitation issue, we solved the allocation problem for age, race, and gender as the sensitive attributes separately in §REF .", "However, our model is able to return a fair-diverse allocation using the intersectional subgroups, given that the real intersectional population is available for each area.", "In fact, applying the same framework on the intersectional data could be a future direction of this work.", "Another future direction with respect to COVID-19 exposure rates, could incorporate a learning approach to obtain the rates.", "Several successful time-series techniques such as traditional ARIMA models or more novel approaches such as RNN-LSTM can be applied on the historical data.", "Time-series analysis would enable one to predict more precise exposure rates for the future time windows.", "Moreover, we only considered single treatment, mainly vaccination, however, the proposed approach can be generalized to incorporate multiple treatments.", "Our proposed fair-diverse allocation can be utilized at different stages of pandemic considering the updated exposure ratio.", "In this paper, we mainly focused on the resource allocation to the centers and facilities.", "Although, designing allocation policy at a granular level of individuals from each center or facility is also critical, high-level policies have been shown to be more effective and feasible for deployment than individual levels ones.", "There are multiple barriers to the individual level deployment of the policies.", "For example, vaccine hesitancy [22], [57], [56] is a widespread problem for many people and enforcing it could be in violation of social values such as Freedom of choice(FoC)[52], [21].", "Moreover, designing strategies at individual level requires an extensive amount of data and medical considerations as vaccines(e.g.", "COVID19 vaccines) can cause serious allergic reaction and people with pre-existing allergies should avoid taking it[29], [50], [19].", "Last but not least, applications of our proposed framework are beyond the allocation policy design use case.", "The solutions obtained for the allocation problem, could be helpful in identifying the vulnerable regions to incentive vaccination and for advertisement programs purposes.", "This will help maximizing the vaccination rate within higher risk communities in the future." ], [ "Conclusion", "In this paper, we propose the idea of fairness in scarce resource allocation problems (e.g, vaccine distribution) in terms of disparities across various population subgroups in different regions.", "To do so, we consider diversity and fairness components to design a Fair-Diverse allocation.", "We first formulate a general multi-objective (MO) problem $P1$ , and propose the weighted sum method together with the LP relaxation to simplify it to $P2$ .", "We then solve the LP-relaxed problem, $P2$ , based on the fairness and diversity requirements as described in § REF .", "For this purpose, we propose a binary search approach, Algorithm REF , to find an optimal range for the trade-off parameter $\\alpha $ in $P2$ such that the obtained solution is feasible.", "Moreover, we have empirically analyzed our proposed methodologies in § using COVID-19 datasets in three major and segregated US cities (New York City, Chicago, and Baltimore).", "We designed three instance problems for each city based on different demographic attributes, race, age, and gender.", "We then implemented our fair-diverse model and compared it with other models (Diverse-only, Fair-only, and $\\alpha =0.5$ ) to investigate the fairness criteria in the solution and highlight the necessity behind our approach.", "We have also discussed the fairness and diversity requirements, $\\epsilon _{d}$ and $\\epsilon _{f}$ , and compared the Fair-Diverse allocation with other allocation solutions under different thresholds.", "Lastly, we have examined the price of fairness based on the associated gaps across different models.", "In brief, our empirical results reveal the paramount role of fairness criteria in decision-making problems involving scarce resource allocation (e.g, a vaccine allocation ).", "While certain minorities and population groups are more vulnerable to the COVID-19 virus, a Diverse-only (population-based) vaccine allocation can lead to higher fatality rates by neglecting the vulnerability of various population subgroups.", "We require to ensure a fair and diverse vaccine allocation to induce lower mortality rates across different regions.", "That is, we aim to find a decent balance between the diversity and fairness measures in different geographical regions and allocate the resources accordingly." ], [ "Acknowledgements", "L. Kang is partially supported by the National Science Foundation grant DMS-1916467.", "APPENDIX" ], [ "Proof of Proposition 1", "Proposition 2.", "Let be $X^{\\prime }=\\lbrace x\|\\mathcal {D}(x)<\\epsilon _d, \\mathcal {F}(x)<\\epsilon _f, \\sum _{j\\in M} x_j=b, x_j \\ge 0\\rbrace $ : If $X^{\\prime }\\ne \\emptyset $ : given $x^ \\in X^{\\prime }$ , $\\exists \\alpha ^* $ such that $x^$ is an optimal solution of $P2$ .", "If $\\lnot \\exists ~\\alpha ^ $ such that $x^$ is an optimal solution of $P2$ that satisfies both conditions $\\mathcal {D}(x)<\\epsilon _d$ and $ \\mathcal {F}(x)<\\epsilon _f$ , then $X^{\\prime } = \\emptyset $ .", "We first argue that the feasibility problem is equivalent to the following optimization model: P': D(x) s.t.", "F(x)= D+j(x)D(x), j M D-j(x )D(x), j M F+i(x)F(x), i I F-i(x)F(x), i I jM xj = b xj 0, j M Consider $\\epsilon \\in [0,\\epsilon _{\\mathcal {F}}]$ .", "Let $x^$ be the optimal solution of $P^{\\prime }$ .", "Hence, $\\mathcal {F}(x^)=\\epsilon \\le \\epsilon _{\\mathcal {F}}$ satisfies the fairness constraint.", "Now, if $\\mathcal {D}(x^) \\ge \\epsilon _{\\mathcal {D}}$ , we conclude that for $\\epsilon $ there does not exist a feasible solution that satisfies the diversity constraint.", "Otherwise, that feasible solution would minimize $P^{\\prime }$ .", "Introducing a Lagrangian multiplier $\\lambda $ , we now define the dual Lagrangian transformation of $P^{\\prime }$ to be: P”: D(x)+(F(x)-) s.t.", "D+j(x)D(x), j M D-j(x )D(x), j M F+i(x)F(x), i I F-i(x)F(x), i I jM xj= b xj 0, j M Now, we argue that given $x^* \\in X^{\\prime }$ there exists an $\\alpha ^$ such that $x^$ is an optimum of $P2$ .", "Let $\\lambda ^$ be the multiplier when $x^$ is an optimum of $P^{\\prime \\prime }$ , then $\\forall x^{\\prime \\prime }\\ne x^$ : $\\mathcal {D}(x^)+\\lambda ^(\\mathcal {F}(x^)-\\epsilon _{\\mathcal {F}}) \\le \\mathcal {D}(x^{\\prime \\prime })+\\lambda ^(\\mathcal {F}(x^{\\prime \\prime })-\\epsilon _{\\mathcal {F}})$ $\\Rightarrow \\mathcal {D}(x^)+\\lambda ^\\mathcal {F}(x^)-\\lambda ^\\epsilon _{\\mathcal {F}} \\le \\mathcal {D}(x^{\\prime \\prime })+\\lambda ^\\mathcal {F}(x^{\\prime \\prime })-\\lambda ^\\epsilon _{\\mathcal {F}}$ $\\Rightarrow \\mathcal {D}(x^)+\\lambda ^\\mathcal {F}(x^)\\le \\mathcal {D}(x^{\\prime \\prime })+\\lambda ^\\mathcal {F}(x^{\\prime \\prime })$ Thus, $x^$ is an optimum of $P2$ where $\\lambda ^=\\frac{(1-\\alpha )}{\\alpha }$ .", "On the other hand, if $\\exists \\alpha ^$ under which $x^$ is optimal of $P2$ , and $\\mathcal {D}(x^)<\\epsilon _d$ and $\\mathcal {F}(x^)<\\epsilon _f$ , then $x^ \\in X^{\\prime }$ and $X^{\\prime }\\ne \\emptyset $ .", "This means if such $\\alpha ^$ does not exist, there is no feasible solution under $\\epsilon _d$ and $\\epsilon _f$ ." ], [ "Proof of Proposition 2", "Proposition 2.", "Let $x_1^$ and $x_2^$ be optimum solutions of $P2$ given $\\alpha _1$ and $\\alpha _2$ , respectively.", "It can be shown that if $\\alpha _1\\le \\alpha _2$ , then $\\mathcal {F}(x_2^)\\le \\mathcal {F}(x_1^)$ and $\\mathcal {D}(x_1^)\\le \\mathcal {D}(x_2^)$ .", "Let $\\beta =\\frac{\\alpha }{(1-\\alpha )}$ .", "Given $x_1^$ and $x_2^$ corresponding to $\\beta _1$ and $\\beta _2$ , where $\\beta _2<\\beta _1$ : $\\mathcal {D}(x_1^)+\\beta _1\\mathcal {F}(x_1^)\\le \\mathcal {D}(x_2^)+\\beta _1\\mathcal {F}(x_2^)$ $\\mathcal {D}(x_2^)+\\beta _2\\mathcal {F}(x_2^)\\le \\mathcal {D}(x_1^)+\\beta _2\\mathcal {F}(x_1^)$ Adding the above Equations, we will have: $\\beta _1\\mathcal {F}(x_1^)+\\beta _2\\mathcal {F}(x_2^)\\le \\beta _1\\mathcal {F}(x_2^)+\\beta _2\\mathcal {F}(x_1^)$ $\\Rightarrow (\\beta _2-\\beta _1)(\\mathcal {F}(x_2^)-\\mathcal {F}(x_1^)) \\le 0$ which implies $\\mathcal {F}(x_2^)\\le \\mathcal {F}(x_1^)$ .", "The monotonicity proof for $\\mathcal {D}(x)$ is the same with $\\beta =\\frac{(1-\\alpha )}{\\alpha }$ .", "If $\\alpha _2 < \\alpha _1$ then $\\mathcal {D}(x_1^)>\\mathcal {D}(x_2^*)$ ." ] ]	2011.14198
[ [ "Solving parametric systems of polynomial equations over the reals\n through Hermite matrices" ], [ "Abstract We design a new algorithm for solving parametric systems having finitely many complex solutions for generic values of the parameters.", "More precisely, let $f = (f_1, \\ldots, f_m)\\subset \\mathbb{Q}[y][x]$ with $y = (y_1, \\ldots, y_t)$ and $x = (x_1, \\ldots, x_n)$, $V\\subset \\mathbb{C}^{t+n}$ be the algebraic set defined by $f$ and $\\pi$ be the projection $(y, x) \\to y$.", "Under the assumptions that $f$ admits finitely many complex roots for generic values of $y$ and that the ideal generated by $f$ is radical, we solve the following problem.", "On input $f$, we compute semi-algebraic formulas defining semi-algebraic subsets $S_1, \\ldots, S_l$ of the $y$-space such that $\\cup_{i=1}^l S_i$ is dense in $\\mathbb{R}^t$ and the number of real points in $V\\cap \\pi^{-1}(\\eta)$ is invariant when $\\eta$ varies over each $S_i$.", "This algorithm exploits properties of some well chosen monomial bases in the algebra $\\mathbb{Q}(y)[x]/I$ where $I$ is the ideal generated by $f$ in $\\mathbb{Q}(y)[x]$ and the specialization property of the so-called Hermite matrices.", "This allows us to obtain compact representations of the sets $S_i$ by means of semi-algebraic formulas encoding the signature of a symmetric matrix.", "When $f$ satisfies extra genericity assumptions, we derive complexity bounds on the number of arithmetic operations in $\\mathbb{Q}$ and the degree of the output polynomials.", "Let $d$ be the maximal degree of the $f_i$'s and $D = n(d-1)d^n$, we prove that, on a generic $f=(f_1,\\ldots,f_n)$, one can compute those semi-algebraic formulas with $O^~( \\binom{t+D}{t}2^{3t}n^{2t+1} d^{3nt+2(n+t)+1})$ operations in $\\mathbb{Q}$ and that the polynomials involved have degree bounded by $D$.", "We report on practical experiments which illustrate the efficiency of our algorithm on generic systems and systems from applications.", "It allows us to solve problems which are out of reach of the state-of-the-art." ], [ "Problem statement and motivations", "In the whole paper, $\\mathbb {Q}$ , $\\mathbb {R}$ and $\\mathbb {C}$ denote respectively the fields of rational, real and complex numbers.", "Let ${\\mathbf {f}}=(f_1,\\ldots ,f_m)$ be a polynomial sequence in $\\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ where the indeterminates ${\\mathbf {y}}=(y_1,\\ldots ,y_t)$ are considered as parameters and ${\\mathbf {x}}=(x_1,\\ldots ,x_n)$ are considered as variables.", "We denote by ${\\mathcal {V}}\\subset \\mathbb {C}^{t}\\times \\mathbb {C}^n$ the (complex) algebraic set defined by $f_1=\\cdots =f_m=0$ and by ${\\mathcal {V}}_{\\mathbb {R}}$ its real trace ${\\mathcal {V}}\\cap \\mathbb {R}^{t+n}$ .", "We consider also the projection on the parameter space ${\\mathbf {y}}$ $\\pi :\\begin{array}{rl}\\mathbb {C}^t \\times \\mathbb {C}^n &\\rightarrow \\mathbb {C}^t, \\\\({\\mathbf {y}},{\\mathbf {x}}) & \\mapsto {\\mathbf {y}}.\\end{array}$ Further, we say that ${\\mathbf {f}}$ satisfies Assumption (REF ) when the following holds.", "AAssumption A There exists a non-empty Zariski open subset $\\mathcal {O} \\subset \\mathbb {C}^t$ such that $\\pi ^{-1}({\\eta })\\cap {\\mathcal {V}}$ is non-empty and finite for any ${\\eta }\\in \\mathcal {O}$ .", "In other words, assuming (REF ) ensures that, for a generic value ${\\eta }$ of the parameters, the sequence ${\\mathbf {f}}({\\eta },\\cdot )$ defines a finite algebraic set and hence finitely many real points.", "Note that, it is easy to prove that one can choose $\\mathcal {O}$ in a way that the number of complex solutions to the entries of ${\\mathbf {f}}({\\eta },\\cdot )$ is invariant when ${\\eta }$ ranges over $\\mathcal {O}$ (e.g.", "using the theory of Gröbner basis).", "This is no more the case when considering real solutions whose number may vary when ${\\eta }$ ranges over $\\mathcal {O}$ .", "By Hardt's triviality theorem [31], there exists a real algebraic proper subset $\\mathcal {R}$ of $\\mathbb {R}^t$ such that, for any non-empty connected open set $\\mathcal {U}$ of $\\mathbb {R}^t\\setminus \\mathcal {R}$ and ${\\eta }\\in \\mathcal {U}$ , $\\pi ^{-1}({\\eta }) \\times \\mathcal {U}$ is homeomorphic with $\\pi ^{-1}(\\mathcal {U})$ .", "This leads us to consider the following real root classification problem.", "Problem 1 (Real root classification) On input ${\\mathbf {f}}$ satisfying Assumption (REF ), compute semi-algebraic formulas (i.e.", "finitely many disjunctions of conjunctions of polynomial inequalities) defining semi-algebraic sets $\\mathcal {S}_1, \\ldots , \\mathcal {S}_\\ell $ such that (i) The number of real points in $\\mathcal {V}\\cap \\pi ^{-1}({\\eta })$ is invariant when ${\\eta }$ ranges over $\\mathcal {S}_i$ , for $1\\le i \\le \\ell $ ; (ii) The union of the $\\mathcal {S}_i$ 's is dense in $\\mathbb {R}^t$ ; as well as at least one sample point ${\\eta }_i$ in each $\\mathcal {S}_i$ and the corresponding number of real points in ${\\mathcal {V}}\\cap \\pi ^{-1}({\\eta }_i)$ .", "A collection of semi-algebraic formulas sets is said to solve Problem (REF ) for the input ${\\mathbf {f}}$ if it defines a collection of semi-algebraic sets ${\\mathcal {S}}_i$ satisfies the above properties (i) and (ii).", "Our output will have the form $\\lbrace (\\Phi _i,{\\eta }_i,r_i)\\; \| \\; 1 \\le i\\le \\ell \\rbrace $ where $\\Phi _i$ is a semi-algebraic formula defining the set ${\\mathcal {S}}_i$ , ${\\eta }_i \\in \\mathbb {Q}^t$ is a sample point of ${\\mathcal {S}}_i$ and $r_i$ is the corresponding number of real roots.", "A weak version of Problem (REF ) would be to compute only a set $\\lbrace {\\eta }_1, \\ldots , {\\eta }_{\\ell }\\rbrace $ of sample points for a collection of semi-algebraic sets ${\\mathcal {S}}_i$ solving Problem (REF ) and their corresponding numbers of real points in ${\\mathcal {V}}\\cap \\pi ^{-1}({\\eta }_j)$ .", "Example 2 Consider the equation $x^2+y_1x+y_2=0$ where $y_1$ and $y_2$ are the parameters and $x$ is the unique variable.", "While $y_1^2-4y_2\\ne 0$ , this equation always has exactly two distinct complex solutions.", "On the other hand, its number of distinct real solutions can take any value from 0 to 2, depending on the sign of the discriminant $y_1^2-4y_2$ .", "One possible output for Problem (REF ) on this toy example is the following: $\\left\\lbrace \\begin{array}{c c c}y_1^2-4y_2 < 0, & (0, 1), & 0 \\text{ real solution}\\\\y_1^2-4y_2 = 0, & (2, 1), & 1 \\text{ real solution}\\\\y_1^2-4y_2 > 0, & (1, 0), & 2 \\text{ real solutions}\\\\\\end{array}\\right.$ Observe that another possible output is $\\left\\lbrace \\begin{array}{c c c}y_1^2-4y_2 < 0, & (0, 1), & 0 \\text{ real solution}\\\\y_1^2-4y_2 > 0, & (1, 0), & 2 \\text{ real solutions}\\\\\\end{array}\\right.$ as the above two inequalities define semi-algebraic sets whose union is dense in $\\mathbb {R}^2$ .", "Problem (REF ) appears in many areas of engineering sciences such as robotics or medical imagery (see, e.g., [59], [14], [60], [22], [10]).", "In those applications, the behavior of mechanisms or complex systems depends on intrinsic parameters that are related by polynomial equations or inequalities.", "Thus, the polynomial systems arising from those applications are naturally parametric and most of the time the end-user is interested in classifying the number of real roots with respect to parameters' values." ], [ "Prior works", "A first approach to Problem (REF ) would be to compute a cylindrical algebraic decomposition (CAD) of $\\mathbb {R}^t\\times \\mathbb {R}^n$ adapted to ${\\mathbf {f}}$ using e.g.", "Collins' algorithm (and its more recent improvements) ; see [13].", "While, up to our knowledge, there is no clear reference for this fact, the cylindrical structure of the cells of the CAD will imply that their projection on the parameters' space $\\mathbb {R}^t$ define semi-algebraic sets enjoying the properties needed to solve Problem (REF ).", "However, the doubly exponential complexity of CAD both in terms of runtime and output size [18], [11] makes it difficult to use in practice.", "A more popular approach consists in computing polynomials $h_1,\\ldots , h_r$ in $\\mathbb {Q}[{\\mathbf {y}}]$ such that $\\cup _{i=1}^r V(h_i)\\cap \\mathbb {R}^t$ contains the boundaries of semi-algebraic sets $\\mathcal {S}_1, \\ldots ,\\mathcal {S}_\\ell $ enjoying the properties required to solve Problem (REF ).", "Next, one needs to compute semi-algebraic descriptions of the connected components of $\\mathbb {R}^t \\setminus \\cup _{i=1}^r V(h_i)$ as well as sample points in these connected components.", "This is basically the approach followed by [58] (the $h_i$ 's are called border polynomials) and [41] (the set $\\cup _{i=1}^r V(h_i)$ is called discriminant variety) under the assumption that $\\langle {\\mathbf {f}}\\rangle $ is a radical ideal.", "Note that both [58] and [41] provide algorithms that can handle variants of Problem (REF ) allowing inequalities.", "In this paper, we focus on the situation where we only have equations in our input parametric system.", "That being said, when $\\langle {\\mathbf {f}}\\rangle $ is radical and the restriction of $\\pi $ to ${\\mathcal {V}}\\cap \\mathbb {R}^t\\times \\mathbb {R}^n$ is proper, one can easily prove using a semi-algebraic version of Thom's isotopy lemma [15] that one can choose $\\cup _{i=1}^r V(h_i)$ to be the set critical values of the restriction of $\\pi $ to ${\\mathcal {V}}$ (see e.g.", "[10]).", "If ${\\mathbf {f}}$ is a regular sequence (hence $m = n$ ), the critical set of the restriction of $\\pi $ to ${\\mathcal {V}}$ is defined as the intersection of ${\\mathcal {V}}$ with the hypersurface defined by the vanishing of the determinant of the Jacobian matrix of ${\\mathbf {f}}$ with respect to the variables ${\\mathbf {x}}$ .", "When $d$ dominates the degrees of the entries of ${\\mathbf {f}}$ , Bézout's theorem allows us to state that the degree of this set is bounded above by $n(d-1)d^n$ .", "It is worth noticing that, usually, this approach is used only to solve the aforementioned weak version of Problem (REF ) as getting a semi-algebraic description of the connected components of $\\mathbb {R}^t \\setminus \\cup _{i=1}^r V(h_i)$ through CAD is too expensive when $t\\ge 4$ (still, because of the doubly exponential complexity of CAD).", "Under the above assumptions and notation, the output degree of the polynomials in such formulas would be bounded by $\\left(n(d-1)d^n\\right)^{2^{O(t)}}$ .", "An alternative would be to use parametric roadmap algorithms to do such computations using e.g.", "[5] to compute semi-algebraic representations of the connected components of $\\mathbb {R}^t\\setminus \\cup _{i=1}^r V(h_i)$ .", "Under the above extra assumptions, this would result in output formulas involving polynomials of degree bounded by $\\left(n(d-1)d^n\\right)^{O(t^3)}$ using $\\left(n(d-1)d^n\\right)^{O(t^4)}$ arithmetic operations (see [5]).", "Note that the output degrees are by several orders of magnitude larger than $n(d-1)d^n$ which bounds the degree of the set of critical values of the restriction of $\\pi $ to ${\\mathcal {V}}$ .", "Hence, one topical algorithmic issue is to design an efficient algorithm for solving Problem (REF ) which would output semi-algebraic formulas of degree bounded by $n(d-1)d^n$ .", "At this stage of our exposition, this is not clear that it is doable.", "We describe in detail our contributions in the next paragraph but we can already state that, when ${\\mathbf {f}}$ enjoys some genericity properties that are made clear further, the algorithm we design outputs semi-algebraic formulas involving polynomials of degree bounded by $n(d-1)d^n$ and which are computed using $\\left(n(d-1)d^n\\right)^{O(t)}$ arithmetic operations in $\\mathbb {Q}$ .", "To achieve these results, we revisit tools for univariate real root counting, such as Sturm and Sturm-Habicht sequences and Hermite's quadratic form to adapt them in our multivariate setting.", "This leads us to mention [29], [43] or [34] which provide algorithms for classifying the real roots of a univariate polynomial with coefficients in $\\mathbb {Q}[{\\mathbf {y}}]$ , hence restricted to the case where $n=1$ (either using Sturm-based techniques or Hermite's quadratic forms)." ], [ "Main results", "We start by revisiting Sturm-based methods in a multivariate context.", "We basically use the algorithm of [53] to compute a rational parametrization of ${\\mathcal {V}}= V({\\mathbf {f}})$ with respect to the ${\\mathbf {x}}$ -variables.", "More precisely, we compute a sequence of polynomials $({w}, v_1, \\ldots , v_n)$ in $\\mathbb {Q}({\\mathbf {y}})[u]$ where $u$ is a new variable, such that the constructible set $\\mathcal {Z} \\subset \\mathbb {C}^t\\times \\mathbb {C}^n$ of every point $\\left({\\eta },\\frac{v_1}{\\partial {w}/ \\partial u}({\\eta }, \\vartheta ), \\ldots , \\frac{v_n}{\\partial {w}/ \\partial u}({\\eta }, \\vartheta )\\right),$ where $({\\eta },\\vartheta ) \\in \\mathbb {C}^t \\times \\mathbb {C}$ such that $w({\\eta },\\vartheta ) = 0$ and ${\\eta }$ does not cancel $\\partial w/ \\partial u$ and any denominator of $({w}, v_1,\\ldots ,v_n)$ , is Zariski dense in ${\\mathcal {V}}$ , i.e., the Zariski closure of $\\mathcal {Z}$ coincides with ${\\mathcal {V}}$ .", "Then, using the bi-rational equivalence between $\\mathcal {Z}$ and its projection on the $(u, {\\mathbf {y}})$ -space, we establish that semi-algebraic formulas solving Problem (REF ) can be obtained through the computation of the subresultant sequence associated to $\\left({w}, \\frac{\\partial {w}}{\\partial u}\\right)$ .", "This is admittedly folklore in symbolic computation but, as far as we know, is not explicitly written in the literature.", "In particular, the analysis of degree bounds derived from this strategy is one of our contributions.", "Before stating our first complexity result, we need to introduce the complexity model which is used.", "Throughout this paper, we measure only the arithmetic complexity of algorithms, i.e., the number of arithmetic operations $+,-,\\times ,\\div $ , in the base field $\\mathbb {Q}$ .", "We use the Landau notation: Let $f: \\mathbb {R}_+^{\\ell } \\mapsto \\mathbb {R}_+$ be a positive function.", "We let $O(f)$ denote the class of functions $g: \\mathbb {R}_+^{\\ell } \\rightarrow \\mathbb {R}_+$ such that there exist $C,K \\in \\mathbb {R}_+$ such that for all $\\Vert x\\Vert \\ge K$ , $g(x) \\le C f(x)$ , where $\\Vert \\cdot \\Vert $ is a norm of $\\mathbb {R}^{\\ell }$ .", "The notation $O\\ {\\widetilde{~}}$ denotes the class of functions $g:\\mathbb {R}_+^{\\ell } \\rightarrow \\mathbb {R}_+$ such that $ g \\in O(f\\log ^{\\kappa }(f))$ for some $\\kappa > 0 $ .", "Further, the notation $\\omega $ always stands for the exponent constant of the matrix multiplication, i.e., the smallest positive number such that the product of two matrices in $\\mathbb {Q}^{N\\times N}$ can be done using $O\\left(N^{\\omega }\\right)$ arithmetic operations in $\\mathbb {Q}$ .", "The value of $\\omega $ can be bounded from above by $2.3729$ , which is a recent result established in [42].", "Under some genericity assumptions on the input system, Theorem REF establishes the complexity result of our Sturm-based algorithm and also the degree bound for polynomials involved in the semi-algebraic formulas solving Problem (REF ) obtained this way.", "Its proof is given in Section , where all the genericity assumptions are clarified.", "Theorem I Let ${\\mathbf {f}}= (f_1,\\ldots ,f_n) \\subset \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ be a generic parametric system and $d$ be the largest total degree among the $\\deg (f_i)$ 's.", "Then, there exists a probabilistic algorithm that computes semi-algebraic descriptions of a set of semi-algebraic sets solving Problem REF within $O\\ {\\widetilde{~}}\\left(\\binom{t+2d^{2n}}{t}\\ 2^{5t} \\ d^{5nt+3n}\\right)$ arithmetic operations in $\\mathbb {Q}$ in case of success.", "These semi-algebraic formulas computed by this algorithm involve polynomials in $\\mathbb {Q}[{\\mathbf {y}}]$ of degree bounded by $2d^{2n}$ .", "When reporting experimental results, we will see that, even though the complexity bound we obtain lies in $d^{O(nt)}$ , this approach does not allow us to solve problems faster than the state-of-the-art.", "One bottleneck comes from the fact that the polynomials of the output semi-algebraic formulas have degree way higher than the bound $n(d-1)d^n$ which we will prove to apply under the same assumptions as Theorem REF using different algorithmic strategies.", "Note that the above Sturm-based approach as well as the ones which consist in computing polynomials in $\\mathbb {Q}[{\\mathbf {y}}]$ to define a set discriminating semi-algebraic sets in $\\mathbb {R}^t$ enjoying the properties needed to solve Problem (REF ) combine two steps of algebraic elimination.", "The semi-algebraic formulas are obtained through intermediate data who have been obtained through an elimination step.", "The rest of the paper then focuses on an alternative approach which computes semi-algebraic formulas solving Problem (REF ) by avoiding interlaced algebraic elimination steps.", "We will see (as announced earlier) that under genericity assumptions, this allows us to obtain a degree bound and an arithmetic cost which are better than the Sturm-based algorithm by one order of magnitude.", "To do that, we rely on well-known properties of Hermite quadratic forms to count the real roots of zero-dimensional ideals ; see [36].", "Basically, given a zero-dimensional ideal $I\\subset \\mathbb {Q}[{\\mathbf {x}}]$ , Hermite's quadratic form operates on the finite dimensional $\\mathbb {Q}$ -vector space $A \\mathbb {Q}[{\\mathbf {x}}]/I$ as follows $A \\times A & \\rightarrow \\mathbb {Q}\\\\(h,k) & \\mapsto \\mathrm {trace}({\\mathcal {L}}_{h\\cdot k}),$ where ${\\mathcal {L}}_{h\\cdot k}$ denotes the endomorphism $p\\mapsto h\\cdot k\\cdot p$ of $A$ .", "The number of distinct real (resp.", "complex) roots of the algebraic set defined by $I$ equals the signature (resp.", "rank) of Hermite's quadratic form ; see e.g.", "[5].", "Recall that such quadratic form is represented by a symmetric matrix of size $\\delta \\times \\delta $ , where $\\delta $ is the degree of $I$ , once a basis of the finite dimensional vector space on which the form operates is fixed.", "Hence, the signature of a Hermite quadratic form can be computed once a matrix representation, which we call Hermite's matrix, of this quadratic form is known [5].", "We first slightly extend the definition of Hermite's quadratic forms and Hermite's matrices to the context of parametric systems; we call them parametric Hermite quadratic forms and parametric Hermite matrices.", "This is easily done since the ideal of $\\mathbb {Q}({\\mathbf {y}})[{\\mathbf {x}}]$ generated by ${\\mathbf {f}}$ , considering $\\mathbb {Q}({\\mathbf {y}})$ as the base field, has dimension zero.", "We also establish natural specialization properties for these parametric Hermite matrices.", "Hence, a parametric Hermite matrix, similar to its zero-dimensional counterpart, allows one to count respectively the number of distinct real and complex roots at any parameters outside a strict algebraic sets of $\\mathbb {R}^t$ by evaluating the signature and rank of its specialization.", "Based on this specialization property, we design two algorithms for solving Problem (REF ) and also its weak version for the input system ${\\mathbf {f}}$ which satisfies Assumption (REF ) and generates a radical ideal.", "Our algorithm for the weak version of Problem (REF ) reduces to the following main steps.", "(a) Compute a parametric Hermite matrix ${\\mathcal {H}}$ associated to ${\\mathbf {f}}\\subset \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ .", "(b) Compute a set of sample points $\\lbrace {\\eta }_1, \\ldots ,{\\eta }_{\\ell }\\rbrace $ in the connected components of the semi-algebraic set of $\\mathbb {R}^t$ defined by $\\mathbf {w} \\ne 0$ where $\\mathbf {w}$ is derived from ${\\mathcal {H}}$ .", "This is done through the so-called critical point method (see e.g.", "[5] and references therein) which are adapted to obtain practically fast algorithms following [49].", "We will explain in detail this step in Section .", "This algorithm takes as input $s$ polynomials of degree $D$ involving $t$ variables and computes sample points per connected components in the semi-algebraic set defined by the non-vanishing of these polynomials using $ O\\ \\widetilde{~} \\left(\\binom{D + t}{t}(2t)^4s^{t+1}2^{3t}D^{2t+1} \\right) .$ (c) Compute the number $r_i$ of real points in ${\\mathcal {V}}\\cap \\pi ^{-1}({\\eta }_i)$ for $1\\le i \\le \\ell $ .", "This is done by simply evaluating the signature of the specialization of ${\\mathcal {H}}$ at each ${\\eta }_i$ .", "It is worth noting that, in the algorithm above, we obtain through parametric Hermite matrices a polynomial $\\mathbf {w}$ that plays the same role as the discriminant varieties of [41] or the border polynomials of [57].", "We will see in the section reporting experiments that our approach outperforms the other twos on every example we consider.", "To return semi-algebraic formulas, we follow a slightly different routine: (a) Compute a parametric Hermite matrix ${\\mathcal {H}}$ associated to ${\\mathbf {f}}\\subset \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ .", "(b) Compute a set of sample points $\\lbrace {\\eta }_1, \\ldots ,{\\eta }_{\\ell }\\rbrace $ in the connected components of the semi-algebraic set of $\\mathbb {R}^t$ defined by $\\wedge _{i=1}^{\\delta } M_i \\ne 0$ where the $M_i$ 's are the leading principal minors of ${\\mathcal {H}}$ .", "Again, this is done by the algorithm given in Section .", "(c) For $1\\le i \\le \\ell $ , evaluate the sign pattern of $(M_1,\\ldots ,M_{\\delta })$ at the sample point ${\\eta }_i$ .", "From this sign pattern, we obtain a semi-algebraic formula representing the connected component corresponding to ${\\eta }_i$ .", "(d) Compute the number $r_i$ of real points in ${\\mathcal {V}}\\cap \\pi ^{-1}({\\eta }_i)$ for $1\\le i \\le \\ell $ .", "Another contribution of this paper is to make clear how to perform the step (a).", "For this, we rely on the theory of Gröbner bases.", "More precisely, we use specialization properties of Gröbner bases, similar to those already proven in [38].", "This leaves some freedom when running the algorithm: since we rely on Gröbner bases, one may choose monomial orderings which are more convenient for practical computations.", "In particular, the monomial basis of the quotient ring $\\mathbb {Q}({\\mathbf {y}})[{\\mathbf {x}}]/I$ where $I$ is the ideal generated by ${\\mathbf {f}}$ in $\\mathbb {Q}({\\mathbf {y}})[{\\mathbf {x}}]$ depends on the choice of the monomial ordering used for Gröbner bases computations.", "We describe the behavior of our algorithm when choosing the graded reverse lexicographical ordering whose interest for practical computations is explained in [9], [8].", "Further, we denote by ${grevlex}({\\mathbf {x}})$ the graded reverse lexicographical ordering applied to the sequence of the variables ${\\mathbf {x}}= (x_1, \\ldots , x_n)$ (with $x_1\\succ \\cdots \\succ x_n$ ).", "Further, we also denote by $\\succ _{{lex}}$ the lexicographical ordering.", "We report, at the end of the paper, on the practical behavior of this algorithm.", "In particular, it allows us to solve instances of Problem (REF ) which were not tractable by the state-of-the-art as well as the actual degrees of the polynomials in the output formula which are bounded by $n(d-1)d^n$ .", "We actually prove such a statement under some generic assumptions.", "Our main complexity result is stated below.", "Its proof is given in Subsection REF , where the generic assumptions in use are given explicitly.", "Theorem II Let $\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}}]_d$ be the set of polynomials in $\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}}]$ having total degree bounded by $d$ and set $\\mathfrak {D} =n(d-1)d^n$ .", "There exists a non-empty Zariski open set ${F}\\subset \\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}}]_d^n$ such that for ${\\mathbf {f}}=(f_1,\\ldots ,f_n) \\in {F} \\cap \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]^n$ , the following holds: i) There exists an algorithm that computes a solution for the weak-version of Problem (REF ) within $O\\ {\\widetilde{~}}\\left(\\binom{t+\\mathfrak {D}}{t} \\ 2^{3t}\\ n^{2t+1} d^{2nt+n+2t+1} \\right).$ arithmetic operations in $\\mathbb {Q}$ .", "ii) There exists a probabilistic algorithm that returns the formulas of a collection of semi-algebraic sets solving Problem (REF ) within $O\\ {\\widetilde{~}}\\left( \\binom{t+\\mathfrak {D}}{t}\\ 2^{3t}\\ n^{2t+1} d^{3nt+2(n+t)+1} \\right)$ arithmetic operations in $\\mathbb {Q}$ in case of success.", "iii) The semi-algebraic descriptions output by the above algorithm involves polynomials in $\\mathbb {Q}[{\\mathbf {y}}]$ of degree bounded by $\\mathfrak {D}$ .", "We note that the binomial coefficient $\\binom{t+\\mathfrak {D}}{t}$ is bounded from above by $\\mathfrak {D}^t \\simeq n^td^{nt+t}$ .", "Therefore, the complexities given in the items i) and ii) of Theorem REF can be bounded by $O\\ {\\widetilde{~}}\\left(2^{3t}\\ n^{3t} d^{3nt} \\right)$ and $O\\ {\\widetilde{~}}\\left( 2^{3t}\\ n^{3t} d^{4nt} \\right)$ respectively." ], [ "Organization of the paper", "This paper is structured as follows.", "Section reviews fundamental notions of algebraic geometry and the theory of Gröbner bases that we use further.", "In Section , we discuss an algorithm based on Sturm's theorem for computing semi-algebraic formulas of the set ${\\mathcal {S}}_i$ .", "This provides an overview on the drawbacks and potential improvements of this approach.", "Section lies the definition and some useful properties of parametric Hermite matrices.", "There, we also present an algorithm with some optimizations to compute such a matrix.", "In Section , we describe our algorithm for solving the real root classification problem using this parametric Hermite matrix.", "The complexity analysis of the algorithms mentioned above is given in Section .", "Finally, in Section , we report on the practical behavior of our algorithms and illustrate its practical capabilities." ], [ "Preliminaries", "In the first paragraph, we fix some notations on ideals and algebraic sets and recall the definition of critical points associated to a given polynomial map.", "This notion is the foundation of many algorithms in semi-algebraic geometry such as computing sample points of connected components [1], [49], [2], polynomial optimization [47], [30] or answering connectivity queries using roadmaps [50], [6], [7], [51].", "Next, we give the definitions of regular sequences, Hilbert series, Noether position and proper maps, which are used later in Subsection REF for the complexity analysis of our algorithms.", "The fourth paragraph recalls some basic properties of Gröbner bases and quotient algebras of zero-dimensional ideals.", "We refer to [16] for an introductory study on the algorithmic theory of Gröbner bases.", "In the last paragraphs, we recall respectively the definitions of zero-dimensional parametrizations and rational parametrizations.", "The zero-dimensional parametrization goes back to [39] and is widely used in computer algebra (see e.g.", "[26], [28], [27]) to represent finite algebraic sets.", "In Section , the output of our algorithm for computing points per connected components of the non-vanishing locus of a given set of polynomials is encoded by this parametrization.", "On the other hand, the rational parametrization, which generalizes the notion of zero-dimensional parametrizations, is introduced in [53] as the data representation for the parametric geometric resolution algorithm.", "In Section , we compute a rational parametrization of the input system using the algorithm of [53] to reduce Problem (REF ) to the univariate case." ], [ "Algebraic sets and critical points", "We consider a sub-field ${\\mathbb {F}}$ of $\\mathbb {C}$ .", "Let $I$ be a polynomial ideal of ${\\mathbb {F}}[x_1,\\ldots ,x_n]$ , the algebraic subset of $\\mathbb {C}^n$ at which the elements of $I$ vanish is denoted by $V(I)$ .", "Conversely, for an algebraic set ${\\mathcal {V}}\\subset \\mathbb {C}^n$ , we denote by $I({\\mathcal {V}})\\subset \\mathbb {C}[x_1,\\ldots ,x_n]$ the radical ideal associated to ${\\mathcal {V}}$ .", "Given any subset $\\mathcal {A}$ of $\\mathbb {C}^n$ , we denote by $\\overline{\\mathcal {A}}$ the Zariski closure of $\\mathcal {A}$ , i.e., the smallest algebraic set containing $\\mathcal {A}$ .", "A map $\\varphi $ between two algebraic sets ${\\mathcal {V}}\\subset \\mathbb {C}^n$ and ${\\mathcal {W}}\\subset \\mathbb {C}^s$ is a polynomial map if there exist $\\varphi _1,\\ldots ,\\varphi _t \\in \\mathbb {C}[x_1,\\ldots ,x_n]$ such that the $\\varphi ({\\eta })=(\\varphi _1({\\eta }),\\ldots ,\\varphi _s({\\eta }))$ for ${\\eta }\\in {\\mathcal {V}}$ .", "An algebraic set ${\\mathcal {V}}$ is equi-dimensional of dimension $t$ if it is the union of irreducible algebraic sets of dimension $t$ .", "Let $\\varphi $ be a polynomial map from ${\\mathcal {V}}$ to another algebraic set ${\\mathcal {W}}$ .", "The morphism $\\varphi $ is dominant if and only if the image of every irreducible component ${\\mathcal {V}}^{\\prime }$ of ${\\mathcal {V}}$ by $\\varphi $ is Zariski dense in ${\\mathcal {W}}$ , i.e.", "$\\overline{\\varphi ({\\mathcal {V}}^{\\prime })} = {\\mathcal {W}}$ .", "Let $\\phi \\in \\mathbb {C}[x_1,\\ldots ,x_n]$ which defines the polynomial function $\\phi :\\begin{array}{rl}\\mathbb {C}^n & \\rightarrow \\mathbb {C}, \\\\(x_1, \\ldots , x_n) & \\mapsto \\phi (x_1,\\ldots ,x_n)\\end{array}$ and ${\\mathcal {V}}\\subset \\mathbb {C}^n$ be a smooth equi-dimensional algebraic set.", "We denote by $\\mathrm {crit}(\\phi , {\\mathcal {V}})$ the set of critical points of the restriction of $\\phi $ to ${\\mathcal {V}}$ .", "If $c$ is the codimension of ${\\mathcal {V}}$ and $(f_1,\\ldots ,f_m)$ generates the vanishing ideal associated to ${\\mathcal {V}}$ , then $\\mathrm {crit}(\\phi , {\\mathcal {V}})$ is the subset of ${\\mathcal {V}}$ at which the Jacobian matrix associated to $(f_1, \\ldots , f_m, \\phi )$ has rank less than or equal to $c$ (see, e.g., [51])." ], [ "Regular sequences & Hilbert series", "Let ${\\mathbb {F}}$ be a field and $(f_1,\\ldots ,f_m) \\subset {\\mathbb {F}}[{\\mathbf {x}}]$ where ${\\mathbf {x}}= (x_1,\\ldots ,x_n)$ and $m\\le n$ be a homogeneous polynomial sequence.", "We say that $(f_1,\\ldots ,f_m)\\subset {\\mathbb {F}}[{\\mathbf {x}}]$ is a regular sequence if for any $i\\in \\lbrace 1,\\ldots ,m\\rbrace $ , $f_i$ is not a zero-divisor in ${\\mathbb {F}}[{\\mathbf {x}}]/\\langle f_1,\\ldots ,f_{i-1}\\rangle $ .", "The notion of regular sequences is the algebraic analogue of complete intersection.", "In this paper, we focus particularly on the Hilbert series of homogeneous regular sequences, which are recalled below.", "Let $I \\subset {\\mathbb {F}}[{\\mathbf {x}}]$ be a homogeneous ideal.", "We denote by ${\\mathbb {F}}[{\\mathbf {x}}]_{r}$ the set of every homogeneous polynomial whose degree is equals to $r$ .", "Then ${\\mathbb {F}}[{\\mathbf {x}}]_{r}$ and $I\\cap {\\mathbb {F}}[{\\mathbf {x}}]_{r}$ are two ${\\mathbb {F}}$ -vector spaces of dimensions $\\dim _{{\\mathbb {F}}}({\\mathbb {F}}[{\\mathbf {x}}]_r)$ and $\\dim _{{\\mathbb {F}}}(I\\cap {\\mathbb {F}}[{\\mathbf {x}}]_r)$ respectively.", "The Hilbert series of $I$ is defined as ${\\rm HS}_{I}(z) = \\sum _{r=0}^{\\infty }(\\dim _{{\\mathbb {F}}}({\\mathbb {F}}[{\\mathbf {x}}]_{r})-\\dim _{{\\mathbb {F}}}(I\\cap {\\mathbb {F}}[{\\mathbf {x}}]_r)) \\cdot z^r.$ When $I$ can be generated by a homogeneous regular sequence $(f_1,\\ldots ,f_m)$ , the explicit form of the Hilbert series of $I$ is known (see, e.g., [44]): ${\\rm HS}_{I}(z) = \\frac{\\prod _{i=1}^m \\left( 1 - z^{\\deg (f_i)}\\right)}{(1 - z)^n}.$ We now consider the affine polynomial sequences.", "Note that one can define affine regular sequences by simply removing the homogeneity assumption of $(f_1,\\ldots ,f_m)$ from the above definition.", "However, as explained in [3], many important properties that hold for homogeneous regular sequences are no longer valid for the affine ones.", "Therefore, in this paper, we use [3] of affine regular sequences, which is more restrictive but allows us to preserve similar results as the homogeneous case.", "We recall that definition below.", "For $p\\in {\\mathbb {F}}[x_1,\\ldots ,x_n]$ , we denote by ${}^Hp$ the homogeneous component of largest degree of $p$ .", "A polynomial sequence $(f_1,\\ldots ,f_m) \\subset {\\mathbb {F}}[x_1,\\ldots ,x_n]$ , not necessarily homogeneous, is called a regular sequence if and only if $({}^Hf_1,\\ldots ,{}^Hf_m)$ is a homogeneous regular sequence." ], [ "Noether position & Properness", "Let ${\\mathbb {F}}$ be a field and ${\\mathbf {f}}=(f_1,\\ldots ,f_n)\\subset {\\mathbb {F}}[x_1,\\ldots ,x_{n+t}]$ .", "The variables $(x_1,\\ldots ,x_n)$ are in Noether position with respect to the ideal $\\langle {\\mathbf {f}}\\rangle $ if their canonical images in the quotient algebra ${\\mathbb {F}}[x_1,\\ldots ,x_{n+t}]/\\langle {\\mathbf {f}}\\rangle $ are algebraic integers over ${\\mathbb {F}}[x_{n+1}, \\ldots , x_{n+t}]$ and, moreover, ${\\mathbb {F}}[x_{n+1},\\ldots ,x_{n+t}]\\cap \\langle {\\mathbf {f}}\\rangle = \\langle 0\\rangle $ .", "From a geometric point of view, Noether position is strongly related to the notion of proper map below (see [4]).", "Let ${\\mathcal {V}}$ be the algebraic set defined by ${\\mathbf {f}}\\in \\mathbb {R}[y_1,\\ldots ,y_t,x_1,\\ldots ,x_n]$ .", "The restriction of the projection $\\pi :({\\mathbf {y}},{\\mathbf {x}}) \\mapsto {\\mathbf {y}}$ to ${\\mathcal {V}}\\cap \\mathbb {R}^{t+n}$ is said to be proper if the inverse image of every compact subset of $\\pi ({\\mathcal {V}}\\cap \\mathbb {R}^{t+n})$ is compact.", "If the variables ${\\mathbf {x}}=(x_1,\\ldots ,x_n)$ is in Noether position with respect to $\\langle {\\mathbf {f}}\\rangle $ , then the projection $\\pi : {\\mathcal {V}}\\cap \\mathbb {R}^{t+n} \\rightarrow \\mathbb {R}^t, \\; ({\\mathbf {y}},{\\mathbf {x}}) \\mapsto {\\mathbf {y}}$ is proper.", "A point ${\\eta }\\in \\mathbb {R}^t$ is a non-proper point of the restriction of $\\pi $ to ${\\mathcal {V}}$ if and only $\\pi ^{-1}(\\mathcal {U}) \\cap {\\mathcal {V}}\\cap \\mathbb {R}^{t+n} $ is not compact for any compact neighborhood $\\mathcal {U}$ of ${\\eta }$ in $\\mathbb {R}^t$ .", "Example 3 We consider the ideal $\\langle x^2+y^2-1\\rangle $ .", "One can easily see that $x$ is in Noether position with respect to this ideal as the equation $x^2+y^2-1$ is monic in $x$ .", "On the other hand, the variable $x$ is not in Noether position with respect to the ideal $\\langle xy-1\\rangle $ .", "This can be observed geometrically as the fiber at $y=0$ of the projection of $V(xy-1)$ to the $y$ -space lies in infinity.", "Another example is the ideal $\\langle yx^2+2x-1 \\rangle $ .", "The variable $x$ is not in Noether position with respect to this ideal.", "The fiber at $y=0$ of the projection of $V(yx^2+2x-1)$ to the $y$ -space contains a point $(1/2,0)$ and a point at infinity.", "So, this projection is not proper." ], [ "Gröbner bases and zero-dimensional ideals", "Let $\\mathbb {F}$ be a field and $\\overline{\\mathbb {F}}$ be its algebraic closure.", "We denote by $\\mathbb {F}[\\mathbf {x}]$ the polynomial algebra in the variables $\\mathbf {x}=(x_1,\\ldots ,x_n)$ .", "We fix an admissible monomial ordering $\\succ $ (see Section 2.2, [16]) over $\\mathbb {F}[{\\mathbf {x}}]$ .", "For a polynomial $p\\in \\mathbb {F}[\\mathbf {x}]$ , the leading monomial of $p$ with respect to $\\succ $ is denoted by $\\mathrm {lm}_{\\succ }(p)$ .", "Given an ideal $I\\subset \\mathbb {F}[\\mathbf {x}]$ , the initial ideal of $I$ with respect to the ordering $\\succ $ is the ideal $\\langle \\mathrm {lm}_{\\succ }(p) \\; \| \\; p \\in I \\rangle $ .", "A Gröbner basis $G$ of $I$ with respect to the ordering $\\succ $ is a generating set of $I$ such that the set of leading monomials $\\lbrace \\mathrm {lm}_{\\succ }(g) \\; \| \\; g\\in G\\rbrace $ generates the initial ideal $\\langle \\mathrm {lm}_{\\succ }(p) \\; \|\\; p\\in I\\rangle $ .", "For any polynomial $p \\in \\mathbb {F}[\\mathbf {x}]$ , the remainder of the division of $p$ by $G$ using the monomial ordering $\\succ $ is uniquely defined.", "It is called the normal form of $p$ with respect to $G$ and is denoted by $\\mathrm {NF}_{G}(p)$ .", "A polynomial $p$ is reduced by $G$ if $p$ coincides with its normal form in $G$ .", "A Gröbner basis $G$ is said to be reduced if, for any $g\\in G$ , all terms of $g$ are reduced modulo the leading terms of $G$ .", "An ideal $I$ is said to be zero-dimensional if the algebraic set $V(I)\\subset \\overline{{\\mathbb {F}}}^n$ is finite and non-empty.", "By [16], the quotient ring $\\mathbb {F}[\\mathbf {x}]/I$ is a $\\mathbb {F}$ -vector space of finite dimension.", "The dimension of this vector space is also called the algebraic degree of $I$ ; it coincides with the number of points of $V(I)$ counted with multiplicities [5].", "For any Gröbner basis of $I$ , the set of monomials in $\\mathbb {F}[\\mathbf {x}]$ which are irreducible by $G$ forms a monomial basis, which we call $B$ , of this vector space.", "For any $p\\in \\mathbb {F}[{\\mathbf {x}}]$ , the normal form of $p$ by $G$ can be interpreted as the image of $p$ in ${\\mathbb {F}}[{\\mathbf {x}}]/I$ and is a linear combination of elements of $B$ (with coefficients in $\\mathbb {F}$ ).", "Therefore, the operations in the quotient algebra $\\mathbb {F}[\\mathbf {x}]/I$ such as vector additions or scalar multiplications can be computed explicitly using the normal form reduction.", "In this article, while working with polynomial systems depending on parameters in $\\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ , we frequently take $\\mathbb {F}$ to be the rational function field $\\mathbb {Q}({\\mathbf {y}})$ and treat polynomials in $\\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ as elements of $\\mathbb {Q}({\\mathbf {y}})[{\\mathbf {x}}]$ ." ], [ "Zero-dimensional parametrizations", "A zero-dimensional parametrization ${R}$ of coefficients in $\\mathbb {Q}$ consists of a sequence of polynomials $(w,v_1,\\ldots ,v_n) \\in (\\mathbb {Q}[u])^{n+1}$ with a new variable $u$ and $(a_1,\\ldots ,a_n) \\in \\mathbb {Q}^n$ such that $w$ is square-free and $u=(\\sum _{i=1}^n a_i\\cdot v_i/w^{\\prime }) \\bmod w$ .", "The solution set of ${R}$ , defined as $\\left\\lbrace \\left(\\frac{v_1(\\vartheta )}{w^{\\prime }(\\vartheta )}, \\ldots ,\\frac{v_n(\\vartheta )}{w^{\\prime }(\\vartheta )}\\right) \\in \\mathbb {C}^n\\;\|\\;\\vartheta \\in \\mathbb {C}\\text{ such that } w(\\vartheta ) = 0\\right\\rbrace ,$ is finite and is denoted by $Z({R})$ .", "A finite algebraic set ${\\mathcal {V}}\\in \\mathbb {C}^n$ is said to be represented by a zero-dimensional parametrization ${R}$ if and only if ${\\mathcal {V}}$ coincides with $Z({R})$ .", "Note that it is possible to retrieve a polynomial parametrization by inverting the derivative $w^{\\prime }$ modulo $w$ .", "Still, this rational parametrization whose denominator is the derivative of $w$ is known to be better for practical computations as it usually involves coefficients with smaller bit size (see [17])." ], [ "Rational parametrizations", "We consider now a parametric system ${\\mathbf {f}}\\in \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ where ${\\mathbf {y}}= (y_1,\\ldots ,y_t)$ are parameters and ${\\mathbf {x}}= (x_1,\\ldots ,x_n)$ are variables.", "Under some extra assumptions on the system ${\\mathbf {f}}$ , [53] proves the existence of a sequence $(w,v_1,\\ldots ,v_n) \\subset \\left(\\mathbb {Q}({\\mathbf {y}})[u]\\right)^n$ with a new variable $u$ and a Zariski open subset $\\mathcal {Y} \\subset \\mathbb {C}^t$ such that $w$ is a square-free polynomial in $\\mathbb {Q}({\\mathbf {y}})[u]$ .", "$u = \\sum _{i=1}^n a_ix_i$ for some $(a_1,\\ldots ,a_n)\\in \\mathbb {Q}^n$ .", "For ${\\eta }\\in \\mathcal {Y}$ , ${\\eta }$ does not cancel any denominator of $(w,v_1,\\ldots ,v_n)$ and $V({\\mathbf {f}}({\\eta },\\cdot )) = \\left\\lbrace \\left(\\frac{v_1}{\\partial {w}/\\partial u}({\\eta },\\vartheta ), \\ldots , \\frac{v_n}{\\partial {w}/\\partial u}({\\eta },\\vartheta )\\right)\\; \\left\| \\;{w}({\\eta },\\vartheta ) = 0, \\frac{\\partial {w}}{\\partial u}({\\eta },\\vartheta ) \\ne 0 \\right.\\right\\rbrace .$ The sequence $(w,v_1,\\ldots ,v_n)$ is called a rational parametrization of ${\\mathbf {f}}$ .", "It can be computed using the parametric geometric resolution algorithm which is described in [53].", "Intuitively, this parametrization provides a generic description for the solutions of ${\\mathbf {f}}({\\eta },\\cdot )$ when ${\\eta }$ ranges over $\\mathbb {C}^t$ .", "It generalizes the notion of zero-dimensional parametrizations to the parametric setting." ], [ "Computing sample points in semi-algebraic sets defined by the non-vanishing of polynomials", "In this section, we study the following algorithmic problem.", "Given $(g_1, \\ldots , g_s)$ in $\\mathbb {Q}[y_1, \\ldots , y_t]$ , compute at least one sample point per connected component of the semi-algebraic set $\\mathcal {S}\\subset \\mathbb {R}^t$ defined by $g_1\\ne 0, \\ldots , g_s\\ne 0.$ Such sample points will be encoded with zero-dimensional parametrizations which we described in Section .", "The main result of this section which will be used in the sequel of this paper is the following.", "Theorem III Let $(g_1, \\ldots , g_s)$ in $\\mathbb {Q}[y_1, \\ldots , y_t]$ with $D \\ge \\max _{1\\le i \\le s} \\deg (g_i)$ and $\\mathcal {S}\\subset \\mathbb {R}^t$ be the semi-algebraic set defined by $g_1\\ne 0, \\ldots , g_s\\ne 0.$ There exists a probabilistic algorithm which on input $(g_1, \\ldots , g_s)$ outputs a finite family of zero-dimensional parametrizations ${R}_1, \\ldots , {R}_k$ encodes at most $\\left( 2sD\\right)^t $ points such that $\\cup _{i=1}^kZ({R}_i)$ meets every connected component of $\\mathcal {S}$ using $O\\ {\\widetilde{~}}\\left(\\binom{D+t}{t} (2t)^4 s^{t+1} 2^{3t} D^{2t+1}\\right).$ arithmetic operations in $\\mathbb {Q}$ .", "The rest of this section is devoted to the proof of this theorem.", "By [22], there exists a non-empty Zariski open set $\\mathcal {A}\\times \\mathcal {E}\\subset \\mathbb {C}^s\\times \\mathbb {C}$ such that for $(\\mathbf {a} = (a_1, \\ldots , a_s), e)\\in \\mathcal {A}\\times \\mathcal {E}\\cap \\mathbb {R}^s\\times \\mathbb {R}$ , the following holds.", "For $\\mathcal {I} = \\lbrace i_1,\\ldots , i_\\ell \\rbrace \\subset \\lbrace 1, \\ldots , s\\rbrace $ and $\\sigma = (\\sigma _1, \\ldots , \\sigma _s)\\in \\lbrace -1, 1\\rbrace ^s$ , the algebraic sets $V^{\\mathcal {I}, \\sigma }_{\\mathbf {a}, e}\\subset \\mathbb {C}^t$ defined by $g_{i_1}+ \\sigma _{i_1} a_{i_1} e = \\cdots =g_{i_\\ell }+\\sigma _{i_\\ell } a_{i_\\ell }e = 0$ are, either empty, or $(t-\\ell )$ -equidimensional and smooth, and the ideal generated by their defining equations is radical.", "Note that by the transfer principle, one can choose instead of a scalar $e$ an infinitesimal $\\varepsilon $ so that the algebraic sets $V^{\\mathcal {I}, \\sigma }_{\\mathbf {a}, \\varepsilon }$ and their defining set of equations satisfy the above properties.", "When, in the above equations, one leaves $\\varepsilon $ as a variable, one obtains equations defining an algebraic set in $\\mathbb {C}^{t+1}$ .", "We denote by $\\mathfrak {V}^{\\mathcal {I}, \\sigma }_{\\mathbf {a}, \\varepsilon }$ the union of the $(t+1-\\ell )$ -equidimensional components of this algebraic set.", "Further we also assume that the $a_i$ 's are chosen positive.", "Denote by $\\mathcal {S}^{(\\varepsilon )}$ the extension of the semi-algebraic set $\\mathcal {S}$ to $\\mathbb {R}\\langle \\varepsilon \\rangle ^t$ ; similarly, the extension of any connected component $C$ of $\\mathcal {S}$ to $\\mathbb {R}\\langle \\varepsilon \\rangle ^t$ is denoted by $C^{(\\varepsilon )}$ .", "Now, remark that any connected component $C^{(\\varepsilon )}$ of $\\mathcal {S}^{(\\varepsilon )}$ contains a connected component of the semi-algebraic set $\\mathcal {S}_{\\mathbf {a}}^{(\\varepsilon )}$ defined by: $\\left( -a_1\\varepsilon \\ge g_1 \\vee g_1\\ge a_1\\varepsilon \\right)\\wedge \\cdots \\wedge \\left( -a_s\\varepsilon \\ge g_s \\vee g_s\\ge a_s\\varepsilon \\right)$ Hence, we are led to compute sample points per connected component of $\\mathcal {S}_{\\mathbf {a}}^{(\\varepsilon )}$ .", "These will be encoded with zero-dimensional parametrizations with coefficients in $\\mathbb {Q}[\\varepsilon ]$ .", "By [5], in order to compute sample points per connected component in $\\mathcal {S}_{\\mathbf {a}}^{(\\varepsilon )}$ , it suffices to compute sample points in the real algebraic sets $V^{\\mathcal {I}, \\sigma }_{\\mathbf {a}, \\varepsilon }\\cap \\mathbb {R}^t$ .", "To do that, since the algebraic sets $V^{\\mathcal {I}, \\sigma }_{\\mathbf {a},\\varepsilon }$ satisfy the above regularity properties, we can use the algorithm and geometric results of [49].", "To state these results, one needs to introduce some notation.", "Let $\\mathfrak {Q}$ be a real field, $\\mathfrak {R}$ be a real closure of $\\mathfrak {Q}$ and $\\mathfrak {C}$ be an algebraic closure of $\\mathfrak {R}$ .", "For an algebraic set $V\\subset \\mathfrak {C}^t$ defined by $h_1=\\cdots =h_\\ell =0$ ($h_i\\in \\mathfrak {Q}[\\mathbf {y}]$ with $\\mathbf {y} =(y_1, \\ldots , y_t)$ ) and $M\\in \\mathrm {GL}_t(\\mathfrak {R})$ , we denote by $V^{M}$ the set $\\lbrace M^{-1}\\cdot {\\mathbf {x}}\\mid {\\mathbf {x}}\\in V\\rbrace $ and, for $1\\le i\\le \\ell $ , by ${h_i}^M$ the polynomial $h_i(M\\cdot \\mathbf {y})$ and by $\\pi _i$ the canonical projection $(y_1, \\ldots , y_t)\\rightarrow (y_1, \\ldots ,y_i)$ ($\\pi _0$ will simply denote $(y_1, \\ldots , y_t)\\rightarrow \\lbrace \\bullet \\rbrace $ ).", "By slightly abusing notation, we will also denote by $\\pi _i$ projections from $\\mathfrak {V}^{\\mathcal {I}, \\sigma }_{\\mathbf {a},\\varepsilon }$ to the first $i$ coordinates $(y_1, \\ldots , y_i)$ .", "We will consider the set of critical points of the restriction of $\\pi _i$ to $V$ and will denote this set by $\\mathrm {crit}(\\pi _i, V)$ for $1\\le i \\le \\ell $ .", "By [49], for a generic choice of ${M}\\in \\textrm {GL}_t(\\mathfrak {R})$ , the union of $V^{{M}}\\cap \\pi _{t-\\ell }^{-1}(0)$ with the sets $\\textrm {crit}(\\pi _i, V^{{M}})\\cap \\pi _{i-1}^{-1}(0)$ (for $1\\le i \\le t-\\ell $ ) is finite and meets all connected components of $V^{{M}}\\cap \\mathfrak {R}^t$ .", "Because $V$ satisfies the aforementioned regularity assumptions, $\\textrm {crit}(\\pi _i, V^{{M}})\\cap \\pi _{i-1}^{-1}(0)$ is defined as the projection on the $\\mathbf {y}$ -space of the solution set to the polynomials $\\mathbf {h}^M, \\quad (\\lambda _1, \\ldots , \\lambda _\\ell ).jac(\\mathbf {h}^M, i), \\quad u_1\\lambda _1+\\cdots +u_\\ell \\lambda _\\ell = 1, \\quad y_1 = \\cdots = y_{i-1} = 0,$ where $\\mathbf {h} = (h_1, \\ldots , h_\\ell )$ , $\\lambda _1, \\ldots ,\\lambda _\\ell $ are new variables (called Lagrange multipliers), $jac(\\mathbf {h}^M, i)$ is the Jacobian matrix associated to $\\mathbf {h}^M$ truncated by forgetting its first first $i$ columns and the $u_i$ 's are generically chosen (see also [51]).", "Recall that $D$ denotes the maximum degree of the $h_j$ 's and let $E$ be the length of a straight-line program evaluating $\\mathbf {h}$ .", "Observe now that, setting the $y_j$ 's to 0 (for $1\\le j \\le i-1$ ), and using [52] combined with the degree estimates in [52], we obtain that such systems can be solved using $O\\left(\\left(\\binom{t-i}{\\ell } D^\\ell (D-1)^{t-(i-1)-\\ell }\\right)^2(E+(t+\\ell )D+(t+\\ell )^2)(t+\\ell )\\right)$ arithmetic operations in $\\mathfrak {Q}$ and have at most $\\binom{t-i}{\\ell } D^\\ell (D-1)^{t-(i-1)-\\ell }$ solutions.", "Going back to our initial problem, one then needs to solve polynomial systems which encode the set $\\mathrm {crit}(\\pi _i, V^{\\mathcal {I},\\sigma }_{\\mathbf {a}, \\varepsilon })$ of critical points of the restriction of $\\pi _i$ to $V^{\\mathcal {I}, \\sigma }_{\\mathbf {a}, \\varepsilon }$ .", "Note that these systems have coefficients in $\\mathbb {Q}[\\varepsilon ]$ .", "To solve such systems, we rely on [53], which consists in specializing $\\varepsilon $ to a generic value $e\\in \\mathbb {Q}$ and compute a zero-dimensional parametrization of the solution set to the obtained system (within the above arithmetic complexity over $\\mathbb {Q}$ ) and next use Hensel lifting and rational reconstruction to deduce from this parametrization a zero-dimensional parametrization with coefficients in $\\mathbb {Q}(\\varepsilon )$ .", "By [53] and multi-homogeneous bounds on the degree of the critical points of $\\pi _i$ to $\\mathfrak {V}^{\\mathcal {I}, \\sigma }_{\\mathbf {a}, \\varepsilon }$ as in [52], this lifting step has a cost $O\\ {\\widetilde{~}}\\left( ((t+\\ell )^4+ (t+\\ell +1)E)\\left(\\binom{t-i}{\\ell } D^\\ell (D-1)^{t-(i-1)-\\ell }\\right)^2 \\right)$ Hence, all in all computing one zero-dimensional parametrization for one critical locus uses $O\\ {\\widetilde{~}}\\left( ((t+\\ell )^4D+ (t+\\ell +1)E)\\left(\\binom{t-i}{\\ell } D^\\ell (D-1)^{t-(i-1)-\\ell }\\right)^2 \\right)$ arithmetic operations in $\\mathbb {Q}$ .", "Note that following [53], the degrees in $\\varepsilon $ of the numerators and denominators of the coefficients of these parametrizations are bounded by $\\binom{t}{\\ell }D^\\ell (D-1)^{t-\\ell }$ .", "Summing up for all critical loci and using $\\sum _{i=0}^{t-\\ell } \\binom{t-i}{\\ell } = \\binom{t+1}{\\ell +1}$ we need to compute for a fixed ${V}^{\\mathcal {I}, \\sigma }_{\\mathbf {a}, \\varepsilon }$ uses $O\\ {\\widetilde{~}}\\left( ((t+\\ell )^4D+ (t+\\ell +1)E )\\binom{t+1}{\\ell +1}^2\\left( D^\\ell (D-1)^{t-\\ell }\\right)^2 \\right)$ arithmetic operations in $\\mathbb {Q}$ .", "Also, the number of points computed this way is dominated by $\\binom{t+1}{\\ell + 1}\\left( D^\\ell (D-1)^{t-\\ell } \\right).$ Taking the sum for all possible algebraic sets ${V}^{\\mathcal {I},\\sigma }_{\\mathbf {a}, \\varepsilon }$ and remarking that the sum of number of indices of cardinality $\\ell $ for $0 \\le \\ell \\le t$ is bounded by $s^t$ ; the number of sets $\\sigma $ for a given $\\ell $ is bounded by $2^t$ ; the sum $\\sum _{\\ell =0}^t \\binom{t+1}{\\ell +1}^2$ equals $2\\binom{2t+1}{t}-1$ one deduces that all these zero-dimensional parametrizations can be computed within $O\\ {\\widetilde{~}}\\left(s^t 2^t\\binom{2t+1}{t} \\left((2t)^4D+ (2t+1)\\Gamma \\right)\\ D^{2t}\\right)$ arithmetic operations in $\\mathbb {Q}$ (recall that $\\Gamma $ bounds the length of a straight line program evaluating all the polynomials defining our semi-algebraic set $\\mathcal {S}$ ) which we simplify to $O\\ {\\widetilde{~}}\\left(\\Gamma \\ (2t)^4\\ s^t\\ 2^{3t}\\ D^{2t+1}\\right).$ Similarly, using the above simplifications, the total number of points encoded by these zero-dimensional parametrizations is bounded above by $(2sD)^t$ .", "At this stage, we have just obtained zero-dimensional parametrizations with coefficients in $\\mathbb {Q}(\\varepsilon )$ .", "The above bound on the number of returned points is done but it remains to show how to specialize $\\varepsilon $ in order to get sample points per connected components in $\\mathcal {S}$ .", "To do that, given a parametrization ${R}_\\varepsilon = (w, v_1, \\ldots , v_t)\\subset \\mathbb {Q}(\\varepsilon )[u]^{t+1}$ , we need to find a specialization value $e$ for $\\varepsilon $ to obtain a parametrization ${R}_e$ such that the number of real roots of the zero set associated to ${R}_e$ is the same as the number of real roots of the zero set associated to ${R}_\\varepsilon $ ; when $\\eta $ ranges over the interval $]0, e]$ the signs of the $g_i$ 's at the zero set associated to $\\eta $ does not vary.", "To do that, it suffices to choose $e$ such that it is smaller than the smallest positive root of the resultant associated to $\\left(w,\\frac{\\partial w}{\\partial u}\\right)$ and the smallest positive roots of the resultant associated to $w$ and $g_i\\left(\\frac{v_1}{\\partial w / \\partial u}, \\ldots , \\frac{v_t}{\\partial w / \\partial u}\\right)$ .", "The algebraic cost (i.e.", "the resultant computations) are dominated by the complexity estimates of the previous step.", "Finally, note that $\\Gamma $ can be bounded by $s~\\binom{D+t}{t}$ when the $g_i$ 's are given in an expanded form in the monomial basis.", "Therefore, the arithmetic complexity for computing sample points of the semi-algebraic set defined by $g_1\\ne 0 ,\\ldots ,g_s\\ne 0$ can be bounded by $O\\ {\\widetilde{~}}\\left(\\binom{D+t}{t}\\ (2t)^4\\ s^{t+1}\\ 2^{3t}\\ D^{2t+1}\\right).$ We end this section with a Corollary which is a consequence of the proof of [5].", "Basically, once we have the parametrizations computed by the algorithm on which Theorem REF relies, one can compute sample points per connected components of the semi-algebraic set $\\mathcal {S}$ within the same arithmetic complexity bounds.", "The idea is just to evaluate the $g_i$ 's at these rational parametrizations and use bounds on the minimal distance between two roots of a univariate polynomial such as [5].", "Hence, the proof of the corollary below follows mutatis mutandis the same steps as the one of [5].", "Corollary 4 Let $(g_1, \\ldots , g_s)$ in $\\mathbb {Q}[y_1, \\ldots , y_t]$ with $D \\ge \\max _{1\\le i \\le s} \\deg (g_i)$ and $\\mathcal {S}\\subset \\mathbb {R}^t$ be the semi-algebraic set defined by $g_1\\ne 0, \\ldots , g_s\\ne 0.$ There exists a probabilistic algorithm which on input $(g_1, \\ldots ,g_s)$ outputs a finite set of points ${P}$ in $\\mathbb {Q}^t$ of cardinality at most $\\left( 2sD \\right)^t $ points such that ${P}$ meets every connected component of $\\mathcal {S}$ using $O\\ {\\widetilde{~}}\\left(\\binom{D+t}{t} (2t)^4 s^{t+1} 2^{3t} D^{2t+1}\\right).$ arithmetic operations in $\\mathbb {Q}$ ." ], [ "Sturm based classical algorithm", "In this section, we describe an algorithm based on Sturm's theorem for solving Problem (REF ) and discuss its shortcomings.", "We consider a sequence ${\\mathbf {f}}=(f_1,\\ldots ,f_m)\\subset \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ where ${\\mathbf {y}}=(y_1,\\ldots ,y_t)$ and ${\\mathbf {x}}=(x_1,\\ldots ,x_n)$ .", "Let $d$ be an upper bound of the total degree of the $f_i$ 's.", "We require that the input system ${\\mathbf {f}}$ satisfies the properties below.", "BAssumption B Let ${\\mathbf {f}}$ be the above parametric polynomial system and ${\\mathcal {V}}$ be the algebraic set defined by ${\\mathbf {f}}$ .", "We say that ${\\mathbf {f}}$ satisfies Assumptions () if the following properties hold.", "The ideal generated by ${\\mathbf {f}}$ is radical.", "The algebraic set ${\\mathcal {V}}$ is equi-dimensional of dimension $t$ .", "The restriction of $\\pi :({\\mathbf {y}},{\\mathbf {x}})\\mapsto {\\mathbf {y}}$ to ${\\mathcal {V}}$ is dominant.", "It is well-known that the above assumptions are satisfied by sufficiently generic systems (see e.g.", "[52]).", "In what follows, we rely on the existence of a parametric geometric resolution [53] to reduce our initial multivariate problem to a univariate one.", "Using [53] with Assumption (B1), there exists a non-empty open Zariski set $\\mathcal {A}$ of $\\mathbb {C}^n$ such that, for $(a_1,\\ldots ,a_n) \\in \\mathbb {Q}^n\\cap \\mathcal {A}$ , there exists a parametric geometric resolution $({w}_{\\mathbf {a}},v_1,\\ldots ,v_n)\\subset \\left( \\mathbb {Q}({\\mathbf {y}})[{\\mathbf {x}}] \\right)^n$ of ${\\mathbf {f}}$ that satisfies the following properties.", "${w}_{\\mathbf {a}}$ is a square-free polynomial in $\\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ .", "$u = \\sum _{i=1}^n a_ix_i$ .", "There exists a non-empty Zariski open subset $\\mathcal {Y}_{\\mathbf {a}} \\subset \\mathbb {C}^t$ such that, for ${\\eta }\\in \\mathcal {Y}_{\\mathbf {a}}$ , we have that $V({\\mathbf {f}}({\\eta },\\cdot )) = \\left\\lbrace \\left(\\frac{v_1}{\\partial {w}_{\\mathbf {a}}/\\partial u}({\\eta },\\vartheta ), \\ldots , \\frac{v_n}{\\partial {w}_{\\mathbf {a}}/\\partial u}({\\eta },\\vartheta )\\right)\\; \\left\| \\;{w}_{\\mathbf {a}}({\\eta },\\vartheta ) = 0, \\frac{\\partial {w}_{\\mathbf {a}}}{\\partial u}({\\eta },\\vartheta ) \\ne 0 \\right.\\right\\rbrace .$ The set $\\mathcal {Y}_{\\mathbf {a}}$ can be chosen as the set where the leading coefficient of ${w}_{\\mathbf {a}}$ , the resultant of ${w}_{\\mathbf {a}}$ and $\\partial {w}_{\\mathbf {a}}/\\partial u$ , and the denominators appearing in the coefficients of $v_1,\\ldots ,v_n$ do not vanish.", "As a consequence, for ${\\eta }\\in \\mathcal {Y}_{\\mathbf {a}}$ , the number of complex solutions of ${\\mathbf {f}}({\\eta },\\cdot )$ is invariant and equals to the partial degree of ${w}_{\\mathbf {a}}$ in $u$ .", "We denote by $D$ the partial degree of ${w}_{\\mathbf {a}}$ in $u$ .", "By Bézout's inequality (see e.g.", "[33]), $D$ is bounded above by $d^n$ .", "Let $\\eta \\in \\mathbb {C}^t$ and ${w}_{\\mathbf {a}}({\\eta },\\cdot )$ be the specialization of the ${\\mathbf {y}}$ variables in ${w}_{\\mathbf {a}}$ at $\\eta $ .", "From the existence of such a parametric resolution, we deduce that, for ${\\eta }\\in \\mathcal {Y}_{\\mathbf {a}}$ , the map $\\varphi : (x_1,\\ldots ,x_n)\\mapsto \\sum _{i=1}^n a_ix_i$ is a bijection between the complex roots of ${\\mathbf {f}}({\\eta },\\cdot )$ and ${w}_{\\mathbf {a}}({\\eta },\\cdot )$ .", "Lemma 5 Let ${\\mathbf {f}}$ be a parametric system satisfying Assumption () and ${w}_{\\mathbf {a}}$ be the eliminating polynomial in the parametric geometric resolution of ${\\mathbf {f}}$ as above.", "Then, we have $V(\\langle f_1,\\ldots ,f_m,u-\\sum _{i=1}^n a_ix_i\\rangle \\cap \\mathbb {Q}[{\\mathbf {y}}][u]) = V({w}_{\\mathbf {a}}).$ Consequently, the total degree of ${w}_{\\mathbf {a}}$ is at most $d^n$ .", "We prove that, under Assumption (), there exists a square-free polynomial ${w}\\in \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ satisfying $V(\\langle f_1,\\ldots ,f_m,u-\\sum _{i=1}^n a_ix_i\\rangle \\cap \\mathbb {Q}[{\\mathbf {y}}][u]) = V({w}).$ Let $\\pi _u: \\mathbb {C}^{t+n+1} \\rightarrow \\mathbb {C}^{t+1}, ({\\mathbf {y}},{\\mathbf {x}},u) \\mapsto ({\\mathbf {y}},u)$ and ${\\mathcal {V}}_u$ be the algebraic set defined by $\\langle {\\mathbf {f}},u-\\sum _{i=1}^n a_ix_i\\rangle $ .", "Note that ${\\mathcal {V}}$ and ${\\mathcal {V}}_u$ are isomorphic taking the map $({\\mathbf {y}},{\\mathbf {x}}) \\mapsto ({\\mathbf {y}},{\\mathbf {x}},\\sum _{i=1}^na_ix_i)$ as an isomorphism between them.", "Then, as the algebraic set ${\\mathcal {V}}$ satisfies Assumption (), ${\\mathcal {V}}_u$ is equi-dimensional of dimension $t$ and the restriction of $\\Pi :\\mathbb {C}^{t+n+1} \\rightarrow \\mathbb {C}^t, ({\\mathbf {y}},{\\mathbf {x}},u) \\mapsto {\\mathbf {y}}$ to ${\\mathcal {V}}_u$ is dominant.", "Therefore, the Zariski closure of $\\pi _u({\\mathcal {V}}_u)$ is an equi-dimensional algebraic set of dimension $t$ .", "Hence, there exists a square-free polynomial ${w}\\in \\mathbb {Q}[{\\mathbf {y}}][u]$ such that $V({w}) =\\overline{\\pi _u({\\mathcal {V}}_u)}$ .", "Therefore, we obtain $V(\\langle f_1,\\ldots ,f_m,u-\\sum _{i=1}^n a_ix_i\\rangle \\cap \\mathbb {Q}[{\\mathbf {y}}][u]) =\\overline{\\pi _u({\\mathcal {V}}_u)} = V({w})$ .", "It remains to show that ${w}_{\\mathbf {a}}$ equals to ${w}$ up to a constant.", "By the definition of parametric geometric resolution, for ${\\eta }\\in \\mathcal {Y}_{\\mathbf {a}}$ , then ${w}_{\\mathbf {a}}({\\eta },\\cdot )$ and ${w}({\\eta },\\cdot )$ share the same complex roots.", "Therefore, ${w}_{\\mathbf {a}}$ equals to ${w}$ up to a factor in $\\mathbb {Q}[{\\mathbf {y}}]$ .", "However, both ${w}_{\\mathbf {a}}$ and ${w}$ do not contain such kind of factor by Assumption ().", "By Bézout's inequalities, the degree of $V(f_1,\\ldots ,f_n,u-\\sum _{i=1}^n a_ix_i)$ is at most $d^n$ .", "Hence, the degree of $\\overline{\\pi _u({\\mathcal {V}}_u)}$ is also bounded by $d^n$ .", "Therefore, the total degree of ${w}_{\\mathbf {a}}$ is bounded by $d^n$ .", "Recall that $\\mathcal {Y}_{\\mathbf {a}}$ is the non-empty Zariski open subset of $\\mathbb {C}^t$ where the leading coefficient of ${w}_{\\mathbf {a}}$ , the resultant of ${w}_{\\mathbf {a}}$ and $\\partial {w}_{\\mathbf {a}}/\\partial u$ , and the denominators appearing in the coefficients of $v_1,\\ldots ,v_n$ do not vanish.", "Lemma REF shows that the numbers of real roots of ${\\mathbf {f}}({\\eta },\\cdot )$ and ${w}_{\\mathbf {a}}({\\eta },\\cdot )$ also coincide over $\\mathcal {Y}_{\\mathbf {a}}$ .", "Lemma 6 Let $\\mathcal {Y}_{\\mathbf {a}}$ as above.", "Then, for ${\\eta }\\in \\mathcal {Y}_{\\mathbf {a}}\\cap \\mathbb {R}^t$ , the numbers of real solutions of ${w}_{\\mathbf {a}}({\\eta },\\cdot )$ and ${\\mathbf {f}}({\\eta },\\cdot )$ are equal.", "Let ${\\eta }\\in \\mathbb {R}^t\\cap \\mathcal {Y}_{\\mathbf {a}}$ .", "By definition of $\\mathcal {Y}_{\\mathbf {a}}$ , the restriction of $\\varphi : (x_1,\\ldots ,x_n)\\mapsto \\sum _{i=1}^n a_ix_i$ to $V({\\mathbf {f}}({\\eta },\\cdot ))$ is a bijection of between the complex roots of ${\\mathbf {f}}({\\eta },\\cdot )$ and ${w}_{\\mathbf {a}}({\\eta },\\cdot )$ .", "As the sequence ${\\mathbf {f}}({\\eta },\\cdot )$ contains polynomials of coefficients in $\\mathbb {R}$ , the non-real complex roots of ${\\mathbf {f}}({\\eta },\\cdot )$ appears as pairs of conjugate complex points of $\\mathbb {C}^n$ .", "Assume that there exists a complex root whose image by $\\varphi $ is a real root of ${w}_{\\mathbf {a}}({\\eta },\\cdot )$ , then its conjugate is also mapped to the same real root.", "This contradicts the bijectivity of $\\varphi $ .", "Therefore, the numbers of real solutions of ${\\mathbf {f}}({\\eta },\\cdot )$ and ${w}_{\\mathbf {a}}({\\eta },\\cdot )$ coincide.", "For $h\\in \\mathbb {Q}[{\\mathbf {y}}][u]$ of degree $D$ in $u$ , we denote by $ \\Sigma \\left(h, \\frac{\\partial h}{\\partial u}\\right) = (s_0, \\ldots , s_D)\\subset \\mathbb {Q}[{\\mathbf {y}}]$ the leading coeffients of the subresultant sequence associated to $\\left(h, \\partial h/\\partial u\\right)$ (see [5]).", "Here we enumerate this sequence in a way such that $s_0$ is the leading coefficient of $h$ as a polynomial in $u$ and $s_D$ is the resultant of $h$ and $\\partial h/ \\partial u$ .", "We recall the specialization property of subresultant coefficients (see e.g.", "[5]).", "For ${\\eta }\\in \\mathbb {R}^t$ that does not cancel the leading coefficient of $h$ as a polynomial in $u$ , then the subresultant coefficients of $h({\\eta },\\cdot )$ and $\\partial h({\\eta },\\cdot )/\\partial u$ are exactly the evaluation of $(s_0,\\ldots ,s_D)$ at ${\\eta }$ .", "By [5], the number of real roots of $h({\\eta },\\cdot )$ equals the generalized permanences minus variations (see [5]) of $(s_0,\\ldots ,s_D)_{{\\eta }}$ .", "Note that this value is uniquely defined upon a sign pattern of $(s_0,\\ldots ,s_D)_{{\\eta }}$ .", "We can now describe Algorithm which takes as input a sequence ${\\mathbf {f}}= (f_1, \\ldots , f_m)\\subset \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ satisfying Assumption () and it outputs semi-algebraic formulas solving Problem (REF ).", "It uses the following subroutines: EliminatingPolynomial which takes as input ${\\mathbf {f}}$ and outputs an eliminating polynomial ${w}_{\\mathbf {a}}$ , i.e., the first polynomial in a parametric geometric resolution of ${\\mathbf {f}}$ .", "Such an algorithm can be derived from the probabilistic algorithm given in [53] that computes parametric geometric resolutions.", "SubresultantCoefficients which takes as input ${w}_{\\mathbf {a}}$ and outputs $\\Sigma ({w}_{\\mathbf {a}},\\partial {w}_{\\mathbf {a}}/\\partial u) = (s_0,\\ldots ,s_D)$ .", "We refer to [5] for the explicit description of such an algorithm.", "SamplePoints which takes as input the subresultant coefficients $(s_0,\\ldots ,s_D)\\subset \\mathbb {Q}[{\\mathbf {y}}]$ and outputs at least one point per connected components of the semi-algebraic set defined by $\\lbrace s_i \\ne 0 \\; \| \\; 1\\le i \\le D,\\; s_i \\text{ is not aconstant}\\rbrace $ .", "We refer to Section for the explicit description of such an algorithm.", "PermanencesMinusVariations which takes as input a sequence $(s_0,\\ldots , s_D)_{{\\eta }}$ and return its generalized permanences minus variations.", "Using [5], we can easily design such a subroutine.", "[H] A parametric system ${\\mathbf {f}}\\subset \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ satisfying Assumption () Semi-algebraic descriptions solving Problem (REF ) for the input ${\\mathbf {f}}$ ${w}_{\\mathbf {a}} \\leftarrow {\\sf EliminatingPolynomial}({\\mathbf {f}})$ $(s_0,\\ldots ,s_D) \\leftarrow {\\sf SubresultantCoefficients}\\left({w}_{\\mathbf {a}},\\partial {w}_{\\mathbf {a}}/\\partial u, u\\right)$ $L \\leftarrow {\\sf SamplePoints}(\\lbrace s_i \\ne 0 \\; \| \\; 1\\le i \\le D,\\; s_i\\text{ is not a constant}\\rbrace )$ ${\\eta }\\in L$ $r_{{\\eta }} \\leftarrow {\\sf PermanencesMinusVariations}((s_0,\\ldots ,s_D)_{{\\eta }})$ $\\lbrace (\\mathrm {sign}\\;(s_0,\\ldots ,s_D)_{{\\eta }},{\\eta },r_{{\\eta }})\\; \| \\;{\\eta }\\in L\\rbrace $ RRC-Sturm Theorem REF Let ${\\mathbf {f}}= (f_1,\\ldots ,f_n) \\subset \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ be a parametric system and $d$ be the largest total degree among the $\\deg (f_i)$ 's.", "We assume that ${\\mathbf {f}}$ satisfies Assumption ().", "Then, Algorithm , which is probabilistic, computes semi-algebraic formulas solving Problem REF within $O\\ {\\widetilde{~}}\\left(\\binom{t+2d^{2n}}{t}\\ 2^{5t} \\ d^{5nt+3n}\\right)$ arithmetic operations in $\\mathbb {Q}$ .", "These semi-algebraic formulas contains polynomials in $\\mathbb {Q}[{\\mathbf {y}}]$ of degree bounded by $2d^{2n}$ .", "We start with the correctness statement.", "Recall that $s_0$ is the leading coefficient of ${w}_{\\mathbf {a}}$ as a polynomial in $u$ .", "By [5], for ${\\eta }\\in \\mathbb {C}^t$ that does not cancel $s_0$ , the subresultant coefficients of ${w}_{\\mathbf {a}}({\\eta },\\cdot )$ and $\\partial {w}_{\\mathbf {a}}({\\eta },\\cdot )/\\partial u$ is the specialization of $(s_0,\\ldots ,s_D)$ at ${\\eta }$ .", "Therefore, from [5], the number of real roots of ${w}_{\\mathbf {a}}({\\eta },\\cdot )$ can be derived from the sign of the sequence $(s_0,\\ldots ,s_D)_{{\\eta }}$ for ${\\eta }\\notin V(s_0)$ .", "On the other hand, the semi-algebraic set ${\\mathcal {S}}$ defined by $\\lbrace s_i\\ne 0\\; \| \\; 1\\le i \\le D,\\; s_i\\text{ is not a constant}\\rbrace $ is composed of open semi-algebraic connected components, namely ${\\mathcal {S}}_1,\\ldots ,{\\mathcal {S}}_{\\ell }$ .", "Over each of them, the subresultant coefficients $s_i$ are sign-invariant.", "Thus, the number of distinct real roots of ${w}_{\\mathbf {a}}({\\eta },\\cdot )$ is invariant when ${\\eta }$ varies in ${\\mathcal {S}}_i$ for each $1 \\le i \\le \\ell $ .", "Recall that $\\mathcal {Y}_{\\mathbf {a}}\\subset \\mathbb {C}^t$ is the non-empty Zariski open set in Lemma REF such that for ${\\eta }\\in \\mathcal {Y}_{\\mathbf {a}}$ , the numbers of real roots of ${\\mathbf {f}}({\\eta },\\cdot )$ and ${w}_{\\mathbf {a}}({\\eta },\\cdot )$ coincide.", "Therefore, the number of real solutions of ${\\mathbf {f}}({\\eta },\\cdot )$ is also invariant when ${\\eta }$ varies in ${\\mathcal {S}}_i\\cap \\mathcal {Y}_{\\mathbf {a}}$ .", "Let $L$ be the set of sample points of ${\\mathcal {S}}$ .", "We deduce from the above arguments that the semi-algebraic sets defined by $\\wedge _{i=1}^D\\mathrm {sign}\\;(s_i) = \\mathrm {sign}\\;(s_i({\\eta }))$ for ${\\eta }\\in L$ satisfy the requirement of Problem REF .", "The correctness of our algorithm is then proven.", "By Lemma REF , the total degree of ${w}_{\\mathbf {a}}$ is bounded above by $d^n$ .", "Using [5] on the polynomial ${w}_{\\mathbf {a}}$ and $\\partial {w}_{\\mathbf {a}}/\\partial u$ , we obtain the bound $\\deg s_j \\le d^n(2j-1) \\le 2d^n$ for $0\\le j\\le D$ .", "We are now able to analyze the complexity of Algorithm .", "By [53], running EliminatingPolynomial on input ${\\mathbf {f}}=(f_1,\\ldots ,f_n)$ where the total degree of each $f_i$ is bounded by $d$ takes $O\\ {\\widetilde{~}}\\left( \\binom{4d^n+t}{t} d^n \\right)$ arithmetic operations in $\\mathbb {Q}$ .", "The subresultant coefficients of ${w}_{\\mathbf {a}}$ and $\\partial {w}_{\\mathbf {a}}/\\partial u$ can be computed using an evaluation-interpolation scheme as follows.", "As the degree of $s_i$ is bounded by $2d^{2n}$ , we need to compute the subresultant coefficients of the evaluation of $({w}_{\\mathbf {a}},\\partial {w}_{\\mathbf {a}}/\\partial u)$ at $\\binom{t+2d^{2n}}{t}$ distinct points.", "Note that $\\binom{t+2d^{2n}}{t}$ is bounded by $2^t d^{2nt}$ .", "Using [5], it yields an arithmetic complexity $O\\left(d^{2n} \\right)$ for each of those subresultant computations.", "Hence, in total, the specialized subresultant coefficients can be computed by $O \\left(2^t d^{2nt+2n} \\right)$ .", "Next, the cost of interpolating the $s_i$ 's can be bounded by $O\\ \\widetilde{~}\\left(D\\ 2^td^{2nt} \\log \\left(2^td^{2nt} \\right)\\right)$ using the interpolation given in [12].", "Thus, the arithmetic complexity of SubresultantCoefficients lies in $O\\ \\widetilde{~} \\left(D\\ 2^td^{2nt}\\log \\left(2^td^{2nt} \\right) \\right).$ We rely on Corollary REF for estimating the complexity of SamplePoints.", "Using the algorithm of Section (see Theorem REF and Corollary REF ) on the sequence $(s_0,\\ldots ,s_D)$ , one can compute sample points per connected components of the semi-algebraic set defined by $\\lbrace s_i\\ne 0\\; \| \\; 1\\le i \\le D,\\;s_i\\text{ is not a constant}\\rbrace $ in time $O\\ {\\widetilde{~}}\\left(\\binom{t+2d^{2n}}{t} \\ t^4d^{nt+n} 2^{3t}\\ \\left(2d^{2n}\\right)^{2t+1} \\right) \\simeq O\\ {\\widetilde{~}}\\left(\\binom{t+2d^{2n}}{t}\\ 2^{5t} \\ d^{5nt+3n}\\right).$ By Corollary REF , this subroutine outputs a finite subset of $\\mathbb {Q}^t$ whose cardinal is bounded by $4^td^{3nt}$ .", "Using [5] to compute the permanences minus variations leads to a complexity of $O \\left(4^td^{3nt+n} \\right)$ .", "Summing up all the partial costs, we conclude that Algorithm runs within $O\\ {\\widetilde{~}}\\left(\\binom{t+2d^{2n}}{t}\\ 2^{5t} \\ d^{5nt+3n} \\right)$ arithmetic operations in $\\mathbb {Q}$ .", "Example 7 We will illustrate the algorithms of this paper using the sequence ${\\mathbf {f}}=(x_1^2+x_2^2-y_1,\\; x_1x_2+y_2x_2+y_3x_1).$ We choose $u=x_2$ when running Algorithm (in a reasonable implementation, one would pick randomly a linear form but we choose this one to obtain smaller data).", "We obtain the following rational parametrization $({w}, v_1, v_2)$ with ${w}&= u^4+2y_3u^3+(y_2^2+y_3^2-y_1)u^2-2y_1y_3u-y_1y_3^2,\\\\v_2& = 2\\,y_3\\,{u}^{3}+ \\left( 2\\,{y_2}^{2}+2\\,{y_3}^{2}-2\\,y_1\\right) {u}^{2}-6\\,y_1\\,y_3\\,u-4\\,y_1\\,{y_3}^{2}, \\\\v_1&=2\\,y_2\\,{u}^{3}+2\\,y_1\\,y_3\\,y_2.$ The subresultant coefficients associated to $\\left({w},\\frac{\\partial {{w}}}{\\partial {u}}\\right)$ are: $s_0 & = 1, s_1 = 1, s_2 = -2y_2^2+y_3^2+2y_1, \\\\s_3 & = -y_2^6-2y_2^4y_3^2-y_2^2y_3^4+3y_1y_2^4-14y_1y_2^2y_3^2+y_1y_3^4-3y_1^2y_2^2- 2y_1^2 y_3^2+y_1^3, \\\\s_4 & = (y_2y_3)^2 y_1(-y_2^6-3y_2^4y_3^2-3y_2^2y_3^4-y_3^6+3y_1y_2^4-21y_1y_2^2y_3^2+3y_1y_3^4-3y_1^2y_2^2-3y_1^2y_3^2+y_1^3).$ Since $s_0$ and $s_1$ are constants, we then compute at least one point per connected component of the semi-algebraic set defined by $s_2\\ne 0 \\wedge s_3 \\ne 0 \\wedge s_4 \\ne 0.$ This is done using e.g.", "the RAGlib (Real Algebraic Geometry library) [48].", "We obtain 35 points and find that the realizable sign conditions for $(s_2, s_3, s_4)$ are $[-1, -1, -1], [-1, -1, 1], [-1, 1, 1], [1, -1, -1], [1, -1, 1], [1, 1, -1], [1, 1, 1].$ Applying [5], we deduce the corresponding numbers of real roots to these sign patterns $0 \\text{ real root } & \\rightarrow (s_2<0 \\wedge s_3 <0 \\wedge s_4 >0) \\vee (s_2<0\\wedge s_3 >0 \\wedge s_4 >0) \\vee (s_2>0 \\wedge s_3 <0 \\wedge s_4 >0) \\\\2 \\text{ real roots } & \\rightarrow (s_2<0 \\wedge s_3 <0 \\wedge s_4 <0) \\vee (s_2>0 \\wedge s_3 <0 \\wedge s_4 <0) \\vee (s_2>0 \\wedge s_3 >0 \\wedge s_4 <0) \\\\4 \\text{ real roots } & \\rightarrow s_2>0 \\wedge s_3 >0 \\wedge s_4 >0$ Note that the maximum degree of the polynomials involved in the above formulas is 11.", "By contrast, observe that the restriction of the projection $\\pi :({\\mathbf {y}}, {\\mathbf {x}})\\rightarrow {\\mathbf {y}}$ to the real algebraic set defined by ${\\mathbf {f}}$ is proper.", "Hence, applying a semi-algebraic variant of Thom's isotopy lemma as in [10], one can deduce that the set of critical values of this map discriminates the regions of the parameters' space over which the number of real roots of ${\\mathbf {f}}$ remains invariant.", "Using immediate Gröbner bases computations, one obtains that the Zariski closure of this set of critical values is defined by the vanishing of $y_1(-y_2^6-3y_2^4y_3^2-3y_2^2y_3^4-y_3^6+3y_1y_2^4-21y_1y_2^2y_3^2+3y_1y_3^4-3y_1^2y_2^2-3y_1^2y_3^2+y_1^3)$ which has only degree 7." ], [ "Parametric Hermite matrices", "In this section, we adapt the construction encoding Hermite's quadratic forms, also known as Hermite matrices to the context of parametric systems and describe an algorithm for computing those parametric Hermite matrices." ], [ "Definition", "Let $\\mathbb {K}$ be a field and $I\\subset \\mathbb {K}[{\\mathbf {x}}]$ be a zero-dimensional ideal.", "Recall that the quotient ring $A_{\\mathbb {K}} = \\mathbb {K}[{\\mathbf {x}}]/I$ is a $\\mathbb {K}$ -vector space of finite dimension [16].", "The multiplication maps of $A_{\\mathbb {K}}$ are defined as follows.", "Definition 8 For any $p\\in \\mathbb {K}[{\\mathbf {x}}]$ , the multiplication map ${\\mathcal {L}}_p$ is defined as ${\\mathcal {L}}_p :\\begin{array}{rl}A_{\\mathbb {K}} &\\rightarrow A_{\\mathbb {K}},\\\\\\overline{q} & \\mapsto \\overline{p\\cdot q},\\end{array}$ where $\\overline{q}$ and $\\overline{p\\cdot q}$ are respectively the classes of $q$ and $p\\cdot q$ in the quotient ring $A_{\\mathbb {K}}$ .", "Note that the map ${\\mathcal {L}}_p$ is an endomorphism of $A_{\\mathbb {K}}$ as a $\\mathbb {K}$ -vector space.", "The Hermite quadratic form associated to $I$ is defined as the bilinear form that sends $(\\overline{p},\\overline{q})\\in A_{\\mathbb {K}}\\times A_{\\mathbb {K}}$ to the trace of ${\\mathcal {L}}_{p\\cdot q}$ as an endomorphism of $A_{\\mathbb {K}}$ .", "We refer to [5] for more details about Hermite quadratic forms.", "Now, let ${\\mathbf {f}}=(f_1,\\ldots ,f_m)$ be a polynomial sequence in $\\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ .", "We take the rational function field $\\mathbb {Q}({\\mathbf {y}})$ as the base field $\\mathbb {K}$ and denote by $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ the ideal generated by ${\\mathbf {f}}$ in $\\mathbb {K}[{\\mathbf {x}}]$ .", "We require that the system ${\\mathbf {f}}$ satisfies Assumption (REF ).", "This leads to the following well-known lemma, which is the foundation for the construction of our parametric Hermite matrices.", "Lemma 9 Assume that ${\\mathbf {f}}$ satisfies Assumption (REF ).", "Then the ideal $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ is zero-dimensional.", "Assume that there exists a coordinate $x_i$ for $1\\le i \\le n$ such that $\\langle {\\mathbf {f}}\\rangle \\cap \\mathbb {C}[{\\mathbf {y}},x_i] = \\langle 0\\rangle $ .", "We denote respectively by $\\pi _{i}$ and $\\tilde{\\pi }_{i}$ the projections $({\\mathbf {y}},{\\mathbf {x}}) \\mapsto ({\\mathbf {y}},x_i)$ and $({\\mathbf {y}},x_i)\\mapsto {\\mathbf {y}}$ .", "By the assumption above, $\\overline{\\pi _{i}({\\mathcal {V}})}$ is the whole space $\\mathbb {C}^{t+1}$ .", "Then, we have the identity $\\mathbb {C}^{t+1}=\\overline{\\tilde{\\pi _i}^{-1}(\\mathcal {O})\\cup \\tilde{\\pi _i}^{-1}(\\mathbb {C}^t\\setminus \\mathcal {O})},$ where $\\mathcal {O}$ be the open Zariski subset of $\\mathbb {C}^t$ required in Assumption (REF ).", "As Assumption (REF ) holds, $\\dim \\overline{\\tilde{\\pi _i}^{-1}(\\mathcal {O})}=t$ .", "Since $\\tilde{\\pi }_i$ is a map from $\\mathbb {C}^{t+1}$ to $\\mathbb {C}^t$ , its fibers are of at most dimension 1.", "Therefore, we have that $\\tilde{\\pi _i}^{-1}(\\mathbb {C}^t\\setminus \\mathcal {O})\\le 1+\\dim (\\mathbb {C}^t\\setminus \\mathcal {O}) \\le t$ .", "This contradicts to the above identity above.", "We conclude that, for $1 \\le i \\le n$ , $\\langle {\\mathbf {f}}\\rangle \\cap \\mathbb {C}[{\\mathbf {y}},x_i] \\ne \\langle 0 \\rangle $ .", "On the other hand, by Assumption (REF ), the Zariski-closure of $\\pi ({\\mathcal {V}})$ is the whole parameter space $\\mathbb {C}^t$ .", "Thus, we have that $\\langle {\\mathbf {f}}\\rangle \\cap \\mathbb {C}[{\\mathbf {y}}] =\\langle 0 \\rangle $ .", "Since $\\langle {\\mathbf {f}}\\rangle \\cap \\mathbb {C}[{\\mathbf {y}}] =(\\langle {\\mathbf {f}}\\rangle \\cap \\mathbb {C}[{\\mathbf {y}},x_i])\\cap \\mathbb {C}[{\\mathbf {y}}]$ for every $1\\le i \\le n$ , there exists a polynomial $p_i\\in \\langle {\\mathbf {f}}\\rangle \\cap \\mathbb {C}[{\\mathbf {y}},x_i]$ whose degree with respect to $x_i$ is non-zero.", "Clearly, $p_i$ is an element of the ideal $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ .", "Thus, there exists $d_i$ such that $x_i^{d_i}$ is a leading term in $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ .", "Hence, $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ is a zero-dimensional ideal.", "Lemma REF allows us to apply the construction of Hermite matrices described in [5] to parametric systems as follows.", "Since the ideal $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ is zero-dimensional by Lemma REF , its associated quotient ring $A_\\mathbb {K}=\\mathbb {K}[{\\mathbf {x}}]/\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ is a finite dimensional $\\mathbb {K}$ -vector space.", "Let $\\delta $ denote the dimension of $A_\\mathbb {K}$ as a $\\mathbb {K}$ -vector space.", "We consider a basis $B=\\lbrace b_1,\\ldots ,b_{\\delta }\\rbrace $ of $A_\\mathbb {K}$ , where the $b_i$ 's are taken as monomials in the variables ${\\mathbf {x}}$ .", "Such a basis can be derived from Gröbner bases as follows.", "We fix an admissible monomial ordering $\\succ $ over the set of monomials in the variables ${\\mathbf {x}}$ and compute a Gröbner basis $G$ with respect to the ordering $\\succ $ of the ideal $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ .", "Then, the monomials that are not divisible by any leading monomial of elements of $G$ form a basis of $A_{\\mathbb {K}}$ .", "Recall that, for an element $p\\in \\mathbb {K}[{\\mathbf {x}}]$ , we denote by $\\overline{p}$ the class of $p$ in the quotient ring $A_{\\mathbb {K}}$ .", "A representative of $\\overline{p}$ can be derived by computing the normal form of $p$ by the Gröbner basis $G$ , which results in a linear combination of elements of $B$ with coefficients in $\\mathbb {Q}({\\mathbf {y}})$ .", "Assume now the basis $B$ of $A_{\\mathbb {K}}$ is fixed.", "For any $p\\in \\mathbb {K}[{\\mathbf {x}}]$ , the multiplication map ${\\mathcal {L}}_p$ is an endomorphism of $A_{\\mathbb {K}}$ .", "Therefore, it admits a matrix representation with respect to $B$ , whose entries are elements in $\\mathbb {Q}({\\mathbf {y}})$ .", "The trace of ${\\mathcal {L}}_p$ can be computed as the trace of the matrix representing it.", "Similarly, the Hermite's quadratic form of the ideal $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ can be represented by a matrix with respect to $B$ .", "This leads to the following definition.", "Definition 10 Given a parametric polynomial system ${\\mathbf {f}}=(f_1,\\ldots ,f_m)\\subset \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ satisfying Assumption (REF ).", "We fix a basis $B=\\lbrace b_1,\\ldots ,b_{\\delta }\\rbrace $ of the vector space $\\mathbb {K}[{\\mathbf {x}}]/\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ .", "The parametric Hermite matrix associated to ${\\mathbf {f}}$ with respect to the basis $B$ is defined as the symmetric matrix $H=(h_{i,j})_{1\\le i,j\\le \\delta }$ where $h_{i,j} =\\mathrm {trace}({\\mathcal {L}}_{b_i\\cdot b_j})$ .", "It is important to note that the definition of parametric Hermite matrices depends both on the input system ${\\mathbf {f}}$ and the choice of the monomial basis $B$ .", "Example 11 We consider the same system ${\\mathbf {f}}=(x_1^2+x_2^2-y_1,\\;x_1x_2+y_2x_2+y_3x_1)$ as in Example REF .", "The parametric Hermite matrix associated to ${\\mathbf {f}}$ with respect to the basis $B_1=\\lbrace 1,x_2,x_1,x_2^2\\rbrace $ is $\\begin{bmatrix}4 & -2y_3 & -2y_2 & -2(y_2^2+y_3^2+y_1) \\\\* & -2(y_2^2+y_3^2+y_1) & 4y_2y_3 & 2(3y_2^2y_3-y_3^3) \\\\* & * & 2(y_2^2-y_3^2+y_1) & 2(y_2^3-3y_2y_3^2-y_1y_2) \\\\* & * & * & 2y_2^4-12y_2^2y_3^2+2y_3^4-4y_1y_2^2+2y_1^2\\end{bmatrix}.$ Whereas, using the lexicographical ordering $x_1 \\succ x_2$ , we obtain the basis $B_2=\\lbrace 1,x_2,x_2^2,x_2^3\\rbrace $ .", "The matrix associated to ${\\mathbf {f}}$ with respect to $B_2$ is the following Hankel matrix: $\\begin{bmatrix}4 & -2y_3 & -2y_2^2+2y_3^2+2y_1 & 6y_2^2y_3-2y_3^3 \\\\* & * & * & 2y_2^4-12y_2^2y_3^2+2y_3^4-4y_1y_2^2+2y_1^2 \\\\* & * & * & -10y_2^4y_3+20y_2^2y_3^3-2y_3^5+10y_1y_2^2y_3\\\\* & * & * &-2y_2^6+30y_2^4y_3^2-30y_2^2y_3^4+2y_3^6+6y_1y_2^4-18y_1y_2^2y_3^2-6y_1^2y_2^2+2y_1^3\\end{bmatrix}.$ We remark that all the entries of the matrices above lie in $\\mathbb {Q}[{\\mathbf {y}}]$ and that the entries of the second matrix are of higher degree than the first one's." ], [ "Gröbner bases and parametric Hermite matrices", "In the previous subsection, we have defined parametric Hermite matrices assuming one knows a Gröbner basis $G$ with respect to some monomial ordering of the ideal $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ where $\\mathbb {K}=\\mathbb {Q}({\\mathbf {y}})$ and $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ is the ideal of $\\mathbb {K}[{\\mathbf {x}}]$ generated by ${\\mathbf {f}}$ .", "Computing such a Gröbner basis may be costly as this would require to perform arithmetic operations over the field $\\mathbb {Q}({\\mathbf {y}})$ (or $\\mathbb {Z}/p\\mathbb {Z}({\\mathbf {y}})$ where $p$ is a prime when tackling this computational task through modular computations).", "In this paragraph, we show that one can obtain parametric Hermite matrices by considering some Gröbner bases of the ideal $\\langle {\\mathbf {f}}\\rangle \\subset \\mathbb {Q}[{\\mathbf {y}},{\\mathbf {x}}]$ (hence, enabling the use of efficient implementations of Gröbner bases such as the $F_4/ F_5$ algorithms [20], [21]).", "Since the graded reverse lexicographical ordering (grevlex for short) is known for yielding Gröbner bases of relatively small degree comparing to other orders, we prefer using this ordering to construct our parametric Hermite matrices.", "Further, we will use the notation ${grevlex}({\\mathbf {x}})$ for the grevlex ordering among the variables ${\\mathbf {x}}$ (with $x_1\\succ \\cdots \\succ x_n$ ) and ${grevlex}({\\mathbf {x}}) \\succ {grevlex}({\\mathbf {y}})$ (with $y_1\\succ \\cdots \\succ y_t$ ) for the elimination ordering.", "We denote respectively by $\\mathrm {lm}_{{\\mathbf {x}}}(p)$ and $\\mathrm {lc}_{{\\mathbf {x}}}(p)$ the leading monomial and the leading coefficient of $p \\in \\mathbb {K}[{\\mathbf {x}}]$ with respect to the ordering ${grevlex}({\\mathbf {x}})$ .", "Lemma 12 Let ${\\mathcal {G}}$ be the reduced Gröbner basis of $\\langle {\\mathbf {f}}\\rangle $ with respect to the elimination ordering ${grevlex}({\\mathbf {x}}) \\succ {grevlex}({\\mathbf {y}})$ .", "Then ${\\mathcal {G}}$ is also a Gröbner basis of $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ with respect to the ordering ${grevlex}({\\mathbf {x}})$ .", "Since ${\\mathcal {G}}$ is a Gröbner basis of the ideal $\\langle {\\mathbf {f}}\\rangle $ , every polynomial $f_i$ of ${\\mathbf {f}}$ can be written as $f_i=\\sum _{g\\in {\\mathcal {G}}}c_g\\cdot g$ where $c_g \\in \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]$ .", "Therefore, any element of $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ can also be written as a combination of elements of ${\\mathcal {G}}$ with coefficients in $\\mathbb {Q}({\\mathbf {y}})[{\\mathbf {x}}]$ .", "In other words, ${\\mathcal {G}}$ is a set of generators of $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ .", "Let $p$ be a polynomial in $\\mathbb {K}[{\\mathbf {x}}]$ , $p$ is contained $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ if and only if there exists a polynomial $q\\in \\mathbb {Q}[{\\mathbf {y}}]$ such that $q \\cdot p \\in \\langle {\\mathbf {f}}\\rangle $ .", "Thus, the leading monomial of $p$ as an element of $\\mathbb {K}[{\\mathbf {x}}]$ with respect to the grevlex ordering ${grevlex}({\\mathbf {x}})$ is contained in the ideal $\\langle \\mathrm {lm}_{{\\mathbf {x}}}(g)\\; \|\\; g \\in {\\mathcal {G}}\\rangle $ .", "Therefore, ${\\mathcal {G}}$ is a Gröbner basis of $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ .", "Hereafter, we denote by ${\\mathcal {G}}$ the reduced Gröbner basis of $\\langle {\\mathbf {f}}\\rangle $ with respect to the elimination ordering ${grevlex}({\\mathbf {x}}) \\succ {grevlex}({\\mathbf {y}})$ .", "Let $\\mathcal {B}$ be the set of all monomials in ${\\mathbf {x}}$ that are not reducible by ${\\mathcal {G}}$ , which is finite by Lemmas REF and REF .", "The set $\\mathcal {B}$ actually forms a basis of the $\\mathbb {K}$ -vector space $\\mathbb {K}[{\\mathbf {x}}]/\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ .", "Then, we denote by $\\mathcal {H}$ the parametric Hermite matrix associated to ${\\mathbf {f}}$ with respect to this basis $\\mathcal {B}$ .", "We consider the following assumption on the input system ${\\mathbf {f}}$ .", "CAssumption C For $g\\in {\\mathcal {G}}$ , the leading coefficient ${\\rm lc}_{{\\mathbf {x}}}(g)$ does not depend on the parameters ${\\mathbf {y}}$ .", "As the computations in the quotient ring $A_{\\mathbb {K}}$ are done through normal form reductions by ${\\mathcal {G}}$ , the lemma below is straight-forward.", "Lemma 13 Under Assumption (REF ), the entries of the parametric Hermite matrix ${\\mathcal {H}}$ are elements of $\\mathbb {Q}[{\\mathbf {y}}]$ .", "Since Assumption (REF ) holds, the leading coefficients $\\mathrm {lc}_{{\\mathbf {x}}}(g)$ do not depend on parameters ${\\mathbf {y}}$ for $g\\in {\\mathcal {G}}$ .", "The normal form reduction in $A_{\\mathbb {K}}$ of any polynomial in $\\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ returns a polynomial in $\\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ .", "Thus, each normal form can be written as a linear combination of $\\mathcal {B}$ whose coefficients lie in $\\mathbb {Q}[{\\mathbf {y}}]$ .", "Hence, the multiplication map ${\\mathcal {L}}_{b_i\\cdot b_j}$ for $1\\le i,j \\le \\delta $ can be represented by polynomial matrices in $\\mathbb {Q}[{\\mathbf {y}}]$ with respect to the basis $\\mathcal {B}$ .", "As an immediate consequence, the entries of ${\\mathcal {H}}$ , as being the traces of those multiplication maps, are polynomials in $\\mathbb {Q}[{\\mathbf {y}}]$ .", "The next proposition states that Assumption (REF ) is satisfied by a generic system ${\\mathbf {f}}$ .", "It implies that the entries of the parametric Hermite matrix of a generic system with respect to the basis $\\mathcal {B}$ derived from ${\\mathcal {G}}$ completely lie in $\\mathbb {Q}[{\\mathbf {y}}]$ .", "We postpone the proof of Proposition REF to Subsection REF where we prove a more general result (see Proposition REF ).", "Proposition 14 Let $\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}}]_d$ be the set of polynomials in $\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}}]$ having total degree bounded by $d$ .", "There exists a non-empty Zariski open subset ${F}_C$ of $\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}}]_d^n$ such that Assumption (REF ) is satisfied by any ${\\mathbf {f}}\\in {F}_C \\cap \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]^n$ ." ], [ "Specialization property of parametric Hermite matrices", "Recall that ${\\mathcal {G}}$ is the reduced Gröbner basis of $\\langle {\\mathbf {f}}\\rangle $ with respect to the ordering ${grevlex}({\\mathbf {x}}) \\succ {grevlex}({\\mathbf {y}})$ and $\\mathcal {B}$ is the basis of $\\mathbb {K}[{\\mathbf {x}}]/\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ derived from ${\\mathcal {G}}$ as discussed in the previous subsection.", "Then, $\\mathcal {H}$ is the parametric Hermite matrix associated to ${\\mathbf {f}}$ with respect to the basis $\\mathcal {B}$ .", "Let ${\\eta }\\in \\mathbb {C}^t$ and $\\phi _{{\\eta }}:\\mathbb {C}({\\mathbf {y}})[{\\mathbf {x}}] \\rightarrow \\mathbb {C}[{\\mathbf {x}}]$ , $p({\\mathbf {y}},{\\mathbf {x}})\\mapsto p({\\eta },{\\mathbf {x}})$ be the specialization map that evaluates the parameters ${\\mathbf {y}}$ at ${\\eta }$ .", "Then ${\\mathbf {f}}({\\eta },\\cdot ) =(\\phi _{{\\eta }}(f_1),\\ldots ,\\phi _{{\\eta }}(f_m))$ .", "We denote by ${\\mathcal {H}}({{\\eta }})$ the specialization $(\\phi _{{\\eta }}(h_{i,j}))_{1\\le i,j\\le \\delta }$ of ${\\mathcal {H}}$ at ${\\eta }$ .", "Recall that, for a polynomial $p \\in \\mathbb {C}({\\mathbf {y}})[{\\mathbf {x}}]$ , the leading coefficient of $p$ considered as a polynomial in the variables ${\\mathbf {x}}$ with respect to the ordering ${grevlex}({\\mathbf {x}})$ is denoted by $\\mathrm {lc}_{{\\mathbf {x}}}(p)$ .", "In this subsection, for $p \\in \\mathbb {C}[{\\mathbf {x}}]$ , we use $\\mathrm {lm}(p)$ to denote the leading monomial of $p$ with respect to the ordering ${grevlex}({\\mathbf {x}})$ .", "Let ${\\mathcal {W}}_{\\infty }\\subset \\mathbb {C}^t$ denote the algebraic set $\\cup _{g\\in {\\mathcal {G}}} V({\\rm lc}_{{\\mathbf {x}}}(g))$ .", "In Proposition REF , we prove that, outside ${\\mathcal {W}}_{\\infty }$ , the specialization ${\\mathcal {H}}({\\eta })$ coincides with the classic Hermite matrix of the zero-dimensional ideal ${\\mathbf {f}}({\\eta },\\cdot ) \\subset \\mathbb {Q}[{\\mathbf {x}}]$ .", "This is the main result of this subsection.", "Since the operations over the $\\mathbb {K}$ -vector space $A_{\\mathbb {K}}$ rely on normal form reductions by the Gröbner basis ${\\mathcal {G}}$ of $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ , the specialization property of ${\\mathcal {H}}$ depends on the specialization property of ${\\mathcal {G}}$ .", "Lemma REF below, which is a direct consequence of [38], provides the specialization property of ${\\mathcal {G}}$ .", "We give here a more elementary proof for this lemma than the one in [38].", "Lemma 15 Let ${\\eta }\\in \\mathbb {C}^t \\setminus {\\mathcal {W}}_{\\infty }$ .", "Then the specialization ${\\mathcal {G}}({\\eta },\\cdot )\\lbrace \\phi _{{\\eta }}(g) \\; \| \\; g\\in {\\mathcal {G}}\\rbrace $ is a Gröbner basis of the ideal $\\langle {\\mathbf {f}}({\\eta },\\cdot )\\rangle \\subset \\mathbb {C}[{\\mathbf {x}}]$ generated by ${\\mathbf {f}}({\\eta },\\cdot )$ with respect to the ordering ${grevlex}({\\mathbf {x}})$ .", "Since ${\\eta }\\in \\mathbb {C}^t \\setminus {\\mathcal {W}}_{\\infty }$ , the leading coefficient $\\mathrm {lc}_{{\\mathbf {x}}}(g)$ does not vanish at ${\\eta }$ for every $g\\in {\\mathcal {G}}$ .", "Thus, $\\mathrm {lm}_{{\\mathbf {x}}}(g) = \\mathrm {lm}(\\phi _{{\\eta }}(g))$ .", "We denote by $\\mathcal {M}$ the set of all monomials in the variables ${\\mathbf {x}}$ and $\\mathcal {M}_{{\\mathcal {G}}} \\lbrace m\\in \\mathcal {M}\\;\|\\; \\exists g\\in {\\mathcal {G}}\\; : \\; \\mathrm {lm}_{{\\mathbf {x}}}(g)\\text{ divides }m\\rbrace =\\lbrace m\\in \\mathcal {M}\\;\|\\;\\exists g\\in {\\mathcal {G}}\\; : \\; \\mathrm {lm}(\\phi _{{\\eta }}(g)) \\text{ divides }m\\rbrace .$ For any $p \\in \\langle {\\mathbf {f}}\\rangle \\subset \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]$ , we prove that $\\mathrm {lm}(\\phi _{{\\eta }}(f)) \\in \\mathcal {M}_G$ .", "If $p$ is identically zero, there is nothing to prove.", "So, we assume that $p \\ne 0$ , $p$ is then expanded in the form below: $p = \\sum _{m\\in \\mathcal {M}_{G}} c_m \\cdot m + \\sum _{m\\in \\mathcal {M}\\setminus \\mathcal {M}_{G}} c_m \\cdot m,$ where the $c_m$ 's are elements of $\\mathbb {Q}[{\\mathbf {y}}]$ .", "Since $p$ is not identically zero, there exists $m\\in \\mathcal {M}_{{\\mathcal {G}}}$ such that $c_m\\ne 0$ .", "Since ${\\mathcal {G}}$ is a Gröbner basis of $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ , any monomial in $\\mathcal {M}_{{\\mathcal {G}}}$ can be reduced by ${\\mathcal {G}}$ to a unique normal form in $\\mathbb {K}[{\\mathbf {x}}]$ .", "These divisions involve denominators, which are products of some powers of the leading coefficients of ${\\mathcal {G}}$ with respect to the variables ${\\mathbf {x}}$ .", "We write $\\mathrm {NF}_{{\\mathcal {G}}}(p) = \\sum _{m\\in \\mathcal {M}_{{\\mathcal {G}}}} c_m\\cdot \\mathrm {NF}_{{\\mathcal {G}}}(m) + \\sum _{m\\in \\mathcal {M}\\setminus \\mathcal {M}_{{\\mathcal {G}}}}c_m \\cdot m .$ As $p \\in \\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ , we have that $\\mathrm {NF}_{{\\mathcal {G}}}(p) = 0$ , which implies $\\sum _{m\\in \\mathcal {M}\\setminus \\mathcal {M}_{{\\mathcal {G}}}} c_m \\cdot m =-\\sum _{m\\in \\mathcal {M}_{{\\mathcal {G}}}} c_m\\cdot \\mathrm {NF}_{{\\mathcal {G}}}(m).$ Therefore, we have the identity $p=\\sum _{m \\in \\mathcal {M}_{{\\mathcal {G}}}} c_m \\cdot (m-\\mathrm {NF}_{{\\mathcal {G}}}(m))$ Since ${\\eta }$ does not cancel any denominator appearing in $\\mathrm {NF}_{{\\mathcal {G}}}(m)$ , we can specialize the identity above without any problem: $\\phi _{{\\eta }}(p)=\\sum _{m \\in \\mathcal {M}_{{\\mathcal {G}}}} \\phi _{{\\eta }}(c_m) \\cdot (m-\\phi _{{\\eta }}(\\mathrm {NF}_{{\\mathcal {G}}}(m))).$ If at least one of the $\\phi _{{\\eta }}(c_m)$ does not vanish, then the leading monomial of $\\phi _{{\\eta }}(f)$ is in $\\mathcal {M}_{{\\mathcal {G}}}$ .", "Otherwise, if all the $\\phi _{{\\eta }}(c_m)$ are canceled, then $\\phi _{{\\eta }}(p)$ is identically zero, and there is not any new leading monomial appearing either.", "So, the leading monomial of any $p \\in \\langle {\\mathbf {f}}_{{\\eta }}\\rangle $ is contained in $\\mathcal {M}_{{\\mathcal {G}}}$ , which means ${\\mathcal {G}}({\\eta },\\cdot )$ is a Gröbner basis of $\\langle {\\mathbf {f}}({\\eta },\\cdot )\\rangle $ with respect to ${grevlex}({\\mathbf {x}})$ .", "Proposition 16 For any ${\\eta }\\in \\mathbb {C}^t \\setminus {\\mathcal {W}}_{\\infty }$ , the specialization ${\\mathcal {H}}({\\eta })$ coincides with the classic Hermite matrix of the zero-dimensional ideal $\\langle {\\mathbf {f}}({\\eta },\\cdot ) \\rangle \\subset \\mathbb {C}[{\\mathbf {x}}]$ .", "As a consequence of Lemma REF , each computation in $A_{\\mathbb {K}}$ derives a corresponding one in $\\mathbb {C}[{\\mathbf {x}}]/\\langle {\\mathbf {f}}({\\eta },\\cdot )\\rangle $ by evaluating ${\\mathbf {y}}$ at ${\\eta }$ in every normal form reduction by ${\\mathcal {G}}$ .", "This evaluation is allowed since ${\\eta }$ does not cancel any denominator appearing during the computation.", "Therefore, we deduce immediately the specialization property of the Hermite matrix.", "Using Proposition REF and [5], we obtain immediately the following corollary that allows us to use parametric Hermite matrices to count the root of a specialization of a parametric system.", "Corollary 17 Let ${\\eta }\\in \\mathbb {C}^t\\setminus {\\mathcal {W}}_{\\infty }$ , then the rank of $H({\\eta })$ is the number of distinct complex roots of ${\\mathbf {f}}({\\eta },\\cdot )$ .", "When ${\\eta }\\in \\mathbb {R}^t \\setminus {\\mathcal {W}}_{\\infty }$ , the signature of $H({\\eta })$ is the number of distinct real roots of ${\\mathbf {f}}({\\eta },\\cdot )$ .", "By Proposition REF , ${\\mathcal {H}}({\\eta })$ is a Hermite matrix of the zero-dimensional ideal $\\langle {\\mathbf {f}}({\\eta },\\cdot )\\rangle $ .", "Then, [5] implies that the rank (resp.", "the signature) of ${\\mathcal {H}}({\\eta })$ equals to the number of distinct complex (resp.", "real) solutions of ${\\mathbf {f}}({\\eta },\\cdot )$ .", "We finish this subsection by giving some explanation for what happens above ${\\mathcal {W}}_{\\infty }$ , where our parametric Hermite matrix ${\\mathcal {H}}$ does not have good specialization property.", "Lemma 18 Let ${\\mathcal {W}}_{\\infty }$ defined as above.", "Then ${\\mathcal {W}}_{\\infty }$ contains all the following sets: The non-proper points of the restriction of $\\pi $ to ${\\mathcal {V}}$ (see Section for this definition).", "The set of points ${\\eta }\\in \\mathbb {C}^t$ such that the fiber $\\pi ^{-1}({\\eta }) \\cap {\\mathcal {V}}$ is infinite.", "The image by $\\pi $ of the irreducible components of ${\\mathcal {V}}$ whose dimensions are smaller than $t$ .", "The claim for the set of non-properness of the restriction of $\\pi $ to ${\\mathcal {V}}$ is already proven in [41].", "We focus on the two remaining sets.", "Using the Hermite matrix, we know that for ${\\eta }\\in \\mathbb {C}^t\\setminus {\\mathcal {W}}_{\\infty }$ , the system ${\\mathbf {f}}({\\eta },\\cdot )$ admits a non-empty finite set of complex solutions.", "On the other hand, for any ${\\eta }\\in \\mathbb {C}^t$ such that $\\pi ^{-1}({\\eta })\\cap {\\mathcal {V}}$ is infinite, ${\\mathbf {f}}({\\eta },\\cdot )$ has infinitely many complex solutions.", "Therefore, the set of such points ${\\eta }$ is contained in ${\\mathcal {W}}_{\\infty }$ .", "Let ${\\mathcal {V}}_{> t}$ be the union of irreducible components of ${\\mathcal {V}}$ of dimension greater than $t$ .", "By the fiber dimension theorem [54], the fibers of the restriction of $\\pi $ to ${\\mathcal {V}}_{>t}$ must have dimension at least one.", "Similarly, the components of dimension $t$ whose images by $\\pi $ are contained in a Zariski closed subset of $\\mathbb {C}^t$ also yield infinite fibers.", "Therefore, as proven above, all of these components are contained in $\\pi ^{-1}({\\mathcal {W}}_{\\infty })$ .", "We now consider the irreducible components of dimension smaller than $t$ .", "Let ${\\mathcal {V}}_{\\ge t}$ and ${\\mathcal {V}}_{<t}$ be respectively the union of irreducible components of ${\\mathcal {V}}$ of dimension at least $t$ and at most $t-1$ .", "We have that ${\\mathcal {V}}= {\\mathcal {V}}_{\\ge t} \\cup {\\mathcal {V}}_{< t}$ .", "Let $I \\subset \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]$ denote the ideal generated by ${\\mathbf {f}}$ .", "Using the primary decomposition of $I$ (see e.g.", "[16]), we have that $I$ is the intersection of two ideals $I_{\\ge t}$ and $I_{< t}$ such that $V(I_{\\ge t}) = {\\mathcal {V}}_{\\ge t}$ and $V(I_{< t}) = {\\mathcal {V}}_{< t}$ .", "We write $I = I_{\\ge t} \\cap I_{< t}.$ We denote by $R$ the polynomial ring $\\mathbb {Q}({\\mathbf {y}})[{\\mathbf {x}}]$ .", "Then, the above identity is transferred into $R$ : $I\\cdot R = (I_{\\ge t}\\cdot R) \\cap (I_{<t} \\cdot R).$ Since $\\dim (\\overline{\\pi ({\\mathcal {V}}_{<t})}) \\le t-1$ , then there exists a non-zero polynomial $p \\in I_{<t}\\cap \\mathbb {Q}[{\\mathbf {y}}]$ .", "As $p$ is a unit in $\\mathbb {Q}({\\mathbf {y}})$ , the ideal $I_{<t}\\cdot R$ is exactly $R$ .", "So, $I\\cdot R = I_{\\ge t} \\cdot R.$ Note that, by Lemma REF , ${\\mathcal {G}}$ is a Gröbner basis of $I\\cdot R$ , then it is also a Gröbner basis of $I_{\\ge t} \\cdot R$ .", "Therefore, the Hermite matrices associated to $I$ and $I_{\\ge t}$ (with respect to the basis derived from ${\\mathcal {G}}$ ) coincide.", "So, for ${\\eta }\\notin {\\mathcal {W}}_{\\infty }$ , we have that $\\pi ^{-1}({\\eta })\\cap {\\mathcal {V}}=\\pi ^{-1}({\\eta }) \\cap {\\mathcal {V}}_{\\ge t}$ .", "This leads to $\\pi ^{-1}(\\mathbb {C}^t\\setminus {\\mathcal {W}}_{\\infty })\\cap {\\mathcal {V}}_{\\ge t} =\\pi ^{-1}(\\mathbb {C}^t\\setminus {\\mathcal {W}}_{\\infty })\\cap {\\mathcal {V}}.$ Then, $\\pi ^{-1}(\\mathbb {C}^t\\setminus {\\mathcal {W}}_{\\infty }) \\cap {\\mathcal {V}}_{< t} = \\emptyset $ or equivalently, ${\\mathcal {V}}_{<t} \\subset \\pi ^{-1}({\\mathcal {W}}_{\\infty })$ , which concludes the proof." ], [ "Computing parametric Hermite matrices", "Given ${\\mathbf {f}}= (f_1,\\ldots ,f_m) \\in \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ satisfying Assumption (REF ).", "We keep denoting $\\mathbb {K}= \\mathbb {Q}({\\mathbf {y}})$ .", "Let ${\\mathcal {G}}$ be the reduced Gröbner basis of $\\langle {\\mathbf {f}}\\rangle $ with respect to the ordering ${grevlex}({\\mathbf {x}}) \\succ {grevlex}({\\mathbf {y}})$ and $\\mathcal {B}$ be the set of all monomials in the variables ${\\mathbf {x}}$ which are not reducible by ${\\mathcal {G}}$ .", "The set $\\mathcal {B}$ then forms a basis of the $\\mathbb {K}$ -vector space $\\mathbb {K}[{\\mathbf {x}}] / \\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ .", "In this subsection, we focus on the computation of the parametric Hermite matrix associated to ${\\mathbf {f}}$ with respect to the basis $\\mathcal {B}$ .", "Note that one can design an algorithm using only the definition of parametric Hermite matrices given in Subsection REF .", "More precisely, for each $b_i\\cdot b_j \\in \\mathcal {B}$ ($1\\le i, j \\le \\delta $ ), one computes the matrix representing ${\\mathcal {L}}_{b_i\\cdot b_j}$ in the basis $\\mathcal {B}$ by computing the normal form of every $b_i\\cdot b_j\\cdot b_k$ for $1 \\le k \\le \\delta $ .", "Therefore, in total, this direct algorithm requires $O(\\delta ^3)$ normal form reductions of polynomials in $\\mathbb {K}[{\\mathbf {x}}]$ .", "In Algorithm REF below, we present another algorithm for computing ${\\mathcal {H}}$ .", "We call to the following subroutines successively: $\\bullet $ GrobnerBasis that takes as input the system ${\\mathbf {f}}$ and computes the reduced Gröbner basis ${\\mathcal {G}}$ of $\\langle {\\mathbf {f}}\\rangle $ with respect to the ordering ${grevlex}({\\mathbf {x}}) \\succ {grevlex}({\\mathbf {y}})$ and the basis $\\mathcal {B} =\\lbrace b_1,\\ldots ,b_{\\delta }\\rbrace \\subset \\mathbb {Q}[{\\mathbf {x}}]$ derived from ${\\mathcal {G}}$ .", "Such an algorithm can be obtained using any general algorithm for computing Gröbner basis, which we refer to F4/F5 algorithms [20], [21].", "$\\bullet $ ReduceGB that takes as input the Gröbner basis ${\\mathcal {G}}$ and outputs a subset ${\\mathcal {G}}^{\\prime }$ of ${\\mathcal {G}}$ which is still a Gröbner basis of $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ with respect to the ordering ${grevlex}({\\mathbf {x}})$ .", "This subroutine aims to remove the elements in ${\\mathcal {G}}$ that we do not need.", "Even though ${\\mathcal {G}}$ is reduced as a Gröbner basis of $\\langle {\\mathbf {f}}\\rangle $ with respect to ${grevlex}({\\mathbf {x}}) \\succ {grevlex}({\\mathbf {y}})$ , it is not necessarily the reduced Gröbner basis of $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ with respect to ${grevlex}({\\mathbf {x}})$ .", "Using [16], we can design ReduceGB to remove all the elements of ${\\mathcal {G}}$ which have duplicate leading monomials (in ${\\mathbf {x}}$ ).", "We obtain as output a subset ${\\mathcal {G}}^{\\prime }$ of ${\\mathcal {G}}$ which is also a Gröbner basis ${\\mathcal {G}}^{\\prime }$ for $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ with respect to ${grevlex}({\\mathbf {x}})$ .", "Note that this tweak reduces not only the cardinal of the Gröbner basis in use but also the size of the set ${\\mathcal {W}}_{\\infty }$ introduced in Subsection REF (as we have less leading coefficients).", "$\\bullet $ XMatrices that takes as input $({\\mathcal {G}}^{\\prime },\\mathcal {B})$ and computes the matrix representation of the multiplication maps ${\\mathcal {L}}_{x_i}$ ($1\\le i \\le n$ ) with respect to $\\mathcal {B}$ .", "This computation is done directly by reducing every $x_i\\cdot b_j$ ($1\\le i \\le n$ , $1\\le j \\le \\delta $ ) to its normal form in $\\mathbb {K}[{\\mathbf {x}}]/\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ using ${\\mathcal {G}}^{\\prime }$ .", "$\\bullet $ BMatrices that takes as input the matrices representing $({\\mathcal {L}}_{x_1},\\ldots ,{\\mathcal {L}}_{x_n})$ and $\\mathcal {B}$ and computes the matrices representing the ${\\mathcal {L}}_{b_i}$ 's ($1\\le i\\le \\delta $ ) in the basis $\\mathcal {B}$ .", "We design BMatrices in a way that it constructs the matrices of ${\\mathcal {L}}_{b_i}$ 's inductively in the degree of the $b_i$ 's as follows.", "At the beginning, we have the multiplication matrices of 1 and the $x_i$ 's; those are the matrices of the elements of degree zero and one.", "Note that, for any element $b$ of $\\mathcal {B}$ .", "At the step of computing the matrix of an element $b\\in \\mathcal {B}$ , we remark that there exist a variable $x_i$ and a monomial $b^{\\prime }\\in \\mathcal {B}$ such that $b=x_i\\cdot b^{\\prime }$ and the matrix of $b^{\\prime }$ is already computed (as $\\deg (b^{\\prime }) <\\deg (b)$ .", "Therefore, we simply multiply the matrices of ${\\mathcal {L}}_{x_i}$ and ${\\mathcal {L}}_{b^{\\prime }}$ to obtain the matrix of ${\\mathcal {L}}_{b}$ .", "$\\bullet $ TraceComputing that takes as input the multiplication matrices ${\\mathcal {L}}_{b_1},\\ldots ,{\\mathcal {L}}_{b_{\\delta }}$ and computes the matrix $(\\mathrm {trace}({\\mathcal {L}}_{b_i\\cdot b_j}))_{1\\le i \\le j \\le \\delta }$ .", "This matrix is in fact the parametric Hermite matrix ${\\mathcal {H}}$ associated to ${\\mathbf {f}}$ with respect to the basis $\\mathcal {B}$ .", "To design this subroutine, we use the following remark given in [46].", "Let $p,q \\in \\mathbb {K}[{\\mathbf {x}}]$ .", "The normal form $\\overline{p}$ of $p$ by ${\\mathcal {G}}$ can be written as $\\overline{p} = \\sum _{i=1}^{\\delta } c_i\\cdot b_i$ where the $c_i$ 's lie in $\\mathbb {K}$ .", "Then, we have the identity ${\\rm trace}({\\mathcal {L}}_{p\\cdot q})= \\sum _{i=1}^{\\delta } c_i\\cdot \\mathrm {trace}({\\mathcal {L}}_{p\\cdot b_i}),$ Hence, by choosing $p = b_i\\cdot b_j$ and $q =1$ , we can compute $h_{i,j}$ using the normal form $\\overline{b_i\\cdot b_j}$ and $\\mathrm {trace}({\\mathcal {L}}_{b_1}),\\ldots ,\\mathrm {trace}({\\mathcal {L}}_{b_{\\delta }})$ .", "Note that $\\mathrm {trace}({\\mathcal {L}}_{b_i})$ is easily computed from the matrix of the map ${\\mathcal {L}}_{b_i}$ .", "On the other hand, the normal form $\\overline{b_i\\cdot b_j}$ can be read off from the $j$ -th row of the matrix representing ${\\mathcal {L}}_{b_i}$ , which is already computed at this point.", "It is also important to notice that there are many duplicated entries in ${\\mathcal {H}}$ .", "Thus, we should avoid all the unnecessary re-computation.", "This is done easily be keeping a list for tracking distinct entries of ${\\mathcal {H}}$ .", "The pseudo-code of Algorithm REF is presented below.", "Its correctness follows simply from our definition of parametric Hermite matrices.", "Beside the parametric Hermite matrix ${\\mathcal {H}}$ , we return a polynomial $\\mathbf {w}_{\\infty }$ which is the square-free part of ${\\rm lcm}_{g\\in {\\mathcal {G}}}(\\mathrm {lc}_{{\\mathbf {x}}}(g))$ for further usage.", "Note that $V(\\mathbf {w}_{\\infty }) = {\\mathcal {W}}_{\\infty }$ .", "[H] A parametric polynomial system ${\\mathbf {f}}=(f_1,\\ldots ,f_m)$ A parametric Hermite matrix ${\\mathcal {H}}$ associated to ${\\mathbf {f}}$ with respect to the basis $\\mathcal {B}$ ${\\mathcal {G}}, \\mathcal {B} \\leftarrow {\\sf GrobnerBasis}({\\mathbf {f}}, {grevlex}({\\mathbf {x}}) \\succ {grevlex}({\\mathbf {y}}))$ ${\\mathcal {G}}^{\\prime } \\leftarrow {\\sf ReduceGB}({\\mathcal {G}})$ $\\mathbf {w}_{\\infty } \\leftarrow {\\sf sqfree}({\\rm lcm}_{g\\in {\\mathcal {G}}}(\\mathrm {lc}_{{\\mathbf {x}}}(g)))$ $({\\mathcal {L}}_{x_1},\\ldots ,{\\mathcal {L}}_{x_n})\\leftarrow {\\sf XMatrices}({\\mathcal {G}}^{\\prime },\\mathcal {B})$ $({\\mathcal {L}}_{b_1},\\ldots ,{\\mathcal {L}}_{b_{\\delta }}) \\leftarrow {\\sf BMatrices}(({\\mathcal {L}}_{x_1},\\ldots ,{\\mathcal {L}}_{x_n}),\\mathcal {B})$ ${\\mathcal {H}}\\leftarrow {\\sf TraceComputing}({\\mathcal {L}}_{b_1},\\ldots ,{\\mathcal {L}}_{b_{\\delta }})$ $[{\\mathcal {H}},\\mathbf {w}_{\\infty }]$ DRL-Matrix" ], [ "Removing denominators", "Note that, through the computation in the quotient ring $A_{\\mathbb {K}}$ , the entries of our parametric Hermite matrix possibly contains denominators that lie in $\\mathbb {Q}[{\\mathbf {y}}]$ .", "As the algorithm that we introduce in Section will require us to manipulate the parametric Hermite matrix that we compute, these denominators can be a bottleneck to handle the matrix.", "Therefore, we introduce an extra subroutine RemoveDenominator that returns a parametric Hermite matrix ${\\mathcal {H}}^{\\prime }$ of ${\\mathbf {f}}$ without denominator.", "$\\bullet $ RemoveDenominator that takes as input the matrix ${\\mathcal {H}}$ computed by DRL-Matrix and outputs a matrix ${\\mathcal {H}}^{\\prime }$ which is the parametric Hermite matrix associated to ${\\mathbf {f}}$ with respect to a basis $\\mathcal {B^{\\prime }}$ that will be made explicit below.", "As we can freely choose any basis of form $\\lbrace c_i\\cdot b_i \\; \| \\;1\\le i\\le \\delta \\rbrace $ where the $c_i$ 's are elements of $\\mathbb {Q}[{\\mathbf {y}}]$ , we should use a basis that leads to a denominator-free matrix.", "To do this, we choose $c_i$ as the denominator of $\\mathrm {trace}({\\mathcal {L}}_{b_i})$ (which lies in the first row of the matrix ${\\mathcal {H}}$ computed by TraceComputing).", "Then, for the entry of ${\\mathcal {H}}$ that corresponds to $b_i$ and $b_j$ , we can multiply it with $c_i\\cdot c_j$ .", "The output matrix ${\\mathcal {H}}^{\\prime }$ is the parametric Hermite matrix associated to ${\\mathbf {f}}$ with respect to the basis $\\lbrace c_i\\cdot b_i\\; \| \\; 1\\le i \\le \\delta \\rbrace $ .", "It usually does not contain any denominator and is handled easier in practice." ], [ "Evaluation & interpolation scheme for generic systems", "Here we assume that the input system ${\\mathbf {f}}$ satisfies Assumption (REF ).", "By Lemma REF , the entries of ${\\mathcal {H}}$ are polynomials in $\\mathbb {Q}[{\\mathbf {y}}]$ .", "Suppose that we know beforehand a value $\\Lambda $ that is larger than the degree of any entry of ${\\mathcal {H}}$ , we can compute ${\\mathcal {H}}$ by an evaluation & interpolation scheme as follows.", "We start by choosing randomly a set $\\mathcal {E}$ of $\\binom{t +\\Lambda }{t}$ distinct points in $\\mathbb {Q}^t$ .", "Then, for each ${\\eta }\\in \\mathcal {E}$ , we use DRL-Matrix (Algorithm REF ) on the input ${\\mathbf {f}}({\\eta },\\cdot )$ to compute the classic Hermite matrix associated to ${\\mathbf {f}}({\\eta },\\cdot )$ with respect to the ordering ${grevlex}({\\mathbf {x}})$ .", "These computations involve only polynomials in $\\mathbb {Q}[{\\mathbf {x}}]$ and not in $\\mathbb {Q}({\\mathbf {y}})[{\\mathbf {x}}]$ .", "Finally, we interpolate the parametric Hermite matrix ${\\mathcal {H}}$ from its specialized images ${\\mathcal {H}}({\\eta })$ computed previously.", "Since Assumption (REF ) holds, then ${\\mathcal {W}}_{\\infty }$ is empty.", "By Proposition REF , the Hermite matrix of ${\\mathbf {f}}({\\eta },\\cdot )$ with respect to ${grevlex}({\\mathbf {x}})$ is the image ${\\mathcal {H}}({\\eta })$ of ${\\mathcal {H}}$ .", "Therefore, the above scheme computes correctly the parametric Hermite matrix ${\\mathcal {H}}$ .", "We also remark that, in the computation of the specializations ${\\mathcal {H}}({\\eta })$ , we can replace the subroutine XMatrices in DRL-Matrix by a linear-algebra-based algorithm described in [19].", "That algorithm constructs the Macaulay matrix and carries out matrix reductions to obtain simultaneously the normal forms that XMatrices requires.", "Assume a degree bound $\\Lambda $ is known, we estimate the arithmetic complexity for computing the parametric Hermite matrix in Proposition REF below.", "We postpone to Subsection REF for proving an explicit bound for $\\Lambda $ when the input system satisfies some extra generic assumptions.", "Proposition 19 Assume that the system ${\\mathbf {f}}= (f_1,\\ldots ,f_m) \\subset \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ satisfying Assumptions (REF ) and (REF ).", "Let $\\delta $ be the dimension of the $\\mathbb {K}$ -vector space $\\mathbb {K}[{\\mathbf {x}}]/\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ where $\\mathbb {K}=\\mathbb {Q}({\\mathbf {y}})$ .", "Let ${\\mathcal {H}}$ be the parametric Hermite matrix associated to ${\\mathbf {f}}$ constructed using ${grevlex}({\\mathbf {x}})$ ordering.", "Then, by Lemma REF , the entries of the parametric Hermite matrix ${\\mathcal {H}}$ lie in $\\mathbb {Q}[{\\mathbf {y}}]$ .", "Let $\\Lambda $ be an upper degree bound of the entries of ${\\mathcal {H}}$ .", "Using the evaluation & interpolation scheme, one can compute ${\\mathcal {H}}$ within $O \\ {\\widetilde{~}} \\left(\\binom{t+\\Lambda }{t}\\left(m\\ \\binom{d+n+t}{n+t} + \\delta ^{\\omega +1} + \\delta ^2\\log ^2\\binom{t+\\Lambda }{t} \\right)\\right)$ arithmetic operations in $\\mathbb {Q}$ , where, by Bézout's bound, $\\delta $ is bounded by $d^m$ .", "As the degrees of the entries of ${\\mathcal {H}}$ are bounded by $\\Lambda $ .", "Following the evaluation & interpolation scheme requires one to compute $\\binom{t+\\Lambda }{t}$ specialized Hermite matrices.", "We first analyze the complexity for computing each of those specialized Hermite matrices.", "The evaluation of ${\\mathbf {f}}$ at each point ${\\eta }\\in \\mathbb {Q}^t$ costs $O\\left(m\\ \\binom{d+n+t}{n+t}\\right)$ arithmetic operations in $\\mathbb {Q}$ .", "Next, we compute the matrices representing the ${\\mathcal {L}}_{x_i}$ 's using the linear algebra approach given in [19].", "It yields an arithmetic complexity of $O(n \\delta ^{\\omega })$ , where $\\omega $ is the exponential constant for matrix multiplication.", "The traces of those matrices are then computed using $n\\delta $ additions in $\\mathbb {Q}$ .", "The subroutine BMatrices consists of essentially $\\delta $ multiplication of $\\delta \\times \\delta $ matrices (with entries in $\\mathbb {Q}$ ).", "This leads to an arithmetic complexity $O(\\delta ^{\\omega +1})$ .", "Next, the computation of each entries $h_{i,j}$ is simply a vector multiplication of length $\\delta $ , whose complexity is $O(\\delta )$ .", "Thus, TraceComputing takes in overall $O(\\delta ^3)$ arithmetic operations in $\\mathbb {Q}$ .", "Thus, every specialized Hermite matrix can be computed using $O(\\delta ^{\\omega +1})$ arithmetic operations in $\\mathbb {Q}$ .", "In total, the complexity of the evaluation step lies in $O\\left(\\binom{t+\\Lambda }{t}\\left(m\\ \\binom{d+n+t}{n+t} + \\delta ^{\\omega +1}\\right)\\right)$ .", "Finally, we interpolation $\\delta ^2$ entries which are polynomials in $\\mathbb {Q}[{\\mathbf {y}}]$ of degree at most $\\Lambda $ .", "Using the multivariate interpolation algorithm of [12], the complexity of this step therefore lies in $O\\ {\\widetilde{~}} \\left( \\delta ^2\\ \\binom{t+\\Lambda }{t} \\log ^2\\binom{t+\\Lambda }{t} \\right)$ .", "Summing up the both steps, we conclude that the parametric Hermite matrix ${\\mathcal {H}}$ can be obtained within $O \\ {\\widetilde{~}} \\left(\\binom{t+\\Lambda }{t}\\left(m\\ \\binom{d+n+t}{n+t} + \\delta ^{\\omega +1} + \\delta ^2\\log ^2\\binom{t+\\Lambda }{t} \\right)\\right)$ arithmetic operations in $\\mathbb {Q}$ ." ], [ "Algorithms for real root classification", "We present in this section two algorithms targeting the real root classification problem through parametric Hermite matrices.", "The one described in Subsection REF aims to solve the weak version of Problem (REF ).", "The second algorithm, given in Subsection REF outputs the semi-algebraic formulas of the cells ${\\mathcal {S}}_i$ that solves Problem (REF ).", "Further, in Section , we will see that, for a generic sequence ${\\mathbf {f}}$ , the semi-algebraic formulas computed by this algorithm consist of polynomials of degree bounded by $n(d-1)d^n$ , which is better than the degree bound $2d^{2n}$ obtained by Algorithm and all previously known bounds.", "Throughout this section, our input is a parametric polynomial system ${\\mathbf {f}}=(f_1,\\ldots ,f_m) \\subset \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ .", "We require that ${\\mathbf {f}}$ satisfies Assumptions (REF ) and that the ideal $\\langle {\\mathbf {f}}\\rangle $ is radical.", "Let ${\\mathcal {G}}$ be the reduced Gröbner basis of the ideal $\\langle {\\mathbf {f}}\\rangle \\subset \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]$ with respect to the ordering ${grevlex}({\\mathbf {x}})\\succ {grevlex}({\\mathbf {y}})$ .", "Let $\\mathbb {K}$ denote the rational function field $\\mathbb {Q}({\\mathbf {y}})$ .", "We recall that $\\mathcal {B}\\subset \\mathbb {Q}[{\\mathbf {x}}]$ is the basis of $\\mathbb {K}[{\\mathbf {x}}]/\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ derived from ${\\mathcal {G}}$ and ${\\mathcal {H}}$ is the parametric Hermite matrix associated to ${\\mathbf {f}}$ with respect to the basis $\\mathcal {B}$ ." ], [ "Algorithm for the weak-version of Problem\n(", "From Subsection REF , we know that, outside the algebraic set ${\\mathcal {W}}_{\\infty } \\cup _{g\\in {\\mathcal {G}}}V(\\mathrm {lc}_{{\\mathbf {x}}}(g))$ , the parametric matrix ${\\mathcal {H}}$ possesses good specialization property (see Proposition REF ).", "We denote by $\\mathbf {w}_{\\infty }$ the square-free part of ${\\rm lcm}_{g\\in {\\mathcal {G}}}\\mathrm {lc}_{{\\mathbf {x}}}(g)$ .", "This polynomial $\\mathbf {w}_{\\infty }$ is returned as an output of Algorithm REF .", "Note that $V(\\mathbf {w}_{\\infty }) = {\\mathcal {W}}_{\\infty }$ .", "Lemma 20 When Assumption (REF ) holds and the ideal $\\langle {\\mathbf {f}}\\rangle $ is radical, the determinant of ${\\mathcal {H}}$ is not identically zero.", "Recall that $\\mathbb {K}$ denotes the rational function field $\\mathbb {Q}({\\mathbf {y}})$ .", "We prove that the ideal $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}} \\subset \\mathbb {K}[{\\mathbf {x}}]$ is radical.", "Let $p \\in \\mathbb {K}[{\\mathbf {x}}]$ such that there exists $n\\in \\mathbb {N}$ satisfying $p^n \\in \\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ .", "Therefore, there exists a polynomial $q \\in \\mathbb {Q}[{\\mathbf {y}}]$ such that $q\\cdot p^n \\in \\langle {\\mathbf {f}}\\rangle $ .", "Then, $(q\\cdot p)^n \\in \\langle {\\mathbf {f}}\\rangle $ .", "As $\\langle {\\mathbf {f}}\\rangle $ is radical, we have that $q\\cdot p \\in \\langle {\\mathbf {f}}\\rangle $ .", "Thus, $p \\in \\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ , which concludes that $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ is radical.", "By Lemma REF , $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ is a radical zero-dimensional ideal in $\\mathbb {Q}({\\mathbf {y}})$ .", "Since ${\\mathcal {H}}$ is also a Hermite matrix (in the classic sense) of $\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ , ${\\mathcal {H}}$ is full rank.", "Therefore, $\\det ({\\mathcal {H}})$ is not identically zero.", "Let $\\mathbf {w}_{{\\mathcal {H}}} \\mathfrak {n}/\\gcd (\\mathfrak {n},\\mathbf {w}_{\\infty })$ where $\\mathfrak {n}$ is the square-free part of the numerator of $\\det ({\\mathcal {H}})$ .", "We denote by ${\\mathcal {W}}_{{\\mathcal {H}}}$ the vanishing set of $\\mathbf {w}_{{\\mathcal {H}}}$ .", "By Lemma REF , ${\\mathcal {W}}_{{\\mathcal {H}}}$ is a proper Zariski closed subset of $\\mathbb {C}^t$ .", "Our algorithm relies on the following proposition.", "Proposition 21 Assume that Assumption (REF ) holds and the ideal $\\langle {\\mathbf {f}}\\rangle $ is radical.", "Then, for each connected component ${\\mathcal {S}}$ of the semi-algebraic set $\\mathbb {R}^t \\setminus ({\\mathcal {W}}_{\\infty } \\cup {\\mathcal {W}}_{{\\mathcal {H}}})$ , the number of real solutions of ${\\mathbf {f}}({\\eta },\\cdot )$ is invariant when ${\\eta }$ varies over ${\\mathcal {S}}$ .", "By Lemma REF , ${\\mathcal {W}}_{\\infty }$ contains the following sets: The non-proper points of the restriction of $\\pi $ to ${\\mathcal {V}}$ .", "The point ${\\eta }\\in \\mathbb {C}^t$ such that the fiber $\\pi ^{-1}({\\eta })\\cap {\\mathcal {V}}$ is infinite.", "The image by $\\pi $ of the irreducible components of ${\\mathcal {V}}$ whose dimensions are smaller than $t$ .", "Now we consider the set $K(\\pi ,{\\mathcal {V}}) \\mathrm {sing}({\\mathcal {V}}) \\cup \\mathrm {crit}(\\pi ,{\\mathcal {V}})$ .", "Let $\\Delta {\\rm jac}({\\mathbf {f}},{\\mathbf {x}})$ be the Jacobian matrix of ${\\mathbf {f}}$ with respect to the variables ${\\mathbf {x}}$ .", "The ideal generated by the $n\\times n$ -minors of $\\Delta $ is denoted by $I_{\\Delta }$ .", "Note that, since ${\\mathbf {f}}$ is radical, $K(\\pi ,{\\mathcal {V}})$ is the algebraic set defined by the ideal $\\langle {\\mathbf {f}}\\rangle + I_{\\Delta }$ .", "By Proposition REF , for ${\\eta }\\in \\mathbb {C}^t\\setminus {\\mathcal {W}}_{\\infty }$ , $\\langle {\\mathbf {f}}\\rangle $ is a zero-dimensional ideal and the quotient ring $\\mathbb {C}[{\\mathbf {x}}]/\\langle {\\mathbf {f}}({\\eta },\\cdot )\\rangle $ has dimension $\\delta $ .", "Moreover, if ${\\eta }\\in \\mathbb {C}^t \\setminus ({\\mathcal {W}}_{\\infty }\\cup {\\mathcal {W}}_{{\\mathcal {H}}})$ , the system ${\\mathbf {f}}({\\eta },\\cdot )$ has $\\delta $ distinct complex solutions as the rank of ${\\mathcal {H}}({\\eta })$ is $\\delta $ .", "Therefore, every complex root of ${\\mathbf {f}}({\\eta },\\cdot )$ is of multiplicity one (we use the definition of multiplicity given in [5]).", "Now we prove that, for such a point ${\\eta }$ , the fiber $\\pi ^{-1}({\\eta })$ does not intersect $K(\\pi ,{\\mathcal {V}})$ .", "Assume by contradiction that there exists a point $({\\eta },\\chi ) \\in \\mathbb {C}^{t+n}$ lying in $\\pi ^{-1}({\\eta })\\cap K(\\pi ,{\\mathcal {V}})$ .", "Note that $\\chi $ is a solution of ${\\mathbf {f}}({\\eta },\\cdot )$ , i.e., ${\\mathbf {f}}({\\eta },\\chi ) = 0$ .", "As $({\\eta },\\chi )\\in K(\\pi ,{\\mathcal {V}})$ , then it is contained in $V(I_{\\Delta })$ .", "Hence, as the derivation in $\\Delta $ does not involve ${\\mathbf {y}}$ , $\\chi $ cancels all the $n \\times n$ -minors of the Jacobian matrix ${\\rm jac}({\\mathbf {f}}({\\eta },\\cdot ),{\\mathbf {x}})$ .", "[5] implies that $\\chi $ has multiplicity greater than one.", "This contradicts to the claim that ${\\mathbf {f}}({\\eta },\\cdot )$ admits only complex solutions of multiplicity one.", "Therefore, we conclude that, for ${\\eta }\\in \\mathbb {C}^t\\setminus ({\\mathcal {W}}_{\\infty } \\cup {\\mathcal {W}}_{{\\mathcal {H}}})$ , $\\pi ^{-1}({\\eta })$ does not intersect $K(\\pi ,{\\mathcal {V}})$ .", "So, using what we prove above and Lemma REF , we deduce that, for ${\\eta }\\in \\mathbb {R}^t \\setminus ({\\mathcal {W}}_{\\infty }\\cup {\\mathcal {W}}_{{\\mathcal {H}}})$ , then there exists an open neighborhood $O_{{\\eta }}$ of ${\\eta }$ for the Euclidean topology such that $\\pi ^{-1}(O_{{\\eta }})$ does not intersect $K(\\pi ,{\\mathcal {V}}) \\cup \\pi ^{-1}({\\mathcal {W}}_{\\infty })$ .", "Therefore, by Thom's isotopy lemma [15], the projection $\\pi $ realizes a locally trivial fibration over $\\mathbb {R}^t\\setminus ({\\mathcal {W}}_{\\infty }\\cup {\\mathcal {W}}_{{\\mathcal {H}}})$ .", "So, for any connected component ${\\mathcal {C}}$ of $\\mathbb {R}^t \\setminus ({\\mathcal {W}}_{\\infty } \\cup {\\mathcal {W}}_{{\\mathcal {H}}})$ and any ${\\eta }\\in {\\mathcal {C}}$ , we have that $\\pi ^{-1}({\\mathcal {C}}) \\cap {\\mathcal {V}}\\cap \\mathbb {R}^{t+n}$ is homeomorphic to ${\\mathcal {C}}\\times (\\pi ^{-1}({\\eta })\\cap {\\mathcal {V}}\\cap \\mathbb {R}^{t+n})$ .", "As a consequence, the number of distinct real solutions of ${\\mathbf {f}}({\\eta },\\cdot )$ is invariant when ${\\eta }$ varies over each connected component of $\\mathbb {R}^t \\setminus ({\\mathcal {W}}_{\\infty }\\cup {\\mathcal {W}}_{{\\mathcal {H}}})$ .", "To describe Algorithm REF , we need to introduce the following subroutines: $\\bullet $ CleanFactors which takes as input a polynomial $p \\in \\mathbb {Q}[{\\mathbf {y}},{\\mathbf {x}}]$ and the polynomial $\\mathbf {w}_{\\infty }$ .", "It computes the square-free part of $p$ with all the common factors with $\\mathbf {w}_{\\infty }$ removed.", "$\\bullet $ Signature which takes as input a symmetric matrix with entries in $\\mathbb {Q}$ and evaluates its signature.", "$\\bullet $ SamplePoints which takes as input a set of polynomials $g_1,\\ldots ,g_s \\in \\mathbb {Q}[{\\mathbf {y}}]$ and computes a finite subset $\\mathcal {R}$ of $\\mathbb {Q}^t$ that intersects every connected component of the semi-algebraic set defined by $\\wedge _{i=1}^s g_i \\ne 0$ .", "An explicit description of SamplePoints is given in the proof of Theorem REF in Section .", "The pseudo-code of Algorithm REF is below.", "Its proof of correctness follows immediately from Proposition REF and Corollary REF .", "[H] A polynomial sequence ${\\mathbf {f}}\\in \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ such that $\\langle {\\mathbf {f}}\\rangle $ is radical and Assumptions (REF ) holds.", "A set of sample points and the corresponding numbers of real solutions solving the weak version of Problem (REF ) $[{\\mathcal {H}}, \\mathbf {w}_{\\infty }] \\leftarrow \\textsf {DRL-Matrix}({\\mathbf {f}})$ $\\mathbf {w}_{{\\mathcal {H}}}\\leftarrow {\\sf CleanFactors}({\\rm numer}(\\det ({\\mathcal {H}})),\\mathbf {w}_{\\infty })$ $L \\leftarrow {\\sf SamplePoints}(\\mathbf {w}_{{\\mathcal {H}}}\\ne 0 \\wedge \\mathbf {w}_{\\infty }\\ne 0)$ ${\\eta }\\in L$ $r_{{\\eta }} \\leftarrow {\\sf Signature}({\\mathcal {H}}({\\eta }))$ $\\lbrace ({\\eta }, r_{{\\eta }})\\;\|\\; {\\eta }\\in L\\rbrace $ Weak-RRC-Hermite Example 22 We continue with the system in Example REF .", "The determinant of its parametric Hermite matrix is $\\mathbf {w}_{{\\mathcal {H}}} = 16 y_1 (-y_2^6-3y_2^4y_3^2-3y_2^2y_3^4-y_3^6+3y_1y_2^4-21y_1y_2^2y_3^2+3y_1y_3^4-3y_1^2y_2^2-3y_1^2y_3^2+y_1^3).$ We notice that $\\mathbf {w}_{{\\mathcal {H}}}$ coincides exactly with the output returned by the procedure DiscriminantVariety of Maple's package RootFinding[Parametric] that computes a discriminant variety [41].", "Computing at least one point per connected component of the semi-algebraic set $\\mathbb {R}^3\\setminus V(\\mathbf {w}_{{\\mathcal {H}}})$ using RAGlib gives us 28 points.", "We evaluate the signatures of ${\\mathcal {H}}$ specialized at those points and find that the input system can have 0, 2 or 4 distinct real solutions when the parameters vary.", "Remark 23 As we have seen, Algorithm REF obtains a polynomial which serves similarly as discriminant varieties [41] or border polynomials [58] through computing the determinant of parametric Hermite matrices.", "Whereas, the two latter strategies rely on algebraic elimination based on Gröbner bases to compute the projection of $\\mathrm {crit}(\\pi ,{\\mathcal {V}})$ on the ${\\mathbf {y}}$ -space.", "Since it is well-known that the computation of such Gröbner basis could be heavy, our algorithm has a chance to be more practical.", "In Section , we provide experimental results to support this claim.", "Remark 24 It is worth noticing that, even though the design of Algorithm REF employs the grevlex monomial ordering where $x_1 \\succ \\cdots \\succ x_n$ , we can replace it by any grevlex ordering with another lexicographical order among the ${\\mathbf {x}}$ 's.", "For instance, we can use the monomial ordering ${grevlex}(x_n \\succ \\cdots \\succ x_1)$ .", "While every theoretical claim still holds for this ordering, the practical behavior could be different.", "We demonstrate this remark in Example REF below.", "Example 25 We consider the polynomial sequence $(f_1,f_2,f_3) \\subset \\mathbb {Q}[y_1,y_2,y_3][x_1,x_2,x_3]$ $f_1 & = x_1x_2-x_3, \\\\f_2 & = x_1^3+4x_1^2x_3+2x_2^3-x_2^2x_3+x_2x_3^2-2x_3^3+3x_1^2-x_1x_3-3x_2^2-3 x_3^2-x_2+4x_3+4, \\\\f_3 & = y_3x_1x_2+y_1x_1+y_2x_2+1.$ By computing the reduced Gröbner basis of the ideal generated by $f_1,f_2,f_3$ with respect to the ordering ${grevlex}(x_1\\succ x_2\\succ x_3) \\succ {grevlex}(y_1\\succ y_2 \\succ y_3)$ , one note that this system above does not satisfy Assumption (REF ).", "Hence, the algebraic set ${\\mathcal {W}}_{\\infty }$ defining the locus over which our parametric Hermite matrix does not well specialize is non-empty.", "The polynomials $w_{\\infty }$ and $w_{{\\mathcal {H}}}$ computed in Algorithm REF with respect to the monomial ordering ${grevlex}(x_1 \\succ x_2 \\succ x_3)$ have respectively the degrees 13 and 18.", "On the other hand, using the monomial ordering ${grevlex}(x_3 \\succ x_2\\succ x_1)$ in Algorithm REF , one obtains a polynomial $\\tilde{w}_{\\infty }$ of degree 7 and the same polynomial $w_{{\\mathcal {H}}}$ as above.", "Therefore, the degree of the input given into the subroutine SamplePoints is reduced by using the second ordering (25 compared with 31).", "In practice, this choice of ordering accelerates significantly the computation of sample points." ], [ "Computing semi-algebraic formulas", "By Corollary REF , the number of real roots of the system ${\\mathbf {f}}({\\eta },\\cdot )$ for a given point ${\\eta }\\in \\mathbb {R}^t\\setminus {\\mathcal {W}}_{\\infty }$ can be obtained by evaluating the signature of the parametric Hermite matrix ${\\mathcal {H}}$ .", "We recall that the signature of a matrix can be deduced from the sign pattern of its leading principal minors.", "More precisely, we recall the following criterion, introduced by [55] and [37] (see [25] for a summary on these works).", "Lemma 26 [25] Let $S$ be a $\\delta \\times \\delta $ symmetric matrix in $\\mathbb {R}^{\\delta \\times \\delta }$ and, for $1\\le i\\le \\delta $ , $S_i$ be the $i$ -th leading principal minor of $S$ , i.e., the determinant of the sub-matrix formed by the first $i$ rows and $i$ columns of $S$ .", "By convention, we denote $S_0=1$ .", "We assume that $S_i\\ne 0$ for $0\\le i \\le \\delta $ .", "Let $k$ be the number of sign variations between $S_i$ and $S_{i+1}$ .", "Then, the numbers of positive and negative eigenvalues of $S$ are respectively $\\delta -k$ and $k$ .", "Thus, the signature of $S$ is $\\delta -2k$ .", "This criterion leads us to the following idea.", "Assume that none of the leading principal minors of ${\\mathcal {H}}$ is identically zero.", "We consider the semi-algebraic subset of $\\mathbb {R}^t$ defined by the non-vanishing of those leading principal minors.", "Over a connected component ${\\mathcal {S}}^{\\prime }$ of this semi-algebraic set, each leading principal minor is not zero and its sign is invariant.", "As a consequence, by Lemma REF and Corollary REF , the number of distinct real roots of ${\\mathbf {f}}({\\eta },\\cdot )$ when ${\\eta }$ varies over ${\\mathcal {S}}^{\\prime }\\setminus {\\mathcal {W}}_{\\infty }$ is invariant.", "However, this approach does not apply directly if one of the leading principle minors of ${\\mathcal {H}}$ is identically zero.", "We bypass this obstacle by picking randomly an invertible matrix $A\\in {\\rm GL}_{\\delta }(\\mathbb {Q})$ and working with the matrix ${\\mathcal {H}}_A A^T\\cdot {\\mathcal {H}}\\cdot A$ .", "The lemma below states that, with a generic matrix $A$ , all of the leading principal minors of ${\\mathcal {H}}_A$ are not identically zero.", "Lemma 27 There exists a Zariski dense subset $\\mathcal {A}$ of ${\\rm GL}_{\\delta }(\\mathbb {Q})$ such that for $A \\in \\mathcal {A}$ , all of the leading principal minors of ${\\mathcal {H}}_AA^T\\cdot {\\mathcal {H}}\\cdot A$ are not identically zero.", "For $1\\le r\\le \\delta $ , we denote by $\\mathfrak {M}_r$ the set of all $r\\times r$ minors of ${\\mathcal {H}}$ .", "Let ${\\eta }\\in \\mathbb {Q}^t\\setminus {\\mathcal {W}}_{\\infty }\\cup {\\mathcal {W}}_{{\\mathcal {H}}}$ .", "We have that ${\\mathcal {H}}({\\eta })$ is a full rank matrix in $\\mathbb {Q}^{\\delta \\times \\delta }$ and, for $A\\in {\\rm GL}_{\\delta }(\\mathbb {R})$ , ${\\mathcal {H}}_A({\\eta }) = A^T\\cdot {\\mathcal {H}}({\\eta })\\cdot A$ .", "We prove that there exists a Zariski dense subset $\\mathcal {A}$ of ${\\rm GL}_{\\delta }(\\mathbb {Q})$ such that, for $A\\in \\mathcal {A}$ , all of the leading principal minors of ${\\mathcal {H}}_A({\\eta })$ are not zero.", "Then, as an immediate consequence, all the leading principal minors of ${\\mathcal {H}}_A$ are not identically zero.", "We consider the matrix $A = (a_{i,j})_{1\\le i, j \\le \\delta }$ where $\\mathbf {a}=(a_{i,j})$ are new variables.", "Then, the $r$ -th leading principal minor $M_r(\\mathbf {a})$ of $A^T \\cdot {\\mathcal {H}}({\\eta }) \\cdot A$ can be written as $M_r(\\mathbf {a}) = \\sum _{\\mathfrak {m} \\in \\mathfrak {M}_r}a_{\\mathfrak {m}} \\cdot \\mathfrak {m}({\\eta }),$ where the $a_{\\mathfrak {m}}$ 's are elements of $\\mathbb {Q}[\\mathbf {a}]$ .", "As ${\\mathcal {H}}({\\eta })$ is a full rank symmetric matrix by assumption, there exists a matrix $Q \\in {\\rm GL}_{\\delta }(\\mathbb {R})$ such that $Q^T\\cdot {\\mathcal {H}}({\\eta }) \\cdot Q$ is a diagonal matrix with no zero on its diagonal.", "Hence, the evaluation of $\\mathbf {a}$ at the entries of $Q$ gives $M_r(\\mathbf {a})$ a non-zero value.", "As a consequence, $M_r(\\mathbf {a})$ is not identically zero.", "Let $\\mathcal {A}_r$ be the non-empty Zariski open subset of ${\\rm GL}_{\\delta }(\\mathbb {Q})$ defined by $M_r(\\mathbf {a}) \\ne 0$ .", "Then, the set of the matrices $A \\in \\mathcal {A}_r$ such that the $r\\times r$ leading principal minor of $A^T\\cdot {\\mathcal {H}}({\\eta })\\cdot A$ is not zero.", "Taking $\\mathcal {A}$ as the intersection of $\\mathcal {A}_r$ for $1\\le r\\le \\delta $ , then, for $A \\in \\mathcal {A}$ , none of the leading principal minors of $A^T\\cdot {\\mathcal {H}}({\\eta }) \\cdot A$ equals zero.", "Consequently, each leading principal minor of $A^T \\cdot {\\mathcal {H}}\\cdot A$ is not identically zero.", "Our algorithm (Algorithm REF ) for solving Problem (REF ) through parametric Hermite matrices is described below.", "As it depends on the random choice of the matrix $A$ , Algorithm REF is probabilistic.", "One can easily modify it to be a Las Vegas algorithm by detecting the cancellation of the leading principal minors for each choice of $A$ .", "[H] A polynomial sequence ${\\mathbf {f}}\\subset \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ such that the ideal $\\langle {\\mathbf {f}}\\rangle $ is radical and ${\\mathbf {f}}$ satisfies Assumption (REF ) The descriptions of a collection of semi-algebraic sets ${\\mathcal {S}}_i$ solving Problem (REF ) ${\\mathcal {H}},\\mathbf {w}_{\\infty } \\leftarrow \\textsf {DRL-Matrix}({\\mathbf {f}})$ Choose randomly a matrix $A$ in $\\mathbb {Q}^{\\delta \\times \\delta }$ ${\\mathcal {H}}_A \\leftarrow A^T\\cdot {\\mathcal {H}}\\cdot A$ $(M_1,\\ldots ,M_{\\delta }) \\leftarrow {\\sf LeadingPrincipalMinors}({\\mathcal {H}}_A)$ $L \\leftarrow {\\sf SamplePoints}\\left( \\mathbf {w}_{\\infty }\\wedge \\left(\\wedge _{i=1}^{\\delta } M_i \\ne 0 \\right) \\right)$ ${\\eta }\\in L$ $r_{{\\eta }} \\leftarrow {\\sf Signature}({\\mathcal {H}}({\\eta }))$ $\\lbrace (\\mathrm {sign}\\;(M_1({\\eta }),\\ldots ,M_{\\delta }({\\eta })),{\\eta },r_{{\\eta }})\\;\|\\;{\\eta }\\in L\\rbrace $ RRC-Hermite Proposition 28 Assume that ${\\mathbf {f}}$ satisfies Assumptions (REF ) and that the ideal $\\langle {\\mathbf {f}}\\rangle $ is radical.", "Let $A$ be a matrix in ${\\rm GL}_{\\delta }(\\mathbb {Q})$ such that all of the leading principal minors $M_1,\\ldots ,M_{\\delta }$ of ${\\mathcal {H}}_AA^T\\cdot {\\mathcal {H}}\\cdot A$ are not identically zero.", "Then, Algorithm REF computes correctly a solution for Problem (REF ).", "Note that for ${\\eta }\\in \\mathbb {R}^t\\setminus {\\mathcal {W}}_{\\infty }$ , we have that ${\\mathcal {H}}_A({\\eta }) = A^T\\cdot {\\mathcal {H}}({\\eta }) \\cdot A$ .", "Therefore, the signature of ${\\mathcal {H}}({\\eta })$ equals to the signature of ${\\mathcal {H}}_A({\\eta })$ .", "Let $M_1,\\ldots ,M_{\\delta }$ be the leading principal minors of ${\\mathcal {H}}_A$ and ${\\mathcal {S}}$ be the algebraic set defined by $\\wedge _{i=1}^{\\delta } M_i\\ne 0$ .", "Over each connected component ${\\mathcal {S}}^{\\prime }$ of ${\\mathcal {S}}$ , the sign of each $M_i$ is invariant and not zero.", "Therefore, by Lemma REF , the signature of ${\\mathcal {H}}_A({\\eta })$ , and therefore of ${\\mathcal {H}}({\\eta })$ , is invariant when ${\\eta }$ varies over ${\\mathcal {S}}^{\\prime } \\setminus {\\mathcal {W}}_{\\infty }$ .", "As a consequence, by Corollary REF , the number of distinct real roots of ${\\mathbf {f}}({\\eta },\\cdot )$ is also invariant when ${\\eta }$ varies over ${\\mathcal {S}}^{\\prime }\\setminus {\\mathcal {W}}_{\\infty }$ .", "We finish the proof of correctness of Algorithm REF .", "Example 29 From the parametric Hermite matrix ${\\mathcal {H}}$ computed in Example REF , we obtain the sequence of leading principal minors below: $M_1 & = 4, \\\\M_2 & = 4(-2y_2^2+y_3^2+2y_1), \\\\M_3 & = 8(-y_2^4-2y_2^2y_3^2-y_3^4-y_1y_2^2-y_1y_3^2+2y_1^2), \\\\M_4 & = 16 y_1 (-y_2^6-3y_2^4y_3^2-3y_2^2y_3^4-y_3^6+3y_1y_2^4-21y_1y_2^2y_3^2+3y_1 y_3^4-3y_1^2y_2^2-3y_1^2y_3^2+y_1^3).", "\\\\$ Since $M_1$ is constant, we compute at least one point per connected component of the semi-algebraic set defined by $M_2\\ne 0 \\wedge M_3 \\ne 0 \\wedge M_4 \\ne 0.$ The computation using RAGlib outputs a set of 48 sample points and finds the following realizable sign conditions of $(M_2, M_3, M_4)$ : $[-1,1,1],[-1,-1,1],[1,-1,-1],[-1,-1,-1],[1,1,-1].$ By evaluating the signature of ${\\mathcal {H}}$ at each of those sample points, we deduce the semi-algebraic formulas corresponding to every possible number of real solutions $0 \\text{ real root } \\rightarrow & (M_2 < 0 \\wedge M_3 > 0 \\wedge M_4 > 0)\\vee (M_2 < 0 \\wedge M_3 < 0 \\wedge M_4 > 0)\\\\2 \\text{ real roots } \\rightarrow & (M_2 > 0 \\wedge M_3 < 0 \\wedge M_4 < 0)\\vee (M_2 < 0 \\wedge M_3 < 0 \\wedge M_4< 0)& \\\\& \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\vee (M_2 > 0 \\wedge M_3 > 0 \\wedge M_4 < 0) \\\\4 \\text{ real roots } \\rightarrow & (M_2 > 0 \\wedge M_3 > 0 \\wedge M_4 > 0).$ We recall that the semi-algebraic formulas obtained in Example REF involve the subresultant coefficients $s_2$ , $s_3$ and $s_4$ of degree 2, 6 and 11 respectively.", "Whereas, the degrees of the minors $M_2$ , $M_3$ and $M_4$ that we obtain from the parametric Hermite matrix are only 2, 4 and 7." ], [ "Degree bound of parametric Hermite matrices on generic\ninput", "In this subsection, we consider an affine regular sequence ${\\mathbf {f}}=(f_1, \\ldots , f_n) \\subset \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ according to the variables ${\\mathbf {x}}$ , i.e., the homogeneous components of largest degree in ${\\mathbf {x}}$ of the $f_i$ 's form a homogeneous regular sequence (see Section ).", "Additionally, we require that ${\\mathbf {f}}$ satisfies Assumptions (REF ) and (REF ).", "Let $d$ be the highest value among the total degrees of the $f_i$ 's.", "Since the homogeneous regular sequences are generic among the homogeneous polynomial sequences (see, e.g., [3] or [45]), the same property of genericity holds for affine regular sequences (thanks to the definition we use).", "As in previous sections, ${\\mathcal {G}}$ denotes the reduced Gröbner basis of $\\langle {\\mathbf {f}}\\rangle $ with respect to the ordering ${grevlex}({\\mathbf {x}}) \\succ {grevlex}({\\mathbf {y}})$ .", "Let $\\delta $ be the dimension of the $\\mathbb {K}$ -vector space $\\mathbb {K}[{\\mathbf {x}}]/\\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ where $\\mathbb {K}= \\mathbb {Q}({\\mathbf {y}})$ .", "By Bézout's inequality, $\\delta \\le d^n$ .", "We derive from ${\\mathcal {G}}$ a basis $\\mathcal {B} = \\lbrace b_1,\\ldots ,b_{\\delta }\\rbrace $ of $\\mathbb {K}[{\\mathbf {x}}]/ \\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ consisting of monomials in the variables ${\\mathbf {x}}$ .", "Finally, the parametric Hermite matrix of ${\\mathbf {f}}$ with respect to $\\mathcal {B}$ is denoted by ${\\mathcal {H}}= (h_{i,j})_{1\\le i,j \\le \\delta }$ .", "For a polynomial $p\\in \\mathbb {Q}[{\\mathbf {y}},{\\mathbf {x}}]$ , we denote by $\\deg (p)$ the total degree of $p$ in $({\\mathbf {y}},{\\mathbf {x}})$ and $\\deg _{{\\mathbf {x}}}(p)$ the partial degree of $p$ in the variables ${\\mathbf {x}}$ .", "As Assumption (REF ) holds, by Lemma REF , the entries of the parametric Hermite matrix ${\\mathcal {H}}$ associated to ${\\mathbf {f}}$ with respect to the basis $\\mathcal {B}$ are elements of $\\mathbb {Q}[{\\mathbf {y}}]$ .", "To establish a degree bound on the entries of ${\\mathcal {H}}$ , we need to introduce the following assumption.", "DAssumption D For any $g\\in {\\mathcal {G}}$ , we have that $\\deg (g)=\\deg _{\\mathbf {x}}(g)$ .", "Proposition REF below states that Assumption (REF ) is generic.", "Its direct consequence is a proof for Proposition REF .", "Proposition 30 Let $\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}}]_d$ be the set of polynomials in $\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}}]$ having total degree bounded by $d$ .", "There exists a non-empty Zariski open subset ${F}_D$ of $\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}}]_d^n$ such that Assumption (REF ) holds for ${\\mathbf {f}}\\in {F}_D\\cap \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]^n$ .", "Consequently, for ${\\mathbf {f}}\\in {F}_D \\cap \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]^n$ , ${\\mathbf {f}}$ satisfies Assumption (REF ).", "Let $y_{t+1}$ be a new indeterminate.", "For any polynomial $p \\in \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]$ , we consider the homogenized polynomial $p_h\\in \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}},y_{t+1}]$ of $p$ defined as follows: $p_h = y_{t+1}^{\\deg (p)} p \\left(\\frac{x_1}{y_{t+1}},\\ldots ,\\frac{x_n}{y_{t+1}},\\frac{y_1}{y_{t+1}},\\ldots ,\\frac{y_t}{y_{t+1}}\\right).$ Let $\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}},y_{t+1}]^h_d$ be the set of homogeneous polynomials in $\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}},y_{t+1}]$ whose degrees are exactly $d$ .", "By [56], there exists a non-empty Zariski subset ${F}^h_D$ of $\\left( \\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}},y_{t+1}]^h_d \\right)^n$ such that the variables ${\\mathbf {x}}$ is in Noether position with respect to ${\\mathbf {f}}_h$ for every ${\\mathbf {f}}_h \\in {F}^h_D$ .", "For ${\\mathbf {f}}_h \\in {F}^h_D$ , let $G_h$ be the reduced Gröbner basis of ${\\mathbf {f}}_h$ with respect to the grevlex ordering ${grevlex}({\\mathbf {x}}\\succ {\\mathbf {y}}\\succ y_{t+1})$ .", "By [4], if the variables ${\\mathbf {x}}$ is in Noether position with respect to ${\\mathbf {f}}_h$ , then the leading monomials appearing in $G_h$ depend only on ${\\mathbf {x}}$ .", "Let ${\\mathbf {f}}$ and $G$ be the image of ${\\mathbf {f}}_h$ and $G_h$ by substituting $y_{t+1}=1$ .", "We show that $G$ is a Gröbner basis of ${\\mathbf {f}}$ with respect to the ordering ${grevlex}({\\mathbf {x}}\\succ {\\mathbf {y}})$ .", "Since $G_h$ generates $\\langle {\\mathbf {f}}_h\\rangle $ , $G$ is a generating set of $\\langle {\\mathbf {f}}\\rangle $ .", "As the leading monomials of elements in $G_h$ do not depend on $y_{t+1}$ , the substitution $y_{t+1}=1$ does not affect these leading monomials.", "For a polynomial $p\\in \\langle {\\mathbf {f}}\\rangle \\subset \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]$ , then $p$ writes $p = \\sum _{i=1}^n c_i \\cdot f_i,$ where the $c_i$ 's lie in $\\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]$ .", "We homogenize the polynomials $c_i\\cdot f_i$ on the right hand side to obtain a homogeneous polynomial $P_h \\in \\langle {\\mathbf {f}}_h\\rangle $ .", "Note that $P_h$ is not necessarily the homogenization $p_h$ of $p$ but only the product of $p_h$ with a power of $y_{t+1}$ .", "Then, there exists a polynomial $g_h\\in G_h$ such that the leading monomial of $g_h$ divides the leading monomial of $P_h$ .", "Since the leading monomial of $g_h$ depends only on ${\\mathbf {x}}$ , it also divides the leading monomial of $p_h$ , which is the leading monomial of $p$ .", "So, the leading monomial of the image of $g_h$ in $G$ divides the leading monomial of $p$ .", "We conclude that $G$ is a Gröbner basis of ${\\mathbf {f}}$ with respect to the ordering ${grevlex}({\\mathbf {x}}\\succ {\\mathbf {y}})$ and the set of leading monomials in $G$ depends only on the variables ${\\mathbf {x}}$ .", "Let ${F}_D$ be the subset of $\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}}]_d^n$ such that for every ${\\mathbf {f}}\\in {F}_D$ , its homogenization ${\\mathbf {f}}_h$ is contained in ${F}^h_D$ .", "Since the two spaces $\\left(\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}},y_{t+1}]_d^h\\right)^n$ and $\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}}]_d^n$ are both exactly $\\mathbb {C}^{\\binom{d+n+t}{n+t} \\times n}$ (by considering each monomial coefficient as a coordinate), ${F}_D$ is also a non-empty Zariski open subset of $\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}}]_d^n$ .", "Assume now that the polynomial sequence ${\\mathbf {f}}$ belongs to ${F}_D$ .", "We consider the two monomial orderings over $\\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]$ below: $\\bullet $ The elimination ordering ${grevlex}({\\mathbf {x}}) \\succ {grevlex}({\\mathbf {y}})$ is abbreviated by $O_1$ .", "The leading monomial of $p\\in \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]$ with respect to $O_1$ is denoted by $\\mathrm {lm}_{1}(p)$ .", "The reduced Gröbner basis of ${\\mathbf {f}}$ with respect to $O_1$ is ${\\mathcal {G}}$ .", "$\\bullet $ The grevlex ordering ${grevlex}({\\mathbf {x}}\\succ {\\mathbf {y}})$ is abbreviated by $O_2$ .", "The leading monomial of $p\\in \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]$ with respect to $O_2$ is denoted by $\\mathrm {lm}_2(p)$ .", "The reduced Gröbner basis of ${\\mathbf {f}}$ with respect to $O_2$ is denoted by ${\\mathcal {G}}_2$ .", "As proven above, the set $\\lbrace \\mathrm {lm}_2(g_2)\\; \| \\; g_2\\in {\\mathcal {G}}_2\\rbrace $ does not depend on ${\\mathbf {y}}$ .", "With this property, we will show, for any $g_2\\in {\\mathcal {G}}_2$ , there exists a polynomial $g \\in {\\mathcal {G}}$ such that $\\mathrm {lm}_{1}(g)$ divides $\\mathrm {lm}_{2}(g_2)$ .", "By definition, $\\mathrm {lm}_{2}(g_2)$ is greater than any other monomial of $g_2$ with respect to the ordering $O_2$ .", "Since $\\mathrm {lm}_{2}(g_2)$ depends only on the variables ${\\mathbf {x}}$ , it is then greater than any monomial of $g_2$ with respect to the ordering $O_1$ .", "Hence, $\\mathrm {lm}_{2}(g_2)$ is also $\\mathrm {lm}_{1}(g_2)$ .", "Consequently, since ${\\mathcal {G}}$ is a Gröbner basis of ${\\mathbf {f}}$ with respect to $O_1$ , there exists a polynomial $g\\in {\\mathcal {G}}$ such that $\\mathrm {lm}_{1}(g)$ divides $\\mathrm {lm}_{1}(g_2) = \\mathrm {lm}_{2}(g_2)$ .", "Next, we prove that for every $g \\in {\\mathcal {G}}$ , $\\mathrm {lm}_1(g)$ is also $\\mathrm {lm}_2(g)$ .", "For this, we rely on the fact that ${\\mathcal {G}}$ is reduced.", "Assume by contradiction that there exists a polynomial $g \\in {\\mathcal {G}}$ such that $\\mathrm {lm}_1(g) \\ne \\mathrm {lm}_2(g)$ .", "Thus, $\\mathrm {lm}_2(g)$ must contain both ${\\mathbf {x}}$ and ${\\mathbf {y}}$ .", "Let $t_{{\\mathbf {x}}}$ be the part in only variables ${\\mathbf {x}}$ of $\\mathrm {lm}_2(g)$ .", "Note that $\\mathrm {lm}_1(g)$ is greater than $t_{{\\mathbf {x}}}$ with respect to $O_1$ .", "There exists an element $g_2\\in {\\mathcal {G}}_2$ such that $\\mathrm {lm}_2(g_2)$ divides $\\mathrm {lm}_2(g)$ .", "Since $\\mathrm {lm}_2(g_2)$ depends only on the variables ${\\mathbf {x}}$ , we have that $\\mathrm {lm}_2(g_2)$ divides $t_{{\\mathbf {x}}}$ .", "Then, by what we proved above, there exists $g^{\\prime } \\in {\\mathcal {G}}$ such that $\\mathrm {lm}_{1}(g)$ divides $\\mathrm {lm}_2(g_2)$ , so $\\mathrm {lm}_1(g)$ divides $t_{{\\mathbf {x}}}$ .", "This implies that ${\\mathcal {G}}$ is not reduced, which contradicts the definition of ${\\mathcal {G}}$ .", "So, $\\mathrm {lm}_1(g) = \\mathrm {lm}_2(g)$ for every $g \\in {\\mathcal {G}}$ and, consequently, $\\deg (g) = \\deg _{{\\mathbf {x}}}(g)$ .", "We conclude that there exists a non-empty Zariski open subset ${F}_D$ (as above) of $\\mathbb {C}[{\\mathbf {x}},{\\mathbf {y}}]_d^n$ such that Assumption (REF ) holds for every ${\\mathbf {f}}\\in {F}_D\\cap \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]^n$ .", "Additionally, one easily notices that Assumption (REF ) implies Assumption (REF ).", "As a consequence, ${\\mathbf {f}}$ also satisfies Assumption (REF ) for any ${\\mathbf {f}}\\in {F}_D\\cap \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]^n$ .", "Recall that, when Assumption (REF ) holds, by Lemma REF , the trace of any multiplication map ${\\mathcal {L}}_p$ is a polynomial in $\\mathbb {Q}[{\\mathbf {y}}]$ where $p\\in \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ .", "We now estimate the degree of $\\mathrm {trace}({\\mathcal {L}}_p)$ .", "Since the map $p \\mapsto \\mathrm {trace}({\\mathcal {L}}_p)$ is linear, it is sufficient to consider $p$ as a monomial in the variables ${\\mathbf {x}}$ .", "Proposition 31 Assume that Assumption (REF ) holds.", "Then, for any monomial $m$ in the variables ${\\mathbf {x}}$ , the degree in ${\\mathbf {y}}$ of $\\mathrm {trace}({\\mathcal {L}}_m)$ is bounded by $\\deg (m)$ .", "As a consequence, the total degree of the entry $h_{i,j}=\\mathrm {trace}({\\mathcal {L}}_{b_i\\cdot b_j})$ of ${\\mathcal {H}}$ is at most the sum of the total degrees of $b_i$ and $b_j$ , i.e., $\\deg (h_{i,j}) \\le \\deg (b_i) + \\deg (b_j).$ Let $m$ be a monomial in $\\mathbb {Q}[{\\mathbf {x}}]$ .", "The multiplication matrix ${\\mathcal {L}}_m$ is built as follows.", "For $1\\le i \\le \\delta $ , the normal form of $b_i \\cdot m$ as a polynomial in $\\mathbb {Q}({\\mathbf {y}})[{\\mathbf {x}}]$ writes $\\mathrm {NF}_{{\\mathcal {G}}}(b_i\\cdot m) = \\sum _{j=1}^{\\delta } c_{i,j}\\cdot b_j.$ Note that this normal form is the remainder of the successive divisions of $b_i\\cdot m$ by polynomials in ${\\mathcal {G}}$ .", "As Assumption (REF ) holds, Assumption (REF ) also holds.", "Therefore, those divisions do not introduce any denominator.", "So, every term appearing during these normal form reductions are polynomials in $\\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ .", "Let $p \\in \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ .", "For any $g\\in {\\mathcal {G}}$ , by Assumption (REF ), the total degree in $({\\mathbf {y}},{\\mathbf {x}})$ of every term of $g$ is at most the degree of $\\mathrm {lm}_{{\\mathbf {x}}}(g)$ .", "Thus, a division of $p$ by $g$ involves only terms of total degree $\\deg (p)$ .", "Thus, during the polynomial division of $p$ to ${\\mathcal {G}}$ , only terms of degree at most $\\deg (p)$ will appear.", "Hence the degree of $\\mathrm {NF}_{{\\mathcal {G}}}(p)$ is bounded by $\\deg (p)$ .", "Note that $\\mathrm {trace}({\\mathcal {L}}_{m}) = \\sum _{i=1}^{\\delta } c_{i,i}$ .", "As the degree of $c_{i,i}\\cdot b_i$ is bounded by $\\deg (b_i) +\\deg (m)$ , the degree of $c_{i,i}$ is at most $\\deg (m)$ .", "Then, we obtain that $\\deg (\\mathrm {trace}({\\mathcal {L}}_m)) \\le \\deg (m)$ .", "Finally, the degree bound of $h_{i,j}$ follows immediately: $\\deg (h_{i,j}) =\\deg (\\mathrm {trace}({\\mathcal {L}}_{b_i\\cdot b_j}))\\le \\deg (b_i\\cdot b_j) = \\deg (b_i) + \\deg (b_j).$ Lemma 32 Assume that ${\\mathbf {f}}$ satisfies Assumption (REF ).", "Then the degree of a minor $M$ consisting of the rows $(r_1,\\ldots ,r_{\\ell })$ and the columns $(c_1,\\ldots ,c_{\\ell })$ of ${\\mathcal {H}}$ is bounded by $\\sum _{i=1}^{\\ell }\\left(\\deg (b_{r_i})+\\deg (b_{c_i})\\right).$ Particularly, the degree of $\\det ({\\mathcal {H}})$ is bounded by $2\\sum _{i=1}^{\\delta } \\deg (b_i)$ .", "We expand the minors $M$ into terms of the form $(-1)^{\\mathrm {sign}\\;(\\sigma )}h_{r_1,\\sigma (c_1)}\\ldots h_{r_{\\ell },\\sigma (c_{\\ell })}$ , where $\\sigma $ is a permutation of $\\lbrace c_1,\\ldots ,c_{\\ell }\\rbrace $ and $\\mathrm {sign}\\;(\\sigma )$ is its signature.", "We then bound the degree of each of those terms as follows using Proposition REF : $\\deg \\left(\\prod _{i=1}^{\\ell } h_{r_i,\\sigma (c_i)} \\right) =\\sum _{i=1}^{\\ell } \\deg (h_{r_i,\\sigma (c_i)})\\le \\sum _{i=1}^{\\ell }\\left(\\deg (b_{r_i}) +\\deg (b_{\\sigma (c_i)})\\right) = \\sum _{i=1}^{\\ell }\\left(\\deg (b_{r_i})+\\deg (b_{c_i})\\right).$ Hence, taking the sum of all those terms, we obtain the inequality: $\\deg (M_i) \\le \\sum _{i=1}^{\\ell }\\left(\\deg (b_{r_i})+\\deg (b_{c_i})\\right).$ When $M$ is taken as the determinant of ${\\mathcal {H}}$ , then $\\deg (\\det ({\\mathcal {H}})) \\le 2\\sum _{i=1}^{\\delta } \\deg (b_i).$ Proposition REF implies that, when Assumption (REF ) holds, the degree pattern of ${\\mathcal {H}}$ depends only on the degree of the elements of $\\mathcal {B}=\\lbrace b_1,\\ldots ,b_\\delta \\rbrace $ .", "We rearrange $\\mathcal {B}$ in the increasing order of degree, i.e., $\\deg (b_i) \\le \\deg (b_j)$ for $1\\le i < j \\le \\delta $ .", "So, $b_1=1$ and $\\deg (b_1) = 0$ .", "The degree bounds of the entries of ${\\mathcal {H}}$ are expressed by the matrix below $\\begin{bmatrix}0 & \\deg (b_2) & \\ldots & \\deg (b_{\\delta })\\\\\\deg (b_2) & 2\\deg (b_2) & \\ldots & \\deg (b_{\\delta }) + \\deg (b_2)\\\\\\vdots & \\vdots & \\ddots & \\vdots \\\\\\deg (b_{\\delta }) & \\deg (b_{\\delta }) + \\deg (b_2) &\\ldots &2\\deg (b_{\\delta })\\end{bmatrix}.$ Moreover, using the regularity of ${\\mathbf {f}}$ , we are able to establish explicit degree bounds for the elements of $\\mathcal {B}$ and then, for the minors of ${\\mathcal {H}}$ .", "Lemma 33 Assume that ${\\mathbf {f}}$ is an affine regular sequence and let $\\mathcal {B}$ be the basis defined as above.", "Then the highest degree among the elements of $\\mathcal {B}$ is bounded by $n(d-1)$ and $2\\sum _{i=1}^{\\delta } \\deg (b_i) \\le n(d-1)d^n.$ For $p\\in \\mathbb {K}[{\\mathbf {x}}]$ , let $p_h\\in \\mathbb {K}[x_1,\\ldots ,x_{n+1}]$ be the homogenization of $p$ with respect to the variable $x_{n+1}$ , i.e., $ p_h = x_{n+1}^{\\deg _{{\\mathbf {x}}}(p)} p\\left(\\frac{x_1}{x_{n+1}},\\ldots ,\\frac{x_n}{x_{n+1}}\\right).$ The dehomogenization map ${\\rm \\alpha }$ is defined as: $\\alpha : \\; \\mathbb {K}[x_1,\\ldots ,x_{n+1}] & \\rightarrow \\mathbb {K}[x_1,\\ldots ,x_n], \\\\p(x_1,\\ldots ,x_{n+1}) & \\mapsto p(x_1,\\ldots ,x_n,1).$ Also, the homogeneous component of largest degree of $p$ with respect to the variables ${\\mathbf {x}}$ is denoted by ${}^Hp$ .", "Throughout this proof, we use the following notations: $I = \\langle {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ and ${\\mathcal {G}}$ is the reduced Gröbner basis of $I$ w.r.t ${grevlex}(x_1\\succ \\cdots \\succ x_{n})$ .", "$I_h = \\langle p_h \\; \| \\; p\\in {\\mathbf {f}}\\rangle _{\\mathbb {K}}$ and ${\\mathcal {G}}_h$ is the reduced Gröbner basis of $I_h$ w.r.t ${grevlex}(x_1\\succ \\cdots \\succ x_{n+1})$ .", "The Hilbert series of the homogeneous ideal $I_h$ writes ${\\rm HS}_{I_h}(z) = \\sum _{r = 0}^{\\infty } \\left(\\dim _{\\mathbb {K}} \\mathbb {K}[{\\mathbf {x}}]_r -\\dim _{\\mathbb {K}} (I_h \\cap \\mathbb {K}[{\\mathbf {x}}]_r) \\right) \\cdot z^r,$ where $\\mathbb {K}[{\\mathbf {x}}]_{r} = \\lbrace p \\; \| \\; p\\in \\mathbb {K}[{\\mathbf {x}}]:\\ \\deg _{{\\mathbf {x}}}(p) = r\\rbrace $ Since ${\\mathbf {f}}$ is an affine regular sequence, by definition (see Section ), ${}^H{\\mathbf {f}}=({}^Hf_1,\\ldots ,{}^Hf_n)$ forms a homogeneous regular sequence.", "Equivalently, by [56], the homogeneous polynomial sequence $((f_1)_h,\\ldots ,(f_n)_h,x_{n+1})$ is regular.", "Particularly, $((f_1)_h,\\ldots , (f_n)_h)$ is a homogeneous regular sequence and, by [44], we obtain ${\\rm HS}_{I_h}(z) = \\frac{\\prod _{i=1}^n \\left(1-z^{\\deg (f_i)}\\right)}{\\left(1-z \\right)^{n+1}} = \\frac{\\prod _{i=1}^n \\left(1+\\ldots +z^{\\deg (f_i)-1} \\right)}{1-z} .$ On the other hand, as $((f_1)_h,\\ldots ,(f_n)_h,x_{n+1})$ is a homogeneous regular sequence, by [4], the leading terms of ${\\mathcal {G}}_h$ w.r.t ${grevlex}(x_1 \\succ \\cdots \\succ x_{n+1})$ do not depend on the variables $x_{n+1}$ .", "Thus, the dehomogenization map $\\alpha $ does not affect the set of leading terms of ${\\mathcal {G}}_h$ .", "Besides, $\\alpha ({\\mathcal {G}}_h)$ is a Gröbner basis of $I$ with respect to ${grevlex}({\\mathbf {x}})$ (see, e.g., the proof of [23]).", "Hence, the leading terms of ${\\mathcal {G}}_h$ coincides with the leading terms of ${\\mathcal {G}}$ .", "As a consequence, the set of monomials in $(x_1,\\ldots ,x_{n+1})$ which are not contained in the initial ideal of $I_h$ with respect to ${grevlex}(x_1 \\succ \\cdots \\succ x_{n+1})$ is exactly $\\lbrace b \\cdot x_{n+1}^j\\; \| \\; b \\in \\mathcal {B}, j\\in \\mathbb {N} \\rbrace .$ As a consequence, $\\dim _{\\mathbb {K}} \\mathbb {K}[{\\mathbf {x}}]_r -\\dim _{\\mathbb {K}} ( I_h\\cap \\mathbb {K}[{\\mathbf {x}}]_r) =\\sum _{j=0}^{r} \|\\mathcal {B} \\cap \\mathbb {K}[{\\mathbf {x}}]_{j}\|.$ Let $H(z) = \\sum _{r=0}^{\\infty } \| \\mathcal {B} \\cap \\mathbb {K}[{\\mathbf {x}}]_r\| \\cdot z^r$ .", "We have that $ (1-z)\\cdot {\\rm HS}_{I_h}(z) = (1-z)\\sum _{r=0}^{\\infty }\\sum _{j=0}^{r}\|\\mathcal {B} \\cap \\mathbb {K}[{\\mathbf {x}}]_j\| \\cdot z^r =\\sum _{r=0}^{\\infty } \|\\mathcal {B} \\cap \\mathbb {K}[{\\mathbf {x}}]_r\| \\cdot z^r = H(z).$ Then, $H(z) = \\prod _{i=1}^n \\left( 1+\\ldots +z^{\\deg (f_i)-1} \\right).$ As a direct consequence, $\\max _{1\\le i \\le \\delta } \\deg (b_i)$ is bounded by $ \\sum _{i=1}^n \\deg (f_i) - n \\le n(d-1)$ .", "Let $G_1$ and $G_2$ be two polynomials in $\\mathbb {Z}[z]$ .", "We write $G_1 \\le G_2$ if and only if for any $r \\ge 0$ , the coefficient of $z^r$ in $G_2$ is greater than or equal to the one in $G_1$ .", "Since $\\deg (f_i) \\le d$ for every $1 \\le i \\le n$ , then $H(z) = \\prod _{i=1}^n \\left( 1+\\ldots +z^{\\deg (f_i)-1} \\right) \\le \\prod _{i=1}^n \\left( 1+\\ldots +z^{d-1} \\right).$ As a consequence, $H^{\\prime }(z) = \\sum _{r=1}^{\\infty } (r\\ \|\\mathcal {B} \\cap \\mathbb {K}[{\\mathbf {x}}]_r \|) \\cdot z^{r-1} \\le \\left( \\prod _{i=1}^n \\left(1+\\ldots +z^{d-1} \\right)\\right)^{\\prime }.$ Expanding $G^{\\prime }(z)$ , we obtain $H^{\\prime }(z) & \\le \\frac{n \\left(1+\\ldots +z^{d-1} \\right)^{n-1} \\left(1+\\ldots +z^{d-1}-dz^{d-1} \\right)}{1-z} = n \\left(1+\\ldots +z^{d-1}\\right)^{n-1}\\sum _{i=0}^{d-2}\\frac{z^i - z^{d-1}}{1-z}\\\\& = n \\left(1+\\ldots +z^{d-1} \\right)^{n-1}\\sum _{i=0}^{d-2}z^i\\left(1+\\ldots +z^{d-i-2} \\right).$ By substituting $z=1$ in the above inequality, we obtain $H^{\\prime }(1) \\le nd^{n-1}\\sum _{i=0}^{d-2}(d-i-1) =\\frac{n(d-1)d^{n}}{2}.$ Thus, we have that $\\sum _{i=1}^{\\delta }\\deg (b_i) = \\sum _{r=0}^{\\infty } r\\ \|\\mathcal {B}\\cap \\mathbb {K}[{\\mathbf {x}}]_r\| = H^{\\prime }(1) \\le \\frac{n(d-1)d^{n}}{2}.$ Corollary REF below follows immediately from Lemmas REF and REF .", "Corollary 34 Assume that ${\\mathbf {f}}$ is a regular sequence that satisfies Assumption (REF ).", "Then the degree of any minor of ${\\mathcal {H}}$ is bounded by $n(d-1)d^n$ .", "Example 35 We consider again the system ${\\mathbf {f}}=(x_1^2+x_2^2-y_1,\\;x_1x_2+y_2x_2+y_3x_1)$ in Example REF .", "Note that ${\\mathbf {f}}$ forms a regular sequence.", "The Gröbner basis ${\\mathcal {G}}$ of ${\\mathbf {f}}$ with respect to the ordering ${grevlex}({\\mathbf {x}}) \\succ {grevlex}({\\mathbf {y}})$ is ${\\mathcal {G}}=\\lbrace x_2^3+y_3x_2^2+(y_2^2-y_1)x_2+y_2y_3x_1-y_1y_3, x_1^2+x_2^2-y_1,x_1x_2+x_1y_3+x_2y_2\\rbrace .$ So, ${\\mathbf {f}}$ satisfies Assumption (REF ).", "The matrix with respect to the basis $B_1=\\lbrace 1,x_2,x_1,x_2^2\\rbrace $ has the following degree pattern: $\\begin{bmatrix}0 & 1 & 1 & 2 \\\\1 & 2 & 2 & 3 \\\\1 & 2 & 2 & 3 \\\\2 & 3 & 3 & 4\\end{bmatrix}$ This degree pattern agrees with the result of Proposition REF .", "The determinant of this matrix is of degree 7, which is indeed smaller than $n (d-1) d^n = 8$ Whereas, using the basis $B_2=\\lbrace 1,x_2,x_2^2,x_2^3\\rbrace $ leads to another parametric Hermite matrix of different degrees.", "For $1\\le i,j\\le 4$ , the degree of its $(i,j)$ -entry, which is equals to $\\mathrm {trace}({\\mathcal {L}}_{x_2^{i+j-2}})$ , is bounded by $\\deg (x_2^{i-1})+\\deg (x_2^{j-1}) = i+j-2$ using Proposition REF .", "Applying Lemma REF , the determinant is bounded by $2\\sum _{i=0}\\deg (x_2^i) = 12$ .", "By computing the parametric Hermite matrix of ${\\mathbf {f}}$ with respect to $B_2$ , we obtain the degree pattern $\\begin{bmatrix}0 & 1 & 2 & 3 \\\\1 & 2 & 3 & 4 \\\\2 & 3 & 4 & 5 \\\\3 & 4 & 5 & 6\\end{bmatrix}$ on its entries and a determinant of degree 11.", "Again, both of our theoretical bounds hold for this matrix.", "Remark 36 Note that Assumption (REF ) requires a condition on the degrees of polynomials in the Gröbner basis ${\\mathcal {G}}$ of $\\langle {\\mathbf {f}}\\rangle $ .", "We remark that it is possible to establish similar bounds for the degrees of entries of our parametric Hermite matrix and its minors when the system ${\\mathbf {f}}$ satisfies a weaker property than Assumption (REF ) (we still keep the regularity assumption).", "Indeed, we only need to assume that, for any $g\\in {\\mathcal {G}}$ , the homogeneous component of the highest degree in ${\\mathbf {x}}$ of $g$ does not depend on the parameters ${\\mathbf {y}}$ .", "Let $d_{{\\mathbf {y}}}$ be an upper bound of the partial degrees in ${\\mathbf {y}}$ of elements of ${\\mathcal {G}}$ .", "Under the change of variables $x_i\\mapsto x_i^{d_{{\\mathbf {y}}}}$ , ${\\mathbf {f}}$ is mapped to a new polynomial sequence that satisfies Assumption (REF ).", "Therefore, we easily deduce the two following bounds, which are similar to the ones of Proposition REF and Corollary REF .", "$\\deg (h_{i,j}) \\le d_{{\\mathbf {y}}} (\\deg (b_i) + \\deg (b_j))$ ; The degree of any minor of ${\\mathcal {H}}$ is bounded by $d_{{\\mathbf {y}}}~n(d-1)d^n$ .", "Even though these bounds are not sharp anymore, they still allow us to compute the parametric Hermite matrices using evaluation & interpolation scheme and control the complexity of this computation in the instances where Assumption (REF ) does not hold." ], [ "Complexity analysis of our algorithms", "In this subsection, we analyze the complexity of our algorithms on generic systems.", "Let ${\\mathbf {f}}= (f_1,\\ldots ,f_n) \\subset \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]$ be a regular sequence, where ${\\mathbf {y}}= (y_1,\\ldots ,y_t)$ and ${\\mathbf {x}}= (x_1,\\ldots ,x_n)$ , satisfying Assumptions (REF ) and (REF ).", "We denote by ${\\mathcal {G}}$ be the reduced Gröbner basis of ${\\mathbf {f}}$ with respect to the ordering ${grevlex}({\\mathbf {x}}) \\succ {grevlex}({\\mathbf {y}})$ .", "The basis $\\mathcal {B}$ is taken as all the monomials in ${\\mathbf {x}}$ that are irreducible by ${\\mathcal {G}}$ .", "Then, ${\\mathcal {H}}$ is the parametric Hermite matrix associated of ${\\mathbf {f}}$ with respect to $\\mathcal {B}$ .", "We start by estimating the arithmetic complexity for computing the parametric Hermite matrix ${\\mathcal {H}}$ and its minors.", "We denote $\\lambda n(d-1)$ and $\\mathfrak {D} n(d-1)d^n$ .", "Proposition 37 Assume that ${\\mathbf {f}}= (f_1,\\ldots ,f_n) \\subset \\mathbb {Q}[{\\mathbf {y}}][{\\mathbf {x}}]$ is a regular sequence that satisfies Assumptions (REF ) and (REF ).", "Then, the following holds.", "i) The parametric Hermite matrix ${\\mathcal {H}}$ can be computed using $O\\ \\widetilde{~} \\left( \\binom{t+2\\lambda }{t} \\left( n\\binom{d+n+t}{n+t} + \\delta ^{\\omega +1} + \\delta ^2 \\log ^2\\binom{t+2\\lambda }{t} \\right) \\right)$ arithmetic operations in $\\mathbb {Q}$ .", "ii) Each minor (including the determinant) of ${\\mathcal {H}}$ can be computed using $O\\ \\widetilde{~} \\left( \\binom{t+\\mathfrak {D}}{t} \\left( \\delta ^2\\binom{t + 2\\lambda }{t} + \\delta ^{\\omega } + \\log ^2\\binom{t+\\mathfrak {D}}{t} \\right) \\right)$ arithmetic operations in $\\mathbb {Q}$ .", "For the computation of the matrix, we rely on Proposition REF which estimates the complexity of the evaluation & interpolation scheme described in Subsection REF .", "By Lemma REF and Proposition REF , the highest degree among the entries of ${\\mathcal {H}}$ is bounded by $2 \\lambda =2n(d-1)$ .", "Therefore, we replace $\\Lambda $ in Proposition REF by $2 \\lambda $ in the complexity statement of Proposition REF to obtain $O\\ \\widetilde{~} \\left( \\binom{t+2\\lambda }{t} \\left( n\\binom{d+n+t}{n+t} + \\delta ^{\\omega +1} + \\delta ^2 \\log ^2\\binom{t+2\\lambda }{t} \\right) \\right).$ Similarly, the minors of ${\\mathcal {H}}$ can be computed using the technique of evaluation & interpolation.", "By Corollary REF , the degree of every minor of ${\\mathcal {H}}$ is bounded by $\\mathfrak {D}$ .", "We specialize ${\\mathcal {H}}$ at $\\binom{t+\\mathfrak {D}}{t}$ points in $\\mathbb {Q}^t$ and compute the corresponding minor of each specialized Hermite matrix.", "This step takes $O\\left(\\binom{t+\\mathfrak {D}}{t} \\left( \\delta ^2\\binom{t + 2\\lambda }{t} + \\delta ^{\\omega } \\right) \\right)$ arithmetic operations in $\\mathbb {Q}$ .", "Finally, using the multivariate interpolation algorithm of [12], it requires $O\\ \\widetilde{~} \\left( \\binom{t+\\mathfrak {D}}{t} \\log ^2\\binom{t+\\mathfrak {D}}{t} \\right)$ arithmetic operations in $\\mathbb {Q}$ to interpolate the final minor.", "Therefore, the whole complexity for computing each minor of ${\\mathcal {H}}$ lies within $O\\ \\widetilde{~} \\left( \\binom{t+\\mathfrak {D}}{t} \\left( \\delta ^2\\binom{t + 2\\lambda }{t} + \\delta ^{\\omega } + \\log ^2\\binom{t+\\mathfrak {D}}{t} \\right) \\right).$ Finally, we state our main result, which is Theorem REF below.", "It estimates the arithmetic complexity of Algorithms REF and REF .", "Theorem REF Let ${\\mathbf {f}}\\subset \\mathbb {Q}[{\\mathbf {x}},{\\mathbf {y}}]$ be a regular sequence such that the ideal $\\langle {\\mathbf {f}}\\rangle $ is radical and ${\\mathbf {f}}$ satisfies Assumptions (REF ) and (REF ).", "Recall that $\\mathfrak {D}$ denotes $n(d-1)d^n$ .", "Then, we have the following statements: i) The arithmetic complexity of Algorithm REF lies in $O\\ {\\widetilde{~}}\\left(\\binom{t+\\mathfrak {D}}{t} \\ 2^{3t}\\ n^{2t+1} d^{2nt+n+2t+1} \\right).$ ii) Algorithm REF , which is probabilistic, computes a set of semi-algebraic descriptions solving Problem (REF ) within $O\\ {\\widetilde{~}}\\left( \\binom{t+\\mathfrak {D}}{t}\\ 2^{3t}\\ n^{2t+1} d^{3nt+2(n+t)+1} \\right)$ arithmetic operations in $\\mathbb {Q}$ in case of success.", "iii) The semi-algebraic descriptions output by Algorithm REF consist of polynomials in $\\mathbb {Q}[{\\mathbf {y}}]$ of degree bounded by $\\mathfrak {D}$ .", "As Assumption (REF ) holds, we have that $\\mathbf {w}_{\\infty } = 1$ and $\\mathbf {w}_{{\\mathcal {H}}}$ is the square-free part of $\\det ({\\mathcal {H}})$ .", "Therefore, after computing the parametric Hermite matrix ${\\mathcal {H}}$ and its determinant, whose complexity is given by Proposition REF , Algorithm REF essentially consists of computing sample points of the connected components of the algebraic set $\\mathbb {R}^t \\setminus V(\\det ({\\mathcal {H}}))$ .", "By Corollary REF , the degree of $\\det ({\\mathcal {H}})$ is bounded by $\\mathfrak {D}$ .", "Applying Corollary REF , we obtain the following arithmetic complexity for this computation of sample points $O\\ {\\widetilde{~}}\\left( \\binom{t+\\mathfrak {D}}{t}\\ t^4\\ 2^{3t}\\mathfrak {D}^{2t+1} \\right) \\simeq O\\ {\\widetilde{~}}\\left(\\binom{t+\\mathfrak {D}}{t}\\ 2^{3t}\\ n^{2t+1} d^{2nt+n+2t+1}\\right).$ Also by Corollary REF , the finite subset of $\\mathbb {Q}^t$ output by SamplePoints has cardinal bounded by $2^t\\mathfrak {D}^t$ .", "Thus, evaluating the specializations of ${\\mathcal {H}}$ at those points and their signatures costs in total $O \\left( 2^t\\mathfrak {D}^t \\left( \\delta ^2 \\binom{2\\lambda + t}{t} +\\delta ^{\\omega +1/2} \\right) \\right)$ arithmetic operations in $\\mathbb {Q}$ using [5].", "Therefore, the complexity of SamplePoints dominates the whole complexity of the algorithm.", "We conclude that Algorithm REF runs within $O\\ {\\widetilde{~}}\\left( \\binom{t+\\mathfrak {D}}{t}\\ 2^{3t}\\ n^{2t+1} d^{2nt+n+2t+1} \\right)$ arithmetic operations in $\\mathbb {Q}$ .", "For Algorithm REF , we start by choosing randomly a matrix $A$ and compute the matrix ${\\mathcal {H}}_{A} = A^T \\cdot {\\mathcal {H}}\\cdot A$ .", "Then, we compute the leading principal minors $M_1,\\ldots ,M_{\\delta }$ of ${\\mathcal {H}}_{A}$ .", "Using Proposition REF , this step admits the arithmetic complexity bound $O\\ \\widetilde{~} \\left( \\delta \\ \\binom{t+\\mathfrak {D}}{t} \\left(\\delta ^2 \\binom{t + 2\\lambda }{t} + \\delta ^{\\omega } + \\log ^2\\binom{t+\\mathfrak {D}}{t} \\right) \\right).$ Next, Algorithm REF computes sample points for the connected components of the semi-algebraic set defined by $\\wedge _{i=1}^{\\delta }M_i \\ne 0$ .", "Since the degree of each $M_i$ is bounded by $\\mathfrak {D}$ , Corollary REF gives the arithmetic complexity $O\\ {\\widetilde{~}}\\left(\\binom{t+\\mathfrak {D}}{t} \\ t^4 \\ d^{nt+n}\\ 2^{3t}\\ \\mathfrak {D}^{2t+1} \\right) \\simeq O\\ {\\widetilde{~}}\\left( \\binom{t+\\mathfrak {D}}{t}\\ 2^{3t}\\ n^{2t+1} d^{3nt+2(n+t)+1} \\right).$ It returns a finite subset of $\\mathbb {Q}^t$ whose cardinal is bounded by $\\left( 2\\delta \\mathfrak {D} \\right) ^t $ .", "The evaluation of the leading principal minors' sign patterns at those points has the arithmetic complexity lying in $O \\left( 2^t\\delta ^{t+1}\\mathfrak {D}^{2t} \\right) \\simeq O\\left( 2^t n^{2t} d^{3nt+n+2t}\\right)$ .", "Again, the complexity of SamplePoints dominates the whole complexity of Algorithm REF .", "The proof of Theorem REF is then finished." ], [ "Remark on the implementation of\nAlgorithm ", "Recall that Algorithm REF leads us to compute sample points per connected components of the non-vanishing set of the leading principal minors $(M_1,\\ldots ,M_{\\delta })$ .", "Comparing to Algorithm REF in which we only compute sample points for $\\mathbb {R}^t \\setminus V(M_{\\delta })$ , the complexity of Algorithm REF contains an extra factor of $d^{nt}$ due to the higher number of polynomials given as input to the subroutine SamplePoints.", "Even though the complexity bounds of these two algorithms both lie in $d^{O(nt)}$ , the extra factor $d^{nt}$ mentioned above sometimes becomes the bottleneck of Algorithm REF for tackling practical problems.", "Therefore, we introduce the following optimization in our implementation of Algorithm REF .", "We start by following exactly the steps (REF -REF ) of Algorithm REF to obtain the leading principal minors $(M_1,\\ldots ,M_{\\delta })$ and the polynomial $\\mathbf {w}_{\\infty }$ .", "Then, by calling the subroutine SamplePoints on the input $M_{\\delta } \\ne 0 \\wedge \\mathbf {w}_{\\infty } \\ne 0$ , we compute a set of sample points (and their corresponding numbers of real roots) $\\lbrace ({\\eta }_1,r_1),\\ldots ,({\\eta }_{\\ell },r_{\\ell })\\rbrace $ that solves the weak-version of Problem (REF ).", "We obtain from this output all the possible numbers of real roots that the input system can admit.", "For each value $0 \\le r \\le \\delta $ , we define $\\Phi _r = \\lbrace \\sigma = (\\sigma _1,\\ldots ,\\sigma _{\\delta }) \\in \\lbrace -1,1\\rbrace ^{\\delta } \\; \| \\;\\text{ the sign variation of }\\sigma \\text{ is}(\\delta -r)/2 \\rbrace .$ If $r \\lnot \\equiv \\delta \\pmod {2}$ , $\\Phi _r = \\emptyset $ .", "For $\\sigma \\in \\Phi _r$ and ${\\eta }\\in \\mathbb {R}^t\\setminus V(\\mathbf {w}_{\\infty })$ such that $\\mathrm {sign}\\;(M_i({\\eta })) = \\sigma _i$ for every $1\\le i \\le \\delta $ , the signature of ${\\mathcal {H}}({\\eta })$ is $r$ .", "As a consequence, for any ${\\eta }$ in the semi-algebraic set defined by $(\\mathbf {w}_{\\infty } \\ne 0) \\wedge (\\vee _{\\sigma \\in \\Phi _r}(\\wedge _{i=1}^{\\delta } \\mathrm {sign}\\;(M_i) = \\sigma _i)),$ the system ${\\mathbf {f}}({\\eta },.", ")$ has exactly $r$ distinct real solutions.", "Therefore, $(\\mathcal {S}_{r_i})_{1\\le i \\le \\ell }$ is a collection of semi-algebraic sets solving Problem (REF ).", "Then, we can simply return $\\lbrace (\\Phi _{r_i},{\\eta }_i, r_i)\\; \| \\; 1\\le i \\le \\ell \\rbrace $ as the output of Algorithm REF without any further computation.", "Note that, by doing so, we may return sign conditions which are not realizable.", "We discuss now about the complexity aspect of the steps described above.", "For $r \\equiv \\delta \\pmod {2}$ , the cardinal of $\\Phi _r$ is $\\binom{\\delta }{(\\delta -r-2)/2}$ .", "In theory, the total cardinal of all the $\\Phi _{r_i}$ 's ($1\\le i\\le \\ell $ ) can go up to $2^{\\delta -1}$ , which is doubly exponential in the number of variables $n$ .", "However, in the instances that are actually tractable by the current state of the art, $2^{\\delta }$ is still smaller than $\\delta ^{3t}$ .", "And when it is the case, following this approach has better performance than computing the sample points of the semi-algebraic set defined by $\\wedge _{i=1}^{\\delta } M_i \\ne 0$ .", "Otherwise, when $2^{\\delta }$ exceeds $\\delta ^{3t}$ , we switch back to the computation of sample points.", "This implementation of Algorithm REF does not change the complexity bound given in Theorem REF ." ], [ "Experiments", "This subsection provides numerical results of several algorithms related to the real root classification.", "We report on the performance of each algorithm for different test instances.", "The computation is carried out on a computer of Intel(R) Xeon(R) CPU E7-4820 2GHz and 1.5 TB of RAM.", "The timings are given in seconds (s.), minutes (m.) and hours (h.).", "The symbol $\\infty $ means that the computation cannot finish within 120 hours.", "Throughout this subsection, the column hermite reports on the computational data of our algorithms based on parametric Hermite matrices described in Section .", "It uses the notations below: - mat: the timing for computing a parametric Hermite matrix ${\\mathcal {H}}$ .", "- det: the runtime for computing the determinant of ${\\mathcal {H}}$ .", "- min: the timing for computing the leading principal minors of ${\\mathcal {H}}$ .", "- sp: the runtime for computing at least one points per each connected component of the semi-algebraic set $\\mathbb {R}^t\\setminus V(\\det ({\\mathcal {H}}))$ .", "- deg: the highest degree among the leading principal minors of ${\\mathcal {H}}$ ." ], [ "Generic systems", "In this paragraph, we report on the results obtained with generic inputs, i.e., randomly chosen dense polynomials $(f_1,\\ldots ,f_n)\\subset \\mathbb {Q}[y_1,\\ldots ,y_t][x_1,\\ldots ,x_n]$ .", "The total degrees of input polynomials are given as a list $d =[\\deg (f_1),\\ldots ,\\deg (f_n)]$ .", "We first compare the algorithms using Hermite matrices (Section ) with the Sturm-based algorithm (Section ) for solving Problem (REF ).", "The column sturm of Fig.", "(REF ) shows the experimental results of the Sturm-based algorithm.", "It contains the following sub-columns: - elim: the timing for computing the eliminating polynomial.", "- sres: the timing for computing the subresultant coefficients in the Sturm-based algorithm.", "- sp-s: the timing for computing sample points per connected components of the non-vanishing set of the last subresultant coefficient.", "- deg-s: the highest degree among the subresultant coefficients.", "We observe that the sum of mat-h and min-h is smaller than the sum of elim and sres.", "Hence, obtaining the input for the sample point computation in hermite strategy is easier than in sturm strategy.", "We also remark that the degree deg-h is much smaller than deg-s, that explains why the computation of sample points using Hermite matrices is faster than using the subresultant coefficients.", "We conclude that the parametric Hermite matrix approach outperforms the Sturm-based one both on the timings and the degree of polynomials in the output formulas.", "Figure: Generic random dense systemsIn Fig.", "(REF ), we compare our algorithms using parametric Hermite matrices with two Maple packages for solving parametric polynomial systems: RootFinding[Parametric] [24] and RegularChains[ParametricSystemTools] [57].", "The new notations used in Fig.", "(REF ) are explained below.", "The column rf stands for the RootFinding[Parametric] package.", "To solve a parametric polynomial systems, it consists of computing a discriminant variety $\\mathcal {D}$ and then computing an open CAD of $\\mathbb {R}^t\\setminus \\mathcal {D}$ .", "This package does not return explicit semi-algebraic formulas but an encoding based on the real roots of some polynomials.", "This column contains: - dv : the runtime of the command DiscriminantVariety that computes a set of polynomials defining a discriminant variety $\\mathcal {D}$ associated to the input system.", "- cad : the runtime of the command CellDecomposition that outputs semi-algebraic formulas by computing an open CAD for the semi-algebraic set $\\mathbb {R}^t\\setminus \\mathcal {D}$ .", "$\\bullet $ The column rc stands for the RegularChains[ParametricSystemTools] package of Maple.", "The algorithms implemented in this package is given in [57].", "It also contains two sub-columns: - bp : the runtime of the command BorderPolynomial that returns a set of polynomials.", "- rrc : the runtime of the command RealRootClassification.", "We call this command with the option output=`samples' to compute at least one point per connected component of the complementary of the real algebraic set defined by border polynomials.", "Note that, in a strategy for solving the weak-version of Problem (REF ), DiscriminantVariety and BorderPolynomial can be completely replaced by parametric Hermite matrices.", "On generic systems, the determinant of our parametric Hermite matrix coincides with the output of DiscriminantVariety, which we denote by $\\mathbf {w}$ .", "Whereas, because of the elimination BorderPolynomial returns several polynomials, one of them is $\\mathbf {w}$ .", "In Fig.", "(REF ), the timings for computing a parametric Hermite matrix is negligible.", "Comparing the columns det, dv and bp, we remark that the time taken to obtain $\\mathbf {w}$ through the determinant of parametric Hermite matrices is much smaller than using DiscriminantVariety or BorderPolynomial.", "For computing the polynomial $\\mathbf {w}$ , using parametric Hermite matrices allows us to reach the instances that are out of reach of DiscriminantVariety, for example, the instances $\\lbrace t=3,\\; d =[2,2,2]\\rbrace $ , $\\lbrace t=3\\; d = [4,2]\\rbrace $ , $\\lbrace t=3,\\; d=[3,3]\\rbrace $ and $\\lbrace t =4,\\; d = [2,2]\\rbrace $ in Fig.", "(REF ) below.", "Moreover, we succeed to compute the semi-algebraic formulas for $\\lbrace t=3,\\; d =[2,2,2]\\rbrace $ , $\\lbrace t=3\\; d = [4,2]\\rbrace $ and $\\lbrace t = 4,\\; d = [2,2]\\rbrace $ .", "Using the implementation in Subsection REF , we obtain the semi-algebraic formulas of degrees bounded by $\\deg (\\mathbf {w})$ .", "Therefore, for these generic systems, our algorithm based on parametric Hermite matrices outperforms DiscriminantVariety and BorderPolynomial for obtaining a polynomial that defines the boundary of semi-algebraic sets over which the number of real solutions are invariant.", "Moreover, using the minors of parametric Hermite matrices, we can compute semi-algebraic formulas of problems that are out of reach of CellDecomposition and RealRootClassification.", "Figure: Generic random dense systemsIn what follows, we consider the systems coming from some applications as test instances.", "These examples allow us to observe the behavior of our algorithms on non-generic systems." ], [ "Kuramoto model", "This application is introduced in [40], which is a dynamical system used to model synchronization among some given coupled oscillators.", "Here we consider only the model constituted by 4 oscillators.", "The maximum number of real solutions of steady-state equations of this model was an open problem before it is solved in [32] using numerical homotopy continuation methods.", "However, to the best of our knowledge, there is no exact algorithm that is able to solve this problem.", "We present in what follows the first solution using symbolic computation.", "Moreover, our algorithm can return the semi-algebraic formulas defining the regions over which the number of real solutions is invariant.", "As explained in [32], we consider the system ${\\mathbf {f}}$ of the following equations $\\left\\lbrace \\begin{array}{ll}y_i-\\sum _{j=1}^4(s_ic_j-s_jc_i) & = 0 \\\\s_i^2+c_i^2 & = 1\\end{array}\\right.\\text{ for }1\\le i \\le 3,$ where $(s_1,s_2,s_3)$ and $(c_1,c_2,c_3)$ are variables and $(y_1,y_2,y_3)$ are parameters.", "We are asked to compute the maximum number of real solutions of ${\\mathbf {f}}({\\eta },.", ")$ when ${\\eta }$ varies over $\\mathbb {R}^3$ .", "This leads us to solve the weak version of Problem (REF ) for this parametric system.", "We first construct the parametric Hermite matrix ${\\mathcal {H}}$ associated to this system.", "This matrix is of size $14\\times 14$ .", "The polynomial $\\mathbf {w}_{\\infty }$ has the factors $y_1+y_2$ , $y_2+y_3$ , $y_3+y_1$ and $y_1+y_2+y_3$ .", "The polynomial $\\mathbf {w}_{{\\mathcal {H}}}$ has degree 48 (c.f.", "[32]).", "We denote by $\\mathbf {w}$ the polynomial $\\mathbf {w}_{\\infty } \\cdot \\mathbf {w}_{{\\mathcal {H}}}$ .", "Note that the polynomial system has real roots only if $\|y_i\|\\le 3$ (c.f.", "[32]).", "So we only need to consider the compact connected components of $\\mathbb {R}^3\\setminus V(\\mathbf {w})$ .", "Since the polynomial $\\mathbf {w}$ is invariant under any permutation acting on $(y_1,y_2,y_3)$ , we exploit this symmetry to accelerate the computation of sample points.", "Following the critical point method, we compute the critical points of the map $(y_1,y_2,y_3) \\mapsto y_1+y_2+y_3$ restricted to $\\mathbb {R}^3\\setminus V(\\mathbf {w})$ ; this map is also symmetric.", "We apply the change of variables $(y_1,y_2,y_3) \\mapsto (e_1,e_2,e_3),$ where $e_1=y_1+y_2+y_3$ , $e_2=y_1y_2+y_2y_3+y_3y_1$ and $e_3=y_1y_2y_3$ are elementary symmetric polynomials of $(y_1,y_2,y_3)$ .", "This change of variables reduces the number of distinct solutions of zero-dimensional systems involved in the computation and, therefore, reduces the computation time.", "From the sample points obtained by this computation, we derive the possible number of real solutions and conclude that the system ${\\mathbf {f}}$ has at most 10 distinct real solutions when $(y_1,y_2,y_3)$ varies over $\\mathbb {R}^3$ .", "Fig.", "(REF ) reports on the timings for computing the parametric Hermite matrix (mat), for computing its determinant (det) and for computing the sample points (sp).", "We stop both of the commands DiscriminantVariety and BorderPolynomial after 240 hours without obtaining the polynomial $\\mathbf {w}$ .", "Figure: Kuramoto model for 4 oscillators" ], [ "Static output feedback", "The second non-generic example comes from the problem of static output feedback [35].", "Given the matrices $A \\in \\mathbb {R}^{\\ell \\times \\ell }$ , $B\\in \\mathbb {R}^{\\ell \\times 2}$ , $C \\in \\mathbb {R}^{1\\times \\ell }$ and a parameter vector $P = \\begin{bmatrix} y_1 \\\\ y_2\\end{bmatrix} \\in \\mathbb {R}^2$ , the characteristic polynomial of $A+BPC$ writes $f(s,{\\mathbf {y}}) = \\det (sI_l - A - BKC) = f_0(s)+y_1f_1(s)+y_2f_2(s),$ where $s$ is a complex variable.", "We want to find a matrix $P$ such that all the roots of $f(s,{\\mathbf {y}})$ must lie in the open left half-plane.", "By substituting $s$ by $x_1+ix_2$ , we obtain the following system of real variables $(x_1,x_2)$ and parameters $(y_1,y_2)$ : $\\left\\lbrace \\begin{array}{ll} \\Re (f(x_1+ix_2,{\\mathbf {y}})) & = 0 \\\\\\Im (f(x_1+ix_2,{\\mathbf {y}})) & = 0 \\\\ x_1 & < 0\\end{array}\\right.$ Note that the total degree of these equations equals $\\ell $ .", "We are now interested in solving the weak-version of Problem (REF ) on the system $\\Re (f) = \\Im (f)=0$ .", "We observe that this system satisfies Assumptions (REF ) and (REF ).", "Let ${\\mathcal {H}}$ be the parametric Hermite matrix ${\\mathcal {H}}$ of this system with respect to the usual basis we consider in this paper.", "This matrix ${\\mathcal {H}}$ behaves very differently from generic systems.", "Computing the determinant of ${\\mathcal {H}}$ (which is an element of $\\mathbb {Q}[{\\mathbf {y}}]$ ) and taking its square-free part allows us to obtain the same output $\\mathbf {w}$ as DiscriminantVariety.", "However, this direct approach appears to be very inefficient as the determinant appears as a large power of the output polynomial.", "For example, for a value $\\ell $ , we observe that the system consists of two polynomials of degree $\\ell $ .", "The determinant of ${\\mathcal {H}}$ appears as $\\mathbf {w}^{2\\ell }$ , where $\\mathbf {w}$ has degree $2(\\ell -1)$ .", "The bound we establish on the degree of this determinant is $2(\\ell -1)\\ell ^2$ , which is much larger than what happens in this case.", "Therefore, we need to introduce the optimization below to adapt our implementation of Algorithm REF to this problem.", "We observe that, on these examples, the polynomial $\\mathbf {w}$ can be extracted from a smaller minor instead of computing the determinant ${\\mathcal {H}}$ .", "To identify such a minor, we reduce ${\\mathcal {H}}$ to a matrix whose entries are univariate polynomials with coefficients lying in a finite field $\\mathbb {Z}/p \\mathbb {Z}$ as follow.", "Let $u$ be a new variable.", "We substitute each $y_i$ by random linear forms in $\\mathbb {Q}[u]$ in ${\\mathcal {H}}$ and then compute ${\\mathcal {H}}\\bmod p$ .", "Then, the matrix ${\\mathcal {H}}$ is turned into a matrix ${\\mathcal {H}}_u$ whose entries are elements of $\\mathbb {Z}/p\\mathbb {Z}[u]$ .", "The computation of the leading principal minors of ${\\mathcal {H}}_u$ is much easier than the one of ${\\mathcal {H}}$ since it involves only univariate polynomials and does not suffer from the growth of bit-sizes as for the rational numbers.", "Next, we compute the sequence of the leading principal minors of ${\\mathcal {H}}_u$ in decreasing order, starting from the determinant.", "Once we obtain a minor, of some size $r$ , that is not divisible by $\\overline{\\mathbf {w}}_u$ , we stop and take the index $r+1$ .", "Then, we compute the square-free part of the $(r+1)\\times (r+1)$ leading principal minor of ${\\mathcal {H}}$ , which can be done through evaluation-interpolation method.", "This yields a Monte Carlo implementation that depends on the choice of the random linear forms in $\\mathbb {Q}[u]$ and the finite field to compute the polynomial $\\mathbf {w}$ .", "In Fig.", "(REF ), we report on some computational data for the static output feedback problem.", "Here we choose the prime $p$ to be 65521 so that the elements of the finite field $\\mathbb {Z}/p\\mathbb {Z}$ can be represented by a machine word of 32 bits.", "We consider different values of $\\ell $ and the matrices $A,B,C$ are chosen randomly.", "On these examples, our algorithm returns the same output as the one of DisciminantVariety.", "Whereas, BorderPolynomial (bp) returns a list of polynomials which contains our output and other polynomials of higher degree.", "The timings of our algorithm are given by the two following columns: The column mat shows the timings for computing parametric Hermite matrices ${\\mathcal {H}}$ .", "The column comp-w shows the timings for computing the polynomials $\\mathbf {w}$ from ${\\mathcal {H}}$ using the strategy described as above.", "We observe that our algorithm (mat + comp-w) wins some constant factor comparing to DiscriminantVariety (dv).", "On the other hand, BorderPolynomial (bp) performs less efficiently than the other two algorithms in these examples.", "Since the degrees of the polynomials $\\mathbf {w}$ here (given as deg-w) are small comparing with the bounds in the generic case.", "Hence, unlike the generic cases, the computation of the sample points in these problems is negligible as being reported in the column sp.", "Figure: Static output feedback" ] ]	2011.14136
[ [ "Suppression of epitaxial thin film growth by mixing" ], [ "Abstract We consider following fourth-order parabolic equation with gradient nonlinearity on the two-dimensional torus with and without advection of an incompressible vector field in the case $2<p<3$: \\begin{equation} \\partial_t u + (-\\Delta)^2 u = -\\nabla\\cdot(\|\\nabla u\|^{p-2}\\nabla u).", "\\end{equation} The study of this form of equations arises from mathematical models that simulate the epitaxial growth of the thin film.", "We prove the local existence of mild solutions for any initial data lies in $L^2$ in both cases.", "Our main result is: in the advective case, if the imposed advection is sufficiently mixing, then the global existence of solution can be proved, and the solution will converge exponentially to a homogeneous mixed state.", "While in the absence of advection, there exist initial data in $H^2\\cap W^{1,\\infty}$ such that the solution will blow up in finite time." ], [ "Introduction", "The global well-posedness and finite-time blow-up of the solution to a nonlinear parabolic PDE is of widespread interest and arises in different applications in many different areas, especially in mathematical biology and fluid dynamics.", "Advection is often discussed in the corresponding papers, and the presence of it sometimes may stabilize the singularity.", "We refer the interested reader FannjiangKiselevEA06,BerestyckiKiselevEA10,KiselevXu16,BedrossianHe17,He18,HeTadmor19,feng2020global and the references therein for more details.", "As a remark, the stabilization of the fluid flow was proposed to study Keller-Segel type equations with some constraints.", "Later, in feng2020phase,feng2020global, the authors considered such a phenomenon for more general types of equations such as Cahn-Hilliard equations and Kuramoto-Sivashinsky equations.", "One common ingredient for most of these previous results is to explore the special convective motions with mixing flows, whose dissipation-enhancing properties have been understood well, and we refer the interested reader ConstantinKiselevEA08, YaoZlatos17, AlbertiCrippaEA16, ElgindiZlatos19, Zlatos10,CotiZelatiDelgadinoEA18,FengIyer19, iyer2019convection for more examples about such flows.", "In this article, we consider the following fourth-order parabolic equations with gradient nonlinearity, which presents one of the continuum models for epitaxial thin film growth (see, e.g., [22], [29]): for any $p>2$ , $ h_t + A_1h + A_2^2 h + A_3\\cdot ({h}^{p-2}h) = g,$ where $h(t, x)$ denotes the height of a film in epitaxial growth with $g(t,x)$ being the deposition flux and $A_1, A_2, A_3 \\in $ .", "The spatial derivatives in the above equation have the following physical interpretations: (1).", "$A_1h$ : diffusion due to evaporation-condensation edwards1982surface,mullins1957theory; (2).", "$A_2^2 h$ : capillarity-driven surface diffusion herring1999surface,mullins1957theory; (3).", "$A_3\\cdot ({h}^{p-2}h)$ : (upward) hopping of atoms [26].", "These models simulate the complex process of making a thin film layer on a substrate by chemical vapor deposition, and one of the most interesting questions is to understand these growth processes quantitatively on the correct scale so that people can optimize the particular properties of the film.", "These thin-film type equations have been studied by many authors (see, e.g., ishige2020blowup,king2003fourth,sandjo2014space,li2003thin,sandjo2015solutions).", "One can also refer to ortiz1999continuum,schulze1999geometric for more development of such equations, which concentrates on the background of material science.", "In this paper, we will concentrate on the analytical properties of the model (REF ) on two dimensional torus $2=\\left[0,1 \\right]^2$ .", "One typical situation for such models is the case when $A_1=0$ , $A_2=A_3=1$ and $g=0$ : $\\partial _t u + (-)^2 u = -\\cdot ({u}^{p-2}u), \\qquad u(0,x)= u_0(x),$ where $p>2$ and $\\partial _t\\partial /\\partial t$ .", "For simplicity, we denote $N(u)\\cdot F(u)$ , where $F(\\xi )={\\xi }^{p-2}\\xi $ .", "It is worth to mention that (REF ) can be regarded as the $L^2$ -gradient flow for the energy functional $E(\\phi )\\frac{1}{2}\\int _{d}(\\phi )^2\\,dx - \\frac{1}{p}\\int _{d}{\\phi }^{p}\\,dx= \\frac{1}{2}{\\phi }_2^2 - \\frac{1}{p}{\\phi }_p^p,$ which satisfies $\\lim _{\\eta \\rightarrow \\infty } E(\\eta \\phi )=-\\infty $ for $\\phi \\in H^2(2)\\cap W^{1,p}(2)\\setminus \\lbrace 0\\rbrace $ .", "For the nonlinear term in (REF ), the local existence and singularity behavior was first studied in [14] on $\\mathbb {R}^N$ .", "In this paper, we study (REF ) with and without advection is given by an incompressible vector field.", "We will prove that the advection term with good mixing property would prevent the finite time blowup for non-classical solutions to this equation.", "In this article, we focus on the case $2<p<3$ on 2, however, a parallel argument can extend our results to the three-dimension torus with a smaller range of $p$ , we leave the details to the interested reader.", "Throughout this paper, $C$ denotes a generic constant that may change from line to line.", "This paper is organized as follows: Section is devoted to studying the local well-posedness of the solution to (REF ); in Section , we show that adding advection term to (REF ) may enable the existence of global solution if we choose the velocity field carefully with respect to the initial data; finally, in Section we give examples of the blow-up of the solutions, and also characterize the blow-up behavior by providing a quantitative blow-up rate of the $L^2$ norm." ], [ "Acknowledgments", "The authors would like to thank Yuanyuan Feng for useful discussions.", "We would also like to thank the anonymous referees for the helpful comments." ], [ "Epitaxial thin film growth equation without advection", "In this section we study (REF ) without advection on 2 under the regime $2<p<3$ .", "We begin with introducing some notation that will be used throughout this paper.", "Given a function $f\\in L^{p}(2)$ with $p\\ge 1$ , we denote $\\hat{f}(\\mathbf {k})$ to be the Fourier coefficient of $f$ at frequency $\\mathbf {k}\\in \\mathbb {Z}^2$ .", "Now for $s \\in \\mathbb {R}_{+}$ , we can define the Sobolev space $H^{s}(2)$ and the homogeneous Sobolev space $\\dot{H}^{s}(2)$ by the collection of measurable functions $f$ on 2, with ${f}_{H^s}^2 \\sum _{\\mathbf {k}\\in \\mathbb {Z}^2}(1+{\\mathbf {k}}^2)^s{\\hat{f}(\\mathbf {k})}^2={(I-)^{s/2}f}_{L^2}^2$ and ${f}_{\\dot{H}^s}^2 \\sum _{\\mathbf {k}\\in \\mathbb {Z}^2}{\\mathbf {k}}^{2s}{\\hat{f}(\\mathbf {k})}^2={(-)^{s/2}f}_{L^2}^2,$ respectively, where $(-)^{s/2}$ agrees with the Fourier multiplier with symbol ${\\mathbf {k}}^s$ , $\\mathbf {k}\\ne 0$ and $I$ is the identity operator.", "Note that for $f \\in L^2(2)$ , $f \\in H^s(2)$ if and only if $f \\in \\dot{H}^s(2)$ .", "Let $\\mathcal {L}=^2$ , which maps $H^4$ to $L^2$ , and $e^{-t\\mathcal {L}}$ be the strongly continuous semigroup generated by $\\mathcal {L}$ on $L^2$ given by $e^{-t\\mathcal {L}} f \\mathcal {F}^{-1}\\Big (e^{-t{\\mathbf {k}}^4}\\hat{f}\\Big ),$ where $\\mathcal {F}^{-1}$ denotes the inverse Fourier transform on $^2$ .", "For the reader's convenience, we will recall some useful properties of $e^{-t \\mathcal {L}}$ in Lemma REF and Lemma REF .", "To this end, for any measurable function $\\phi (t, x)$ defined on $_{\\ge 0} \\times 2$ , we denote $\\phi (t)$ to be the function $\\phi (t, \\cdot )$ , that is, for any $x \\in 2$ , $\\phi (t)(x)=\\phi (t,x)$ .", "Next we recall the notion of mild and weak solutions in the following definitions.", "For $p>2$ , a function $ u\\in C([0,T];L^{2}(\\mathbb {T}^2)),T>0$ , such that $u$ is locally integrable, is called a mild solution of (REF ) on $[0,T]$ with initial data $ u_0\\in L^{2}(2)$ , if for any $0 \\le t \\le T$ , $u(t)=\\mathcal {T}(u)(t)e^{-t\\mathcal {L}} u_0-\\int _0^t\\, e^{-(t-s)\\mathcal {L}}\\Big ({u}^{p-2}u\\Big )\\,ds$ holds pointwisely in time with values in $L^2$ , where the integral is defined in the Böchner sense.", "For $p>2$ , a function $ u\\in L^{\\infty }([0,T];L^{2}({2}))\\cap L^2([0,T];H^2({2}))$ is called a weak solution of (REF ) on $[0,T)$ with initial data $u_0\\in L^{2}({2})$ if for all $\\phi \\in C_c^{\\infty }([0,T)\\times {2})$ , $&\\int _{2}\\,u_0\\phi (0)\\,dx\\, +\\, \\int _0^T\\,\\int _{2}u\\,\\partial _t \\phi \\,dx dt \\\\&=\\int _0^T\\,\\int _{2}\\,u\\phi dx dt\\, -\\, \\int _0^T \\int _{2}\\,{u}^{p-2}u\\cdot \\phi \\,dx dt,$ and $\\partial _t u\\in L^{2}([0,T]; H^{-2}(2))$ .", "Mild solutions are formally fixed points of the non-linear map $\\mathcal {T}$ , and (REF ) is in the form of a Volterra integral equation.", "The mild solution with rough initial data ($L^2$ ) established above is quite different from the ones considered in the previous studies (see ishige2020blowup,sandjo2014space,sandjo2015solutions).", "In Section , we show that this type of mild solution can be easily extended to be a global one by adding an advection term with specific mixing property compared to the mild solutions considered before." ], [ "Local existence with $L^2$ initial data", "When $3<p<4$ , the local existence of the mild solution to (REF ) on $\\mathbb {R}^N$ was studied in sandjo2014space,sandjo2015solutions.", "In the recent paper [14], the authors provided the local existence result of the mild solution for a better range $2<p\\le 4$ on $\\mathbb {R}^N$ with initial data that has better regularity (rather than $L^2(2)$ ).", "While in this paper, we focus on the case $2<p< 3$ on $\\mathbb {T}^2$ with rough initial data.", "We show for any initial data $u_0\\in L^2(2)$ , there exists $0<T<1$ depends on ${u_0}_{L^2}$ and $p$ such that (REF ) admits a mild solution on $[0,T]$ .", "Moreover, in Proposition REF , we show that this mild solution is also a weak solution to (REF ).", "Finally, it's worth mentioning that the existence of weak solution and classical solution to (REF ) for $p>2$ was studied in [16].", "A standard way to prove the local existence of a mild solution is to apply the Banach contraction mapping theorem, hence it suffices to argue that $\\mathcal {T}$ is a contraction map in a suitable adapted Banach space $\\tilde{\\mathcal {S}}_T$ , which is defined as follows.", "Given $0<T<1$ , we let $\\mathcal {S}_T\\lbrace u:\\mathbb {R}_+\\times 2\\rightarrow \\mathbb {R}\\vert \\sup _{0<t\\le T}t^{1/4}{u}_{L^2}<\\infty \\rbrace ,$ and $\\tilde{\\mathcal {S}}_TC([0,T];L^2(2))\\cap \\mathcal {S}_T,$ It is clear that $\\tilde{\\mathcal {S}}_T$ is a Banach space equipped with the norm: ${ u}_{\\tilde{\\mathcal {S}}_T} \\max \\left\\lbrace \\sup _{0\\le t\\le T}{ u}_{L^2},\\sup _{0< t\\le T}t^{1/4}{u}_{L^{2}}\\right\\rbrace .$ For completeness, we first recall and prove two useful estimates for the semigroup operator $e^{-t\\mathcal {L}}$ in the following two lemmas.", "Then we use these estimates to verify that $\\mathcal {T}$ is a contraction map on a ball in $\\tilde{\\mathcal {S}}_T$ .", "For $2<p \\le 3$ , there exists a constant $C$ such that ${e^{-t\\mathcal {L}}f}_{L^2}\\le Ct^{-\\frac{p-2}{4}}{f}_{L^\\frac{2}{p-1}}.$ We begin with the case when $2<p<3$ .", "By definition of operator $e^{-t\\mathcal {L}}$ and Plancherel's identity, we can easily get: ${e^{-t\\mathcal {L}}f}_{L^2}^2&=\\sum _{\\mathbf {k}\\in \\mathbb {Z}^2}e^{-2t{\\mathbf {k}}^4} {\\hat{f}(\\mathbf {k})}^2\\le \\left(\\sum _{k \\in ^2} e^{-\\frac{2t\|\\mathbf {k}\|^4}{p-2}} \\right)^{p-2} \\left(\\sum _{k \\in ^2} \\left\|\\hat{f}(\\mathbf {k}) \\right\|^{\\frac{2}{3-p}} \\right)^{3-p} \\\\&\\le C{f}_{L^\\frac{2}{p-2}}^2\\Big (\\int _{\\mathbb {R}^2}\\, e^{-\\frac{2t\|x\|^4}{p-2}} dx\\Big )\\le C t^{-\\frac{p-2}{2}} {f}_{L^\\frac{2}{p-2}}^2,$ where in the second last estimate above, we have used the assumption $p>2$ and the Hausdorff–Young inequality on 2.", "While for the case when $p=3$ , we simply bound $\\Vert e^{-t\\mathcal {L}} f\\Vert _{L^2}^2$ by $\\left\\Vert \\hat{f} \\right\\Vert ^2_{\\ell ^\\infty (^2)} \\cdot \\sum _{\\mathbf {k}\\in ^2} e^{-2t\|\\mathbf {k}\|^4}.$ The rest of the proof follows in a similar fashion and hence we omit it here.", "For any $s>0$ , there exists a constant $C$ such that ${(-)^{s/2}e^{-t\\mathcal {L}}f}_{L^2}\\le Ct^{-\\frac{s}{4}}{f}_{L^2}.$ The proof of Lemma REF is similar to the one of Lemma REF and hence we would like to leave the detail to the interested reader.", "The local existence of mild solution is summarized in the following main theorem.", "Let $u_0\\in L^{2}(2)$ and $2<p <3$ .", "Then there exists $0< T\\le 1$ depending only on ${u_0}_{L^2}$ such that (REF ) admits a mild solution $u$ on $[0,T]$ , which is unique in $\\tilde{\\mathcal {S}}_T$ .", "With the same assumptions in Theorem REF , if $T^$ is the maximal time of existence of the mild solution $u$ , then $\\limsup _{t\\rightarrow T_-^}{u(t)}_{L^2(2)}=\\infty .$ Otherwise, $T^=\\infty $ .", "We divide the proof of main Theorem REF into Lemma REF and Lemma REF .", "In Lemma REF , we show that the solution operator $\\mathcal {T}$ is bounded.", "In Lemma REF , we show the map $\\mathcal {T}$ is Lipshitz continuous on $\\tilde{\\mathcal {S}}_T$ with some constant that depends on $T$ .", "Which state as follows: When $2<p < 3$ and $0<T\\le 1$ , the map $\\mathcal {T}$ maps from $ \\tilde{\\mathcal {S}}_T$ to itself, and there exists $C_1>0$ , such that ${\\mathcal {T}( u)}_{\\tilde{\\mathcal {S}}_T}\\leqslant C_1\\Big ({ u_0}_{L^2}+T^{\\frac{3-p}{2}}{ u}_{\\tilde{\\mathcal {S}}_T}^{p-1}\\Big ).$ When $2<p < 3$ and $0<T\\le 1$ , there exists a constant $C_2$ such that, for any $ u_1$ , $ u_2\\in \\tilde{\\mathcal {S}}_T$ , ${\\mathcal {T}( u_1)-\\mathcal {T}( u_2)}_{\\tilde{\\mathcal {S}}_T}\\le C_2\\,T^{\\frac{3-p}{2}}\\Big ({ u_1}_{\\tilde{\\mathcal {S}}_T}^{p-2}+{ u_2}_{\\tilde{\\mathcal {S}}_T}^{p-2}\\Big ){ u_1- u_2}_{\\tilde{\\mathcal {S}}_T}.$ Assuming Lemma REF and Lemma REF , we prove Theorem REF first, then use it to show Corollary REF .", "Let $\\mathbb {B}_{R}(0)$ denotes the closed ball centered at origin with radius $R$ in $\\tilde{\\mathcal {S}}_T$ space.", "Choose $R\\ge 2 C_0{u_0}_{L^2}$ , where $C_0=\\max \\lbrace 1, C_1,C_2\\rbrace $ , and $C_1, C_2$ are the constants in (REF ) and (REF ), and we assume that $T\\le \\min \\Big \\lbrace 1, (4C_0 R^{p-2})^{\\frac{-2}{3-p}}\\Big \\rbrace .$ Then, Lemma REF implies ${\\mathcal {T}(u)}_{\\tilde{\\mathcal {S}}_T}\\le R,\\qquad \\forall u\\in \\mathbb {B}_{R}(0).$ On the other hand, Lemma REF yields for any $u_1,u_2\\in \\mathbb {B}_{R}(0)$ we have ${\\mathcal {T}(u_1)-\\mathcal {T}(u_2)}_{\\tilde{\\mathcal {S}}_T}\\le \\frac{{u_1-u_2}_{\\tilde{\\mathcal {S}}_T}}{2}.$ By Banach contraction mapping theorem, there is a unique fixed point of $\\mathcal {T}$ in $\\mathbb {B}_{R}(0)$ .", "By Definition , $u$ is a mild solution to (REF ) with initial data $u_0$ , which is also unique in $\\tilde{\\mathcal {S}}_T$ .", "We prove the result via contradiction.", "Assume $T^<\\infty $ and $\\limsup \\limits _{t\\rightarrow T_-^}{u(t)}_{L^2(2)}=\\alpha <\\infty $ .", "By continuity, there exists $t^$ such that for $\\forall t^<t<T^$ , we have ${u(t)}_{L^2(2)}\\le 3\\alpha $ .", "Take $T\\le \\min \\Big \\lbrace 1, (4C_0 (3\\alpha )^{p-2})^{\\frac{-2}{3-p}}\\Big \\rbrace ,$ where $C_0$ is defined in the proof of Theorem REF .", "Now we pick up any $t_0$ belonging to $[t^,T^]$ s.t.", "$t_0+T\\ge T^$ .", "Then by Theorem REF , there exists a mild solution $\\bar{u}$ on $[t_0,t_0+T]$ with initial data $u(t_0)$ and $u=\\bar{u}$ on $[t_0,T^)$ , according to the uniqueness of mild solution.", "Hence, the solution can be extended past $T^$ , which yields a contradiction.", "It now remains to check the lemmas.", "To prove Lemma REF , it suffices to check following two claims: Claim 1: If $ u\\in \\tilde{\\mathcal {S}}_T$ , then $\\mathcal {T}( u)\\in C([0,T];L^2(2))$ ; Claim 2: If $ u\\in \\tilde{\\mathcal {S}}_T$ , then $\\sup \\limits _{0<t\\le T} t^{1/4}{(\\mathcal {T}( u))}_{L^2}<\\infty $ .", "Utilizing the estimates in Lemma REF and Lemma REF , and the assumption $2<p \\le 3$ , we check Claim 1 first: $\\Vert \\mathcal {T}(u)\\Vert _{L^2}&\\le & \\Vert u_0\\Vert _{L^2}+\\int _0^t \\left\\Vert \\nabla e^{-(t-s)\\mathcal {L}} \\left(\|\\nabla u\|^{p-2} \\nabla u \\right) \\right\\Vert _{L^2} ds \\\\&\\le & \\Vert u_0\\Vert _{L^2}+\\int _0^t \\left\\Vert \\nabla e^{-\\frac{(t-s)\\mathcal {L}}{2}} \\right\\Vert _{L^2 \\rightarrow L^2} \\left\\Vert e^{-\\frac{(t-s)\\mathcal {L}}{2}} \\left(\|\\nabla u\|^{p-2} \\nabla u \\right) \\right\\Vert _{L^2} ds \\\\&\\le & C \\left(\\Vert u_0\\Vert _{L^2}+\\int _0^t (t-s)^{-\\frac{1}{4}} \\cdot (t-s)^{-\\frac{p-2}{4}}\\Vert \\nabla u\\Vert _{L^2}^{p-1} ds \\right) \\\\&\\le & C\\Big ({ u_0}_{L^2}+\\int _0^t\\,(t-s)^{-\\frac{p-1}{4}}s^{-\\frac{p-1}{4}}\\big (s^{\\frac{1}{4}}{u}_{L^2}\\big )^{p-1}\\,ds\\Big )\\\\&\\le & C\\Big ({ u_0}_{L^2} + t^{\\frac{3-p}{2}}\\int _0^{1}\\,(1-\\tau )^{-\\frac{1}{2}}\\tau ^{-\\frac{p-1}{4}} { u}_{\\tilde{\\mathcal {S}}_T}^{p-1}\\,d\\tau \\Big )\\\\&\\le & C\\Big ({ u_0}_{L^2}+t^{\\frac{3-p}{2}}{ u}_{\\tilde{\\mathcal {S}}_T}^{p-1}\\Big ),$ Similarly, for Claim 2, we have $\\left\\Vert \\nabla (\\mathcal {T}(u))(t) \\right\\Vert _{L^2}&\\le & \\left\\Vert \\nabla e^{-t\\mathcal {L}} u_0\\right\\Vert _{L^2} \\\\&& \\quad \\quad + \\int _0^t \\left\\Vert e^{-\\frac{(t-s)\\mathcal {L}}{2}} \\right\\Vert _{L^2 \\rightarrow L^2} \\left\\Vert e^{-\\frac{(t-s)\\mathcal {L}}{2}} \\left(\|\\nabla u\|^{p-2} \\nabla u \\right) \\right\\Vert _{L^2} ds \\\\&\\le & C \\left( t^{-\\frac{1}{4}}\\Vert u_0\\Vert _{L^2}+\\int _0^t (t-s)^{-\\frac{1}{2}} \\cdot (t-s)^{-\\frac{p-2}{4}}\\Vert \\nabla u\\Vert _{L^2}^{p-1} ds \\right) \\\\&=& C \\left( t^{-\\frac{1}{4}} \\Vert u_0\\Vert _{L^2}+\\int _0^t (t-s)^{-\\frac{p}{4}} s^{-\\frac{p-1}{4}}\\big (s^{\\frac{1}{4}}{u}_{L^2}\\big )^{p-1}\\,ds\\right)\\\\&\\le & C \\left(t^{-\\frac{1}{4}} \\Vert u_0\\Vert _{L^2}+t^{\\frac{5-2p}{4}} \\int _0^1 (1-\\tau )^{-\\frac{p}{4}} \\tau ^{-\\frac{p-1}{4}} d\\tau \\cdot \\Vert u\\Vert _{\\tilde{\\mathcal {S}}_T}^{p-1} \\right) \\\\&\\le & C \\left(t^{-\\frac{1}{4}} \\Vert u_0\\Vert _{L^2}+t^{\\frac{5-2p}{4}} \\Vert u\\Vert _{\\tilde{\\mathcal {S}}_T}^{p-1} \\right)$ Multiplying $t^{\\frac{1}{4}}$ on both sides of the above estimate, we get $t^{\\frac{1}{4}} {(\\mathcal {T}(u)(t))}_{L^2}\\le C\\Big ({ u_0}_{L^2}+t^{\\frac{3-p}{2}}{ u}_{\\tilde{\\mathcal {S}}_T}^{p-1}\\Big ),$ for any $t\\in [0,T]$ .", "Combining the proof of two claims, the proof of Lemma REF is completed.", "Next, we check that the map $\\mathcal {T}$ is Lipschitz on $\\tilde{\\mathcal {S}}_T$ with a constant that depends on $T$ .", "Similar to the proof of previous lemma, it is sufficient for us to check the following two claims.", "Let $ u_1, u_2\\in \\tilde{\\mathcal {S}}_T$ .", "For all $0\\le t\\le T$ : Claim 1.", "${\\mathcal {T}(u_1)(t)-\\mathcal {T}( u_2)(t)}_{L^2}\\leqslant C t^{\\frac{3-p}{2}}\\big ({ u_1}_{\\tilde{\\mathcal {S}}_T}^{p-2}+{ u_2}_{\\tilde{\\mathcal {S}}_T}^{p-2}\\big ){ u_1- u_2}_{\\tilde{\\mathcal {S}}_T};$ Claim 2.", "$\\sup _{t\\in [0,T]}t^{\\frac{1}{4}}{\\mathcal {T}( u_1)(t)-\\mathcal {T}( u_2)(t)}_{L^2}\\leqslant C t^{\\frac{3-p}{2}}\\big ({ u_1}_{\\tilde{\\mathcal {S}}_T}^{p-2}+{ u_2}_{\\tilde{\\mathcal {S}}_T}^{p-2}\\big ){ u_1- u_2}_{\\tilde{\\mathcal {S}}_T}.$ Proof of Claim 1: For any $0 \\le t \\le T$ , $&&\\left\\Vert \\mathcal {T}(u_1)-\\mathcal {T}(u_2) \\right\\Vert _{L^2}\\le \\int _0^t \\left\\Vert \\nabla e^{-(t-s) \\mathcal {L}} \\left(\|\\nabla u_1\|^{p-2} \\nabla u_1-\|\\nabla u_2\|^{p-2} \\nabla u_2 \\right) \\right\\Vert _{L^2} ds \\\\&& \\quad \\quad \\quad \\quad \\le C \\int _0^t (t-s)^{-\\frac{1}{4}} \\cdot (t-s)^{-\\frac{p-2}{4}} \\left\\Vert \|\\nabla u_1\|^{p-2} \\nabla u_1-\|\\nabla u_2\|^{p-2} \\nabla u_2 \\right\\Vert _{L^{\\frac{2}{p-1}}} ds \\\\&& \\quad \\quad \\quad \\quad \\le C \\int _0^t (t-s)^{-\\frac{p-1}{4}} \\left\\Vert \\left\|\\nabla u_1-\\nabla u_2 \\right\| \\left(\|\\nabla u_1\|^{p-2}+\|\\nabla u_2\|^{p-2} \\right) \\right\\Vert _{L^{\\frac{2}{p-1}}} ds \\\\&& \\quad \\quad \\quad \\quad \\le C \\int _0^t (t-s)^{-\\frac{p-1}{4}} \\Vert \\nabla u_1-\\nabla u_2\\Vert _{L^2} \\left( \\Vert \\nabla u_1\\Vert _{L^2}^{p-2}+\\Vert \\nabla u_2 \\Vert _{L^2}^{p-2} \\right)ds \\\\&& \\quad \\quad \\quad \\quad \\le C\\int _0^t (t-s)^{-\\frac{p-1}{4}} s^{-\\frac{p-1}{4}} \\Vert u_1-u_2\\Vert _{\\tilde{\\mathcal {S}}_T} \\left(\\Vert u_1\\Vert _{\\tilde{\\mathcal {S}}_T}^{p-2}+\\Vert u_2\\Vert ^{p-2}_{\\tilde{\\mathcal {S}}_T} \\right) ds \\\\&& \\quad \\quad \\quad \\quad \\le Ct^{\\frac{3-p}{2}} \\left(\\Vert u_1\\Vert _{\\tilde{\\mathcal {S}}_T}^{p-2}+\\Vert u_2\\Vert ^{p-2}_{\\tilde{\\mathcal {S}}_T} \\right)\\Vert u_1-u_2\\Vert _{\\tilde{\\mathcal {S}}_T},$ where in the above estimate, we have used the following elementary estimate: for any $a, b \\in ^2$ and $p \\ge 2$ , $\\left\| \|a\|^{p-2}a-\|b\|^{p-2} b\\right\| \\le C_p (\|a\|^{p-2}+\|b\|^{p-2})\|a-b\|.$ Proof of Claim 2: The proof is similar to the one of Claim 2 in the previous lemma: $&&\\left\\Vert \\nabla \\mathcal {T}(u_1)-\\nabla T(u_2) \\right\\Vert _{L^2} \\le \\int _0^t \\left\\Vert e^{-(t-s)\\mathcal {L}} \\left(\|\\nabla u_1\|^{p-2} \\nabla u_1-\|\\nabla u_2\|^{p-2}\\nabla u_2 \\right)\\right\\Vert _{L^2} ds \\\\&& \\quad \\quad \\quad \\quad \\quad \\quad \\le \\int _0^t (t-s)^{-\\frac{p}{4}} \\Vert \\nabla u_1-\\nabla u_2\\Vert _{L^2} \\left(\\Vert \\nabla u_1\\Vert _{L^2}^{p-2}+\\Vert \\nabla u_2\\Vert _{L^2}^{p-2} \\right) ds \\\\&& \\quad \\quad \\quad \\quad \\quad \\quad \\le Ct^{\\frac{5-2p}{4}} \\left(\\Vert u_1\\Vert _{\\tilde{\\mathcal {S}}_T}^{p-2}+\\Vert u_2\\Vert ^{p-2}_{\\tilde{\\mathcal {S}}_T} \\right)\\Vert u_1-u_2\\Vert _{\\tilde{\\mathcal {S}}_T}.$ Again, multiplying $t^{\\frac{1}{4}}$ on the both sides of the above estimate yields $t^{\\frac{1}{4}}{\\mathcal {T}(u_1)-\\mathcal {T}(u_2)}_{L^2}\\le C t^{\\frac{3-p}{2}}\\big ({ u_1}_{\\tilde{\\mathcal {S}}_T}^{p-2}+{ u_2}_{\\tilde{\\mathcal {S}}_T}^{p-2}\\big ){ u_1- u_2}_{\\tilde{\\mathcal {S}}_T},$ for any $t\\in [0,T]$ .", "The proof of Lemma REF is complete.", "Finally, we show that the mild solution given in Theorem REF is also a weak solution in the sense of Definition .", "Let $2<p<3$ and $u$ be the mild solution on $[0,T]$ given in Theorem REF .", "Then it is also a weak solution of (REF ) on $[0,T]$ .", "Moreover, the following energy identity holds: for any $t\\in [0,T)$ , ${u(t)}_{L^2}^2 + 2\\int _0^t\\,{u(s)}_{L^2}^2\\,ds={u_0}_{L^2}^2 + 2\\int _0^t\\,{u(s)}_{L^p}^p ds.$ First of all, observe that for any $0<\\varepsilon \\le t \\le T$ , one can rewrite the mild solution $u(t)$ with following expression on $[\\epsilon , T]$ : $u(t)=e^{-(t-\\epsilon )\\mathcal {L}}u(\\epsilon )\\,-\\,\\int _{\\epsilon }^{t}\\,\\nabla e^{-(t-s)\\mathcal {L}}\\, F(u(s))\\,ds,$ where we recall that $F(\\xi )=\|\\xi \|^{p-2}\\xi , \\xi \\in ^2$ .", "Here the equality holds as functions in $C([\\epsilon ,T]; L^2(2))$ .", "Indeed, by the property of semigroup and Definition , it follows that: $&u(t)-u(\\epsilon )=\\left(e^{-(t-\\epsilon )\\mathcal {L}}-I\\right)\\left(e^{-\\epsilon \\mathcal {L}}u_0-\\int _0^{\\epsilon }e^{-(\\epsilon -s)\\mathcal {L}}F\\left(u\\right)\\left(s\\right)ds\\right)\\\\& \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad -\\int _{\\epsilon }^{t}e^{-(t-s)\\mathcal {L}}F\\left(u\\right)\\left(s\\right)ds \\\\&\\quad \\quad \\quad \\quad \\quad =\\left(e^{-(t-\\epsilon )\\mathcal {L}}-I\\right)u(\\epsilon )-\\int _{\\epsilon }^{t}e^{-(t-s)\\mathcal {L}}F\\left(u\\right)\\left(s\\right)ds,$ where $I$ is the identity map.", "Next, we show that this actually implies $u\\in L^2([\\epsilon ,T]; H^2(2))$ .", "Since $u\\in C([\\epsilon ,T]; L^2(2))$ , it is enough to check $u\\in L^2([\\epsilon ,T]\\times 2)$ , therefore we take $$ to both sides of (REF ), and use the decay estimates of the semigroup $e^{-t\\mathcal {L}}$ also the fact that $u\\in \\tilde{\\mathcal {S}}_T$ on $[\\epsilon ,T]$ : ${u(t)}_{L^2}&\\le C\\Big ((t-\\epsilon )^{-\\frac{1}{4}}{u(\\epsilon )}_{\\dot{H}^1}+\\int _{\\epsilon }^{t}\\,(t-s)^{-\\frac{3}{4}} \\cdot (t-s)^{-\\frac{p-2}{4}} \\cdot {u}_{L^{2}}^{p-1}\\,ds\\Big )\\\\&= C \\left( (t-\\varepsilon )^{-\\frac{1}{4}} \\Vert u(\\varepsilon )\\Vert _{\\dot{H}^1}+\\int _\\varepsilon ^t (t-s)^{-\\frac{p+1}{4}} \\Vert \\nabla u\\Vert _{L^2}^{p-1} ds\\right) \\\\&\\le C\\Big ((t-\\epsilon )^{-\\frac{1}{4}}\\epsilon ^{-\\frac{1}{4}}{u}_{\\tilde{\\mathcal {S}}_T}+t^{\\frac{2-p}{2}} {u}_{\\tilde{\\mathcal {S}}_T}^{p-1}\\Big ).$ Note for the last inequality, we implicitly use the fact that $p<3$ .", "Also notice that the right hand side is square integrable in time for $p<3$ , $\\epsilon >0$ .", "As some consequence, we have the following claims.", "Claim 1: $F(\\nabla u) \\in L^2 \\left( \\left[\\varepsilon , T \\right]; L^{\\frac{1}{p-2}}(2) \\right)$ .", "Claim 2: $N(u) \\in L^1 \\left( \\left[\\varepsilon , T\\right] \\times 2 \\right)$ .", "The first claim is a consequence of the Gagliardo-Nirenberg's inequality (note that $\\frac{1}{2}< \\frac{1}{p-1} < 1$ ): $ &&\\int _{\\epsilon }^{T}\\,{F(u)}^2_{L^{\\frac{1}{p-2}}}\\,ds= \\int _{\\epsilon }^{T}\\,{u}^{2(p-1)}_{L^{\\frac{p-1}{p-2}}}\\,ds \\nonumber \\\\&& \\quad \\quad \\le C \\int _\\varepsilon ^T \\left( \\left\\Vert u \\right\\Vert _{L^2}^2 \\left\\Vert u \\right\\Vert _{L^2}^{2(p-2)}+\\Vert u\\Vert _{L^2}^{2(p-1)} \\right) ds \\nonumber \\\\&& \\quad \\quad \\leqslant C \\left( {u}_{C([\\epsilon ,T]; L^2(2))}^{2(p-2)}{u}_{L^2([\\epsilon ,T]; H^2(2))}^2+T {u}_{C([\\epsilon ,T]; L^2(2))}^{2(p-1)} \\right)$ While for the second claim, we first compute that $\\left\|N(u) \\right\|=\\left\| \\nabla \\cdot \\left(\|\\nabla u\|^{p-2} \\nabla u \\right) \\right\| \\le \\left\|\\nabla u \\right\|^{p-2} \\left\|u \\right\|+(p-2)\\left\|\\nabla u \\right\|^{p-4} \\left\| \\left(\\nabla u \\right)^T \\nabla ^2 u \\nabla u \\right\|,$ where $v^T$ denotes the transpose of a vector $v \\in ^2$ .", "This gives $&&\\int _{\\varepsilon }^T \\left\\Vert N(u) \\right\\Vert _{L^1}ds \\le \\int _\\varepsilon ^T \\left\\Vert \\left\|\\nabla u \\right\|^{p-2} \\left\|u \\right\| \\right\\Vert _{L^1} ds \\\\&& \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad + (p-2)\\int _\\varepsilon ^T \\left\\Vert \\left\|\\nabla u \\right\|^{p-4} \\left\| \\left(\\nabla u \\right)^T \\nabla ^2 u \\nabla u \\right\| \\right\\Vert _{L^1} ds \\\\&& \\quad \\quad \\quad = \\int _\\varepsilon ^T \\left\\Vert \\left\|\\nabla u \\right\|^{p-2} \\left\|u \\right\| \\right\\Vert _{L^1} ds+ (p-2)\\int _\\varepsilon ^T \\left\\Vert \\left\|\\nabla u \\right\|^{p-2} \\cdot \\frac{\\left\| \\left(\\nabla u \\right)^T \\nabla ^2 u \\nabla u \\right\|}{\|\\nabla u\|^2} \\right\\Vert _{L^1} ds \\\\&& \\quad \\quad \\quad \\le C \\int _\\varepsilon ^T {u}_{L^{2(p-2)}}^{p-2}{u}_{L^2}\\,ds \\\\&& \\quad \\quad \\quad \\le C \\int _\\varepsilon ^T \\Vert u \\Vert _{L^2}^2 ds + C \\int _\\varepsilon ^T \\Vert \\nabla u\\Vert _{L^{2(p-2)}}^{2(p-2)} ds \\\\&& \\quad \\quad \\quad \\le C\\Vert u\\Vert _{L^2([\\varepsilon , T]; H^2(2)}^2+C \\Vert u\\Vert _{\\tilde{S}_T}^{2(p-2)}<\\infty .$ Hence, the second claim is verified.", "Let $\\phi \\in C_c^{\\infty }([\\epsilon ,T)\\times 2)$ and consider the $L^2$ pairing of $\\phi $ with $u$ .", "Observe that $e^{-t\\mathcal {L}}$ is a self-adjoint operator on $L^2$ , and $$ operator commutes with $e^{-t\\mathcal {L}}$ .", "This gives $&\\int _{2}\\,\\phi (t)u(t)dx=\\int _{2}\\,e^{-(t-\\epsilon )\\mathcal {L}}\\phi (t)u(\\epsilon )dx\\, \\\\& \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad +\\,\\int _{\\epsilon }^t\\,\\int _{2}\\,(e^{-(t-s)\\mathcal {L}}\\phi (t))\\cdot F(u(s))\\,dx ds.$ Note that $\\phi (t)$ and $\\phi (t)$ are in the domain of $\\mathcal {L}$ for all $t\\in [\\epsilon ,T)$ , and $\\mathcal {L}$ generates an analytic semigroup.", "Therefore, we have $\\lim _{h\\rightarrow 0}\\frac{e^{-(t+h-\\epsilon )\\mathcal {L}}\\phi (t)-e^{-(t-\\epsilon )\\mathcal {L}}\\phi (t)}{h}=-e^{-(t-\\epsilon )\\mathcal {L}}\\mathcal {L}\\phi (t),$ $\\lim _{h\\rightarrow 0}\\frac{e^{-(t-\\epsilon )\\mathcal {L}}\\phi (t+h)-e^{-(t-\\epsilon )\\mathcal {L}}\\phi (t)}{h}=e^{-(t-\\epsilon )\\mathcal {L}}\\partial _t \\phi (t),$ both strongly in $L^2(2)$ , similarly if $\\phi $ is replaced by $\\phi $ .", "Since $e^{-t\\mathcal {L}}$ is a strongly continuous semigroup on $L^2(2)$ , it follows that $\\frac{d}{dt}e^{-(t-\\epsilon )\\mathcal {L}}\\phi (t)\\in C([\\epsilon ,T);L^2(2))$ and by Leibniz rule: $\\frac{d}{dt}e^{-(t-\\epsilon )\\mathcal {L}}\\phi (t)=e^{-(t-\\epsilon )\\mathcal {L}}\\partial _t\\phi (t)-\\mathcal {L}e^{-(t-\\epsilon )\\mathcal {L}}\\phi (t).$ In particular, the pairing of $e^{-(t-\\epsilon )\\mathcal {L}}\\phi (t)$ with any function $f(t)$ that is absolutely continuous in $t\\in [\\epsilon ,T)$ with values in $L^2(2)$ , is differentiable a.e.", "in $t$ , the derivative is integrable in time, and $&&\\frac{d}{dt}\\int _{2}\\,e^{-(t-\\epsilon )\\mathcal {L}}\\phi (t)f(t)\\,dx =\\int _{2}\\,\\Big (-\\mathcal {L}e^{-(t-\\epsilon )\\mathcal {L}}\\phi (t)f(t)+e^{-(t-\\epsilon )\\mathcal {L}}\\partial _t\\phi (t)f(t) \\nonumber \\\\&& \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad +e^{-(t-\\epsilon )\\mathcal {L}}\\phi (t)f^{\\prime }(t)\\Big )\\,dx.$ Similarly with $\\phi $ being replaced by $\\nabla \\phi $ , $&&\\frac{d}{dt}\\int _{2}\\,e^{-(t-\\epsilon )\\mathcal {L}}\\phi (t)\\cdot f(t)\\,dx =\\int _{2}\\,\\Big (-\\mathcal {L}e^{-(t-\\epsilon )\\mathcal {L}}\\phi (t)\\cdot f(t) \\nonumber \\\\&& \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad +e^{-(t-\\epsilon )\\mathcal {L}}\\partial _t\\phi (t)\\cdot f(t) +e^{-(t-\\epsilon )\\mathcal {L}}\\phi (t)\\cdot f^{\\prime }(t)\\Big )\\,dx.$ Recall the following property from functional analysis: if $X$ is a separable and reflexive Banach space, then for a $X$ -valued function of time is absolute continuous if and only if it has weak derivative for almost every time, and is integrable in the Böchner sense.", "By using (REF ) and (REF ), we have $&\\frac{d}{dt}\\int _{2}\\,\\phi (t)u(t)\\,dx=\\int _{2}\\,\\Big (e^{-(t-\\epsilon )\\mathcal {L}}\\partial _t\\phi (t)-e^{-(t-\\epsilon )\\mathcal {L}}\\mathcal {L}\\phi (t)\\Big )u(\\epsilon )\\,dx\\\\& \\quad \\quad +\\int _{2}\\,\\int _{\\epsilon }^{t}\\,\\Big (e^{-(t-s)\\mathcal {L}}\\partial _t\\big (\\phi (t)\\big )-\\mathcal {L}e^{-(t-s)\\mathcal {L}}\\phi (t)\\Big )\\cdot \\Big ({u(s)}^{p-2}u(s)\\Big )\\,dt dx\\\\& \\quad \\quad +\\int _{2}\\,\\phi (t)\\cdot \\Big ({u(t)}^{p-2}u(t)\\Big )\\,dx.$ Rearranging the terms, we see that $&\\frac{d}{dt}\\int _{2}\\,\\phi (t)u(t)\\,dx\\\\&\\quad =\\int _{2}\\,\\partial _t\\phi (t)\\Big (e^{-(t-\\epsilon )\\mathcal {L}}u(\\epsilon )-\\int _{\\epsilon }^t\\,\\nabla e^{-(t-s)\\mathcal {L}} \\big ({u}^{p-2}u\\big )(s)\\,ds\\Big )\\,dx\\\\&\\quad \\quad \\quad -\\int _{2}\\,\\mathcal {L}\\phi (t)\\Big (e^{-(t-\\epsilon )\\mathcal {L}}u(\\epsilon )-\\int _{\\epsilon }^t\\,\\nabla e^{-(t-s)\\mathcal {L}} \\big ({u}^{p-2}u\\big )(s)\\,ds\\Big )\\,dx\\\\&\\quad \\quad \\quad +\\int _{2}\\,\\phi (t)\\cdot \\Big ({u(t)}^{p-2}u(t)\\Big )\\,dx\\\\&\\quad =\\int _{2}\\,\\big (\\partial _t\\phi (t)-\\mathcal {L}\\phi (t)\\big )u(t)dx\\,+\\,\\int _{2}\\,\\phi (t)\\cdot \\Big ({u(t)}^{p-2}u(t)\\Big )\\,dx.$ Integrating the above identity from $\\epsilon $ to $t$ for $t\\in (\\epsilon ,T)$ yields: $&&\\int _{2}\\,\\phi (t)u(t)\\,dx\\,-\\,\\int _{2}\\,\\phi (\\epsilon )u(\\epsilon )\\,dx =\\int _{\\epsilon }^t\\int _{2}\\,\\partial _{t}\\phi (s)u(s)\\,dx ds\\, \\nonumber \\\\&& \\quad \\quad \\quad \\quad -\\,\\int _{\\epsilon }^t\\int _{2}\\,\\phi (s)u(s)dx ds +\\,\\int _{\\epsilon }^t\\int _{2}\\,\\phi (s)\\cdot \\Big ({u(s)}^{p-2}u(s)\\Big )\\,dx ds.$ We claim that $\\partial _t u \\in L^2 \\left([\\varepsilon , T); H^{-2} \\right)$ .", "To see this, we note that it suffices to show that (REF ) is well-defined for $\\phi \\in L^2 \\left( \\left[ \\varepsilon , t \\right]; H^2 \\right)$ with $\\partial _t \\phi \\in L^2 \\left( \\left[ \\varepsilon , t \\right]; H^{-2} \\right)$ .", "The key observation here is to show that last term in (REF ) is well-defined under such an assumption.", "Indeed, $&& \\,\\int _{\\epsilon }^t\\int _{2}\\,\\phi (s)\\cdot \\Big ({u(s)}^{p-2}u(s)\\Big )\\,dx ds=\\int _\\varepsilon ^t \\int _{2} \\nabla \\phi (s) \\cdot F \\left(\\nabla u(s) \\right) dxds \\\\&& \\quad \\quad \\quad \\le \\int _\\varepsilon ^t \\left\\Vert \\nabla \\phi (s)\\right\\Vert _{L^{\\frac{1}{3-p}}} \\cdot \\left\\Vert F(\\nabla u(s)) \\right\\Vert _{L^{\\frac{1}{p-2}}} ds \\\\&& \\quad \\quad \\quad \\le \\left(\\int _\\varepsilon ^t \\left\\Vert \\nabla \\phi \\right\\Vert _{L^{\\frac{1}{3-p}}}^2 ds \\right)^{\\frac{1}{2}} \\cdot \\left(\\int _\\varepsilon ^t \\left\\Vert F(\\nabla u(s) ) \\right\\Vert _{L^{\\frac{1}{p-2}}}^2 ds \\right)^{\\frac{1}{2}} \\\\&& \\quad \\quad \\quad \\le C\\left(\\int _\\varepsilon ^t \\left\\Vert \\phi \\right\\Vert _{L^2}^2 +\\Vert \\phi \\Vert _{L^2}^2ds \\right)^{\\frac{1}{2}} \\cdot \\left\\Vert F(\\nabla u) \\right\\Vert _{L^2 \\left( \\left[ \\varepsilon , T \\right]; L^{\\frac{1}{p-2}}(2) \\right)} \\\\&& \\quad \\quad \\quad \\le C\\Vert \\phi \\Vert _{L^2 \\left(\\left[\\varepsilon , t \\right]; H^2 \\right)} \\left\\Vert F(\\nabla u) \\right\\Vert _{L^2 \\left( \\left[ \\varepsilon , T \\right]; L^{\\frac{1}{p-2}}(2) \\right)}<\\infty ,$ where we use (REF ) (see, Claim 1) and Gagliardo-Nirenberg’s inequality in the second last inequality.", "Since $u\\in L^2([\\epsilon ,T);H^2)$ with $\\partial _t u \\in L^2 \\left([\\varepsilon , T); H^{-2} \\right)$ , there exists a sequence $\\lbrace u_{m}\\rbrace \\subset C^1([\\epsilon ,T);H^2)$ such that $u_{m}\\rightarrow u$ strongly in $C((\\epsilon ,T);L^2)\\cap L^2((\\epsilon ,T);H^2)$ and $\\partial _t u_{m}\\rightarrow \\partial _t u$ weakly in $L^2((\\epsilon ,T);H^{-2})$ as $m \\rightarrow \\infty $ .", "This in particular gives $&& \\Big \\vert \\int _{\\epsilon }^t\\int _{2}\\,(u_m-u)F(u(s))\\,dx ds\\Big \\vert \\\\&& \\quad \\quad \\quad \\quad \\le {u_{m}-u}_{L^2\\left((\\epsilon ,T),L^{\\frac{1}{3-p}}\\right)}{F(u)}_{L^2\\left([\\epsilon ,T),L^{\\frac{1}{p-2}}\\right)}\\\\&& \\quad \\quad \\quad \\quad \\leqslant C{u_{m}-u}_{L^2\\left([\\epsilon , T);H^2\\right)}{F(u)}_{L^2\\left([\\epsilon ,T); L^{\\frac{1}{p-2}}\\right)},$ which converges to 0 as $m \\rightarrow \\infty $ .", "Therefore, by plugging $u_m$ as test functions in (REF ), and letting $m\\rightarrow \\infty $ , we derive that $ {u(t)}_{L^2}^2-{u(\\epsilon )}_{L^2}^2=\\int _{\\epsilon }^t\\int _{2}\\,u(s)\\partial _t u(s)dxds\\,-\\,\\int _{\\epsilon }^{t}\\,{u(s)}_{L^2}^2\\,ds\\,+\\,\\int _{\\epsilon }^t{u}_{L^p}^p\\,ds.$ Since $u\\in L^2([\\epsilon ,T);H^2)$ with $\\partial _t u\\in L^2([\\epsilon ,T);H^{-2})$ , we have $t\\rightarrow {u(t)}_{L^2}^2$ is absolutely continuous and an application of the Fundamental Theorem of Calculus gives: $\\int _{\\epsilon }^t\\int _{2}\\,u(s)\\partial _t u(s)\\,dx ds =\\frac{1}{2}\\int _{\\epsilon }^t\\int _{2}\\,\\partial _t (u(s))^2\\,dx ds=\\frac{1}{2}\\left({u(t)}_{L^2}^2-{u(\\epsilon )}_{L^2}^2\\right),$ which together with (REF ) implies ${u(t)}_{L^2}^2\\,-\\,{u(\\epsilon )}_{L^2}^2\\,+\\,2\\int _{\\epsilon }^t\\,{u(s)}_{L^2}^2\\,ds = 2\\int _{\\epsilon }^t\\,{u}_{L^p}^p\\,ds.$ Next, we show that we can take $\\epsilon \\rightarrow 0$ in the above identity and obtain an energy inequality valid on $[0,T)$ .", "Indeed, the Gagliardo-Nirenberg inequality yields $\\int _{\\epsilon }^t\\,{u}_{L^p}^p\\,ds\\le \\int _{\\epsilon }^t\\,{u(s)}_{L^2}^2\\,ds\\,+\\, C\\int _{\\epsilon }^t\\ \\left({u(s)}_{L^2}^{\\frac{2}{3-p}}+\\Vert u(s)\\Vert _{L^2}^p \\right)ds.$ Plugging the above estimate into (REF ), we see that $\\int _{\\epsilon }^t\\,{u(s)}_{L^2}^2\\,ds&\\le & {u(\\epsilon )}_{L^2}^2\\,-\\,{u(t)}_{L^2}^2\\,+\\,C\\int _{\\epsilon }^t\\left({u(s)}_{L^2}^{\\frac{2}{3-p}}+\\Vert u(s)\\Vert _{L^2}^p \\right)ds \\\\&\\le & C \\left(\\Vert u\\Vert _{\\tilde{\\mathcal {S}}_T}+T\\Vert u\\Vert _{\\tilde{\\mathcal {S}}_T}^{\\frac{2}{3-p}}+T\\Vert u\\Vert _{\\tilde{\\mathcal {S}}_T}^p \\right),$ where the right hand side in the above estimate above is uniform in $\\varepsilon $ .", "Note that this also implies $u\\in L^2([0,T);H^2)$ .", "Therefore, by the Dominated convergence theorem, we are able to let $\\epsilon \\rightarrow 0$ in (REF ) to recover (REF ).", "Here, in particular we have used the continuity of the function $\\Vert u(t)\\Vert _{L^2}$ at $t=0$ (one can easily verify this by the definition of mild solutions).", "Similarly, since $u\\in L^2([0,T);H^2)$ , the right hand side of (REF ) is uniformly bounded with respect to $\\epsilon $ with $\\phi \\in C_c^\\infty ((0,T);H^2)$ .", "Thus, $(\\partial _t u)\\chi _{[\\epsilon ,T)}$ converges weakly to $\\partial _t u$ in $L^2((0,T);H^{-2})$ .", "Finally, by taking $\\epsilon \\rightarrow 0$ in (REF ) with a test function $\\phi \\in C_c^{\\infty }([0,T]\\times 2)$ (by Dominated convergence theorem again), we get the weak formulation in (REF ).", "Hence the mild solution $u$ is also a weak solution on $[0,T]$ ." ], [ "Epitaxial thin film growth equation with advection", "In this section, we study the thin film equation with advection on 2.", "Accordingly, we add an advection term to the equation (REF ): $\\partial _t u + v\\cdot u = -^2 u -\\cdot ({u}^{p-2}u),\\qquad u(0,x)= u_0(x)$ where $v$ is a given divergence free vector field.", "The plan of this section is as follow.", "To begin with, we establish the local existence of solution to equation (REF ) in Section REF .", "Next, in Section REF we show that if the velocity field $v$ has small dissipation time (see Definition REF ), then the corresponding solution to (REF ) can be extended to be a global one.", "Further, the $L^2$ norm of the global solution will decay exponentially fast on the whole time interval." ], [ "Local existence with $L^2$ initial data", "As in the non-advective case, equation (REF ) admits an unique local mild solution in $\\tilde{\\mathcal {S}}_T$ if $v\\in L^{\\infty }([0,\\infty );L^2(2))$ is provided.", "Moreover, one can also check that the mild solution is equivalent to a weak solution and satisfies certain energy identity.", "The proofs are very similar to the non-advective case.", "Therefore, we only sketch the proof to these results and leave the details to the interested reader.", "For $p>2$ and $v\\in L^{\\infty }([0,\\infty );L^2(2))$ , a function $ u\\in C([0,T];L^{2}(\\mathbb {T}^2))$ , $T>0$ , such that $u$ is locally integrable, is called a mild solution of (REF ) on $[0,T]$ with initial data $ u_0\\in L^{2}(2)$ , if for any $0\\le t\\le T$ , $u(t)=\\mathcal {T}_{v}(u)(t)e^{-t\\mathcal {L}} u_0\\,-\\,\\int _0^t\\,e^{-(t-s)\\mathcal {L}}({u}^{p-2}u)\\,ds\\,-\\, \\int _0^t\\,e^{-(t-s)\\mathcal {L}}(v\\cdot u)\\,ds,$ holds pointwisely in time with values in $L^2$ , where the integral is defined in the Böchner sense.", "For $p>2$ and $v\\in L^{\\infty }([0,\\infty );L^2(2))$ , a function $ u\\in L^{\\infty }([0,T];L^{2}({2}))\\cap L^{2}([0,T];H^{2}({2}))$ is called a weak solution of (REF ) on $[0,T)$ with initial value $ u(0)= u_0\\in L^{2}({2})$ if for all $\\phi \\in C_c^{\\infty }([0,T)\\times {2})$ , $&\\int _{T^2}\\,u_0\\phi (0)\\,dx\\, +\\, \\int _0^T\\,\\int _{2}u\\,\\partial _t \\phi \\,dx dt \\\\&=\\int _0^T\\,\\int _{2}\\,u\\phi dx dt\\, -\\, \\int _0^T \\int _{2}\\,{u}^{p-2}u\\cdot \\phi \\,dx dt \\,+\\,\\int _0^T \\int _{2}\\,\\phi v\\cdot u\\,dx dt ,$ and $\\partial _t u\\in L^{2}([0,T]; H^{-2}(2))$ .", "Let $u_0\\in L^{2}(2)$ , $2<p< 3$ , and $v\\in L^{\\infty }([0,\\infty );L^2(2))$ .", "Then there exists $0<T\\leqslant 1$ , which only depends on ${u_0}_{L^2}$ and $\\sup \\limits _{t\\in [0,\\infty )}{v}_{L^2}$ such that equation (REF ) admits a mild solution $u$ on $[0,T]$ , which is unique in $\\tilde{\\mathcal {S}}_T$ .", "Observe that if $v\\in L^{\\infty }([0,\\infty );L^2(2))$ , then for $u\\in \\tilde{\\mathcal {S}}_T$ , we have the following estimates for the advection term: $\\Big \\Vert \\int _0^t\\,e^{-\\left(t-s\\right)\\mathcal {L}}(v\\cdot u)\\,ds\\Big \\Vert _{L^2}&\\leqslant \\int _{0}^{t}(t-s)^{-\\tfrac{1}{4}}{v\\cdot u}_{L^1}ds\\\\&\\leqslant {v}_{L^\\infty ([0,\\infty );L^2)}\\int _{0}^{t}\\left(t-s\\right)^{-\\tfrac{1}{4}}\\cdot s^{-\\tfrac{1}{4}}\\left(s^{\\tfrac{1}{4}}{u\\left(s\\right)}_{L^2}\\right)\\,ds\\\\&\\leqslant t^{\\tfrac{1}{2}} \\cdot {v}_{L^\\infty \\left([0,\\infty );L^2\\right)}{ u}_{\\tilde{\\mathcal {S}}_T},$ where we used Lemma REF with $p=3$ in the first inequality above.", "Similarly, $t^{\\tfrac{1}{4}} \\cdot \\Big \\Vert \\int _0^t\\,e^{-\\left(t-s\\right)\\mathcal {L}}\\left(v\\cdot u\\right)\\,ds\\Big \\Vert _{L^2}\\leqslant t^{\\tfrac{1}{2}} \\cdot {v}_{L^\\infty \\left([0,\\infty );L^2\\right)} \\cdot { u}_{\\tilde{\\mathcal {S}}_T}.$ Moreover, for $u_1,u_2\\in \\tilde{\\mathcal {S}}_T$ we have $\\Big \\Vert \\int _0^t\\,e^{-(t-s)\\mathcal {L}}\\left(v\\cdot \\left(u_1-u_2\\right)\\right)\\,ds\\Big \\Vert _{L^2}\\leqslant t^{\\tfrac{1}{2}}\\cdot {v}_{L^\\infty ([0,\\infty );L^2)} \\cdot {u_1-u_2}_{\\tilde{\\mathcal {S}}_T},\\\\t^{\\tfrac{1}{4}} \\cdot \\Big \\Vert \\int _0^t\\,e^{-(t-s)\\mathcal {L}}\\left(v\\cdot \\left(u_1-u_2\\right)\\right)\\,ds\\Big \\Vert _{L^2}\\leqslant t^{\\tfrac{1}{2}} \\cdot {v}_{L^\\infty ([0,\\infty );L^2)} \\cdot {u_1-u_2}_{\\tilde{\\mathcal {S}}_T}.$ By using the above estimates to modify the proof of Theorem REF , one can check that for $0< T\\le 1$ , and $u,u_1,u_2\\in \\tilde{\\mathcal {S}}_T$ there exists constant $C_0$ such that ${\\mathcal {T}_{v}( u)}_{\\tilde{\\mathcal {S}}_T}\\le C_0\\Big ({ u_0}_{L^2}+T^{\\tfrac{3-p}{2}}{ u}_{\\tilde{\\mathcal {S}}_T}^{p-1}+T^{\\tfrac{1}{2}}{v}_{L^{\\infty }([0,T];L^2)}{ u}_{\\tilde{\\mathcal {S}}_T}\\Big ),$ and $&{\\mathcal {T}_{v}(u_1)-\\mathcal {T}_{v}(u_2)}_{\\tilde{\\mathcal {S}}_T}\\\\&\\le C_0 T^{\\tfrac{3-p}{2}}\\left({ u_1}_{\\tilde{\\mathcal {S}}_T}^{p-2}+{ u_2}_{\\tilde{\\mathcal {S}}_T}^{p-2}+T^{\\tfrac{p-2}{2}}{v}_{L^{\\infty }([0,\\infty );L^2)}\\right){ u_1- u_2}_{\\tilde{\\mathcal {S}}_T}.$ Therefore, for $R\\ge 2C_0{u_0}_{L^2}$ , if $T\\leqslant \\min \\Big \\lbrace 1, (4 C_0 R^{p-2}+1)^{\\frac{-2}{3-p}},\\big (16C_0^2{v}_{L^{\\infty }([0,\\infty );L^2)}^2+1\\big )^{-1}\\Big \\rbrace ,$ then ${\\mathcal {T}_{v}(u)}_{\\tilde{\\mathcal {S}}_T}&\\le R,\\qquad \\forall u\\in \\mathcal {B}_{R}(0);\\\\{\\mathcal {T}_{v}(u_1)-\\mathcal {T}(u_2)}_{\\tilde{\\mathcal {S}}_T}&\\le q {u_1-u_2}_{\\tilde{\\mathcal {S}}_T},\\qquad \\text{with}\\quad q<1.$ By the Banach contraction mapping theorem, there is a unique fixed point of $\\mathcal {T}_{v}$ in $\\mathcal {B}_{R}(0)$ , which is the unique mild solution to (REF ) with initial data $u_0$ .", "With the same assumptions in Theorem REF , if $T^$ is the maximal time of existence of the mild solution to (REF ), then $\\limsup _{t\\rightarrow T_-^}{u(t)}_{L^2(2)}=\\infty .$ Otherwise, $T^=\\infty $ .", "Let $u$ be the mild solution on $[0,T]$ given in Theorem REF .", "Then $u$ is also a weak solution of (REF ) on $[0,T]$ , and satisfies the energy identity for any $t\\in [0,T)$ : ${u(t)}_{L^2}^2 + 2\\int _0^t\\,{u(s)}_{L^2}^2\\,ds={u_0}_{L^2}^2 + 2\\int _0^t\\,{u(s)}_{L^p}^p\\, ds.$ The proof of Corollary REF and Proposition REF are similar to the proof of non-advective case, and hence we omit it.", "By (REF ), we can conclude that the length of $T$ is non-decreasing as ${u_0}_{L^2}$ decreases.", "In the proof of Proposition REF , we showed that when $2<p<3$ , $N(u)$ actually belongs to $L^1([0,T]\\times 2)$ , and this conclusion remains hold for the advective case.", "Therefore, in this regime we are allowed to rewrite the mild solution (REF ) into the following way: $u(t) &=& e^{-t\\mathcal {L}} u_0\\,-\\,\\int _0^t\\,e^{-(t-s)\\mathcal {L}}\\cdot \\left({u}^{p-2}u\\right)\\,ds \\\\&& \\quad \\quad \\quad \\quad \\quad -\\, \\int _0^t\\,e^{-(t-s)\\mathcal {L}}\\left(v\\cdot u\\right)\\,ds.$ Note that this alternative form plays a crucial role in the sequel (see, e.g., Lemma REF )." ], [ "Global existence with advecting flows with small dissipation time", "In the second part of this section, our goal is to show the existence of global mild solution in $\\tilde{\\mathcal {S}}_T$ space to equation (REF ) when $v\\in L^{\\infty }\\left([0,\\infty ), W^{1,\\infty }(d)\\right)$ and the dissipation time of it satisfies certain constraint (see condition (REF )).", "Let $\\bar{u}$ denotes the mean of the solution, then observe that equation (REF ) is mean conserved, and $u-\\bar{u}$ satisfies equation (REF ) as well.", "Therefore, without loss of generality, we may assume $u_0\\in L_0^2$ , where $L_0^2$ is the space of all mean-zero, square integrable functions on the torus.", "Our main result of this section is in the same spirit and can be regarded as certain counterparts to those in [15], in which the authors proved exponential decay of solutions to a large class of nonlinear second order parabolic equations.", "However, the assumptions required to apply the result in [15] do not apply to the nonlinear term $N(u)$ in (REF ), more precisely, $N(u)$ belongs to $L^1(d)$ instead of $L^2(d)$ , and it cannot be bounded in terms of ${u}_{L^2}$ and ${u}_{L^2}^2$ .", "To overcome this difficulty, we estimate the difference between the solution of (REF ) and (REF ) with $L^1$ initial data for short time intervals, and show that both the difference and the solution of (REF ) can be controlled by ${u}_{L^2}$ and ${u}_{L^2}^2$ .", "Finally, combining all these ingredients, we show that the $L^2$ norm of the solution to (REF ) will decay exponentially fast, if the dissipation time of $v$ satisfies specific constraint.", "Now we turn to the detail.", "Let $\\mathcal {S}_{s, s+t}$ be the solution operator for advective hyper diffusion equation, which is the linear part of (REF ): $\\partial _t\\phi + v\\cdot \\phi + \\Delta ^2\\phi = 0.$ That is, the function $\\phi (s+t)=S_{s, s+t}\\phi (s)$ solves (REF ) with initial data $\\phi (s)$ and periodic boundary condition.", "Now we are ready to introduce the term dissipation time.", "Heuristically, it is the smallest time that the system halves the energy ($L^2$ norm) of the solution to (REF ).", "The dissipation time associated to the solution operator $\\mathcal {S}_{s, s+t}$ is defined by $\\tau ^(v) \\inf [\\Big ]{ t \\ge 0 { \\mathcal {S}_{s, s+t} }_{L_0^2 \\rightarrow L_0^2 } \\le \\tfrac{1}{2} \\text{ for all $s\\ge 0$} }\\,.$ We show that the mild solution to (REF ) in space $\\tilde{\\mathcal {S}}_T$ can be extended over any finite time interval $[0,T]$ if we choose $v$ carefully enough with respect to the initial data.", "Indeed, Theorem REF indicates that as long as the $L^2$ norm remains finite, the solution can always be extended to a longer time interval.", "Therefore, it suffices for us to show when the dissipation time of $v$ is small enough, then the $L^2$ norm of the solution will not increase over the interval $[0,T]$ .", "We are ready to state the main theorem of this section.", "For $2<p<3$ , $v\\in L^{\\infty }\\left([0,\\infty ), W^{1,\\infty }(d)\\right)$ , and $\\mu >0$ , let $u$ be the mild solution of (REF ) with initial data $u(0)=u_0\\in L_0^2$ .", "Then there exists a threshold value $T_1=T_1({u_0}_{L^2}, \\mu , p)$ such that if $\\left({v}_{L^\\infty ([0,\\infty );L^{\\infty })}\\left(\\tau ^(v)\\right)^{\\frac{5}{4}}+\\left(\\tau ^(v)\\right)^{\\frac{3}{4}}\\right)\\leqslant T_1({u_0}_{L^2}, \\mu , p),$ then there exist a constant $\\beta >0$ , such that for any $t>0$ it holds ${u(t)}_{L^2}\\le \\beta e^{-\\mu t}{u_0}_{L^2}.$ The dependence of $T_1$ can be computed explicitly, as can be seen from equation (REF ).", "It indicates that $T_1$ is a decreasing function of ${u_0}_{L^2}$ , and it also implies $\\tau ^(v)\\leqslant \\min \\left\\lbrace \\frac{1}{10\\mu },T_0^2({u_0}_{L^2})\\right\\rbrace $ .", "Before proving the theorem, we discuss the existence of flow that satisfies the assumption (REF ).", "Let $v\\in L^{\\infty }([0,\\infty );C^2(d))$ , and define $v_{A}=Av(x,At)$ .", "If $v$ is weakly mixing flow with rate function $h$ , then $\\tau ^{}(v_A) \\xrightarrow{} 0.$ If further $v$ is strongly mixing with rate function $h$ , and $t^4 h(t) \\xrightarrow{} 0,$ then ${v_A}_{L^\\infty ([0,\\infty );L^{\\infty })}^{\\frac{4}{5}}\\tau ^(v_A) \\xrightarrow{} 0.$ (1).", "For the definition of weakly mixing flow and strongly mixing flow we refer reader to Definition 3.1 of [10].", "(2).", "The proof of Proposition REF is in the same spirit of the proof of Proposition 1.4 of [10], therefore we omit it here.", "(3).", "There are other families of flows that we can manipulate and gain arbitrary small dissipation time, one typical type of flows are so-called mixing flows, which are the flows that can produce small scale structures.", "To the best of our knowledge, the existence of smooth time-independent or even time-periodic, mixing flows remains an open question (see, for example [6]).", "Nevertheless, there are many examples of (spatially) smooth, time-dependent flows can be used or be expected to provide small dissipation times, for example the alternating horizontal/vertical sinusodial shear flows with randomized phases.", "One can check [10], [23], [20], [21] for more discussion.", "Now we turn to the proofs, we divide the proof of the main theorem into several lemmas.", "In the following lemma, we first show that the $L^2$ norm of $u$ grows continuously.", "When $2<p<3$ , there is a constant $A_p>0$ .", "For any $B>0$ define $T_0^2(B)=\\int _{B}^{2B}\\,\\frac{y}{A_p y^{2/(3-p)}}\\,dy.$ Let $u$ be a mild solution of (REF ) on $[0,T]$ , and $t_0\\in [0,T]$ .", "Then for any $t\\in [t_0,t_0+T_0^2({u(t_0)}_{L^2})] \\cap [0,T]$ , the following estimate holds: ${u(t)}_{L^2}\\le 2{u(t_0)}_{L^2}.$ Note that for $B>0$ and $2<p<3$ , $T_0^2(B)$ is a decreasing function.", "Since we can check that $\\frac{d}{dB}(T_0^2(B))=\\frac{1}{A_p} \\left(2^{\\frac{2(2-p)}{3-p}}-1 \\right)B^{\\tfrac{1-p}{3-p}}<0.$ In the next two lemmas, we show that if the time average of ${u}_{L^2}^2$ is large, then ${u}_{L^2}$ will decrease exponentially.", "Conversely, if the time average of ${u}_{L^2}^2$ is small, then the dissipation enhancement phenomenon will still cut down ${u}_{L^2}$ exponentially with a comparable rate.", "We start with the first case in the following lemma.", "When $2<p<3$ , let $\\mu >0$ and $u$ be a mild solution of (REF ) on $[0,T]$ with $t_0\\in [0,T]$ .", "If for some $\\tau \\in \\left(0, T_0^2({u(t_0)}_{L^2})\\right)$ , with $[t_0, t_0+\\tau ]\\subset [0,T]$ , and further $\\frac{1}{\\tau }\\int _{t_0}^{t_0+\\tau }\\,{u(t)}_{L^2}^2\\,dt\\ge 2A_p\\left(2{u(t_0)}_{L^2}\\right)^{\\tfrac{2}{3-p}}+2\\mu {u(t_0)}_{L^2}^2,$ where $A_p$ is the constant in Lemma REF , then ${u(t_0+\\tau )}_{L^2}\\le e^{-\\mu \\tau }{u(t_0)}_{L^2}.$ For the second case, if the time average of ${u}_{L^2}^2$ is small, we show that if $\\tau ^(v)$ is small enough, then ${u}_{L^2}^2$ will still decrease with a comparable rate.", "When $2<p<3$ , let $v\\in L^\\infty ([0,\\infty );L^{\\infty })$ , $\\mu >0$ , and $u$ be a mild solution of (REF ) on $[0,T]$ .", "Assume $t_0\\in [0,T]$ , then there exists a threshold value $T_1=\\min \\left\\lbrace \\frac{2}{5C\\left(A_p B^{\\frac{2(p-2)}{3-p}}+\\mu \\right)^{\\frac{p}{4}}B^{p-2}},\\left(\\frac{1}{10\\mu }\\right)^{\\frac{3}{4}}, \\left(T_0^2({u(t_0)}_{L^2})\\right)^{\\frac{3}{4}}\\right\\rbrace $ such that if $\\left({v}_{L^\\infty ([0,\\infty );L^{\\infty })}\\left(\\tau ^(v)\\right)^{\\frac{5}{4}}+\\left(\\tau ^(v)\\right)^{\\frac{3}{4}}\\right)\\leqslant T_1({u_0}_{L^2}, \\mu , p),$ and $t_0+\\tau ^\\leqslant T$ such that $\\frac{1}{\\tau ^}\\int _{t_0}^{t_0+\\tau ^}\\,{u(t)}_{L^2}^2\\,dt\\le 2A_p\\left(2{u(t_0)}_{L^2}\\right)^{\\tfrac{2}{3-p}}+2\\mu {u(t_0)}_{L^2}^2,$ where $A_p$ is the constant in Lemma REF .", "Then (REF ) still holds at time $\\tau =\\tau ^$ .", "Temporally assuming Lemma REF , Lemma REF and Lemma REF , we prove Theorem REF first.", "By Lemma REF and Lemma REF , estimate (REF ) always hold for $\\tau =2\\tau ^$ : ${u(\\tau ^)}_{L^2}\\le e^{-\\mu \\tau ^}{u_0}_{L^2}.$ Repeating this argument, we have ${u(n\\tau ^)}_{L^2}\\le e^{-\\mu n\\tau ^}{u_0}_{L^2}$ for any $n\\in \\mathbb {N}$ .", "For any $t>0$ , there exists $n\\in \\mathbb {N}$ such that $t\\in [n\\tau ^, (n+1)\\tau ^)$ .", "Since $t-n\\tau ^\\le \\tau ^\\le T_0^2({u_0}_{L^2})$ , further we have ${u(t)}_{L^2}\\le 2e^{-\\mu n\\tau ^}{u_0}_{L^2}\\le 2e^{-\\mu t+\\mu \\tau ^}{u_0}_{L^2}\\le \\beta e^{-\\mu t}{u_0}_{L^2},$ where $\\beta =2e^{\\frac{1}{10}}$ .", "Now, we remain to prove these lemmas.", "Let $B{u(t_0)}_{L^2}$ .", "In Proposition REF we showed that the mild solution is also a weak solution, and it satisfies the energy identity (REF ) on $[0, T]$ .", "Taking derivative with respect to $t$ on the both hand sides of identity (REF ), we get ${u(t)}_{L^2} \\cdot \\frac{d}{dt}{u(t)}_{L^2}=-{u(t)}_{L^2}^2 + {u(t)}_{L^p}^p.$ By Gagliardo-Nirenberg's inequality and Young's inequality with $\\epsilon $ , we further get ${u}_{L^2} \\cdot \\frac{d}{dt}{u}_{L^2}\\le -\\frac{1}{2}{u}_{L^2}^2 + A_p{u}_{L^2}^{\\tfrac{2}{3-p}}\\leqslant A_p{u}_{L^2}^{\\tfrac{2}{3-p}},$ where $A_p$ is a constant that only depends on $p$ .", "Thus, for a.e.", "$t\\in (0,T)$ we have $\\frac{d}{dt}\\int _{B}^{{u(t)}_{L^2}}\\,\\frac{y}{A_p y^{\\tfrac{2}{3-p}}}\\,dy=\\frac{{u}_{L^2}\\frac{d}{dt}{u}_{L^2}}{A_p{u}_{L^2}^{\\frac{2}{3-p}}}\\le 1,$ which further implies for all $t\\in [t_0,t_0+T_0^{2}(B)]$ , we have $\\int _{B}^{{u(t)}_{L^2}}\\,\\frac{y}{A_p y^{\\tfrac{2}{3-p}}}\\,dy\\le t-t_0\\le T_0^2(B)= \\int _{B}^{2B}\\,\\frac{y}{A_p y^{\\tfrac{2}{3-p}}}\\,dy.$ Since the integrands are strictly positive and identical, which forces ${u(t)}_{L^2}\\le 2B$ in $[t_0,t_0+T_0^{2}(B)]$ and the proof is complete.", "By Lemma REF , the $L^2$ norm of $u$ is uniformly bounded by $2{u(t_0)}_{L^2}$ on the interval $[t_0,t_0+\\tau ]$ .", "Integrating (REF ) from $t_0$ to $t_0+\\tau $ and apply the assumption (REF ), we get ${u(t_0+\\tau )}_{L^2}^2&\\le {u(t_0)}_{L^2}^2-\\int _{t_0}^{t_0+\\tau }\\,{u(t)}_{L^2}^2 dt + 2\\int _{t_0}^{t_0+\\tau }A_p{u(t)}_{L^2}^{\\tfrac{2}{3-p}}\\,dt\\\\&\\le {u(t_0)}_{L^2}^2-\\int _{t_0}^{t_0+\\tau }\\,{u(t)}_{L^2}^2\\,dt + 2\\tau A_p\\left(2{u(t_0)}_{L^2}\\right)^{\\tfrac{2}{3-p}}\\\\&\\le (1-2\\mu \\tau ){u(t_0)}_{L^2}^2\\le e^{-2\\mu \\tau }{u(t_0)}_{L^2}^2,$ Finally, we take square root on the both sides of the inequality to complete the proof.", "In order to simplify the proof of Lemma REF , we introduce and prove following lemma.", "For any $f\\in L^1$ , $v\\in L^\\infty ([0,\\infty );L^{\\infty })$ , consider: (1).", "Hyper-diffusion equation with initial data $f$ : $\\theta _t + ^2 \\theta = 0\\quad \\text{with}\\quad \\theta _0=f;$ (2).", "Advective hyper-diffusion equation with initial data $f$ : $\\phi _t + v\\cdot \\phi + ^2 \\phi = 0\\quad \\text{with}\\quad \\phi _0=f.$ Then ${e^{-t\\mathcal {L}}f-S_{0, t}f}_{L^2}={\\theta (t,\\cdot )-\\phi (t,\\cdot )}_{L^2}\\leqslant C t^{1/4}{v}_{L^\\infty ([0,\\infty );L^{\\infty })}{f}_{L^1}.$ Let $w=\\theta -\\phi $ , subtracting (REF ) with (REF ) yields $w_t+^2 w+v\\cdot \\nabla w=v\\cdot \\nabla \\theta \\quad \\text{with}\\quad w(0)=0.$ Multiply $w$ on the both to get following energy identity: $\\frac{d}{dt}{w(t)}_{L^2}^2+2{w(t)}_{L^2}^2=2\\int _{2} v\\cdot \\nabla \\theta \\cdot w dx$ Since $v$ is divergence free, we have $\\frac{d}{dt}{w(t)}_{L^2}^2+2{w(t)}_{L^2}^2&=2\\int _{2} \\theta v\\cdot \\nabla w dx\\\\&\\leqslant 2{v}_{L^\\infty ([0,\\infty );L^{\\infty })}{\\theta }_{L^2}{w}_{L^2}\\\\&\\leqslant C{v}_{L^\\infty ([0,\\infty );L^{\\infty })}^2{\\theta }_{L^2}^2+{w}_{L^2}^2,$ where we applied Poincare and Young's inequalities in the last step.", "Therefore, $\\frac{d}{dt}{w}_{L^2}^2\\leqslant C{v}_{L^\\infty ([0,\\infty );L^{\\infty })}^2{\\theta }_{L^2}^2.$ Integrating the above inequality from 0 to $t$ on the both sides gives ${w(t)}_{L^2}^2&\\leqslant C{v}_{L^\\infty ([0,\\infty );L^{\\infty })}^2\\int _0^t{e^{-s\\mathcal {L}}f}_{L^2}^2 ds\\\\&\\leqslant C{v}_{L^\\infty ([0,\\infty );L^{\\infty })}^2\\int _0^t{e^{-s\\mathcal {L}}}_{L^1\\rightarrow L^2}^2{f}_{L^1}^2 ds\\\\&\\leqslant C{v}_{L^\\infty ([0,\\infty );L^{\\infty })}^2{f}_{L^1}^2\\int _0^t s^{-\\frac{1}{2}}ds\\\\&\\leqslant Ct^{\\frac{1}{2}}{v}_{L^\\infty ([0,\\infty );L^{\\infty })}^2 {f}_{L^1}^2$ Finally, take square root on the both to complete the proof.", "Now we are ready to prove Lemma REF .", "For simplicity, let $B$ denote ${u(t_0)}_{L^2}$ .", "Since $u$ is a mild solution of $(\\ref {e:adv 4 order eqn})$ , and by Remark REF we can write $u(t_0+\\tau ^)$ in the following Duhamel's form: $u(t_0+\\tau ^)=\\mathcal {S}_{t_0, t_0+\\tau ^}u(t_0)\\,-\\,\\int _{t_0}^{t_0+\\tau ^}\\,\\mathcal {S}_{s, t_0+\\tau ^}N(u(s))\\,ds.$ Take the $L^2$ norm on the both hand sides to get: $&{u(t_0+\\tau ^)}_{L^2}\\\\& \\quad \\leqslant {\\mathcal {S}_{t_0, t_0+\\tau ^}u(t_0)}_{L^2}\\,+\\,\\int _{t_0}^{t_0+\\tau ^}\\,{\\mathcal {S}_{s, t_0+\\tau ^}N(u(s))}_{L^2}\\,ds\\\\&\\quad \\leqslant {\\mathcal {S}_{t_0, t_0+\\tau ^}u(t_0)}_{L^2}\\,+\\,\\int _{t_0}^{t_0+\\tau ^}\\, \\left\\Vert \\left(\\mathcal {S}_{s, t_0+\\tau ^}-e^{(t_0+\\tau ^-s)}\\right)N(u(s))\\right\\Vert _{L^2}\\,ds \\\\ &\\quad \\quad +\\int _{t_0}^{t_0+\\tau ^}\\,{e^{(t_0+\\tau ^-s)}N(u(s))}_{L^2}\\,ds\\\\& \\quad I_1+I_2+I_3.$ Estimate of $I_1$ .", "By the definition of dissipation time, for the term ${\\mathcal {S}_{t_0, t_0+\\tau ^}u(t_0)}_{L^2}$ we have: ${\\mathcal {S}_{t_0, t_0+\\tau ^}u(t_0)}_{L^2}\\leqslant \\frac{1}{2}{u(t_0)}_{L^2}=\\frac{B}{2}.$ Estimate of $I_2$ .", "Firstly, recall that for a.e.", "$s>0$ we have $N(u(s))\\in L^1(d)$ , and ${N(u)}_{L^1}\\leqslant C{u}_{L^{2(p-2)}}^{p-2}{u}_{L^2}\\leqslant C{u}_{L^2}^{\\frac{p}{2}}{u}_{L^2}^{\\frac{p-2}{2}}.$ Therefore, for the second term above we can apply Lemma REF to get: $&&\\int _{t_0}^{t_0+\\tau ^}\\, \\left\\Vert \\left(\\mathcal {S}_{s, t_0+\\tau ^}-e^{(t_0+\\tau ^-s)}\\right)N(u(s))\\right\\Vert _{L^2}\\,ds\\\\&& \\quad \\leqslant C{v}_{L^\\infty ([0,\\infty );L^{\\infty })}\\int _{t_0}^{t_0+\\tau ^}\\left(t_0+\\tau ^-s\\right)^{\\frac{1}{4}}{N(u)}_{L^1}ds\\\\&& \\quad \\leqslant C{v}_{L^\\infty ([0,\\infty );L^{\\infty })}\\int _{t_0}^{t_0+\\tau ^}\\left(t_0+\\tau ^-s\\right)^{\\frac{1}{4}}{u}_{L^2}^{\\frac{p}{2}}{ u}_{L^2}^{\\frac{p-2}{2}}ds\\\\&& \\quad \\leqslant C{v}_{L^\\infty ([0,\\infty );L^{\\infty })}\\left(2B\\right)^{\\frac{p-2}{2}}\\left(\\int _{t_0}^{t_0+\\tau ^}\\left(t_0+\\tau ^-s\\right)^{\\frac{1}{4-p}}ds\\right)^{\\frac{4-p}{4}}\\left(\\int _{t_0}^{t_0+\\tau ^}{u(s)}_{L^2}^2 ds\\right)^{\\frac{p}{4}}\\\\&& \\quad \\leqslant C{v}_{L^\\infty ([0,\\infty );L^{\\infty })}\\left(2B\\right)^{\\frac{p-2}{2}}(\\tau ^)^{\\frac{5-p}{4}}\\left(\\tau ^\\left(2A_p(2B)^{\\frac{2}{3-p}}+2\\mu B^2\\right)\\right)^{\\frac{p}{4}}\\\\&& \\quad \\leqslant C{v}_{L^\\infty ([0,\\infty );L^{\\infty })}(\\tau ^)^{\\frac{5}{4}}B^{p-1}\\left(A_p B^{\\frac{2(p-2)}{3-p}}+\\mu \\right)^{\\frac{p}{4}}$ Estimate of $I_3$ .", "Similarly, for the last term.", "By applying Lemma REF we have $&\\int _{t_0}^{t_0+\\tau ^}\\,{e^{(t_0+\\tau ^-s)}N(u(s))}_{L^2}\\,ds\\\\& \\quad \\leqslant \\int _{t_0}^{t_0+\\tau ^}\\,\\left(t_0+\\tau ^-s\\right)^{-\\frac{1}{4}}{N(u)}_{L^1}\\,ds\\\\& \\quad \\leqslant \\int _{t_0}^{t_0+\\tau ^}\\left(t_0+\\tau ^-s\\right)^{-\\frac{1}{4}}{u}_{L^2}^{\\frac{p}{2}}{ u}_{L^2}^{\\frac{p-2}{2}}ds\\\\& \\quad \\leqslant \\left(2B\\right)^{\\frac{p-2}{2}}\\left(\\int _{t_0}^{t_0+\\tau ^}\\left(t_0+\\tau ^-s\\right)^{-\\frac{1}{4-p}}ds\\right)^{\\frac{4-p}{4}}\\left(\\int _{t_0}^{t_0+\\tau ^}{u(s)}_{L^2}^2 ds\\right)^{\\frac{p}{4}}\\\\& \\quad \\leqslant (\\tau ^)^{\\frac{3}{4}}B^{p-1}\\left(A_p B^{\\frac{2(p-2)}{3-p}}+\\mu \\right)^{\\frac{p}{4}}$ Finally, combine the estimations above and the assumption (REF ), we obtain $&{u(t_0+\\tau ^)}_{L^2}\\\\& \\quad \\leqslant \\frac{B}{2}+ C\\left({v}_{L^\\infty ([0,\\infty );L^{\\infty })}(\\tau ^)^{\\frac{5}{4}}+(\\tau ^)^{\\frac{3}{4}}\\right)\\left(A_p B^{\\frac{2(p-2)}{3-p}}+\\mu \\right)^{\\frac{p}{4}}B^{p-1}\\\\& \\quad \\leqslant (1-\\mu \\tau ^)B\\leqslant e^{-\\mu \\tau ^}B.$ The proof is complete." ], [ "Finite time blow-up of the solutions", "The goal of this section is to study the blow-up behavior corresponding to equation (REF ), in particular, we prove parallel results of [14] on the 2-torus.", "Moreover, instead of using Picard iteration as in [14] we utilize Banach contraction mapping theorem to construct solutions, which makes the proof much easier.", "On the other hand, it is worth to mention that the proof in [14] makes use of the dilation structure of (REF ), which does not hold in our setting.", "Here is the plan of this section: in Section REF , we first recall one more semigroup estimate, which will play an essential role of our construction.", "Next, in Section REF , we construct mild solutions induced by an $W^{1,\\infty }$ initial data.", "In Section REF , we first show that the mild solutions constructed in Section REF possess better regularity, if the initial data belongs to $H^2\\cap W^{1,\\infty }$ .", "Then we provide non-trivial $H^2\\cap W^{1,\\infty }$ initial data such that the corresponding mild solution must blow up in a finite time.", "Finally, in Section REF , we show that in the previous example of blow-up, the $L^2$ norm of the solution must blow up, moreover, we characterize the blow-up behaviour by providing a quantitative blow-up rate of the $L^2$ norm." ], [ "Revisit of Fourier analysis on torus.", "A main ingredient in the proof of the main theorem of this section is to estimate the $L^r$ -$L^{\\infty }$ -norm of $e^{-t\\mathcal {L}}$ .", "Recall for any $f \\in L^p(2)$ , $p \\ge 2$ , $\\hat{f}=\\left\\lbrace \\hat{f}(k) \\right\\rbrace _{k \\in ^2} \\in \\ell ^2$ and for any $\\lbrace a_n\\rbrace _{n \\in ^2} \\in \\ell ^2$ and $x \\in 2$ , $\\mathcal {F}^{-1} \\left( \\left( a_n \\right)_{k \\in ^2} \\right)(x)=\\sum \\limits _{k \\in ^2} a_k e^{2\\pi i k \\cdot x} \\in L^2(2)$ .", "Let $1\\leqslant r\\leqslant \\infty $ and $j=\\lbrace 0,1,2,\\cdots \\rbrace $ , then for any $f\\in L^{r}(2)$ we have ${^{j}e^{-t\\mathcal {L}} f}_{L^{\\infty }}\\leqslant C_{j}t^{-\\tfrac{1}{2r}-\\tfrac{j}{4}}{f}_{L^{r}},$ for some $C_j>0$ .", "For any $x\\in 2$ , $\\left\| ^{j}e^{-t\\mathcal {L}} f(x)\\right\|=\\left\| \\mathcal {F}^{-1} \\widehat{\\left( ^{j}e^{-t\\mathcal {L}} f \\right)}(x) \\right\|=\\left\|\\sum _{\\mathbf {k}\\in ^2}\\left\|\\mathbf {k}\\right\|^{j}e^{-t\|\\mathbf {k}\|^4} \\hat{f}(k)e^{2\\pi i\\mathbf {k}\\cdot x}\\right\|$ Thus for $r=1$ , we have $\\left\| ^{j}e^{-t\\mathcal {L}} f(x)\\right\|\\ &\\le & {f}_{L^1}\\cdot \\sum _{\\mathbf {k}\\in ^2} \\left\|\\mathbf {k}\\right\|^je^{-t\|\\mathbf {k}\|^4}\\\\&\\le & {f}_{L^1}\\cdot \\int _{^2}\\left\|x\\right\|^{j} e^{-t\|x\|^4}dx\\\\&\\le & t^{-\\tfrac{1}{2}-\\tfrac{j}{4}}{f}_{L^1}.$ On the other hand, for $r=\\infty $ , $\\left\| ^{j}e^{-t\\mathcal {L}} f(x)\\right\|\\ &\\le & \\sup _{x\\in ^2}\\left\|x\\right\|^{j} e^{-\\tfrac{t\|x\|^4}{2}}\\left\|\\sum _{\\mathbf {k}\\in Z^2}\\left\|\\hat{f}(\\mathbf {k})\\right\|e^{-\\tfrac{t\|\\mathbf {k}\|^4}{2}}\\right\|\\\\&\\le & C t^{-\\tfrac{j}{4}}{f}_{L^{\\infty }}.$ Finally, for $1<r<\\infty $ , the desired estimate follows form real interpolation." ], [ "Mild solutions with $W^{1,\\infty }$ initial data.", "We now turn to the construction of the $L^2$ blow up example.", "In contrast to the mild solutions defined in Definition , we begin with constructing mild solutions induced by $W^{1,\\infty }$ initial data.", "We first update the definition of mild solution to the equation (REF ) with an $W^{1,\\infty }$ initial data.", "For $2<p<3$ , a function $ u\\in C((0,T]; W^{1,\\infty }(2)),0<T$ is called a mild solution of (REF ) on $(0,T]$ with initial data $ u_0\\in W^{1,\\infty }(2)$ , if for any $0 < t \\le T$ , $u(t)=\\mathcal {T}(u)(t)e^{-t\\mathcal {L}} u_0-\\int _0^t\\, e^{-(t-s)\\mathcal {L}}\\Big ({u}^{p-2}u\\Big )\\,ds$ holds pointwisely in time with values in $W^{1,\\infty }$ , where the integral is defined in the Böchner sense.", "Note that we have removed the endpoint 0 in the above definition for mild solutions.", "It turns out when the initial data $u_0 \\in W^{1, \\infty }(2)$ , this does not quite distinguish the mild solutions if we require $u \\in C\\left([0, T]; W^{1, \\infty }(2) \\right)$ (see, Theorem REF ); however, it will play a major role if the initial data merely belongs to $L^2(2)$ (see, Proposition REF ), which leads to the desired $L^2$ blow-up example (see, Theorem REF ).", "We begin with the following local existence result.", "Let $u_0 \\in W^{1,\\infty }(2)$ and $2<p<3$ .", "Then there exists $0< \\breve{T} \\le 1$ , depending only on $\\Vert u_0\\Vert _{W^{1,\\infty }}$ and $p$ such that (REF ) admits a mild solution $\\breve{u}$ on $(0, \\breve{T}]$ (in the sense of Definition REF ), which is unique in $C\\left([0,\\breve{T}]; W^{1,\\infty }(2)\\right)$ and hence in $C\\left((0,\\breve{T}]; W^{1,\\infty }(2)\\right)$ .", "The proof of Theorem REF is similar to the one of Theorem REF .", "We only sketch the proof here and would like to leave the details to the interested reader.", "First of all, we show that for any $0<T \\leqslant 1$ , the map $\\mathcal {T}$ is bounded on $C\\left([0, T]; W^{1,\\infty }(2)\\right)$ .", "More precisely, by using the semigroup estimate given in Lemma REF we have the following quantitative bound: there exists a constant $C_3>0$ , such that for any $0<T \\le 1$ and $u \\in C\\left([0,T]; W^{1,\\infty }(2)\\right)$ , $ \\left\\Vert \\mathcal {T}(u) \\right\\Vert _{C\\left([0, T]; W^{1,\\infty }\\right)} \\le C_3 \\left(\\left\\Vert u_0\\right\\Vert _{W^{1,\\infty }}+T^{\\tfrac{1}{2}} \\left\\Vert u \\right\\Vert _{C\\left([0, T]; W^{1,\\infty }\\right)}^{p-1} \\right),$ (this is the counterpart of Lemma REF ) which follows from the following estimates: $\\sup \\limits _{0<t \\le T} \\Vert \\nabla \\mathcal {T}(u) \\Vert _{L^{\\infty }} \\le C_3 \\left( \\Vert u_0\\Vert _{ W^{1,\\infty }}+T^{\\frac{1}{2}} \\Vert u\\Vert _{C\\left([0, T]; W^{1,\\infty }\\right)}^{p-1} \\right)$ ; $\\sup \\limits _{0<t \\le T} \\Vert \\mathcal {T}(u) \\Vert _{L^{\\infty }} \\le C_3 \\left( \\Vert u_0\\Vert _{ W^{1,\\infty }}+T^{\\frac{3}{4}} \\Vert u\\Vert _{C\\left([0, T]; W^{1,\\infty }\\right)}^{p-1} \\right)$ ; Next, we show that $\\mathcal {T}$ is a Lipschitz mapping on $C\\left([0, T]; W^{1,\\infty }(2)\\right)$ for any $0<T \\le 1$ , that is: there exists a constant $C_4>0$ , such that for any $0<T \\le 1$ , $&\\left\\Vert \\mathcal {T}(u_1)-\\mathcal {T}(u_2) \\right\\Vert _{C\\left([0, T]; W^{1,\\infty }\\right)} \\\\&\\le C_4 T^{\\frac{1}{2}} \\left( \\Vert u_1\\Vert _{C\\left([0, T]; W^{1,\\infty }\\right)}^{p-2}+\\Vert u_2\\Vert _{C\\left([0, T]; W^{1,\\infty }\\right)}^{p-2} \\right) \\left\\Vert u_1-u_2 \\right\\Vert _{C\\left([0, T]; W^{1,\\infty }\\right)} ,$ (this is the counterpart of Lemma REF ), which, similarly, can be achieved by showing the following estimates $\\sup \\limits _{0<t \\le T}\\left\\Vert \\nabla \\mathcal {T}(u_1)-\\nabla \\mathcal {T}(u_2) \\right\\Vert _{L^{\\infty }} \\\\\\le C_4 T^{\\frac{1}{2}} \\left( \\Vert u_1\\Vert _{C\\left([0, T]; W^{1,\\infty }\\right)}^{p-2}+\\Vert u_2\\Vert _{C\\left([0, T]; W^{1,\\infty }\\right)}^{p-2} \\right) \\left\\Vert u_1-u_2 \\right\\Vert _{C\\left([0, T]; W^{1,\\infty }\\right)} $ ; $\\sup \\limits _{0<t \\le T}\\left\\Vert \\mathcal {T}(u_1)-\\mathcal {T}(u_2) \\right\\Vert _{L^{\\infty }} \\\\\\le C_4 T^{\\frac{3}{4}} \\left( \\Vert u_1\\Vert _{C\\left([0, T]; W^{1,\\infty }\\right)}^{p-2}+\\Vert u_2\\Vert _{C\\left([0, T]; W^{1,\\infty }\\right)}^{p-2} \\right) \\left\\Vert u_1-u_2 \\right\\Vert _{C\\left([0, T]; W^{1,\\infty }\\right)} $ .", "The final step is to apply the Banach contraction mapping theorem in a ball $\\mathbb {B}_{\\breve{R}}(0)$ in $C\\left([0, T]; W^{1,\\infty }\\right)$ , where, we can take $\\breve{R} \\ge 2C_0^{\\prime }\\Vert u_0\\Vert _{W^{1,\\infty }}$ with $C_0^{\\prime }=\\max \\lbrace 1, C_3, C_4\\rbrace $ and $\\breve{T} \\le \\min \\left\\lbrace 1, \\left(2C_0^{\\prime }\\breve{R}^{p-2} \\right)^{-2} \\right\\rbrace $ .", "One may notice that in addition to the existence result, we also have the following estimate: $\\Vert u\\Vert _{C([0,T],W^{1,\\infty })}\\leqslant 2C_0^{\\prime }\\Vert u_0\\Vert _{W^{1,\\infty }},$ whenever $T\\in [0,\\breve{T}]$ .", "With the same assumptions in Theorem REF , if $\\breve{T}^$ is the maximal time of existence of the mild solution $\\breve{u}$ (in the sense of Definition REF ), then $\\limsup _{t\\rightarrow \\breve{T}_-^}{\\breve{u}(t)}_{W^{1,\\infty }(2)}=\\infty .$ Otherwise, $\\breve{T}^=\\infty $ .", "We omit the detail of the proof of Corollary REF since it follows from an easy modification of Corollary REF .", "More precisely, one can easily show that if $\\limsup \\limits _{t\\rightarrow T_-^}{\\breve{u}(t)}_{W^{1,\\infty }(2)}<\\infty $ , then the corresponding mild solution can be extended by surpassing $\\breve{T}^$ , which contradicts to the assumption that $\\breve{T}^$ is the maximal time of existence of the mild solution." ], [ "Main result.", "In the third part of this section, we first show that the mild solutions constructed in Section REF possess better regularity if the initial data further belongs to $H^2\\cap W^{1,\\infty }$ .", "Then our main theorem shows that there exists non-trivial initial data $u_0\\in H^2\\cap W^{1,\\infty }$ such that the corresponding mild solution must blow up in a finite time.", "This extra assumption that we need is that the energy of $u_0$ is negative, which we recall is given by $E(u_0)=\\frac{1}{2} \\left\\Vert u_0 \\right\\Vert _{L^2}^2-\\frac{1}{p} \\left\\Vert \\nabla u_0\\right\\Vert _{L^p}^p.$ We begin by showing that the mild solutions constructed in Section REF own better regularity if $u_0\\in H^2$ is provided.", "Let $T>0$ , and $u$ is a mild solution with respect to Definition REF on $(0,T]$ .", "If further $u_0\\in H^2$ , then in addition to Theorem REF , the mild solution has a better regularity: $u\\in L^{\\infty }([0, T];H^2),$ For any $0< t\\leqslant T$ we have $&&{u(t)}_{L^2} = {\\mathcal {T}u(t)}_{L^2}\\\\&& \\quad \\quad \\quad \\le \\left\\Vert e^{-t\\mathcal {L}}u_0 \\right\\Vert _{L^2}+\\int _0^t\\left\\Vert \\left(-\\right)^{\\tfrac{3}{2}}e^{-(t-s)\\mathcal {L}}\\left(\\vert u\\vert ^{p-2}u\\right)\\right\\Vert _{L^2}ds \\\\&& \\quad \\quad \\quad \\le {u_0}_{H^2}+\\int _0^t\\left(t-s\\right)^{-\\tfrac{p+1}{4}}{u}_{L^2}^{p-1}ds\\\\&& \\quad \\quad \\quad \\le {u_0}_{H^2}+ T^{\\tfrac{3-p}{4}}{u}_{C\\left([0, T]; W^{1,\\infty }\\right)}^{p-1}.$ This calculation together with Theorem REF and Remark REF gives ${u}_{L^{\\infty }([0, \\breve{T}];H^2)}\\leqslant {u_0}_{H^2}+ C\\breve{T}^{\\tfrac{3-p}{4}}{u_0}_{ W^{1,\\infty }}^{p-1}.$ Here $\\breve{T}$ is the same one defined on Theorem REF .", "We are ready to state the main result in this section.", "Let $2<p<3$ and $u_0\\in H^2(2)\\cap W^{1,\\infty }(2)$ with $E(u_0)<0$ , then the corresponding mild solution to (REF ) (namely, the mild solution defined in Definition REF ) must blow up in finite time.", "To prove this theorem, we need to consider the following approximated problem to (REF ): for $\\rho \\in (0,1)$ and $2<p<3$ , ${\\left\\lbrace \\begin{array}{ll}\\partial _t u^{\\rho } + (-)^2 u^{\\rho } = -\\cdot \\big (({u^{\\rho }}^{2}+\\rho )^{\\frac{p-2}{2}}u^{\\rho }\\big ); \\\\\\\\u^{\\rho }(0,x)= u_0(x) \\in H^2(2)\\cap W^{1,\\infty },\\end{array}\\right.", "}$ which can be regarded as the $L^2$ -gradient flow of the energy $E_\\rho $ : $E_{\\rho }(\\phi )\\frac{1}{2}\\int _{2}\\,(\\phi )^2\\,dx - \\frac{1}{p}\\int _{2}\\,\\left(({\\phi }^2+\\rho )^{\\frac{p}{2}}-\\rho ^{\\frac{p}{2}}\\right)\\,dx.$ We formulate a definition of the mild solution $u^{\\rho }$ to problem (REF ) in the same manner as in Definition REF .", "The existence and uniqueness of the solution is established in the following theorem.", "Let $2<p<3$ and $u_0\\in H^2(2)\\cap W^{1,\\infty }(2)$ with ${u_0}_{H^2}+{u_0}_{W^{1,\\infty }}<D$ .", "Then there exists $0< T\\le 1$ depending on $D$ such that for any $0<\\rho <1$ equation (REF ) admits a unique mild solution $u^{\\rho }$ on $[0,T]$ with the representation $u^{\\rho }(t)= e^{-t\\mathcal {L}}u_0\\,-\\,\\int _0^t\\, \\nabla e^{-(t-s)\\mathcal {L}}\\,\\left(({u^{\\rho }}^{2}+\\rho )^{\\frac{p-2}{2}}u^{\\rho }\\right)\\,ds,\\qquad 0\\le t\\le T.$ In particular, $ \\sup _{0<\\rho <1}{u^{\\rho }}_{L^{\\infty }([0,T];H^2)}\\le L_1 (D+D^{p-1}),$ for some $L_1>0$ that is independent on $D$ .", "The proof of Theorem REF follows from a proper modification of Theorem REF , Proposition REF and Remark REF , and hence we omit it here.", "Nevertheless, we would like to make a remark that the upper bound in the estimate (REF ) follows from the choice of $\\breve{T}$ in the Remark REF .", "We now divide the proof of Theorem REF into following lemmas.", "We start with arguing that there exists a short time interval that $u^{\\rho }$ converges to $u$ in $L^{\\infty }([0,T];H^2(2))$ as $\\rho \\rightarrow 0$ .", "Let $u$ and $u^{\\rho }$ be the corresponding mild solutions with $u_0\\in H^2(2)\\cap W^{1,\\infty }(2)$ of (REF ) and (REF ), respectively.", "If there is a $T_1>0$ such that $\\max \\left\\lbrace {u}_{L^{\\infty }\\left([0,T_1];H^2\\right)},\\sup _{0<\\rho <1}{u^{\\rho }}_{L^{\\infty }\\left([0,T_1];H^2\\right)}\\right\\rbrace <D$ for some $D>0$ , then there exists $T_2<T_1$ , which only depends on $D$ and $T_1$ , such that ${u-u^{\\rho }}_{L^{\\infty }([0,T_2];H^2)}\\rightarrow 0$ as $\\rho \\rightarrow 0$ .", "For simplicity, let $F_{\\rho }(\\xi )({\\xi }^{2}+\\rho )^{\\frac{p-2}{2}}\\xi $ .", "By Lemma REF and Lemma REF with $s=3$ , we have for any $0<t<T_1$ , $&& {u(t) - u^{\\rho }(t)}_{L^2} \\le \\left\\Vert \\int _0^t\\,\\left(-\\right)^{\\frac{3}{2}} e^{-(t-s)\\mathcal {L}}\\big (F_{\\rho }(u^{\\rho })-F(u^{\\rho })\\big )\\,ds \\right\\Vert _{L^2}\\\\&& \\quad \\quad \\quad \\quad \\quad \\quad +\\left\\Vert \\int _0^t\\,\\left(-\\right)^{\\frac{3}{2}} e^{-(t-s)\\mathcal {L}} \\big (F(u^{\\rho })-F(u)\\big )\\,ds \\right\\Vert _{L^2}\\\\&& \\quad \\le C\\int _0^t\\,(t-s)^{-\\frac{p+1}{4}}{F_{\\rho }(u^{\\rho })-F(u^{\\rho })}_{L^{\\frac{2}{p-1}}}\\,ds\\\\ && \\quad \\quad \\quad \\quad \\quad \\quad +C\\int _0^t\\,(t-s)^{-\\frac{p+1}{4}}{F(u^{\\rho })-F(u)}_{L^\\frac{2}{p-1}}\\,ds\\\\&& \\quad \\le C\\int _0^t\\,(t-s)^{-\\frac{p+1}{4}}\\left\\Vert \\rho ^{\\frac{p-2}{2}}{u^{\\rho }}\\right\\Vert _{L^\\frac{2}{p-1}}\\, ds\\\\ &&\\quad \\quad \\quad \\quad \\quad \\quad +C\\int _0^t\\,(t-s)^{-\\frac{p+1}{4}}\\left\\Vert ({u^{\\rho }}^{p-2}+{u}^{p-2}){u^{\\rho }-u}\\right\\Vert _{L^\\frac{2}{p-1}}\\,ds\\\\&& \\quad \\le C\\rho ^{\\frac{p-2}{2}}\\int _0^t\\,(t-s)^{-\\frac{p+1}{4}}{u^{\\rho }}_{L^2}\\, ds\\\\&& \\quad \\quad \\quad \\quad \\quad \\quad +C\\int _0^t\\,(t-s)^{-\\frac{p+1}{4}}\\big ({u^{\\rho }}_{L^2}^{p-2}+{u}_{L^2}^{p-2}\\big ){u^{\\rho }-u}_{L^2}\\,ds\\\\&&\\quad \\le C\\Big (\\rho ^{\\frac{p-2}{2}} Dt^{\\frac{3-p}{4}}+D^{p-2}t^{\\frac{3-p}{4}}{u^{\\rho }-u}_{L^\\infty ((0,T_1);H^2)}\\Big ).$ Similarly, we can show that for any $0<t<T_1$ , $\\left\\Vert \\nabla u(t)-\\nabla u^{\\rho }(t) \\right\\Vert _{L^2} \\le C \\left(\\rho ^{\\frac{p-2}{2}}Dt^{\\frac{4-p}{4}}+D^{p-2}t^{\\frac{4-p}{4}} \\Vert u^\\rho -u \\Vert _{L^\\infty \\left((0, T_1); H^2\\right)} \\right)$ and $\\left\\Vert u(t)- u^{\\rho }(t) \\right\\Vert _{L^2} \\le C \\left(\\rho ^{\\frac{p-2}{2}}Dt^{\\frac{5-p}{4}}+D^{p-2}t^{\\frac{5-p}{4}} \\Vert u^\\rho -u \\Vert _{L^\\infty \\left((0, T_1); H^2\\right)} \\right).$ Combining these estimates we have for any $0<t<T_1$ , ${u - u^{\\rho }}_{L^\\infty ((0, t);H^2)}\\le C (1+t^{\\frac{1}{2}}) \\Big (\\rho ^{\\frac{p-2}{2}} Dt^{\\frac{3-p}{4}}+D^{p-2} t^{\\frac{3-p}{4}} {u^{\\rho }-u}_{L^\\infty ((0,T_1);H^2)}\\Big ).$ Therefore, there exists some $T_2<T_1$ sufficiently small, which only depends on $T_1$ and $D$ , such that $C(1+T_2^{\\frac{1}{2}})D^{p-2} T_2^{\\frac{3-p}{4}}<\\frac{1}{2},$ which further gives ${u - u^{\\rho }}_{L^\\infty ((0,T_2);H^2)}< C(1+T_2^{\\frac{1}{2}}) \\rho ^{\\frac{p-2}{2}} D T_2^{\\frac{3-p}{4}}.$ This clearly suggests that for such choice of $T_2$ , one has ${u - u^{\\rho }}_{L^\\infty ((0,T_2);H^2)}\\rightarrow 0$ as $\\rho \\rightarrow 0$ .", "With the same assumptions in Lemma REF , if further we have $E(u_0)<0$ , then there exists some $0<T_3<1$ which only depends on $D$ , such that for any $0<t<T_3$ , the following energy estimates hold $E(u(t))\\le E(u_0)<0,$ $\\frac{E(u(t))}{{u(t)}_{L^2}^{p}} \\leqslant \\frac{ E(u_0)}{{u_0}_{L^2}^{p}},$ and $ \\frac{1}{{u(t)}_{L^2}^{p-2}}-\\frac{1}{{u_0}_{L^2}^{p-2}}\\le \\frac{p(p-2)E(u_0) t}{{u_0}_{L^2}^{p}}.$ We divide the proof of Lemma REF into two parts.", "Step 1: We start with considering the case when $u_0 \\in C^\\infty $ .", "More precisely, we show that if $u_0 \\in C^\\infty $ , then the three estimates (REF ), (REF ) and (REF ) hold.", "To begin with, we note that since $u_0 \\in C^\\infty $ , $u^{\\rho }$ is indeed a classical solution of (REF ) as long as $\\Vert u^{\\rho }\\Vert _{L^2}<\\infty $ .", "To see this, one may refer to, for example, [17] for a similar argument.", "Using the above argument, we multiply $\\partial _t u^{\\rho }$ on both sides of (REF ) and then apply integration by parts (see, also [14]) to obtain $0\\le \\int _s^t\\,\\int _{2}\\,(\\partial _t u^{\\rho })^2\\,dx\\,d\\tau \\,=\\,E_{\\rho }(u^{\\rho }(s))\\,-\\,E_{\\rho }(u^{\\rho }(t))$ for $0<s<t<T_1$ .", "Here $T_1$ is the same one as in Lemma REF .", "On the other hand, multiplying (REF ) by $u^{\\rho }$ and integrating by parts, we have $\\frac{d}{dt}{u^{\\rho }(t)}_{L^2}^2&=&-2\\int _{2}\\,u^{\\rho }\\Big ((-)^2 u^{\\rho }+\\cdot F_{\\rho }(u^{\\rho })\\Big )\\,dx \\nonumber \\\\&=& -2pE_{\\rho }(u^{\\rho }(t))+G_\\rho (u^{\\rho }(t))$ for $t\\in [0,T_1]$ , where $G_{\\rho }(\\phi ) (p-2){\\phi }_{L^2}^2-2\\rho \\int _{2}\\,\\Big (({\\phi }^2+\\rho )^{\\frac{p-2}{2}}-\\rho ^{\\frac{p-2}{2}}\\Big )\\,dx.$ Combining (REF ) and (REF ), we have ${u^{\\rho }(t)}_{L^2}^2\\frac{d}{dt}\\big (-E_{\\rho }(u^{\\rho }(t))\\big )&={u^{\\rho }(t)}_{L^2}^2{\\partial _t u^{\\rho }(t)}_{L^2}^2\\\\ &\\ge \\frac{1}{4}\\Big (\\frac{d}{dt}{u^{\\rho }(t)}_{L^2}^2\\Big )^2\\\\ &=\\frac{1}{4}\\Big (-2p E_{\\rho }(u^{\\rho }(t))+G_{\\rho }(u^{\\rho }(t))\\Big )\\frac{d}{dt}{u^{\\rho }(t)}_{L^2}^2,$ which further yields $&&\\frac{d}{dt}\\Big (-\\frac{E_{\\rho }(u^{\\rho }(t))}{{u^{\\rho }(t)}_{L^2}^p}\\Big )=\\frac{{u^{\\rho }}_{L^2}^2\\frac{d}{dt}\\left(-E_{\\rho }\\left(u^{\\rho }\\right)\\right)}{{u^{\\rho }}_{L^2}^{p+2}}+\\frac{\\frac{p}{2}E_{\\rho }\\left(u^{\\rho }\\right)\\frac{d}{dt}{u^{\\rho }}_{L^2}^2}{{u^{\\rho }}_{L^2}^{p+2}}\\\\&& \\quad \\quad \\quad \\ge \\frac{\\frac{1}{4} \\cdot \\Big (-2p E_{\\rho }(u^{\\rho }(t))+G_{\\rho }(u^{\\rho }(t))\\Big )\\frac{d}{dt}{u^{\\rho }(t)}_{L^2}^2}{{u^{\\rho }}_{L^2}^{p+2}}+\\frac{\\frac{p}{2} \\cdot E_{\\rho }\\left(u^{\\rho }\\right)\\frac{d}{dt}{u^{\\rho }}_{L^2}^2}{{u^{\\rho }}_{L^2}^{p+2}}\\\\&& \\quad \\quad \\quad = {\\frac{G_{\\rho }(u^{\\rho }(t))}{4{u^{\\rho }(t)}_{L^2}^{p+2}} \\cdot \\frac{d}{dt}{u^{\\rho }(t)}_{L^2}^2}\\\\ && \\quad \\quad \\quad =\\frac{G_{\\rho }(u^{\\rho }(t))}{4{u^{\\rho }(t)}_{L^2}^{p+2}} \\cdot \\Big (-2p E_{\\rho }(u^{\\rho }(t))+G_{\\rho }(u^{\\rho }(t))\\Big )$ for $t\\in [0,T_1]$ .", "Integrating both sides of above inequality from 0 to $\\tau $ along the time variable, where $\\tau \\in [0,T_1]$ , we get $&-\\frac{E_{\\rho }(u^{\\rho }(\\tau ))}{{u^{\\rho }(\\tau )}_{L^2}^p}+\\frac{E_{\\rho }(u_0)}{{u_0}_{L^2}^p} \\\\& \\quad \\quad \\quad \\quad \\ge \\frac{1}{4} \\cdot \\int _0^{\\tau }\\,\\frac{G_{\\rho }(u^{\\rho }(t))}{{u^{\\rho }(t)}_{L^2}^{p+2}} \\Big (-2p E_{\\rho }(u^{\\rho }(t))+G_{\\rho }(u^{\\rho }(t))\\Big )\\,dt.$ By (REF ), we have $-\\frac{1}{p(p-2)} \\cdot \\frac{d}{dt} \\left(\\frac{1}{{u^{\\rho }(t)}_{L^2}^{p-2}} \\right)&=\\frac{1}{2p {u^{\\rho }(t)}_{L^2}^{p}} \\cdot \\frac{d}{dt}{u^{\\rho }(t)}_{L^2}^2\\\\ &=\\frac{1}{2p} \\cdot \\frac{-2pE_{\\rho }(u^{\\rho }(t))+G_\\rho (u^{\\rho }(t))}{{u^{\\rho }(t)}_{L^2}^{p}}.$ Thus, $\\frac{1}{{u^{\\rho }(\\tau )}_{L^2}^{p-2}}-\\frac{1}{{u_0}_{L^2}^{p-2}}=-\\frac{p-2}{2} \\cdot \\int _0^{\\tau } \\frac{-2pE_{\\rho }(u^{\\rho }(t))+G_\\rho (u^{\\rho }(t))}{{u^{\\rho }(t)}_{L^2}^{p}}dt$ for $\\tau \\in [0,T_1]$ .", "Now we use Lemma REF to see that $ E_{\\rho }(u^{\\rho }(t))\\rightarrow E(u(t))\\qquad \\text{uniformly on}\\quad [0,T_2]\\quad \\text{as}\\quad \\rho \\rightarrow 0,$ and $ G_{\\rho }(u^{\\rho }(t))\\rightarrow (p-2){u(t)}_{L^2}^2\\qquad \\text{uniformly on}\\quad [0,T_2]\\quad \\text{as}\\quad \\rho \\rightarrow 0.$ Plugging (REF ) in (REF ), we have $E(u(t))\\le E(u(s))\\qquad \\text{for}\\quad 0\\le s\\le t\\le T_2,$ which in particular verifies (REF ).", "While if we apply (REF ) in (REF ), one can check that $&-\\frac{E(u(t))}{{u(t)}_{L^2}^{p}}+ \\frac{E(u_0)}{{u_0}_{L^2}^{p}} \\\\& \\quad \\quad \\ge \\frac{1}{4}\\int _0^t\\,\\frac{(p-2){u(t)}_{L^2}^2}{{u(t)}_{L^2}^{p+2}} \\cdot \\Big (-2pE(u(t))+(p-2){u(t)}_{L^2}^2\\Big )\\,ds$ for $t\\in [0,T_2]$ .", "Since $E(u_0)<0$ , by (REF ) we get $E(u(t))<0$ for $0\\le t\\le T_2$ .", "Using this fact in (REF ), we see the right hand side of (REF ) is non-negative and therefore $-\\frac{E(u(t))}{{u(t)}_{L^2}^{p}}+ \\frac{E(u_0)}{{u_0}_{L^2}^{p}} \\ge 0$ for $t\\in [0,T_2]$ , which is (REF ).", "Finally, we prove (REF ).", "Letting $\\rho \\rightarrow 0$ in (REF ), we have $\\frac{1}{{u(t)}_{L^2}^{p-2}}-\\frac{1}{{u_0}_{L^2}^{p-2}}=-\\frac{p-2}{2} \\cdot \\int _0^{t}\\,\\frac{-2p E(u(s))+(p-2){u(s)}_{L^2}^2}{{u(s)}_{L^2}^p}ds$ for $t\\in [0,T_2]$ .", "Combining (REF ) with (REF ) yields $\\frac{1}{{u(t)}_{L^2}^{p-2}}-\\frac{1}{{u_0}_{L^2}^{p-2}}\\le \\frac{p(p-2)E(u_0)t}{{u_0}_{L^2}^{p}}$ for $t\\in [0,T_2]$ .", "The proof of (REF ) is complete.", "Step II: We now prove the desired estimates (REF ), (REF ) and (REF ) under the assumption when $u_0 \\in H^2\\cap W^{1,\\infty }$ .", "Note that it suffices to prove the following: there exists some $0<T_3<1$ , which only depends on $D$ such that for any $0<\\varepsilon <1$ and $0<t<T_3$ , the following estimates hold $ E(u(t))<\\varepsilon E(u_0)<0,\\qquad \\mathrm {(\\ref {20210710eq01}^{\\prime })}$ $ \\frac{E(u(t))}{{u(t)}_{L^2}^{p}}<\\frac{ \\varepsilon E(u_0)}{{u_0}_{L^2}^{p}},\\qquad \\mathrm {(\\ref {20210710eq02}^{\\prime })}$ and $ \\frac{1}{{u(t)}_{L^2}^{p-2}}-\\frac{1}{{u_0}_{L^2}^{p-2}}<\\frac{\\varepsilon p(p-2)E(u_0) t}{{u_0}_{L^2}^{p}}.\\qquad \\mathrm {(\\ref {20210710eq03}^{\\prime })}$ Indeed, if we have already proved estimates (REF ), (REF ) and (REF ), then it suffices to let $\\varepsilon \\rightarrow 1$ and this then recovers the desired estimates that we need.", "Note that it is important for our analysis that the choice of $T_3$ is independent of the choice of $\\varepsilon $ .", "Let us pick a sequence of smooth function $\\left\\lbrace \\varphi _m\\right\\rbrace _{m=1}^\\infty \\subset C^\\infty (2)$ such that $ \\varphi _m \\rightarrow u_0 \\quad \\textrm {in} \\quad H^2\\cap W^{1,\\infty }.$ Note that for $2<p<3$ , the Gagliardo–Nirenberg's inequality suggests $\\left\\Vert \\nabla u\\right\\Vert _{L^q} \\le C \\left\\Vert u \\right\\Vert _{H^2},$ for any $1<q<\\infty $ , which together with (REF ) gives $E(\\varphi _m) \\rightarrow E(u_0) \\quad \\textrm {as} \\quad m \\rightarrow \\infty .$ Let $u_m$ be the mild solution to the equation (REF ) with initial data $\\varphi _m$ .", "By (REF ), we may assume $ \\sup _{1 \\le m<+\\infty } \\Vert \\varphi _m\\Vert _{H^2\\cap W^{1,\\infty }} \\le 2D.$ Therefore, by Proposition REF and Remark REF , there exists some $0<T^{\\prime }_1<1$ and $L>0$ (both parameters are independent of the choice of $m$ ), such that $ \\sup _{1 \\le m<+\\infty } \\Vert u_m\\Vert _{L^\\infty \\left([0, T^{\\prime }_1]; H^2 \\right)} \\le L\\left( D+D^{p-1}\\right).$ Without loss of generality, we may assume $T_1=T^{\\prime }_1$ , otherwise, we simply replace both numbers by $\\min \\lbrace T^{\\prime }_1, T_1\\rbrace $ , which only depends on $D$ .", "We have the following claim: there exists some $0<T_3 \\le T_1$ and some $C>0$ , such that for any $0<t<T_3$ and $m \\ge 1$ , $ \\left\\Vert u_m-u \\right\\Vert _{L^\\infty \\left([0, t]; H^2 \\right)} \\le C \\left\\Vert \\varphi _m-u_0 \\right\\Vert _{H^2},$ which further guarantees that $ u_m \\rightarrow u \\quad \\textrm {in} \\quad C \\left([0, T_3]; H^2 \\right).$ Proof of the claim: First of all, we note that for any $0<t<T_1$ , there exists some constant $C>0$ , such that $\\left\\Vert u_m(t)-u(t)\\right\\Vert _{L^2}&\\le C \\Vert \\varphi _m-u_0\\Vert _{H^2} \\\\& \\quad +Ct^{\\frac{3-p}{4}} \\Vert u_m-u\\Vert _{L^\\infty \\left([0, t]; H^2 \\right)} \\cdot (DL)^{p-2} \\cdot \\left(1+D^{p-2}\\right)^{p-2};$ $\\left\\Vert \\nabla u_m(t)-\\nabla u(t)\\right\\Vert _{L^2}&\\le C \\Vert \\varphi _m-u_0\\Vert _{H^2} \\\\&\\quad +Ct^{\\frac{4-p}{4}} \\Vert u_m-u\\Vert _{L^\\infty \\left([0, t]; H^2 \\right)} \\cdot (DL)^{p-2} \\cdot \\left(1+D^{p-2}\\right)^{p-2};$ $\\left\\Vert u_m(t)- u(t)\\right\\Vert _{L^2}&\\le C \\Vert \\varphi _m-u_0\\Vert _{H^2} \\\\& \\quad +Ct^{\\frac{5-p}{4}} \\Vert u_m-u\\Vert _{L^\\infty \\left([0, t]; H^2 \\right)} \\cdot (DL)^{p-2} \\cdot \\left(1+D^{p-2}\\right)^{p-2}.$ The proof for these estimates are similar to the proof of Lemma REF (see, also (REF )), and hence we would like to leave it to the interested reader.", "Now combining all these estimates, we see that there exists some $C>0$ , such that for any $0<t<T_1$ and $m \\ge 1$ , $ &&\\Vert u_m-u\\Vert _{L^\\infty \\left([0, t]; H^2 \\right)} \\le C \\left\\Vert \\varphi _m-u_0 \\right\\Vert _{H^2} \\nonumber \\\\&& \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad +C t^{\\frac{3-p}{4}} \\Vert u_m-u\\Vert _{L^\\infty \\left([0, t]; H^2 \\right)} \\cdot (DL)^{p-2} \\cdot \\left(1+D^{p-2}\\right)^{p-2}.$ This suggest that we can find some $T_3 \\in (0, T_1)$ , such that for any $0<t<T_3$ , $Ct^{\\frac{3-p}{4}} \\cdot (DL)^{p-2} \\cdot \\left(1+D^{p-2}\\right)^{p-2}<\\frac{1}{2},$ which together with (REF ), gives the desired claim (REF ).", "From now on, let us fix the choice of $T_3$ .", "Note that here we might assume $T_3 \\le T_2$ , otherwise, we just replace $T_3$ by $T_2$ .", "We show that with this particular choice of $T_3$ , the estimates (REF ), (REF ) and (REF ) hold.", "Here we only prove (REF ), while the proofs for the other two estimates are similar and hence we omit them here.", "Take and fix any $0<\\varepsilon <1$ and $0<t<T_3$ from now on.", "Our goal is to show that $ \\varepsilon E(u_0)-E(u(t))>0.$ We prove this via a standard approximation argument.", "We make a remark here that the choice of $m$ is allowed to be depended on $\\varepsilon , t, E(u_0)$ and $p$ in this case.", "We now turn to the detail.", "For any $m \\ge 1$ , we have $\\textrm {LHS of (\\ref {20210711eq23})}&=& E(u_0)-E(u(t))-(1-\\varepsilon ) E(u_0) \\\\&=& \\left(E(u_0)-E(\\varphi _m) \\right)- \\left(E(u(t))-E(u_m(t)) \\right) \\\\&& +\\left(E(\\varphi _m)-E(u_m(t)) \\right)+ \\left(\\varepsilon -1 \\right) E(u_0).$ Now by (REF ) and (REF ), we can find some $M=M(t, \\varepsilon ,E(u_0),p)$ sufficiently large, such that for any $m>M$ , $\\left\|E(u_0)-E(\\varphi _m) \\right\|+ \\left\|E(u(t))-E(u_m(t)) \\right\|<\\frac{(\\varepsilon -1)E(u_0)}{2}.$ This, together with the fact that $E(\\varphi _m)-E(u_m(t)) \\ge 0$ , suggest that when $m \\ge M$ , $\\textrm {LHS of (\\ref {20210711eq23})}>0.$ The proof is complete.", "We are ready to prove our main result in this section.", "Let $[0,T_{\\max })$ be the maximum time interval that (REF ) admits a mild solution (in the sense of Definition REF ) with $u_0\\in H^2\\cap W^{1,\\infty }$ and $E(u_0)<0$ .", "We prove $T_{\\max } \\le -\\frac{\\Vert u_0\\Vert _{L^2}^2}{p(p-2)E(u_0)}$ by showing contradiction.", "Assume (REF ) admits a mild solution on $[0,T_{\\max })$ with $T_{\\max }>-\\frac{\\Vert u_0\\Vert _{L^2}^2}{p(p-2)E(u_0)}$ .", "This in particular means $u\\in L^\\infty ([0,T];H^2(2)\\cap W^{1,\\infty }(2)),$ for any $T \\in \\left(-\\frac{\\Vert u_0\\Vert _{L^2}^2}{p(p-2)E(u_0)}, T_{\\max } \\right)$ by Proposition REF .", "Fix such a choice of $T$ , we denote $D \\Vert u\\Vert _{L^\\infty \\left([0, T]; H^2\\cap W^{1,\\infty } \\right)}$ .", "Then by Theorem REF and Theorem REF , there exists some $T_1$ which depends on $D$ , such that $\\max \\left\\lbrace {u}_{L^{\\infty }([0,T_1];H^2)},\\sup _{0\\le \\rho \\le 1}{u^{\\rho }}_{L^{\\infty }([0,T_1];H^2)}\\right\\rbrace \\le L(D+D^{p-1})$ for some $L>0$ independent of the choice of $D$ .", "Then by Lemma REF , there exits $T_3>0$ , which only depends on $L$ , $D$ , and $T_1$ , such that $\\frac{1}{{u(t)}_{L^2}^{p-2}}-\\frac{1}{{u_0}_{L^2}^{p-2}}\\le \\frac{p(p-2)E(u_0) t}{{u_0}_{L^2}^{p}}$ for $t\\in [0, T_3]$ .", "Next, by the choice of $D$ and (REF ), we have ${u(T_3)}_{H^2\\cap W^{1,\\infty }}\\leqslant D$ and $E(u(T_3))<0$ .", "This allows us to repeat the argument above by considering the initial data $u(T_3)$ to get $\\frac{1}{{u(t+T_3)}_{L^2}^{p-2}}-\\frac{1}{{u(T_3)}_{L^2}^{p-2}}\\le \\frac{p(p-2)E(u(T_3))t}{{u(T_3)}_{L^2}^{p}}$ for $t\\in [0,T_3]$ .", "Adding (REF ) and (REF ), and using (REF ), we get $\\frac{1}{{u(t)}_{L^2}^{p-2}}-\\frac{1}{{u_0}_{L^2}^{p-2}}\\le \\frac{p(p-2)E(u_0)t}{{u_0}_{L^2}^{p}}$ for $t\\in [0,2 T_3]$ .", "Repeating the above argument, we are able to get for any $t\\in [0,T]$ : $\\frac{1}{{u(t)}_{L^2}^{p-2}}-\\frac{1}{{u_0}_{L^2}^{p-2}}\\le \\frac{p(p-2)E(u_0)t}{{u_0}_{L^2}^{p}}$ equivalently, $\\frac{1}{{u(t)}_{L^2}^{p-2}}\\leqslant \\frac{{u(t)}_{L^2}^2+p(p-2)E(u_0)t}{{u_0}_{L^2}^{p}}.$ Note that since the left hand side of (REF ) is positive, thus the right hand side of (REF ) must be positive as well.", "Therefore, the above inequality can be further rewritten into: ${u(t)}_{L^2}\\ge \\Big (\\frac{{u_0}_{L^2}^{p}}{{u_0}_{L^2}^2+p(p-2)E(u_0)t}\\Big )^{\\frac{1}{p-2}}.$ Since $u$ is a mild solution on $[0,T]$ by assumption, Corollary REF yields $\\Big (\\frac{{u_0}_{L^2}^{p}}{{u_0}_{L^2}^2+p(p-2)E(u_0)t}\\Big )^{\\frac{1}{p-2}}<\\infty $ for any $t\\in [0,T]$ .", "This clearly gives a contradiction as $t\\rightarrow -\\frac{\\Vert u_0\\Vert _{L^2}^2}{p(p-2)E(u_0)}$ (since $T>-\\frac{\\Vert u_0\\Vert _{L^2}^2}{p(p-2)E(u_0)}$ ).", "The proof is complete." ], [ "Blow-up rate of $L^2$ norm", "In this section, we show that in our previous example (Theorem REF ), the $L^2$ -norm of the solution must blow up (see, Theorem REF ).", "This in particular gives an example which satisfies the assumption of Corollary REF .", "Moreover, we characterize the blow-up rate of the $L^2$ norm near the maximal time of existence.", "We begin with establishing the following local existence result.", "Let $2<p<3$ .", "There exists a $\\delta _{}>0$ sufficiently small, only depending on $p$ , such that if $\\varrho ^{\\tfrac{3-p}{2(p-2)}}{u_0}_{L^2}\\leqslant \\delta ,$ for some $\\delta \\in \\left(0,\\delta _{}\\right)$ and $\\varrho >0$ , then there is a mild solution $u$ (in the sense of Definition REF ) to the problem (REF ) in $2\\times \\left(0, 2\\varrho \\right]$ , such that ${u(t)}_{L^{\\infty }}\\leqslant Ct^{-\\tfrac{1}{2}}\\left(\\delta _{}\\varrho ^{-\\tfrac{3-p}{2(p-2)}}\\right)$ for some $C>0$ which only depends on $p$ .", "Before we prove the above theorem, we need to define a proper Banach space so that one can apply the Banach contraction mapping theorem.", "Let $\\tilde{Y}_{T}$ be the collection of all measurable functions on $2 \\times [0, T]$ with $\\sup \\limits _{0\\le t\\le T}t^{1/4}{u(t)}_{L^{\\infty }}<\\infty $ ; $\\sup \\limits _{0\\le t\\le T}t^{1/2}{u(t)}_{L^{\\infty }}<\\infty $ ; ${u(0)}_{L^2}<\\infty $ , and define: $\\left[u\\right]_{\\tilde{Y}_{T}}\\max \\left\\lbrace \\sup _{0\\le t\\le T}t^{1/4}{u(t)}_{L^{\\infty }}, \\sup _{0\\le t\\le T}t^{1/2}{u(t)}_{L^{\\infty }}\\right\\rbrace .$ Then $\\tilde{Y}_{T}$ is a Banach space equipping with the following norm: ${u}_{\\tilde{Y}_{T}}{u(0)}_{L^2}+ \\left[u\\right]_{\\tilde{Y}_{T}}.$ The proof of Proposition REF is similar to the one of Theorem REF and Theorem REF .", "Therefore, we only sketch the proof here and omit the details.", "First of all, we show that for any $T>0$ , the map $\\mathcal {T}$ is bounded on $\\tilde{Y}_{T}$ .", "More precisely, we have the following quantitative bound: there exists a constant $C_5>0$ , such that for any $T>0$ and $u \\in \\tilde{Y}_{T}$ , $\\left\\Vert \\mathcal {T}(u) \\right\\Vert _{\\tilde{Y}_{T}} \\le C_5 \\left(\\left\\Vert u_0\\right\\Vert _{L^2}+T^{\\tfrac{3-p}{2}} \\left\\Vert u \\right\\Vert _{\\tilde{Y}_{T}}^{p-1} \\right),$ which follows from the following estimates: $\\sup \\limits _{0<t \\le T} t^{\\tfrac{1}{4}}\\Vert \\mathcal {T}(u) \\Vert _{L^{\\infty }} \\le C_5 \\left( \\Vert u_0\\Vert _{L^2}+T^{\\frac{3-p}{2}} \\Vert u\\Vert _{\\tilde{Y}_{T}}^{p-1} \\right)$ ; $\\sup \\limits _{0<t \\le T} t^{\\tfrac{1}{2}}\\Vert \\mathcal {T}(u) \\Vert _{L^{\\infty }} \\le C_5 \\left( \\Vert u_0\\Vert _{L^2}+T^{\\frac{3-p}{2}} \\Vert u\\Vert _{\\tilde{Y}_{T}}^{p-1} \\right)$ ; Next, we show that $\\mathcal {T}$ is a Lipschitz mapping on $\\tilde{Y}_{T}$ for any $T>0$ , that is: there exists a constant $C_6>0$ , such that for any $T>0$ , $\\left\\Vert \\mathcal {T}(u_1)-\\mathcal {T}(u_2) \\right\\Vert _{\\tilde{Y}_{T}}\\le C_6 T^{\\frac{3-p}{2}} \\left( \\Vert u_1\\Vert _{\\tilde{Y}_{T}}^{p-2}+\\Vert u_2\\Vert _{\\tilde{Y}_{T}}^{p-2} \\right) \\left\\Vert u_1-u_2 \\right\\Vert _{\\tilde{Y}_{T}} ,$ which, similarly, can be achieved by showing the following estimates $\\sup \\limits _{0<t \\le T} t^{\\tfrac{1}{4}}\\left\\Vert \\mathcal {T}(u_1)-\\mathcal {T}(u_2) \\right\\Vert _{L^{\\infty }}\\le C_6 T^{\\frac{3-p}{2}} \\left( \\Vert u_1\\Vert _{\\tilde{Y}_{T}}^{p-2}+\\Vert u_2\\Vert _{\\tilde{Y}_{T}}^{p-2} \\right) \\left\\Vert u_1-u_2 \\right\\Vert _{\\tilde{Y}_{T}} $ ; $\\sup \\limits _{0<t \\le T} t^{\\tfrac{1}{2}}\\left\\Vert \\mathcal {T}(u_1)-\\nabla \\mathcal {T}(u_2) \\right\\Vert _{L^{\\infty }}\\le C_6 T^{\\frac{3-p}{2}} \\left( \\Vert u_1\\Vert _{\\tilde{Y}_{T}}^{p-2}+\\Vert u_2\\Vert _{\\tilde{Y}_{T}}^{p-2} \\right) \\left\\Vert u_1-u_2 \\right\\Vert _{\\tilde{Y}_{T}} $ .", "Finally, we apply the Banach contraction mapping theorem in a ball $\\mathbb {B}_{R}(0)$ in $\\tilde{Y}_{\\tilde{T}}$ , where, we can take $R \\ge 2C_0\\Vert u_0\\Vert _{L^2}$ with $C_0=\\max \\lbrace 1, C_5, C_6\\rbrace $ and $\\tilde{T} \\le \\left(4C_0 R^{p-2}\\right)^{-\\tfrac{2}{3-p}}$ .", "By assumption (REF ), we may take $R=2C_0\\delta \\varrho ^{-\\tfrac{3-p}{2(p-2)}}$ for instance, and this means $\\tilde{T}\\leqslant \\left(4C_0\\right)^{-\\tfrac{2}{3-p}}\\cdot \\left(2C_0\\delta \\right)^{-\\tfrac{2(p-2)}{3-p}}\\cdot \\varrho .$ Now we take $\\delta _{}$ small enough, such that $\\left(4C_0\\right)^{-\\tfrac{2}{3-p}}\\cdot \\left(2C_0\\delta \\right)^{-\\tfrac{2(p-2)}{3-p}}>2$ since $2<p<3$ .", "This allows us to take $\\tilde{T}=2\\varrho $ .", "Finally, the estimate (REF ) follows from the choice of $\\tilde{T}$ , $R$ and ${u_0}_{L^2}$ .", "Finally, we show that the $L^2$ -norm of the solution constructed in Theorem REF must blow up.", "Under the same assumption of Theorem REF , and let $T_{\\max }$ be the maximal time of existence of the mild solution to (REF ), then the $L^2$ norm of the solution must blow up at $T_{\\max }$ .", "Moreover, we have following quantitative blow-up rate: $\\liminf _{t\\nearrow T_{\\max }}\\left(T_{\\max }-t\\right)^{\\tfrac{3-p}{2(p-2)}}{u(t)}_{L^2}>0.$ It suffices to show (REF ), and we prove it by showing contradiction.", "Assume (REF ) does not hold.", "Then there exists $C_{}>0$ small enough, such that ${u(t_{})}_{L^2}\\le C_{}\\left(T_{\\max }-t_{}\\right)^{-\\tfrac{3-p}{2(p-2)}}$ for some $t_{} \\in (0, T_{\\max })$ .", "Here we let $C_{}\\leqslant \\frac{\\delta _{}}{2}$ .", "Now let $\\tilde{u}$ be the solution to (REF ) with initial data $u(t_{})\\in H^2\\cap W^{1,\\infty }$ (this is guaranteed by Theorem REF and Proposition REF ).", "By Theorem REF , we see that the maximal existence time $\\tilde{T}_{\\max }$ of $\\tilde{u}$ satisfies $\\tilde{T}_{\\max } \\le T_{\\max }-t_{}$ .", "However, Theorem REF indicates that $\\tilde{T}_{\\max } \\ge 2\\left(T_{\\max }-t_{}\\right)>T_{\\max }-t_{}.$ This is a contradiction." ] ]	2011.14088
[ [ "Brownian motion can feel the shape of a drum" ], [ "Abstract We study the scenery reconstruction problem on the $d$-dimensional torus, proving that a criterion on Fourier coefficients obtained by Matzinger and Lember (2006) for discrete cycles applies also in continuous spaces.", "In particular, with the right drift, Brownian motion can be used to reconstruct any scenery.", "To this end, we prove an injectivity property of an infinite Vandermonde matrix." ], [ "Background", "In its most general formulation, the scenery reconstruction problem asks the following: Let $C$ be a set, let be $f$ a function on $C$ , and $\\left(X_{t}\\right)_{t\\ge 0}$ a stochastic process taking values in $C$ .", "What information can we learn about $f$ from the (infinite) trace $f\\left(X_{t}\\right)_{t\\ge 0}$ ?", "Can $f$ be completely reconstructed from this trace?", "In one of the most common settings, $C$ is taken to be the discrete integer graph $\\mathbb {Z}$ , the function $f$ maps $C$ to $\\left\\lbrace 0,1\\right\\rbrace $ , and $X_{t}$ is a discrete-time random walk.", "For this model, numerous results exist in the literature for a variety of cases, e.g reconstruction when $f$ is random [1] and when $f$ is periodic.", "In the latter case, $f$ is essentially defined on a cycle of length $\\ell $ .", "Matzinger and Lember showed the following: Theorem 1 ([5]) Let $f$ be a 2-coloring of the cycle of length $\\ell $ , and let $X_{t}$ be a random walk with step distribution $\\gamma \\left(x\\right)$ .", "If the Fourier coefficients $\\left\\lbrace \\hat{\\gamma }\\left(k\\right)\\right\\rbrace _{k=0}^{\\ell -1}$ are all distinct, then $f$ can be reconstructed from the trace $f\\left(X_{t}\\right)$ .", "Finucane, Tamuz and Yaari [3] considered the problem for finite Abelian groups, and showed that in many such cases, the above condition on the Fourier coefficients is necessary.", "The problem can also be posed for a continuous space $C$ , such as $\\mathbb {R}^{d}$ or the torus $\\mathbb {T}^{d}$ , with $X_{t}$ a continuous-time stochastic process.", "Here there has been considerably less work; to the best of our knowledge, at the time of writing this paper there are only two published results: Detecting “bells” [7] and reconstructing iterated Brownian motion [2].", "See [4], [6] and references therein for an overview of the reconstruction problem, with a focus on $\\mathbb {Z}$ and $\\mathbb {Z}^{d}$ ." ], [ "Results", "In this paper, we extend Theorem REF from the discrete cycle to the continuous $d$ -dimensional torus $\\mathbb {T}^{d}=\\left(\\mathbb {R}/2\\pi \\mathbb {Z}\\right)^{d}$ .", "The discrete-time random walks are replaced by continuous-time processes $X_{t}$ (such as Brownian motion), and the 2-colorings are replaced by the indicators $f$ of open sets.", "For an example of how the sample paths $f\\left(X_{t}\\right)$ might look like, see Figure REF , where $f$ is the indicator of a union of three intervals on the circle, and $X_{t}$ is Brownian motion.", "The goal is to reconstruct the size and position of the intervals, up to rotations, from the trace $f\\left(X_{t}\\right)$ .", "Figure: Left: Scenery reconstruction in one dimension.", "The blackcurve is a polar depiction of one-dimensional Brownian motion, withθ=X t \\theta =X_{t} and r=tr=\\sqrt{t} (so that points near the centerrepresent times close 0, and points near the edge represent largertimes).", "The function ff is represented by the three shaded sectors.Right: The trace fX t f\\left(X_{t}\\right).", "It is equal to 1precisely when the curve on the left is inside one of the shaded sectors.Definition 2 Let $\\mathcal {F}$ be a family of functions on the torus.", "The family $\\mathcal {F}$ is said to be reconstructible by $X_{t}$ if there is a function $A:\\mathbb {R}^{\\mathbb {R}^{+}}\\rightarrow \\mathbb {R}^{\\mathbb {T}^{d}}$ such that for every $f\\in \\mathcal {F}$ , with probability 1 there exists a (random) shift $\\theta \\in \\mathbb {T}^{d}$ such that $A\\left(f\\left(X_{t}\\right)\\right)\\left(x\\right)=f\\left(x+\\theta \\right)$ for almost all $x$ .", "For the particular case of $d=1$ , we also deal with reconstruction up to reflections: Definition 3 The family $\\mathcal {F}$ is said to be reconstructible up to reflections by $X_{t}$ , if for all $f\\in \\mathcal {F}$ , with probability 1 either $A\\left(f\\left(X_{t}\\right)\\right)\\left(x\\right)=f\\left(x+\\theta \\right)$ for almost all $x$ , or $A\\left(f\\left(X_{t}\\right)\\right)\\left(x\\right)=f\\left(-x+\\theta \\right)$ for almost all $x$ .", "In order to analyze the trace $f\\left(X_{t}\\right)$ , we must of course have some control over the behavior of $X_{t}$ .", "In this paper, we assume that $X_{t}$ is an infinitely-divisible process with independent increments (this is the natural analog of a discrete-time random walk with independent steps).", "That is, there is a time-dependent distribution $D_{t}$ on $\\mathbb {T}^{d}$ such that $X_{t_{2}}-X_{t_{1}}\\sim D_{t_{2}-t_{1}}$ for every $t_{2}\\ge t_{1}$ ; For all $0\\le t_{1}\\le \\ldots \\le t_{n}$ , the increments $X_{t_{2}}-X_{t_{1}},\\ldots ,X_{t_{n}}-X_{t_{n-1}}$ are independent; $D_{t+s}=D_{t}\\star D_{s}$ for every $s,t\\ge 0$ , where $\\star $ is the convolution operator.", "We will also assume that $D_{t}$ is either continuous, or that it is a mixture of an atom at 0 and a continuous distribution.", "In other words, writing $D_{t}$ as a function of $x$ for simplicity, we have $D_{t}\\left(x\\right)=\\beta _{t}\\delta \\left(x\\right)+\\left(1-\\beta _{t}\\right)\\gamma _{t}\\left(x\\right),$ where $\\delta \\left(x\\right)$ is the Dirac $\\delta $ -distribution, $\\gamma _{t}$ is a probability density function, and $\\beta _{t}\\in \\left[0,1\\right]$ is a time dependent factor.", "We will also assume that $\\gamma _{t}$ is not too wild: $\\gamma _{t}\\in \\mathrm {L}^{2}\\left(\\mathbb {T}^{d}\\right)$ for all $t>0$ .", "Remark 4 This class of distributions includes Brownian motion, and any Poisson process whose steps have an $\\mathrm {L}^{2}$ probability density function.", "It also contains the sum of Brownian motion and any arbitrary independent Poisson process, since the diffusion smooths out any irregularities in the jumps.", "It does not, however, contain general jump processes with atoms, even if the atoms are dense in $\\mathbb {T}^{d}$ (e.g a Poisson process on $\\mathbb {T}$ which jumps by a step size $\\alpha $ rationally independent from $\\pi $ ).", "The functions we reconstruct will be the indicators of open sets, whose boundary has 0 measure in $\\mathbb {R}^{d}$ .", "Let $\\mathcal {F}_{d}=\\left\\lbrace \\mathbf {1}_{x\\in \\Omega }\\left(x\\right)\\mid \\Omega \\subseteq \\mathbb {T}^{d}\\text{ is open, }\\mathrm {Lebesgue}_{d}\\left(\\partial \\Omega \\right)=0\\right\\rbrace .$ Our main result is as follows: Theorem 5 (General reconstruction) Let $X_{t}$ be a stochastic process on $\\mathbb {T}^{d}$ as above.", "If there exists a time $t_{0}$ such that the Fourier coefficients $\\left\\lbrace \\hat{\\gamma }_{t_{0}}\\left(k\\right)\\right\\rbrace _{k\\in \\mathbb {Z}^{d}}$ are all distinct and nonzero, then $\\mathcal {F}_{d}$ is reconstructible by $X_{t}$ .", "In one dimension, we show that symmetric distributions can reconstruct $\\mathcal {F}_{1}$ up to reflections: Theorem 6 (Symmetric reconstruction) Let $X_{t}$ be a stochastic process on $\\mathbb {T}$ as above, and suppose that $\\gamma _{t}$ is symmetric, i.e $\\gamma _{t}\\left(y\\right)=\\gamma _{t}\\left(-y\\right)$ for all $y$ .", "If there exists a time $t_{0}$ such that the positive-indexed Fourier coefficients $\\left\\lbrace \\hat{\\gamma }_{t_{0}}\\left(k\\right)\\right\\rbrace _{k\\ge 0}$ are all distinct and nonzero, then $\\mathcal {F}_{1}$ is reconstructible up to reflections by $X_{t}$ .", "One corollary of Theorem REF , is that with the right drift, Brownian motion can be used to reconstruct $\\mathcal {F}_{d}$ .", "Corollary 7 (Brownian motion can feel the shape of a drum) Let $X_{t}$ be Brownian motion with drift $v\\in \\mathbb {R}^{d}$ , such that $\\left\\lbrace v_{1},\\ldots ,v_{d}\\right\\rbrace $ are rationally independent.", "Then $\\mathcal {F}_{d}$ is reconstructible by $X_{t}$ .", "Remark 8 The condition on the drift $v$ is natural: If the components of $v$ are rationally independent, then the geodesic flow defined by $v$ is dense in $\\mathbb {T}^{d}$ .", "Reconstructing a set $\\Omega $ from this geodesic is immediate.", "In this sense, Corollary REF states that reconstruction is possible also in the presence of noise which pushes us out of the trajectory.", "See Section for a question on a related model.", "The starting point for our results is a relation, introduced in [5], between two types of $n$ -point correlations related to $f$ - one known, and one unknown.", "After inverting the relation, the latter correlation can be used to reconstruct the function $f$ .", "In the discrete case, the relation is readily inverted using a finite Vandermonde matrix.", "In the continuous setting, additional difficulties arise due to both the more complicated nature of the distribution $D_{t}$ , which mixes together different correlations, and the fact that the function space on $\\mathbb {T}^{d}$ is infinite-dimensional.", "To address the latter issue, we prove an injectivity result for infinite Vandermonde matrices, which may be of independent interest.", "Lemma 9 (Infinite Vandermonde) Let $p,q\\in \\left[1,\\infty \\right]$ be such that $\\frac{1}{p}+\\frac{1}{q}=1$ .", "Let $\\left(z_{n}\\right)_{n}\\in \\ell ^{p}\\left(\\mathbb {C}\\right)$ be a sequence of distinct complex numbers such that $z_{n}\\rightarrow 0$ , and $z_{n}\\ne 0$ for all $n$ .", "Let $V$ be the infinite Vandermonde matrix with $z_{n}$ as generators, i.e $V_{ij}=z_{j}^{i}.$ If $x\\in \\ell ^{q}\\left(\\mathbb {C}\\right)$ is a zero of the infinite system of equations $Vx=0,$ then $x=0$ .", "Remark 10 The matrix equation $Vx=0$ means that for every index $i\\in \\mathbb {N}$ we have $0=\\sum _{j=1}^{\\infty }V_{ij}x_{j}=\\sum _{j=1}^{\\infty }z_{j}^{i}x_{j}.$ By HÃ¶lder's inequality, since $z\\in \\ell ^{p}\\left(\\mathbb {C}\\right)$ and $x\\in \\ell ^{q}\\left(\\mathbb {C}\\right)$ , the series $\\left(z_{j}^{i}x_{j}\\right)_{j}$ is absolutely convergent, and the left hand side of (REF ) is well defined.", "Basic preliminaries and the proof of Theorem REF are given in the next section.", "Section gives the outline of Theorem , relying on the same techniques described in Section .", "Brownian motion is discussed in Section , and Lemma REF is proved in Section .", "We conclude the paper with some open questions." ], [ "Acknowledgments", "The author thanks Itai Benjamini, Ronen Eldan, Shay Sadovsky, Ori Sberlo and Ofer Zeitouni for their stimulating discussions and suggestions." ], [ "Notation and simple properties of $D_{t}$", "We write $d$ -dimensional vectors in standard italics, e.g $k\\in \\mathbb {Z}^{d}$ .", "Tuples of $n$ vectors are written in boldface, e.g $k\\in \\mathbb {Z}^{nd}$ , with $k=\\left(k_{1},\\ldots ,k_{n}\\right)$ and $k_{i}\\in \\mathbb {Z}^{d}$ .", "The Fourier series of a function $g:\\mathbb {T}^{d}\\rightarrow \\mathbb {C}$ is a function $\\hat{g}:\\mathbb {Z}^{d}\\rightarrow \\mathbb {C}$ , given by $\\hat{g}\\left(k\\right)=\\int _{\\mathbb {T}^{d}}g\\left(x\\right)e^{-ik\\cdot x}dx.$ Note that we do not divide by the customary $1/\\left(2\\pi \\right)^{d}$ .", "This simplifies the statement of the convolution theorem: In this setting, we have $\\widehat{f\\star g}=\\hat{f}\\cdot \\hat{g},$ without any leading factor in the right hand side.", "This definition also extends to the Dirac $\\delta $ -distribution, even though it is not a function, so that for all $k\\in \\mathbb {Z}^{d}$ , $\\hat{\\delta }\\left(k\\right)=\\int _{-\\pi }^{\\pi }\\delta \\left(x\\right)e^{-ik\\cdot x}dx=1.$ Recall that $D_{t}=\\beta _{t}\\delta +\\left(1-\\beta _{t}\\right)\\gamma _{t}$ .", "Using the fact that $D_{s+t}=D_{s}\\star D_{t}$ , a short calculation shows that if the parameter $\\beta _{t}$ is not identically 0, it must decay exponentially: $\\beta _{t}=e^{-ct}$ for some constant $c$ .", "We implicitly assume that $\\beta _{t}$ is not identically 1, as in this case $X_{t}$ does not move and is uninteresting.", "We thus have that $\\beta _{\\alpha t}=\\beta _{t}^{\\alpha }\\,\\,\\,\\,\\,\\,\\,\\forall \\alpha ,t>0.$ Since $D_{t}$ has a probability density function $\\gamma _{t}$ , we have that $D_{t}\\rightarrow U\\left(\\mathbb {T}^{d}\\right)$ in distribution as $t\\rightarrow \\infty $ , i.e $X_{t}$ converges to the uniform distribution no matter its starting point.", "The distribution $D_{t}$ has a Fourier representation $\\hat{D}_{t}$ , given by $\\hat{D}_{t}\\left(k\\right)=\\beta _{t}\\hat{\\delta }\\left(k\\right)+\\left(1-\\beta _{t}\\right)\\hat{\\gamma }_{t}\\left(k\\right)=\\beta _{t}+\\left(1-\\beta _{t}\\right)\\hat{\\gamma }_{t}\\left(k\\right)\\,\\,\\,\\,\\,k\\in \\mathbb {Z}^{d}.$ By the convolution theorem, $\\hat{D}_{t+s}=\\widehat{D_{t}\\star D_{s}}=\\hat{D}_{t}\\hat{D}_{s}$ .", "From this it follows that for any $\\alpha ,t>0$ , we have $\\hat{D}_{\\alpha t}=\\left(\\hat{D}_{t}\\right)^{\\alpha }.$ Plugging this into (REF ) gives $\\beta _{\\alpha t}+\\left(1-\\beta _{\\alpha t}\\right)\\hat{\\gamma }_{\\alpha t}\\left(k\\right)=\\left(\\beta _{t}+\\left(1-\\beta _{t}\\right)\\hat{\\gamma }_{t}\\left(k\\right)\\right)^{\\alpha },$ and so by (REF ), $\\hat{\\gamma }_{\\alpha t}=\\frac{\\left(\\beta _{t}+\\left(1-\\beta _{t}\\right)\\hat{\\gamma }_{t}\\right)^{\\alpha }-\\beta _{t}^{\\alpha }}{1-\\beta _{t}^{\\alpha }}.$" ], [ "Proof of Theorem ", "The proofs of Theorems REF and REF use the relation between the spatial correlation and temporal correlation introduced in [5].", "Let $f$ be the indicator of an open set $\\Omega $ .", "For every integer $n\\ge 0$ , define the $n$ -th spatial correlation of $f$ , denoted $S_{n}:\\mathbb {T}^{nd}\\rightarrow \\mathbb {R}$ , as $S_{n}\\left(y\\right) & =\\frac{1}{\\left(2\\pi \\right)^{d}}\\int _{\\mathbb {T}^{d}}f\\left(x\\right)f\\left(x+y_{1}\\right)\\cdots f\\left(x+\\sum _{i=1}^{n}y_{i}\\right)dx\\\\& =\\frac{1}{\\left(2\\pi \\right)^{d}}\\int _{\\mathbb {T}^{d}}f\\left(x\\right)\\prod _{k=1}^{n}\\left[f\\left(x+\\sum _{i=1}^{k}y_{i}\\right)\\right]dx,$ and the $n$ -th temporal correlation of $f\\left(X_{t}\\right)$ , denoted $T_{n}:\\mathbb {R}_{+}^{n}\\rightarrow \\mathbb {R}$ , as $T_{n}\\left(t\\right)=\\mathbb {E}_{X_{0}\\sim U\\left(\\mathbb {T}^{d}\\right)}\\left[f\\left(X_{0}\\right)f\\left(X_{t_{1}}\\right)\\cdots f\\left(X_{\\sum _{i=1}^{n}t_{i}}\\right)\\right]$ (note that $S_{0}$ and $T_{0}$ are just constants, equal to the measure of $\\Omega $ relative to $\\mathbb {T}^{d}$ ).", "The proof involves two parts: The first shows that $S_{n}\\left(y\\right)$ can be calculated from our knowledge of $T_{n}\\left(t\\right)$ .", "The second uses $S_{n}\\left(y\\right)$ to reconstruct $f$ with better and better precision as $n\\rightarrow \\infty $ .", "Proposition 11 Let $n\\in \\mathbb {N}$ .", "Under the conditions of Theorem REF , the function $S_{n}$ can be calculated from $f\\left(X_{t}\\right)$ with probability 1.", "We intend to prove Proposition REF using the Vandermonde lemma (Lemma REF ).", "To this end, we first show the following: Proposition 12 For every positive integers $n,m\\in \\mathbb {N}$ , and every set of times $t_{1},\\ldots ,t_{n}>0$ , the value of the sum $\\sum _{k\\in \\mathbb {Z}^{nd}}\\left(\\prod _{i=1}^{n}\\hat{\\gamma }_{t_{i}}\\left(k_{i}\\right)\\right)^{m}\\hat{S}\\left(k\\right)$ can be calculated from $f\\left(X_{t}\\right)$ with probability 1.", "By (REF ), for every $m\\in \\mathbb {N}$ and $t>0$ we have $\\hat{\\gamma }_{mt}=\\frac{\\left(\\beta _{t}+\\left(1-\\beta _{t}\\right)\\hat{\\gamma }_{t}\\right)^{m}-\\beta _{t}^{m}}{1-\\beta _{t}^{m}}.$ Rearranging, we can write $\\hat{\\gamma }_{t}^{m}$ as a sum of smaller powers of $\\hat{\\gamma }_{t}$ : $\\hat{\\gamma }_{t}^{m}=\\sum _{j=1}^{m-1}c_{j}\\hat{\\gamma }_{t}^{j}+\\hat{\\gamma }_{mt},$ where $c_{j}$ are some coefficients (in particular, when $D_{t}$ has no atom, i.e when $\\beta _{t}=0$ for all $t>0$ , this sum is rather simple: $\\hat{\\gamma }_{t}^{m}=\\hat{\\gamma }_{mt}$ ).", "Reiterating this process, we find that the $m$ -th power of $\\hat{\\gamma }_{t}$ can be written as some linear combination $\\hat{\\gamma }_{t}^{m}=\\sum _{j=1}^{m-1}c_{j}^{\\prime }\\hat{\\gamma }_{jt}.$ Thus, the product $\\left(\\prod _{i=1}^{n}\\hat{\\gamma }_{t_{i}}\\left(k_{i}\\right)\\right)^{m}$ is itself a sum of multilinear monomials in $\\left\\lbrace \\hat{\\gamma }_{jt_{i}}\\right\\rbrace _{j=1}^{m-1}$ , and so to prove the proposition it suffices to prove it for $m=1$ .", "The temporal correlation $T_{n}\\left(t\\right)$ can be computed by the process $f\\left(X_{t}\\right)$ : Since $X_{t}$ approaches the uniform distribution on $\\mathbb {T}^{d}$ as $t\\rightarrow 0$ , for any fixed times $t_{1},\\ldots ,t_{n}$ , we can choose sampling times $\\tau _{1},\\tau _{2},\\ldots $ so that $\\left\\lbrace f\\left(X_{\\tau _{j}}\\right)f\\left(X_{\\tau _{j}+t_{1}}\\right)\\ldots f\\left(X_{\\tau _{j}+\\sum _{i=1}^{n}t_{i}}\\right)\\right\\rbrace _{j=1}^{\\infty }$ have pairwise correlations that are arbitrarily small, and are arbitrarily close in distribution to $f\\left(X_{0}\\right)f\\left(X_{t_{1}}\\right)\\cdots f\\left(X_{\\sum _{i=1}^{n}t_{i}}\\right)$ with $X_{0}\\sim U\\left(\\mathbb {T}^{d}\\right)$ .", "The temporal correlation $T_{n}\\left(t\\right)$ is then given, with probability 1, by the sample average at times $\\tau _{j}$ .", "A relation between the spatial and temporal correlation can be obtained as follows.", "First, since $X_{0}$ is uniform on $\\mathbb {T}^{d}$ in the definition of $T_{n}\\left(t\\right)$ , $T_{n}\\left(t\\right)=\\frac{1}{\\left(2\\pi \\right)^{d}}\\int _{\\mathbb {T}^{d}}f\\left(x\\right)\\mathbb {E}\\left[\\prod _{k=1}^{n}f\\left(X_{\\sum _{i=1}^{k}t_{i}}\\right)\\big \|X_{0}=x\\right]dx.$ By conditioning on the event that between times $t_{i-1}$ and $t_{i}$ the process took a step of size $y_{i}$ , this is equal to $T_{n}\\left(t\\right) & =\\int _{\\mathbb {T}^{d}}\\int _{\\mathbb {T}^{nd}}\\left(\\prod _{i=1}^{n}D_{t_{i}}\\left(y_{i}\\right)\\right)\\left(f\\left(x\\right)\\prod _{k=1}^{n}\\left[f\\left(x+\\sum _{i=1}^{k}y_{i}\\right)\\right]\\right)dydx\\nonumber \\\\& =\\int _{\\mathbb {T}^{nd}}\\left(\\prod _{i=1}^{n}D_{t_{i}}\\left(y_{i}\\right)\\right)S_{n}\\left(y\\right)dy.$ Since $D_{t}\\left(y\\right)=\\beta _{t}\\delta \\left(y\\right)+\\left(1-\\beta _{t}\\right)\\gamma _{t}\\left(y\\right)$ , the product inside the integral breaks into a sum, where, for each time step $t_{i}$ , we have to choose whether the process stayed in place (corresponding the $\\delta \\left(y_{i}\\right)$ ), or moved according to the density $\\gamma _{t_{i}}$ .", "Whenever we choose to stay in place, we shrink the number of spatial variables in our correlation, since $S_{n}\\left(y_{1},\\ldots ,y_{k},0,y_{k+2},\\ldots ,y_{n}\\right)=S_{n-1}\\left(y_{1},\\ldots ,y_{k},y_{k+2},\\ldots ,y_{n}\\right)$ .", "We can thus go over all choices $A\\subseteq \\left[n\\right]$ of indices of times when the process moved according to $\\gamma _{t}$ , giving $T_{n}\\left(t\\right)=\\sum _{A\\subseteq \\left[n\\right]}\\prod _{i\\notin A}\\beta _{t_{i}}\\prod _{i\\in A}\\left(1-\\beta _{t_{i}}\\right)\\int _{\\mathbb {T}^{\\left\|A\\right\|d}}\\left(\\prod _{i\\in A}\\gamma _{t_{i}}\\left(y_{i}\\right)\\right)S_{\\left\|A\\right\|}\\left(y\\text{ restricted to $A$}\\right)\\left(\\prod _{i\\in A}dy_{i}\\right).$ The integral in this expression can be seen as an inner product between $\\prod _{i\\in A}\\gamma _{t_{i}}\\left(y_{i}\\right)$ and $\\overline{S_{\\left\|A\\right\|}\\left(y\\right)}$ over the torus $\\mathbb {T}^{\\left\|A\\right\|d}$ .", "Since both $\\gamma _{t}$ and $S_{\\left\|A\\right\|}$ are in $\\mathrm {L}^{2}$$\\left(\\mathbb {T}^{d}\\right)$ , by Parseval's theorem, we can therefore replace it by a sum over all Fourier coefficients $k\\in \\mathbb {Z}^{\\left\|A\\right\|d}$ : $T_{n}\\left(t\\right)=\\frac{1}{\\left(2\\pi \\right)^{d}}\\sum _{A\\subseteq \\left[n\\right]}\\prod _{i\\notin A}\\beta _{t_{i}}\\prod _{i\\in A}\\left(1-\\beta _{t_{i}}\\right)\\sum _{k\\in \\mathbb {Z}^{\\left\|A\\right\|d}}\\left(\\prod _{i\\in A}\\hat{\\gamma }_{t_{i}}\\left(k_{i}\\right)\\right)\\hat{S}_{\\left\|A\\right\|}\\left(k\\text{ restricted to $A$}\\right).$ With the above display, Proposition REF follows quickly by induction on $n$ .", "For the case $n=1$ , we have $\\left(2\\pi \\right)^{d}T_{1}\\left(t\\right)=\\beta _{t}S_{0}+\\sum _{k\\in \\mathbb {Z}^{d}}\\hat{\\gamma }_{t}\\left(k\\right)\\hat{S}_{1}\\left(k\\right).$ Since $S_{0}=T_{0}$ , the value of $\\sum _{k\\in \\mathbb {Z}^{d}}\\hat{\\gamma }_{t}\\left(k\\right)\\hat{S}_{1}\\left(k\\right)$ is known.", "The induction step for general $n$ is now immediate, since by (REF ) it is evident that $T_{n}\\left(t\\right)$ is a multilinear polynomial in $\\hat{\\gamma }_{t_{i}}$ , and all terms with degree strictly smaller than $n$ are known by the induction hypothesis.", "[Proof of Proposition REF ] We will now show that for every $n\\in \\mathbb {N}$ , the values $\\left\\lbrace \\sum _{k\\in \\mathbb {Z}^{nd}}\\left(\\prod _{i=1}^{n}\\hat{\\gamma }_{t_{i}}\\left(k_{i}\\right)\\right)^{m}\\hat{S}_{n}\\left(k\\right)\\mid m\\in \\mathbb {N},t_{1},\\ldots ,t_{n}>0\\right\\rbrace $ uniquely determine $\\hat{S}_{n}\\left(k\\right)$ .", "Proposition REF then follows, since (as a quick, omitted, calculation shows) $S_{n}\\left(y\\right)$ is continuous on $\\mathbb {T}^{d}$ , and is thus completely determined by its Fourier coefficients.", "Suppose that there exists a bounded function $Q_{n}\\left(y\\right)$ on $\\mathbb {T}^{nd}$ such that for all $m\\in \\mathbb {N}$ and all times $t_{1},\\ldots ,t_{n}>0$ , $\\sum _{k\\in \\mathbb {Z}^{nd}}\\left(\\prod _{i=1}^{n}\\hat{\\gamma }_{t_{i}}\\left(k_{i}\\right)\\right)^{m}\\hat{S}_{n}\\left(k\\right)=\\sum _{k\\in \\mathbb {Z}^{nd}}\\left(\\prod _{i=1}^{n}\\hat{\\gamma }_{t_{i}}\\left(k_{i}\\right)\\right)^{m}\\hat{Q}_{n}\\left(k\\right).$ Then, denoting $x_{k}=\\hat{S}_{n}\\left(k\\right)-\\hat{Q}_{n}\\left(k\\right)$ , we have that $\\sum _{k\\in \\mathbb {Z}^{nd}}\\left(\\prod _{i=1}^{n}\\hat{\\gamma }_{t_{i}}\\left(k_{i}\\right)\\right)^{m}x_{k}=0.$ Note that $x_{k}\\in \\ell ^{2}\\left(\\mathbb {C}\\right)$ , since both $S_{n}$ and $Q_{n}$ are in $\\mathrm {L}^{2}\\left(\\mathbb {T}^{d}\\right)$ .", "We wish to use Lemma REF to show that necessarily $x_{k}=0$ .", "To do this, we must choose the times $t_{i}$ so that the products $\\prod _{i=1}^{n}\\hat{\\gamma }_{t_{i}}\\left(k_{i}\\right)$ are all non-zero and distinct.", "Then, setting $z_{k}=\\prod _{i=1}^{n}\\hat{\\gamma }_{t_{i}}\\left(k_{i}\\right)$ , we will have that $z_{k}\\in \\ell ^{2}\\left(\\mathbb {C}\\right)$ (since $\\hat{\\gamma }_{t_{i}}$ are the Fourier coefficients of a square integrable function $\\gamma _{t_{i}}$ ), $z_{k}$ are all distinct, and $z_{k}\\ne 0$ for all $k\\in \\mathbb {Z}^{nd}$ , exactly meeting the requirements of the lemma with $p=q=1/2$ .", "To choose the times $t_{i}$ , recall that by assumption, there exists a time $t_{0}$ such that $\\left\\lbrace \\hat{\\gamma }_{t_{0}}\\left(k\\right)\\right\\rbrace _{k\\in \\mathbb {Z}^{d}}$ are all distinct and nonzero.", "We will show that there exist numbers $\\alpha _{1},\\ldots ,\\alpha _{n}>0$ , so that if $t_{i}=\\alpha _{i}t_{0}$ , then the products $\\left\\lbrace \\prod _{i=1}^{n}\\hat{\\gamma }_{t_{i}}\\left(k_{i}\\right)\\right\\rbrace _{k\\in \\mathbb {Z}^{nd}}$ satisfy the above requirements.", "By (REF ), for every $\\alpha _{i}$ we have $\\hat{\\gamma }_{\\alpha _{i}t_{0}}=\\frac{\\left(\\beta _{t_{0}}+\\left(1-\\beta _{t_{0}}\\right)\\hat{\\gamma }_{t_{0}}\\right)^{\\alpha _{i}}-\\beta _{t_{0}}^{\\alpha _{i}}}{1-\\beta _{t_{0}}^{\\alpha _{i}}}.$ For a particular $k\\in \\mathbb {Z}^{d}$ , let $B_{k}=\\left\\lbrace \\alpha >0\\mid \\hat{\\gamma }_{\\alpha t_{0}}\\left(k\\right)=0\\right\\rbrace $ be the set of “bad” multipliers for $\\hat{\\gamma }_{t_{0}}\\left(k\\right)$ .", "The coefficient $\\hat{\\gamma }_{\\alpha t_{0}}\\left(k\\right)$ can be 0 only if $\\left(\\beta _{t_{0}}+\\left(1-\\beta _{t_{0}}\\right)\\hat{\\gamma }_{t_{0}}\\left(k\\right)\\right)^{\\alpha _{i}}-\\beta _{t_{0}}^{\\alpha _{i}}=0,$ and since $\\hat{\\gamma }_{t_{0}}\\left(k\\right)\\ne 0$ , the function $z\\mapsto \\left(\\beta _{t_{0}}+\\left(1-\\beta _{t_{0}}\\right)\\hat{\\gamma }_{t_{0}}\\left(k\\right)\\right)^{z}-\\beta _{t_{0}}^{z}$ is a non-constant holomorphic function of $z$ .", "The set of zeros $B_{k}$ is thus isolated, and in particular countable.", "Now let $k\\ne k^{\\prime }\\in \\mathbb {Z}^{nd}$ be two different vectors, and let $B_{k,k^{\\prime }}=\\left(\\cup _{i=1}^{n}B_{k_{i}}\\right)\\bigcup \\left(\\cup _{i=1}^{n}B_{k_{i}^{\\prime }}\\right)$ be the set of bad $\\alpha _{i}$ 's which cause one of the $\\hat{\\gamma }_{\\alpha _{i}t_{0}}$ to be 0.", "Let $R_{k,k^{\\prime }}=\\left\\lbrace \\alpha \\in \\mathbb {R}_{>0}^{n}\\mid \\forall j\\,\\alpha _{j}\\notin B_{k,k^{\\prime }}\\text{ and }\\prod _{i=1}^{n}\\hat{\\gamma }_{\\alpha _{i}t_{0}}\\left(k_{i}\\right)=\\prod _{i=1}^{n}\\hat{\\gamma }_{\\alpha _{i}t_{0}}\\left(k_{i}^{\\prime }\\right)\\right\\rbrace .$ By (REF ), this means that for every $\\alpha \\in R_{k,k^{\\prime }}$ , $\\prod _{i=1}^{n}\\frac{\\left(\\beta _{t_{0}}+\\left(1-\\beta _{t_{0}}\\right)\\hat{\\gamma }_{t_{0}}\\left(k_{i}\\right)\\right)^{\\alpha _{i}}-\\beta _{t_{0}}^{\\alpha _{i}}}{\\left(1-\\beta _{t_{0}}^{\\alpha _{i}}\\right)}-\\prod _{i=1}^{n}\\frac{\\left(\\beta _{t_{0}}+\\left(1-\\beta _{t_{0}}\\right)\\hat{\\gamma }_{t_{0}}\\left(k_{i}^{\\prime }\\right)\\right)^{\\alpha _{i}}-\\beta _{t_{0}}^{\\alpha _{i}}}{\\left(1-\\beta _{t_{0}}^{\\alpha _{i}}\\right)}=0.$ Let $i^{}$ be an index so that $k_{i^{}}\\ne k_{i^{}}^{\\prime }$ .", "Since none of the factors $\\hat{\\gamma }_{\\alpha _{i}t_{0}}\\left(k_{i}\\right)$ or $\\hat{\\gamma }_{\\alpha _{i}t_{0}}\\left(k_{i}^{\\prime }\\right)$ are 0 by choice of $R_{k,k^{\\prime }}$ , we can rearrange the above, yielding $\\frac{1-\\left(1+\\frac{1-\\beta _{t_{0}}}{\\beta _{t_{0}}}\\hat{\\gamma }_{t_{0}}\\left(k_{i^{}}\\right)\\right)^{\\alpha _{i^{}}}}{1-\\left(1+\\frac{1-\\beta _{t_{0}}}{\\beta _{t_{0}}}\\hat{\\gamma }_{t_{0}}\\left(k_{i^{}}^{\\prime }\\right)\\right)^{\\alpha _{i^{}}}}-\\prod _{i\\ne i^{}}\\frac{\\left(\\beta _{t_{0}}+\\left(1-\\beta _{t_{0}}\\right)\\hat{\\gamma }_{t_{0}}\\left(k_{i}^{\\prime }\\right)\\right)^{\\alpha _{i}}-\\beta _{t_{0}}^{\\alpha _{i}}}{\\left(\\beta _{t_{0}}+\\left(1-\\beta _{t_{0}}\\right)\\hat{\\gamma }_{t_{0}}\\left(k_{i}\\right)\\right)^{\\alpha _{i}}-\\beta _{t_{0}}^{\\alpha _{i}}}=0.$ For fixed $\\left\\lbrace \\alpha _{i}\\right\\rbrace _{i\\ne i^{}}$ , the expression on the left-hand side is a function of the form $z\\mapsto \\frac{1-a^{z}}{1-b^{z}}-c$ .", "Since $\\left\\lbrace \\hat{\\gamma }_{t_{0}}\\left(k\\right)\\right\\rbrace _{k\\in \\mathbb {Z}^{d}}$ are all distinct, $a\\ne b$ .", "Thus this map is a non-constant holomorphic function on $\\mathbb {C}\\backslash B_{k,k^{\\prime }}$ , and so for any fixed choice of $\\left\\lbrace \\alpha _{i}\\right\\rbrace _{i\\ne i^{}}$ , has only countably many zeros, i.e only countably many choices for $\\alpha _{i^{}}$ .", "The Lebesgue measure of $R_{k,k^{\\prime }}$ in $\\mathbb {R}^{n}$ is therefore 0.", "But the set of all $\\alpha _{1},\\ldots ,\\alpha _{n}>0$ such that either there are two equal nonzero products in $\\left\\lbrace \\prod _{i=1}^{n}\\hat{\\gamma }_{\\alpha _{i}t_{0}}\\left(k_{i}\\right)\\right\\rbrace _{k\\in \\mathbb {Z}^{nd}}$ or one of the products is itself 0 is the countable union $\\bigcup _{k,k^{\\prime }\\in \\mathbb {Z}^{nd}}R_{k,k^{\\prime }}\\cup \\left(\\bigcup _{k,k^{\\prime }\\in \\mathbb {Z}^{nd}}\\left\\lbrace \\left(\\alpha _{1},\\ldots ,\\alpha _{k}\\right)\\mid \\alpha _{i}\\in \\left(B_{k_{i}}\\cup B_{k_{i}^{\\prime }}\\right)\\right\\rbrace \\right),$ and so too has measure 0.", "In particular, there must exist $\\alpha _{1},\\ldots ,\\alpha _{n}>0$ so that the products are distinct and nonzero, as needed.", "[Proof of Theorem REF ] We now show that knowledge of $S_{n}\\left(y\\right)$ is enough to reconstruct $f$ up to a translation of the torus.", "The main idea is this.", "Suppose that $S_{n}\\left(y\\right)>0$ for some given $y\\in \\mathbb {R}^{nd}$ .", "Then $\\int _{\\mathbb {T}^{d}}\\prod _{k=0}^{n}f\\left(x+\\sum _{i=1}^{k}y_{i}\\right)dx>0$ , which means that there exists a point $x_{0}\\in \\mathbb {T}^{d}$ such that $f\\left(x_{0}+\\sum _{i=1}^{k}y_{i}\\right)=1$ for all $k=0,\\ldots ,n$ .", "By considering only $y_{i}$ 's which partition $\\mathbb {T}^{d}$ into a grid, we can get an approximation of $\\Omega $ by taking the union of the grid blocks.", "Let $\\delta _{m}=2\\pi /m$ , let $n=m^{d}$ , and consider the set $\\mathcal {Y}=\\left\\lbrace y\\in \\delta \\cdot \\mathbb {N}^{nd}\\mid S_{n}\\left(y\\right)>0\\right\\rbrace $ .", "Note that $\\left(2\\pi \\right)^{d}S_{n}\\left(0\\right)=\\int _{\\mathbb {T}^{d}}f\\left(x\\right)dx=\\mu \\left(\\Omega \\right)$ (where $\\mu $ is the Lebesgue measure), so if $\\mathcal {Y}=\\left\\lbrace \\emptyset \\right\\rbrace $ then $\\Omega =\\emptyset $ , and $f$ is identically zero.", "We may therefore assume that $\\mathcal {Y\\ne \\emptyset }$ .", "Each vector $y\\in \\mathcal {Y}$ defines a set of points $G_{y}=\\left\\lbrace \\sum _{i=1}^{k}y_{i}\\mid k=0,\\ldots ,n\\right\\rbrace $ , where the sum $\\sum _{i=1}^{k}y_{i}$ is taken to be in the torus $\\mathbb {T}^{d}$ .", "Since $\\delta _{m}$ divides the side-length of the torus, $G_{y}$ can be viewed as a subset of a $d$ -dimensional grid in $\\mathbb {T}^{d}$ , with the individual $y_{i}$ serving as “pointer vectors” to the next point in the grid.", "The number of points in $G_{y}$ depends on $y$ : If $y=0$ , for example, then $G_{y}=\\left\\lbrace 0\\right\\rbrace $ ; however, we can also choose $y$ such that $G_{y}=\\delta \\mathbb {Z}^{d}\\cap \\mathbb {T}^{d}$ .", "Let $y^{}\\in \\mathcal {Y}$ be such that $\\left\|G_{y}\\right\|\\ge \\left\|G_{y}\\right\|$ for all $y\\in \\mathcal {Y}$ , and let $G_{m}^{}=G_{y^{}}$ , so that $G_{m}^{}$ is a largest possible subset when the pointer vectors are taken from $\\mathcal {Y}$ .", "Using $G_{m}^{}$ , we can now define a rough, shifted approximation $\\Omega _{m}$ to the domain $\\Omega $ : Letting $C_{d}=\\left[-\\frac{1}{2},\\frac{1}{2}\\right]^{d}$ be the unit $d$ -dimensional cube, we cover each point $x\\in G_{m}^{}$ by the scaled cube $\\delta _{m}C_{d}$ : $\\Omega _{m}=G_{m}^{}+\\delta _{m}C_{d}$ (here we use the Minkowski sum for the addition of two sets / the addition of a point and a set).", "See Figure REF for an example of this procedure in 2 dimensions.", "Figure: Left: The domain Ω\\Omega (cyan), together with a maximalgrid G m G_{m}^{} (black dots).", "The origin is shifted so that allgrid points fall in Ω\\Omega .", "The yellow vectors represent a possiblechoice for y i y_{i}.", "Right: The resultant approximation Ω m \\Omega _{m}.We now claim that $\\Omega _{m}\\rightarrow \\Omega $ up to translations, in the sense that there is a shift $\\theta \\in \\mathbb {T}^{d}$ such that the symmetric difference vanishes: $\\mu \\left(\\left(\\Omega _{m}+\\theta \\right)\\Delta \\Omega \\right)\\rightarrow 0$ as $m\\rightarrow \\infty $ .", "To see this, let's look separately at the contribution of $\\left(\\Omega _{m}+\\theta \\right)\\backslash \\Omega $ and the contribution of $\\Omega \\backslash \\left(\\Omega _{m}+\\theta \\right)$ .", "First, as mentioned above, since $S_{n}\\left(y\\right)>0$ , there is a point $x_{m}$ such that $G_{m}^{}+x_{m}\\subseteq \\Omega $ .", "Adding $\\delta _{m}C_{d}$ to both sides gives $\\Omega _{m}+x_{m}\\subseteq \\Omega +\\delta _{m}C_{d}$ .", "We then have $\\mu \\left(\\left(\\Omega _{m}+x_{m}\\right)\\backslash \\Omega \\right)\\le \\mu \\left(\\left(\\Omega +\\delta _{m}C_{d}\\right)\\backslash \\Omega \\right)\\le \\mu \\left(\\partial \\Omega +\\delta _{m}C_{d}\\right).$ The latter expression goes to 0 as $\\delta _{m}\\rightarrow 0$ , since $\\lim _{m\\rightarrow \\infty }\\mu \\left(\\partial \\Omega +\\delta _{m}C_{d}\\right)=\\mu \\left(\\bigcap _{m}\\left(\\partial \\Omega +\\delta _{m}C_{d}\\right)\\right)\\stackrel{\\text{$\\partial \\Omega $ is closed}}{=}\\mu \\left(\\partial \\Omega \\right)=0.$ Suppose now that $x\\in \\Omega \\backslash \\left(\\Omega _{m}+x_{m}\\right)$ , and let $z\\in \\delta _{m}\\mathbb {Z}^{d}\\cap \\mathbb {T}^{d}$ be a grid-point closest to $x-x_{m}$ .", "The point $z$ cannot be in $G_{m}^{}$ : If it were, then the cube $z+\\delta _{m}C_{d}$ (which contains $x-x_{m}$ ) would be contained in $\\Omega _{m}$ , contradicting the fact that $x\\notin \\Omega _{m}+x_{m}$ .", "Since $\\left\|G_{m}^{}\\right\|$ is maximal, we necessarily have $z+x_{m}\\notin \\Omega $ (otherwise we could add it to $G_{m}^{}$ ).", "So every point not in $\\Omega _{m}+x_{m}$ can be covered by placing the cube $\\delta _{m}C_{d}$ on some point in $\\left(\\Omega +x_{m}\\right)^{c}$ .", "Thus $\\mu \\left(\\Omega \\backslash \\left(\\Omega _{m}+x_{m}\\right)\\right) & =\\mu \\left(\\left(\\Omega _{m}+x_{m}\\right)^{c}\\backslash \\Omega ^{c}\\right)\\\\& \\le \\mu \\left(\\left(\\Omega ^{c}+\\delta _{m}C_{d}\\right)\\backslash \\Omega ^{c}\\right),$ and again the latter goes to 0 as $\\delta _{m}\\rightarrow 0$ .", "We thus have a sequence of vectors $x_{m}\\in \\mathbb {T}^{d}$ such that $\\mu \\left(\\left(\\Omega _{m}+x_{m}\\right)\\Delta \\Omega \\right)\\rightarrow 0$ as $m\\rightarrow \\infty $ .", "Since $\\mathbb {T}^{d}$ is compact, $x_{m}$ has a subsequence converging to some $\\theta $ , and it follows that $\\mu \\left(\\left(\\Omega _{m}+\\theta \\right)\\Delta \\Omega \\right)\\rightarrow 0$ as well." ], [ "Symmetric reconstruction", "The proof of Theorem REF is similar to that of Theorem REF .", "The main difference is that since $\\hat{\\gamma }_{t}\\left(k\\right)=\\hat{\\gamma }_{t}\\left(-k\\right)$ for all $k$ , we cannot immediately use Lemma REF to recover $S_{n}\\left(y\\right)$ anymore.", "This can be overcome by working with a completely symmetric version of $S_{n}\\left(y\\right)$ , denoted $\\sigma _{n}\\left(y\\right)$ and defined as $\\sigma _{n}\\left(y\\right)=\\sum _{\\varepsilon \\in \\left\\lbrace -1,1\\right\\rbrace ^{n}}S_{n}\\left(\\varepsilon _{1}y_{1},\\ldots ,\\varepsilon _{n}y_{n}\\right).$ Proposition 13 Under the conditions of Theorem REF , $\\sigma _{n}\\left(y\\right)$ can be calculated from $f\\left(X_{t}\\right)$ with probability 1.", "The proof uses the same techniques as that of Proposition REF ; we highlight the differences here.", "Starting with the temporal-spatial relation (REF ), $T_{n}\\left(t\\right) & =\\int _{\\mathbb {T}^{n}}\\left(\\prod _{i=1}^{n}D_{t_{i}}\\left(y_{i}\\right)\\right)S_{n}\\left(y_{1},\\ldots ,y_{n}\\right)dy,$ observe that each integral of the form $\\int _{-\\pi }^{\\pi }D_{t_{i}}\\left(y_{i}\\right)S_{n}\\left(y_{1},\\ldots ,y_{n}\\right)dy_{i}$ can be split into two parts: $\\int _{-\\pi }^{\\pi }D_{t_{i}}\\left(y_{i}\\right)S_{n}\\left(y_{1},\\ldots ,y_{n}\\right)dy_{i}=\\int _{-\\pi }^{0}D_{t_{i}}\\left(y_{i}\\right)S_{n}\\left(y_{1},\\ldots ,y_{n}\\right)dy_{i}+\\int _{0}^{\\pi }D_{t_{i}}\\left(y_{i}\\right)S_{n}\\left(y_{1},\\ldots ,y_{n}\\right)dy_{i},$ where we use the convention that $\\int _{-\\pi }^{0}\\delta \\left(y\\right)g\\left(y\\right)dy=\\frac{1}{2}\\lim _{\\varepsilon \\rightarrow 0^{+}}\\int _{-\\pi }^{\\varepsilon }\\delta \\left(y\\right)g\\left(y\\right)dy=\\frac{1}{2}g\\left(0\\right)$ .", "Making the change of variables $y_{i}\\rightarrow -y_{i}$ in the first integral and using the fact that $D_{t}$ is symmetric, we thus have $\\int _{-\\pi }^{\\pi }D_{t_{i}}\\left(y_{i}\\right)S_{n}\\left(y_{1},\\ldots ,y_{n}\\right)dy_{i} & =\\frac{1}{2}\\int _{-\\pi }^{\\pi }D_{t_{i}}\\left(y_{i}\\right)\\left[S_{n}\\left(y_{1},\\ldots ,-y_{i},\\ldots ,y_{n}\\right)+S_{n}\\left(y_{1},\\ldots ,y_{i},\\ldots ,y_{n}\\right)\\right]dy_{i}.$ Performing this $n$ times yields $T_{n}\\left(t\\right) & =\\frac{1}{2^{n}}\\int _{\\mathbb {T}^{n}}\\left(\\prod _{i=1}^{n}D_{t_{i}}\\left(y_{i}\\right)\\right)\\sigma _{n}\\left(y\\right)dy.$ As in the proof of Proposition REF , this allows us to calculate the sum $\\sum _{k\\in \\mathbb {Z}^{n}}\\left(\\prod _{i=1}^{n}\\hat{\\gamma }_{t_{i}}\\left(k_{i}\\right)\\right)^{m}\\hat{\\sigma }_{n}\\left(k\\right)$ for all $n,m\\in \\mathbb {N}$ and every set of times $t_{1},\\ldots ,t_{n}>0$ .", "Now, both $\\left(\\prod _{i=1}^{n}\\gamma _{t_{i}}\\left(y_{i}\\right)\\right)$ and $\\sigma _{n}\\left(y\\right)$ are symmetric in every coordinate $y_{i}$ , and so the Fourier coefficients are invariant under flipping of individual entries.", "We can therefore restrict the sum to non-negative $k$ vectors: Defining $\\alpha _{n}\\left(k\\right)=2^{\\#\\left\\lbrace i\\mid k_{i}\\ne 0\\right\\rbrace }\\hat{\\sigma }_{n}\\left(k\\right),$ we can calculate the sum $\\sum _{k\\in \\mathbb {Z}_{+}^{n}}\\left(\\prod _{i=1}^{n}\\hat{\\gamma }_{t_{i}}\\left(k_{i}\\right)\\right)^{m}\\alpha _{n}\\left(k\\right)$ for all $n,m\\in \\mathbb {N}$ and times $t_{1},\\ldots ,t_{n}>0$ .", "As in the proof of Proposition REF , $\\alpha _{n}$ (and therefore $\\sigma _{n}$ ) can be recovered from these quantities using Lemma REF , since by assumption there is a time $t_{0}$ such that $\\left\\lbrace \\hat{\\gamma }_{t_{0}}\\left(k\\right)\\right\\rbrace _{k\\ge 0}$ are all distinct and non-zero.", "Proposition 14 Let $\\delta >0$ divide $2\\pi $ .", "Given $\\sigma _{n}\\left(y\\right)$ for all $n\\in \\mathbb {N}$ and $y\\in \\mathbb {T}^{n}$ , it is possible to calculate $S_{n}\\left(y\\right)+S_{n}\\left(-y\\right)$ for all $n\\in \\mathbb {N}$ and all $y\\in \\left\\lbrace \\left(k_{1}\\delta ,\\ldots ,k_{n}\\delta \\right)\\mid k_{i}\\in \\mathbb {Z}_{+}\\right\\rbrace $ .", "The proof is by induction.", "For $n=1$ , we just have $\\sigma _{1}\\left(y\\right)=S_{1}\\left(y\\right)+S_{1}\\left(-y\\right)$ , and for general $n$ and $k=0$ , we just have $\\sigma _{n}\\left(0\\right)=2^{n}S_{n}\\left(0\\right)$ .", "Now let $n\\in \\mathbb {N}$ and $k\\in \\mathbb {Z}_{+}^{n}$ .", "Assume that the statement holds true for $n$ for all vectors $k^{\\prime }$ with $\\sum _{i=1}^{n}k_{i}^{\\prime }<\\sum _{i=1}^{n}k_{i}$ , and also for all $m<n$ .", "We have $\\sigma _{n}\\left(k_{1}\\delta ,\\ldots ,k_{n}\\delta \\right) & =\\sum _{\\varepsilon \\in \\left\\lbrace -1,1\\right\\rbrace ^{n}}S_{n}\\left(\\varepsilon _{1}k_{1}\\delta ,\\ldots ,\\varepsilon _{n}k_{n}\\delta \\right)\\\\& =S_{n}\\left(k_{1}\\delta ,\\ldots ,k_{n}\\delta \\right)+S_{n}\\left(-k_{1}\\delta ,\\ldots ,-k_{n}\\delta \\right)+\\sum _{\\varepsilon \\in \\left\\lbrace -1,1\\right\\rbrace ^{n},\\,\\varepsilon _{i}\\text{ not all equal}}S_{n}\\left(\\ldots \\right).$ If not all $\\varepsilon _{i}$ are equal, then $S_{n}\\left(\\varepsilon _{1}k_{1}\\delta ,\\ldots ,\\varepsilon _{n}k_{n}\\delta \\right)$ is equal to some $S_{m}\\left(k^{\\prime }_{1}\\delta ,\\ldots ,k_{m}^{\\prime }\\delta \\right)$ for $m\\le n$ : The total sum $\\sum _{i}\\varepsilon _{i}k_{i}$ is strictly smaller than $\\sum _{i}k_{i}$ , and so the partial sums can be seen as the forward-only jumps of some $y^{\\prime }$ with corresponding $k^{\\prime }$ such that $\\sum _{i}k_{i}^{\\prime }<\\sum _{i}k_{i}$ (the strict $m<n$ case is when the partial sums $\\sum \\varepsilon _{i}k_{i}$ themselves are not unique).", "See Figure REF for a visualization.", "Similarly, $S_{n}\\left(-\\varepsilon _{1}k_{1}\\delta ,\\ldots ,-\\varepsilon _{n}k_{n}\\delta \\right)=-S_{m}\\left(-k^{\\prime }_{1}\\delta ,\\ldots ,-k_{m}^{\\prime }\\delta \\right)$ .", "Since the sum over all $\\varepsilon _{i}$ can be split into polar pairs, by the induction hypothesis we can calculate the sum $\\sum _{\\varepsilon \\in \\left\\lbrace -1,1\\right\\rbrace ^{n},\\,\\varepsilon _{i}\\text{ not all equal}}S_{n}$ , and therefore also $S_{n}\\left(k_{1}\\delta ,\\ldots ,k_{n}\\delta \\right)+S_{n}\\left(-k_{1}\\delta ,\\ldots ,-k_{n}\\delta \\right).$ Figure: S 3 y 1 ,y 2 ,y 3 =S 3 y 1 ' ,y 2 ' ,y 3 ' S_{3}\\left(y_{1},y_{2},y_{3}\\right)=S_{3}\\left(y_{1}^{\\prime },y_{2}^{\\prime },y_{3}^{\\prime }\\right).The choice of origin does not matter, since S n S_{n} integrates overall 𝕋\\protect \\mathbb {T} anyway.", "The total length y 1 ' +y 2 ' +y 3 ' y_{1}^{\\prime }+y_{2}^{\\prime }+y_{3}^{\\prime }is smaller than y 1 +y 2 +y 3 y_{1}+y_{2}+y_{3}.", "[Proof sketch of Theorem REF ] Similarly to the proof of Theorem REF , once we know $S_{n}\\left(k_{1}\\delta ,\\ldots ,k_{n}\\delta \\right)+S_{n}\\left(-k_{1}\\delta ,\\ldots ,-k_{n}\\delta \\right)$ , for every $\\delta _{n}=2\\pi /n$ we can construct a set $\\Omega _{n}$ according a $y\\in \\left\\lbrace y\\in \\delta \\cdot \\mathbb {N}^{n}\\mid S_{n}\\left(y\\right)+S_{n}\\left(-y\\right)>0\\right\\rbrace $ which maximizes $G_{y}$ .", "The resulting $f_{n}$ contains a subsequence which converges to either $f\\left(x\\right)$ or $f\\left(-x\\right)$ , where the ambiguity is because we do not know which of the two of $S_{n}\\left(y\\right)$ and $S_{n}\\left(-y\\right)$ was greater than 0." ], [ "Proof of Corollary ", "For $d=1$ , the step distribution $\\gamma _{t}$ of Brownian motion on $\\mathbb {T}$ is that of a wrapped normal distribution with drift, and is given by $\\gamma _{t}\\left(y\\right)=\\frac{1}{\\sqrt{2t\\pi }}\\sum _{m=-\\infty }^{\\infty }e^{-\\left(y-vt+2\\pi m\\right)^{2}/2t}.$ The Fourier coefficients of $\\gamma _{t}\\left(y\\right)$ can readily be calculated, by observing that the wrap-around gives the continuous Fourier transform evaluated at integer points: $\\hat{\\gamma }_{t}\\left(k\\right) & =\\int _{-\\pi }^{\\pi }e^{-iky}\\frac{1}{\\sqrt{2t\\pi }}\\sum _{m=-\\infty }^{\\infty }e^{-\\left(y-vt+2\\pi m\\right)^{2}/2t}dy\\\\& =\\int _{-\\infty }^{\\infty }e^{-iky}\\frac{1}{\\sqrt{2t\\pi }}e^{-\\left(y-vt\\right)^{2}/2t}dy\\\\& =e^{-ivtk}e^{-2t\\pi ^{2}k^{2}}.$ For standard Brownian motion, without drift, the coefficients are symmetric, but $\\left\\lbrace \\hat{\\gamma }_{t}\\left(k\\right)\\right\\rbrace _{k\\ge 0}$ are all distinct, and so by Theorem REF , reconstruction is possible using Brownian motion up to rotations and reflections.", "The drift, however, can make the coefficients distinct.", "For $d=1$ , any non-zero drift will do, and reconstruction is possible up to rotations.", "In the general case, the Fourier coefficients of $\\gamma _{t}$ are $\\hat{\\gamma }_{t}\\left(k\\right)=e^{-it\\sum _{j=1}^{d}v_{j}\\cdot k_{j}}e^{-2t\\pi ^{2}\\sum _{j}k_{j}^{2}}\\,\\,\\,\\,;\\,\\,\\,k\\in \\mathbb {Z}^{d}.$ In order for the coefficients to be distinct, it suffices to make the factors $e^{-it\\sum _{j=1}^{d}v_{j}\\cdot k_{j}}$ all distinct, i.e there should exist a time $t$ such that for every $k\\ne k^{\\prime }\\in \\mathbb {Z}^{d}$ and every $m\\in \\mathbb {Z}$ , $\\sum _{j=1}^{d}v_{j}\\left(k_{j}-k_{j}^{\\prime }\\right)\\ne 2\\pi m/t.$ Choosing $t$ so that $\\left\\lbrace v_{1},\\ldots ,v_{d},\\frac{2\\pi }{t}\\right\\rbrace $ are all rationally independent completes the proof." ], [ "An explicit inversion", "As a side note, we would like to mention that for Brownian motion, inverting the symmetric integral (REF ) (note the changed range of integration), $T_{n}\\left(t\\right)=\\int _{\\left[0,\\pi \\right]^{n}}\\left(\\prod _{i=1}^{n}\\gamma _{t_{i}}\\left(y_{i}\\right)\\right)\\sigma _{n}\\left(y\\right)dy,$ is possible without resorting to Lemma REF : There exists an explicit relation between $\\sigma _{n}\\left(y\\right)$ and $T_{n}\\left(t\\right)$ .", "The relation appears in (e.g) [8].", "We repeat the arguments here for completeness.", "If we treat $\\sigma _{n}\\left(y\\right)$ as a periodic function over all $\\mathbb {R}^{n}$ where every coordinate has period $2\\pi $ , we can replace the folded normal distribution $\\gamma _{t}$ with a normal distribution over all $\\mathbb {R}_{+}$ : $T_{n}\\left(t\\right)=\\int _{y\\in \\mathbb {R}_{+}^{n}}\\left(\\prod _{i=1}^{n}\\frac{1}{\\sqrt{2\\pi t_{i}}}e^{-\\frac{y_{i}^{2}}{2t_{i}}}\\right)\\sigma _{n}\\left(y\\right)dy.$ Let $p\\in \\mathbb {R}_{+}^{n}$ .", "Multiplying both sides by $e^{-p_{1}t_{1}-\\ldots -p_{n-1}t_{n-1}}$ and integrating all $t_{i}$ s from 0 to $\\infty $ , we get $\\mathcal {L}\\left\\lbrace T_{n}\\right\\rbrace \\left(p\\right)=\\int _{t\\in \\mathbb {R}_{+}^{n}}\\int _{y\\in \\mathbb {R}_{+}^{n}}\\left(\\prod _{i=1}^{n}\\frac{1}{\\sqrt{2\\pi t_{i}}}e^{-\\frac{y_{i}^{2}}{2t_{i}}}e^{-p_{i}t_{i}}\\right)\\sigma _{n}\\left(y\\right)dydt,$ where $\\mathcal {L}\\left\\lbrace f\\right\\rbrace $ is the Laplace transform of $f$ .", "The individual integrals over $t_{i}$ in the right hand side can be readily calculated to be: $\\int _{0}^{\\infty }e^{-pt-y^{2}/2t}\\frac{1}{\\sqrt{2\\pi t}}dt=\\frac{1}{\\sqrt{2p}}e^{-\\sqrt{2p}y}.$ This gives $\\mathcal {L}\\left\\lbrace T_{n}\\right\\rbrace \\left(p\\right)=\\int _{\\mathbb {R}_{+}^{n}}\\left(\\prod _{i=1}^{n}\\frac{1}{\\sqrt{2p_{i}}}e^{-\\sqrt{2p_{i}}y}\\right)\\sigma _{n}\\left(y\\right)dy.$ Up to a change of variables $s_{i}=\\sqrt{2p_{i}}$ , the right hand side is the Laplace transform of $\\sigma _{n}\\left(y\\right)$ .", "Thus $\\sigma _{n}=\\mathcal {L}^{-1}\\left\\lbrace \\left(\\prod _{i=1}^{n}s_{i}\\right)\\mathcal {L}\\left\\lbrace T_{n}\\right\\rbrace \\left(\\frac{1}{2}s_{1}^{2},\\ldots ,\\frac{1}{2}s_{n}^{2}\\right)\\right\\rbrace .$" ], [ "Proof of Lemma ", "As noted in Remark REF , the equality $Vx=0$ means that for every index $i\\in \\mathbb {N}$ we have $0=\\sum _{j=1}^{\\infty }V_{ij}x_{j}=\\sum _{j=1}^{\\infty }z_{j}^{i}x_{j}.$ Since $z\\in \\ell ^{p}\\left(\\mathbb {C}\\right)$ and $x\\in \\ell ^{q}\\left(\\mathbb {C}\\right)$ , by HÃ¶lder's inequality the series $\\left(z_{j}^{i}x_{j}\\right)_{j}$ is absolutely convergent for all $i$ , and we can change the order of summation; without loss of generality we can assume that $z_{n}$ are ordered so that $\\left\|z_{1}\\right\|\\ge \\left\|z_{2}\\right\|\\ge \\ldots $ .", "Assume by induction that $x_{1},\\ldots ,x_{k}$ have already been shown to be equal to 0, and let $\\ell >0$ be such that $\\left\|z_{k+1}\\right\|,\\ldots ,\\left\|z_{k+\\ell }\\right\|$ are all of equal magnitudes, but $\\left\|z_{k+\\ell }\\right\|>\\left\|z_{k+\\ell +1}\\right\|$ .", "Since $z_{n}\\rightarrow 0$ , $\\ell $ is necessarily finite.", "For any fixed $i$ , the sum (REF ) can then be split into two parts: $\\sum _{s=1}^{\\ell }z_{k+s}^{i}x_{k+s}+\\sum _{j>k+\\ell }z_{j}^{i}x_{j}=0.$ Dividing by $z_{k+\\ell }^{i}$ and denoting $\\omega _{s}=\\frac{z_{k+s}}{z_{k+\\ell }}$ , we have $\\sum _{s=1}^{\\ell }\\omega _{s}^{i}x_{k+s}+\\sum _{j>k+\\ell }\\left(\\frac{z_{j}}{z_{k+\\ell }}\\right)^{i}x_{j}=0.$ Consider the $\\ell $ equations of the above form for $i=r\\cdot m$ , where $r=1,\\ldots ,\\ell $ and $m$ is a large number.", "In matrix form, this system of $\\ell $ equations can be written as $V_{m}\\tilde{x}+u=0,$ where: $V_{m}$ is a finite $\\ell \\times \\ell $ Vandermonde matrix with generators $\\omega _{s}^{m}=\\left(\\frac{z_{k+s}}{z_{k+\\ell }}\\right)^{m}$ , $s=1,\\ldots ,\\ell $ .", "$\\tilde{x}$ is a vector of size $\\ell $ with $\\tilde{x}_{s}=x_{k+s}$ for $s=1,\\ldots ,\\ell $ .", "$u$ is a vector of length $\\ell $ with entries $u_{r}=\\sum _{j>k+\\ell }\\left(\\frac{z_{j}}{z_{k+\\ell }}\\right)^{rm}x_{j}$ for $r=1,\\ldots ,\\ell $ .", "By our ordering and choice of $\\ell $ , $\\left\|z_{k+\\ell }\\right\|>\\left\|z_{j}\\right\|$ for $j>k+\\ell $ , and we can factor out an exponentially decreasing term from each summand: $\\left\|u_{r}\\right\| & \\le \\left\|\\frac{z_{k+\\ell +1}}{z_{k+\\ell }}\\right\|^{rm}\\sum _{j>k+\\ell }\\left\|\\frac{z_{j}}{z_{k+\\ell +1}}\\right\|^{rm}\\left\|x_{j}\\right\|\\\\\\left(\\text{HÃ¶lder's inequality}\\right) & \\le \\left\|\\frac{z_{k+\\ell +1}}{z_{k+\\ell }}\\right\|^{rm}\\left(\\sum _{j>k+\\ell }\\left\|\\frac{z_{j}}{z_{k+\\ell +1}}\\right\|^{prm}\\right)^{1/p}\\left\\Vert x\\right\\Vert _{q}.$ The sum $\\sum _{j>k+\\ell }\\left\|\\frac{z_{j}}{z_{k+\\ell +1}}\\right\|^{prm}$ is finite since $z\\in \\ell ^{p}\\left(\\mathbb {C}\\right)$ , and is in fact uniformly bounded as a function of $m$ , since every summand has magnitude less than or equal to 1.", "Since $\\left\|z_{k+\\ell +1}\\right\|<\\left\|z_{k+\\ell }\\right\|$ , the term $\\left\|\\frac{z_{k+\\ell +1}}{z_{k+\\ell }}\\right\|^{rm}$ goes to 0 as $m\\rightarrow \\infty $ , and so the entries of $u$ also decrease to 0 as $m\\rightarrow \\infty $ .", "The generators of $V_{m}$ are all distinct, and so $V_{m}$ is invertible.", "Equation (REF ) thus gives $\\tilde{x}=-V_{m}^{-1}u.$ As mentioned in item (REF ), the entries of $u$ decay to 0 in $m$ .", "To show that $\\tilde{x}=0$ (and thus all of $x_{k+s}=0$ for $s=1,\\ldots ,\\ell $ ), it therefore suffices to uniformly bound the infinity-norm $\\left\\Vert V_{m}^{-1}\\right\\Vert _{\\infty }$ for infinitely many $m$ .", "Recall that the inverse of the $\\ell \\times \\ell $ Vandermonde matrix with generators $\\omega _{s}^{m}$ has entries $\\left(V_{m}^{-1}\\right)_{ij}=\\left(-1\\right)^{j-1}\\dfrac{C_{ij}}{\\omega _{i}^{m}\\prod _{\\stackrel{1\\le k\\le \\ell }{k\\ne i}}\\left(\\omega _{i}^{m}-\\omega _{k}^{m}\\right)},$ where $C_{ij}={\\left\\lbrace \\begin{array}{ll}\\sum _{\\stackrel{1\\le k_{1}<\\ldots <k_{n-j}\\le \\ell }{m_{1},\\ldots ,m_{\\ell -j}\\ne i}}\\omega _{m_{1}}\\cdots \\omega _{m_{\\ell -j}} & 1\\le j<\\ell \\\\1 & j=\\ell .\\end{array}\\right.", "}$ Since the generators $\\omega _{s}^{m}$ are all on the unit circle, the numerator $C_{ij}$ is always bounded by a constant independent of $m$ .", "For the denominator, it suffices to show that we can choose infinitely many $m$ such that $\\left\|\\omega _{i}^{m}-\\omega _{j}^{m}\\right\|$ is uniformly bounded away from 0 for all $i\\ne j$ .", "Lemma 15 (Recurrent rotations) Let $\\ell >0$ be an integer and let $\\varepsilon >0$ .", "There exists a constant $C_{\\varepsilon ,\\ell }$ such that for every set of distinct points $\\omega _{1},\\ldots ,\\omega _{\\ell }$ on the unit circle, there is an integer $1\\le m\\le C_{\\varepsilon ,\\ell }$ such that $\\frac{1}{\\pi }\\left\|\\arg \\left(\\omega _{i}^{m}\\right)\\right\|\\le \\varepsilon \\,\\,\\,\\,i=1,\\ldots ,\\ell ,$ where $\\arg :\\mathbb {C}\\rightarrow \\left(-\\pi ,\\pi \\right]$ returns the angle with the origin.", "The proof is by induction on $\\ell $ .", "For $\\ell =1$ , denote $\\omega =e^{\\pi i\\alpha }$ for $\\alpha \\in \\left[-1,1\\right]$ .", "If $\\left\|\\alpha \\right\|\\le \\varepsilon $ , then $m=1$ will do.", "Otherwise, let $k>0$ be the smallest integer such that $\\alpha \\in \\left[-2^{k}\\varepsilon ,2^{k}\\varepsilon \\right]$ .", "Then one of the points $y\\in \\left\\lbrace \\omega ,\\omega ^{2},\\ldots ,\\omega ^{\\left\\lceil 1/\\alpha \\right\\rceil +1}\\right\\rbrace $ satisfies $y=e^{\\pi i\\beta }$ with $\\beta \\in \\left[-2^{k-1}\\varepsilon ,2^{k-1}\\varepsilon \\right]$ : The points $\\omega ,\\omega ^{2},\\ldots ,\\omega ^{\\left\\lceil 1/\\alpha \\right\\rceil +1}$ make at least one complete revolution around the unit circle, but since the angle between two consecutive points is at most $2^{k}\\varepsilon $ , one of these points must fall in the interval $\\left[-2^{k-1}\\varepsilon ,2^{k-1}\\varepsilon \\right]$ .", "Repeating this procedure iteratively, we obtain a sequence of points $y_{1}=e^{\\pi i\\beta _{1}},\\ldots ,y_{q}=e^{\\pi i\\beta _{q}}$ , where: $y_{1}=\\omega =e^{\\pi i\\alpha }.$ $y_{i+1}=y_{i}^{\\ell _{i}}$ for some $\\ell _{i}\\le \\left\\lceil 1/\\left\|\\beta _{i}\\right\|\\right\\rceil +1$ .", "$\\frac{1}{\\pi }\\left\|\\arg \\left(y_{q}\\right)\\right\|\\le \\varepsilon $ The number of iterations $q$ is bounded by $\\log _{2}1/\\varepsilon +1$ .", "Apart from $\\beta _{q}$ , the magnitude of each $\\beta _{i}$ is larger than $\\varepsilon $ .", "Thus each $\\ell _{i}\\le \\left\\lceil 1/\\varepsilon \\right\\rceil +1$ , and we have $\\varepsilon \\ge \\frac{1}{\\pi }\\left\|\\arg \\left(y_{q}\\right)\\right\|=\\frac{1}{\\pi }\\left\|\\arg \\omega ^{\\ell _{1}^{\\ldots ^{\\ell _{q-1}}}}\\right\|=\\frac{1}{\\pi }\\left\|\\arg \\omega ^{\\ell _{1}\\cdot \\ldots \\cdot \\ell _{q-1}}\\right\|,$ yielding an $m\\le \\prod _{j=1}^{q-1}\\ell _{j}\\le \\left(\\left\\lceil 1/\\varepsilon \\right\\rceil +1\\right)^{\\log _{2}1/\\varepsilon }$ .", "We therefore have $C_{\\varepsilon ,1}=\\left(\\left\\lceil 1/\\varepsilon \\right\\rceil +1\\right)^{\\log _{2}1/\\varepsilon }$ .", "Now assume by induction that the statement holds for all $n<\\ell $ .", "By applying the induction hypothesis on the first $\\ell -1$ points with $\\varepsilon ^{\\prime }=\\varepsilon /C_{\\varepsilon ,1}$ , we obtain an integer $m_{1}\\le C_{\\varepsilon ^{\\prime },\\ell -1}$ such that for $i=1,\\ldots ,\\ell -1$ , $\\frac{1}{\\pi }\\left\|\\arg \\left(\\omega _{i}^{m_{1}}\\right)\\right\|\\le \\frac{\\varepsilon }{C_{\\varepsilon ,1}}.$ This does not give any bound on $\\arg \\left(\\omega _{\\ell }^{m_{1}}\\right)$ .", "However, we can now apply the lemma for a single point $\\omega _{\\ell }^{m_{1}}$ and $\\varepsilon $ , obtaining an $m_{2}\\le C_{\\varepsilon ,1}$ such that $\\frac{1}{\\pi }\\left\|\\arg \\left(\\left(\\omega _{\\ell }^{m_{1}}\\right)^{m_{2}}\\right)\\right\|\\le \\varepsilon .$ Choosing $m=m_{1}m_{2}\\le C_{\\varepsilon ,1}C_{\\varepsilon ^{\\prime },\\ell -1}$ yields the required bound on $\\omega _{\\ell }$ ; as for $i<\\ell ,$ using (REF ), we have $\\frac{1}{\\pi }\\left\|\\arg \\left(\\left(\\omega _{i}^{m_{1}}\\right)^{m_{2}}\\right)\\right\|\\le \\frac{\\varepsilon }{C_{\\varepsilon ,1}}\\cdot m_{2}\\le \\varepsilon $ as well.", "We can now finish the proof of Lemma REF .", "If all the ratios $\\left\\lbrace \\arg \\left(\\omega _{s}\\right)/\\pi \\right\\rbrace _{s=1}^{\\ell }$ are rational, then there are infinitely many $m$ 's such that $\\omega _{s}^{m}=\\omega _{s}$ for all $s=1,\\ldots ,\\ell $ .", "The denominator in (REF ) stays the same in this case for all such $m$ .", "Otherwise, there is an $s^{}$ such that $\\arg \\left(\\omega _{s^{}}\\right)$ is an irrational multiple of $\\pi $ .", "Let $\\varepsilon _{n}\\rightarrow 0$ be a positive sequence and let $m_{n}$ be the number of rotations given by Lemma REF applied to $\\left\\lbrace \\omega _{s}\\right\\rbrace _{s=1}^{\\ell }$ with $\\varepsilon _{n}$ .", "Then $m_{n}$ has a subsequence which diverges to infinity, since for any finite set of values of $m$ , $\\arg \\left(\\omega _{s^{}}^{m}\\right)$ is bounded below.", "For all large enough $n$ , we necessarily have $\\left\|\\omega _{i}^{m_{n}+1}-\\omega _{j}^{m_{n}+1}\\right\|\\ge \\frac{1}{2}\\left\|\\omega _{i}-\\omega _{j}\\right\|$ for all $i\\ne j$ , and so the denominator in (REF ) is bounded." ], [ "Other directions and open questions", "At least for processes with continuous paths on the circle $\\mathbb {T}$ (such as Brownian motion), it seems reasonable that it's possible to reconstruct functions which are more complicated than indicators.", "Question 16 What classes of functions are reconstructible from Brownian motion, with or without drift?", "Given a nice enough function $f:\\mathbb {T}\\rightarrow \\mathbb {R}$ , is it perhaps possible to stitch together its level sets $\\left\\lbrace f\\ge \\alpha \\right\\rbrace $ , which we know are reconstructible due to Theorems REF / REF , to gain knowledge about the entire function?", "Question 17 Find another algorithm which shows directly that Brownian motion can reconstruct $\\mathcal {F}_{d}$ , using its local properties.", "Theorem REF suggests that stochastic processes with symmetries should allow reconstruction, up to some symmetry of $\\mathbb {T}^{d}$ itself.", "Question 18 Suppose that some coordinates of $X_{t}$ are independent from others, or that $X_{t}$ is invariant under some orthogonal transformation in $O_{\\mathbb {R}}\\left(d\\right)$ .", "What can be said about reconstruction on $\\mathbb {T}^{d}$ ?", "It is natural to consider larger spaces of functions, larger spaces, and more general distributions.", "Question 19 What can be said for sets $\\Omega $ with fat boundary, i.e $\\mu \\left(\\partial \\Omega \\right)>0$ ?", "Question 20 Can Theorem REF be extended to general compact Riemannian manifolds?", "Question 21 How do the results extend to processes whose step sizes are allowed to contain atoms at $x\\ne 0$ ?", "(consider, for example, a Poisson process which, when its clock fires, jumps either by $\\alpha _{1}$ or $\\alpha _{2}$ , with $\\left\\lbrace \\alpha _{1},\\alpha _{2},\\pi \\right\\rbrace $ rationally independent).", "Finally, the independence condition of the drift in Corollary REF gives rise to a slightly different model for reconstruction on the torus, where we try to learn $f$ from its values on a random (irrational) geodesic.", "Question 22 Let $v$ be a uniformly random unit vector in $\\mathbb {R}^{d}$ .", "Which classes of functions $f$ can be reconstructed (with probability 1) from $f\\left(t\\cdot v\\right)_{t\\in \\mathbb {R}}$ ?", "How about $f\\left(X_{t}\\right)$ , where $X_{t}$ is Brownian motion with random drift $v_{t}$ ?" ] ]	2011.14153
[ [ "Approximation and localized polynomial frame on conic domains" ], [ "Abstract Highly localized kernels constructed by orthogonal polynomials have been fundamental in recent development of approximation and computational analysis on the unit sphere, unit ball and several other regular domains.", "In this work we first study homogeneous spaces that are assumed to contain highly localized kernels and establish a framework for approximation and localized tight frame in such spaces, which extends recent works on bounded regular domains.", "We then show that the framework is applicable to homogeneous spaces defined on bounded conic domains, which consists of conic surfaces and the solid domains bounded by such surfaces and hyperplanes.", "The main results provide a construction of semi-discrete localized tight frame in weighted $L^2$ norm and a characterization of best approximation by polynomials on conic domains.", "The latter is achieved by using a $K$-functional, defined via the differential operator that has orthogonal polynomials as eigenfunctions, as well as a modulus of smoothness defined via a multiplier operator that is equivalent to the $K$-functional.", "Several intermediate results are of interest in their own right, including the Marcinkiewicz-Zygmund inequalities, positive cubature rules, Christoeffel functions, and several Bernstein type inequalities.", "Moreover, although the highly localizable kernels hold only for special families of weight functions on each domain, many intermediate results are shown to hold for doubling weights defined via the intrinsic distance on the domain." ], [ "Introduction", "In recent years highly localized kernels constructed via orthogonal polynomials have become important tools in approximation theory, both computational and theoretical harmonic analysis, and functional analysis.", "We start with an introduction on the background of this development in the first subsection, describe our main results in the second subsection and state the organization of the work in the third subsection." ], [ "Background", "Let $\\Omega $ be a set in ${\\mathbb {R}}^d$ , either an algebraic surface or a domain with non-empty interior, and let $\\varpi $ be a nonnegative weight function defined on $\\Omega $ , normalized with unit integral with respect to the Lebesgue measure $\\mathrm {d}x$ on $\\Omega $ , so that the bilinear form ${\\langle }f, g{\\rangle }_{\\varpi } = \\int _{\\Omega } f(x) g(x) \\varpi (x) \\mathrm {d}x$ is a well defined inner product on the space of polynomials restricted to $\\Omega $ .", "Let ${\\mathcal {V}}_n^d(\\varpi )$ be the space of orthogonal polynomials of degree $n$ with respect to this inner product.", "Assume that $\\varpi $ is regular so that the orthogonal decomposition $L^2(\\Omega , \\varpi ) = \\bigoplus _{n=0}^\\infty {\\mathcal {V}}_n^d(\\varpi )$ holds.", "Let $P_n(\\varpi ; \\cdot ,\\cdot )$ be the reproducing kernel of the space ${\\mathcal {V}}_n^d(\\varpi )$ .", "The projection operator $\\operatorname{proj}_n: L^2(\\Omega , \\varpi )\\mapsto {\\mathcal {V}}_n^d(\\varpi )$ can be written as $(\\operatorname{proj}_n f)(x) =\\int _\\Omega P_n(\\varpi ; x,y)f(y)\\varpi (y) \\mathrm {d}y, \\qquad f\\in L^2(\\Omega , \\varpi ).$ If $\\widehat{a}$ is a cut-off function, defined as a compactly supported function in $C^\\infty ({\\mathbb {R}}_+)$ and $\\widehat{a}(t) = 1$ for $t$ near zero, then our highly localized kernels are of the form $L_n(\\varpi ;x,y) := \\sum _{j=0}^\\infty \\widehat{a} \\Big (\\frac{j}{n}\\Big ) P_j(\\varpi ; x,y).$ For orthogonal systems on several regular domains, the kernel $L_n(\\varpi ;x,y)$ for a suitable $\\widehat{a}$ satisfies a localization principle, meaning that the kernel decays at rates faster than any inverse polynomial rate away from the main diagonal $y=x$ in $\\Omega \\times \\Omega $ with respect to an intrinsic distance in $\\Omega $ .", "To get a sense of the highly localized estimate, we specify to two cases.", "The first one is the unit sphere ${\\mathbb {S}^{d-1}}$ of ${\\mathbb {R}}^d$ with $\\varpi (x) \\mathrm {d}x = \\mathrm {d}\\sigma $ is the surface measure, for which the kernel $L_n(\\cdot ,\\cdot ) =L_n(\\varpi ; \\cdot ,\\cdot )$ is highly localized in the sense that [34], for every ${\\kappa }> 0$ , $ \|L_n(x,y)\| \\le c_{\\kappa }\\frac{n^{d-1}}{(1+n \\,{\\mathsf {d}}_{\\mathbb {S}}(x,y))^{\\kappa }}, \\qquad x,y \\in {\\mathbb {S}^{d-1}},$ where ${\\mathsf {d}}_{\\mathbb {S}}$ denotes the geodesic distance $ {\\mathsf {d}}_{\\mathbb {S}}(x,y) = \\arccos {\\langle }x,y{\\rangle }, \\qquad x, y \\in {\\mathbb {S}^{d-1}}.$ For the unit ball, the classical weight function is $\\varpi _\\mu (x) = (1-\\Vert x\\Vert ^2)^{\\mu -\\frac{1}{2}}$ , $\\mu > -\\frac{1}{2}$ .", "For $\\mu \\ge 0$ , the kernel $L_n(\\varpi _\\mu ; \\cdot ,\\cdot )$ is highly localized in the sense that [38] $ \\left\|L_n(\\varpi _\\mu ;x,y)\\right\| \\le c_{\\kappa }\\frac{n^d}{\\sqrt{\\varpi _\\mu (n;x)}\\sqrt{\\varpi _\\mu (n;y)}(1+n\\, {\\mathsf {d}}_{\\mathbb {B}}(x,y))^{\\kappa }}, \\qquad x,y \\in {\\mathbb {B}}^d,$ where $\\varpi _\\mu (n;x) = (1-\\Vert x\\Vert ^2 + n^{-2})^\\mu $ and ${\\mathsf {d}}_{\\mathbb {B}}$ is the distance on ${\\mathbb {B}}^d$ defined by $ {\\mathsf {d}}_{\\mathbb {B}}(x,y) = \\arccos \\left({\\langle }x, y{\\rangle }+ \\sqrt{1-\\Vert x\\Vert ^2}\\sqrt{1-\\Vert y\\Vert ^2}\\right).$ By choosing ${\\kappa }$ large, the kernel decays away form $y =x$ by (REF ) and (REF ) faster than any polynomial rate.", "Furthermore, under a technical restriction on the cut-off function, both estimates can be improved to sub-exponential rate [28].", "The proof of these estimates relies on closed-form formula of the reproducing kernel $P_n(\\varpi ; \\cdot ,\\cdot )$ on the unit sphere and the unit ball; see the next subsection.", "Let us denote the integral operator with $L_n(\\varpi )$ as its kernel by $L_n * f$ ; that is, $ L_n * f (x): = \\int _\\Omega f(y)L_n(\\varpi ; x,y) \\varpi (y) \\mathrm {d}y.$ If $\\widehat{a}$ has the support $[0,2]$ and satisfies $\\widehat{a}(t) =1$ for $0\\le t \\le 1$ , then the operator $L_nf$ is a polynomial of degree $2n$ and it reproduces polynomials of degree $n$ .", "Furthermore, in many cases, it provides the near best polynomial approximation to $f \\in L^p(\\Omega , \\varpi )$ in the sense that $\\Vert L_n f -f \\Vert _{L^p(\\Omega , \\varpi )} \\le c \\inf _{\\deg g \\le n} \\Vert f - g \\Vert _{L^p(\\Omega , \\varpi )}, \\quad 1 \\le p \\le \\infty .$ The rapid decay of the kernel makes it a powerful tool for studying polynomial approximation on $\\Omega $ .", "Moreover, the operator $L_n* f$ plays an important role in the characterization of best approximation, which is a central problem in approximation theory.", "For the unit sphere, for example, it is used in the classical result that characterizes the error of best polynomial approximation by a modulus of smoothness defied via spherical means or by its equivalent $K$ -functional defined via the Laplace-Beltrami operator on the sphere.", "The theory has a long history that spans from [4], [36] to [42] with contributions from many researchers in between.", "It has also been extended, more recently, to weight approximation on the sphere, the unit ball, and the simplex in [52].", "The highly localized kernels can also be used to construct localized frames via a semi-continuous Calderón type decomposition based on the decomposition, $f = \\sum _{j=0}^\\infty L_{2^j} \\ast L_{2^j} \\ast f, \\qquad f\\in L^2(\\Omega , \\varpi ),$ which holds if the cut-off function $\\widehat{a}$ satisfies $\\sum _{k=0}^\\infty \|\\widehat{a}(2^{-k} t)\|^2 = 1$ .", "To define the frame elements, the integrals in the right-hand side of the above expansion need to be appropriately discretized by a cubature rule, established with the help of the highly localized kernels, which states that $\\int _\\Omega f(x)\\mathrm {d}x = \\sum _{\\xi \\in \\Xi _j} {\\lambda }_{\\xi } f(\\xi ), \\qquad f \\in \\Pi _{2^j}(\\Omega ),$ where ${\\lambda }_{\\xi } > 0$ , $\\Xi _j$ is a finite subset in $\\Omega $ , and $\\Pi _n(\\Omega )$ denotes the space of polynomials, restricted on $\\Omega $ , of degree at most $n$ .", "The discretization leads to the tight frame $\\lbrace \\psi _\\xi \\rbrace _{\\xi \\in \\Xi }$ of $L^2(\\Omega , \\varpi )$ , where $\\Xi $ is a discrete set of well separated, with respect to the intrinsic distance, points in $\\Omega $ , and the frame elements $\\psi _\\xi $ , also called needlets, are of the form $\\psi _\\xi (x) = \\sqrt{{\\lambda }_{\\xi }} L_{2^j} (x,\\xi ), \\qquad \\xi \\in \\Xi ,$ which is highly localized with its center at $\\xi $ since ${\\lambda }_\\xi $ can be quantized and $L_{2^j} (x,\\xi )$ is highly localized.", "The tight frame means that, for all $f \\in L^2(\\Omega , \\varpi )$ , $f = \\sum _{\\xi \\in \\Xi } {\\langle }f,\\psi _\\xi {\\rangle }_{\\varpi } \\psi _\\xi \\quad \\hbox{and} \\quad \\int _\\Omega \|f(x)\|^2 \\varpi (x) \\mathrm {d}x = \\sum _{\\xi \\in \\Xi } \|{\\langle }f, \\psi _\\xi {\\rangle }\|^2.$ Because of their rapid decay, needlets can be used for decomposition of spaces of functions and distributions in various settings, including $L^p$ , Sobolev, Besov, and Triebel- Lizorkin spaces.", "The above narrative has been carried out in the setting of the regular domains.", "The localized kernels are most heavily studied and used when $\\Omega $ is the unit sphere ${\\mathbb {S}^{d-1}}$ of ${\\mathbb {R}}^d$ , for which our outline of the localized frames is initiated in [34], [35].", "They are further explored in, for example, [12], [28], on the sphere, and also studied on the interval $[-1,1]$ with the Jacobi weight [5], [24], [37], the unit ball with the Gegenbauer weight [25], [38], and the simplex with the multivariable Jacobi weight [28] as well as for ${\\mathbb {R}}_+^d$ with the product Laguerre weight and ${\\mathbb {R}}^d$ with the Hermite weight [15], [18], [20], [39] and for product domains [29].", "The subexponential decay of the kernel was established in [28], following earlier work in [16].", "The needlets on the sphere and the ball have found applications in computational harmonics analysis, mathematical physics and statistics; see, for example, [2], [3], [26], [27], [31], [49] and references in them." ], [ "New contributions", "Our starting point is the recent study of orthogonal structures on conic domains of ${\\mathbb {R}}^{d+1}$ in [53], [54], which shows that many properties of spherical harmonics and classical orthogonal polynomials on the unit ball have analogs on conic domains.", "Most importantly, there are families of orthogonal polynomials on conic domains that possess properties akin to the addition formula and the Laplace-Beltrami operator for the spherical harmonics, the former provides a closed-form formula for the reproducing kernels and the latter is a second order differential operator that has orthogonal polynomials as eigenfunctions.", "This paves the way for carrying out the narrative outlined in the previous subsection on conic domains, which has been a topic largely untouched hitherto.", "In the present paper, we will deal with two types of conic domains and they are standardized as: conic surface ${\\mathbb {V}}_0^{d+1}$ defined by ${\\mathbb {V}}_0^{d+1}: = \\left\\lbrace (x,t) \\in {\\mathbb {R}}^{d+1}: \\Vert x\\Vert = t, \\, 0 \\le t \\le 1, \\, x\\in {\\mathbb {R}}^d\\right\\rbrace ;$ solid cone ${\\mathbb {V}}^{d+1}$ bounded by ${\\mathbb {V}}_0^{d+1}$ and the hyperplane $t =1$ .", "Two other conic domains, double hyperbolic surface and solid hyperboloid, will be dealt with elsewhere because of the page limitation.", "On each domain, a closed-form formula of the reproducing kernels is uncovered for a family of weight functions, which allows us to study highly localized kernels and carries out the program described in Subsection REF , and a differential operator is defined that has orthogonal polynomials as eigenfunctions, which leads to a $K$ -functional that is used to characterize the best approximation by polynomials on the domain.", "Instead of dealing with each case individually, we will develop a unified theory for all homogeneous spaces that contain highly localized kernels that we describe in the first two subsections below.", "The unified theory is applicable to conic domains, which is described in the third and the fourth subsections." ], [ "Localizable homogeneous space", "A homogeneous space is a measure space $(\\Omega , \\mathrm {d}\\mu , {\\mathsf {d}})$ , where $\\mathrm {d}\\mu $ is a nonnegative doubling measure with respect to the metric ${\\mathsf {d}}(\\cdot ,\\cdot )$ on $\\Omega $ .", "When $\\mathrm {d}\\mu = {\\mathsf {w}}(x) \\mathrm {d}x$ , where ${\\mathsf {w}}$ is a doubling weight function on $\\Omega $ , we denote the homogenous space by $(\\Omega , {\\mathsf {w}}, {\\mathsf {d}})$ .", "For example, the space $({\\mathbb {S}^{d-1}}, \\mathrm {d}\\sigma , {\\mathsf {d}}_{\\mathbb {S}})$ is a homogeneous space, so is $({\\mathbb {B}}^d, \\varpi _\\mu , {\\mathsf {d}}_{\\mathbb {B}})$ for $\\mu > -\\frac{1}{2}$ .", "We call a homogeneous space $(\\Omega , \\varpi , {\\mathsf {d}})$ localizable if the orthogonal polynomials with respect to $\\varpi $ on $\\Omega $ admit highly localized kernels, which requires a fast decay estimate on $\|L_n(\\varpi ; x, y)\|$ as well as on $\|L_n(\\varpi ; x_1,y) -L_n(\\varpi ; x_2,y)\|$ ; see Subsection REF for precise definition.", "Our first main result is to show that the narrative on the localized tight frame in the previous subsection can be established in a localizable homogenous space.", "This provides a unified theory that is applicable to conical domains to be studied in later sections.", "Furthermore, it allows us to establish several intermediate results that are of interest in their own right for doubling weight functions, including Marcinkiewicz-Zygmund inequalities and positive cubature rules.", "The development generalizes and follows closely recent studies on regular domains, especially those on the unit sphere and the unit ball.", "We follow the approach on the unit sphere in [7], [12], which is formulated in the homogeneous space on the sphere.", "While some of the results for the unit sphere can be extended easily, others need to be dealt with more carefully.", "Amongst various reasons, we mention that, on the unit sphere, it is the Lebesgue measure $\\mathrm {d}\\sigma $ that admits the highly localized kernels.", "It is, however, not the case in general.", "For example, the highly localized kernels on the conic surface ${\\mathbb {V}}_0^{d+1}$ are established for the weight function ${\\mathsf {w}}_{-1,{\\gamma }}(t)= t^{-1}(1-t)^{\\gamma }$ , which does not include the Lebesgue measure.", "While spherical caps of the same radius on ${\\mathbb {S}^{d-1}}$ have the same constant surface area, this is not the case on ${\\mathbb {V}}_0^{d+1}$ , where we need to deal with conical caps defined via the intrinsic distance ${\\mathsf {d}}_{{\\mathbb {V}}_0}$ , defined in (REF ) below, and their volumes are measured by integrals over the caps against the weight function ${\\mathsf {w}}_{-1,{\\gamma }}$ ." ], [ "Best approximation by polynomials", "The characterization of best approximation is at the core of approximation theory (cf.", "[17]).", "The classical approach on the unit sphere relies essentially on multiplier operators of the Fourier harmonic series and the convolution structure arising from the addition formula.", "We provide a general framework on localized homogeneous space by assuming that the reproducing kernel $P_n(\\varpi ; \\cdot ,\\cdot )$ satisfies an addition formula of a specific form satisfied by all known cases, including those on conic domains, and that there is a second order differential operator that has orthogonal polynomials as eigenfunctions.", "The former allows us to introduce a convolution structure that can be used to define a modulus of smoothness via a multiplier operator, and the latter is used to define a $K$ -functional.", "The two quantities are then shown to be equivalent and either one can be used to characterize the error of the best polynomial approximation on the domain.", "The framework generalizes the classical result on the unit sphere and is applicable to other regular domains satisfying our assumptions, which are met by all four types of conic domains." ], [ "Conic surface ${\\mathbb {V}}_0^{d+1}$ and cone {{formula:2236a5f1-946c-49dc-ba2d-403836ff020b}}", "On the conic surface ${\\mathbb {V}}_0^{d+1}$ , the orthogonal structure is studied in [53] for the weight function ${\\mathsf {w}}_{{\\beta },{\\gamma }}(t) = t^{\\beta }(1-t)^{\\gamma }$ for ${\\beta }> - d$ and ${\\gamma }> -1$ .", "For our study we need an intrinsic metric on the conic surface.", "In contrast to the unit sphere with its geodesic distance ${\\mathsf {d}}_{\\mathbb {S}}(\\cdot ,\\cdot )$ that depend only on relative positions of the points, the conic surface ${\\mathbb {V}}_0^{d+1}$ has a boundary at $t =1$ and an apex point at $t=0$ that its metric needs to take into account.", "The distance that we will use is defined by ${\\mathsf {d}}_{{\\mathbb {V}}_0}( (x,t), (y,s)) = \\arccos \\left[\\sqrt{\\frac{{\\langle }x ,y {\\rangle }+ t s}{2}} + \\sqrt{1-t} \\sqrt{1-s} \\right].$ Using this distance function and the closed-form formula of the reproducing kernel proved in [53], we shall show that ${\\mathsf {w}}_{{\\beta },{\\gamma }}$ is a doubling weight and that the space $(\\Omega , {\\mathsf {w}}_{-1,{\\gamma }}, {\\mathsf {d}}_{{\\mathbb {V}}_0})$ is a localizable homogeneous space.", "These results are established through delicate estimates, partly because the distance ${\\mathsf {d}}_{{\\mathbb {V}}_0}(\\cdot ,\\cdot )$ is not comparable to the Euclidean distance at around the apex of ${\\mathbb {V}}_0^{d+1}$ .", "With the highly localized kernel, we can then carry out our program and establish positive cubature rules, and use them to establish the highly localized frame.", "Furthermore, using the second order differential operator that has orthogonal polynomials with respect to ${\\mathsf {w}}_{-1,{\\gamma }}$ as eigenfunctions, which is an analog of the Laplace-Beltrami operator in the unit sphere, we can define a $K$ -functional and use it to establish a characterization of the best approximation by polynomials in the weighted $L^p$ norm on ${\\mathbb {V}}_0^{d+1}$ .", "Our study on the cone ${\\mathbb {V}}^{d+1}$ follows along the similar line.", "The distance function is defined by identifying ${\\mathbb {V}}^{d+1}$ with a subset of ${\\mathbb {V}}_0^{d+2}$ .", "The orthogonal structure is studied in [53] for the weight function $W_{{\\beta },{\\gamma },\\mu }(x,t) = t^{\\beta }(1-t)^{\\gamma }(t^2-\\Vert x\\Vert ^2)^{\\mu -\\frac{1}{2}}$ , which has a more involved close form formula for its reproducing kernels.", "While part of the estimate can be carried out for the kernel associated with $W_{0,{\\gamma },\\mu }$ , the localizable homogeneous space is established for $W_{0,{\\gamma },0}$ , which nevertheless allows us to complete our program on the solid cone ${\\mathbb {V}}^{d+1}$ .", "Several remarks are in order over the above description of our main results.", "First of all, analysis on conical domains have not been seriously studied in the literature as far as we are aware, and it does pose new phenomena and new challenges.", "In particular, while the framework for localizable homogeneous space is applicable to conic domains, the verification of assertions under which the framework holds possess considerable difficulty in each case.", "Second, there are many intermediate results that are of interests.", "For example, to quantize the coefficients in positive cubature rule, we need bounds on the Christoeffel functions $ {\\lambda }_n({\\mathsf {w}};x) = \\inf _{\\begin{array}{c}g(x) =1 \\\\ g \\in \\Pi _n(\\Omega )\\end{array}} \\int _{\\Omega } \|g(x)\|^2 {\\mathsf {w}}(x) \\mathrm {d}x,$ which are of interest in their own right.", "See, for example, [6], [14], [22], [23], [40], [41], [50] and their references for recent works on Christoffel functions in several variables.", "While the lower bound of the Christoeffel function on a homogenous space can be derived using highly localized kernels when ${\\mathsf {w}}$ admits such kernels, we will establish the upper bound for all doubling weight under the assumption that certain fast decaying polynomials exist on $\\Omega $ .", "The latter requires the construction of such polynomials in each conic domain.", "Third, there have been several recent works that deal with multivariate approximation by polynomials or Marcinkiewicz-Zygmund inequalities and positive cubature rules on either polyhedra domain or $C^2$ domain [8], [9], [47], [48].", "They do not cover conical domains and our approach is different and relies on a specific orthogonal structure on the domain, akin to those on the unit sphere and the unit ball.", "Finally, we should mention the important book Analysis on Symmetric Cones [19] and the literature around it.", "While the topic works in the theory of Euclidean Jordan algebras and lies in a more abstract setting, our study requires very specific structures of Fourier orthogonal series and is far less abstract.", "We have not been able to discern a connection between the two topics duo to our lack of background in Jordan algebras.", "It would be of great interest if a connection could be identified." ], [ "Organization and convention", "The paper is organized as follows.", "The localizable homogeneous space will be defined and studied in the next section, which consists of results discussed in Subsection 1.2.1.", "The best approximation by polynomials in the homogeneous space is discussed in the third section, which consists of the results described in Subsection 1.2.2.", "The results on the conic surface and solid cone will be discussed in Sections 4 and 5, respectively.", "Each section will contain several subsections and its organization will be described in the preamble of the section.", "Throughout this paper, we will denote by $L^p(\\Omega , {\\mathsf {w}})$ the weighted $L^p$ space with respect to the weight function ${\\mathsf {w}}$ defined on the domain $\\Omega $ for $1 \\le p \\le \\infty $ .", "Its norm will be denote by $\\Vert \\cdot \\Vert _{p,{\\mathsf {w}}}$ for $1 \\le p \\le \\infty $ with the understanding that the space is $C(\\Omega )$ with the uniform norm when $p= \\infty $ .", "For conical domains, in an attempt to distinguish conic surfaces and solid conic bodies, we shall denote the operator on the surface in sans serif font, such as ${\\mathsf {P}}_n$ and ${\\mathsf {L}}_n$ , and the operator on the solid domains in bold font, such as ${\\mathbf {P}}_n$ and ${\\mathbf {L}}_n$ .", "Finally, we shall use $c$ , $c^{\\prime }$ , $c_1$ , $c_2$ etc.", "to denote positive constants that depend on fixed parameters and their values may change from line to line.", "Furthermore, we shall write $A \\sim B$ if $ c^{\\prime } A \\le B \\le c A$ ." ], [ "Homogeneous spaces with highly localized kernels", "Much of recent work on regular domains, such as the sphere, the ball, and the regular simplex falls in the framework of homogeneous space.", "What distinguishes these regular domains is the existence of highly localized kernels constructed via orthogonal polynomials.", "From the prior work on these regular domains and what we will prove for the conic domains latter sections, it becomes clear that the desired estimates for the highly localized kernels share a common formation.", "In this section, we work with homogeneous spaces that are assumed to contain highly localized kernels and carry out our analysis on such spaces.", "The first subsection contains basics on homogeneous space and orthogonal polynomials, and it contains three assertions that define highly localized kernels precisely.", "Three examples are given in the second subsection, which also serves as a review of the Jacobi polynomials, spherical harmonics, and orthogonal polynomials on the ball that will be used later.", "An auxiliary maximal function is introduced in the third subsection and, with the help of highly localized kernels for $\\varpi $ , is shown to be bounded in weighted $L^p$ space with respect to a doubling weight.", "Using this maximal function, Marcinkiewicz-Zygmund inequalities are established in the fourth subsection for all doubling weights.", "The fifth subsection contains the fourth assertion that is used to obtain an upper bound for the Christoeffel function.", "The latter is needed in the sixth subsection, where the positive cubature rules are established.", "With these preparations, the localized tight frame is defined and studied in the seventh subsection.", "The development of this section follows and generalizes the study on the unit sphere.", "Some parts of the work follow those on the sphere with little extra effort but by no means all.", "Since the geodesic distance on the sphere and the surface measure on the unit sphere will be replaced by a distance function and a weight function that offer no specific geometric information, we will provide complete proofs in the most part and will be brief only when the proof follows that on the unit sphere fairly straightforwardly." ], [ "Homogeneous spaces", "A homogeneous space is a measure space $(\\Omega ,\\mu , {\\mathsf {d}})$ with a positive measure $\\mu $ and a metric ${\\mathsf {d}}$ such that all open balls $B(x,r)=\\lbrace y \\in \\Omega : {\\mathsf {d}}(x,y) < r\\rbrace ,$ are measurable and $\\mu $ is a regular measure satisfying the doubling property $\\mu (B(x,2r)) \\le c \\mu (B(x,r)), \\qquad \\forall x \\in \\Omega , \\quad \\forall r >0,$ where $c$ is independent of $x$ and $r$ .", "Such a measure $\\mu $ is called a doubling measure.", "Let $\\Omega $ be a domain in ${\\mathbb {R}}^d$ .", "We assume that $\\Omega $ is equipped with an intrinsic distance ${\\mathsf {d}}$ , so that the Lebesuge measure ${\\mathsf {m}}$ on $\\Omega $ is a doubling measure and, for each ${\\varepsilon }> 0$ , $\\min _{x \\in \\Omega } {\\mathsf {m}}( B(x,{\\varepsilon })) \\ge c_{\\varepsilon }> 0$ .", "Let ${\\mathsf {w}}$ be a nonnegative integrable function defined on $\\Omega $ .", "For a given set $E \\subset \\Omega $ , we define ${\\mathsf {w}}(E) = \\int _{E} {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x).$ The weight function ${\\mathsf {w}}$ is called a doubling weight if its satisfies the doubling condition: there exists a constant $L > 0$ such that ${\\mathsf {w}}(B(x,2 r)) \\le L {\\mathsf {w}}(B(x,r)), \\qquad \\forall x \\in \\Omega , \\quad r \\in (0, r_0),$ where $r_0$ is the largest positive number such that $B(x,r) \\subset \\Omega $ .", "If ${\\mathsf {w}}$ is a doubling weight, then ${\\mathsf {d}}\\mu = {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}$ is a doubling measure.", "We are particularly interested in the homogeneous space with its measure so defined, which we denote by $(\\Omega , {\\mathsf {w}}, {\\mathsf {d}})$ .", "For a doubling weight ${\\mathsf {w}}$ on $\\Omega $ with respect to ${\\mathsf {d}}$ , we denote by $L({\\mathsf {w}})$ the least constant $L$ for the doubling condition ${\\mathsf {w}}(B(x,2 r)) \\le L {\\mathsf {w}}(B(x,r))$ .", "Let ${\\alpha }({\\mathsf {w}})$ be a positive number such that $ \\sup _{B(x, r) \\subset \\Omega } \\frac{ {\\mathsf {w}}(B(x,2^m r))}{ {\\mathsf {w}}(B(x, r))} \\le c_{L({\\mathsf {w}})} 2^{{\\alpha }({\\mathsf {w}})m}, \\quad m=1,2,\\ldots .$ Then ${\\alpha }({\\mathsf {w}}) \\le \\log _2 L({\\mathsf {w}})$ , as can be seen by iteration.", "We call $L({\\mathsf {w}})$ the doubling constant and ${\\alpha }({\\mathsf {w}})$ the doubling index.", "Lemma 2.1 Let ${\\mathsf {w}}$ be a doubling weight on $\\Omega $ .", "If $0 < r < t$ and $x \\in \\Omega $ , then ${\\mathsf {w}}(B(x,t)) \\le c_{L({\\mathsf {w}})} \\left( \\frac{t}{r} \\right)^{{\\alpha }({\\mathsf {w}})} {\\mathsf {w}}(B(x,r)).$ For $x,y \\in \\Omega $ and $n=1,2,\\ldots $ , ${\\mathsf {w}}(B(x, n^{-1})) \\le c_{L({\\mathsf {w}})} (1+n {\\mathsf {d}}(x,y))^{\\alpha ({\\mathsf {w}})} {\\mathsf {w}}(B(y, n^{-1})).$ Assume $2^{m-1} \\le t/r \\le 2^m$ , (i) follows immediately from the definition of ${\\alpha }({\\mathsf {w}})$ .", "By the triangle inequality of the distance function, ${\\mathsf {w}}(B(x,n^{-1})) \\le {\\mathsf {w}}(B(y, {\\mathsf {d}}(x,y)+n^{-1}))$ , applying (i) to bound it by ${\\mathsf {w}}(B(y, n^{-1}))$ proves (ii).", "Many fundamental results that hold for the Euclidean space can be extended to homogeneous spaces [44].", "For example, the Hardy-Littlewood maximal function is defined, for a locally integrable function $f$ , by $M_{\\mathsf {w}}f(x) := \\sup _{r> 0} \\frac{1}{{\\mathsf {w}}(B(x,r))} \\int _{B(x,r)} \|f(y)\| {\\mathsf {w}}(y) \\mathrm {d}{\\mathsf {m}}, \\qquad x \\in \\Omega ,$ which possesses the usual properties of the maximal function [44].", "For example, for every $f \\in L^p(\\Omega , {\\mathsf {w}})$ , $\\Vert M_{\\mathsf {m}}f\\Vert _{p,{\\mathsf {w}}} \\le c_p \\Vert f\\Vert _{p,{\\mathsf {w}}}, \\qquad 1 < p \\le \\infty ,$ where $\\Vert f\\Vert _{p,{\\mathsf {w}}}$ denotes the $L^p$ norm with respect to the measure ${\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}$ .", "We study homogeneous spaces that contain highly localized kernels, to be defined below.", "Such spaces are known to hold for certain weight functions on several regular domains.", "We adopt the convention of denoting the weight function that admits highly localized kernels by $\\varpi $ and we also assume that $\\varpi $ is normalized by $\\int _\\Omega \\varpi (x) \\mathrm {d}{\\mathsf {m}}=1$ .", "The kernels are defined via orthogonal polynomials with respect to the inner product ${\\langle }\\cdot ,\\cdot {\\rangle }_\\varpi $ defined in terms of $\\varpi $ in (REF )." ], [ "Orthogonal polynomials", "Let $\\Pi (\\Omega )$ denote the space of polynomials restricted to $\\Omega $ and let $\\Pi _n(\\Omega )$ denote its subspace of polynomials of degree at most $n$ .", "If the interior of $\\Omega $ is open, then $\\Pi (\\Omega )$ contains all polynomials of degree $n$ in $d$ variables and we write $\\Pi _n = \\Pi _n(\\Omega )$ .", "If $\\Omega $ is an algebraic surface, such as the sphere ${\\mathbb {S}^{d-1}}$ , then $\\Pi _n(\\Omega )$ contains all polynomials restricted to the surface.", "Let $\\varpi $ be a normalized weight function on $\\Omega $ .", "Then the bilinear form ${\\langle }\\cdot ,\\cdot {\\rangle }_\\varpi $ ${\\langle }f, g{\\rangle }_{\\varpi } := \\int _{\\Omega } f(x) g(x) \\varpi (x) \\mathrm {d}x$ is a well defined inner product on $\\Pi _n(\\Omega )$ .", "Let ${\\mathcal {V}}_n^d(\\Omega ,\\varpi )$ be the space of orthogonal polynomials of degree $n$ with respect to this inner product.", "We are particularly interested in the case when $\\Omega $ is either a quadratic surface, such as the unit sphere ${\\mathbb {S}^{d-1}}$ or the conical surface ${\\mathbb {V}}_0^{d}$ , for which $ \\dim {\\mathcal {V}}_n^d(\\Omega ,\\varpi ) = \\binom{n+d-2}{n} + \\binom{n+d-3}{n-1}, \\quad n = 1,2,3,\\ldots ,$ where we assume $\\binom{n}{k} =0$ if $k < 0$ , or a solid domain bounded by a quadratic surface, such as the unit ball ${\\mathbb {B}}^d$ and the solid cone ${\\mathbb {V}}^d$ , for which $ \\dim {\\mathcal {V}}_n^d(\\Omega ,\\varpi ) = \\binom{n+d-1}{n}, \\quad n = 0, 1,2,\\ldots ,$ The reproducing kernel of ${\\mathcal {V}}_n^d(\\Omega ,\\varpi )$ , denoted by $P_n(\\varpi ; \\cdot ,\\cdot )$ , is uniquely determined by $\\int _\\Omega f(y) P_n(\\varpi ; x, y) \\varpi (y)\\mathrm {d}{\\mathsf {m}}(y) = f(x), \\qquad \\forall f\\in {\\mathcal {V}}_n^d(\\Omega ,\\varpi ).$ If $\\lbrace P_{\\nu ,n}: 1 \\le \\nu \\le \\dim {\\mathcal {V}}_n^d(\\Omega ,\\varpi )\\rbrace $ is an orthogonal basis of ${\\mathcal {V}}_n^d(\\Omega ,\\varpi )$ , then $P_n(\\varpi ; x, y) = \\sum _{\\nu =1}^{\\dim {\\mathcal {V}}_n^d(\\Omega ,\\varpi )} \\frac{P_{\\nu ,n}(x) P_{\\nu ,n}(y)}{{\\langle }P_{\\nu ,n},P_{\\nu ,n} {\\rangle }_\\varpi }.$ The kernel plays an essential role in the study of the Fourier orthogonal series, since it is the kernel of the orthogonal projection operator $\\operatorname{proj}_n: L^2(\\Omega , \\varpi )\\mapsto {\\mathcal {V}}_n^d(\\Omega , \\varpi )$ ; more precisely, $\\operatorname{proj}_n(\\varpi ; f, x) = \\int _\\Omega f(y) P_n(\\varpi ; x,y) \\varpi (y) \\mathrm {d}{\\mathsf {m}}(y), \\quad f\\in L^2(\\Omega , \\varpi ).$ The Fourier orthogonal series of $f \\in L^2(\\Omega ,\\varpi )$ is defined by $f = \\sum _{n=0}^\\infty \\operatorname{proj}_n(\\varpi ; f) = \\sum _{n=0}^\\infty \\sum _{\\nu =1}^{\\dim {\\mathcal {V}}_n^d(\\Omega ,\\varpi )}\\widehat{f}_{\\nu ,n} P_{\\nu ,n},\\quad \\widehat{f}_{\\nu ,n} = \\frac{{\\langle }f, P_{\\nu ,n}{\\rangle }_\\varpi }{{\\langle }P_{\\nu ,n},P_{\\nu ,n} {\\rangle }_\\varpi }.$ For these definitions and orthogonal polynomials of several variables in general, we refer to [14].", "Our highly localized kernels are defined by a sampling of the kernels of the Fourier series through a smooth cut-off function, which we define first.", "Definition 2.2 A nonnegative function $\\widehat{a} \\in C^\\infty ({\\mathbb {R}})$ is said to be admissible if it obeys either one of the conditions $\\mathrm {supp}\\, \\widehat{a} \\subset [0, 2]$ and $\\widehat{a}(t) = 1$ , $t\\in [0, 1]$ ; or $\\mathrm {supp}\\, \\widehat{a} \\subset [1/2, 2]$ .", "Let $\\widehat{a}$ be an admissible cut-off function.", "For $n = 0,1,2,\\ldots $ , we define the kernel function $L_n(\\varpi )$ on $\\Omega \\times \\Omega $ by $ L_n (\\varpi ; x,y) = \\sum _{k=0}^{\\infty } \\widehat{a} \\left(\\frac{k}{n}\\right) P_n(\\varpi ; x,y), \\qquad x,y \\in \\Omega ,$ Since $\\widehat{a}$ is supported on $[0,2]$ , this is a kernel of polynomials of degree at most $2n$ in either $x$ or $y$ variable." ], [ "Localizable homogeneous spaces", "The main purpose of this section is to study homogeneous spaces $(\\Omega , \\varpi , {\\mathsf {d}})$ for which the kernels $L_n(\\varpi ; \\cdot ,\\cdot )$ are assumed to satisfy the following assertions.", "Definition 2.3 The kernels $L_n(\\varpi ; \\cdot ,\\cdot )$ , $n=1,2,\\ldots $ , are called highly localized if they satisfy the following assertions: Assertion 1.", "For ${\\kappa }> 0$ and $x, y\\in \\Omega $ , $\|L_n(\\varpi ; x ,y)\| \\le c_{\\kappa }\\frac{1}{\\sqrt{\\varpi \\!\\left(B(x,n^{-1})\\right)} \\sqrt{\\varpi \\!\\left(B(y,n^{-1})\\right)}\\left(1+n {\\mathsf {d}}(x,y)\\right)^{\\kappa }}.$ Assertion 2.", "For $0 < \\delta \\le \\delta _0$ with some $\\delta _0<1$ and $x_1 \\in B(x_2, \\frac{\\delta }{n})$ , $\\left\|L_n(\\varpi ; x_1,y) - L_n(\\varpi ; x_2,y)\\right\| \\le c_{\\kappa }\\frac{n {\\mathsf {d}}(x_1,x_2)}{\\sqrt{\\varpi \\!\\left(B(x_1,n^{-1})\\right)} \\sqrt{\\varpi \\!\\left(B(x_2,n^{-1})\\right)} \\left(1+n {\\mathsf {d}}(x_2,y)\\right)^{\\kappa }}.$ Assertion 3.", "For sufficient large ${\\kappa }>0$ , $\\int _{\\Omega } \\frac{ \\varpi (y)}{\\varpi \\!\\left(B(y,n^{-1})\\right)\\big (1 + n {\\mathsf {d}}(x,y) \\big )^{{\\kappa }}} \\mathrm {d}{\\mathsf {m}}(y) \\le c.$ The third assertion affirms the sharpness of the first two assertions.", "It implies the following more general inequality.", "Lemma 2.4 Let $\\varpi $ be a doubling weight that satisfies Assertion 3 with ${\\kappa }>0$ .", "For $ 0< p < \\infty $ , let $\\tau ={\\kappa }- \\frac{p}{2} {\\alpha }(\\varpi ) \|1-\\frac{p}{2}\| > 0$ .", "Then, for $x\\in \\Omega $ , $\\int _{\\Omega } \\frac{ \\varpi (y)}{\\varpi \\!\\left(B\\left(y,n^{-1}\\right)\\right)^{\\frac{p}{2}} \\big (1 + n {\\mathsf {d}}(x,y) \\big )^{\\tau }} \\mathrm {d}{\\mathsf {m}}(y) \\le c \\, \\varpi \\!\\left(B\\left(x,n^{-1}\\right)\\right)^{1-\\frac{p}{2}}.$ Denote the left-hand side of (REF ) by $J_{p,\\tau }$ .", "When $p =2$ , the inequality (REF ) is precisely the Assertion 3.", "The case $p \\ne 2$ can be deduced from the case $d=2$ .", "Indeed, by Lemma REF , it is easy to see that, $\\big (1+ n {\\mathsf {d}}(x,y) \\big )^{-{\\alpha }(\\varpi )} \\le \\frac{ \\varpi \\!\\left(B\\left(y,n^{-1}\\right)\\right)}{ \\varpi \\!\\left(B\\left(x,n^{-1}\\right)\\right)}\\le \\big (1+ n {\\mathsf {d}}(x,y) \\big )^{{\\alpha }(\\varpi )}.$ For $ p \\ne 2$ , using the right-hand side inequality if $0 < p < 2$ and the left-hand side inequality if $p >2$ , we obtain $\\varpi \\!\\left(B\\left(y,n^{-1}\\right)\\right)^{\\frac{p}{2}} \\ge \\frac{ \\varpi \\!\\left(B\\left(y,n^{-1}\\right)\\right)\\varpi \\!\\left(B\\left(x,n^{-1}\\right)\\right)^{\\frac{p}{2} -1}}{\\big (1+ n {\\mathsf {d}}(x, y) \\big )^{{\\alpha }(w) \|1-\\frac{p}{2}\|}},$ which gives $J_{p,\\tau } \\le \\varpi \\!\\left(B\\left(x,n^{-1}\\right)\\right)^{1-\\frac{p}{2}} J_{2,{\\kappa }}$ .", "Hence, (REF ) for $p \\ne 2$ follows from the case $p =2$ .", "The proof is completed.", "Corollary 2.5 For $0 < p < \\infty $ and $x \\in \\Omega $ , the highly localized kernel satisfies $\\int _{\\Omega } \\left\|L_n (\\varpi ;x,y) \\right\|^p \\varpi (y) \\mathrm {d}{\\mathsf {m}}(y)\\le c \\left[ \\varpi \\!\\left(B\\!\\left(x,n^{-1}\\right)\\right)\\right]^{1-p}.$ In particular, for $p =1$ , it ensures that the Assertion 1 is sharp in the sense that the estimate guarantees the boundedness of $\\left\|L_n (\\varpi ;x,y) \\right\|$ and it ensures that Assertion 2 is sharp in the same sense.", "Definition 2.6 The homogeneous space $(\\Omega , \\varpi , {\\mathsf {d}})$ is called localizable if $\\varpi $ is a doubling weight that admits highly localized kernels.", "For convenience, we also say that the domain $\\Omega $ , or the weight $\\varpi $ , admits a localizable homogeneous space without specifying $\\varpi $ , or $\\Omega $ .", "Throughout the rest of this section, we assume that the weight function $\\varpi $ admits a localizable homogeneous space on $\\Omega $ ." ], [ "Examples of localizable homogeneous spaces", "In this subsection, we give three examples of localizable homogeneous spaces: the interval $[-1,1]$ with the Jacobi weight, the unit sphere ${\\mathbb {S}^{d-1}}$ with the surface measure, the unit ball ${\\mathbb {B}}^d$ with its classical weight function.", "These cases have been thoroughly studied fairly recently in the literature, and they motivate our definition of the localizable homogeneous spaces.", "Moreover, their corresponding orthogonal polynomials serve as the building blocks for orthogonal polynomials on the conic domains." ], [ "The interval $[-1,1]$ with the Jacobi weight", "For ${\\alpha }, {\\beta }> -1$ , the Jacobi weight function is defined by $w_{{\\alpha },{\\beta }}(t):=(1-t)^{\\alpha }(1+t)^{\\beta }, \\qquad -1 < x <1.$ Its normalization constant $c^{\\prime }_{{\\alpha },{\\beta }}$ , defined by $c^{\\prime }_{{\\alpha },{\\beta }} \\int _{-1}^1 w_{{\\alpha },{\\beta }} (x)dx = 1$ , is given by $c^{\\prime }_{{\\alpha },{\\beta }} = \\frac{1}{2^{{\\alpha }+{\\beta }+1}} c_{{\\alpha },{\\beta }} \\quad \\hbox{with} \\quad c_{{\\alpha },{\\beta }} := \\frac{\\Gamma ({\\alpha }+{\\beta }+2)}{\\Gamma ({\\alpha }+1)\\Gamma ({\\beta }+1)}.$ The intrinsic distance of the interval $[-1,1]$ is defined by ${\\mathsf {d}}_{[-1,1]}(t,s) = \\arccos \\left(t s + \\sqrt{1-t^2} \\sqrt{1-s^2}\\right),$ which is the projection of the distance $\|{\\theta }-\\phi \|$ on the upper half of the unit circle if we set $t =\\cos {\\theta }$ and $s = \\cos \\phi $ .", "The space $([-1,1], w_{{\\alpha },{\\beta }}, {\\mathsf {d}}_{[-1,1]})$ is a localizable homogeneous space if ${\\alpha }, {\\beta }\\ge -\\frac{1}{2}$ .", "The orthogonal polynomials with respect to $w_{{\\alpha },{\\beta }}$ are the Jacobi polynomials $P_n^{({\\alpha },{\\beta })}$ , which are given by the hypergeometric function as [46] $P_n^{({\\alpha },{\\beta })}(x) = \\frac{({\\alpha }+1)_n}{n!}", "{}_2F_1 \\left(\\begin{matrix} -n, n+{\\alpha }+{\\beta }+1 \\\\{\\alpha }+1 \\end{matrix}; \\frac{1-x}{2} \\right).$ These polynomials satisfy the orthogonal relations $c_{{\\alpha },{\\beta }}^{\\prime } \\int _{-1}^1 P_n^{({\\alpha },{\\beta })}(x) P_m^{({\\alpha },{\\beta })}(x) w_{{\\alpha },{\\beta }}(x) \\mathrm {d}x = h_n^{({\\alpha },{\\beta })} \\delta _{m,n},$ where $h_n^{({\\alpha },{\\beta })}$ is the square of the $L^2$ norm that satisfies $h_n^{({\\alpha },{\\beta })} = \\frac{({\\alpha }+1)_n ({\\beta }+1)_n({\\alpha }+{\\beta }+n+1)}{n!", "({\\alpha }+{\\beta }+2)_n({\\alpha }+{\\beta }+2 n+1)}.$ The Jacobi polynomials are eigenfunctions of a second order differential operator: $\\left[(1-x) \\partial ^2 - ({\\alpha }-{\\beta }+({\\alpha }+{\\beta }+2)x ) \\partial \\right] u = - n (n+{\\alpha }+ {\\beta }+1)u,$ where $\\partial f= f^{\\prime }$ and $u = P_n^{({\\alpha },{\\beta })}$ for $n =0,1,2,\\ldots $ .", "The highly localized kernel for the Jacobi polynomials will be denoted by $L_n^{({\\alpha },{\\beta })}(x,y)=\\sum _{j=0}^\\infty \\widehat{a} \\Big (\\frac{j}{n}\\Big )\\frac{P_j^{({\\alpha },{\\beta })}(x) P_j^{({\\alpha },{\\beta })}(y)}{h_j^{({\\alpha },{\\beta })}},$ where $\\widehat{a}$ is an admissible cutoff function.", "This kernel decays rapidly in terms of the distance ${\\mathsf {d}}_{[-1,1]}(\\cdot ,\\cdot )$ .", "Let the ball $B(x,r)$ be defined with respect to this distance.", "Then the Jacobi weight is a doubling weight and satisfies, $w_{{\\alpha }, {\\beta }}(B(x, n^{-1})) \\sim n^{-1} w_{{\\alpha },{\\beta }}(n; t),$ where $w_{{\\alpha },{\\beta }}(n; t)$ is defined by $ w_{{\\alpha },{\\beta }}(n; t) = (1-t + n^{-2})^{{\\alpha }+\\frac{1}{2}} (1+t + n^{-2})^{{\\beta }+\\frac{1}{2}}.$ The following estimate of the localized kernel can be found in [37].", "Theorem 2.7 Let $\\alpha , \\beta \\ge -1/2$ and $0<{\\varepsilon }\\le 1$ .", "Then, for any ${\\kappa }>0$ and $n \\ge 1$ , $\|L_n^{({\\alpha },{\\beta })} (t,s)\| \\le \\frac{c\\,n}{\\sqrt{w_{{\\alpha },{\\beta }}(n; t)}\\sqrt{w_{{\\alpha },{\\beta }}(n; s)}}\\left(1+n {\\mathsf {d}}_{[-1,1]}(t, s)\\right)^{- {\\kappa }},$ This verifies Assertion 1 of the Definition REF .", "The Assertions 2 and 3 can be found in [37], [24], respectively.", "The proof of the estimate relies on the estimate for the special case $L_n^{({\\alpha },{\\beta })}(t) = L_n^{({\\alpha },{\\beta })}(t,1), \\qquad -1 \\le t \\le 1.$ If $s =1$ and $t= \\cos {\\theta }$ , then ${\\mathsf {d}}_{[-1,1]}(t,1) = {\\theta }\\sim \\sin \\theta /2 \\sim \\sqrt{1-t}$ and we obtain [5], $ \\left\|L_n^{({\\alpha },{\\beta })}(t) \\right\| \\le c\\, \\frac{n^{2 {\\alpha }+2}}{\\left(1+ n \\sqrt{1-t}\\right)^{\\kappa }}, \\quad -1 \\le t \\le 1,$ which is used to establish the estimate in Theorem REF .", "There is also a more general estimate stated under weaker assumption on the cut-off function [5] and [12], which will be useful in later development.", "Theorem 2.8 Let $\\ell $ be a positive integer and let $\\eta $ be a function that satisfy, $\\eta \\in C^{3\\ell -1}({\\mathbb {R}})$ , $\\mathrm {supp}\\, \\eta \\subset [0,2]$ and $\\eta ^{(j)} (0) = 0$ for $j = 0,1,2,\\ldots , 3 \\ell -2$ .", "Then, for ${\\alpha }\\ge {\\beta }\\ge -\\frac{1}{2}$ , $t \\in [-1,1]$ and $n\\in {\\mathbb {N}}$ , $ \\left\| \\frac{d^m}{dt^m} L_n^{({\\alpha },{\\beta })}(t) \\right\| \\le c_{\\ell ,m,{\\alpha }}\\left\\Vert \\eta ^{(3\\ell -1)}\\right\\Vert _\\infty \\frac{n^{2 {\\alpha }+ 2m+2}}{(1+n\\sqrt{1-t})^{\\ell }}, \\quad m=0,1,2,\\ldots .$ The Jacobi polynomials with equal parameters are the Gegenbauer polynomials $C_n^{\\lambda }$ , which are of particular interest for our study.", "These polynomials are orthogonal with respect to the the weight function $w_{\\lambda }(x) = (1-x^2)^{{\\lambda }-\\frac{1}{2}}, \\quad -1< x<1, \\quad {\\lambda }> -\\tfrac{1}{2}.$ More precisely, the polynomials $C_n^{\\lambda }$ satisfy the orthogonal relation $ c_{{\\lambda }} \\int _{-1}^1 C_n^{{\\lambda }}(x) C_m^{{\\lambda }}(x) w_{\\lambda }(x) \\mathrm {d}x = h_n^{{\\lambda }} \\delta _{n,m},$ where the normalization constant $c_{\\lambda }$ of $w_{\\lambda }$ and the norm square $h_n^{\\lambda }$ are given by $c_{{\\lambda }} = \\frac{\\Gamma ({\\lambda }+1)}{\\Gamma (\\frac{1}{2})\\Gamma ({\\lambda }+\\frac{1}{2})} \\qquad \\hbox{and} \\qquad h_n^{\\lambda }= \\frac{{\\lambda }}{n+{\\lambda }}C_n^{\\lambda }(1) = \\frac{{\\lambda }}{n+{\\lambda }}\\frac{(2{\\lambda })_n}{n!", "}.$ The polynomial $C_n^{\\lambda }$ is a constant multiple of the Jacobi polynomial $P_n^{({\\lambda }-\\frac{1}{2},{\\lambda }-\\frac{1}{2})}$ and it is also related to the Jacobi polynomial by a quadratic transform [46] $ C_{2n}^{\\lambda }(x) = \\frac{({\\lambda })_n}{(\\frac{1}{2})_n} P_n^{({\\lambda }-\\frac{1}{2},{\\lambda }-\\frac{1}{2})}(2x^2-1).$" ], [ "The unit sphere ${\\mathbb {S}^{d-1}}$", "With the unit weight function, or the Lebesgue measure $\\mathrm {d}\\sigma $ , the space $({\\mathbb {S}^{d-1}}, \\mathrm {d}\\sigma , \\mathrm {d}_{\\mathbb {S}})$ is a localizable homogeneous space, where the metric $\\mathrm {d}_{\\mathbb {S}}= \\mathrm {d}_{{\\mathbb {S}^{d-1}}}$ is the geodesic distance defined in (REF ).", "The orthogonal polynomials are spherical harmonics.", "Let ${\\mathcal {P}}_n^d$ denote the space of homogeneous polynomials of degree $n$ in $d$ variables.", "A spherical harmonics $Y$ of degree $n$ is an element of ${\\mathcal {P}}_n^d$ that satisfies $\\Delta Y =0$ , where $\\Delta $ is the Laplace operator of ${\\mathbb {R}}^d$ .", "If $Y \\in {\\mathcal {P}}_n^d$ , then $Y(x) = \\Vert x\\Vert ^n Y(x^{\\prime })$ , $x^{\\prime } = x/\\Vert x\\Vert \\in {\\mathbb {S}^{d-1}}$ , so that $Y$ is determined by its restriction on the unit sphere.", "Let ${\\mathcal {H}}_n({\\mathbb {S}^{d-1}})$ denote the space of spherical harmonics of degree $n$ .", "Its dimension $\\dim {\\mathcal {H}}_n({\\mathbb {S}^{d-1}})$ is given by (REF ).", "Spherical harmonics of different degrees are orthogonal on the sphere.", "For $n \\in {\\mathbb {N}}_0$ let $\\lbrace Y_\\ell ^n: 1 \\le \\ell \\le \\dim {\\mathcal {H}}_n({\\mathbb {S}^{d-1}})\\rbrace $ be an orthonormal basis of ${\\mathcal {H}}_n({\\mathbb {S}^{d-1}})$ ; then $\\frac{1}{{\\omega }_d} \\int _{\\mathbb {S}^{d-1}}Y_\\ell ^n (\\xi ) Y_{\\ell ^{\\prime }}^m (\\xi )\\mathrm {d}\\sigma (\\xi ) = \\delta _{\\ell ,\\ell ^{\\prime }} \\delta _{m,n},$ where ${\\omega }_d$ denotes the surface area ${\\omega }_d = 2 \\pi ^{\\frac{d}{2}}/\\Gamma (\\frac{d}{2})$ of ${\\mathbb {S}^{d-1}}$ .", "Let ${\\mathsf {P}}_n(\\cdot ,\\cdot )$ denote the reproducing kernel of the space ${\\mathcal {H}}_n({\\mathbb {S}^{d-1}})$ .", "Among many properties of the spherical harmonics (see, for example, [12], [45]), one characteristic property is a closed-formula for this kernel [12], $ {\\mathsf {P}}_n(x,y) = \\sum _{\\ell =1}^{\\dim {\\mathcal {H}}_n({\\mathbb {S}^{d-1}})} Y_\\ell ^n (x) Y_\\ell ^n(y) = Z_n^{\\frac{d-2}{2}} ({\\langle }x,y{\\rangle }), \\quad x, y \\in {\\mathbb {S}^{d-1}},$ where $Z_n^{\\lambda }$ is defined in terms of the Gegenbauer polynomial by $ Z_n^{\\lambda }(t) = \\frac{n+{\\lambda }}{{\\lambda }} C_n^{\\lambda }(t), \\qquad {\\lambda }= \\frac{d-2}{2}.$ Because of the second equal sign, the identity (REF ) is often called the addition formula of spherical harmonics.", "The function in its right-hand side is often called zonal harmonic, since it depends only on ${\\langle }x,y{\\rangle }$ .", "This formula allows us to use (REF ) to derive the estimate (REF ), which is Assertion 1, for the highly localized kernel ${\\mathsf {L}}_n(x,y) = \\sum _{k=0}^\\infty \\widehat{a} \\left(\\frac{k}{n}\\right) {\\mathsf {P}}_k(x,y), \\qquad x, y \\in {\\mathbb {S}^{d-1}}.$ The other two assertions also hold; see, for example, [12] and [12].", "Another characteristic property of the spherical harmonics is that they are eigenfunctions of the Laplace-Beltrami operator $\\Delta _0$ on the sphere, which is the restriction of the Laplace operator $\\Delta $ on the unit sphere; see, for example, [12] for its explicit expression.", "More precisely, it is known [12] that $ \\Delta _0 Y = -n(n+d-2) Y, \\qquad Y \\in {\\mathcal {H}}_n^d,$ so that the eigenvalues depend only on $n$ .", "This relation, as well as the operator $\\Delta _0$ , plays an important role in the approximation theory on the unit sphere." ], [ "The unit ball ${\\mathbb {B}}^d$ with classical weight function", "The classical weight function on the unit ball ${\\mathbb {B}}^d$ of ${\\mathbb {R}}^d$ is defined by $W_\\mu (x) = (1-\\Vert x\\Vert )^{\\mu -\\frac{1}{2}}, \\qquad x\\in {\\mathbb {B}}^d, \\quad \\mu > -\\tfrac{1}{2}.$ Its normalization constant is $b_\\mu ^{\\mathbb {B}}= \\Gamma (\\mu +\\frac{d+1}{2}) /(\\pi ^{\\frac{d}{2}}\\Gamma (\\mu +\\frac{1}{2}))$ .", "For the ball $B(x, r)$ defined via the distance function ${\\mathsf {d}}_{\\mathbb {B}}$ in (REF ) on the unit ball [38], $W_\\mu ( B(x,r)) \\sim r^d \\left(\\sqrt{1-\\Vert x\\Vert ^2} + r \\right)^{2\\mu }, \\qquad 0 < r \\le 1,$ so that $W_\\mu $ is a doubling weight.", "The space $({\\mathbb {B}}^d, W_\\mu , {\\mathsf {d}}_{\\mathbb {B}})$ is a localizable homogeneous space for $\\mu \\ge 0$ , where ${\\mathsf {d}}_{\\mathbb {B}}(\\cdot ,\\cdot )$ is the distance defined in (REF ) on the unit ball.", "Let ${\\mathcal {V}}_n^d({\\mathbb {B}}^d,W_\\mu )$ be the space of orthogonal polynomials of degree $n$ with respect to $W_\\mu $ .", "An orthogonal basis of ${\\mathcal {V}}_n^d({\\mathbb {B}}^d, W_\\mu )$ can be given explicitly in terms of the Jacobi polynomials and spherical harmonics.", "For $ 0 \\le m \\le n/2$ , let $\\lbrace Y_\\ell ^{n-2m}: 1 \\le \\ell \\le \\dim {\\mathcal {H}}_{n-2m}({\\mathbb {S}^{d-1}})\\rbrace $ be an orthonormal basis of ${\\mathcal {H}}_{n-2m}^d$ .", "Define [14] $P_{\\ell , m}^n (x) = P_m^{(\\mu -\\frac{1}{2}, n-2m+\\frac{d-2}{2})} \\left(2\\Vert x\\Vert ^2-1\\right) Y_{\\ell }^{n-2m}(x).$ Then $\\lbrace P_{\\ell ,m}^n: 0 \\le m \\le n/2, 1 \\le \\ell \\le \\dim {\\mathcal {H}}_{n-2m}({\\mathbb {S}^{d-1}})\\rbrace $ is an orthogonal basis of ${\\mathcal {V}}_n^d(W_\\mu )$ .", "Let ${\\mathbf {P}}_n(W_\\mu ; \\cdot , \\cdot )$ denote the reproducing kernel of the space ${\\mathcal {V}}_n({\\mathbb {B}}^d, W_\\mu )$ .", "This kernel satisfies an analog of the addition formula [51]: for $x,y \\in {\\mathbb {B}}^d$ , $ {\\mathbf {P}}_n(W_\\mu ;x,y) = c_{\\mu -\\frac{1}{2}}\\int _{-1}^1 Z_n^{\\mu +\\frac{d-1}{2}} & \\Big ({\\langle }x, y {\\rangle }+ u \\sqrt{1-\\Vert x\\Vert ^2}\\sqrt{1-\\Vert y\\Vert ^2} \\Big ) \\\\& \\times (1-u^2)^{\\mu -1}\\mathrm {d}u, $ where $\\mu > 0$ and it holds for $\\mu =0$ under the limit $\\lim _{\\mu \\rightarrow 0+} c_{\\mu -\\frac{1}{2}} \\int _{-1}^1 f(t) (1-t^2)^{\\mu -1} \\mathrm {d}u = \\frac{f(1) + f(-1)}{2}.$ This is an analog of the addition formula for the unit ball and it allows us to use (REF ) to derive the estimate (REF ), which is Assertion 1, for the highly localized kernel now denoted by ${\\mathbf {L}}_n(W_\\mu ; x,y) = \\sum _{k=0}^\\infty \\widehat{a} \\left(\\frac{k}{n}\\right) {\\mathbf {P}}_k(W_\\mu ; x,y).$ All three assertions were proved in [38].", "For the first one, see also [12].", "The orthogonal polynomials with respect to $W_\\mu $ on the unit ball are eigenfunctions of a second order differential operator [14]: $\\left( \\Delta - {\\langle }x,\\nabla {\\rangle }^2 - (2\\mu +d-1) {\\langle }x ,\\nabla {\\rangle }\\right)u = - n(n+2\\mu + d-1) u$ for all $u \\in {\\mathcal {V}}_n({\\mathbb {B}}^d, W_\\mu )$ .", "The differential operator in the left-hand side is the analog of the Laplace-Beltrami operator on the sphere." ], [ "An auxiliary maximal function", "Some of the polynomial inequalities in the latter sections will be established for norms defined via a doubling weight.", "The study of approximation and polynomial inequality with doubling weights was pioneered in [32] for the interval $[-1,1]$ and developed subsequently in [7] for the unit sphere ${\\mathbb {S}^{d-1}}$ .", "The main tool for the latter is an auxiliary maximal function.", "We define its analog on the domain $\\Omega $ that admits a localizable homogeneous space.", "Definition 2.9 For ${\\beta }> 0$ , $n \\in {\\mathbb {N}}$ and $f \\in C(\\Omega )$ , a maximal function $f_{{\\beta },n}^\\ast $ is defined by $f_{{\\beta },n}^\\ast (x) := \\max _{y \\in \\Omega } \|f(y)\| \\big (1+ n {\\mathsf {d}}(x,y) \\big )^{-{\\beta }}, \\quad x \\in \\Omega .$ The development below follows closely [12] but with a somewhat more streamlined proof.", "We clearly have $\|f(x)\| \\le f_{{\\beta },n}^\\ast (x)$ for all $x \\in \\Omega $ .", "The maximal function $f_{{\\beta },n}^\\ast $ has an upper bound in the Hardy-Littlewood maximum function $M_\\mu $ defined in (REF ).", "To see this, we first prove a lemma.", "For $f\\in C(\\Omega )$ and $r>0$ , we define ${\\rm osc } (f) (x, r) := \\sup _{x_1,x_2 \\in B(x, r)} \|f(x_1)-f(x_2)\|, \\qquad x\\in \\Omega .$ Lemma 2.10 If $f\\in \\Pi _n(\\Omega )$ and $0 < \\delta \\le 1$ , then for any ${\\beta }>0$ , ${\\rm osc } (f) \\left(x, \\tfrac{\\delta }{n}\\right) \\le c_{\\beta }\\delta f_{{\\beta }, n}^\\ast (x), \\qquad x\\in \\Omega ,$ where the constant $c_{\\beta }$ depends only on $d$ and ${\\beta }$ when ${\\beta }$ is large.", "We use the highly localized kernel $L_n$ with an admissible cut-off function of type (a).", "For $\\delta >0$ and $x,y \\in \\Omega $ , set $A_{n,\\delta } (x,y) := \\sup _{z \\in B (x,\\frac{\\delta }{n})} \\left\| L_n (\\varpi ; x,y) - L_n(\\varpi ; z,y)\\right\|.$ Let $L_n(\\varpi )f$ be defined as in (REF ).", "Then $L_n(\\varpi )f = f$ for $f \\in \\Pi _n(\\Omega )$ .", "If $x_1,x_2 \\in B(x, r)$ , then by the triangle inequality $\\left\| L_n (\\varpi ; x_1,y) - L_n(\\varpi ; x_2,y)\\right\| \\le 2 A_{n,\\delta }(x,y).$ Hence, it follows by (REF ) that ${\\rm osc} (f) \\left(x, \\tfrac{\\delta }{n} \\right) & \\le 2 \\int _{\\Omega } \|f(y)\|A_{n,\\delta }(x,y) \\varpi (y) \\mathrm {d}{\\mathsf {m}}(y)\\\\& \\le 2 f_{{\\beta }, n}^\\ast (x) \\int _{\\Omega } (1+n {\\mathsf {d}}(x,y))^{{\\beta }} A_{n,\\delta } (x,y)\\, \\mathrm {d}{\\mathsf {m}}(y)\\\\& \\le c_{\\beta }\\delta f_{{\\beta }, n}^\\ast (x),$ where the last step follows from Assertion 1 and Assertion 3.", "Theorem 2.11 Let ${\\mathsf {w}}$ be a doubling weight.", "For $f\\in \\Pi _n(\\Omega )$ , ${\\beta }>0$ and setting ${\\gamma }= {\\alpha }({\\mathsf {w}})/{\\beta }$ , then $f_{{\\beta },n}^\\ast (x) \\le c_{{\\beta },L({\\mathsf {w}})} \\left( M_{\\mathsf {w}}\\big (\|f\|^{\\gamma }\\big ) (x) \\right)^{1/{\\gamma }}, \\qquad x \\in \\Omega .$ We apply Lemma REF with $c_{\\beta }\\delta = \\frac{1}{4}$ to obtain, by $(a+b)^{\\gamma }\\le 2^{\\gamma }(a^{\\gamma }+b^{\\gamma })$ , $\\frac{\|f(y)\|^{{\\gamma }} }{( 1+ n {\\mathsf {d}}(x, y))^{{\\alpha }({\\mathsf {w}})}} \\le 2^{{\\gamma }} \\frac{\\min _{ z\\in B(y,\\frac{\\delta }{n})} \|f(z)\|^{{\\gamma }} }{( 1+ n {\\mathsf {d}}(x, y))^{{\\alpha }({\\mathsf {w}})}} +\\frac{1}{2^{\\gamma }} \\left( f_{{\\beta }, n}^\\ast (x) \\right)^{\\gamma },$ Taking maximum over $x \\in \\Omega $ , the left-hand side becomes $\\big ( f_{{\\beta }, n}^\\ast (x) \\big )^{\\gamma }$ using ${\\alpha }({\\mathsf {w}}) = {\\beta }{\\gamma }$ , so that the inequality can be rearranged to give $ \\left( f_{{\\beta }, n}^\\ast (x) \\right)^{\\gamma } \\le c ( 1+ n {\\mathsf {d}}(x, y))^{-{\\alpha }({\\mathsf {w}})}\\min _{ z\\in B( y,\\frac{\\delta }{n})} \|f(z)\|^{\\gamma },$ where $c = 2^{\\gamma }/(1 - 2^{-{\\gamma }})$ .", "We now estimate $\\min _{ z\\in B( y,\\frac{\\delta }{n})}\|f(z)\|^{\\gamma }$ , following the proof in the case of the unit sphere [12].", "Let ${\\theta }= \\max \\lbrace \\frac{\\delta }{n}, {\\mathsf {d}}(x,y)\\rbrace $ .", "Then $B(y, \\frac{\\delta }{n})\\subset B(x, 2{\\theta })\\subset B(y,3{\\theta })$ .", "By the doubling condition for ${\\mathsf {w}}$ and (ii) of Lemma REF , ${\\mathsf {w}}\\big (B (y,\\tfrac{\\delta }{n})\\big ) \\ge \\frac{1}{c_{L({\\mathsf {w}})}} \\left( \\frac{ 3{\\theta }n}{\\delta } \\right)^{-{\\alpha }({\\mathsf {w}})}{\\mathsf {w}}(B (y, 3 {\\theta }) ) \\ge \\frac{1}{c_{L({\\mathsf {w}})}} \\left( \\frac{3{\\theta }n}{\\delta } \\right)^{-{\\alpha }({\\mathsf {w}})} {\\mathsf {w}}(B(x, 2 {\\theta })),$ which implies, together with $n {\\theta }\\le 1+n{\\mathsf {d}}(x,y)$ , the estimate $\\min _{z\\in B(y,\\frac{\\delta }{n})} \|f(z)\|^{\\gamma }& \\le \\frac{1}{{\\mathsf {w}}(B(y, \\frac{\\delta }{n}))} \\int _{B(y,\\frac{\\delta }{n})} \| f(z)\|^{ {\\gamma }} {\\mathsf {w}}(z)\\, \\mathrm {d}{\\mathsf {m}}(z) \\\\& \\le c_{L({\\mathsf {w}})} \\left( \\frac{ 3{\\theta }n}{\\delta } \\right)^{{\\alpha }({\\mathsf {w}})} \\frac{1}{{\\mathsf {w}}(B(x, 2 {\\theta }))}\\int _{B (x,2{\\theta })} \|f(z)\|^{\\gamma } {\\mathsf {w}}(z)\\, \\mathrm {d}{\\mathsf {m}}(z)\\\\& \\le c_{L({\\mathsf {w}})} \\left( \\frac{3}{\\delta } \\right)^{{\\alpha }({\\mathsf {w}})}(1+n {\\mathsf {d}}(x,y))^{{\\alpha }({\\mathsf {w}})} M_{{\\mathsf {w}}} \\left(\|f\|^{\\gamma }\\right)(x).$ Together with (REF ), this completes the proof.", "Let $\\Vert \\cdot \\Vert _{p, {\\mathsf {w}}}$ denote the norm of $L^p(\\Omega , {\\mathsf {w}})$ and we adopt this notation for $0 < p <1$ , even though it is no longer a norm.", "The $L^p$ boundedness of $f_{{\\beta },n}^\\ast $ follows from that of $M_{\\mu }(\|f\|^{\\gamma })$ for ${\\gamma }p > 1$ or ${\\beta }> {\\alpha }({\\mathsf {w}})/p$ .", "Corollary 2.12 If $ 0< p\\le \\infty $ , $ f\\in \\Pi _n(\\Omega )$ and ${\\beta }> {\\alpha }({\\mathsf {w}})/p$ , then $\\Vert f\\Vert _{p, {\\mathsf {w}}} \\le \\Vert f_{{\\beta },n}^\\ast \\Vert _{p,{\\mathsf {w}}} \\le c \\Vert f\\Vert _{p,{\\mathsf {w}}},$ where $c$ depends also on $L({\\mathsf {w}})$ and ${\\beta }$ when ${\\beta }$ is either large or close to ${\\alpha }({\\mathsf {w}})/p$ ." ], [ "Marcinkiewicz-Zygmund inequality", "These inequalities are between the $L^p$ norm and a discrete $L^p$ norm defined with respect to a well-distributed set of points.", "We start with the definition of well-separated sets of points.", "Definition 2.13 Let $\\Xi $ be a discrete set in $\\Omega $ .", "A finite collection of subsets $\\lbrace S_z: z \\in \\Xi \\rbrace $ is called a partition of $\\Omega $ if $S_z^\\circ \\cap S_y^\\circ = \\emptyset $ when $z \\ne y$ and $\\Omega = \\bigcup _{z \\in \\Xi } S_z$ .", "Let ${\\varepsilon }>0$ .", "A discrete subset $\\Xi $ of $\\Omega $ is called ${\\varepsilon }$ -separated if ${\\mathsf {d}}(x,y) \\ge {\\varepsilon }$ for every two distinct points $x, y \\in \\Xi $ .", "$\\Xi $ is called maximal if there is a constant $c_d > 1$ such that $1 \\le \\sum _{z\\in \\Xi } \\chi _{B(z, {\\varepsilon })}(x) \\le c_d, \\qquad \\forall x \\in \\Omega ,$ where $\\chi _E$ denotes the characteristic function of the set $E$ .", "Evidently (REF ) implies $\\Omega = \\bigcup _{z \\in \\Xi } B(z,{\\varepsilon })$ .", "Moreover, it implies that the cardinality $\\# \\Xi $ of $\\Xi $ satisfies $c_1 \\frac{{\\mathsf {m}}(\\Omega )}{\\max _{z \\in \\Xi } {\\mathsf {m}}(B(z,{\\varepsilon }))} \\le \\# \\Xi \\le c_2\\frac{{\\mathsf {m}}(\\Omega )}{\\min _{z \\in \\Xi } {\\mathsf {m}}(B(z,{\\varepsilon }))}.$ For the unit sphere, all balls of the same radius have the same volume; that is, $\\sigma (B(x,r)) = \\sigma (B(0,1)) r^{d-1}$ for all $x \\in {\\mathbb {S}^{d-1}}$ , where $\\sigma $ is the surface measure on ${\\mathbb {S}^{d-1}}$ .", "This allows us to deduce (REF ) from $\\Omega = \\bigcup _{x \\in \\Xi } B(x,{\\varepsilon })$ and the cardinality of the maximal ${\\varepsilon }$ -separated set $\\Xi _{\\mathbb {S}}$ on the sphere ${\\mathbb {S}^{d-1}}$ satisfies $c_d^{\\prime } {\\varepsilon }^{-d+1} \\le \\# \\Xi _{\\mathbb {S}}\\le c_d {\\varepsilon }^{-d+1}$ ; see, for example, [12].", "As we shall show in the next section that the volume of the ball $B(x,r)$ on the conic surface depends on the position of $x$ .", "In particular, we no longer have $\\varpi (B(x,{\\varepsilon })) \\sim \\varpi (B(y,{\\varepsilon }))$ for all $x, y$ in the domain in general.", "Theorem 2.14 Let ${\\mathsf {w}}$ be a doubling weight on $\\Omega $ .", "Let ${\\varepsilon }= \\frac{\\delta }{n}$ for $n = 1,2, \\ldots $ and $\\delta \\in (0,1)$ .", "If $\\Xi $ is a maximal ${\\varepsilon }$ -separated subset of $\\Omega $ , then for all $f\\in \\Pi _n(\\Omega )$ and $0 < p < \\infty $ , $\\bigg ( \\sum _{y \\in \\Xi } \\left\|{\\rm osc} (f)\\left(y, \\tfrac{\\delta }{n}\\right)\\right\|^p{\\mathsf {w}}\\!\\left(B(y,\\tfrac{\\delta }{n})\\right)\\bigg )^{\\frac{1}{p}} \\le c_p \\delta \\Vert f\\Vert _{p, {\\mathsf {w}}},$ where $c_p$ depends on $p$ , when $p$ is close to 0, and on $d$ and $L({\\mathsf {w}})$ .", "If $ y \\in B(x, \\frac{ \\delta }{n})$ , then $f_{2 {\\alpha }({\\mathsf {w}})/p, n}^\\ast (y) \\sim f_{2 {\\alpha }({\\mathsf {w}})/p, n}^\\ast (x)$ .", "Hence, it follows from Lemma REF that $\\sum _{y \\in \\Xi } \\left\|{\\rm osc} (f)\\left(y, \\tfrac{\\delta }{n}\\right)\\right\|^p {\\mathsf {w}}\\!\\left(B(y,\\tfrac{\\delta }{n})\\right)& \\le c (c_p \\delta )^p \\sum _{y \\in \\Xi } \\int _{B(y, \\frac{\\delta }{n})}\\left( f_{2{\\alpha }({\\mathsf {w}})/p, n}^\\ast (x)\\right)^p {\\mathsf {w}}(x) \\, \\mathrm {d}{\\mathsf {m}}(x) \\\\& \\le ( c_p \\delta )^p \\int _{\\Omega } \\left( f_{2{\\alpha }({\\mathsf {w}})/p,n}^\\ast (x)\\right)^p {\\mathsf {w}}(x)\\, \\mathrm {d}{\\mathsf {m}}(x).$ The last integral is bounded by $\\Vert f\\Vert _{p, {\\mathsf {w}}}^p$ by Corollary REF .", "We now prove the Marcinkiewicz-Zygmund inequality for polynomials.", "Theorem 2.15 Let ${\\mathsf {w}}$ be a doubling weight on $\\Omega $ .", "Let $\\Xi $ be a maximal $\\frac{\\delta }{n}$ -separated subset of $\\Omega $ and let $\\delta > 0$ and $\\delta \\le 1$ .", "For all $0<p<\\infty $ and $f\\in \\Pi _m(\\Omega )$ with $n \\le m \\le c n$ , $\\Bigg (\\sum _{z \\in \\Xi } \\Big ( \\max _{x\\in B(z, \\frac{\\delta }{n})} \|f(x)\|^p \\Big ){\\mathsf {w}}\\!\\left(B(z, \\tfrac{\\delta }{n}) \\right)\\Bigg )^{\\frac{1}{p}} \\le c_{{\\mathsf {w}},p} \\Vert f\\Vert _{p,{\\mathsf {w}}},$ where $c_{{\\mathsf {w}},p}$ depends on $L({\\mathsf {w}})$ and on $p$ when $p$ is close to 0.", "For $0 < r < 1$ , there is a $\\delta _r > 0$ such that for $\\delta \\le \\delta _r$ , $r \\le p < \\infty $ and $f \\in \\Pi _n(\\Omega )$ , $\\Vert f\\Vert _{p,{\\mathsf {w}}} \\le c_{{\\mathsf {w}},r} \\bigg (\\sum _{z \\in \\Xi }\\bigg (\\min _{x\\in B\\bigl (z, \\tfrac{\\delta }{n}\\bigr )} \|f(x)\|^p\\bigg ){\\mathsf {w}}\\bigl (B(z, \\tfrac{\\delta }{n})\\bigr )\\Bigg )^{\\frac{1}{p}},$ where $c_{{\\mathsf {w}},r}$ depends only on $L({\\mathsf {w}})$ and on $r$ when $r$ is close to 0.", "Furthermore, $\\Vert f\\Vert _\\infty \\le c \\max _{z\\in \\Xi } \|f(z)\|.$ For every $y \\in \\Xi $ , choose $z_y \\in B(y,\\frac{\\delta }{n})$ such that $\|f(z_y)\| = \\sup _{x \\in B(y, \\frac{\\delta }{n})} \|f(x)\|$ .", "For $f \\in \\Pi _m(\\Omega )$ , we have $\|f(z_y)\| \\le f_{{\\beta },m}^\\ast (z_y) \\le c f_{{\\beta },n}^\\ast (y)$ .", "Hence, $\\sum _{z \\in \\Xi } \\Big ( \\max _{x\\in B(z, \\frac{\\delta }{n})} \|f(x)\|^p \\Big ) {\\mathsf {w}}\\!", "\\left(B(z, \\tfrac{\\delta }{n}) \\right)& \\le c \\sum _{z \\in \\Xi } \\big (f_{{\\beta },n}^\\ast (y)\\big )^p {\\mathsf {w}}\\!", "\\left(B(z, \\tfrac{\\delta }{n}) \\right) \\\\& \\le c \\int _\\Omega \\big ( f_{{\\beta },n}^\\ast (x)\\big )^p {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x),$ from which (i) follows from the bound in Corollary REF .", "We now prove (ii).", "Since $\\Xi $ is maximal $\\frac{\\delta }{n}$ -separated, $\\Vert f\\Vert _{p, w}^p & \\le \\sum _{z \\in \\Xi } \\int _{B(z,\\frac{\\delta }{n})} \|f(x)\|^p {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x) \\\\& \\le 2^p \\sum _{z \\in \\Xi } \|\\mathrm {osc}(f)\\left(z, \\tfrac{\\delta }{n}\\right)\|^p {\\mathsf {w}}\\left(B\\big (z,\\tfrac{\\delta }{n}\\big )\\right)+ 2^p \\sum _{z \\in \\Xi } \\bigg (\\min _{x\\in B\\bigl (z, \\tfrac{\\delta }{n}\\bigr )} \|f(x)\|^p\\bigg ){\\mathsf {w}}\\left(B\\big (z,\\tfrac{\\delta }{n}\\big )\\right).$ The first term in the right-hand side is bounded by $( 2 c_r \\delta )^p \\Vert f\\Vert _{p,w}^p$ by Lemma REF .", "Hence, choosing $\\delta _r = 1/(4 c_r)$ so that $(2 c_r \\delta )^p \\le 2^{-p}$ for $\\delta \\le \\delta _r$ , we conclude that $(1- 2^{-p}) \\Vert f\\Vert _{p,{\\mathsf {w}}}^p \\le 2^p \\sum _{z \\in \\Xi } \\bigg (\\min _{x\\in B\\bigl (z, \\tfrac{\\delta }{n}\\big )} \|f(x)\|^p\\bigg ){\\mathsf {w}}\\big (B(z,\\tfrac{\\delta }{n})\\big ).$ Taking the power of $1/p$ proves (REF ).", "In particular, since the constant in (REF ) is independent of $p$ , it readily implies (REF ).", "This completes the proof.", "For the unit sphere without weight and the unit ball with the classical weight, the MZ inequality was established in [5], [33], [34], [38].", "For the unit sphere with the doubling weight, it was established in [7]." ], [ "Christoffel function for doubling weight", "Let ${\\mathsf {w}}$ be a doubling weight on $\\Omega $ .", "The Christoffel function ${\\lambda }_n({\\mathsf {w}};\\cdot )$ is defined by ${\\lambda }_n({\\mathsf {w}};x): = \\inf _{\\begin{array}{c}g(x) =1 \\\\ g \\in \\Pi _n(\\Omega )\\end{array}} \\int _{\\Omega } \|g(x)\|^2 {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x).$ It is known that the Christoffel function is closely related to the kernel $K_n({\\mathsf {w}};\\cdot ,\\cdot )$ of the partial sum operator $K_n({\\mathsf {w}};x,y) = \\sum _{k=0}^n P_k({\\mathsf {w}}; x,y), \\qquad x, y \\in \\Omega .$ More precisely, it is known [14] that, for $n=0,1,2,\\ldots $ , ${\\lambda }_n({\\mathsf {w}};x) = \\frac{1}{K_n({\\mathsf {w}}; x,x)}, \\qquad x \\in \\Omega .$ The Christoffel function encodes essential information on weighted approximation by polynomials on the domain $\\Omega $ .", "It will also be useful in the study of cubature rules in the next subsection, for which we need an upper bound of ${\\lambda }_n({\\mathsf {w}}; x)$ .", "We establish this bound for all doubling weights on the localizable homogeneous space under the assumption that there exist fast decaying polynomials on $\\Omega $ .", "More precisely, we make the following assertion.", "Assertion 4.", "Let $\\Omega $ be compact.", "For each $x \\in \\Omega $ , there is a nonnegative polynomial $T_x$ of degree at most $n$ that satisfies $T_x(x) =1$ , $T_x(y) \\ge \\delta > 0$ for $y \\in B(x,\\frac{1}{n})$ for some $\\delta $ independent of $n$ , and, for each ${\\gamma }> 1$ , $0 \\le T_x(y) \\le c_{\\gamma }(1+ n {\\mathsf {d}}(x,y))^{-{\\gamma }}, \\qquad y \\in \\Omega ;$ there is a polynomial $q_n$ such that $q_n(x) T_x(y)$ is a polynomial of degree at most $r n$ , for some positive integer $r$ , in $x$ -variable and $c_1 \\le q_n(x) \\le c_2$ for $x \\in \\Omega $ for some positive numbers $c_1$ and $c_2$ .", "The fast decaying polynomials of one variable are studied in [30], see also [43], which leads to fast decaying polynomials on the unit sphere by the addition formula (REF ).", "For the unit ball, such polynomials are constructed in [38].", "We use these polynomials to derive an upper bound for the Christoffel function.", "First, we need a lemma.", "Lemma 2.16 Assume Assertion 4.", "Let ${\\alpha }> 0$ be a positive number and let $x \\in \\Omega $ be fixed.", "For a doubling weight ${\\mathsf {w}}$ on $\\Omega $ , there is a polynomial $Q_x$ of degree $n$ such that, for all $y \\in \\Omega $ , $ c_1 (1+n {\\mathsf {d}}(x,y))^{{\\alpha }} {\\mathsf {w}}(B(y,\\tfrac{1}{n}))\\le Q_x(y) \\le c_2 (1+n {\\mathsf {d}}(x,y))^{{\\alpha }} {\\mathsf {w}}(B(y,\\tfrac{1}{n})),$ where $c_1$ and $c_2$ are positive constant independent of $n$ , $x$ and $y$ .", "Let $T_x$ and $q$ be polynomials in Assertion 4 of degree at most $\\lfloor \\frac{n}{2}\\rfloor $ .", "For fixed $x$ , we define $Q_x$ by $Q_x(y) = q_n(y) \\int _\\Omega T_y(v) (1+n {\\mathsf {d}}(x,v))^{{\\alpha }} {\\mathsf {w}}(v)\\mathrm {d}{\\mathsf {m}}(v), \\qquad y \\in \\Omega .$ Since $q_n(y)T_y(v)$ is a polynomial of degree at most $n$ in $y$ , it is easy to see that $Q_x$ is a polynomial of degree at most $n$ .", "Using Assertion 4 with ${\\gamma }> {\\alpha }+ 1$ and using the triangle inequality $1+n {\\mathsf {d}}(x,v)\\le (1+n {\\mathsf {d}}(x,y))(1+n {\\mathsf {d}}(y,v))$ , we obtain $Q_x(y) \\, & \\le c (1+n {\\mathsf {d}}(x,y))^{\\alpha }\\int _\\Omega \\frac{1}{(1+n {\\mathsf {d}}(y,v))^{{\\gamma }-{\\alpha }}} {\\mathsf {w}}(v) \\mathrm {d}{\\mathsf {m}}(v)\\\\& \\le c (1+n {\\mathsf {d}}(x,y))^{\\alpha }{\\mathsf {w}}\\big (B (y,\\tfrac{1}{n})\\big ),$ since, by the doubling property of ${\\mathsf {m}}$ and choosing ${\\gamma }$ sufficiently large, $ \\int _\\Omega \\frac{ {\\mathsf {w}}(v)\\mathrm {d}{\\mathsf {m}}(v)}{(1+n {\\mathsf {d}}(y,v))^{{\\gamma }-{\\alpha }}} \\, & \\le \\sum _{k=1}^\\infty \\int _{B(y,\\frac{2^k}{n})\\setminus B(y,\\frac{2^{k-1}}{n})} \\frac{ {\\mathsf {w}}(v)\\mathrm {d}{\\mathsf {m}}(v)}{(1+n {\\mathsf {d}}(y,v))^{{\\gamma }-{\\alpha }}} \\\\& \\le c \\sum _{k=1}^\\infty \\frac{1}{(1+ 2^{k-1})^{{\\gamma }-{\\alpha }} } {\\mathsf {w}}\\!\\left( B\\big (y,\\tfrac{2^{k}}{n}\\big ) \\right)\\\\& \\le c\\, {\\mathsf {w}}\\!", "\\left( B\\big (y, \\tfrac{1}{n} \\big ) \\right) \\sum _{k=1}^\\infty \\frac{ L({\\mathsf {w}})^{k-1} }{(1+ 2^{k-1})^{{\\gamma }-{\\alpha }} } \\\\& \\le c\\, {\\mathsf {w}}\\!", "\\left( B\\big (y, \\tfrac{1}{n} \\big ) \\right).", "$ This establishes the upper bound of $Q_x$ .", "In the other direction, we use $q_n(y) \\ge c_1$ , $ T_y(v) \\ge \\delta > 0$ for $v \\in B(y,\\frac{1}{n})$ and $ T_y(v) \\ge 0$ to obtain $Q_x(y) \\, & \\ge (1+n {\\mathsf {d}}(x,y))^{\\alpha }\\int _{\\Omega } T_y(v) (1+n {\\mathsf {d}}(y,v))^{- {\\alpha }} {\\mathsf {w}}(v) \\mathrm {d}{\\mathsf {m}}(v) \\\\& \\ge c\\delta (1+n {\\mathsf {d}}(x,y))^{\\alpha }\\int _{B(y,\\frac{1}{n})} (1+n {\\mathsf {d}}(y,v))^{- {\\alpha }} {\\mathsf {w}}(v) \\mathrm {d}{\\mathsf {m}}(v) \\\\& \\ge c \\delta (1+n {\\mathsf {d}}(x,y))^{\\alpha }{\\mathsf {w}}\\!", "\\left( B\\big (y, \\tfrac{1}{n} \\big ) \\right),$ since the last integral is trivially bounded below by $c {\\mathsf {w}}\\!", "\\left( B\\big (y, \\tfrac{1}{n} \\big ) \\right)$ .", "This completes the proof.", "Proposition 2.17 Assume Assertion 4 holds.", "Let ${\\mathsf {w}}$ be a doubling weight on the domain $\\Omega $ .", "Then ${\\lambda }_n({\\mathsf {w}}; x) \\le c\\, {\\mathsf {w}}\\!", "\\left(B(x,\\tfrac{1}{n}) \\right), \\qquad x\\in \\Omega .$ Let $m = \\lfloor \\frac{n}{3} \\rfloor $ .", "Let $ T_x(y)$ be the polynomial of degree $m$ in Assertion 4.", "Let $\\Xi $ be a maximal $\\frac{\\delta }{n}$ -separated set in $\\Omega $ with $\\delta \\le \\delta _0$ as in Theorem REF .", "By the inequality (REF ) with $p = 1$ and (ii) of Lemma REF for both ${\\mathsf {m}}$ and ${\\mathsf {w}}$ , we obtain $\\int _\\Omega [ T_x(y)]^2 {\\mathsf {w}}(y)\\mathrm {d}{\\mathsf {m}}(y) \\, &\\le c \\sum _{z \\in \\Xi } [ T_x(z)]^2 {\\mathsf {w}}\\big (B(z,\\tfrac{\\delta }{n})\\big ) \\\\& \\le c \\frac{{\\mathsf {w}}\\big (B(x,\\tfrac{\\delta }{n})\\big )}{[{\\mathsf {m}}\\big (B(x,\\tfrac{\\delta }{n})\\big )]^2}\\sum _{z \\in \\Xi } [ T_x(z)]^2 (1+ n {\\mathsf {d}}(x,z))^{{\\alpha }} [{\\mathsf {m}}\\big (B(z,\\tfrac{\\delta }{n})\\big )]^2,$ where ${\\alpha }= {\\alpha }({\\mathsf {w}})+ 2 {\\alpha }({\\mathsf {m}})$ .", "Let $Q_x$ be the polynomial of degree $m$ that satisfies (REF ) with ${\\mathsf {m}}$ in place of ${\\mathsf {w}}$ .", "Using the lower bound of $Q_x$ in (REF ) and applying the inequality (REF ) with $p=1$ on the polynomial $[ T_x(z)]^2 Q_y(z)$ of degree $n$ with respect to the measure $\\mathrm {d}{\\mathsf {m}}$ , we obtain $\\int _\\Omega [ T_x(y)]^2 {\\mathsf {w}}(y)\\mathrm {d}{\\mathsf {m}}(y)& \\le c \\frac{{\\mathsf {w}}\\big (B(x,\\tfrac{\\delta }{n})\\big )}{[{\\mathsf {m}}\\big (B(x,\\tfrac{\\delta }{n})\\big )]^2}\\sum _{z \\in \\Xi } [ T_x(z)]^2 Q_x(z) {\\mathsf {m}}(B(z,\\tfrac{\\delta }{n})) \\\\& \\le c\\, \\frac{{\\mathsf {w}}\\big (B(x,\\tfrac{\\delta }{n})\\big )}{[{\\mathsf {m}}\\big (B(x,\\tfrac{\\delta }{n})\\big )]^2}\\int _\\Omega [ T_x(v)]^2 Q_x(v) \\mathrm {d}{\\mathsf {m}}(v).$ Using now the upper bound of $Q_x$ in (REF ), again with ${\\mathsf {m}}$ in place of ${\\mathsf {w}}$ , and the upper bound of $T_x(v)$ in Assertion 4, and (ii) of Lemma REF for the measure $\\mathrm {d}{\\mathsf {m}}$ , we further deduce that the last integral is bounded by $\\int _\\Omega [ T_x(v)]^2 Q_x(v) \\mathrm {d}{\\mathsf {m}}(v) & \\le c \\int _\\Omega \\frac{{\\mathsf {m}}(B(v,\\tfrac{\\delta }{n}))}{ (1+ n {\\mathsf {d}}(x,v))^{2 {\\gamma }- {\\alpha }}} \\mathrm {d}{\\mathsf {m}}(v) \\\\& \\le c {\\mathsf {m}}(B(x,\\tfrac{\\delta }{n})) \\int _\\Omega \\frac{1}{ (1+ n {\\mathsf {d}}(x,z))^{2 {\\gamma }- {\\alpha }- {\\alpha }({\\mathsf {m}})}} \\mathrm {d}{\\mathsf {m}}(v) \\\\& \\le c \\big [{\\mathsf {m}}\\big (B(x,\\tfrac{\\delta }{n})\\big )\\big ]^2,$ where the last step follows from (REF ).", "Together, from the last two displayed inequalities follows that $\\int _\\Omega [ T_x(y)]^2 {\\mathsf {w}}(y)\\mathrm {d}{\\mathsf {m}}(y) \\le c {\\mathsf {w}}\\big (B(x,\\tfrac{\\delta }{n})\\big ).$ Since $ T_x(x) =1$ , this completes the proof by (REF ).", "Proposition 2.18 If ${\\mathsf {w}}$ is a weight function for which Assertion 1 holds, then ${\\lambda }_n({\\mathsf {w}}; x) \\ge c\\, {\\mathsf {w}}\\!", "\\left(B(x,\\tfrac{1}{n}) \\right), \\qquad x\\in \\Omega .$ Let $L_n({\\mathsf {w}}; \\cdot ,\\cdot )$ be the localized kernel defined with a cut-off function of type (a).", "It follows immediately that $K_n({\\mathsf {w}}; x,x) \\le L_n({\\mathsf {w}}; x,x)$ .", "Hence, if ${\\mathsf {w}}$ admits Assertion 1, then $K_n({\\mathsf {w}}; x,x) \\le L_n({\\mathsf {w}};x,x) \\le \\frac{c}{{\\mathsf {w}}(B(x,\\tfrac{1}{n}))},$ which is equivalent to the lower bound ${\\lambda }_n({\\mathsf {w}}; x)$ by (REF ).", "Together, the last two propositions give both upper and lower bounds for the Christoffel functions.", "Such bounds are known for regular domains, see for example [14], [23], [50] and the references therein, and it has been established recently for fairly general convex domains [22], [40] and planar domains with piecewise boundary [41].", "None of the previous results, however, apply to conic domains that will come out as special cases of the above theorem." ], [ "Positive cubature rules", "Let ${\\mathsf {w}}$ be a doubling weight on $\\Omega $ .", "A cubature rule of degree $n$ for ${\\mathsf {w}}$ is a finite linear combination of point evaluations that satisfies $\\int _\\Omega f(x) {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x) = \\sum _{k=1}^N {\\lambda }_k f(x_k), \\qquad \\forall f \\in \\Pi _n(\\Omega ),$ where ${\\lambda }_k \\in {\\mathbb {R}}$ and $x_k \\in \\Omega $ .", "When ${\\lambda }_k > 0$ for all $k$ , the cubature rule is called positive.", "Given a maximal ${\\varepsilon }$ -separated subset $\\Xi $ on $\\Omega $ , we establish the existence of a positive cubature rule based on the points in $\\Xi $ , which plays an essential role for discretizing our near best approximation $L_n * f$ .", "The proof will use the Farkas lemma (cf.", "[12]) stated below.", "Lemma 2.19 Let $V$ be a finite dimensional real Hilbert space with inner product ${\\langle }\\cdot ,\\cdot {\\rangle }$ .", "Then for any points $a^1, a^2, \\dots , a^m$ and $\\zeta \\in V$ , exactly of the following two systems has a solution: $\\displaystyle {\\sum _{j=1}^m \\mu _j a^j = \\zeta }, \\qquad 0 \\le \\mu _1,\\mu _2, \\ldots , \\mu _m \\in {\\mathbb {R}}$ .", "${\\langle }a^j,x{\\rangle }\\ge 0$ , $j =1,2,\\ldots , m$ and ${\\langle }\\zeta , x {\\rangle }< 0$ for some $x \\in V$ .", "We use the lemma to prove the existence of the positive cubature rules.", "For the unit sphere and the unit ball, this was established in [33] and later in [5], [34], [38].", "We follow the proof in [12] for the doubling weight on the unit sphere.", "Theorem 2.20 Let ${\\mathsf {w}}$ be a doubling weight on $\\Omega $ .", "Let $\\Xi $ be a maximum $\\frac{\\delta }{n}$ -separated subset of $\\Omega $ .", "There is a $\\delta _0 > 0$ such that for $0 < \\delta < \\delta _0$ there exist positive numbers ${\\lambda }_z$ , $z \\in \\Xi $ , so that $\\int _{\\Omega } f(x) {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x) = \\sum _{z \\in \\Xi }{\\lambda }_z f(z), \\qquad \\forall f \\in \\Pi _n(\\Omega ).$ Moreover, ${\\lambda }_z \\ge c_1 {\\mathsf {w}}\\!\\left(B(z, \\tfrac{\\delta }{n})\\right)$ and, if Assertion 4 holds, then ${\\lambda }_z \\sim {\\mathsf {w}}\\!\\left(B(z, \\tfrac{\\delta }{n})\\right)$ .", "Let $\\varpi $ be the weight function that admits highly localized kernels on $\\Omega $ .", "We use Lemma REF with the inner product defined by ${\\langle }\\cdot ,\\cdot {\\rangle }_{\\varpi }$ .", "Recall that $K_n(\\varpi ; \\cdot ,\\cdot )$ denotes the reproducing kernel of $\\Pi _n(\\Omega )$ .", "Let $\\lbrace z_k: 1 \\le k \\le N\\rbrace $ be an enumeration of $\\Xi $ .", "We define the functions $a^j$ and $\\zeta $ by $a^j(x) \\, & = K_n\\big (\\varpi ; x, z_j\\big ), \\qquad 1 \\le j \\le N, \\\\\\zeta (x) & = \\int _{\\Omega } K_n\\big (\\varpi ; x, y\\big ) {\\mathsf {w}}(y) \\mathrm {d}{\\mathsf {m}}(y) -\\frac{1}{2 c_d} \\sum _{j=1}^N K_n\\big (\\varpi ; x, z_j\\big ) {\\mathsf {w}}\\big (B(z_j, \\tfrac{\\delta }{n}) \\big ),$ where $c_d$ is the constant in (REF ).", "For every $f \\in \\Pi _n(\\Omega )$ , the reproducing property of $K_n(\\varpi ; \\cdot ,\\cdot )$ implies that ${\\langle }f, a^j{\\rangle }_{\\varpi } & = f(z_j), \\qquad 1 \\le j \\le N, \\\\{\\langle }f, \\zeta {\\rangle }_{\\varpi } & = \\int _{\\Omega }f(y) {\\mathsf {w}}(y) \\mathrm {d}{\\mathsf {m}}(y)- \\frac{1}{2 c_0} \\sum _{j=1}^N f (z_j) {\\mathsf {w}}\\!\\big (B(z_j), \\tfrac{\\delta }{n}\\big ).$ Assume ${\\langle }f, a^j{\\rangle }_\\varpi \\ge 0$ for all $j$ , so that $f$ is nonnegative on $\\Xi $ .", "Then, by (REF ), $\\int _\\Omega f(x) & {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x) = \\sum _{j=1}^N \\int _{B(z_j,\\frac{\\delta }{n})} f(x)\\frac{{\\mathsf {w}}(x)}{\\sum _{z \\in \\Xi } \\chi _{B(y,\\frac{\\delta }{n})}(x)} \\mathrm {d}{\\mathsf {m}}(x) \\\\& \\ge \\frac{1}{c_d} \\sum _{j=1}^N f(z_j) \\int _{B(z_j,\\frac{\\delta }{n})} {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x)- \\sum _{j=1}^N \\int _{B(z_j,\\frac{\\delta }{n})} \\mathrm {osc}(f)(x,\\tfrac{\\delta }{n}) {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x) \\\\& \\ge \\frac{1}{c_d} \\sum _{j=1}^N f(z_j) {\\mathsf {w}}\\big (B(z_j,\\tfrac{\\delta }{n})\\big ) - c_1\\delta \\int _{\\Omega } \|f(x)\|{\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x)$ by (REF ).", "Hence, using the Marcinkiewicz-Zygmund inequality in (ii) of Theorem REF with $p = 1$ , we conclude that ${\\langle }f, \\zeta {\\rangle }_{\\varpi } \\ge \\left((2 c_d)^{-1} - c_{{\\mathsf {w}},1} c_1 \\delta \\right)\\sum _{j=1}^N f(z_j){\\mathsf {w}}\\big (B(z_j,\\tfrac{\\delta }{n})\\big ) \\ge 0$ if we choose $\\delta \\le (2 c_d)^{-1} /(c_{{\\mathsf {w}},1} c_1)$ .", "Consequently, (2) of Farkas lemma does not hold so that (1) must hold, which shows that there exist $\\mu _j \\ge 0$ such that $\\int _\\Omega f(x) {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x) - \\frac{1}{2 c_0} \\sum _{j=1}^N f (z_j) {\\mathsf {w}}\\!\\big (B(z_j, \\tfrac{\\delta }{n})\\big )= \\sum _{j=1}^N \\mu _j f(z_j).$ This establishes the cubature rule (REF ) with ${\\lambda }_{z_j} = (2c_0)^{-1}{\\mathsf {w}}(B(z_j, \\tfrac{\\delta }{n}))+\\mu _j$ .", "It follows immediately that ${\\lambda }_z \\ge c {\\mathsf {w}}(B(z, \\tfrac{\\delta }{n}))$ for all $z \\in \\Xi $ .", "To prove the upper bound of this inequality, we set $m = \\lfloor \\frac{n}{2} \\rfloor $ and use the reproducing kernel $K_m({\\mathsf {w}}; \\cdot ,\\cdot )$ of $\\Pi _n(\\Omega )$ with respect to ${\\mathsf {w}}$ .", "Since $K_m({\\mathsf {w}}; z,z) > 0$ , we define a polynomial $q_z(x) = \\frac{K_m({\\mathsf {w}}; x,z)}{K_m({\\mathsf {w}}; z,z)}, \\qquad z \\in \\Xi .$ For each $z$ , the polynomial $q_z^2$ is a polynomial of degree at most $n$ and $q_z(z) =1$ .", "Hence, by (REF ), ${\\lambda }_z \\le \\sum _{y \\in \\Xi } {\\lambda }_y [q_z(y)]^2= \\int _{\\Omega } [ q_z(x) ]^2 {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x) = \\int _{\\Omega } \\frac{K_m({\\mathsf {w}}; x,z)^2}{K_m({\\mathsf {w}}; z,z)^2} {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x).$ Hence, using the reproducing property of the kernel $K_m({\\mathsf {w}}; \\cdot ,\\cdot )$ , $\\int _\\Omega K_m({\\mathsf {w}}; x,z)^2 {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x) =K_m({\\mathsf {w}}; z,z),$ we conclude that ${\\lambda }_z \\le c [K_m({\\mathsf {w}}; z,z)]^{-1}= {\\lambda }_n({\\mathsf {w}}; z)$ .", "Hence, ${\\lambda }_z \\le c\\, {\\mathsf {w}}(B(z, \\tfrac{\\delta }{n}))$ by Proposition REF ." ], [ "Tight polynomial frames", "The highly localized kernels are powerful tools when they exist.", "We use them to study approximation behavior of the integral operator $L_n(\\varpi )f$ that has $L_n\\big (\\varpi ; \\cdot ,\\cdot )$ as its kernel, or more precisely, $L_n(\\varpi ) f (x) = \\int _{\\Omega } f(y) L_n\\big (\\varpi ; x, y) \\varpi (y) \\mathrm {d}{\\mathsf {m}}(y).$ The fast decaying of the kernel indicates that $L_n(\\varpi )f$ provides a good approximation to the function $f$ .", "Its approximation property will be studied in the next section.", "For now, we observe that $L_n(\\varpi )f$ is a polynomial of degree at most $2n$ by the property of the cut-off function.", "We now use the operator $L_n(\\varpi )* f$ and the positive cubature rules based on the discrete set $\\Xi $ to construct tight polynomial frames in $L^2(\\Omega , \\varpi )$ .", "The construction is fairly standard by now, so is the proof of the tight frame.", "See, for example, [12], [34], [37], [38] for the case of the interval, the unit sphere and the unit ball.", "We recall the procedure and will be brief in proof.", "Let $(\\Omega , \\varpi , {\\mathsf {d}})$ be a localizable homogeneous space.", "For the highly localized kernels $L_n(\\varpi ;x,y)$ , we assume its cut-off function $\\widehat{a}$ is of type (b) and satisfies $ \\begin{split}\\widehat{a}(t)\\ge \\rho > 0, \\qquad & \\mbox{if $t \\in [3/5, 5/3]$},\\\\[\\widehat{a}(t)]^2 + [\\widehat{a}(2t)]^2 =1, \\qquad & \\mbox{if $t \\in [1/2, 1]$.", "}\\end{split}$ If $g$ is a nonnegative even function in $C^\\infty ({\\mathbb {R}})$ , so that $\\mathrm {supp}\\, g = [-1, 1]$ , $g(0) =1$ and $\|g(t)\|^2 + \|g(t+1)\|^2 =1$ on $[-1, 0]$ , then $\\widehat{a}(t) := g(\\log _2t)$ has the desired properties.", "The last assumption on $\\widehat{a}$ implies $\\sum _{j=0}^\\infty \\left[ \\widehat{a}\\left( \\frac{t}{2^{j} } \\right) \\right]^2 = 1, \\quad t \\in [1, \\infty ).$ Let ${\\mathsf {w}}$ be the doubling weight that admits Assertion 4.", "Then the cubature rules in Theorem REF are well established for ${\\mathsf {w}}$ .", "For $j =0,1,\\ldots ,$ let ${\\varepsilon }_j = \\frac{\\delta }{2^{j}}$ and let $\\Xi _j$ be a maximal ${\\varepsilon }_j$ -separated subset in $\\Omega $ , where $\\delta $ is chosen so that the cubature rule in Theorem REF holds; that is, there are ${\\lambda }_{z,j} > 0$ for $z \\in \\Xi _j$ such that $\\int _\\Omega f(x) {\\mathsf {w}}(x) \\mathrm {d}{\\mathsf {m}}(x) = \\sum _{z \\in \\Xi _j} \\Xi _{z,j} f(z), \\qquad f \\in \\Pi _{2^j} (\\Omega );$ moreover, ${\\lambda }_{z,j} \\sim {\\mathsf {w}}(B(z, 2^{-j}))$ since we assume that Assertion 4 holds for ${\\mathsf {w}}$ .", "In terms of the kernel $L_n({\\mathsf {w}};\\cdot ,\\cdot )$ defined via the cut-off function $\\widehat{a}$ , we define $F_0(x,z):=1, \\quad \\hbox{and}\\quad F_j(x,z): = L_{2^{j-1}}({\\mathsf {w}}; x, z), \\quad j = 1, 2, 3, \\ldots .$ Accordingly, we define $F_j * f = L_{2^{j-1}} * f$ , that is, $F_j * f (x): = \\int _\\Omega f(y) F_j(x,y) {\\mathsf {w}}(y) \\mathrm {d}{\\mathsf {m}}(y), \\qquad j = 0,1,2,\\ldots .$ Now, for $z \\in \\Xi _j$ and $j = 1, 2,\\ldots $ , we define our frame elements by $\\psi _{z,j}(x):= \\sqrt{{\\lambda }_{z,j}} F_j(x, z).$ Then $\\Psi := \\big \\lbrace \\psi _{z,j}: z \\in \\Xi _j, \\quad 1 \\le j \\le \\infty \\big \\rbrace $ is a frame system.", "By the orthogonality of the reproducing kernel, it follows readily that $F_jF_j f = \\sum _{k=1}^{2^{j}} \\left\|\\widehat{a}\\left(\\frac{k}{2^{j-1}}\\right)\\right\|^2 \\operatorname{proj}_k({\\mathsf {w}}; f).$ Hence, the following semi-discrete Calderón type decomposition follows from (REF ), $f = \\sum _{k=0}^\\infty \\operatorname{proj}_k({\\mathsf {w}};f) = \\sum _{j=0}^\\infty F_j* F_j * f, \\qquad f\\in L^2(\\Omega ,{\\mathsf {w}}).$ Furthermore, the system $\\Psi $ is a tight frame in $L^2(\\Omega , {\\mathsf {w}})$ norm.", "Theorem 2.21 Assume that $\\Omega $ admits a localizable homogenous space.", "Let ${\\mathsf {w}}$ be a doubling weight satisfying Assertion 4.", "If $f\\in L^2(\\Omega , {\\mathsf {w}})$ , then $ f =\\sum _{j=0}^\\infty \\sum _{z \\in \\Xi _j}\\langle f, \\psi _{z, j} \\rangle _{\\mathsf {w}}\\psi _{z,j} \\qquad \\mbox{in $L^2(\\Omega , {\\mathsf {w}})$}$ and $ \\Vert f\\Vert _{2, {\\mathsf {w}}} = \\Big (\\sum _{j=0}^\\infty \\sum _{z \\in \\Xi _j} \|\\langle f, \\psi _{z,j} \\rangle _{\\mathsf {w}}\|^2\\Big )^{1/2}.$ Since $F_j(x,\\cdot ) F_j(\\cdot ,y)$ is a polynomial of degree $2^j$ , we can apply the cubature rule (REF ) to discretize $F_jF_j$ , which gives $F_j F_j(x,y) = \\int _\\Omega F_j(x,z)F_j(z,y) {\\mathsf {w}}(z) \\mathrm {d}{\\mathsf {m}}(z) = \\sum _{z \\in \\Xi _j}\\psi _{z,j}(x)\\psi _{z,j}(y),$ from which the identity (REF ) follows from (REF ) right away.", "Furthermore, let $S_R f$ be the sum $S_R f = \\sum _{j=0}^R \\sum _{z \\in \\Xi _j} \\langle f, \\psi _{z, j} \\rangle _{\\mathsf {w}}\\psi _{z,j}$ .", "Then, it follows immediately ${\\langle }f, S_R f {\\rangle }_{\\mathsf {w}}= \\sum _{j=0}^R \\sum _{z \\in \\Xi _j} \\left\|\\langle f, \\psi _{z, j} \\rangle _{\\mathsf {w}}\\right\|^2.$ Taking the limit $R\\rightarrow \\infty $ we obtain the tight frame identity (REF ).", "For the weight function $\\varpi $ that admits the highly localized kernels, the frame element $\\psi _{z,j}$ has near exponential rate of decay away from its center with respect to the distance ${\\mathsf {d}}(\\cdot , \\cdot )$ on $\\Omega $ .", "Theorem 2.22 Let $(\\Omega ,\\varpi , {\\mathsf {d}})$ be a localizable homogeneous space.", "There, there is a constant $c_{\\kappa }>0$ depending only on ${\\kappa }$ , $d$ , $\\varpi $ and $\\widehat{a}$ such that for $z \\in \\Xi _j$ , $j=0, 1, \\dots $ , $ \|\\psi _{z,j}(x)\| \\le c_{\\kappa }\\frac{1}{\\sqrt{\\varpi (B(z, 2^{-j}))} (1+ 2^j {\\mathsf {d}}(x,z))^{\\kappa }}, \\quad x\\in \\Omega .$ This follows readily from the fast decaying of the kernel and ${\\lambda }_{z,j} \\sim {\\mathsf {w}}(B(z, 2^{-j}))$ .", "It shows that the frame $\\Psi $ is highly localized, which makes it a powerful tool for various applications, such as decompositions of functions spaces and computational harmonic analysis, on more specific domains." ], [ "Polynomial Approximation on homogeneous spaces", "We consider approximation by polynomials in the space $L^p(\\Omega ,\\varpi )$ when the weight function $\\varpi $ posses two additional properties that are analogs of two characteristics of spherical harmonics: addition formula for the reproducing kernels and a differential operator that has orthogonal polynomials as eigenfunctions.", "With appropriate weight functions, these properties are shared by orthogonal polynomials on the unit sphere, on the unit ball, and on the simplex.", "The addition formula premises that the reproducing kernels possess a one-dimensional character, which leads to a convolution structure that allows us to reduce much of the study of the Fourier orthogonal series to that of the Fourier-Jacobi series of one variable.", "The differential operator that has orthogonal polynomials as eigenfunctions leads to a $K$ -functional.", "Together they provide us with tools for characterizing the best approximation by polynomials, which is a central problem in approximation theory.", "The goal of this section is to develop this framework, which extends the results on the unit sphere and the unit ball and it is applicable on the conic domains.", "The two characteristic properties are defined and discussed in the first subsection.", "The first one is used to define the convolution structure that leads to the definition of the modulus of smoothness in the second subsection.", "The second one is used to define a $K$ -functional in the third subsection, where the main results on the characterization of the best polynomial approximation are stated and discussed.", "The near-best approximation operator, used to provide a direct estimate, is studied in the fourth subsection with the help of the differential operator, where some of the results are established with norms defined via a doubling weight.", "The Bernstein inequality, essential for proving the inverse estimate, is proved in the fifth subsection, together with the Nikolskii inequality.", "Finally, the proof of the main results is given in the seventh subsection." ], [ "Addition formula and differential operator", "Let $\\varpi $ be a weight function defined on $\\Omega $ .", "For studying approximation by polynomials, we require two more properties on the orthogonal structure of $L^2(\\Omega ,\\varpi )$ .", "These properties are analogs to the two characteristics of spherical harmonics on the unit sphere: the Laplace-Beltrami operator (REF ) and the addition formula (REF ), which serve as the quintessential examples in the study [53], [54] as well as in this subsection.", "These properties are shared by several other domains, including the unit ball, the simplex, and conic domains.", "Below we assume these properties and use them to carry out our subsequent study.", "The first property that we assume is an analog of the Laplace-Beltrami operator.", "Recall that ${\\mathcal {V}}_n(\\Omega , \\varpi )$ denote the space of orthogonal polynomials of degree $n$ with respect to $\\varpi $ on $\\Omega $ .", "Definition 3.1 Let $\\varpi $ be a weight function on $\\Omega $ .", "We denote by ${\\mathfrak {D}}_\\varpi $ the second order derivation operator that has orthogonal polynomials with respect to $\\varpi $ as eigenfunctions; more precisely, ${\\mathfrak {D}}_{\\varpi } Y = - \\mu (n) Y, \\qquad \\forall \\, Y \\in {\\mathcal {V}}_n(\\Omega ,\\varpi ),$ where $\\mu $ is a nonnegative quadratic polynomial.", "For the unit sphere ${\\mathbb {S}^{d-1}}$ , the operator ${\\mathfrak {D}}_\\varpi $ is the Laplace-Beltrami operator given by (REF ).", "For the unit ball ${\\mathbb {B}}^d$ , ${\\mathfrak {D}}_\\varpi $ is the operator given by (REF ).", "In both cases, the operator is a differential operator.", "For the unit sphere and the unit ball with reflection invariant weight functions, ${\\mathfrak {D}}_\\varpi $ is a differential-difference operator, called Dunkl Laplacian [13], [14].", "For $d =2$ and $\\Omega $ is a domain with non-empty interior, the existence of a second order differential operator that satisfies (REF ) was characterized in [21].", "There are essentially five cases with positive weight functions on regular domains but only two are compact domains: disk and triangle.", "The characterization for $d > 2$ remains an open problem.", "The operators for the conic domains are the recent additions, discovered in [53], [54] and will be recalled in later sections.", "The second property that we assume is the addition formula, which gives a closed-form formula for the reproducing kernel $P_n(\\varpi ; \\cdot ,\\cdot )$ of ${\\mathcal {V}}_n(\\Omega , \\varpi )$ .", "Definition 3.2 Let $\\varpi $ be a weight function on $\\Omega $ .", "The reproducing kernel $P_n(\\varpi ; \\cdot ,\\cdot )$ is said to satisfy an addition formula if, for some ${\\alpha }\\ge {\\beta }\\ge - \\frac{1}{2}$ , $P_n (\\varpi ; x, y) = \\int _{[-1,1]^m} Z_n^{({\\alpha },{\\beta })} \\big (\\xi (x, y; u) \\big ) \\mathrm {d}\\tau (u), \\qquad Z_n^{({\\alpha },{\\beta })}(t) = \\frac{P_n^{({\\alpha },{\\beta })}(1) P_n^{({\\alpha },{\\beta })}(t)}{h_n^{({\\alpha },{\\beta })}},$ where $m$ is a positive integer; $\\xi (x, y; u)$ is a function of $u \\in [-1,1]^m$ , symmetric in $x$ and $y$ , and $\\xi (x, y; u) \\in [-1,1]$ ; moreover, $\\mathrm {d}\\tau $ is a probability measure on $[-1,1]^m$ , which can degenerate to have a finite support.", "The classical addition formula is the one for the unit sphere ${\\mathbb {S}^{d-1}}$ , given in (REF ), which is the degenerate case, with ${\\alpha }={\\beta }= \\frac{d-3}{2}$ .", "The one for the classical weight $\\varpi _\\mu $ on the unit ball ${\\mathbb {B}}^d$ is given by (REF ), which has $m =1$ and ${\\alpha }= {\\beta }= \\mu +\\frac{d-2}{2}$ .", "Similar formula holds for reflection invariant weight functions on the unit sphere and on the unit ball (cf.", "[14]).", "Furthermore, for the simplex in ${\\mathbb {R}}^d$ (cf.", "[14]) and the conic domains ${\\mathbb {V}}_0^{d+1}$ and ${\\mathbb {V}}^{d+1}$ , the polynomial $Z_n^{({\\alpha },{\\beta })}$ is given by $Z_{2n}^{\\lambda }$ for some ${\\lambda }\\ge 0$ , so that, by the quadratic transform (REF ), it is given by the Jacobi polynomial with ${\\alpha }= {\\lambda }-\\frac{1}{2}$ and ${\\beta }= -\\frac{1}{2}$ .", "The addition formula premises a one-dimensional structure for the reproducing kernel.", "In all known cases, the estimates of the highly localized kernel $L_n(\\varpi ; \\cdot ,\\cdot )$ are established with the help of this formula, since the addition formula implies $L_n(\\varpi ; x, y) = \\int _{[-1,1]^m} L_n^{({\\alpha },{\\beta })} \\big (\\xi (x, y; u) \\big ) \\mathrm {d}\\tau (u),$ where the function $t \\mapsto L_n^{({\\alpha },{\\beta })}(t) = L_n^{({\\alpha },{\\beta })}(t,1)$ is the kernel defined in (REF ) for the Jacobi polynomials.", "In the rest of this section, we assume that the orthogonal polynomials with respect to $\\varpi $ posses both the derivation operator and the addition formula.", "We do not, however, require that $\\varpi $ admits highly localized kernels that satisfy Assertions 1–3.", "In fact, some of our results hold if $\\varpi $ satisfies Assertions 1 and 3, but need not satisfy Assertion 2." ], [ "Convolution structure", "Making use of the one-dimensional structure premised by the addition formula, we define a convolution operator on $\\Omega $ .", "Definition 3.3 Assume $\\varpi $ admits the addition formula.", "For $f \\in L^1(\\Omega , \\varpi )$ and $g \\in L^1([-1,1],w_{{\\alpha },{\\beta }})$ , we define the convolution of $f$ and $g$ by $(f \\ast _\\varpi g)(x) := \\int _{\\Omega } f(y) T^{({\\alpha },{\\beta })} g (x,y) \\varpi (y) \\mathrm {d}{\\mathsf {m}}(y),$ where the operator $g\\mapsto T^{({\\alpha },{\\beta })} g$ is defined by $T^{({\\alpha },{\\beta })} g(x,y) := \\int _{[-1,1]^m} g \\big ( \\xi (x, y; u)\\big ) \\mathrm {d}\\tau (u).$ The addition formula shows that the projection operator $\\operatorname{proj}_n (\\varpi ; f)$ and the reproducing kernel $P_n(\\varpi ; \\cdot ,\\cdot )$ satisfy $\\operatorname{proj}_n(\\varpi ; f, x) = f_\\varpi Z_n^{({\\alpha },{\\beta })} \\quad \\hbox{and}\\quad P_n(\\varpi ; x,y) = T^{({\\alpha },{\\beta })} \\left( Z_n^{({\\alpha },{\\beta })}\\right).$ The operator $T^{({\\alpha },{\\beta })}$ is defined in [54] with a more generic orthogonal polynomial in place of $P_n^{({\\alpha },{\\beta })}$ , where the following boundedness of the operator is established [54].", "Lemma 3.4 Let $g \\in L^1([-1,1],w_{{\\alpha },{\\beta }})$ .", "Then, for each $Q_n \\in {\\mathcal {V}}_n (\\Omega , \\varpi )$ , $\\int _{\\Omega } T^{({\\alpha },{\\beta })} g (x,y) Q_n (y) \\varpi (y) \\mathrm {d}y = \\Lambda _n^{({\\alpha },{\\beta })} (g) Q_n(x),$ where $\\Lambda _n^{({\\alpha },{\\beta })} (g) = c_{{\\alpha },{\\beta }} \\int _{-1}^1 g(t) R_n^{({\\alpha },{\\beta })}(t) w_{{\\alpha },{\\beta }}(t)\\mathrm {d}t\\quad \\hbox{with} \\quad R_n^{({\\alpha },{\\beta })}(t) : = \\frac{P_n^{({\\alpha },{\\beta })}(t)}{P_n^{({\\alpha },{\\beta })}(1)}$ Furthermore, for $1\\le p \\le \\infty $ and $x \\in \\Omega $ , $\\left\\Vert T^{({\\alpha },{\\beta })} g (x, \\cdot )\\right\\Vert _{L^p(\\Omega , \\varpi )} \\le \\Vert g \\Vert _{L^p([-1,1],w_{{\\alpha },{\\beta }})}.$ The convolution operator $f \\ast _\\varpi g$ is also defined in [54] and shown to satisfy the usual Young's inequality.", "Theorem 3.5 Let $p,q,r \\ge 1$ and $p^{-1} = r^{-1}+q^{-1}-1$ .", "For $f \\in L^q(\\Omega , \\varpi )$ and $g \\in L^r([-1,1]; w_{{\\alpha },{\\beta }})$ , $ \\Vert f \\ast _\\varpi g\\Vert _{L^p (\\Omega , \\varpi )} \\le \\Vert f\\Vert _{L^q(\\Omega ,\\varpi )}\\Vert g\\Vert _{L^r( [-1,1];\\varpi )}.$ As an application of these results, we consider the Cesàro $(C,\\delta )$ means $S_n^\\delta (\\varpi ;f)$ of the Fourier orthogonal series with respect to $\\varpi $ .", "For $\\delta > 0$ , the operator $S_n^\\delta (\\varpi )$ is defined by $S_n^\\delta (\\varpi ;f) := \\frac{1}{\\binom{n+\\delta }{n}} \\sum _{k=0}^n \\binom{n-k+\\delta }{n-k} \\operatorname{proj}_k(\\varpi ; f).$ It can be written as an integral operator with the kernel $K_n^\\delta (\\varpi ; \\cdot ,\\cdot )$ being the $(C,\\delta )$ mean of the reproducing kernel $P_n(\\varpi ;\\cdot ,\\cdot )$ .", "By the addition formula, the kernel $K_n^\\delta (\\varpi ; \\cdot ,\\cdot )$ can be written as $K_n^\\delta (\\varpi ; x,y) = \\int _{[-1,1]^m} k_n^{({\\alpha },{\\beta }), \\delta }(\\xi (x,y;u), 1) \\mathrm {d}\\tau (u),$ where $k_n^{({\\alpha },{\\beta }), \\delta }(s,t)$ denotes the kernel of the $(C,\\delta )$ mean of the Fourier-Jacobi series.", "In particular, using the positivity of the kernel $k_n^{({\\alpha },{\\beta }), \\delta }$ [1] and the boundedness of the kernel [46], we obtain the following theorem.", "Theorem 3.6 Let $\\varpi $ be a weight function that admits the addition formula.", "The Cesàro $(C,\\delta )$ means of the Fourier orthogonal series with respect to $\\varpi $ satisfy If $\\delta \\ge {\\alpha }+{\\beta }+2$ , then $S_n^\\delta (\\varpi ; f)$ is a nonnegative operator; If $\\delta > {\\alpha }+\\frac{1}{2}$ , then for $n= 0,1,2,\\ldots $ , $\\left\\Vert S_n^\\delta (\\varpi ; f)\\right\\Vert _{p,\\varpi } \\le \\Vert f\\Vert _{p,\\varpi }, \\qquad 1 \\le p \\le \\infty .$ To further explore the one-dimensional structure that leads to the definition of the convolution, we define an operator $S_{{\\theta },\\varpi } f$ as follows.", "Definition 3.7 Assume the addition formula holds for the weight function $\\varpi $ .", "For $0\\le {\\theta }\\le \\pi $ , the translation operator $S_{{\\theta },\\varpi }$ is defined by $\\operatorname{proj}_n (\\varpi ; S_{{\\theta }, \\varpi } f) = R_n^{({\\alpha },{\\beta })}(\\cos {\\theta }) \\operatorname{proj}_n(\\varpi ; f),\\quad n = 0,1,2,\\ldots .$ As a corollary of Theorem REF , a function $f \\in L^1(\\Omega ,\\varpi )$ is uniquely determined by its orthogonal projections $\\operatorname{proj}_n (\\varpi ; f)$ , $n \\ge 0$ .", "Hence, the operator $S_{{\\theta },\\varpi }$ is well defined for all $f\\in L^1(\\Omega , \\varpi )$ .", "For the unit sphere with the Lebesgue measure $\\mathrm {d}\\sigma $ , the operator $S_{\\theta }$ is an integral operator given by [12] $S_{{\\theta }, \\mathrm {d}\\sigma } f(x) = \\frac{1}{{\\omega }_{d-1}(\\sin {\\theta })^{d-1}} \\int _{c(x,{\\theta })} f(y) \\mathrm {d}\\ell (y),$ where $c(x,{\\theta }) = \\lbrace y: {\\langle }x,y{\\rangle }=\\cos {\\theta }\\rbrace $ is the spherical cap and $\\mathrm {d}\\ell $ denotes the Lebesgue measure on the cap.", "For the unit ball ${\\mathbb {B}}^d$ with the classical weight $W_\\mu $ , the operator $S_{{\\theta },W_\\mu }$ can also be written as an explicit integral operator [12].", "Proposition 3.8 The operator $S_{{\\theta },\\varpi }$ satisfies the following properties: For $f \\in L^2(\\Omega , \\varpi )$ and $g \\in L^1([-1,1], w_{{\\alpha },{\\beta }})$ , $(f_\\varpi g)(x) = c_{{\\alpha },{\\beta }} \\int _0^\\pi S_{{\\theta }, \\varpi } f(x) g(\\cos {\\theta }) w_{{\\alpha },{\\beta }}(\\cos {\\theta }) \\sin {\\theta }\\mathrm {d}{\\theta }.$ $S_{{\\theta }, \\varpi } f$ preserves positivity; that is, $S_{{\\theta }, \\varpi } f \\ge 0$ if $f \\ge 0$ .", "For $f\\in L^p(\\varpi ; \\Omega )$ , if $1 \\le p \\le \\infty $ , or $f \\in C(\\Omega )$ if $p =\\infty $ , $\\Vert S_{{\\theta }, \\varpi } f \\Vert _{p,\\varpi } \\le \\Vert f\\Vert _{p, \\varpi } \\quad \\hbox{and} \\quad \\lim _{{\\theta }\\rightarrow 0} \\Vert S_{{\\theta }, \\varpi } f - f\\Vert _{\\varpi ,p} =0.$ From the Fubini theorem and (REF ), it is easy to see that $\\operatorname{proj}_n (\\varpi ; f_\\varpi g) \\, & = \\Lambda _n^{({\\alpha },{\\beta })}(g) \\operatorname{proj}_n (\\varpi ; f) \\\\& = c_{{\\alpha },{\\beta }} \\int _0^\\pi \\operatorname{proj}_n (\\varpi ; S_{{\\theta },\\varpi } f) g(\\cos {\\theta }) w_{{\\alpha },{\\beta }}(\\cos {\\theta }) \\sin {\\theta }\\mathrm {d}{\\theta },$ from which (i) follows.", "To prove (ii), we let $g_n$ be a non-negative function such that $g_n(\\cos {\\theta })$ is supported on $[-\\frac{1}{n}, \\frac{1}{n}]$ and $\\int _0^\\pi g_n(\\cos {\\theta }) w_{{\\alpha },{\\beta }}({\\theta }) \\sin {\\theta }\\mathrm {d}{\\theta }= 1$ .", "Then, using the expression in (i), $f \\ast _\\varpi g_n$ converges to $S_{{\\theta },\\varpi } f$ , which proves (ii).", "Moreover, by (REF ), $\\Vert f\\ast _\\varpi g_n \\Vert _{p, \\varpi } \\le \\Vert f\\Vert _{p,\\varpi }$ , so that by the Fatou lemma, $\\Vert S_{{\\theta }, \\varpi } f \\Vert _{p,\\varpi } \\le \\Vert f\\Vert _{p, \\varpi }$ .", "Finally, if $f_n = S_n^\\delta (\\varpi ;f)$ with $\\delta \\ge {\\alpha }+{\\beta }+2$ , then $f_n \\ge 0$ and $f_n \\rightarrow f$ in $L^p(\\Omega , \\varpi )$ ; furthermore, by (REF ), $S_{{\\theta },\\varpi } f_n$ converges to $f_n$ for ${\\theta }\\rightarrow 0$ ; consequently, by the triangle inequality, $\\Vert S_{{\\theta }, \\varpi } f - f\\Vert _{p,\\varpi }$ converges to 0 when ${\\theta }\\rightarrow 0$ .", "This completes the proof.", "The operator $S_{{\\theta },\\varpi }$ can be used to define a modulus of smoothness.", "For $r > 0$ , we defined the $r$ -th difference operator $\\triangle _{{\\theta },\\varpi }^r = \\left(I - S_{{\\theta },\\varpi }\\right)^{r/2} =\\sum _{n=0}^\\infty (-1)^n \\binom{r/2}{n} (S_{{\\theta },\\varpi })^n,$ where $I$ denote the identity operator, in the distribution sense, by $\\operatorname{proj}_n \\left(\\varpi ; \\triangle _{{\\theta },\\varpi }^r f\\right)= \\left(1- R_n^{({\\alpha },{\\beta })}(\\cos \\theta )\\right)^{r/2} \\operatorname{proj}_n(\\varpi ; f),\\quad n=0,1,2,\\cdots .$ Definition 3.9 Let $r >0$ and $0 < {\\theta }< \\pi $ .", "For $f\\in L^p(\\Omega ,\\varpi )$ and $1\\le p<\\infty $ or $f\\in C(\\Omega )$ and $p=\\infty $ , the weighted $r$ th order modulus of smoothness is defined by $\\omega _r(f,t)_{p,\\varpi }:= \\sup _{0< \\theta \\le t} \\left\\Vert \\triangle _{{\\theta },\\varpi }^r f\\right\\Vert _{p,\\varpi }, \\quad 0 < t <\\pi .$ For the unit sphere with the Lebesuge measure, the definition of this modulus of smoothness is classical.", "For weighed approximation on the unit sphere, the unit ball, and the simplex, it is given in [52].", "The proof of the following lemma is standard, see [12] for example, and will be omitted.", "Proposition 3.10 Let $f\\in L^p(\\Omega ,\\varpi )$ if $1\\le p<\\infty $ and $f\\in C(\\Omega )$ if $p=\\infty $ .", "Then ${\\omega }_r(f,t)_{p,\\varpi }\\le 2^{r+2} \\Vert f\\Vert _{p,\\varpi }$ ; $\\omega _r(f,t)_{p,\\varpi } \\rightarrow 0$ if $t \\rightarrow 0+$ ; $\\omega _r(f,t)_{p,\\varpi }$ is monotone nondecreasing on $(0,\\pi )$ ; $\\omega _r(f+g,t)_{p,\\varpi } \\le \\omega _r(f,t)_{\\kappa ,p}+ \\omega _r(g,t)_{p,\\varpi }$ ; For $0 < s < r$ , $\\omega _r(f,t)_{p,\\varpi } \\le 2^{(r-s)+2} \\omega _s(f,t)_{p,\\varpi }.$" ], [ "Characterization of best approximation", "Let ${\\mathsf {w}}$ be a doubling weight on $\\Omega $ .", "For $f\\in L^p(\\Omega , {\\mathsf {w}})$ , we denote by $E_n(f)_{p, {\\mathsf {w}}}$ the best approximation to $f$ from $\\Pi _n(\\Omega )$ in the norm $\\Vert \\cdot \\Vert _{p, {\\mathsf {w}}}$ of $L^p(\\Omega , {\\mathsf {w}})$ ; that is, $E_n(f)_{p, {\\mathsf {w}}}:= \\inf _{g \\in \\Pi _n(\\Omega )} \\Vert f - g\\Vert _{p, {\\mathsf {w}}}, \\qquad 1 \\le p \\le \\infty .$ A central problem of approximation theory is to character this quantity by the smoothness of the functions, usually in terms of a modulus of smoothness or a $K$ -functional, which are often equivalent.", "In this subsection, we state our main results on the characterization of the best approximation.", "The modulus of smoothness is already defined.", "We now define the $K$ -functional via the differential operator ${\\mathfrak {D}}_\\varpi $ in (REF ).", "Since the operator $-{\\mathfrak {D}}_\\varpi $ has nonnegative eigenvalues, it is a non-negative operator.", "A function $f \\in L^p(\\Omega ;\\varpi )$ belongs to the Sobolev space $W_p^r (\\Omega ; \\varpi )$ if there is a function $g \\in L^p(\\Omega ; \\varpi )$ , which we denote by $(-{\\mathfrak {D}}_\\varpi )^{\\frac{r}{2}}f$ , such that $ \\operatorname{proj}_n\\left(\\varpi ; (-{\\mathfrak {D}}_\\varpi )^{\\frac{r}{2}}f\\right) = \\mu (n)^{\\frac{r}{2}} \\operatorname{proj}_n(\\varpi ; f),$ where we assume that $f, g \\in C(\\Omega )$ when $p = \\infty $ .", "The fractional differential operator $(-{\\mathfrak {D}}_\\varpi )^{\\frac{r}{2}}f$ is a linear operator on the space $W_p^r (\\Omega ; \\varpi )$ defined by (REF ).", "Let ${\\mathsf {w}}$ be a doubling weight.", "We denote by $W_p^r(\\Omega , {\\mathsf {w}})$ the Sobolev space that consists of $f \\in L^p(\\Omega , {\\mathsf {w}})$ such that $(-{\\mathfrak {D}}_\\varpi )^{\\frac{r}{2}}f \\in L^p(\\Omega , {\\mathsf {w}})$ .", "The space is well defined as we shall see later in Theorem REF .", "We define our $K$ -functional as follows.", "Definition 3.11 Let $r > 0$ and $1 \\le p \\le \\infty $ .", "The $r$ -th $K$ -functional of $f\\in L^p(\\Omega , {\\mathsf {w}})$ is defined by $K_r(f,t)_{p,{\\mathsf {w}}} : = \\inf _{g \\in W_p^r(\\Omega , {\\mathsf {w}})}\\left\\lbrace \\Vert f-g\\Vert _{p,{\\mathsf {w}}} + t^r\\left\\Vert (-{\\mathfrak {D}}_\\varpi )^{\\frac{r}{2}}g \\right\\Vert _{p,{\\mathsf {w}}} \\right\\rbrace .$ We first state of our characterization theorem in terms of the $K$ -functional, which contains two parts, the first part is the direct estimate or the Jackson inequality, and the second part is the inverse estimate.", "For the first part, we need another assertion to ensure that a Bernstein inequality holds.", "This is Assertion 5 and it will be stated in Subsection REF .", "Theorem 3.12 Let $\\varpi $ be a weight function that admits Assertions 1, 3 and 5.", "Let $f \\in L^p(\\Omega , \\varpi )$ if $1 \\le p < \\infty $ and $f\\in C(\\Omega )$ if $p = \\infty $ .", "Then for $r > 0$ and $n =1,2,\\ldots $ , the direct estimate $E_n(f)_{p,\\varpi } \\le c K_r (f;n^{-1})_{p,\\varpi };$ the inverse estimate $K_r(f;n^{-1})_{p,\\varpi } \\le c n^{-r} \\sum _{k=0}^n (k+1)^{r-1}E_k(f)_{p, \\varpi }.$ It is worth mentioning that the direct estimate holds for the weight norm of $\\varpi $ , whereas the inverse estimate holds for all doubling weight.", "We can also replace the above characterization by the modulus of smoothness $\\varpi _r (f;t)_{p,\\varpi }$ , since the two quantities will be shown to be equivalent.", "Theorem 3.13 Let $\\varpi $ be a weight function that admits Assertions 1, 3 and 5.", "Let $f \\in L^p(\\Omega ,\\varpi )$ if $1 \\le p < \\infty $ and $f\\in C(\\Omega )$ if $p = \\infty $ .", "If $0 < {\\theta }\\le \\pi /2$ and $r > 0$ , then $c_1 K_r(f;{\\theta })_{p,\\varpi } \\le {\\omega }_r(f;{\\theta })_{p,\\varpi } \\le c_2 K_r(f;{\\theta })_{p,\\varpi }.$ For the unit sphere and the unit ball, these theorems are known to hold when $\\varpi $ being reflection invariant weight functions.", "For the unit sphere with the Lebesgue measure, the characterization of best approximation has a long history.", "Starting from [4], [36] for $r =2$ , the problem was studied by a number of authors and finally completed in [42]; we refer to [12] for historical notes.", "These results are extended to the weighted case, with reflection invariant weight functions, on the unit sphere, the unit ball, and the simplex, in [52].", "Another pair of $K$ -functional and modulus of smoothness was used to characterize the best approximation for the unit sphere with the Lebesgue measure in [11].", "The approach appears to be very much domain-specific and relies on the geometric and the differential structure on the unit sphere.", "We will give the proof of these two theorems in Subsection REF , after proving several preliminary results that are of interest in their own right." ], [ "Near best approximation operator", "In this subsection we study the approximation property of the operator $L_n(\\varpi )f$ , defined in (REF ), that has the highly localized kernel as its kernel.", "By (REF ), the operator can be written in terms of the convolution operator by $L_n(\\varpi ) f = f \\ast _\\varpi L_n^{({\\alpha },{\\beta })}, \\qquad n =0,1,2,\\ldots .$ We now show that $L_n(\\varpi )f$ provides near-best approximation to $f$ .", "Theorem 3.14 Let $\\widehat{a}$ be admissible of type $(a)$ .", "Assume $\\varpi $ admits Assertions 1 and 3.", "Then the operator $L_n f$ satisfies $L_n(\\varpi )f$ is a polynomial of degree at most $2n$ ; $L_n(\\varpi )g = g$ for any polynomial $g \\in \\Pi _n(\\Omega )$ of degree $n$ ; For $f \\in L^p(\\Omega , \\varpi )$ , $1 \\le p \\le \\infty $ , $ \\Vert L_n(\\varpi )f\\Vert _{p, \\varpi } \\le c \\Vert f\\Vert _{p, \\varpi } \\quad \\hbox{and} \\quad \\Vert L_n(\\varpi )f - f \\Vert _{p, \\varpi } \\le c\\, E_n (f)_{p, \\varpi }.$ The first two properties are immediate consequences of the definition.", "For the third property, in the case $p=1$ we obtain $\\Vert L_n(\\varpi )f\\Vert _{1,\\varpi } \\le c \\Vert f\\Vert _{1,\\varpi } \\max _{x\\in \\Omega } \\int _{\\Omega } \\left\| L_n\\big (\\varpi ; x,y)\\right\|\\varpi (y) \\mathrm {d}{\\mathsf {m}}(y).$ By Assertion 1 and (REF ) with $p =1$ and a large $\\tau $ , the integral in the right-hand side is bound by $\\max _{x\\in \\Omega } \\int _{\\Omega } \\frac{ \\varpi (y) \\mathrm {d}{\\mathsf {m}}(y)}{\\sqrt{\\varpi \\left(B\\left(y,n^{-1}\\right)\\right)}\\sqrt{\\varpi \\left(B\\left(x,n^{-1}\\right)\\right)}\\big (1 + n {\\mathsf {d}}(x,y) \\big )^{\\tau }} \\le c,$ which shows that $\\Vert L_n(\\varpi ) f\\Vert _{p, {\\mathsf {w}}}$ is bounded for $p =1$ .", "The same inequality also shows that the boundedness holds for $p = \\infty $ .", "The case $1< p < \\infty $ follows from the Riesz-Thorin interpolation theorem.", "By item (b), the boundedness of $\\Vert L_n(\\varpi )f\\Vert _{p,\\varpi }$ implies, by the triangle inequality, $\\Vert L_n(\\varpi )f - f \\Vert _{p, \\varpi } \\le \\Vert L_n(\\varpi ) (f - g) \\Vert _{p, \\varpi } + \\Vert f - g \\Vert _{p, \\varpi } \\le c \\Vert f-g\\Vert _{p,\\varpi }$ for any polynomial $g \\in \\Pi _n(\\Omega )$ .", "Taking infimum over $g$ completes the proof.", "Because of property (c), we call the operator $L_n(\\varpi )f$ near best approximation polynomial.", "Such operators have been used extensively in approximation theory and computational harmonic analysis on the unit sphere and the unit ball (see [12], for example), and in various other domains.", "Theorem 3.15 Let the assumption be the same as in the previous theorem.", "For $f \\in W_p^r(\\Omega , \\varpi )$ , $1\\le p\\le \\infty $ , $ \\Vert f - L_n(\\varpi )f\\Vert _{p,\\varpi } \\le c n^{-r} \\left\\Vert (-{\\mathfrak {D}}_{\\varpi })^{\\frac{r}{2}}f \\right\\Vert _{p,\\varpi },\\qquad n =1,2,\\ldots .$ Let $\\ell $ be a positive integer.", "Without loos of generality, we can assume $n > \\ell $ .", "Since $L_n(\\varpi )f = f$ if $f$ is a polynomial of degree $n$ , we can write $f - L_n(\\varpi )f \\, & = \\sum _{k=n+1}^\\infty \\left(1 - \\widehat{a} \\left(\\frac{k}{n} \\right) \\right)\\operatorname{proj}_k (\\varpi ; f) \\\\& = \\sum _{k=n+1}^\\infty \\!", "\\left(1 - \\widehat{a} \\!\\left(\\frac{k}{n} \\right) \\right) \\mu (k)^{-\\frac{r}{2}}\\operatorname{proj}_k\\!", "\\left(\\varpi ;F\\right),$ where $F =\\left(-{\\mathfrak {D}}_\\varpi \\right)^{\\frac{r}{2}} f$ .", "Summation by parts $\\ell +1$ times, we obtain $f -L_n(\\varpi )f = \\sum _{k=n+1}^\\infty b_k \\operatorname{proj}_k\\!", "\\left(\\varpi ;F\\right)= \\sum _{k=n+1}^\\infty \\left( \\triangle ^{\\ell +1} b_k\\right) A_k^\\ell S_k^\\ell \\left(\\varpi ; F\\right),$ where $b_k = (1 - \\widehat{a}(\\frac{k}{n} )) \\mu (k)^{-\\frac{r}{2}}$ , $A_k^\\ell = \\binom{k+\\ell }{k} \\sim k^\\ell $ and $S_k^\\ell \\left(\\varpi ; F\\right)$ denotes the $k$ -th Cesàro $(C,\\ell )$ mean of the Fourier orthogonal series with respect to $\\varpi $ .", "Since $\\widehat{a}$ is in $C^\\infty $ and its support is $[0,2]$ , it is easy to see that $\|\\triangle ^{\\ell +1} b_k\| \\le c k^{-r-\\ell -1}$ .", "Hence, it follows that $\\sum _{k=n+1}^\\infty \\left\| \\triangle ^{\\ell +1} b_k\\right\| A_k^\\ell \\le c \\sum _{k=n+1}^\\infty k^{-r-\\ell -1} k^\\ell \\le c n^r.$ Since the choice of $\\ell $ implies that $\\Vert S_k^\\ell \\left(\\varpi ; F\\right)\\Vert _{p, \\varpi } \\le \\Vert F\\Vert _{p, \\varpi }$ by the convergence of the Cèsaro means, we obtain $\\Vert f - L_n(\\varpi )f\\Vert _{p, \\varpi } \\le cn^{-r} \\Vert F\\Vert _{p,\\varpi },$ which is what we need to prove." ], [ "Bernstein and Nikolskii inequalities", "The Bernstein inequality is essential for studying approximation by polynomials.", "We establish such an inequality for the deviation operator ${\\mathfrak {D}}_\\varpi $ defined by (REF ).", "First we prove a proposition that is of independent interest.", "Proposition 3.16 Let $(\\Omega , \\varpi , {\\mathsf {d}})$ be a localizable homogeneous space and assume Assertion 4 holds.", "Let $G_n(\\cdot ,\\cdot ): \\Omega \\times \\Omega \\mapsto {\\mathbb {R}}$ be a kernel such that $G_n(x,y) = G_n(y,x)$ for all $x, y\\in \\Omega $ and $G_n$ is a polynomial of degree $n$ in either of its variables.", "Let $T: f\\mapsto Tf$ be the operator defined by $T f(x) = \\int _\\Omega f(y) G_n(x,y) \\varpi (y) \\mathrm {d}{\\mathsf {m}}(y).$ If $f$ is a polynomial of degree $n$ , then for $1 \\le p \\le \\infty $ and any doubling weight ${\\mathsf {w}}$ , $\\Vert T f\\Vert _{p,{\\mathsf {w}}} \\le c \\Vert f\\Vert _{p,{\\mathsf {w}}}\\max _{z\\in \\Omega } \\int _\\Omega \| G_n(x,z)\| (1+ n {\\mathsf {d}}(z,x))^{{\\alpha }({\\mathsf {w}})+{\\alpha }(\\varpi )} \\varpi (x) \\mathrm {d}{\\mathsf {m}}(x).$ Since $T$ is a linear integral operator, if $f$ is a polynomial of degree $n$ then $T f$ is a polynomial of the same degree.", "Let $\\Xi $ be an ${\\varepsilon }$ -separated set of $\\Omega $ for ${\\varepsilon }= \\frac{\\delta }{2n}$ so that the Marcinkiewicz-Zygmund inequalities in Theorem REF hold for all polynomials of degree $2n$ .", "The inequality is trivial for $p = \\infty $ since it holds without $ (1+ n {\\mathsf {d}}(z,x))^{{\\alpha }({\\mathsf {w}})}$ term.", "For $1\\le p < \\infty $ , by (REF ), $\\Vert T f\\Vert _{p,{\\mathsf {w}}}^p \\le c \\sum _{z \\in \\Xi } \| T f(z)\|^p {\\mathsf {w}}\\left(B(z,\\tfrac{\\delta }{2n})\\right).$ Since $f(y) G_n(z,y)$ is a polynomial of degree $2n$ in $y$ variable, applying (REF ) again, we obtain $\| T f(z)\| \\le c \\sum _{u \\in \\Xi } \|f(u)\| \|G_n(u, z)\| \\varpi \\left(B(u,\\tfrac{\\delta }{2n})\\right).$ Assume $1\\le p < \\infty $ .", "By the Hölder's inequality, $\|Tf(z)\|^p \\,& \\le c \\sum _{u \\in \\Xi } \|f(u)\|^p \|G_n(z,u)\|\\varpi \\left(B(u,\\tfrac{\\delta }{2n})\\right)\\left( \\sum _{u \\in \\Xi } \|G_n(z,u)\| \\varpi \\left(B(u,\\tfrac{\\delta }{2n})\\right)\\right)^{\\frac{p}{q}} \\\\& \\le c \\sum _{u \\in \\Xi } \|f(u)\|^p \|G_n(z,u)\|\\varpi \\left(B(u,\\tfrac{\\delta }{2n})\\right)\\Vert G_n(z,\\cdot )\\Vert _1^{\\frac{p}{q}},$ where we used (REF ) in the second step.", "Using (ii) of Lemma REF for both $\\varpi $ and ${\\mathsf {w}}$ , we see that $& \\sum _{z\\in \\Xi } \|T f(z)\|^p {\\mathsf {w}}\\!\\left(B(z,\\tfrac{\\delta }{2n})\\right)\\le c \\max _{z \\in \\Xi } \\Vert G_n(z,\\cdot )\\Vert _1^{\\frac{p}{q}} \\\\& \\qquad \\times \\sum _{u \\in \\Xi } \|f(u)\|^p {\\mathsf {w}}\\!\\left(B(u,\\tfrac{\\delta }{2n})\\right)\\sum _{z\\in \\Xi } \|G_n(z,u)\| (1+ n {\\mathsf {d}}(z,u))^{{\\alpha }({\\mathsf {w}})+{\\alpha }(\\varpi )} \\varpi \\!\\left(B(z,\\tfrac{\\delta }{2n})\\right).$ By Lemma REF , there is a nonnegative polynomial $Q_u$ of degree $n$ that satisfies (REF ) with ${\\alpha }={\\alpha }({\\mathsf {w}})+ {\\alpha }(\\varpi )$ , which allows us to apply (REF ) on the polynomial $z \\mapsto G(z,u) Q_u(z)$ to show that the last sum in the right-hand side is bounded by $c \\sum _{z\\in \\Xi } \|G_n(z,u)\| Q_u(z) \\varpi \\!", "\\left(B(z,\\tfrac{\\delta }{2n})\\right)& \\le c \\int _\\Omega \| G_n(x,u)\| Q_u(x) \\varpi (x) \\mathrm {d}{\\mathsf {m}}(x) \\\\& \\le c \\int _\\Omega \| G_n(x,u)\| (1+ n {\\mathsf {d}}(x,u))^{{\\alpha }({\\mathsf {w}})+{\\alpha }(\\varpi )} \\varpi (x) \\mathrm {d}{\\mathsf {m}}(x).$ Putting these together and using (REF ) and (REF ), we have proved that $\\Vert T f\\Vert _{p,{\\mathsf {w}}}^p \\,& \\le c \\Vert f\\Vert _{p,{\\mathsf {w}}}^p \\max _{z \\in \\Xi } \\Vert G_n(z,\\cdot )\\Vert _1^{\\frac{p}{q}} \\\\& \\qquad \\times \\max _{u\\in \\Xi } \\int _\\Omega \| G_n(x,u)\| (1+ n {\\mathsf {d}}(x,u))^{{\\alpha }({\\mathsf {w}})+{\\alpha }(\\varpi )} \\varpi (x) \\mathrm {d}{\\mathsf {m}}(x)\\\\& \\le c \\Vert f\\Vert _{p,{\\mathsf {w}}}^p \\left(\\max _{u \\in \\Xi }\\int _\\Omega \| G_n(x,u)\| (1+ n {\\mathsf {d}}(x,u))^{{\\alpha }({\\mathsf {w}})+{\\alpha }(\\varpi )} \\varpi (x) \\mathrm {d}{\\mathsf {m}}(x) \\right)^{1+\\frac{p}{q}},$ where we have used the assumption that $G_n$ is symmetric in its variables.", "Since $1+ \\frac{p}{q} = p$ , this proves the stated inequality.", "For $r > 0$ , we denote by $L_{n}^{(r)}(\\varpi ; \\cdot ,\\cdot )$ the kernel defined by $ L_{n}^{(r)}(\\varpi ; x,y) = \\sum _{n=0}^\\infty \\widehat{a}\\left( \\frac{k}{n} \\right) [\\mu (k)]^{r/2} P_k(\\varpi ; x,y),$ which is the kernel ${\\mathfrak {D}}_\\varpi ^{r/2} L_n(x,y)$ with ${\\mathfrak {D}}_\\varpi ^{r/2}$ applying on $x$ variable.", "Our Bernstein inequality is proved under the following assumption on the decaying of this kernel.", "Assertion 5.", "For $r > 0$ and ${\\kappa }> 0$ , the kernel $ L_{n}^{(r)}(\\varpi )$ satisfies, for $x,y \\in \\Omega $ , $\\left\| L_{n}^{(r)}(\\varpi ; x,y) \\right\| \\le c_{\\kappa }\\frac{n^r}{\\sqrt{\\varpi (B(x,n^{-1}))}\\sqrt{\\varpi (B(y,n^{-1}))}}(1+n{\\mathsf {d}}(x,y))^{-{\\kappa }}.$ Theorem 3.17 Let $\\varpi $ be a weight functions that admits Assertions 1–5.", "Let ${\\mathsf {w}}$ be a doubling weight on $\\Omega $ .", "If $r > 0$ , $1 \\le p \\le \\infty $ and $f\\in \\Pi _n(\\Omega )$ , then $\\Vert (-{\\mathfrak {D}}_\\varpi )^{\\frac{r}{2}}f \\Vert _{p,{\\mathsf {w}}} \\le c n^r \\Vert f\\Vert _{p,{\\mathsf {w}}}.$ Since ${\\mathsf {L}}_n(\\varpi )* f$ reproduces polynomials of degree $n$ , we can write $(-{\\mathfrak {D}}_\\varpi )^{\\frac{r}{2}}f = (-{\\mathfrak {D}}_\\varpi )^{\\frac{r}{2}} \\left( L_n(\\varpi ) f \\right) =\\int _{\\Omega } f(y) L_{n}^{(r)}(\\varpi ; x,y)\\varpi (y) \\mathrm {d}{\\mathsf {m}}(y),$ where $ L_{n}^{(r)} (\\varpi )$ is defined by (REF ).", "Applying Proposition REF , we conclude that $\\left\\Vert (-{\\mathfrak {D}})^{\\frac{r}{2}}f \\right\\Vert _{p,{\\mathsf {w}}} \\, & \\le c \\Vert f\\Vert _{p,{\\mathsf {w}}}\\max _{x \\in \\Omega } \\int _{\\Omega } \\left\| L_{n}^{(r)} (\\varpi ; x,y)\\right\| (1+n{\\mathsf {d}}(x,y))^{{\\alpha }({\\mathsf {w}})+{\\alpha }(\\varpi )}\\varpi (y) \\mathrm {d}{\\mathsf {m}}(y).$ By Assertion 5, the integral in the right-hand side is bounded by $c_{\\kappa }n^{r} \\int _{\\Omega } \\frac{\\varpi (s)}{\\sqrt{\\varpi (B(x,n^{-1}))}\\sqrt{\\varpi (B(y,n^{-1}))}\\left(1+n{\\mathsf {d}}(x,y)\\right)^{{\\kappa }- {\\alpha }({\\mathsf {w}})-{\\alpha }(\\varpi )}} \\mathrm {d}{\\mathsf {m}}(y) \\le c\\, n^r$ using (REF ) with $p=1$ and ${\\kappa }> {\\alpha }({\\mathsf {w}})+\\frac{5}{4}{\\alpha }(\\varpi )$ .", "For the Laplace-Beltrami operator on the unit sphere, this Bernestin inequality is classical if ${\\mathsf {w}}$ is the Lebesgue measure, whereas the version with the doubling weight was proved in [7].", "For the unit ball, the inequality was established in [52] for the classical weight function with a different proof that applies to reflection invariant weight functions on the unit sphere and the unit ball.", "We can also prove a Nikolskii type inequality for the doubling weight on $\\Omega $ that admits a localizable homogeneous space.", "Theorem 3.18 Let ${\\mathsf {w}}$ be a doubling weight on $\\Omega $ .", "If $0 < p < q \\le \\infty $ and $f \\in \\Pi _n(\\Omega )$ , then $ \\Vert f\\Vert _{q,{\\mathsf {w}}} \\le c n^{ (\\frac{1}{p} - \\frac{1}{q}) {\\alpha }({\\mathsf {w}})} \\Vert f\\Vert _{p,{\\mathsf {w}}}.$ The proof is another one that follows as in the case of the unit sphere.", "The main work lies in proving the case $q = \\infty $ , for which we choose a maximal $\\frac{\\delta }{n}$ -separated subset of $\\Omega $ , so that $\\Vert f\\Vert _\\infty \\le c \\max _{z\\in \\Xi } \|f(z)\|\\, & \\le c \\left(\\min _{z\\in \\Xi } B\\Big (z, \\frac{\\delta }{n}\\Big )\\right)^{-\\frac{1}{p}}\\Bigg (\\sum _{z\\in \\Xi } {\\mathsf {w}}\\left(B\\Big (z,\\frac{\\delta }{n}\\Big ) \\right)\|f(z)\|^p \\Bigg )^{\\frac{1}{p}} \\\\& \\le c \\Vert f\\Vert _{p,{\\mathsf {w}}}\\max _{z\\in \\Xi } \\left( B\\Big (z, \\frac{1}{n}\\Big )\\right)^{-\\frac{1}{p}}.$ Since $\\Omega $ is compact, there is a positive number $\\rho _0 > 0$ such that, for each $z \\in \\Xi $ , $1= {\\mathsf {w}}(\\Omega ) = {\\mathsf {w}}(B(z, \\rho ))$ for some positive number $\\rho \\le \\rho _0$ .", "Let $m$ be a positive integer such that $2^{m-1} \\le \\rho n \\le 2^m$ .", "Then, by (i) of Lemma REF , $1 = {\\mathsf {w}}(B(z,\\rho )) \\le c_{L({\\mathsf {w}})} 2^{m{\\alpha }({\\mathsf {w}})} {\\mathsf {w}}(B(z,\\tfrac{1}{n})) \\le c n^{{\\alpha }({\\mathsf {w}})} {\\mathsf {w}}(B(z,\\tfrac{1}{n})),$ which implies that ${\\mathsf {w}}(B(z,\\tfrac{1}{n})) \\le c n^{-{\\alpha }({\\mathsf {w}})}$ .", "Combing the two inequalities, we conclude that $\\Vert f\\Vert _\\infty \\le c \\Vert f\\Vert _{p,{\\mathsf {w}}} n^{\\frac{1}{p}{\\alpha }({\\mathsf {w}})}$ .", "This proves (REF ) for $q = \\infty $ .", "The case $q < \\infty $ reduces to that of $q= \\infty $ since, using (REF ) for $q =\\infty $ , $\\Vert f\\Vert _{q,{\\mathsf {w}}}^q \\le \\Vert f\\Vert _\\infty ^{q-p} \\Vert f\\Vert _{q,{\\mathsf {w}}}^p \\le c \\left(\\Vert f\\Vert _{p,{\\mathsf {w}}} n^{\\frac{1}{p}{\\alpha }({\\mathsf {w}})}\\right)^{q-p} \\Vert f\\Vert _{q,{\\mathsf {w}}}^p= c n^{q(\\frac{1}{p} - \\frac{1}{q}){\\alpha }({\\mathsf {w}})}\\Vert f\\Vert _{p,\\sigma }^q,$ which is (REF ) for $q < \\infty $ .", "For the unit sphere, this inequality was established in [10].", "The above proof uses essentially the same argument as seen in [12].", "For the unit ball with the classical weight, the inequality was proved in [25]." ], [ "Proof of the main results", "We are now ready to prove Theorems REF and REF .", "The proof follows the same procedure used in [42], [52], which is summarized in [12].", "Since most technical parts are essentially the same, we shall be brief.", "Proof of Theorem REF.", "To prove the direct estimate, we use Theorems REF and REF to obtain $\\left\\Vert f- L_n\\left(\\varpi ; f\\right) \\right\\Vert _{p,\\varpi }\\,& \\le 2 \\Vert f- g\\Vert _{p,\\varpi } + \\left\\Vert g- L_n\\left(\\varpi ; g\\right) \\right\\Vert _{p,\\varpi } \\\\& \\le 2 \\Vert f- g\\Vert _{p,\\varpi } + c n^{-r}\\left\\Vert (-{\\mathfrak {D}}_\\varpi )^{\\frac{r}{2}}g \\right\\Vert _{p,\\varpi }.$ Taking infimum over $g$ proves (i).", "To prove the inverse estimate, we choose $m$ such that $2^{m-1} \\le n < 2^m$ .", "Let us set $L_{2^{-1}}(\\varpi )f =0$ .", "Then, choosing $g = L_{2^m}(\\varpi )f$ , we obtain by Theorem REF , $K_r\\left(f,n^{-1}\\right)_{p,{\\mathsf {w}}}&\\, \\le \\Vert f - L_{2^m} (\\varpi )f\\Vert _{p,{\\mathsf {w}}} +2^{- (m-1) r} \\left\\Vert (-{\\mathfrak {D}}_\\varpi )^{\\frac{r}{2}} L_{2^m} (\\varpi )f \\right\\Vert _{p,{\\mathsf {w}}} \\\\& \\le c E_n(f)_{p,{\\mathsf {w}}} + 2^{- (m-1) r} \\sum _{j=0}^m \\left\\Vert (-{\\mathfrak {D}}_\\varpi )^{\\frac{r}{2}}\\left[ L_{2^j} ({\\mathsf {w}})f- L_{2^{j-1}} ({\\mathsf {w}})f \\right]\\right\\Vert _{p,{\\mathsf {w}}}.$ Applying the Bernstein inequality Theorem REF and using the triangle inequality with Theorem REF , we conclude that $K_r\\left(f,n^{-1}\\right)_{p,\\varpi }&\\, \\le c E_n(f)_{p,\\varpi } + 2^{- m r}\\sum _{j=0}^m 2^{-j r} \\left\\Vert L_{2^j} (\\varpi )f- L_{2^{j-1}} (\\varpi )f\\right\\Vert _{p,{\\mathsf {w}}} \\\\& \\le c\\, 2^{-m r} \\sum _{j=0}^m 2^{j r} E_{2^{j-1}}(f)_{p,\\varpi }\\le c\\, n^{-r} \\sum _{k=0}^m (k+1)^{r-1}E_k(f)_{p, \\varpi }.$ This completes the proof.", "$\\Box $ The proof of the equivalence of the $K$ -functional and the modulus of smoothness relies on two technical lemmas.", "Recall that $\\widehat{a}$ is a cut-off function and $\\mu (j)$ is the eigenvalue in (REF ).", "Lemma 3.19 Let ${\\alpha }\\ge {\\beta }\\ge -\\frac{1}{2}$ .", "For $r>0$ , $0 \\le {\\theta }\\le 3n^{-1}$ , and any $\\ell \\in {\\mathbb {N}}$ , $\\sum _{ j=0}^{2n} \\Biggl \| \\triangle ^{\\ell +1}\\biggl [\\biggl ( \\frac{ 1-R^{({\\alpha },{\\beta })}_j(\\cos {\\theta })}{ \\mu (j) {\\theta }^2 }\\biggr )^r \\widehat{a} \\left( \\frac{j}{n}\\right)\\biggr ]\\Biggr \| ( j+1)^\\ell \\le c,$ and $\\sum _{ j=0}^{2n}\\Biggl \| \\triangle ^{\\ell +1} \\biggl [\\biggl ( \\frac{ \\mu (j) {\\theta }^2 }{ 1-R_j^{({\\alpha },{\\beta })}(\\cos {\\theta })}\\biggr )^r \\widehat{a} \\left( \\frac{j}{n}\\right)\\biggr ]\\Biggr \| ( j+1)^\\ell \\le c,$ where the difference $\\triangle ^{\\ell +1}$ is acting on $j$ and $c$ depends only on ${\\alpha }, {\\beta }, \\ell , r$ .", "Lemma 3.20 Let ${\\alpha }\\ge {\\beta }\\ge -\\frac{1}{2}$ and let $\\ell = \\lceil {\\alpha }\\rceil $ .", "If $r>0$ , $k^{-1}\\le {\\theta }\\le \\frac{\\pi }{2}$ , $j\\in {\\mathbb {N}}_0$ and $0\\le j\\le \\ell +1$ , then for any $m\\in {\\mathbb {N}}$ , $\\biggl \| \\triangle ^j \\biggl ( \\frac{ ( 1-(1-R_k^{({\\alpha },{\\beta })}(\\cos {\\theta }))^r)^{m+\\ell +1}}{(1-R_k^{({\\alpha },{\\beta })}(\\cos {\\theta }))^r}\\biggr ) \\biggr \| \\le c_{m,r} (k{\\theta })^{-m{\\alpha }}{\\theta }^j.$ For ${\\alpha }= {\\beta }= {\\lambda }$ , these two lemma are Lemma 10.4.3 and Lemma 10.4.4 in [12].", "The proof relies on properties of the Jacobi polynomials and holds readily for ${\\alpha }\\ge {\\beta }\\ge -\\frac{1}{2}$ .", "Proof of Theorem REF.", "Choose $n$ such that ${\\theta }^{-1} \\le n \\le 3 {\\theta }^{-1}$ .", "Let $g$ be a polynomial in $\\Pi _n(\\Omega )$ .", "Then so is $\\left(I - S_{{\\theta },\\varpi }\\right)^{r/2} g$ in $\\Pi _n(\\Omega )$ by (REF ).", "Let $\\widehat{a}$ be a cut-off function of type (a), so that $L_n(\\varpi )$ reproduces polynomials of degree $n$ .", "Using (REF ), it follows readily that $\\left(I - S_{{\\theta },\\varpi }\\right)^{r/2} g = {\\theta }^r \\sum _{j=0}^{2n} \\biggl (\\frac{1-R^{({\\alpha },{\\beta })}_j (\\cos {\\theta })}{ \\mu (j){\\theta }^2}\\biggr )^{r/2}\\widehat{a} \\Big (\\frac{j}{n}\\Big ) \\operatorname{proj}_j\\left(\\varpi ; (-{\\mathfrak {D}}_\\varpi )^{r/2}g\\right).$ Following the proof of Theorem REF to take a summation by parts and use the boundedness of the Cesàro means, we can use (REF ) to obtain $\\left\\Vert \\left(I - S_{{\\theta },\\varpi }\\right)^{r/2} g \\right\\Vert _{p,\\varpi } \\le c n^{-r} \\left\\Vert (-{\\mathfrak {D}}_{\\varpi })^{r/2} g\\right\\Vert _{p,\\varpi }.$ Using this inequality with $g = L_{\\lfloor n/2 \\rfloor } (\\varpi ) f$ and the Jackson estimate in Theorem REF , we obtain $\\left\\Vert ( I-S_{{\\theta },\\varpi })^{r/2} f \\right\\Vert _{p,\\varpi } & \\le c \\left\\Vert f-L_{\\lfloor \\frac{n}{2} \\rfloor } (\\varpi )* f\\right\\Vert _{p, \\varpi } +\\left\\Vert ( I-S_{{\\theta },\\varpi })^{r/2} L_{\\lfloor \\frac{n}{2} \\rfloor } (\\varpi )* f \\right\\Vert _{p, \\varpi } \\\\& \\le c K_r(f; n^{-1})_{p,\\varpi } + c n^{-r} \\left\\Vert (-{\\mathfrak {D}})^{r/2} L_{\\lfloor \\frac{n}{2} \\rfloor } (\\varpi )* f\\right\\Vert _{p,\\varpi }.$ By the triangle inequality and applying the Bernstein inequality on $L_n(\\varpi )(f-g)$ , we obtain $n^{-r} \\left\\Vert (-{\\mathfrak {D}})^{r/2} L_{\\lfloor \\frac{n}{2} \\rfloor } (\\varpi ) f\\right\\Vert _{p,\\varpi }& \\le c \\left\\Vert f-g\\right\\Vert _{p,\\varpi }+ n^{-r} \\left\\Vert (-{\\mathfrak {D}})^{r/2} L_{\\lfloor \\frac{n}{2} \\rfloor } (\\varpi )* g\\right\\Vert _{p,\\varpi } \\\\& \\le c \\left( \\left\\Vert f-g\\right\\Vert _{p,\\varpi }+ n^{-r} \\left\\Vert (-{\\mathfrak {D}})^{r/2} g\\right\\Vert _{p,\\varpi }\\right),$ where we used, by (REF ), $(-{\\mathfrak {D}})^{r/2} L_{\\lfloor \\frac{n}{2} \\rfloor } (\\varpi )* g= L_{\\lfloor \\frac{n}{2} \\rfloor } (\\varpi )* (-{\\mathfrak {D}})^{r/2}g$ in the second step.", "Hence, taking infimum over $g$ , it follows from the above two inequalities and ${\\theta }\\sim n^{-1}$ that $\\left\\Vert (I-S_{{\\theta },\\varpi })^{r/2} f \\right\\Vert _{p, \\varpi } \\le c K_{r}(f, {\\theta })_{p,\\varpi },$ from which the right-hand inequality of (REF ) follows.", "In the other direction, we can follow the proof of inequality (REF ) and use (REF ) instead of (REF ), to establish $n^{-r} \\left\\Vert (-{\\mathfrak {D}}_{\\varpi })^{r/2} L_n(\\varpi )f \\right\\Vert _{p,\\varpi } \\le c \\left\\Vert \\left(I - S_{{\\theta },\\varpi }\\right)^{r/2} f \\right\\Vert _{p,\\varpi }.$ Hence, in order to prove the left-hand inequality of (REF ), it suffices to prove that $\\left\\Vert f- L_n(\\varpi )f \\right\\Vert _{p,\\varpi } \\le c \\left\\Vert \\left(I - S_{{\\theta },\\varpi }\\right)^{r/2} f \\right\\Vert _{p,\\varpi }.$ Using $(1-r)^{-1} = \\sum _{i=0}^{m+\\ell } r^i + r^{m+\\ell }(1-r)^{-1}$ with $r = 1- \\big (1-R^{({\\alpha },{\\beta })}_j (\\cos {\\theta })\\big )^{r/2}$ , one can write, as seen in [12], that $f - L_n(\\varpi )* f &\\, = \\sum _{i=0}^{m+\\ell } (I-L_n(\\varpi )) * \\left(I-(I-S_{{\\theta },\\varpi })^{r/2}\\right)^i F\\\\& + \\sum _{j=n}^\\infty \\left(1-\\widehat{a} \\left(\\frac{j}{n} \\right) \\right)\\frac{ \\big (1- \\big (1-R^{({\\alpha },{\\beta })}_j (\\cos {\\theta })\\big )^{r/2}\\big )^{m+\\ell +1}F}{(1-R_j^{({\\alpha },{\\beta })}(\\cos {\\theta }))^{r/2}} \\operatorname{proj}_j(\\varpi ; F),$ where $F = \\left(I - S_{{\\theta },\\varpi }\\right)^{r/2} f$ .", "While the first term is bounded by $\\Vert F\\Vert _{p,\\varpi }$ by the boundedness of $L_n(\\varpi )f$ and $(I-(I-S_{{\\theta },\\varpi })^{r/2})F$ , the second one can be shown to be bounded by $\\Vert F\\Vert _{p,\\varpi }$ by using (REF ) with the summation by parts and the boundedness of Cesàro means that we have used several times.", "This proves (REF ) and completes the proof.", "$\\Box $" ], [ "Homogeneous space on conic surfaces", "In this section we work in the setting of homogeneous space on the conic surface ${\\mathbb {V}}_{0}^{d+1}= \\lbrace (x,t): \\Vert x\\Vert = t, \\, x \\in {\\mathbb {R}}^d, \\, 0 \\le t \\le 1\\rbrace .$ We shall verify that the framework in the previous two sections is applicable on this domain for the weight function $t^{-1} (1-t)^{\\gamma }$ , which has a singularity at the apex.", "The verification is highly non-trivial because of new phenomena and obstacles encountered.", "While the structure of orthogonal polynomials on the conic surface shares characteristic features of spherical harmonics, the conic surface is markedly different from that of the unit sphere because of its apex and its boundary.", "Our first task is to understand, in the first subsection, the intrinsic distance function on the conic surface, which turns out to be incomparable to the Euclidean distance around the apex.", "In the second subsection, we show that the Jacobi weight function ${\\mathsf {w}}_{{\\beta },{\\gamma }}(t) = t^{\\beta }(1-t)^{\\gamma }$ is a doubling weight with respect to the intrinsic distance.", "The orthogonal structure with respect to the Jacobi weight is reviewed in the third subsection, which is used to verify Assertions 1-3 of the highly localized kernels for the weight ${\\mathsf {w}}_{-1,{\\gamma }}$ in the fourth subsection.", "Construction of ${\\varepsilon }$ -separated set of ${\\mathbb {V}}_0^{d+1}$ is provided in the fifth subsection and used to state the Marcinkiewicz-Zygmund inequality.", "Assertion 4 is verified in the sixth subsection, which ensures that the positive cubature rules and the tight localized frames can both be stated for the conic surface.", "In the seventh subsection, Assertion 5 is verified and the characterization of the best approximation by polynomials is stated." ], [ "Distance on the surface of the cone", "Our first task is to define an appropriate distance function on the surface of the cone.", "Unlike the sphere, the surface ${\\mathbb {V}}_0^{d+1}$ has a boundary at $t =1$ and a singularity at the apex $t=0$ .", "Our distance function should measure the distance between points near the boundary or the apex and the distance between interior points differently.", "This is a well-known phenomenon as we have already seen for the interval $[-1,1]$ .", "More generally, the distance on the interval $[a,b]$ is given by the change of variables $x \\in [a,b] \\mapsto y \\in [-1,1]$ , ${\\mathsf {d}}_{[a,b]}(x_1,x_2) = \\frac{b-a}{2} \\arccos \\left( y_1 y_2 + \\sqrt{1-y_1^2}\\sqrt{1-y_2^2}\\right),$ where $y_i = -1+2 \\frac{x_i-a}{b-a}$ .", "In particular, the distance function for $[0,1]$ is given by ${\\mathsf {d}}_{[0,1]}(x_1,x_2) \\, & = \\frac{1}{2} \\arccos \\left( (2x_1-1) (2y_1-1) + 4 \\sqrt{x_1(1-x_1)} \\sqrt{y_1(1-y_1)}\\right)\\\\& = \\arccos \\left(\\sqrt{x_1 x_2} + \\sqrt{(1-x_1)(1-x_2)} \\right),$ where the second identity follows from $\\arccos ({\\alpha }) = \\frac{1}{2} \\arccos (2{\\alpha }^2-1)$ .", "In particular, setting $x_i = \\cos ^2 \\frac{{\\theta }_i}{2} \\in [0,1]$ , $0\\le {\\theta }_i \\le \\pi $ , we obtain ${\\mathsf {d}}_{[0,1]}(x_1,x_2) = \\frac{1}{2}\|{\\theta }_1-{\\theta }_2\|$ .", "Definition 4.1 For $(x,t)$ and $(y,s)$ on ${\\mathbb {V}}_0^{d+1}$ , define ${\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)): = \\arccos \\left(\\sqrt{\\frac{{\\langle }x,y{\\rangle }+ t s}{2}} + \\sqrt{1-t}\\sqrt{1-s}\\right).$ Proposition 4.2 The function ${\\mathsf {d}}_{{\\mathbb {V}}_0^{d+1}} (\\cdot , \\cdot )={\\mathsf {d}}_{{\\mathbb {V}}_0}(\\cdot ,\\cdot )$ defines a distance on the surface of the cone ${\\mathbb {V}}_0^{d+1}$ .", "Evidently ${\\mathsf {d}}_{{\\mathbb {V}}_0}(\\cdot ,\\cdot )$ is symmetric.", "Since $\\Vert x\\Vert = t$ and $\\Vert y\\Vert = s$ , it follows readily that $0 \\le \\sqrt{\\frac{{\\langle }x,y{\\rangle }+ t s}{2}} + \\sqrt{1-t}\\sqrt{1-s} \\le 1$ , so that ${\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)) \\ge 0$ and, furthermore, ${\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (x,t))= \\arccos 1 =0$ .", "Hence, we only need to prove that it satisfies the triangle inequality.", "For $d =2$ , we write $(x,t) = (x_1,x_2,t) \\in {\\mathbb {V}}_0^3$ with $x_1^2+x_2^2 = t^2$ .", "For $(x,t), (y,s) \\in {\\mathbb {V}}_0^3$ , it is easy to verify the identity $\\left( \\sqrt{ (t+x_1)(s+y_1) } + \\mathrm {sign}(x_2 y_2) \\sqrt{ (t-x_1)(s-y_1)}\\right)^2= \\frac{t s + x_1 y_1 + x_2 y_2}{2}.$ Hence, setting $z_{x,t} = (\\sqrt{t+x_1}, \\mathrm {sign}(x_2) \\sqrt{t-x_1}, \\sqrt{1-t}$ ) and define $z_{y,s}$ similarly, then they are elements in ${\\mathbb {S}}^2$ and it follows that ${\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)) = \\arccos ({\\langle }z_{x,t}, z_{y,s}{\\rangle }) = {\\mathsf {d}}_{{\\mathbb {S}}^2} (z_{x,t}, z_{y,s}),$ where ${\\mathsf {d}}_{{\\mathbb {S}}^2}(\\cdot ,\\cdot )$ is the geodesic distance of the unit sphere ${\\mathbb {S}}^2$ .", "In particular, $ {\\mathsf {d}}_{{\\mathbb {V}}_0} (\\cdot ,\\cdot )$ satisfies the triangle inequality.", "For $d> 2$ , given three distinct point in ${\\mathbb {V}}_0^{d+1}$ , written as $(t_i \\xi _i, t_i)$ , $1 \\le i \\le 3$ , where $\\xi \\in {\\mathbb {S}^{d-1}}$ , we can find a rotation in ${\\mathbb {R}}^d$ so that $\\xi _i = (\\eta _i, 0)$ with $\\eta _i \\in {\\mathbb {S}}^2$ .", "Hence, the triangle inequality for $d > 2$ follows from the triangle inequality for $d =2$ .", "This completes the proof.", "The distance function ${\\mathsf {d}}_{{\\mathbb {V}}_0}(\\cdot ,\\cdot )$ is closely related to the distance function ${\\mathsf {d}}_{[0,1]}(\\cdot ,\\cdot )$ of the interval $[0,1]$ and the geodesic distance ${\\mathsf {d}}_{{\\mathbb {S}}}(\\cdot ,\\cdot )$ of the unit sphere ${\\mathbb {S}^{d-1}}$ .", "Proposition 4.3 For $d \\ge 2$ and $(x,t), (y,s) \\in {\\mathbb {V}}_0^{d+1}$ , write $x = t\\xi $ and $y = s \\eta $ with $\\xi ,\\eta \\in {\\mathbb {S}^{d-1}}$ .", "Then $1- \\cos {\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)) =1-\\cos {\\mathsf {d}}_{[0,1]}(t,s) +\\sqrt{t}\\sqrt{s} \\left[1-\\cos \\left(\\tfrac{1}{2} {\\mathsf {d}}_{{\\mathbb {S}}}(\\xi ,\\eta ) \\right)\\right].$ In particular, $ c_1 {\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)) \\le {\\mathsf {d}}_{[0,1]}(t,s) + (t s )^{\\frac{1}{4}} {\\mathsf {d}}_{{\\mathbb {S}}}(\\xi ,\\eta ) \\le c_2 {\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)).$ Using $\\arccos ({\\alpha }) = \\frac{1}{2} \\arccos (2{\\alpha }^2-1)$ , we deduce $\\sqrt{\\frac{ts +{\\langle }x, y{\\rangle }}{2}} = \\sqrt{t s} \\sqrt{\\frac{1+{\\langle }\\xi , \\eta {\\rangle }}{2}} = \\sqrt{ts} \\cos \\left(\\tfrac{1}{2} \\arccos {\\langle }\\xi , \\eta {\\rangle }\\right).$ Consequently, in terms of the geodesic distance on the unit sphere, we can write $ {\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)) = \\arccos \\left[\\sqrt{t}\\sqrt{s} \\cos \\left(\\tfrac{1}{2} {\\mathsf {d}}_{{\\mathbb {S}}}(\\xi ,\\eta ) \\right)+ \\sqrt{1-t}\\sqrt{1-s} \\right].$ In particular, it follows that $1- \\cos {\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)) = 1- \\sqrt{t}\\sqrt{s}- \\sqrt{1-t}\\sqrt{1-s} +\\sqrt{t}\\sqrt{s}\\left[1- \\cos \\left(\\tfrac{1}{2} {\\mathsf {d}}_{{\\mathbb {S}}}(\\xi ,\\eta ) \\right)\\right],$ which is the identity (REF ).", "From this identity, (REF ) follows from $1-\\cos {\\theta }= 2 \\sin ^2\\frac{{\\theta }}{2}$ , $\\frac{1}{\\pi } {\\theta }\\le \\sin \\frac{{\\theta }}{2} \\le \\frac{{\\theta }}{2}$ for $0 \\le {\\theta }\\le \\pi $ , and $(a+b)^2/2 \\le a^2+b^2 \\le (a+b)^2$ for $a,b \\ge 0$ .", "The line segment from the apex to a point $(\\xi ,1)$ , $\\xi \\in {\\mathbb {S}^{d-1}}$ , on the top boundary of the cone ${\\mathbb {V}}_0^{d+1}$ can be parametrized by $l_\\xi = \\lbrace (t\\xi ,t): 0 \\le t \\le 1\\rbrace $ .", "For two points $(t\\xi ,t)$ and $(s \\xi , s)$ on $l_\\xi $ , the identity (REF ) shows that ${\\mathsf {d}}_{{\\mathbb {V}}_0} ((t\\xi ,t), (s\\xi ,s)) = \\tfrac{\|{\\theta }-\\phi \|}{2} = {\\mathsf {d}}_{[0,1]}(t,s)$ if $t = \\cos ^2 \\frac{{\\theta }}{2}$ and $s = \\cos ^2 \\frac{\\phi }{2}$ .", "Moreover, the top boundary of ${\\mathbb {V}}_0^{d+1}$ is the unit sphere ${\\mathbb {S}^{d-1}}$ , or $\\lbrace (\\xi ,1): \\xi \\in {\\mathbb {S}^{d-1}}\\rbrace $ .", "For $(\\xi _1,1)$ and $(\\xi _2,1)$ on this boundary, (REF ) gives ${\\mathsf {d}}_{{\\mathbb {V}}_0}( (\\xi _1,1),(\\xi _2,1)) = \\frac{1}{2} {\\mathsf {d}}_{{\\mathbb {S}}}(\\xi _1,\\xi _2),$ or half of the geodesic distance on the unit sphere.", "Remark 4.1 It is well-known that the geodesic distance ${\\mathsf {d}}_{\\mathbb {S}}(\\cdot ,\\cdot )$ of ${\\mathbb {S}^{d-1}}$ is proportional to the Euclidean distance; that is, $\\Vert \\xi - \\eta \\Vert \\sim {\\mathsf {d}}_{{\\mathbb {S}}}(\\xi ,\\eta )$ for all $\\xi ,\\eta \\in {\\mathbb {S}^{d-1}}$ .", "For the surface of the cone ${\\mathbb {V}}_0^{d+1}$ , however, this is no longer true when the points are near the apex.", "Indeed, for $(x,t), (y,s)\\in {\\mathbb {V}}_0^{d+1}$ with $x = t\\xi $ , $y =s \\eta $ , $\\xi ,\\eta \\in {\\mathbb {S}^{d-1}}$ , $\\Vert (x,t) - (y,s)\\Vert ^2 = 2 (t-s)^2 + 2 t s \\big (1- {\\langle }\\xi ,\\eta {\\rangle }\\big ) =2 (t-s)^2 + 2 t s (1-\\cos {\\mathsf {d}}_{{\\mathbb {S}}}(\\xi ,\\eta )).$ In particular, if $t = s$ , we see that $\\Vert (t\\xi ,t) - (t\\eta ,t)\\Vert \\sim t {\\mathsf {d}}_{{\\mathbb {S}}}(\\xi ,\\eta )$ , whereas we have $d_{{\\mathbb {V}}_0} \\big ( (t \\xi ,t),(t\\eta ,t) \\big ) \\sim \\sqrt{t} {\\mathsf {d}}_{{\\mathbb {S}}}(\\xi ,\\eta )$ by (REF ).", "Hence the two distances are not compatible when $t$ is small.", "We will also need the following lemma in the estimate of the kernels.", "Lemma 4.4 For $(x,t), (y,s) \\in {\\mathbb {V}}_0^{d+1}$ , $\\big \| \\sqrt{t} - \\sqrt{s} \\big \|\\le {\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)) \\quad \\hbox{and} \\quad \\big \| \\sqrt{1-t} - \\sqrt{1-s} \\big \| \\le {\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)).$ Let $t = \\cos ^2 \\frac{{\\theta }}{2}$ and $s = \\cos ^2 \\frac{\\phi }{2}$ , $0 \\le {\\theta }, \\phi \\le \\pi $ .", "Since $\|{\\langle }x,y{\\rangle }\| \\le ts$ , we obtain $\\cos {\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)) \\le \\sqrt{t s} + \\sqrt{1-t}\\sqrt{1-s} = \\cos \\tfrac{{\\theta }-\\phi }{2},$ so that ${\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)) \\ge \\frac{1}{2} \|{\\theta }-\\phi \|$ .", "Elementary trigonometric identities shows that $\\big \| \\sqrt{t} - \\sqrt{s} \\big \| = \\big \|\\cos \\tfrac{{\\theta }}{2} - \\cos \\tfrac{\\phi }{2} \\big \| \\le 2 \\sin \\tfrac{\|{\\theta }-\\phi \|}{4}\\le \\tfrac{\|{\\theta }-\\phi \|}{2} \\le {\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)).$ The inequality for $\\big \| \\sqrt{1-t} - \\sqrt{1-s} \\big \|$ follows from $\\big \|\\sin \\tfrac{{\\theta }}{2} - \\sin \\tfrac{\\phi }{2} \\big \|\\le 2 \\sin \\tfrac{\|{\\theta }-\\phi \|}{4}$ ." ], [ "A family of doubling weights", "For the conic surface, balls are conic caps.", "For $r > 0$ and $(x,t)$ on ${\\mathbb {V}}_0^{d+1}$ , we denote the conic cap centered at $(x,t)$ with radius $r$ by ${\\mathsf {c}}((x,t), r): = \\left\\lbrace (y,s) \\in {\\mathbb {V}}_0^{d+1}: {\\mathsf {d}}_{{\\mathbb {V}}_0} \\big ((x,t),(y,s)\\big )\\le r \\right\\rbrace .$ A weight function ${\\mathsf {w}}$ is a doubling weight if it satisfies ${\\mathsf {w}}\\big ({\\mathsf {c}}((x,t), 2 r)\\big ) \\le L \\, {\\mathsf {w}}\\big ({\\mathsf {c}}((x,t), r)\\big ), \\quad r >0.$ In comparison with the spherical cap on ${\\mathbb {S}^{d-1}}$ , the geometry of ${\\mathsf {c}}(x,t)$ is more complicated.", "Denote the surface measure on ${\\mathbb {S}^{d-1}}$ by $\\mathrm {d}\\sigma _{\\mathbb {S}}$ .", "Lemma 4.5 For $r > 0$ , $t, s \\in [0,1]$ , define $\\tau _r(t,s) = (\\cos r - \\sqrt{1-t}\\sqrt{1-s} )/\\sqrt{ts}$ and ${\\theta }_r(t,s) = \\arccos \\tau _r(t,s)$ .", "Then, for $(x,t) \\in {\\mathbb {V}}_0^{d+1}$ with $x = t \\xi $ , $\\xi \\in {\\mathbb {S}^{d-1}}$ , ${\\mathsf {w}}\\big ({\\mathsf {c}}((x,t), r)\\big )& = \\int _{{\\mathsf {d}}_{[0,1]}(t, s)\\le r} s^{d-1}\\int _{{\\mathsf {d}}_{{\\mathbb {S}}}(\\xi ,\\eta ) \\le \\tfrac{1}{2} {\\theta }_r(t,s)} {\\mathsf {w}}(s\\eta , s) \\mathrm {d}\\sigma _{{\\mathbb {S}}}(\\eta )\\mathrm {d}s.$ From ${\\mathsf {d}}_{{\\mathbb {V}}_0}((x,t),(y,s)) \\le r$ , we obtain ${\\mathsf {d}}_{[0,1]}(t, s) \\le r$ by (REF ) and, with $x = t \\xi $ and $y = s \\eta $ , it follows from (REF ) that ${\\mathsf {d}}_{{\\mathbb {S}}} (\\xi ,\\eta ) \\le \\arccos \\left(2 [\\tau _r(t,s)]^2 -1\\right) = \\tfrac{1}{2} \\arccos \\tau _r(t,s) = \\tfrac{1}{2} {\\theta }_r(t,s).$ Hence, the stated identity follows from $\\mathrm {d}\\sigma (y,s) = s^{d-1} \\mathrm {d}\\sigma _{\\mathbb {S}}(\\eta ) \\mathrm {d}s$ .", "For ${\\beta }> -d$ and ${\\gamma }> -1$ , consider the Jacobi weight function defined on the cone ${\\mathsf {w}}_{{\\beta },{\\gamma }} (t) = t^{\\beta }(1-t)^{\\gamma }, \\quad 0 < t <1.$ Let ${\\mathsf {b}}_{{\\beta },{\\gamma }}$ be the normalization constant so that ${\\mathsf {b}}_{{\\beta },{\\gamma }} {\\mathsf {w}}_{{\\beta },{\\gamma }}$ has unit integral on ${\\mathbb {V}}_0^{d+1}$ .", "Setting $y = s \\eta $ , $\\eta \\in {\\mathbb {S}^{d-1}}$ , then ${\\mathsf {b}}_{{\\beta },{\\gamma }}^{-1} = \\int _0^1 s^{d+{\\beta }-1}(1-s)^{\\gamma }\\mathrm {d}s \\int _{{\\mathbb {S}^{d-1}}}\\mathrm {d}\\sigma _{\\mathbb {S}^{d-1}}(\\xi )= \\omega _d \\frac{\\Gamma ({\\beta }+d) \\Gamma ({\\gamma }+1)}{\\Gamma ({\\beta }+{\\gamma }+d+1)},$ where $\\omega _d$ is the surface are of ${\\mathbb {S}^{d-1}}$ .", "Proposition 4.6 Let $r > 0$ and $(x,t) \\in {\\mathbb {V}}_0^{d+1}$ .", "Then for ${\\beta }> - d$ and ${\\gamma }> -1$ , ${\\mathsf {w}}_{{\\beta },{\\gamma }}\\big ({\\mathsf {c}}((x,t), r)\\big ):= \\, & {\\mathsf {b}}_{{\\beta },{\\gamma }} \\int _{ {\\mathsf {c}}((x,t), r)} {\\mathsf {w}}_{{\\beta },{\\gamma }}(s) \\mathrm {d}\\sigma (y,s) \\\\\\sim &\\, r^d (t+ r^2)^{{\\beta }+\\frac{d}{2}} (1-t+ r^2)^{{\\gamma }+\\frac{1}{2}}.", "$ In particular, ${\\mathsf {w}}_{{\\beta },{\\gamma }}$ is a doubling weight and the doubling index ${\\alpha }( {\\mathsf {w}}_{{\\beta },{\\gamma }})$ , defined in (REF ), is give by ${\\alpha }({\\mathsf {w}}_{{\\beta },{\\gamma }}) = d + 2 \\max \\lbrace 0, {\\beta }+\\frac{d}{2}\\rbrace + 2 \\max \\lbrace 0,{\\gamma }+\\frac{1}{2}\\rbrace $ .", "Without loss of generality, we assume $r$ is bounded by a small positive number $r \\le \\delta $ ; for example, $\\delta = \\frac{\\pi }{12}$ will do.", "By rotation symmetry, we could choose $x = t e_1$ , where $e_1=(1,0,\\ldots ,0)$ .", "Then, by Lemma REF , ${\\mathsf {w}}_{{\\beta },{\\gamma }}\\big ({\\mathsf {c}}((x,t), r)\\big ) = {\\omega }_{d-1} \\int _{{\\mathsf {d}}_{[0,1]}(t, s)\\le r} s^{d-1} {\\mathsf {w}}_{{\\beta },{\\gamma }}(s) \\int _0^{\\tfrac{1}{2} {\\theta }_r(t,s)}(\\sin {\\theta })^{d-2} \\mathrm {d}{\\theta }\\mathrm {d}s,$ where we have used the identity (cf.", "[12]) $ \\int _{{\\mathbb {S}^{d-1}}} g({\\langle }\\xi ,\\eta {\\rangle }) d\\sigma (\\eta ) = {\\omega }_{d-1} \\int _0^\\pi g (\\cos {\\theta }) (\\sin {\\theta })^{d-2} \\mathrm {d}{\\theta }$ with ${\\omega }_{d-1}$ being the surface are of ${\\mathbb {S}}^{d-2}$ .", "Since ${\\theta }\\sim \\sin {\\theta }\\sim \\sqrt{1-\\cos {\\theta }}$ , it follows that $ {\\mathsf {w}}_{{\\beta },{\\gamma }}\\big ({\\mathsf {c}}((x,t), r)\\big )\\, &\\sim \\int _{{\\mathsf {d}}_{[0,1]}(t, s)\\le r} s^{d-1} {\\mathsf {w}}_{{\\beta },{\\gamma }}(s)\\big (1-\\tau _r(t,s) \\big )^{\\frac{d-1}{2}} \\mathrm {d}s.$ By its definition, $\\tau _r(t,s) \\ge 0$ and, furthermore, $\\tau _r(t,s) \\le 1$ since we can write $ 1-\\tau _r(t,s) = \\frac{\\cos {\\mathsf {d}}_{[0,1]}(t,s) - \\cos r}{\\sqrt{t}\\sqrt{s}}.$ We need to consider three cases.", "Case 1.", "Assume $3 r^2 \\le t \\le 1- 3 r^2$ .", "By Lemma REF , this implies that $s \\sim t + r^2$ and $1-s \\sim 1-t + r^2$ , which allows us to conclude that ${\\mathsf {w}}_{{\\beta },{\\gamma }}\\big ({\\mathsf {c}}((x,t), r)\\big ) & \\, \\sim (t+ r^2)^{{\\beta }+\\frac{d}{2}} (1-t+ r^2)^{{\\gamma }+\\frac{1}{2}} \\\\& \\times \\int _{{\\mathsf {d}}_{[0,1]}(t, s)\\le r} \\big (\\cos ({\\mathsf {d}}_{[0,1]} (t,s)) - \\cos r \\big )^{\\frac{d-1}{2}}\\frac{\\mathrm {d}s}{\\sqrt{s (1-s)}}.$ Setting $t = \\sin ^2 \\frac{{\\theta }}{2}$ and $s = \\sin ^2 \\frac{\\phi }{2}$ so that ${\\mathsf {d}}_{[0,1]} (t,s) = \|{\\theta }-\\phi \|/2$ and the last integral is easily seen to be $\\int _{\|{\\theta }-\\phi \|\\le 2 r} \\big (\\cos \\tfrac{{\\theta }-\\phi }{2} - \\cos r \\big )^{\\frac{d-1}{2}} \\mathrm {d}\\phi = c \\int _{\|\\zeta \|\\le 2 r} \\big (\\sin \\tfrac{\\zeta -r}{2} \\sin \\tfrac{\\zeta +r}{2} \\big )^{\\frac{d-1}{2}} \\mathrm {d}\\zeta \\sim r^d.$ This completes the proof of the first case.", "Case 2.", "$0 \\le t \\le 3 r^2$ .", "For $(y,s) \\in {\\mathsf {c}}((x,t),r)$ , we also have $\\sqrt{s} \\le \\sqrt{t}+ r \\le (1+ \\sqrt{3})r$ by Lemma REF , which shows, in particular, that $s \\le c (t +r^2)$ .", "Evidently $1-s \\sim 1-t \\sim 1$ in this case.", "Furthermore, let $t = \\sin ^2 \\frac{{\\theta }}{2}$ and $s = \\sin ^2 \\frac{\\phi }{2}$ ; then $\|s - t\| = \\left\|\\sin \\frac{{\\theta }-\\phi }{2} \\sin \\frac{{\\theta }+\\phi }{2} \\right\| \\le c r^2$ since $\|{\\theta }-\\phi \| \\le 2 {\\mathsf {d}}_{[0,1]}(t,s)\\le r$ and ${\\theta }\\le c r$ .", "Now, we have a trivial upper bound $1-\\tau _r(t,s) \\le 2$ , which leads to, by (REF ), ${\\mathsf {w}}_{{\\beta },{\\gamma }}\\big ({\\mathsf {c}}((x,t), r)\\big ) \\le c \\int _{{\\mathsf {d}}_{[0,1]}(t, s)\\le r/2 } s^{d+ {\\beta }-1} \\mathrm {d}s\\sim \\int _0^{r^2} \\phi ^{d+ {\\beta }-1} d\\phi \\sim r^{2{\\beta }+ 2 d},$ which proves the upper bound in (REF ).", "For the lower bound, we consider a subset of ${\\mathsf {c}}((x,t),r)$ with $\\mathrm {d}_{[0,1]}(t,s) \\le r/2$ .", "Using the upper bound of $s$ and $t$ , we then deduce $1-\\tau _r(t,s)= \\frac{\\cos d_{[0,1]}(t,s) - \\cos r}{\\sqrt{t}\\sqrt{s}} \\ge \\frac{\\cos \\tfrac{r}{2} - \\cos r}{ (3+\\sqrt{3}) r^2} \\ge \\frac{2}{\\pi ^2},$ where in the last step we have used the monotonicity of the function over $0 \\le r \\le \\pi /12$ , which shows then ${\\mathsf {w}}_{{\\beta },{\\gamma }}\\big ({\\mathsf {c}}((x,t), r)\\big )\\,& \\ge c \\int _{{\\mathsf {d}}_{[0,1]}(t, s)\\le r/2 } s^{d+ {\\beta }-1} \\mathrm {d}s\\sim r^{2{\\beta }+ 2 d}.$ This completes the proof of this second case.", "Case 3.", "$1- t \\le 3 r^2$ .", "In this case, we clearly have $s \\sim t \\sim 1$ for $(y,s) \\in {\\mathsf {c}}((x,t),r)$ .", "Since ${\\mathsf {d}}_{[0,1]}(t,s) = {\\mathsf {d}}_{[0,1]}(1-t,1-s)$ , changing variable $s\\mapsto 1-s$ in (REF ), we obtain ${\\mathsf {w}}_{{\\beta },{\\gamma }}\\big ({\\mathsf {c}}((x,t), r)\\big ) \\sim c \\int _{{\\mathsf {d}}_{[0,1]}(t, s)\\le r} (1-s)^{\\gamma }\\mathrm {d}s= c \\int _{{\\mathsf {d}}_{[0,1]}(1-t, s)\\le r} s^{\\gamma }\\mathrm {d}s,$ where the last integral can be estimated as in Case 2.", "This completes the proof of (REF ).", "It is worthwhile to mention that the proof relies on the geometry of ${\\mathsf {c}}((x,t),r)$ when $t \\le c r^2$ , which we describe in the following remark.", "Remark 4.2 By (REF ), $(y,s) \\in {\\mathsf {c}}((x,t),r)$ in equivalent to $2 \\sin ^2 \\frac{{\\mathsf {d}}_{{\\mathbb {S}}}(\\xi ,\\eta )}{2} = \\frac{\\cos d_{[0,1]}(t,s) - \\cos d_{{\\mathbb {V}}_0}\\big ((x,t), (y,s)\\big )}{\\sqrt{t}\\sqrt{s} }\\le \\frac{\\cos d_{[0,1]}(t,s) - \\cos r}{\\sqrt{t}\\sqrt{s}}.$ If $t \\le c r^2$ , then the proof of the Case 2 shows, by $\\sin {\\theta }\\le {\\theta }$ , that the above inequality holds whenever ${\\mathsf {d}}_{{\\mathbb {S}}}(\\xi ,\\eta ) \\le \\frac{4}{\\pi ^2}$ .", "In other words, for any $s$ that satisfies ${\\mathsf {d}}_{[0,1]}(t,s) \\le r$ , the set ${\\mathsf {c}}((x,t), r)$ contains a large spherical cap $\\lbrace \\eta : {\\mathsf {d}}_{{\\mathbb {S}}} (\\xi ,\\eta ) \\le \\frac{4}{\\pi ^2}\\rbrace $ .", "Corollary 4.7 For $d\\ge 2$ , ${\\beta }> -d$ and ${\\gamma }> -1$ , the space $({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }}, {\\mathsf {d}}_{{\\mathbb {V}}_0})$ is a homogeneous space.", "For convenience, we will introduce and use the function ${\\mathsf {w}}_{{\\beta },{\\gamma },d}(n;t)$ defined by ${\\mathsf {w}}_{{\\beta },{\\gamma },d} (n; t):= n^{d} {\\mathsf {w}}_{{\\beta },{\\gamma }}\\big ({\\mathsf {c}}((x,t), n^{-1})\\big )= \\big (t+n^{-2}\\big )^{{\\beta }+\\frac{d}{2}} \\big (1-t+n^{-2}\\big )^{{\\gamma }+\\frac{1}{2}}.$ For any $\\xi $ in the unit sphere ${\\mathbb {S}}^{m-1}$ , a spherical cap ${\\mathsf {c}}(\\xi ,r)$ has $\\sigma ({\\mathsf {c}}(\\xi ,r)) = c r^{m-1}$ so that, for a doubling weight $w$ on the unit sphere ${\\mathbb {S}}^{m-1}$ , the function $w(n;\\xi ) = n^{m-1} \\sigma ({\\mathsf {c}}(\\xi ,\\tfrac{1}{n}))$ is an approximation to $w(\\xi )$ by the Lebesgue differentiation theorem.", "The function ${\\mathsf {w}}_{{\\beta },{\\gamma },d}(n; t)$ in (REF ), however, is not an approximation to ${\\mathsf {w}}_{{\\beta },{\\gamma }}(t)$ on the conic surface since for the Lebesgue measure $\\mathrm {d}\\sigma $ on ${\\mathbb {V}}_0^{d+1}$ , $\\sigma \\big ({\\mathsf {c}}((x,t), n^{-1})\\big ) \\sim n^{-d} (t+ n^{-2})^{\\frac{d}{2}} (1 - t + n^{-2})^{\\frac{1}{2}}$ by (REF ), whereas it is the ratio ${\\mathsf {w}}_n(x,t) = \\frac{{\\mathsf {w}}({\\mathsf {c}}((x,t), n^{-1}))}{\\sigma ({\\mathsf {c}}((x,t), n^{-1}))}$ that provides an approximation to ${\\mathsf {w}}(t)$ on ${\\mathbb {V}}_0^{d+1}$ .", "For the unit sphere, it can be verified that $w_n(\\xi ) = w(n;\\xi )$ is a doubling weight on ${\\mathbb {S}}^{m-1}$ whenever $w$ is, and this property has been used to show, for example, that $\\Vert f\\Vert _{p,w} \\sim \\Vert f\\Vert _{p, w_n}$ for all polynomials of degree at most $n$ .", "For a doubling measure ${\\mathsf {w}}$ on the conic surface, it is not clear, however, if ${\\mathsf {w}}_n$ defined above is itself a doubling weight on ${\\mathbb {V}}_0^{d+1}$ ." ], [ "Orthogonal polynomials on the conic surface", "Orthogonal structure with respect to ${\\mathsf {w}}_{{\\beta },{\\gamma }}$ on the conic surface was studied in [53].", "For ${\\beta }> - d$ and ${\\gamma }> -1$ , define the inner product ${\\langle }f, g{\\rangle }_{{\\mathsf {w}}} ={\\mathsf {b}}_{{\\beta },{\\gamma }} \\int _{{\\mathbb {V}}_0^{d+1}} f(x,t) g(x,t) {\\mathsf {w}}_{{\\beta },{\\gamma }} \\mathrm {d}\\sigma (x,t),$ where $\\mathrm {d}\\sigma $ denote the surface measure on ${\\mathbb {V}}_0^{d+1}$ , which is well-defined for all polynomials restricted on the conic surface.", "Let ${\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1},{\\mathsf {w}}_{{\\beta },{\\gamma }})$ be the space of orthogonal polynomials of degree $n$ .", "Since ${\\mathbb {V}}_0^{d+1}$ is a quadratic surface in ${\\mathbb {R}}^{d+1}$ , the dimension of the space ${\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }})$ is the same as the dimension of the spherical harmonics of degree $n$ on ${\\mathbb {S}}^d$ ; that is, $\\dim {\\mathcal {V}}_0({\\mathbb {V}}_0^{d+1},{\\mathsf {w}}_{{\\beta },{\\gamma }}) =1$ and $\\dim {\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1},{\\mathsf {w}}_{{\\beta },{\\gamma }}) = \\binom{n+d-1}{n}+\\binom{n+d-2}{n-1},\\quad n=1,2,3,\\ldots .$ Furthermore, let $\\Pi _n({\\mathbb {V}}_0^{d+1})$ denote the space of polynomials of degree at most $n$ restricted on ${\\mathbb {V}}_0^{d+1}$ , then it is the union of ${\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1},{\\mathsf {w}}_{{\\beta },{\\gamma }})$ and $\\dim \\Pi _n({\\mathbb {V}}_0^{d+1}) = \\binom{n+d}{n}+\\binom{n+d-1}{n-1}.$ An orthogonal basis of ${\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1}, \\varphi _{{\\beta },{\\gamma }})$ can be given in terms of the Jacobi polynomials and spherical harmonics.", "Let ${\\mathcal {H}}_m({\\mathbb {S}^{d-1}})$ be the space of spherical harmonics of degree $m$ in $d$ variables.", "Let $\\lbrace Y_\\ell ^m: 1 \\le \\ell \\le \\dim {\\mathcal {H}}_m({\\mathbb {S}^{d-1}})\\rbrace $ denote an orthonormal basis of ${\\mathcal {H}}_m({\\mathbb {S}^{d-1}})$ .", "Then the polynomials $ {\\mathsf {S}}_{m, \\ell }^n (x,t) = P_{n-m}^{(2m + {\\beta }+ d-1,{\\gamma })} (1-2t) Y_\\ell ^m (x), \\quad 0 \\le m \\le n, \\,\\,1 \\le \\ell \\le \\dim {\\mathcal {H}}_m({\\mathbb {S}^{d-1}}),$ consist of an orthogonal basis of ${\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1}, \\varphi _{{\\beta },{\\gamma }})$ .", "More precisely, ${\\langle }{\\mathsf {S}}_{m, \\ell }^n, {\\mathsf {S}}_{m^{\\prime }, \\ell ^{\\prime }}^{n^{\\prime }} {\\rangle }_{{\\mathsf {w}}_{b,{\\gamma }}} = {\\mathsf {H}}_{m,n}^{{\\beta },{\\gamma }} \\delta _{n,n^{\\prime }} \\delta _{m,m^{\\prime }} \\delta _{\\ell ,\\ell ^{\\prime }}$ where the norm ${\\mathsf {H}}_{m,n}^{{\\beta },{\\gamma }}$ of ${\\mathsf {S}}_{m,\\ell }^n$ is given by ${\\mathsf {H}}_{m,n}^{{\\beta },{\\gamma }} = \\frac{c_{{\\beta }+d-1,{\\gamma }}}{c_{2m + {\\beta }+d-1,{\\gamma }}} h_{n-m}^{(2m+{\\beta }+d-1,{\\gamma })},$ where $c_{{\\alpha },{\\beta }}$ is the normalization constant in (REF ) and $h_m^{({\\alpha },{\\beta })}$ is the norm square of the Jacobi polynomial.", "We call $S_{m, \\ell }^n$ the Jacobi polynomials on the conic surface.", "The reproducing kernel of the space ${\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{{\\beta },{\\gamma }})$ is denoted by ${\\mathsf {P}}_n({\\mathsf {w}}_{{\\beta },{\\gamma }};\\cdot ,\\cdot )$ , which can be written in terms of the above basis, ${\\mathsf {P}}_n\\big ({\\mathsf {w}}_{{\\beta },{\\gamma }}; (x,t),(y,s) \\big ) = \\sum _{m=0}^n \\sum _{k=1}^{\\dim {\\mathcal {H}}_m^d}\\frac{ {\\mathsf {S}}_{m, \\ell }^n(x,t) {\\mathsf {S}}_{m, \\ell }^n(y,s)}{{\\mathsf {H}}_{m,n}^{{\\beta },{\\gamma }}},$ and it is the kernel of the orthogonal projection operator $\\operatorname{proj}_n({\\mathsf {w}}_{{\\beta },{\\gamma }}): L^2({\\mathbb {V}}_0^{d+1},{\\mathsf {w}}_{{\\beta },{\\gamma }}) \\rightarrow {\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1}, \\varphi _{{\\beta },{\\gamma }})$ , $\\operatorname{proj}_n({\\mathsf {w}}_{{\\beta },{\\gamma }};f) = \\int _{{\\mathbb {V}}_0^{d+1}} f(y,s) {\\mathsf {P}}_n\\big ({\\mathsf {w}}_{{\\beta },{\\gamma }}; \\,\\cdot , (y,s) \\big ) {\\mathsf {w}}_{{\\beta },{\\gamma }}(s) \\mathrm {d}\\sigma (y,s).$ The case ${\\beta }= -1$ turns out to be the most interesting case, for which there is a second order differential operator that has orthogonal polynomials as eigenfunctions, akin the Laplace-Beltrami operator for the spherical harmonics.", "Theorem 4.8 Let $\\Delta _0^{(\\xi )}$ denote the Laplace-Beltrami operator in $\\xi \\in {\\mathbb {S}^{d-1}}$ .", "Define $\\Delta _{0,{\\gamma }}:= \\left(t(1-t)\\partial _t^2 + \\big ( d-1 - (d+{\\gamma })t \\big ) \\partial _t+ t^{-1} \\Delta _0^{(\\xi )}\\right)$ for ${\\gamma }> -1$ .", "Then the polynomials in ${\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{-1,{\\gamma }})$ are eigenfunctions of $\\Delta _{0,{\\gamma }}$ , $\\Delta _{0,{\\gamma }} u = -n (n+{\\gamma }+d-1) u, \\qquad \\forall u \\in {\\mathcal {V}}_n({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{-1,{\\gamma }}).$ The reproducing kernel enjoys an addition formula that is a mixture of the addition formula for the spherical harmonics and the Jacobi polynomials.", "The formula has the most elegant form when ${\\beta }= -1$ , which is stated below.", "Theorem 4.9 Let $d \\ge 2$ and ${\\gamma }\\ge -\\frac{1}{2}$ .", "Then, for $(x,t), (y,s) \\in {\\mathbb {V}}_0^{d+1}$ , $ {\\mathsf {P}}_n \\big ({\\mathsf {w}}_{-1,{\\gamma }}; (x,t), (y,s)\\big ) = b_{{\\gamma },d} \\int _{[-1,1]^2} & Z_{2n}^{{\\gamma }+d-1} \\big ( \\zeta (x,t,y,s; v) \\big ) \\\\& \\times (1-v_1^2)^{\\frac{d-4}{2}} (1-v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}v, $ where $b_{{\\gamma },d}$ is a constant so that ${\\mathsf {P}}_0\\big ({\\mathsf {w}}_{-1,{\\gamma }}; (x,t), (y,s)\\big ) =1$ and $\\zeta (x,t,y,s; v) = v_1 \\sqrt{\\tfrac{st + {\\langle }x,y {\\rangle }}{2}}+ v_2 \\sqrt{1-t}\\sqrt{1-s};$ moreover, the identity holds under the limit (REF ) when ${\\gamma }= -\\frac{1}{2}$ and/or $d = 2$ .", "In particular, the orthogonal structure for the weight function ${\\mathsf {w}}_{-1,{\\gamma }}$ on the conic surface satisfies both characteristic properties specified in Definition REF and Definition REF .", "We show that ${\\mathsf {w}}_{-1,{\\gamma }}$ admits highly localized kernels in the next subsection." ], [ "Highly localized kernels", "Let $\\widehat{a}$ be an admissible cut-off function.", "For $(x,t)$ , $(y,s) \\in {\\mathbb {V}}_0^{d+1}$ , define the kernel ${\\mathsf {L}}_n({\\mathsf {w}}_{-1,{\\gamma }})$ by ${\\mathsf {L}}_n\\big ({\\mathsf {w}}_{-1,{\\gamma }}; (x,t),(y,s)\\big ) = \\sum _{j=0}^\\infty \\widehat{a}\\left( \\frac{j}{n} \\right) {\\mathsf {P}}_j\\big ({\\mathsf {w}}_{-1,{\\gamma }}; (x,t), (y,s)\\big ).$ We show that this kernel is highly localized.", "It is worthwhile to point out that ${\\mathsf {w}}_{-1,{\\gamma }}$ does not include the constant weight function, or the Lebesgue measure on the conic surface.", "We shall need ${\\mathsf {w}}_{-1,{\\gamma }}\\big ({\\mathsf {c}}((x,t), r)\\big )$ with $r = n^{-2}$ .", "Following the notation (REF ), we define ${\\mathsf {w}}_{{\\gamma },d} (n; t) = \\big (1-t+n^{-2}\\big )^{{\\gamma }+\\frac{1}{2}}\\big (t+n^{-2}\\big )^{\\frac{d-2}{2}}.$ Theorem 4.10 Let $d\\ge 2$ and ${\\gamma }\\ge -\\frac{1}{2}$ .", "Let $\\widehat{a}$ be an admissible cutoff function.", "Then for any ${\\kappa }> 0$ and $(x,t), (y,s) \\in {\\mathbb {V}}_0^{d+1}$ , $\\left\|{\\mathsf {L}}_n ({\\mathsf {w}}_{-1,{\\gamma }}; (x,t), (y,s))\\right\|\\le \\frac{c_{\\kappa }n^d}{\\sqrt{ {\\mathsf {w}}_{{\\gamma },d} (n; t) }\\sqrt{ {\\mathsf {w}}_{{\\gamma },d} (n; s) }}\\big (1 + n {\\mathsf {d}}_{{\\mathbb {V}}_0}( (x,t), (y,s)) \\big )^{-{\\kappa }}.$ We only prove the case when $d > 2$ and ${\\gamma }> -\\frac{1}{2}$ .", "The remaining cases ${\\gamma }= -\\frac{1}{2}$ and/or $d =2$ follow similarly and are easier.", "Using the addition formula for the reproducing kernel and, by the quadratic transform (REF ), $ Z_{2n}^{\\lambda }(x) = \\frac{P_n^{({\\lambda }-\\frac{1}{2},-\\frac{1}{2})}(1)P_n^{({\\lambda }-\\frac{1}{2},-\\frac{1}{2})}(2x^2-1)}{h_n^{({\\lambda }-\\frac{1}{2},-\\frac{1}{2})}}, $ we can write ${\\mathsf {L}}({\\mathsf {w}}_{{\\beta },{\\gamma }})$ in terms of the kernel $L_n ^{({\\lambda }-\\frac{1}{2},-\\frac{1}{2})}$ of the Jacobi polynomials, where ${\\lambda }= {\\gamma }+d -1$ .", "Then ${\\mathsf {L}}_n ({\\mathsf {w}}_{-1,{\\gamma }}; (x,t), (y,s) )=c_{{\\gamma }} \\int _{[-1,1]^2} & L_n ^{({\\lambda }-\\frac{1}{2},-\\frac{1}{2})}\\big (2 \\zeta (x,t,y,s; v)^2-1 \\big )\\\\& \\times (1-v_1^2)^{\\frac{d-2}{2}-1}(1-v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}v. $ Let $\\theta (x,t,y,s;v)= \\arccos (2\\zeta (x,t,y,s; v)^2 -1)$ .", "Then $1- \\zeta (x,t,y,s;t)^2 = \\frac{1}{2} (1 - \\cos \\theta (x,t,y,s;v)) = \\sin ^2 \\frac{\\theta (x,t,y,s;v)}{2}\\sim \\theta (x,t,y,s;v)^2.$ We apply the estimate (REF ) for $L_n^{{\\alpha },{\\beta }}$ with $\\alpha ={\\lambda }-1/2$ , $\\beta = -1/2$ to obtain $\\left\| {\\mathsf {L}}_n ({\\mathsf {w}}_{-1,{\\gamma }}; (x,t), (y,s)) \\right\| \\le c n^{2 {\\lambda }+1} \\int _{[-1,1]^2}& \\frac{1}{ \\left(1+ n\\sqrt{1- \\zeta (x,t,y,s; v)^2}\\right)^{{\\kappa }+2{\\gamma }+d+1} }\\\\& \\times (1-v_1^2)^{\\frac{d-2}{2}-1}(1-v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}v.$ From its definition, it is easy to verify that $\|\\zeta (x,t,y,s;v)\|\\le 1$ and $1- \\zeta (x,t,y,s; v) =\\, & 1- \\cos {\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)) \\\\+ & \\sqrt{\\frac{{\\langle }x,y{\\rangle }+ t s}{2}}(1 - v_1)+\\sqrt{1-s}\\sqrt{1-t} (1 - v_2).", "$ Consequently, since $t s + {\\langle }x,y{\\rangle }\\ge 0$ , it follows that $1 - \\zeta (x,t,y,s; v) \\, &\\ge 1- \\cos {\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)) \\\\& = 2 \\sin ^2 \\frac{ {\\mathsf {d}}_{{\\mathbb {V}}_0} ((x,t), (y,s)) }{2} \\ge \\frac{2}{\\pi ^2} [{\\mathsf {d}}_{{\\mathbb {V}}_0}((x,t),(y,s))]^2.$ Applying this inequality, we obtain the estimate $\\left\| {\\mathsf {L}}_n ({\\mathsf {w}}_{-1,{\\gamma }}; (x,t), (y,s) )\\right\| \\, & \\le c n^{2 {\\lambda }+1}\\frac{1}{\\left(1+ n {\\mathsf {d}}_{{\\mathbb {V}}_0}((x,t),(y,s))\\right)^{\\kappa }} \\\\& \\times b_{{\\gamma },d} \\int _{[-1,1]^2}\\frac{(1-v_1^2)^{\\frac{d-2}{2}-1}(1-v_2^2)^{{\\gamma }-\\frac{1}{2}}}{\\left(1+n\\sqrt{1- \\zeta (x,t,y,s; v)}\\right)^{2{\\gamma }+d+1}} \\mathrm {d}v,$ where $b_{{\\gamma },d} = c_{{\\gamma }-\\frac{1}{2},{\\gamma }-\\frac{1}{2}} c_{\\frac{d-4}{2},\\frac{d-4}{2}}$ .", "The estimate of the last expression is the crux of the proof and it is contained in the lemma below.", "Lemma 4.11 Let $d \\ge 2$ and ${\\gamma }> -\\frac{1}{2}$ .", "Then, for ${\\beta }\\ge 2{\\gamma }+ d+1$ , $b_{{\\gamma },d} \\int _{[-1,1]^2} & \\frac{(1-v_1)^{\\frac{d-2}{2}-1}(1-v_2)^{{\\gamma }-\\frac{1}{2}}}{\\big (1+n\\sqrt{1- \\zeta (x,t,y,s; v)}\\,\\big )^{{\\beta }}} \\mathrm {d}v\\\\& \\qquad \\le \\frac{cn^{- (2{\\gamma }+d-1)}}{\\sqrt{{\\mathsf {w}}_{{\\gamma },d}(n; t)}\\sqrt{{\\mathsf {w}}_{{\\gamma },d}(n; s)}\\big (1+n {\\mathsf {d}}_{{\\mathbb {V}}_0}((x,t),(y,s))\\big )^{{\\beta }- 3{\\gamma }- \\frac{3d+1}{2}}} .$ Using the lower bound of $1- \|\\zeta (x,t,y,s;v)\|$ in the proof of the previous theorem, the left-hand side of the stated inequality has the upper bound $\\frac{c}{\\big (1+n {\\mathsf {d}}_{{\\mathbb {V}}_0}((x,t),(y,s)) \\big )^{{\\beta }- 2 {\\gamma }- d-1}}\\int _{[-1,1]^2}\\frac{(1-v_1)^{\\frac{d-2}{2}-1}(1-v_2)^{{\\gamma }-\\frac{1}{2}}}{(1+n \\sqrt{1- \\zeta (x,t,y,s; v)})^{2{\\gamma }+d+1}} \\mathrm {d}v.$ Denote the above integral by $I(x,t,y,s)$ .", "To complete the proof, we need to show that $\\frac{ I(x,t,y,s)}{\\big (1+ n {\\mathsf {d}}_{{\\mathbb {V}}_0}((x,t),(y,s))\\big )^{{\\gamma }+\\frac{d-1}{2}}} \\le \\frac{cn^{- (2{\\gamma }+d-1)}}{\\sqrt{{\\mathsf {w}}_{{\\gamma }}(n; t)}\\sqrt{{\\mathsf {w}}_{{\\gamma }}(n; s)}}.$ We need a lower bound for $1-\\zeta $ .", "Let ${\\alpha }= \\sqrt{\\frac{{\\langle }x,y{\\rangle }+ts}{2}}$ and ${\\beta }= \\sqrt{1-t}\\sqrt{1-s}$ .", "Then $1- \\zeta (x,t,y,s; v) \\,& = 1- {\\alpha }+ (1-v_1) {\\alpha }- v_2 {\\beta }\\ge 1 -{\\alpha }+ \\frac{1}{2}(1-v_1){\\alpha }- v_2 {\\beta }\\\\& = 1- \\frac{1+v_1}{2} {\\alpha }- \\frac{1-v_1}{2} \\sqrt{ts} + \\frac{1-v_1}{2}\\sqrt{ts} -v_2 {\\beta }.$ Using ${\\alpha }\\le \\sqrt{ts}$ and $1- \\sqrt{t s} - \\sqrt{1-t}\\sqrt{1-s} \\ge 0$ , we then obtain $1- \\zeta (x,t,y,s; v) \\, & \\ge 1-\\sqrt{ts} - {\\beta }+ \\frac{1}{2}(1-v_1)\\sqrt{ts} + (1-v_2) {\\beta }\\\\& \\ge \\frac{1}{2} (1 - v_1)\\sqrt{t}\\sqrt{s} +(1 - v_2) \\sqrt{1-s}\\sqrt{1-t}.", "$ Using this inequality in $I(x,t,y,s)$ and then making a change of variable $v_1 \\mapsto 2 u_1-1$ and $v_2 \\mapsto 2 u_2-1$ , we obtain $I(x,t,y,s) \\le c \\int _{[0,1]^2} & \\frac{(1-v_1)^{\\frac{d-2}{2}-1}(1-v_2)^{{\\gamma }-\\frac{1}{2}}}{\\big (1+n\\sqrt{1- (1 - v_1)\\sqrt{t}\\sqrt{s} - 2 (1 - v_2) \\sqrt{1-s}\\sqrt{1-t}}\\,\\big )^{2{\\gamma }+d+1}} \\mathrm {d}v.$ This integral can be estimated by using the inequality [12] $\\int _0^1 \\frac{(1-t)^{a-1} \\mathrm {d}t}{(1+n \\sqrt{B+A(1-t)})^b} \\le c \\frac{n^{-2 a}}{A^a (1+n\\sqrt{B})^{b-2a-1}},$ which holds for $A>0$ , $B\\ge 0$ , $a>0$ and $b\\ge 2 a +1$ .", "Applying this inequality with $A = \\sqrt{1-t}\\sqrt{1-s}$ , $a = {\\gamma }+\\frac{1}{2}$ , we obtain $I(x,t,y,s) \\le \\frac{cn^{-2{\\gamma }-1}}{(\\sqrt{1-t}\\sqrt{1-s})^{{\\gamma }+\\frac{1}{2}}} \\int _0^1\\frac{(1-v_1^2)^{\\frac{d-2}{2}-1}}{\\left(1+n\\sqrt{\\sqrt{t}\\sqrt{s}(1 - v_1)}\\right)^{d-1}} \\mathrm {d}v_1.$ Applying (REF ) one more time with $B=0$ , $a = \\frac{d-2}{2}$ and $b= d-1$ , we see that the last integral is bounded by $c n^{-(d-2)}$ , hence, $I(x,t,y,s) \\,& \\le \\frac{cn^{- (2{\\gamma }+d-1)}}{(\\sqrt{1-t}\\sqrt{1-s})^{{\\gamma }+\\frac{1}{2}} (\\sqrt{t}\\sqrt{s})^{\\frac{d-2}{2}}} \\\\& \\le \\frac{cn^{- (2{\\gamma }+d-1)}}{(\\sqrt{1-t}\\sqrt{1-s}+n^{-2} )^{{\\gamma }+\\frac{1}{2}} (\\sqrt{t}\\sqrt{s}+n^{-2})^{\\frac{d-2}{2}}},$ where the second inequality follows since $I(x,t,y,s) \\le 1$ holds trivially by the choice of $b_{{\\gamma },\\mathrm {d}}$ .", "Now, using the elementary identity [12] $ (a+n^{-1})(b+n^{-1}) \\le 3 (ab+n^{-2})(1+n\|b-a\|)$ with $a = \\sqrt{t}$ and $b=\\sqrt{s}$ as well as with $a = \\sqrt{1-t}$ and $b=\\sqrt{1-s}$ , we obtain $I(x,t,y,s) \\le \\frac{cn^{- (2{\\gamma }+d-1)}\\big (1+ n \\big \|\\sqrt{t}-\\sqrt{s}\\big \|\\big )^{\\frac{d-2}{2}} \\big (1+ n \\big \|\\sqrt{1-t}-\\sqrt{1-s}\\big \|\\big )^{{\\gamma }+\\frac{1}{2}}}{\\sqrt{{\\mathsf {w}}_{{\\gamma },d}(n; t)}\\sqrt{{\\mathsf {w}}_{{\\gamma },d}(n; s)}}.$ Finally, by Lemma REF , we conclude that $I(x,t,y,s) \\le \\frac{cn^{- (2{\\gamma }+d-1)}(1+ n {\\mathsf {d}}_{{\\mathbb {V}}_0}((x,t),(y,s)))^{{\\gamma }+\\frac{d-1}{2}}}{\\sqrt{{\\mathsf {w}}_{{\\gamma },d}(n; t)}\\sqrt{{\\mathsf {w}}_{{\\gamma },d}(n; s)}},$ which is what we need to complete the proof.", "The following corollary, following immediately from (REF ), will be used later.", "Corollary 4.12 Let $d \\ge 2$ and ${\\gamma }> -\\frac{1}{2}$ .", "Then, for ${\\beta }\\ge 2{\\gamma }+ d+1$ , $b_{{\\gamma },d} \\int _{[-1,1]^2} & \\frac{(1-v_1)^{\\frac{d-2}{2}-1}(1-v_2)^{{\\gamma }-\\frac{1}{2}}}{\\left(1+n\\sqrt{1- (1 - v_1)\\sqrt{t}\\sqrt{s} - (1 - v_2) \\sqrt{1-s}\\sqrt{1-t}}\\,\\right)^{2{\\gamma }+\\mathrm {d}+1}} \\mathrm {d}v\\\\& \\qquad \\le \\frac{cn^{- (2{\\gamma }+d-1)}\\big (1+n {\\mathsf {d}}_{[0,1]}((x,t),(y,s))\\big )^{{\\gamma }+ \\frac{d-1}{2}}}{\\sqrt{{\\mathsf {w}}_{{\\gamma },d}(n; t)}\\sqrt{{\\mathsf {w}}_{{\\gamma },d}(n; s)}} .$ By (REF ), we have shown that ${\\mathsf {w}}_{-1,{\\gamma }}$ admits Assertion 1 of the highly localized kernel.", "Our next result shows that it also admits Assertion 2.", "Theorem 4.13 Let $\\widehat{a}$ be an admissible cutoff function.", "Let $d\\ge 2$ and ${\\gamma }\\ge -\\frac{1}{2}$ .", "For $(x_i,t_i), (y,s) \\in {\\mathbb {V}}_0^{d+1}$ and $(x_1,t_1) \\in {\\mathsf {c}}\\big ((x_2,t_2), c^ n^{-1}\\big )$ with $c^$ small and for ${\\kappa }> 0$ , $& \\left\|{\\mathsf {L}}_n ({\\mathsf {w}}_{-1,{\\gamma }}; (x_1,t_1), (y,s))-{\\mathsf {L}}_n ({\\mathsf {w}}_{-1,{\\gamma }}; (x_2,t_2), (y, s))\\right\| \\\\& \\qquad \\le c_{\\kappa }\\frac{ n^{d+1} {\\mathsf {d}}_{{\\mathbb {V}}_0}((x_1,t_1), (x_2, t_2))}{\\sqrt{ {\\mathsf {w}}_{{\\gamma },d} (n; s) }\\sqrt{ {\\mathsf {w}}_{{\\gamma },d} (n; t_2) }\\big (1 + n {\\mathsf {d}}_{{\\mathbb {V}}_0}( (y,s), (x_2, t_2)) \\big )^{{\\kappa }}}.", "$ Denote the left-hand side of (REF ) by $K$ .", "Let $\\partial L(u) = L^{\\prime }(u)$ .", "Using the integral expression (REF ) of ${\\mathsf {L}}_n ({\\mathsf {w}}_{-1,{\\gamma }})$ , we obtain $ K &\\le 2 \\int _{[-1,1]^2} \\big \\Vert \\partial L_n^{{\\lambda }-\\frac{1}{2},-\\frac{1}{2}} \\big (2(\\cdot )^2-1\\big )\\big \\Vert _{L^\\infty (I_v)}\\big \|\\zeta _1(v)^2 - \\zeta _2(v)^2\| \\\\ & \\qquad \\qquad \\qquad \\times (1-v_1^2)^{\\frac{d-2}{2}-1} (1-v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}v, $ where $\\zeta _i(v) = \\zeta (x_i,t_i,y,s;v)$ , and $I_v$ is the interval with end points $\\zeta _1(v)$ and $\\zeta _2(v)$ .", "Since $\|\\zeta (\\cdot )\| \\le 1$ , $\|\\zeta _1(v)^2 - \\zeta _2(v)^2\| \\le 2 \|\\zeta _1(v)- \\zeta _2(v)\|$ .", "We claim that $ \|\\zeta _1(v)- \\zeta _2(v)\| \\le c\\, {\\mathsf {d}}_{{\\mathbb {V}}_0}\\big ((x_1,t_1),(x_2,t_2)\\big )\\big ( \\Sigma _1 + \\Sigma _2(v_1) + \\Sigma _3 (v_2)\\big ),$ where $\\Sigma _1 \\, & = {\\mathsf {d}}_{{\\mathbb {V}}_0}\\big ((x_i, t_i),(y,s)\\big ) + {\\mathsf {d}}_{{\\mathbb {V}}_0}\\big ((x_1,t_1),(x_2,t_2)\\big ), \\\\\\Sigma _2(v_1) \\, & = (1-v_1)\\sqrt{s}, \\\\\\Sigma _3 (v_2)\\, & = (1 - v_2)\\sqrt{1-s}.$ To see this, we use (REF ) and writing $x_i = t_i \\xi _i$ and $y = s \\eta $ to obtain $\\zeta _1(v)- \\zeta _2(v) \\, & = \\cos {\\mathsf {d}}_{{\\mathbb {V}}_0}\\big ((x_1,t_1),(y,s)\\big ) - \\cos {\\mathsf {d}}_{{\\mathbb {V}}_0}\\big ((x_2,t_2),(y,s)\\big ) \\\\& + (1-v_1) \\left(\\sqrt{ t_2 s} \\cos \\frac{{\\mathsf {d}}_{{\\mathbb {S}}}(\\xi _2,\\eta )}{2} -\\sqrt{ t_1 s} \\cos \\frac{{\\mathsf {d}}_{{\\mathbb {S}}}(\\xi _1,\\eta )}{2} \\right) \\\\& + (1- v_2) \\left(\\sqrt{1- t_2} - \\sqrt{1-t_1}\\right) \\sqrt{1-s}.$ Denote temporarily ${\\alpha }_i = {\\mathsf {d}}_{{\\mathbb {V}}_0}((x_1,t_1),(y,s))$ for $i =1, 2$ .", "Hence, using the identity $\\cos {\\alpha }_1 - \\cos {\\alpha }_2 \\, &= 2 \\sin \\frac{{\\alpha }_1 - {\\alpha }_2}{2} \\sin \\frac{{\\alpha }_1+{\\alpha }_2}{2} \\\\& = 2 \\sin \\frac{{\\alpha }_1 - {\\alpha }_2}{2} \\left( 2 \\sin \\frac{{\\alpha }_i}{2} + \\sin \\frac{\|{\\alpha }_1-{\\alpha }_2\|}{2}\\right),$ it follows readily that $\|\\cos {\\alpha }_1 - \\cos {\\alpha }_2\| \\le \|{\\alpha }_1-{\\alpha }_2\| \\left( \|{\\alpha }_1\| + \\tfrac{1}{2} \|{\\alpha }_1 - {\\alpha }_2\| \\right).$ By the triangle inequality of ${\\mathsf {d}}_{{\\mathbb {V}}_0}$ , $\|{\\alpha }_1-{\\alpha }_2\| \\le {\\mathsf {d}}_{{\\mathbb {V}}_0}((x_1,t_1),(x_2,t_2))$ , this gives the estimate for the $\\Sigma _1$ term.", "Moreover, assuming $t_2 \\ge t_1$ , for example, and applying similar argument for ${\\mathsf {d}}_{\\mathbb {S}}$ , we then obtain, using Lemma REF and (REF ), that $\\bigg \| \\sqrt{ t_1 } \\cos \\frac{{\\mathsf {d}}_{{\\mathbb {S}}}(\\xi _1,\\eta )}{2} - & \\sqrt{ t_2 } \\cos \\frac{{\\mathsf {d}}_{{\\mathbb {S}}}(\\xi _2,\\eta )}{2} \\bigg \| \\\\& \\le \\left\| \\sqrt{t_2} - \\sqrt{t_1} \\right\| + \\sqrt{t_1}\\left\| \\cos \\frac{{\\mathsf {d}}_{{\\mathbb {S}}}(\\xi _1,\\eta )}{2} -\\cos \\frac{{\\mathsf {d}}_{{\\mathbb {S}}}(\\xi _2,\\eta )}{2} \\right\| \\\\& \\le {\\mathsf {d}}_{{\\mathbb {V}}_0}((x_1,t_1),(x_2,t_2)) + \\frac{1}{2} (t_1 t_2)^{\\frac{1}{4}} {\\mathsf {d}}_{{\\mathbb {S}}}(\\xi _1,\\xi _2) \\\\& \\le c\\, {\\mathsf {d}}_{{\\mathbb {V}}_0}((x_1,t_1),(x_2,t_2)),$ which verifies the $\\Sigma _2(v_1)$ term.", "The third term with $\\Sigma _3(v_2)$ follows from using Lemma REF one more time.", "This verifies the claim (REF ).", "Since $\\max _{r\\in I_v} \|1+n \\sqrt{1- r}\|^{-\\sigma }$ is attained at one of the end points of the interval, it follows from (REF ) with $m =1$ .", "$\\big \\Vert \\partial L_n^{{\\lambda }-\\frac{1}{2},-\\frac{1}{2}} \\big (2(\\cdot )^2-1\\big )\\big \\Vert _{L^\\infty (I_v)}\\le c \\left[ \\frac{n^{2 {\\lambda }+ 3}}{\\big (1+n\\sqrt{1-\\zeta _1(v)^2} \\big )^{{\\kappa }}} + \\frac{n^{2 {\\lambda }+ 3}}{\\big (1+n\\sqrt{1-\\zeta _2(v)^2}\\big )^{{\\kappa }}} \\right].$ Consequently, we see that $K$ is bounded by a sum of integrals $K \\le c \\, {\\mathsf {d}}_{{\\mathbb {V}}_0}\\big ((x_1,t_1),(x_2,t_2)\\big ) & \\int _{[-1,1]^2}\\left[ \\frac{n^{2 {\\lambda }+ 3}}{\\big (1+n\\sqrt{1-\\zeta _1(v)^2} \\big )^{{\\kappa }}} + \\frac{n^{2 {\\lambda }+ 3}}{\\big (1+n\\sqrt{1-\\zeta _2(v)^2}\\big )^{{\\kappa }}} \\right] \\\\& \\quad \\times \\big (\\Sigma _1+\\Sigma _2(v) + \\Sigma _3(v) \\big ) (1-v_1^2)^{\\frac{d-2}{2}-1} (1-v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}v.$ Since $(x_1,t_1) \\in {\\mathsf {c}}\\big ((x_2,t_2), c^ n^{-1}\\big )$ , $\\Sigma _1$ is bounded by $\\Sigma _1 \\le c n^{-1} \\big (1 + n {\\mathsf {d}}_{{\\mathbb {V}}_0}\\big ((x_i, t_i),(y,s)\\big ) \\big )$ .", "Hence, we obtain from Lemma REF that $\\int _{[-1,1]^2} & \\frac{n^{2 {\\lambda }+ 3}}{\\big (1+n\\sqrt{1-\\zeta _i(v)^2} \\big )^{{\\kappa }}} \\Sigma _1(1-v_1^2)^{\\frac{d-2}{2}-1} (1- v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}v\\\\& \\le c \\frac{ n^{d+1} }{\\sqrt{ {\\mathsf {w}}_{{\\gamma },d} (n; s) }\\sqrt{ {\\mathsf {w}}_{{\\gamma },d} (n; t_i) }\\big (1 + n {\\mathsf {d}}_{{\\mathbb {V}}_0}( (y,s), (x_i, t_i)) \\big )^{{\\kappa }({\\gamma },d)-1}},$ where ${\\kappa }({\\gamma },d) = {\\kappa }- 3 {\\gamma }- \\frac{3d+1}{2}$ for either $i=1$ or $i=2$ .", "Since ${\\mathsf {w}}_{{\\gamma },d}(n,t_1) \\sim {\\mathsf {w}}_{{\\gamma },d}(n,t_2)$ and ${\\mathsf {d}}_{{\\mathbb {V}}_0}((x_1, t_1),(y,s))+ n^{-1} \\sim {\\mathsf {d}}_{{\\mathbb {V}}_0}((x_2, t_2),(y,s)) + n^{-1}$ by Lemma REF , we can replace $(x_1,t_1)$ in the right-hand side by $(x_2,t_2)$ .", "This shows that the integral containing $\\Sigma _1$ has the desired estimate.", "For the remaining integrals, the same consideration shows that we only need to consider those containing $\\zeta _2(v)$ .", "For the integral that contains $\\Sigma _2(v_1)= (1-v_1)\\sqrt{s}$ , the factor $(1-v_1)$ increases the power of the weight to $(1-v_1)^{\\frac{d}{2}}$ , so that we can apply Lemma REF with $\\frac{d-2}{2}$ replaced by $\\frac{d}{2}$ , which leads to $\\int _{-1}^1 & \\frac{n^{2 {\\lambda }+ 3}}{\\big (1+n\\sqrt{1-\\zeta _2(v)^2} \\big )^{{\\kappa }}} \\Sigma _2(v_1)(1-v_1^2)^{\\frac{d-2}{2}-1} (1- v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}v\\\\& \\le c \\frac{ n^{d+1} n^{-1}\\sqrt{s} }{\\sqrt{ {\\mathsf {w}}_{{\\gamma },d+1} (n; s) }\\sqrt{ {\\mathsf {w}}_{{\\gamma },d+1} (n; t_2) }\\big (1 + n {\\mathsf {d}}_{{\\mathbb {V}}_0}( (y,s), (x_2, t_2)) \\big )^{{\\kappa }({\\gamma },d+1)}} \\\\& \\le c \\frac{ n^{d+1} }{\\sqrt{ {\\mathsf {w}}_{{\\gamma },d} (n; s) }\\sqrt{ {\\mathsf {w}}_{{\\gamma },d} (n; t_2) }\\big (1 + n {\\mathsf {d}}_{{\\mathbb {V}}_0}( (y,s), (x_2, t_2)) \\big )^{{\\kappa }({\\gamma },d+1)}},$ where the last step follows from the inequality $n^{-1} \\sqrt{s}\\le (\\sqrt{t_2}+n^{-1}) (\\sqrt{s}+n^{-1})$ .", "The integral that contains $\\Sigma _3(v_2)$ can be estimated similarly by applying Lemma REF with ${\\gamma }$ replaced by ${\\gamma }+1$ and using $n^{-1} \\sqrt{1-s}\\le (\\sqrt{1-t_2}+n^{-1}) (\\sqrt{1-s}+n^{-1})$ .", "This completes the proof.", "The first two assertions for the highly localized kernels are established for ${\\mathsf {w}}_{-1,{\\gamma }}$ when ${\\gamma }\\ge -\\frac{1}{2}$ .", "The case of $p=1$ of the following lemma establishes Assertion 3.", "Recall that ${\\mathsf {w}}_{{\\beta },{\\gamma },d}(n; t)$ is defined in (REF ).", "Lemma 4.14 Let $d\\ge 2$ , ${\\beta }> -d$ and ${\\gamma }> -1$ .", "For $0 < p < \\infty $ , assume ${\\kappa }> \\frac{2d}{p} + ({\\gamma }+{\\beta }+\\frac{d+1}{2}) \|\\frac{1}{p}-\\frac{1}{2}\|$ .", "Then for $(x,t) \\in {\\mathbb {V}}_0^{d+1}$ , $\\int _{{\\mathbb {V}}_0^{d+1}} \\frac{ {\\mathsf {w}}_{{\\beta },{\\gamma }}(s) \\mathrm {d}\\sigma (y,s) }{ {\\mathsf {w}}_{{\\beta },{\\gamma },d} (n; s)^{\\frac{p}{2}}\\big (1 + n {\\mathsf {d}}_{{\\mathbb {V}}_0}( (x,t), (y,s)) \\big )^{{\\kappa }p}}\\le c n^{-d} {\\mathsf {w}}_{{\\beta },{\\gamma },d} (n; t)^{1-\\frac{p}{2}}.$ Let $J_{p}$ denote the left-hand side of (REF ).", "By Lemma REF , it is sufficient to estimate $J_{2}$ .", "Let $x = t \\xi $ and $y = s\\eta $ .", "The definition of ${\\mathsf {d}}_{{\\mathbb {V}}_0}(\\cdot ,\\cdot )$ shows that it is a function of ${\\langle }\\xi ,\\eta {\\rangle }$ .", "Hence, by (REF ), we deduce that $J_{2,{\\kappa }} \\, &\\le c \\int _0^1 \\int _{-1}^1 \\frac{ s^{d-1} {\\mathsf {w}}_{{\\beta },{\\gamma }}(s) (1-u^2)^{\\frac{d-3}{2}} }{{\\mathsf {w}}_{{\\beta },{\\gamma },d} (n; s)\\big (1 + n \\arccos \\big ( \\sqrt{ts} \\sqrt{\\frac{1+u}{2}} + \\sqrt{1-t}\\sqrt{1-s} \\big )\\big )^{2{\\kappa }} } \\mathrm {d}u \\mathrm {d}s \\\\& \\le c \\int _0^1 \\int _{0}^1 \\frac{s^{d-1} {\\mathsf {w}}_{{\\beta },{\\gamma }}(s) v^{d-2}(1-v^2)^{\\frac{d-3}{2}} }{{\\mathsf {w}}_{{\\beta },{\\gamma },d} (n; s)\\left(1 + n \\sqrt{1- \\sqrt{ts} v - \\sqrt{1-t}\\sqrt{1-s}} \\right)^{2k}} \\mathrm {d}v \\mathrm {d}s,$ where the second step follows from changing variable $ \\sqrt{\\frac{1+u}{2}} \\mapsto v$ and the relation ${\\theta }\\sim \\sin \\frac{{\\theta }}{2} \\sim \\sqrt{1-\\cos {\\theta }}$ .", "Making a further changing of variable $v \\mapsto z/\\sqrt{s}$ gives $J_{2} \\,& \\le c \\int _0^1 \\int _{0}^{\\sqrt{s}} \\frac{s\\, {\\mathsf {w}}_{{\\beta },{\\gamma }}(s) z^{d-2}(s- z^2)^{\\frac{d-3}{2}} }{{\\mathsf {w}}_{{\\beta },{\\gamma },d} (n; s)\\left(1 + n \\sqrt{1- \\sqrt{t}\\,z - \\sqrt{1-t}\\sqrt{1-s}} \\right)^{2{\\kappa }} } \\mathrm {d}z \\mathrm {d}s \\\\& \\le c \\int _0^1 \\int _{0}^{\\sqrt{s}} \\frac{(s- z^2)^{\\frac{d-3}{2}} }{ (1-s+n^{-2})^{\\frac{1}{2}}\\left(1 + n \\sqrt{1- \\sqrt{t}\\, z - \\sqrt{1-t}\\sqrt{1-s}} \\right)^{2{\\kappa }} } \\mathrm {d}z \\mathrm {d}s,$ where we have used $s z^{d-2} \\le s^{\\frac{d}{2}} \\le (s+n^{-2})^{\\frac{d}{2}}$ .", "One more change of variable $s\\mapsto 1-w^2$ with $\\mathrm {d}s = w \\mathrm {d}w$ and $w \\le (w^2+n^{-2})^\\frac{1}{2}$ , we obtain $J_{2} \\le c \\int _0^1 \\int _{0}^{\\sqrt{1-w^2}} \\frac{(1-w^2- z^2)^{\\frac{d-3}{2}} }{\\left(1 + n \\sqrt{1- \\sqrt{t} z - \\sqrt{1-t} \\, w} \\right)^{2{\\kappa }}} \\mathrm {d}z \\mathrm {d}w,$ which is an integral over the positive quadrant $\\lbrace (z,w) \\in {\\mathbb {B}}^2: w \\ge 0, z \\ge 0\\rbrace $ of the unit disk ${\\mathbb {B}}^2$ .", "Setting $p = \\sqrt{t} z + \\sqrt{1-t} \\, w$ and $q = - \\sqrt{1-t} z + \\sqrt{t}w$ in the integral, which is an orthogonal transformation, and enlarging the integral domain while taking into account that $p \\ge 0$ , it follows that $J_{2} \\, & \\le c \\int _0^1 \\frac{1}{\\left(1 + n \\sqrt{1- p} \\right)^{2{\\kappa }} }\\int _{- \\sqrt{1-p^2}}^{\\sqrt{1-p^2}}(1-p^2- q^2)^{\\frac{d-3}{2}} \\mathrm {d}q \\mathrm {d}p \\\\& \\le c \\int _0^1 \\frac{(1-p^2)^{\\frac{d-2}{2}} }{\\left(1 + n \\sqrt{1- p} \\right)^{2{\\kappa }} }\\le c n^{-d} \\int _0^n \\frac{ r^{d-1} }{ (1 + r )^{\\kappa }} \\mathrm {d}r \\le c n^{-d}$ by setting $r = n \\sqrt{1-p}$ and recalling that ${\\kappa }> d$ .", "This completes the proof.", "Proposition 4.15 Let $d\\ge 2$ and ${\\gamma }\\ge - \\frac{1}{2}$ .", "For $0 < p < \\infty $ and $(x,t) \\in {\\mathbb {V}}_0^{d+1}$ , $\\int _{{\\mathbb {V}}_0^{d+1}} \\left\| {\\mathsf {L}}_n\\big ({\\mathsf {w}}_{-1,{\\gamma }};(x,t),(y,s)\\big ) \\right\|^p {\\mathsf {w}}_{-1,{\\gamma }}(s) \\mathrm {d}\\sigma (y,s)\\le c \\left(\\frac{n^d}{{\\mathsf {w}}_{{\\gamma },d} (n; t)}\\right)^{p-1}.$ The proof is immediate by (i) of Theorem REF and Lemma REF .", "Corollary 4.16 For ${\\gamma }\\ge -\\frac{1}{2}$ , the space $({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{-1,{\\gamma }}, {\\mathsf {d}}_{{\\mathbb {V}}_0})$ is a localizable homogeneous space." ], [ "Maximal ${\\varepsilon }$ -separated sets and MZ inequality", "Let ${\\varepsilon }> 0$ .", "An ${\\varepsilon }$ -separated set is defined in Definition REF .", "In this subsection, we provide a construction of some examples of maximal ${\\varepsilon }$ -separated points on the conic surface.", "For our construction, we shall need ${\\varepsilon }$ -separated points on the unit sphere ${\\mathbb {S}^{d-1}}$ .", "We adopt the following notation.", "For ${\\varepsilon }> 0$ , we denote by $\\Xi _{{\\mathbb {S}}}({\\varepsilon })$ a maximal ${\\varepsilon }$ -separated set on the unit sphere ${\\mathbb {S}^{d-1}}$ and we let ${\\mathbb {S}}_\\xi ({\\varepsilon })$ be the subsets in ${\\mathbb {S}^{d-1}}$ so that the collection $\\lbrace {\\mathbb {S}}_\\xi ({\\varepsilon }): \\xi \\in \\Xi _{\\mathbb {S}}({\\varepsilon })\\rbrace $ is a partition of ${\\mathbb {S}^{d-1}}$ , and we assume ${\\mathsf {c}}_{{\\mathbb {S}}}(\\xi , c_1 {\\varepsilon }) \\subset {\\mathbb {S}}_\\xi ({\\varepsilon }) \\subset {\\mathsf {c}}_{{\\mathbb {S}}}(\\xi , c_2 {\\varepsilon }), \\qquad \\xi \\in \\Xi _{{\\mathbb {S}}}({\\varepsilon }),$ where ${\\mathsf {c}}_{{\\mathbb {S}}}(\\xi ,{\\varepsilon })$ denotes the spherical cap centered at $\\xi $ with radius ${\\varepsilon }$ , $c_1$ and $c_2$ depending only on $d$ .", "Such a $\\Xi _{\\mathbb {S}}({\\varepsilon })$ exists for all ${\\varepsilon }> 0$ , see for example [12], and its cardinality satisfies $c_d^{\\prime } {\\varepsilon }^{-d+1} \\le \\# \\Xi _{{\\mathbb {S}}}({\\varepsilon }) \\le c_d {\\varepsilon }^{-d+1}.$ We now consider the subsets of points on the cone ${\\mathbb {V}}_0^{d+1}$ .", "For ${\\varepsilon }> 0$ , we denote by $\\Xi _{{\\mathbb {V}}_0} = \\Xi _{{\\mathbb {V}}_0}({\\varepsilon })$ an ${\\varepsilon }$ -separated set and denote by $\\lbrace {\\mathbb {V}}_0(\\xi ,t): (t\\xi ,t) \\in \\Xi _{{\\mathbb {V}}_0}\\rbrace $ a partition of ${\\mathbb {V}}_0^{d+1}$ .", "We start with an explicit construction of such $\\Xi _{{\\mathbb {V}}_0}$ and ${\\mathbb {V}}_0(\\xi ,t)$ .", "Let ${\\varepsilon }> 0$ and let $N = \\lfloor \\frac{\\pi }{2}{\\varepsilon }^{-1} \\rfloor $ .", "For $1\\le j \\le N$ we define ${\\theta }_j:= \\frac{(2j-1)\\pi }{2 N}, \\qquad {\\theta }_j^- := {\\theta }_j- \\frac{\\pi }{2 N} \\quad \\hbox{and} \\quad {\\theta }_j^+ := {\\theta }_j +\\frac{\\pi }{2 N}.$ Let $t_j = \\sin ^2 \\frac{{\\theta }_j}{2}$ and define $t_j^-$ and $t_j^+$ accordingly.", "In particular, $t_1^- = 0$ and $t_N^+ = 1$ .", "Then ${\\theta }_{j+1}^- ={\\theta }_j^+$ and ${\\mathbb {V}}_0^{d+1}$ can be partitioned by ${\\mathbb {V}}_0^{d+1} = \\bigcup _{j=1}^N {\\mathbb {V}}_0^{(j)}, \\quad \\hbox{where}\\quad {\\mathbb {V}}_0^{(j)}:=\\left\\lbrace (x,t) \\in {\\mathbb {V}}_0^{d+1}: t_j^- < t \\le t_j^+ \\right\\rbrace .$ Let ${\\varepsilon }_j := (2 \\sqrt{t_j})^{-1} \\pi {\\varepsilon }$ .", "Then $\\Xi _{\\mathbb {S}}({\\varepsilon }_j)$ is the maximal ${\\varepsilon }_j$ -separated set of ${\\mathbb {S}^{d-1}}$ such that $\\lbrace {\\mathbb {S}}_\\xi ({\\varepsilon }_j): \\xi \\in \\Xi _{\\mathbb {S}}({\\varepsilon }_j)\\rbrace $ is a partition ${\\mathbb {S}^{d-1}}= \\bigcup _{\\eta \\in \\Xi _{\\mathbb {S}}({\\varepsilon }_j)} {\\mathbb {S}}_\\eta ({\\varepsilon }_j)$ , and $\\# \\Xi _{\\mathbb {S}}({\\varepsilon }_j) \\sim {\\varepsilon }_j^{-d+1}.$ For each $j =1,\\ldots , N$ , we decompose ${\\mathbb {V}}_0^{(j)}$ by ${\\mathbb {V}}_0^{(j)} = \\bigcup _{\\xi \\in \\Xi _{\\mathbb {S}}({\\varepsilon }_j)} {\\mathbb {V}}_0(\\xi ,t_j), \\quad \\hbox{where}\\quad {\\mathbb {V}}_0(\\xi ,t_j):= \\left\\lbrace (t\\eta ,t): t_j^- < t \\le t_j^+, \\, \\eta \\in {\\mathbb {S}}_\\xi ({\\varepsilon }_j) \\right\\rbrace .$ Finally, we define the subset $\\Xi _{{\\mathbb {V}}_0}$ of ${\\mathbb {V}}_0^{d+1}$ by $\\Xi _{{\\mathbb {V}}_0} = \\big \\lbrace (t_j \\xi , t_j): \\, \\xi \\in \\Xi _{\\mathbb {S}}({\\varepsilon }_j), \\, 1\\le j \\le N \\big \\rbrace .$ Proposition 4.17 Let ${\\varepsilon }> 0$ and $N = \\lfloor \\frac{\\pi }{2} {\\varepsilon }^{-1} \\rfloor $ .", "Then $\\Xi _{{\\mathbb {V}}_0}$ is a maximal ${\\varepsilon }$ -separated set of ${\\mathbb {V}}_0^{d+1}$ and $\\lbrace {\\mathbb {V}}_0(\\xi , t_j): \\xi \\in \\Xi _{\\mathbb {S}}({\\varepsilon }_j), \\, 1\\le j \\le N \\rbrace $ is a partition ${\\mathbb {V}}_0^{d+1} = \\bigcup _{j=1}^N \\bigcup _{\\xi \\in \\Xi _{\\mathbb {S}}({\\varepsilon }_j)} {\\mathbb {V}}_0(\\xi ,t_j).$ Moreover, there are positive constants $c_1$ and $c_2$ depending only on $d$ such that ${\\mathsf {c}}\\big ((t_j\\xi ,t_j), c_1 {\\varepsilon }\\big ) \\subset {\\mathbb {V}}_0(\\xi ,t_j) \\subset {\\mathsf {c}}\\big ( (t_j \\xi ,t_j), c_2 {\\varepsilon }\\big ),$ and $c_d^{\\prime }$ and $c_d$ such that $c_d^{\\prime } {\\varepsilon }^{-d} \\le \\# \\Xi _{{\\mathbb {V}}_0} \\le c_d {\\varepsilon }^{-d}.$ Let $(t_j \\xi , t_j)$ and $(t_k \\eta , t_k)$ be two distinct points in $\\Xi _{{\\mathbb {V}}_0}$ .", "If $t_j \\ne t_k$ , then ${\\mathsf {d}}_{{\\mathbb {V}}_0}\\big ((t_j \\xi , t_j), (t_k \\eta , t_k) \\big ) \\ge {\\mathsf {d}}_{[0,1]}(t_j,t_k) = \\frac{1}{2} \|{\\theta }_j - {\\theta }_k\| \\ge \\frac{\\pi }{2N} \\ge {\\varepsilon }.$ If $j = k$ , then $\\xi $ and $\\eta $ are both elements of ${\\mathbb {S}}({\\varepsilon }_j)$ , so that ${\\mathsf {d}}_{{\\mathbb {S}}}(\\xi ,\\eta ) \\ge {\\varepsilon }_j$ .", "Hence, using $\\frac{2}{\\pi }\\phi \\le \\sin \\phi \\le \\phi $ , we deduce from (REF ) that ${\\mathsf {d}}_{{\\mathbb {V}}_0}\\big ((t_j \\xi , t_j), (t_j \\eta , t_j)\\big ) \\ge \\frac{2}{\\pi } \\sqrt{t_j} {\\mathsf {d}}_{{\\mathbb {S}}}(\\xi ,\\eta ) \\ge \\frac{2}{\\pi }\\sqrt{t_j}{\\varepsilon }_j = {\\varepsilon }.$ Hence, $\\Xi _{{\\mathbb {V}}_0}$ is ${\\varepsilon }$ -separated.", "Moreover, since $\\#\\Xi _{\\mathbb {S}}({\\varepsilon }_j) \\sim {\\varepsilon }_j^{-d+1}$ and ${\\varepsilon }_j \\sim {\\varepsilon }/ {\\theta }_j$ , it follows that $\\#\\Xi _{{\\mathbb {V}}_0} = \\sum _{j=1}^N \\#\\Xi _{\\mathbb {S}}({\\varepsilon }_j)\\sim \\sum _{j=1}^N {\\varepsilon }_j^{-d+1} \\sim {\\varepsilon }^{-d+1} \\sum _{j=1}^N {\\theta }_j^{d-1} \\sim {\\varepsilon }^{-d+1} N \\sim {\\varepsilon }^{-d}.$ For the proof of (REF ), we first consider the ball ${\\mathsf {c}}_{[0,1]}(t_j, r) = \\lbrace s: {\\mathsf {d}}_{[0,1]}(s,t_j) \\le r\\rbrace $ on $[0,1]$ .", "For $0< \\delta < \\pi /2$ , it is easy to see that ${\\mathsf {c}}_{[0,1]}(t_j, \\delta /N) \\subset \\lbrace s: (s\\eta , s)\\in {\\mathbb {V}}_0(\\xi ,t_j)\\rbrace \\subset {\\mathsf {c}}_{[0,1]}(t_j, \\pi /N).$ We further choose $\\delta $ so that $4 \\delta (1+\\delta ) < \\frac{1}{2}$ .", "For $s \\in {\\mathsf {c}}_{[0,1]}(t_j, \\delta /N)$ , write $ s= \\sin ^2 \\frac{\\phi }{2}$ , then $\|t_j-s\| = \\frac{1}{2} \|\\cos {\\theta }-\\cos \\phi \| = \\left\|\\sin \\frac{{\\theta }_j-\\phi }{2} \\sin \\frac{{\\theta }_j+\\phi }{2}\\right\| \\le \\frac{\\delta }{N}\\left( 2 \\sqrt{t_j} + \\frac{\\delta }{N}\\right),$ where we have used $\|\\sin \\frac{{\\theta }_j+\\phi }{2}\| \\le 2 \\sin \\frac{{\\theta }_j}{2} + \|\\sin \\frac{{\\theta }_j-\\phi }{2}\| \\le 2 \\sqrt{t_j} + {\\mathsf {d}}_{[0,1]}(s,t_j)$ .", "For $j \\ge 1$ , $N^{-1} = \\frac{2}{(2j-1)\\pi } {\\theta }_j \\le 2 \\sin \\frac{{\\theta }_j}{2} = 2 \\sqrt{t_j}$ , it follows that $\|t_j-s\| \\le 2 \\delta ( 2 + \\delta ) t_j \\le \\tfrac{1}{2} t_j,$ which implies in particular that $\\frac{1}{2}t_j \\le s \\le \\frac{3}{2} t_j$ .", "Furthermore, the same proof shows if $s \\in {\\mathsf {c}}_{[0,1]}(t_j, \\pi /N)$ , then $s \\le c_* t_j$ .", "By definition, there are constants $b_1 > 0$ and $b_2> 0$ such that ${\\mathsf {c}}_{\\mathbb {S}}(\\xi ,b_1{\\varepsilon }_j)\\subset {\\mathbb {S}}_\\xi ({\\varepsilon }_j) \\subset {\\mathsf {c}}_{\\mathbb {S}}(\\xi ,b_2 {\\varepsilon }_j)$ .", "We claim that (REF ) holds for some $c_1 < \\delta $ and some $c_2 > b_2$ .", "Indeed, if $(y,\\eta ) \\subset {\\mathsf {c}}\\big ((t_j\\xi ,t_j), c_1 {\\varepsilon }\\big )$ , then ${\\mathsf {d}}_{[0,1]}(s,t_j) \\le c_1 {\\varepsilon }\\le \\delta /N$ so that $s \\ge t_j/2$ , and $(s t)^{\\frac{1}{4}} {\\mathsf {d}}_{{\\mathbb {S}}}(\\xi ,\\eta ) \\le c c_1 {\\varepsilon }$ by (REF ) so that ${\\mathsf {d}}_{\\mathbb {S}}(\\xi ,\\eta ) \\le 2^{\\frac{1}{4}} c c_1 {\\varepsilon }/\\sqrt{t_j} \\le b_1 {\\varepsilon }_j$ by choosing $c_1$ small.", "This establishes the left-hand side inclusion of (REF ).", "The right-hand side inclusion can be similarly established.", "The proof is completed.", "The above construction establishes the existence of maximal ${\\varepsilon }$ -separated set on the conic surface.", "Since $(\\Omega , {\\mathsf {w}}_{-1,{\\gamma }},{\\mathsf {d}}_{{\\mathbb {V}}_0})$ is a localizable homogeneous space, we can then deduce the Marcinkiewicz-Zygmund inequality on such sets for all doubling weights on the conic surface.", "The weight ${\\mathsf {w}}$ on ${\\mathbb {V}}_0^{d+1}$ is in general a function of both $x$ and $t$ .", "Theorem 4.18 Let ${\\mathsf {w}}$ be a doubling weight on ${\\mathbb {V}}_0^{d+1}$ .", "Let $\\Xi _{{\\mathbb {V}}_0}$ be a maximal $\\frac{\\delta }{n}$ -separated subset of ${\\mathbb {V}}_0^{d+1}$ and $0 < \\delta \\le 1$ .", "For all $0<p< \\infty $ and $f\\in \\Pi _m({\\mathbb {V}}_0^{d+1})$ with $n \\le m \\le c n$ , $\\sum _{z \\in \\Xi _{{\\mathbb {V}}_0}} \\Big ( \\max _{(x,t)\\in {\\mathsf {c}}((z,r), \\frac{\\delta }{n})} \|f(x,t)\|^p \\Big ){\\mathsf {w}}\\!\\left({\\mathsf {c}}((z, r), \\tfrac{\\delta }{n}) \\right) \\le c_{{\\mathsf {w}},p} \\Vert f\\Vert _{p,{\\mathsf {w}}}^p$ where $c_{{\\mathsf {w}},p}$ depends on the doubling constant $L({\\mathsf {w}})$ and on $p$ when $p$ is close to 0.", "For $0 < r < 1$ , there is a $\\delta _r > 0$ such that for $\\delta \\le \\delta _r$ , $r \\le p < \\infty $ and $f \\in \\Pi _n({\\mathbb {V}}_0^{d+1})$ , $\\Vert f\\Vert _{p,{\\mathsf {w}}}^p \\le c_{{\\mathsf {w}},r} \\sum _{z \\in \\Xi }\\Big (\\min _{(x,t)\\in {\\mathsf {c}}\\bigl ((z,r), \\tfrac{\\delta }{n}\\bigr )} \|f(x,t)\|^p\\Big ){\\mathsf {w}}\\bigl ({\\mathsf {c}}((z,r), \\tfrac{\\delta }{n})\\bigr )$ where $c_{{\\mathsf {w}},r}$ depends only on $L({\\mathsf {w}})$ and on $r$ when $r$ is close to 0." ], [ "Cubature rules and localized tight frames", "We now turn our attention to Assertion 4 in Subsection REF and construct fast decaying polynomials on the conic surface.", "Lemma 4.19 Let $d\\ge 2$ .", "For each $(x,t) \\in {\\mathbb {V}}_0^{d+1}$ , there is a polynomial $T_{x,t}$ of degree $n$ that satisfies $T_{x,t}(x,t) =1$ , $T_{x,t}(y,s) \\ge c > 0$ if $(y,s) \\in {\\mathsf {c}}( (x,t), \\frac{\\delta }{n})$ , and for every ${\\kappa }> 0$ , $0 \\le T_{x,t}(y,s) \\le c_{\\kappa }\\left(1+ {\\mathsf {d}}_{{\\mathbb {V}}_0}\\big ((x,t),(y,s)\\big ) \\right)^{-{\\kappa }}, \\quad (y,s) \\in {\\mathbb {V}}_0^{d+1}.$ there is a polynomial $q(t)$ of degree $n$ such that $q(t) T_{x,t}$ is a polynomial of degree $2 n$ in $(x,t)$ variables and $1 \\le q_n(t) \\le c$ .", "Let $r$ be a positive integer such that ${\\kappa }\\le 2r$ .", "For positive integer $n$ , let $m = \\lfloor \\frac{n}{r} \\rfloor +1$ and define $S_n(\\cos {\\theta }) = \\left( \\frac{\\sin (m+\\frac{1}{2})\\frac{{\\theta }}{2}}{(m+\\frac{1}{2}) \\sin \\frac{{\\theta }}{2}} \\right)^{2r}, \\qquad 0 \\le {\\theta }\\le \\pi .$ By considering $m {\\theta }\\ge 1$ and $m {\\theta }\\le 1$ separately if necessary, it follows readily that $ S_n(1) = 1, \\qquad 0 \\le S_n(\\cos {\\theta }) \\le c \\big (1+ n {\\theta }\\big )^{-2r}, \\quad 0 \\le {\\theta }\\le \\pi .$ Moreover, $S_n(z)$ is an even algebraic polynomial of degree at most $2 n$ .", "For a fixed $(x,t) \\in {\\mathbb {V}}_0^{d+1}$ , we define $T_{(x,t)}(y,s) := \\frac{S_n\\Big (\\sqrt{ \\frac{{\\langle }x,y{\\rangle }+ t s}{2}}+ \\sqrt{1-t}\\sqrt{1-s}\\Big ) +S_n\\Big (\\sqrt{ \\frac{{\\langle }x,y{\\rangle }+ t s}{2}}- \\sqrt{1-t}\\sqrt{1-s}\\Big )}{1 + S_n(2 t-1)}.$ As an even polynomial, $S_n(z)$ is a sum of even monomials.", "Since, by binomial formula, $(\\sqrt{a} + \\sqrt{b})^{2k} + (\\sqrt{a} - \\sqrt{b})^{2k}$ is a polynomial of degree $k$ in $a$ and $b$ , it follows that $T_{(x,t)}$ is indeed a polynomial in $(y,s)$ of degree at most $n$ .", "Moreover, it satisfies $T_{(x,t)} (x,t) = 1$ since ${\\langle }x,x{\\rangle }= t^2$ .", "Furthermore, using $\\frac{2}{\\pi } {\\theta }\\le \\sin {\\theta }\\le {\\theta }$ for $0 \\le {\\theta }\\le \\pi /2$ , we see that $S_n$ satisfies $S_n (\\cos {\\theta }) \\ge \\left(\\frac{2}{\\pi }\\right)^{2r}, \\qquad \\hbox{if \\, $0 \\le {\\theta }\\le \\frac{2\\pi }{2m+1}$}.$ Hence, since $0 \\le S_n(2t-1) \\le 1$ , it follows that $T_{(x,t)}(y,s) \\ge \\frac{S_n\\Big (\\sqrt{ \\frac{{\\langle }x,y{\\rangle }+ t s}{2}}+ \\sqrt{1-t}\\sqrt{1-s}\\Big )}{1 + S_n(2 t-1)}\\ge \\frac{1}{2} S_n(\\cos {\\mathsf {d}}_{{\\mathbb {V}}_0}((x,t),(y,s))) \\ge \\frac{1}{2} \\left(\\frac{2}{\\pi }\\right)^{2r}$ for $(y,s) \\in {\\mathsf {c}}((x,t), \\frac{2\\pi }{2m+1})$ .", "Finally, we can use the fast decay (REF ) of $S_n$ with ${\\theta }\\sim \\sin \\frac{{\\theta }}{2} = \\sqrt{\\frac{1}{2} (1-\\cos {\\theta })}$ to derive an upper bound of $T_{(x,t)}$ .", "Indeed, since evidently $1- \\sqrt{\\frac{{\\langle }x,y{\\rangle }+ t s}{2}} + \\sqrt{1-t}\\sqrt{1-s} \\ge 1- \\sqrt{\\frac{{\\langle }x,y{\\rangle }+ t s}{2}} - \\sqrt{1-t}\\sqrt{1-s},$ we obtain then $0 \\le T_{(x,t)}(y,s) \\, & \\le c \\frac{1}{\\left(1+ n \\sqrt{1 - \\sqrt{ \\frac{{\\langle }x,y{\\rangle }+ t s}{2}} - \\sqrt{1-t}\\sqrt{1-s}}\\right)^{2r}}\\\\& = c \\frac{1}{\\left(1+ n \\sqrt{1 -\\cos {\\mathsf {d}}_{{\\mathbb {V}}_0}((x,t),(y,s))}\\right)^{2r}}.$ Hence, by the definition of ${\\mathsf {d}}_{{\\mathbb {V}}_0}$ and $1 - \\cos {\\theta }\\sim {\\theta }^2$ , we obtain $0 \\le T_{(x,t)}(y,s) \\le c \\frac{1}{\\left(1+ n {\\mathsf {d}}_{{\\mathbb {V}}_0} ( (x,t),(y, s))\\right)^{2r}}.$ This completes the proof of (1).", "Finally, $q_n(t) = 1 + S_n(2t-1)$ is a polynomial of degree $n$ and it satisfies $1 \\le q_n(t) \\le c$ for all $t \\in [0,1]$ .", "Moreover, $q_n(t)T_{x,t}$ is a polynomial of degree $2n$ in $(x,t)$ variable.", "This completes the proof.", "From Propositions REF and REF , we have established the following result.", "Corollary 4.20 Let ${\\mathsf {w}}$ be a doubling weight function on ${\\mathbb {V}}_0^{d+1}$ .", "Then $ {\\lambda }_n \\big ({\\mathsf {w}}; (x,t) \\big ) \\le c \\, {\\mathsf {w}}\\left({\\mathsf {c}}\\left((x,t), \\tfrac{1}{n} \\right) \\right).$ Moreover, for ${\\gamma }\\ge -\\frac{1}{2}$ , ${\\lambda }_n \\big ({\\mathsf {w}}_{-1,{\\gamma }}; (x,t) \\big ) \\ge c \\, {\\mathsf {w}}_{-1,{\\gamma }} \\left({\\mathsf {c}}\\left((x,t), \\tfrac{1}{n} \\right) \\right)= c n^{-d} {\\mathsf {w}}_{-1,{\\gamma }}(n; t).$ This implies, in particular, that the cubature rule in Theorem REF holds for all doubling weights on the conic surface.", "Theorem 4.21 Let ${\\mathsf {w}}$ be a doubling weight on ${\\mathbb {V}}_0^{d+1}$ .", "Let $\\Xi $ be a maximum $\\frac{\\delta }{n}$ -separated subset of ${\\mathbb {V}}_0^{d+1}$ .", "There is a $\\delta _0 > 0$ such that for $0 < \\delta < \\delta _0$ there exist positive numbers ${\\lambda }_{z,r}$ , $(z,r) \\in \\Xi $ , so that $\\int _{{\\mathbb {V}}_0^{d+1}} f(x,t) {\\mathsf {w}}(x,t) \\mathrm {d}\\sigma (x,t) = \\sum _{(z,r) \\in \\Xi }{\\lambda }_{z,r} f(z,r), \\qquad \\forall f \\in \\Pi _n({\\mathbb {V}}_0^{d+1}).$ Moreover, ${\\lambda }_{z,r} \\sim {\\mathsf {w}}\\!\\left({\\mathsf {c}}((z,r), \\tfrac{\\delta }{n})\\right)$ for all $(z,r) \\in \\Xi $ .", "We can now construct localized frame on the conic surface.", "Let us recapitulate necessary definitions in the setting of ${\\mathbb {V}}_0^{d+1}$ .", "For $j =0,1,\\ldots ,$ let $\\Xi _j$ be a maximal $ \\frac{\\delta }{2^{j}}$ -separated subset in ${\\mathbb {V}}_0^{d+1}$ , so that $\\int _{{\\mathbb {V}}_0^{d+1}} f(x,t) {\\mathsf {w}}(x,t) \\mathrm {d}\\sigma (x,t) = \\sum _{(z,r) \\in \\Xi _j} {\\lambda }_{(x,r),j} f(z,r),\\qquad f \\in \\Pi _{2^j} ({\\mathbb {V}}_0^{d+1}).$ Let $\\widehat{a}$ be an admissible cut-off function satisfying (REF ).", "Let ${\\mathsf {L}}_{2^j}({\\mathsf {w}})f$ denote the near best approximation operator defined by ${\\mathsf {L}}_{2^j}({\\mathsf {w}}) f (x): = \\int _{{\\mathbb {V}}_0^{d+1}}f(y,s) {\\mathsf {L}}_{2^j}({\\mathsf {w}}; (x,t),(y,s)) {\\mathsf {w}}(y,s) \\mathrm {d}\\sigma (y,s).$ For $j=1,\\ldots ,$ we introduce the notation $F_j({\\mathsf {w}}) = {\\mathsf {L}}_{2^j}({\\mathsf {w}})$ for both the kernel and the operator.", "More precisely, $F_0({\\mathsf {w}}; \\cdot ,\\cdot ) =1$ and for $j \\ge 1$ , $F_j ({\\mathsf {w}}; \\cdot ,\\cdot ) = {\\mathsf {L}}_{2^{j-1}}({\\mathsf {w}}; \\cdot ,\\cdot ) \\quad \\hbox{and}\\quad F_j({\\mathsf {w}}) * f = F_j ({\\mathsf {w}}) * f$ and define the frame elements $\\psi _{(z,r),j}$ for $(z,r) \\in \\Xi _j$ by $\\psi _{(z,r),j}(x,t):= \\sqrt{{\\lambda }_{(z,r),j}} F_j({\\mathsf {w}}; (x,t), (z,r)), \\qquad (x,t) \\in {\\mathbb {V}}_0^{d+1},$ which are well defined for all doubling weight by Theorem REF .", "By Theorem REF , the system $\\Phi =\\lbrace \\psi _{(z,r),j}: (z,r) \\in \\Xi _j, \\, j =1,2,3,\\ldots \\rbrace $ is a tight frame of $L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}})$ .", "Theorem 4.22 Let ${\\mathsf {w}}$ be a doubling weight on ${\\mathbb {V}}_0^{d+1}$ .", "If $f\\in L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}})$ , then $f =\\sum _{j=0}^\\infty \\sum _{(z,r) \\in \\Xi _j}\\langle f, \\psi _{(z,r), j} \\rangle _{\\mathsf {w}}\\psi _{(z,r),j} \\qquad \\mbox{in $L^2({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}})$}$ and $\\Vert f\\Vert _{2, {\\mathsf {w}}} = \\Big (\\sum _{j=0}^\\infty \\sum _{(z,r) \\in \\Xi _j} \|\\langle f, \\psi _{(z,r),j} \\rangle _{\\mathsf {w}}\|^2\\Big )^{1/2}.$ For the weight function ${\\mathsf {w}}_{-1,{\\gamma }}$ , the frame elements are highly localized.", "Theorem 4.23 For ${\\gamma }\\ge -\\frac{1}{2}$ , the frame for ${\\mathsf {w}}_{-1,{\\gamma }}$ is highly localized in the sense that, for every ${\\kappa }>0$ , there exists a constant $c_{\\kappa }>0$ such that $ \|\\psi _{(z,r),j}(x,t)\| \\le c_\\sigma \\frac{2^{j d/2}}{\\sqrt{ {\\mathsf {w}}_{{\\gamma },d}(2^{j}; t)} \\left(1+ 2^j {\\mathsf {d}}_{{\\mathbb {V}}_0}((x,t),(z,r))\\right)^{\\kappa }},\\quad (x,t)\\in {\\mathbb {V}}_0^{d+1}.$ The inequality (REF ) follows from the highly localized estimate of the kernel in Theorem REF and ${\\lambda }_{(z,r),j} \\sim 2^{- jd} {\\mathsf {w}}_{{\\gamma }}(2^j;r)$ which holds for ${\\mathsf {w}}_{-1,{\\gamma }}$ according to Corollary REF .", "It is worth pointing out that the localization of the frame elements is established for ${\\mathsf {w}}_{-1,{\\gamma }}(t) = t^{-1}(1-t)^{{\\gamma }}$ but not for the Lebesgue measure on ${\\mathbb {V}}_0^{d+1}$ ." ], [ "Characterization of best approximation", "For $f\\in L_p({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}})$ , $1 \\le p < \\infty $ , and $f\\in C({\\mathbb {V}}_0^{d+1})$ if $p=\\infty $ , the error of best approximation by polynomials of degree at most $n$ is defined by ${\\mathsf {E}}_n(f)_{p,{\\mathsf {w}}} = \\inf _{g \\in \\Pi _n({\\mathbb {V}}_0^{d+1})} \\Vert f - g \\Vert _{p,{\\mathsf {w}}}, \\quad 1 \\le p \\le \\infty .$ Following the study in Section 3, we can give a characterization for this quantity by the modulus of smoothness defined via the operator ${\\mathsf {S}}_{{\\theta }, {\\mathsf {w}}}$ and the $K$ -functional defined via the differential operator $\\Delta _{0,{\\gamma }}$ for ${\\mathsf {w}}_{-1,{\\gamma }}$ .", "More specifically, for $f\\in L^p({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{-1,{\\gamma }})$ and $r > 0$ , the modulus of smoothness is defined by ${\\omega }_r(f; \\rho )_{p,{\\mathsf {w}}_{-1,{\\gamma }}} = \\sup _{0 \\le {\\theta }\\le \\rho }\\left\\Vert \\left(I - {\\mathsf {S}}_{{\\theta },{\\mathsf {w}}_{-1,{\\gamma }}}\\right)^{r/2} f\\right\\Vert _{p,{\\mathsf {w}}_{-1,{\\gamma }}}, \\quad 1 \\le p \\le \\infty ,$ where the operator ${\\mathsf {S}}_{{\\theta },{\\mathsf {w}}_{-1,{\\gamma }}}$ is defined by, for $n = 0,1,2,\\ldots $ and ${\\lambda }= {\\gamma }+d-1$ , $\\operatorname{proj}_n({\\mathsf {w}}_{-1,{\\gamma }}; {\\mathsf {S}}_{{\\theta },{\\mathsf {w}}_{-1,{\\gamma }}}f) = R_n^{({\\lambda }-\\frac{1}{2}, -\\frac{1}{2})} (\\cos {\\theta }) \\operatorname{proj}_n({\\mathsf {w}}_{-1,{\\gamma }}; f).$ Moreover, in terms of the fractional differential operator $(-\\Delta _{0,{\\gamma }})^{\\frac{r}{2}}$ , the $K$ -functional is defined for a weight ${\\mathsf {w}}$ by ${\\mathsf {K}}_r(f, t)_{p,{\\mathsf {w}}} : = \\inf _{g \\in W_p^r({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}})}\\left\\lbrace \\Vert f-g\\Vert _{p,{\\mathsf {w}}} + t^r\\left\\Vert (-\\Delta _{0,{\\gamma }})^{\\frac{r}{2}}f \\right\\Vert _{p,{\\mathsf {w}}} \\right\\rbrace .$ Both these quantities are well defined as shown in Section 3.", "Since Assertions 1–4 hold for ${\\mathsf {w}}_{-1,{\\gamma }}$ , we only have to verify that the Assertion 5 in Subsection REF holds for the differential operator ${\\mathfrak {D}}_\\varpi = \\Delta _{0,{\\gamma }}$ .", "By Theorem REF , the kernel $L_n^{(r)}(\\varpi )$ in Assertion 5 becomes ${\\mathsf {L}}_n^{(r)}\\big ({\\mathsf {w}}_{-1,{\\gamma }}; (x,t),(y,s)\\big ) =\\sum _{k=0}^\\infty \\widehat{a}\\left(\\frac{k}{n} \\right) (k(k+{\\gamma }+d-1))^{\\frac{r}{2}}{\\mathsf {P}}_k\\big ({\\mathsf {w}}_{-1,{\\gamma }}; (x,t),(y,s)\\big ).$ Lemma 4.24 Let ${\\gamma }\\ge -\\frac{1}{2}$ and ${\\kappa }> 0$ .", "Then, for $r > 0$ and $(x,t), (y,s) \\in {\\mathbb {V}}_0^{d+1}$ , $\| {\\mathsf {L}}_n^{(r)}\\big ({\\mathsf {w}}_{-1,{\\gamma }}; (x,t),(y,s)\\big )\| \\le c_{\\kappa }\\frac{n^{d+r}}{\\sqrt{ {\\mathsf {w}}_{{\\gamma },d} (n; t) }\\sqrt{ {\\mathsf {w}}_{{\\gamma },d} (n; s) }}\\big (1 + n {\\mathsf {d}}_{{\\mathbb {V}}_0}( (x,t), (y,s)) \\big )^{-{\\kappa }}.$ By (REF ), the kernel can be written as ${\\mathsf {L}}_n^{(r)}\\big ({\\mathsf {w}}_{-1,{\\gamma }};(x,t),(y,s)) = b_{{\\gamma },d} \\int _{[-1,1]^2} & L_{n,r}\\left(2 \\zeta (x,t,y,s;v)^2-1\\right) \\\\& \\times (1-v_1^2)^{\\frac{d-2}{2} -1} (1-v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}v,$ in which $L_{n,r}$ is defined by, with ${\\lambda }= {\\gamma }+d -1$ , $L_{n,r}(t) = \\sum _{k=0}^\\infty \\widehat{a}\\left(\\frac{k}{n} \\right) (k(k+{\\gamma }+d-1))^{\\frac{r}{2}}\\frac{P_n^{({\\lambda }-\\frac{1}{2}, -\\frac{1}{2})}(1)P_n^{({\\lambda }-\\frac{1}{2}, -\\frac{1}{2})}(t)}{h^{({\\lambda }-\\frac{1}{2},-\\frac{1}{2})}}.$ Applying (REF ) with $\\eta (t) = \\widehat{a}(t) \\left( t( t + n^{-1} ({\\gamma }+d-1))\\right)^{\\frac{r}{2}}$ and $m=0$ , it follows that $\\left\| L_{n,r}(t) \\right\| \\le c n^{r} \\frac{n^{2{\\lambda }+1}}{(1+n\\sqrt{1-t})^\\ell }.$ Consequently, it follows readily that $\|{\\mathsf {L}}_n^{(r)}\\big ({\\mathsf {w}}_{-1,{\\gamma }};(x,t),(y,s))\| \\le c n^{2{\\lambda }+r+1}\\int _{[-1,1]^2}& \\frac{1}{ \\left(1+n \\sqrt{1-\\zeta (x,t,y,s;v)^2}\\right)^{{\\kappa }+2{\\gamma }+d+1}} \\\\& \\times (1-v_1^2)^{\\frac{d-2}{2} -1} (1-v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}v.$ Apart from $n^r$ , this estimate is the same as that of $L_n({\\mathsf {w}}_{-1,{\\gamma }})$ appeared in the proof of Theorem REF , so that the desired upper bound follows from the estimate there.", "With all assertions hold for the ${\\mathsf {w}}_{-1,{\\gamma }}$ , the characterization of the best approximation by polynomials in Subsection REF holds on the conic surface, which we state below.", "Theorem 4.25 Let $f \\in L^p({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}})$ if $1 \\le p < \\infty $ and $f\\in C({\\mathbb {V}}_0^{d+1})$ if $p = \\infty $ .", "Le $r > 0$ and $n =1,2,\\ldots $ .", "For ${\\mathsf {w}}= {\\mathsf {w}}_{-1,{\\gamma }}$ with ${\\gamma }\\ge -\\frac{1}{2}$ , there holds direct estimate ${\\mathsf {E}}_n(f)_{p,{\\mathsf {w}}_{-1,{\\gamma }}} \\le c {\\mathsf {K}}_r (f;n^{-1})_{p,{\\mathsf {w}}_{-1,{\\gamma }}};$ inverse estimate ${\\mathsf {K}}_r(f;n^{-1})_{p,{\\mathsf {w}}_{-1,{\\gamma }}} \\le c n^{-r} \\sum _{k=0}^n (k+1)^{r-1}{\\mathsf {E}}_k(f)_{p, {\\mathsf {w}}_{-1,{\\gamma }}}.$ Furthermore, the characterization can be given via the modulus of smoothness, since it is equivalent to the $K$ -functional.", "Theorem 4.26 Let ${\\gamma }\\ge -\\frac{1}{2}$ and $f \\in L_p^r({\\mathbb {V}}_0^{d+1}, {\\mathsf {w}}_{-1,{\\gamma }})$ , $1 \\le p \\le \\infty $ .", "Then for $0 < {\\theta }\\le \\pi /2$ and $r >0$ $c_1 {\\mathsf {K}}_r(f; {\\theta })_{p,{\\mathsf {w}}_{-1,{\\gamma }}} \\le {\\omega }_r(f;{\\theta })_{p,{\\mathsf {w}}_{-1,{\\gamma }}} {\\mathsf {K}}_r(f;{\\theta })_{p,{\\mathsf {w}}_{-1,{\\gamma }}}.$" ], [ "Homogeneous space on solid cones", "In this section we work in the setting of homogeneous space on the solid cone ${\\mathbb {V}}^{d+1} = \\lbrace (x,t): \\Vert x\\Vert \\le t, \\, x \\in {\\mathbb {B}}^d, \\, 0 \\le t \\le 1\\rbrace ,$ which is bounded by ${\\mathbb {V}}_0^{d+1}$ and the hyperplane $t =1$ in ${\\mathbb {R}}^{d+1}$ .", "As in the previous section, we shall verify that the framework we developed for approximation and tight frame on homogeneous space is applicable on this domain with respect to the Jacobi weight function $W_{{\\gamma },\\mu }(x,t) = (1-t)^{{\\gamma }} (t^2-\\Vert x\\Vert ^2)^{\\mu -\\frac{1}{2}}$ .", "The structure of the section is parallel to that of the conic surface.", "We again need to define and understand an intrinsic distance and doubling weights on the cone and do so in the first two subsections.", "The orthogonal structure with respect to the Jacobi weight is reviewed in the third subsection.", "The highly localized kernels are established in the fourth subsection, where the Assertions 1 and 3 are verified for $W_{{\\gamma },\\mu }$ for all $\\mu \\ge 0$ but the Assertion 2 is established only for $W_{{\\gamma },0}$ .", "We also provide construction of $ve$ -separated set on the cone and state the Marcinkiewicz-Zygmund inequality in the fifth subsection.", "Assertion 4 is verified in the sixth subsection, which allows us to state the positive cubature rules and the tight frame on the cone.", "While the tight localized frame holds for the Jacobi wight with $\\mu =0$ , the characterization of the best approximation by polynomials works out for all $\\mu \\ge 0$ in the seventh subsection." ], [ "Distance on solid cone", "The distant function ${\\mathsf {d}}_{{\\mathbb {B}}}(\\cdot , \\cdot )$ on the unit ball ${\\mathbb {B}}^d$ can be deduced from regarding ${\\mathbb {B}}^d$ as the projection of the upper hemisphere of ${\\mathbb {S}}^{d+1}$ , or the distant function ${\\mathsf {d}}_{\\mathbb {S}}(\\cdot ,\\cdot )$ applied on the points $X= (x,\\sqrt{1-\\Vert x\\Vert ^2})$ and $Y= (y,\\sqrt{1-\\Vert y\\Vert ^2})$ with $x \\in {\\mathbb {B}}^d$ .", "The same holds for the solid cone ${\\mathbb {V}}^{d+1}$ .", "Definition 5.1 For $(x,t)$ and $(y,s)$ on ${\\mathbb {V}}^{d+1}$ , let $X=\\big (x,\\sqrt{t^2-\\Vert x\\Vert ^2}\\big )$ and $Y=\\big (y,\\sqrt{s^2-\\Vert y\\Vert ^2}\\big )$ .", "Define $ {\\mathsf {d}}_{{\\mathbb {V}}} ((x,t), (y,s)) : = \\arccos \\left(\\sqrt{\\frac{{\\langle }X,Y{\\rangle }+ts}{2}} + \\sqrt{1-t}\\sqrt{1-s}\\right).$ Proposition 5.2 The function ${\\mathsf {d}}_{{\\mathbb {V}}}(\\cdot ,\\cdot )$ defines a distance function on the solid cone ${\\mathbb {V}}^{d+1}$ .", "Since $\\Vert X \\Vert = t$ and $\\Vert Y\\Vert = s$ , both $(X,t)$ and $(Y,s)$ are elements of ${\\mathbb {V}}_0^{d+2}$ .", "Furthermore, ${\\mathsf {d}}_{{\\mathbb {V}}^{d+1}} ((x,t), (y,s)) = {\\mathsf {d}}_{{\\mathbb {V}}_0^{d+2}} ((X,t), (Y,s)),$ from which it is easy to see that ${\\mathsf {d}}_{{\\mathbb {V}}}(\\cdot ,\\cdot )$ is a distance function on ${\\mathbb {V}}^{d+1}$ .", "The distance function ${\\mathsf {d}}_{{\\mathbb {V}}}(\\cdot ,\\cdot )$ is closely related to the distance function ${\\mathsf {d}}_{[0,1]}(\\cdot ,\\cdot )$ of the interval $[0,1]$ and the distance ${\\mathsf {d}}_{{\\mathbb {B}}}(\\cdot ,\\cdot )$ of the unit ball ${\\mathbb {B}}^d$ .", "Proposition 5.3 For $d \\ge 2$ and $(x,t), (y,s) \\in {\\mathbb {V}}^{d+1}$ , write $x = t x^{\\prime }$ and $y = s y^{\\prime }$ with $x^{\\prime }, y^{\\prime } \\in {\\mathbb {B}}^d$ .", "Then $1- \\cos {\\mathsf {d}}_{{\\mathbb {V}}} ((x,t), (y,s)) =1-\\cos {\\mathsf {d}}_{[0,1]}(t,s) + \\sqrt{t}\\sqrt{s} \\left[1-\\cos \\left(\\tfrac{1}{2} {\\mathsf {d}}_{{\\mathbb {B}}}(x^{\\prime },y^{\\prime }) \\right)\\right].$ In particular, $ c_1 {\\mathsf {d}}_{{\\mathbb {V}}} ((x,t), (y,s)) \\le {\\mathsf {d}}_{[0,1]}(t,s) + (t s)^{\\frac{1}{4}} {\\mathsf {d}}_{{\\mathbb {B}}}(x^{\\prime },y^{\\prime }) \\le c_2 {\\mathsf {d}}_{{\\mathbb {V}}} ((x,t), (y,s)).$ With $x= t x^{\\prime }$ , we see that $X = t (x^{\\prime }, \\sqrt{1-\\Vert x^{\\prime }\|^2})$ and $Y = s (y^{\\prime }, \\sqrt{1-\\Vert y^{\\prime }\|^2})$ .", "Hence, it follows as in the case of ${\\mathbb {V}}_0^{d+1}$ that $ {\\mathsf {d}}_{{\\mathbb {V}}} ((x,t), (y,s)) = \\arccos \\left[\\sqrt{t}\\sqrt{s} \\cos \\left(\\tfrac{1}{2} {\\mathsf {d}}_{{\\mathbb {B}}}(x^{\\prime }, y^{\\prime }) \\right)+ \\sqrt{1-t}\\sqrt{1-s} \\right].$ The rest of the proof follows from that of Proposition REF almost verbatim.", "Like the distance function on the surface, we see that the distance on the line segment from the apex to $(x^{\\prime },1)$ on the top boundary of the cone ${\\mathbb {V}}^{d+1}$ is exactly the distance function on $[0,1]$ ; that is, for $x^{\\prime }\\in {\\mathbb {B}}^d$ , ${\\mathsf {d}}_{{\\mathbb {V}}}((x^{\\prime },t), (x^{\\prime },s)) = {\\mathsf {d}}_{[0,1]}(t,s), \\qquad 0 \\le t, s \\le 1.$ The following lemma is an analog of Lemma REF and will be needed in the next subsection.", "Lemma 5.4 For $(x,t), (y,s) \\in {\\mathbb {V}}^{d+1}$ , $\\big \| \\sqrt{t} - \\sqrt{s} \\big \|\\le {\\mathsf {d}}_{{\\mathbb {V}}} ((x,t), (y,s)) \\quad \\hbox{and} \\quad \\big \| \\sqrt{1-t} - \\sqrt{1-s} \\big \| \\le {\\mathsf {d}}_{{\\mathbb {V}}} ((x,t), (y,s)),$ and $\\big \| \\sqrt{t^2-\\Vert x\\Vert ^2} - \\sqrt{s^2-\\Vert y\\Vert ^2} \\big \|\\le c \\max \\lbrace \\sqrt{s},\\sqrt{t} \\rbrace \\, {\\mathsf {d}}_{{\\mathbb {V}}} ((x,t), (y,s)).$ The first two inequalities are immediate consequence of (REF ) and Lemma REF .", "Let $x= t x^{\\prime }$ and $y =s y^{\\prime }$ with $x^{\\prime },y^{\\prime } \\in {\\mathbb {B}}^d$ .", "We know [12] that $ \\left\| \\sqrt{1-\\Vert x^{\\prime }\\Vert ^2} - \\sqrt{1-\\Vert y^{\\prime }\\Vert ^2}\\right\| \\le \\sqrt{2} {\\mathsf {d}}_{\\mathbb {B}}(x^{\\prime },y^{\\prime }).$ Without loss of generality, assuming $t \\ge s$ .", "Then $s \\le \\sqrt{s}\\sqrt{t}$ , so that $\\left\| \\sqrt{t^2-\\Vert x\\Vert ^2} - \\sqrt{s^2-\\Vert y\\Vert ^2}\\right\| & = \\left\| t \\sqrt{1-\\Vert x^{\\prime }\\Vert ^2} - s\\sqrt{1-\\Vert y^{\\prime }\\Vert ^2}\\right\| \\\\& \\le \|t-s\| \\sqrt{1-\\Vert x^{\\prime }\\Vert ^2} + s \\left\| \\sqrt{1-\\Vert x^{\\prime }\\Vert ^2} - \\sqrt{1-\\Vert y^{\\prime }\\Vert ^2}\\right\| \\\\& \\le \\sqrt{2} \\max \\lbrace \\sqrt{s},\\sqrt{t} \\rbrace \\left( \\big \|\\sqrt{t} - \\sqrt{s}\\big \| + (ts)^{\\frac{1}{4}} \\,{\\mathsf {d}}_{\\mathbb {B}}(x^{\\prime },y^{\\prime })\\right),$ which is bounded by $c \\max \\lbrace \\sqrt{s},\\sqrt{t} \\rbrace \\, {\\mathsf {d}}_{{\\mathbb {V}}} ((x,t), (y,s))$ by the first inequality in the statement and the inequality in Proposition REF ." ], [ "A family of doubling weights", "For $d \\ge 1$ , $\\mu \\ge 0$ , ${\\beta }+2\\mu > -d$ and ${\\gamma }> -1$ , we consider the Jacobi weight function defined on the solid cone by $W_{{\\beta },{\\gamma },\\mu }(x,t) := (t^2-\\Vert x\\Vert ^2)^{\\mu -\\frac{1}{2}} t^{\\beta }(1-t)^{\\gamma }, \\quad (x,t) \\in {\\mathbb {V}}^{d+1}.$ Let ${\\mathbf {b}}_{{\\beta },{\\gamma },\\mu }$ be the normalization constant so that ${\\mathbf {b}}_{{\\beta },{\\gamma },\\mu } W_{{\\beta },{\\gamma },\\mu }$ has unit integral on ${\\mathbb {V}}^{d+1}$ .", "Setting $x= t x^{\\prime }$ with $x^{\\prime } \\in {\\mathbb {B}}^d$ , then ${\\mathbf {b}}_{{\\beta },{\\gamma },\\mu } =\\bigg (\\int _0^1 t^{d +{\\beta }+ 2\\mu -1} (1-t)^{\\gamma }\\mathrm {d}t \\int _{{\\mathbb {B}}^d} (1-\\Vert x^{\\prime }\\Vert ^2)^{\\mu -\\frac{1}{2}} \\mathrm {d}x^{\\prime } \\bigg )^{-1}= c_{{\\beta }+2\\mu +d-1, {\\gamma }} b_\\mu ^{\\mathbb {B}},$ where $c_{{\\alpha },{\\beta }}$ is defined in (REF ) and $b_\\mu ^{\\mathbb {B}}$ is the normalization constant of the classical weight function on the unit ball.", "The case ${\\beta }=0$ is of particular interest and will be denoted by $W_{{\\gamma },\\mu }$ ; that is, $W_{{\\gamma },\\mu }(x,t):= W_{0,{\\gamma },\\mu } (x,t) = (t^2-\\Vert x\\Vert ^2)^{\\mu -\\frac{1}{2}}(1-t)^{\\gamma }, \\quad (x,t) \\in {\\mathbb {V}}^{d+1}.$ We show that this is a doubling weight with respect to the distance ${\\mathsf {d}}_{\\mathbb {V}}$ .", "For $r > 0$ and $(x,t)$ in ${\\mathbb {V}}^{d+1}$ , we denote the ball centered at $(x,t)$ with radius $r$ by ${\\mathbf {c}}((x,t), r): = \\big \\lbrace (y,s) \\in {\\mathbb {V}}^{d+1}: {\\mathsf {d}}_{{\\mathbb {V}}} \\big ((x,t),(y,s)\\big )\\le r \\big \\rbrace .$ Proposition 5.5 Let $r > 0$ and $(x,t) \\in {\\mathbb {V}}^{d+1}$ .", "Then for ${\\beta }> -d$ , ${\\gamma }> -1$ and $\\mu \\ge 0$ , $W_{{\\beta },{\\gamma },\\mu }\\big ({\\mathbf {c}}((x,t), r)\\big ) & := \\int _{ {\\mathbf {c}}((x,t), r)} W_{{\\beta },{\\gamma },\\mu }(y,s) \\mathrm {d}y \\mathrm {d}s \\\\& \\sim r^{d+1} (t+ r^2)^{{\\beta }+\\frac{d-1}{2}} (1-t+ r^2)^{{\\gamma }+\\frac{1}{2}} (t^2-\\Vert x\\Vert ^2 + r^2)^\\mu .", "$ In particular, $W_{{\\beta },{\\gamma },\\mu }$ is a doubling weight and the doubling index ${\\alpha }(W_{{\\beta },{\\gamma },\\mu })$ is give by ${\\alpha }(W_{{\\beta },{\\gamma },\\mu }) = 2\\mu +d+1 + 2 \\max \\lbrace 0, {\\beta }+\\frac{d-1}{2}\\rbrace + 2 \\max \\lbrace 0,{\\gamma }+\\frac{1}{2}\\rbrace $ .", "Let $\\tau _r(t,s)$ and ${\\theta }_r(t,s) = \\arccos \\tau _r(t,s)$ be as in the proof of Proposition REF .", "From ${\\mathsf {d}}_{{\\mathbb {V}}}((x,t),(y,s)) \\le r$ , we obtain ${\\mathsf {d}}_{[0,1]}(t, s) \\le r$ and, by (REF ), ${\\mathsf {d}}_{{\\mathbb {B}}} (x^{\\prime }, y^{\\prime }) \\le \\arccos \\left(2 [\\tau _r(t,s)]^2 -1\\right) = \\tfrac{1}{2} \\arccos \\tau _r(t,s) = \\tfrac{1}{2} {\\theta }_r(t,s),$ where ${\\mathsf {d}}_{\\mathbb {B}}(\\cdot ,\\cdot )$ is the distance on the unit ball ${\\mathbb {B}}^d$ .", "Hence, it follows that $& W_{{\\beta },{\\gamma },\\mu }\\big ({\\mathbf {c}}((x,t), r)\\big ) = \\int _{{\\mathsf {d}}_{[0,1]}(t, s)\\le r} s^{d}\\int _{{\\mathsf {d}}_{{\\mathbb {B}}}(x^{\\prime },y^{\\prime }) \\le \\tfrac{1}{2} {\\theta }_r(t,s)} W_{{\\beta },{\\gamma },\\mu }(sy^{\\prime },s) \\mathrm {d}y^{\\prime } \\mathrm {d}s\\\\& \\qquad \\sim \\int _{{\\mathsf {d}}_{[0,1]}(t, s)\\le r} s^{d+{\\beta }+ 2\\mu -1}(1-s)^{\\gamma }\\int _{{\\mathsf {d}}_{{\\mathbb {B}}}(x^{\\prime },y^{\\prime }) \\le \\tfrac{1}{2} {\\theta }_r(t,s)} (1-\\Vert y^{\\prime }\\Vert ^2)^{\\mu -\\frac{1}{2}} \\mathrm {d}y^{\\prime } \\mathrm {d}s$ For $\\mu \\ge 0$ and $0 < \\rho \\le 1$ , it is known [38] or [12] that $\\int _{{\\mathsf {d}}_{{\\mathbb {B}}}(x^{\\prime },y^{\\prime }) \\le \\rho } (1-\\Vert y^{\\prime }\\Vert ^2)^{\\mu -\\frac{1}{2}} \\mathrm {d}y^{\\prime } \\sim (1-\\Vert x^{\\prime }\\Vert ^2 + \\rho ^2)^\\mu \\rho ^d,$ which can also be established by following the approach in Proposition REF .", "Consequently, using ${\\theta }_r(t,s) \\sim \\sqrt{1-\\tau _r(t,s)}$ , we conclude that $W_{{\\beta },{\\gamma },\\mu }\\big ({\\mathbf {c}}((x,t), r)\\big )\\sim \\,& \\int _{{\\mathsf {d}}_{[0,1]}(t, s)\\le r} s^{d+{\\beta }+ 2\\mu -1}(1-s)^{\\gamma }\\\\& \\quad \\times \\big (1-\\Vert x^{\\prime }\\Vert ^2 + 1-\\tau _r(t,s) \\big )^\\mu \\big (1-\\tau _r(t,s)\\big )^{\\frac{d}{2}} \\mathrm {d}s.$ If $ t \\ge 3 r^2$ , then $s \\sim t +r^2$ and, by (REF ), it follows that $s^\\mu \\big (1-\\Vert x^{\\prime }\\Vert ^2 + 1-\\tau _r(t,s) \\big )^\\mu \\sim (t^2-\\Vert x\\Vert ^2 + r^2)^\\mu .$ With this term removed, the integral of the remaining integrand can be estimated by following the estimates of Case 1 and Case 3 of the proof of Proposition REF .", "If $t \\le 3 r^2$ , then $t^2 - \\Vert x\\Vert ^2 + r^2 \\sim r^2$ .", "We use $1-\\Vert x^{\\prime }\\Vert ^2 + 1-\\tau _r(t,s) \\le 2$ for the upper bound and $1-\\tau _r(t,s) \\ge 2/\\pi ^2$ on the subset ${\\mathsf {d}}_{[0,1]}(t,s)\\le r/2$ , proved in the Case 2 of the proof of Proposition REF , to remove the term $ \\big (1-\\Vert x^{\\prime }\\Vert ^2 + 1-\\tau _r(t,s) \\big )^\\mu $ from the integral.", "The rest of the proof then follows from that of the Case 2 of the proof of Proposition REF .", "This completes the proof.", "Corollary 5.6 For $d\\ge 2$ , ${\\beta }> -d$ , ${\\gamma }> -1$ and $\\mu \\ge 0$ , $({\\mathbb {V}}^{d+1}, W_{{\\beta },{\\gamma },\\mu }, {\\mathsf {d}}_{{\\mathbb {V}}})$ is a homogeneous space." ], [ "Orthogonal polynomials on the solid cone", "These polynomials are studied in [53] and they are orthogonal with respect to the inner product ${\\langle }f, g {\\rangle }_{{\\beta },{\\gamma },\\mu } ={\\mathbf {b}}_{{\\beta },{\\gamma },\\mu }\\int _{{\\mathbb {V}}^{d+1}} f(x,t) g(x,t)W_{{\\beta },{\\gamma },\\mu }(x,t) \\mathrm {d}x \\mathrm {d}t.$ Let ${\\mathcal {V}}_n({\\mathbb {V}}^{d+1},W_{{\\beta },{\\gamma },\\mu })$ be the space of orthogonal polynomials of degree $n$ , which has dimension $\\binom{n+d+1}{n}$ .", "An orthogonal basis of ${\\mathcal {V}}_n({\\mathbb {V}}^{d+1}, W_{{\\beta },{\\gamma },\\mu })$ can be given in terms of the Jacobi polynomials and the orthogonal polynomials on the unit ball.", "For $m =0,1,2,\\ldots $ , let $\\lbrace P_{\\mathbf {k}}^m(W_\\mu ): \|{\\mathbf {k}}\| = m, {\\mathbf {k}}\\in {\\mathbb {N}}_0^d\\rbrace $ be an orthonormal basis of ${\\mathcal {V}}_n({\\mathbb {B}}^d, W_\\mu )$ on the unit ball.", "Let $\\alpha : = \\mu +\\frac{{\\beta }+d-1}{2}$ and $ {\\mathbf {J}}_{m,{\\mathbf {k}}}^n(x,t):= P_{n-m}^{(2{\\alpha }+2m, {\\gamma })}(1- 2t) t^m P_{\\mathbf {k}}^m\\left(W_\\mu ; \\frac{x}{t}\\right), \\quad $ Then $\\lbrace {\\mathbf {J}}_{m,{\\mathbf {k}}}^n(x,t): \|{\\mathbf {k}}\| = m, \\, 0 \\le m\\le n\\rbrace $ is an orthogonal basis of ${\\mathcal {V}}_n({\\mathbb {V}}^{d+1},W_{{\\beta },{\\gamma },\\mu })$ and the norm of ${\\mathbf {J}}_{m,{\\mathbf {k}}}^n$ is given by $ {\\mathbf {H}}_{m,n}^{({\\alpha },{\\gamma })}:= {\\langle }{\\mathbf {J}}_{m, {\\mathbf {k}}}^n, {\\mathbf {J}}_{m, {\\mathbf {k}}}^{n} {\\rangle }_{W_{\\mu ,{\\beta },{\\gamma }}}= \\frac{ c_{2{\\alpha },{\\gamma }}}{c_{2{\\alpha }+2m,{\\gamma }} } h_{n-m}^{(2{\\alpha }+2m, {\\gamma })},$ where $c_{{\\alpha },{\\beta }}$ is as in (REF ) and $h_m^{({\\alpha },{\\gamma })}$ is the norm square of the Jacobi polynomial.", "We call ${\\mathbf {J}}_{m, {\\mathbf {k}}}^n$ the Jacobi polynomials on the cone.", "The reproducing kernel of the space ${\\mathcal {V}}_n({\\mathbb {V}}^{d+1}, W_{{\\beta },{\\gamma },\\mu })$ , denoted by ${\\mathbf {P}}_n(W_{{\\beta },{\\gamma },\\mu };\\cdot ,\\cdot )$ , can be written in terms of the above basis, ${\\mathbf {P}}_n\\big (W_{{\\beta },{\\gamma },\\mu }; (x,t),(y,s) \\big ) = \\sum _{m=0}^n \\sum _{\|{\\mathbf {k}}\|=n}\\frac{ {\\mathbf {J}}_{m, {\\mathbf {k}}}^n(x,t) {\\mathbf {J}}_{m, {\\mathbf {k}}}^n(y,s)}{{\\mathbf {H}}_{m,n}^{{\\beta },{\\gamma }}}.$ It is the kernel of the projection $\\operatorname{proj}_n(W_{{\\beta },{\\gamma },\\mu }): L^2({\\mathbb {V}}^{d+1},W_{{\\beta },{\\gamma },\\mu }) \\rightarrow {\\mathcal {V}}_n({\\mathbb {V}}^{d+1}, W_{{\\beta },{\\gamma },\\mu })$ , $\\operatorname{proj}_n(W_{{\\beta },{\\gamma },\\mu };f) = \\int _{{\\mathbb {V}}^{d+1}} f(y,s) {\\mathbf {P}}_n\\big (W_{{\\beta },{\\gamma },\\mu }; \\,\\cdot , (y,s) \\big ) W_{{\\beta },{\\gamma },\\mu }(s) \\mathrm {d}y \\mathrm {d}s.$ In contrast to the conic surface, it is the case ${\\beta }= 0$ that turns out to be the most interesting one.", "For ${\\beta }=0$ , there is a second order differential operator that has orthogonal polynomials as eigenfunctions.", "Recall that $W_{0,{\\gamma },\\mu } = W_{{\\gamma },\\mu }$ .", "Theorem 5.7 Let $\\mu > -\\tfrac{1}{2}$ , ${\\gamma }> -1$ and $n \\in {\\mathbb {N}}_0$ .", "Define the second order differential operator ${\\mathfrak {D}}_{\\mu ,{\\gamma }}$ by ${\\mathfrak {D}}_{{\\gamma },\\mu } : = & \\, t(1-t)\\partial _t^2 + 2 (1-t) {\\langle }x,\\nabla _x {\\rangle }\\partial _t + \\sum _{i=1}^d(t - x_i^2) \\partial _{x_i}^2- 2 \\sum _{i<j } x_i x_j \\partial _{x_i} \\partial _{x_j} \\\\& + (2\\mu +d)\\partial _t - (2\\mu +{\\gamma }+d+1)( {\\langle }x,\\nabla _x{\\rangle }+ t \\partial _t),$ where $\\nabla _x$ and $\\Delta _x$ denote the gradient and the Laplace operator in $x$ -variable.", "Then the polynomials in ${\\mathcal {V}}_n({\\mathbb {V}}^{d+1},W_{{\\gamma },\\mu })$ are eigenfunctions of ${\\mathfrak {D}}_{{\\gamma },\\mu }$ , ${\\mathfrak {D}}_{{\\gamma },\\mu } u = -n (n+2\\mu +{\\gamma }+d) u, \\qquad \\forall u \\in {\\mathcal {V}}_n({\\mathbb {V}}^{d+1},W_{{\\gamma },\\mu }).$ The reproducing kernel enjoys an addition formula that is a mixture of the addition formula for orthogonal polynomials with respect to the classical weight function $W_\\mu $ on the unit ball and the Jacobi polynomials.", "The addition formula is complicated and has the most elegant form when ${\\beta }=0$ .", "Theorem 5.8 Let $d \\ge 2$ .", "For $\\mu \\ge 0$ and ${\\gamma }\\ge -\\frac{1}{2}$ , let ${\\alpha }= \\mu + \\frac{d-1}{2}$ ; then ${\\mathbf {P}}_n \\big (W_{{\\gamma },\\mu }; (x,t), (y,s)\\big ) =\\, &c_{\\mu ,{\\gamma },d} \\int _{[-1,1]^3} Z_{2n}^{2 {\\alpha }+{\\gamma }+1} (\\xi (x, t, y, s; u, v)) \\\\&\\times (1-u^2)^{\\mu -1} (1-v_1^2)^{{\\alpha }- 1}(1-v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}u \\mathrm {d}v, $ where $c_{\\mu ,{\\gamma },d}$ is a constant, so that ${\\mathbf {P}}_0 =1$ and $\\xi (x,t, y,s; u, v) \\in [-1,1]$ is defined by $ \\xi (x,t, y,s; u, v) = &\\, v_1 \\sqrt{\\tfrac{1}{2} \\left(ts+{\\langle }x,y {\\rangle }+ \\sqrt{t^2-\\Vert x\\Vert ^2} \\sqrt{s^2-\\Vert y\\Vert ^2} \\, u \\right)}\\\\& + v_2 \\sqrt{1-t}\\sqrt{1-s}.", "$ When $\\mu = 0$ or ${\\gamma }= -\\frac{1}{2}$ , the identity (REF ) holds under the limit (REF ).", "In particular, ${\\mathbf {P}}_n \\big (W_{{\\gamma }, 0}; & (x,t), (y,s)\\big ) = c_{0,{\\gamma },d} \\\\& \\times \\int _{[-1,1]^2} \\left[ Z_{2n}^{{\\gamma }+d} (\\xi (x, t, y, s; 1, v))+Z_{2n}^{{\\gamma }+d} (\\xi (x, t, y, s; -1, v))\\right] \\\\&\\qquad \\qquad \\quad \\times (1-v_1^2)^{\\frac{d-1}{2} - 1}(1-v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}v. $ As a consequence, the orthogonal structure for the weight function $W_{{\\gamma },\\mu }$ on the cone ${\\mathbb {V}}^{d+1}$ satisfies both characteristic properties specified in Definition REF and Definition REF .", "In the next subsection, we use the addition formula to establish Assertions 1–3 for highly localized kernels." ], [ "Highly localized kernels", "Let $\\widehat{a}$ be an admissible cut-off function.", "For $(x,t)$ , $(y,s) \\in {\\mathbb {V}}^{d+1}$ , the localized kernel ${\\mathbf {L}}_n(W_{{\\gamma },\\mu }; \\cdot ,\\cdot )$ is defined by ${\\mathbf {L}}_n\\left(W_{{\\gamma },\\mu }; (x,t),(y,s)\\right) = \\sum _{j=0}^\\infty \\widehat{a}\\left( \\frac{j}{n} \\right){\\mathbf {P}}_j\\left(W_{{\\gamma },\\mu }; (x,t), (y,s)\\right).$ For ${\\gamma }> -1$ and $\\mu \\ge 0$ , we need the function $n^{d+1}W_{{\\gamma },\\mu }\\big ({\\mathbf {c}}((x,t),n^{-2})\\big )$ , which we denote by $W_{{\\gamma },\\mu } (n; x, t)$ ; that is, $W_{{\\gamma },\\mu } (n; x, t) := \\big (t+n^{-2}\\big )^{\\frac{d-1}{2}} \\big (1-t+n^{-2}\\big )^{{\\gamma }+\\frac{1}{2}}\\big (t^2-\\Vert x\\Vert ^2+n^{-2}\\big )^{\\mu }.$ Theorem 5.9 Let $d\\ge 2$ , ${\\gamma }\\ge -\\frac{1}{2}$ and $\\mu \\ge 0$ .", "Then for any $k>0$ and $(x,t), (y,s) \\in {\\mathbb {V}}^{d+1}$ , $\\left\|{\\mathbf {L}}_n \\left(W_{{\\gamma },\\mu }; (x,t), (y,s)\\right)\\right\|\\le \\frac{c_{\\kappa }n^{d+1}}{\\sqrt{ W_{{\\gamma },\\mu } (n; x, t) }\\sqrt{ W_{{\\gamma },\\mu } (n; y, s) }}\\big (1 + n {\\mathsf {d}}_{{\\mathbb {V}}}( (x,t), (y,s)) \\big )^{-{\\kappa }}.$ The proof follows along the line of the estimate on the surface of the cone, but it will be more involved as can be seen in the lower bound of $1- \\xi (x,t,y,s; u,v)$ at (REF ) below.", "The kernel ${\\mathbf {L}}_n(W_{\\mu ,{\\gamma }})$ can be written in terms of the kernel of the Jacobi polynomials by the addition formula in Theorem REF , which gives, with ${\\alpha }= \\mu +\\frac{d-1}{2}$ and ${\\lambda }= 2\\mu + {\\gamma }+d$ , that ${\\mathbf {L}}_n \\left(W_{{\\gamma },\\mu }; (x,t), (y,s) \\right)=c_{\\mu ,{\\gamma },d} \\int _{[-1,1]^3} & L_n ^{({\\lambda }-\\frac{1}{2},-\\frac{1}{2})}\\big (2 \\xi (x,t,y,s; u, v)^2-1 \\big )\\\\& \\times (1-u^2)^{\\mu -1} (1-v_1^2)^{{\\alpha }-1}(1-v_2^2)^{{\\gamma }-\\frac{1}{2}}\\mathrm {d}u \\mathrm {d}v.$ Then, as in the proof of Theorem REF , we can apply the estimate (REF ) for $L_n^{{\\alpha },{\\beta }}$ with $\\alpha ={\\lambda }-1/2$ , $\\beta = -1/2$ to obtain $\\left\| {\\mathbf {L}}_n \\left(W_{{\\gamma },\\mu }; (x,t), (y,s)\\right) \\right\| \\le c n^{2 {\\lambda }+1} \\int _{[-1,1]^3}& \\frac{1}{ \\left(1+n \\sqrt{1- \\xi (x,t,y,s; u,v)^2} \\right)^{{\\kappa }+5\\mu +3{\\gamma }+\\frac{3d}{2}+1} }\\\\& \\times (1-u^2)^{\\mu -1} (1-v_1^2)^{{\\alpha }-1}(1-v_2^2)^{{\\gamma }-\\frac{1}{2}}\\mathrm {d}u \\mathrm {d}v.$ We first need an lower bound for $\\xi (x,t,y,s; u,v)$ .", "We simplify the notation and write $\\xi (x,t, y,s; u, v) = v_1 \\sqrt{\\eta (x,t,y,s;u)}+ v_2 \\sqrt{1-t}\\sqrt{1-s}$ by introducing the notation $\\eta (x,t,y,s;u) = \\tfrac{1}{2} \\left(ts+{\\langle }x,y {\\rangle }+ \\sqrt{t^2-\\Vert x\\Vert ^2} \\sqrt{s^2-\\Vert y\\Vert ^2} \\, u\\right).$ Let $x = t x^{\\prime }$ and $y = s y^{\\prime }$ , where $x^{\\prime },y^{\\prime }\\in {\\mathbb {B}}^d$ .", "Then by $\|u\|\\le 1$ , it is easy to verify that $0 \\le \\eta (x,t,y,s;u)\\le t s$ and, consequently, $\|\\xi (x,t,y,s;u,v)\| \\le \\sqrt{t s} + \\sqrt{1-s^2}\\sqrt{1-t^2} \\le 1$ .", "Furthermore, we can write $1- \\xi (x,t,y,s;u,v) = \\, & 1 - \\sqrt{\\eta (x,t,y,s;u)} - \\sqrt{1-t}\\sqrt{1-s} \\\\& + (1-v_1) \\sqrt{\\eta (x,t,y,s;u)} + (1-v_2) \\sqrt{1-t}\\sqrt{1-s} \\\\\\ge \\, & 1 - \\sqrt{\\eta (x,t,y,s;1)} - \\sqrt{1-t}\\sqrt{1-s} \\\\= \\,& 1 - \\cos {\\mathsf {d}}_{{\\mathbb {V}}}((x,t),(y,s)) = 2 \\sin ^2 \\frac{ {\\mathsf {d}}_{{\\mathbb {V}}} ((x,t), (y,s)) }{2}\\\\\\ge \\, & \\frac{2}{\\pi ^2} [{\\mathsf {d}}_{{\\mathbb {V}}}((x,t),(y,s))]^2.", "$ As in the case of ${\\mathbb {V}}_0^{d+1}$ , we use this inequality to obtain the estimate $\\left\| {\\mathbf {L}}_n \\left(W_{{\\gamma },\\mu }; (x,t), (y,s)\\right) \\right\| \\le c n^{2 {\\lambda }+1}\\frac{1}{(1+ n {\\mathsf {d}}_{{\\mathbb {V}}}((x,t),(y,s)))^{{\\kappa }+\\mu +{\\gamma }+\\frac{d}{2}}} I(x,t,y,s)$ with the integral $I(x,t, y,s)$ defined by $I(x,t,y,s) = c_{{\\gamma },\\mu } \\int _{[-1,1]^3}\\frac{(1-u^2)^{\\mu -1} (1-v_1^2)^{{\\alpha }-1}(1-v_2^2)^{{\\gamma }-\\frac{1}{2}}}{(1+n\\sqrt{1- \\xi (x,t,y,s;u,v)})^{4\\mu +2{\\gamma }+d+1}} \\mathrm {d}u \\mathrm {d}v,$ where $c_{{\\gamma },\\mu }$ is the normalization constant so that $I(x,t,y,s)=1$ when $n =0$ .", "To complete the proof, we need to show that $\\frac{ I(x,t,y,s)}{(1+ n {\\mathsf {d}}_{{\\mathbb {V}}}((x,t),(y,s)))^{\\mu +{\\gamma }+\\frac{d}{2}}} \\le \\frac{cn^{- (4\\mu +2{\\gamma }+d)}}{\\sqrt{W_{{\\gamma },\\mu }(n;x, t)}\\sqrt{W_{{\\gamma },\\mu }(n; y, s)}}.$ This is a special case of Lemma REF established below.", "Lemma 5.10 Let $d \\ge 2$ and ${\\gamma }\\ge -\\frac{1}{2}$ and $\\mu \\ge 0$ .", "Then for ${\\beta }\\ge 4\\mu +2{\\gamma }+d+1$ , $c_{{\\gamma },\\mu }& \\int _{[-1,1]^3}\\frac{(1-u^2)^{\\mu -1} (1-v_1^2)^{{\\alpha }-1}(1-v_2^2)^{{\\gamma }-\\frac{1}{2}}}{(1+n\\sqrt{1- \\xi (x,t,y,s;u,v)})^{{\\beta }}} \\mathrm {d}u \\mathrm {d}v \\\\&\\le \\frac{cn^{- (4\\mu +2{\\gamma }+d)}}{\\sqrt{W_{{\\gamma },\\mu }(n;x, t)}\\sqrt{W_{{\\gamma },\\mu }(n; y, s)}(1+ n {\\mathsf {d}}_{{\\mathbb {V}}}((x,t),(y,s)))^{{\\beta }- 5\\mu -3{\\gamma }- \\frac{3d}{2}-1}}.", "$ Let $I(x,t,y,s)$ be defined as in the proof of the previous proof.", "Using the lower bound of $\\xi $ in (REF ), we see that the left-hand side of (REF ) is bounded by $\\frac{1}{ (1+ n {\\mathsf {d}}_{{\\mathbb {V}}}((x,t),(y,s)))^{{\\beta }- 4\\mu -2{\\gamma }-d-1} } I(x,t,y,s).$ In order to estimate $I(x,t,y,s)$ we need a refined lower bound for $1- \\xi (x,t,y,s;u,v)$ .", "Let $x = t x^{\\prime }$ and $y = s y^{\\prime }$ with $x^{\\prime }, y^{\\prime } \\in {\\mathbb {B}}^d$ .", "We claim that for $-1 \\le v_1, v_2,u \\le 1$ $ 1 - \\xi (x,t,y,s;u,v) \\, & \\ge \\frac{1}{4} \\sqrt{t s}(1-v_1) + \\sqrt{1-s}\\sqrt{1-t} (1-v_2)\\\\& + \\frac{1}{4} \\sqrt{ts} \\sqrt{1-\\Vert x^{\\prime }\\Vert ^2} \\sqrt{1-\\Vert y^{\\prime }\\Vert ^2}(1-u).", "$ To prove this lower bound, we write $A(x^{\\prime },y^{\\prime }) = \\sqrt{1-\\Vert x^{\\prime }\\Vert ^2} \\sqrt{1-\\Vert y^{\\prime }\\Vert ^2}$ so that the notation can be further simplified as $\\eta (x,t,y,s;u) =\\tfrac{1}{2} t s \\big (1+{\\langle }x^{\\prime },y^{\\prime } {\\rangle }+ A(x^{\\prime },y^{\\prime }) u\\big ).$ We shall use $\\xi $ and $\\eta $ without their variables below.", "Our first step is to write $1- \\xi \\, & = 1- \\sqrt{ts} - \\sqrt{1-t}\\sqrt{1-s} + \\sqrt{ts} - \\sqrt{\\eta } + (1-v_1) \\sqrt{\\eta } + (1-v_2) \\sqrt{1-t}\\sqrt{1-s}\\\\& \\ge \\sqrt{ts} - \\sqrt{\\eta } + (1-v_1) \\sqrt{\\eta } + (1-v_2) \\sqrt{1-t}\\sqrt{1-s}.$ Next, since $\\eta (x,t,y,s;u) \\le t s$ by the Cauchy's inequality, we obtain $\\sqrt{ts} - \\sqrt{\\eta } \\, &= \\frac{ ts - \\eta }{ \\sqrt{ts} + \\sqrt{\\eta }} \\ge \\frac{ ts}{4\\sqrt{ts}} \\left(1 -{\\langle }x^{\\prime },y^{\\prime } {\\rangle }- A(x^{\\prime },y^{\\prime })\\, u\\right)\\\\& = \\frac{ \\sqrt{ts}}{4} \\big (1 -{\\langle }x^{\\prime },y^{\\prime } {\\rangle }- A(x^{\\prime },y^{\\prime }) + (1- u) A(x^{\\prime },y^{\\prime })\\big ).$ Since $1 -{\\langle }x^{\\prime },y^{\\prime } {\\rangle }- A(x^{\\prime },y^{\\prime }) \\ge 0$ and $1 \\ge 1- v_1$ , this leads to $1- \\xi \\, & \\ge B \\sqrt{ts} (1-v_1)+ \\sqrt{1-s}\\sqrt{1-t} (1-v_2)+ \\frac{\\sqrt{ts}}{4} A(x^{\\prime },y^{\\prime }) (1-u),$ where $B$ is given by $B = \\frac{1}{4} \\big (1 -{\\langle }x^{\\prime },y^{\\prime } {\\rangle }- A(x^{\\prime },y^{\\prime })\\big ) + \\sqrt{\\tfrac{1}{2} \\big (1+{\\langle }x^{\\prime },y^{\\prime } {\\rangle }+ A(x^{\\prime },y^{\\prime }) u\\big )},$ which can be rewritten to give $B \\, & = 1- \\frac{1}{4} (1-u) A(x^{\\prime },y^{\\prime }) - \\frac{1}{2} \\left(1- \\sqrt{\\tfrac{1}{2} \\big (1+{\\langle }x^{\\prime },y^{\\prime } {\\rangle }+ A(x^{\\prime },y^{\\prime }) u\\big )}\\right)^2\\\\& \\ge 1- \\frac{1}{4} - \\frac{1}{2} = \\frac{1}{4}.$ This proves (REF ).", "We are now ready to prove (REF ).", "Denote its left-hand side by $J(x,t,u,v)$ .", "Using the inequality (REF ), we obtain $J(x,t,y,s) \\le c \\int _{[0,1]^3}\\frac{(1-u^2)^{\\mu -1} (1-v_1^2)^{\\mu +\\frac{d-1}{2}-1}(1-v_2^2)^{{\\gamma }-\\frac{1}{2}}}{\\left(1+n\\sqrt{B_1 (1-v_1)+B_2(1-v_2)+B_3(1-u)}\\right)^{4\\mu +2{\\gamma }+d+1}} \\mathrm {d}u \\mathrm {d}v,$ where $B_1 = \\frac{1}{4}\\sqrt{t}\\sqrt{s}$ , $B_2 = \\sqrt{1-s}\\sqrt{1-t}$ and $B_3 = \\frac{1}{4} \\sqrt{ts} \\sqrt{1-\\Vert x^{\\prime }\\Vert ^2} \\sqrt{1-\\Vert y^{\\prime }\\Vert ^2}$ .", "We can estimate the integral as in the proof of Theorem REF by applying (REF ), which we now need to apply three times, and we obtain $J(x,t,y,s) \\, &\\le c \\frac{n^{-2\\mu }}{B_3^{\\mu }} \\int _{[0,1]^2}\\frac{ (1-v_1^2)^{\\mu +\\frac{d-1}{2}-1}(1-v_2^2)^{{\\gamma }-\\frac{1}{2}}}{\\left(1+n\\sqrt{B_1 (1-v_1)+B_2(1-v_2)}\\right)^{2\\mu +2{\\gamma }+d}}\\mathrm {d}v \\\\&\\le c \\frac{n^{-2\\mu -2{\\gamma }-1}}{B_3^{\\mu }B_2^{{\\gamma }+\\frac{1}{2}}} \\int _0^1\\frac{ (1-v_1^2)^{\\mu +\\frac{d-1}{2}-1}}{\\left(1+n\\sqrt{B_1 (1-v_1)}\\right)^{2\\mu +d-1}}\\mathrm {d}v_1 \\\\&\\le c \\frac{n^{-4\\mu -2{\\gamma }-d}}{B_3^{\\mu }B_2^{{\\gamma }+\\frac{1}{2}}B_1^{\\mu +\\frac{d-1}{2}}}.$ Combining part of $B_1$ and $B_3$ , it follows from $\\sqrt{t}\\sqrt{s} B_3 = \\frac{1}{4} \\sqrt{t^2-\\Vert x\\Vert ^2} \\sqrt{s^2-\\Vert y\\Vert ^2}$ that $& J(x,t,y,s) \\le c \\frac{n^{-4\\mu -2{\\gamma }-d}}{\\left(\\sqrt{t^2-\\Vert x\\Vert ^2} \\sqrt{s^2-\\Vert y\\Vert ^2}\\right)^{\\mu }\\left(\\sqrt{1-t}\\sqrt{1-s}\\right)^{{\\gamma }+\\frac{1}{2}}\\left(\\sqrt{t}\\sqrt{s}\\right)^{\\frac{d-1}{2}}}\\\\& \\le c \\frac{n^{-4\\mu -2{\\gamma }-d}}{ \\left(\\sqrt{t^2-\\Vert x\\Vert ^2} \\sqrt{s^2-\\Vert y\\Vert ^2}+n^{-2}\\right)^{\\mu }\\left(\\sqrt{1-t}\\sqrt{1-s}+n^{-2}\\right)^{{\\gamma }+\\frac{1}{2}} \\left(\\sqrt{t}\\sqrt{s}+n^{-2}\\right)^{\\frac{d-1}{2}}},$ where the second inequality follows since $I(x,t,y,s) \\le 1$ holds trivially.", "We can now apply (REF ) three times to obtain $J(x,t,y,s) \\le &\\, c \\frac{n^{-4\\mu -2{\\gamma }-d}}{\\sqrt{W_{\\mu ,{\\gamma }}(n; x,t)}\\sqrt{W_{\\mu ,{\\gamma }}(n; y,s)}}\\big (1+ n \\big \|\\sqrt{t^2-\\Vert x\\Vert ^2}-\\sqrt{s^2-\\Vert y\\Vert ^2}\\big \|\\big )^{\\mu } \\\\& \\times \\big (1+ n \\big \|\\sqrt{t}-\\sqrt{s}\\big \|\\big )^{\\frac{d-1}{2}} \\big (1+ n \\big \|\\sqrt{1-t}-\\sqrt{1-s}\\big \|\\big )^{{\\gamma }+\\frac{1}{2}}.$ Finally, by Lemma REF , we conclude $J(x,t,y,s) \\le c \\frac{n^{-4\\mu -2{\\gamma }-d}(1+ n {\\mathsf {d}}_{{\\mathbb {V}}}((x,t),(y,s)))^{\\mu + {\\gamma }+\\frac{d}{2}}}{\\sqrt{W_{{\\gamma },\\mu }(n; x,t)}\\sqrt{W_{{\\gamma },\\mu }(n; y,s)}}$ which is what we need to complete the proof.", "The theorem we just proved verifies Assertion 1 for $W_{{\\gamma },\\mu }$ .", "Our next theorem verifies Assertion 2 but only for $W_{{\\gamma },0}$ .", "Recall that $W_{{\\gamma },0}$ depends only on $t$ , $W_{{\\gamma },0} (n; x, t) = \\big (t+n^{-2}\\big )^{\\frac{d-1}{2}} \\big (1-t+n^{-2}\\big )^{{\\gamma }+\\frac{1}{2}},$ Theorem 5.11 Let $d\\ge 2$ , ${\\gamma }\\ge -\\frac{1}{2}$ .", "For $(x_i,t_i), (y,s) \\in {\\mathbb {V}}^{d+1}$ , and $(x_1,t_1) \\in {\\mathsf {c}}((x_2,t_2), c^* n^{-1})$ with $c^$ small, and any ${\\kappa }> 0$ , $&\\left\|{\\mathbf {L}}_n \\left(W_{{\\gamma },0}; (x_1,t_1), (y,s)\\right)-{\\mathbf {L}}_n \\left(W_{{\\gamma },0}; (x_2,t_2), (y,s)\\right)\\right\| \\\\& \\qquad \\qquad \\le \\frac{c_{\\kappa }n^{d+2} {\\mathsf {d}}_{{\\mathbb {V}}}( (x_1,t_1), (x_2,t_2))}{\\sqrt{ W_{{\\gamma },0} (n; x_2, t_2) }\\sqrt{ W_{{\\gamma },0} (n; y, s) } \\big (1 + n {\\mathsf {d}}_{{\\mathbb {V}}}( (x_2,t_2), (y,s)) \\big )^{{\\kappa }}}.", "$ Let $K$ denote the left-hand side of (REF ).", "As in the proof of Theorem REF , we use the integral expression of ${\\mathbf {L}}_n (W_{{\\gamma },0})$ and (REF ) to obtain $K \\le K_1 + K_{-1}$ , where $K_u & \\le 2 \\int _{[-1,1]^2} \\big \\Vert \\partial L_n^{{\\gamma }+d-\\frac{1}{2},-\\frac{1}{2}} \\big (2(\\cdot )^2-1\\big )\\big \\Vert _{L^\\infty (I_{u,v})}\\big \|\\xi _1(u,v)^2 - \\xi _2(u,v)^2\| \\\\& \\qquad \\qquad \\qquad \\times (1-v_1^2)^{\\frac{d-1}{2}-1} (1-v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}v,$ where $u = 1$ or $-1$ , $\\partial f = f^{\\prime }$ , $\\xi _i(u,v) = \\xi (x_i,t_i,y,s;u,v)$ and $I_{u,v}$ is the interval with end points $\\xi _1(u,v)$ and $\\xi _2(u,v)$ .", "We first consider the case $u =1$ .", "Clearly $\|\\xi _1(1,v)^2 - \\xi _2(1,v)^2\| \\le 2 \|\\xi _1(1,v)- \\xi _2(1,v)\|$ .", "We claim that it has the upper bound $ \|\\xi _1(u,v)- \\xi _2(u,v)\| \\le c\\, {\\mathsf {d}}_{{\\mathbb {V}}}\\big ((x_1,t_1),(x_2,t_2)\\big )\\big [ \\Sigma _1 + \\Sigma _2(v_1) + \\Sigma _3 (v_2)\\big ],$ where $\\Sigma _1 \\, & = {\\mathsf {d}}_{{\\mathbb {V}}}\\big ((x_i,t_i),(y,s)\\big )+{\\mathsf {d}}_{{\\mathbb {V}}}\\big ((x_1,t_1),(x_2,t_2)\\big ), \\\\\\Sigma _2(v_1) \\, & = (1-v_1)\\sqrt{s}, \\\\\\Sigma _3 (v_2)\\, & = (1 - v_2)\\sqrt{1-s}.$ To see this, we follow the notation used in the proof of Lemma REF , writing $\\xi _i = v_1 \\sqrt{\\eta _i (1)} + v_2 \\sqrt{1-t}\\sqrt{1-s}$ , where $\\eta _i(1) =\\eta (x_i,t_i,s_i; 1)$ .", "Using (REF ) and writing $x_i = t_i x_i^{\\prime }$ and $y = s y^{\\prime }$ , we obtain $\\xi _1(1,v)- \\xi _2(1,v) \\, & = \\xi _1(1,\\mathbf {1}) - \\xi _2(1,\\mathbf {1}) \\\\& + (1-v_1) \\big (\\sqrt{\\eta _2(1)} - \\sqrt{\\eta _1(1)}\\big ) \\\\& + (1- v_2) \\big (\\sqrt{1- t_2} - \\sqrt{1-t_1}\\big ) \\sqrt{1-s}, $ where $\\mathbf {1} = (1,1)$ .", "For $u = 1$ , as in Proposition REF , $\\eta _i(1) = t_i s \\cos \\frac{{\\mathsf {d}}_{{\\mathbb {B}}}(x_i^{\\prime }, y^{\\prime })}{2} \\quad \\hbox{and}\\quad \\xi _i (1,\\mathbf {1}) = \\cos {\\mathsf {d}}_{{\\mathbb {V}}}\\big ((x_i,t_i),(y,s)\\big ).$ Substituting these into (REF ), the resulted identity is similar to the corresponding expression in the proof of Theorem REF and can be analogously estimated using Proposition REF and Lemma REF , so that (REF ) follows.", "As in the proof of Theorem REF , it follows from (REF ) with $m =1$ that $&\\big \\Vert \\partial L_n^{{\\lambda }-\\frac{1}{2},-\\frac{1}{2}} \\big (2(\\cdot )^2-1\\big )\\big \\Vert _{L^\\infty (I_{1,v})} \\\\& \\qquad \\qquad \\le c \\left[ \\frac{n^{2 {\\lambda }+ 3}}{\\big (1+n\\sqrt{1-\\xi _1(1,v)^2} \\big )^{{\\kappa }}}+ \\frac{n^{2 {\\lambda }+ 3}}{\\big (1+n\\sqrt{1-\\xi _2(1,v)^2}\\big )^{{\\kappa }}} \\right].$ Hence, $K_1$ is bounded by, with ${\\lambda }= {\\gamma }+d$ , $K_1\\le c\\, {\\mathsf {d}}_{{\\mathbb {V}}}\\big ((x_1,t_1),(x_2,t_2)\\big ) & \\int _{[-1,1]^2}\\left[ \\frac{n^{2 {\\lambda }+ 3}}{\\big (1+n\\sqrt{1-\\xi _1(1,v)^2} \\big )^{{\\kappa }}} + \\frac{n^{2 {\\lambda }+ 3}}{\\big (1+n\\sqrt{1-\\xi _2(1,v)^2}\\big )^{{\\kappa }}} \\right] \\\\& \\times \\big (\\Sigma _1+\\Sigma _2(v_1) + \\Sigma _3(v_2) \\big ) (1-v_1^2)^{\\frac{d-1}{2}-1} (1-v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}v.$ Using the assumption that $(x_1,t_1) \\in {\\mathsf {c}}((x_2,t_2), c^ n^{-1})$ , the integral that contains $\\Sigma _1$ term can be handled as in the proof of Theorem REF , which leads to $\\int _{[-1,1]^2} & \\frac{n^{2 {\\lambda }+ 3}}{\\big (1+n\\sqrt{1-\\xi _i(1,v)^2} \\big )^{{\\kappa }}}\\Sigma _1 (1-v_1^2)^{\\frac{d-1}{2}-1} (1-v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}v \\\\& \\le \\frac{n^{2 {\\lambda }+ 2}}{\\big (1+n{\\mathsf {d}}_{\\mathbb {V}}((x,t), (y,s)) \\big )^{{\\kappa }- 2{\\gamma }-d-2}}\\int _{[-1,1]^2} \\frac{ (1-v_1^2)^{\\frac{d-1}{2}-1} (1-v_2^2)^{{\\gamma }-\\frac{1}{2}}}{\\big (1+n\\sqrt{1-\\xi _i(1,v)^2} \\big )^{2{\\gamma }+d+2}}\\mathrm {d}v.$ Now, the inequality (REF ) shows that $1-\\xi _i(1,v) \\ge \\frac{1}{4} \\sqrt{t_i s}(1- v_1)+\\sqrt{1-t_i}\\sqrt{1-s}(1-v_2).$ Hence, applying Corollary REF with $d-1$ replaced by $d$ and Lemma REF , we conclude that the integral in the right-hand side is bound by $\\frac{c\\, n^{d+2}}{\\sqrt{ W_{{\\gamma },0} (n; x_i, t_i) }\\sqrt{ W_{{\\gamma },0} (n; y, s)}\\big (1 + n {\\mathsf {d}}_{{\\mathbb {V}}}( (x_i,t_i), (y,s)) \\big )^{{\\kappa }({\\gamma },d)}},$ where ${\\kappa }({\\gamma },d) = {\\kappa }- 3 {\\gamma }- \\frac{3d+4}{2}$ , since $W_{{\\gamma },0} (n; x, t) = {\\mathsf {w}}_{{\\gamma }, d+1}(n;t)$ .", "Putting together, we see that the integral containing $\\Sigma _1$ has the desired upper bound.", "The two integrals containing $\\Sigma _2(v_1)$ and $\\Sigma _3(v_2)$ can be estimated similarly and more straightforwardly by using Corollary REF .", "Moreover, the estimate is parallel to the two corresponding terms in the proof of Theorem REF and can be carried out similarly.", "This completes the proof of the case $u =1$ .", "For $u =-1$ , we observe that if we define $Y_* = (y, -\\sqrt{s^2-\\Vert y\\Vert ^2})$ and ${\\mathsf {d}}_((x,t), (y,s)) = \\arccos \\left( \\sqrt{\\frac{{\\langle }X, Y_ {\\rangle }+ t s}{2} } + \\sqrt{1-t}\\sqrt{1-s} \\right),$ then for $u =-1$ and $v = \\mathbf {1} = (1,1)$ , $\\xi (x,t,y,s;-1,\\mathbf {1}) = \\cos {\\mathsf {d}}_((x,t), (y,s)).$ Since $(Y_, s) \\in {\\mathbb {V}}_0^{d+2}$ and ${\\mathsf {d}}_((x,t), (y,s)) = {\\mathsf {d}}_{{\\mathbb {V}}_0^{d+2}}( (X,t), (Y_,s))$ , it follows that ${\\mathsf {d}}_$ is also a distance function and satisfies, in particular, the triangle inequality.", "As a result, we can repeat the proof for $u=1$ to estimate $K_{-1}$ , so that $K_{-1}$ is bounded by the right-hand side of (REF ) with $(1+n {\\mathsf {d}}_((x,t),(y,s)))^{\\kappa }$ in place of $(1+n {\\mathsf {d}}_{{\\mathbb {V}}}((x,t),(y,s)))^{\\kappa }$ in the denominator.", "Since ${\\langle }X,Y{\\rangle }\\ge {\\langle }X,Y_{\\rangle }$ , $\\cos {\\mathsf {d}}_{{\\mathbb {V}}} ((x,t), (y,s)) \\ge \\cos {\\mathsf {d}}_((x,t),(y,s))$ and, consequently, ${\\mathsf {d}}_{\\mathbb {V}}((x,t), (y,s)) \\le {\\mathsf {d}}_* ((x,t), (y,s))$ .", "Hence, the upper bound of $K_{-1}$ is bounded by the right-hand side of (REF ).", "This completes the proof.", "Remark 5.1 It is not clear if this theorem also holds for $W_{{\\gamma },\\mu }$ when $\\mu > 0$ .", "The main task lies in establishing an analog of (REF ), perhaps with a multiple of $1-u$ as one more term.", "This turns out to be elusive, not because of lack of trying, and there appears to be a real obstacle for $(x_i,t_i)$ around the apex of the cone.", "Our next lemma is an analog of Lemma REF , which establishes Assertion 3 when $p=1$ for the weight function $W_{{\\gamma },\\mu }$ on the solid cone.", "Lemma 5.12 Let $d\\ge 2$ , ${\\gamma }> -1$ and $\\mu > -\\frac{1}{2}$ .", "For $0 < p < \\infty $ , assume ${\\kappa }> \\frac{2d+2}{p} + ({\\gamma }+\\mu +\\frac{d}{2}) \|\\frac{1}{p}-\\frac{1}{2}\|$ .", "Then for $(x,t) \\in {\\mathbb {V}}^{d+1}$ , $\\int _{{\\mathbb {V}}^{d+1}} \\frac{ W_{{\\gamma },\\mu }(s)}{ W_{{\\gamma },\\mu } (n; s)^{\\frac{p}{2}}\\big (1 + n {\\mathsf {d}}_{{\\mathbb {V}}}( (x,t), (y,s)) \\big )^{{\\kappa }p}} \\mathrm {d}y \\mathrm {d}s\\le c n^{-d-1}\\, W_{{\\gamma },\\mu } (n; t)^{1-\\frac{p}{2}}.$ Let $J_{p}$ denote the left-hand side of (REF ).", "Using the doubling property of $W_{{\\beta },{\\gamma }}$ and by Lemma REF , it is sufficient to estimate $J_{2,{\\kappa }}$ .", "Setting $y = s y^{\\prime }$ , $y^{\\prime } \\in {\\mathbb {B}}^d$ , we obtain $J_2 = \\int _0^1 s^d \\int _{{\\mathbb {B}}^d} \\frac{ W_{{\\gamma },\\mu }(s y^{\\prime },s)}{W_{{\\gamma },\\mu } (n; s y^{\\prime }, s)(1 + n {\\mathsf {d}}_{{\\mathbb {V}}}( (x,t), (s y^{\\prime },s)) )^{2{\\kappa }}} \\mathrm {d}y^{\\prime } \\mathrm {d}s .$ Using ${\\mathsf {w}}_{{\\gamma },d} (n; t)$ defined in (REF ), it follows readily that $\\frac{W_{{\\gamma },\\mu }(y,s)}{W_{{\\gamma },\\mu }(n; y,s)} = c \\frac{s {\\mathsf {w}}_{-1,{\\gamma }}(s)}{(s^2-\\Vert y\\Vert ^2+n^{-2})^{\\frac{1}{2}} {\\mathsf {w}}_{{\\gamma },d+1}(n;s)}\\le c \\frac{ {\\mathsf {w}}_{-1,{\\gamma }} (n; s)}{{\\mathsf {w}}_{{\\gamma },d+1}(n;s) \\sqrt{1-\\Vert y^{\\prime }\\Vert ^2}},$ which leads to $J_2 \\le c \\int _0^1 s^d \\int _{{\\mathbb {B}}^d} \\frac{{\\mathsf {w}}_{-1,{\\gamma }} (n; s)}{{\\mathsf {w}}_{{\\gamma },d+1}(n;s) (1 + n {\\mathsf {d}}_{{\\mathbb {V}}}( (x,t), (s y^{\\prime },s)) )^{2{\\kappa }}}\\frac{\\mathrm {d}y^{\\prime }}{\\sqrt{1-\\Vert y^{\\prime }\\Vert ^2} }\\mathrm {d}s.$ Setting $x= t x^{\\prime }$ , $X = (x^{\\prime }, \\sqrt{1-\\Vert x^{\\prime }\\Vert ^2})$ and $Y = (y^{\\prime }, \\sqrt{1-\\Vert y^{\\prime }\\Vert ^2})$ , so that ${\\mathsf {d}}_{{\\mathbb {B}}^d}(x^{\\prime },y^{\\prime })= {\\mathsf {d}}_{{\\mathbb {S}}^d}(X,Y)$ , we use the identity $\\int _{{\\mathbb {B}}^d} g\\left(y^{\\prime }, \\sqrt{1-\\Vert y^{\\prime }\\Vert ^2} \\right) \\frac{d y^{\\prime }}{\\sqrt{1-\\Vert y^{\\prime }\\Vert ^2}} = \\int _{{\\mathbb {S}}_+^d} g(y) \\mathrm {d}\\sigma (y),$ where ${\\mathbb {S}}_+^d$ denotes the upper hemisphere of ${\\mathbb {S}}^d$ , which allows us to follow the proof of Proposition REF to obtain $J \\, & \\le c \\int _0^1 \\int _{-1}^1 \\frac{s^{d} {\\mathsf {w}}_{-1,{\\gamma }}(s)(1-u^2)^{\\frac{d-2}{2}} }{{\\mathsf {w}}_{{\\gamma },d+1}(n; s) \\left(1 + n \\arccos \\left(\\sqrt{t s} \\sqrt{\\frac{1+u}{2}}+ \\sqrt{1-t}\\sqrt{1-s}\\right) \\right)^{2 {\\kappa }}} \\mathrm {d}u\\mathrm {d}s.$ Comparing with the proof of Proposition REF , we see that the above integral with $d$ replaced by $d-1$ has already appeared in that proof, so that it is bounded by $c n^{-d-1}$ as the proof of Proposition REF shows.", "The proof is completed.", "Proposition 5.13 Let $d\\ge 2$ , $\\mu \\ge 0$ and ${\\gamma }\\ge -\\frac{1}{2}$ .", "For $0<p<\\infty $ and $(x,t) \\in {\\mathbb {V}}^{d+1}$ , $\\int _{{\\mathbb {V}}^{d+1}} \\left\| {\\mathbf {L}}_n\\big (W_{{\\gamma },\\mu };(x,t),(y,s)\\big ) \\right\| ^p W_{{\\gamma },\\mu }(y,s) \\mathrm {d}y \\mathrm {d}s\\le c \\left(\\frac{n^{d+1}}{W_{{\\gamma },\\mu }(n;x,t)}\\right)^{p-1}.$ This follows immediately from applying Lemma REF on (i) of Theorem REF .", "Corollary 5.14 For $d\\ge 2$ and ${\\gamma }\\ge -\\frac{1}{2}$ , the space $({\\mathbb {V}}^{d+1}, W_{{\\gamma },0}, {\\mathsf {d}}_{{\\mathbb {V}}})$ is a localizable homogeneous space." ], [ "Maximal ${\\varepsilon }$ -separated sets and MZ inequality", "We provide a construction of maximal ${\\varepsilon }$ -separated sets on the solid cone as defined in Definition REF .", "Our construction follows the approach in Subsection REF .", "Instead of starting with ${\\varepsilon }$ -separated sets on the unit sphere, we now need such sets on the unit ball ${\\mathbb {B}}^d$ .", "We adopt the following notation.", "For ${\\varepsilon }> 0$ , we denote by $\\Xi _{{\\mathbb {B}}}({\\varepsilon })$ a maximal ${\\varepsilon }$ -separated set on the unit ball ${\\mathbb {B}}^d$ and we let ${\\mathbb {B}}_u({\\varepsilon })$ be the subsets in ${\\mathbb {B}}^d$ so that the collection $\\lbrace {\\mathbb {B}}_u({\\varepsilon }): u \\in \\Xi _{\\mathbb {B}}({\\varepsilon })\\rbrace $ is a partition of ${\\mathbb {B}}^d$ , and we assume ${\\mathbf {c}}_{{\\mathbb {B}}}(u, c_1 {\\varepsilon }) \\subset {\\mathbb {B}}_u({\\varepsilon }) \\subset {\\mathbf {c}}_{{\\mathbb {B}}}(u, c_2 {\\varepsilon }), \\qquad u \\in \\Xi _{{\\mathbb {B}}}({\\varepsilon }),$ where ${\\mathbf {c}}_{{\\mathbb {B}}}(u,{\\varepsilon })$ denotes the ball centered at $u$ with radius ${\\varepsilon }$ in ${\\mathbb {B}}^d$ , $c_1$ and $c_2$ depend only on $d$ .", "It is known (see, for example, [38]) that such a $\\Xi _{\\mathbb {B}}({\\varepsilon })$ exists for all ${\\varepsilon }> 0$ and its cardinality satisfies $c_d^{\\prime } {\\varepsilon }^{-d} \\le \\# \\Xi _{{\\mathbb {B}}}({\\varepsilon }) \\le c_d {\\varepsilon }^{-d}.$ On the solid cone ${\\mathbb {V}}^{d+1}$ we denote by $\\Xi _{{\\mathbb {V}}} = \\Xi _{{\\mathbb {V}}}({\\varepsilon })$ a maximum ${\\varepsilon }$ -separated set and denote by $\\lbrace {\\mathbb {V}}(u,t): (tu,t) \\in \\Xi _{{\\mathbb {V}}}\\rbrace $ a partition of ${\\mathbb {V}}^{d+1}$ .", "We give a construction of these sets.", "Let ${\\varepsilon }> 0$ and let $N = \\lfloor \\frac{\\pi }{2}{\\varepsilon }^{-1} \\rfloor $ .", "We define $t_j = \\sin ^2 \\frac{{\\theta }_j}{2}$ , $t_j^+$ and $t_j^-$ , $1 \\le j \\le N$ , as in Subsection REF , so that ${\\mathbb {V}}^{d+1} = \\bigcup _{j=1}^N {\\mathbb {V}}^{(j)}, \\quad \\hbox{where}\\quad {\\mathbb {V}}^{(j)}:=\\left\\lbrace (x,t) \\in {\\mathbb {V}}^{d+1}: t_j^- < t \\le t_j^+ \\right\\rbrace .$ Let ${\\varepsilon }_j := (2 \\sqrt{t_j})^{-1} \\pi {\\varepsilon }$ .", "Then $\\Xi _{\\mathbb {B}}({\\varepsilon }_j)$ is the maximal ${\\varepsilon }_j$ -separated set of ${\\mathbb {B}}^d$ such that, for each $j \\ge 1$ , $\\lbrace {\\mathbb {B}}_u({\\varepsilon }_j): u \\in \\Xi _{\\mathbb {B}}({\\varepsilon }_j)\\rbrace $ is a partition of ${\\mathbb {B}}^d$ and $\\# \\Xi _{\\mathbb {B}}({\\varepsilon }_j) \\sim {\\varepsilon }_j^{-d}.$ For each $j =1,\\ldots , N$ , we decompose ${\\mathbb {V}}^{(j)}$ by ${\\mathbb {V}}^{(j)} = \\bigcup _{u \\in \\Xi _{\\mathbb {B}}({\\varepsilon }_j)} {\\mathbb {V}}(u,t_j), \\quad \\hbox{where}\\quad {\\mathbb {V}}(u,t_j):= \\left\\lbrace (t v,t): t_j^- < t \\le t_j^+, \\, v \\in {\\mathbb {B}}_u({\\varepsilon }_j) \\right\\rbrace .$ Finally, we define the set $\\Xi _{{\\mathbb {V}}}$ of ${\\mathbb {V}}^{d+1}$ by $\\Xi _{{\\mathbb {V}}} = \\big \\lbrace (t_j u, t_j): \\, u \\in \\Xi _{\\mathbb {B}}({\\varepsilon }_j), \\, 1\\le j \\le N \\big \\rbrace .$ Proposition 5.15 Let ${\\varepsilon }> 0$ and $N = \\lfloor \\frac{\\pi }{2} {\\varepsilon }^{-1} \\rfloor $ .", "Then $\\Xi _{{\\mathbb {V}}}$ is a maximal ${\\varepsilon }$ -separated set of ${\\mathbb {V}}^{d+1}$ and $\\lbrace {\\mathbb {V}}(t_j u, t_j): u \\in \\Xi _{\\mathbb {B}}({\\varepsilon }_j), \\, 1\\le j \\le N \\rbrace $ is a partition ${\\mathbb {V}}^{d+1} = \\bigcup _{j=1}^N \\bigcup _{u \\in \\Xi _{\\mathbb {B}}({\\varepsilon }_j)} {\\mathbb {V}}(u,t_j).$ Moreover, there are positive constants $c_1$ and $c_2$ depending only on $d$ such that ${\\mathbf {c}}\\big ((t_j u,t_j), c_1 {\\varepsilon }\\big ) \\subset {\\mathbb {V}}(u,t_j) \\subset {\\mathbf {c}}\\big ( (t_j u,t_j), c_2 {\\varepsilon }\\big ), \\qquad (t_j u, t_j) \\in \\Xi _{\\mathbb {V}},$ and $c_d^{\\prime }$ and $c_d$ depending only on $d$ such that $c_d^{\\prime } {\\varepsilon }^{-d-1} \\le \\# \\Xi _{{\\mathbb {V}}} \\le c_d {\\varepsilon }^{-d-1}.$ The proof is parallel to that of Proposition REF .", "In fact, it can be carried out almost verbatim under obvious modifications such as replacing ${\\mathsf {d}}_{\\mathbb {S}}$ by ${\\mathsf {d}}_{\\mathbb {B}}$ and ${\\mathbb {V}}_0$ by ${\\mathbb {V}}$ .", "We shall omit it.", "Even though we only established Assertion 2 for $W_{{\\gamma },0}$ for ${\\gamma }\\ge 0$ , it is sufficient for applying Theorem REF to conclude that the Marcinkiewicz-Zygmund inequality on a maximal ${\\varepsilon }$ -separated set holds for any doubling weight on ${\\mathbb {V}}^{d+1}$ .", "Theorem 5.16 Let $W$ be a doubling weight on ${\\mathbb {V}}^{d+1}$ .", "Let $\\Xi _{{\\mathbb {V}}}$ be a maximal $\\frac{\\delta }{n}$ -separated subset of ${\\mathbb {V}}^{d+1}$ and $0 < \\delta \\le 1$ .", "For all $0<p< \\infty $ and $f\\in \\Pi _m^{d+1}$ with $n \\le m \\le c n$ , $\\sum _{z \\in \\Xi _{{\\mathbb {V}}}} \\Big ( \\max _{(x,t)\\in {\\mathbf {c}}((z,r), \\frac{\\delta }{n})} \|f(x,t)\|^p \\Big )W\\!\\left({\\mathbf {c}}((z, r), \\tfrac{\\delta }{n}) \\right) \\le c_{W,p} \\Vert f\\Vert _{p,W}^p$ where $c_{W,p}$ depends on $p$ when $p$ is close to 0 and on the doubling constant ${\\alpha }(W)$ .", "For $0 < r < 1$ , there is a $\\delta _r > 0$ such that for $\\delta \\le \\delta _r$ , $r \\le p < \\infty $ and $f \\in \\Pi _n^{d+1}$ , $\\Vert f\\Vert _{p,W}^p \\le c_{W,r} \\sum _{z \\in \\Xi }\\Big (\\min _{(x,t)\\in {\\mathbf {c}}\\bigl ((z,r), \\tfrac{\\delta }{n}\\bigr )} \|f(x,t)\|^p\\Big )W\\bigl ({\\mathbf {c}}((z,r), \\tfrac{\\delta }{n})\\bigr )$ where $c_{W,r}$ depends only on $L(W)$ and on $r$ when $r$ is close to 0." ], [ "Cubature rules and localized tight frames", "We first verify Assertion 4 in Subsection REF by constructing fast decaying polynomials on the solid cone.", "Lemma 5.17 Let $d\\ge 2$ .", "For each $(x,t) \\in {\\mathbb {V}}^{d+1}$ , there is a polynomial ${\\mathcal {T}}_{x,t}$ of degree $n$ that satisfies ${\\mathcal {T}}_{x,t}(x,t) =1$ , ${\\mathcal {T}}_{x,t}(y,s) \\ge c > 0$ if $(y,s) \\in {\\mathbf {c}}( (x,t), \\frac{\\delta }{n})$ , and for every ${\\kappa }> 0$ , $0 \\le {\\mathcal {T}}_{x,t}(y,s) \\le c_{\\kappa }\\left(1+ {\\mathsf {d}}_{{\\mathbb {V}}}\\big ((x,t),(y,s)\\big ) \\right)^{-{\\kappa }}, \\quad (y,s) \\in {\\mathbb {V}}^{d+1}.$ there is a polynomial $q$ of degree $n$ such that $q(x,t) {\\mathcal {T}}_{x,t}$ is a polynomial of degree $3 n$ in $(x,t)$ and $1 \\le q(x,t) \\le c$ .", "For $(x,t), (y,s) \\in {\\mathbb {V}}^{d+1}$ , we introduce the notation $X = (x, \\sqrt{t^2-\\Vert x\\Vert ^2})$ and $Y = (y, \\sqrt{s^2-\\Vert y\\Vert ^2})$ .", "Moreover, we also denote $Y_* = (y, - \\sqrt{t^2-\\Vert y\\Vert ^2})$ .", "Then $(X,t), (Y,s)$ and $(Y_, s)$ are all elements of ${\\mathbb {V}}_0^{d+2}$ .", "Let $T_{(X,t)}$ denote the polynomial of degree $n$ on ${\\mathbb {V}}_0^{d+2}$ defined in Lemma REF .", "We now define ${\\mathcal {T}}_{x,t}(y,s): = \\frac{T_{(X,t)} (Y,s)+ T_{(X,t)} (Y_ ,s)}{1 + T_{(X,t)}(X_,t)}.$ Since $T_{(X,t)}(X,t) =1$ , it follows that ${\\mathcal {T}}_{(x,t)}(x,t) = 1$ .", "Since $\\sqrt{\\frac{{\\langle }X_, X{\\rangle }+t^2}{2}} = \\Vert x\\Vert $ , we have $T_{(X,t)}(X_,t) = \\frac{S_n(\\Vert x\\Vert + (1-t))+ S_n(\\Vert x\\Vert -(1-t))}{1 + S_n(2 t-1)}.$ By $0 \\le S_n(t) \\le c$ , we see that $0 \\le T_{(X,t)}(X_,s) \\le c$ .", "In particular, it follows that ${\\mathcal {T}}_{x,t}(y,s) \\ge c \\,T_{(X,t)} (Y,s) \\ge c > 0, \\qquad (y,s) \\in {\\mathbf {c}}\\big ((x,t), \\tfrac{\\delta }{n}\\big ),$ since ${\\mathsf {d}}_{{\\mathbb {V}}}((x,t),(y,s)) = {\\mathsf {d}}_{{\\mathbb {V}}_0^{d+2}}((X,t),(Y,s))$ .", "Furthermore, since $\\cos {\\mathsf {d}}_{{\\mathbb {V}}_0} ((X,t), (Y,s)) \\ge \\cos {\\mathsf {d}}_{{\\mathbb {V}}_0} ((X,t), (Y_,s))$ , we obtain ${\\mathsf {d}}_{{\\mathbb {V}}_0} ((X,t), (Y_,s)) \\ge {\\mathsf {d}}_{{\\mathbb {V}}_0} ((X,t), (Y,s)) = {\\mathsf {d}}_{{\\mathbb {V}}}((x,t),(y,s)).$ Hence, using the estimate of $T_{(X,t)}$ in Lemma REF , we conclude that $0 \\le {\\mathcal {T}}_{(x,t)}(y,s) \\,& \\le c\\left[\\big (1+n {\\mathsf {d}}_{{\\mathbb {V}}_0}((X,t),(Y,s) ) \\big )^{-{\\kappa }}+ \\big (1+n {\\mathsf {d}}_{{\\mathbb {V}}_0}((X,t),(Y_,s)) \\big )^{-{\\kappa }}\\right] \\\\\\,& \\le c \\big (1+n {\\mathsf {d}}_{{\\mathbb {V}}}((x,t),(y,s)) \\big )^{-{\\kappa }}.$ Finally, let $q(x,t) = (1+S_n(2t-1)) T_{(X,t)}(X_,t)$ .", "Then $q(x,t)$ is a polynomial of degree at most $2n$ , so that $q(x,t) {\\mathcal {T}}_{x,t}$ is a polynomial of degree at most $3 n$ and $1 \\le q(x,t) \\le c$ .", "This completes the proof.", "By Propositions REF and REF and using Theorem REF , we have established the following result on the Christoffel function ${\\lambda }_n(W)$ .", "Corollary 5.18 Let $W$ be a doubling weight function on ${\\mathbb {V}}^{d+1}$ .", "Then $ \\lambda _n \\big (W; (x,t) \\big ) \\le c \\, W\\!\\left({\\mathbf {c}}\\left((x,t), \\tfrac{1}{n} \\right) \\right).$ Moreover, for ${\\gamma }\\ge -\\frac{1}{2}$ and $\\mu \\ge 0$ , $\\lambda _n \\big (W_{{\\gamma },\\mu }; (x,t) \\big ) \\ge c \\,W_{{\\gamma },\\mu } \\left({\\mathbf {c}}\\left((x,t), \\tfrac{1}{n} \\right) \\right)= c n^{-d-1} W_{{\\gamma },\\mu }(n; t).$ Since the lower bound of the Christoffel function holds for all doubling weight, we see that the cubature rules in Theorem REF holds for all doubling weights on the solid cone.", "Theorem 5.19 Let $W$ be a doubling weight on ${\\mathbb {V}}^{d+1}$ .", "Let $\\Xi $ be a maximum $\\frac{\\delta }{n}$ -separated subset of ${\\mathbb {V}}^{d+1}$ .", "There is a $\\delta _0 > 0$ such that for $0 < \\delta < \\delta _0$ there exist positive numbers ${\\lambda }_{z,r}$ , $(z,r) \\in \\Xi $ , so that $\\int _{{\\mathbb {V}}^{d+1}} f(x) W(x,t) \\mathrm {d}x \\mathrm {d}t = \\sum _{(z,r) \\in \\Xi }{\\lambda }_{z,r} f(z,r), \\qquad \\forall f \\in \\Pi _n^{d+1}.$ Moreover, ${\\lambda }_{z,r} \\sim W\\!\\left({\\mathbf {c}}((z,r), \\tfrac{\\delta }{n})\\right)$ for all $(z,r) \\in \\Xi $ .", "We can now state our local frame on the solid cone.", "For $j =0,1,\\ldots ,$ let $\\Xi _j$ be a maximal $ \\frac{\\delta }{2^{j}}$ -separated subset in ${\\mathbb {V}}^{d+1}$ , so that $\\int _{{\\mathbb {V}}^{d+1}} f(x,t) W(x,t) \\mathrm {d}x \\mathrm {d}t = \\sum _{(z,r) \\in \\Xi _j} {\\lambda }_{(z,r),j} f(z,r),\\qquad f \\in \\Pi _{2^j}^{d+1}.$ Denote by ${\\mathbf {L}}_n(W)f$ the near best approximation operator defined by ${\\mathbf {L}}_n(W)f (x,t) = \\int _{{\\mathbb {V}}^{d+1}}f(y,s) {\\mathbf {L}}_n(W; (x,t),(y,s)) W(y,s) \\mathrm {d}y \\mathrm {d}s.$ For $j= 0,1,\\ldots ,$ define the operator $F_j(W)$ by $F_j(W) * f = {\\mathbf {L}}_{2^{j-1}}(W) * f$ and define the frame elements $\\psi _{(z,r),j}$ for $(z,r) \\in \\Xi _j$ by $\\psi _{(z,r),j}(x,t):= \\sqrt{{\\lambda }_{(z,r),j}} F_j((x,t), (z,r)), \\qquad (x,t) \\in {\\mathbb {V}}^{d+1}.$ Then $\\Phi =\\lbrace \\psi _{(z,r),j}: (z,r) \\in \\Xi _j, \\, j =1,2,3,\\ldots \\rbrace $ is a tight frame.", "Theorem 5.20 Let $W$ be a doubling weight on ${\\mathbb {V}}^{d+1}$ .", "If $f\\in L^2({\\mathbb {V}}^{d+1}, W)$ , then $f =\\sum _{j=0}^\\infty \\sum _{(z,r) \\in \\Xi _j}\\langle f, \\psi _{(z,r), j} \\rangle _W \\psi _{(z,r),j} \\qquad \\mbox{in $L^2({\\mathbb {V}}^{d+1}, W)$}$ and $\\Vert f\\Vert _{2,W} = \\Big (\\sum _{j=0}^\\infty \\sum _{(z,r) \\in \\Xi _j} \\left\|\\langle f, \\psi _{(z,r),j} \\rangle _W\\right\|^2\\Big )^{1/2}.$ Furthermore, for $W_{{\\gamma },\\mu }$ with ${\\gamma }\\ge -\\frac{1}{2}$ and $\\mu \\ge 0$ , the frame is highly localized in the sense that, for every ${\\kappa }>0$ , there exists a constant $c_{\\kappa }>0$ such that $ \|\\psi _{(z,r),j}(x,t)\| \\le c_\\sigma \\frac{2^{j(d+1)/2}}{\\sqrt{ W_{{\\gamma },\\mu }(2^{j}; x,t)} (1+ 2^j {\\mathsf {d}}_{{\\mathbb {V}}}((x,t),(z,r)))^{\\kappa }},\\quad (x,t)\\in {\\mathbb {V}}^{d+1}.$ The frame elements are well defined for all doubling weight by Theorem REF .", "The decomposition is the consequence of Theorem REF .", "Moreover, the localization (REF ) follows from Theorem REF and ${\\lambda }_{(z,r),j} \\sim 2^{- j(d+1)} W_{{\\gamma },\\mu }(2^j;t)$ that holds for $W_{{\\gamma },\\mu }$ as see from Corollary REF and (REF )." ], [ "Characterization of best approximation", "For $f\\in L^p({\\mathbb {V}}^{d+1}, W)$ , we denote by ${\\mathbf {E}}_n(f)_{p, W}$ the error of best approximation to $f$ from $\\Pi _n^{d+1}$ , the space of polynomials of degree at most $n$ , in the norm $\\Vert \\cdot \\Vert _{p, W}$ , ${\\mathbf {E}}_n(f)_{p, W}:= \\inf _{g \\in \\Pi _n^{d+1}} \\Vert f - g\\Vert _{p, W}, \\qquad 1 \\le p \\le \\infty .$ We give a characterization of this quantity in terms of the modulus of smoothness defined via the operator ${\\mathsf {S}}_{{\\theta },W}$ and the $K$ -functional defined via the differential operator ${\\mathfrak {D}}_{{\\gamma },\\mu }$ for $W_{{\\gamma },\\mu }$ .", "For $f\\in L^p({\\mathbb {V}}^{d+1}, W_{{\\gamma },\\mu })$ and $r > 0$ , the modulus of smoothness is defined by ${\\omega }_r(f; \\rho )_{p,W_{{\\gamma },\\mu }} = \\sup _{0 \\le {\\theta }\\le \\rho }\\left\\Vert \\left(I - {\\mathbf {S}}_{{\\theta },W_{{\\gamma },\\mu }}\\right)^{r/2} f\\right\\Vert _{p,W_{{\\gamma },\\mu }}, \\quad 1 \\le p \\le \\infty ,$ where the operator ${\\mathbf {S}}_{{\\theta },W_{{\\gamma },\\mu }}$ is defined by, for $n = 0,1,2,\\ldots $ and ${\\lambda }= 2\\mu +{\\gamma }+d$ , $\\operatorname{proj}_n(W_{{\\gamma },\\mu }; {\\mathbf {S}}_{{\\theta },W_{{\\gamma },\\mu }}f) = R_n^{({\\lambda }-\\frac{1}{2}, -\\frac{1}{2})} (\\cos {\\theta }) \\operatorname{proj}_n(W_{{\\gamma },\\mu }; f).$ Moreover, in terms of the fractional differential operator $(-{\\mathfrak {D}}_{{\\gamma },\\mu })^{\\frac{r}{2}}$ , the $K$ -functional is defined for a weight $W$ on ${\\mathbb {V}}^{d+1}$ by ${\\mathbf {K}}_r(f,\\rho )_{p,W} : = \\inf _{g \\in {\\mathcal {W}}_p^r({\\mathbb {V}}^{d+1}, W)}\\left\\lbrace \\Vert f-g\\Vert _{p,W} + \\rho ^r\\left\\Vert (-{\\mathfrak {D}}_{{\\gamma },\\mu })^{\\frac{r}{2}}f \\right\\Vert _{p,W} \\right\\rbrace ,$ where ${\\mathcal {W}}_p^r({\\mathbb {V}}^{d+1},W)$ denotes the Sobolev space consisting of functions in $L^p({\\mathbb {V}}^{d+1}, W)$ with finite $\\left\\Vert (-{\\mathfrak {D}}_{{\\gamma },{\\beta }})^{\\frac{r}{2}}f \\right\\Vert _{p,W}$ .", "The weight function $W_{{\\gamma },\\mu }$ admits Assertion 1 and 3 by Theorem REF and Lemma REF .", "We now verify that the Assertion 5 in Subsection REF holds.", "By Theorem REF , the kernel $L_n^{(r)}(\\varpi )$ in Assertion 5 becomes ${\\mathbf {L}}_n^{(r)}\\big (W_{{\\gamma },\\mu }; (x,t),(y,s)\\big )=\\sum _{k=0}^\\infty \\widehat{a}\\left(\\frac{k}{n} \\right) (k(k+2\\mu +{\\gamma }+d))^{\\frac{r}{2}}{\\mathbf {P}}_k\\big (W_{{\\gamma },\\mu }; (x,t),(y,s)\\big ).$ Lemma 5.21 Let ${\\gamma }\\ge -\\frac{1}{2}$ and $\\mu \\ge 0$ .", "Let ${\\kappa }> 0$ .", "Then, for $r > 0$ and $(x,t), (y,s) \\in {\\mathbb {V}}^{d+1}$ , $\| {\\mathbf {L}}_n^{(r)}\\big (W_{{\\gamma },\\mu }; (x,t),(y,s)\\big )\| \\le c_{\\kappa }\\frac{n^{r+d}}{\\sqrt{ W_{{\\gamma },\\mu } (n;x,t) }\\sqrt{ W_{{\\gamma },\\mu } (n; y,s)}\\left(1 + n {\\mathsf {d}}_{{\\mathbb {V}}}( (x,t), (y,s)) \\right)^{{\\kappa }}}.$ By (REF ), the kernel can be written as ${\\mathbf {L}}_n^{(r)}\\big (W_{{\\gamma },\\mu };(x,t),(y,s)) = \\, & c_{\\mu ,{\\gamma },d} \\int _{[-1,1]^3} L_{n,r}\\left(2 \\xi (x,t,y,s;u,v)^2-1\\right) \\\\& \\times (1-u^2)^{\\mu -1}(1-v_1^2)^{{\\alpha }-1} (1-v_2^2)^{{\\gamma }-\\frac{1}{2}} \\mathrm {d}u \\mathrm {d}v,$ in which ${\\alpha }= \\mu + \\frac{d-1}{2}$ and $L_{n,r}$ is defined by, with ${\\lambda }= 2 {\\alpha }+ {\\gamma }+1$ , $L_{n,r}(t) = \\sum _{k=0}^\\infty \\widehat{a}\\left(\\frac{k}{n} \\right) (k(k+{\\gamma }+d-1))^{\\frac{r}{2}}\\frac{P_n^{({\\lambda }-\\frac{1}{2}, -\\frac{1}{2})}(1)P_n^{({\\lambda }-\\frac{1}{2}, -\\frac{1}{2})}(t)}{h^{({\\lambda }-\\frac{1}{2},-\\frac{1}{2})}}.$ Applying (REF ) with $\\eta (t) = \\widehat{a}(t) \\left( t( t + n^{-1} (2\\mu +{\\gamma }+d))\\right)^{\\frac{r}{2}}$ and $m=0$ , it follows that $\\left\| L_{n,r}(t) \\right\| \\le c n^{r} \\frac{n^{2{\\lambda }+1}}{(1+n\\sqrt{1-t})^\\ell }.$ Using this estimate, we can then deduce the proof to one that has already appeared for ${\\mathbf {L}}_n(W_{{\\gamma },\\mu })$ in the proof of Theorem REF .", "With Assertions 1, 3 and 5 verified for $W_{{\\gamma },\\mu }$ , the characterization of the best approximation by polynomials in Subsection REF holds on the solid cone, which we state below.", "Theorem 5.22 Let $f \\in L^p({\\mathbb {V}}^{d+1}, W)$ if $1 \\le p < \\infty $ and $f\\in C({\\mathbb {V}}^{d+1})$ if $p = \\infty $ .", "Le $r > 0$ and $n =1,2,\\ldots $ .", "For $W = W_{{\\gamma },\\mu }$ with ${\\gamma }\\ge -\\frac{1}{2}$ and $\\mu \\ge 0$ , there holds direct estimate ${\\mathbf {E}}_n(f)_{p,W_{{\\gamma },\\mu }} \\le c \\, {\\mathbf {K}}_r (f;n^{-1})_{p,W_{{\\gamma },\\mu }}.$ inverse estimate, for $\\mu = 0$ , ${\\mathbf {K}}_r(f;n^{-1})_{p,W_{{\\gamma },0}} \\le c n^{-r} \\sum _{k=0}^n (k+1)^{r-1}{\\mathbf {E}}_k(f)_{p, W_{{\\gamma },0}}.$ The direct estimate follows from Theorem REF , which requires only Assertions 1, 3, 5 and holds for $W_{{\\gamma },\\mu }$ by Theorem REF and Lemma REF .", "The inverse estimate follows from Theorem REF , which requires one weigh function that admits all Assertions 1–3 and 5 and that holds for $W_{{\\gamma },0}$ on the cone.", "Both the direct and the inverse estimates hold for the weight function $W_{{\\gamma },\\mu }$ in the above theorem.", "However, it should be noted that the inverse estimate uses the $K$ -functional ${\\mathbf {K}}_r(f,\\rho )_{p,W} = \\inf _{g \\in {\\mathcal {W}}_p^r({\\mathbb {V}}^{d+1}, W)}\\left\\lbrace \\Vert f-g\\Vert _{p,W} + \\rho ^r\\left\\Vert (-{\\mathfrak {D}}_{{\\gamma },0})^{\\frac{r}{2}}f \\right\\Vert _{p,W} \\right\\rbrace $ defined via the operator ${\\mathfrak {D}}_{{\\gamma },0}$ for the weight $W_{{\\gamma },0}$ .", "For $W = W_{{\\gamma },\\mu }$ , both direct and inverse estimates can be given via the modulus of smoothness, since it is equivalent to the $K$ -functional.", "Theorem 5.23 Let ${\\gamma }\\ge -\\frac{1}{2}$ , $\\mu \\ge 0$ and $f \\in L_p^r({\\mathbb {V}}^{d+1}, W_{{\\gamma },\\mu })$ , $1 \\le p \\le \\infty $ .", "Then for $0 < {\\theta }\\le \\pi /2$ and $r >0$ $c_1 {\\mathbf {K}}_r(f; {\\theta })_{p,W_{{\\gamma },\\mu }} \\le {\\omega }_r(f;{\\theta })_{p,W_{{\\gamma },\\mu }} {\\mathbf {K}}_r(f;{\\theta })_{p,W_{{\\gamma },\\mu }}.$" ] ]	2011.14180
[ [ "A Closer Look at the Tropical Cryptography" ], [ "Abstract We examine two public key exchange protocols proposed recently by Grigoriev and Shpilrain (arXiv:1811.06386), which use tropical algebra.", "We introduce a fast attack on the first protocol, and we show that the second protocol cannot be implemented as described." ], [ "Introduction", "In this paper we analyse the two novel key exchange protocols proposed in , which are both based on tropical matrix algebra.", "These protocols utilise the semidirect products of semigroups , in an attempt to avoid exploitable patterns, which were exhibited by previous protocols based on tropical matrix algebra .", "It has already been shown that a private parameter of these novel protocols can be recovered in about 15 minutes using a binary search .", "We will describe a significantly faster method of attack on the first protocol, which recovers the same private parameter.", "We will also show that the second protocol cannot be implemented as described, due to its reliance on the associativity of an operation that is not associative." ], [ "Tropical Matrix Algebra", "The tropical algebra, $\\overline{\\mathbb {R}}$ , is given by equipping the extended set of real numbers, $\\mathbb {R} \\cup \\lbrace \\infty \\rbrace $ , with the addition operation, $\\oplus $ , and the multiplication operation, $\\otimes $ , defined by: $x\\oplus y=\\min (x, y)$ $x\\otimes y=x+y$ $\\overline{\\mathbb {R}}$ satisfies all the axioms of a semiring.", "On top of satisfying the semiring axioms, $\\overline{\\mathbb {R}}$ is idempotent under addition (and therefore an idempotent semiring) and commutative under multiplication.", "A tropical matrix algebra, $\\overline{T}$ can be defined by equipping the set of square matrices of order $\\omega $ , that contains elements from $\\overline{\\mathbb {R}}$ , with the addition operation, $\\oplus $ , and the multiplication operation, $\\otimes $ .", "The elements of the matrix $X$ , produced by the matrix addition $Y\\oplus Z$ , are defined by: $X_{ij}=Y_{ij}\\oplus Z_{ij}$ The elements of the matrix $X$ , produced by the matrix multiplication $Y\\otimes Z$ , are defined by: $X_{ij}=\\bigoplus _{k=1}^\\omega Y_{ik}\\otimes Z_{kj}$ Like $\\overline{\\mathbb {R}}$ , $\\overline{T}$ is an idempotent semiring, however $\\overline{T}$ is not commutative under multiplication.", "Note that $\\overline{\\mathbb {R}}$ is also known as min-plus algebra.", "For further reading on tropical algebra see and ." ], [ "Almost Linear Periodicity", " define a sequence of matrices, $H^n, n \\in \\mathbb {N}$ , as almost linear periodic if there exists a period $\\rho $ , a linear factor $\\xi $ , and some defect $d$ , such that for all $n>d$ and all indices $i, j$ the following equation holds: $H^{n+\\rho }_{ij} =\\xi +H^{n}_{ij}$ This property is exhibited by the first protocol, and exploited to create our attack." ], [ "Overview of the first protocol", "NB Throughout this paper we will use $M_n$ and $H_n$ to represent the results of $(M,H)^n$ : that is to say $(M,H)^n=(M_n,H_n)$ .", "The protocol is based on a semigroup given by equipping the set of pairs, containing entries from $\\overline{T}$ , with the operation defined by: $(X,G)(Y,H)=((X\\circ H)\\oplus Y, G\\circ H)$ where $X\\circ H$ is defined by: $X\\circ H = X\\oplus H\\oplus (X\\otimes H)$ Alice and Bob agree upon public matrices $M$ and $H$ , with elements in $\\mathbb {Z}$ .", "Alice generates a private positive integer $a$ , while Bob generates a private positive integer $b$ .", "Alice calculates $(M,H)^a=(M_a,H_a)$ , and shares $M_a$ with Bob, keeping $H_a$ private.", "Bob calculates $(M,H)^b=(M_b,H_b)$ , and shares $M_b$ with Alice, keeping $H_b$ private.", "Alice calculates $K=(M_b\\circ H_a)\\oplus M_a$ to get the secret key.", "Bob calculates $K=(M_a\\circ H_b)\\oplus M_b$ to get the same key as Alice.", "When implementing the protocol state that $(M,H)^a$ should be calculated using the square-and-multiply method.", "The associativity of the semigroup operation makes this possible." ], [ "Attack on the first protocol", "The sequence of matrix powers over $\\overline{T}$ is shown to be almost linear periodic in .", "We observed that the sequence $H_n$ also behaves in an almost linear periodic manner.", "Since $H_a$ is a private matrix, this behaviour does not compromise the protocol in an obvious manner.", "However, we also observed the sequence $M_n$ exhibit almost linear periodic behaviour.", "As $M_a$ is public, assuming that the $d$ and $\\rho $ for $M_n$ are sufficiently small such that $M_{d+\\rho }$ can be enumerated in polynomial time, it is possible to use this behaviour to derive $a$ .", "$d$ and $\\rho $ were sufficiently small in all combinations of $M$ and $H$ that we tested, as shown in Table REF .", "Below we describe an attack on the protocol that utilises the almost linear periodicity of $M_n$ .", "The attack uses the public matrices $M$ , $H$ and $M_a$ to derive the private exponent $a$ .", "Once $a$ is obtained, it is trivial to use $M$ , $H$ , $M_b$ , and $a$ with the procedures described in the protocol to obtain the secret key.", "The attack is split into two parts; first, finding $d$ and $\\rho $ for $M_n$ , and second, finding $a$ using $d$ and $\\rho $ ." ], [ "Finding $d$ and {{formula:f2447bfc-258b-4830-a10f-bcc393ff78d6}}", "The sequence $M_n$ is defined by: $\\begin{split}M_1&=M\\\\M_n&=(M_{n-1} \\circ H) \\oplus M \\mbox{, for } n > 1\\\\\\end{split}$ $M$ and $H$ are public allowing for the enumeration of this sequence.", "Assuming $M_n$ is almost linear periodic, all terms after the defect that are a period apart differ by the linear factor.", "It follows that, after the defect, consecutive terms will change in a pattern that repeats every period.", "This implies that the sequence containing the differences between successive elements of $M_n$ will be periodic in nature after the defect.", "We will refer to this sequence of differences between successive elements of $M_n$ as $D_n$ : $D_n=M_{n+1} - M_n,\\mbox{ for }n\\ge 1$ To find $d$ and $\\rho $ , we enumerate the terms of $D_n$ (by enumerating the terms of $M_n$ and calculating $D_n$ ) and compare the current term to previously enumerated terms.", "If a previously enumerated term is equal to the current term, it is possible that the previously enumerated term marks the beginning of periodicity, and its index is $d+1$ .", "The difference in indexes of the previously enumerated term and the current term would, therefore, be equal to $\\rho $ .", "It is possible that terms could repeat before the defect, resulting in false values for $d$ and $\\rho $ .", "This is covered in more detail in REF ." ], [ "Finding $a$", " From the definition of $D_n$ : $D_n=M_{n+1} - M_n,\\mbox{ for }n\\ge 1$ it follows that: $M_n = M_1 + \\sum _{i=1}^{n-1}D_i$ Let $Y=M_a-M_{d+1}$ .", "It follows that: $\\begin{split}Y&=M_a-M_{d+1} \\\\&=(M_1 + \\sum _{i=1}^{a-1}D_i) - (M_1 + \\sum _{i=1}^{d}D_i) \\\\&=\\sum _{i=d+1}^{a-1}D_i \\\\\\end{split}$ Due to the periodic nature of $D_i$ , for $i>d$ , this sum can be decomposed into two parts: the sum of the differences within the period, $\\rho $ , multiplied $x$ times, and the sum of the differences within the period in which $M_a$ is located up until $D_{a-1}$ : $Y=x\\sum _{i=d+1}^{d+\\rho }D_i + \\sum _{i=a-k}^{a-1}D_i$ where $x$ is some positive integer, $1\\le k\\le \\rho $ and $d+x\\rho +k=a$ .", "The periodic nature of $D_i$ after the defect implies that any sum of a number of consecutive elements in $D_i$ , where the first term occurs after the defect, is equal to the sum of the same number of consecutive elements that occur any multiple of the period further along the sequence.", "Therefore the above can be rewritten as: $Y=x\\sum _{i=d+1}^{d+\\rho }D_i + \\sum _{i=d+1}^{d+k}D_i$ $k$ can be found by testing all possible values from 1 to $\\rho $ .", "For a value to be $k$ the following must hold, for all indices $u,v$ : $(Y_{uv} - \\sum _{i=d+1}^{d+k}D_{iuv})\\mod {\\sum }_{i=d+1}^{d+\\rho }D_{iuv}=0$ Once $k$ has been obtained, $x$ can be found through the equation: $\\frac{Y_{uv} - \\sum _{i=d+1}^{d+k}}{D}_{iuv}{\\sum _{i=d}^{d+\\rho }D_{iuv}}=x$ Now that we have $d$ , $x$ , and $k$ , we can solve $d+x\\rho +k=a$ to find $a$ ." ], [ "Special cases", "There may be repeated elements in $D_n$ , where $n\\le d$ , leading to false values for $d$ and $\\rho $ .", "This will often be detected when searching for $k$ , as no values for $k$ will satisfy the equation: $(Y_{uv}-\\sum _{i=a-k}^{a-1}D_{iuv})\\mod {\\sum }_{i=d}^{d+\\rho }D_{iuv}=0$ There is a small chance that a value could satisfy the equation, resulting in an incorrect derivation of $a$ .", "This can be handled by checking that the derived $a$ satisfies the equation: $(M,H)^{derived\\_a} = (M_a, Z)$ where the variable $Z$ can be ignored, because if the first term is correct, it follows that the second term is correct.", "If an incorrect $a$ is detected, the search for $d$ and $\\rho $ can be resumed.", "A second special case which should be accounted for in the attack, is when $D_n$ becomes the zero matrix for $n>d$ .", "This results in a division by zero when finding $k$ and $x$ .", "This special case is simple to account for, as it implies that for all $M_n$ , such that $n>d$ , $M_n=M_{d+1}$ .", "Therefore, although it is impossible to find $a$ , this has no bearing on the success of the attack, as $d+1$ can be substituted for $a$ ." ], [ "Experimental Results", "The success of the attack against 10000 instances of the protocol, with the parameters suggested by , is detailed in Table REF .", "$d$ is the number of elements of the sequence $M^n$ that were enumerated before periodic behaviour was observed.", "$p$ is the period.", "The attack times give the time taken to find the private parameter $a$ .", "The protocol and attack were implemented in Python and can be found in .", "All tests were performed on a single core of an i7 CPU at 2.9GHz, with 8GB of RAM, running Windows 10, and interpreted using Python 3.7.6." ], [ "Overview of the second protocol", "The protocol is based on a supposed semigroup (which we will show is not a semigroup in the next section) given by equipping the set of pairs, containing entries from $\\overline{T}$ , with the operation defined by: $(M,G)(S,H)=((H\\otimes M^T)\\oplus (M^T\\otimes H)\\oplus S, G\\otimes H)$ Alice and Bob agree upon public matrices $M$ and $H$ , with elements in $\\mathbb {Z}$ .", "Alice generates a private positive integer $a$ , while Bob generates a private positive integer $b$ .", "Alice calculates $(M,H)^a=(M_a,H_a)$ , and shares $M_a$ with Bob, keeping $H_a$ private.", "Bob calculates $(M,H)^b=(M_b,H_b)$ , and shares $M_b$ with Alice, keeping $H_b$ private.", "Alice calculates $K=(M_b\\otimes H_a)\\oplus M_a$ to get the secret key.", "Bob calculates $K=(M_a\\otimes H_b)\\oplus M_b$ to get the same key as Alice.", "As with the first protocol, state that $(M,H)^a$ should be calculated using the square-and-multiply method." ], [ "Proof that the second protocol cannot be implemented", "This protocol cannot be implemented as the operation that the protocol is based upon is not associative.", "Consider the example below: $\\mbox{Let }A=\\begin{pmatrix}0 & -1\\\\0 & 0\\end{pmatrix},B=\\begin{pmatrix}0 & -2\\\\0 & 0\\end{pmatrix},$ $(A,B)^2=\\left(\\begin{pmatrix}-3 & -2\\\\-1 & -3\\end{pmatrix},\\begin{pmatrix}-2 & -2\\\\0 & -2\\end{pmatrix}\\right)$ $(A,B)(A,B)^2=\\left(\\begin{pmatrix}-3 & -2\\\\-3 & -3\\end{pmatrix},\\begin{pmatrix}-2 & -4\\\\-2 & -2\\end{pmatrix}\\right)$ $(A,B)^2(A,B)=\\left(\\begin{pmatrix}-4 & -5\\\\-3 & -4\\end{pmatrix},\\begin{pmatrix}-2 & -4\\\\-2 & -2\\end{pmatrix}\\right)$ $(A,B)(A,B)^2\\ne (A,B)^2(A,B)$ It follows, from the operation's lack of associativity, that it is not possible to calculate $(M,H)^a$ by utilising the square-and-multiply method.", "Consequently, the protocol cannot be successfully implemented." ], [ "Conclusion", "The first protocol we analysed is insecure when using the proposed parameters for key generation.", "It is unclear how to modify the protocol such that it resists the attack we describe.", "Our attack is significantly faster than the binary search attack given in , requiring about 0.5% of the time to find $a$ when using proposed protocol parameters.", "The longest the attack took to break the protocol was 200 seconds, which was still considerably faster than the binary search.", "It is not possible to implement the second protocol we analysed, since the operation it relies upon is not associative.", "This prevents the use of the square-and-multiply method for exponentiation, which is a fundamental aspect of the general protocol on which these protocols are based .", "We encourage interested readers to examine our implementation of tropical matrix algebra, the protocols, and the attack, and perform their own experiments using it ." ] ]	2011.14163
[ [ "Matter and forces in quantum field theory -- an attempt at a\n philosophical elucidation" ], [ "Abstract This is a translation into English of my Masters thesis (hovedoppgave) from 1991.", "The main topic of the thesis is the relation between fundamental physics and philosophy, and a discussion of several possible ontologies of quantum field theory." ], [ "Preface", "This thesis arose from one primary motivation.", "I felt there was little sense in delving straight into theoretical calculations within e.g.", "particle physics without any overarching idea of what the theory was about and what would be the value of such work — I had little interest in doing something I might myself consider next to worthless for my thesis.", "But to judge what it could be worth working on it was necessary to take a step back and evaluate the theory as a whole.", "This feeling was further strengthened by the time I spent at CERN in the summer of 1989.", "It was an interesting stay and an exciting environment, but I found myself asking whether much of the activity might not be driven too much by prestige, and whether perhaps there was not so much scientific insight to be gained from the large accelerator experiments compared to the investments.", "It could be useful to conduct a general evaluation of where we stood and what was the aim of the experiments.", "In addition, I have always (for as long as I have been interested in physics) been interested in the connections between philosophy and physics.", "After also having studied some philosophy in addition to physics, I believed it could be good to attempt to do some proper work in this area.", "It has been good for me — I have personally got a lot out of my work with this thesis.", "I have not managed to do as much as I had hoped — there is for example a half-finished section on particle species which did not make it in because I ran out of time, and there are several other questions I would have liked to discuss given more time.", "I have however got a greater interest in and understanding of both physics and philosophy.", "In particular got an idea of which areas of research within fundamental physics can be of interest — which was part of the aim of doing this work in the first place.", "I would like to thank my supervisors Audun Øfsti and Kåre Olaussen, who have read through the manuscript and given me good advice along the way.", "Thank you also to everyone else who has provided opinions and encouragement.", "Finally, thank you to my father, who has given me access to his computers for writing the thesis, and also given me much support.", "Trondheim, 6 May 1991 jon ivar skullerud In this revised edition I have made some minor changes to the text in several places, and included more references.", "I have also corrected some typographical errors.", "Trondheim, 13 July 1991 j.i.s.", "This thesis should be read as the opinions of a budding physicist in 1991.", "I have resisted any temptation to add anything save a small number of footnotes, and have also kept the style, including the idiosyncratic use of punctuation and quote marks, mostly intact.", "My opinions (and my writing style) have obviously evolved in the intervening years, but that will have to be for another day.", "This translation has been a very long time in the making.", "Already in the first few years after it was written several people were asking me if I would translate it into English so that they could read it.", "I thought that would be a good idea, but it was never a priority among so many other things to do.", "Anyway, what would I do with it?", "The thought arose again after I put the original thesis on my web page in 1995, but again time and lack of any urgency meant nothing happened.", "Eventually, many years later, I started regularly reading the History and Philosophy of Physics eprints on arXiv [hist-ph], and concluded there would indeed be a potential repository where the thesis could reach a wider audience, justifying the effort that would go into a translation.", "Thus started a slow process, in my spare time, not being sure if I could justify this being part of my work or not.", "Then, when I went on a one-semester sabbatical to Florence in autumn 2019 I decided this was when I was going to complete the translation, and I stuck to that.", "I wish to express my great appreciation to the Galileo Galilei Institute for Theoretical Physics for their hospitality during this time, and especially to the organisers of the mini-workshop “Beyond Standard Model: Historical-Critical Perspectives”, which was the highlight of my stay at the GGI and gave me an additional spur to finish this work.", "I wish to thank Máire O'Dwyer for her careful proofreading of the English translation.", "The Feynman diagrams were drawn using the FeynGame package.R. V.", "Harlander, S. Y. Klein and M. Lipp, Comput.", "Phys.", "Commun.", "256, 107465 (2020) [arXiv:2003.00896].", "Firenze, 25 January 2020 / Dublin, 28 November 2020 j.i.s." ] ]	2011.14181
[ [ "Preclinical Stage Alzheimer's Disease Detection Using Magnetic Resonance\n Image Scans" ], [ "Abstract Alzheimer's disease is one of the diseases that mostly affects older people without being a part of aging.", "The most common symptoms include problems with communicating and abstract thinking, as well as disorientation.", "It is important to detect Alzheimer's disease in early stages so that cognitive functioning would be improved by medication and training.", "In this paper, we propose two attention model networks for detecting Alzheimer's disease from MRI images to help early detection efforts at the preclinical stage.", "We also compare the performance of these two attention network models with a baseline model.", "Recently available OASIS-3 Longitudinal Neuroimaging, Clinical, and Cognitive Dataset is used to train, evaluate and compare our models.", "The novelty of this research resides in the fact that we aim to detect Alzheimer's disease when all the parameters, physical assessments, and clinical data state that the patient is healthy and showing no symptoms" ], [ "Introduction", "Alzheimer's disease (AD) is a type of brain disease, which is degenerative, and its symptoms worsen over the years.", "According to the Alzheimer’s Association , 5.8 million Americans are living with AD.", "Statistics have also shown that annual AD related death rates by age are increasing every year .", "According to , AD could begin twenty years or more before symptoms are perceptible, and between one and six years with changes in the brain that are unnoticeable to the person affected.", "As the disease evolves, more and more brain neurons stop functioning, lose connection, even die.", "At first, AD affects the entorhinal cortex and hippocampus brain regions that are involved in memory .", "Eventually, it affects the cerebral cortex, which is responsible for language, reasoning, and social behavior, and many other areas of the brain.", "This pathology, according to the Alzheimer's Association [], is the most common cause of dementia, especially among older people.", "AD accounts for 60-80% of the total dementia cases .", "AD significantly reduces the quality of life and life expectancy of patients, and is considered as the most expensive disease in USA .", "When the economic impact of this disease is considered, just in 2019, the total combined payments from all AD patients are estimated at $290 billion [].", "https://www.alz.org/media/documents/alzheimers-facts-and-figures-2019-r.pdf While there is no cure for Alzheimer’s disease or a way to stop its progression after diagnosis, and the treatment of AD is still an open research question, there are drug and non-drug options that may help treat symptoms.", "Studies have demonstrated that early stage intervention of AD can significantly impact the degeneration process, and treatment of symptoms , , .", "The early detection of AD through conventional MRI scanning will facilitate effective and in-time interventions/treatments, that would expand the life expectancy and quality of life of patients.", "National Institute on Aging has defined three stages of AD: (1) Preclinical, when the patients do not exhibit any symptoms, but brain neuronal structure has started to deteriorate; (2) Mild cognitive impairment (MCI), when patients start to exhibit cognitive impairments, but still can perform all activities of daily living (ADL); and (3) Alzheimer’s dementia, when symptoms of dementia are severe enough to interfere with ADLs.", "With the advancement of MRI technology , and the recent development of deep learning based computer vision approaches, several studies have addressed the detection of AD in MCI and dementia stages (stage 2 and 3) from MRI brain scans , .", "While these studies have improved our understanding of Alzheimer’s disease, they do not contribute in early stage interventions, since stage 2 and 3 are accurately identifiable through clinical diagnosis.", "Accurate detection or indication of preclinical AD is a major interest in the medical community , .", "However, to the best of our knowledge, existing studies have not addressed the challenge of preclinical AD detection from MRI brain scans yet.", "Recent reports on AD suggest that change in brain may be evident 20 years before the stage of dementia (stage 3), and that there is already substantial neuronal loss by the stage of mild cognitive impairment (MCI) (stage 2).", "Hence, the goal of this study is the development of an effective machine learning approach that can identify the latent patterns due to preclinical AD from MRI brain scans, which can significantly improve intervention and treatment of AD patients.", "MRI brain scans are 3 dimensional (3D) image representation of the brain structure.", "Several sequential classifiers such as, 3D Convolutional Neural Network (CNN) and 3D recurrent visual attention (RVN) model , have been implemented to detect disease related patterns from the 3D brain scans.", "Recently, Transformer model has been demonstrated to outperform all the existing sequential classifiers.", "In this study, we employ and adapt two attention network models (3D recurrent visual attention model and Transformer model ) to our problem of preclinical AD detection from 3D MRI brain scan data, and compare their performances with a baseline model that is based on 3D CNNs.", "We evaluated these approaches for differentiating individuals with`preclinical AD' from `others'.", "The `others' class includes healthy individuals or patients suffering from other dementia problems not related with AD.", "According to our evaluation, the 3D CNN model achieves an F1 score of $0.83$ and accuracy of 88.34%, the 3D RVN model achieves an F1 score of $0.87$ , and 90.65% accuracy, and the developed transformer model achieves an F1 score of $0.90$ and 91.18% accuracy in preclinical (i.e., prior to MCI and dementia stage) Alzheimer disease detection.", "Experimental results demonstrate that, by using the MRI images effectively, it is possible to detect preclinical stage Alzheimer's disease with a very promising accuracy." ], [ "Related Work", "Deep learning algorithms perform very well in identifying complex structures in high dimensional data, and this is why there is a very rich literature on detecting diseases using magnetic resonance images (MRI) and positron emission tomography (PET).", "studied the presence of Alzheimer's disease using two different 3D CNN approaches, namely VoxCNN and residual neural networks (RNN), on MRI data.", "Also aiming for binary classification, Alzheimer's/Non Alzheimer's, used two 3D CNNs on MRI and PET scans, combining them with a fully connected layer and a softmax classification.", "Over the past few years, there has been a trend in applying attention based approaches to medical problems.", "With these kind of models, we can not only detect a certain pathology, but also represent which part of the data is more important to make accurate predictions.", "proposed a novel method, referred to as attention gated networks, for medical image analysis.", "This algorithm learns to focus on target structures, and can be used for leveraging certain regions for classification purposes.", "With a similar goal, proposed Neuro-Dram, a 3D recurrent visual attention, for explainable neuroimaging classification.", "used a layer-wise relevance propagation (LRP) to visualize CNN decisions based on MRI data.", "This algorithm attributes relevance to every input node, and studies the contribution of each node.", "It can output a heat map to highlight the most informative parts of every input image.", "Early detection of AD can be critical when developing an optimal treatment for each patient .", "showed how early stage AD patients, improved their general cognitive abilities after 12 weeks of paper-based cognitive training.", "Along these lines, there has been a lot of work trying to differentiate between mild cognitive impairment (MCI) and Alzheimer's disease (AD).", "According to , MCI causes a slight and measurable decline in cognitive abilities and it is the earliest clinically detectable stage before AD.", "Suffering from MCI increases the risk of developing AD.", "proposed a four-class SVM classifier: AD, MCI stable (patients with MCI who do not develop AD), MCI converted (patients whose MCI develop into AD), and healthy patients.", "Similarly, to study MCI to AD conversion, proposed a multi-modal recurrent neural network.", "In this case, they used MRI, demographic information, cerebrospinal fluid (CSF) biomarkers and cognitive performance reports as their inputs to the GRU units.", "With a slightly different approach, used a model inspired by Inception-V4 network to detect between non demented, very mild dementia, mild dementia and moderate dementia.", "The goal of all the discussed approaches is detecting AD (stage 3) or predicting MCI cases (stage 2) that will develop AD, i.e.", "existing studies focus on stages 2 and 3.", "To the best of our knowledge, there is no previous work on detecting preclinical stage AD (stage 1) when all the indicators, including the clinical assessments by doctors, ensure that the patient is healthy and no symptoms of the disease are present.", "Our main goal is to detect future AD from the latent brain scan patterns even before MCI develops, when the disease is in a preclinical stage." ], [ "Dataset", "In this work, we employ the recently published longitudinal neuroimaging, clinical and cognitive dataset, called OASIS-3 .", "It consists of MRI and PET imaging from 1098 individuals collected across several studies over the course of 15 years.", "There are 605 cognitively normal adults and 493 individuals at different stages of cognitive decline.", "Ages of the participants range from 42 to 95 years.", "The dataset contains over 2000 MRI sessions.", "For every patient, the aging and disability resource center (ADRC) clinical data is available.", "This data includes patient's age at entry, height, weight, and clinical dementia rating (CDR).", "Diagnoses for this datatype include “cognitively normal\", “AD dementia\", “vascular dementia\" and factors that could be contributing such as vitamin deficiency, alcoholism, and mood disorders.", "The goal of this study is to identify potential AD patients even when they were clinically diagnosed as “cognitively normal” from the latent patterns of their brain scans.", "Hence, in this work, we are only using brain imaging and the affiliated clinical diagnoses, matching each scan with the closest clinical diagnosis available.", "The presented classifiers perform a binary classification task: differentiating preclinical AD individuals from others.", "More specifically, our presented binary classifiers use `0' for the patients diagnosed as healthy or with other non-Alzheimer's related pathologies, and `1' for the patients with `preclinical stage AD'.", "We define a `preclinical stage AD' patient as a person who is currently diagnosed by a doctor as healthy but we know that in the future he or she will develop AD.", "For this, we label each MRI session independently, matching every clinical data that we have with the closest MRI session available.", "In this dataset, MRI brain scans are available from two different views, the axial or horizontal plane, and sagittal or longitudinal plane .", "We decided to use the axial plane since only a smaller portion of the individual's data contain sagittal scans.", "Since not all the images had the same resolution, as part of our image pre-processing, we resized and normalized all the images.", "One of the initial challenge we encountered was the significant class imbalance.", "More specifically, for class `0' (healthy subjects), there were 2181 scans, whereas for class `1' (patients with preclinical stage AD), we had only 176 scans.", "To address this issue, we down-sampled our class `0' and over-sampled the class `1', by randomly rotating and mirroring some of the images (as proposed in and ).", "We also implemented a balanced sampling process.", "This method creates a sampler in the data loader based on the number of images of each class that yields the next index/key set to fetch.", "This is useful to ensure that we have an even number of labels on each training batch." ], [ "Proposed Approach", "As mentioned above, existing works addressed stage 2 and 3 AD detection.", "One of the significant differences of our work, compared to the state-of-the-art, is that we address the more challenging problem of preclinical AD (stage 1) detection.", "To ensure the latent brain pattern extraction for preclinical AD detection task, we only considered the 3D brain scans as input.", "This makes the classification task more challenging, since we cannot adapt the models to different factors that could be critical when predicting AD, such as the age or the sex of the patient.", "To address this challenging problem, we adapt and employ two different attention mechanisms, namely 3D recurrent visual attention model, and attention transformer.", "We compare our two attention mechanisms with a baseline model, which is based on 3D CNNs." ], [ "Baseline - 3D CNN Model", "The baseline model is based on a 3D CNN model, which was initially used for video classification tasks .", "This model uses 3D kernels and channels to convolve video input, where the videos are viewed as 3D data (2D images over time dimension).", "For our baseline model, we stack all the images in a brain scan, turn them into 3D input data, and then feed it to the network.", "The model we developed consists of five convolutional layers and three fully connected layers.", "Each convolution layer is followed by batch normalization, ReLu, dropout, and pooling layers.", "We consider this model as our baseline." ], [ "3D Recurrent Visual Attention Model", "For our first model shown in Fig.", "REF , we employ a recently proposed 3D recurrent visual attention model, which is tailored for neuroimaging classification and focuses on already developed AD detection task.", "This model uses a recurrent attention mechanism that tries to find relevant locations of brain scan indicative of AD.", "The model consists of an agent that is trained with reinforcement learning.", "It is built around a two-layer recurrent neural network (RNN).", "At each timestamp, the agent receives a small portion of the entire image, which is a glimpse, centered around a position $l$ , and decides which location to select at the next timestamp.", "After a fixed number of steps, a classification decision is made.", "The aim of using an agent is to maximize the rewards along the timestamps, and then decide to attend the most informative regions of the images.", "We define our reward, $r_t$ , as 1 for all timestamp $t$ if the classification is correct, or 0 if it is not.", "Overall, the model consists of four different networks: the glimpse network, the recurrent network, the location network and the classification network.", "The glimpse network takes a small 3D image fraction $x_t$ , and its location coordinate $l_t$ as input, and outputs a vector $g_t$ .", "It generates a representation of the glimpse (i.e., 3D image fraction) summarizing the `what' $g_{x_t}$ , and `where' $g_{l_t}$ information.", "The glimpse network consists of 3D convolutional layers (with batch normalization and max pooling) that generates the `what' representation, and a single-layer fully connected layer that converts the location coordinated to the `where' representation.", "Final $g_t$ is obtained by an element-wise multiplication of these representations: $ g_t = {g_{x_t}} \\odot {g_{l_t}}$ The recurrent network is used to obtain the agent's internal representation encapsulating the information extracted from past timestamps.", "This network consists of two stacked LSTM units .At each timestamps, hidden-layer representation generated by the LSTM units is fed into the location network to obtain the next timestamp's glimpse location, $l_{t+1}$ .", "At the last timestamp (i.e., last iteration of the LSTM sequential analysis) the hidden layer representation is fed into the classification network.", "The location network consists of a single-layer fully connected layer, which maps $r_{(t)}^2$ to a 3D vector in the range [-1,1] which is an isotropic 3D normal distribution.", "Next location $l_{t+1}$ is then produced by sampling from this distribution.", "Finally, the classification network consists of a single fully connected layer with a sigmoid activation function, which is used for binary classification.", "As previously stated, the input to this network is the final timestamp hidden layer representation of the LSTM unit.", "Total number of timestamps/iteration $T$ in the recurrent network, and the glimpse size are hyper-parameters.", "In our evaluation we identified $T=6$ , and the glimpse size (3D image fraction) $40\\times 40\\times 40$ as the beneficial value, based on the grid search on possible values." ], [ "Attention Transformer", "As our second model, we employed a transformer network for the task of preclinical AD detection.", "Transformer models have been used for different tasks such as human action recognition from videos and text translation .", "Although transformer networks have been used for other tasks and applications, we firmly believe that this is the first work that employs a transformer network on MRI images of brain for preclinical stage Alzheimeir's disease detection.", "Slices from a brain scan are fed to the network, and the network is expected to detect whether any sign of dementia is observable or not, even the subject is showing no signs nor symptoms of the disease yet.", "Implemented Transformer model consists of a base network and a head network similar to .", "Base network extracts feature-representations from each of the brain MRI image slices, and the head network generates the binary inferences (i.e., preclinical stage AD or others).", "Both the base and the head networks are described along with the modifications we have made in the following subsections." ], [ "Base Network.", "Since we deal with binary classification for preclinical stage AD detection, we use a VGG16 network as our base network.", "We extract frames from brain MRI scans (96 frames) and feed them into the base network.", "Due to the nature of the base network, it accepts $224\\times 224 \\times 3$ images, where $224\\times 224$ represents image height and weight, and 3 represents the channel size.", "On the other hand, slices from brain MRI scans differ in size, and are gray scale, which has only one channel.", "After resizing our input images to $224\\times 224$ , the first modification we do for the base network is adding an extra convolutional layer at the beginning to get the desired input for the base network.", "We set in_channels=1, out_channels=3, kernel_size=3, stride=1, padding=1 and dilation=1 for this new convolutional layer.", "We have experimented with two different approaches, namely training the base network from scratch and using a pre-trained model (trained on conventional RGB images).", "As detailed in the experiments section, we obtained better results when the base network was trained from scratch.", "Instead of stacking images, (compared to 3D CNN and 3D recurrent visual attention model), we feed each of the brain scan images/frames in a sequence, like a video.", "The base network portion of the Fig.", "REF illustrates the network described so far.", "We then send the output of the base network to the head network." ], [ "Transformer Unit.", "The original transformer architecture was proposed in for sequence to sequence tasks to overperform recurrents models.", "It is done by selecting a feature frame and comparing it with all features in a sequence in order to compute attention.", "To do that, features are mapped to a query (Q) and memory (K for key and V for value) embeddings using linear projections.", "Since the original transformer is designed for language to language translation, Q is the word that is being translated and K and V are the linear projections of the generated input and output sequences.", "On the other hand, our transformer network consists of a positional encoder, three head units, shown as Block Head Units in Fig.", "REF , and a classifier.", "This unit takes the brain MRI image sequence features and also the positional embeddings for attention proposal regions and maps them into query (Q) and memory features (K, V).", "Query (Q) represents the region with signs of preclinical stage AD.", "Frames around that region are projected into K and V. Block Head Units as in Fig.", "REF process the query and memory embeddings to update the query vector.", "By doing so, they aggregate the information over the brain MRI scans to classify whether the given sequence belongs to preclinical stage AD class or not.", "We evaluated using different optimizers, and SGD was the most beneficial choice.", "We set learning rate as $1e^{-4}$ , momentum as $9e^{-4}$ and nesterov as True .", "We use cross-entropy loss for our classifier." ], [ "Experiments", "We first split our dataset into three person-disjoint subsets: training, validation and test subsets.", "When trying to detect a certain pathology using patients' scans, it is crucial to split the data correctly.", "Different scans from the same person cannot appear among different datasets.", "If so, there is a risk that the model gets trained to correctly label those repeated patients, and it will perform poorly on scans from never-before-seen patients.", "This is why we decided to use person disjoint datasets.", "Each participant's MRI scan consists of 256 images.", "After the downsampling and oversampling process previously described, we used 65% of the available data for training, 20% for validation and 15% for testing.", "It should be emphasized that the goal of this study is to detect scans, which were confirmed to be healthy by the doctors at the time of the scan, and which we know that in the future will develop Alzheimer's disease.", "Hence, we developed binary classifiers with preclinical stage Alzheimer's patients labeled as class `1', and the healthy patients or patients suffering from other dementia problems not related with Alzheimer's labeled as `0'.", "The evaluations discussed in this section follow the data labeling scheme shown in Fig.", "REF .", "In line with the study goal, we exclude the scans of patients, who have already developed and been diagnosed with AD, from the `other' category (i.e., class 0), in order to focus on better differentiating between healthy and preclinical AD patients.", "As previously stated, to the best of our knowledge, there is no previous work on detecting preclinical stage AD.", "Figure: Overview of our data labeling approachFor each model, we calculate the confusion matrix, F1 score, false negative rate, precision, recall, and accuracy.", "One of the best metrics to compare models when dealing with an imbalanced classification problem is the F1 score https://towardsdatascience.com/metrics-for-imbalanced-classification-41c71549bbb5.", "In the following subsections, we will present the obtained results for each model, and compare them in terms of the F1 score.", "Also, since our main task is being able to detect possible AD patients when they are still healthy, the false negative rate will also be considered as an important comparative metric." ], [ "Results for the 3D CNN model.", "We used our 3D CNN model as our baseline.", "The best version of this model is trained with a learning rate of $1e^{-5}$ for 50 epochs.", "To avoid overfitting the training data, we chose AdamW as our optimizer with a weight decay of $0.1$ .", "We used cross-entropy as our loss function.", "With these configurations, we obtained the confusion matrix shown in Table REF .", "Table: Confusion matrix for the baseline model.According to the evaluation, the 3D CNN model achieves an F1 score of $0.83$ on preclinical stage AD detection.", "As shown in Table REF , we have a false negative (FN) rate of 25%.", "By taking a closer look at some of the patients wrongly labeled as healthy or other dementia when they were preclinical AD patients, we identified that they are the patients whose clinical data is not very consistent.", "More specifically, there are some patients who are diagnosed with AD at some point of the data collection process, and then they recover their cognitive abilities.", "Therefore, these patients are diagnosed as “Cognitive normal\" for the rest of their life.", "We find that this type of patients are the hardest to correctly classify.", "Additionally, OASIS 3 dataset contains brain scans from patients with non-Alzheimer (non-AD) related dementia, that 3D CNN baseline model finds difficult to classify correctly.", "On the other hand, the baseline model performs well in classifying subjects who, once diagnosed with AD dementia, do not fully recover their cognitive abilities.", "As an example, the clinical data of patient ID 30205 (on OASIS 3 dataset) is shown in Table REF .", "The patient was clinically diagnosed as “Cognitively normal’’ on day 0000, 0406, and 0773.", "OASIS 3 Dataset contains a complete MRI brain scan of the patient performed at day 61.", "Our baseline 3D CNN model can detect `preclinical stage AD’ from these scans performed on day 61.", "Thus, the baseline algorithm is able to detect `preclinical stage AD' 1064 days (1125 - 61 = 1064) before the patient is diagnosed with uncertain dementia.", "If we just consider when the patient is actually diagnosed with Alzheimer's disease, we are able to detect that pathology 1,776 days (1837-61) before it is diagnosed.", "Table: Clinical data for patient 30205." ], [ "Results for the 3D recurrent visual attention model.", "With the attention mechanism models incorporated, our goal is to outperform the baseline.", "For the 3D recurrent visual attention (3D RVN) model, the best performance is achieved with a learning rate of $1e^{-4}$ , trained for 200 epochs using AdamW .", "We also evaluated different glimpse sizes (i.e., image fraction size).", "More specifically, we experimented with $20\\times 20 \\times 20$ , $40\\times 40 \\times 40$ , and $60\\times 60\\times 60$ .", "The best results presented in Table REF were obtained with the $40\\times 40 \\times 40$ glimpse size.", "According to the evaluation, the 3D RVN model achieves an F1 score of $0.87$ for `preclinical stage AD' detection, which is 5% higher compared to the baseline 3D CNN model.", "Additionally, according to the results, the false negative (FN) rate is $19.2$ %, which is lower than the 3D CNN model.", "Table: Confusion matrix for the 3D RVN model.As an example, the clinical data of patient ID 30025 (on OASIS 3 dataset) is shown in Table REF .", "The patient was clinically diagnosed as “Cognitively normal’’ from day 0000 to day 2608, and diagnosed as “AD dementia’’ on day 2933.", "OASIS 3 Dataset contains two complete MRI brain scans of the patient performed on day 0210 and 2298.", "Our 3D RVN model correctly infers `preclinical stage AD’ by taking the brain scan taken on day 0210 as input.", "This result demonstrates that the presented 3D RVN can detect AD 2723 days before it is clinically diagnosed by doctors.", "Table: Clinical data for patient 30025.One of the important benefits of this model is that not only we can detect AD before any symptoms, but also we can plot the part of the data that the algorithm interprets as more important when making accurate decisions.", "In our implementation, the initial (timestamp 0) glimpse location $l_t$ is at the center of the brain.", "After that, the glimpse moves in a direction that maximizes the reward function.", "Figure: Trajectory taken by the glimpse algorithm.", "The green point is the first location, followed by the red, blue, yellow, brown, and pink points.", "We draw a line between them for visualization purposes.Figure REF shows the extracted glimpse locations by the 3D RVN model for a random participant with preclinical AD.", "The figure demonstrates how the 3D RVN agent is exploring the brain regions.", "The green point shows the first glimpse, which is located at the center of the brain at the first timestamp.", "Since we are using a total of six timestamps ($T=6$ ) in the current 3D RVN, there are six different areas where the model agent focuses on to make an inference.", "By plotting the glimpse's location, we can visualize which parts of the brain are more informative when detecting preclinical stage Alzheimer's disease.", "By examining the testset brain scans, and the areas where the 3D RVN model is paying more attention (producing location $l_t$ to extract glimpse/3D image fraction), we identified that it is focusing on areas like: locus coeruleus, hippocampus, entorhinal cortex, and amygdala, parts of the brain that are important for memory and very relevant when detecting MCI .", "Another key brain region for detecting AD is the lateral ventricles , and the 3D RVN model is focusing on that area as well." ], [ "Results for the attention transformer model.", "The transformer model is trained in a different way compared to other networks.", "First, instead of feeding stacked 3D images to the network, we feed images in a sequence.", "As mentioned in the Proposed Approach section, we use a VGG16 network as our base to extract features.", "We tried a pre-trained VGG16 model to see the effects of pre-trained weights, and also trained the network from scratch.", "As shown in Table REF , the transformer model when trained from scratch achieved $4.6$ % higher F1 score.", "Table: Comparison between pre-trained transformer and our transformer.Due to memory limitations, we are not able to use all frames from brain MRI scans.", "We started by selecting 48 frames from the middle and then went up to 96 frames, which is the highest amount we can select with our current GPU configuration.", "We achieved the best results with 96 frames, and trained the network with same amount of images from every scan.", "We also trained the network for different epochs.", "Due to the nature of the transformer networks, we need to select higher number of epochs for an optimal training.", "Thus, we trained our transformer model for 200 epochs.", "As for the optimizer, we started with Adam , and set the learning rate (lr) to $1e^{-4}$ and $amsgrad$ parameter to true .", "Then, we trained the network with AdamW with $lr = 1e^{-4}$ and $amsgrad$ set as true.", "Finally, we trained the transformer network with SGD , and set $lr=1e^{-4}$ , momentum as $9e^{-4}$ , and nesterov as True .", "Among all the optimizers, we obtained better results, presented in Table REF , with SGD.", "Table: Confusion matrix for the transformer model.According to the evaluation results shown in Table REF , the transformer model achieves an F1 score of $0.90$ , not only improving the baseline model but also outperforming the 3D RVN model.", "In terms of the false negative (FN) rate, the transformer model also achieves the lowest rate with 15%.", "The transformer model also achieves the best performance in terms of the earliest detection.", "The clinical data of patient ID 30557 is shown in Table REF .", "The patient was clinically diagnosed as “Cognitively normal’’ from day 0000 to day 3816, and first diagnosed as “AD dementia’’ on day 4222.", "OASIS 3 Dataset contains two complete MRI brain scans of the patient performed on days 1448 and 2185.", "Transformer model infers `preclinical stage AD’ by taking the brain scan taken on day 1448 as input.", "This means that the algorithm is detecting `preclinical AD' 2774 days (4222 - 1448 = 2774) before the patient is clinically diagnosed.", "Table: Clinical data for patient 30025.Table: Scores on test data for binary classification." ], [ "Comparison between models.", "In Table REF , we compare the three models by measuring accuracy, precision, recall, and F1 score on our class 1 (i.e., `preclinical stage AD').", "As can be seen, the transformer model outperforms others in terms of all the measured parameters.", "It is worth pointing out how measuring just accuracy could not always be a good metric when comparing models.", "In terms of accuracy, if we compare the baseline model with our transformer algorithm, the improvement is just 3.21%.", "However, with F1 score e.g., the improvement is much higher (e.g., $8.4$ % higher F1 score).", "This is why, to have a clear picture of which model is performing better, other statistical measures, such as F1 score, precision and recall, have to be taken into consideration." ], [ "Discussion", "It is important to note that a subset of individuals with biomarker evidence of preclinical AD will not progress to developed AD dementia during their lifetime .", "Hence, a fraction of the `cognitively normal' labeled/diagnosed participants in the OASIS 3 dataset have preclinical AD biomarker, even though their disease never progressed to clinicaly diagnosed dementia stage.", "These individuals may contribute to the false positive (misclassification to AD) rate of our evaluation results.", "Figure: Overview of data labeling approach that includes AD dementiaIn addition to using the labeling scheme shown in Fig.", "REF , we performed further experiments by following the data labeling scheme shown in Fig.", "REF .", "In this case, the `other' class included patients with diagnosed AD, and we performed classification to differentiate individuals with `preclinical AD' from others, including the patients with developed AD dementia.", "We compared the performance of 3D CNN and 3D RVN models.", "As shown in Table REF , the 3D RVN model achieves an F1 score of $0.78$ and accuracy of $84.6\\%$ .", "Even though these results are not as high compared to the previously evaluated `preclinical AD’ assessment models (using the labeling scheme in Fig.", "REF ), the glimpse locations identified by this RVN agent carry potential.", "More specifically, they highlight the brain regions responsible for differentiating `preclinical AD' versus developed AD (stages 2 and 3).", "These findings will enhance our understanding on the deterioration of the brain structure with progression of AD (from preclinical to AD stages 2 and 3).", "Table: Binary approach to differentiate preclinical AD versus others, including developed AD." ], [ "Conclusion", "A lot of work has been done detecting MCI (stage 2), the earliest form of detectable Alzheimer's disease .", "The novelty and significance of our work is that we focus on and are able to predict AD at preclinical stage (stage 1), when all the parameters, physical assessments, and clinical data state that the patient is healthy.", "With that goal, we adapted two different attention-based network models, and compared their performances with a 3D CNN-based baseline that we implemented.", "The baseline model uses stacked MRI scans for classification.", "The first attention model is a recurrent attention model, which extracts glimpses from stacked images, and feeds them into recurrent attention units to get a classification result.", "The second model is a modified and re-purposed transformer network, which first extracts features of a sequence of images from a pre-trained network, and then feeds these features to a transformer structure to be able to classify the sequence of images.", "Among these three approaches, the transformer model outperformed others achieving an F1 score of $0.90$ and accuracy of 91.18% in preclinical (i.e., prior to MCI and dementia stage) Alzheimer disease detection.", "Experimental results demonstrate that, by using the MRI images effectively, it is possible to detect preclinical stage Alzheimer's disease with a very promising accuracy." ] ]	2011.14139
[ [ "The deformed Hermitian-Yang-Mills equation on almost Hermitian manifolds" ], [ "Abstract In this paper, we consider the deformed Hermitian-Yang-Mills equation on closed almost Hermitian manifolds.", "In the case of hypercritical phase, we derive a priori estimates under the existence of an admissible $\\mathcal{C}$-subsolution.", "As an application, we prove the existence of solutions for the deformed Hermitian-Yang-Mills equation under the condition of existence of a supersolution." ], [ "Introduction", "Motivated by Mirror Symmetry and Mathematical Physics, on a Kähler manifold $(M,\\chi )$ of complex dimension $\\dim _{\\mathbb {C}}M=n$ , the deformed Hermitian-Yang-Mills equation, which has been studied extensively, can be written as the following form: $\\textrm {Im}(\\chi +\\sqrt{-1}\\omega _{u})^{n}=\\tan (\\hat{\\theta })\\textrm {Re}(\\chi +\\sqrt{-1}\\omega _{u})^{n}$ where $\\hat{\\theta }$ is a constant and $\\omega $ is a smooth real $(1,1)$ -form, $\\omega _{u}=\\omega +\\sqrt{-1} \\partial \\overline{\\partial }{u}$ .", "Assume that $(\\lambda _{1}(u),\\lambda _{2}(u), \\cdots ,\\lambda _{n}(u))$ are the eigenvalues of $\\omega _{u}$ with respect to $\\chi .$ Without confusion, we also denote $\\lambda _{i}(u)$ by $\\lambda _{i}$ , $i=1,\\cdots , n.$ The equation (REF ) can be rewritten as $\\sum _{i}\\arctan \\lambda _{i}=\\hat{\\theta }.$ By solving this equation, we can find a Hermitian metric on the line bundle over $M$ such that the argument of Chern curvature is constant [20].", "For the dimension $n=2$ , one only need to solve a Monge-Ampère equation [20].", "In addition, Jacob-Yau [20] used a parabolic flow to prove the existence of solution when $(M,\\chi )$ has non-negative othogonal bisectional curvature and $\\hat{\\theta }$ satisfies the hypercritical phase condition, i.e., $n\\frac{\\pi }{2}> \\hat{\\theta }>(n-1)\\frac{\\pi }{2}$ for general dimensions.", "V. Pingali [22], [23] proved the existence of solution when $n=3$ .", "Collins-Jacob-Yau [10] gave the existence theorem of (REF ) under the condition of existence of subsolution in general dimensions.", "R. Takahashi [28] introduced the tangent Lagrangian phase flow and used it to proved the existence of solution of the deformed-Hermitian-Yang-Mills equation, assuming the existence of a $\\mathcal {C}$ -subsolution.", "For more details, we refer to [4], [5], [6], [9], [11], [12], [13] and the references therein.", "In this paper, we give a priori estimates on an almost Hermitian manifold $(M, \\chi , J)$ with real dimension $2n$ for the deformed Hermitian-Yang-Mills equation.", "Consider the following general equation $F(\\omega _{u})=f(\\lambda _{1},\\cdots , \\lambda _{n})=\\sum _{i}\\arctan \\lambda _{i}=h.$ where $h:M\\rightarrow ((n-1)\\frac{\\pi }{2}, n\\frac{\\pi }{2})$ is a given function on $M$ and $(\\lambda _{1}, \\cdots ,\\lambda _{n}) $ are the eigenvalues of $\\omega _{u}$ with respect to $\\chi $ .", "We now state our main result.", "Theorem 1.1 Assume $h:M\\rightarrow ((n-1)\\frac{\\pi }{2}, \\frac{n\\pi }{2})$ is a smooth function and $\\underline{u}:M\\rightarrow \\mathbb {R}$ is a smooth $\\mathcal {C}$ -subsolution (Definition REF ).", "Suppose $\\omega $ is a smooth real $(1, 1)$ -form.", "Let $u$ be a solution of (REF ), for each $0<\\beta <1$ , then we have $\\Vert u\\Vert _{C^{k,\\beta }}\\le C,$ where $C$ depends on $\\Vert h\\Vert _{C^{k+1}(M)}$ , $\\inf _{M} h$ , $\\underline{u}$ , $k$ , $\\beta $ , $(M,\\chi , J)$ and $\\omega $ .", "Almost Hermitian manifolds have been studied extensively motivated by differential geometry and mathematical physics (see [2], [17], [18], [19], [25] and references therein).", "The theory of fully nonlinear elliptic equations were developed, such as [7], [8], [32], etc.", "On the other hand, the geometry of complex vector bundles over almost Hermitian manifolds were researched in [1], [31], [33] and references therein.", "Note that Lau-Tseng-Yau [21] studied SYZ mirror symmetry in the context of non-Kähler Calabi-Yau manifolds.", "Generalized complex geometry, proposed by Hitchin [18], is closed related to flux compactifications in string theory [16].", "In fact, the deformed Hermitian-Yang-Mills equation plays an important role in mirror symmetry and string theory [12].", "It is natural to consider the equation (REF ) to non-Kähler case.", "Motivated by these works, we prove this result.", "In order to prove the theorem above, we will use the maximum principle.", "A crucial ingredient of the proof need the equation (REF ) is concave.", "To prove that the equation is concave, we need to assume $h$ satisfies the hypercritical phase condition, i.e., $n\\frac{\\pi }{2}> \\hat{\\theta }>(n-1)\\frac{\\pi }{2}$ .", "In addition, to use the Proposition REF provided by $\\mathcal {C}$ -subsolution, the equation have to satisfy some properties in Lemma REF .", "We can obtain these properties when $h$ satisfies the hypercritical phase condition.", "In the proof of the second order estimates, we apply the maximum principle.", "We will use the argument of [7] and [32].", "To deal with the bad third order terms, we need to give a lower bound for the third order terms from the concavity of the equation (see Lemma REF ).", "In the proof, we give a positive lower bound for the complex eigenvalues $\\omega _{u}$ provided by $\\inf _{M} h>\\frac{(n-1)\\pi }{2}$ .", "When $h\\in ((n-1)\\frac{\\pi }{2}, n\\frac{\\pi }{2})$ is a constant, we can prove the existence under the condition of existence of supersolution.", "Theorem 1.2 Assume there exists a supersolution $\\hat{u}$ of (REF ), i.e.", "$F(\\omega _{\\hat{u}})\\le h$ .", "Suppose $F(\\omega _{\\hat{u}})>\\frac{(n-1)\\pi }{2}$ and $h\\in (\\frac{(n-1)\\pi }{2}, \\frac{n\\pi }{2})$ is a constant.", "If there exists a $\\mathcal {C}$ -subsolution $\\underline{u}$ for (REF ), we have a function $u$ and a constant $c$ such that $\\sum _{i}\\arctan \\lambda _{i}=h+c,$ where $ h+c>\\frac{n-1}{2}\\pi $ .", "Remark 1.1 If $(M,\\chi )$ is a Kähler manifold, then the supersolution $\\hat{u}$ is a solution of the equation (REF ), using $\\hat{\\theta }$ is an invariant.", "However, $\\hat{\\theta }$ is not an invariant on almost Hermitian manifolds.", "Let $\\pi :L\\rightarrow M $ be a complex line bundle on $M$ and $\\varpi $ be a Hermitian metric on $L$ (for more details, see [1] or [31]).", "There exists a unique type $(1, 0)$ Hermitian connection $D_{\\varpi }$ which is called the canonical Hermitian connection.", "Let $F(\\varpi )$ be the curvature form of connection $D_{\\varpi }$ and $F^{1,1}(\\varpi )$ be the $(1,1)$ part of the curvature of $\\varpi $ .", "Denote $\\varpi (u)=e^{-u}\\varpi $ .", "We have (see [1] or [31]) $F^{1,1}(\\varpi (u))=F^{1,1}(\\varpi )+\\partial \\bar{\\partial } u.", "$ We assume $\\int _{M}\\chi ^{n}=1.$ Set $\\omega =\\sqrt{-1} F^{1,1}(\\varpi )$ and $\\hat{\\theta }(\\varpi (u))=\\text{Arg}\\int _{M}\\frac{(\\chi +\\sqrt{-1}(\\omega +\\sqrt{-1} \\partial \\overline{\\partial }u))^{n}}{\\chi ^{n}}\\chi ^{n}.$ Here $\\text{Arg}\\, \\varphi $ means the argument of a complex function $\\varphi $ .", "Note that (REF ) is equivalent to (REF ).", "We immediately obtain Corollary 1.3 Suppose there exists a supersolution $\\hat{u}$ of (REF ).", "Assume $F(\\omega _{\\hat{u}})>\\frac{(n-1)\\pi }{2}$ and $h\\in (\\frac{(n-1)\\pi }{2}, \\frac{n\\pi }{2})$ is a constant.", "Suppose there exists $\\varpi (\\underline{u})=e^{-\\underline{u}}\\varpi $ such that $\\underline{u}$ is a $\\mathcal {C}$ -subsolution of (REF ).", "There exists a Hermitian metric $\\varpi (u)$ on line bundle $L$ such that the argument of $\\frac{\\big (\\chi +\\sqrt{-1}F^{1,1}((\\varpi )(u))\\big )^{n}}{\\chi ^{n}}$ is constant.", "In Kähler case, the proof in [10] relied on the argument $\\hat{\\theta }$ of the integral is independent of the choice of $\\omega _{u}$ .", "They used this fact to prove the $\\mathcal {C}$ -subsolution is preserved along the family of equations used in the continuity method.", "However, $\\hat{\\theta }(\\varpi (u))$ depend on $u$ in our case.", "Under the existence of supersolution, we prove the $\\mathcal {C}$ -subsolution and hypercritical condition are preserved when we use the continuity method, by using the arguments of [10] and [26].", "The organization of the paper is as follows.", "In section 2, we recall the definition of $\\mathcal {C}$ -subsolution and some properties of equation (REF ).", "In section 3, we give the $C^{0}$ estimates.", "We use the argument of Székelyhidi [27] (see also [3]).", "In section 4, the gradient estimates are proved.", "In Section 5, we will give the second order estimates and complete the proof of Theorem REF .", "In section 6, by the continuity method, we prove Theorem REF under the condition of the existence of the supersolution.", "Acknowledgment: The authors would like to thank the referees for many useful suggestions and comments." ], [ "preliminaries", "On an almost Hermitian manifold $(M,\\chi ,J)$ with real dimension $2n$ , for any $(p,q)$ -form $\\beta $ , we can define $\\partial $ and $\\overline{\\partial }$ operators (cf.", "[18], [7]).", "Denote by $A^{1,1}(M)$ the space of smooth real (1,1) forms on $(M,\\chi ,J)$ .", "Then, for any $\\varphi \\in C^{2}(M)$ , $\\sqrt{-1} \\partial \\overline{\\partial }\\varphi =\\frac{1}{2}(dJd\\varphi )^{(1,1)}$ is a real $(1,1)$ -form in $A^{1,1}(M)$ .", "Let $\\lbrace e_{i}\\rbrace _{i=1}^{n}$ be a local frame for $T_{\\mathbb {C}}^{(1,0)}M$ and $\\lbrace \\theta ^{1}, \\cdots , \\theta ^{n}\\rbrace $ be a dual coframe associated to the metric $\\chi $ on $(M,\\chi ,J)$ (cf.", "[18]).", "Denote $\\chi _{i\\overline{j}}=\\chi (e_{i},\\overline{e}_{j})$ and $g_{i\\overline{j}}=\\omega (e_{i},\\overline{e}_{j})$ .", "Then $\\chi =\\chi _{i\\overline{j}}\\sqrt{-1}\\theta ^{i}\\wedge \\overline{\\theta }^{j}$ and $\\omega =g_{i\\overline{j}}\\sqrt{-1}\\theta ^{i}\\wedge \\overline{\\theta }^{j}$ .", "We have $\\varphi _{i\\overline{j}}=(\\sqrt{-1} \\partial \\overline{\\partial }\\varphi )(e_{i},\\overline{e}_{j})=e_{i}\\overline{e}_{j}(\\varphi )-[e_{i},\\overline{e}_{j}]^{(0,1)}(\\varphi ),$ where $[e_{i}, \\bar{e}_{j}]^{(0,1)}$ is the $(0,1)$ part of the Lie bracket $[e_{i}, \\bar{e}_{j}]$ .", "We use the following notation $F^{i\\overline{j}}=\\frac{\\partial \\sum _{k}\\arctan \\lambda _{k}(\\tilde{g})}{\\partial \\tilde{g}_{i\\overline{j}}},$ where $\\tilde{g}_{i\\bar{j}}=g_{i\\bar{j}}+u_{i\\bar{j}}.$ For any point $x_{0}\\in M$ , let $\\lbrace e_{i}\\rbrace _{i=1}^{n}$ be a local unitary frame (with respect to $\\chi $ ) such that $\\tilde{g}_{i\\overline{j}}(x_{0})=\\delta _{ij}\\tilde{g}_{i\\overline{i}}(x_{0})$ .", "We denote $\\tilde{g}_{i\\overline{i}}(x_{0})$ by $\\lambda _{i}$ .", "It is useful to order $\\lbrace \\lambda _{i}\\rbrace $ such that $\\lambda _{1}\\ge \\lambda _{2}\\ge \\cdots \\ge \\lambda _{n}.$ Then at $x_{0}$ , we have $F^{i\\overline{j}}=F^{i\\overline{i}}\\delta _{ij}=\\frac{1}{1+\\lambda _{i}^{2}}\\delta _{ij}.$ By [27] or [14], we deduce $F^{i\\bar{k},j\\bar{l}}=f_{ij}\\delta _{ik}\\delta _{jl}+\\frac{f_{i}-f_{j}}{\\lambda _{i}-\\lambda _{j}}(1-\\delta _{ij})\\delta _{il}\\delta _{jk}.$ It follows that, at $x_{0}$ , $F^{i\\overline{k},j\\overline{l}}=\\left\\lbrace \\begin{array}{ll}F^{i\\overline{i},i\\overline{i}}, \\text{~~~~if $i=j=k=l$;}\\\\[1mm]F^{i\\overline{k},k\\overline{i}}, \\text{~~~~if $i=l$, $k=j$, $i\\ne k$;}\\\\[1mm]0, \\text{\\quad \\quad \\quad ~otherwise.", "}\\end{array}\\right.$ Moreover, at $x_{0}$ , $\\begin{split}F^{i\\overline{i},i\\overline{i}} & = -\\frac{2\\lambda _{i}}{(1+\\lambda ^{2}_{i})^{2}},\\\\F^{i\\overline{k},k\\overline{i}} & = -\\frac{\\lambda _{i}+\\lambda _{k}}{(1+\\lambda ^{2}_{i})(1+\\lambda ^{2}_{k})}.\\end{split}$ The linearization operator of (REF ) is $L:=\\sum _{i,j}F^{i\\bar{j}}(e_{i}\\bar{e}_{j}-[e_{i},\\bar{e}_{j}]^{0,1}).$ Note that $[e_{i},\\bar{e}_{j}]^{0,1}$ are first order defferential operators.", "By (REF ), $L$ is a second order elliptic operator." ], [ "$\\mathcal {C}$ -subsolution", "Now we recall the definition of $\\mathcal {C}$ -subsolution of (REF ) ([10], [27]).", "Denote $\\Gamma _{n}=\\lbrace \\lambda =(\\lambda _{1}, \\cdots , \\lambda _{n})\\in \\mathbb {R}^{n},\\ \\lambda _{i}>0, 1\\le i \\le n\\rbrace ,$ $\\Gamma =\\lbrace \\lambda =(\\lambda _{1}, \\cdots , \\lambda _{n})\\in \\mathbb {R}^{n},\\ \\sum _{i}\\arctan (\\lambda _{i})> (n-1)\\frac{\\pi }{2}\\rbrace ,$ and $\\Gamma ^{\\sigma }=\\lbrace \\lambda \\in \\Gamma , \\sum _{i}\\arctan (\\lambda _{i})> \\sigma \\rbrace ,$ where $\\sigma \\in ((n-1)\\frac{\\pi }{2}, n\\frac{\\pi }{2})$ .", "Definition 2.1 ([10], [27]) We say that a smooth function $\\underline{u}:M\\rightarrow R$ is a $\\mathcal {C}$ -subsolution of (REF ) if at each point $x\\in M$ , we have $\\left\\lbrace \\lambda \\in \\Gamma : \\sum _{i=1}^{n}\\arctan (\\lambda _{i})=h(x), \\text{\\ and\\ } \\lambda -\\lambda (\\underline{u})\\in \\Gamma _{n}\\right\\rbrace $ is bounded.", "Collins-Jacob-Yau gave an explicit description of $\\mathcal {C}$ -subsolution.", "Lemma 2.1 ([10], Lemma 3.3) A smooth function $\\underline{u}: M\\rightarrow \\mathbb {R} $ is a $\\mathcal {C}$ -subsolution of (REF ) if and only if at each point $x\\in M$ , for all $j=1,\\cdots , n$ , we have $\\sum _{i\\ne j }\\arctan (\\lambda _{i}(\\underline{u}))>h(x)-\\frac{\\pi }{2},$ where $\\lambda _{1}(\\underline{u}), \\cdots , \\lambda _{n}(\\underline{u})$ are the eigenvalues of $\\omega _{\\underline{u}}$ with respect to $\\chi $ .", "Therefore, there are uniform constants $\\delta , R>0$ such that at each $x\\in M$ we have $(\\lambda (\\underline{u})-\\delta \\textbf {1}+\\Gamma _{n})\\cap \\partial \\Gamma ^{h(x)}\\subset B_{R}(0),$ where $B_{R}(0)$ is a $R$ -radius ball in $\\mathbb {R}^{n}$ with center 0, $\\textbf {1}=(1,1,\\cdots ,1)$ .", "We now prove the following lemma: Lemma 2.2 Suppose $h\\in ((n-1)\\frac{\\pi }{2},n\\frac{\\pi }{2})$ , then we have the following properties: $f_{i}=\\frac{\\partial f}{\\partial \\lambda _{i}}>0$ for all $i$ , and the equation (REF ) is concave, $\\sup _{\\partial \\Gamma } f<\\inf _{M}h$ , for any $\\sigma <\\sup _{\\Gamma } f$ and $\\lambda \\in \\Gamma $ we have $\\underset{t\\rightarrow \\infty }{\\lim }f(t\\lambda )>\\sigma .$ Note $f_{i}=\\frac{1}{1+\\lambda _{i}^{2}}$ .", "It is obvious that $f_{i}>0$ and $\\sup _{\\partial \\Gamma } f<\\inf _{M}h$ , if $h>(n-1)\\frac{\\pi }{2}$ .", "When $h>(n-1)\\frac{\\pi }{2}$ , we have $\\lambda _{i}>0, ~ \\textrm {for}~ i=1,2,\\cdots ,n.$ In fact, if there is $\\lambda _{i}\\le 0$ , then we must have $\\sum _{j}\\arctan \\lambda _{j}\\le (n-1)\\frac{\\pi }{2} $ and this is a contradiction.", "Hence, by (REF ), $f$ is concave and $\\underset{t\\rightarrow \\infty }{\\lim }f(t\\lambda )=n\\frac{\\pi }{2}>\\sigma $ .", "Using the above Lemma and [27], we have the following Proposition.", "It plays an important role in the proof.", "Proposition 2.3 Let $[a,b]\\subset ((n-1)\\frac{\\pi }{2}, n\\frac{\\pi }{2})$ and $\\delta , R>0$ .", "There exists $\\theta >0$ , depending on $\\sigma $ and the set in (REF ), with the following property: suppose that $\\sigma \\in [a,b]$ and $B$ is a Hermitian matrix such that $(\\lambda (B)-2\\delta \\textbf {1}+\\Gamma _{n})\\cap \\partial \\Gamma ^{\\sigma }\\subset B_{R}(0).$ Then for any Hermitian matrix A with eigenvalues $\\lambda (A)\\in \\partial \\Gamma ^{\\sigma }$ and $\|\\lambda (A)\|>R$ , we either have $\\sum _{p,q}F^{p\\overline{q}}(A)[B_{p\\overline{q}}-A_{p\\overline{q}}]>\\theta \\sum _{p}F^{p\\overline{p}}(A)$ or $F^{i\\overline{i}}(A)>\\theta \\sum _{p}F^{p\\overline{p}}(A)$ for all $i$ .", "In addition, there exists a constant $\\mathcal {K}$ depending on $\\sigma $ such that $\\sum _{i}F^{i\\overline{i}}>\\mathcal {K}.$ Corollary 2.4 Suppose $h\\in ((n-1)\\frac{\\pi }{2},n\\frac{\\pi }{2})$ .", "Assume $\\underline{u}$ is an admissible $\\mathcal {C}$ -subsolution and $u$ is the smooth solution for (REF ).", "Then there exists a constant $\\theta >0$ (depending only on $h$ and $\\underline{u}$ ) such that either $L(\\underline{u}-u)\\ge \\theta \\sum _{i}F^{i\\bar{i}}(\\tilde{g})$ or $F^{k\\bar{k}}(\\tilde{g})\\ge \\theta \\underset{i}{\\sum }F^{i\\bar{i}}(\\tilde{g}),~ \\textrm {for}~ k=1,2,\\cdots ,n$ if $\\lambda (\\omega _{u})\\in (\\lambda (\\omega _{\\underline{u}})-\\delta \\textbf {1}+\\Gamma _{n})\\cap \\partial \\Gamma ^{h}$ .", "In addition, there is a constant $\\mathcal {K}>0$ depending on $h$ and $\\underline{u}$ such that $\\mathcal {F}:=\\sum _{i}F^{i\\bar{i}}(\\tilde{g})>\\mathcal {K},~ \\textrm {if}~\\lambda (\\omega _{u})\\in \\partial \\Gamma ^{h}.$ By Definition REF , there are uniform constants $\\delta , R>0$ such that at each $x\\in M$ we have $(\\lambda (\\underline{u})-\\delta \\textbf {1}+\\Gamma _{n})\\cap \\partial \\Gamma ^{h(x)}\\subset B_{R}(0).$ If $\|\\lambda (u)\|>R$ , by Proposition REF , the results follow.", "If $\|\\lambda (u)\|\\le R$ , then $1\\ge F^{i\\bar{i}}\\ge \\frac{1}{1+R^{2}}, i=1,2, \\cdots , n$ , which implies (REF ) and (REF ) hold." ], [ "Zero order estimates", "In this seciton we prove the $C^{0}$ estimates.", "We need the following Proposition provided by [7].", "Proposition 3.1 Let $(M, \\chi , J)$ be a compact almost Hermitian manifold.", "Suppose that $\\psi $ satisfies $\\omega +\\sqrt{-1} \\partial \\overline{\\partial }\\psi >0,\\ \\ \\sup _{M}\\psi =0.$ Then there exists a constant $C$ depending only on $(M, \\chi , J)$ and $\\omega $ such that $\\int _{M}(-\\psi )\\chi ^{n}\\le C.$ There exists a constant $C_{0}$ such that $C_{0}\\chi \\ge \\omega .$ Therefore, by (REF ), $C_{0}\\chi +\\sqrt{-1} \\partial \\overline{\\partial }\\psi >0$ .", "Then by [7], we have $\\int _{M}(-\\psi )\\chi ^{n}\\le C.$ Indeed, by (REF ), the assumption in Proposition REF is satisfied in our paper.", "The following variant of the Alexandroff-Bakelman-Pucci maximum principle [27], similarly with Gilbarg-Trudinger [15], is used to prove the $C^{0}$ estimates.", "Proposition 3.2 Let $\\varphi :B_{1}(0)\\rightarrow \\mathbb {R}$ be a smooth function, such that $\\varphi (0)+\\varepsilon \\le \\inf _{\\partial B_{1}(0)}\\varphi $ , where $\\varepsilon >0$ , $B_{1}(0)\\subset \\mathbb {R}^{2n}$ .", "Define the set $P=\\left\\lbrace x\\in B_{1}(0):\\begin{matrix} \|D\\varphi (x)\|<\\frac{\\varepsilon }{2}, \\text{\\ and\\ }\\varphi (y)\\ge \\varphi (x)+D\\varphi (x)\\cdot (y-x)\\\\\\text{ for all } y\\in B_{1}(0)\\end{matrix} \\right\\rbrace .$ Then there exists a constant $c_{0}$ depending only on $n$ such that $c_{0}\\varepsilon ^{2n}\\le \\int _{P}\\det (D^{2}\\varphi ).$ The $C^{0}$ estimates follow the argument of [27] or [7].", "It is similar to [32].", "For the reader's convenience, we include the proof.", "Proposition 3.3 Let $u$ be the solution for (REF ) with $\\sup _{M}(u-\\underline{u})=0$ .", "Then $\\Vert u\\Vert _{L^{\\infty }}\\le C$ for some constant $C>0$ depending on $(M, \\chi , J)$ , $\\omega $ , $h$ and $\\underline{u}$ .", "From the hypothesis, it suffices to estimate the infimum $m_{0}=\\inf _{M}(u-\\underline{u})$ .", "We may assume $m_{0}$ is attained at $x_{0}$ .", "Choose a local coordinate chart $(x^{1},\\cdots , x^{2n})$ in a neighborhood of $x_{0}$ containing the unit ball $B_{1}(0)\\subset \\mathbb {R}^{2n}$ such that the coordinates of $x_{0}$ are the origin $0\\in \\mathbb {R}^{2n}$ .", "Consider the test function $v:= u-\\underline{u}+\\varepsilon \\sum _{i=1}^{2n}(x^{i})^{2}$ for a small $\\varepsilon >0$ determined later.", "Then we have $v(0)=m_{0}~ \\textrm {and}~v\\ge m_{0}+\\varepsilon ~\\textrm {on}~ \\partial B_{1}(0).$ We define the lower contact set of $v$ by $P:=\\Big \\lbrace x\\in B_{1}(0): \|Dv(x)\|\\le \\frac{\\varepsilon }{2},v(y)\\ge v(x)+Dv(x)\\cdot (y-x),\\text{\\ for all\\ } y\\in B_{1}(0)\\Big \\rbrace .$ By Proposition REF , we have $c_{0}\\varepsilon ^{2n}\\le \\int _{P}\\det (D^{2}v).$ Let $\\Big (D^{2}(u-\\underline{u})\\Big )^{J}$ be the $J$ -invariant part of $\\Big (D^{2}(u-\\underline{u})\\Big )$ , i.e., $\\Big (D^{2}(u-\\underline{u})\\Big )^{J}=\\frac{1}{2}(D^{2}(u-\\underline{u})+J^{T}D^{2}(u-\\underline{u})J),$ where $J^{T}$ is the transpose of $J$ .", "Note that $0\\in P$ and $D^{2}v\\ge 0$ on $P$ .", "Then we deduce $\\left(D^{2}(u-\\underline{u})\\right)^{J}(x)\\ge (D^{2}v)^{J}(x)-C\\varepsilon Id\\ge -C\\varepsilon Id, \\text{\\ for\\ } x\\in P.$ Consider the bilinear form $H(v)(X,Y)=\\sqrt{-1} \\partial \\overline{\\partial }v(X, JY)$ .", "In fact, we obtain $H(v)(X,Y)(x)=\\frac{1}{2}\\left(D^{2}v\\right)^{J}(x)+E(v)(x),\\ \\ x\\in M ,$ where $E(v)(x)$ is an error matrix which depends linearly on $Dv(x)$ (see e.g.", "[30]).", "Using $\|D(u-\\underline{u})\|\\le \\frac{5\\varepsilon }{2}$ on $P$ and (REF ), it follows $\\begin{split}H(u)-H(\\underline{u})=&\\Big (D^{2}u+E(Du)\\Big )^{J}-\\Big (D^{2}\\underline{u}+E(D\\underline{u})\\Big )^{J} \\\\=&\\Big (D^{2}(u-\\underline{u})\\Big )^{J}+ \\Big (E(D(u-\\underline{u}))\\Big )^{J}\\\\\\ge & -C\\varepsilon Id.\\end{split}$ Hence $\\omega _{u}-\\omega _{\\underline{u}}\\ge -C\\varepsilon \\chi .$ Therefore, if we choose $\\varepsilon $ sufficient small such that $C\\varepsilon \\le \\delta $ , then $\\lambda (u)\\in \\lambda (\\underline{u})-\\delta \\textbf {1}+\\Gamma _{n}.$ On the other hand, the equation (REF ) implies $\\lambda (u)\\in \\partial \\Gamma ^{h}.$ Consequently, $\\lambda (u)\\in (\\lambda (\\underline{u})-\\delta \\textbf {1}+\\Gamma _{n})\\cap \\partial \\Gamma ^{h} \\subset B_{R}(0)$ for some $R>0$ by the argument (REF ).", "This gives an upper bound for $H(u)$ and hence also for $H(v)-E(Dv)$ on $P$ .", "Note $\\det (A+B)\\ge \\det (A)+\\det (B)$ for positive definite Hermitian matrices $A, B$ .", "Recall the definition of $(D^{2}v)^{J}$ in (REF ).", "Then on $P$ , we have $\\begin{split}\\det (D^{2}v)\\le & 2^{2n-1}\\det ((D^{2}v)^{J}) \\\\= & 2^{2n-1}\\det (H(v)-E(Dv))\\le C.\\end{split}$ Plugging (REF ) into (REF ), we obtain $c_{0}\\varepsilon ^{2n}\\le C\|P\|.$ For each $x\\in P$ , choosing $y=0$ in $(\\ref {P})$ , we have $m_{0}=v(0)\\ge v(x)-\|Dv(x)\|\|x\|\\ge v(x)-\\frac{\\varepsilon }{2}.$ We may and do assume $m_{0}+\\varepsilon \\le 0$ (otherwise we are done), then, on $P$ , $-v\\ge \|m_{0}+\\varepsilon \|.$ Integrating it on $P$ , we get $\|P\|\\le \\frac{\\int _{P}(-v)\\chi ^{n}}{\|m_{0}+\\varepsilon \|}\\le \\frac{C}{\|m_{0}+\\varepsilon \|},$ where in the last inequality we used Proposition REF .", "By (REF ), we get a uniform lower bound for $m_{0}$ ." ], [ "First order estimate", "In this section,we give the proof of the $C^{1}$ estimates.", "Let $\|\\nabla u\|_{\\chi }$ be the norm of gradient $u$ with respect to $\\chi $ .", "For convenience, we use $\|\\nabla u\|$ to denote $\|\\nabla u\|_{\\chi }.$ We denote $\\mathcal {F}=\\sum _{i}F^{i\\bar{i}}.$ Proposition 4.1 $\|\\nabla u\|\\le C$ for some constant $C$ depending on $(M ,\\chi , J)$ , $\\omega $ , $\\Vert h\\Vert _{C^{1}}$ and $\\underline{u}$ .", "Proof.", "Let $\\zeta = Ae^{B_{1}\\eta }$ where $\\eta =\\underline{u}-u-\\inf _{M}(\\underline{u}-u)\\ge 0,$ $A, B_{1}$ are positive constants to be chosen later.", "Consider the test function $\\hat{Q}:= e^{\\zeta }\|\\nabla u\|^{2}.$ Suppose $\\hat{Q}$ achieves maximum at the point $x_{0}$ .", "Near $x_{0}$ , there exists a local unitary frame $\\lbrace e_{i}\\rbrace _{i=1}^{n}$ (with respect to $\\chi $ ) such that at $x_{0}$ , we have $\\text{$\\chi _{i\\bar{j}}=\\delta _{ij}$, $\\tilde{g}_{i\\bar{j}}=\\delta _{ij} \\tilde{g}_{i\\bar{j}}$ and $\\tilde{g}_{1\\bar{1}}\\ge \\tilde{g}_{2\\bar{2}}\\ge \\cdots \\ge \\tilde{g}_{n\\bar{n}}$.", "}$ From now on, we will use the Einstein summation convention, and all the following calculations are done at $x_{0}$ .", "The $C$ below in this section denote the constants that may change from line to line, where $C$ depends on all the allowed data that we determined later.", "Recall that $L$ is defined in (REF ).", "By the maximum principle, it follows that $\\begin{split}0\\ge \\frac{L(\\hat{Q})}{B_1\\zeta e^{\\zeta }\|\\nabla u\|^{2}}=&\\frac{L(\|\\nabla u\|^{2})}{B_1\\zeta \|\\nabla u\|^{2}}+\\frac{L(e^{\\zeta })}{B_1\\zeta e^{\\zeta }}+2F^{i\\bar{i}}\\textrm {Re}\\Big \\lbrace e_{i}(\\zeta )\\frac{\\bar{e}_{i}(\|\\nabla u\|^{2})}{B_1\\zeta \|\\nabla u\|^{2}}\\Big \\rbrace \\\\=&\\frac{L(\|\\nabla u\|^{2})}{B_1\\zeta \|\\nabla u\|^{2}}+L(\\eta )+B_{1}(1+\\zeta )F^{i\\bar{i}}\|e_{i}(\\eta )\|^{2}\\\\&+\\frac{2}{\|\\nabla u\|^{2}}\\sum _{j}F^{i\\bar{i}}\\textrm {Re}\\Big \\lbrace e_{i}(\\eta )\\bar{e}_{i}e_{j}(u)\\bar{e}_{j}(u)+e_{i}(\\eta )\\bar{e}_{i}\\bar{e}_{j}(u)e_{j}(u)\\Big \\rbrace .\\end{split}$ Now we deal with these terms in turn.", "First we have Lemma 4.2 $L(\|\\nabla u\|^{2})\\ge 2\\sum _{j}\\textrm {Re}\\lbrace e_{j}(h)\\bar{e}_{j}u\\rbrace + (1-\\varepsilon ) \\sum _{j}F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})- \\frac{C}{\\varepsilon } \\mathcal {F}\|\\nabla u\|^{2}.$ By direct calculation, $L(\|\\nabla u\|^{2})=F^{i\\bar{i}}\\Big ({e_{i}e_{\\bar{i}}}(\|\\nabla u\|^{2})-[e_{i},\\bar{e}_{i}]^{0,1}(\|\\nabla u\|^{2})\\Big ):= I+II+III,$ where $I=\\sum _{j}F^{i\\bar{i}}(e_{i}\\bar{e}_{i}e_{j}u-[e_{i},\\bar{e}_{i}]^{0,1}e_{j}u)\\bar{e}_{j}u;$ $II=\\sum _{j}F^{i\\bar{i}}(e_{i}\\bar{e}_{i}\\bar{e}_{j}u-[e_{i},\\bar{e}_{i}]^{0,1}\\bar{e}_{j}u)e_{j}u;$ $III=\\sum _{j}F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2}).$ Differentiating (REF ) along $e_{j}$ without summation, we have $ F^{i\\overline{i}}e_{j}(g_{i\\overline{i}})+F^{i\\bar{i}}(e_{j}e_{i}\\bar{e}_{i}u-e_{j}[e_{i},\\bar{e}_{i}]^{0,1}u)=e_{j}(h).$ Recall the definition of Lie bracket $[e_{i},e_{j}]=e_{i}e_{j}-e_{j}e_{i}$ .", "Then we have $\\begin{split}&I+II\\\\=& 2 \\sum _{j}F^{i\\bar{i}}\\textrm {Re}\\Big \\lbrace (e_{i}\\bar{e}_{i}e_{j}u-[e_{i},\\bar{e}_{i}]^{0,1}e_{j}u)\\bar{e}_{j}(u)\\Big \\rbrace \\\\= & 2 \\sum _{j}F^{i\\bar{i}}\\textrm {Re} \\Big \\lbrace (e_{j}e_{i}\\bar{e}_{i}u+e_{i}[\\bar{e}_{i},e_{j}]u+[e_{i},e_{j}]\\bar{e}_{i}u-[e_{i},\\bar{e}_{i}]^{0,1}e_{j}u)\\bar{e}_{j}(u)\\Big \\rbrace \\\\= & 2\\sum _{j}\\textrm {Re}\\lbrace e_{j}(h)\\bar{e}_{j}(u)\\rbrace -2 \\sum _{j}F^{i\\overline{i}}\\textrm {Re}\\lbrace e_{j}(g_{i\\overline{i}})\\bar{e}_{j}(u)\\rbrace +2\\sum _{j}\\textrm {Re}\\lbrace F^{i\\bar{i}}e_{j}[e_{i},\\bar{e}_{i}]^{0,1}u\\bar{e}_{j}(u)\\rbrace \\\\&+2\\textrm {Re}\\sum _{j}\\Big \\lbrace F^{i\\bar{i}}\\left(e_{i}[\\bar{e}_{i},e_{j}]u+[e_{i},e_{j}]\\bar{e}_{i}u-[e_{i},\\bar{e}_{i}]^{0,1}e_{j}u\\right)\\bar{e}_{j}(u)\\Big \\rbrace .\\\\\\end{split}$ We may assume $\|\\nabla u\|>1$ .", "It follows $\\begin{split}I+II\\ge & 2\\sum _{j}\\textrm {Re}\\lbrace e_{j}(h)\\bar{e}_{j}u\\rbrace -C\|\\nabla u\|\\sum _{j}F^{i\\bar{i}}(\|e_{i}e_{j}u\|+\|e_{i}\\bar{e}_{j}u\|)-C\|\\nabla u\|^{2} \\mathcal {F}\\\\\\ge &2\\sum _{j}\\textrm {Re}\\lbrace e_{j}(h)\\bar{e}_{j}u\\rbrace -\\frac{C}{\\varepsilon }\|\\nabla u\|^{2}\\mathcal {F}-\\varepsilon \\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2}).\\end{split}$ Combining (REF ) with (REF ), we obtain $L(\|\\nabla u\|^{2})\\ge 2\\sum _{j}\\textrm {Re}\\lbrace e_{j}(h)\\bar{e}_{j}u\\rbrace + (1-\\varepsilon ) \\sum _{j}F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})- \\frac{C}{\\varepsilon } \\mathcal {F}\|\\nabla u\|^{2}.$ Using the above Lemma, it follows $\\begin{split}\\frac{L(\|\\nabla u\|^{2})}{B_{1}\\zeta \|\\nabla u\|^{2}}\\ge & \\frac{2}{B_1\\zeta \|\\nabla u\|^{2}}\\sum _{j}\\textrm {Re}\\lbrace e_{j}(h)\\bar{e}_{j}u\\rbrace \\\\&+ (1-\\varepsilon ) \\sum _{j}F^{i\\bar{i}}\\frac{\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2}}{B_1\\zeta \|\\nabla u\|^{2}}- \\frac{C}{B_1\\zeta \\varepsilon } \\mathcal {F}.\\end{split}$ Now we estimate the last term of (REF ).", "By the Cauchy-Schwarz inequality, for $0<\\varepsilon \\le \\frac{1}{2}$ , we have $\\begin{split}&2\\sum _{j}F^{i\\bar{i}}\\textrm {Re}\\Big \\lbrace e_{i}(\\eta )\\bar{e}_{i}e_{j}(u)\\bar{e}_{j}(u)\\Big \\rbrace \\\\=& 2\\sum _{j}F^{i\\bar{i}}\\textrm {Re}\\Big \\lbrace e_{i}(\\eta )\\bar{e}_{j}(u)\\big \\lbrace e_{j}\\bar{e}_{i}(u)-[e_{j},\\bar{e}_{i}]^{0,1}(u)-[e_{j},\\bar{e}_{i}]^{1,0}(u)\\big \\rbrace \\Big \\rbrace \\\\=& 2F^{i\\bar{i}}\\lambda _{i}\\textrm {Re}\\big \\lbrace e_{i}(\\eta )\\bar{e}_{i}(u)\\big \\rbrace -2\\sum _{j}F^{i\\bar{i}}\\textrm {Re}\\left\\lbrace e_{i}(\\eta )\\bar{e}_{j}(u) g_{i\\bar{j}}\\right\\rbrace \\\\& -2\\sum _{j}F^{i\\bar{i}}\\textrm {Re}\\big \\lbrace e_{i}(\\eta )\\bar{e}_{j}(u)[e_{j},\\bar{e}_{i}]^{1,0}(u)\\big \\rbrace \\\\\\ge & 2F^{i\\bar{i}}\\lambda _{i}\\textrm {Re}\\lbrace e_{i}(\\eta )\\bar{e}_{i}(u)\\rbrace -\\varepsilon B_1\\zeta \|\\nabla u\|^{2}F^{i\\bar{i}}\|e_{i}(\\eta )\|^{2}-\\frac{C}{B_1\\zeta \\varepsilon }\|\\nabla u\|^{2}\\mathcal {F},\\end{split}$ where in the last inequality we used $\|\\nabla u\|>1.$ When $0<\\varepsilon \\le \\frac{1}{2}$ , we have $1\\le (1-\\varepsilon )(1+2\\varepsilon )$ .", "Using the Cauchy-Schwarz inequality again we obtain $\\begin{split}&2\\sum _{j}F^{i\\bar{i}}\\textrm {Re}\\Big \\lbrace e_{i}(\\eta )\\bar{e}_{i}\\bar{e}_{j}(u)e_{j}(u)\\Big \\rbrace \\\\\\ge & -\\frac{(1-\\varepsilon )}{B_1\\zeta }\\sum _{j}F^{i\\bar{i}}\|\\bar{e}_{i}\\bar{e}_{j}(u)\|^{2}-(1+2\\varepsilon )B_1\\zeta \|\\nabla u\|^{2}F^{i\\bar{i}}\|e_{i}(\\eta )\|^{2}.\\end{split}$ Therefore, $\\begin{split}&\\frac{2}{\|\\nabla u\|^{2}}\\sum _{j}F^{i\\bar{i}}\\textrm {Re}\\Big \\lbrace e_{i}(\\eta )\\bar{e}_{i}e_{j}(u)\\bar{e}_{j}(u)+e_{i}(\\eta )\\bar{e}_{i}\\bar{e}_{j}(u)e_{j}(u)\\Big \\rbrace \\\\\\ge &2F^{i\\bar{i}}\\lambda _{i}\\frac{\\textrm {Re}\\lbrace e_{i}(\\eta )\\bar{e}_{i}(u)\\rbrace }{\|\\nabla u\|^{2}}-(1+3\\varepsilon )B_1\\zeta F^{i\\bar{i}}\|e_{i}(\\eta )\|^{2}-\\frac{C}{B_1\\zeta \\varepsilon }\\mathcal {F}-(1-\\varepsilon )\\sum _{j}F^{i\\bar{i}}\\frac{\|\\bar{e}_{i}\\bar{e}_{j}(u)\|^{2}}{B_1\\zeta \|\\nabla u\|^{2}}.\\end{split}$ Then, using (REF ), (REF ) and (REF ), we obtain $\\begin{split}0 \\ge & L(\\eta )+B_1(1+\\zeta )F^{i\\bar{i}}\|e_{i}(\\eta )\|^{2}-\\frac{2C}{B_1\\zeta \\varepsilon }\\mathcal {F}+\\frac{2}{B_1\\zeta \|\\nabla u\|^{2}}\\sum _{j}\\textrm {Re}\\lbrace e_{j}(h)\\bar{e}_{j}u\\rbrace \\\\&+2F^{i\\bar{i}}\\lambda _{i}\\frac{\\textrm {Re}\\lbrace e_{i}(\\eta )\\bar{e}_{i}(u)\\rbrace }{\|\\nabla u\|^{2}}-(1+3\\varepsilon )B_1\\zeta F^{i\\bar{i}}{\|e_{i}(\\eta )\|^{2}}\\\\\\ge & L(\\eta )+B_1(1-3\\varepsilon \\zeta ) F^{i\\bar{i}}\|e_{i}(\\eta )\|^{2}-\\frac{2C}{B_1\\zeta \\varepsilon }\\mathcal {F}-\\frac{C}{B_1\\zeta \|\\nabla u\|}+2F^{i\\bar{i}}\\lambda _{i}\\frac{\\textrm {Re}\\lbrace e_{i}(\\eta )\\bar{e}_{i}(u)\\rbrace }{\|\\nabla u\|^{2}}.\\end{split}$ We have $\\varepsilon =\\frac{1}{12\\sup _{x\\in M}\\zeta (x)}\\le \\frac{1}{2}$ if $A$ is big enough.", "It follows $B_1(1-3\\varepsilon \\zeta ) F^{i\\bar{i}}\|e_{i}(\\eta )\|^{2}\\ge \\frac{1}{2}B_1F^{i\\bar{i}}\|e_{i}(\\eta )\|^{2}.$ We use the Cauchy-Schwarz inequality to obtain $\\begin{split}F^{i\\bar{i}}\\lambda _{i}\\frac{2\\textrm {Re}\\lbrace e_{i}(u)\\bar{e_{i}}(\\eta )\\rbrace }{\|\\nabla u\|^{2}} \\ge -\\frac{B_{1}}{4}F^{i\\bar{i}}\|e_{i}(\\eta )\|^{2}-\\frac{4}{B_1\|\\nabla u\|^{2}}F^{i\\bar{i}}\\lambda _{i}^{2}.\\end{split}$ Combining with (REF )-(REF ), we have $\\begin{split}\\frac{B_1}{4}F^{i\\bar{i}}\|e_{i}(\\eta )\|^{2}+L(\\eta )&\\le \\frac{C}{B_1\\zeta \|\\nabla u\|}+ \\frac{C}{B_1\\zeta }\\mathcal {F}+\\frac{4}{B_1\|\\nabla u\|^{2}}F^{i\\bar{i}}\\lambda _{i}^{2}\\\\&\\le \\frac{C}{B_1\\zeta \|\\nabla u\|}+ \\frac{C}{B_1\\zeta }\\mathcal {F}+\\frac{4n}{B_1\|\\nabla u\|^{2}},\\end{split}$ where in the last inequality we used $F^{i\\bar{i}}\\lambda _{i}^{2}=\\frac{\\lambda _{i}^{2}}{1+\\lambda _{i}^{2}}<1$ provided by (REF ) for each $i=1,\\cdots ,n$ .", "The proof is divided to two cases, where $\\lambda =(\\lambda _{1},\\cdots , \\lambda _{n})$ .", "Case (a).", "First, suppose (REF ) holds, i.e.", "$L(\\eta )\\ge \\theta \\mathcal {F}.$ Therefore, using (REF ) and (REF ), we have $\\frac{1}{2}\\theta \\mathcal {K}+\\frac{\\theta }{2}\\mathcal {F}\\le \\theta \\mathcal {F}\\le \\frac{C}{B_1\\zeta \|\\nabla u\|}+ \\frac{C}{B_1\\zeta }\\mathcal {F}+\\frac{4n}{B_1\|\\nabla u\|^{2}}.", "$ Note that the terms involving $\\mathcal {F}$ can be discarded if $B_1$ big enough.", "Then we have $\\frac{1}{2}\\theta \\mathcal {K}\\le \\frac{C}{B_1\\zeta \|\\nabla u\|}+\\frac{4}{B_1\|\\nabla u\|^{2}}.$ It follows $\|\\nabla u\|\\le C.$ Case (b).", "Second, suppose (REF ) holds.", "Then, by (REF ), we have $1\\ge F^{i\\bar{i}}\\ge \\theta \\mathcal {F}\\ge \\theta \\mathcal {K}, ~ \\textrm {for}~i=1,2,\\cdots , n.$ Hence, by (REF ), we have $\|\\lambda _{i}\|\\le C$ and $F^{i\\overline{i}}\\le 1.$ It follows $L(\\eta )=F^{i\\bar{i}}(g_{i\\bar{i}}+\\underline{u}_{i\\bar{i}}-\\lambda _{i})\\ge -C.$ By (REF ), (REF ) and (REF ), we obtain $-C+\\frac{1}{C}\|\\nabla \\eta \|^{2}\\le \\frac{C}{B_1\\zeta \|\\nabla u\|}+\\frac{4n}{B_1\|\\nabla u\|^{2}}+C.$ We can assume that $\|\\nabla u\|\\ge 2\|\\nabla \\underline{u}\|$ .", "Hence $\|\\nabla \\eta \|\\ge \\frac{1}{2}\|\\nabla u\|$ .", "It follows $-C+\\frac{1}{C}\|\\nabla u\|^{2}\\le \\frac{C}{B_1\\zeta \|\\nabla u\|}+\\frac{4n}{B_1\|\\nabla u\|^{2}}+C.$ Therefore, $\|\\nabla u\|\\le C.$" ], [ "Second order estimates", "In this section, we prove the following second order estimates.", "Theorem 5.1 There exists a constant $C_{0}>0$ such that $\\Vert \\nabla ^{2}u\\Vert _{C^{0}(M)}\\le C_{0},$ where $C_{0}$ depends on $(M, \\chi , J)$ , $\\omega $ , $\\Vert h\\Vert _{C^{2}}$ , $\\inf _{M} h$ and $\\underline{u}$ and $\\nabla $ is the Levi-Civita connection of $\\chi $ .", "Let $\\mu _{1}(\\nabla ^{2} u)\\ge \\cdots \\ge \\mu _{2n}(\\nabla ^{2}u)$ be the eigenvalues of $\\nabla ^{2}u$ with respect to $\\chi .$ By (REF ), we have $\\sum _{\\beta =1}^{2n}\\mu _{\\beta }=\\Delta u=\\Delta ^{\\mathbb {C}}u+\\tau (du)=\\sum _{i}\\lambda _{i}+\\tau (du)\\ge \\tau (du)\\ge -C,$ (see [7]).", "Then $\\mu _{2n}\\ge -C\\mu _{1}-C,$ which implies $\|\\nabla ^{2}u\|_{g}\\le C\\mu _{1}(\\nabla ^{2}u)+C,$ for a uniform constant $C$ .", "Hence, it suffices to give an upper bound for $\\mu _{1}$ .", "First, we consider the function $\\tilde{Q}:= \\log \\mu _{1}(\\nabla ^{2}u)+\\phi (\|\\nabla u\|^{2})+\\varphi (\\widetilde{\\eta })$ on $\\Omega :=\\lbrace \\mu _{1}(\\nabla ^{2}u)>0\\rbrace \\subset M$ .", "Here $\\varphi $ is a function defined by $\\varphi (\\widetilde{\\eta }):= e^{B\\widetilde{\\eta }},~~\\widetilde{\\eta }:=\\underline{u}-u+\\sup _{M}(u-\\underline{u})+1$ for a real constant $B>0$ to be determined later, and $\\phi $ is defined by $\\phi (s):=-\\frac{1}{2}\\log (1+\\sup _{M}\|\\nabla u\|^{2}-s).$ Set $K=1+\\sup _{M}\|\\nabla u\|^{2}$ .", "Note that $\\frac{1}{2K}\\le \\phi ^{\\prime }(\|\\nabla u\|^{2})\\le \\frac{1}{2},~~\\phi ^{\\prime \\prime }=2(\\phi ^{\\prime })^{2}.$ We may assume $\\Omega $ is a nonempty open set (otherwise we are done).", "Note that when $z$ approaches to $\\partial {\\Omega }$ , then $Q(z)\\rightarrow -\\infty $ .", "Suppose $Q$ achieves a maximum at $x_{0}$ in $\\Omega $ .", "Near $x_{0}$ , choose a local unitary frame $\\lbrace e_{i}\\rbrace _{i=1}^{n}$ (with respect to $\\chi $ ) such that at $x_{0}$ , $\\text{$\\chi _{i\\overline{j}}=\\delta _{ij}$, $\\tilde{g}_{i\\overline{j}}=\\delta _{ij}\\tilde{g}_{i\\overline{j}}$ and $\\tilde{g}_{1\\overline{1}}\\ge \\tilde{g}_{2\\overline{2}}\\ge \\cdots \\ge \\tilde{g}_{n\\overline{n}}$.", "}$ For convenience, we denote $\\tilde{g}_{i\\overline{i}}(x_{0})$ by $\\lambda _{i}$ .", "In addition, there exists a normal coordinate system $(U,\\lbrace x^{\\alpha }\\rbrace _{i=1}^{2n})$ in a neighbourhood of $x_{0}$ such that $e_{i}=\\frac{1}{\\sqrt{2}}(\\partial _{2i-1}-\\sqrt{-1}\\partial _{2i}) \\text{~for~} i=1,2,\\cdots ,n$ and $\\frac{\\partial \\chi _{\\alpha \\beta }}{\\partial x^{\\gamma }}=0 \\text{~for~} \\alpha ,\\beta ,\\gamma =1,2,\\cdots ,2n$ at $x_{0}$ , where $\\chi _{\\alpha \\beta }=\\chi (\\partial _{\\alpha }, \\partial _{\\beta }).$ Suppose that $V_{1},\\cdots , V_{2n}$ are $\\chi $ -unit eigenvectors for $\\Phi $ at $x_{0}$ with eigenvalues $\\mu _{1}(\\nabla ^{2}u)\\ge \\cdots \\ge \\mu _{2n}(\\nabla ^{2} u),$ respectively.", "Assume $V_{\\alpha }=V_{\\alpha }^{\\beta }\\partial _{\\beta }$ at $x_{0}$ and extend vectors $V_{\\alpha }$ to vector fields on $U$ by taking the components $V_{\\alpha }^{\\beta }$ to be constant.", "Since $\\mu _{1}(\\nabla ^{2}u)$ may not be smooth, we apply a perturbation argument as in [7], [27].", "On $U$ , define $\\begin{split}\\Phi & = \\Phi _{\\alpha }^{\\beta }~\\frac{\\partial }{\\partial x^{\\alpha }}\\otimes dx^{\\beta }\\\\& = (g^{\\alpha \\gamma }u_{\\gamma \\beta }-g^{\\alpha \\gamma }B_{\\gamma \\beta })\\frac{\\partial }{\\partial x^{\\alpha }}\\otimes dx^{\\beta },\\end{split}$ where $B_{\\gamma \\beta }=\\delta _{\\gamma \\beta }-V_{1}^{\\gamma }V_{1}^{\\beta }$ .", "Assume that $\\mu _{1}(\\Phi )\\ge \\mu _{2}(\\Phi )\\ge \\cdots \\ge \\mu _{2n}(\\Phi )$ are the eigenvalues of $\\Phi $ .", "Then $V_{1},V_{2},\\cdots ,V_{2n}$ are still eigenvectors of $\\Phi $ , corresponding to eigenvalues $\\mu _{1}(\\Phi ),\\mu _{2}(\\Phi ),\\cdots ,\\mu _{2n}(\\Phi )$ at $x_{0}$ .", "Note that $\\mu _{1}(\\Phi )(x_{0})>\\mu _{2}(\\Phi )(x_{0})$ , which implies $\\mu _{1}(\\Phi )$ is smooth near $x_{0}$ .", "On $U$ , we replace $\\tilde{Q}$ by the following smooth quantity $Q=\\log \\mu _{1}(\\Phi )+\\phi (\|\\nabla u\|^{2})+\\varphi (\\widetilde{\\eta }).$ Since $\\mu _{1}(\\nabla ^{2}u)(x_{0})=\\mu _{1}(\\Phi )(x_{0})$ and $\\mu _{1}(\\nabla ^{2} u)\\ge \\mu _{1}(\\Phi )$ , $x_{0}$ is still the maximum point of $\\hat{Q}$ .", "For convenience, we denote $\\mu _{\\alpha }(\\Phi )$ by $\\mu _{\\alpha }$ for $\\alpha =1,2,\\cdots ,2n$ .", "The proof needs the first and second derivatives of the first eigenvalue $\\mu _{1}$ at $x_{0}$ (See [7] or [24], [27]).", "Lemma 5.2 At $x_{0}$ , we have $\\begin{split}\\frac{\\partial \\mu _{1}}{\\partial \\Phi ^{\\alpha }_{\\beta } }&=V_{1}^{\\alpha }V_{1}^{\\beta };\\\\\\frac{\\partial ^{2} \\mu _{1}}{\\partial \\Phi ^{\\alpha }_{\\beta }\\partial \\Phi ^{\\gamma }_{\\delta }}&=\\sum _{\\kappa >1}\\frac{1}{\\mu _{1}-\\mu _{\\kappa }}(V_{1}^{\\alpha }V_{\\kappa }^{\\beta }V_{\\kappa }^{\\gamma }V_{1}^{\\delta }+V_{\\kappa }^{\\alpha }V_{1}^{\\beta }V_{1}^{\\gamma }V_{\\kappa }^{\\delta }).\\end{split}$ At $x_{0}$ , for each $i=1,2, \\cdots , n$ , we have $\\frac{1}{\\mu _{1}}e_{i}(\\mu _{1})=-\\phi ^{\\prime }e_{i}(\|\\nabla u\|^{2})-Be^{B\\widetilde{\\eta }}e_{i}(\\widetilde{\\eta })$ and $\\begin{split}0\\ge L(Q)=& \\frac{L(\\mu _{1})}{\\mu _{1}}- F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}} +\\phi ^{\\prime \\prime } F^{i\\bar{i}}\|e_{i}(\|\\nabla u\|^{2})\|^{2} \\\\&+\\phi ^{\\prime }L(\|\\nabla u\|^{2})+Be^{B\\widetilde{\\eta }}L(\\widetilde{\\eta })+B^{2}e^{B\\widetilde{\\eta }} F^{i\\bar{i}}\|e_{i}(\\widetilde{\\eta })\|^{2}.\\end{split}$" ], [ "Lower bound for $L(Q)$", "In this subsection, we calculation $L(Q)$ .", "Lemma 5.3 For $\\varepsilon \\in (0,\\frac{1}{2}]$ , at $x_{0}$ , we have $\\begin{split}L(Q) \\ge & (2-\\varepsilon )\\sum _{\\beta >1}F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}(\\mu _{1}-\\mu _{\\beta })}-\\frac{1}{\\mu _{1}} F^{i\\bar{k},j\\bar{l}}V_{1}(\\tilde{g}_{i\\bar{k}})V_{1}(\\tilde{g}_{j\\bar{l}})\\\\&-(1+\\varepsilon )F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}-\\frac{C}{\\varepsilon }\\mathcal {F}+ \\frac{\\phi ^{\\prime }}{2}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})\\\\&+\\phi ^{\\prime \\prime } F^{i\\bar{i}}\|e_{i}(\|\\nabla u\|^{2})\|^{2}+Be^{B\\widetilde{\\eta }}L(\\widetilde{\\eta })+B^{2}e^{B\\widetilde{\\eta }} F^{i\\bar{i}}\|e_{i}(\\widetilde{\\eta })\|^{2}.\\\\\\end{split}$ First, we calculate $L(\\mu _{1})$ .", "Let $u_{ij}=e_{i}e_{j}u-(\\nabla _{e_{i}}e_{j})u $ and $u_{V_{\\alpha }V_{\\beta }}=u_{\\gamma \\delta }V^{\\gamma }_{\\alpha }V^{\\delta }_{\\beta }$ .", "By Lemma REF and (REF ), we have $\\begin{split}L(\\mu _{1})=&F^{i\\bar{i}}\\frac{\\partial ^{2} \\mu _{1}}{\\partial \\Phi ^{\\alpha }_{\\beta }\\partial \\Phi ^{\\gamma }_{\\delta }}e_{i}(\\Phi ^{\\gamma }_{\\delta })\\bar{e}_{i}(\\Phi ^{\\alpha }_{\\beta })+F^{i\\bar{i}}\\frac{\\partial \\mu _{1}}{\\partial \\Phi ^{\\alpha }_{\\beta }}(e_{i}\\bar{e}_{i}-[e_{i},\\bar{e}_{i}]^{0,1})(\\Phi ^{\\alpha }_{\\beta })\\\\=&F^{i\\bar{i}}\\frac{\\partial ^{2} \\mu _{1}}{\\partial \\Phi ^{\\alpha }_{\\beta }\\partial \\Phi ^{\\gamma }_{\\delta }}e_{i}(u_{\\gamma \\delta })\\bar{e}_{i}(u_{\\alpha \\beta })+F^{i\\bar{i}}\\frac{\\partial \\mu _{1}}{\\partial \\Phi ^{\\alpha }_{\\beta }}(e_{i}\\bar{e}_{i}-[e_{i},\\bar{e}_{i}]^{0,1})(u_{\\alpha \\beta })\\\\&+F^{i\\bar{i}}\\frac{\\partial \\mu _{1}}{\\partial \\Phi ^{\\alpha }_{\\beta }}u_{\\gamma \\beta }e_{i}\\bar{e}_{i}(\\chi ^{\\alpha \\gamma })\\\\\\ge & 2\\sum _{\\beta >1}F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}-\\mu _{\\beta }}+ F^{i\\bar{i}}(e_{i}\\bar{e}_{i}-[e_{i},\\bar{e}_{i}]^{0,1})(u_{V_{1}V_{1}})-C\\mu _{1}\\mathcal {F},\\\\\\end{split}$ where $(\\chi ^{\\alpha \\beta })$ is the inverse of the matrix $(\\chi _{\\alpha \\beta })$ .", "Let $W$ be a vector field.", "Differentiating the equation (REF ), we obtain $F^{i\\bar{i}}W(\\tilde{g}_{i\\bar{i}})=W(h)$ and $F^{i\\bar{i}}V_{1}V_{1}(\\tilde{g}_{i\\bar{i}})=- F^{i\\bar{k},j\\bar{l}}V_{1}(\\tilde{g}_{i\\bar{k}})V_{1}(\\tilde{g}_{j\\bar{l}})+V_{1}V_{1}(h).$ Commuting the derivatives and using Proposition REF , we obtain, for any vector field $W$ , $\|L(W(u))\|\\le C+C\\mu _{1}\\mathcal {F}.$ Claim 1.", "If $\\mu _{1}\\gg 1$ , then $\\begin{split}F^{i\\bar{i}}(e_{i}\\bar{e}_{i}-[e_{i},\\bar{e}_{i}]^{0,1})(u_{V_{1}V_{1}})\\ge & - F^{i\\bar{k},j\\bar{l}}V_{1}(\\tilde{g}_{i\\bar{k}})V_{1}(\\tilde{g}_{j\\bar{l}})-C\\mu _{1}\\mathcal {F}\\\\&-2 F^{i\\bar{i}}\\Big \\lbrace [V_{1},\\bar{e}_{i}]V_{1}e_{i}(u)+[V_{1},e_{i}]V_{1}\\bar{e}_{i}(u)\\Big \\rbrace .", "\\\\\\end{split}$ Proof.", "By (REF ), we have $\\begin{split}& F^{i\\bar{i}}(e_{i}\\bar{e}_{i}-[e_{i},\\bar{e}_{i}]^{0,1})(u_{V_{1}V_{1}})\\\\=& F^{i\\bar{i}}e_{i}\\bar{e}_{i}(V_{1}V_{1}(u)-(\\nabla _{V_{1}}V_{1})u)-F^{i\\bar{i}}[e_{i},\\bar{e}_{i}]^{0,1}(V_{1}V_{1}(u)-(\\nabla _{V_{1}}V_{1})u)\\\\\\ge & F^{i\\bar{i}}e_{i}\\bar{e}_{i}(V_{1}V_{1}(u))-F^{i\\bar{i}}[e_{i},\\bar{e}_{i}]^{0,1}V_{1}V_{1}(u)-C\\mu _{1}\\mathcal {F}-C.\\end{split}$ Recall the definition of Lie bracket $[e_{i}, e_{j}]=e_{i}e_{j}-e_{j}e_{i}.$ Then we get (for more details, see [7]) $\\begin{split}& F^{i\\overline{i}}e_{i}\\overline{e}_{i}V_{1}V_{1}(\\varphi )-F^{i\\overline{i}}[e_{i},\\overline{e}_{i}]^{(0,1)}V_{1}V_{1}(\\varphi ) \\\\\\ge ~~& F^{i\\overline{i}}\\left(V_{1}e_{i}\\overline{e}_{i}V_{1}(\\varphi )+[e_{i},V_{1}]\\overline{e}_{i}V_{1}(\\varphi )-[V_{1},\\overline{e}_{i}]e_{i}V_{1}(\\varphi )-V_{1}V_{1}[e_{i},\\overline{e}_{i}]^{(0,1)}(\\varphi )\\right)\\\\& -C\\mu _{1}\\mathcal {F}\\\\\\ge ~~&F^{i\\overline{i}}V_{1}V_{1}\\left(e_{i}\\overline{e}_{i}(\\varphi )-[e_{i},\\overline{e}_{i}]^{(0,1)}(\\varphi )\\right)-2F^{i\\overline{i}}[V_{1},e_{i}]V_{1}\\overline{e}_{i}(\\varphi )\\\\& -2F^{i\\overline{i}}[V_{1},\\overline{e}_{i}]V_{1}e_{i}(\\varphi )-C\\mu _{1}\\mathcal {F}.\\end{split}$ Combining with (REF ), (REF ) and (REF ), the Claim 1 follows if $\\mu _{1}\\gg 1$ .", "$\\Box $ Combining the equalities (REF ) and (REF ) together, it follows that $\\begin{split}L(\\mu _{1})\\ge & 2\\sum _{\\beta >1}F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}-\\mu _{\\beta }}-F^{i\\bar{k},j\\bar{l}}V_{1}(\\tilde{g}_{i\\bar{k}})V_{1}(\\tilde{g}_{j\\bar{l}})\\\\&-2F^{i\\bar{i}}\\textrm {Re}\\Big \\lbrace [V_{1},e_{i}]V\\bar{e}_{i}(u)+[V_{1},\\bar{e}_{i}]Ve_{i}(u)\\Big \\rbrace -C\\mu _{1}\\mathcal {F}.\\end{split}$ By (REF ), Lemma REF and Proposition REF , we have $L(\|\\nabla u\|^{2})\\ge \\frac{1}{2}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})- C\\mathcal {F}.$ Substituting (REF ) and (REF ) into (REF ), we obtain $\\begin{split}L(Q)\\ge & 2\\sum _{\\beta >1}F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}(\\mu _{1}-\\mu _{\\beta })}-\\frac{1}{\\mu _{1}} F^{i\\bar{k},j\\bar{l}}V_{1}(\\tilde{g}_{i\\bar{k}})V_{1}(\\tilde{g}_{j\\bar{l}})+B^{2}e^{B\\widetilde{\\eta }} F^{i\\bar{i}}\|e_{i}(\\widetilde{\\eta })\|^{2}\\\\&+Be^{B\\widetilde{\\eta }}L(\\widetilde{\\eta })-2 F^{i\\bar{i}}\\frac{\\textrm {Re}\\lbrace [V_{1},e_{i}]V_{1}\\bar{e}_{i}(u)+[V_{1},\\bar{e}_{i}]V_{1}e_{i}(u)\\rbrace }{\\mu _{1}}-C\\mathcal {F}\\\\&-F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}+ \\frac{\\phi ^{\\prime }}{2}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})+\\phi ^{\\prime \\prime } F^{i\\bar{i}}\|e_{i}(\|\\nabla u\|^{2})\|^{2}.\\\\\\end{split}$ Now we deal with the third order derivatives of the right hand side of (REF ).", "Claim 2.", "For any $\\varepsilon \\in (0,\\frac{1}{2}]$ , we have $\\begin{split}& 2 F^{i\\bar{i}}\\frac{\\textrm {Re}\\lbrace [V_{1},e_{i}]V_{1}\\bar{e}_{i}(u)+[V_{1},\\bar{e}_{i}]V_{1}e_{i}(u)\\rbrace }{\\mu _{1}}\\\\\\le & \\varepsilon F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{1}V_{1}})\|^{2}}{\\mu _{1}^{2}}+\\varepsilon \\sum _{\\beta >1}F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}(\\mu _{1}-\\mu _{\\beta })}+\\frac{C}{\\varepsilon }\\mathcal {F}.\\end{split}$ Asume $[V_{1},e_{i}]=\\sum _{\\beta =1}^{2n} \\mu _{i\\beta }V_{\\beta },~[V_{1},\\bar{e}_{i}]=\\sum _{\\beta =1}^{2n} \\overline{\\mu _{i\\beta }}V_{\\beta },$ where $\\mu _{i\\beta }\\in \\mathbb {C}$ are constants.", "Thus, $\\textrm {Re}\\lbrace [V_{1},e_{i}]V_{1}\\bar{e}_{i}(u)+[V_{1},\\bar{e}_{i}]V_{1}e_{i}(u)\\rbrace \\le C\\sum _{\\beta =1}^{2n}\|V_{\\beta }V_{1}e_{i}(u)\|.$ Then we are reduced to estimating $\\underset{\\beta }{\\sum } F^{i\\bar{i}}\\frac{\|V_{\\beta }V_{1}e_{i}(u)\|}{\\mu _{1}}$ .", "Using the definition of Lie bracket $e_{i}e_{j}-e_{j}e_{i}=[e_{i},e_{j}]$ , we have $\\begin{split}\\big \|V_{\\beta }V_{1}e_{i}(u)\\big \|= &\\big \|e_{i}V_{\\beta }V_{1}(u)+V_{\\beta }[V_{1},e_{i}](u)+[V_{\\beta },e_{i}]V_{1}(u)\\big \| \\\\=& \\big \|e_{i}(u_{V_{\\beta }V_{1}})+e_{i}(\\nabla _{V_{\\beta }}V_{1})(u)+V_{\\beta }[V_{1},e_{i}](u)+[V_{\\beta },e_{i}]V_{1}(u)\\big \|\\\\\\le &\\big \|e_{i}(u_{V_{\\beta }V_{1}})\\big \|+C\\mu _{1}.\\end{split}$ Therefore, $\\begin{split}\\sum _{\\beta } F^{i\\bar{i}}\\frac{\|V_{\\beta }V_{1}e_{i}(u)\|}{\\mu _{1}}\\le & \\sum _{\\beta } F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|}{\\mu _{1}}+C\\mathcal {F}\\\\= & F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{1}V_{1}})\|}{\\mu _{1}}+\\sum _{\\beta >1} F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|}{\\mu _{1}}+C\\mathcal {F}.\\\\\\end{split}$ By the Cauchy-Schwarz inequality, for $\\varepsilon \\in (0,\\frac{1}{2}]$ , we derive $F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{1}V_{1}})\|}{\\mu _{1}}\\le \\varepsilon F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{1}V_{1}})\|^{2}}{\\mu _{1}^{2}}+\\frac{C}{\\varepsilon }\\mathcal {F}$ and $\\begin{split}\\sum _{\\beta >1} F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|}{\\mu _{1}}\\le & \\varepsilon F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}(\\mu _{1}-\\mu _{\\beta })}+\\sum _{\\beta >1}\\frac{\\mu _{1}-\\mu _{\\beta }}{\\varepsilon \\mu _{1}}\\mathcal {F} \\\\\\le & \\varepsilon \\sum _{\\beta >1} F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}(\\mu _{1}-\\mu _{\\beta })}+\\frac{C}{\\varepsilon }\\mathcal {F},\\end{split}$ where the last inequality we used $\\sum _{\\beta =1}^{2n}\\mu _{\\beta }=\\Delta u=\\Delta ^{\\mathbb {C}}u+\\tau (du)\\ge -C+\\tau (du)\\ge -C$ (see [7]).", "Here $\\tau $ is the torsion vector field of $(\\chi , J)$ (the dual of its Lee form, see e.g.", "[29]).", "Combining with the above three inequalities, we have $\\sum _{\\beta } F^{i\\bar{i}}\\frac{\|V_{\\beta }V_{1}e_{i}(u)\|}{\\mu _{1}}\\le \\varepsilon F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{1}V_{1}})\|^{2}}{\\mu _{1}^{2}}+\\varepsilon \\sum _{\\beta >1} F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}(\\mu _{1}-\\mu _{\\beta })}+\\frac{C}{\\varepsilon }\\mathcal {F}.$ Then by (REF ), it follows (REF ).", "Consequently, Lemma REF follows from (REF ) and (REF ).", "Now we continue to prove Theorem REF ." ], [ "Proof of Theorem ", "The proof can be divided into three cases.", "Case 1: $F^{n\\bar{n}}\\le B^{3}e^{2B\\widetilde{\\eta }(0)}F^{1\\bar{1}}.$ In this case, we can choose $\\varepsilon =\\frac{1}{2}$ .", "Using the elemental inequality $\|a+b\|^{2}\\le 4\|a\|^{2}+\\frac{4}{3}\|b\|^{2}$ for (REF ), we get $-(1+\\varepsilon )F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}\\ge -6\\sup _{M}(\|\\nabla \\widetilde{\\eta }\|^{2})B^{2}e^{2B\\widetilde{\\eta }}\\mathcal {F}-2(\\phi ^{\\prime })^{2} F^{i\\bar{i}}\|e_{i}(\|\\nabla u\|^{2})\|^{2}.$ Plugging (REF ) and (REF ) into (REF ), we get, $\\begin{split}L(Q) \\ge & (2-\\varepsilon )\\sum _{\\beta >1}F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}(\\mu _{1}-\\mu _{\\beta })}-\\frac{1}{\\mu _{1}} F^{i\\bar{k},j\\bar{l}}V_{1}(\\tilde{g}_{i\\bar{k}})V_{1}(\\tilde{g}_{j\\bar{l}})\\\\&-\\Big (\\frac{C}{\\varepsilon }+6\\sup _{M}\\lbrace \|\\nabla \\widetilde{\\eta }\|^{2}\\rbrace B^{2}e^{2B\\widetilde{\\eta }}\\Big )\\mathcal {F}+ \\frac{\\phi ^{\\prime }}{2}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})\\\\&+Be^{B\\widetilde{\\eta }}L(\\widetilde{\\eta })+B^{2}e^{B\\widetilde{\\eta }} F^{i\\bar{i}}\|e_{i}(\\widetilde{\\eta })\|^{2}\\\\\\ge & -\\Big (\\frac{C}{\\varepsilon }+6\\sup _{M}\\lbrace \|\\nabla \\widetilde{\\eta }\|^{2}\\rbrace B^{2}e^{2B\\widetilde{\\eta }}\\Big )\\mathcal {F}+ \\frac{\\phi ^{\\prime }}{2}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})\\\\&+Be^{B\\widetilde{\\eta }}L(\\widetilde{\\eta }),\\end{split}$ where the last inequality we used the concavity of $F$ .", "Note $F^{i\\bar{i}}\\le 1$ .", "We have $\\begin{split}L(\\widetilde{\\eta })=&\\sum _{i}F^{i\\bar{i}}(\\underline{u}_{i\\bar{i}}-u_{i\\bar{i}})=\\sum _{i}F^{i\\bar{i}}(g_{i\\bar{i}}+\\underline{u}_{i\\bar{i}}-\\tilde{g}_{i\\bar{i}})\\\\\\ge &-C-\\sum _{i}F^{i\\bar{i}}\\tilde{g}_{i\\bar{i}}=-C-\\sum _{i}\\frac{\\lambda _{i}}{1+\\lambda _{i}^{2}}\\ge -C.\\end{split}$ Then, by (REF ), we have $0\\ge L(Q)\\ge \\frac{\\phi ^{\\prime }}{2}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})-C_{B}\\mathcal {F}.$ Here $C_{B}$ are positive constants depending on $B$ which might different from line to line.", "By (REF ), we have $\\sum _{i,j}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})\\le C_{B}.$ Then the complex covariant derivatives $u_{ij}=e_{i}e_{j}u-(\\nabla _{e_{i}}e_{j})u, ~u_{i\\bar{j}}=e_{i}\\bar{e}_{j}u-(\\nabla _{e_{i}}\\bar{e}_{j})u$ satisfy $\\sum _{i,j}(\|u_{ij}\|^{2}+\|u_{i\\bar{j}}\|^{2})\\le C_{B}.$ Case 2: $\\frac{\\phi ^{\\prime }}{4}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})>6\\sup _{M}(\|\\nabla \\widetilde{\\eta }\|^{2})B^{2}e^{2B\\widetilde{\\eta }}\\mathcal {F}.$ Note that (REF ) is still true.", "By (REF ) and (REF ), we have $\\begin{split}L(Q) \\ge & \\frac{\\phi ^{\\prime }}{4}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})-\\frac{C}{\\varepsilon }\\mathcal {F}+Be^{B\\widetilde{\\eta }}L(\\widetilde{\\eta }).\\end{split}$ (a).", "If (REF ) holds, $L(\\widetilde{\\eta })\\ge \\theta \\mathcal {F}$ .", "Then, by (REF ), we have $0 \\ge \\frac{\\phi ^{\\prime }}{4}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})+\\Big (\\theta Be^{B\\widetilde{\\eta }}-\\frac{C}{\\varepsilon }\\Big )\\mathcal {F}.$ This yields a contradiction if we further assume $B$ is large enough.", "(b).", "If (REF ) holds, then $F^{k\\bar{k}}\\ge \\theta \\mathcal {F}, k=1,2,\\cdots ,n.$ Then by (REF ) and (REF ) we obtain $\\begin{split}0\\ge &\\frac{\\phi ^{\\prime }}{4}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})-C_{B}\\mathcal {F} \\\\\\ge & \\frac{\\phi ^{\\prime }}{4}\\theta \\mathcal {F}\\sum _{i,j} (\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})-C_{B}\\mathcal {F}.\\end{split}$ Therefore, $\\sum _{i,j}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})\\le C_{B}.$ Case 3: If the Case 1 and Case 2 do not hold, define the index set $I:=\\Big \\lbrace 1\\le i\\le n:~ F^{n\\bar{n}}\\ge B^{3}e^{2B\\widetilde{\\eta }}F^{i\\bar{i}}\\Big \\rbrace .$ Clearly, we have $1\\in I$ and $n\\notin I$ .", "Hence we may write $I=\\lbrace 1,2,\\cdots , p\\rbrace $ for some positive integer $p<n$ .", "Now we deal with the third order term.", "Lemma 5.4 Assume $B\\ge 6n\\sup _{M}\|\\nabla \\widetilde{\\eta }\|^{2}$ .", "At $x_{0}$ , we have $-(1+\\varepsilon )\\sum _{i\\in I}F^{i\\bar{i}} \\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}\\ge -\\mathcal {F}-2(\\phi ^{\\prime })^{2}\\sum _{i\\in I}F^{i\\bar{i}}\|e_{i}(\|\\nabla u\|^{2})\|.$ Using (REF ) and the inequality $\|a+b\|^{2}\\le 4\|a\|^{2}+\\frac{4}{3}\|b\|^{2}$ , we obtain $\\begin{split}&-(1+\\varepsilon )\\sum _{i\\in I} F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}\\\\=&-(1+\\varepsilon )\\sum _{i\\in I} F^{i\\bar{i}}\|\\phi ^{\\prime }e_{i}(\|\\nabla u\|^{2})+Be^{B\\widetilde{\\eta }}e_{i}(\\widetilde{\\eta })\|^{2}\\\\\\ge & -6\\sup _{M}\|\\nabla \\widetilde{\\eta }\|^{2}B^{2}e^{2B\\widetilde{\\eta }}\\sum _{i\\in I} F^{i\\bar{i}}-2(\\phi ^{\\prime })^{2}\\sum _{i\\in I} F^{i\\bar{i}}\|e_{i}(\|\\nabla u\|^{2})\|^{2}\\\\\\ge & -6n\\sup _{M}\|\\nabla \\widetilde{\\eta }\|^{2}B^{-1} F^{n\\bar{n}}-2(\\phi ^{\\prime })^{2}\\sum _{i\\in I} F^{i\\bar{i}}\|e_{i}(\|\\nabla u\|^{2})\|^{2}\\\\\\ge &-\\mathcal {F}-2(\\phi ^{\\prime })^{2}\\sum _{i\\in I}F^{i\\bar{i}}\|e_{i}(\|\\nabla u\|^{2})\|^{2},\\end{split}$ where we used the hypothesis $B\\ge 6n\\sup _{M}\|\\nabla \\widetilde{\\eta }\|^{2}$ in the last second inequality.", "To deal with the bad third order terms, we need to give a lower bound of the good third order terms from the concavity of the equation (REF ).", "Lemma 5.5 We have $\\begin{split}&-\\frac{1}{\\mu _{1}} F^{i\\bar{k},j\\bar{l}}V_{1}(\\tilde{g}_{i\\bar{k}})V_{1}(\\tilde{g}_{j\\bar{l}})\\\\\\ge &\\frac{2}{\\mu _{1}}\\sum _{i\\notin I, k\\in I}F^{i\\bar{i}}\\tilde{g}^{k\\bar{k}}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}+\\frac{C_{B}}{\\varepsilon \\mu _{1}^{4}}\\sum _{i\\notin I,k\\notin I}F^{i\\bar{i}}{\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}},\\\\\\end{split}$ where $(\\tilde{g}^{k\\bar{l}})$ is the inverse of the matrix $(\\tilde{g}_{i\\bar{j}}).$ By (REF ), we have $\\begin{split}& -F^{i\\bar{k},j\\bar{l}}V_{1}(\\tilde{g}_{i\\bar{k}})V_{1}(\\tilde{g}_{j\\bar{l}}) \\\\=& \\sum _{i}\\frac{2\\lambda _{i}}{(1+\\lambda _{i}^{2})^{2}}\|V_{1}(\\tilde{g}_{i\\bar{i}})\|^{2}+\\sum _{i\\ne k}\\frac{\\lambda _{i}+\\lambda _{k}}{(1+\\lambda _{i}^{2})(1+\\lambda _{k}^{2})}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}.\\end{split}$ Now we calculate these two terms.", "We claim that $\\begin{split}&\\sum _{i\\ne k}\\frac{\\lambda _{i}+\\lambda _{k}}{(1+\\lambda _{i}^{2})(1+\\lambda _{k}^{2})}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}\\\\\\ge &2\\sum _{i\\notin I, k\\in I}\\frac{1}{(1+\\lambda _{i}^{2})\\lambda _{k}}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}+\\frac{C_{B}}{\\varepsilon \\mu _{1}^{3}}\\sum _{i\\notin I,k\\notin I,i\\ne k}\\frac{1}{1+\\lambda _{i}^{2}}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}.\\\\\\end{split}$ Denote $\\begin{split}S_{1}:=&\\lbrace (i,k):\\ i\\notin I,\\ k\\in I,\\ 1\\le i,\\ k\\le n\\rbrace ;\\\\[1.5mm]S_{2}:=&\\lbrace (i,k):\\ i\\ne k,\\ i\\notin I,\\ k\\notin I, \\ 1\\le i,\\ k\\le n\\rbrace ;\\\\[1.5mm]S_{3}:=&\\lbrace (i,k):\\ i\\in I,\\ k\\notin I,\\ 1\\le i,\\ k\\le n\\rbrace .\\end{split}$ Note $S_{j}\\subset \\lbrace (i,k):\\ i\\ne k,\\ 1\\le i,\\ k\\le n\\rbrace $ and $S_{j}\\cap S_{l}=\\emptyset $ , for $1\\le j,\\ l\\le 3$ and $j\\ne l$ .", "By the symmetry of $i, k$ , we have $\\begin{split}\\sum _{i\\ne k}\\frac{\\lambda _{i}+\\lambda _{k}}{(1+\\lambda _{i}^{2})(1+\\lambda _{k}^{2})}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}=&2\\sum _{i\\ne k}\\frac{\\lambda _{k}}{(1+\\lambda _{i}^{2})(1+\\lambda _{k}^{2})}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}\\\\\\ge &2\\sum _{l=1}^{3}\\sum _{S_{l}}\\frac{\\lambda _{k}}{(1+\\lambda _{i}^{2})(1+\\lambda _{k}^{2})}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}\\\\=&2\\sum _{S_{1}}\\frac{\\lambda _{i}+\\lambda _{k}}{(1+\\lambda _{i}^{2})(1+\\lambda _{k}^{2})}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}~(\\textrm {reverse}~ i,k~\\textrm {in}~ S_{3})\\\\&+2\\sum _{S_{2}}\\frac{\\lambda _{k}}{(1+\\lambda _{i}^{2})(1+\\lambda _{k}^{2})}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}.\\end{split}$ Hence, to prove (REF ), we only need to prove $\\frac{\\lambda _{i}+\\lambda _{k}}{(1+\\lambda _{i}^{2})(1+\\lambda _{k}^{2})}\\ge \\frac{1}{(1+\\lambda _{i}^{2})\\lambda _{k}},\\ (i,k)\\in S_{1}$ and $\\frac{\\lambda _{k}}{(1+\\lambda _{i}^{2})(1+\\lambda _{k}^{2})}\\ge \\frac{C_{B}}{\\varepsilon \\mu _{1}^{3}}\\frac{1}{1+\\lambda _{i}^{2}},~\\ (i,k)\\in S_{2}.$ Now we give the proof of the inequality (REF ).", "Note $\\begin{split}\\arctan \\lambda _{j}+\\arctan \\lambda _{n}=&h-\\sum _{i\\ne j, n}\\arctan \\lambda _{i}\\\\\\ge &(n-1)\\frac{\\pi }{2} -(n-2)\\frac{\\pi }{2} =\\frac{\\pi }{2}.\\end{split}$ Hence $\\arctan \\lambda _{j}\\ge \\frac{\\pi }{2}-\\arctan \\lambda _{n}=\\arctan \\frac{1}{\\lambda _{n}}.$ By the monotonicity of $\\arctan x $ , we have $\\lambda _{j}\\lambda _{n}\\ge 1,\\ \\ j=1,2, \\cdots , n-1,$ which implies (REF ).", "Next, for (REF ), it suffices to prove $\\frac{\\lambda _{k}}{1+\\lambda _{k}^{2}}\\ge \\frac{C_{B}}{\\varepsilon \\mu _{1}^{3}}, \\ \\ k=1,2, \\cdots , n.$ Since $h>(n-1)\\frac{\\pi }{2}$ , we have $\\arctan \\lambda _{n}\\ge h-\\sum _{1\\le i\\le n-1}\\arctan \\lambda _{i}\\ge h-(n-1)\\frac{\\pi }{2}\\ge C_{1}^{-1}>0$ for some uniform constant $C_{1}$ depending on $\\inf _{M}h$ .", "This implies $\\lambda _{k}\\ge \\lambda _{n}\\ge C_{1}^{-1}.$ Therefore, when $\\lambda _{1}\\gg 1$ , we have $\\frac{\\lambda _{k}}{1+\\lambda _{k}^{2}}\\ge \\frac{1}{C_{1}(1+\\lambda _{k}^{2})} \\ge \\frac{1}{C_{1}\\mu _{1}^{2}}.$ Now if $\\mu _{1}\\ge C_{B}/\\varepsilon $ , then we obtain (REF ).", "It follows (REF ).", "We complete the proof of (REF ) Now we deal with the first term on the right hand side in (REF ).", "By (REF ), we have $\\sum _{i}\\frac{2\\lambda _{i}}{(1+\\lambda _{i}^{2})^{2}}\|V_{1}(\\tilde{g}_{i\\bar{i}})\|^{2}\\ge \\frac{C_{B}}{\\varepsilon \\mu _{1}^{3}}\\sum _{i\\notin I}\\frac{1}{1+\\lambda _{i}^{2}}{\|V_{1}(\\tilde{g}_{i\\bar{i}})\|^{2}}.$ Therefore, by (REF ) and (REF ), we have $\\begin{split}&-\\frac{1}{\\mu _{1}} F^{i\\bar{k},j\\bar{l}}V_{1}(\\tilde{g}_{i\\bar{k}})V_{1}(\\tilde{g}_{j\\bar{l}})\\\\\\ge &\\frac{2}{\\mu _{1}}\\sum _{i\\notin I, k\\in I}F^{i\\bar{i}}\\tilde{g}^{k\\bar{k}}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}+\\frac{C_{B}}{\\varepsilon \\mu _{1}^{4}}\\sum _{i\\notin I,k\\notin I}F^{i\\bar{i}}{\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}}.\\\\\\end{split}$ Define a new (1,0) vector field by $\\widetilde{e}_{1}=\\frac{1}{\\sqrt{2}}(V_{1}-\\sqrt{-1}JV_{1}).$ At $x_{0}$ , we can find a sequence of complex numbers $\\nu _{1},\\cdots , \\nu _{n}$ such that $\\widetilde{e}_{1}:=\\sum _{1}^{n}\\nu _{k}e_{k},~\\sum _{1}^{n}\|\\nu _{k}\|^{2}=1.$ Lemma 5.6 We have $\|\\nu _{k}\|\\le \\frac{C_{B}}{\\mu _{1}} ~\\textrm {for all} ~k\\notin I.$ The idea of the proof is similar to the argument of Lemma 5.6 in [7].", "Since Case 2 does not hold, then we obtain $\\frac{\\phi ^{\\prime }}{4}\\sum _{i\\notin I}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})\\le (6n^{2}\\sup _{M}\|\\nabla \\widetilde{\\eta }\|^{2})B^{2}e^{2B\\widetilde{\\eta }}F^{n\\bar{n}}.$ While $ F^{n\\bar{n}}\\le B^{3}e^{2B\\widetilde{\\eta }}F^{i\\bar{i}}$ for each $i\\notin I$ , it follows that $\\sum _{\\gamma =2p+1}^{2n}\\sum _{\\beta =1}^{2n}\|\\nabla ^{2}_{\\gamma \\beta }u\|\\le C_{B}.$ Therefore, $\|\\Phi _{\\beta }^{\\gamma }\|\\le C_{B}$ for $2p+1\\le \\gamma \\le 2n$ , $1\\le \\beta \\le 2n$ .", "Since $\\Phi (V_{1})=\\mu _{1}V_{1}$ , then $\|V_{1}^{\\gamma }\|=\|\\frac{1}{\\mu _{1}}(\\Phi (V_{1}))^{\\gamma }\|=\\frac{1}{\\mu _{1}}\|\\sum _{\\beta }\\Phi _{\\beta }^{\\gamma }V_{1}^{\\beta }\|\\le \\frac{C_{B}}{\\mu _{1}},~2p+1\\le \\gamma \\le 2n.$ Then, by (REF ), $\|\\nu _{k}\|\\le \|V_{1}^{2k-1}\|+\|V^{2k}_{1}\|\\le \\frac{C_{B}}{\\mu _{1}}, k\\notin I.$ Now we can estimate the first three terms in Lemma REF .", "Since $JV_{1}$ is $\\chi $ -unit and $\\chi $ -orthogonal to $V_{1}$ , then we can find real numbers $\\xi _{2},\\cdots ,\\xi _{2n}$ such that $JV_{1}=\\sum _{\\beta >1}\\xi _{\\beta }V_{\\beta }, ~\\sum _{\\beta >1}\\xi _{\\beta }^{2}=1~ \\textrm {at x_{0}}.$ Lemma 5.7 For any constant $\\tau >0$ , we have $\\begin{split}& (2-\\varepsilon )\\sum _{\\beta >1}F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}(\\mu _{1}-\\mu _{\\beta })}-\\frac{1}{\\mu _{1}} F^{i\\bar{k},j\\bar{l}}V_{1}(\\tilde{g}_{i\\bar{k}})V_{1}(\\tilde{g}_{j\\bar{l}})-(1+\\varepsilon )\\sum _{i\\notin I}F^{i\\bar{i}} \\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}\\\\\\ge &(2-\\varepsilon )\\sum _{i\\notin I}\\sum _{\\beta >1}F^{i\\bar{i}} \\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}(\\mu _{1}-\\mu _{\\beta })}+\\sum _{k\\in I}\\sum _{i\\notin I}\\frac{2}{\\mu _{1}} F^{i\\bar{i}}\\tilde{g}^{k\\bar{k}}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}\\\\&-3\\varepsilon \\sum _{i\\notin I}F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}-2(1-\\varepsilon )(1+\\tau )\\tilde{g}_{\\tilde{1}\\bar{\\tilde{1}}}\\sum _{k\\in I}\\sum _{i\\notin I}F^{i\\bar{i}}\\tilde{g}^{k\\bar{k}}\\frac{\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}}{\\mu _{1}^{2}}\\\\&-\\frac{C}{\\varepsilon }\\mathcal {F}-(1-\\varepsilon )(1+\\frac{1}{\\tau })(\\mu _{1}-\\sum _{\\beta >1}\\mu _{\\beta }\\xi _{\\beta }^{2})\\sum _{i\\notin I}\\sum _{\\beta >1}\\frac{F^{i\\bar{i}}}{\\mu _{1}^{2}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}-\\mu _{\\beta }}\\\\\\end{split}$ if we assume $\\mu _{1}\\ge \\frac{C_{B}}{\\varepsilon }$ , where $\\tilde{g}_{\\tilde{1}\\bar{\\tilde{1}}}=\\sum \\tilde{g}_{i\\bar{i}}\|\\nu _{i}\|^{2}$ .", "Proof.", "First, we can prove $e_{i}(u_{V_{1}V_{1}})=\\sqrt{2}\\sum _{k} \\bar{\\nu }_{k}V_{1}(\\tilde{g}_{i\\bar{k}})-\\sqrt{-1}\\sum _{\\beta >1}\\xi _{\\beta }e_{i}(u_{V_{1}V_{\\beta }})+O(\\mu _{1}),$ where $O(\\mu _{1})$ denotes the terms which can be controlled by $\\mu _{1}$ .", "Indeed, since $\\overline{\\widetilde{e}}_{1}=\\frac{1}{\\sqrt{2}}(V_{1}+\\sqrt{-1}JV_{1})$ , $e_{i}(u_{V_{1}V_{1}})=\\sqrt{2} e_{i}(u_{V_{1}\\overline{\\widetilde{e}}_{1}})-\\sqrt{-1} e_{i}(u_{V_{1}JV_{1}}).$ For the first term, using $\\tilde{g}_{i\\bar{k}}=g_{i\\bar{k}}+u_{i\\bar{k}}$ , $\\begin{split}e_{i}(u_{V_{1}\\overline{\\widetilde{e}}_{1}})= & e_{i}(V_{1}\\overline{\\widetilde{e}}_{1}u-(\\nabla _{V_{1}}\\overline{\\widetilde{e}}_{1})u)=\\overline{\\widetilde{e}}_{1}e_{i}V_{1}u+O(\\mu _{1}) \\\\= & \\sum _{k}\\overline{\\nu }_{k}V_{1}(\\tilde{g}_{i\\bar{k}})+O(\\mu _{1}).\\end{split}$ For the second term, by (REF ), $\\begin{split}e_{i}(u_{V_{1}JV_{1}})=&e_{i}{V_{1}JV_{1}}(u)+O(\\mu _{1})=JV_{1}e_{i}{V_{1}}(u)+O(\\mu _{1}) \\\\=&\\sum _{\\beta >1}\\xi _{\\beta }V_{\\beta }e_{i}{V_{1}}(u)+O(\\mu _{1})=\\sum _{\\beta >1}\\xi _{\\beta }e_{i}(u_{V_{\\beta }{V_{1}}})+O(\\mu _{1}).\\\\\\end{split}$ Thus, (REF ) follows from (REF ) and (REF ).", "Hence, by (REF ), Lemma REF and Cauchy-Schwarz inequality, we have $\\begin{split}&-(1+\\varepsilon )\\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}\\\\=&-(1-2\\varepsilon )\\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|e_{i}(\\sqrt{2}\\sum _{k} \\bar{\\nu }_{k}V_{1}(\\tilde{g}_{i\\bar{k}})-\\sqrt{-1}\\sum _{\\beta >1}\\xi _{\\beta }e_{i}(u_{V_{1}V_{\\beta }})+O(\\mu _{1}))\|^{2}}{\\mu _{1}^{2}}\\\\&-3\\varepsilon \\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}\\\\\\ge &-(1-\\varepsilon )\\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|\\sqrt{2}\\sum _{k\\in I} \\overline{\\nu }_{k}V_{1}(\\tilde{g}_{i\\bar{k}})-\\sqrt{-1}\\sum _{\\beta >1}\\xi _{\\beta }e_{i}(u_{V_{1}V_{\\beta }})\|^{2}}{\\mu _{1}^{2}}\\\\&-3\\varepsilon \\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}-\\frac{C_{B}}{\\varepsilon }\\sum _{i\\notin I,k\\notin I}F^{i\\bar{i}}\\frac{\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}}{\\mu _{1}^{4}}-\\frac{C}{\\varepsilon }\\mathcal {F}.\\end{split}$ In addition, using the Cauchy-Schwarz inequality, we have $\\Big \|\\sum _{\\beta >1}\\xi _{\\beta }e_{i}(u_{V_{1}V_{\\beta }})\\Big \|^{2}\\le \\sum _{\\beta >1}(\\mu _{1}-\\mu _{\\beta }\\xi _{\\beta }^{2})\\sum _{\\beta >1}\\frac{\|e_{i}(u_{V_{1}V_{\\beta }})\|^{2}}{\\mu _{1}-\\mu _{\\beta }},$ and $\\begin{split}\\Big \|\\sum _{k\\in I}\\overline{\\nu }_{k}V_{1}(\\tilde{g}_{i\\bar{k}})\\Big \|^{2}&\\le \\Big (\\sum _{i}\\tilde{g}_{i\\bar{i}}\|\\nu _{i}\|^{2}\\Big )\\sum _{k\\in I}\\tilde{g}^{k\\bar{k}}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}\\\\&= \\tilde{g}_{\\tilde{1}\\bar{\\tilde{1}}}\\sum _{k\\in I}\\tilde{g}^{k\\bar{k}}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}.\\\\\\end{split}$ Then for each $\\gamma >0$ , using the Cauchy-Schwarz inequality again, we get $\\begin{split}& (1-\\varepsilon )\\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|\\sqrt{2}\\sum _{k\\in I} \\overline{\\nu _{k}}V_{1}(\\tilde{g}_{i\\bar{k}})-\\sqrt{-1}\\sum _{\\beta >1}\\xi _{\\beta }e_{i}(u_{V_{1}V_{\\beta }})\|^{2}}{\\mu _{1}^{2}}\\\\\\le & 2(1-\\varepsilon )(1+\\tau )\\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|\\sum _{k\\in I} \\overline{\\nu _{k}}V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}}{\\mu _{1}^{2}}\\\\&+(1-\\varepsilon )(1+\\frac{1}{\\tau })\\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|\\sum _{\\beta >1}\\xi _{\\beta }e_{i}(u_{V_{1}V_{\\beta }})\|^{2}}{\\mu _{1}^{2}}\\\\\\le & 2(1-\\varepsilon )(1+\\tau )\\tilde{g}_{\\tilde{1}\\bar{\\tilde{1}}}\\sum _{i\\notin I}\\sum _{k\\in I} \\frac{F^{i\\bar{i}}}{\\mu _{1}^{2}} \\tilde{g}^{k\\bar{k}}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}\\\\&+(1-\\varepsilon )(1+\\frac{1}{\\tau })(\\mu _{1}-\\sum _{\\beta >1} \\mu _{\\beta }\\xi _{\\beta }^{2})\\sum _{i\\notin I}\\sum _{\\beta >1}\\frac{F^{i\\bar{i}}}{\\mu _{1}^{2}} \\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}-\\mu _{\\beta }}.\\end{split}$ Combining with (REF ), we have $\\begin{split}&-(1+\\varepsilon )\\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}\\\\\\ge &-2(1-\\varepsilon )(1+\\tau )\\tilde{g}_{\\tilde{1}\\bar{\\tilde{1}}}\\sum _{i\\notin I}\\sum _{k\\in I} \\frac{F^{i\\bar{i}}}{\\mu _{1}^{2}} \\tilde{g}^{k\\bar{k}}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}\\\\&-(1-\\varepsilon )(1+\\frac{1}{\\tau })(\\mu _{1}-\\sum _{\\beta >1} \\mu _{\\beta }\\xi _{\\beta }^{2})\\sum _{i\\notin I}\\sum _{\\beta >1}\\frac{F^{i\\bar{i}}}{\\mu _{1}^{2}} \\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}-\\mu _{\\beta }}\\\\&-3\\varepsilon \\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}-\\frac{C_{B}}{\\varepsilon }\\sum _{i\\notin I,k\\notin I}F^{i\\bar{i}}\\frac{\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}}{\\mu _{1}^{4}}-\\frac{C}{\\varepsilon }\\mathcal {F}.\\end{split}$ Then the lemma follows from it and Lemma REF .", "$\\Box $ Lemma 5.8 If we assume $\\mu _{1}\\ge C/{\\varepsilon ^{3}}$ , then $\\begin{split}&(2-\\varepsilon )\\sum _{\\beta >1}F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}(\\mu _{1}-\\mu _{\\beta })}-\\frac{1}{\\mu _{1}} F^{i\\bar{k},j\\bar{l}}V_{1}(\\tilde{g}_{i\\bar{k}})V_{1}(\\tilde{g}_{j\\bar{l}})-(1+\\varepsilon )\\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}\\\\\\ge & -6\\varepsilon B^{2}e^{2B\\widetilde{\\eta }}\\sum _{i} F^{i\\bar{i}}\|e_{i}(\\widetilde{\\eta })\|^{2}-6\\varepsilon (\\phi ^{\\prime })^{2}\\sum _{i\\notin I}F^{i\\bar{i}}\|e_{i}(\|\\nabla u\|^{2})\|^{2}-\\frac{C}{\\varepsilon }\\mathcal {F}.\\end{split}$ By (REF ), it suffices to prove $\\begin{split}(2-\\varepsilon )&\\sum _{\\beta >1}F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}(\\mu _{1}-\\mu _{\\beta })}-\\frac{1}{\\mu _{1}} F^{i\\bar{k},j\\bar{l}}V_{1}(\\tilde{g}_{i\\bar{k}})V_{1}(\\tilde{g}_{j\\bar{l}})\\\\&-(1+\\varepsilon )\\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}\\ge -3\\varepsilon \\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}-\\frac{C}{\\varepsilon } \\mathcal {F}.\\end{split}$ We divide the proof into two cases.", "Case I: Assume that $\\mu _{1}+\\sum _{\\beta >1}\\mu _{\\beta }\\xi _{\\beta }^{2}\\ge 2(1-\\varepsilon )\\tilde{g}_{\\tilde{1}\\bar{\\tilde{1}}}>0.$ It follows from Lemma REF and (REF ) that $\\begin{split}& (2-\\varepsilon )\\sum _{\\beta >1}F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}(\\mu _{1}-\\mu _{\\beta })}-\\frac{1}{\\mu _{1}} F^{i\\bar{k},j\\bar{l}}V_{1}(\\tilde{g}_{i\\bar{k}})V_{1}(\\tilde{g}_{j\\bar{l}})\\\\& -(1+\\varepsilon )\\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}\\\\\\ge &\\sum _{i\\notin I}\\sum _{\\beta >1} \\frac{F^{i\\bar{i}}}{\\mu _{1}}\\left(\\frac{(2-\\varepsilon )\\mu _{1}}{\\mu _{1}-\\mu _{\\beta }}\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}\\right)+\\sum _{k\\in I}\\sum _{i\\notin I}\\frac{2}{\\mu _{1}} F^{i\\bar{i}}\\tilde{g}^{k\\bar{k}}\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}\\\\&-3\\varepsilon \\sum _{i\\notin I}F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}-(1+{\\tau })(\\mu _{1}+\\sum _{\\beta >1}\\mu _{\\beta }\\xi _{\\beta }^{2})\\sum _{k\\in I}\\sum _{i\\notin I}F^{i\\bar{i}}\\tilde{g}^{k\\bar{k}}\\frac{\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}}{\\mu _{1}^{2}}\\\\&-\\frac{C}{\\varepsilon }\\mathcal {F}-(1-\\varepsilon )(1+\\frac{1}{\\tau })(\\mu _{1}-\\sum _{\\beta >1}\\mu _{\\beta }\\xi _{\\beta }^{2})\\sum _{i\\notin I}\\sum _{\\beta >1}\\frac{F^{i\\bar{i}}}{\\mu _{1}^{2}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}-\\mu _{\\beta }}.\\end{split}$ Choose $\\tau =\\frac{\\mu _{1}-\\underset{\\beta >1}{\\sum }\\mu _{\\beta }\\xi _{\\beta }^{2}}{\\mu _{1}+\\underset{\\beta >1}{\\sum }\\mu _{\\beta }\\xi _{\\beta }^{2}}.$ Therefore, $\\begin{split}& (1+{\\tau })(\\mu _{1}+\\sum _{\\beta >1}\\mu _{\\beta }\\xi _{\\beta }^{2})\\sum _{k\\in I}\\sum _{i\\notin I}F^{i\\bar{i}}\\tilde{g}^{k\\bar{k}}\\frac{\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}}{\\mu _{1}^{2}} \\\\& +(1-\\varepsilon )(1+\\frac{1}{\\tau })(\\mu _{1}-\\sum _{\\beta >1}\\mu _{\\beta }\\xi _{\\beta }^{2})\\sum _{i\\notin I}\\sum _{\\beta >1}\\frac{F^{i\\bar{i}}}{\\mu _{1}^{2}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}-\\mu _{\\beta }}\\\\=& 2\\sum _{k\\in I}\\sum _{i\\notin I}F^{i\\bar{i}}\\tilde{g}^{k\\bar{k}}\\frac{\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}}{\\mu _{1}}+2(1-\\varepsilon )\\sum _{i\\notin I}\\sum _{\\beta >1}\\frac{F^{i\\bar{i}}}{\\mu _{1}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}-\\mu _{\\beta }}.\\end{split}$ Then (REF ) follows from (REF ).$\\Box $ Case II: Assume that ${\\mu _{1}+\\sum _{\\beta >1}\\mu _{\\beta }\\xi _{\\beta }^{2}}< 2(1-\\varepsilon )\\tilde{g}_{\\tilde{1}\\bar{\\tilde{1}}}.$ By a directly calculation, $\\begin{split}&\\tilde{g}(\\tilde{e},\\overline{\\tilde{e}})\\\\=& g(\\tilde{e},\\overline{\\tilde{e}})+\\tilde{e}\\overline{\\tilde{e}}(u)-[\\tilde{e},\\overline{\\tilde{e}}]^{(0,1)}(u)\\\\=&g(\\tilde{e},\\overline{\\tilde{e}})+\\frac{1}{2}(V_{1}V_{1}(u)+(JV_{1})(JV_{1})(u)+\\sqrt{-1}[V_{1},JV_{1}](u))\\\\&\\quad \\, -[\\tilde{e},\\overline{\\tilde{e}}]^{(0,1)}(u)\\\\=& \\frac{1}{2}\\big (\\mu _{1}+\\sum _{\\alpha >1}\\mu _{\\alpha }\\xi _{\\alpha }^{2}\\big )+g(\\tilde{e},\\overline{\\tilde{e}})+(\\nabla _{V_{1}}V_{1})(u)+(\\nabla _{JV_{1}}JV_{1})(u)\\\\& \\quad \\, +\\sqrt{-1}[V_{1},JV_{1}](u) -[\\tilde{e},\\overline{\\tilde{e}}]^{(0,1)}(u)\\\\\\le & \\frac{1}{2}\\big (\\mu _{1}+\\sum _{\\alpha >1}\\mu _{\\alpha }\\xi _{\\alpha }^{2}\\big )+C.\\end{split}$ Plugging (REF ) into (REF ), then $\\tilde{g}_{\\tilde{1}\\bar{\\tilde{1}}}\\le C/{\\varepsilon }.$ By (REF ) we have $\\mu _{1}+\\sum _{\\beta >1} \\mu _{\\beta }\\xi _{\\beta }^{2}\\ge -C$ .", "Hence, $0<\\mu _{1}-\\sum _{\\beta >1} \\mu _{\\beta }\\xi _{\\beta }^{2}\\le 2\\mu _{1}+C\\le (2+2\\varepsilon ^{2})\\mu _{1}$ provided by $\\mu _{1}\\ge C/{\\varepsilon ^{2}}$ .", "Choose $\\tau =1/{\\varepsilon ^{2}}$ .", "It follows that $\\begin{split}(1-\\varepsilon )(1+\\frac{1}{\\tau })(\\mu _{1}-\\sum _{\\beta >1}\\mu _{\\beta }\\xi _{\\beta }^{2})\\le & 2(1-\\varepsilon )(1+\\varepsilon ^{2})^{2}\\mu _{1}\\\\\\le &(2-\\varepsilon )\\mu _{1},\\end{split}$ when $\\varepsilon $ small enough.", "Then, by Lemma REF , we have $\\begin{split}& (2-\\varepsilon )\\sum _{\\beta >1}F^{i\\bar{i}}\\frac{\|e_{i}(u_{V_{\\beta }V_{1}})\|^{2}}{\\mu _{1}(\\mu _{1}-\\mu _{\\beta })}-\\frac{1}{\\mu _{1}} F^{i\\bar{k},j\\bar{l}}V_{1}(\\tilde{g}_{i\\bar{k}})V_{1}(\\tilde{g}_{j\\bar{l}}) -(1+\\varepsilon )\\sum _{i\\notin I} F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}\\\\\\ge &2\\sum _{k\\in I}\\sum _{i\\notin I} F^{i\\bar{i}}\\tilde{g}^{k\\bar{k}}\\frac{\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}}{\\mu _{1}}-3\\varepsilon \\sum _{i\\notin I}F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}\\\\&-2(1-\\varepsilon )(1+\\frac{1}{\\varepsilon ^{2}})\\tilde{g}_{\\tilde{1}\\bar{\\tilde{1}}}\\sum _{k\\in I}\\sum _{i\\notin I}F^{i\\bar{i}}\\tilde{g}^{k\\bar{k}}\\frac{\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}}{\\mu _{1}^{2}} -\\frac{C}{\\varepsilon }\\mathcal {F}\\\\\\stackrel{(\\ref {3..42})}{\\ge } &2\\sum _{k\\in I}\\sum _{i\\notin I} F^{i\\bar{i}}\\tilde{g}^{k\\bar{k}}\\frac{\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}}{\\mu _{1}} -3\\varepsilon \\sum _{i\\notin I}F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}\\\\&-(1-\\varepsilon )(1+\\frac{1}{\\varepsilon ^{2}})\\frac{C}{\\varepsilon }\\sum _{k\\in I}\\sum _{i\\notin I}F^{i\\bar{i}}\\tilde{g}^{k\\bar{k}}\\frac{\|V_{1}(\\tilde{g}_{i\\bar{k}})\|^{2}}{\\mu _{1}^{2}} -\\frac{C}{\\varepsilon }\\mathcal {F}\\\\\\ge &-3\\varepsilon \\sum _{i\\notin I}F^{i\\bar{i}}\\frac{\|e_{i}(\\mu _{1})\|^{2}}{\\mu _{1}^{2}}-\\frac{C}{\\varepsilon }\\mathcal {F},\\\\\\end{split}$ if we assume $\\mu _{1}\\ge C/{\\varepsilon ^{3}}$ in the last inequality.", "This proves (REF ).", "We now complete the proof of second order estimates.", "By Lemma REF , Lemma REF and (REF ), we have $\\begin{split}L(Q) \\ge & -6\\varepsilon B^{2}e^{2B\\widetilde{\\eta }} F^{i\\bar{i}}\|e_{i}(\\widetilde{\\eta })\|^{2}-6\\varepsilon (\\phi ^{\\prime })^{2}\\sum _{i\\notin I}F^{i\\bar{i}}\|e_{i}(\|\\nabla u\|^{2})\|^{2}-\\frac{C}{\\varepsilon }\\mathcal {F}\\\\&+ \\frac{\\phi ^{\\prime }}{2}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})+B^{2}e^{B\\widetilde{\\eta }} F^{i\\bar{i}}\|e_{i}(\\widetilde{\\eta })\|^{2}+Be^{B\\widetilde{\\eta }}L(\\widetilde{\\eta })\\\\& +\\phi ^{\\prime \\prime }F^{i\\bar{i}}\|e_{i}(\|\\nabla u\|^{2})\|^{2}-{2(\\phi ^{\\prime })^{2}\\sum _{i\\in I} F^{i\\bar{i}}\|e_{i}(\|\\nabla u\|^{2})\|^{2}}.\\end{split}$ Choose $\\varepsilon < \\min \\lbrace \\frac{1}{6n},\\theta /6\\rbrace $ such that $e^{B\\widetilde{\\eta }(0)}=\\frac{1}{6\\varepsilon }$ .", "By $\\phi ^{\\prime \\prime }=2(\\phi ^{\\prime })^{2}$ , then $\\begin{split}0 \\ge &-\\frac{C}{\\varepsilon }\\mathcal {F}+ \\frac{\\phi ^{\\prime }}{2}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})\\\\&+(B^{2}e^{B\\widetilde{\\eta }}-6\\varepsilon B^{2}e^{2B\\widetilde{\\eta }}) \\sum _{i\\notin I}F^{i\\bar{i}}\|e_{i}(\\widetilde{\\eta })\|^{2}+Be^{B\\widetilde{\\eta }}L(\\widetilde{\\eta })\\\\=&-\\frac{C}{\\varepsilon }\\mathcal {F}+ \\frac{\\phi ^{\\prime }}{2}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})+Be^{B\\widetilde{\\eta }}L(\\widetilde{\\eta }).\\end{split}$ In other words, $\\frac{B}{6\\varepsilon }L(\\widetilde{\\eta })-\\frac{C}{\\varepsilon }\\mathcal {F}+ \\frac{\\phi ^{\\prime }}{2}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})\\le 0.$ (a).", "Suppose (REF ) holds.", "Then we have $(\\frac{B\\theta }{6\\varepsilon }-\\frac{C}{\\varepsilon })\\mathcal {F}+ \\frac{\\phi ^{\\prime }}{2}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})\\le 0.$ Choose $B$ sufficiently large and $\\varepsilon <\\theta /6$ small enough such that $B\\theta /6-C\\ge B\\varepsilon $ .", "Then at $x_{0}$ we have $0\\ge B\\mathcal {F}+ \\frac{\\phi ^{\\prime }}{2}\\sum _{j} F^{i\\bar{i}}(\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2}).$ This yields a contradiction.", "(b).", "Suppose (REF ) holds.", "That is, $1\\ge F^{i\\bar{i}}\\ge \\theta \\mathcal {F}\\ge \\theta \\mathcal {K},~ \\textrm {for}~ i=1,2,\\cdots ,n.$ Combining (REF ) with (REF ), we have $\\frac{\\theta \\phi ^{\\prime }}{2}\\mathcal {K}\\sum _{i,j} (\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})\\le \\frac{nB}{12\\varepsilon }+\\frac{C}{\\varepsilon }.$ Therefore, $\\sum _{i,j} (\|e_{i}e_{j}u\|^{2}+\|e_{i}\\bar{e}_{j}u\|^{2})\\le C_{B}.$ Then this proves the Case 3.", "In conclusion, we obtain the second order estimates.", "$\\Box $ By (REF ) and Theorem REF , the equation (REF ) is uniformly elliptic.", "By the $C^{2,\\alpha }$ estimates (e.g.", "[30]), it follows $\\Vert u\\Vert _{C^{2,\\alpha }}\\le C.$ Then, by a standard bootstrapping argument, we complete the proof of Theorem REF ." ], [ "Proof of Theorem ", "We use the arguments of [10] and [26] to prove the Theorem REF .", "We will use the continuity method.", "First we give the openness." ], [ "Openness", "Consider the family of equations $\\sum _{i}\\arctan \\lambda _{i}(u_{t})=(1-t)h_{0}+th_{1}+c_{t},$ where $h_{0}, h_{1}\\in ((n-1)\\frac{\\pi }{2}, n\\frac{\\pi }{2} )$ are smooth functions on $M$ .", "Proposition 6.1 Suppose $u_{t_{0}}$ satisfies $\\sum _{i}\\arctan \\lambda _{i}(u_{t_{0}})=(1-t_{0})h_{0}+t_{0}h_{1}+c_{t_{0}}.$ Then there exists $\\epsilon >0$ such that when $\|t-t_{0}\|\\le \\epsilon $ , we can find $(u_{t}, c_{t})$ solving (REF ).", "Set $L=F^{i\\bar{j}}(e_{i}\\bar{e}_{j}-[e_{i},\\bar{e}_{j}]^{0,1}).$ Since the operator is homotopic to the canonical Laplacian operator $\\Delta ^{C}$ , the index of $L$ is zero, where $\\Delta ^{C} \\psi =\\frac{n\\chi ^{n-1}\\wedge \\sqrt{-1} \\partial \\overline{\\partial }\\psi }{\\chi ^{n}}$ .", "By the maximum principle, $\\text{Ker}(L)=\\lbrace \\text{Constants} \\rbrace .$ Denote by $L^{}$ the $L^{2}$ -adjoint operator of $L$ .", "By the Fredholm theorem, there is a smooth function $\\varphi _{0}$ such that $\\text{Ker}(L^{})=\\text{Span}\\lbrace \\varphi _{0}\\rbrace .$ Now we prove that $\\varphi _{0}$ does not change the sign.", "If $\\varphi _{0}$ changes sign, there is a positive function $\\tilde{f}$ perpendicular to it, but cannot be the image of $L$ by the maximum principle.", "It is a contradiction.", "Without loss of generality, assume that $\\varphi _{0}$ is non-negative.", "By the strong maximum principle for the elliptic operator $L^{*}$ , the function $\\varphi _{0}>0$ .", "Denote $\\tilde{B}=\\Big \\lbrace \\psi \\in C^{2,\\alpha }(M): \\int _{M}\\psi \\varphi _{0}\\chi ^{n}=0 \\text{\\ and\\ } \\omega +\\sqrt{-1} \\partial \\overline{\\partial }\\psi >0 \\Big \\rbrace \\times \\mathbb {R}.$ Consider the map $G$ , from $\\tilde{B}$ to $C^{\\alpha }(M)$ , where $G(\\psi ,c):=\\sum _{i}\\arctan \\lambda _{i}(\\psi )-c.$ Then the linear operator of $G$ at $(u_{0}, 0)$ is $L-c: \\left\\lbrace \\xi \\in C^{2,\\alpha }(M): \\int _{M}\\xi \\varphi _{0}\\chi ^{n}=0\\right\\rbrace \\times \\mathbb {R}\\rightarrow C^{\\alpha }(M).$ For any $\\hat{h}\\in C^{\\alpha }(M)$ , there exists a unique constant $c$ such that $\\int _{M}(\\hat{h}+c)\\varphi _{0}\\chi ^{n}=0$ .", "Then by (REF ) and the Fredholm theorem, there exists $\\xi $ such that $L(\\xi )-c=\\hat{h}.$ Hence $L-c$ is surjective.", "Let $(\\xi _{0}, \\hat{c})$ be the solution of $L-c=0$ .", "By (REF ) and the Fredholm theorem, $ \\hat{c}=0$ .", "Using $(\\ref {ker L})$ and (REF ), we obtain $\\xi _{0}=0.$ Hence, $ L-c$ is injective.", "Then by the implicit function theory, when $\|t-t_{0}\|$ small enough, there exist $u_{t}$ and a constant $c_{t}$ satisfying $\\sum _{i}\\arctan \\lambda _{i}(u_{t})=(1-t)h_{0}+th_{1}+c_{t}.$" ], [ "Existence", "Suppose $\\underline{u}$ is a $\\mathcal {C}$ -subsolution to the deformed Hermitian-Yang-Mills equation (REF ).", "Recall $\\lambda _{i}(\\hat{u})$ are the eigenvalues of $\\omega _{\\hat{u}}$ .", "Denote $\\theta _{0}=\\sum _{i}\\arctan \\lambda _{i}(\\hat{u}).$ Now we use the continuity method to prove that there exists a solution when the right hand side $h$ of (REF ) is a constant.", "Proposition 6.2 Under the assumption of Theorem REF , there exists a function $u$ on $M$ and a constant $c$ such that $\\sum _{i}\\arctan \\lambda _{i}=h+c,$ where $h+c>\\frac{(n-1)\\pi }{2}$ .", "Consider the family of equations $\\sum _{i}\\arctan \\lambda _{i}(u_{t})=(1-t)\\theta _{0}+th+c_{t}.$ Define $I=\\lbrace t\\in [0,1]: \\text{\\ there exist\\ } (u_{t}, c_{t})\\in \\tilde{B} \\text{\\ solving \\ } (\\ref {c1})\\rbrace .$ Note $(\\hat{u},0)$ is the solution of (REF ) at $t=0$ .", "Then $I$ is non-empty.", "By Proposition REF , $I$ is open.", "To prove $I$ is closed, by (REF ), it suffices to prove $\\underline{u}$ is still a $\\mathcal {C}$ -subsolution of (REF ) for any $t\\in [0,1]$ and $(1-t)\\theta _{0}+th+c_{t}\\ge \\inf _{M} \\theta _{0}>(n-1)\\frac{\\pi }{2}.$ First, assume $u_{t}-\\hat{u}$ achieves its maximum at the point $q$ .", "Then $\\sqrt{-1} \\partial \\overline{\\partial }(u_{t}-\\hat{u})(q)\\le 0.$ It follows that $F(\\omega _{u_{t}})(q)-F(\\omega _{\\hat{u}})(q)=\\int _{0}^{1}F^{i\\overline{j}}(\\omega _{\\hat{u}}+s\\sqrt{-1} \\partial \\overline{\\partial }(u_{t}-\\hat{u}))\\ ds\\, (u_{t}-\\hat{u})_{i\\overline{j}}(q)\\le 0.$ Then at $q$ , $\\theta _{0}(q)\\ge \\sum _{i}\\arctan \\lambda _{i}(u_{t})(q)=(1-t)\\theta _{0}(q)+th+c_{t},$ which implies $c_{t}\\le t(\\theta _{0}(q)-h)\\le 0.$ Here we used $h\\ge \\theta _{0}.$ Then, we have $\\begin{split}\\sum _{i\\ne j}\\arctan (\\lambda _{i}(\\underline{u}))&\\ge h-\\frac{\\pi }{2}\\\\&\\ge (1-t)\\theta _{0}+th+c_{t}-\\frac{\\pi }{2} .\\end{split}$ By Lemma REF , $\\underline{u}$ is a $\\mathcal {C}$ -subsolution for $t\\in [0,1]$ .", "Assume $u_{t}-\\hat{u}$ achieves its minimum at the point $q^{\\prime }$ .", "Similarly with (REF ), we have $c_{t}\\ge - t(h-\\theta _{0}(q^{\\prime }))\\ge -\\sup _{M} t(h-\\theta _{0}).$ Assume $\\theta _{0}$ achieves its minimum at the point $p$ .", "Note that $h$ is a constant.", "Then, we obtain $\\begin{split}\\inf _{M}\\left((1-t)\\theta _{0}+th+c_{t}\\right)&=(1-t)\\theta _{0}(p)+th+c_{t}\\\\&=\\theta _{0}(p)+t(h-\\theta _{0}(p))+c_{t}\\\\&=\\theta _{0}(p)+t\\sup _{M}(h-\\theta _{0})+c_{t}\\\\&\\ge \\theta _{0}(p)= \\inf _{M}\\theta _{0}>(n-1)\\frac{\\pi }{2}.\\end{split}$ By Theorem REF , we conclude $I$ is closed." ] ]	2011.14091
[ [ "Soft x-ray detection for small satellites with a commercial CMOS sensor\n at room temperature" ], [ "Abstract Recently CMOS (complementary metal-oxide semiconductor) sensors have progressed to a point where they may offer improved performance in imaging x-ray detection compared to the CCDs often used in x-ray satellites.", "We demonstrate x-ray detection in the soft x-ray band (250-1700 eV) by a commercially available back-illuminated Sony IMX290LLR CMOS sensor using the Advanced Photon Source at the Argonne National Laboratory.", "While operating the device at room temperature, we measure energy resolutions (FWHM) of 48 eV at 250 eV and 83 eV at 1700 eV which are comparable to the performance of the Chandra ACIS and the Suzaku XIS.", "Furthermore, we demonstrate that the IMX290LLR can withstand radiation up to 17.1 krad, making it suitable for use on spacecraft in low earth orbit." ], [ "Introduction", "The availability of low-cost small satellites for astrophysics drives a need for low-cost detectors that place minimal demands on the limited spacecraft resources available[1].", "The charge-coupled device (CCD) detectors often used for x-ray detection on space-based x-ray observatories have high cost and require cooling to temperatures near -100 K.[2], [3], [4] Complementary Metal-Oxide-Semiconductor (CMOS) sensors are designed for operation at room temperature and many commercial devices are available at low cost.", "The active pixel design of CMOS sensors enables lower power consumption, faster readout rates, and comparable noise levels to CCDs[5].", "Here, we investigate the candidacy of a low-cost commercial CMOS sensor as an x-ray spectrometer in low earth orbit (LEO) for future small satellite missions.", "In Sec.", "2, we describe the selected CMOS sensor, initial results on x-ray detection, and our x-ray event processing algorithm.", "We characterize the soft x-ray performance of the sensor using measurements obtained at an x-ray synchrotron beamline at the Argonne National Laboratory in Sec. 3.", "We then discuss the electron noise (Sec.", "4), radiation testing (Sec.", "5), and our conclusions (Sec. 6).", "Back-illuminated (BI) sensors can provide superior performance for soft x-ray detection because the photons do not pass through the passivation, metallization, and inter-dielectric layers that form the pixel electronics.", "We evaluated several candidate BI sensors and selected the Sony IMX290LLR CMOS sensor for additional study due to its low electronic noise which is important in achieving good spectral resolution for soft x-rays.", "The IMX290LLR contains a 1936$\\times $ 1096 array of 2.9 $\\mu $ m square back-side-illuminated pixels and can achieve frame rates of up to 135 fps.", "To minimize development costs, we used an IDS Imaging Development Systems (https://en.ids-imaging.com/) UI-3860-LE-M-GL camera, that includes an IMX290 sensor and circuitry to provide a USB interface, and the associated Software Development Kit (SDK).", "For x-ray detection, we removed the coverglass designed to protect the sensor's surface.", "Our initial x-ray tests were performed using an $^{55}$ Fe radioactive source and an x-ray tube fluorescing a teflon target and the walls of the vacuum chamber where the sensor was mounted.", "This produces characteristic emission lines from several elements including F, Cr, Mn, Fe, and bremsstraulung continuum radiation.", "The chamber was evacuated to $\\approx 10^{-2}$ Torr and the sensor was mounted on a chiller plate maintained at a constant temperature of 21$^{\\circ }$ C. We acquired 4,000 images each with 300 ms exposure under x-ray illumination and 100 frames with no x-rays present for dark and bias measurements.", "Cutouts of x-ray events taken from single frames are shown in Fig REF .", "x-rays produce events in which charge is deposited in one, two, or multiple pixels.", "To select an appropriate gain for the sensor, we collected a series of spectra using the $^{55}$ Fe radioactive source at different gain settings.", "We measured the centroid of the pulse height distribution for the Mn K-$\\alpha $ emission line (corresponding to 1620 $e^{-}$ for an average hole-pair creation energy of $w = 3.64 \\rm \\, eV$ in silicon at room temperature [6]) using the event processing methodology outlined in Sec REF , and adopted settings producing a gain and associated fitting error of 0.991 $\\pm $ 0.006 ADU/e$^-$ .", "These gain settings were used for all of the data described below.", "Figure: Cutout images of the three types of events we consider for event processing.", "Shown from left to right are single pixel, double pixel, and extended pixel events.", "These events correspond to energies of 3.3 keV, 3.1 keV, and 5.4 keV respectively." ], [ "Event Processing", "Individual x-rays may deposit charge in multiple pixels due to the finite extent of the track traversed by primary photoelectron, the ejection of Auger electrons, and charge diffusion.", "To cope with the variable spread of charge across pixels, the event processing algorithms used for x-ray CCDs, such as Chandra's Advanced CCD imaging spectrometer (ACIS), search for pixels with charge greater than the `event threshold' and then record values in the surrounding 3$\\times $ 3 pixel grid.", "Pixels above the `split pixel threshold' and sharing a side with the central pixel or in a 2$\\times $ 2 square including the central pixel are included in the sum used to reconstruct the energy of the incident photon [3].", "As shown in Fig.", "REF , the x-ray events from the IMX290 sometimes extend over a larger range of pixels.", "This is likely due to the relatively small size of the pixels.", "We have developed an event processing algorithm utilizing a 5$\\times $ 5 grid of pixels.", "Before processing the images, we first identified anomalous (hot) pixels.", "We started by collecting 100 dark frames of the same exposure time as the data.", "Setting a threshold equal to 2 standard deviations above the median value of the 100 dark frames, we marked all pixels that are above the threshold in 10% or more of the frames as hot.", "These pixels were excluded in further processing.", "For the data described below, 0.47% of the pixels were marked as hot.", "A master dark frame was compiled by taking the mean of the dark frames.", "The master dark frame was subtracted from each of the x-ray frames.", "The dark-subtracted x-ray frame is then scanned.", "For each pixel above the event threshold, a surrounding 5$\\times $ 5 grid is defined.", "The set of pixels above the split pixel threshold (`hit pixels') that are connected to the central pixel by a chain of adjoining hit pixels are included in the sum used to estimate the photon energy and the position of the central pixel is recorded as the x-ray position.", "The event and split pixel thresholds were optimized by examining pulse height spectra for x-rays with energies between 250 eV and 1700 eV (see Sec. 3).", "The thresholds were adjusted to minimize the width and skew of the pulse height distributions.", "We found an optimal event threshold of 26 ADU and split pixel threshold of 5 ADU." ], [ "Experimental Setup", "In December 2019, we tested the sensor at the Intermediate Energy x-ray (IEX) beamline 29-ID of the Advanced Photon Source (APS) at the Argonne National Laboratory in Illinois.", "This beamline utilizes an electromagnetic, variably polarizing undulator source followed by a monochromator and is capable of producing monochromatic x-rays between 250-2500 eV[7].", "Images of the experimental setup are shown in Fig REF and Fig REF .", "Our sensor is not capable of handling the intensity of the direct beam in photon counting mode, so it was mounted on a port at an angle 35$^{\\circ }$ with respect to the incident beam.", "A pump near the sensor maintained a high vacuum ($\\approx 10^{-9}$ Torr).", "A Si wafer was used to reflect the incident beam thereby reducing the beam flux ($\\sim 10^{12}~\\gamma $ /s) to a level that could be handled by the camera ($< 100~\\gamma $ /s).", "Some experimentation was required to focus the beam in the center of the sensor and the optimal graze angle was 17.5$^{\\circ }$ .", "With the beam centered on the sensor and the vacuum chamber covered to minimize ambient light on the sensor, we accumulated sensor images with an exposure time of 300 ms at energies from 250 eV to 1700 eV (see Table REF ).", "At each energy, we accumulated frames until at least 20,000 photon events were registered.", "The sensor maintained an average temperature of 25$^\\circ $ C during this continuous run.", "Both before and after taking beamline data, we turned off the beam and acquired 100 dark frames with 300 ms exposure.", "Figure: Example of a fitted spectrum from a beamline energy of 274 eV with the best fit Gaussian model." ], [ "Spectral Fitting", "Using the algorithm described in Sec.", "REF , x-ray events were extracted from the sensor images.", "The list of x-ray events was then used to produce an energy spectrum at each beam energy using a bin width of 2 ADU/bin.", "We fitted each spectrum with a Gaussian model over a pulse height band covering the peak of the Gaussian, as seen in the solid line in Fig REF .", "The error on the counts in each pulse height bin was calculated as the quadrature sum of the Gehrels variance function[8] and a linear term due to differential non-linearity (DNL) proportional to the counts in each bin.", "The same DNL factor was used for all spectra and was adjusted until the fits produced $\\chi ^{2}_{\\nu } \\approx 1$ giving a value of 0.045.", "We interpret this as the DNL of the analog-to-digital converters in the IMX290.", "Table REF shows the best fit parameters for each spectrum.", "Table: Summary of fit values obtained with a Gaussian plus a constant model." ], [ "Energy Calibration", "We performed a linear regression of the best fitted Gaussian centroids versus beam energy as shown in Fig.", "REF .", "We inverted the relation to find the pulse height to energy calibration.", "We obtained a best fit slope of 3.6877 $\\pm $ 0.0008 eV/ADU and intercept of 1.840 $\\pm $ 0.005 eV yielding the following conversion function: $\\text{Energy (eV)} = \\frac{{3.69} \\text{\\, eV}}{\\text{ADU}} (\\text{ADU value} + {1.8} \\ \\text{eV}).", "\\ \\qquad \\mathrm {(1)}$ The slope is equivalent to a camera gain of $0.9870 \\pm 0.0002 \\, {\\rm ADU}/e^{-}$ .", "This is more accurately determined than our initial gain calibration done with the radioactive $^{55}$ Fe source and consistent within the uncertainties of that measurement.", "With active pixel technology, some variations of gain across the regions of the sensor can be expected.", "To quantify the variations in gain across the sensor, we divided the sensor area into a 5x5 grid, each grid 387 by 219 pixels.", "We then calculated the gain within each grid by fitting only the events that occurred within the grid and performing the same linear regression described above.", "We determined an average gain weighted by the slope error in each grid region, which we find to be 3.686 $\\pm $ 0.005 ADU/eV.", "The gain varies by 0.14% across the sensor and we choose to use a single gain for the whole sensor.", "Figure: Best fit linear regression to the centroid of the Argonne IEX APS lines with residuals plotted in the lower figure.In Fig.", "REF we display calibrated spectra from various beam energies on the same plot.", "The Gaussians were normalized to have the same area under their curves.", "Figure: Energy calibrated spectra with a y-axis normalization at beamline energies of 274 eV, 950 eV, and 1700 eV.", "There are no significant counts below 400 eV and 1150 eV for the 950 eV and 1700 eV lines respectively.", "We do not plot these spectra below these thresholds for visual clarity.Figure: Energy resolution (FWHM) modeled by Eq ." ], [ "Energy Resolution", "The IEX resolving power is $10^4$ ; thus, the finite widths of the spectral peaks are due to the sensor.", "Fig.", "REF shows the energy resolution in terms of the Gaussian full width half max (FWHM) as a function of beam energy.", "We modeled the energy resolution as the quadrature sum of the Fano limited Poisson fluctuations in the number of photoelectrons produced per photon and the electronic noise per pixel, $\\text{FWHM} = 2.355 w \\; \\sqrt{N \\sigma ^2 + \\frac{f E}{w}} \\ \\qquad \\mathrm {(2)}$ where $w = 3.64$ eV is the average ionization energy of silicon, $f= 0.115$ is the Fano factor for silicon [9], $N$ is the number of pixels used to calculated the pulse height sum for each event, and $\\sigma $ is the root-mean-square (RMS) electronic noise per pixel.", "Figure: Mean number of summed pixels versus energy.Fig.", "REF shows the average number of pixels used in the pulse height summation as a function of energy.", "The error bars indicate the standard deviation.", "We attempted two different variations of $N$ for fitting, the first uses a constant median $N$ value of 2 and the second uses the variable mean number of pixels as a function of energy as shown in Fig.", "REF .", "We allowed the electronic noise $\\sigma $ parameter to vary and fit the two models to the resolution curve in Fig.", "REF .", "We found best fit values of $3.64 \\pm 0.05 \\, e^{-}$ and $3.59 \\pm 0.10 \\, e^{-}$ for the variable and constant pixel models, respectively.", "Our FWHM resolution values of 48.1 eV at 274 eV and 75.6 eV at 1450 eV are competitive with the values of $\\sim $ 50 eV and $\\sim $ 70 eV near 0.3 keV and 1.5 keV, respectively, for the back-side illuminated CCDs aboard the Suzaku x-ray Imaging Spectrometer [10], [4], and also with the resolution of 96 eV at 1.49 keV reported for the ACIS CCDs [11].", "We fitted the Mn K-$\\alpha $ and Mn K-$\\beta $ lines in the spectrum mentioned in Sec .", "The energy resolution values are larger than those estimated using Eq REF and the best fit noise for energies below 2 keV.", "The degradation of the energy resolution above 2 keV may be due to incomplete charge collection.", "Significant degradation of energy resolution at high energies has been found in a front-illuminated sensor and attributed to a reduction in charge collection efficiency for X-ray that penetrate into the $p^+$ substrate beneath the epitaxial layer [12].", "This preferentially occurs for higher energy X-rays since they have longer absorption depth.", "The $p^+$ layer is absent in back-illuminated devices, but incomplete charge collection may occur in the pixel circuitry layer or the carrier wafer.", "Figure: RMS noise distribution of the IMX290LLR chip at 300 ms.", "The vertically and non-vertically dashed lines correspond to the mean and median of the distribution respectively.The sensor noise can be measured by examining the average variance in each pixel when a sensor is subject to no light at short exposures.", "Due to the construction of the CMOS pixel, namely that each pixel contains its own circuitry, the RMS noise varies between pixels.", "This is different than a CCD detector which traditionally has a single amplifier and analog-to-digital converter for all pixels and therefore maintains uniform noise for all pixels.", "We evaluated the noise of the IMX290 by taking a series of 100 frames with the sensor in the dark in a vacuum chamber.", "Each image had an exposure 300 ms and the sensor was held at a constant temperature of 21$^\\circ $ C. We constructed a master dark frame by taking the mean ADU value in each pixel of the 100 images and then subsequently subtracting the master image from each individual one.", "We constructed a distribution of the per pixel noise by taking the RMS of the 100 master subtracted images (Fig.", "REF ).", "We find mean and median read noise values of $2.18 \\, e^{-}$ and $2.17 \\, e^{-}$ , respectively.", "The tail extending to large read noise of the distribution is well known for CMOS sensors and has been characterized as Random Telegraph Signal (RTS) noise.", "RTS is generated from defects within the silicon that lead to increased electron entrapment and signal variance upon readout.", "[13] Fewer than 0.2% of the pixels have an RMS noise greater than $4 \\, e^{-}$ .", "Figure: The IMX290LLR sensor at RadCORE.", "(Left) The sensor underneath the Cs-137 sample.", "(Right) Close up of mount securing sensor" ], [ "Radiation Testing", "We irradiated a IMX290LLR sensor with gamma rays to study its performance in a radiation environment.", "The testing was done at the RadCore facility at the University of Iowa, see Fig.REF .", "The camera was exposed to gamma-rays from Cs-137 (main peak at 662 keV) at a rate of 0.054 krad(Si)/minute.", "We operated the camera, recording 10 ms frames at the same gain settings as discussed above, while monitoring the current drawn by the camera via its USB connection from a radiation shielded location.", "While the camera was operating correctly, the current was 0.16 A.", "We ended the test after the current dropped to 0.02 A and the camera simultaneously ceased to function.", "The camera functioned correctly while exposed to radiation for a total of 314.8 minutes, giving a total exposure of 17.1 krad before failure.", "This is a lower bound on the radiation tolerance of the sensor since some other component on the camera may have failed.", "Figure: Median ADU value versus accumulated doseFig.", "REF shows the median ADU value versus accumulated dose.", "The ADU value at the start of the test corresponds to charge deposition of $\\sim 20,000 \\, e^{-}/{\\rm s}$ .", "The median ADU drops by $\\approx 15\\%$ during accumulation of the first 6 krad, recovers by $\\approx 4\\%$ during a 13 minute interval during which the irradiation was stopped, and then shows a more gradual decrease during the remaining testing.", "The variations may be due to ionization damage of the gate oxide in the active pixel circuity.", "[14] Such damage anneals at room temperature, consistent with the recovery while irradiation was stopped.", "Ionization damage depends on rate of irradiation which was much higher during the test than expected in a typical low Earth orbit (LEO), e.g.", "$\\sim $ 5 krad over 2 years.", "Thus, testing at a low dose rate would be of interest.", "We note that the sensor continued to operate as the median ADU decreased.", "Thus, the sensor would likely continue to function, although perhaps with some change in the energy calibration, over a moderate duration mission of several years in LEO." ], [ "Conclusion", "We have characterized the performance of a commercial Sony IMX290LLR CMOS sensor as an imaging x-ray spectrometer intended for small satellites.", "Radiation testing shows that the sensor and accompanying electronics is operational for well beyond the radiation doses expected for missions of two years duration in low Earth orbit.", "Testing at a synchrotron x-ray beamline demonstrates the back-illuminated CMOS sensor operated at temperatures near +20$^{\\circ }$ C has an energy resolution competitive with the cooled CCD sensors on current x-ray observatories, enabling use of such sensors in the resource-limited environment of small spacecraft." ], [ "Acknowledgments", "We thank Amanda Kalen of RadCore for her assistance with the radiation testing.", "This research was supported in part by the Iowa Space Grant Consortium under NASA Award No.", "NNX16AL88H.", "This research used resources of the Advanced Photon Source, a U.S. Department of Energy (DOE) Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No.", "DE-AC02-06CH11357; additional support by National Science Foundation under Grant No.", "DMR-0703406." ] ]	2011.14223
[ [ "Wideband Monitoring Observations of PSR J1803-3002A in the Globular\n Cluster NGC 6522" ], [ "Abstract We report the first wideband monitoring observations of PSR J1803-3002A, a relatively bright millisecond pulsar in the globular cluster NGC 6522 with a spin period of 7.1 ms and no known binary companion.", "These observations were performed using the Parkes 64-m radio telescope with the Ultra-Wideband Low (UWL) receiver system, which covers 704 to 4032 MHz.", "We confirm that PSR J1803-3002A is an isolated millisecond pulsar located near the cluster center and probe the emission properties of the pulsar over the wide observed band.", "The mean pulse profile consists of three components, with the outer components becoming more prominent at higher frequencies, and a mean spectral index for the pulsed emission of -1.66+/-0.07 over the observed band.", "The fractional linear and circular polarization increase with increasing frequency, which is unusual for pulsars.", "We determine a Faraday rotation measure of -107+/-6 rad m-2 for the pulsar.", "PSR J1803-3002A is a distant pulsar in the Galactic plane, but there is no evidence of pulse broadening due to interstellar scattering in our observations.", "These results demonstrate the power of ultra-wideband receiving and signal processing systems." ], [ "Introduction", "Millisecond pulsars (MSPs) in globular clusters (GCs) are important diagnostic tools for a suite of astrophysical problems.", "GC MSPs can be used to study the evolution of binary systems [11], the gravitational potential of the GC [10], the dynamical interactions in the GC core [39], the interstellar medium (ISM) and its magnetization [3], the intracluster medium, e.g., in 47 Tucanae, [2] and more generally [23], the equation of state of nuclear matter [26] and neutron star retention in GCs [27].", "Radio searches for and/or studies of the pulsars in GCs can be challenging, because many of them are at large distances, which makes their flux density typically weak and signals strongly distorted by propagation through the ISM.", "Moreover, the pulsars in GCs are often members of tight binary systems, causing large changes in their observed spin period and sometimes periodic eclipsing of the radio signal.", "Baade's Window is an area of the sky close to the Galactic Center direction with relatively low amounts of interstellar dust and gas along the line of sight from the Earth.", "The globular cluster NGC 6522 lies near the center of Baade's Window and, with an estimated distance of 7.7 kpc [13]http://physwww.mcmaster.ca/$\\sim $ harris/Databases.html, is relatively close to the Galactic Center.", "This GC is well studied and, with a stellar age of $>$ 12 Gyr, is possibly the oldest GC in our Galaxy [4].", "It is a core-collapsed cluster [18] and searching for pulsars in such clusters has the potential to identify exotic systems.", "Although the majority of the pulsars are expected to be isolated because of the high interaction rate [37], eccentric MSP binaries with massive companions (neutron stars as well as white dwarfs) can be produced through exchange encounters, e.g.", "B2127$+$ 11C in M15, J1807$-$ 2500B in NGC 6544, and J1835$-$ 3259A in NGC 6652 [29], [21], [8].", "Another such system, J0514-4002A, is found in NGC 1851, which possesses a massive and dense core [34], although not a core-collapsed cluster.", "The ultimate reward of such searches would be a MSP – black-hole binary, which may result from similar conditions.", "Three MSPs (PSRs J1803$-$ 3002A, J1803$-$ 3002B and J1803$-$ 3002C) have been found in the cluster by previous pulsar surveys.", "PSR J1803$-$ 3002A is relatively strong and was discovered in the Parkes Globular Cluster survey at 20 cm [28].", "The other two pulsars were discovered using the Green Bank Telescope [9].", "To date, there have been no published long-term timing measurements, polarization observations or flux density measurements for the pulsars in NGC 6522.", "Their approximate positions, pulse periods, and dispersion measures (DMs) were obtained from pulsar-search data.", "We have carried out the first wide-bandwidth observations, timing and polarization analyses for pulsars in NGC 6522 using the Ultra-Wideband Low (UWL) receiver system installed on the Parkes 64m radio telescope [17].", "In Section , we describe the details of our observations and processing methods.", "We then present a wideband timing solution for PSR J1803$-$ 3002A in Section REF , pulse polarization profiles and rotation measure (RM) in Section REF , and flux densities and spectral properties in Section REF .", "In Section REF we discuss our search results for other pulsars in the direction of NGC 6522.", "A summary is provided in Section ." ], [ "Observations and Processing", "We have carried out 17 observations of PSR J1802$-$ 3002A in NGC 6522 between 2019 August 4 (MJD 58669.28) and 2020 July 4 (MJD 59034.69) using the UWL receiver system on the Parkes 64-m radio telescope.", "The telescope was pointed at the nominal cluster center: 18$^{\\rm h}\\;03^{\\rm m}\\;3402$ , $-30\\;02\\;023$ [13].", "Table REF lists each observation, the start time of the observation in Coordinated Universal Time (UTC) and modified Julian date (MJD), the observation length, the observing mode (fold or search), and the project ID.", "For the search-mode observations, total intensity data were recorded with 2-bit sampling every 64 $\\mu s$ in each of the 1 MHz-wide frequency channels (3328 channels across the whole band from 704 MHz to 4032 MHz).", "Within each 1 MHz channel, the data were coherently de-dispersed at a DM of 192.37 pc cm$^{-3}$ (corresponding to the mean of the values of the known pulsars in the GC).", "We carried out a periodicity search for pulsars in the GC within a DM range of 182–202 pc cm$^{-3}$ using a relatively wide band (970–3018 MHz) which was cleaned of radio frequency interference (RFI).", "We also carried out a periodicity search for pulsars in the GC direction with a DM range of 0–1000 pc cm$^{-3}$ using a narrower RFI-clean band: 1216–1472 MHz.", "For the periodicity search, we split each search mode observation into 4400 s blocks and carried out searches for pulsar signals in the Fourier domain using the prestohttp://www.cv.nrao.edu/sransom/presto/ software suite [32], with a Fourier drift-rate $z$ range [1] of $\\pm 200$ to give sensitivity to pulsars in short-period binary orbits [24].", "Pulsars with narrow pulse profiles have many significant harmonics of the spin frequency in the power spectrum and so we summed up to 16 harmonics to increase the significance of the final detection.", "We were only able to detect PSR J1803$-$ 3302A, which has a pulse period of $\\sim $ 7.1 ms, in any of our searches.", "We subsequently folded each search-mode observation with 60 s subintegrations and 256 phase bins using dspsr [36].", "We removed 5 MHz at each edge of each of the 26 sub-bands which comprise the full bandwidth of the UWL in order to avoid the effects of aliasing [17] and then manually removed data affected by RFI in frequency and time for each channel and sub-integration.", "The folded search-mode data were helpful in building a timing solution for PSR J1803$-$ 3302A.", "In order to obtain a phase-connected timing solution for PSR J1803$-$ 3002A and accurately calibrate both its flux density and polarization properties, we carried out additional follow-up fold-mode observations that cover the entire band of the UWL receiver.", "For these fold-mode observations, the data was coherently de-dispersed at a DM of 192.37 pc cm$^{-3}$ with full Stokes information recorded, channelized with 1 MHz channels, reduced to 512 phase bins per period, and written to disk with 20 s subintegrations.", "We also recorded a short (1 minute) observation of a switched calibration noise source [17] before each observation in order to allow for polarisation calibration of each observation.", "To calibrate the polarization response of the UWL feed, we used multiple observations of the bright MSP PSR J0437$-$ 4715 that covered a wide range of parallactic angles [35], taken during the commissioning of UWL in 2018 November.", "To get an initial timing solution of PSR J1803$-$ 3002A, we measured pulse times of arrival (ToAs) using programs from the psrchive package [14] to sum each observation in both frequency and polarisation, before partially summing in time to form 20-min sub-integrations and cross-correlating each summed profile against a standard reference pulse profile.", "The tempo2 software package [15] was then used to derive an initial timing solution for the pulsar from the ToAs.", "We then re-fitted each observation by using this initial timing solution to generate more precise ToAs, which in turn enabled us to iteratively obtain more precise timing solutions.", "For a final solution, the data were again partially summed in time to form 20-min sub-integrations and in frequency to form 11 unequal sub-bands defined by three groups: five equal sub-bands from 704 to 1344 MHz (RF Band 1) four sub-bands from 1344 to 2368 MHz (RF Band 2), and two sub-bands from 2368 to 4032 MHz (RF Band 3).", "ToAs were then formed using different noise-free templates for the three RF bands obtained by fitting to mean pulse profiles for these bands using the program paas.", "The final phase-coherent timing solution was then obtained using tempo2 with jumps between the RF bands to allow for the different templates used.", "To probe the pulsed emission properties of PSR J1803$-$ 3002A, we again used programs from the psrchive package.", "We constructed average polarization profiles by summing the pulsar's calibrated observations in time using the timing solution of the pulsar to ensure phase alignment.", "To obtain a high signal-to-noise ratio (S/N) polarization profile for further analysis, we selected the average polarization profiles of the pulsar in the three RF bands for each observation for which the S/N was more than 10.", "They were then included in the summation using psrwt.", "Since the pulse profile has no measurable linear polarisation in RF Band 1 (see Section REF below) the rotation measure (RM) was determined across the two higher-frequency bands using the rmfit program.", "The measured RM was then used to refer all measured position angles (PAs) to the overall band center, 2688 MHz, before summing in frequency to form the average polarization profiles.", "The pulse widths at $50\\%$ (W$_{50}$ ) and $10\\%$ (W$_{10}$ ) of the pulse peak were measured using the program pdv from noise-free profiles obtained using the program paas.", "Flux densities were also measured from the summed, calibrated observations using the program pdv, summing the data across phase bins as appropriate.", "The flux density scale was set using observations of the radio quasar 0407-658 assuming a flux density of 14.4 Jy at 1400 MHz with a power-law spectral index of $-1.189$ .", "A separate flux density measurement was made for each of the 26 sub-bands comprising the whole bandwidth from (704 to 4032 MHz) of the UWL receiver.", "These sub-band flux densities were then used to derive the spectral properties (see Section REF below).", "Conditional on data embargoesTypically 18 months following the observation., the data from these observations are available from the CSIRO's data archivehttps://data.csiro.au/ [16].", "We have also produced a publicly downloadable data collection containing our processed data fileshttps://doi.org/10.25919/5f45d801827d6 [40].", "This data collection contains the integrated and RFI-removed file for each observation, and the adopted timing model for PSR J1803$-$ 3002A.", "Table: Observations of PSR J1803--3002A" ], [ "Timing Solution", "We used the tempo2 software package [15] with the DE436 solar system ephemeris and the TT(TAI) time standard to obtain a phase-connected timing solution extending over 335 days.", "The timing residuals for PSR J18003$-$ 3002A are shown in Figure REF .", "Residuals have an approximately Gaussian distribution about the zero line, with no trends in either time or frequency, indicating that there is neither detectable red timing noise nor non-dispersive frequency dependence in the ToAs.", "Although the data span is slightly less than one year, the pulsar position is well determined.", "The timing solution of the pulsar is presented in Table REF .", "As discussed in Section , since the pulse profile evolves significantly with frequency (see Section REF below), we use different profile templates for the three RF bands and fit for jumps between the bands, with Band 2 as reference.", "Timing residuals are shown in Figure REF with the different RF bands identified by color.", "We also present derived parameters in Table REF .", "According to our timing solution, the pulsar's position is offset from the nominal cluster centre by 145 which, for a cluster distance of 7.7 kpc, corresponds to a perpendicular offset of 0.54 pc.", "Table: Parameters for PSR J1803--3002A" ], [ "Polarization profiles and RM", "In Figure REF , we present the average polarization profile for PSR J1803$-$ 3002A for the entire UWL observation band centred at 2368 MHz in the top panel.", "We also show the average polarization profiles of the pulsar in the three RF bands, centered at 1024, 1856, and 3200 MHz, respectively.", "In all cases, the position angles are referred to the overall band center, 2688 MHz, using the derived RM (see below).", "The bottom panel of the figure shows the phase-resolved power-law spectral indices and their uncertainties.", "Since searches over a wide range of RM detected no measurable linear polarization in RF Band 1, we determine the RM using data from just RF Bands 2 and 3.", "The derived RM is $-107\\pm 5$ rad m$^{-2}$ .", "We have not corrected this result for the ionospheric contribution, expected to be relatively small, between $-0.2$ and $-2$ rad m$^{-2}$ , at the latitude of Parkes [12].", "However, to allow for the additional uncertainty in the measured value, we increase quoted uncertainty to 6 rad m$^{-2}$ as given in Table REF .", "With the wide-band and coherently de-dispersed observations, we are able to investigate changes of the observed pulse shape relating to intrinsic profile changes and propagation effects in the ISM over a wide frequency range.", "We measured the width of the profiles for PSR J1803$-$ 3002A at 50% and 10% of the peak flux density in the three RF bands and have listed these results in Table REF .", "We notice that there is a systematic increase in pulse width with increasing frequency.", "This is entirely due to the flatter spectrum of the outer components relative to the central component.", "Like many other wide-profile MSPs [7], the pulse components in PSR J1803$-$ 3002A have a central pulse phase that is frequency-independent.", "In general, MSPs do not follow the radius-to-frequency mapping commonly inferred for normal pulsars [6].", "For MSPs, the emission region may be close to the light cylinder and caustic effects are likely to be important in defining the observed pulse profile [33].", "For PSR J1803$-$ 3002A, the observed pulse profiles indicate three components (labeled as C1, C2 and C3 in the top panel of Figure REF ), with the leading and trailing components becoming relatively stronger compared to the central component as the observing frequency increases.", "As shown by [7], many MSPs show variations in relative spectral index between different profile components.", "Linear polarization is commonly seen in MSPs [7] although the observed PA variations seldom follow a rotating-vector-model pattern [30].", "Compared to other MSPs, PSR J1803$-$ 3002A has a relatively small amount of linear polarization.", "Furthermore, it is unusual in that the fractional linear polarisation decreases with decreasing frequency (Figure REF ).", "As mentioned above, no significant linear polarisation was observed for Band 1 data in a search over a wide range of RMs and variations in ionospheric RM are too small to depolarize the profile.", "Depolarisation can also arise in highly scattered pulsars due to propagation through turbulent plasma components that are irregularly magnetised.", "For example, PSRs B2111$+$ 46 [25] and J0742$-$ 2822 [38] show a rapid decrease in linear polarisation with decreasing frequency at frequencies $<300$ MHz where there is significant scattering by the ISM.", "However, we have not seen any evidence for interstellar scattering even in the low-frequency band (scattering timescale $<$ 0.5 ms).", "Consequently, the observed low polarisation cannot be attributed to stochastic Faraday rotation across the scattering disk.", "We therefore conclude that the observed low degree of linear polarization for PSR J1803$-$ 3002A at low frequencies is intrinsic.", "Figure: Average polarization profiles for PSR J1803--3002A in the three RF bands between 704 and 4032 MHz and the sum over the whole band.", "The red line is the linear polarization profile, the blue line is the circular polarization profile and the black line is the mean flux density profile.", "Black dots in the upper panel give the linear position angle (PA) referred to the overall band center of the averaged profiles, 2688 MHz.", "The three main pulse components are labeled on the whole-band profile.", "The bottom panel shows phase-resolved spectral index with uncertainties (blue points with error bars) for the pulsar.", "The horizontal dashed line marks the fitted power-law spectral index for the whole profile (--1.66) and the gray area shows its uncertainty (±\\pm 0.07).", "The horizontal bars in the top panel mark the region of each component used to calculate its spectral index." ], [ "Flux densities and spectral properties", "We list the calibrated flux densities averaged over all observations for the 26 sub-bands (each with a 128 MHz bandwidth) across the UWL band in Table REF .", "Figure REF shows the flux density values separately from observations taken from 2019 November to 2020 February (labeled “obs1\"), and from 2020 April to May (labeled “obs2\"), together with the average of all fold-mode observations (labeled “All\").", "We first fit a simple power law: $\\text{S}_{\\nu }=\\beta x^{\\alpha },$ where $x = {\\nu }/{\\nu _{0}}$ , $\\alpha $ is the spectral index and $\\beta $ is a scaling constant.", "The results of the power-law fit are given in Table REF and shown in Figure REF .", "The fitted spectral index is $-1.66\\pm 0.07$ .", "Figure REF suggests that there may be some curvature in the spectrum although it is of marginal significance.", "We have also fitted the measured flux densities with a log-parabolic spectrum (LPS): $\\log _{10}\\text{S}_{\\nu } = ay^{2}+by+c,$ where $y=\\log _{10}({\\nu }/{\\nu _{0}})$ , $a$ is the curvature parameter, $b$ is an effective spectral index and $c$ is a constant.", "The LPS fit result is presented in Table REF and shown in Figure REF .", "Overall, we believe the simple power-law fit to be a more reliable representation of the spectral properties of PSR J1803$-$ 3002A.", "The ATNF Pulsar Cataloguehttps://www.atnf.csiro.au/research/pulsar/psrcat/ [22] lists spectral indices for 55 MSPs and these lie in the range $-$ 1.1 to $-$ 3.8, with a mean value of $-2.0$ .", "The measured spectral index for PSR J1802$-$ 3002A, $-1.66$ , is well within this range, and somewhat flatter than the mean value.", "The bottom panel in Figure REF shows the phase-resolved power-law spectral index across the PSR J1803$-$ 3002A profile, clearly showing variations related to the pulse components.", "We calculated the mean spectral indices for the central region of each component where the spectral index is relatively phase-independent to avoid the effects of overlap of the different components.", "The derived spectral indices are $-0.79\\pm 0.10$ , $-2.20\\pm 0.05$ and $-1.01\\pm 0.13$ for the leading (C1), central (C2) and trailing (C3) components, respectively.", "Steep-spectrum central or “core\" components and flatter-spectrum outer or “conal\" components are common in normal pulsars [5], [31], [20] and are usually interpreted in the context of the magnetic-pole model.", "In general, MSP profiles are more complex and do not have “core-conal\" structure, but such structures are seen in a few MSPs, e.g., PSR J1730$-$ 2304 [7].", "This pulsar is similar to PSR J1803$-$ 3002A in having a central component with a steeper spectrum.", "Figure: Flux density as a function of frequency with 26 frequency sub-bands (each sub-band with 128 MHz bandwidth) over the whole band of the UWL for PSR J1803--3002A.", "The legend indicates the color/line scheme used to represent specific fold-mode observations or the average of all fold-mode observations.", "The dot-dashed and dashed lines represent the spectral fitting result of the average of all fold-mode observations for a power-law and LPS spectrum, respectively.", "See the text in Section for more details." ], [ "Search for pulsars", "Globular clusters have proved to be fruitful for pulsar searches with 157 radio pulsars in 30 GCs discovered to date [22], [9].", "Almost all of these pulsars are MSPs and 85 are known to be in binary systems.", "A Fourier-domain implementation of the acceleration search in PRESTO [32] has been use to find the vast majority of the binary pulsars in GCs.", "Such searches are effective when $T \\lesssim P_b/10$ , where $T$ and $P_b$ are the observation duration and the binary-system orbital period, respectively.", "On the other hand, the sensitivity to fainter pulsars improves as $\\sqrt{T}$ .", "As described in Section , we split long search-mode observations into 4400-second blocks as a compromise between sensitivity and computational feasibility giving sensitivity to binary systems with orbital period greater than about 0.5 d. The nominal ($8\\sigma $ ) sensitivity of our searches to a 4 ms pulsar with DM $\\sim $ 192 pc cm$^{-3}$ and spectral index about $-$ 2.0 is 0.03 to 0.04 mJy at 1400 MHz.", "Only PSR J1803$-$ 3002A was detected in our pulsar search.", "Specifically, we were not able to detect either of two previously reported pulsars in the cluster (PSRs J1803$-$ 3002B and J1803$-$ 3002C), nor any other pulsar lying within the telescope beam.", "Even folding our search data at the known period and DM of PSRs J1803$-$ 3002B and J1803$-$ 3002C [9] gave no significant detection of either pulsar.", "As mentioned in the Introduction, there are no published flux densities for any of the three known pulsars in NGC 6522.", "In this work, we present the previously unpublished timing solution (Section REF ) and emission properties (Sections REF and REF ) of PSR J1803$-$ 3002A in NGC 6522.", "A search for pulsars in the direction of NGC 6522 detected only PSR J1803$-$ 3002A, confirming that it is the strongest of the three known pulsars in the cluster (Section REF ).", "The mean pulse profile of PSR J1803$-$ 3002A is rather weakly polarized and no linear polarization was detectable in the lowest RF band (704–1344 MHz) – the pulsar RM was measured using data from the two higher RF bands.", "The overall radio spectral index for PSR J1803$-$ 3002A, $-1.66\\pm 0.07$ , is somewhat flatter than $-2.0$ , the mean spectral index for MSPs.", "The three identifiable components of the pulse profile have significantly different spectral indices, with the central component having a steeper spectrum than the mean and the outer components having flatter spectra.", "We will continue to observe this pulsar with the UWL receiver to improve its timing parameters and to build up higher-S/N pulse polarization profiles.", "Searches for pulsars in Baade's Window with the UWL receiver are on-going and will help us to probe the poorly understood distribution of distant pulsars.", "The observations described here were in either search mode or fold mode.", "The facility for simultaneous pulsar search as well as fold-mode observations and HI spectroscopy [19] with the UWL system has been recently implemented.", "Furthermore, simultaneous folding at multiple pulsar periods is planned.", "These improvements will substantially enhance the observational efficiency for searches, confirmation, and timing.", "This work is supported by National Key R&D Program of China No.", "2017YFA0402600, the National Natural Science Foundation of China (Grant No.", "11988101, 11725313, 11690024, 11743002, 11873067), the Strategic Priority Research Program of the Chinese Academy of Sciences Grant No.", "XDB23000000 and the Foundation of Guizhou Provincial Education Department (No.", "KY(2020)003).", "QJZ is supported by the National Natural Science Foundation of China (U1731218) and the Science and Technology Fund of Guizhou Province ((2016)-4008, (2017)5726-37).", "JBW is supported by the Youth Innovation Promotion Association of Chinese Academy of Sciences.", "The Parkes radio telescope is part of the Australia Telescope National Facility which is funded by the Australian Government for operation as a National Facility managed by CSIRO.", "We thank the Parkes team for their great effort to install and commission the UWL receiver system.", "This paper includes archived data obtained through the CSIRO Data Access Portal$^4$ ." ] ]	2011.14092
[ [ "Medium-band observation of the neutrino emitting blazar, TXS 0506+056" ], [ "Abstract TXS 0506+056 is a blazar that has been recently identified as the counterpart of the neutrino event IceCube-170922A.", "Understanding blazar type of TXS 0506+056 is important to constrain the neutrino emission mechanism, but the blazar nature of TXS 0506+056 is still uncertain.", "As an attempt to understand the nature of TXS 0506+056, we report the medium-band observation results of TXS 0506+056, covering the wavelength range of 0.575 to 1.025 $\\mu$m.", "The use of the medium-band filters allow us to examine if there were any significant changes in its spectral shapes over the course of one month and give a better constraint on the peak frequency of synchrotron radiation with quasi-simultaneous datasets.", "The peak frequency is found to be $10^{14.28}$ Hz, and our analysis shows that TXS 0506+056 is not an outlier from the blazar sequence.", "As a way to determine the blazar type, we also analyzed if TXS 0506+056 is bluer-when-brighter (BL Lac type and some flat spectrum radio quasars, FSRQs) or redder-when-brighter (found only in some FSRQs).", "Even though we detect no significant variability in the spectral shape larger than observational error during our medium-band observation period, the comparison with a dataset taken at 2012 shows a possible redder-when-brighter behavior of FSRQs.", "Our results demonstrate that medium-band observations with small to moderate-sized telescopes can be an effective way to trace the spectral evolution of transients such as TXS 0506+056." ], [ "Introduction", "There has been a significant amount of efforts to understand the origin of high-energy cosmic neutrinos with energies $> 100$ TeV detected by the IceCube collaboration (e.g., Aartsen et al.", "2013, 2014, 2019, 2020; IceCube Collaboration, 2013; Kankare et al.", "2019), and their connection with high-energy astrophysical phenomena such as cosmic rays.", "In this regard, the recent identification of TXS 0506+056, a blazar at $z=0.3365$ (Paiano et al.", "2018), as the likely site of the high-energy neutrino emission phenomenon IceCube-170922A marked the first time that enhanced electromagnetic (EM) radiation from an astronomical source is found to coincide with a high-energy cosmic neutrino event, and thus provided an exciting opportunity to understand where and how high-energy cosmic neutrinos are produced.", "Studies to convincingly associate TXS 0506+056 with the neutrino event is still ongoing (e.g., Lipunov et al.", "2020), and so as the efforts to explain the physical mechanism for the observed neutrino and EM radiation event all together (e.g., Ansoldi et al.", "2018; Padovani et al.", "2018; Sahakyan.", "2018; Banik & Bhadra.", "2019; Cerruti et al.", "2019; Righi et al.", "2019; Xue et al.", "2019; Cao et al.", "2020; Li et al.", "2020; Morokuma et al.", "2020; Petropoulou et al.", "2020; Zhang et al.", "2020).", "One of key ingredients for such efforts to succeed is to understand the physical properties of the neutrino emission site, namely the blazar TXS 0506+056.", "The leading candidate of high-energy neutrino site has been blazars.", "Therefore, it is not very surprising that IceCube-170922A was found to be associated with one.", "Blazars are active galactic nuclei (AGNs) with their relativistic jets pointing toward us.", "The production of high-energy neutrinos is thought to be a result of the interaction of protons and nuclei accelerated in the jet with the surrounding medium that triggers chain reactions and produce high-energy neutrinos and cosmic rays.", "The same environment provides relativistically traveling leptons that give rise to the prominent two components in the spectral energy distribution (SED) of blazars, the low-energy peaked component (the infrared to X-ray) through synchrotron radiation of high-speed electrons, and the high-energy peaked component (GeV to TeV gamma-rays) through inverse Compton scattering of low-energy photons by electrons.", "However, there is a wide variety of blazars and different types of blazars often represent very different physical environments, making it important to pin down the dominant type of neutrino emitting blazars for constraining the high-energy neutrino emission mechanism.", "Blazars can be broadly divided into two types, BL Lac objects and Flat Spectrum Radio Quasars (FSRQs).", "Phenomenologically speaking, BL Lacs are blazars with featureless optical spectra, while FSRQs have strong, quasar-like emission lines in their optical spectra (Padovani et al.", "2017).", "Physically speaking, FSRQs are intrinsically high-luminosity AGNs with their jets pointing toward us, and BL Lacs are intrinsically low-luminosity AGNs such as radio galaxies with beamed jets very closely aligned to us.", "Therefore, in terms of energetics, FSRQs are more likely to be the site of high-energy neutrino emission (e.g., Murase et al.", "2014; Dermer et al.", "2014), although there are proposed mechanisms that can produce high-energy neutrinos from BL Lacs (Tavecchio et al.", "2014; Righi et al.", "2017; Ansoldi et al.", "2018) or even from radio galaxies (e.g., Hooper 2016).", "Interestingly, the optical spectrum of TXS 0506+056 shows no notable features and a smooth continuum indicative of the emission being dominated by synchrotron radiation (Halpern et al.", "2003; Paiano et al.", "2018).", "Therefore, TXS 0506+056 is a BL Lac object from the spectral classification point of view.", "However, Padovani et al.", "(2019, hereafter, P19) recently suggested that TXS 0506+056 is a masquerading BL Lac with properties in line with FSRQs based on several accounts: (i) its radio and [OIII] luminosities are consistent with those of jetted luminous quasars rather than radio galaxies; (ii) it has the emission line ratios of Seyfert 2 galaxies; and (iii) the Eddington ratio is $> 0.01$ , which is too high for BL Lacs and low luminosity AGNs.", "Furthermore, they showed that TXS 0506+056 is an outlier of the blazar sequence, a sequence defined in the 2-D plane of the peak frequency of the synchrotron radiation of blazar SED, $\\nu ^{S}_{\\rm {peak}}$ , versus the gamma-ray luminosity (Ghisellini et al.", "1998; Fossati et al.", "1998; Giommi et al.", "2012; Padovani et al.", "2012).", "Therefore, it is necessary to examine in detail the observational characteristics of TXS 0506+056 to see how it fits into the blazar types.", "One way to better constrain the blazar nature of TXS 0506+056 is to study its SED in the optical-infrared (IR), and spectral variability as a function of time.", "The $\\nu ^{S}_{\\rm {peak}}$ has been found to be $\\lesssim 10^{14}$ Hz for FSRQs, and $\\gtrsim 10^{14}$ Hz for masquerading BL Lacs.", "Previous studies looked into this issue already, finding $\\nu ^{S}_{\\rm {peak}} \\sim 10^{14.5}$ Hz using available archival data from various epochs and sparsely sampled photometric information at around the IceCube-170922A event epoch (e.g., P19).", "However, re-examination of the $\\nu ^{S}_{\\rm {peak}}$ value using quasi-simultaneous, densely sampled (in wavelength) datasets are desirable to better constrain $\\nu ^{S}_{\\rm {peak}}$ since TXS 0506+056 is known to show a rapid variability and the existing data are sometimes too sparse in wavelength coverage.", "Optical spectral variability of blazars have been investigated extensively in the past, and it has been found that the optical spectra of BL Lacs are “bluer-when-brighter” (Ikejiri et al.", "2011; Bonning et al.", "2012; Gaur et al.", "2012; Wierzcholska et al.", "2015), while a significant portion of FSRQs ($\\sim $ tens of %) have the variability of “redder-when-brighter” (Gu et al.", "2006; Hu et al.", "2006; Rani et al.", "2010; Meng et al.", "2018).", "Although the fraction of “redder-when-brighter” FSRQs may be lower and some FSRQs show “bluer-when-brighter” or no variable behavior, virtually no BL Lacs are found to be “redder-when-brighter”.", "Therefore, the behavior of the optical spectral variability can be another indicator to tell if TXS 0506+056 is a BL Lac or a FSRQ.", "In this respect, low resolution spectroscopy with narrow or medium-band (MB) filters can be a powerful way to gain insight on the characteristics of the blazar.", "Although moderate- to high-resolution spectra can give us rich information about the object in study, a wider spectral width sampled by each MB filter increases signal-to-noise (S/N) per spectral element in comparison to conventional moderate- to high-resolution spectroscopy.", "At the same time, MB filter widths are many times finer than broad-band filters, so that shapes of SEDs can be better traced with MB filters than broad-band filters.", "With these advantages, small ($\\lesssim 1$ m) and mid-sized telescopes ($\\sim 2$ m) can be utilized for MB-based low resolution spectroscopy, even for faint objects for which spectroscopy has been considered only possible with large telescopes.", "Since observing time is more readily available for small and mid-sized telescopes than large telescopes, time-intensive spectroscopy of many targets, such as long-term monitoring observations, can be done with MB filters on a small telescope.", "Several examples that show the power of the MB observations are the identification of faint, high-redshift quasars (Jeon et al.", "2017; Kim et al.", "2018a, 2019b; Shin et al.", "2020) and the AGN reverberation mapping (Kim et al.", "2019a).", "To better understand the nature of the optical variability associated with IceCube-170922A, we conducted a series of MB filter observations using SED camera for Quasars in Early Universe (SQUEAN, Kim et al.", "2016) and the Wide-field Integral-Field Unit (IFU) Telescope (WIT) from 2017 October 14 (UT), or about 23 days after the IceCube-170922A event, until 2017 November 6.", "Here, WIT is a 0.25 m wide-field imaging telescope, equipped with multiple MB filters covering a field view of 2.34 $\\times $ 2.34 deg$^2$ .", "Since it can produce MB-based low-resolution spectra at each pixel in the entire field of view, we name the telescope as the Wide-field “IFU” Telescope.", "This paper presents results from our mostly MB-based observations of TXS 0506+056, focusing on demonstration of the effectiveness of MB-based data for the study of the SED temporal variation and $\\nu ^{S}_{\\rm {peak}}$ of the blazar emission.", "We introduce WIT in Section 2.", "The observation and data reduction are introduced in Section 3.", "The variability of TXS 0506+056 during our monitoring period is given in Section 4, along with the time-series SEDs of the blazar.", "In Section 5, we derive $\\nu ^{S}_{\\rm {peak}}$ and discuss the spectral variability of this object based on our observations.", "Finally, we present our conclusion in Section 6." ], [ "Wide-field Integral-Field-Unit Telescope (WIT)", "WIT is a 0.25 m, f/3.6 Takahashi CCA-250 telescope equipped with a 4k $\\times $ 4k CCD camera and a series of MB filters.", "The telescope is a modified Cassegrain telescope with a focal reducer.", "The CCD camera is the Finger Lake Instrumentation (FLI)'s ML16803 model, which has 4096 $\\times $ 4096 pixels with a pixel size of 9 $\\mu $ m. Therefore, the pixel scale is 206, and the camera covers a field of view of 2.34 $\\times $ 2.34 deg$^2$ .", "Figure: Wide-field IFU Telescope (middle), attached on the 0.8 m telescope (left) of the McDonald Observatory.The telescope was mounted on the 0.8 m telescope of the McDonald Observatory on 2017 February 14 (Figure 1).", "The system includes a filter wheel that can house eight 50 mm square filters, and a suite of softwares that allows remote operation of the telescope.", "Currently, five medium-band filters, $m575, m625, m675, m725$ , and $m775$ are installed, where the alphabet $m$ in the filter name stands for “medium-band”, and the number represents the central wavelength of the filter in nm.", "The filter widths are 50 nm each.", "The combined transmission curves of the filters and the quantum efficiency of the camera are shown in Figure 2.", "The effective wavelength and the full-width at half-maximum (FWHM) of the combined filter throughput of each filter of WIT are computed following Choi & Im (2017), and they are presented in Table 1. c\|c\|c Effective wavelength and FWHM of MB on WIT Filter $\\lambda _{eff}$ FWHM $m575$ 573.7 48.4 $m625$ 623.6 48.7 $m675$ 674.6 48.6 $m725$ 724.8 48.8 $m775$ 774.2 49.2 (1) - Filter name; (2) - Effective wavelength of the filter (nm); (3) - Full-width at half-maximum (nm) Figure: The throughput curves for each filter of SQUEAN (Top) and WIT (Bottom).", "The black lines represent the throughput of MBs and the red, dashed lines represent the throughput of broad-bands.", "The names of filters are shown on the throughput curve.In addition to these MB filters, WIT can be equipped with the standard Johnson $B, V, R, I$ filters and holographic diffraction gratings for slitless spectroscopy.", "WIT has been operational since 2017 February 16, and mostly operated remotely from Korea, for monitoring observation of nearby galaxies to discover supernovae (Im et al.", "2019), and to trace the spectral variation of AGNs in the field around each nearby galaxy." ], [ "Observation", "We observed TXS 0506+056 with the Otto Struve 2.1 m telescope and WIT at the McDonald Observatory.", "The 2.1 m Otto Struve telescope is equipped with SQUEAN.", "SQUEAN has 10 MB and 10 broad-band filters, allowing us to trace SED shapes of targets with a moderate investment of observational resources (see, e.g.", "Kim et al.", "2019b).", "The observations were conducted at 2017 October 14 through 2017 November 6 (UT) for a total of 10 nights.", "We used 9 MBs and 4 broad-bands of SQUEAN, and 5 MBs for the WIT observation.", "The exposure time per frame was set at 150 seconds for the WIT observations.", "For the case of SQUEAN, the exposure times were set at 10 secs to 60 secs depending on the weather condition and the passband.", "The observation logs are given in Tables 5 and 6 in Appendix A.", "Figure: The VV-band light curve of TXS 0506+056 from CRTS and ASAS-SN surveys.", "Only the points that have magnitude error less than 0.1 are shown.", "The VV-band magnitude of the object increased about 0.5 mag from 50 days before the neutrino event.", "The time of the neutrino detection is shown with a red vertical line.", "The epoch of the Rau et al.", "(2012) data (inactive phase) is indicated with the green arrow and the corresponding VV-band magnitude is marked with a green circle, while our observation period is indicated with the blue arrow and the corresponding m575m575-band magnitudes are overplotted with blue circles.Figure 3 indicates $m575$ data from our observations, overplotted on the Catalina Real-time Transient Survey (CRTS, Drake et al.", "2009) and the All-Sky Automated Survey of Supernovae (ASAS-SN, Shappee et al.", "2014; Jayasinghe et al.", "2019) $V$ -band light curve.", "Epochs of our observations correspond to the time shortly after the most active phase near the IceCube-170922A event, but still during the active phase." ], [ "Data reduction and calibration", "We reduced the data in a standard manner, for the bias, dark, and flat-field corrections.", "For flat-field correction, we used both sky flat-fields and dome flat-fields.", "The IRAF (Tody 1986) and custom IDL codes were used for these reductions.", "Figure: Light curves of TXS 0506+056 using different filters.", "The blue points are the WIT data and the black points are those from SQUEAN.", "It shows the mild variabilities in the filters m575m575 through m775m775.", "The filter name is shown in the upper left corner and the excess variability is shown in the upper right corner in each plot.Photometric calibration was done using objects near the target in the image, in a manner similar to the method described in Jeon et al.", "(2016) and Choi & Im (2017).", "In this procedure, we first identified point sources in the field to be used as photometry reference stars by running the object detection software SExtractor (Bertin & Arnouts 1996) and matched them to $B, V, g, r, i$ photometry from the AAVSO Photometric All Sky Survey (APASS) catalog (Henden et al.", "2016).", "For the case of WIT, we selected 10 to 40 photometry reference stars within 36$$ from the target, where the point sources are chosen to be those with the probability of being point source greater than 0.98, which is calculated by SExtractor.", "For the 2.1 m telescope data, we selected about 10 stars in the SQUEAN field of view.", "For both broad-band and MB data, we fitted spectral templates of 175 stars (Gunn & Stryker, 1983; Strecker et al., 1979) to the multi-band APASS magnitudes of the reference stars, and derived the best-fit template.", "After identifying the best-fit template, we used the template to synthetically calculate the magnitude of each band, $m_{\\rm AB}$ , of the reference stars using the following equation: $m_{\\rm {AB}}= -2.5 \\, \\rm {log}( \\frac{\\textit {\\int } \\textit {d}\\textit {\\lambda \\, \\lambda } \\, \\textit {F}_{\\lambda } \\, \\textit {R}_{\\lambda }}{\\textit {c} \\, \\textit {\\int } \\textit {d}\\textit {\\lambda } \\, \\textit {R}_{\\lambda } \\, \\textit {/} \\, \\textit {\\lambda }})-48.6 ,$ where $R_{\\lambda }$ is the throughput function of each filter including the detector quantum efficiency, and $F_{\\lambda }$ is the specific flux of the best-fit stellar template.", "Using these magnitudes of the reference stars, we derived the zero point of each image in each filter.", "The error is taken as the standard deviation of the zero points from all the reference stars.", "To derive the magnitude of TXS 0506+056, we performed aperture photometry using SExtractor.", "The aperture size was chosen as 1.5 times of the seeing value to obtain the maximal signal-to-noise (S/N).", "The magnitudes of the reference stars were obtained in the same way, therefore, the aperture magnitude zero points include aperture correction.", "The photometric error of each magnitude is computed as root mean square (rms) of the zero point error and the photometric error from SExtractor.", "Note that the magnitudes in the tables are corrected for the Galactic extinction, by adopting E(B-V) = 0.094 from E(B-V) map of Schlafly $\\&$ Finkbeiner (2011) with the Cardelli et al.", "(1989) law assuming $\\it R_{V}$ = 3.1 and the dust extinction function of Fitzpatrick (1999)." ], [ "Variability of TXS 0506+056", "Figure 4 shows the MB and broad-band light curves of TXS 0506+056.", "At a glance, TXS 0506+056 seems to be somewhat variable during the explored period, showing a decrease in flux, and then increase in flux again.", "To quantify the significance of variability, we performed two analyses.", "First, we performed a $\\chi ^{2}$ test to see how variable the light curves are (de Diego 2010; Villforth et al.", "2010; Kim et al.", "2018b).", "The $\\chi ^{2}$ value of each light curve is $\\chi ^2 =\\sum _{i=1}^{N}\\frac{(m_{i}-\\mu _{mag})^2}{\\sigma _{mag,i}^{2}} ,$ where $m_{i}$ is the magnitude at the epoch $i$ , $\\mu _{mag}$ is the mean value of the magnitudes, and $\\sigma _{mag,i}$ is the magnitude error at the epoch $i$ .", "The null hypothesis is that the object in the filter is not variable.", "We can calculate the confidence level of rejection, $C_{reject}$ as $C_{reject} =1 - \\int _{\\chi ^2}^{\\infty }P(\\chi ^2 ,dof) \\, d\\chi ^2,$ where $P(\\chi ^2 ,dof)$ is the $\\chi ^{2}$ distribution fuction of the degree of freedom $dof$ .", "The results are presented in Table 2, and the $C_{reject}$ values are about 1 for all the filters except for the filters with only 3 epochs and the mean photometric error $\\langle \\sigma _{mag} \\rangle > 0.045$ Therefore, the $\\chi ^2$ result shows that the object is variable.", "Second, in order to confirm the variability of the object, C-test value ($C_{test}$ ) was calculated as (e.g., Kim et al.", "2018b) $C_{test} = \\frac{\\sigma _{\\rm {LC}}}{\\langle \\sigma _{mag}\\rangle },$ where $\\langle \\sigma _{mag}\\rangle $ is the average photometric error over all the epochs, and $\\sigma _{\\rm {LC}}$ is the standard deviation of the light curve magnitudes.", "If $C_{test}$ is greater than 1.96 (2.56), the object is variable with 95% (99%) confidence level.", "We find that the $C_{test}$ values are above 1.96 and some exceed 3, except for those for the light curves of the filters at $\\ge 875$ nm with only 3 epochs and $\\langle \\sigma _{mag}\\rangle > 0.05$ mag (Table 2).", "The $C_{test}$ value for the $m975$ filter light curve is the lowest value of 1.25, and even for this case, we can say that the confidence level is 79% in favor of variability.", "With the results from the $\\chi ^2$ test and C-test, we conclude that the blazar has been variable during our observation.", "To quantify how much the object was variable, we derived the excess variability $\\sigma _{ex}$ defined as (Kim et al.", "2018b, 2019a), $\\sigma _{ex} =\\sqrt{\\sigma _{\\rm {LC}}^{2} - \\langle \\sigma _{mag} \\rangle ^{2}} \\, .$ The excess variabilities are found to be at 0.04 to 0.12 mag, and they are positive for all the MB filters.", "In particular, for the filters with more than 8 epochs of data ($m575$ through $m775$ ), suggesting inter-day variability of TXS 0506+056 at the level of 0.1 mag level during the monitored period.", "c\|c\|c\|c\|c\|c\|c\|c Variability Test Results Filter $\\chi ^2$ $\\nu $ $\\sigma _{LC}$ $\\langle \\sigma _{mag}\\rangle $ $C_{reject}$ $C_{test}$ $\\sigma _{ex}$ $m575$ 34.8 8 0.101 0.045 1.00 2.24 0.09 $m625$ 394.8 11 0.114 0.029 1.00 3.93 0.11 $m675$ 249.4 11 0.103 0.040 1.00 2.58 0.09 $m725$ 392.9 11 0.122 0.032 1.00 3.81 0.12 $m775$ 94.1 11 0.107 0.041 1.00 2.61 0.10 $m825$ 23.3 3 0.083 0.042 1.00 1.98 0.07 $m875$ 4.7 3 0.090 0.057 0.80 1.58 0.07 $m925$ 4.2 3 0.069 0.045 0.76 1.53 0.05 $m975$ 4.1 3 0.075 0.060 0.75 1.25 0.05 $m1025$ 6.5 3 0.061 0.048 0.91 1.27 0.04 $r$ 129.2 3 0.087 0.023 1.00 3.78 0.08 $i$ 28.1 3 0.062 0.031 1.00 2.00 0.05 $z$ 11.8 3 0.078 0.031 1.00 2.52 0.07 $Y$ 2.9 3 0.074 0.058 0.59 1.28 0.05 (1) - Filter name; (2) - Chi-squared value; (3) - Degree of freedom; (4) - Standard deviation of the light curve (mag); (5) - Mean photometric error of the light curve (mag); (6) - Confidence of rejection; (7) - C-statistic; (8) - Excess variability (mag); Figure: The optical SEDs at each epoch and the fitting results.", "The best-fit spectral slope α λ \\alpha _{\\lambda } and χ ν 2 \\chi _{\\nu }^2 values are shown in the upper right corner of each plot.", "UT date of the observation is shown as well." ], [ "SED temporal variation", "c\|c\|c\|c Power-law fitted parameters Date $\\alpha _{\\lambda }$ $\\sigma _{\\alpha _{\\lambda }}$ $\\chi _{\\nu }^2$ 2017-10-14 -0.55 0.03 0.58 2017-10-16 -0.60 0.03 1.02 2017-10-17 -0.78 0.16 0.31 2017-10-18 -0.51 0.07 0.22 2017-10-21 -0.45 0.19 0.09 2017-10-28 -0.71 0.15 1.05 2017-10-29 -0.57 0.19 0.58 2017-10-30 -0.81 0.12 0.24 2017-11-03 -0.65 0.18 0.45 2017-11-06 -0.62 0.17 0.60 (1) - Date of observation (UT); (2) - Power-law slope; (3) - Error of the power-law slope; (4) - Reduced chi-squared value The time-series MB data allow us to examine in detail how the SED of TXS 0506+056 evolves with time.", "To quantify the variability of the SED shape, we modeled the specific flux of TXS 0506+056 with a power-law shape of $F_{\\lambda } \\sim \\lambda ^{\\alpha _{\\lambda }}$ .", "We did so because typical blazars show a power-law shape of SEDs indicative of the synchrotron radiation (e.g., Zacharias & Schlickeiser 2010).", "In particular, TXS 0506+056 has been reported as having a featureless power-law continuum in optical with the $\\alpha _{\\lambda }$ value at around $\\alpha _{\\lambda } \\sim -1$ to 0 (Rau et al.", "2012; IceCube Collaboration et al.", "2018; Paiano et al.", "2018).", "We fitted the observed SEDs at each epoch with this power law model, $\\rm {log} \\, F_{\\lambda }=\\alpha \\,\\rm {log}\\,\\lambda +K ,$ where $K$ is the logarithm of the proportional constant.", "Figure 5 shows SEDs of TXS 0506+056 at different epochs.", "From the figure, we confirm that the power law model can represent the SED shape of the blazar well, with the reduced $\\chi ^{2}$ value of each fit showing the value around one or less.", "Table 3 lists the fitted parameters of each epoch SED.", "The spectral slopes $\\alpha _{\\lambda }$ are plotted as a function of time in Figure 6.", "Figure: The spectral slope as a function of time.", "The power-law index α λ \\alpha _{\\lambda } varies during our observation period.To examine if the power-law index shows a strong variability, we applied the $\\chi ^2$ test, similarly to what we did in Section 3.1.", "We find the value of $\\chi ^2$ = 13.88 with the degree of freedom of 9 for $\\alpha _{\\lambda }$ , and a modest value of $C_{reject}$ = 0.87 for the null hypothesis that $\\alpha _{\\lambda }$ is not variable.", "This result shows that $\\alpha _{\\lambda }$ , or the spectral shape is not strongly varying above the level that goes beyond the measured uncertainty of $\\sim 0.15$ during the monitoring period.", "Combining with the result from Section 4.1, we conclude that TXS 0506+056 is variable in short-term ($\\sim $ days), but the variability is rather achromatic during the same period." ], [ "Discussion", "As we mentioned in the introduction, the blazar nature of TXS 0506+056 is an interesting topic, to better understand the neutrino emission mechanism.", "We discuss here two spectral properties of TXS 0506+056, its synchrotron peak frequency, $\\nu _{peak}^S$ , and the spectral variability as a function of brightness." ], [ "Synchrotron peak frequency, $\\nu ^{S}_{peak}$", "The synchrotron peak frequency has been used as a rough way to sort blazars, with FSRQs having log($\\nu _{peak}^S$ /Hz) $\\lesssim $ 14, and BL Lacs having any kind of $\\nu _{peak}^S$ .", "Masquerading BL Lacs are known to have log($\\nu _{peak}^S$ /Hz) $\\gtrsim $ 14 (Padovani et al.", "2019).", "It is also a key quantity that defines the blazar sequence.", "Therefore, we examine its value in detail using our new data.", "Due to the moderate but rapid variability of TXS 0506+056, we need to construct its SED with data taken nearly simultaneously ($\\lesssim $ 1 day).", "The $\\nu _{peak}^S$ value can be in near-infrared or mid-infrared, therefore it is desirable to have IR data too.", "Our MB photometry data satisfy the criteria of being taken nearly simultaneously, and we also identify the quasi-simultaneous multi-band photometry data from UV through NIR in Rau et al.", "(2012, hereafter, R12).", "For the R12 dataset, the $Swift$ UV and $u, b$ , and $v$ and the GROND data ($g,r,i,z,J,H$ , and $K$ ) were taken at about 2 days apart.", "Therefore, the $Swift$ data and the optical/NIR data are not exactly simultaneous.", "There is indeed a small offset in the $v$ and $b$ magnitudes versus their GROND $g$ and $r$ magnitudes.", "Therefore, we shifted the $Swift$ magnitude to match the epoch of the GROND data by deriving the magnitude offset with respect to a reference epoch and applying it to all the $Swift$ data.", "The achromatic nature of the short-term variability (Section 4.2) justifies this approach for constructing a quasi-simultaneous dataset.", "The offset was found by deriving the interpolated $v$ magnitude from the GROND data and comparing it to the Swift $v$ mag, where quasi-simultaneous $v$ magnitude is obtained by linearly interpolating the GROND $g$ and $r$ magnitudes and taking the flux at the effective wavelength of the filter.", "The offset value is found to be 0.14 mag.", "Additionally, we estimated the WISE $W1$ (3.36 $\\mu $ m) and $W2$ (4.61 $\\mu $ m) data corresponding to the two epochs.", "Searching for the WISE and NEOWISE data, we find the WISE data taken about 36 and 91 days near our MB observaion and the R12 GROND epochs not close enough in time to treat as quasi-simultaneous.", "Therefore, we took an alternative approach of finding the correlation between $V$ -band data from ASAS-SN, and WISE $W1$ and $W2$ data.", "We matched the ASAS-SN and WISE data in time domain, allowing a maximum time gap to be 0.2 days.", "12 epochs of data are fitted within the criterion.", "As shown in Figure 7, there is good correlation between $V$ and $W1$ or $W2$ , and we derived the relation between the $V$ magnitude and the WISE $W1, W2$ magnitudes as below.", "$W1= 1.30 \\, V - 8.63 \\,\\, (\\pm 0.14, 1\\,\\sigma )$ $W2= 1.18 \\, V - 7.73 \\,\\, (\\pm 0.16, 1\\,\\sigma )$ Using the above relations, we can calculate the WISE $W1, W2$ magnitude at any epoch using a given $V$ magnitude at a given epoch.", "The $V$ -band derived $W1$ or $W2$ magnitude errors can be obtained as the rms of the offset between the WISE data and the best-fit correlations, which are found to be 0.14 mag and 0.16 mag respectively.", "Figure: The VV - W1W1 and VV - W2W2 magnitude relation.", "VV magnitudes are from ASAS-SN survey and W1W1, W2W2 magnitudes are from WISE survey.", "We matched the data collected at 57200 and 58300 MJD with the time difference of observation between WISE and ASAS-SN data less than 0.2 days.", "The correlation between VV vs. W1W1 and W2W2 and the standard deviation in the WISEWISE magnitudes are shown in each plot.To calculate the WISE $W1$ , $W2$ magnitude at the epoch of our observation, we extrapolated the MB SED at the corresponding epoch and took the flux at 550 nm which is the effective central wavelength of the $V$ -band filter.", "For the R12 epoch, the offset-corrected $v$ photometry was adopted to calculate the $W1$ and $W2$ data.", "In Figure 8, we show these quasi-simultaneous SEDs of the TXS 0506+056.", "The $V$ , $W1$ , and $W2$ photometry values are given in Table 4.", "For our MB data, we used the epoch of 2017 October 14, since the optical SED shape does not vary dramatically during our observation period (Section 4.2).", "We note that our observation epoch corresponds to the active phase, while the R12 epoch corresponds to a rather inactive phase.", "This is evident in the long-term $V$ -band light curve (Figure 3).", "The $\\nu _{peak}^S$ values were obtained by fitting the SEDs with polynomial functions.", "We obtained the best-fit $\\nu _{peak}^S$ values for the 2nd-, 3rd-, 4th- order polynomial functions, since the derivation of $\\nu _{peak}^S$ can vary significantly what fitting function we use.", "Then, we calculated $\\nu _{peak}^S$ and its error to be the mean value and the standard deviation of the three $\\nu _{peak}^S$ values obtained from the three fitting methods.", "We find log($\\nu _{peak}^S$ )= 14.69 $\\pm $ 0.11 for MJD 55540 (the R12 epoch, the darker state) and log($\\nu _{peak}^S$ ) = 14.28 $\\pm $ 0.09 for MJD 58040 (our epoch, the brighter state).", "If we use the magnitudes of Swift and WISE for the darker state that are closest in time (open circles in Figure 8), instead of the quasi-simultaneous magnitudes, we get log($\\nu _{peak}^S$ ) = 14.59 $\\pm $ 0.10, which is slightly smaller than the log($\\nu _{peak}^S$ ) from the quasi-simultaneous magnitudes.", "The $\\nu _{peak}^S$ values are consistent with the values presented in P19, and the expected $\\nu _{peak}^S$ of a masquerading BL Lac, or an intermediate energy BL Lac.", "We also note that the $\\nu ^{S}_{peak}$ determination could have imporved if there were NIR data, since the estimated $\\nu ^{S}_{peak}$ occurs at NIR for MJD 58040. c\|c\|c\|c Photometry values of $V$ , $W1$ and $W2$ filters Date $V$ (error) $W1$ (error) $W2$ (error) 56911.555 14.838 (0.065) 10.457 (0.015) 9.569 (0.017) 56911.556 14.697 (0.052) 10.457 (0.015) 9.569 (0.017) 57276.530 15.065 (0.048) 11.041 (0.019) 10.068 (0.019) 57276.531 15.133 (0.046) 11.041 (0.019) 10.068 (0.019) 57276.532 15.062 (0.046) 11.041 (0.019) 10.068 (0.019) 57441.298 14.520 (0.051) 10.478 (0.018) 9.589 (0.013) 57441.299 14.600 (0.056) 10.478 (0.018) 9.589 (0.013) 57441.301 14.478 (0.048) 10.478 (0.018) 9.589 (0.013) 57805.117 14.690 (0.041) 10.193 (0.019) 9.282 (0.014) 57805.118 14.595 (0.040) 10.193 (0.019) 9.282 (0.014) 57805.119 14.640 (0.041) 10.193 (0.019) 9.282 (0.014) 58004.559 14.542 (0.048) 10.023 (0.021) 9.087 (0.015) (1) - Date of observation of $V$ -band (Modified Julian Date).", "; (2) - $V$ -band magnitude from ASAS-SN survey; (3) - $W1$ -band magnitude from WISE survey matched with $V$ -band in time within 0.2 days; (4) - $W2$ -band magnitude from WISE survey matched with $V$ -band in time within 0.2 days; Figure: SED of the TXS 0506+056 at two different epochs, one near the neutrino emission (brighter phase, the blue line and points), and another at darker phase (the black dashed line and points).", "The Swift u,b,v,UVW1,UVM2,UVW2u, b, v, UVW1, UVM2, UVW2 fluxes presented in black filled circles are corrected for the 2-days of time gap between GROND and Swift observation.", "Uncorrected Swift fluxes are also shown in gray empty circles.", "The WISE data that are closest in time to the epoch of each observation are also shown as open symbols.", "The best-fit of the data with 2nd-order polynomial are overploted as solid and dashed lines.Figure: The blazar sequence with gamma-ray luminosity (left) and 5 GHz radio luminosity (right).", "The data of FSRQs and BL Lacs from Fossati et al.", "(1998, right) and Fermi 3LAC (Ackermann et al.", "(2015, left) are plotted with black squares and green triangles respectively.", "The TXS 0506+056 data from P19 is presented with open blue circles and those from our observation is plotted with red circles.", "The blazar model from Ghisellini et al.", "(2017) is plotted with cyan circles." ], [ "Is TXS 0506+056 an outlier of the blazar sequence?", "Figure 9 shows how this object fits in the blazar sequence, in both gamma-ray luminosity vs. $\\nu _{peak}^S$ and radio luminosity vs. $\\nu _{peak}^S$ spaces.", "For the case of the gamma-ray luminosity vs. $\\nu _{peak}^S$ plot, the FSRQ and BL Lac data are taken from the Fermi 3LAC catalog (Ackermann et al.", "2015), while the Fossati et al.", "(1998) data are used for the radio luminosity vs. $\\nu _{peak}^S$ plot.", "The gamma-ray luminosity of TXS 0506+056 is taken from P19.", "The 5 GHz radio flux of TXS 0506+056 is derived from the interpolation of 4.68 GHz and 7.02 GHz data (rest-frame) of IceCube Collaboration et al.", "(2018) obtained at MJD = 58038.35, which is the closest in time to our MB observation.", "The luminosity ($\\rm {log}\\,L_{5GHz}$ ) is calculated from flux, assuming a $\\Lambda $ CDM concordance cosmology of $H_{0} = 70$ km s$^{-1}$ Mpc$^{-1}$ , $\\Omega _{M} = 0.3$ and $\\Omega _{\\Lambda } = 0.7$ which has been supported by observations in the past decades (e.g., Im et al.", "1997; Planck Collaboration et al.", "2018).", "We found $\\rm {log}\\,L_{5GHz}$ = 43.01.", "On contrary to the suggestion by P19, Figure 9 shows that TXS 0506+056 sits in the blazar sequence, meaning that TXS 0506+056 is not an exceptional blazar in the blazar sequence.", "Compared to the blazar sequence constructed from the blazar model of Ghisellini et al.", "(2017, cyan circles), TXS 0506+056 appears as an outlier, but TXS 0506+056 sits near the Ackermann et al.", "(2015, black squares and green triangles) data, which is also used to construct the model of Ghisellini et al.", "(2017).", "We note that the blazar sequence points taken from Ghisellini et al.", "(2017) deviates from the Fermi 3LAC SEDs and the Fossati et al.", "(1998) points.", "By plotting the Fermi 3LAC SEDs of blazars, we confirm that the values listed in the Fermi 3LAC catalogs.", "Therefore, it is not clear the origin of the discrepancy, but we suspect that it is due to the model function used to determine the $\\nu _{peak}^S$ by Ghisellini et al.", "(2017)." ], [ "Redder-when-brighter or bluer-when-brighter?", "The next interesting point in Figure 8 is the difference between the spectral shapes during the active and inactive phases.", "We find that the R12 spectral shape in the inactive phase is flatter in the optical than the active phase (our data), suggesting a redder-when-brighter behavior found in FSRQs.", "The redder-when-brighter behavior is likely to be due to the thermal component of the SED at optical-IR range as suggested by many previous studies (Rani et al.", "2010; Bonning et al.", "2012).", "The SED of the blazars at optical-IR range can be broken into two main components.", "One is the non-thermal emission by the jet and the other is the thermal emission by the accretion disk.", "Intrinsically bright quasars such as FSRQs, have a strong thermal emission component with a blue SED, and their rapid variability is controlled by a non-thermal component which is dominant at longer wavelength.", "Therefore, when the jet activity is high, the non-thermal component brightens (SED at longer wavelength) causing the redder-when-brighter behavior.", "Therefore, the redder-when-brigther behavior of TXS 0506+056 suggests that it is an intrinsically bright AGN.", "Obviously, one should not draw a definitive conclusion from the comparison of SEDs at only the two epochs.", "Therefore, we examined how the optical spectral slope changes with the apparent magnitude, assuming that the apparent magnitude is a good proxy of the AGN activity.", "Figure 10 shows $\\alpha _{\\lambda }$ versus $m575$ and $m775$ magnitudes which are the bluest and reddest passbands covered by WIT.", "When SQUEAN MB photometry is available we use it instead of the WIT photometry since it provides smaller errors.", "The Pearson's coefficient ($\\rho $ ) and its p-value were 0.38 and 0.35 for the case of $m575$ -band and -0.269 and 0.45 for the case of $m775$ -band.", "Therefore, the significance of the correlation is only 65% and 55% each.", "The results shown in Figure 10 indicate that the change in $\\alpha _{\\lambda }$ is consistent with null hypothesis within the measurement error, agreeing with the achromatic behavior of the SED shape over a short period found in Section 4.2.", "It appears that a longer time and larger magnitude base line is desirable to better characterize the spectral variability.", "If there was such a long-term data, since the measurement accuracy for the $\\alpha _{\\lambda }$ of MB SED is $\\delta \\alpha _{\\lambda }$ of 0.05 - 0.2 (Table 3), the SED slope change of $\\alpha _{\\lambda } \\sim -0.6$ (our result) to $\\alpha _{\\lambda } \\sim -1.0$ (R12) would have been easily explored." ], [ "Implications on the neutrino emission mechanism", "TXS 0506+056 being an intrinsically bright AGN is in line with the conventional view of high energy neutrino production in AGN jets (e.g., see discussion in Ansoldi et al.", "2018).", "The typical production channel of high energy neutrinos is the decay of charged pions that were produced in collisions between high energy protons in jets and ambient matters (Halzen 2017; Meszaros 2017), such as matter ($pp$ interaction) or low energy photons ($p\\gamma $ interaction).", "In AGN jets, with their low-density plasma, the $p\\gamma $ interaction has been a more favored channel (Mannheim 1995) where the target photon energies are in the range of UV to soft X-ray photons.", "Hot, luminous accretion disks of intrinsically bright AGNs are an excellent source for the UV to soft X-ray photons (Ansoldi et al.", "2018).", "Hence, FSRQ-type blazars have been favored to be high energy neutrino sources (Mannheim et al.", "1992; Atoyan & Dermer 2001; Murase et al.", "2014; Dermer et al.", "2014), and our observation is consistent with such an expectation.", "Figure: Upper pannel shows the α λ \\alpha _{\\lambda } versus m575m575 band magnitude plot and lower pannel shows the α λ \\alpha _{\\lambda } versus m775m775 band magnitude plot.", "The Pearson's coefficient (ρ\\rho ) and its p-value are shown as well." ], [ "Discussion summary", "The above analyses teach us that TXS 0506+056 is not an extraordinary outlier from the blazar sequence.", "From the long-term spectral variability analysis, we find that TXS 0506+056 varies as redder-when-brighter which is consistent with the earlier P19 report that TXS 0506+056 is not a BL Lac.", "However, this was limited to only two epochs.", "A more definitive answer requires data at multiple long-term epochs.", "Figure 10 and the spectral slope measurement accuracy achieved from our observation gives a great promise that spectral variability study with MB data at more extended epochs could firmly establish if the object is redder-when-brighter or bluer-when-brighter." ], [ "Conclusion", "We conducted the time-series, MB observational study of the blazar, TXS 0506+056, to understand its nature as the host galaxy of the neutrino event IceCube-170922A.", "Our observation reveals the variable nature of TXS 0506+056, and the MB data obtained using WIT and SQUEAN are found to excellently trace the optical SED shape of the blazar allowing us to measure the spectral index, $\\alpha _{\\lambda }$ to be $\\sim -0.6$ to $\\sim -0.8$ to the accuracy of 2 % to 10 %.", "Our MB data were combined with published dataset and archival data to produce quasi-simultaneous SED at two epochs (one active, another inactive).", "The multi-wavelength SEDs show that $\\nu _{peak}^S$ is of order of $\\nu _{peak}^S \\sim 10^{14.5}$ Hz, which is consistent with previous results.", "Considering that the $\\nu _{peak}^{S}$ value is in NIR, NIR monitoring observation can help better determine $\\nu _{peak}^{S}$ .", "However, the comparison with the blazar sequence data in both gamma-ray and radio luminosities shows that TXS 0506+056 is not an outlier of the blazar sequence.", "The comparison of the 2012 data (inactive) and the 2017 data (active) shows a flatter SED at 2012, possibly indicating the redder-when-brighter nature of FSRQ for TXS 0506+056, but the spectral variability analysis using our data show no clear spectral change as a function of the brightness.", "A more extensive study using a larger set of data is desirable, using a longer time-baseline and larger phase difference.", "In conclusion, our study demonstrates that the MB photometry observation with small and medium-sized telescopes can be a powerful way to study spectral variability of many kinds of astrophysical objects such as neutrino sources, AGNs, gravitational-wave sources, gamma-ray bursts, and supernovae, to name a few (e.g., Abbott et al.", "2017; Im et al.", "2015, 2019; Kim et al.", "2019a).", "Additional NIR monitoring data will be also helpful for better determining the spectral shape, especially $\\nu _{peak}^{S}$ of blazars like TXS 0506+056.", "Wide-field telescopes with MB filters, such as WIT with a field of view of 5.5 deg$^2$ , can be a great force for multi-messenger astronomy to search for and characterize optical counterparts and host galaxies of neutrino and gravitaional-wave sources (e.g., Abbott et al.", "2017; Im et al.", "2017) for which initial positional estimates can be very uncertain ($\\gg 1$ deg$^2$ ).", "We thank John Kuehne and the staffs at the McDonald Observatory for their support during the installation and the operation of WIT.", "This work was supported by the National Research Foundation of Korea (NRF) grant (2020R1A2C3011091) by the Korean government (MSIT) and the Korea Astronomy and Space Science Institute under the R&D program (Project No.", "2020-1-600-05) supervised by MSIT.", "S.S. acknowledges support from Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No.2020R1A6A3A13069198).", "This paper includes data taken at the McDonald Observatory of the University of Texas at Austin.", "Struve (SQUEAN) Astronomy.net (Lang et al.", "2010), IRAF (Tody 1986), SExtractor (Bertin & Arnouts 1996), SWarp (Bertin 2010)" ], [ "Photometry Information Tables", "c\|c\|c\|c\|c\|c\|c\|c\|c TXS 0506+056 photometry with WIT WIT Date Weather Filter Seeing Exposure time Zeropoint (error) Detection limit mag $\\sigma _{m}$ 2017-10-14 08:31 clear $m575$ 4.06 150s$\\times $ 4 18.409 (0.045) 17.58 14.38 0.05 2017-10-14 08:51 clear $m625$ 4.39 150s$\\times $ 4 18.212 (0.036) 17.37 14.35 0.04 2017-10-14 09:13 clear $m675$ 4.84 150s$\\times $ 4 17.800 (0.048) 16.90 14.17 0.05 2017-10-14 09:25 clear $m725$ 4.55 150s$\\times $ 4 17.306 (0.041) 16.58 14.07 0.04 2017-10-14 09:40 clear $m775$ 4.95 150s$\\times $ 4 16.759 (0.046) 16.29 14.01 0.05 2017-10-17 07:35 clear $m575$ 5.18 150s$\\times $ 4 18.412 (0.042) 17.34 14.28 0.04 2017-10-17 07:46 clear $m625$ 5.09 150s$\\times $ 4 18.195 (0.033) 17.34 14.16 0.03 2017-10-17 07:56 clear $m675$ 4.95 150s$\\times $ 4 17.8073 (0.046) 17.17 14.04 0.05 2017-10-17 08:07 clear $m725$ 5.86 150s$\\times $ 4 17.386 (0.032) 16.66 13.94 0.04 2017-10-17 08:19 clear $m775$ 5.64 150s$\\times $ 4 16.768 (0.033) 16.18 13.90 0.04 2017-10-21 09:47 clear $m575$ 3.70 150s$\\times $ 4 18.404 (0.050) 17.78 14.55 0.05 2017-10-21 09:58 clear $m625$ 4.12 150s$\\times $ 4 18.177 (0.034) 17.58 14.40 0.04 2017-10-21 10:10 clear $m675$ 4.46 150s$\\times $ 4 17.786 (0.049) 17.19 14.26 0.05 2017-10-21 10:21 clear $m725$ 3.99 150s$\\times $ 4 17.312 (0.042) 16.95 14.17 0.05 2017-10-21 10:33 clear $m775$ 4.17 150s$\\times $ 4 16.685 (0.045) 16.39 14.03 0.05 2017-10-28 08:27 clear $m575$ 3.62 150s$\\times $ 5 18.388 (0.044) 14.81 14.49 0.05 2017-10-28 08:54 clear $m625$ 3.94 150s$\\times $ 5 18.099 (0.026) 15.20 14.46 0.03 2017-10-28 09:09 clear $m675$ 4.20 150s$\\times $ 5 17.770 (0.036) 14.83 14.31 0.04 2017-10-28 09:24 clear $m725$ 3.86 150s$\\times $ 5 17.280 (0.029) 14.44 14.22 0.03 2017-10-28 09:40 clear $m775$ 4.14 150s$\\times $ 5 16.688 (0.027) 13.84 14.12 0.05 2017-10-29 07:49 clear $m575$ 4.01 150s$\\times $ 5 18.426 (0.045) 17.91 14.60 0.04 2017-10-29 08:05 clear $m625$ 4.38 150s$\\times $ 5 18.203 (0.038) 17.60 14.47 0.04 2017-10-29 08:20 clear $m675$ 4.55 150s$\\times $ 5 17.798 (0.051) 17.34 14.29 0.05 2017-10-29 08:36 clear $m725$ 4.25 150s$\\times $ 5 17.360 (0.044) 17.16 14.26 0.05 2017-10-29 08:51 clear $m775$ 4.68 150s$\\times $ 5 16.768 (0.049) 16.62 14.12 0.05 2017-10-30 06:17 clear $m575$ 6.05 150s$\\times $ 4 18.298 (0.033) 16.81 14.51 0.04 2017-10-30 06:30 clear $m625$ 5.89 150s$\\times $ 4 18.097 (0.020) 16.91 14.37 0.02 2017-10-30 06:42 clear $m675$ 5.22 150s$\\times $ 4 17.686 (0.036) 16.83 14.26 0.04 2017-10-30 06:53 clear $m725$ 7.63 150s$\\times $ 4 17.440 (0.009) 16.30 14.19 0.02 2017-10-30 07:12 clear $m775$ 6.53 150s$\\times $ 4 16.756 (0.012) 15.97 14.11 0.03 2017-11-03 05:58 clear $m575$ 4.37 150s$\\times $ 4 18.440 (0.045) 16.27 14.50 0.05 2017-11-03 06:10 clear $m625$ 4.69 150s$\\times $ 4 18.201 (0.030) 16.28 14.36 0.03 2017-11-03 06:22 clear $m675$ 4.51 150s$\\times $ 4 17.829 (0.044) 16.35 14.22 0.05 2017-11-03 06:33 clear $m725$ 4.65 150s$\\times $ 4 17.376 (0.030) 16.23 14.18 0.04 2017-11-03 06:45 clear $m775$ 4.87 150s$\\times $ 4 16.777 (0.047) 15.92 14.02 0.05 2017-11-06 06:01 clear $m575$ 6.64 150s$\\times $ 4 18.488 (0.035) 15.86 14.42 0.04 2017-11-06 06:16 clear $m625$ 5.37 150s$\\times $ 5 18.195 (0.026) 16.16 14.31 0.03 2017-11-06 06:39 clear $m675$ 5.80 150s$\\times $ 5 17.850 (0.040) 15.97 14.14 0.04 2017-11-06 07:00 clear $m725$ 4.87 150s$\\times $ 4 17.369 (0.039) 15.93 14.11 0.04 2017-11-06 07:22 clear $m775$ 5.07 150s$\\times $ 5 16.774 (0.047) 15.60 13.96 0.05 (1) - Date of observation (UT); (2) - Weather; (3) - Name of filter; (4) - Seeing value ($$ ); (5) - Exposure time (sec); (6) - Zeropoint (mag); (7) - Point source detection limit at 5-$\\sigma $ with exposure time and zeropoint in same row (mag); (8) - Magnitude; (9) - Error of the magnitude c\|c\|c\|c\|c\|c\|c\|c\|c TXS 0506+056 photometry with 2.1m telescope 2.1m telescope Date Weather Filter Seeing Exposure time Zeropoint (error) Detection limit mag $\\sigma _{m}$ 2017-10-14 09:07 clear $Y$ 1.73 60s$\\times $ 3 22.414 (0.067) 19.55 13.54 0.07 2017-10-14 09:22 clear $i$ 2.53 10s$\\times $ 36 24.979 (0.012) 20.72 13.97 0.01 2017-10-14 09:02 clear $m1025$ 1.86 60s$\\times $ 3 21.046 (0.032) 18.61 13.51 0.03 2017-10-14 08:29 clear $m625$ 1.49 30s$\\times $ 5 23.087 (0.012) 20.52 14.28 0.01 2017-10-14 08:32 clear $m675$ 1.47 30s$\\times $ 5 23.436 (0.015) 20.74 14.14 0.02 2017-10-14 08:36 clear $m725$ 1.51 15s$\\times $ 12 23.675 (0.012) 20.64 14.05 0.01 2017-10-14 08:40 clear $m775$ 1.66 15s$\\times $ 12 23.670 (0.021) 20.49 13.95 0.02 2017-10-14 08:45 clear $m825$ 1.63 15s$\\times $ 12 23.642 (0.019) 20.43 13.86 0.02 2017-10-14 08:49 clear $m875$ 1.77 30s$\\times $ 6 23.459 (0.056) 20.44 13.75 0.06 2017-10-14 08:54 clear $m925s$ 1.48 30s$\\times $ 6 22.375 (0.053) 19.30 13.64 0.05 2017-10-14 08:58 clear $m975$ 1.66 60s$\\times $ 3 22.113 (0.060) 19.65 13.60 0.06 2017-10-14 09:29 clear $r$ 2.15 15s$\\times $ 12 24.318 (0.008) 20.80 14.30 0.01 2017-10-14 09:12 clear $z$ 1.31 15s$\\times $ 12 23.708 (0.033) 20.38 13.71 0.03 2017-10-16 08:54 clear $Y$ 1.30 30s$\\times $ 10 21.849 (0.046) 19.29 13.42 0.05 2017-10-16 09:08 clear $i$ 2.46 10s$\\times $ 30 24.908 (0.047) 20.47 13.92 0.05 2017-10-16 08:45 clear $m1025$ 1.52 60s$\\times $ 5 20.600 (0.020) 18.55 13.42 0.02 2017-10-16 07:51 clear $m625$ 2.52 60s$\\times $ 5 23.056 (0.009) 20.51 14.16 0.01 2017-10-16 07:58 clear $m675$ 1.89 30s$\\times $ 10 23.311 (0.010) 20.69 14.02 0.01 2017-10-16 08:06 clear $m725$ 1.71 15s$\\times $ 20 23.595 (0.009) 20.65 13.95 0.01 2017-10-16 08:12 clear $m775$ 1.61 15s$\\times $ 20 23.538 (0.035) 20.51 13.83 0.04 2017-10-16 08:19 clear $m825$ 1.58 15s$\\times $ 20 23.373 (0.054) 20.40 13.70 0.05 2017-10-16 08:28 clear $m875$ 1.59 15s$\\times $ 20 23.165 (0.046) 20.11 13.61 0.05 2017-10-16 08:33 clear $m925s$ 1.86 60s$\\times $ 5 22.494 (0.044) 19.38 13.52 0.04 2017-10-16 08:39 clear $m975$ 1.44 30s$\\times $ 10 21.802 (0.026) 19.44 13.48 0.03 2017-10-16 09:15 clear $r$ 1.84 10s$\\times $ 30 23.994 (0.030) 20.55 14.13 0.03 2017-10-16 09:00 clear $z$ 1.30 10s$\\times $ 30 23.327 (0.026) 19.92 13.59 0.03 2017-10-18 11:53 cloudy $Y$ 1.13 60s$\\times $ 8 21.475 (0.059) 19.27 13.41 0.06 2017-10-18 12:07 cloudy $i$ 1.23 10s$\\times $ 18 23.336 (0.034) 18.72 13.85 0.03 2017-10-18 11:43 cloudy $m1025$ 1.05 60s$\\times $ 10 18.749 (0.089) 17.34 13.39 0.09 2017-10-18 10:24 cloudy $m625$ 1.27 60s$\\times $ 5 19.307 (0.028) 17.81 14.16 0.03 2017-10-18 10:31 cloudy $m675$ 1.18 60s$\\times $ 12 20.156 (0.038) 18.89 14.05 0.04 2017-10-18 10:42 cloudy $m725$ 1.09 60s$\\times $ 5 19.284 (0.018) 17.68 13.91 0.02 2017-10-18 10:53 cloudy $m775$ 1.11 60s$\\times $ 8 19.306 (0.028) 17.72 13.83 0.03 2017-10-18 11:02 cloudy $m825$ 1.46 60s$\\times $ 10 18.948 (0.049) 17.01 13.76 0.05 2017-10-18 11:15 cloudy $m875$ 1.04 60s$\\times $ 10 22.212 (0.069) 20.03 13.58 0.07 2017-10-18 11:23 cloudy $m925s$ 1.84 60s$\\times $ 5 21.462 (0.037) 18.75 13.52 0.04 2017-10-18 11:31 cloudy $m975$ 1.03 60s$\\times $ 10 20.042 (0.094) 18.47 13.46 0.09 2017-10-18 12:11 cloudy $r$ 1.87 30s$\\times $ 10 22.270 (0.031) 18.33 14.19 0.03 2017-10-18 12:01 cloudy $z$ 1.30 15s$\\times $ 12 22.833 (0.032) 19.16 13.57 0.03 (1) - Date of observation (UT); (2) - Weather; (3) - Name of filter; (4) - Seeing value ($$ ); (5) - Exposure time (sec); (6) - Zeropoint (mag); (7) - Point source detection limit at 5-$\\sigma $ with exposure time and zeropoint in same row (mag); (8) - Magnitude; (9) - Error of the magnitude" ] ]	2011.14049
[ [ "Adaptive Inference in Multivariate Nonparametric Regression Models Under\n Monotonicity" ], [ "Abstract We consider the problem of adaptive inference on a regression function at a point under a multivariate nonparametric regression setting.", "The regression function belongs to a H\\\"older class and is assumed to be monotone with respect to some or all of the arguments.", "We derive the minimax rate of convergence for confidence intervals (CIs) that adapt to the underlying smoothness, and provide an adaptive inference procedure that obtains this minimax rate.", "The procedure differs from that of Cai and Low (2004), intended to yield shorter CIs under practically relevant specifications.", "The proposed method applies to general linear functionals of the regression function, and is shown to have favorable performance compared to existing inference procedures." ], [ "Introduction", "We consider the problem of inference on a regression function at a point under the nonparametric regression model $y_{i} = f(x_{i}) + u_{i}, \\quad u_{i} \\overset{\\mathrm {i.i.d.", "}}{\\sim } N(0, \\sigma ^{2}(x_{i})),$ where $f$ is assumed to lie in a Hölder class with exponent $\\gamma \\in (0,1]$ .", "Procedures based on $\\gamma $ is conservative (or suboptimal) when the true regression function in fact lies in a smoother Hölder class with $\\gamma ^{\\prime } > \\gamma $ .", "Adaptive procedures try to overcome this issue by automatically adjusting to the (unknown) underlying smoothness class.", "However, unlike in the case of estimation, where adaptation to the unknown smoothness class is in general possible with an additional logarithmic term ([23]), adaptation is impossible in the case of inference without further restrictions on the function class ([24]) .", "Two shape restrictions that can be used to overcome this impossibility have been discussed in the literature, convexity and monotonicity.", "In this paper, we impose monotonicity on the regression function to construct a CI that adapts to the underlying smoothness of the regression function.", "The main difference with other papers that consider adaptation under a monotonicity condition ([9]; [1]) is our general treatment of the dimension of $x_{i}$ .", "To our knowledge, this is the first paper to construct adaptive CIs, under a multivariate nonparametric regression setting.", "We consider coordinate-wise monotonicity with respect to all or some of the coordinates.", "A function $f$ is coordinate-wise monotone with respect to $\\mathcal {V} \\subseteq \\lbrace 1, \\dots , k\\rbrace $ if $x_{j}\\ge z_{j}$ for all $j \\in \\mathcal {V}$ and $x_{j}= z_{j}$ for all $j\\notin \\mathcal {V}$ imply $f(z) \\ge f(z)$ .", "The minimax expected length of a CI over the Hölder class with exponent $\\gamma $ converges to 0 at the well-known rate of $n^{-1/(2 + k/\\gamma )}$ .", "When the regression function is monotone in all variables, i.e., $\\mathcal {V} = \\lbrace 1, \\dots , k\\rbrace $ , we can construct a CI that achieves this minimax rate over all $\\gamma \\in (0,1]$ just as in the univariate case.", "Also, again as in the univariate case, if the regression is not monotone to any of the variables so that $\\mathcal {V} = \\emptyset $ , there is no scope for adaptation.", "An interesting case is when the function is monotone with respect to only some of the variables so that $k_{+} := \\vert \\mathcal {V} \\vert < k$ , which can arise due to the multivariate nature of the problem.", "In this case, we show that for a CI that maintains coverage over the Hölder class with exponent $\\gamma $ , the minimax expected length over a smoother class $\\gamma ^{\\prime } > \\gamma $ converges to 0 at the rate $n^{-1/(2 + k_{+}/\\gamma ^{\\prime } +(k-k_{+})/\\gamma )}$ .", "The denominator of the exponent can be written as $2 + k/\\gamma - k_{+}( 1/\\gamma - 1/\\gamma ^{\\prime })$ .", "This is the sum of a term that comes from the minimax rate over $\\gamma $ , $2 + k/\\gamma $ , and $- k_{+}( 1/\\gamma - 1/\\gamma ^{\\prime })$ .", "In this sense, $k_{+}( 1/\\gamma - 1/\\gamma ^{\\prime })$ exactly quantifies the possible gain from monotonicity, indicating larger gains if the regression function is monotone in more variables and/or smoother.", "We propose a CI that obtains this minimax rate (of adaptation) for a sequence of Hölder exponents $\\lbrace \\gamma _{j}\\rbrace _{j=1}^{J} \\subset (0, 1]$ .", "While the method provided by [7] can be used to construct such a CI, we provide an alternative method that builds upon the one-sided CI proposed by [3].", "Their one-sided CI “directs power” to a smoother class while maintaining coverage over a larger class of functions.", "Our CI is constructed by combining the lower and upper versions of their one-sided CI to create a two-sided CI, and then taking the intersection of a sequence of such two-sided CIs that direct power to each $\\gamma _{j}$ .", "An appropriate Bonferroni correction is used to obtain correct coverage.", "This CI can be used in more general nonparametric regression settings, as long as the parameter of interest is a linear functional of the regression function and the regression functions lies in a convex function class.", "While the proposed CI obtains the minimax length over $\\gamma _{j}$ for each $j$ up to a constant factor that does not depend on the sample size, this constant does depend on the number of parameter spaces $J$ the CI adapts to.", "This is in contrast with the CI of [7], which gives a multiplicative constant that does not depend on $J$ .", "However, the multiplicative constant of our CI grows slowly with $J$ at a $(\\log J)^{1/2}$ rate, and is smaller than the constant given by [7] for any reasonable specification of $J$ .", "Even if one wishes to adapt to $J= 10^{3}$ parameter spaces, our CI obtains the minimax expected length of each parameter space within a multiplicative constant of $4.14$ , whereas this constant is 16 for the CI by [7].", "A simulation study confirms that our CI can be significantly shorter in practice as well.", "Nonetheless, the uniform constant that [7] obtain is theoretically attractive and allows one to adapt to the continuum of Hölder exponents $(0,1]$ in this context.", "Related literature.", "An adaptation theory for CIs in a nonparametric regression setting was developed by [7].", "[9] provide a procedure for constructing adaptive CIs that adapt to each individual function under monotonicity and convexity.", "[1] provides an inference method for the regression function at a point, possibly on the boundary of the support, that adapts to the underlying Hölder classes under a monotonicity assumption.", "As noted earlier, the main difference of our paper is that we consider a multivariate regression setting where there is no restriction on the dimension of the independent variable as long as it is fixed and finite.", "The adaptation theory for CIs builds upon the more classical minimax theory for CIs, which has been developed in [11] and [24].", "[5] provides an excellent review on the theory of minimax and adaptive CIs, along with the minimax and adaptive estimation problems.", "While the focus of this paper is on adaptive CIs, there are other forms of confidence sets that are of interest in the context of nonparametric regression setting.", "Adaptive confidence balls have been considered in [14], [8] and [25].", "An adaptation theory for confidence bands has been considered in, for example, [12], [13], and [6].", "In the context of density estimation, adaptive confidence bands have also been considered in [17], [15], and [18].", "Recently, there has been interest in isotonic regression in general dimensions.", "The monotonicity condition imposed in such models is the same as the one we impose here with $\\mathcal {V} = \\lbrace 1, \\dots , k\\rbrace $ .", "[16] derive minimax rates for the least squares estimation problem.", "[10] provide a method for constructing CIs at a point based on block max-min and min-max estimators.", "Outline.", "Section describes the nonparametric regression model and the function class we consider.", "Section introduces the notion of adaptivity in more detail and describes our procedure for constructing adaptive CIs.", "Section presents the main result of the paper, the minimax rate of adaptation, and an adaptive CI that obtains this rate by solving the corresponding modulus problem.", "Section provides a simulation study, and Section illustrates our method in the context of production function estimation.", "Any proof omitted in the main text can be found in the appendix.", "Appendix collects the proofs for lemmas and corollaries.", "Appendix contains the proof for our main theoretical result, Theorem REF ." ], [ "Nonparametric Regression Under Monotonicity", "We observe $\\left\\lbrace \\left(y_{i},x_{i}\\right)\\right\\rbrace _{i=1}^{n}$ and consider a nonparametric regression model, $y_{i}=f(x_{i})+u_{i},$ where $x_{i}\\mathcal {\\in X}\\subset \\mathbb {R}^{k}$ is a (fixed) regressor, $f:\\mathbb {R}^{k}\\rightarrow \\mathbb {R}$ is the unknown regression function that lies in some function class $\\mathcal {F}$ , and $u_{i}$ 's are independent with $u_{i}\\sim N(0,\\sigma ^{2}(x_{i}))$ and $\\sigma ^{2}(\\cdot )$ known.", "The parameter of interest is $f(x_{0})$ .", "For the rate results provided in Section REF , we require that $x_{0} \\in \\mathrm {Int}\\,\\mathcal {X}$ .", "However, we note that the solution to the modulus problem given in Section REF does not depend on whether $x_0$ is on the boundary or not.", "Without loss of generality, we normalize $x_{0}$ to be 0.", "We take the $\\mathcal {F}$ to be the class of functions that are Hölder continuous and nondecreasing in all or some of the variables.", "Let $\\Lambda (\\gamma ,C)$ denote the set of functions from $\\mathbb {R}^{k}$ to $\\mathbb {R}$ that are Hölder continuous with Hölder constants $(\\gamma ,C)$ , $\\Lambda (\\gamma ,C):=\\left\\lbrace f\\in \\mathcal {\\mathcal {F}}(\\mathbb {R}^{k},\\mathbb {R}):\\left\|f(x)-f(z)\\right\|\\le C\\left\\Vert x-z \\right\\Vert ^{\\gamma }\\text{ for all }x,z\\in \\mathcal {X}\\right\\rbrace ,$ where $\\mathcal {F}\\left(\\mathbb {R}^{k}, \\mathbb {R}\\right)$ is the set of functions from $\\mathbb {R}^{k}$ to $\\mathbb {R}$ , $\\gamma \\in [0,1],$ $C \\ge 0$ and $\\left\\Vert \\cdot \\right\\Vert $ is a norm on $\\mathbb {R}^{k}$ .", "For notational simplicity, we omit the dependence of the function class on the choice of the norm $\\Vert \\cdot \\Vert $ .", "We impose the following restriction that $\\Vert \\cdot \\Vert $ is monotone in the magnitude of each element, which is satisfied by most norms used in practice.", "such as the $\\ell _{p}$ norm or a weighted version of it.", "We discuss the relationship between this assumption and the monotonicity of the regression function in Remark REF .", "Assumption 2.1 $\\left\\Vert \\cdot \\right\\Vert $ is a norm on $\\mathbb {R}^{k}$ such that $\\left\\Vert z \\right\\Vert $ is nondecreasing in $\\left\|z_{j}\\right\|$ for each $j\\in \\lbrace 1,...,k\\rbrace .$ We now define the (coordinate-wise) monotone Hölder class.", "For a subset of the covariate indices $\\mathcal {V}\\subset \\left\\lbrace 1,\\dots ,k\\right\\rbrace $ , write $\\Lambda _{+,\\mathcal {V}}(\\gamma ,C):=\\left\\lbrace f\\in \\Lambda (\\gamma ,C):f(x)\\ge f(z)\\,\\,\\text{if }x_{j}\\ge z_{j}\\text{ }\\forall j \\in \\mathcal {V} \\text{ and }x_{j}= z_{j}\\,\\,\\forall j\\notin \\mathcal {V}\\right\\rbrace .$ This is the set of Hölder continuous functions that are nondecreasing, coordinate-wise, with respect to the $j$ th element for $j \\in \\mathcal {V}$ .", "Define $k_{+}:=\\left\|\\mathcal {V}\\right\|.$ By a relabeling argument, it is without loss of generality to write $\\mathcal {V}:=\\left\\lbrace 1,\\dots ,k_{+}\\right\\rbrace $ .", "If $k_{+}=k$ , then $\\Lambda _{+,\\mathcal {V}}(\\gamma ,C)$ is the set of nondecreasing and Hölder continuous functions where the monotonicity is with respect to the coordinate-wise partial ordering on $\\mathbb {R}^{k}$ ." ], [ "Notion of Adaptivity", "In this section, we discuss the problem of inference for a general linear functional of the regression function, $Lf$ .", "Consider a sequence of convex parameter spaces $\\mathcal {F}_{1}$ , ..., $\\mathcal {F}_{J}$ , with the requirement that $\\mathcal {F}_{j} \\subset \\mathcal {F}_{J}$ for all $j \\le J$ .", "Note that the parameter spaces are not necessarily nested, but there is a largest convex parameter space that nests all the other parameter spaces.", "Here, $\\mathcal {F}_{J}$ reflects a conservative choice of the parameter space where the researcher believes the true regression function to lie in.", "Hence, the CI we construct will be required to maintain correct coverage over this space.", "An adaptive CI maintains this correct coverage over the largest parameter space $\\mathcal {F}_{J}$ while having good performance (e.g.", "shorter expected length) when the true function happens to lie in the smaller parameter space $\\mathcal {F}_{j}$ , simultaneously for all $j \\le J$ .", "Then, a natural question is how well a CI that maintains coverage over $\\mathcal {F}_{J}$ can perform over $\\mathcal {F}_{j}$ , which is one of the main questions that [7] raise and address in detail in the context of two-sided CIs.", "The case of one-sided CIs has been considered by [3], along with other questions." ], [ "Two-sided Adaptive CIs", "Let $\\mathcal {I}_{\\alpha ,2}^{J}$ denote the set of all two-sided CIs that have coverage at least $1-\\alpha $ over $\\mathcal {F}_{J}.$ Following [7], the performance criterion we consider for two-sided CIs is the worst-case expected length.", "That is, the performance of a CI, $CI$ , over the parameter space $\\mathcal {F}_{j}$ is measured by $\\sup _{f\\in \\mathcal {F}_{j}}\\operatorname{\\mathbf {E}}_{f}\\mu (CI)$ with smaller values of this quantity meaning better performance.", "Here, $\\operatorname{\\mathbf {E}}_{f}$ denotes the expectation when the true regression function is $f$ and $\\mu $ is the Lebesgue measure on the real line.", "Then, the shortest possible worst-case expected length a CI can achieve over $\\mathcal {F}_{j}$ (while maintaining correct coverage over $\\mathcal {F}_{J}$ ) is characterized by the quantity $L_{j,J}^{\\ast }:=\\inf _{CI\\in \\mathcal {I}_{\\alpha ,2}^{J}}\\sup _{f\\in \\mathcal {F}_{j}}\\operatorname{\\mathbf {E}}_{f}\\mu (CI).$ Following [7], we say a CI is adaptive if it achieves $L_{j,J}^{\\ast }$ for all $j \\le J$ up to a multiplicative constant that does not depend on the sample size.", "Let $z_{q}$ denote the $q$ –quantile of the standard normal distribution.", "[7] show that $L_{j,J}^{\\ast } \\asymp \\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{J}),$ with $\\asymp $ denoting asymptotic equivalenceWe write $a_{n}\\asymp b_{n}$ if $0<\\underset{n\\rightarrow \\infty }{\\lim \\inf }({a_{n}/}{b_{n}})\\le \\underset{n\\rightarrow \\infty }{\\lim \\sup }({a_{n}}/{b_{n}})<\\infty .$ and $\\omega _{+}(\\delta ,\\mathcal {F}_{j},\\mathcal {F}_{J})$ is the between class modulus of continuity defined as $&\\omega _{+}(\\delta ,\\mathcal {F}_{j},\\mathcal {F}_{J}) \\\\:=&\\sup \\left\\lbrace \\left\|Lf_{J}-Lf_{j}\\right\|:\\textstyle \\sum _{i=1}^{n}\\left((f_{J}(x_{i})-f_{j}(x_{i}))/\\sigma (x_{i})\\right)^{2}\\le \\delta ^{2},f_{j}\\in \\mathcal {F}_{j},f_{J}\\in \\mathcal {F}_{J}\\right\\rbrace ,$ for $\\delta \\ge 0.$Note that the definition is slightly different with [7] due to the $\\sigma (x_{i})$ term that appears in the denominator of the summand.", "This is because we divide both sides of (REF ) by the (known) $\\sigma (x_{i})$ to convert the model into the same form as that of [7].", "In general, $\\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{J})$ is more tractable than $L_{j,J}^{\\ast }$ , and thus the strategy is to construct a CI that has worst case length over $\\mathcal {F}_{j}$ bounded by $\\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{J})$ , up to a multiplicative constant.", "We refer to the rate at which $\\omega _{+}(z_{1-\\alpha }, \\mathcal {F}_{j}, \\mathcal {F}_{J}) $ converges to 0 as the minimax rate of adaptation (of $\\mathcal {F}_{j}$ over $\\mathcal {F}_{J}$ ).", "If $\\mathcal {F}_{j}=\\mathcal {F}_{J}$ , this is the minimax rate over $\\mathcal {F}_{J}$ , which is the fastest rate at which the worst-case expected length over $\\mathcal {F}_{J}$ of a CI that maintains correct coverage over the same space $\\mathcal {F}_{J}$ can achieve." ], [ "One-sided CIs", "While our main focus is on adaptive two-sided CIs, the construction of our adaptive CI relies heavily on the one-sided CI proposed by [3].", "Hence, we briefly describe the notion of adaptivity in the context of one-sided CIs.", "For one-sided CIs, we follow [3] and consider the $\\beta $ th quantile of excess length as the performance criterion.", "More specifically, for a one sided lower CI, $[\\hat{c},\\infty )$ , we denote the $\\beta $ th quantile of the excess length at $f$ as $q_{\\beta ,f}(Lf-\\hat{c})$ , where $q_{\\beta ,f}(\\cdot )$ denotes the $\\beta $ th quantile function when the true regression function is $f$ .", "Under this criterion, the best possible performance over $\\mathcal {F}_{j}$ is quantified by $\\ell _{j,J}^{\\ast }:=\\inf _{\\hat{c}:[\\hat{c},\\infty )\\in \\mathcal {I}_{\\alpha ,\\ell }^{J}}\\sup _{f\\in \\mathcal {F}_{j}}q_{\\beta ,f}(Lf-\\hat{c}),$ where $\\mathcal {I}_{\\alpha , \\ell }^{J}$ denotes the set of all one-sided lower CIs that have coverage at least $1-\\alpha $ over $\\mathcal {F}_{J}.$ [3] showed that $\\ell _{j,J}^{\\ast }=\\omega (z_{1-\\alpha }+z_{\\beta },\\mathcal {\\mathcal {F}}_{J},\\mathcal {\\mathcal {F}}_{j})$ , where $\\omega (z_{1-\\alpha }+z_{\\beta },\\mathcal {\\mathcal {F}}_{J},\\mathcal {\\mathcal {F}}_{j})$ is the ordered class modulus of continuity defined as $&\\omega (\\delta ,\\mathcal {F}_{j},\\mathcal {F}_{k}) \\\\:=&\\sup \\left\\lbrace Lf_{k}-Lf_{j}:\\textstyle \\sum _{i=1}^{n}\\left((f_{k}(x_{i})-f_{j}(x_{i}))/\\sigma (x_{i})\\right)^{2}\\le \\delta ^{2},f\\in \\mathcal {F}_{j},f_{J}\\in \\mathcal {F}_{J}\\right\\rbrace ,$ for any $\\delta \\ge 0$ and $j,k \\le J$ .", "We refer to the optimization problem in the definition as the ordered modulus problem.", "Naturally, an analogous result holds for upper one-sided CIs so that $u_{j,J}^{\\ast }=\\omega (z_{1-\\alpha }+z_{\\beta },\\mathcal {\\mathcal {F}}_{j},\\mathcal {\\mathcal {F}}_{J})$ , where $u_{j,J}^{\\ast }:=\\inf _{\\hat{c}:(-\\infty ,\\hat{c}]\\in \\mathcal {I}_{\\alpha ,u}^{J}}\\sup _{f\\in \\mathcal {F}_{j}}q_{\\beta ,f}(\\hat{c}-Lf),$ with $\\mathcal {I}_{\\alpha , u}^{J}$ denoting the set of all one-sided upper CIs that have coverage at least $1-\\alpha $ over $\\mathcal {F}_{J}.$ We say a one-sided lower CI, $[\\hat{c}^{\\ast },\\infty )$ , is adaptive if there exists some $c>0$ that does not depend on $n$ such that $\\sup _{f\\in \\mathcal {F}_{j}}q_{\\beta ,f}(Lf-\\hat{c}^{\\ast })\\le c\\,\\omega (z_{1-\\alpha }+z_{\\beta },\\mathcal {\\mathcal {F}}_{j},\\mathcal {\\mathcal {F}}_{J})$ for all $j \\le J$ , and similarly for one-sided upper CIs." ], [ "Modes of Adaptation", "Note that it must be the case that $\\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{j}) \\le \\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{J})$ (and similarly for the ordered moduli) because $ \\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{j})$ takes the supremum over a smaller set.", "However, if it happens to be the case that $\\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{j})\\asymp \\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{J})$ , an adaptive CI, $CI^{\\ast }$ , satisfies $\\sup _{f\\in \\mathcal {F}_{j}}\\operatorname{\\mathbf {E}}_{f}\\mu (CI^{\\ast })\\le \\overline{c}\\,L_{j,j}^{\\ast }$ for all $j\\le J$ .", "[7] define such CI to be strongly adaptive.", "This is an ideal case because we obtain $L_{j,j}^{\\ast }$ , up to a multiplicative constant, which is the minimax length we could have achieved if we “knew” that our true regression function lied in the smaller class $\\mathcal {F}_{j}$ (i.e., if we made a stronger assumption that the true regression function lies in this smaller class).", "While adaptive CIs exist in general, strong adaptation is possible only when $\\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{J})\\asymp \\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{j})$ for all $j \\le J$ .", "This is not a property of a given procedure, but of the given statistical model.", "The least desirable case is when $\\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{J})\\asymp \\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{J},\\mathcal {F}_{J})$ , because this leaves no scope of adaptation.", "An intermediate case is when $\\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{j})\\prec \\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{J})\\prec \\omega _{+}(z_{1-\\alpha },\\mathcal {F}_{J},\\mathcal {F}_{J}),$ so that the minimax rate of adaptation is better than the worst-case minimax rate over $\\mathcal {F}_{J}$ but not as good as the minimax rate over $\\mathcal {F}_{j}$ .For positive sequences $\\left\\lbrace a_{n}\\right\\rbrace $ and $\\left\\lbrace b_{n}\\right\\rbrace $ , we write $a_{n} \\prec b_{n}$ if $\\underset{n\\rightarrow \\infty }{\\lim \\inf }({b_{n}/}{a_{n}}) = \\infty $ .", "That is, one can do better than simply taking the most conservative parameter space as the true space but not quite as good as knowing that the true function actually lies in the smaller parameter space.", "Hence, the minimax adaptation rate plays an important role in determining whether sharp adaptation is possible.", "In Section REF , we derive the minimax rates of adaptation under the model given in Section ." ], [ "Construction of Adaptive CIs", "[7] provide a general method of constructing adaptive CIs of $Lf$ under the general model (REF ).", "Here, we provide an alternative method that is intuitive and gives smaller constants in the case of non-nested parameter spaces.For a given adaptive CI, $CI^{\\ast }$ , we refer to the positive number $c$ (that does not depend on $n$ ) such that $ \\sup _{f\\in \\mathcal {F}_{j}}E\\mu (CI^{\\ast })\\le c\\,\\omega _{+}\\left(z_{\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{J}\\right), $ as the “constant” of $CI^{\\ast }$ .", "For the nested case, the CI of [7] has a bounded constant even as $J \\rightarrow \\infty $ , which is an attractive theoretical property.", "For the CI we propose, the constant will grow with $J$ in general.", "In practice, however, one can only adapt to finitely many parameter spaces due to computational constraints.", "The proposed procedure gives a smaller constant than that of [7] even for unrealistically large values of $J$ (e.g., $J = 10^{10}$ ).", "The main building block for our adaptive CI is the minimax one-sided CI proposed by [3], which relies on the ordered modulus.", "We say that $(f_{j}, f_{k}) \\in \\mathcal {F}_{j} \\times \\mathcal {F}_{k} $ is a solution to $\\omega (\\delta , \\mathcal {F}_{j}, \\mathcal {F}_{k})$ if $(f_{j}, f_{k})$ solves the optimization problem corresponding to $\\omega (\\delta , \\mathcal {F}_{j}, \\mathcal {F}_{k})$ .", "Let $(f_{J,{\\delta }}^{,Jj},g_{j,{\\delta }}^{,Jj})\\in \\mathcal {F}_{J}\\times \\mathcal {F}_{j}$ be a solution to the ordered modulus $\\omega \\left({\\delta },\\mathcal {F}_{J},\\mathcal {F}_{j}\\right),$ and define the estimator $\\begin{aligned}\\hat{L}_{\\delta }^{\\ell ,j} = &\\frac{1}{2}\\, L({g_{j,\\delta }^{,Jj}+f_{J,\\delta }^{,Jj}}) \\\\&+\\frac{\\omega ^{\\prime }\\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right)}{\\delta }\\sum _{i=1}^{n}(g_{j,\\delta }^{,Jj}(x_{i})-f_{J,\\delta }^{,Jj}(x_{i}))\\Bigg (\\frac{y_{i}}{\\sigma (x_{i})}-\\frac{g_{j,\\delta }^{,Jj}(x_{i})+f_{J,\\delta }^{,Jj}(x_{i})}{2}\\Bigg ),\\end{aligned}$ where $\\omega ^{\\prime }(\\cdot , \\mathcal {F}_{J}, \\mathcal {F}_{j})$ is the derivative of $\\omega (\\cdot , \\mathcal {F}_{J}, \\mathcal {F}_{j})$ .", "Based on this estimator, define a lower one-sided CI by subtracting the maximum bias and an appropriately scaled normal quantile: $\\hat{c}_{\\alpha , \\delta }^{\\ell ,j}:= \\hat{L}_{\\delta }^{\\ell ,j}-\\frac{1}{2}\\omega \\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right)+\\frac{1}{2}\\delta \\omega ^{\\prime }\\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right)-z_{1-\\alpha }\\omega ^{\\prime }\\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right).$ The following theorem from [3] shows that for a specific choice of $\\delta $ , this CI is optimal in the sense that it achieves $\\ell ^{\\ast }_{j,J}.$ Lemma 3.1 (Theorem 3.1 of [3]) Let $\\underline{\\delta }=z_{\\beta }+z_{1-\\alpha }$ .", "Then, $\\underset{f\\in \\mathcal {F}_{j}}{\\sup }q_{f,\\beta }(Lf-\\hat{c}_{\\alpha ,\\underline{\\delta }}^{\\ell ,j})= \\ell ^{\\ast }_{j,J}= \\omega (\\underline{\\delta },\\mathcal {F}_{J},\\mathcal {F}_{j}).$ The excess length $Lf-\\hat{c}_{\\alpha , \\underline{\\delta }}^{\\ell ,j}$ follows a Gaussian distribution because it is a affine transformation of the data, which follows a Gaussian distribution by assumption.", "Hence, the median and mean of the excess length are the same.", "Taking $\\beta =1/2$ , we can replace $q_{f,\\beta }$ with the expectation under $f$ , which gives $\\underset{f\\in \\mathcal {F}_{j}}{\\sup }\\operatorname{\\mathbf {E}}_{f}\\left(Lf-\\hat{c}_{\\alpha }^{\\ell ,j}\\right)= \\omega \\left(z_{1-\\alpha },\\mathcal {F}_{J},\\mathcal {F}_{j}\\right),$ where we define $\\hat{c}_{\\alpha }^{\\ell ,j}:= \\hat{c}_{\\alpha , z_{1-\\alpha }}^{\\ell ,j}.$ Likewise, we can define an optimal upper one-sided CI $(-\\infty , \\hat{c}_{\\alpha , \\underline{\\delta }}^{\\ell ,j}]$ such that $\\sup q_{f,\\beta }(\\hat{c}_{\\alpha , \\underline{\\delta }}^{u,j}-Lf) =u^{\\ast }_{j,J} = \\omega (\\underline{\\delta },\\mathcal {F}_{j},\\mathcal {F}_{J}),$ where the precise definition of $\\hat{c}_{\\alpha , \\underline{\\delta }}^{\\ell ,j}$ is given in Appendix .", "Similarly, let $\\hat{c}_{\\alpha }^{u,j}$ denote the upper counterpart of $\\hat{c}_{\\alpha }^{\\ell ,j}.$ Using the optimal one-sided CIs, we first show how a naive Bonferroni procedure leads to a two-sided adaptive CI.", "We then provide a method that improves upon this naive Bonferroni CI by taking into account the correlation among the CIs.", "The naive Bonferroni CI is defined as $CI_{\\alpha }^{Bon,J}:=\\cap _{j=1}^{J}[\\hat{c}_{\\alpha /2J}^{\\ell ,j},\\hat{c}_{\\alpha /2J}^{u,j}].$ This has coverage at least $1-\\alpha $ over $\\mathcal {F}_{J}$ because each $[\\hat{c}_{\\alpha /2J}^{\\ell ,j},\\hat{c}_{\\alpha /2J}^{u,j}]$ has coverage $1 - \\alpha /J$ over $\\mathcal {F}_{J}$ and $CI_{\\alpha }^{Bon,J}$ is simply the intersection of such CIs.", "The following theorem shows that this CI is indeed adaptive.", "Theorem 3.1 For any $j=1,\\dots ,J$ , we have $\\sup _{f\\in \\mathcal {F}_{j}}\\operatorname{\\mathbf {E}}\\mu (CI_{\\alpha }^{Bon,\\mathcal {J}})\\le \\frac{2z_{1-\\frac{\\alpha }{2J}}}{z_{1-\\frac{\\alpha }{2}}}\\,\\omega _{+}(z_{1-\\frac{\\alpha }{2}},\\mathcal {F}_{j},\\mathcal {F}_{J}).$ The constant $2z_{1-\\frac{\\alpha }{2J}}/z_{1-\\frac{\\alpha }{2}}$ increases with the number of parameter spaces $J$ .The constant, $z_{1-\\frac{\\alpha }{2J}}/z_{1-\\frac{\\alpha }{2}}$ , grows with $J$ at the rate $(\\log J)^{1/2}$ .", "This is the same rate that [7] find in their analysis of the case with non-nested parameter spaces.", "Their constant is at least eight times greater than what we provide here, but does not require that the largest space in consideration is convex.", "On the other hand, the constant given in [7] is 16 and thus does not depend on the number of parameter spaces.", "However, we note that $2z_{1-\\frac{\\alpha }{2J}}/z_{1-\\frac{\\alpha }{2}}$ is not too large, in fact smaller than 16, for reasonable specifications of $J.$ For example, when $\\alpha =0.05$ and $J=50$ , we get $2z_{1-\\frac{\\alpha }{2J}}/z_{1-\\frac{\\alpha }{2}}\\approx 3.36$ , which is considerably smaller than the constant given in [7].", "Even for unrealistically large $J$ such as $J=10^{10}$ , we have $2z_{1-\\frac{\\alpha }{2J}}/z_{1-\\frac{\\alpha }{2}}<8$ , which is still less than half of the constant given by [7].", "Simulation results given in Section confirm that not only the upper bound, but also the actual length itself is often much shorter for our CI.", "Remark 3.1 Suppose one is interested in constructing the one-sided CI in an adaptive way.", "Note that Lemma REF implies that any one-sided CI $[\\hat{c}_{\\alpha }^{\\ell ,J},\\infty )$ with coverage probability $1-\\alpha $ should satisfy $\\underset{f\\in \\mathcal {F}_{j}}{\\sup }\\operatorname{\\mathbf {E}}(Lf-\\hat{c}_{\\alpha }^{\\ell ,J})\\ge \\omega \\left(z_{1-\\alpha },\\mathcal {F}_{J},\\mathcal {F}_{j}\\right).$ Define $\\hat{c}_{\\alpha }^{\\ell ,J}=\\max _{j}\\hat{c}_{\\alpha /J}^{\\ell ,j}.$ Then, by an analogous argument to Theorem REF , we have $\\underset{f\\in \\mathcal {F}_{j}}{\\sup }\\operatorname{\\mathbf {E}}(Lf-\\hat{c}_{\\alpha }^{\\ell ,J})\\le \\frac{z_{1-\\frac{\\alpha }{J}}}{z_{1-\\alpha }}\\omega \\left(z_{1-\\alpha },\\mathcal {F}_{J},\\mathcal {F}_{j}\\right).$ Therefore, $\\left[\\hat{c}_{\\alpha }^{\\ell ,J},\\infty \\right)$ is an adaptive one-sided CI in a similar sense with the two-sided case.", "The naive CI given in (REF ) does not take into account the possible correlation among the CIs that we take the intersection of.", "However, if parameter spaces are “close” to each other, the corresponding CIs will be correlated, implying that there is room for improvement over the Bonferroni procedure.", "Consider the CIs of the form $CI^{\\tau ,\\mathcal {J}}=\\cap _{j=1}^{J}[\\hat{c}_{\\tau }^{\\ell ,j},\\hat{c}_{\\tau }^{u,j}]$ .", "If we take $\\tau = \\alpha /(2J)$ , this is precisely the CI given in (REF ).", "The CI that gives the smallest constant among CIs of such forms is $CI^{\\tau ^{\\ast },\\mathcal {J}},$ where $\\tau ^{\\ast }$ is the largest possible $\\tau $ such that $CI^{\\tau ,\\mathcal {J}}$ has correct coverage over $\\mathcal {F}_{J}$ : $\\tau ^{\\ast }:= \\sup _{\\tau } \\tau \\,\\, \\text{ s.t. }", "\\inf _{f \\in \\mathcal {F}_{J}}\\operatorname{\\mathbf {P}}_{f}(Lf \\in CI^{\\tau ,\\mathcal {J}}) \\ge 1- \\alpha .$ We know that $\\tau = \\alpha /(2J)$ satisfies the constraint, and also that any $ \\tau > \\alpha $ does not because then $[\\hat{c}_{\\tau }^{\\ell ,j}, \\infty )$ will have coverage probability $1-\\tau < 1 - \\alpha $ .", "Hence, we can restrict $\\tau $ to lie in $[\\alpha /(2J), \\alpha ]$ .", "However, the coverage probability $ \\inf _{f \\in \\mathcal {F}_{J}}\\operatorname{\\mathbf {P}}_{f}(Lf \\in CI^{\\tau ,\\mathcal {J}})$ is unknown in general, rendering $CI^{\\tau ^{\\ast },\\mathcal {J}}$ infeasible.", "Instead, we replace this coverage probability with a lower bound that we can calculate either analytically or via simulation.", "Then, we take $\\tau ^{\\ast }$ as the largest value that makes this lower bound at least $1-\\alpha $ .", "As we show later, using $\\tau ^{\\ast }$ rather than $\\alpha /(2J)$ can only make the resulting CI shorter.", "Let $(V(\\tau )^{\\prime }, W(\\tau )^{\\prime })^{\\prime }$ be a centered Gaussian random vector with unit variance.", "The covariance terms for $V(\\tau )=\\left(V_{1}(\\tau ),...,V_{J}(\\tau )\\right)^{\\prime } $ is given by $\\text{Cov}\\left(V_{j}(\\tau ),V_{\\ell }(\\tau )\\right)=\\frac{1}{z_{1-\\tau }^{2}}\\sum _{i=1}^{n}\\big (g_{j,z_{1-\\tau }}^{,Jj}(x_{i})-f_{J,z_{1-\\tau }}^{,J j}(x_{i})\\big )\\big (g_{\\ell ,z_{1-\\tau }}^{,J\\ell }(x_{i})-f_{J,z_{1-\\tau }}^{, J \\ell }(x_{i})\\big ).$ Likewise, the covariance terms for $ W(\\tau )=\\big (W_{1}(\\tau ),...,W_{J}(\\tau )\\big )^{\\prime }$ is given by $\\text{Cov}\\big (W_{j}(\\tau ),W_{\\ell }(\\tau )\\big )=\\frac{1}{z_{1-\\tau }^{2}}\\sum _{i=1}^{n}\\big (g_{j,z_{1-\\tau }}^{,jJ}(x_{i})-f_{J,z_{1-\\tau }}^{,jJ}(x_{i})\\big )\\big (g_{\\ell ,z_{1-\\tau }}^{, \\ell J}(x_{i})-f_{J,z_{1-\\tau }}^{, \\ell J}(x_{i})\\big ).$ Finally, the covariance terms across $V(\\tau )$ are $W(\\tau )$ given as $\\text{Cov}\\big (V_{j}(\\tau ),W_{\\ell }(\\tau )\\big )=\\frac{1}{z_{1-\\tau }^{2}}\\sum _{i=1}^{n}\\big (g_{j,z_{1-\\tau }}^{,Jj}(x_{i})-f_{J,z_{1-\\tau }}^{,Jj}(x_{i})\\big )\\big (g_{\\ell ,z_{1-\\tau }}^{,\\ell J}(x_{i})-f_{J,z_{1-\\tau }}^{, \\ell J}(x_{i})\\big ).$ This Gaussian random vector can be used to tune the critical value, as the following lemma implies.", "Lemma 3.2 Let $\\tau ^{}\\in \\left[\\frac{\\alpha }{2J},\\alpha \\right]$ to be the largest value of $\\tau $ such that $\\operatorname{\\mathbf {P}}\\left(\\max \\left\\lbrace V\\left(\\tau \\right)^{\\prime },W\\left(\\tau \\right)^{\\prime }\\right\\rbrace >z_{1-\\tau }\\right) \\le \\alpha .$ Then, we have $\\sup _{f\\in \\mathcal {F}_{J}}\\operatorname{\\mathbf {P}}\\left(f(0)\\notin CI_{\\tau ^{}}^{\\mathcal {J}}\\right)\\le \\alpha $ .", "Such a $\\tau ^{\\ast }$ always exists because the inequality (REF ) holds with $\\tau = \\alpha /(2J)$ due to the union bound.", "A solution $\\tau ^{}$ can be found via numerical simulation.", "By construction, its length will be also bounded by (REF ).", "In Section REF , we show that as $n \\rightarrow \\infty $ the distribution of $(V(\\tau )^{\\prime }, W(\\tau )^{\\prime })^{\\prime }$ does not depend on $\\tau $ , under our setting of $Lf = f(0)$ with $f$ belonging to a Hölder class.", "Hence, finding $\\tau ^{}$ boils down to simply finding the $1-\\alpha $ quantile of the maximum of a Gaussian vector in this case." ], [ "Adaptive Inference for $f(0)$", "In this section, we provide an adaptive inference procedure for $f(0)$ .", "To construct the adaptive CI introduced in Section , we first solve the corresponding modulus problem.", "By using this solution to the modulus problem, we derive the minimax rate of adaptation.", "Finally, we provide a CI that obtain this rate, using the method described in Section ." ], [ "Solution to the Modulus Problem", "Let $\\Lambda _{+,\\mathcal {V}}(\\gamma _{j},C_{j})\\subset \\Lambda _{+,\\mathcal {V}}(\\gamma _{J},C_{J})$ with $\\gamma _{j}\\ge \\gamma _{J}$ and $C_{j}\\le C_{J}$ .", "To construct the adaptive CI, we first calculate the ordered moduli, $\\omega \\left(\\delta ,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{j},C_{j}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{J},C_{J}\\right)\\right)$ and $\\omega \\left(\\delta ,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{J},C_{J}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{j},C_{j}\\right)\\right),$ for each $j=1,\\dots ,J.$ For notational simplicity, we consider the case with $J=2$ and solve $\\omega _{+}\\left(\\delta ,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right)\\right)$ , from which the general solution follows immediately.", "Recall the definition of the ordered modulus of continuity $&\\sup f_{2}(0)-f_{1}(0) \\\\\\text{s.t.", "}&\\sum _{i=1}^{n}\\left(\\left(f_{2}(x_{i})-f_{1}(x_{i})\\right)/\\sigma (x_{i})\\right)^{2}\\le \\delta ^{2}, \\,\\,f_{j}\\in \\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{j},C_{j}\\right) \\text{ for }j = 1,2,$ with the maximized value denoted by $\\omega \\left(\\delta ,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right)\\right).$ It is convenient to solve the inverse modulus problem instead, which is defined as $\\begin{aligned}\\inf \\,&\\sum _{i=1}^{n}\\left(\\left(f_{2}(x_{i})-f_{1}(x_{i})\\right)/\\sigma (x_{i})\\right)^{2} \\\\\\text{s.t.", "}& f_{2}(0)-f_{1}(0)=b,\\,\\,f_{j}\\in \\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{j},C_{j}\\right) \\text{ for }j =1,2,\\end{aligned}$ for $b>0$ , with the square root of the maximized value denoted by the inverse (ordered) modulus $\\omega ^{-1}\\left(b,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right)\\right).$ We provide a closed form solution for the this inverse problem, from which we can recover the solution to the original problem by finding $b$ such that $\\omega ^{-1}\\left(b,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right)\\right)=\\delta $ .", "Note that this is simply a search problem on the positive real line.", "To characterize the solution to (REF ), we show two simple lemmas about the properties of the class $\\Lambda _{+,\\mathcal {V}}\\left(\\gamma ,C\\right)$ .", "For $z=(z_{1},\\dots ,z_{k})\\in \\mathbb {R}^{k}$ , define $\\left(z\\right)_{\\mathcal {V}+}={\\left\\lbrace \\begin{array}{ll}\\max \\left\\lbrace z_{i},0\\right\\rbrace & i\\in \\mathcal {V}\\\\z_{i} & i\\notin \\mathcal {V}\\end{array}\\right.", "}$ and $\\left(z\\right)_{\\mathcal {V}-}=\\left(-z\\right)_{\\mathcal {V}+}.$ Lemma 4.1 Suppose Assumption REF holds, and let $\\gamma \\in [0,1]$ and $C>0$ .", "Define $h_{+}(x) = C\\left\\Vert (x)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma }\\quad \\text{and}\\quad h_{-}(x) = -C\\left\\Vert (x)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma }.$ Then, $h_{+},\\ h_{-}\\in \\Lambda _{+,\\mathcal {V}}\\left(\\gamma ,C\\right)$ .", "The following lemma asserts that the class of functions we consider is closed under the maximum operator.", "Lemma 4.2 Suppose $h_{1},h_{2}\\in \\Lambda _{+,\\mathcal {V}}\\left(\\gamma ,C\\right)$ .", "Then, $\\max \\left\\lbrace h_{1},h_{2}\\right\\rbrace \\in \\Lambda _{+,\\mathcal {V}}\\left(\\gamma ,C\\right)$ .", "The next lemma can be used to establish the solutions to the problem (REF ).", "This is a generalization of Proposition 4.1 of [4], which gives the same result for the special case of $\\gamma =1$ .", "Lemma 4.3 Given $f_{0}\\in \\mathbb {R}$ and $0<\\gamma \\le 1$ , define $\\Lambda _{+,\\mathcal {V}}^{f_{0}}\\left(\\gamma ,C\\right)=\\left\\lbrace f\\in \\Lambda _{+,\\mathcal {V}}\\left(\\gamma ,C\\right):\\ f(0)=f_{0}\\right\\rbrace .$ Then, for any $x\\in \\mathbb {R}^{k}$ , we have $\\underset{f\\in \\Lambda _{+,\\mathcal {V}}^{f_{0}}\\left(\\gamma ,C\\right)}{\\max }f(x) & = & f_{0}+C\\left\\Vert (x)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma }\\\\\\underset{f\\in \\Lambda _{+,\\mathcal {V}}^{f_{0}}\\left(\\gamma ,C\\right)}{\\min }f(x) & = & f_{0}-C\\left\\Vert (x)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma }.$ We are now ready to characterize the solution to the inverse modulus problem (REF ).", "For $r\\in \\mathbb {R}$ , define $\\left(r\\right)_{+}:= \\max \\left\\lbrace r,0\\right\\rbrace $ .", "Proposition 4.1 Suppose Assumption REF holds, and define $f_{1}^{}(x) & = & C_{1}\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}\\\\f_{2}^{}(x) & = & \\max \\left\\lbrace b-C_{2}\\left\\Vert \\left(x\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}},\\ C_{1}\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}\\right\\rbrace .$ Then, $\\left(f_{1}^{},f_{2}^{}\\right)$ solves the inverse modulus problem (REF ), and the inverse modulus is given by $\\begin{aligned}&\\omega ^{-1}\\left(b,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right)\\right) \\\\= &\\Big ( \\sum \\nolimits _{i=1}^{n}\\left(\\left(b-C_{1}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}-C_{2}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right)/\\sigma \\left(x_{i}\\right)\\right)_{+}^{2} \\Big )^{1/2}.\\end{aligned}$ To solve (REF ), note that it is without loss of generality to restrict attention to the functions with $f_{1}(0)=0$ and $f_{2}(0)=b$ , which is satisfied by $f_{1}^{}$ and $f_{2}^{}$ .", "To simplify notation, write $\\mathcal {F}_{1}^{0}=\\Lambda _{+,\\mathcal {V}}^{0}\\left(\\gamma _{1},C_{1}\\right)$ and $\\mathcal {F}_{2}^{b}=\\Lambda _{+,\\mathcal {V}}^{b}\\left(\\gamma _{2},C_{2}\\right)$ .", "Since $f_{2}(0)>f_{1}(0)$ , we want $ f_{1}(x) = \\max _{f\\in \\mathcal {F}_{1}^{0}}f(x)$ and $ f_{2}(x) = \\min _{f\\in \\mathcal {F}_{2}^{b}}f(x)$ as long as $x\\in \\mathcal {X}$ satisfies $\\min _{f\\in \\mathcal {F}_{2}} f(x)\\ge \\max _{f\\in \\mathcal {F}_{1}}f(x)$ , and $f_{1}(x)=f_{2}(x)$ otherwise.", "Note that $f_{1}^{}$ and $f_{2}^{}$ are designed exactly to achieve this goal, which follows by Lemma REF .", "It remains to check whether $f_{1}^{}\\in \\mathcal {F}_{1}$ and $f_{2}^{}\\in \\mathcal {F}_{2}$ .", "The former case is trivial.", "For the latter case, note that $f_{1}^{\\ast }\\in \\Lambda _{+,\\mathcal {V}}(\\gamma _{1},C_{1})\\subseteq \\Lambda _{+,\\mathcal {V}}(\\gamma _{2},C_{2})$ .", "Now, by Lemma REF , we have $f_{2}^{}\\in \\mathcal {F}_{2}.$ The following corollary states an analogous result regarding the inverse modulus $\\omega ^{-1}\\left(b,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right)\\right)$ .", "Corollary 4.1 Define $g_{1}^{}(x) & = & b-C_{1}\\left\\Vert \\left(x\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{1}}\\\\g_{2}^{}(x) & = & \\min \\left\\lbrace b-C_{1}\\left\\Vert \\left(x\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{1}},\\ C_{2}\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{2}}\\right\\rbrace .$ Then, $\\left(g_{1}^{},g_{2}^{}\\right)$ solves the inverse modulus $\\omega ^{-1}\\left(b,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right)\\right)$ .", "Remark 4.1 (Role of Assumption REF ) Proposition REF requires Assumption REF due to the specific form of monotonicity we consider.", "By considering coordinate-wise monotonicity, we must take a norm that is “aligned” with this direction of monotonicity.", "The assumption precisely imposes this.", "This is a unique feature that arises in the multivariate setting.", "To allow for more general norms, let $\\mathcal {B}$ be an orthonormal basis of $\\mathbb {R}^{k}$ , and denote by $z^{\\mathcal {B}}$ the coordinate vector of $z \\in \\mathbb {R}^{k}$ with respect to $\\mathcal {B}$ and $z^{\\mathcal {B}}_{j}$ its $j$ th component.", "Suppose the regression function is monotone in the coefficients with respect to this basis $\\mathcal {B}$ , so that the monotone Hölder class is given as $\\Lambda _{+,\\mathcal {V}}(\\gamma ,C):=\\left\\lbrace f\\in \\Lambda (\\gamma ,C):f(x)\\ge f(z)\\,\\,\\text{if }x^{\\mathcal {B}}_{j}\\ge z^{\\mathcal {B}}_{j}\\text{ }\\forall j \\in \\mathcal {V} \\text{ and }x^{\\mathcal {B}}_{j}\\ge z^{\\mathcal {B}}_{j}\\,\\,\\forall j\\notin \\mathcal {V}\\right\\rbrace .$ Then, the condition we want to impose on the norm $\\Vert \\cdot \\Vert $ is monotonicity with respect to the magnitude of $z^{\\mathcal {B}}_{j}$ .", "A special case is the Mahalanobis distance." ], [ "Minimax Rate of Adaptation", "Using this solution to the inverse modulus, we derive the rate of convergence of the between class of modulus, which characterizes how fast the worst-case expected length of the adaptive CIs can go to 0 as $n\\rightarrow \\infty $ .", "We derive the rates under the assumption that the sequence of design points $\\lbrace x_{i}\\rbrace _{i=1}^{\\infty }$ is a realization of a sequence of independent and identically distributed random vectors $\\lbrace X_{i} \\rbrace _{i=1}^{\\infty }$ drawn from a distribution that satisfies some mild regularity conditions.", "This gives an intuitive restriction on the design points, and also shows that the result applies under random design points as well.Consider the model $y_{i}=f(X_{i})+\\varepsilon _{i}$ , for $i=1,\\dots ,n$ , with the $X_{i}\\overset{\\mathrm {i.i.d.", "}}{\\sim }p_{X}$ with $\\varepsilon _{i}\|X_{i}\\sim N(0,\\sigma ^{2}(X_{i}))$ .", "Then, conditional on $\\lbrace X_{i} \\rbrace _{i=1}^{n}=\\lbrace x_{i}\\rbrace _{i=1}^{n}$ , this model is equivalent with our model.", "Define $r(\\gamma _{1}, \\gamma _{2}) =({2+k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}})^{-1}$ .", "The following theorem fully characterizes the minimax rate of adaptation.", "Theorem 4.1 Let $\\left\\lbrace X_{i}\\right\\rbrace _{i=1}^{\\infty }$ be an i.i.d.", "sequence of random vectors with support $\\mathcal {X}$ .", "Suppose $X_{i}$ admits a probability density function $p_{X}(\\cdot )$ that is continuous at 0 with $p_{X}(0)>0,$ and assume $\\sigma (\\cdot ) = 1$ .", "Then, for almost all realizations $\\left\\lbrace x_{i}\\right\\rbrace _{i=1}^{\\infty }$ of $\\left\\lbrace X_{i}\\right\\rbrace _{i=1}^{\\infty }$ and for all $\\delta > 0$ , we have $\\lim _{n\\rightarrow \\infty }n^{r(\\gamma _{1},\\gamma _{2})}\\omega \\left(\\delta ,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right)\\right)& =\\left(\\delta ^{2}/c_{1,2}^{\\ast }\\right)^{r(\\gamma _{1}, \\gamma _{2})},\\text{ and}\\\\\\lim _{n\\rightarrow \\infty }n^{r(\\gamma _{1},\\gamma _{2})}\\omega \\left(\\delta ,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right)\\right) & =\\left(\\delta ^{2}/c_{2,1}^{\\ast }\\right)^{r(\\gamma _{1}, \\gamma _{2})},$ where $c_{1,1}^{\\ast }$ and $c_{2,1}^{\\ast }$ are constants that depend only on the function spaces.", "Remark 4.2 The result immediately implies the rate of convergence for the between class modulus $& \\lim _{n\\rightarrow \\infty }n^{r(\\gamma _{1},\\gamma _{2})}\\omega _{+}\\left(\\delta ,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right)\\right)\\\\= & \\max \\left\\lbrace \\delta ^{2}/c_{1,2}^{\\ast },\\delta ^{2}/c_{2,1}^{\\ast }\\right\\rbrace ^{r(\\gamma _{1},\\gamma _{2})}.$ Hence, if a CI maintains coverage over $\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right)$ , the best possible worst-case length of this CI over $\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right)$ goes to 0 at the same rate as $n^{-r(\\gamma _{1},\\gamma _{2})}$ .", "Remark 4.3 (Heteroskedasticity) For simplicity, the theorem imposes a homoskedasticity condition (i.e., $\\sigma (\\cdot ) = 1$ ).", "However, allowing for general $\\sigma (\\cdot )$ is straightforward and requires only weak regularity conditions on $\\sigma (\\cdot )$ .", "See Appendix for details.", "Theorem REF shows how the monotonicity restriction plays a role in determining the minimax rates of adaptation to Hölder coefficients under the multivariate nonparametric regression setting.", "When $k_{+}=k$ , the minimax rate of adaptation is $n^{-\\frac{1}{2+k/\\gamma _{1}}}$ , which equals the minimax convergence rate over $\\omega \\left(\\delta ,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right)\\right)$ .", "This shows that strong adaptation is possible if the regression function is monotone with respect to all the variables, just like in the univariate case.", "On the other hand, when $k_{+}=0$ , the rate becomes $n^{-\\frac{1}{2+k/\\gamma _{2}}}$ , consistent with the previous findings that there is no scope of adaptation for general Hölder classes without any shape constraint.", "Importantly, Theorem REF characterizes the convergence rate for the case where $0<k_{+}<k$ , where it gives an intuitive intermediate rate between the two extreme." ], [ "Construction of the Adaptive CI", "Here, we give the explicit formula of the CIs for our parameters of interest, now that we have derived the form of the moduli of continuity and the solutions to the modulus problems in the previous section.", "We first consider $L_{0}f$ .", "Before stating the result, it is convenient to define the following functions $D_{Jj,\\delta }(x_{i}) := &(\\omega (\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j})-C_{j}\\Vert (x_{i})_{\\mathcal {V}-}\\Vert ^{\\gamma _{j}}-C_{J}\\Vert (x_{i})_{\\mathcal {V}+}\\Vert ^{\\gamma _{J}})_{+},\\text{ and}\\\\D_{jJ, \\delta }(x_{i}) := &( \\omega (\\delta ,\\mathcal {F}_{j},\\mathcal {F}_{J})-C_{J}\\Vert (x_{i})_{\\mathcal {V}-}\\Vert ^{\\gamma _{J}}-C_{j}\\Vert (x_{i})_{\\mathcal {V}+}\\Vert ^{\\gamma _{j}})_{+}.$ Corollary 4.2 For $Lf=f(0)$ and $\\beta =1/2$ , the lower CI defined in (REF ) is given by $\\hat{c}_{\\delta }^{\\ell ,j}=\\hat{L}_{\\delta }^{\\ell ,j}-\\frac{1}{2}\\left( \\omega \\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right)+\\frac{\\delta ^{2}}{\\sum _{i=1}^{n}D_{Jj,\\delta }\\left(x_{i}\\right)} \\right),$ where $\\hat{L}_{\\delta }^{\\ell ,j} = &\\, \\frac{\\sum _{i=1}^{n}D_{Jj,\\delta }\\left(x_{i}\\right)y_{i}}{\\sum _{i=1}^{n}D_{Jj,\\delta }\\left(x_{i}\\right)}+\\frac{\\omega \\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right)}{2}\\\\& -\\frac{\\sum _{i=1}^{n}D_{Jj,\\delta }\\left(x_{i}\\right)(\\omega \\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right)-C_{j}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{j}}+C_{J}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{J}})}{2\\sum _{i=1}^{n}D_{Jj,\\delta }\\left(x_{i}\\right)}.$ Likewise, the upper bound of the CI is given by $\\hat{c}_{\\delta }^{u,j}=\\hat{L}_{\\delta }^{u,j}+\\frac{1}{2}\\left(\\omega \\left(\\delta ,\\mathcal {F}_{j},\\mathcal {F}_{J}\\right)+\\frac{\\delta ^{2}}{\\sum _{i=1}^{n}D_{jJ, \\delta }\\left(x_{i}\\right)}\\right),$ where $\\hat{L}_{\\delta }^{u,j} = & \\, \\frac{\\sum _{i=1}^{n}D_{jJ, \\delta }\\left(x_{i}\\right)y_{i}}{\\sum _{i=1}^{n}D_{jJ, \\delta }\\left(x_{i}\\right)}+\\frac{\\omega \\left(\\delta ,\\mathcal {F}_{j},\\mathcal {F}_{J}\\right)}{2}\\\\& -\\frac{\\sum _{i=1}^{n}D_{jJ, \\delta }\\left(x_{i}\\right)(\\omega \\left(\\delta ,\\mathcal {F}_{j},\\mathcal {F}_{J}\\right)-C_{J}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{J}}+C_{j}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{j}})}{2\\sum _{i=1}^{n}D_{jJ, \\delta }\\left(x_{i}\\right)}.$ The first terms in the formula of $\\hat{L}_{\\delta }^{\\ell ,j}$ and $\\hat{L}_{\\delta }^{u,j}$ are the random terms linear in $y_{i}$ while the remaining terms are non-random fixed terms.", "If $\\mathcal {V}=\\lbrace 1,...,k\\rbrace $ (so the function is monotone in every coordinate), the random terms can be viewed as a kernel estimator with a data-dependent bandwidth.", "Too see this, if we define $k(x)=\\left[1-C_{j}\\left\\Vert \\left(x\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{j}}-C_{J}\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{J}}\\right]_{+},$ and $h_{mn}\\left(x\\right)={\\left\\lbrace \\begin{array}{ll}\\omega \\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right){}^{1/\\gamma _{J}} & \\text{if the }m\\text{th coordinate of }x\\ge 0\\\\\\omega \\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right){}^{1/\\gamma _{j}}& \\text{otherwise},\\end{array}\\right.", "}$ we have $\\frac{\\sum _{i=1}^{n}D_{Jj,\\delta }\\left(x_{i}\\right)y_{i}}{\\sum _{i=1}^{n}D_{Jj,\\delta }\\left(x_{i}\\right)}=\\frac{\\sum _{i=1}^{n}k\\left(x_{1i}/h_{1n}\\left(x_{i}\\right),...,x_{ki}/h_{kn}\\left(x_{i}\\right)\\right)y_{i}}{\\sum _{i=1}^{n}k\\left(x_{1i}/h_{1n}\\left(x_{i}\\right),...,x_{ki}/h_{kn}\\left(x_{i}\\right)\\right)}.$ Hence, the CI can be considered to be based on a Nadaraya-Watson type estimator, correcting for the bias.", "As described in Section REF , the proposed CI is given by $ \\cap _{j=1}^{J} [\\hat{c}_{z_{1-\\tau ^{\\ast }}}^{\\ell ,j},\\hat{c}_{z_{1-\\tau ^{\\ast }}}^{\\ell ,j}]$ , where $\\tau ^{\\ast }$ is defined in Lemma REF .", "Here, we show that the distribution of $(V(\\tau )^{\\prime },W(\\tau )^{\\prime })^{\\prime }$ does not depend on $\\tau $ as $n \\rightarrow \\infty $ .", "The implication of this invariance with respect to $\\tau $ , is that calculating $\\tau ^{\\ast }$ boils down to calculating the quantile of the maximum of Gaussian vectors.", "The variance matrix of this limiting Gaussian random vector is known, and thus the said quantile can be easily simulated.", "Moreover, when $\\gamma _{1} = \\cdots = \\gamma _{J}$ so that the parameters spaces differs only in $C_{j}$ , $\\tau ^{\\ast }$ can be shown to be bounded away from zero by a constant that does not depend on $J$ , for large $n$ .", "Hence, the constant of the CI does not grow to infinity as $J \\rightarrow \\infty $ in this case.This is especially useful when one wishes to adapt to $C$ while keeping $\\gamma $ fixed.", "For example, [22] take $\\gamma _{j} = 1$ and consider the problem of adapting to the Lipschitz constant in a regression discontinuity setting.", "Lemma 4.4 Under the same set of conditions of Theorem REF , $(V\\left(\\tau \\right)^{\\prime },W\\left(\\tau \\right)^{\\prime } )^{\\prime } \\overset{d}{\\rightarrow }(V_{\\infty }^{\\prime },W_{\\infty }^{\\prime })^{\\prime }$ as $n \\rightarrow \\infty $ , where $(V_{\\infty }^{\\prime },W_{\\infty }^{\\prime })^{\\prime }$ is a Gaussian random vector that does not depend on $\\tau .$ Moreover, if $\\gamma _{1}= \\cdots = \\gamma _{J}$ , then, for large $n$ , $\\tau ^{\\ast } > \\eta $ for some $\\eta > 0$ that does not depend on $J$ .", "Remark 4.4 (Dependence on $J$ ) The proof reveals that when all $J$ parameter spaces correspond to different Hölder exponents (i.e., $\\gamma _{1} > \\cdots > \\gamma _{J}$ ), the dependence of $\\tau ^{\\ast }$ on $J$ does not vanish and in fact results in CIs whose constants grow at the same rate as the naive Bonferroni CI, $(\\log J)^{1/2}$ .", "However, some finite sample improvement in terms of the length of the resulting CI compared to the naive Bonferroni CI is shown in the empirical exercise.", "When some of the parameter spaces have the same Hölder exponent, the improvement can be significant.", "As an extreme case, when $\\gamma _{1}= \\cdots = \\gamma _{J}$ , $\\tau ^{\\ast }$ can is bounded away from 0 by a constant that does not depend on $J$ , which is exactly what the second part of the lemma asserts." ], [ "Simulation Results", "In this section, we compare the performances of the adaptive CI of [7] and the adaptive CI constructed using the naive Bonferroni procedure described in Section REF .", "As a benchmark, we also provide the lengths of the shortest fixed length confidence intervals of [11], referred to as minimax CIs.", "We consider inference for $f(0)$ , given some regression function $f$ .", "We consider the case where the researcher is uncertain about the value of the Hölder exponent $\\gamma $ , and thus tries to adapt to its value.", "First, we construct adaptive CIs with respect to two smoothness parameters $\\left(\\gamma _{1},\\gamma _{2}\\right)=\\left(1,10^{-3}\\right)$ while fixing $C = 1$ , which gives $J = 2$ .", "We vary $n$ over $\\lbrace 10^2,\\ 5 \\times 10^2,\\ 10^3,\\ 5 \\times 10^3,\\ 10^4 \\rbrace $ to investigate the rate of adaptation as the sample size grows.", "The true regression function is over $\\mathbb {R}^2$ and given by either $f_{1}$ or $f_{2}$ , defined as $f_{1}(x_{1},x_{2}) = 0, \\quad f_{2}(x_{1},x_{2}) = \\left\\Vert \\left(x_{1},x_{2}\\right)_{\\mathcal {V}+}\\right\\Vert _{2}^{\\gamma _{2}},\\quad \\mathcal {V}=\\lbrace 1,2\\rbrace .$ By construction, we have $f_{j}\\in \\Lambda _{+,\\mathcal {V}}(\\gamma _{j},1)$ .", "The covariates are drawn from a uniform distribution over $[-1/(2\\sqrt{2}),1/(2\\sqrt{2})]^{2}$ , and the noise terms, $\\lbrace u_{i}\\rbrace _{i=1}^{n}$ , are drawn from a standard normal distribution.", "The outcome variable is given as $y_{i} = f(x_{i}) + u_{i}$ , for $f \\in \\lbrace f_1, f_2\\rbrace $ .", "We fix the draw of $\\left\\lbrace x_{i} \\right\\rbrace _{i=1}^{n}$ within each simulation iteration.", "We run 500 iterations to calculate the average lengths and coverage probabilities of CIs.", "The nominal coverage probability is $.95$ for all CIs.", "Table: Lengths of CIs when f=f 1 f=f_{1} with J=2J = 2Table: Lengths of CIs when f=f 2 f=f_{2} with J=2J = 2Table: Coverage probabilities of adaptive CIs (J=2J = 2)Table REF shows the results for the case where $f=f_{1}$ .", "Each column corresponds to 1) our proposed (naive) Bonferroni adaptive procedure (AdaptBonf), 2) the adaptive CI of [7] (CL, henceforth), 3) the minimax CI with respect to $\\Lambda _{+,\\mathcal {V}}(\\gamma _2, 1)$ , and 4) the minimax CI with respect to $\\Lambda _{+,\\mathcal {V}}(\\gamma _1, 1)$ .", "Regarding the last two minimax procedures, we refer to them as the “conservative minimax CI” and the “oracle minimax CI”, respectively.", "Note that the oracle minimax CI is an optimal benchmark, which is only feasible when we actually know the true regression function is in the smaller parameter space $\\Lambda _{+,\\mathcal {V}}(\\gamma _1, 1)$ .", "In Table REF , the average lengths of both adaptive confidence intervals decrease considerably as $n$ increases from 100 to 10,000.", "In comparison, the length of the conservative minimax CI (column 3) decreases only about 28% for the same change in the sample size.", "This shows the lengths of the adaptive confidence intervals decrease more sharply when the true function is smooth, as predicted by the theory.", "To compare the performances of different adaptive inference procedures, note that the average lengths of the CI of CL adapting to the Hölder exponents (column 2) are often wider than the conservative minimax CI (column 3).", "When $n = 100$ , the former is more than three times wider than the latter, and the adaptive procedure starts to dominate the minimax procedure only when $n$ is greater than 5,000.", "In comparison, our proposed Bonferroni adaptive procedure (column 1) yields shorter CIs than those by CL, as predicted in Section REF .", "To compare the Bonferroni adaptive CI with the conservative minimax CI, the lengths of the former are always exceeded by those of the minimax CI, even for the relatively small sample size of $n=100$ .", "Moreover, the length of the adaptive CI becomes only 20% of the length of the conservative minimax CI for the sample size of $n=10^4$ .", "The Bonferroni procedure also performs well even when compared to the infeasible oracle minimax CI (column 4), with the length of the former only 13% wider than the latter when $n = 10^4$ .", "This demonstrates the strong adaptivity property of the adaptive procedure when the regression function is monotone with respect to all variables, as shown in Section REF .", "Table REF demonstrates the analogous simulation results when $f=f_{2}$ .", "In this case, the minimax CI with respect to $\\Lambda _{+,\\mathcal {V}}(\\gamma _2, 1)$ (column 3) is referred to as the oracle minimax CI.", "While the lengths of the oracle minimax procedure are considerably shorter than the CIs of CL for various values of $n$ , the performance of the Bonferroni CIs almost matches that of the oracle minimax procedure.", "Especially, the performance of the Bonferroni adaptive procedure becomes extremely close to the oracle minimax procedure when $n$ is greater than 500.", "Table REF shows the coverage probabilities of adaptive CIs for both of the cases when $f = f_1$ and $f = f_2$ .", "While all the CIs achieve the correct coverage, none of those CIs exactly achieves the nominal coverage of $.95$ , reflecting the conservative nature of the adaptive CIs.", "We can see that the adaptive procedure of CL is particularly conservative, almost always yielding 100% coverage probabilities.", "So far we considered adapting to the smoothness parameters at two extremes, $\\gamma \\in (0.001,1)$ .", "Since the multiplicative constant for the Bonferroni procedure increases with $J$ , a concern is that the performance of the Bonferroni procedure relative to the CL procedure might get worse when $J$ is larger.", "To investigate the possibility, we consider adapting to a wider set of parameters, $\\lbrace \\gamma _j\\rbrace _{j = 1}^6$ , where $\\gamma _j = 1 - (j - 1)/5$ for $j = 1,...,5$ and $\\gamma _6 = 10^{-3}$ .", "Moreover, rather than taking the extreme value of $\\gamma $ as the true parameter, we consider the case where $\\gamma $ takes an intermediate value, $\\gamma =1/2$ .", "The true regression function is given by $f_3(x_{1},x_{2})=\\left\\Vert \\left(x_{1},x_{2}\\right)_{\\mathcal {V}+}\\right\\Vert _{2}^{1/2},\\quad \\mathcal {V} = \\lbrace 1,2\\rbrace ,$ so that $f_3 \\in \\Lambda _{+,\\mathcal {V}}(1/2,1)$ .", "Table REF displays the simulation results corresponding to this specification.", "Each column corresponds to 1) our proposed Bonferroni adaptive procedure, 2) the adaptive CI of CL, 3) the minimax CI with respect to $\\Lambda _{+,\\mathcal {V}}(\\gamma _6, 1)$ , and 4) the minimax CI with respect to $\\Lambda _{+,\\mathcal {V}}(1/2, 1)$ .", "As before, we refer to the last two CIs as the conservative minimax CI and the oracle minimax CI, respectively.", "We observe the same pattern as in the case of adapting to two parameters—adaptive CIs shrink faster than the conservative minimax CI as the sample size increases, and the Bonferroni adaptive CIs are shorter than the ones of CL.", "While the ratio of the length of the Bonferroni CI to that of the CI of CL is larger in this case compared to the case where $J = 2$ , especially when $n$ is large, the Bonferroni CI is still more than 50 % narrower than the CI of CL, and not much wider than the oracle minimax CI.", "Table: Lengths of CIs when f=f 3 f=f_{3} with J=6J = 6" ], [ "Empirical Illustration", "In this section, we apply our procedure to the production function estimation problem for the Chinese chemical industry.", "Specifically, we use the firm-level data of [21] for the year 2001, which was also used by [19] to illustrate their method of constructing the uniform confidence band for the production function under shape restrictions.", "In the dataset, the dependent variable is the logarithm of value-added real output ($y$ ), and the explanatory variables are the logarithms of the net value of the real fixed asset ($k$ ) and the number of employees ($\\ell $ ).", "After removing the outliers for $y,k$ and $\\ell $ , the remaining sample size was $n=1,636$ .We used the conventional way of outlier detection, removing the observations that are greater than the third quantile plus IQR times 1.5, or less than the first quantile minus IQR times 1.5.", "Our resulting sample size is close to [19], who have $n=1,638$ .", "Table REF shows the brief summary of the variables used in our analysis.", "We are interested in construction of the confidence interval for $f(k_{0},\\ell _{0}) := \\operatorname{\\mathbf {E}}\\left[y\|k=k_{0},\\ell =\\ell _{0}\\right]$ .", "We take $\\left(k_{0},\\ell _{0}\\right)$ to be medians of each variable.", "Table: Summary statistics - Chinese chemical industry dataset for the year2001The first step is to estimate the variance of the error term.", "We assume homoskedastic errors for simplicity.", "The variance estimator is defined as $\\hat{\\sigma }^{2}=\\frac{\\sum _{i=1}^{n}\\left(y_{i}-\\hat{r}(k_{i},\\ell _{i})\\right)}{n-2\\nu _{1}+\\nu _{2}},$ where $\\hat{r}(k_{i},\\ell _{i})$ is the estimator for the conditional mean using kernel regression, $\\nu _{1}=\\text{tr}(L)$ , $\\nu _{2}=\\text{tr}(L^{\\prime }L)$ , where $L$ is the weight matrix for the kernel estimator.", "Refer to [27] for a justification for this variance estimator.", "We used the Gaussian kernel with the bandwidth chosen by expected Kullback-Leibler cross validation as in [20].", "For the function space, we consider adapting to a sequence of parameter spaces $\\lbrace \\Lambda _{+,\\mathcal {V}}(\\gamma _{j},C) \\rbrace _{j = 1}^6$ with $\\gamma _j = 1 - (j - 1)/5$ for $j = 1,...,5$ and $\\gamma _6 = 10^{-3}$ .", "We take $\\mathcal {V}=\\lbrace 1,2\\rbrace $ , assuming that the production function is nondecreasing in both fixed assets and labor, which is consistent with economic theory.", "To make $\\Lambda _{+,\\mathcal {V}}(\\gamma _{j},C) \\subset \\Lambda _{+,\\mathcal {V}}(\\gamma _{6},C)$ hold for all $j = 1,...,5$ , we only use observations in a restricted support, and the effective sample size is given by $n_{\\text{eff}} = 272$ .", "For the norm, we use the Euclidean norm weighted by the inverse of the standard deviation of each input, $\\left\\Vert (k,\\ell )\\right\\Vert =(({k}/{s_{k}})^{2}+({\\ell }/{s_{\\ell }})^{2})^{1/2}$ where $s_{k}$ and $s_{\\ell }$ are standard deviations of $k$ and $\\ell $ , respectively.", "We take conservative values of $C$ by setting $C=2\\times \\underset{(i,j)\\in \\left\\lbrace 1,...,n_{\\text{eff}}\\right\\rbrace ^{2}}{\\max }\\frac{\\left\|y_{j}-y_{i}\\right\|}{\\left\\Vert (k_{j},\\ell _{j})-(k_{i},\\ell _{i})\\right\\Vert ^{\\gamma _{6}}}.$ Table: 95% confidence intervals for f(k 0 ,ℓ 0 )f(k_{0},\\ell _{0})We compare different procedures to construct CIs.", "The methods in comparison are the minimax CI with respect to the largest space $\\Lambda _{+,\\mathcal {V}}(\\gamma _{6},C)$ (row 1), the restricted minimax CI with respect to the smallest space $\\Lambda _{+,\\mathcal {V}}(\\gamma _{1},C)$ (row 2), the adaptive Bonferroni CI adapting to $\\lbrace \\gamma _j\\rbrace _{j = 1}^6$ (row 3), the same adaptive CI, but taking into account the correlations between different CIs (fourth row), and the adaptive CI of [7] (henceforth CL) adapting to $\\lbrace \\gamma _j\\rbrace _{j = 1}^6$ (fifth row).", "Note that all the CIs maintain correct coverage over the largest space $\\Lambda _{+,\\mathcal {V}}(\\gamma _{6},C)$ , except for the second one, which is valid only over the smallest space $\\Lambda _{+,\\mathcal {V}}(\\gamma _{1},C)$ .", "We refer to the first minimax CI as the conservative minimax CI.", "Table REF demonstrates the 95% confidence intervals for $f(k_{0},\\ell _{0})$ produced by different inference methods.", "First of all, the lengths of the adaptive Bonferroni CIs are much shorter than the conservative minimax CI, while the procedure of CL yields a wider CI, almost as long as the conservative minimax CI.", "We can also observe that the adaptive Bonferroni CI using the calibrated value of $\\tau ^$ (fourth row) is relatively narrower than its naive version taking $\\tau = 0.05/2J$ (third row).", "Lastly, while the length of the second minimax procedure (second row) is the shortest, it is only valid when we are confident that the true regression function is in the smallest function space we consider, $ \\Lambda _{+,\\mathcal {V}}(\\gamma _{1},C)$ .", "Together with the simulation results in the previous section, our empirical analysis demonstrates the advantage of using an adaptive procedure when the monotonicity restriction is plausible as well as good finite sample performance of our proposed Bonferroni adaptive procedure." ], [ "Proof of Corollary ", "Suppose $\\hat{c}_{\\alpha ,-}^{\\ell ,j}$ solves $\\min _{\\hat{c}:[\\hat{c},\\infty )\\in \\mathcal {I}_{\\alpha ,1,-}^{J}}\\sup _{f\\in \\mathcal {F}_{j}}q_{\\beta ,f}(-Lf-\\hat{c}),$ where $\\mathcal {I}_{\\alpha ,1,-}^{J}$ denotes the set of one-sided CIs that covers $-Lf$ with probability at least $1-\\alpha $ over $\\mathcal {F}_{J}$ .", "Then, taking $\\hat{c}_{\\alpha }^{u,J}=-\\hat{c}_{\\alpha ,-}^{\\ell ,J}$ , we have $\\left(-\\infty ,\\hat{c}_{\\alpha }^{u,J}\\right]\\in \\mathcal {I}_{\\alpha ,1}^{J}$ and $\\hat{c}_{\\alpha }^{u,J}$ solves (REF ).", "Applying Theorem 3.1 of [3] with $\\widetilde{L}f=-Lf$ , we get the desired result." ], [ "Proof of Theorem ", "Consider the CI $[\\hat{c}_{\\alpha /2J}^{u,j},\\hat{c}_{\\alpha /2J}^{\\ell ,j}]$ , and observe that $\\operatorname{\\mathbf {E}}[\\hat{c}_{\\alpha /2J}^{u,j}-\\hat{c}_{\\alpha /2J}^{\\ell ,j}]=\\operatorname{\\mathbf {E}}[\\hat{c}_{\\alpha /2J}^{u,j}-Lf]+\\operatorname{\\mathbf {E}}[Lf-\\hat{c}_{\\alpha /2J}^{\\ell ,j}],$ for any $f\\in \\mathcal {F}_{j}$ .", "Then, by (REF ) and (REF ), we have $& \\sup _{f\\in \\mathcal {F}_{j}}\\operatorname{\\mathbf {E}}[\\hat{c}_{\\alpha /2J}^{u,j}-\\hat{c}_{\\alpha /2J}^{\\ell ,j}]\\le \\omega (z_{1-\\frac{\\alpha }{2J}},\\mathcal {F}_{J},\\mathcal {F}_{j})+\\omega (z_{1-\\frac{\\alpha }{2J}},\\mathcal {F}_{j},\\mathcal {F}_{J})\\le 2\\omega _{+}(z_{1-\\frac{\\alpha }{2J}},\\mathcal {F}_{j},\\mathcal {F}_{J}).$ It follows that $\\sup _{f\\in \\mathcal {F}_{j}}\\operatorname{\\mathbf {E}}\\mu (CI_{\\alpha }^{Bon,J}) & =\\sup _{f\\in \\mathcal {F}_{j}}\\operatorname{\\mathbf {E}}[\\min _{j}\\hat{c}_{\\alpha /2J}^{u,j}-\\max _{j}\\hat{c}_{\\alpha /2J}^{\\ell ,j}]\\\\& \\le \\sup _{f\\in \\mathcal {F}_{j}}\\operatorname{\\mathbf {E}}[\\hat{c}_{\\alpha /2J}^{u,j}-\\hat{c}_{\\alpha /2J}^{\\ell ,j}]\\\\& \\le 2\\omega _{+}(z_{1-\\frac{\\alpha }{2J}},\\mathcal {F}_{j},\\mathcal {F}_{J})$ for any $j=1,...,J$ .", "Noting that $2\\omega _{+}(z_{1-\\frac{\\alpha }{2J}},\\mathcal {F}_{j},\\mathcal {F}_{J})\\le \\frac{2z_{1-\\frac{\\alpha }{2J}}}{z_{1-\\frac{\\alpha }{2}}}\\omega _{+}(z_{1-\\frac{\\alpha }{2}},\\mathcal {F}_{j},\\mathcal {F}_{J}),$ which follows from the concavity of the ordered modulus of continuity, we obtain the desired result." ], [ "Proof of Lemma ", "First, note that we can write $& \\operatorname{\\mathbf {P}}\\left(Lf<\\hat{c}_{\\tau }^{L,j}\\right)\\\\= & \\operatorname{\\mathbf {P}}\\left(\\hat{c}_{\\tau }^{L,j}-Lf>0\\right)\\\\= & \\operatorname{\\mathbf {P}}\\left(\\frac{\\hat{c}_{\\tau }^{L,j}-Lf}{\\omega ^{\\prime }\\left(z_{1-\\tau },\\Lambda _{+,\\mathcal {V}}\\left(\\gamma ,C_{J}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma ,C_{j}\\right)\\right)}>0\\right)\\\\= & \\operatorname{\\mathbf {P}}\\left(\\frac{\\hat{c}_{\\tau }^{L,j}-Lf}{\\omega ^{\\prime }\\left(z_{1-\\tau },\\Lambda _{+,\\mathcal {V}}\\left(\\gamma ,C_{J}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma ,C_{j}\\right)\\right)}+z_{1-\\tau }>z_{1-\\tau }\\right)\\\\\\equiv & \\operatorname{\\mathbf {P}}(\\widetilde{V_{j}}(\\tau )>z_{1-\\tau }).$ Likewise, we can write $& \\operatorname{\\mathbf {P}}\\left(Lf>\\hat{c}_{\\tau }^{U,j}\\right)\\\\= & \\operatorname{\\mathbf {P}}\\left(\\frac{Lf-\\hat{c}_{\\tau }^{U,j}}{\\omega ^{\\prime }\\left(z_{1-\\tau },\\Lambda _{+,\\mathcal {V}}\\left(\\gamma ,C_{j}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma ,C_{J}\\right)\\right)}+z_{1-\\tau }\\ge z_{1-\\tau }\\right)\\\\\\equiv & \\operatorname{\\mathbf {P}}(\\widetilde{W_{j}}(\\tau )>z_{1-\\tau }).$ Therefore, writing $\\widetilde{V}(\\tau )=\\left(\\widetilde{V_{1}}(\\tau ),...,\\widetilde{V_{J}}(\\tau )\\right)^{\\prime }$ and similarly for $\\widetilde{W}(\\tau )$ , we have $\\operatorname{\\mathbf {P}}(Lf\\notin CI_{\\tau }^{\\mathcal {J}})=\\operatorname{\\mathbf {P}}(\\max \\lbrace \\widetilde{V}\\left(\\tau \\right)^{\\prime },\\widetilde{W}\\left(\\tau \\right)^{\\prime }\\rbrace >z_{1-\\tau }).$ Now, we want to find an upper bound on $\\sup _{f\\in \\mathcal {F}_{J}}\\operatorname{\\mathbf {P}}(\\max \\lbrace \\widetilde{V}(\\tau )^{\\prime },\\widetilde{W}(\\tau )^{\\prime }\\rbrace >z_{1-\\tau }).$ Note that the quantile of $\\max \\lbrace \\widetilde{V}(\\tau )^{\\prime },\\widetilde{W}(\\tau )^{\\prime }\\rbrace $ is increasing in the mean of each $\\widetilde{V_{j}}(\\tau )$ 's and $\\widetilde{W_{j}}(\\tau )$ 's.", "Moreover, the variances and covariances of $(\\widetilde{V}(\\tau )^{\\prime },\\widetilde{W}(\\tau )^{\\prime })^{\\prime }$ do not depend on the true regression function $f$ , by the construction of $\\hat{c}_{\\tau }^{L,j}$ and $\\hat{c}_{\\tau }^{U,j}$ .", "Therefore, it is useful to consider $\\sup _{f\\in \\mathcal {F}_{J}}\\operatorname{\\mathbf {E}}\\widetilde{V_{j}}(\\tau )$ and $\\sup _{f\\in \\mathcal {F}_{J}}\\operatorname{\\mathbf {E}}\\widetilde{W_{j}}(\\tau )$ .", "Actually, Lemma A.1 in AK can be used to show $\\sup _{f\\in \\mathcal {F}_{J}}\\operatorname{\\mathbf {E}}\\widetilde{V_{j}}(\\tau )=\\sup _{f\\in \\mathcal {F}_{J}}\\operatorname{\\mathbf {E}}\\widetilde{W_{j}}(\\tau )=0.$ Moreover, it is straightforward to show that the variance matrix of $(\\widetilde{V}(\\tau )^{\\prime },\\widetilde{W}(\\tau )^{\\prime })^{\\prime }$ is given by the formula in the statement of Lemma REF .", "Therefore, we have $\\sup _{f\\in \\mathcal {F}_{J}}\\operatorname{\\mathbf {P}}(\\max \\lbrace \\widetilde{V}(\\tau )^{\\prime },\\widetilde{W}(\\tau )^{\\prime }\\rbrace >z_{1-\\tau })\\\\\\le \\operatorname{\\mathbf {P}}(\\max \\lbrace V(\\tau )^{\\prime },W(\\tau )^{\\prime }\\rbrace >z_{1-\\tau }),$ and by setting $\\tau ^{}$ so that the latter term becomes $\\alpha $ , we get the desired result." ], [ "Proof of Lemma ", "First, we note that $h(x)\\equiv C\\left\\Vert x\\right\\Vert ^{\\gamma }$ satisfies the Hölder continuity condition.", "This is because for any $x,z\\in \\mathbb {R}^{k}$ , such that (without loss of generality) $\\left\\Vert x\\right\\Vert \\ge \\left\\Vert z \\right\\Vert $ , we have $\\left\|h(x)-h(z)\\right\|= C\\left(\\left\\Vert x\\right\\Vert ^{\\gamma }-\\left\\Vert z\\right\\Vert ^{\\gamma }\\right)\\le C\\left\\Vert x-z\\right\\Vert ^{\\gamma }.$ The inequality holds because we have $\\left\\Vert x\\right\\Vert ^{\\gamma }\\le \\left(\\left\\Vert x-z\\right\\Vert +\\left\\Vert z\\right\\Vert \\right)^{\\gamma },$ by the triangle inequality, and thus $\\left\\Vert x-z\\right\\Vert +\\left\\Vert z\\right\\Vert \\le \\left(\\left\\Vert x-z\\right\\Vert ^{\\gamma }+\\left\\Vert z\\right\\Vert ^{\\gamma }\\right)^{1/\\gamma },$ using the fact that $\\gamma \\in (0,1]$ Next, we show that $h_{+}(x)\\equiv C\\Vert \\left(x\\right)_{\\mathcal {V}+}\\Vert ^{\\gamma }$ also satisfies Hölder continuity.", "For $x,z\\in \\mathbb {R}^{k}$ , define $\\widetilde{x}=(x)_{\\mathcal {V}+}$ and $\\widetilde{z}=(z)_{\\mathcal {V}_{+}}$ .", "Then, we can see that $\\Vert x-z \\Vert \\ge \\left\\Vert \\widetilde{x}-\\widetilde{z}\\right\\Vert ,$ since $\\left\|x_{m}-z_{m}\\right\|\\ge \\left\|\\widetilde{x}_{m}-\\widetilde{z}_{m}\\right\|$ for $m\\in \\mathcal {V}$ and $\\left\|x_{m}-z_{m}\\right\|=\\left\|\\widetilde{x}_{m}-\\widetilde{z}_{m}\\right\|$ otherwise.", "Therefore, for any $x,z\\in \\mathbb {R}^{k}$ with $x\\ne z$ , we have $\\frac{\\left\|h_{+}(x)-h_{+}(z)\\right\|}{\\Vert x-z \\Vert ^{\\gamma }}=\\frac{\\left\|h(\\widetilde{x})-h(\\widetilde{z})\\right\|}{\\Vert x-z \\Vert ^{\\gamma }}\\le \\frac{\\left\|h(\\widetilde{x})-h(\\widetilde{z})\\right\|}{\\Vert \\widetilde{x}-\\widetilde{z} \\Vert ^{\\gamma }}\\le C,$ where the last inequality follows from the Hölder continuity of $h$ .", "Lastly, for monotonicity, note that for any $x,z\\in \\mathbb {R}^{k}$ such that $z_{i}\\ge x_{i}$ for some $i\\in \\mathcal {V}$ and $z_{j}=x_{j}$ for all $j\\ne i$ , we have $\\left\|\\widetilde{z}_{i}\\right\|\\ge \\left\|\\widetilde{x}_{i}\\right\|$ .", "Therefore, we have $h_{+}(z)\\ge h_{+}(x)$ .", "For $h_{-}(x)$ , note that $h_{-}(x)=-h_{+}(-x)$ .", "So Hölder continuity and monotonicity follows." ], [ "Proof of Lemma ", "First of all, monotonicity easily follows from the monotonicity of each $h_{1}$ and $h_{2}$ .", "For the Hölder continuity, fix some $x,z\\in \\mathbb {R}^{k}$ , and suppose $h_{1}(x)\\ge h_{2}(x)$ without loss of generality.", "Then, we have $\\left\|\\max \\left\\lbrace h_{1}(x),h_{2}(x)\\right\\rbrace -\\max \\left\\lbrace h_{1}(z),h_{2}(z)\\right\\rbrace \\right\|={\\left\\lbrace \\begin{array}{ll}\\left\|h_{1}(x)-h_{1}(z)\\right\|& \\text{if }h_{1}(z)\\ge h_{2}(z)\\\\\\left\|h_{1}(x)-h_{2}(z)\\right\|& \\text{if }h_{1}(z)<h_{2}(z).\\end{array}\\right.", "}$ For the former case, $\\left\|h_{1}(x)-h_{1}(z)\\right\|\\le C\\Vert x-z \\Vert ^{\\gamma }$ .", "For the latter case, note that if $h_{1}(x)\\ge h_{2}(x)$ $\\left\|h_{1}(x)-h_{2}(z)\\right\|<\\left\|h_{1}(x)-h_{1}(z)\\right\|\\le C\\Vert x-z \\Vert ^{\\gamma }.$ Moreover, if $h_{1}(x)<h_{2}(x)$ , we have $\\left\|h_{1}(x)-h_{2}(z)\\right\|<\\left\|h_{2}(x)-h_{2}(z)\\right\|\\le C\\Vert x-z \\Vert ^{\\gamma },$ which proves our claim." ], [ "Proof of Lemma ", "We only prove the claim about the maximum, since the proof for the minimum is analogous.", "First, note that due to Lemma REF , $f^{}(x)=f_{0}+C\\left\\Vert (x)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma }$ is in $\\Lambda _{+,\\mathcal {V}}^{f_{0}}\\left(\\gamma ,C\\right)$ .", "Now, for some $x\\in \\mathbb {R}^{k}$ , suppose there exists some $f^{\\dagger }\\in \\Lambda _{+,\\mathcal {V}}^{c}\\left(\\gamma ,C\\right)$ such that $f^{\\dagger }(x)>f^{}(x)$ .", "Then, we have $f^{\\dagger }(x)-f^{\\dagger }(0) > f^{}(x)-f^{}(0) = & C\\left\\Vert (x)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma }.$ Define $z=\\left(z_{1},...,z_{k}\\right)$ such that $z_{i}={\\left\\lbrace \\begin{array}{ll}\\max \\left\\lbrace 0,x_{i}\\right\\rbrace & \\text{if }i\\in \\mathcal {V}\\\\x_{i} & \\text{otherwise.}\\end{array}\\right.", "}$ Then, we have $z_{i}\\ge x_{i}$ for all $i\\in \\mathcal {V}$ , so we must have $f^{\\dagger }(z)\\ge f^{\\dagger }(x)$ .", "Similarly, we also have $f^{\\dagger }(z)\\ge f^{\\dagger }(0)=f_{0}$ .", "Moreover, by definition of $z$ , we have $\\Vert (x)_{\\mathcal {V}+}\\Vert =\\Vert (z)_{\\mathcal {V}+}\\Vert =\\Vert z \\Vert .$ Then, we can see that $\\left\|f^{\\dagger }(z)-f^{\\dagger }(0)\\right\|= & f^{\\dagger }(z)-f_{0}\\\\\\ge & f^{\\dagger }(x)-f_{0}\\\\> & C\\left\\Vert (x)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma }\\\\= & C\\Vert z \\Vert ^{\\gamma },$ which violates Hölder continuity.", "Therefore, $f^{}(x)$ attains the maximum." ], [ "Proof of Corollary ", "We first note that the function classes $\\Lambda _{+,\\mathcal {V}}(\\gamma _{j},C_{j})$ 's are translation invariant as defined in [2].", "Definition 1 For some linear functional $L$ on $\\mathcal {F},$ the function class $\\mathcal {F}$ is translation invariant if there exists a function $\\iota \\in \\mathcal {F}$ such that $L\\iota =1$ and $f+c\\iota \\in \\mathcal {F}$ for all $c\\in \\mathbb {R}$ and $f\\in \\mathcal {F}$ .", "In our case, by taking $\\iota =1$ , we can easily see that the function class $\\mathcal {F}_{j}=\\Lambda _{+,\\mathcal {V}}(\\gamma _{j},C_{j})$ satisfies translation invariance for our linear function $Lf=f(0)$ for all $j=1,...,J$ .", "Let $f_{j,\\delta }^{}\\in \\mathcal {F}_{j}$ and $f_{J,\\delta }^{}\\in \\mathcal {F}_{J}$ solve the the modulus of continuity problem with respect to $\\omega \\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right)$ .", "Then, by Lemma B.3 in [2], we have $\\omega ^{\\prime }\\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right) =\\frac{\\delta }{\\sum _{i=1}^{n}\\left(f_{j,\\delta }^{}(x_{i})-f_{J,\\delta }^{}(x_{i})\\right)}.$ Therefore, we can rewrite $\\hat{L}_{\\delta }^{\\ell ,j}$ in (REF ) as $\\hat{L}_{\\delta }^{\\ell ,j} = \\frac{f_{j,\\delta }^{}(0)+f_{J,\\delta }^{}(0)}{2}+\\frac{\\sum _{i=1}^{n}\\left(f_{j,\\delta }^{}(x_{i})-f_{J,\\delta }^{}(x_{i})\\right)\\left(y_{i}-\\frac{f_{j,\\delta }^{}(x_{i})+f_{J,\\delta }^{}(x_{i})}{2}\\right)}{\\sum _{i=1}^{n}\\left(f_{j,\\delta }^{}(x_{i})-f_{J,\\delta }^{}(x_{i})\\right)}.$ Next, using Corollary REF , we have $\\hat{L}_{\\delta }^{\\ell ,j} = &\\frac{\\sum _{i=1}^{n}D_{Jj,\\delta }\\left(x_{i}\\right)y_{i}}{\\sum _{i=1}^{n}D_{Jj,\\delta }\\left(x_{i}\\right)}+\\frac{\\omega \\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right)}{2} \\\\&-\\frac{\\sum _{i=1}^{n}D_{Jj,\\delta }\\left(x_{i}\\right)\\left[\\omega \\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right)-C_{j}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{j}}\\right]}{2\\sum _{i=1}^{n}D_{Jj,\\delta }\\left(x_{i}\\right)}\\\\& -\\frac{\\sum _{i=1}^{n}D_{Jj,\\delta }\\left(x_{i}\\right)\\min \\left\\lbrace \\omega \\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right)-C_{j}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{j}},C_{J}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{J}}\\right\\rbrace }{2\\sum _{i=1}^{n}D_{Jj,\\delta }\\left(x_{i}\\right)}.$ Noting that $& D_{Jj,\\delta }(x_{i})\\min \\lbrace \\omega (\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j})-C_{j}\\Vert (x_{i})_{\\mathcal {V}-}\\Vert ^{\\gamma _{j}},C_{J}\\Vert (x_{i})_{\\mathcal {V}+}\\Vert ^{\\gamma _{J}}\\rbrace \\\\= & D_{Jj,\\delta }(x_{i})\\min \\lbrace \\omega (\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j})-C_{j}\\Vert (x_{i})_{\\mathcal {V}-}\\Vert ^{\\gamma _{j}}-C_{J}\\Vert (x_{i})_{\\mathcal {V}+}\\Vert ^{\\gamma _{J}},0\\rbrace +C_{J}\\Vert (x_{i})_{\\mathcal {V}+}\\Vert ^{\\gamma _{J}}\\\\= & C_{J}\\Vert (x_{i})_{\\mathcal {V}+}\\Vert ^{\\gamma _{J}},$ by the definition of $D_{Jj,\\delta }\\left(x_{i}\\right)$ , we can rewrite the formula for $\\hat{L}_{\\delta }^{\\ell ,j}$ as in the statement of the corollary.", "To get the lower end of the CI, we subtract from $\\hat{L}_{\\delta }^{\\ell ,j}$ $& \\frac{1}{2}(\\omega (\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j})+\\delta \\omega ^{\\prime }(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}))\\\\= & \\frac{1}{2}\\left(\\omega (\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j})+\\frac{\\delta ^{2}}{\\sum _{i=1}^{n}D_{Jj,\\delta }(x_{i})}\\right),$ where the equality is from the equation (REF ).", "The upper end of the CI can be derived in an analogous way, this time using Proposition REF ." ], [ "Proof of Lemma ", "We first show that the limiting distribution is invariant with respect to $\\tau $ .", "For notational simplicity, we write $\\delta = z_{1-\\tau }$ and show invariance with respect to $\\delta $ .", "With some abuse of notation, we write $(V(\\delta )^{\\prime }, W(\\delta )^{\\prime })^{\\prime }$ as this reparametrized version whose value is the same with $(V(\\tau )^{\\prime }, W(\\tau )^{\\prime })^{\\prime }$ if $\\delta = z_{1-\\tau }$ .", "Because $(V(\\delta )^{\\prime }, W(\\delta )^{\\prime })^{\\prime }$ is centered and has unit variance, it suffices to show that the covariance terms converge to a limit that does not depend on $\\delta $ .", "We show that this is the case for the covariance terms of $V(\\delta ):= (V_{1}(\\delta ), \\dots , V_{J}(\\delta ))^{\\prime }$ .", "The same invariance for other covariance terms (covariance between elements of $W(\\delta )$ and the covariance between an element of $V(\\delta )$ and of $W(\\delta )$ ) follows by an analogous calculation.", "Again, we consider the case where $\\sigma (\\cdot ) =1$ .", "However, this can be relaxed (with more notation) under mild regularity conditions given in Appendix .", "Define $b_{Jj,\\delta } := \\omega \\left(\\delta ,\\mathcal {F}_{J},\\mathcal {F}_{j}\\right)$ for $j \\le J$ .", "Note that $\\begin{aligned}\\mathrm {cov}\\left(V_{j}(\\delta ),V_{\\ell }(\\delta )\\right) = &\\frac{\\sum _{i=1}^{n}D_{Jj,\\delta }(x_{i})D_{J\\ell }(x_{i})}{\\delta ^{2}}\\\\= & \\frac{\\sum _{i=1}^{n}(D_{Jj,\\delta }(x_{i})/b_{Jj,\\delta })(D_{J\\ell }(x_{i})/b_{Jj,\\delta })}{\\delta ^{2}/(b_{Jj,\\delta }b_{J\\ell , \\delta })}.\\end{aligned}$ The numerator of the right-hand side is $\\sum _{i=1}^{n}\\Bigg (1-\\frac{C_{j}}{b_{Jj,\\delta }}\\Vert (x_{i})_{\\mathcal {V}-}\\Vert ^{\\gamma _{j}}-\\frac{C_{J}}{b_{Jj,\\delta }}\\Vert (x_{i})_{\\mathcal {V}+}\\Vert ^{\\gamma _{J}}\\Bigg )_{+}\\Bigg (1-\\frac{C_{\\ell }}{b_{J\\ell ,\\delta }}\\Vert (x_{i})_{\\mathcal {V}-}\\Vert ^{\\gamma _{\\ell }}-\\frac{C_{J}}{b_{J\\ell ,\\delta }}\\Vert (x_{i})_{\\mathcal {V}+}\\Vert ^{\\gamma _{J}}\\Bigg )_{+}.$ We investigate the term $\\int \\left(1-\\frac{C_{j}}{b_{Jj,\\delta }}\\Vert (x)_{\\mathcal {V}-}\\Vert ^{\\gamma _{j}}-\\frac{C_{J}}{b_{Jj,\\delta }}\\Vert (x)_{\\mathcal {V}+}\\Vert ^{\\gamma _{J}}\\right)_{+}\\left(1-\\frac{C_{\\ell }}{b_{J\\ell ,\\delta }}\\Vert (x)_{\\mathcal {V}-}\\Vert ^{\\gamma _{\\ell }}-\\frac{C_{J}}{b_{J\\ell ,\\delta }}\\Vert (x)_{\\mathcal {V}+}\\Vert ^{\\gamma _{J}}\\right)_{+}dx.$ We consider the case $ \\gamma _{j} > \\gamma _{\\ell } $ , but the case where $\\gamma _{j} = \\gamma _{\\ell }$ can be dealt with by taking analogous steps.", "By a similar argument made in the proof of Theorem REF , showing that this integral term and $b_{J\\ell ,\\delta }^{1+{k_{+}}/{\\gamma _{\\ell }}+{(k-k_{+})}/{\\gamma _{j}}}b_{Jj,\\delta }n$ are both $o(1)$ will establish $ \\mathrm {cov}\\left(V_{j}(\\delta ),V_{\\ell }(\\delta )\\right) \\rightarrow 0$ .", "By Theorem REF , we have $b_{Jj,\\delta } << b_{J\\ell ,\\delta }$ , with both going to 0 as $n \\rightarrow \\infty .$ By applying a change of variable $(x_{[1,m]}/b_{J\\ell ,\\delta }^{1/\\gamma _{\\ell }},x_{[m+1,k]}/b_{J\\ell ,\\delta }^{1/\\gamma _{J}})=z,$ we have for a given orthant $O \\in \\mathcal {O}$ $& \\int _{ O}\\left(1-\\frac{C_{j}}{b_{Jj,\\delta }}\\Vert (x)_{\\mathcal {V}-}\\Vert ^{\\gamma _{j}}-\\frac{C_{J}}{b_{Jj,\\delta }}\\Vert (x)_{\\mathcal {V}+}\\Vert ^{\\gamma _{J}}\\right)_{+}\\left(1-\\frac{C_{\\ell }}{b_{J\\ell ,\\delta }}\\Vert (x)_{\\mathcal {V}-}\\Vert ^{\\gamma _{\\ell }}-\\frac{C_{J}}{b_{J\\ell ,\\delta }}\\Vert (x)_{\\mathcal {V}+}\\Vert ^{\\gamma _{J}}\\right)_{+}dx \\\\= &b_{J\\ell ,\\delta }^{{m}/{\\gamma _{\\ell }}+{(k-m)}/{\\gamma _{J}}}\\int _{O}I_{j\\ell J, \\delta }(z)\\, dz.$ Here, $I_{j\\ell J, \\delta }(z)$ is defined as $&I_{j\\ell J, \\delta }(z)\\\\=&\\left(1-C_{j}\\Vert (z_{[1,m]}b_{J\\ell ,\\delta }^{1/{\\gamma _{\\ell }}} b_{Jj,\\delta }^{-1/{\\gamma _{j}}},0,z_{[k_{+}+1,k]}b_{J\\ell ,\\delta }^{1/{\\gamma _{J}}}b_{Jj,\\delta }^{-1/{\\gamma _{j}}})\\Vert ^{\\gamma _{j}}-C_{J}\\Vert (0,-z_{[m+1,k]}b_{J\\ell ,\\delta }^{1/{\\gamma _{J}}}b_{Jj,\\delta }^{-1/{\\gamma _{J}}})\\Vert ^{\\gamma _{J}}\\right)_{+}\\\\&\\cdot \\left(1-C_{\\ell }\\Vert (z_{[1,m]},0,z_{[k_{+}+1,k]}b_{J\\ell ,\\delta }^{1/{\\gamma _{J}}-1/{\\gamma _{\\ell }}})\\Vert ^{\\gamma _{\\ell }}-C_{J}\\Vert (0,-z_{[m+1,k]})\\Vert ^{\\gamma _{J}}\\right)_{+}$ The limit behavior of this term depends on the limit of the following four quantities: 1) $b_{J\\ell ,\\delta }^{1/{\\gamma _{\\ell }}} b_{Jj,\\delta }^{-1/{\\gamma _{j}}}$ , 2) $b_{J\\ell ,\\delta }^{1/{\\gamma _{J}}}b_{Jj,\\delta }^{-1/{\\gamma _{j}}}$ , 3) $b_{J\\ell ,\\delta }^{1/{\\gamma _{J}}}b_{Jj,\\delta }^{-1/{\\gamma _{J}}}$ , and 4) $b_{J\\ell ,\\delta }^{1/{\\gamma _{J}}-1/{\\gamma _{\\ell }}}.$ Since $\\gamma > \\gamma _{\\ell } \\ge \\gamma $ and $b_{J\\ell ,\\delta } \\asymp n^{-1/{2 + k_{+}/\\gamma _{\\ell } + (k-k_{+})/\\gamma }}$ , we have $b_{Jj,\\delta }<<b_{J\\ell ,\\delta }$ and $ b_{J\\ell ,\\delta }^{1/{\\gamma }} << b_{J\\ell ,\\delta }^{1/{\\gamma _{\\ell }}} <<b_{Jj,\\delta }^{ 1/{\\gamma }}$ .", "This gives 1) $b_{J\\ell ,\\delta }^{1/{\\gamma _{\\ell }}} b_{Jj,\\delta }^{-1/{\\gamma _{j}}} \\rightarrow 0$ , 2) $b_{J\\ell ,\\delta }^{1/{\\gamma _{J}}}b_{Jj,\\delta }^{-1/{\\gamma _{j}}} \\rightarrow 0$ , 3) $b_{J\\ell ,\\delta }^{1/{\\gamma _{J}}}b_{Jj,\\delta }^{-1/{\\gamma _{J}}} \\rightarrow \\infty $ , and 4) $b_{J\\ell ,\\delta }^{1/{\\gamma _{J}}-1/{\\gamma _{\\ell }}} \\rightarrow 0.$ Hence, we have $\\int _{O}I_{j\\ell J, \\delta }(z) \\,dz = o(1),$ by a dominated convergence argument, and the convergence rate is the slowest on the orthant where $m = k_{+}$ .", "Now, it remains to show that $b_{J\\ell ,\\delta }^{{k_{+}}/{\\gamma _{\\ell }}+{(k-k_{+})}/{\\gamma _{J}}}b_{J\\ell ,\\delta }b_{Jj,\\delta }n = o(1).$ Note that the order of the expression on the left-hand side is $n^{r}$ where $r$ is $\\frac{1}{2+k_{+}/\\gamma _{\\ell }+(k-k_{+})/\\gamma _{J}} -\\frac{1}{2+k_{+}/\\gamma _{j}+(k-k_{+})/\\gamma _{J}} < 0.$ This establishes that $\\text{cov}\\left(V_{j}(\\delta ),V_{\\ell }(\\delta )\\right) \\rightarrow 0$ for any $j \\ne \\ell $ and for any $\\delta > 0.$ Now, to establish the second half of the lemma, consider the case when $\\gamma _{j} = \\gamma $ for all $j$ .", "In such case, we have $& \\int _{O}\\left(1-\\frac{C_{j}}{b_{Jj,\\delta }}\\Vert (x)_{\\mathcal {V}-}\\Vert ^{\\gamma }-\\frac{C_{J}}{b_{Jj,\\delta }}\\Vert (x)_{\\mathcal {V}+}\\Vert ^{\\gamma }\\right)_{+}\\left(1-\\frac{C_{\\ell }}{b_{J\\ell ,\\delta }}\\Vert (x)_{\\mathcal {V}-}\\Vert ^{\\gamma }-\\frac{C_{J}}{b_{J\\ell ,\\delta }}\\Vert (x)_{\\mathcal {V}+}\\Vert ^{\\gamma }\\right)_{+}dx \\\\= &b_{J\\ell ,\\delta }^{\\frac{m}{\\gamma }+\\frac{k-m}{\\gamma }}\\int _{O} \\Bigg ( \\Big (1-C_{\\ell }\\Vert (z_{[1,m]},0,z_{[k_{+}+1,k]}b_{J\\ell ,\\delta }^{\\frac{1}{\\gamma }-\\frac{1}{\\gamma }})\\Vert ^{\\gamma }-C_{J}\\Vert (0,-z_{[m+1,k]})\\Vert ^{\\gamma }\\Big )_{+}\\cdot \\\\&\\Big (1-C_{j}\\Vert (z_{[1,m]}b_{J\\ell ,\\delta }^{\\frac{1}{\\gamma }} b_{Jj,\\delta }^{-\\frac{1}{\\gamma }},0,z_{[k_{+}+1,k]}b_{J\\ell ,\\delta }^{\\frac{1}{\\gamma }}b_{Jj,\\delta }^{-\\frac{1}{\\gamma }})\\Vert ^{\\gamma }-C_{J}\\Vert (0,-z_{[m+1,k]}b_{J\\ell ,\\delta }^{\\frac{1}{\\gamma }}b_{Jj,\\delta }^{-\\frac{1}{\\gamma }})\\Vert ^{\\gamma }\\Big )_{+}\\Bigg )dx\\\\= & b_{J\\ell ,\\delta }^{\\frac{k}{\\gamma }}\\int _{O}\\Bigg ( \\left(1-C_{\\ell }\\Vert (z_{[1,m]},0,z_{[k_{+}+1,k]})\\Vert ^{\\gamma }-C_{J}\\Vert (0,-z_{[m+1,k]})\\Vert ^{\\gamma }\\right)_{+}\\cdot \\\\& \\Big (1-C_{j}\\Vert (z_{[1,m]}b_{J\\ell ,\\delta }^{\\frac{1}{\\gamma }} b_{Jj,\\delta }^{-\\frac{1}{\\gamma }},0,z_{[k_{+}+1,k]}b_{J\\ell ,\\delta }^{\\frac{1}{\\gamma }}b_{Jj,\\delta }^{-\\frac{1}{\\gamma }})\\Vert ^{\\gamma }-C_{J}\\Vert (0,-z_{[m+1,k]}b_{J\\ell ,\\delta }^{\\frac{1}{\\gamma }}b_{Jj,\\delta }^{-\\frac{1}{\\gamma }})\\Vert ^{\\gamma }\\Big )_{+}\\Bigg )dx$ We know that $ b_{J\\ell ,\\delta } \\asymp n^{-\\frac{1}{2+k/\\gamma }}\\left(\\delta ^{2}/c_{J\\ell }^{\\ast }\\right)^{\\frac{1}{2+k/\\gamma }}$ for some constant $c^{\\ast }_{J\\ell }$ , by Theorem REF .", "It follows that $(b_{J\\ell ,\\delta }/b_{Jj,\\delta })^{1/\\gamma } \\asymp (c^{\\ast }_{Jj}/c^{\\ast }_{J\\ell })^{\\frac{1/\\gamma }{2+k/\\gamma }},$ which then implies $& \\int _{O}\\left(1-C_{1}\\Vert (z_{[1,m]}b_{J\\ell ,\\delta }^{\\frac{1}{\\gamma _{\\ell }}} b_{Jj,\\delta }^{-\\frac{1}{\\gamma }},0,z_{[k_{+}+1,k]}b_{J\\ell ,\\delta }^{\\frac{1}{\\gamma }}b_{Jj,\\delta }^{-\\frac{1}{\\gamma }})\\Vert ^{\\gamma _{1}}-C_{2}\\Vert (0,-z_{[m+1,k]}b_{J\\ell ,\\delta }^{\\frac{1}{\\gamma }}b_{Jj,\\delta }^{-\\frac{1}{\\gamma }})\\Vert ^{\\gamma _{2}}\\right)_{+}\\cdot \\\\& \\hspace{80.0pt}\\left(1-C_{\\ell }\\Vert (z_{[1,m]},0,z_{[k_{+}+1,k]})\\Vert ^{\\gamma _{1}}-C_{J}\\Vert (0,-z_{[m+1,k]})\\Vert ^{\\gamma _{2}}\\right)_{+}dx\\\\= & \\int _{O} \\Big (\\left(1-C_{\\ell }\\Vert (z_{[1,m]},0,z_{[k_{+}+1,k]})\\Vert ^{\\gamma }-C_{J}\\Vert (0,-z_{[m+1,k]})\\Vert ^{\\gamma }\\right)_{+}\\cdot \\\\& \\Big (1-C_{j} \\Big (\\frac{c^{\\ast }_{Jj}}{c^{\\ast }_{J\\ell }}\\Big )^{\\frac{1}{2+k/\\gamma }}\\Vert (z_{[1,m]},0,z_{[k_{+}+1,k]})\\Vert ^{\\gamma }-C_{J}\\Big (\\frac{c^{\\ast }_{Jj}}{c^{\\ast }_{J\\ell }}\\Big )^{\\frac{1}{2+k/\\gamma }}\\Vert (0,-z_{[m+1,k]})\\Vert ^{\\gamma }\\Big )_{+} \\Big )dx + o(1).$ Denote the integral term following the last equality as $B_{j\\ell , O}$ , and $B_{j\\ell } = \\sum _{O \\in \\mathcal {O}}B_{j\\ell , O}$ .", "Plugging this result back into (REF ), we have that $\\text{cov}\\left(V_{j}(\\delta ),V_{\\ell }(\\delta )\\right) =b_{J\\ell ,\\delta }^{k/\\gamma }b_{Jj,\\delta }b_{J\\ell ,\\delta }n B_{j\\ell } (1+ o(1))/\\delta ^{2}.$ Now, note that $b_{J\\ell ,\\delta }^{k/\\gamma }b_{Jj,\\delta }b_{J\\ell ,\\delta }n B_{j\\ell } /\\delta ^{2}= & \\left(\\delta ^{2}/c_{J\\ell }^{\\ast }\\right)^{\\frac{k/\\gamma }{2+k/\\gamma }}\\left(\\delta ^{2}/c_{J\\ell }^{\\ast }\\right)^{\\frac{1}{2+k/\\gamma }}\\left(\\delta ^{2}/c_{Jj}^{\\ast }\\right)^{\\frac{1}{2+k/\\gamma }} B_{j\\ell }/\\delta ^{2} + o(1) \\\\= & c_{J\\ell }^{\\ast }{}^{-\\frac{k/\\gamma }{2+k/\\gamma }}c_{J\\ell }^{\\ast }{}^{-\\frac{1}{2+k/\\gamma }}c_{Jj}^{\\ast }{}^{-\\frac{1}{2+k/\\gamma }} B_{j\\ell } + o(1).$ While this calculation is sufficient to show the invariance of the limiting covariance with respect to $\\delta $ , we further simplify the term by some additional calculations.", "By changing the role of $j$ and $\\ell $ in the above change of variables, we know that $& b_{J\\ell ,\\delta }^{\\frac{k}{\\gamma }}\\int _{O} \\left(1-C_{\\ell }\\Vert (z_{[1,m]},0,z_{[k_{+}+1,k]})\\Vert ^{\\gamma }-C_{J}\\Vert (0,-z_{[m+1,k]})\\Vert ^{\\gamma }\\right)_{+}\\cdot \\\\& \\hspace{20.0pt}\\left(1-C_{j} (c^{\\ast }_{Jj}/c^{\\ast }_{J\\ell })^{\\frac{1}{2+k/\\gamma }}\\Vert (z_{[1,m]},0,z_{[k_{+}+1,k]})\\Vert ^{\\gamma }-C_{J}(c^{\\ast }_{Jj}/c^{\\ast }_{J\\ell })^{\\frac{1}{2+k/\\gamma }}\\Vert (0,-z_{[m+1,k]})\\Vert ^{\\gamma }\\right)_{+}dx\\\\ = & b_{Jj,\\delta }^{\\frac{k}{\\gamma }}\\int _{O} \\left(1-C_{j} \\Vert (z_{[1,m]},0,z_{[k_{+}+1,k]})\\Vert ^{\\gamma }-C_{J}\\Vert (0,-z_{[m+1,k]})\\Vert ^{\\gamma }\\right)_{+}\\cdot \\\\& \\hspace{20.0pt} \\left(1-C_{\\ell }(c^{\\ast }_{J\\ell }/c^{\\ast }_{Jj})^{\\frac{1}{2+k/\\gamma }}\\Vert (z_{[1,m]},0,z_{[k_{+}+1,k]})\\Vert ^{\\gamma }-C_{J}(c^{\\ast }_{J\\ell }/c^{\\ast }_{Jj})^{\\frac{1}{2+k/\\gamma }}\\Vert (0,-z_{[m+1,k]})\\Vert ^{\\gamma }\\right)_{+}dx.$ Now, consider the change of variables given by $(x_{[1,m]}/(b_{J\\ell ,\\delta }^{1/(2\\gamma )}b_{Jj,\\delta }^{1/(2\\gamma )}),x_{[m+1,k]}/(b_{J\\ell ,\\delta }^{1/(2\\gamma )}b_{Jj,\\delta }^{1/(2\\gamma )}))=z.$ We have $& \\int _{ O}\\left(1-\\frac{C_{j}}{b_{Jj,\\delta }}\\Vert (x)_{\\mathcal {V}-}\\Vert ^{\\gamma }-\\frac{C_{J}}{b_{Jj,\\delta }}\\Vert (x)_{\\mathcal {V}+}\\Vert ^{\\gamma }\\right)_{+}\\left(1-\\frac{C_{\\ell }}{b_{J\\ell ,\\delta }}\\Vert (x)_{\\mathcal {V}-}\\Vert ^{\\gamma }-\\frac{C_{J}}{b_{J\\ell ,\\delta }}\\Vert (x)_{\\mathcal {V}+}\\Vert ^{\\gamma }\\right)_{+}dx \\\\= &b_{Jj,\\delta }^{\\frac{k}{2\\gamma }}b_{J\\ell ,\\delta }^{\\frac{k}{2\\gamma }}\\,\\, \\cdot \\\\&\\int _{O}\\left(1-C_{j}(c^{\\ast }_{Jj}/c^{\\ast }_{J\\ell })^{\\frac{1/(2\\gamma )}{2+k/\\gamma }}\\Vert (z_{[1,m]},0,z_{[k_{+}+1,k]})\\Vert ^{\\gamma }-C_{J}(c^{\\ast }_{Jj}/c^{\\ast }_{J\\ell })^{\\frac{1/(2\\gamma )}{2+k/\\gamma }} \\Vert (0,-z_{[m+1,k]})\\Vert ^{\\gamma }\\right)_{+}\\cdot \\\\& \\hspace{20.0pt}\\left(1-C_{\\ell }(c^{\\ast }_{J\\ell }/c^{\\ast }_{Jj})^{\\frac{1/(2\\gamma )}{2+k/\\gamma }}\\Vert (z_{[1,m]},0,z_{[k_{+}+1,k]})\\Vert ^{\\gamma }-C_{J}(c^{\\ast }_{J\\ell }/c^{\\ast }_{Jj})^{\\frac{1/(2\\gamma )}{2+k/\\gamma }}\\Vert (0,-z_{[m+1,k]})\\Vert ^{\\gamma }\\right)_{+}dx \\\\& + o\\big ( b_{Jj,\\delta }^{\\frac{k}{2\\gamma }}b_{J\\ell ,\\delta }^{\\frac{k}{2\\gamma }}\\big )$ Here, we used $(b_{J\\ell ,\\delta }/b_{Jj,\\delta })^{1/(2\\gamma )} =(c^{\\ast }_{Jj}/c^{\\ast }_{J\\ell })^{\\frac{1/(2\\gamma )}{2+k/\\gamma }} + o(1)$ Now, write the integral in the last term as $B^{\\ast }_{j\\ell , O}$ , and $B^{\\ast }_{j\\ell } = \\sum _{O \\in \\mathcal {O}}B^{\\ast }_{j\\ell ,O}$ Finally, by similar calculations as above $\\text{cov}\\left(V_{j}(\\delta ),V_{\\ell }(\\delta )\\right)& = n b_{Jj,\\delta }^{\\frac{k}{2\\gamma }}b_{J\\ell ,\\delta }^{\\frac{k}{2\\gamma }} b_{Jj,\\delta }b_{J\\ell ,\\delta } B^{\\ast }_{j\\ell }/\\delta ^{2} + o(1) \\\\& = c_{Jj}^{\\ast }{}^{-\\frac{k/(2\\gamma )}{2+k/\\gamma }} c_{J\\ell }^{\\ast }{}^{-\\frac{k/(2\\gamma )}{2+k/\\gamma }}c_{Jj}^{\\ast }{}^{-\\frac{1}{2+k/\\gamma }} c_{J\\ell }^{\\ast }{}^{-\\frac{1}{2+k/\\gamma }}B^{\\ast }_{j\\ell } + o(1) \\\\& = c_{Jj}^{\\ast }{}^{-1/2} c_{J\\ell }^{\\ast }{}^{-1/2}B^{\\ast }_{j\\ell } + o(1).$ This shows that $ \\text{cov}\\left(V_{j}(\\delta ),V_{\\ell }(\\delta )\\right) \\rightarrow c_{Jj}^{\\ast }{}^{-1/2} c_{J\\ell }^{\\ast }{}^{-1/2} B^{\\ast }_{j\\ell }$ as $n \\rightarrow \\infty $ .", "Note that the limiting covariance term does not depend on $\\delta $ .", "Note that we have $V_{j}(\\delta ) \\overset{d}{=} \\sum _{i=1}^{n}D_{Jj,\\delta }(x_{i})Z_{i}/\\delta $ where $Z_{i}$ 's are i.i.d standard normal random variables.", "Furthermore, this equivalence holds jointly for $V_{j}(\\delta )$ , $j = 1, \\dots , J$ .", "Let $\\lbrace x_{i}\\rbrace _{i=1}^{\\infty }$ be a sequence where the under which where Theorem REF holds.", "Define for $C \\in [C_{1}, C_{J}]$ $Z_{ni}(C) = ( \\omega (\\delta ,\\mathcal {F}_{J},\\Lambda _{+,\\mathcal {V}}(\\gamma , C))-C\\Vert (x_{i})_{\\mathcal {V}-}\\Vert ^{\\gamma }-C_{J}\\Vert (x_{i})_{\\mathcal {V}+}\\Vert ^{\\gamma })_{+}Z_{i}/\\delta ,$ and consider the stochastic process $\\sum _{i=1}^{n}Z_{ni}(C)$ indexed by $C \\in [C_{1}, C_{J}]$ .", "We show that this process weakly converges to a tight Gaussian process, from which the fact that the quantile of the maximum of $V(\\delta )$ does not depend on $J$ follows.", "We use Theorem 2.11.1 of [26] to establish this convergence.", "Specifically, we use the result given by Example 2.11.13.", "Given the results we already have, it suffices to show that $\\sum _{i=1}^{n }\\left\|\\frac{\\partial }{\\partial C} ( \\omega (\\delta ,\\mathcal {F}_{J},\\Lambda _{+,\\mathcal {V}}(\\gamma , C))-C\\Vert (x_{i})_{\\mathcal {V}-}\\Vert ^{\\gamma }-C_{J}\\Vert (x_{i})_{\\mathcal {V}+}\\Vert )_{+} \\right\|^{2} = O(1),$ and that a Lindeberg condition is satisfied.", "With some abuse of notation, we write $D_{C,n,\\delta }(x_{i}) := ( \\omega (\\delta ,\\mathcal {F}_{J},\\Lambda _{+,\\mathcal {V}}(\\gamma , C))-C\\Vert (x_{i})_{\\mathcal {V}-}\\Vert ^{\\gamma }-C_{J}\\Vert (x_{i})_{\\mathcal {V}+}\\Vert )_{+},$ and $\\omega (\\delta , C_{J}, C) =\\omega (\\delta ,\\mathcal {F}_{J},\\Lambda _{+,\\mathcal {V}}(\\gamma , C))$ with $\\omega ^{-1}(b, C_{J}, C)$ defined similarly.", "Recall that $\\omega ^{-1}(b, C_{J}, C) = \\left( \\sum _{i=1}^{n} ( b -C \\Vert (x_{i})_{\\mathcal {V}-} \\Vert ^{\\gamma } - C_{J} \\Vert (x_{i})_{\\mathcal {V}+} \\Vert ^{\\gamma }) ^{2}_{+} \\right)^{\\frac{1}{2}}.$ From the identity $ \\delta = \\omega ^{-1}(\\omega (\\delta , C_{J}, C), C_{J}, C)$ , we have $0 = \\frac{\\partial }{\\partial b} \\omega ^{-1}(\\omega (\\delta , C_{J}, C),C_{J}, C)\\frac{\\partial }{\\partial C}\\omega (\\delta , C_{J}, C) +\\frac{\\partial }{\\partial C}\\omega ^{-1}(\\omega (\\delta , C_{J}, C), C_{J}, C)$ so that $\\frac{\\partial }{\\partial C}\\omega (\\delta , C_{J}, C) & = -\\frac{ \\frac{\\partial }{\\partial C}\\omega ^{-1}(\\omega (\\delta , C_{J}, C), C_{J}, C)}{\\frac{\\partial }{\\partial b} \\omega ^{-1}(\\omega (\\delta , C_{J}, C),C_{J}, C)} \\\\& = \\frac{ \\sum _{i=1}^{n} \\Vert (x_{i})_{\\mathcal {V}-} \\Vert ^{\\gamma }\\left[ \\omega (\\delta , C_{J}, C) -C \\Vert (x_{i})_{\\mathcal {V}-} \\Vert ^{\\gamma } - C_{J} \\Vert (x_{i})_{\\mathcal {V}+} \\Vert ^{\\gamma } \\right]_{+}}{\\sum _{i=1}^{n}\\left[ \\omega (\\delta , C_{J}, C) -C \\Vert (x_{i})_{\\mathcal {V}-} \\Vert ^{\\gamma } - C_{J} \\Vert (x_{i})_{\\mathcal {V}+} \\Vert ^{\\gamma } \\right]_{+}}.$ This gives $\\frac{\\partial }{\\partial C} D_{C,n,\\delta }(x_{i}) =&\\left( \\frac{ \\sum _{i=1}^{n} \\Vert (x_{i})_{\\mathcal {V}-} \\Vert ^{\\gamma } \\left[ \\omega (\\delta , C_{J}, C) -C \\Vert (x_{i})_{\\mathcal {V}-} \\Vert ^{\\gamma } - C_{J} \\Vert (x_{i})_{\\mathcal {V}+} \\Vert ^{\\gamma } \\right]_{+}}{\\sum _{i=1}^{n}\\left[ \\omega (\\delta , C_{J}, C) -C \\Vert (x_{i})_{\\mathcal {V}-} \\Vert ^{\\gamma } - C_{J} \\Vert (x_{i})_{\\mathcal {V}+} \\Vert ^{\\gamma } \\right]_{+}} - \\Vert (x_{i})_{\\mathcal {V}-} \\Vert ^{\\gamma } \\right) \\\\&\\cdot {1}\\left( \\omega (\\delta , C_{J}, C) -C \\Vert (x_{i})_{\\mathcal {V}-} \\Vert ^{\\gamma } - C_{J} \\Vert (x_{i})_{\\mathcal {V}+} \\Vert ^{\\gamma } \\ge 0 \\right)\\\\=&\\left( \\frac{ \\sum _{k=1}^{n} (\\Vert (x_{k})_{\\mathcal {V}-}\\Vert ^{\\gamma } - \\Vert (x_{i})_{\\mathcal {V}-} \\Vert ^{\\gamma })\\left[ \\omega (\\delta , C_{J}, C) -C \\Vert (x_{k})_{\\mathcal {V}-} \\Vert ^{\\gamma } - C_{J} \\Vert (x_{k})_{\\mathcal {V}+} \\Vert ^{\\gamma } \\right]_{+}}{\\sum _{k=1}^{n}\\left[ \\omega (\\delta , C_{J}, C) -C \\Vert (x_{k})_{\\mathcal {V}-} \\Vert ^{\\gamma } - C_{J} \\Vert (x_{k})_{\\mathcal {V}+} \\Vert ^{\\gamma } \\right]_{+}} \\right) \\\\&\\cdot {1}\\left( \\omega (\\delta , C_{J}, C) -C \\Vert (x_{i})_{\\mathcal {V}-} \\Vert ^{\\gamma } - C_{J} \\Vert (x_{i})_{\\mathcal {V}+} \\Vert ^{\\gamma } \\ge 0 \\right),$ with the understanding that the fraction equals 0 if the denominator is $0.$ We have $&\\left\|\\frac{\\partial }{\\partial C}D_{C,n,\\delta }(x_{i})(x_{i}) \\right\|^{2} \\\\\\le & \\left( \\frac{ \\omega (\\delta , C_{J}, C)\\sum _{k=1}^{n} \\left[ \\omega (\\delta , C_{J}, C) -C \\Vert (x_{k})_{\\mathcal {V}-} \\Vert ^{\\gamma } - C_{J} \\Vert (x_{k})_{\\mathcal {V}+} \\Vert ^{\\gamma } \\right]_{+}}{C\\sum _{k=1}^{n}\\left[ \\omega (\\delta , C_{J}, C) -C \\Vert (x_{k})_{\\mathcal {V}-} \\Vert ^{\\gamma } - C_{J} \\Vert (x_{k})_{\\mathcal {V}+} \\Vert ^{\\gamma } \\right]_{+}} \\right)^{2} \\\\&\\cdot {1}\\left( \\omega (\\delta , C_{J}, C) -C \\Vert (x_{i})_{\\mathcal {V}-} \\Vert ^{\\gamma } - C_{J} \\Vert (x_{i})_{\\mathcal {V}+} \\Vert ^{\\gamma } \\ge 0 \\right) \\\\\\le & (\\omega (\\delta , C_{J}, C)/C)^{2} {1}\\left( \\omega (\\delta , C_{J}, C) -C \\Vert (x_{i})_{\\mathcal {V}-} \\Vert ^{\\gamma } - C_{J} \\Vert (x_{i})_{\\mathcal {V}+} \\Vert ^{\\gamma } \\ge 0 \\right),$ so that $& \\sum _{i=1}^{n} \\left\|\\frac{\\partial }{\\partial C}D_{C,n,\\delta }(x_{i})(x_{i}) \\right\|^{2} \\\\\\le & (\\omega (\\delta , C_{J}, C)/C)^{2}\\sum _{i=1}^{n} {1}\\left( \\omega (\\delta , C_{J}, C) -C \\Vert (x_{i})_{\\mathcal {V}-} \\Vert ^{\\gamma } - C_{J} \\Vert (x_{i})_{\\mathcal {V}+} \\Vert ^{\\gamma } \\ge 0 \\right) \\\\\\asymp & n^{-\\frac{2}{2 + k/\\gamma }} \\cdot n^{1-\\frac{k/\\gamma }{2+\\gamma /k}}= O(1).$ To check the Lindeberg condition, note that $\\Vert Z_{ni} \\Vert := \\sup _{C \\in [C_{1}, C_{J}]}\\vert Z_{ni}(C) \\vert \\le \\frac{\\omega (\\delta , C_{J}, C_{J})}{\\delta }\\vert Z_{i} \\vert $ so that $& \\sum _{i=1}^{n}\\mathbf {E} \\Vert Z_{ni} \\Vert {1}( \\Vert Z_{ni} \\Vert > \\eta ) \\\\\\le & \\frac{\\omega _{n}(\\delta , C_{J}, C_{J})}{\\delta }\\sum _{i=1}^{n}\\mathbf {E} \\vert Z_{i} \\vert {1}(\\frac{\\omega _{n}(\\delta , C_{J}, C_{J})}{\\delta }\\vert Z_{i} \\vert > \\eta ) \\\\\\le & \\frac{n \\omega _{n}(\\delta , C_{J}, C_{J})}{\\delta } \\mathbf {E} \\vert Z_{i} \\vert {1}( \\frac{\\omega _{n}(\\delta , C_{J},C_{J})}{\\delta }\\vert Z_{i} \\vert > \\eta ) \\\\= & 2 \\frac{n \\omega _{n}(\\delta , C_{J}, C_{J})}{\\delta } \\mathbf {E}Z_{i} {1}( \\frac{\\omega _{n}(\\delta , C_{J},C_{J})}{\\delta } Z_{i} > \\eta ) \\\\= & 2 \\frac{n \\omega _{n}(\\delta , C_{J},C_{J})}{\\delta }\\phi ( \\frac{\\eta \\delta }{\\omega _{n}(\\delta ,C_{J}, C_{J})} ) \\rightarrow 0.$ We have already shown that the covariance function converges pointwise.", "Hence, we conclude that $\\sum _{i=1}^{n}Z_{ni} $ converges in distribution in $\\ell ^{\\infty }([C_{1}, C_{J}])$ to a tight Gaussian process.", "Moreoever, this limiting distribution does not depend on $\\delta $ ." ], [ "Proof of Theorem ", "For simplicity, write $b_{n} = b n^{-r(\\gamma _{1}, \\gamma _{2})}$ , where $b > 0$ is arbitrary, and define $W_{i,n}(\\gamma _{1}, C_{1}, \\gamma _{2},C_{2}):=\\left(1-\\frac{C_{1}}{b_{n}}\\left\\Vert \\left(X_{i}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}-\\frac{C_{2}}{b_{n}}\\left\\Vert \\left(X_{i}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right)_{+}^{2}.$ Note that $b_{n} \\rightarrow 0$ and $n^{1-\\eta }b_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}} \\rightarrow \\infty $ for some $ \\eta > 0 $ .", "First, we show that, for constants $c_{2,1}^{\\ast }$ and $c_{2,1}^{\\ast }$ that do not depend on $b$ , $\\mathrm {(a)}\\,\\,\\, &\\lim _{n\\rightarrow \\infty }\\frac{1}{nb_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}}\\sum _{i=1}^{n}W_{i,n}(\\gamma _{1},C_{1},\\gamma _{2},C_{2})=c_{1,2}^{\\ast }>0,\\\\&\\lim _{n\\rightarrow \\infty }\\frac{1}{nb_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}}\\sum _{i=1}^{n}W_{i,n}(\\gamma _{2},C_{2},\\gamma _{1},C_{1})=c_{2,1}^{\\ast }>0 \\,\\, , \\text{ and} \\\\\\mathrm {(b)}\\,\\,\\, &\\lim _{n\\rightarrow \\infty }b_{n}^{-1}\\min _{i\\le n}\\left\\lbrace C_{1}\\left\\Vert \\left(X_{i}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}+C_{2}\\left\\Vert \\left(X_{i}\\right)_{\\mathcal {V}c-}\\right\\Vert ^{\\gamma _{2}}\\right\\rbrace = 0 \\\\& \\lim _{n\\rightarrow \\infty }b_{n}^{-1}\\min _{i\\le n}\\left\\lbrace C_{2}\\left\\Vert \\left(X_{i}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{2}}+C_{2}\\left\\Vert \\left(X_{i}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{1}}\\right\\rbrace =0,$ where all equalities hold in an almost sure sense.", "To show (a), take an arbitrary $\\varepsilon >0$ .", "Due to the regularity conditions on $p_{X}(\\cdot )$ and $\\sigma (\\cdot )$ , there exists a neighborhood $\\mathcal {N}_{\\varepsilon }$ of 0 such that $\\vert p_{X}(x) - p_{X}(0)\\vert \\le \\varepsilon $ for all $x \\in \\mathcal {N}_{\\varepsilon }$ .", "Writing $ B_{n}:=\\left\\lbrace x\\in \\mathbb {R}^{k}:b_{n}-C_{1}\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}-C_{2}\\left\\Vert \\left(x\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}>0\\right\\rbrace $ , there exists $N_{\\varepsilon }$ such that for all $n\\ge N_{\\varepsilon }$ we have $B_{n}\\subset \\mathcal {N_{\\varepsilon }} \\cap \\mathcal {X}$ because $b_{n}\\rightarrow 0$ and the interior of $ \\mathcal {X}$ contains $0.$ Hence, for $n\\ge N_{\\varepsilon }$ , we have $\\begin{aligned}& \\,\\,(p_{X}(0)-\\varepsilon )\\int _{B_{n}}\\left(1-\\frac{C_{1}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}-\\frac{C_{2}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right)^{2}dx \\\\\\le & \\,\\,\\operatorname{\\mathbf {E}}W_{i,n}\\\\\\le &\\,\\,\\left(p_{X}(0)+\\varepsilon \\right)\\int _{B_{n}}\\left(1-\\frac{C_{1}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}-\\frac{C_{2}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right)^{2}dx.\\end{aligned}$ Let $\\mathcal {O}$ be the collection of the $2^{k}$ orthants on $\\mathbb {R}^{k}.$ Then, we can write $\\begin{aligned}& \\int _{B_{n}}\\left(1-\\frac{C_{1}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}-\\frac{C_{2}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right)^{2}dx \\\\= & \\sum _{O\\in \\mathcal {O}}\\int _{B_{n}\\cap O}\\left(1-\\frac{C_{1}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}-\\frac{C_{2}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right)^{2}dx.\\end{aligned}$ Now, consider an orthant $O$ and let $O_{+}\\subset \\left\\lbrace 1,\\dots ,k\\right\\rbrace $ be the index set for those elements that take positive values on $O.$ Without loss of generality, suppose $O_{+}\\cap \\mathcal {V}=\\left\\lbrace 1,\\dots ,m\\right\\rbrace $Here, we are implicitly assuming that we modify the definition of the norm in a way that corresponds to the relabeling.", "More formally, we could write the modified norm as $\\Vert \\cdot \\Vert _{O}$ , which we do not do for succinctness.", "Note that this modification is unnecessary when $\\Vert z \\Vert $ is invariant with respect to permutations of $z$ , which is the case for (unweighted) $\\ell _{p}$ norms.", "for $m=0,\\dots ,k$ , where we take $O_{+}\\cap \\mathcal {V}=\\emptyset $ if $m=0.$ For $k_{1}\\le k_{2},$ define the subvector $z_{[k_{1},k_{2}]}=(z_{k_{1},}z_{k_{1}+1},\\dots ,z_{k_{2}})$ for any $z:=(z_{1},\\dots ,z_{k})\\in \\mathbb {R}^{k}$ .", "It follows that $& \\int _{B_{n}\\cap O}\\left(1-\\frac{C_{1}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}-\\frac{C_{2}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right)^{2}dx\\\\= & \\int _{B_{n}\\cap O}\\left(1-\\frac{C_{1}}{b_{n}}\\left\\Vert (x_{[1,m]},0,x_{[k_{+}+1,k]})\\right\\Vert ^{\\gamma _{1}}-\\frac{C_{2}}{b_{n}}\\left\\Vert (0,-x_{[m+1,k_{+}]},-x_{[k_{+}+1,k]})\\right\\Vert ^{\\gamma _{2}}\\right)^{2}dx.$ By applying a changes of variables with $(x_{[1,m]}/b_{n}^{1/\\gamma _{1}},x_{[m+1,k]}/b^{1/\\gamma _{2}})=z$ , the last equation becomes $\\begin{aligned}& \\int _{B_{n}\\cap O}\\left(1-\\frac{C_{1}}{b_{n}}\\left\\Vert (x_{[1,m]},0,x_{[k_{+}+1,k]})\\right\\Vert ^{\\gamma _{1}}-\\frac{C_{2}}{b_{n}}\\left\\Vert (0,-x_{[m+1,k]})\\right\\Vert ^{\\gamma _{2}}\\right)^{2}dx \\\\= &b_{n}^{\\frac{m}{\\gamma _{1}}+\\frac{k-m}{\\gamma _{2}}}\\int _{O}\\left(1-C_{1}\\left\\Vert (z_{[1,m]},0,z_{[k_{+}+1,k]}b_{n}^{\\frac{1}{\\gamma _{2}}-\\frac{1}{\\gamma _{1}}})\\right\\Vert ^{\\gamma _{1}}-C_{2}\\left\\Vert (0,-z_{[m+1,k]})\\right\\Vert ^{\\gamma _{2}}\\right)_{+}^{2}dx.\\end{aligned}$ Note that by Lebesgue's dominated convergence theorem the integral in the last expression can be written as $c_{O}(C_{1,}C_{2})+o(1)$ where $&c_{O}(C_{1},C_{2}) \\\\:=&{\\left\\lbrace \\begin{array}{ll}\\int _{O}\\left(1-C_{1}\\left\\Vert (z_{[1,m]},0)\\right\\Vert ^{\\gamma _{1}}-C_{2}\\left\\Vert (0,-z_{[m+1,k]})\\right\\Vert ^{\\gamma _{2}}\\right)_{+}^{2}dz &\\text{if }\\gamma _{1}>\\gamma _{2}\\\\\\int _{O}\\left(1-C_{1}\\left\\Vert (z_{[1,m]},0,z_{[k_{+}+1,k]})\\right\\Vert ^{\\gamma _{1}}-C_{2}\\left\\Vert (0,-z_{[m+1,k]})\\right\\Vert ^{\\gamma _{2}}\\right)_{+}^{2}dz &\\text{if }\\gamma _{1}=\\gamma _{2}.\\end{array}\\right.", "}$ Hence, we have $\\int _{O}\\left(1-\\frac{C_{1}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}-\\frac{C_{2}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right)_{+}^{2}dx=b_{n}^{\\frac{m}{\\gamma _{1}}+\\frac{k-m}{\\gamma _{2}}}\\left(c_{O}(C_{1},C_{2})+o(1)\\right).$ Moreover, note that $c_{O}(C_{1},C_{2})>0.$ If $\\gamma _{1}>\\gamma _{2}$ , the integrals that correspond to the orthants where $m=k_{+}$ determine the rate at which the entire integral goes to 0.", "If $\\gamma _{1}=\\gamma _{2}$ note that the exponent of $b_{n}$ is always $k/\\gamma _{1}$ and thus the integral is of the same order (in terms of $b_{n}$ ) on all the orthants.", "Let $\\mathcal {O_{+}}$ denote the collection of those orthants with $m=k_{+},$ and write $c_{+}(C_{1},C_{2})=\\sum _{O\\in \\mathcal {O}_{+}}c_{O}(C_{1},C_{2})$ if $\\gamma _{1}>\\gamma _{2}$ and $c_{+}(C_{1},C_{2})=\\sum _{O\\in \\mathcal {O}}c_{O}(C_{1},C_{2})$ if $\\gamma _{1}=\\gamma _{2}$ .", "Then, it follows that $\\int \\left(1-\\frac{C_{1}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}-\\frac{C_{2}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right)_{+}^{2}dx=b_{n}^{\\frac{k_{+}}{\\gamma _{1}}+\\frac{k-k_{+}}{\\gamma _{2}}}\\left(c_{+}(C_{1},C_{2})+o(1)\\right).$ Combining this with (REF ), it follows that $\\begin{aligned}&\\left(c_{+}(C_{1},C_{2})+o(1)\\right)(p_{X}(0)-\\varepsilon )b_{n}^{\\frac{k_{+}}{\\gamma _{1}}+\\frac{k-k_{+}}{\\gamma _{2}}} \\\\\\le & \\operatorname{\\mathbf {E}}W_{i,n} \\\\[-1.5ex]\\le &\\left(c_{+}(C_{1},C_{2})+o(1)\\right)(p_{X}(0)+\\varepsilon )b_{n}^{\\frac{k_{+}}{\\gamma _{1}}+\\frac{k-k_{+}}{\\gamma _{2}}} \\end{aligned}$ for large $n$ .", "Dividing all sides by $b_{n}^{k_{+}/\\gamma _{1}+\\left(k-k_{+}\\right)/\\gamma _{2}},$ taking $n\\rightarrow \\infty ,$ and then taking $\\varepsilon \\rightarrow 0$ , we have $\\lim _{n\\rightarrow \\infty }\\operatorname{\\mathbf {E}}\\frac{W_{i,n}}{b_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}}=c_{+}(C_{1},C_{2})p_{X}(0).$ Now, consider the term $EW^{2}_{i,n}$ .", "We have $& \\,\\,(p_{X}(0)-\\varepsilon )\\int _{B_{n}}\\left(1-\\frac{C_{1}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}-\\frac{C_{2}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right)^{4}dx\\nonumber \\\\\\le & \\,\\,\\operatorname{\\mathbf {E}}W^{2}_{i,n}\\nonumber \\\\\\le & \\,\\,\\left(p_{X}(0)+\\varepsilon \\right)\\int _{B_{n}}\\left(1-\\frac{C_{1}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}-\\frac{C_{2}}{b_{n}}\\left\\Vert \\left(x\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right)^{4}dx.$ Hence, repeating the exact same steps that we went through for $\\operatorname{\\mathbf {E}}W_{i,n}$ , we have $\\lim _{n\\rightarrow \\infty }\\operatorname{\\mathbf {E}}\\frac{W_{i,n}^{2}}{b_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}}=c^{^{\\dagger }}p_{X}(0),$ for some $c^{\\dagger }>0$ , which shows that $(\\operatorname{\\mathbf {E}}{W_{i,n}^{2}})^{1/2} \\asymp {b_{n}^{(k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2})/2}}.$ Now, define $\\widetilde{W}_{n}:= \\frac{1}{n} \\sum _{i=1}^{n} \\left( W_{i,n} - \\operatorname{\\mathbf {E}}W_{i,n}\\right)$ and $\\varepsilon _{n} = \\varepsilon b_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}.$ By Bernstein's inequality, we have $\\operatorname{\\mathbf {P}}( \\vert \\widetilde{W}_{n} \\vert > \\varepsilon _{n} )\\le 2 \\exp \\left( - \\frac{1}{2} \\frac{n \\varepsilon _{n}^{2}}{ \\operatorname{\\mathbf {E}}W_{i,n}^{2}+ \\varepsilon _{n}/3} \\right) \\\\\\le 2 \\exp \\left( - \\frac{1}{2} \\frac{n \\varepsilon _{n}}{ K+ 1/3} \\right)$ where the last inequality holds for large enough $n$ and some constant $K > 0$ .", "It follows that, for large $n$ , $\\exp \\left( - \\frac{1}{2} \\frac{n \\varepsilon _{n}}{ K+ 1/3} \\right) = \\exp \\left( - n^{\\eta } \\frac{1}{2} \\frac{n^{1-\\eta } b_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}} \\varepsilon }{ K+ 1/3} \\right) \\le \\exp \\left( - n^{\\eta } \\right),$ where the inequality follows from the fact that $n^{1-\\eta } b_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}} \\rightarrow \\infty $ .", "This shows that $\\sum _{n=1}^{\\infty } \\operatorname{\\mathbf {P}}( \\vert \\widetilde{W}_{n} \\vert >\\varepsilon _{n} ) < \\infty $ .", "By the Borel-Cantelli lemma, we have $\\frac{1}{nb_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}}\\sum _{i=1}^{n}\\left(W_{i,n}-\\operatorname{\\mathbf {E}}W_{i,n}\\right)\\overset{a.s.}{\\rightarrow }0.$ Combining (REF ) and (REF ), we have $\\lim _{n\\rightarrow \\infty }\\frac{1}{nb_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}}\\sum _{i=1}^{n}W_{i,n}=c_{+}(C_{1},C_{2})p_{X}(0)$ almost surely, which establishes the desired result with $c_{1,2}^{\\ast }=c_{+}(C_{1},C_{2})p_{X}(0)$ .", "Note that $c^{\\ast }_{1,2}$ does not depend on $b$ .", "The proof for $\\lim _{n\\rightarrow \\infty }\\frac{1}{nb_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}}\\sum _{i=1}^{n}W_{i,n}(\\gamma _{2},C_{2},\\gamma _{1},C_{1})=c_{2,1}^{\\ast }>0$ is essentially the same, with some minor modifications.", "The change of variables we previously used should be modified to $(x_{[1,m]}/b_{n}^{1/\\gamma _{2}},x_{[m+1,k_{+}]}/b_{n}^{1/\\gamma _{1}},x_{[k_{+}+1,k]}/b_{n}^{1/\\gamma _{2}})=z,$ and, the constant $c_{+}(C_{1},C_{2})$ should be changed to $c_{-}(C_{1},C_{2}):=\\sum _{O\\in \\mathcal {O}_{-}}c_{O}(C_{2},C_{1})$ where $\\mathcal {O}_{-}$ is the collection of orthants with $m=0.$Again, the norms must be redefined to be consistent with the “relabeling”.", "Hence, here we get the desired result with $c_{2,1}^{\\ast }=c_{-}(C_{1},C_{2})p_{X}(0),$ which again does not depend on $b$ .", "Now, we prove (b).", "We only give the proof for $\\lim _{n\\rightarrow \\infty }b_{n}^{-1}\\min _{i\\le n}\\left\\lbrace C_{1}\\left\\Vert \\left(X_{i}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}+C_{2}\\left\\Vert \\left(X_{i}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right\\rbrace =0 \\,\\, a.s.,$ since the other half of the statement can be proved analogously.", "Let $\\varepsilon >0$ be an arbitrary constant, and denote the event $A_{n,\\varepsilon }:=\\left\\lbrace b_{n}^{-1}\\min _{i\\le n}\\left\\lbrace C_{1}\\left\\Vert \\left(X_{i}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}+C_{2}\\left\\Vert \\left(X_{i}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right\\rbrace \\ge \\varepsilon \\right\\rbrace .$ Note that it is enough to show $\\Sigma _{n=1}^{\\infty }P\\left(A_{n,\\varepsilon }\\right)<\\infty ,$ since then the result follows from the Borel-Cantelli lemma.", "We have $\\operatorname{\\mathbf {P}}\\left(A_{n,\\varepsilon }\\right) & =\\operatorname{\\mathbf {P}}\\left(\\min _{i\\le n}\\left\\lbrace C_{1}\\left\\Vert \\left(X_{i}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}+C_{2}\\left\\Vert \\left(X_{i}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right\\rbrace \\ge b_{n}\\varepsilon \\right)\\\\& =\\operatorname{\\mathbf {P}}(C_{1}\\Vert (X_{i})_{\\mathcal {V}+}\\Vert ^{\\gamma _{1}}+C_{2}\\Vert (X_{i})_{\\mathcal {V}-}\\Vert ^{\\gamma _{2}}\\ge b_{n}\\varepsilon )^{n}\\\\&=(1-\\operatorname{\\mathbf {P}}(C_{1}\\Vert (X_{i})_{\\mathcal {V}+}\\Vert ^{\\gamma _{1}}+C_{2}\\Vert (X_{i})_{\\mathcal {V}-}\\Vert ^{\\gamma _{2}}< b_{n}\\varepsilon ))^{n}.$ By an analogous calculation as in (a), we can show $&\\operatorname{\\mathbf {P}}(C_{1}\\left\\Vert \\left(X_{1}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}+C_{2}\\left\\Vert \\left(X_{2}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}< b_{n}\\varepsilon )\\\\= & b_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}\\left(c+o(1)\\right),$ where $c>0$ and the $o(1)$ term is also positive.", "This gives, for large $n$ and from some positive constant $K > 0$ , $\\operatorname{\\mathbf {P}}(A_{n,\\varepsilon })\\le &\\left(1-cb_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}\\right)^{n}\\\\\\le & \\exp \\left(-cnb_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}\\right) \\\\= &\\exp \\left(-cn^{\\eta }n^{1-\\eta }b_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}\\right) \\\\\\le & \\exp \\left(-cn^{\\eta }K\\right)$ This shows that $ \\sum _{n=1}^{\\infty } \\operatorname{\\mathbf {P}}(A_{n,\\varepsilon }) \\le \\sum _{n=1}^{\\infty }\\exp (-cnb_{n}^{-(k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2})})<\\infty $ , which establishes (b).", "Now, using (a) and (b), we prove the given rate result.", "Let $\\lbrace x_{i}\\rbrace _{i=1}^{\\infty }$ be a realization of $\\left\\lbrace X_{i} \\right\\rbrace _{i=1}^{\\infty }$ such that (a) and (b) hold, which is the case for almost all realizations.", "We prove the result for only $\\omega \\left(\\delta ,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right)\\right)$ because the proof for $\\omega \\left(\\delta ,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right)\\right)$ is essentially the same.", "Throughout the proof, we write $w_{i,n}:=w_{i,n}(\\gamma _{1},C_{1},\\gamma _{2},C_{2})$ for simplicity.", "Define $\\widetilde{\\omega }_{n}(\\delta ):=n^{r(\\gamma _{1},\\gamma _{2})}\\omega \\left(\\delta ,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right)\\right),$ and $\\widetilde{\\omega }_{\\infty }(\\delta )=(\\delta ^{2}/c^{\\ast })^{\\frac{1}{2+k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}}.$ We want to show $\\widetilde{\\omega }_{n}(\\delta )\\rightarrow \\widetilde{\\omega }_{\\infty }(\\delta )$ for all $\\delta >0.$ On the range of $\\widetilde{\\omega }_{n}(\\cdot )$ , define its inverse $\\widetilde{\\omega }_{n}^{-1}(b)$ for $b>0$ : $\\widetilde{\\omega }_{n}^{-1}(b)=\\omega ^{-1}\\left(n^{-r(\\gamma _{1},\\gamma _{2})}b,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right)\\right),$ and let ${b}_{n}=n^{-r(\\gamma _{1}, \\gamma _{2})}b$ .", "It follows that $\\widetilde{\\omega }_{n}^{-1}(b) & =\\Big ({b}_{n}^{2}\\textstyle \\sum \\limits _{i=1}^{n}w_{i,n}\\Big )^{1/2}\\\\& =\\Big (n{b}_{n}^{2+k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}\\frac{1}{n{b}_{n}^{k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}}\\textstyle \\sum \\limits _{i=1}^{n}w_{i,n}\\Big )^{1/2}\\\\& \\rightarrow \\left(b^{2+k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}c_{1,2}^{\\ast }\\right)^{1/2},$ where the last line follows by (a).", "Defining $\\widetilde{\\omega }_{\\infty }^{-1}(b)=\\left(b^{2+k_{+}/\\gamma _{1}+(k-k_{+})/\\gamma _{2}}c_{1,2}^{\\ast }\\right)^{1/2}$ , which is the precisely the inverse function of $\\widetilde{\\omega }_{\\infty }(\\cdot )$ , on an appropriately defined domain.", "Now, if we can show that any $b>0$ is in the range of $\\widetilde{\\omega }_{n}(\\cdot )$ for large enough $n$ , we can apply Lemma F.1 of [2] to establish that $\\widetilde{\\omega }_{n}(\\delta )\\rightarrow \\widetilde{\\omega }_{\\infty }(\\delta )$ for all $\\delta >0$ .", "To this end, it is enough to show $\\lim _{n\\rightarrow \\infty }n^{r(\\gamma _{1},\\gamma _{2})}\\omega \\left(0,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right)\\right)\\rightarrow 0.$ Following the derivation of the solution to the inverse modulus problem, it is easy to check that $\\omega \\left(0,\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{1},C_{1}\\right),\\Lambda _{+,\\mathcal {V}}\\left(\\gamma _{2},C_{2}\\right)\\right)=\\min _{i\\le n}\\left\\lbrace C_{1}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}+C_{2}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right\\rbrace .$ It remains only to show $\\lim _{n\\rightarrow \\infty }n^{r(\\gamma _{1},\\gamma _{2})}\\min _{i\\le n}\\left\\lbrace C_{1}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma _{1}}+C_{2}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma _{2}}\\right\\rbrace =0,$ which is immediate from (b)." ], [ "Heteroskedasticity ", "In Theorem REF , we assume $\\sigma (\\cdot )=1$ .", "However, allowing for general heteroskedasticity do not change the result as long as $\\sigma (\\cdot )$ is continuous at 0 and $\\sigma (0)>0.$ All proofs follow with minor changes.", "The solution to the inverse modulus problem remain unchanged.", "For Theorem REF , we can take $\\varepsilon \\in (0,\\sigma (0))$ and replace the terms $p_{X}(0)-\\varepsilon $ and $p_{X}(0)+\\varepsilon $ by $\\left(p_{X}(0)-\\varepsilon \\right)/$$\\left(\\sigma (0)+\\varepsilon \\right)$ and $\\left(p_{X}(0)+\\varepsilon \\right)/$$\\left(\\sigma (0)-\\varepsilon \\right)$ in (REF ).", "Accordingly, we replace the right-hand side of (REF ) by $cp_{X}(0)/\\sigma (0),$ and the result of the theorem remains the same with a slightly modified definition of the constant terms." ], [ "Adaptation Under Only Monotonicity", "Define the $\\Lambda _{+,\\mathcal {V}}(0,\\infty )$ the space of monotone functions with respect to those variables whose indices lie in $\\mathcal {V}.$ Specifically, $\\Lambda _{+,\\mathcal {V}}(0,\\infty ):=\\left\\lbrace f\\in \\mathcal {F}(\\mathbb {R}^{k}):f(x)\\ge f(z)\\,\\,\\text{if }x_{i}\\ge z_{i}\\text{ }\\forall i\\in \\mathcal {V}\\text{ and}x_{i}=z_{i}\\,\\,\\forall i\\notin \\mathcal {V}\\right\\rbrace .$ Here, we consider the problem of adapting to $\\Lambda _{+,\\mathcal {V}}(\\gamma ,C)$ while maintaining coverage over $\\Lambda _{+,\\mathcal {V}}(0,\\infty ).$ The corresponding inverse (ordered) modulus problem $& \\inf _{f_{1},f_{2}}\\,\\,\\left(\\sum _{i=1}^{n}\\left(f_{2}(x_{i})-f_{1}(x_{i})\\right)^{2}\\right)^{1/2}\\\\\\text{s.t. }", "& \\text{}f_{2}(0)-f_{1}(0)=b,\\,\\,f_{1}\\in \\Lambda _{+,\\mathcal {V}}(\\gamma _{,}C),f_{2}\\in \\Lambda _{+,\\mathcal {V}}(0,\\infty ).$ Let $f_{1}^{}(x)=\\min \\left\\lbrace C\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma },b\\right\\rbrace $ , and $f_{2}^{\\ast }(x) & ={\\left\\lbrace \\begin{array}{ll}b & \\text{if }x_{j}=0\\,\\,\\forall j\\notin \\mathcal {V}\\text{ and }x_{j}\\ge 0\\,\\,\\forall j\\in \\mathcal {V}\\\\\\min \\left\\lbrace C\\left\\Vert \\left(x\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma },b\\right\\rbrace & \\text{otherwise}.\\end{array}\\right.", "}$ First, we argue that $f_{2}^{\\ast }\\in \\Lambda _{+,\\mathcal {V}}(0,\\infty )$ .", "To show this, we must show that for any $x,z\\in \\mathbb {R}^{k}$ , $f_{2}^{\\ast }(x)\\ge f_{2}^{\\ast }(z)\\,\\,\\text{if }x_{j}\\ge z_{j}\\text{}\\forall j\\in \\mathcal {V}\\text{ and }x_{j}=z_{j}\\,\\,\\forall j\\notin \\mathcal {V}.$ Note that this clearly holds if both $x$ and $z$ fall into the first case or second case, respectively, in the definition of $f_{2}^{\\ast }$ .", "Now, suppose $x$ falls into the first case and $z$ into the second.", "Then, it must be the case that $z_{j}\\ne 0$ for some $j\\notin \\mathcal {V}$ or $z_{j}<0$ for some $j\\in \\mathcal {V}.$ If $z_{j}\\ne 0$ for some $j\\notin \\mathcal {V}$ , then the monotonicity condition holds vacuously.", "Suppose $z_{j}=0$ for all $j\\notin \\mathcal {V}$ and $z_{j}<0$ for some $j\\in \\mathcal {V}$ .", "If $x_{j}<z_{j}$ for some $j\\in \\mathcal {V},$ then again the monotonicity condition holds vacuously.", "If $x_{j}\\ge z_{j}$ for all $j\\in \\mathcal {V},$ then the monotonicity condition holds only if $f_{2}^{\\ast }(x)\\ge f_{2}^{\\ast }(z)$ , which is always the case because $f_{2}^{\\ast }(z)\\le b.$ Define $A_{\\mathcal {V}}:=\\left\\lbrace x\\in \\mathbb {R}^{k}:x_{j}=0\\,\\,\\forall j\\notin \\mathcal {V}\\text{ and }x_{j}\\ge 0\\,\\,\\forall j\\in \\mathcal {V}\\right\\rbrace .$ If $\\mathcal {V}\\subsetneq \\left\\lbrace 1,\\dots ,k\\right\\rbrace ,$ then $A_{\\mathcal {V}}$ is a measure zero set under the Lebesgue measureNote that this is not the case when $\\mathcal {V}=\\left\\lbrace 1,\\dots ,k\\right\\rbrace $.", "Hence, under the assumption that the design points are a realization of a random variable that admits a pdf with respect to the Lebesgue measure, we may assume that $x_{i}\\notin A_{\\mathcal {V}}$ for all $i=1,\\dots ,n$ .", "That is, we have $\\omega ^{-1}\\left(b,\\Lambda _{+,\\mathcal {V}}(\\gamma _{,}C),\\Lambda _{+,\\mathcal {V}}(0,\\infty )\\right)=0$ for all $b\\ge 0.$ On the other hand, if $\\mathcal {V}=\\left\\lbrace 1,\\dots ,k\\right\\rbrace $ , we have $& \\omega ^{-1}\\left(b,\\Lambda _{+,\\mathcal {V}}(\\gamma _{,}C),\\Lambda _{+,\\mathcal {V}}(0,\\infty )\\right)\\\\= & \\sum _{i=1}^{n}\\left(1-\\frac{C}{b}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma }\\right)^{2}{1}\\left(b-C\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}+}\\right\\Vert ^{\\gamma }>0,x_{i}\\in O_{+}\\right),$ where $O_{+}=\\left\\lbrace x\\in \\mathbb {R}^{k}:x_{j}>0\\,\\,\\forall j\\right\\rbrace .$ Likewise, we have $& \\omega ^{-1}\\left(b,\\Lambda _{+,\\mathcal {V}}(0,\\infty ),\\Lambda _{+,\\mathcal {V}}(\\gamma _{,}C)\\right)\\\\= & \\sum _{i=1}^{n}\\left(1-\\frac{C}{b}\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma }\\right)^{2}{1}\\left(b-C\\left\\Vert \\left(x_{i}\\right)_{\\mathcal {V}-}\\right\\Vert ^{\\gamma }>0,x_{i}\\in O_{-}\\right),$ where $O_{-}=\\left\\lbrace x\\in \\mathbb {R}^{k}:x_{j}<0\\,\\,\\forall j\\right\\rbrace .$ Hence, in this case, adaptation is possible and resulting CIs end up using only those data with design points that lie in either the positive or negative orthant." ], [ "Definition of the optimal upper CI", "The following corollary summarizes an analogous result for the upper CI.", "Corollary 11.1 Let $\\left(f_{j,\\delta }^{},g_{J,\\delta }^{}\\right)\\in \\mathcal {F}_{j}\\times \\mathcal {F}_{J}$ solve the inverse modulus $\\omega \\left(\\delta ,\\mathcal {F}_{j},\\mathcal {F}_{J}\\right):$ $& \\sum _{i=1}^{n}\\left(g_{J,\\delta }^{}(x_{i})-f_{j,\\delta }^{}(x_{i})\\right)^{2}=\\delta ^{2},\\text{ and}\\\\& Lg_{J,\\delta }^{}-Lf_{j,\\delta }^{}=\\omega \\left(\\delta ,\\mathcal {F}_{j},\\mathcal {F}_{J}\\right)$ with $\\delta =z_{\\beta }+z_{1-\\alpha }$ , and define $\\hat{L}_{\\delta }^{u,j} = & \\frac{Lf_{j,\\delta }^{}+Lg_{J,\\delta }^{}}{2} \\\\& +\\frac{\\omega ^{\\prime }\\left(\\delta ,\\mathcal {F}_{j},\\mathcal {F}_{J}\\right)}{\\delta }\\times \\sum _{i=1}^{n}\\left(g_{J,\\delta }^{}(x_{i})-f_{j,\\delta }^{}(x_{i})\\right)\\left(y_{i}-\\frac{f_{j,\\delta }^{}(x_{i})+g_{J,\\delta }^{*}(x_{i})}{2}\\right).$ Then, $\\hat{c}_{\\alpha }^{u,j}:=\\hat{L}_{\\delta }^{u,j}+\\frac{1}{2}\\omega \\left(\\delta ,\\mathcal {F}_{j},\\mathcal {F}_{J}\\right)-\\frac{1}{2}\\delta \\omega ^{\\prime }\\left(\\delta ,\\mathcal {F}_{j},\\mathcal {F}_{J}\\right)+z_{1-\\alpha }\\omega ^{\\prime }\\left(\\delta ,\\mathcal {F}_{j},\\mathcal {F}_{J}\\right)$ solves $\\underset{\\hat{c}:\\left(-\\infty ,\\hat{c}\\right]\\in \\mathcal {I}_{\\alpha ,1,+}^{J}}{\\min }\\underset{f\\in \\mathcal {F}_{j}}{\\sup }\\,q_{f,\\beta }(\\hat{c}^{U}-Lf).$ Moreover, we have $\\underset{f\\in \\mathcal {F}_{j}}{\\sup }q_{f,\\beta }\\left(Lf-\\hat{c}_{\\alpha }^{\\ell ,j}\\right)\\le \\omega \\left(\\delta ,\\mathcal {F}_{j},\\mathcal {F}_{J}\\right).$ Especially, when $\\beta =1/2$ , we have $\\underset{f\\in \\mathcal {F}_{j}}{\\sup }\\operatorname{\\mathbf {E}}_{f}\\left(\\hat{c}_{\\alpha }^{u,j}-Lf\\right)\\le \\omega \\left(z_{1-\\alpha },\\mathcal {F}_{j},\\mathcal {F}_{J}\\right).$" ] ]	2011.14219
[ [ "Simultaneous recovery of surface heat flux and thickness of a solid\n structure by ultrasonic measurements" ], [ "Abstract This paper is concerned with a practical inverse problem of simultaneously reconstructing the surface heat flux and the thickness of a solid structure from the associated ultrasonic measurements.", "In a thermoacoustic coupling model, the thermal boundary condition and the thickness of a solid structure are both unknown, while the measurements of the propagation time by ultrasonic sensors are given.", "We reformulate the inverse problem as a PDE-constrained optimization problem by constructing a proper objective functional.", "We then develop an alternating iteration scheme which combines the conjugate gradient method and the deepest decent method to solve the optimization problem.", "Rigorous convergence analysis is provided for the proposed numerical scheme.", "By using experimental real data from the lab, we conduct extensive numerical experiments to verify several promising features of the newly developed method." ], [ "Introduction", "The heat conduction is a ubiquitous phenomenon which forms the basis for many practical applications.", "Given the geometrical and material configurations of a material structure as well as the heat source including the initial and boundary temperature distributions, finding the temperature distribution as well as its change on the material structure constitutes the so-called direct or forward heat conduction problem.", "In many practical applications, one is more interested in the so-called inverse heat conduction problem which reverses the above forward problem through direct or indirect measurement data; see e.g.", "[12], [17], [8], [9], [11], [1], [10], [16], [19], [13], [5], [7] and the references cited therein for some related studies in the literature.", "In this paper, motivated by practical applications (with experimental real data from the lab), we are mainly concerned with the reconstruction of the surface heat flux and the thickness of a solid structure by using the associated ultrasonic measurements.", "The reconstruction of the surface heat flux is one of the most typical inverse heat conduction problems, and is widely encountered in aerospace, nuclear physics, metallurgy, and other industrial fields; see [9], [11] and the references cited therein for more related discussions.", "Ultrasonic thickness measurement is a commonly used non-destructive testing method, and is widely used in petroleum, machinery, ship, chemical industry and other fields [13], [5].", "For most of existing results in the literature, one either recovers the surface heat flux by assuming the thickness of the material structure is a priori known, or recovers the thickness of the material structure by assuming the surface heat flux is a priori known.", "However, it is a more practical scenario that both the surface heat flux and the thickness of the material structure are unknown and one recovers both of them.", "In this paper, based on the ultrasonic echo method and the inverse analysis method of the heat conduction, we propose a novel scheme for simultaneously recovering the surface heat flux and the thickness of the material structure through the pulse-echo measurements by the ultrasonic probe.", "The study is posed as an inverse problem associated with a thermoacoustic coupling model.", "We recast the inverse problem as a PDE (partial differential equation)-constrained optimization problem by constructing a proper objective functional.", "We then develop an alternating iteration scheme which combines the conjugate gradient method and the deepest decent method to solve the optimization problem.", "Rigours convergence analysis is provided for the proposed numerical scheme.", "Finally, by using experimental real data from the lab, we conduct extensive numerical experiments to verify effectiveness and efficiency of the method.", "The rest of this paper is organized as follows.", "In Section 2, we present the mathematical formulation of the direct and inverse problems for our study and also briefly discuss the corresponding physical setup.", "In Section 3, we give the optimization formulation of the inverse problem and then derive the alternating iteration scheme for solving the optimization problem.", "Sections 4 and 5 are, respectively, devoted to the theoretical convergence analysis and numerical experiments." ], [ "Mathematical and physical setups", "The physical principle of the ultrasonic thickness measurement is to use the propagation time of the ultrasonic waves in the medium to infer the thickness of the underlying solid structure.", "The propagation time is mainly determined by the thickness, material properties, and internal temperature field of the solid structure; see Figure REF for a schematic illustration of the physical setup.", "The propagation time of the ultrasonic wave in the solid structure can be expressed as (see [18]): Figure: A one-dimensional model based on ultrasonic detection.$\\Lambda _L(t)=2\\int _{0}^{L}\\frac{1}{V(T(x,t))}\\,\\mathrm {d}x,\\quad t\\in (0,+\\infty ),$ where $L\\in \\mathbb {R}_{+}$ denotes the unidirectional propagation distance of the ultrasonic wave in the medium, i.e.", "the thickness of the material structure being under detection.", "Here, $V$ is the propagation velocity of the acoustic wave in the solid medium and is related to the material properties and the structure temperature.", "Usually, it has an approximately linear relationship with the temperature, i.e., $V(T)=aT+b, \\quad a,\\, b \\in \\mathbb {R},$ where $a$ and $b$ are determined by the properties of the material and calibrated by experiments.", "$T(x,t)$ is the internal temperature of the structure, which satisfies the following heat conduction system for $T\\in W_2^{2,1}\\left(\\left[0,L \\right]\\times (0, \\tau )\\right)$ : $\\left\\lbrace \\begin{split}&\\rho c\\frac{\\partial T}{\\partial t}=\\frac{\\partial }{\\partial x}\\left( k\\frac{\\partial T}{\\partial x} \\right),\\qquad \\qquad \\qquad \\qquad \\ \\, (x,t)\\in [0,L]\\times [0,\\tau ],\\\\&-k\\frac{\\partial T}{\\partial x}\\Big \|_{x=0}=q, \\quad -k\\frac{\\partial T}{\\partial x}\\Big \|_{x=L}=0,\\qquad \\quad q(t)\\in L^{2}\\left[0,\\tau \\right],\\\\&T\\mid _{t=0}=T_{0},\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\quad x\\in [0,L],\\end{split}\\right.$ where $k(x,t)$ , $c(x,t)$ and $\\rho (x)$ are the thermal conductivity, specific heat capacity and density of the material, respectively, and $q(t)$ denotes the heat flux density on the boundary.", "In this paper, the inverse problem that we are concerned with is described as follows: Problem 1 Given the measured propagation time of the ultrasonic wave $\\Lambda _m(t)$ and the measured boundary temperature $T_m(L,t)$ , determine the surface heat flux $q(t)$ and the thickness $L$ , i.e., $\\left\\lbrace \\Lambda _{m}\\left(t\\right),T_m(L,t)\\right\\rbrace \\rightarrow \\lbrace q\\left(t\\right),L\\rbrace ,\\quad t\\in \\left[0,\\tau \\right].$ We would like to point out that in the measured data, the temperature at the end of the solid structure can be measured.", "However the thickness $L$ of the structure is unknown in the practical application of our interest.", "It can be directly verified that the inverse problem (REF ) is nonlinear." ], [ "An alternating iteration scheme for the inverse problem", "In this section, we first recast the inverse problem (REF ) as an optimization problem by following the general framework of Tikhonov regularization for inverse problems; see e.g.", "[2] and the references cited therein.", "Then we present the newly proposed alternating iteration scheme.", "To that end, we introduce the following objective functional with respect to the unknown heat flux $q(t)$ and thickness $L$ : $J\\left(q,L\\right)=\\frac{1}{2}\\int _{0}^{\\tau }\\left(\\Lambda _L\\left(t\\right)-\\Lambda _m\\left(t\\right) \\right)^2\\mathrm {d}t +\\frac{\\alpha }{2}\\int _{0}^{\\tau }\\left(T(L,t)-T_m(L,t)\\right)^2\\mathrm {d}t,$ where $\\alpha \\in \\mathbb {R}_+$ signifies a regularization parameter.", "We then recast the inverse problem (REF ) as the following PDE-constrained optimization problem: $\\min _{q\\in L^2(0, \\tau ),\\, L\\in \\mathbb {R}_+} J\\left(q, L\\right)\\ \\ \\mbox{subject to} \\ T\\in W_2^{2,1}\\left(\\left[0,L \\right]\\times (0, \\tau )\\right) \\ \\mbox{satisfying} \\quad (\\ref {eq:master}).$ We next convert the constrained optimization problem into an unconstrained one by using the Lagrange multiplier method.", "Using the heat conduction equation (REF ), with the boundary conditions and the initial condition, the augmented functional is given as follows: $\\begin{aligned}J\\left(q,L\\right)=&\\frac{1}{2} \\int _{0}^{\\tau }\\left(\\Lambda _L\\left(t\\right)-\\Lambda _m\\left(t\\right) \\right)^2\\mathrm {d}t+\\frac{\\alpha }{2} \\int _{0}^{\\tau }\\left(T(L,t)-T_m(L,t)\\right)^2\\mathrm {d}t\\\\&- \\int _{0}^{L} \\int _{0}^{\\tau }\\left\\lbrace \\rho c\\frac{\\partial T(x,t)}{\\partial t}-\\frac{\\partial }{\\partial x}\\left( k\\frac{\\partial T(x,t)}{\\partial x} \\right)\\right\\rbrace \\lambda _1(x,t)\\,\\mathrm {d}t \\mathrm {d}x\\\\&+ \\int _{0}^{\\tau }\\left( k\\frac{\\partial T(x,t)}{\\partial x}+q(t) \\right)\\bigg \|_{x=0} \\lambda _2(t)\\, \\mathrm {d}t+\\int _{0}^{\\tau }\\left(k\\frac{\\partial T(x,t)}{\\partial x}\\right) \\bigg \|_{x=L} \\lambda _3(t)\\, \\mathrm {d}t\\\\&-\\int _{0}^{L} \\bigg (T(x,t)-T_0(x,t)\\bigg )\\bigg \|_{t=0} \\ \\lambda _4(x) \\, \\mathrm {d}x.\\end{aligned}$ where $\\lambda _1(x,t)$ , $\\lambda _2(t)$ , $\\lambda _3(t)$ and $\\lambda _4(x)$ denote the Lagrange multipliers." ], [ "Gradients with respect to the parameters", "To solve the optimization problem (REF ), the gradients of the objective functional $J$ with respective to $q$ and $L$ are required.", "However, it is difficult to solve the gradients directly.", "Thus, we refer to [15] and use the adjoint state method to derive them.", "Noting that $\\begin{aligned}&\\int _{0}^{L} \\int _{0}^{\\tau } \\rho c \\frac{\\partial T(x,t)}{\\partial t} \\lambda _1(x,t) \\,\\mathrm {d}t \\mathrm {d}x\\\\&\\quad =-\\int _{0}^{L} \\int _{0}^{\\tau } \\rho c \\frac{\\partial \\lambda _1(x,t)}{\\partial t} T(x,t) \\,\\mathrm {d}t \\mathrm {d}x+\\int _{0}^{L} \\bigg (\\rho c\\lambda _1(x,t) T\\left(x,t\\right)\\bigg )\\bigg \|_{0}^{\\tau }\\,\\mathrm {d}x.\\end{aligned}$ Similarly, one can deduce that $\\begin{aligned}&\\int _{0}^{L} \\int _{0}^{\\tau } \\frac{\\partial }{\\partial x}\\left( k\\frac{\\partial T(x,t)}{\\partial x} \\right) \\lambda _1(x,t)\\,\\mathrm {d}t \\mathrm {d}x=\\int _{0}^{L} \\int _{0}^{\\tau } \\frac{\\partial }{\\partial x}\\left( k\\frac{\\partial \\lambda _1(x,t)}{\\partial x} \\right) T(x,t) \\,\\mathrm {d}t \\mathrm {d}x\\\\&\\qquad \\qquad \\qquad +\\int _{0}^{\\tau } \\left(k\\lambda _1(x,t)\\frac{\\partial T(x,t)}{\\partial x}\\right)\\bigg \|_{0}^L\\,\\mathrm {d}t\\quad - \\int _{0}^{\\tau } \\left(k\\frac{\\partial \\lambda _1(x,t)}{\\partial x}T(x,t)\\right)\\bigg \|_{0}^L\\,\\mathrm {d}t.\\end{aligned}$ Finally, the equation (REF ) can be rewritten as $\\begin{aligned}J(q,L)=&\\frac{1}{2}\\int _{0}^{\\tau }\\left(\\Lambda _L\\left(t\\right)-\\Lambda _m\\left(t\\right) \\right)^2\\mathrm {d}t+\\frac{\\alpha }{2} \\int _{0}^{\\tau }\\left(T(L,t)-T_m(L,t)\\right)^2\\mathrm {d}t\\\\&+\\int _{0}^{L} \\int _{0}^{\\tau }\\left\\lbrace \\rho c\\frac{\\partial \\lambda _1(x,t)}{\\partial t}+ \\frac{\\partial }{\\partial x}\\left( k\\frac{\\partial \\lambda _1(x,t)}{\\partial x} \\right)\\right\\rbrace T(x,t)\\,\\mathrm {d}t \\mathrm {d}x\\\\&+\\int _{0}^{\\tau } \\left(k\\lambda _1(x,t)\\frac{\\partial T(x,t)}{\\partial x}\\right)\\bigg \|_{0}^L\\,\\mathrm {d}t-\\int _{0}^{\\tau } \\left(k\\frac{\\partial \\lambda _1(x,t)}{\\partial x}T(x,t)\\right)\\bigg \|_{0}^L\\,\\mathrm {d}t\\\\&-\\int _{0}^{L} \\bigg (\\rho c\\lambda _1(x,t) T(x,t)\\bigg )\\bigg \|_{0}^{\\tau }\\,\\mathrm {d}x+ \\int _{0}^{\\tau }\\left( k\\frac{\\partial T(x,t)}{\\partial x}+q(t) \\right)\\bigg \|_{x=0} \\lambda _2(t)\\, \\mathrm {d}t\\\\&+\\int _{0}^{\\tau }\\left(k\\frac{\\partial T(x,t)}{\\partial x}\\right) \\bigg \|_{x=L} \\lambda _3(t)\\, \\mathrm {d}t-\\int _{0}^{L} \\bigg (T(x,t)-T_0(x,t)\\bigg )\\bigg \|_{t=0} \\ \\lambda _4(x) \\, \\mathrm {d}x.\\end{aligned}$ To obtain the adjoint state equation, we set $\\frac{\\partial J}{\\partial T}=0,$ and it yields $\\left\\lbrace \\begin{aligned}& \\rho c\\frac{\\partial \\lambda _1(x,t)}{\\partial t}+ \\frac{\\partial }{\\partial x}\\left( k\\frac{\\partial \\lambda _1(x,t)}{\\partial x} \\right)=S(x,t),\\\\& -k\\frac{\\partial \\lambda _1(x,t)}{\\partial {x}}\\bigg \|_{x=0}=0,\\quad -k\\frac{\\partial {\\lambda _1(x,t)}}{\\partial {x}}\\bigg \|_{x=L}=0,\\\\& \\lambda _1(x,\\tau )=0,\\\\& \\lambda _2(t)=\\lambda _1(0,t),\\\\& \\lambda _3(t)=-\\lambda _1(L,t),\\\\& \\lambda _4(x)=-\\rho c \\lambda _1(x,0).\\end{aligned}\\right.$ Here the source term is given by $S(x,t)=2\\left(\\Lambda _L\\left(t\\right)-\\Lambda _m\\left(t\\right)\\right)\\frac{a}{(V(x,t))^2}+\\frac{\\alpha }{L}\\left(T(L,t)-T_m(L,t)\\right) .$ From (REF ), through a straightforward calculation, the gradients with respect to the model parameters are given by $\\left\\lbrace \\begin{aligned}&\\frac{\\partial J}{\\partial q}(t)=\\lambda _2(t)=\\lambda _1(0,t),\\\\&\\frac{\\partial J}{\\partial L}=\\int _{0}^{\\tau } \\left(\\Lambda _L\\left(t\\right)-\\Lambda _m\\left(t\\right)\\right)\\frac{\\partial \\Lambda _L\\left(t\\right)}{\\partial L}\\, \\mathrm {d}t= \\int _{0}^{\\tau } \\frac{2\\left(\\Lambda _L\\left(t\\right)-\\Lambda _m\\left(t\\right)\\right)}{V(L,t)}\\,\\mathrm {d}t .\\end{aligned}\\right.$ In order to change the final condition to initial condition, a change of variables can be employed : $\\mu (x,t)=\\lambda _1(x,\\tau -t).$ Consequently, the adjoint state equation is rewritten as $\\left\\lbrace \\begin{aligned}& \\rho c\\frac{\\partial \\mu (x,t)}{\\partial t}+ \\frac{\\partial }{\\partial x}\\left( k\\frac{\\partial \\mu (x,t)}{\\partial x} \\right)=S(x,\\tau -t),\\\\& -k\\frac{\\partial \\mu (x,t)}{\\partial {x}}\\bigg \|_{x=0}=0,\\quad -k\\frac{\\partial {\\mu (x,t)}}{\\partial {x}}\\bigg \|_{x=L}=0,\\\\& \\mu (x,0)=0.\\end{aligned}\\right.$ Therefore, according to (REF ), the gradients with respect to $q(t)$ and $L$ can be represented by $\\left\\lbrace \\begin{aligned}&\\frac{\\partial J}{\\partial q}(t)=\\lambda _1(0,t)=\\mu (0,\\tau -t),\\\\&\\frac{\\partial J}{\\partial L}= \\int _{0}^{\\tau } \\left(\\Lambda _L\\left(t\\right)-\\Lambda _m\\left(t\\right)\\right)\\frac{\\partial \\Lambda \\left(t\\right)}{\\partial L}\\, \\mathrm {d}t= \\int _{0}^{\\tau } \\frac{2\\left(\\Lambda _L\\left(t\\right)-\\Lambda _m\\left(t\\right)\\right)}{V(L,t)}\\,\\mathrm {d}t .\\end{aligned}\\right.$ Next, we use the conjugate gradient method and the steepest descent method to identify the heat flux $q(t)$ and the thickness $L$ , respectively." ], [ "Update $q$ with the conjugate gradient method ", "To numerically reconstruct the heat flux $q$ , we shall discretize the heat flux with respect to the time $t$ .", "Suppose that $[0, \\tau ]$ is discretized as follows $0=t_0\\le t_1 \\le \\cdots \\le t_i\\le t_{i+1}\\le \\cdots \\le t_{N}=\\tau .$ The reconstruction schemes of the heat flux based on the conjugate gradient (CG) method is described as follows $q_i^{n+1}=q_i^{n}+\\beta ^{n}p_i^n,$ where the subscript $i$ indicates the discretization of the heat flux in time, and the superscripts $n$ and $n+1$ denote the iteration steps.", "$p_i^n$ signifies the conjugate direction and it is generated by the rule $p_i^n={\\left\\lbrace \\begin{array}{ll}&-g_i^n, \\qquad \\qquad \\quad \\ n=1,\\medskip \\\\&-g_i^{n}+\\alpha ^{n}p_{i}^{n-1}, \\quad n\\ge 2,\\\\\\end{array}\\right.", "}$ where $\\alpha ^{n}$ is the CG update parameter given by $\\alpha ^{n}=\\frac{\\displaystyle \\sum _{i=0}^{N}(g_i^{n})^{T}(g_i^{n}-g_i^{n-1})}{\\displaystyle \\sum _{i=0}^{N}\\left\\Vert g_i^{n-1} \\right\\Vert ^2},$ with $\\left\\Vert \\cdot \\right\\Vert $ denoting the Euclidean norm, and $g_i^{n}=\\frac{\\partial J}{\\partial q}\\Big \|_{t=t_i}^n.$ In addition, the step size $\\beta ^n$ is obtained by the exact line search and can be described as $\\beta ^{n}=\\frac{\\displaystyle \\sum _{i=0}^{N} \\left(\\Lambda ^n_L\\left(t_i\\right)-\\Lambda _m\\left(t_i\\right)\\right) \\Delta t_{in}\\,}{ \\displaystyle \\sum _{i=0}^{N}{\\left[ \\Delta t_{in} \\right]^2}\\,},$ where $\\Lambda ^n_L\\left(t\\right)$ is the solution of the forward problem and $\\Delta t_n$ is the change in transmission time and can be expressed as: $\\Delta t_{n}(t)=\\int _{0}^{L}{\\frac{1}{V_{g^{n}}(x,t)}}\\mathrm {d}x.$ Here $V_{g^n}=aT_{g^n}+b$ and $T_{g^n}(x,t)$ is the solution of the following sensitivity equation $\\left\\lbrace \\begin{aligned}& \\rho c\\frac{\\partial T}{\\partial t}=\\frac{\\partial }{\\partial x}\\left( k\\frac{\\partial T}{\\partial x} \\right),\\\\& -k\\frac{\\partial T}{\\partial x}\\bigg \|_{x=0}=g^n,\\quad -k\\frac{\\partial T}{\\partial x}\\bigg \| _{x=L}=0,\\\\& T\|_{t=0}=T_{0}.\\end{aligned}\\right.$" ], [ "Update $L$ with the steepest descent method ", "The reconstruction of the thickness based on the steepest descent method is described as follows: $L^{n+1}=L^n+\\lambda ^nd^n,$ where the superscripts $n$ and $n+1$ denote the iteration steps, and $d^n$ denotes the negative gradient direction respect to $L$ , $d^n=- \\frac{\\partial J}{\\partial L}\\bigg \|_{L=L^n}=\\int _{0}^{\\tau } \\frac{2\\left(\\Lambda _m(t)-\\Lambda _L(t)\\right)}{V(L^n,t)}\\,\\mathrm {d}t .$ And the step size $\\lambda ^n$ is determined by an inexact line search technique called Wolfe-Powell search method.", "Assuming that $f(L)=J(q,L)$ is differentiable, the Wolfe-Powell search method is used to find $\\lambda ^n$ along $d^n$ such that $\\triangledown f(L^n+\\lambda ^{n}d^{n})^{T}d^{n}\\ge \\sigma \\triangledown f(L^n)^{T}d^{n},$ $f(L^n+\\lambda ^{n}d^{n})\\le f(L^n)+\\rho \\triangledown f(L^n)^{T}d^{n},\\quad \\rho \\in (0,1/2),\\sigma \\in (\\rho ,1).$ Assuming that $\\varphi (\\lambda ^n)=f(L^n+\\lambda ^nd^n)$ , the strategy for computing the step length $\\lambda ^n$ can be described as follows: Step1: Let $\\lambda ^0=0,\\lambda ^{max}>0$ , and choose $\\lambda ^1 \\in [\\lambda ^0,\\lambda ^{max}], \\rho \\in (0,1/2), \\sigma \\in (\\rho ,1)$ .", "Evaluate $\\varphi (\\lambda ^0)$ and $\\varphi ^{^{\\prime }}(\\lambda ^0)$ .", "Let $a_0=\\lambda ^0,b_0=\\lambda ^{max},n=0$ .", "Step2: Evaluate $\\varphi (\\lambda ^n)$ .", "If $\\varphi (\\lambda ^{n})\\le \\varphi (\\lambda ^0)+\\rho \\lambda ^{n}\\varphi ^{^{\\prime }}(\\lambda ^0),$ go to Step3.", "Else, go to Step4, set $a_{n+1}=a_{n},b_{n+1}=\\lambda ^{n}$ .", "Step3: Evaluate $\\varphi ^{^{\\prime }}(\\lambda ^n)$ .", "If $\\varphi ^{^{\\prime }}(\\lambda ^{n})\\ge \\sigma \\varphi ^{^{\\prime }}(\\lambda ^0)$ , stop.", "Else, set $a_{n+1}=\\lambda ^{n},b_{n+1}=b_{n}$ , go to Step4.", "Step4: Let $\\lambda ^{n+1}=\\frac{\\displaystyle a_{n+1}+b_{n+1}}{\\displaystyle 2 }$ , set $n=n+1$ , go to Step2." ], [ "Optimize algorithm iteration format ", "In this paper, we iterate the heat flux $q$ and the thickness $ L$ alternatively, and the proposed algorithm is listed as follows: Step1: Choose an initial point $q_i^0$ , $L^0$ , $\\varepsilon \\in (0,1)$ .", "Step2: Fixed $L^n$ .", "Update $q_i$ using the formular (REF ).", "Step3: Fixed $q_i^{n+1}$ update $L$ using the formular (REF ).", "Step4: Evaluate $J(q_i^{n+1},L^{n+1})$ .", "If $J(q_i^{n+1},L^{n+1})<\\varepsilon $ , stop.", "Else, set $n=n+1$ , go to Step2.", "[h] Alternating iteration algorithm [1] $q^0(N), L^0, crl,n_{max},\\varepsilon $ .", "$q(N)$ , $L$ , $T(nl)$ .", "$q^n\\leftarrow q^0$ , $L^n\\leftarrow L^0$ , $J\\leftarrow J^0$ .", "$abs(J)>crl .AND.", "n<n_{max}$ call gradient $g^n\\leftarrow g1$ , call cgm $p^n\\leftarrow p1$ , call bet $\\beta ^n\\leftarrow \\beta 1$ , $i=1,N$ $q^n(i)\\leftarrow q^n(i)-betp^n(i) $ $g2(i)\\leftarrow g1(i)$ $p2(i)\\leftarrow p1(i)$ $//$ update $q$ $abs(L-j1)>\\varepsilon $ $j1\\leftarrow L$ call wolfe(j) $//$ update $L$ $aa\\leftarrow aa/10$ $//$ update regularization parameter compute objective function $J$ ." ], [ "Convergence Analysis", "In this section, we shall analyze the convergence of the reconstruction scheme proposed in the previous section.", "Let $\\left( q_{i}^{},L^{} \\right)$ be the optimal solution to the optimization problem (REF ), i.e., $J\\left(q_{i}^{},L^{} \\right) \\le J\\left(q_{i}^n, L^n \\right),\\quad \\forall \\, q_{i}^n \\in \\mathbb {R}^{N},\\ L^n\\in \\mathbb {R}.$ It is clear that the necessary condition of (REF ) is: $\\triangledown J\\left( q_{i}^{},L^{}\\right) = \\left( g_{i}^{}, -d^{*}\\right)^{T}=0,$ and hence it is sufficient for us to prove $\\lim _{n\\rightarrow \\infty } \\inf \\Vert \\left( g_{i}^{n}, d^{n} \\right)\\Vert =0.$ Next, we prove that the optimization algorithm that consists of (REF ) and (REF ) satisfies the convergence condition (REF ).", "Before we discuss the convergence, we introduce some notations and important lemmas.", "Definition 4.1 $Polka-Ribi\\acute{e}re-Polyak (PRP)$ method is a nonlinear conjugate gradient method, and it has the following form: $\\begin{aligned}&q^{n+1}=q^{n}+\\beta ^{n}p^{n},\\\\&p^{n}=\\left\\lbrace \\begin{aligned}&-g^{n} , &\\quad n=1, \\\\&-g^{n}+\\alpha ^{n}p^{n-1} , &\\quad n\\ge 2,\\end{aligned}\\right.\\end{aligned}$ where $\\alpha ^{n}_{PRP}=\\frac{g^{nT}(g^{n}-g^{n-1})}{\\left\\Vert g^{n-1} \\right\\Vert ^2}.$ Definition 4.2 Exact line search: at each iteration, the step size $\\beta ^n$ is selected so that $f(q^n+\\beta ^n p^n)=\\underset{\\beta }{\\min } f(q^n+\\beta p^n),$ where the objective functional is $f(q)=J(q,L), q(t)\\in L^{2}\\left(\\left[0,\\tau \\right]\\right)$ .", "Remark 4.1 Iteration algorithm (REF ) is a PRP conjugate method with the exact line search.", "Next, we prove the convergence of the PRP conjugate method with an exact line search as well as its convergence condition.", "To that end, we first derive several auxiliary lemmas.", "Lemma 4.1 [14] Let $\\theta _n$ be the angle between the searching direction $p^n$ and the negative gradient direction $-g^n$ .", "Then $\\cos \\theta _n =\\frac{-g^{nT}p^n}{\\Vert g^n\\Vert \\Vert p^n\\Vert }.$ When the line search is the exact line search, the angle $\\theta _n$ is represented by Figure REF .", "If $\\alpha $ is given by (REF ), we have $\\tan \\theta _{n+1}\\le \\sec \\theta _n \\frac{\\Vert g^{n+1}-g^n\\Vert }{\\left\\Vert g^n\\right\\Vert }.$ Figure: The definition of figureFigure REF gives the equation $\\left\\Vert p^n\\right\\Vert =\\sec \\theta _n\\left\\Vert g^n\\right\\Vert .$ Further, if $n$ is replaced by $n+1$ in Figure REF , we find the identity $\\alpha ^{n+1}\\left\\Vert p^{n}\\right\\Vert =\\tan \\theta _{n+1}\\Vert g^{n+1}\\Vert .$ By (REF ), one has $\\alpha ^{n+1}=\\frac{(g^{n+1})^{T}(g^{n+1}-g^{n})}{\\left\\Vert g^{n}\\right\\Vert ^2},$ and hence by the Cauchy-Schwarz inequality, $\\alpha ^{n+1}\\leqslant \\frac{\\left\\Vert g^{n+1}\\right\\Vert \\left\\Vert g^{n+1}-g^n\\right\\Vert }{\\left\\Vert g^{n}\\right\\Vert ^2}.$ Next, by the elimination of $\\left\\Vert g^{n}\\right\\Vert $ from (REF ) and(REF ), one can show the following equality, $\\alpha ^{n+1}=\\frac{\\tan \\theta _{n+1}\\left\\Vert g^{n+1}\\right\\Vert }{\\sec \\theta _n\\left\\Vert g^n\\right\\Vert },$ which in combination with (REF ) and (REF ) readily yields that $\\tan \\theta _{n+1}\\leqslant \\sec \\theta _n \\frac{\\left\\Vert g^{n+1}-g^n\\right\\Vert }{\\left\\Vert g^n\\right\\Vert }.$ The proof is complete.", "Lemma 4.2 [3] Consider the iterative method of the form $q^{n+1}=q^n+\\beta ^n p^n$ , where $p^n$ satisfies the descent condtion $p^{nT}g^n\\le 0$ , the step size $\\beta ^n$ is obtained by the exact line search, the objective functional $f(q)$ is bounded below, and its gradient $\\triangledown f(q)$ satisfies the Lipschitz condition, then $\\sum _{n\\geqslant 1}\\frac{(g^{nT}p^n)^2}{\\left\\Vert p^n\\right\\Vert ^2}<\\infty ,$ and according to the definition of $\\theta _n$ , (REF ) is equivalent to $\\sum _{n\\geqslant 1}\\left\\Vert g^n\\right\\Vert ^2\\cos \\theta _n <\\infty .$ Lemma 4.3 [4] If the step size $\\beta ^n$ is obtained by the exact line search and the objective function $f(q)$ is uniformly convex, then $f(q^n)-f(q^n+s^n)\\geqslant c\\left\\Vert s^n\\right\\Vert ^2$ holds, where $s^n=q^{n+1}-q^n$ , $c>0$ is a constant, and further there is $\\left\\Vert s^n\\right\\Vert \\rightarrow 0$ as $n\\rightarrow \\infty $ .", "Lemma 4.4 Assume that the objective function $f(q)$ is uniformly convex and bounded from below, and its gradient $\\triangledown f(q)$ satisfies the Lipschitz condition.", "Consider the algorithm (REF )–(REF ), if the step size $\\beta ^n$ is obtained by the exact line search, then $\\lim _{n\\rightarrow \\infty }\\inf \\left\\Vert g^n \\right\\Vert =0.$ The proof follows a similar spirit to that of Theorem 1 in [14].", "By an absurdity argument, we assume that the theorem does not hold.", "Then there is a constant $\\gamma $ , such that for any $n\\geqslant 1$ , $\\left\\Vert g^n\\right\\Vert \\geqslant \\gamma .$ By Lemma REF , there is $\\left\\Vert s^n\\right\\Vert \\rightarrow 0, n\\rightarrow \\infty $ , which combines with the fact that $\\triangledown f(q)$ is Lipschitz continuous implies that there exists a positive integer $m$ , such that $\\left\\Vert g^{n+1}-g^n\\right\\Vert \\geqslant \\frac{1}{2}\\gamma ,$ which holds for any $n\\geqslant m$ .", "Noticing that for any $\\theta _n\\in [0,\\pi /2)$ , there is $\\sec \\theta _n\\leqslant 1+\\tan \\theta _n,$ which together with (REF ), (REF ) and (REF ) further implies that $\\begin{aligned}\\tan \\theta _{n+1}&\\leqslant \\frac{1}{2}+\\frac{1}{4}+\\cdots +(\\frac{1}{2})^{n-1-m}(1+\\tan \\theta _{m})&=1+\\tan {\\theta _m},\\quad \\forall n\\geqslant m.\\end{aligned}$ Therefore, the angle $\\theta _n$ between the searching direction $p_n$ and the negative gradient direction $-g^n$ is always smaller than an angle bounded above by ${\\pi }/{2}$ .", "Therefore, by (REF ), we have $\\left\\Vert g^n\\right\\Vert \\rightarrow 0$ , which is a contradiction to (REF ).", "Thus the theorem must hold true.", "The proof is complete.", "Next, we establish the convergence of the steepest descent algorithm with the Wolfe-Powell step search method.", "Lemma 4.5 [6] Let the objective function $f(L)=J(q,L),L\\in \\mathbb {R}$ be differentiable and bounded from below, and $g(L)=\\triangledown f(L)$ satisfy the Lipschitz condition.", "Suppose that the steepest descent method generates a sequence $L^n, n\\ge 1$ , using the recurrence $L^{n+1}=L^{n}+\\lambda ^nd^n,$ where the direction $d^n$ denotes the negative gradient direction.", "If the step size $\\lambda $ satisfies $\\left\\lbrace \\begin{aligned}&(g^{n+1})^{T}d^{n}\\ge \\sigma (g^n)^{T} d^{n},\\\\&f(L^{n+1})\\le f(L^n)-\\rho \\lambda ^{n}(d^{n})^2,\\quad \\rho \\in (0,1/2),\\sigma \\in (\\rho ,1),\\end{aligned}\\right.$ where $g^{n+1}=g(L^{n+1})=g(L^n+\\lambda ^nd^n)$ , then one has $f(L^n)-f(L^{n+1})\\ge \\frac{\\rho (1-\\sigma )}{M}\\left\\Vert g^n\\right\\Vert ^2\\cos ^2\\theta _n.$ Lemma 4.6 If the function $f(L)$ is continuously differentiable and satisfys the conditions of Lemma REF , then sequence $L^n$ generated by the steepest descent method satisfys: $\\lim _{n\\rightarrow \\infty }\\left\\Vert g^n\\right\\Vert ^2=0.$ By Lemma REF , $f(L^n)-f(L^{n+1})\\ge \\frac{\\rho (1-\\sigma )}{M}\\left\\Vert g^n\\right\\Vert ^2\\cos ^2\\theta _n.$ Thus $\\begin{aligned}&f(L^0)-f(L^{n+1})=[f(L^0-f(L^1)]+[f(L^1)+f(L^2)]+\\cdots +[f(L^{n})-f(L^{n+1})]\\\\&=\\sum _{k=0}^{n}[f(L^{k})-f(L^{k+1})]\\ge \\sum _{k=0}^{n}\\frac{\\rho (1-\\sigma )}{M}\\left\\Vert g^k\\right\\Vert ^2\\cos ^2(d^k,-g^k).\\end{aligned}$ Notice that $\\cos ^2(d_k,-g_k)=1,k=1,2,\\cdots ,{n}$ .", "We therefore have from (REF ) that $\\sum _{k=0}^{n}\\frac{\\rho (1-\\sigma )}{M}\\left\\Vert g_k\\right\\Vert ^2\\le f(L^0)-f(L^{n+1}).$ Since $f(L)$ is bounded from below, one sees that $f(L^0)-f(L^{n+1})< \\infty $ , which together with (REF ) readily implies that $\\lim _{n\\rightarrow \\infty }\\left\\Vert g^n\\right\\Vert ^2=0.$ The proof is complete.", "Next we prove that the optimization algorithm consisting of (REF ) and (REF ) satisfies the convergence condition (REF ).", "Theorem 4.1 Consider the iterative algorithm consisting of (REF ) and (REF ): $\\left\\lbrace \\begin{aligned}&q_i^{n+1}=q_i^n+\\beta ^np^n_i,\\\\&L^{n+1}=L^n+\\lambda ^nd^n,\\end{aligned}\\right.$ Assume that the objective functional $J(q,L)$ satisfies the following conditions: (a): $J(q,L)$ is continuously differentiable with respect to $q(t)$ and $L$ ; (b): $J(q,L)$ is uniformly convex; (c): $J(q,L)$ is bounded from below; (d): Its gradient $\\triangledown J(q,L)$ is Lipschitz continuous.", "Then the optimization algorithm consisting of (REF ) and (REF ) satisfies the convergence condition (REF ), i.e., $\\liminf _{n\\rightarrow \\infty } \\Vert \\left(g_i^n ,d^n\\right)\\Vert _1=0,$ where $\\Vert \\cdot \\Vert _1$ denotes the $l^1$ -norm, namely $\\Vert \\left(g_i^n ,d^n\\right)\\Vert _1=\\Vert g_i^n \\Vert _1+\|d^n\|.$ We first consider the iterative algorithm (REF ).", "Since the step size $\\lambda ^{n}$ is searched by the Powell-Wolfe method, it satisfies the following conditions: $\\begin{aligned}&\\varphi (\\lambda ^{n})\\le \\varphi (0)+\\rho \\lambda ^{n}\\varphi ^{^{\\prime }}(0),\\\\&\\varphi ^{^{\\prime }}(\\lambda ^{n})\\ge \\sigma \\varphi ^{^{\\prime }}(0),\\quad \\rho \\in (0,1/2),\\sigma \\in (\\rho ,1),\\end{aligned}$ where $\\varphi (\\lambda ^{n})=J(q^n, L^n+\\lambda ^{n}d^{n})$ .", "By using conditions (a) (c) (d) and (REF ), we see that Lemma REF holds.", "Hence by Lemma REF , we have $\\lim _{n\\rightarrow \\infty }\\left\| d^n \\right\|=0.$ We proceed to consider the iterative algorithm (REF ).", "By virtue of the conditions (a) (b) (c) (d), we see that the iteration algorithm (REF ) is a PRP conjugate method with the exact line search.", "By Lemma REF , $\\lim _{n\\rightarrow \\infty }\\inf \\left\\Vert g_i^n \\right\\Vert =0,$ According to (REF ) and (REF ), $\\lim _{n\\rightarrow \\infty }\\inf \\left(\\Vert g_i^n \\Vert +\|d^n\|\\right)=0.$ Thus, $\\lim _{n\\rightarrow \\infty }\\inf \\Vert \\left(g_i^n ,d^n\\right)\\Vert _1=\\lim _{n\\rightarrow \\infty }\\inf \\left(\\Vert g_i^n \\Vert +\|d^n\|\\right)=0.$ The proof is complete." ], [ "Numerical Examples", "In this section, we present several numerical examples to verify the effectiveness and robustness of the proposed scheme in simultaneously reconstructing the surface heat flux and the thickness of a solid structure under different acoustic time accuracies, initial fluxes, and initial thicknesses.", "It is emphasized that all the data in our numerical examples are collected by lab experiments following the setup described in Figure REF .", "The specimens with a thickness of $L=50\\, \\rm {mm}$ are heated at one boundary and the surface heat flux is $q(t)=10^5\\, \\rm {J/s}$ .", "The ultrasonic wave probes are stalled on the other boundary of the specimens with the detection frequency set to be $\\omega =1\\, \\mathrm {Hz}$ , and the total detection time set to be $\\tau =500\\,\\rm {s}$ .", "Moreover, the thermal conductivity of the specimens is $k=50\\,\\rm {W/(m\\cdot ^\\circ \\!C)}$ , specific heat is $c=400\\, \\rm {J/(kg\\cdot ^\\circ \\!", "C)}$ and density of the material is $\\rho =7800\\, \\rm {kg/m^3}$ .", "The initial tempareture is chosen as $T_0=\\rm {26\\, ^\\circ C}$ .", "The relationship between the velocity and temperature is given as follows: $V(T)=-0.4521T+3259.9.$ In the following numerical examples, the stopping criterion for the iterations is set to be $J(q, L)<5\\times 10^{-18}$ .", "The Fortran software is used for implementing of Algorithm 1.", "The reconstruction results of thickness under the acoustic time accuracy of $10^{-9}$ , $10^{-10}$ and $10^{-11}$ are respectively shown in the Table REF .", "Table: Convergence of the iteration method with different initial guesses and measurement errors.It can be found that the thickness can be reconstructed effectively under different acoustics time accuracies.", "The error and the number of iterations show that under the same initial value, the accuracy of acoustic time will affect the convergence speed of the algorithm.", "If the accuracy of acoustic time reaches $10^{-10}$ or $10^{-11}$ , one can achieve much accurate reconstruction results.", "Thus, in the following numerical examples, we adopt the measurement data with an acoustic time accuracy of $10^{-10}$ or $10^{-11}$ to study the effect of the initial values on the inversion procedure.", "Figure REF presents the reconstruction results of the heat flux under different initial thicknesses and different initial heat flux conditions.", "The acoustic accuracy is fixed to be $10^{-10}$ .", "Figure: Figure REF shows the reconstruction results of the heat flux under different initial thicknesses and different initial heat flux conditions.", "The acoustic accuracy is fixed to be $10^{-11}$ .", "Figure: By observing the surface heat flux reconstruction results in Figures REF and REF , it can be found that when the acoustic time accuracy is $10^{-10}$ and $10^{-11}$ , under different initial conditions, the inversion value of the heat flux converges to the real value.", "Moreover, when the acoustic time accuracy reaches $10^{-11}$ , the inversion value of heat flux very close to the real value, which achieves a much accurate reconstruction.", "Table REF lists the inversion results and the iteration times of the thickness under different initial thicknesses and different initial heat fluxes initial conditions.", "The acoustic time accuracy is $10^{-10}$ or $10^{-11}$ .", "Table: Convergence of the proposed iteration method with different initial guesses and measurement errors.The results show that the iterative algorithm converges very fast and robust with different initial conditions." ], [ "Conclusion", "Based on the ultrasonic echo method and the inverse problem analysis method of the heat conduction, combined with the optimization model, a method of simultaneously reconstructing the thickness and the surface heat flux of a solid structure is established in this paper.", "This approach is non-destructive and non-contact and it can be used to recover the surface heat flux and the wall thickness at the same time.", "It possesses a high engineering value.", "We provide a rigorous convergence analysis of the proposed numerical scheme.", "By using experimental lab data, we conducted extensive numerical experiments to verify the effectiveness and efficiency of the newly developed method.", "It is found that as long as the acoustic time accuracy reaches $10^{-10}$ or $10^{-11}$ , the proposed iteration method converges very fast and robust with respect to different initial guesses." ], [ "Acknowledgement", "The work of H Liu was supported by a startup fund from City University of Hong Kong and the Hong Kong RGC General Research Fund (projects 12301420, 12302919, 12301218)." ] ]	2011.14228
[ [ "One Explicitly Solvable Model For The Galton-Watson Processes In the\n Random Environment" ], [ "Abstract In this paper, we study the Galton-Watson process in the random environment for the particular case when the number of the offsprings in each generation has the fractional linear generation function with random parameters.", "In this case, the distribution of $N_t$, the number of particles at the moment time $t=0,1,2,\\cdots$ can be calculated explicitly.", "We present the classification of such processes and limit theorems of two types: quenched type which is for the fixed realization of the random environment and annealed type which includes the averaging over the environment." ], [ "Introduction", "Galton-Watson process in the random environments was intensively discussed in modern probability field ([1][2][7][8]).", "In this paper, we will study one particular model with single-type particles, the discrete time $t=0,1,2,\\cdots $ and the fractional linear generating function for the offspring distribution.", "The fractional linear probability generating function for branching process in the random environment first time was discussed probably in 1994 by D.R.", "Grey and Lu Zhunwen([4]),but they focused on the super-critical Smith-Wilkinson branching process and studied the case of 2D environments especially.", "In 2006, A. Joffe and G. Letac ([5]) studied the multitype branching process with linear fractional generating functions and the same denominator.", "In 2019, V.A.", "Vatutin and his group ([9]) investigated the limit behavior of only super-critical multitype branching processes in random environments with linear fractional offspring distributions but this study didn't contain the detailed categorization for such type branching processes.", "Let's give the description of our model.", "The probability space $(\\Omega ,\\mathcal {F},P)$ in our case is the skew product of two spaces $\\Omega =\\Omega _m\\times \\Omega _{br}({\\omega _m})$ .", "$\\Omega _m$ is the space of medium or environment.", "$\\Omega _{br}({\\omega _m})$ is the space containing the information of the branching and annihilating particles for fixed environment ${\\omega _m}$ .", "Let $N_t$ be the number of the particles at the moment $t$ , $t=0,1,2,\\cdots $ .", "$N_0=1$ and transition from $N_{t-1}$ to $N_t$ is given by the standard formula: $N_t=\\sum \\limits _{i=1}^{N_{t-1}}\\xi _{t,i}({\\omega _m})$ Here $\\xi _{t,i}({\\omega _m})$ are independent copies of the random variable $\\xi _t({\\omega _m})$ , which gives the number of offsprings generated by different particles (among $N_{t-1}$ ) at the moment $t$ .", "That is $P(\\xi _t=k)=p_k(\\omega _t).\\,\\,k=0,1,2,\\cdots ,\\,\\,\\sum \\limits _{k=0}^{\\infty }p_k=1$ The distribution of $\\xi _{t,i}$ depends on some parameters $\\vec{\\pi }_t$ selected randomly and independently at the moments $t=1,2,\\cdots $ .", "One can identify ${\\omega _m}$ with the sequence $\\lbrace \\vec{\\pi }_t,t=1,2,3,\\cdots \\rbrace $ .", "The elementary events associated with the non-homogeneous Galton-Watson process $N_t$ for fixed ${\\omega _m}$ , we denote as $\\omega _{br}({\\omega _m})$ .", "They form the second component $\\Omega _{br}({\\omega _m})$ in our skew product $\\Omega =\\Omega _m \\times \\Omega _{br}({\\omega _m})$ .", "For each random variable $\\eta ({\\omega _m},\\omega _{br})$ on the total probability space $\\Omega $ , one can introduce two types expectations and probabilities: Quenched type We call the following integral the quenched expectation: $E\\eta =\\int _{\\Omega _{br}({\\omega _m})}\\eta ({\\omega _m},\\omega _{br})d P(d{\\omega _m})$ Here $P^{{\\omega _m}}$ is the conditional distribution of the process $N_t,t\\ge 0$ for the fixed ${\\omega _m}$ .", "Similarly, say $P(\\eta <x)=E\\mathbf {1}_{\\eta <x}$ the quenched probability.", "The quenched expectations or probabilities are the random variables on $\\Omega _m$ .", "Annealed type Additional integration over $\\Omega _m$ leads to the total expectation and probability, we will use in this case notations $\\mathcal {E}$ and $\\mathcal {P}$ for expectation and probability.", "$\\mathcal {E}\\eta ({\\omega _m},\\omega _{br})=\\int _{\\Omega _m}E\\eta \\mathcal {P}(d {\\omega _m})$ Under the fixed environment realization ${\\omega _m} \\in \\Omega _m$ , the probability generating function of the offspring distribution is $\\varphi _t(z,{\\omega _m})=Ez^{\\xi _t({\\omega _m})}$ and the probability generating function of $N_t$ is defined as $\\Phi _t(z,{\\omega _m})=Ez^{N_t}$ , then $\\Phi _t(z,{\\omega _m})=\\underbrace{\\varphi _1(\\varphi _2\\cdots \\varphi _t(z,{\\omega _m}))}_{t-fold}$ .", "We will study the very particular case when $\\varphi _t(z,{\\omega _m})$ is the fractional linear function: $\\varphi _t(z,{\\omega _m})=\\frac{a_t z+b_t}{c_t z+d_t}$ This fractional linear function is associated with matrix $\\mathbf {I}_t=\\begin{pmatrix}a_t & b_t \\\\c_t & d_t\\end{pmatrix}$ We will assume that $\\det \\mathbf {I}_t>0$ .", "Such fractional linear functions (Möbius transforms) form a group with operation of composition and unity $\\varphi (z)=z$ .", "The mapping $\\varphi (z)=\\displaystyle \\frac{az+b}{cz+d}\\rightarrow \\mathbf {I}=\\begin{pmatrix}a & b \\\\c & d\\end{pmatrix}$ is not one-to-one since $\\mathbf {I}$ and $c\\mathbf {I}$ present the same function $\\varphi (z)$ , but if we impose the condition $\\det \\mathbf {I}=1$ , then the group of the Möbius transforms will be isomorphic to the group $SL(2,R)$ , this group contains all $2\\times 2$ real matrices with determinant 1.", "But we will not use this fact.", "As easy to prove the function $\\varphi (z)=\\displaystyle \\frac{az+b}{cz+d}$ is the generating function of the random variable $\\xi $ with values $0,1,2,\\cdots n$ .", "if and only if it can be presented in the form $\\varphi _t(z)=\\frac{(p-\\alpha )z+\\alpha }{(p-1)z+1}\\,\\,t=1,2,3,\\cdots $ whose associated matrix is $\\mathbf {I}=\\begin{pmatrix}p-\\alpha & \\alpha \\\\p-1 & 1\\end{pmatrix}$ where $(p,\\alpha )\\in [0,1]^2$ .", "We will consider only non-degenerated case $(p,\\alpha )\\in (0,1)^2$ .", "In this case, $\\xi $ has the generalized geometric law: $P(\\xi =0)=\\alpha , P(\\xi =k)=(1-\\alpha )p(1-p)^{k-1}, k\\ge 1$ and $\\varphi _t(z,{\\omega _m})=\\frac{(p_t-\\alpha _t) z+\\alpha _t}{(p_t-1) z+1}$ with the associated independent matrices $\\mathbf {I}=\\begin{pmatrix}p_t-\\alpha _t & \\alpha _t \\\\p_t-1 & 1\\end{pmatrix},\\,\\,\\det \\mathbf {I}=p_t(1-\\alpha _t)$ One can identify ${\\omega _m}$ with sequence of the independent, identically distributed vectors $(p_t,\\alpha _t,t\\ge 1)={\\omega _m}$ .", "In the future, we will denote $\\beta _t=1-\\alpha _t$ , $q_t=1-p_t$ .", "Section 2 in this paper will contain the calculation of the generating function $\\Phi _t(z,{\\omega _m})=Ez^{N_t}$ and related objects such as quenched moments etc.", "In the section 3, we will give the classification of Galton-Watson processes in the random environment.", "It contains five categories, the standard super-critical and the critical cases of the classical homogeneous Galton-Watson theory will be divided into two sub-categories each.", "In the section 4, we will prove several quenched and annealed limit theorems using the asymptotic formulas for the random geometric progressions.", "We discuss the relation of our topic with the problem of the evaluation of the mass of cells in the one cell population (like plankton)." ], [ "Analysis of the fractional linear model", "Let us calculate the generating function $\\Phi _t(z,{\\omega _m})$ .", "In our case,the matrix $\\mathbf {I}_t({\\omega _m})$ associated with the generating function $\\varphi _t(z,{\\omega _m})=\\displaystyle \\frac{(p_t-\\alpha _t)z+\\alpha _t}{(p_t-1)z+1}$ : $\\mathbf {I}_t(\\omega )=\\begin{pmatrix}p_t-\\alpha _t & \\alpha _t \\\\p_t-1 & 1\\end{pmatrix}$ The fractional-linear (Möbius) transform functions form a group and $\\Phi _t(z,{\\omega _m})=Ez^{N_t({\\omega }_m)}=\\underbrace{\\varphi _1(\\varphi _2\\cdots \\varphi _t(z,{\\omega }_m))}_{t-fold}=\\frac{A_tz+B_t}{C_tz+D_t}$ and the associated matrix for $\\Phi _t(z,{\\omega _m})$ is $\\Pi _t({\\omega }_m)=\\begin{pmatrix}A_t & B_t \\\\C_t & D_t\\end{pmatrix}=\\mathbf {I}_1\\mathbf {I}_2\\cdots \\mathbf {I}_t({\\omega }_m)$ Note the determinant of $I_t$ is $det(\\mathbf {I}_t)=p_t(1-\\alpha _t)$ , thus the determinant of $\\Pi _t({\\omega }_m)=\\displaystyle \\prod \\limits _{i=1}^{t}p_i\\prod \\limits _{i-1}^t \\alpha _i$ .", "And $E[\\xi _t({\\omega }_m)]=\\varphi ^{\\prime }_t(1)=\\displaystyle \\frac{1-\\alpha _t}{p_t}$ , from $N_t=\\sum \\limits _{i=1}^{N_{t-1}}\\xi _{t,i}({\\omega _m})$ , we can get $E[N_t]=\\prod \\limits _{i=1}^tE[\\xi _{t,i}({\\omega _m})]=\\displaystyle \\prod \\limits _{i=1}^t \\frac{1-\\alpha _i}{p_i}=e^{\\sum \\limits _{s=0}^t\\ln \\frac{\\beta }{p}(s,{\\omega }_m)}$ for the fixed environment realization ${\\omega _m}$ .", "In the exponent in the last formula, we have the sum of i.i.d random variables on $\\Omega $ .", "Usually, we will assume that $\\mathcal {E}\\ln \\displaystyle \\frac{\\beta }{p}(\\cdot )<\\infty $ , but some exclusions from this assumption are also possible, we will discuss them later as well as the case $\\mathcal {E}\\displaystyle \\ln ^2\\frac{\\beta }{p}(\\cdot )=\\infty $ .", "Now we will calculate the product $\\Pi _t$ ,i.e.", "$\\Phi _t(z,{\\omega }_m)=\\underbrace{\\varphi _1(\\varphi _2\\cdots \\varphi _t(z,{\\omega }_m))}_{t-fold}$ .", "Let us introduce special $2\\times 2$ matrices: $\\mathbf {\\Sigma }_1=\\begin{pmatrix}-1 & 1\\\\-1 & 1\\end{pmatrix}$ , $\\mathbf {\\Sigma }_2=\\begin{pmatrix}1 & 0\\\\1 & 0\\end{pmatrix}$ , $\\mathbf {\\Sigma }_3=\\begin{pmatrix}1 & -1\\\\0 & 0\\end{pmatrix}$ The following table shows the multiplication results of any two matrices of the above.", "Table: NO_CAPTIONwhere $\\mathbf {O}$ is a $2\\times 2$ matrix with all elements 0.", "One can decompose $\\mathbf {A}_t$ into the following form: $\\mathbf {A}_t=\\begin{pmatrix}-1 & 1\\\\-1 & 1\\end{pmatrix}+p_t\\begin{pmatrix}1 & 0\\\\1 & 0\\end{pmatrix}+\\beta _t\\begin{pmatrix}1 & -1\\\\0 & 0\\end{pmatrix}=\\mathbf {\\Sigma }_1+p_t\\mathbf {\\Sigma }_2+\\beta _t\\mathbf {\\Sigma }_3$ Then $\\Pi _t=\\mathbf {A}_1\\mathbf {A}_2\\cdots \\mathbf {A}_t=a_t\\mathbf {\\Sigma }_1+b_t\\mathbf {\\Sigma }_2+c_t\\mathbf {\\Sigma }_3$ and t+1=at+11+bt+12+ct+13 =t(1+pt+12+t+13) =(at1+bt2+ct3)(1+pt+12+t+13) =(bt+att+1-btt+1)1+btpt+12+ctt+13 It leads to the following iterated equations: at+1=bt+att+1-btt+1 bt+1=btpt+1 ct+1=ctt+1 From the () and (), $b_t=\\prod \\limits _{i=1}^tp_i$ $c_t=\\prod \\limits _{i=1}^t\\beta _i$ Substitute (REF ) and (REF ) into (), we can get $a_{t+1}=\\prod \\limits _{i=1}^tp_i+a_t\\beta _{t+1}-\\prod \\limits _{i=1}^tp_i\\beta _{t+1}$ Without loss of generality, we assume $a_1=1$ and $\\beta _1=1$ .", "We can get now the formula for $a_t$ : at=i=1t-1pi(1-t)+t(1-t-1)i=1t-2pi+tt-1(1-t-2)i=1t-3pi +tt-1t-2(1-t-3)i=1t-4pi++tt-1t-22 With (),(REF ) and (REF ), $\\mathbf {\\Pi }_t=\\begin{pmatrix}-a_t+\\displaystyle \\prod \\limits _{i=1}^tp_i+\\displaystyle \\prod \\limits _{i=1}^t\\beta _i & a_t-\\displaystyle \\prod \\limits _{i=1}^t\\beta _i\\\\-a_t+\\displaystyle \\prod \\limits _{i=1}^tp_i & a_t\\end{pmatrix}=\\begin{pmatrix}A_t & B_t\\\\C_t & D_t\\end{pmatrix}$ The determinant of $\\mathbf {\\Pi }_t$ is $det(\\mathbf {\\Pi }_t)=\\displaystyle \\prod \\limits _{i=1}^tp_i\\prod \\limits _{i=1}^t\\beta _i$ We know $\\mathbf {\\Pi }_t$ is the associated matrix for the probability generating function $\\Phi _t(z,\\omega )=\\displaystyle \\frac{A_tz+B_t}{C_tz+D_t}$ .", "The probability of the population to extinct at time $t$ is $\\pi (t)=P(N_t=0)=\\Phi _t(0,\\omega )=\\frac{B_t}{D_t}=1-\\displaystyle \\frac{\\displaystyle \\prod \\limits _{i=1}^t\\beta _i}{a_t}$ In other words, the survival probability of the population at time $t$ is $P(N_t>0)=1-\\pi (t)=\\displaystyle \\frac{\\displaystyle \\prod \\limits _{i=1}^t\\beta _i}{a_t}=\\frac{1}{S_t}$ where $S_t=\\displaystyle 1+\\frac{1-\\beta _1}{\\beta _1}+\\frac{1-\\beta _2}{\\beta _2}\\frac{p_1}{\\beta _1}+\\cdots \\frac{1-\\beta _t}{\\beta _t}\\prod \\limits _{j=1}^{t-1}\\frac{p}{\\beta }(j)$ .", "Using these notations, we can define another matrix associated with extinction probability: $\\widetilde{\\Pi _t}=\\displaystyle \\frac{1}{a(t)}\\Pi _t=\\begin{pmatrix}\\rho -\\pi & \\pi \\\\\\rho -1 & 1\\end{pmatrix}(t)=\\begin{pmatrix}-1+\\displaystyle \\frac{\\displaystyle \\prod \\limits _{i=1}^t \\displaystyle \\frac{p_i}{\\beta _i}}{S_t}+\\displaystyle \\frac{1}{S_t} & 1-\\displaystyle \\frac{1}{S_t}\\\\-1+\\displaystyle \\frac{\\displaystyle \\prod \\limits _{i=1}^t\\displaystyle \\frac{p_i}{\\beta _i}}{S_t} & 1\\end{pmatrix}$ where $\\rho (t)=\\displaystyle \\frac{\\displaystyle \\prod \\limits _{i=1}^t p_i}{a_t}=\\displaystyle \\frac{\\displaystyle \\prod \\limits _{i=1}^t \\displaystyle \\frac{p_i}{\\beta _i}}{S_t}$ Thus $\\rho (t)=\\displaystyle \\frac{1}{\\displaystyle \\sum \\limits _{k=1}^{t-1}\\displaystyle \\frac{1-\\beta _{t-k+1}}{\\beta _{t-k+1}} \\prod \\limits _{i=t-k+1}^{t}\\displaystyle \\frac{\\beta _i}{p_i}+\\prod \\limits _{i=1}^t\\frac{\\beta _i}{p_i}}$ The probability that we have $k$ particles at time $t$ in the fixed random environment $\\omega _t$ is $P(N_t=k)=(1-\\pi (t))\\rho (t)(1-\\rho (t))^{k-1}$ , $k\\ge 1$ .", "Recall $\\Phi _t(z)$ is the probability generating function of $N_t$ under the fixed random environment.", "$\\Phi _t(z)=\\displaystyle \\frac{(\\rho -\\pi )z+\\pi }{(\\rho -1)z+1}=\\sum \\limits _{k=0}^{\\infty }P(N_t=k)z^k$ .", "This power series is convergent absolutely for all $\|z\|<1$ .", "Thus $\\Phi _t(z)-\\pi (t)=\\displaystyle \\frac{\\rho (t)(1-\\pi (t))z}{(\\rho (t)-1)z+1}=\\rho (t)(1-\\pi (t))z\\sum \\limits _{n=0}^{\\infty }(1-\\rho (t))^nz^n$ .", "Thus $\\Phi _t(z)=\\pi (t)+\\sum \\limits _{n=0}^{\\infty }(1-\\pi (t))\\rho (t)(1-\\rho )^nz^{n+1}$ In other words, $P(N_t=k)=(1-\\pi (t))\\rho (t)(1-\\rho (t))^{k-1}$ , $k\\ge 1$ .", "Assume $\\rho (t,{\\omega _m})=\\displaystyle \\frac{\\displaystyle \\prod \\limits _{i=1}^t \\displaystyle \\frac{p_i}{\\beta _i}}{S_t({\\omega _m})}\\xrightarrow[t\\rightarrow \\infty ]{} 0$ $P_m-a.s.$ Then for $a\\ge 0$ $P\\left(\\frac{N(t)}{E[N_t\|N_t\\ge 1]}>a\|N(t) \\ge 1\\right)\\xrightarrow[t\\rightarrow \\infty ]{} e^{-a}\\,\\,\\, P_m-a.s.$ $P(N_t=k\|N_t\\ge 1)=\\frac{(1-\\pi (t))\\rho (t)(1-\\rho (t))^{k-1}}{1-\\pi (t)}=\\rho (t)(1-\\rho (t))^{k-1}$ , $k\\ge 1$ .", "This is the pure geometric law.", "Thus $E[N_t\|N_t\\ge 1]=\\displaystyle \\frac{1}{\\rho (t)}$ and as $\\rho \\xrightarrow[t\\rightarrow \\infty ]{} 0$ P(N(t)E[Nt\|Nt1]>a\|N(t) 1)=P(Nt>a(t))=(1-(t))a(t)e-a" ], [ "Classification of Galton-Watson Process in Random Environment", "In the classical homogeneous theory, the classification of Galton-Watson process has three classes: super-critical process if $E\\xi =a>1$ , where $\\xi $ is the number of the offsprings $E[N_t]=a^t,a>1$ , In other words, $E[N_t] \\rightarrow \\infty $ exponentially fast as $t\\rightarrow \\infty $ critical process if $E[N_t]=1$ sub-critical process if $E\\xi <1$ (we assume that $E\|\\xi \|<\\infty $ ) Then see B. Sevastyanov [6], we have the following classical results: If $a>1$ then $\\displaystyle \\frac{N_t}{EN_t}=\\frac{N_t}{a^t}\\rightarrow N_{\\infty }^$ in law and $P(N_{\\infty }^=0)=\\alpha <1$ , $\\alpha =\\lim \\limits _{t\\rightarrow \\infty }P(N_t=0)$ The extinction probability $\\alpha $ is the single root of the equation $\\alpha =\\varphi (\\alpha )$ , $\\alpha <1$ .", "If $E\\xi =a=1$ (critical case), then $P(N_t=0)\\rightarrow 1$ as $t\\rightarrow \\infty $ and $P(\\displaystyle \\frac{N_t}{t}>a\|N_t\\ge 1)\\rightarrow e^{-ca}$ for an appropriate constant $c>0$ .", "If $E\\xi =a<1$ and for $c\\in (a,1)$ , $EN_t=a^t$ ,$P(N_t > c^t)\\le \\frac{a^t}{c^t}$ , $\\sum \\limits _t P(N_t\\ge c^t)<\\infty $ , that is , population degenerates very fast.", "Similar classfication for the Galton-Watson processes in the random environment is more complicated." ], [ "Super-critical Galton-Watson process in random environment", "Like in the classical situation, we call the branching process $N_t({\\omega _m})$ in the random environment supercritical if $\\mathcal {E}(\\ln \\displaystyle \\frac{\\beta }{p})=\\gamma >0$ .", "Then $E[N_t]=\\prod \\limits _{i=1}^t\\displaystyle \\frac{\\beta _i}{p_i}=e^{\\sum \\limits _{i=1}^t\\ln ( \\displaystyle \\frac{\\beta _i}{p_i})}=e^{t\\mathcal {E}(ln \\displaystyle \\frac{\\beta }{p})+o(t)}$ due to the Strong Law of Large Numbers.", "That is $\\frac{\\ln EN_t}{t}\\xrightarrow[t\\rightarrow \\infty ]{} \\gamma =\\mathcal {E}(\\ln \\displaystyle \\frac{\\beta }{p})>0\\,\\, P_m \\,\\,\\text{a.s.}$ The additional classification depends on the $P(N_t=0), t \\rightarrow \\infty $ .", "In the classical situation, $P(N_t=0)\\xrightarrow[t\\rightarrow \\infty ]{} \\delta <1$ and $\\delta $ is the root of the equation $\\varphi (z)=z,z\\in [0,1]$ .", "In our case, consider now the event $\\lbrace N_t\\ge 1\\rbrace $ .", "Then $P(N_t\\ge 1)=\\displaystyle \\frac{1}{S_t}$ , $S_t=\\displaystyle 1+\\frac{1-\\beta _1}{\\beta _1}+\\frac{1-\\beta _2}{\\beta _2}\\frac{p_1}{\\beta _1}+\\cdots \\frac{1-\\beta _t}{\\beta _t}\\prod \\limits _{j=1}^{t-1}\\frac{p}{\\beta }(j)$ .", "Obviously $P(N_t\\ge 1)\\rightarrow P(N_\\infty \\ge 1)=\\displaystyle \\frac{1}{S_{\\infty }}$ as $t\\rightarrow \\infty $ , where $S_{\\infty }=\\displaystyle 1+\\frac{1-\\beta _1}{\\beta _1}+\\frac{1-\\beta _2}{\\beta _2}\\frac{p_1}{\\beta _1}+\\cdots \\frac{1-\\beta _t}{\\beta _t}\\frac{p_{t-1}p_{t-2}\\cdots p_1}{\\beta _{t-1}\\beta _{t-2}\\cdots \\beta _1}+\\cdots $ .", "Notation $\\lbrace N_{\\infty }\\ge 1\\rbrace $ means that the branching process $N_t$ is non-degenerating.", "$P(N_{\\infty }=0)=1-P(N_{\\infty }\\ge 1)=1-\\displaystyle \\frac{1}{S_{\\infty }}$ .", "When $\\gamma =\\mathcal {E}\\ln \\frac{\\beta }{p}>0$ , the behavior of the series for $S_{\\infty }$ depends on the front coefficient $\\displaystyle \\frac{1-\\beta _i}{\\beta _i}, i=1,2,3\\cdots $ .", "Assume that for some small enough $\\delta >0$ , $\\sum \\limits _{t}\\mathcal {P}(\\beta _t<exp(-(\\gamma -\\delta )t)<\\infty \\,\\, \\text{and}\\,\\, \\gamma =\\mathcal {E}\\ln \\displaystyle \\frac{\\beta }{p}>0$ then the process $N_t\\rightarrow \\infty $ exponentially fast with uniformly positive probability $P(N_t\\ge 1)\\ge \\delta _1$ for some $\\delta _1>0$ In fact, due to the first Borel Cantelli Lemma, $\\beta _t\\ge exp(-(\\gamma -\\delta )t), \\forall (\\delta <\\gamma ), t\\ge t_o({\\omega }_m)$ .", "That is $\\frac{1-\\beta _t}{\\beta _t}\\le e^{(\\gamma -\\delta )t}, t\\ge t_o$ and the series $S_{\\infty }$ converges.", "Due to Chebyshev' s inequality condition $\\mathcal {E}\\ln ^{1+\\delta }\\frac{1}{\\beta }(\\cdot )\\le c < \\infty $ for some $\\delta >0$ is sufficient for condition (REF ).", "We call such process the strong super-critical Galton-Watson process.", "Assume that for some small enough $\\delta >0$ , $\\sum \\limits _{t}\\mathcal {P}(\\beta _t<exp(-(\\gamma +\\delta )t)=\\infty \\,\\, \\text{and}\\,\\, \\gamma =\\mathcal {E}\\ln \\displaystyle \\frac{\\beta }{p}>0$ then $P$ -a.s. the process $N_t\\rightarrow \\infty $ exponentially fast and $P(N_t=0)\\rightarrow 1$ as $t\\rightarrow \\infty $ This is the corollary of the second Borel Cantelli Lemma for independent random variable $\\beta _t$ .", "We call such process the weak super-critical Galton-Watson process.", "Of course, one can prove more precise results considering the central limit theorems for the sums of $\\displaystyle \\sum \\limits _{i=1}^t \\ln \\frac{\\beta }{p}(j)$ ." ], [ "Sub-critical Galton-Watson process in random environment", "We call Galton-Watson process in random environment sub-critical if $\\mathcal {E}(\\ln \\displaystyle \\frac{\\beta }{p})=\\gamma <0$ , then $E[N_t]=\\prod \\limits _{i=1}^t\\displaystyle \\frac{\\beta _i}{p_i}=e^{\\sum \\limits _{i=1}^t\\ln ( \\displaystyle \\frac{\\beta _i}{p_i})}=e^{\\gamma t+o(t)}$ and as easy to understand, $S_{\\infty }=\\infty $ P-a.s.", "If $\\mathcal {E}(\\ln \\displaystyle \\frac{\\beta }{p}) <0$ (sub-critical case), then like in the classical homogeneous situation, $E[N_t]\\rightarrow 0$ as $t\\rightarrow \\infty $ exponentially fast, and $P(N_t\\ge 1)\\rightarrow 0$ exponentially fast.", "The last statement follows from Chebyshev inequality $P(N_t\\ge 1)\\le E[N_t]=e^{\\gamma t+o(t)}$ .", "Define a random variable $\\tau (\\omega _m)=min\\lbrace t:N_t=0\\rbrace $ , this is the extinction moment.", "Its distribution is given by the formula $P(\\tau >t)=P(N_t\\ge 1)=\\frac{1}{S_t}$ Then $E[\\tau ]=\\displaystyle \\sum \\limits _{t=0}^{\\infty }P(\\tau >t)=\\sum \\limits _{t=0}^{\\infty }\\frac{1}{S_t}$ Since $P(\\tau >t)=P(N_t\\ge 1)\\le E[N_t]\\le e^{(\\gamma +\\epsilon )t}$ for $t\\ge t_0({\\omega }_m)$ and any sufficiently small $\\epsilon $ .", "As a result, $E[\\tau ]=\\sum \\limits _{t=0}^{\\infty }\\frac{1}{S_t}<\\infty $ and even $E[\\tau ^k]<\\infty $ for arbitrary $k\\ge 1$ .", "In some sense for the subcritical Galton-Watson process in the random environment, the population is vanishing \"very fast\"." ], [ "Critical Galton-Watson process in random environment", "The most interesting are the critical processes in the random environment.", "We call Galton-Watson process in random environment critical if $\\gamma =\\mathcal {E}(\\ln \\displaystyle \\frac{\\beta }{p})=0 $ and the second moment $\\mathcal {E}(\\ln ^2 \\displaystyle \\frac{\\beta }{p})=Var(\\ln \\displaystyle \\frac{\\beta }{p})=\\sigma ^2<\\infty $ .", "We will return to the case $\\sigma ^2=\\infty $ in the next paper.", "We will consider two special cases for $\\sigma ^2$ .", "Strong critical case: $\\sigma ^2=0$ That is $\\beta (i)=p(i)$ , $i=1,2,3,\\cdots $ and $S_t=\\displaystyle 1+\\frac{1-\\beta _1}{\\beta _1}+\\frac{1-\\beta _2}{\\beta _2}+\\cdots \\frac{1-\\beta _t}{\\beta _t}$ If $0<\\gamma =\\mathcal {E}\\ln \\frac{1-\\beta }{\\beta }<\\infty $ , then $S_t \\sim \\gamma t$ $P$ -a.s. and $P(N_t\\ge 1)= \\displaystyle \\frac{1}{S_t}\\sim \\frac{1}{\\gamma t}$ .", "Like in the homogeneous case, $EN_t=1$ , $P(N_t\\ge 1)\\sim \\displaystyle \\frac{1}{\\gamma t}$ .", "That is, population slowly degenerates.", "Much more important case is the case when $\\sigma ^2>0$ .", "Here $EN_t=\\prod \\limits _{i=1}^t\\displaystyle \\frac{\\beta _i}{p_i}=e^{\\sum \\limits _{i=1}^t \\ln \\displaystyle \\frac{\\beta }{p}(i)}=e^{\\xi (t)}$ , where $\\xi (t)=\\sum \\limits _{i=1}^t \\ln \\displaystyle \\frac{\\beta }{p}(i)$ .", "Due to the Law of the Iterated Logarithm, $\\limsup \\frac{\\xi (t)}{\\sigma \\sqrt{t\\ln \\ln t}}=1$ $\\liminf \\frac{\\xi (t)}{\\sigma \\sqrt{t\\ln \\ln t}}=-1$ That is, the sum $\\xi (t)$ in the exponent has strong oscillations.", "If $\\mathcal {E}(ln \\displaystyle \\frac{\\beta }{p})=0$ and $0<\\mathcal {E}(\\ln ^2\\displaystyle \\frac{\\beta }{p})=\\sigma ^2<\\infty $ , $E[N_t]$ will grow in an oscillating way.", "If $\\mathcal {E}(ln \\displaystyle \\frac{\\beta }{p})=0$ , and $\\mathcal {E}(ln \\displaystyle \\frac{\\beta }{p})^2> <\\infty $ , by Lindeberg–Lévy Central Limit Theorem, $\\frac{1}{\\sqrt{t}}\\sum \\limits _{i=1}^t\\ln ( \\displaystyle \\frac{\\beta _i}{p_i})\\rightarrow N(0,1)$ in distribution.", "Thus $E[N_t]\\sim e^{\\sqrt{t}w}$ as $t\\rightarrow \\infty $ where $w$ is a standard random normal variable.", "This implies the expected number of particles at moment t will oscillates as $t$ goes to infinity.", "Consider now the series $S_t=\\displaystyle 1+\\frac{1-\\beta _1}{\\beta _1}+\\frac{1-\\beta _2}{\\beta _2}e^{\\ln \\frac{p_1}{\\beta _1}}+\\cdots \\frac{1-\\beta _t}{\\beta _t}\\prod \\limits _{j=1}^{t-1}e^{\\ln \\frac{p}{\\beta }(j)}$ If the random variables $\\beta _t,t=1,2,3\\cdots $ are separated from 1 in some sense, say $\\beta _t\\le 1-\\delta _1$ for any $t\\ge 0$ , then from the previous formula, it follows that $S_t \\rightarrow \\infty $ , $t\\rightarrow \\infty $ $P$ -a.s.", "But it is true without additional assumptions.", "If $\\gamma =\\mathcal {E}\\ln \\frac{\\beta }{p}=0$ , $\\sigma ^2=\\mathcal {E}\\ln ^2\\frac{\\beta }{p}<\\infty $ , then $S_{\\infty }=\\infty $ , that is $P(N_{\\infty }=0)=1$ , in other words, $N_t=0$ , for $t\\ge \\tau (\\omega )<\\infty $ Consider the sequences $A_n=2^{n^2}$ , $B_n=2^n$ and $C_n=\\sum \\limits _{k=1}^n(A_k+B_k)$ , then $C_n \\sim 2^{n^2}$ , $n\\rightarrow \\infty $ .", "For $\\xi (t)=\\sum \\limits _{j=1}^t\\ln \\displaystyle \\frac{\\beta }{p}(j)$ , by Kolmogorov's inequality $P(\\max \\limits _{k\\le C_n} \\xi (k)\\ge \\sqrt{C_n}D_n)\\le \\frac{\\sigma ^2}{D^2_n}$ and for $D_n=n$ , due to the Borel Cantelli Lemma, we will get that $\\max \\limits _{k\\le C_n} \\xi (k) \\le \\sqrt{C_n}n=2^{n^2/2}n\\,\\, \\text{for}\\,\\, n\\ge n_0({\\omega _m)}$ And at the same time $P(\\xi (C_n+A_{n+1})-\\xi (C_n)>\\sqrt{A_{n+1}})=\\frac{1}{2}$ and $A_{n+1}=2^{(n+1)^2}>>C_n$ .", "As a result, there is about 50% of $n$ for which we have the inequality in (REF ).", "Here and only here, we used the assumption $A_n\\sim c^n\\sim 2^{n^2}$ (instead of $n^2$ , one can use $n^{1+\\epsilon }$ , $\\epsilon >0$ .)", "Finally, by Kolmogorov's inequality, $P(\\max \\limits _{k<B_{n(t)}}\|\\sum \\limits _{j=C_n+A_{n+1}}^{C_n+A_{n+1}+k} \\ln \\frac{\\beta }{p}(j)\|>B_n^{\\frac{1}{2}(1+\\epsilon )} \\le \\frac{C}{2^{n\\epsilon }}$ Now we have to estimate the random variables $\\frac{1-\\beta _t}{\\beta _t}$ , $t=1,2,3,\\cdots $ since very small values of these random variables can compensate the large values of $e^{-\\xi (t)}=e^{\\sum \\limits _{j=1}^t \\ln \\frac{p}{\\beta }}$ .", "For some $\\delta _1,\\delta _2>0$ , $P(\\frac{1-\\beta _t}{\\beta _t}\\ge \\delta _1)=P(\\beta _t\\le \\frac{1}{1+\\delta _1})\\ge \\delta _2$ Consider the sequence $\\frac{1-\\beta _t}{\\beta _t}, t\\ge 1$ and divide it into series $\\eta $ which collects all terms of $\\frac{1-\\beta _t}{\\beta _t}<\\delta _1$ and complementary series $\\theta $ which collects all the terms $\\frac{1-\\beta _t}{\\beta _t}\\ge \\delta _1$ .", "The distribution of the length of such series is the geometric distribution and due to the Borel Cantelli Lemma $\\max \\limits _{i\\le N}\\theta _i\\sim c\\ln N$ $P$ -a.s.", "The number of $\\theta $ -series between $C_n+A_{n+1}$ and $C_{n+1}$ again due to Borel-Cantelli lemma is growing exponentially in $n$ $P$ -a.s., that is, inside the interval $(C_n+A_{n+1},C_{n+1})$ , there are many random variables $\\frac{1-\\beta _t}{\\beta _t}\\ge \\delta _1$ .", "But in this interval, $\|\\eta (C_n+A_{n+1}+k)\|\\ge 2^{n^2/2}-2^{-(n+1)^{(1+\\epsilon )/2}}$ if only $\\displaystyle \\eta (C_n+A_{n+1})-\\eta (C_n)>2^{(n+1)^2/2}$ .", "This observation complete the proof of Theorem (REF ) Let us now estimate $S_t$ .", "Note that the main contribution to $S_t$ can give to the factors $\\prod \\limits _{i=1}^s\\displaystyle \\frac{p}{\\beta }(i)=e^{\\sqrt{s}\\gamma +o(s)}$ but prefactors $\\displaystyle \\frac{p}{\\beta }(s)$ can be very small if only $\\beta $ can be very large,.", "Under appropriate control of the large values of $\\displaystyle \\frac{1-\\beta }{\\beta }$ , we can find the asymptotic formula.", "If for any $\\delta _0\\ge 0$ , $\\mathcal {E}\|\\ln \\frac{1-\\beta }{\\beta }\|^{2+\\delta }=\\mathcal {E}\|\\ln \\frac{\\beta }{1-\\beta }\|^{2+\\delta }=c_0<\\infty $ , then $\\frac{1}{\\sigma \\sqrt{t}}\\ln S_t \\xrightarrow[law]{} M_1$ where $M_1=\\max \\limits _{s\\in [0,1]} W_s$ and $W_s$ is the standard 1D Brownian motion.", "By the Chebyshev inequality, for any $\\epsilon >0$ , $P(\|\\ln \\frac{1-\\beta _t}{\\beta _t}\|\\ge \\epsilon \\sqrt{t}) \\le \\frac{\\mathcal {E}(\\ln \\frac{1-\\beta _t}{\\beta _t})^{2+\\delta }}{\\epsilon ^{2+\\delta }t^{1+\\delta /2}}$ By Borel Cantelli Lemma, for any $\\epsilon >0 $ , there exists $t_0>0$ such that when $t\\ge t_0$ , $\\displaystyle \\frac{1}{\\sqrt{t}}\|\\ln \\frac{1-\\beta _t}{\\beta _t}\|<\\epsilon $ , that is e-t1•2<1-tt<et12 Put $M_t=\\max \\limits _{s\\le t}\\frac{\\sum \\limits _{i=1}^s \\ln \\frac{p}{\\beta }(i)}{\\sigma \\sqrt{t}}$ .", "Then for large $t$ , $e^{-\\epsilon \\sqrt{t}}e^{\\sigma \\sqrt{t}M_t}\\le S_t\\le e^{\\sigma \\sqrt{t}}te^{\\delta \\sqrt{t}}$ That is $M_t-\\epsilon \\le \\frac{\\ln S_t}{\\sigma \\sqrt{t}} \\le \\frac{\\ln t}{\\sigma \\sqrt{t}}+\\epsilon +M_t$ Due to the functional central limit theorem by Donsker Prokhorov, as $t\\rightarrow \\infty $ , $\\frac{1}{\\sigma \\sqrt{t}}\\ln S_t \\rightarrow M_t \\xrightarrow[law]{} M_1$ where $M_1=\\max \\limits _{s\\in [0,1]} W_s$ and $W_s$ is the standard 1D Brownian motion." ], [ "Analysis of the Geometric Progression", "In the last section of the paper, we will give two examples when the distribution of the random geometric progression can be calculated explicitly.", "Let $S_{\\infty }=\\displaystyle \\frac{1-\\beta _1}{\\beta _1}+\\frac{1-\\beta _2}{\\beta _2}\\frac{p_1}{\\beta _1}+\\cdots $ and $X_t=\\displaystyle \\frac{1-\\beta _t}{\\beta _t}$ and $\\theta _t=\\displaystyle \\frac{p_t}{\\beta _t}$ .", "Assume that $X_t \\in (0,1)$ and $\\theta _t\\in (0,1)$ are independent for fixed $t$ and the pairs $(X_t,\\theta _t)$ are independent for different $t=2,3,4,\\cdots $ .", "Then $S_{\\infty }=X_1+\\theta _1S_{\\infty }^{\\prime }$ where in the right part all three random variables are independent and $S_{\\infty }\\stackrel{law}{=}S_{\\infty }^{\\prime }$ .", "The random geometric series (REF ) appears in many applications, see for example [3].", "We will present now two typical examples.", "For the general discussion on the properties of the sum in (REF ) about infinite divisibility, asymptotes etc., we will present in a separate paper." ], [ "Example", "Let $X_t, t=1,2,3,\\cdots $ have Erlang law with the index $r=1,2,\\cdots $ , i.e.", "the Laplace transform of $X_t$ is $Ee^{-\\lambda X_t}=\\frac{1}{(1+\\lambda )^r}=\\Phi (\\lambda )$ And $\\theta _t, t=1,2,3,\\cdots $ have the degenerated $\\beta $ -density: $\\pi (x)=\\beta x^{\\beta -1}\\mathbf {1}_{[0,1]}(x),\\beta >0$ Then $\\psi (\\lambda )=Ee^{-\\lambda S_{\\infty }}=\\Phi (\\lambda )\\displaystyle \\int _0^1\\psi (\\lambda \\xi )\\pi (\\xi )d \\xi =\\frac{1}{\\lambda ^{\\beta }}\\Phi (\\lambda )\\int _0^{\\lambda } \\psi (y) \\beta y^{\\beta -1}dy$ Thus $\\frac{\\psi (\\lambda )\\lambda ^{\\beta }}{\\Phi (\\lambda )}=\\beta \\int _0^{\\lambda }\\psi (y)y^{\\beta -1}dy$ $(\\frac{\\psi }{\\Phi })^{\\prime }\\lambda ^{\\beta }+\\frac{\\psi }{\\Phi }\\beta \\lambda ^{\\beta -1}=\\beta \\psi (\\lambda )\\lambda ^{\\beta -1}$ $\\psi (\\lambda )=\\Phi (\\lambda )\\displaystyle e^{-\\beta \\int _0^{\\lambda }\\frac{1-\\Phi (s)}{s}ds}$ Or ()=1(1+)re-0(1+z)r-1z(1+z)rdz =1(1+)re-01+(1+z)++(1+z)r-1(1+z)rdz =1(1+)re-((1+)+(11+-1)+12(1(1+)2-1)++1r-1(1(1+)r-1-1)) =1(1+)r+e-(11+-1)+12(1(1+)2-1)++1r-1(1(1+)r-1-1)) The last Laplace transform means that $S_{\\infty }=Y_1+(Z_{11}+\\cdots Z_{1v_1})+(Z_{21}+\\cdots Z_{2v_2})+\\cdots +(Z_{r-1,1}+\\cdots +Z_{r-1,v_{r-1}})$ where $Y_1$ has Gamma distribution with parameter $r+\\alpha $ , and $v_1,v_2,\\cdots ,v_{r-1}$ has Poissonian laws with parameter $\\beta ,\\beta /2,\\cdots ,\\beta /(r-1)$ respectively.", "And $Z_{l,j},l=1,2,\\cdots , r-1, j=1,2,3,\\cdots , r-1$ have Erlang laws of the rank $l$ and the number of the terms in each sum is $v_{l}$ .", "In particular, if $r=1$ , $\\psi (\\lambda )=\\displaystyle \\frac{1}{1+\\lambda }e^{-\\beta \\int _0^{\\lambda }\\frac{1-\\frac{1}{1+s}}{s}ds}=\\frac{1}{(1+\\lambda )^{1+\\beta }}$ , thus $S_{\\infty }$ has Gamma distribution with parameter $1+\\beta $ The density of random variable $S_{\\infty }$ has power asymptotics near 0.", "Note that the random variable $\\theta _t,t=1,2,3,\\cdots $ have positive density on $(0,1)$ .", "The answer will be different if $\\theta _t$ are separated from 0 and 1.", "Assume $\\theta _t\\equiv \\rho \\in (0,1)$ and consider the random geometric progression $S=\\sum \\limits _{n=0}^{\\infty }X_n\\rho ^n$ where $0<\\rho <1$ is a fixed number and $X_n,n=1,2,3\\cdots $ have i.i.d Gamma distribution with parameter $\\beta >0$ .", "Then $E^{-\\lambda X_i}=\\displaystyle \\int _0^{\\infty }e^{-\\lambda x}\\frac{x^{\\beta -1}e^{-x}}{\\Gamma (\\beta )}dx=\\frac{1}{(1+\\lambda )^{\\beta }}$ We want to estimate $P(S<\\delta )$ , $\\delta <<1$ and $P(S>A)$ , $A>>1$ .", "Note that $Ee^{-\\lambda S}=\\frac{1}{(1+\\lambda )^{\\beta }(1+\\rho \\lambda )^{\\beta }\\cdots (1+\\lambda \\rho ^n)}=\\psi (\\lambda )$ The function $\\psi (\\lambda )$ is analytic for $\\lambda >-1$ and using this fact or the exponential Chebyshev's equality, one can find that $P(S>A)\\sim cA^{\\beta -1}e^{-\\beta }$ .", "Much more interesting problem is the estimation of $P(S<\\delta )$ .", "We have $P(S<\\delta )=P(e^{-\\lambda S}>e^{-\\lambda \\delta })\\le \\frac{Ee^{-\\lambda S}}{e^{-\\lambda \\delta }}=\\psi (\\lambda )e^{\\lambda \\delta }$ But $\\psi (\\lambda )=\\prod \\limits _{n=1}^{\\infty }(1+\\rho ^n\\lambda )^{-\\beta }=\\prod \\limits _{n=1}^{n_0}(1+\\rho ^n\\lambda )^{-\\beta }\\prod \\limits _{n=n_0+1}^{\\infty }(1+\\rho ^n\\lambda )^{-\\beta }$ .", "If $\\rho ^{n_0}\\lambda \\simeq 1$ , then $\\ln \\lambda +n_0\\ln \\rho =0$ , then $n_0\\sim \\displaystyle \\frac{\\ln \\lambda }{\\ln \\frac{1}{\\rho }}$ As easy to see, for $\\lambda >>1$ , $\\psi (\\lambda )=\\prod \\limits _{n=1}^{n_0}(1+\\rho ^n\\lambda )^{-\\beta }\\prod \\limits _{n=n_0+1}^{\\infty }(1+\\rho ^n\\lambda )^{-\\beta }\\sim C\\lambda ^{n_0}\\rho ^{n_0(n_0-1)/2}\\sim C\\lambda ^{\\frac{\\ln \\lambda }{\\ln \\frac{1}{\\rho }}}\\rho ^{\\frac{\\ln ^2\\lambda }{2\\ln ^2\\frac{1}{\\rho }}}$ .", "Then (with log accuracy), $P(S<\\delta )\\le e^{-\\frac{\\ln ^2\\lambda }{\\ln \\frac{1}{\\rho }}+\\frac{\\ln ^2\\lambda }{2\\ln \\frac{1}{\\rho }}+\\delta \\lambda }\\sim e^{-\\frac{\\ln ^2\\lambda }{2\\ln \\frac{1}{\\rho }}+\\delta \\lambda }$ Then when $\\lambda =\\displaystyle \\frac{2\\ln \\lambda }{\\delta \\ln \\frac{1}{\\rho }}$ , $P(S<\\delta )=e^{(\\ln \\frac{1}{\\delta }-\\ln \\ln \\frac{1}{\\delta }+C)^2+{2\\ln \\frac{1}{\\rho }}+\\frac{2\\ln \\frac{1}{\\delta }}{\\ln \\frac{1}{\\rho }}} \\sim e^{-\\frac{\\ln ^2\\frac{1}{\\delta }}{2\\ln \\frac{1}{\\rho }}}$ Calculation of exact asymptotics is more difficult problem.", "For additional details, see ([3]).", "Dan Han was supported by University of Louisville EVPRI Grant \"Spatial Population Dynamics with Disease\" and AMS Research Communities \"Survival Dynamics for Contact Process with Quarantine\".", "S. Molchanov was supported by the Russian Science Foundation RSF grant project 17-11-01098 and project 20-11-20119." ] ]	2011.14171
[ [ "Simple Spyware: Androids Invisible Foreground Services and How to\n (Ab)use Them" ], [ "Abstract With the releases of Android Oreo and Pie, Android introduced some background execution limitations for apps.", "Google restricted the execution of background services to save energy and to prevent apps from running endlessly in the background.", "Moreover, access to the device's sensors was changed and a new concept named foreground service has been introduced.", "Apps were no longer allowed to run background services in an idle state, preventing apps from using the device's resources like the camera.", "These limitations, however, would not affect so-called foreground services because they show a permanently visible notification to the user and could therefore be stopped by the user at any time.", "Our research found out that flaws in the API exists, which allows starting invisible foreground services, making the introduced limitations ineffective.", "We will show that the found flaws allow attackers to use foreground services as a tool for spying on users." ], [ "Introduction", "We found out that foreground services do not show any visual notification when the service's execution time is shorter than five secondsThe exact duration is depending on the phone.. We use this loading time and combine it with another flaw in Androids Job Scheduler API to continuously execute tasks from a background context.", "Exploiting these flaws allows apps to use the device's resources, even when the app is closed or on standby.", "Furthermore, we show that we can use these flaws for continually spying on users and allowing malware developers to create spyware without the need for complicated exploitation.", "We start in Section with some basic introduction to Android's components and then we explain how we spawn a foreground service on Android Pie combined with some basic background schedulers.", "We then use this basic example to show how we can use these API's to implement a simple spyware app.", "At the end of the paper in Section we discuss some limitations as well as ideas to prevent such attacks." ], [ "Abusing Foreground Services", "Android defines two basic types of service classes for apps, background and foreground services [3].", "The difference between these services is how they appear in the user interface and under which constraints they are executed.", "We can start Foreground services even when the user interface is closed; in contrast, background services cannot do so.", "The operating system prevents background services to start when the app's user interface is closed by throwing an IllegalStateExeception.", "If we want to run a task when the app is closed, we can use Android's scheduling classes instead.", "For example, the JobScheduler [4] and AlarmManager [5] classes.", "Those schedulers' idea is that apps can synchronize or process data even when the user interface gets closed.", "We can use this, for example, to set an alarm clock or to upload a file when the user interface is not needed.", "If misused, schedulers often use a lot of battery power.", "For example, when an app is continuously uploading data in the background.", "Since Android Oreo and app can no longer run endless background services when its user interface is not shown.", "Usually, the operating system stops all services some minutes after the app was closed.", "Moreover, access to sensors like microphones and cameras should no longer work when the app is closed.", "In case an app tries to access one of the restricted sensors from a scheduler, the operating system throws an IllegalStateException, and the access is thereby not granted.", "However, in order to access the sensors from the background context, foreground services can be used.", "Schedulers are allowed to start foreground services, and as mentioned, foreground services do not have any restrictions when it comes to sensor access.", "The only limitations foreground services have is that they need to show a permanently visible notification and that the app spawning the foreground service needs to have the permission to access the sensor.", "Listing shows an example code to start a foreground service in Java.", "truebtruehtfalsefalsefalselgraylgreenlblue0.5cm0.5cm public class SomeExampleService extends Service { \t// Start service by intent.", "No filtering shown here \t@Override \tpublic int onStartCommand(Intent intent, int flags, int startId){ \t\t// ~4.9999.. seconds to call startForeground(...) \t\tNotification notification = createCustomNotification(); \t\tthis.startForeground(1, notification) // Sensor access not restricted anymore \t\t// Race condtion started. Let's collect some data fast...", "accessCamera(); \t\taccessMicrophone(); \t\t// ... some more malicious code \t\tstopForeground(true); //Stop the service before notification is loaded and \t\t// win the race against the notification manager.", "return START_STICKY; \t} ... \t/** \t* Create a custom notification.", "/ \tprivate Notification createCustomNotification() { \t\tNotificationChannel channel = new NotificationChannel(\"1\", \"Location\", \t\tNotificationManager.IMPORTANCE_LOW); \t\tNotificationManager notificationManager = getSystemService(NotificationManager.class); \t\tif(notificationManager != null){ \t\t\tnotificationManager.createNotificationChannel(channel); \t\t\tNotificationCompat.Builder builder = new NotificationCompat.Builder(this, \"1\") \t\t\t.setSmallIcon(R.drawable.ic_remove_red_eye_black_24dp) \t\t\t.setContentTitle(\"Simple Spyware\") \t\t\t.setContentText(\"Tracking your position!\")", ".setPriority(NotificationCompat.PRIORITY_LOW) \t\t\treturn builder.build(); \t\t} \t} } What we can see in Listing is that we extend the service class and that we define a onStartCommand method in Line 6 as we would do with a regular foreground service.", "To start a foreground service, we have to define a notification.", "On Lines 20 to 31, we define the notification and set an example icon and text.", "As shown, we can customize the appearance of the notification as we would like it to be.", "Figure REF shows how such a foreground service in the user interface could look.", "Figure: Example of foreground notifications in the user interface.The notification is sticky, and we cannot dismiss it until we stop foreground service or the user manually disables the notification.", "Attempting to cancel the notification within Java will not work since the notification manager will not allow it.", "An attacker could hide its intention by showing some typical notification like an update, announcement, or loading screen.", "In many cases, as long as the notification does not stay too long in the notification bar, it will not raise the user's suspicion.", "Nevertheless, users could get suspicious or annoyed when the notification shows them some unwanted content or the notification stays too long in the notification bar.", "In such cases, the user can stop the app or disable the notifications for the app.", "As attackers, we wanted to go a step further and see if we could stop the notification from showing up at all, so that the users would not get alarmed.", "So we search for possibilities to dismiss or cancel the notification but could not find an easy way to do it.", "Instead, we found out that the notification manager does not show a notification when the foreground service's lifetime is shorter than five seconds.", "We can use this race condition to execute any code before the operating system loads the notification.", "We can spawn one invisible foreground service and execute our code once, and then the app will stop.", "To further abuse this approach, we needed a way to persistently spawn new foreground services on demand.", "To do so we use Android's JobScheduler [4] or AlarmManager [5] classes which allows to execute code outside of the app context.", "For decided to use for our example the JobScheduler class but the AlarmManager has more or less the same functionality.", "We create a new Job with the JobInfo.Builder [6] as shown in Listing .", "truebtruehtfalsefalsefalselgraylgreenlblue0.5cm0.5cm public void scheduleJob(){ long interval = 1000 60L;\t// Some interval ComponentName serviceComponent = new ComponentName(this, JobScheduler.class); JobInfo.Builder builder = new JobInfo.Builder(JOB_ID, serviceComponent); builder.setPeriodic(interval);\t// Minimum is 15 minutes builder.setOverrideDeadline(interval * 2);\t// Sets the maximum scheduling // latency builder.setMinimumLatency(interval);\t// Runs a job after a delay JobScheduler jobScheduler = this.getSystemService(JobScheduler.class); jobScheduler.schedule(builder.build());\t// Schedule the job } As attackers, we want to use the JobScheduler to start a job every X seconds or minutes.", "Our idea is that we want to execute any malicious command like taking a picture or uploading a file whenever we need it.", "The JobScheduler class has a method .setPeriodc(long seconds) that offers precisely that.", "The problem with this method is that it has a minimum interval of 15 minutes.", "So if we would use it, we could only execute malicious code every 15 minutes.", "For some malicious commands, this is maybe a too long period, so we wanted to circumvent this limitation.", "Instead of using the .setPeriodic method, we can create a custom scheduler.", "All we have to do so is to use the JobScheduler's .setMinimumLatency method.", "This method allows us to run a job after a given delay and has no limitations in execution time.", "In other words, we can set a delayed job under 15 minutes an circumvent the limitation.", "Consequently, we can build our period job scheduler by chaining jobs with the delay function.", "Whenever we execute a job, we schedule a new job directly with the delay method.", "As long as our job chain is not interrupted, we can use the JobScheduler class to spawn new foreground services at demand.", "The chaining of jobs in combination with foreground services allows us to circumvent the background service limitation introduced in Android Oreo [1] and the background sensor access limitation introduced in Android Pie [2].", "If we want, we can further enhance our scheduling with some other options of the JobScheduler class.", "For example, we could only schedule jobs when the device is charging or connected to wifi.", "We can as well change our scheduling strategy during execution if necessary, to stay undetected.", "setPersisted(boolean isPersisted): This allows that a job persists restarts.", "Needs the received_boot_completed permission to do so.", "setRequiredNetwork(NetworkRequest networkRequest) and setRequiredNetworkType(int networkType): This allows us to define a specific network type to be active before the job is executed.", "setRequiresBatteryNotLow(boolean batteryNotLow): Run a job only if the battery is not low.", "Usually, this is when the phone has more than 15% capacity.", "setRequiresCharging(boolean requiresCharging): Run a job only when the device is charging.", "setRequiresDeviceIdle(boolean requiresDeviceIdle): Run a job only if a device is not used and therefore in idle state.", "See [4] for a complete overview of methods.", "An example of this approach was implemented in our open-source demo app [7].", "This approach also works with the AlarmManager class, and the code is as well available in our git." ], [ "Collecting Data", "After we have set up the chain of jobs and start our foreground service, we can add methods to collect the user's data.", "As shown in Listing on Line 9, it is common for spyware to take camera pictures or to record the microphone audio.", "We have tested if our approach works with these features, and we implemented some examples in our demo app." ], [ "Camera", "We can modify some of the existing open-source libraries [8], [9], [10] for taking hidden camera pictures to demonstrate that we can access the camera API from the background.", "We start an invisible foreground service and use our access to the camera2 API to capture some images.", "In most cases and with the phones we had at hand, this approach works as expected.", "However, during testing, we noticed that our hidden camera implementation does not work on all cameras.", "Some cameras only show black images due to a short exposure time or wrong camera calibration." ], [ "Microphone", "As we have explained, we use short lived foreground services with an execution time of maximal five seconds.", "This execution time is for most audio recordings not sufficient, and we would generate most likely incomplete audio recordings if we tried to record only for five seconds.", "However, it is possible to record audio for a long time due to Android's Media Server architecture [11].", "Usually, if we want to record audio, we use the MediaRecorder class [12], and we can start to record with the .start() method.", "As soon as we do, our app will contact Android's media server, and the server will start recording.", "Since our app's background limitations do not apply to the media server process, it can record the audio even if we close our spyware app.", "Consequently, our app needs only to control when the audio recording needs to be started or stopped, and we can do this within the five seconds execution time of our invisible foreground service." ], [ "Location", "Location tracking is another interesting feature since not only spyware developers are interested in collecting this data.", "Tracking the position with an invisible foreground service has the advantage that there is no limitation on how often we can get position updates from the location API.", "According to [13] Background services are limited to request location updates a few time per hour.", "Consequently, when we use foreground services, we can track the location near real-time with high precision, which is interesting not only for spyware.", "However, If we track the position in real-time, it will consume a lot of battery power, and battery optimization is likely to trigger and alarm the user.", "Figure: Android10 new runtime location permission.When Android10 was released, we wanted to test if our approach still worked on the new Android version.", "Google introduced with Android10 a new permission level called \"Allow only while in use\" for location tracking [14].", "Moreover, since the release, new apps had to add a new permission to their manifest, ACCESS_BACKGROUND_LOCATION, if they want to access the location from a background service.", "However, tracking the location within a foreground services does not need this permission as long as the service type within the manifest is set to location (see Listing ).", "truebtruehtfalsefalsefalselgraylgreenlblue0.5cm0.5cm <service android:name=\".data.location.LocationTrackerService\" \t\t\t android:foregroundServiceType=\"location\"> We added the new type in the manifest and tested it.", "First, we tested it with the \"Allow all the time\" permission, and as expected, we were able to track the location within our invisible foreground service.", "As a second test, we tested if the \"Allow only while in use\" permission would have any effect since we think Google created it to prevent what we were doing precisely.", "However, we could still track the location of the phone.", "The system sees a foreground service by definition to be in the foreground and, therefore, to be used.", "As a consequence, we can use an invisible foreground service and track the location even when the app gets closed and the \"Allow only while in use\" permission is set." ], [ "Files & Others", "As the last attack, we wanted to see if we could use our foreground service to monitor the file storage.", "Similar to the other attacks, we can conduct this one by running a foreground service every X minutes.", "In our case we just tested if it is possible to upload files from the external storage like camera pictures without the notification showing up.", "We found out that it is possible for smaller files if the data-connection is fast enough.", "We could use the .setRequiresCharging and the .setRequiredNetwork method to further improve our uploading strategy and upload larger files when the phone has the resources to do so.", "When an app does not have the permissions to access the camera or the location, it may have the file permission.", "Android Oreo and Pie allow an app to read all the content on the external storage where often private critical files like photos or documents are store.", "An attacker can then read the information stored in the external storage and use a foreground service to upload documents to a remote server.", "We think this is primarily a problem when all pictures from the standard camera app are stored on the external storage by default.", "Attackers then get direct access to all user camera pictures, which is a privacy problem.", "Moreover, if the user has activated the GPS metadata tracking on the camera app, all pictures will have the location information.", "So if we have the file permission, an attacker can get access to the camera pictures and where the user has taken the pictures.", "Anti-virus vendors have reported such attacks in the past [15].", "We decided not to integrate this feature into our demonstration app because we would have needed to add internet permission to our app, and we want it to be safe for testing usages.", "Furthermore, we tested clip-board hijacking and overlay attacks, which also work with invisible foreground services, and decided not to integrate them into our demo app.", "In Sections REF to REF we demonstrated just some examples of possible attacks.", "All of the common attacks are possible by using a combination of background and foreground services.", "Nevertheless, we want to discuss as well the limitations of our approach.", "First of all, we tested our approach only on three Android devicesSamsung Galaxy S9+, Samsung A50, and Huawei P smart and some Android Oreo, Pie, and Android10 emulators.", "Since wide-scale testing is not feasible within this project, the foreground services may behave differently on other phones.", "Some vendors may have additional security measurements implemented, which can defeat or detect invisible foreground services.", "As far as we know, it works on all tested Android Oreo and Pie phones and as well on Android10 devices.", "Second, we know that the access to some sensors is restricted to one app at a time: For example, if the user has already occupied the camera, a foreground service cannot access the camera for spying.", "Depending on the persistence strategy, this can occur more often than one may think and can attract the user's attention.", "For example, if our spyware captures a picture and the user has FaceID activated, it can occur that FaceID cannot access the camera and show an error message.", "Third, schedulers may not run at specific times: Android reschedule jobs and alarms, for example, when the device goes to the idle state.", "If our device is for a more extended period in stand-by, the schedulers will likely not execute our code until we use the device again.", "Furthermore, scheduling strategies can differ from device to device, and therefore the task execution can work correctly on one device but not on another.", "Fourth, some vendors have different behavior for showing notifications.", "For example, the location icon on some phones is shown as soon as the user activates location tracking and is permanently visible in the top menu bar.", "Other vendors only show the location icon whenever an app accesses the phone's location.", "So visibility for some features is different on some devices.", "Fifth, if we choose a spying strategy that uses many phone resources, like taking a camera picture every 10 seconds, it is likely that the operating system's battery optimizations will trigger.", "Depending on the phone vendor, it may show a notification to the user or directly stop the execution of our app.", "Sixth, all the demonstrated malware features work only if the user has already installed the app and has given the app the necessary runtime permissions." ], [ "Results", "Our demo spyware shows that Android's permission model cannot prevent excessive use of permissions and that the limitations do not prevent the collection of the user's sensitive data.", "As we described, the access restriction in Android Pie cannot entirely prevent our access to the critical sensors like the camera over foreground services.", "We think the restrictions, in general, are a good idea and give the user more security, but it still lacks some fundamental points like restricting the file access.", "We can summarize what we have done in the following steps: Background Scheduling: We frequently spawn new jobs with a chain of JobScheduler or AlarmManager jobs.", "With every job we execute, we start a new short-lived foreground service.", "Invisible Foreground Service: As long as our foreground service's execution time is shorter than five secondsFive seconds it just the average.", "Timing may change on other phones., it will not display a notification.", "The operating system allows only foreground service to access critical sensors like cameras and microphones.", "Spying: We can use common spying techniques to collect data from the user during the five-second window.", "We tested taking camera pictures, recording audio, uploading files, or tracking the user's location and other features.", "We have shown that our approach works as well on Android10.", "Patching these issues is not as simple as it may seem.", "Since invisible foreground services only use standard API calls, which are unlikely to be removed soon.", "Therefore we think that such attacks are likely to be seen for a longer time.", "Even if a patch for the foreground service is released and the notification's behavior changes, it seems that an attacker still has some possibilities for workarounds.", "Access and timings may get patched, but we think it will not entirely prevent such attacks since an attacker can still set the notification design.", "Attackers maybe will come up with custom notifications that do not look suspicious to the user." ], [ "Transparency", "If we wanted to prevent such attacks with the current design, we could continuously monitor the apps' permission usage.", "In case we would find an app that misused its permission, we could revoke the permissions.", "Permission monitoring apps exist but are often not accurate or widely used.", "For example, Samsung's \"App Permission Monitor\"[16] logs the access, and the app notifies the user as soon as the monitor detected suspicious permission access.", "Monitoring the permission usage of an app can help detect abusive apps, but monitoring is often error-prone, and automated detection is difficult.", "A malicious app may only use permissions sporadically and, therefore, stay undetected, or some apps need to access some permissions more often than others.", "In general, monitoring is unlikely to solve the problem entirely, but it could give transparency to what is running in the background to security-aware users.", "Another problem we should address is that Android users cannot check which tasks are running in the background.", "Staying hidden from the user allows malicious apps to run code unseen.", "Even if the user does not grant any dangerous permission, it allows an attacker to collect the phone's usage data, such as the installed apps list." ], [ "Revocation", "We can argue that users can deny dangerous permissions to prevent the described attacks in Section .", "However, the common problem is that many users do not fully understand which permissions are essential for an app to work and grant the requested runtime permissions.", "We think this leads to the fact that some apps have too many permissions that they do not need to function correctly.", "Another point is that once we give permission, the app holds access as long as it does not revoke it.", "In many cases, an app needs dangerous permission just a couple of times when the user uses the app.", "We suggest that the operating system automatically revokes permissions when they are not needed and that Android implements one time and time-based permissions.", "We think this could help in mitigating the time-frame a malicious app can collect data of the users.", "We can argue that one-time permissions or automatic permission revocation is not user-friendly in terms of usability, but not giving Android users the possibility to defend themselves against such fraud is it neither." ], [ "Acknowledgment", "I want to thank my professor Dr. Bernhard Tellenbach for encouraging me to publish this work.", "Moreover, I would like to thank the Zurich University of Applied Science for supporting my research on this topic." ] ]	2011.14117
[ [ "Stability of Finite Horizon Optimisation based Control without Terminal\n Weight" ], [ "Abstract This paper presents a stability analysis tool for model predictive control (MPC) where control action is generated by optimising a cost function over a finite horizon.", "Stability analysis of MPC with a limited horizon but without terminal weight is a well known challenging problem.", "We define a new value function based on an auxiliary one-step optimisation related to stage cost, namely optimal one-step value function (OSVF).", "It is shown that a finite horizon MPC can be made to be asymptotically stable if OSVF is a (local) control Lyapunov function (CLF).", "More specifically, by exploiting the CLF property of OSFV to construct a contractive terminal set, a new stabilising MPC algorithm (CMPC) is proposed.", "We show that CMPC is recursively feasible and guarantees stability under the condition that OSVF is a CLF.", "Checking this condition and estimation of the maximal terminal set are discussed.", "Numerical examples are presented to demonstrate the effectiveness of the proposed stability condition and corresponding CMPC algorithm." ], [ "Introduction", "Stability is essential for all control systems.", "Lack of stability of some earlier model predictive control (MPC) algorithms, or also known as receding horizon control (RHC), has triggered extensive research on stability of MPC.", "Stability analysis of MPC has been an active topic in the last three decades and a rich body of knowledge and understanding about stability and related design parameters of various MPC algorithms has been accumulated; e.g.", "[38], [29],[20].", "A number of tools have been developed for stability analysis and design of MPC algorithms with stability guarantee.", "Early MPC stability analysis methods and corresponding MPC schemes with stability guarantee include enforcing the terminal state arriving at the origin [26], using a dual control scheme by switching to a local stabilising control in the end of the horizon [5], adding contractive constraints in the online optimisation to ensure stability [13], or finding explicit (analytical) solutions [9].", "However, the most widely used MPC framework for stability guarantee, or currently regarded as a `standard' MPC framework, was proposed independently by a number of researchers at around the same time; e.g.", "[35] [8] [17].", "Although [35] provided a much comprehensive and insightful treatment, the basic idea and the derived stability conditions in these works are very much the same – the decrease of the terminal cost shall be larger than that of the stage cost with the moving of the horizon.", "In this framework, there are three important integrants that are closely coupled, namely, a local Lyapunov function used as a terminal weight in the performance index, a terminal set constraint on the terminal state, and a terminal control satisfying certain conditions for all state in the terminal set [35].", "The terminal weight included in the performance index for online optimisation must cover the cost induced by applying the terminal control to steer the system from the terminal state within the terminal set to the origin.", "With this setting, the closed-loop system stemming from an MPC algorithm can guarantee its stability.", "This framework triggered phenomenal research in both theory and applications in the last two decades.", "With the guaranteed stability property, it catalyses the widely applications of MPC in many sectors.", "Despite its huge success, this stability guaranteed framework also has shortcomings.", "For example, enforcing terminal constraints in the online optimisation may significantly increase computational burden so researchers look into how to remove terminal constraints under certain conditions; e.g.", "[31] [25].", "Other shortcomings that are related to the topic of this paper are discussed as follows.", "Difficult to search a terminal weight.", "Essentially, finding a terminal weight to cover the cost-to-go is similar to searching a control Lyapunov function candidate for a constrained system, which is, in general, not trivial.", "Attempts have been made to develop more efficient computational tools, e.g.", "[28].", "Performance may be distorted by an inappropriate terminal weight.", "With the addition of a terminal weight, the cost function now consists of two terms: stage cost within the horizon and a terminal weight.", "If the terminal weight is much larger than the summed stage cost, the influence of the summed stage cost would be too small during the optimisation.", "It could easily happen since in order to make sure that a terminal weight covers the cost-to-go, a terminal weight may have to be selected quite conservatively, particularly for systems where it is difficult to calculate or estimate the cost-to-go.", "It is known that a large terminal weight may give a large stability attraction region but poor performance.", "This fact has been exploited by a number of researchers in striking a good balance between the size of the feasible region (or terminal region) and the performance, e.g.", "[30] [11].", "It may be quite conservative in stability analysis.", "It is widely evident that an MPC algorithm is stable in operation but does not satisfy the conditions in the current stability guaranteed framework.", "Another observation is the stability region provided by the framework is often much smaller than the real stability region.", "This is because the stability conditions are sufficient conditions, which could be quite conservative as shown in this paper later.", "Within the same terminal weight based MPC setting, alternative stability conditions have been developed in an attempt to reduce the conservativeness, e.g.", "[10].", "It is not able to cope with MPC with a short horizon but without terminal weight.", "Essentially, the current `standard' MPC framework requires that the decrease of the terminal cost is larger than that of the stage cost as the horizon moves forward.", "As pointed out in [14], most of MPC algorithms widely applied in engineering actually do not have a terminal weight; for example, [37] [3] [33] [21].", "Early MPC schemes also don't have terminal weight; for example, generalised predictive control (GPC) [12], but were proven to be quite successful in practical applications.", "There is a clear gap between available stability analysis tools and practical applications.", "Stability conditions do not depend on the length of the horizon.", "It is well known that the length of horizon plays a significant role in the performance and stability of MPC, and is a key parameter in MPC design and tuning.", "It is often found that an MPC algorithm with a longer horizon can stabilise a system that cannot be stabilised with a shorter horizon.", "Actually in some MPC algorithms such as GPC, the length of the horizon is used as one of the most important tuning parameters; for example, see [12] and [32].", "However the stability conditions in the current stability guaranteed MPC framework is not related to the length of horizon.", "That is, if certain conditions are satisfied, the corresponding MPC algorithms are stable, irrespective of the length of the horizon.", "So it cannot reflect the influence of the length of the horizon on stability.", "It has the advantage of simplicity in analysis and design, but inevitably it is too conservative and cannot serve the purpose of understanding the influence of the length of horizon on MPC performance and stability.", "In the last two decades, researchers have proposed various ways to relax the requirements in the `standard' MPC framework by addressing one or two of the above shortcomings.", "Please refer to books and survey papers for more comprehensive overview of the topic, e.g.", "[38], [20], [14] and extensive references therein.", "Regarding addressing the issues caused by using a terminal weight, broadly speaking, there are two main approaches.", "One is to find a generalised terminal weight that is not necessary to be a control Lyapunov function if certain property holds, e.g.", "[24], [18], [20], [14].", "For instance, stability for unconstrained discrete-time systems under MPC algorithms was established in [18], where it does not require the terminal cost to be a local control Lyapunov function if certain assumptions are satisfied.", "The other is to drop terminal weight completely, e.g.", "[1] [19], [20], [14].", "This normally requires that the horizon is sufficiently large, or equality terminal constraints have to be included.", "There is little work on stability analysis for MPC with a limited horizon and without terminal weight, except [7] where a stability condition for a finite horizon MPC is reported and the horizon could be any length including only one step ahead.", "Table 1 in [14] provides a nice summary of the state-of-the art about MPC stability.", "These theoretic works made important contributions in improving the current stability guaranteed MPC algorithms and providing useful tools for stability analysis of MPC algorithms.", "However, despite of all the effort, the fundamental principle behind the above stability guaranteed MPC framework largely remains unchanged, and algorithms therefore still suffer some or all of the listed shortcomings.", "For example, to avoid the use of a terminal weight, the requirement of a sufficient large horizon [14] is unrealistic to implement for many applications.", "Essentially, the online constrained optimisation problem involved in MPC is a dynamic programming problem.", "Despite all the progress in developing more efficient optimisation algorithms, it is still difficult, if not impossible, to solve it in real-time online for many engineering problems with a long horizon.", "It also demands unacceptable computational resources so increase the cost of a product or service.", "Furthermore, with the widespread application of MPC or similar concepts (e.g.", "receding horizon, or rolling horizon), MPC is penetrating new applications and new sectors where the current framework for stability guarantee is even more difficult to apply, to name a new, air traffic control [21], robotics [3] smart grid [33], autonomous driving [37],[42].", "Quite often it has been found that the above terminal weight based MPC framework is not applicable for these quite complicated and large scale systems in the presence of uncertainty; for example, no terminal weight has been used in [37] [3] [33] [21].", "One significant new direction of applying MPC is in autonomous systems for planning and decision making [40] [42].", "In this space, the boundary between control, decision making and planning are blur or even does not exist.", "Indeed, rolling horizon or receding horizon is not a new concept in operation research and management science, and has been there for several decades [39].", "It provides an intuitive and natural way to cope with dynamic and uncertain operational environment.", "In these applications, it is impossible to find a terminal weight to cover the cost-to-go in the operation due to the complexity, scale, or the nature of the operation.", "Another recent trend is the merge of control and machine learning where MPC is regarded as a promising vehicle.", "For example, there is a renewed interest in the dual control concept [15] [16].", "By incorporating it with MPC, MPC methods with active learning are developed [36].", "However, MPC with active uncertainty learning is very computationally involved.", "It is even difficult to solve a multistage optimisation problem online in this MPC setting so most of research in this area are on developing suboptimal solutions for this problem.", "Therefore, it is almost impossible to estimate the cost-to-go for real engineering applications.", "It is imperative to develop tools that are able to provide proven properties such as stability for these MPC applications.", "Removing a terminal weight from the performance index creates a significant technical challenge.", "With a terminal weight to cover the cost-to-go, the optimisation problem with a finite horizon involved in MPC actually approximates an optimisation problem with an infinite horizon.", "Therefore stability could be established by the virtue of the well known fact that optimality implies stability for infinite horizon optimisation under mild conditions (e.g.", "controllability and observability).", "This promising property is not applicable for MPC without terminal weight.", "So a radical approach is required.", "Indeed, our intuitive idea is that stability may still be able to establish by showing the first stage cost is larger than the immediate future stage cost just outside the horizon so the summed stage cost decreases with the horizon moving forward.", "If the summed stage cost converges to zero, the closed-loop system under MPC must be stable under certain condition.", "In this paper, we also modify the performance index without the change of the optimisation problem and the corresponding optimal control sequence.", "Through this approach, we are able to establish stability of finite horizon MPC without terminal weight (Point 4 listed above).", "The resulted stability conditions are horizon dependent (Point 5) and much less conservative (Point 3).", "A simple first order numerical example shows this condition is actually necessary and sufficient for stability of this specific MPC algorithm.", "A new MPC algorithm without terminal weight is then proposed based on the proposed stability conditions.", "It is shown that similar to terminal weight based MPC algorithms, this algorithm has a nice property, namely, its feasibility implies stability.", "Therefore, this new MPC algorithm is referred to as MPCS (MPC within Stability guarantee) in this paper.", "We then investigate the recursively stability of MPCS by defining a new level set that is calculated through optimising one-step stage cost.", "The condition of the recursive feasibility of MPCS is then established.", "This work avoids the need of the need of searching a terminal weight to achieve stability guarantee (Point 1) and at each step, only the sum of the stage cost that corresponds to the desired performance is optimised (Point 2).", "Interestingly, the proposed approach and the modified value function do not only work for MPC without terminal weight, but also for MPC with terminal weight.", "It turns out to be true that this condition is always satisfied as long as the stability conditions in establishing the classic MPC schemes for stability guarantee (e.g.", "[35], [8], [17]), so-called the descent property of the value function [34], are met (Point 3).", "This paper is organised as follows.", "In Section , an MPC problem for constrained nonlinear systems without terminal weight is described.", "Necessary preliminaries are presented.", "New stability conditions which explicitly depend on the length of the horizon are developed in Section .", "A simple numerical example is used to illustrate the effectiveness of the new conditions.", "In Section , based on the new conditions, MPCS, a new stability guaranteed MPC algorithm without terminal weight is presented.", "Its stability and recursive feasibility are investigated.", "For the sake of completeness, this modified value function is also applied to developing stability conditions for MPC with terminal weight in Section .", "The relationships between new stability condition and the well known existing stability results are discussed to provide more insight.", "Numerical examples are provided in Section which shows for an unstable MPC algorithm, stability could be restored in several ways by using the results in this paper including extending the length of horizon, adjusting state and (or) control weights, or using the proposed MPCS.", "Finally this paper ends up with conclusions in Section" ], [ "MPC for Nonlinear Systems", "Consider a constrained nonlinear system $ x(k+1)=f(x(k),u(k))$ with the input constraints $ u \\in \\mathcal {U} \\triangleq \\lbrace & & u=[u_1,\\ldots ,u_m]^T \\in R^m: \|u_i\| \\le \\bar{u}_i, \\nonumber \\\\& & i=1, \\ldots , m \\rbrace $ and state constraints $ x \\in \\mathcal {X} \\mbox{ and }0 \\in \\mathcal {X}$ The receding horizon performance index for online optimisation is described by $ J^{\\prime }(x(k),U^{\\prime }(k\|k)) & = & \\sum _{i=0}^{N}l^{\\prime }(x(k+i\|k),u(k+i\|i))$ where $U^{\\prime }(k\|k)=[u(k\|k)^T, \\ldots , u(k+N\|k)^T]^T$ denote the calculated control sequence at the time instant $k$ based on the measurement $x(k)$ , and $x(k+i\|k), i=1,\\ldots ,N$ , are corresponding predicted state sequence in the receding horizon.", "Furthermore, $x(k\|k)=x(k)$ .", "This is a finite horizon performance index without terminal weight and our discussion will be extended to MPC with terminal weight in Section .", "For this widely used performance index, it is observed that ion the absence of the product term of $u$ and $x$ , the cost associated with the initial state $x(k)$ can not be changed by the control sequence $U^{\\prime }(k\|k)$ and the optimal control at $k+N$ is not relevant since it does not change the state sequence and other elements in the optimal sequence $U^{\\prime }(k\|k)$ (actually, $u^(k+N\|k)=0$ in most of time).", "Indeed, at $i$ th step, the choice of $u(k+i\|k)$ only affects the cost associated with $u(k+i\|k)$ and $x(k+i+1\|k)$ but not the cost associated with $x(k+i\|k)$ .", "Motivated by this intuitive but important observation, we rewrite the performance index (REF ) in the form of a new stage cost $l(x(k+i+1),u(k+i))$ as $ J(x(k), U(k\|k)) & = & \\sum _{i=0}^{N-1}l(x(k+i+1\|k),u(k+i\|k))$ where $U(k\|k)=[u(k\|k)^T, \\ldots , u(k+N-1\|k)^T]^T$ .", "At each $k$ , the online optimisation problem is solved by minimising the cost (REF ), that is, $ J^(x(k))= \\min _{u(k+i\|k) \\in \\mathcal {U}} J(x(k),U(k\|k))$ subject to system dynamics (REF ), the control constraint (REF ) and the state constraint (REF ).", "The optimal control sequence and the corresponding state trajectory are denoted by $U^(k\|k)=[u^(k\|k)^T, \\ldots , u^(k+N-1\|k)^T]^T$ and $x^(k+1\|k), \\ldots , x^(k+N\|k)$ , respectively.", "We use the subscript $$ to indicate the optimal solution, the optimal control sequence or states under the optimal control sequence, unless otherwise stated.", "It shall be highlighted that minimisation of the cost function (REF ) or (REF ) yields the same control sequence so the resulted MPC algorithms have the same behaviours.", "However, as shown late, this makes a significant difference in stability analysis.", "Since only the first control action is implemented in MPC, one has $u(k)=u^(k\|k)$ and the real system at time $k+1$ as $x(k+1)=f(x(k),u^(k\|k))$ We will investigate stability of the above described MPC without terminal weight scheme.", "To present our results, the following preliminaries are required.", "Assumptions on the nonlinear system (REF ) and the performance index (REF ) are made as follows.", "A1: $f(0,0)=0$ ; A real-valued scalar function $\\phi : R_+ \\rightarrow R_+$ belongs to class $\\mathcal {K}$ if it is continuous, strictly increasing and $\\phi (0)=0$ .", "$\\phi $ belongs to class $\\mathcal {K}_\\infty $ if $\\phi $ belongs to class $\\mathcal {K}$ and is radically bounded.", "We impose the following assumption on the modified value function of the optimisation problem (REF ).", "A2: There exist $\\alpha _i \\in \\mathcal {K}_\\infty , i=1,2,$ such that $\\alpha _1(\\Vert x\\Vert ) \\le J^(x) \\le \\alpha _2(\\Vert x\\Vert )$ for all $x \\in \\mathcal {X}$ .", "To facilitate the discussion, an auxiliary optimisation problem with one-step stage cost is also defined as $ m(x(k)) \\triangleq \\min _{u(k) \\in \\mathcal {U}} l(x(k+1),u(k)); x(k+1)=f(x(k),u(k)) \\in \\mathcal {X}$ This auxiliary value function $m(x)$ will be used extensively in this paper in establishing stability and developing stability guaranteed MPC algorithms.", "Similarly, it is also assumed that the value function $m(x)$ holds the same properties as the value function $J^(x)$ .", "A3: There exist $\\alpha _i \\in \\mathcal {K}_\\infty , i=1,2, $ such that $\\alpha _1(\\Vert x\\Vert ) \\le m(x) \\le \\alpha _2(\\Vert x\\Vert )$ for all $x \\in \\mathcal {X}$ .", "(positively invariant set [41]) A set $\\mathcal {X}_i$ that contains the origin as its interior is called as a positively invariant set for a nonlinear autonomous system $x(k+1)=g(x(k))$ if $g(x)$ belongs to $\\mathcal {X}_i$ for all $x \\in \\mathcal {X}_i$ .", "It can be shown that Assumption A2 and A3 is satisfied if the stage cost function satisfy the following condition (e.g.", "see [27]).", "There exist $\\alpha _1, \\alpha _2 \\in \\mathcal {K}$ , such that $l(x,u) \\ge \\alpha _1(\\Vert x\\Vert )$ for all $x \\in \\mathcal {X}$ and all $u \\in \\mathcal {U}$ , $l(x,0) \\le \\alpha _2(\\Vert x\\Vert )$ for all $x \\in \\mathcal {X}$ .", "One may argue that following the observation that when $u^(k+N\|k)$ is always zero, the associated state related cost at $N$ stage, i.e.", "$l(x(k+N\|k),0)$ can be considered as a terminal weight.", "Then the current terminal weight based stability conditions could be applied.", "As shown in [7], this approach does not work since this terminal weight does not cover the cost-to-go as required." ], [ "Stability for MPC without terminal weight", "With the problem formulation and preliminaries in Section , we are ready to present a stability condition for MPC without terminal weight in Theorem .", "[Stability of MPC without terminal weight] Consider a nonlinear system (REF ) with constraints (REF ), (REF ) and the performance index (REF ) without terminal weight, satisfying Assumption A1-A2.", "Suppose that it is feasible at an initial state $x(0)=x_0$ and there exists a control $u(k+N\|k) \\in \\mathcal {U} $ such that $ l(x(k+N+1\|k),u(k+N\|k))-l(x^(k+1\|k),u^(k\|k)) \\le 0$ where $x(k+N+1\|k)=f(x^(k+N\|k),u(k+N\|k)) \\in \\mathcal {X}$ and $x^(k+N\|k)$ is the terminal state under the optimal control sequence.", "Then the following properties hold the optimisation problem is recursively feasible; the closed-loop system stemming from the MPC algorithm is stable.", "Suppose that the optimisation problem (REF ) is feasible at time $k$ .", "The optimal control sequence is denoted by $u^(k\|k), \\ldots , u^(k+N-1\|k)$ , the corresponding state trajectory $x^(k+1\|k), \\ldots , x^(k+N\|k)$ , and the optimal cost $J^(x(k))$ .", "To establish the stability of the MPC algorithms, the optimal value function of the performance index (REF ) is chosen as a Lyapunov function candidate, i.e.", "$ V(x) =J^(x).$ Stability is established by showing the modified value function $J^(x)$ has a monotonicity property, that is $ V(x(k+1))-V(x(k)) \\le 0$ for all $k \\ge 0$ where $ V(x(k+1)) = J^(x(k+1))$ and $x(k+1)$ denotes the system state at time $k+1$ under $u(k)=u^(k\|k)$ .", "Following the principle of optimality, one has $ & & J^(x(k)) \\nonumber \\\\& = & \\sum _{i=0}^{N-1} l(x^(k+i+1\|k), u^(k+i\|k)) \\nonumber \\\\& = & l(x^(k+1\|k),u^(k\|k)) +\\sum _{i=1}^{N-1}l(x^(k+i+1\|k), u^(k+i\|k)) +\\nonumber \\\\& & l(x(k+N+1\|k),u(x+N\|k))- l(x(k+N+1\|k),u(x+N\|k))$ with $x(k+N+1\|k)=f(x^(k+N\|k),u(k+N\|k))$ for any $u(k+N\|k) \\in \\mathcal {U}$ .", "Condition (REF ) being satisfied implies that there exists a control $\\widetilde{u}(k+N\|k) \\in \\mathcal {U}$ such that $\\widetilde{x}(k+N+1\|k)=f(x^(k+N\|k),\\widetilde{u}(k+N\|k)) \\in \\mathcal {X}$ under the optimal control sequence $U^(k\|k)$ and corresponding state trajectory.", "Invoking this condition into (REF ) yields $ & & J^(x(k)) \\nonumber \\\\ & \\ge &\\sum _{i=1}^{N-1}l(x^(k+i+1\|k), u^(k+i\|k)) + l(\\widetilde{x}(k+N+1\|k),\\widetilde{u}(k+N\|k)) \\nonumber \\\\$ The control sequence $U(k+1\|k+1) \\triangleq [u^(k+1\|k), \\ldots , u^(k+N-1\|k), \\widetilde{u}(k+N\|k)] \\in \\mathcal {U}$ satisfy all the constraints since $x^(k+2\|k), \\ldots , x^(k+N\|k), \\widetilde{x}(k+N+1\|k) \\in \\mathcal {X}$ .", "With $x(k+1)=x^(k+1\|k)$ under $u(k)=u^(k\|k)$ , it forms a feasible control sequence for the optimisation problem (REF ) at time $k+1$ .", "Hence we establish Item (1) about recursive feasibility.", "Combining the principle of optimality at time $k+1$ and condition (REF ) obtains $J^(x(k)) \\ge J^(x(k+1))$ Hence the monotonicity of the modified value function or condition (REF ) is established.", "Stability can be established by following the standard argument in classic stability guaranteed MPC, e.g.", "[35], [38].", "Item (2) is proved.", "Asymptotic stability can be achieved by replacing inequality in condition (REF ) with strict inequality or under other mild conditions [38].", "The similar argument holds for all the following results in this paper so will not be repeated.", "Condition (REF ) states that if there exists a feasible control such that the stage cost $l(k+N+1\|k),u(k+N\|k))$ is less than the first stage cost under the optimal control $u^(k\|k)$ , i.e.", "$l(x^(k+1\|k),u^(k\|k))$ , stability could be established.", "This is because the summed stage cost keeps decreasing with time so our approach establishes stability through the decrease of the summed stage cost and there is no requirement for a terminal cost to cover the cost-to-go.", "Stability is established by the descent property of the value function in the standard MPC framework which essentially approximates an optimisation problem with an infinite horizon.", "So on the philosophical level, our approach is radically different from the current terminal weight based MPC standard framework.", "Another difference is a different Lyapunov function candidate is used in this paper after modifying the stage cost.", "The original value function $J^{\\prime }(x)$ is used as a Lyapunov function candidate in the traditional `standard' MPC framework; e.g.", "[35] [8] [17], [20].", "To establish the stability condition in Theorem , a modified value function $J^(x)$ is employed.", "The difference between them is $l(x,0)$ .", "This is based on the following observation: changing the cost associated with initial state in the online optimisation problem involved in MPC does not alter the solution of the optimisation problem so the optimised control sequence and the behaviour of the MPC algorithm remains the same.", "However, it changes the optimal cost function, i.e.", "the value function.", "So when the value function is employed as a Lyapunov function candidate, this essentially implies a different Lyapunov function candidate is used in this paper in establishing stability conditions.", "This observation has been initially made in [7] and [10] but has not been fully exploited.", "In order to satisfy condition (REF ), its first item shall be made as small as possible over the admissible control set.", "Therefore, the existence of control $u \\in \\mathcal {U}$ such that this condition is satisfied can be checked by minimising the one step stage cost at $k+N$ with $x^(k+N\|k)$ , that is, $ m(x^(k+N\|k))=\\min _{u(k+N\|k) \\in \\mathcal {U}} l(x(k+N+1\|k),u(k+N\|k))$ subject to $x(k+N+1\|k) \\in \\mathcal {X}$ where the notation defined in Eq.", "(REF ) is used.", "Define a sublevel set as $ \\mathcal {X}_T \\triangleq \\left\\lbrace x \\in R^n: m(x) \\le \\alpha , x \\in \\mathcal {X} \\right\\rbrace $ for any $\\alpha >0 $ .", "Under Assumption A3, one has $0 \\in \\mathcal {X}_T$ .", "It is ready to present the following results.", "[Terminal set and constraints] Consider a nonlinear system (REF ) with constraints (REF ), (REF ) and the performance index (REF ) without terminal weight, satisfying Assumption A1-A3.", "The closed-loop system under the MPC algorithm is recursively feasible and stable if it is feasible with respect to an initial state $x_o \\in \\mathcal {X}$ and the terminal state $x^(k+N\|k)$ satisfies $ x^(k+N\|k) \\in \\mathcal {X}_T$ defined in Eq.", "(REF ) with $\\alpha \\triangleq m(x(k))$ at time $k \\ge 0$ .", "At time $k$ , with the state $x(k)$ , the optimisation problem (REF ) is solved, which gives the optimal control sequence and corresponding state.", "Since the terminal state $x^(k+N\|k)$ belongs to the set $\\mathcal {X}_T$ with $\\alpha \\triangleq m(x(k))$ , which implies $ m(x^(k+N\|k)) \\le m(x(k))$ According to the definition of $m(x)$ in Eq.", "(REF ), it is given by minimising one step stage cost at $k$ so we have $ l(x(k+1),u(k)) \\ge m(x(k))$ for any $u(k) \\in \\mathcal {U}$ .", "This also holds for $u^(k\|k)$ and $x^(k+1\|k)$ with $x^(k+1\|k)=f(x(k),u^(k\|k))$ .", "Therefore combining Eq.", "(REF ) and Eq.", "(REF ), one obtains there exists a control $u(k+N\|k) \\in \\mathcal {U}$ such that condition (REF ) is satisfied and the corresponding $x(k+N+1\|k) \\in \\mathcal {X}$ .", "The recursive feasibility and stability follows from Theorem .", "The set $\\mathcal {X}_T \\subseteq \\mathcal {X}$ actually defines a terminal set for the terminal state.", "If the terminal state $x^(k+N\|k)$ arrives at $\\mathcal {X}_T$ , the MPC algorithm yielded by online minimising (REF ) or (REF ) (without terminal weight) is stable.", "Both condition (REF ) and Theorem are horizon dependent.", "That is, they can be used to investigate the influence of the length of horizon on stability and feasibility.", "It shall be noted that a horizon dependent stability condition for MPC of linear and unconstrained systems was presented in [2].", "The condition is derived by using the stabilising property of the solution of algebraic Ricatti equation, and it is obvious that this tool could not be extended to nonlinear systems or systems with constraints.", "We end this section by illustrating the effectiveness of Theorem with a simple example.", "It is shown that our condition is not only sufficient but also necessary for this simple MPC problem." ], [ "An illustration example", "Consider an unconstrained first order linear system [7] $x(k+1)=ax(k)+bu(k)$ and an one step horizon performance index $ J(k)= qx(k+1\|k)^2+ru(k\|k)^2$ is employed with $q>0$ and $r \\ge 0$ .", "This simple example serves three purposes.", "First it is used to check the correctness of the stability condition presented in this section since an analytical solution could be found.", "Secondly, it is used to assess the conservativeness of the proposed condition.", "Thirdly, it is used to compare with the result in [7] where a stability condition for MPC without terminal weight was reported.", "For this unconstrained linear system, the analytical solution for the MPC optimisation problem (REF ) can be obtained with $J^(x(k))=l(x^(k+1\|k),u^(k\|k))=\\frac{a^2qr}{r+bq^2}x(k)^2$ and $ u^(k\|k)=\\frac{-abq}{r+qb^2}x(k);\\qquad x^(k+1\|k)=\\frac{ar}{r+bq^2}x(k)$ With the above $x^(k+1\|k)$ , solving a similar optimisation problem (REF ) with $N=1$ gives the lower bound of $l(x(k+N+1\|k),u(k+N\|k))$ as $ l(x^(k+2\|k),u^(k+1\|k))=\\frac{a^2qr}{r+bq^2}x^(k+1\|k)^2$ Applying Theorem and substituting (REF ) and (REF ) in condition (REF ), one concludes that the closed-loop system under MPC is asymptotically stable if the following condition holds $\\frac{a^2r^2}{(r+qb^2)^2}<1$ i.e.", "$ \\left\| \\frac{ar}{r+b^2q} \\right\|<1$ Since the optimal control $u^(k\|k)$ is given in (REF ), after applying $u(k)=u(k\|k)^$ , the closed-loop system under MPC is given by $x(k+1) = (a-\\frac{b^2q}{r+b^2q}a)x(k) = \\frac{ar}{r+b^2q}x(k)$ It is asymptotically stable if and only if $\\left\| \\frac{ar}{r+b^2q} \\right\|<1$ Therefore, for this simple example, the stability condition provided by Theorem is actually sufficient and necessary.", "This example has been studies in [7] as a finite horizon MPC without terminal weight.", "Stability condition was given by $ \\frac{a^2r}{r+b^2q}<1$ It can be shown that the stability condition proposed in this paper, i.e.", "Eq.", "(REF ), is much less conservative than that in [7].", "This is because for any stable system, i.e.", "$\|a\| \\le 1$ , both Eq.", "(REF ) and Eq.", "(REF ) are satisfied.", "For any unstable open-loop system, i.e.", "$\|a\| >1$ , condition (REF ) implies condition (REF ).", "This is not surprised since, as shown above, condition (REF ) is sufficient and necessary.", "The same result can be derived for this first order example using Theorem ." ], [ "Finite Horizon MPC with stability guarantee (MPCS)", "The stability conditions in Theorem and can be used to analyse stability of an existing MPC algorithm.", "However, they are not very easy to directly check these conditions since the properties of the optimal solution is required.", "Instead, by making the use of these conditions, this section presents a new finite horizon MPC algorithm without terminal weight that can guarantee stability.", "Since stability can be guaranteed for this algorithm as long as it is feasible, it is referred to as MPCS (MPC with Stability guarantee) in this paper for the sake of simplicity.", "MPCS: MPC with Stability guarantee Define the system dynamics (REF ), constraints $\\mathcal {X}$ and $\\mathcal {U}$ , the length of horizon $N$ , and the stage cost $l(x(k+1+i),u(k+i))$ as in the performance index (REF ) without terminal weight.", "For an initial state $x_0$ , let $k=0$ , $x(0)=x_0$ .", "At time $k$ , calculate $\\alpha (k)=m(x(k))$ Solve the following optimisation problem $ J^(x(k))= \\min _{u(k+i\|k) \\in \\mathcal {U}, i=0,\\ldots , N-1} J(x(k),U(k\|k)) \\mbox{ for } N>1$ or $ J^(x(k))= \\min _{u(k\|k), u(k+1\|k) \\in \\mathcal {U}} J(x(k),u(k\|k),u(k+1\|k)) \\mbox{ for } N=1$ subject to system dynamics (REF ), $ l(x(k+2\|k),u(k+1\|k)) \\le \\alpha (k)$ $x(k+i+1\|k) \\in \\mathcal {X} ; \\qquad i=0,\\ldots ,N-1$ Apply $u(k)=u^(k\|k)$ .", "Let $k=k+1$ and go to Step 2.", "The main difference between $N=1$ and $N>1$ is that only $u(k\|k)$ is to be optimised with $N=1$ and condition (REF ) is not applicable.", "It is modified as (REF ).", "However, the optimisation can be kept in the same form as in (REF ) by observing (REF ) reducing to $ m(x(k+1)) \\le \\alpha (k).$ so only $u(k\|k)$ is to be optimised.", "As discussed later, this requires a high computational burden.", "Different from the conventional MPC without terminal weight (REF ), MPCS has an extra constraint (REF ) to ensure stability.", "[Feasibility implies stability] Consider a nonlinear system (REF ) with constraints (REF ), (REF ) and the performance index (REF ), satisfying Assumption A1-A3.", "The closed-loop system stemming from the MPCS algorithm is stable (or asymptotically stable if strictly inequality is imposed in (REF ) ) if it is recursively feasible.", "Define a set $\\mathcal {X}_T(k) \\subseteq \\mathcal {X}$ as in (REF ) with $\\alpha =\\alpha (k) \\ge 0$ .", "Suppose that the optimal control sequence given by solving the optimisation problem (REF ) at time $k$ is denoted by $u^(k\|k), \\ldots , u^(k+N-1\|k)$ .", "Constraint (REF ) implies this control sequence and corresponding state $x^(k+2\|k) \\in \\mathcal {X}$ satisfy $l(x^(k+2\|k),u^(k+1\|k)) \\le \\alpha (k)$ Following the definition of $m(x)$ , for $x^(k+1\|k)$ , we have $l(x^(k+2\|k),u^(k+1\|k)) \\ge m(x^(k+1\|k))$ Since $u^(k\|k)$ is implemented, this gives $x(k+1)=x^(k+1\|k)$ , and $\\alpha (k+1)=m(x(k+1))$ .", "Combining them with the above two inequalities obtains $ \\alpha (k+1) \\le \\alpha (k)$ Let the control law derived from the MPCS be denoted by $u(k)=K_{MPCS}(x(k))$ .", "The closed-loop system under MPCS is given by $x(k+1)=f(x(k),K_{MPCS}(x(k))$ .", "The set $\\mathcal {X}_T \\subseteq \\mathcal {X}$ defined in Eq.", "(REF ) with $\\alpha =m(x(0))=m(x_0)$ is a positively invariant set for the closed loop system under MPCS if it is recursively feasible.", "When a strict inequality is imposed in constraint (REF ) of MPCS by replacing it with $ l(x(k+2\|k),u(k+1\|k)) \\le \\alpha (k) -\\delta $ where $\\delta $ is a small positive number, it can be shown that Eq.", "(REF ) holds with a strict inequality.", "This implies that, the set $\\mathcal {X}_T(k)$ continuously shrinks over time $k$ until reaching the origin with $\\alpha =0$ by the virtue of the strictly increasing property of $m(x)$ in Assumption A3 and $m(x)=0$ only for $x=0$ .", "Hence it is asymptotically stable.", "It can be observed from the stability analysis in Theorem that as long as the optimisation process is feasible, condition (REF ) is met.", "Implementing the corresponding control sequence implies the decrease of the modified value function so stability can be assured.", "With the help of Theorem , stability analysis is converted into a test of feasibility of optimisation (REF ), which can be performed by numerical computation, depending on which optimisation tools are used in solving the optimisation problem.", "In general, the feasibility of the constrained optimisation problem (REF ) needs to be checked using optimisation algorithms.", "Hereafter, we present a systematic (but conservative) way to ensure recursive feasibility.", "We now choose a special form of the feasibility set as $ {\\mathcal {X}}_f (\\alpha ) & = & \\lbrace x(k) \\in R^n:m(x(k)) \\le \\alpha , \\mbox{there exists } u(k) \\in {\\mathcal {U}} \\nonumber \\\\& & \\mbox{ such that } x(k+1) \\in \\mathcal {X}_f(\\alpha ) \\rbrace $ where $\\alpha >0$ .", "Hence $\\mathcal {X}_f$ defines a control invariant set for system (REF ).", "[Recursive feasibility] Consider a nonlinear system (REF ) with constraints (REF ), (REF ) and the performance index (REF ), satisfying Assumption A1-A3.", "Suppose that an initial $x_0 \\in \\mathcal {X}_f^0$ defined in Eq.", "(REF ) with some $\\alpha _0 >0$ .", "For the optimisation problem (REF ), condition (REF ) is replaced with $ x(k+1\|k) \\in \\mathcal {X}_f(\\alpha (k)) \\mbox{ defined in Eq.", "(\\ref {eq:feasibility})}$ The MPCS algorithm is recursively feasible and the closed-loop system under MPCS is stable with respect to the initial set $\\mathcal {X}_f^0(\\alpha _0)$ .", "First calculate $\\alpha (0)=m(x(0))$ with $x(0)=x_0$ .", "Since $x(0) \\in \\mathcal {X}_f(\\alpha (0)) = \\mathcal {X}_f^0(\\alpha _0)$ defined in (REF ) with $\\alpha (0)=\\alpha _0$ , there exists $u(0\|0) \\in \\mathcal {U}$ such that $x(1\|0) \\in \\mathcal {X}_f(\\alpha (0))$ .", "Therefore, the constraint (REF ) is satisfied and the optimisation is feasible at time 0.", "The optimal control sequence also satisfies the condition (REF ).", "Following the same argument as Theorem , we have $x(k+1) \\in \\mathcal {X}_f(\\alpha (1))$ with $\\alpha (1)=m(x(1))$ .", "Recursively applying the above discussion, it can be shown that the optimisation problem is feasible.", "Repeating this process, the recursive feasibility can be established.", "Stability follows from Theorem .", "Theorem establishes a condition for ensuring the recursive feasibility of MPCS.", "However, there is a difference between the algorithm in Theorem and MPCS.", "The constraint (REF ) requires to find $m(x(k+1\|k))$ within the optimisation loop.", "This involves the solution of minimisation of an one-step stage cost function as in (REF ).", "Therefore, it could be quite computational demanding, particularly for a large scale or complicated system.", "This is avoided by replacing it with the constraint (REF ) where there is no such a requirement but just evaluating the stage cost $l(x(k+2\|k),u(k+1\|k))$ for a candidate $u(k+1\|k)$ , which significantly saves the computational burden.", "There are a number of techniques developed for estimating the feasibility/terminal region of the proposed MPC schemes that could be extended to estimate the feasible region $\\mathcal {X}_f^0$ ; e.g.", "[6], [4].", "First order example continued To illustrate the effectiveness of MPCS, we revisit the example studied in Section REF .", "It will be shown that the MPCS algorithm with $N=1$ is stable for any $a, b \\ne 0, q>0, r \\ge 0$ .", "For this simple problem, condition (REF ) reduces to checking if there exists a control $u$ such that $(ax(k)+bu(k))^2=x(k+1)^2 \\le x(k)^2.$ This is always possible, for example, $u(k)=-\\frac{a}{b}x(k)$ .", "This simple example clearly demonstrates that a much larger stability region or wider range of choice of parameters in the performance index can be achieved in MPCS, in comparison with the original finite horizon MPC algorithm.", "For this example, the system parameters and the choice of state and control weights must satisfy (REF ) in order to achieve stability under the original MPC.", "But MPCS can stabilise the system with almost any combination of systems and weights in the cost function." ], [ "Extension to MPC with terminal weight", "For the sake of completeness, we extend our stability conditions for MPC without terminal weight to the case of MPC with terminal weight.", "We first establish a new condition using the same technique as in Section .", "It is shown that the current widely used descent property in establishing stability (e.g.", "[35]) is actually a subset of our new stability condition and the new condition is much less conservative.", "This is because our new condition is met as long as the existing stability conditions are satisfied.", "A classic predictive performance index is given by $ & & J_w^{\\prime }(x(k),U(k\|k)) \\nonumber \\\\&= & \\sum _{i=0}^{N-1}l^{\\prime }(x(k+i),u(k+i))+p(x(k+N)) \\nonumber \\\\$ where the subscript $w$ is used to indicate the cost function with terminal weight and $p(x(k+N))$ denotes the terminal weight.", "In the same fashion as we treat the cost function without terminal weight in Section , the cost function (REF ) is rewritten as $ & & J_w (x(k),U(k\|k)) \\nonumber \\\\&= & \\sum _{i=0}^{N-2}l(x(k+i+1),u(k+i))+l_T(x(k+N), u(k+N-1)) \\nonumber \\\\$ where $l_T$ is composed of the terminal weight $p(x(k+N))$ and the control related cost at the last stage.", "[New stability condition for MPC with terminal weight] Consider a nonlinear system (REF ) with constraints (REF ), (REF ) and the performance index (REF ) satisfying Assumption A1-A2.", "The closed-loop system under the MPC algorithm is asymptotically stable if it is feasible and there exists a control $u(k+N\|k) \\in \\mathcal {U}$ such that $ & & l(x^(k+N\|k),u^(k+N-1\|k)) - l(x^(k+1),u^(k\|k)) \\nonumber \\\\&+ &l_T(x(k+N+1\|k),u(k+N\|k)) -l_T(x^(k+N\|k), u^(k+N-1\|k)) \\le 0 \\nonumber \\\\$ with $x(k+N+1\|k)=f(x^(k+N\|k),u(k+N\|k)) \\in \\mathcal {X}$ .", "The proof follows the same techniques as that of Theorem .", "Taking into account the difference between the stage cost $l$ and $l_T$ , the value function at $k$ could be rewritten as $ & & J^(x(k)) \\nonumber \\\\& = & l(x^(k+1), u^(k\|k))+\\sum _{i=1}^{N-2}l(x^(k+i+1\|k), u^(k+i\|k)) \\nonumber \\\\& & + l_T(x^(k+N\|k), u^(k+N-1\|k)) \\nonumber \\\\& & +l(x^(k+N\|k), u^(k+N-1\|k))- l(x^(k+N\|k), u^(k+N-1\|k)) \\nonumber \\\\& & + l_T(x(k+N+1\|k), u(k+N\|k)) - l_T(x(k+N+1\|k), u(k+N\|k))\\nonumber \\\\& \\ge & \\sum _{i=1}^{N-1}l(x^(k+i+1\|k), u^(k+i\|k)) + l_T(x(k+N+1\|k), u(k+N\|k)) \\nonumber \\\\$ The last inequality is implied by condition (REF ).", "In the same fashion as in Theorem , the monotonicity of the modified value function follows from the principle of optimality at $k+1$ with $x(k+1)=x^(k+1\|k)$ .", "Hence the results.", "When the terminal weight $l_T$ is chosen the same as the stage cost $l$ , stability condition (REF ) reduces to condition (REF ) in Theorem We are now in a position to compare the stability condition in Theorem with the widely used stability conditions, which lays out the foundation in developing stability guaranteed MPC (e.g.", "[35],[8] and [17]).", "The condition is given in terms of the cost function in the form as Eq.", "(REF ).", "It states that a MPC algorithm is asymptotically stable if there exists a terminal control $u(k+N) \\in \\mathcal {U}$ such that the following inequality holds for all $x \\in \\mathcal {X}$ $ && p(x(k+N+1))-p(x(k+N))\\nonumber \\\\& &+l^{\\prime }(x(k+N),u(k+N)) \\le 0$ For the sake of comparison, we now write the control weight and state weight in $l^{\\prime }(x,u)$ of $J_w^{\\prime }$ (REF ) separately as $l^{\\prime }(x(k+i),u(k+i))=q(x(k+i))+r(u(k+i))$ so we have $l(x(k+i+1),u(k+i))=q(x(k+i+1))+r(u(k+i))$ and $l_T(x(k+N),u(k+N-1))=p(x(k+N))+r(u(k+N-1))$ in the performance index (REF ).", "Substituting the above relationships to Eq.", "(REF ) obtains $ & & p(x(k+N+1\|k))-p(x^(k+N\|k))+q(x^(k+N\|k))+r(u(k+N\|k)) \\nonumber \\\\& & -q(x(k+1))-r(u(k)) \\le 0$ with $x(k+1)=x^(k+1\|k)$ and $u(k)=u^(k\|k)$ .", "The last two terms relate to the stage cost for moving from state $x(k)$ to $x(k+1)$ under MPC, which is always larger than zero unless $x=0$ and $u=0$ .", "Condition (REF ) and hence the condition (REF ) is always satisfied if condition (REF ) is met by the change of the notation slightly.", "It is now ready to summarise the above discussion in the following statement.", "For MPC problem with terminal weight as described in (REF ), the existing stability conditions ((REF ), i.e.", "[35]) is a special case of the stability condition in Theorem in the sense that the stability condition in Theorem is always satisfied if condition (REF ) is met.", "Corollary clearly confirms the discussion made in Introduction about the conservativeness of the current stability theory for MPC.", "The term $q(x(k+1))+r(u(k))$ , in general, could be quite significant, at least larger than the value function $m(x)$ given by minimising the first stage cost as in Eq.", "(REF ) (see the inequality in (REF ) and corresponding discussion)." ], [ "Examples for MPC without terminal weight", "MPC is now used widely for planning in robotics and autonomous systems where a lower level controller is designed to control a mobile robot/vehicle to follow a planned path under robot dynamics, disturbance and uncertainty; e.g.", "[3] [42].", "In the planning level, a more detailed dynamic model is ignored.", "Consider an autonomous agent (a mobile robot, or a vehicle) whose behaviour was abstracted as $ x(k+1)=Ax(k)+Bu(k)$ with $x=[x_1,x_2]^T$ is the position of the agent and $u \\in R^1$ is the move.", "This type of model has been widely used in informative path planning [3] (e.g.", "Simultaneous localization and mapping (SLAM) [40], or environment monitoring [22]) where the agent needs to decide the next move in order to maximise a reward or minimising a cost function.", "This can be formulated in an MPC setting through updating the environment belief and then performing an online optimisation repeatedly to decide the best next move [23].", "It shall be highlighted that there is no terminal weight in the cost function in almost all these applications due to the difficulties discussed in Introduction.", "Let's consider an agent with parameter matrices as $ A=\\left[ \\begin{array}{cc}0.7 & 0.1 \\\\0.8 & 0.6\\end{array}\\right]; \\qquad B=\\left[ \\begin{array}{c}0.8 \\\\-0.5\\end{array}\\right]$ and its task is simply to go back to the home.", "Consequently, it is sensible to choose the distance from the agent's position to the origin (home) as the cost function.", "That is, at each step, we choose to take a move to make the agent's distance at the next time to the origin as small as possible, i.e.", "minimising the cost function $ J(x(k))=x_1(k+1)^2+x_2(k+1)^2$ It shall be noted the agent under concern with the parameters in (REF ) is asymptotically stable in the sense that it naturally goes back the home without any control, $u=0$ , as shown in Figure 1.", "However by minimising the cost function (REF ), this MPC scheme drives the agent actually moving away from the home and becomes unstable (see Figure 1).", "This undesirable behaviour is not surprising for the control, particularly MPC, community, but does show the importance of understanding the dynamic interaction between an optimisation algorithm (or AI based decision making algorithms) and system dynamics even for such a simple system and task.", "The results in this paper provide several approaches to modify the above algorithm so guarantees stability: adjusting the state and (or) control weights, simply increasing the length of horizon by taking the advantage that our stability conditions are horizon dependent, or resorting to the MPCS algorithm in Section .", "Case 1: Adjusting state or control weights.", "We still keep $N=1$ .", "In this case, according to Theorem , the MPC algorithm is stable if condition (REF ) is satisfied for $x^(k+1\|k))$ .", "Suppose that we also keep $R=0$ but adjust $Q$ such as this condition is met.", "For this unconstrained linear system, it is well known that optimal control action for one stage cost is given by $u^(k)=-(B^TQB)^{-1}B^TQAx(k)$ where $Q=diag \\lbrace 1,1\\rbrace $ with the minimum stage cost $m(x(k))=x(k)^T(A^TQA-A^TQB(B^TQB)^{-1}B^TQA)x(k)$ Letting $M=A^TQA-A^TQB(B^TQB)^{-1}B^TQA$ and $K=-(B^TQB)^{-1}B^TQA$ , condition (REF ) reduces to that the following condition holds for any $x(k)$ $x(k)^T(A+BK)^TM(A+BK)x(k) \\le x^T(k)Mx(k)$ It can be shown that if Q is chosen as $Q=diag \\lbrace 1, 0.5^2\\rbrace $ , the above condition is met.", "So it follows from Theorem or that the corresponding MPC algorithm stabilises the system.", "The state trajectory under MPC with this state weight is plotted in Figure 1.", "Case 2: Extending the length of horizon.", "One may argue that the physical meaning is lost by changing the state weight in the way of Case 1.", "Now we keep the original state and control weight unchanged but increase the length of horizon to 2 in (REF ).", "Following Theorem , this requires that condition (REF ) is satisfied with $N=2$ .", "Indeed, this is true.", "With the notation as in Case 1 for the optimal control and associated optimal stage cost, it follows from a similar discussion that condition (REF ) is equivalent to $x(k)^TMx(k) - m(x^(k+2)) \\ge 0$ Solving the two stage optimisation involved in and substituting the data obtains $0.646 x_1(k)^2+0.686x_1(k)x_2(k)+0.184x_2(k)^2 \\ge 0$ It can be shown that it is true for any $x(k)$ so the closed-loop system under $N=2$ is stable which is also confirmed by the state trajectory under MPC with $N=2$ in Figure 1.", "Stability of the above algorithm can also be confirmed by Theorem but with a bit more conservativeness.", "Indeed, the condition (REF ) is equivalent to checking $0.415 x_1(k)^2+0.444 x_1(k)x_2(k)+0.119 x_2(k)^2 \\ge 0$ which is also true.", "Case 3: MPCS.", "In this case, we still employ the same cost function as (REF ) but apply MPCS in Section .", "For this special case $N=1$ , condition (REF ) is equivalent to (REF ).", "With the data for this system and the cost function, it reduces to checking if there exists a control $u(k\|k)$ such that $x(k+1\|k)^TMx(k+1\|k)-x(k)^TMx(k) \\le 0$ with $M=\\left[ \\begin{array}{cc}1.1012 & 0.5896 \\\\0.5896 & 0.3156\\end{array}\\right]$ It is ready to show that there exists control $u(k\|k)$ such that the above inequality holds (for example, by minimising the first item over $u(k\|k)$ ).", "Therefore after adding condition (REF ) or (REF ) to the original MPC, not only feasibility but also stability can be guaranteed.", "The agent trajectory under the proposed MPCS with $N=1$ is also depicted in Figure 1.", "Figure: Comparison of Autonomous agent trajectory under different control schemes.", "*: open loop; o: original MPC; diamond: modified state weight; +: with two step horizon; square: MPCS" ], [ "Conclusions", "Despite huge success of MPC in applications and vast progress in establishing its stability theory, there is still a long way to establish less restrictive stability theory and stability guaranteed MPC schemes.", "As observed in Introduction, there is a large gap between the available stability analysis tools (and conditions) and real engineering applications where a very limited length of horizon without terminal weight is quite often used.", "With increasingly penetrating of MPC into new applications and new sectors, the limit of the existing stability analysis tools is becoming even clearer.", "This paper departs from the traditional approach of establishing MPC stability through approximating an infinite horizon optimisation.", "The stability is established by showing the modified summed stage cost decreases with the horizon moving forward; that is, there exists a control such that the immediate stage cost after the terminal state is less than the first stage cost under the MPC.", "To this end, we also modify the value function and the stage costs.", "With this approach and thinking, we establish stability for MPC with a limited horizon and without terminal weight, ans also extend the result to MPC with terminal weight.", "Based on the established stability conditions, we propose a new stability guaranteed MPC scheme (MPCS).", "To establish its recursive feasibility, a new level set is defined by the value function with one step stage cost.", "It is shown that if the initial state within this properly defined sublevel set is feasible, recursive feasibility can be guaranteed.", "Several numerical examples are provided to illustrate the effectiveness and usefulness of the established stability conditions and the proposed new algorithm MPCS.", "It clearly shows that the new approach is much more powerful and less conservative.", "For an unstable MPC problem, numerical examples show our work also provides multiple approaches to achieve stability, e.g.", "through changing the length of horizon, the state or control weights, or embedding an additional condition in the algorithm using MPCS.", "This paper makes a step in addressing the drawbacks discussed at Introduction for the current stability guaranteed MPC framework.", "It is hoped that this work will trigger more research in establishing better stability analysis tools for MPC.", "The needs of less conservative and more effective stability analysis tools for MPC are not only due to their theoretical significance, but also their importance in providing proven properties for practical applications particularly safety critical systems, extending the applicability of MPC, and providing more degrees of freedom for parameter tuning which potentially could improve performance.", "This work will support to develop a new MPC framework with stability guarantee that achieves a better trade-off between optimality in performance, stability and computational burden.", "It could be extended to deal with MPC with different dynamics, complicated performance indexes (e.g.", "economic MPC) or in the presence of different types of disturbance and uncertainty (e.g.", "robust or stochastic MPC).", "This work will promote the application of MPC or similar concepts in new areas where systems are complex, of a large scale, or safety critical (so requires proven properties such as stability or safety).", "In these applications, only an online optimisation with a limited length of horizon is possibly solvable in real-time and it is impossible to finding a suitable terminal weight or an off-line policy to serve as a local controller as required in the existing MPC framework (e.g.", "air traffic control, robotics, and autonomous systems).", "This will be our future research." ] ]	2011.14193
[ [ "The 2-Categorical Structure of Predicate Theories" ], [ "Abstract The equivalence of formal languages with categories has long been expounded, notably by [10.1007/BFb0066201], relating coherent categories with predicate logic, where coherent functors serve as translations between predicate theories.", "This equivalence yielded remarkable results connecting syntax to semantics, but the formalism relied upon category theory.", "Work by [arXiv:1506.04675,arxiv:1507.02302,10.1017/9781316275603] proposed syntactical notions of equivalence and translation of theories.", "In this paper, we demonstrate that these syntactical notions are equivalent to the categorical formalism, allowing us to present Makkai and Reyes' conceptual completeness of coherent logic in a syntactical form and to explain and remove the required hypotheses of the translations in [10.1017/9781316275603].", "Our equivalence elucidates the relation between categorical equivalence and Morita equivalence, and it reveals necessary and sufficient conditions for when a theory is intertranslatable with or Morita equivalent to a propositional theory." ], [ "Introduction", "There are presently two distinct paradigms in symbolic logic: traditional predicate logic and topos-theoretic categorical logic.", "This disparity leads to fragmentation.", "There will be new definitions and new theorems in both paradigms—when is a theorem in predicate logic analogous to one in categorical logic?", "We investigate the fragmentation in two key areas: equivalence of theories and translation.", "Sorting out this fragmentation is as important as ever.", "For example, proves Morita equivalence (originally defined in ) is a “special case of bi-interpretability.” While this is correct, it is only half the story, with the other half lying in the realm of categorical logic.", "From , Morita equivalence is indeed the same as bi-interpretability, assuming that we restrict to proper theories (this is an innocuous assumption which will be defined later in the paper).", "However, Morita equivalence and bi-interpretability are deeply related for any coherent theory: the `proper' hypothesis is not necessary.", "We justify this by proving the following theorem: Theorem REF Let $T_1$ and $T_2$ be theories in coherent logic.", "$T_1$ and $T_2$ are weakly intertranslatable if and only if they are Morita equivalent using only product, subsort, and quotient extensions.", "Getting to this result will require bridging between predicate logic and categorical logic.", "The key engine to all of our results is a couple of theorems summarizing this bridge: Theorem REF The 2-category of coherent theories, where morphisms are equality-preserving translations and 2-morphisms are t-maps, is equivalent to the 2-category of coherent categories.", "Theorem REF The 2-category of coherent theories, where morphisms are weak translations and 2-morphisms are t-maps, is equivalent to the 2-category of exact coherent categories.", "The latter theorem reveals that the relationship between weak translations, as defined in e.g.", ", and categorical logic is non-trivial in the sense that all translations can be faithfully reduced to translations which preserve equality (see Remark REF ).", "Moreover, these theorems greatly simplify questions of coherent logic, particularly questions about the relationship between Morita equivalence and categorical equivalence.", "The authors of sought a condition $\\mathfrak {P}$ of the following form: If $T_1$ and $T_2$ are categorically equivalent and $\\mathfrak {P}$ , then $T_1$ and $T_2$ are Morita equivalent.", "While the authors cite the duality in (Makkai Duality) as a candidate for $\\mathfrak {P}$ , the relationship was only developed at the level of isomorphism.", "Makkai Duality is much stronger than that: there should be a relationship between the morphisms of ultracategories, ultrafunctors, and a notion of morphism between coherent theories, namely translations.", "Theorems REF and REF allow us to extend the connection between Morita equivalence and ultracategories to a correspondence between translations and ultrafunctors.", "That is, we prove the following theorems: Theorem REF Let $F: T_1 \\rightarrow T_2$ be a translation between proper theories.", "If the induced pullbuck functor on models $F^: \\normalfont {\\text{Mod}}(T_2) \\rightarrow \\normalfont {\\text{Mod}}(T_1)$ is an equivalence of categories, then $F$ is half of a homotopy equivalence.", "That is, $T_1$ and $T_2$ are weakly intertranslatable.", "Theorem REF Let $F: T_1 \\rightarrow T_2$ be a translation between proper theories.", "Then $F^: \\normalfont {\\text{Mod}}(T_2) \\rightarrow \\normalfont {\\text{Mod}}(T_1)$ preserves ultraproducts.", "Theorem REF There is a fully faithful (dual) embedding of proper theories with weak translations into the category of ultracategories.", "Finally, we use Theorems REF and REF again to provide a novel characterization of propositional theories (theories with no sorts) with respect to translations and Morita equivalence.", "We prove these characterizations in §REF ." ], [ "Categorical Preliminaries", "We assume the reader is familiar with standard category theory (as treated in ).", "However, we leave open the possibility that the reader is less familiar with more technical definitions of category theory applied to logic, particularly those used in , .", "The fundamental object of categorical logic is, arguably, a coherent category.", "Definition 2.1 A category $C$ is a coherent category if: $C$ admits all finite limits.", "All kernel pairsA kernel pair is a pullback $X \\times _fX$ for a morphism $f: X \\rightarrow Y$ .", "admit coequalizers, and these coequalizers are stable under pullback.", "A coequalizer of a kernel pair is called a regular epimorphism.", "The subobject poset of any object in $C$ has the structure of a (finitary) bounded distributive lattice.", "Joins in this lattice are called unions of subobjects.", "For any morphism $f: X \\rightarrow Y$ , the change of base functor $f^:Sub(Y) \\rightarrow Sub(X)$ preserves meets and joins.", "Given this notion of category, there is a related notion of coherent functor: Definition 2.2 A functor $F: C \\rightarrow D$ between coherent categories is a coherent functor if: $F$ preserves finite limits ($F$ is left-exact).", "$F$ preserves unions.", "$F$ preserves regular epimorphisms.", "Coherent categories and coherent functors are part of a 2-category $\\mathsf {Coh}$ , with the 2-morphisms being natural transformations of functors.", "uses coherent categories to study a generalization of classical predicate logic known as coherent logic.", "Namely, establishes a bijective correspondence between coherent categories and coherent theories (up to homotopy equivalence in $\\mathsf {Coh}$ ).", "For a coherent category $C$ , the corresponding theory is called the internal logic $C$ of $C$ .", "For a coherent theory $T$ , the corresponding category is called the syntactic category $T$ of $T$ .", "We will show that these associations are part of a 2-equivalence of coherent categories and coherent theories with translations (to be defined later).", "While all coherent categories admit finite unions and coequalizers to kernel pairs (i.e.", "coherent categories are regular), coproducts and general quotients typically do not exist.", "The existence of quotients and coproducts play a special role in establishing conceptual completeness per , , so they have their own definitions.", "Definition 2.3 Let $C$ be a coherent category.", "A congruence $K$ (or internal equivalence relation) of an object $X$ in $C$ is a subobject $i_K: K\\hookrightarrow X \\times X$ (with projections $p_1,p_2: K \\rightarrow X$ ) satisfying three properties: (Reflexivity) The diagonal map $X \\rightarrow X \\times X$ factors through $i_K$ .", "(Symmetry) $i_K$ factors through $i_K \\circ (p_2,p_1)$ , where $(p_2,p_1): X \\times X \\rightarrow X \\times X$ is the order flipping map.", "(Transitivity) $K \\times _X K \\rightrightarrows X$ factors through $K \\rightrightarrows X$ .", "The coequalizer of $p_1$ and $p_2$ (if it exists) is called the quotient $X/K$ of $K$ .", "If $K$ is a kernel pair, $X \\times _f X$ , then $K$ is said to be effective.", "Finally, $C$ is an exact category if all congruences are effective (all congruences admit quotients).", "Remark 2.4 As shown in (Definition 3.3.6), there is a natural one-to-one correspondence between congruences of $C$ and equivalence relations in the internal logic of $C$ .", "For any coherent category $C$ , we can extend $C$ to an exact category.", "This is called the exact completion $C^{ex}$ of $C$ , and is derived in Chapter 8 of using the internal logic of $C$ .", "theorem 2.5 (Makkai and Reyes Theorem 8.4.3) For a coherent category $C$ , there exists an exact coherent category $C^{ex}$ and a coherent functor $I: C \\rightarrow C^{ex}$ universal to all coherent functors $F: C \\rightarrow D$ where $D$ is exact.", "That is, for any such coherent functor $F$ , there exists a functor $F^{{ex}}: C^{ex}\\rightarrow D$ making the following diagram commute: $\\begin{tikzcd}C^{ex}[dr,\"F^{ex}\",dotted] & \\\\C [u,\"I\"] [r,\"F\",swap] & D.\\end{tikzcd}$ Furthermore, $I$ is a conservative functor: if $IX \\leqslant IY$ as subobjects in $C^{ex}$ , then $X \\leqslant Y$ in $C$ .", "$I$ is also a full functor.", "We define a category $\\mathsf {ExactCoh}$ consisting of exact (coherent) categories where morphisms are coherent functors between them.", "One may expect that any sensible choice of morphism for $\\mathsf {ExactCoh}$ should preserve quotients.", "This is a property of coherent functors.", "Proposition 2.6 Let $F: C \\rightarrow D$ be a coherent functor between exact categories, and let $K$ be a congruence of an object $X$ of $C$ .", "Then $FK$ is a congruence of $FX$ , and $F(X/K) = FX/FK$ .", "That $FK$ is a congruence of $FX$ follows from the left-exactness of $F$ .", "Since $X/K$ is the coequalizer of a kernel pair $p: K \\rightarrow X$ , it is a regular epimorphism; $F$ preserves regular epimorphisms, so $F(X/K)$ is the coequalizer $FX/FK$ of $Fp$ .", "The existence of coproducts in a coherent category $C$ is also notable.", "We say that $C$ is extensive if finite disjoint coproducts exist in $C$ and coproducts are pullback-stable.", "Definition 2.7 A category $C$ is a pretopos if it is exact and extensive.", "In a pretopos $C$ , the coproduct $X \\amalg Y$ of objects $X$ and $Y$ is the join of $X$ and $Y$ when interpreted as subobjects of $X \\amalg Y$ .", "Thus, coherent functors between pretopoi (more generally, between extensive categories) preserve coproducts.", "With this in mind, we define the category $\\mathsf {Pretopos}$ to be the 2-category with objects pretopoi, morphisms coherent functors, and 2-morphisms natural transformations.", "In particular, all morphisms in $\\mathsf {Pretopos}$ preserve quotients and coproducts.", "Similar to the exact completion of a coherent category $C$ , there also exists a pretopos completion proven in , among other publications.", "theorem 2.8 (Makkai and Reyes Theorem 8.4.1) Let $C$ be a coherent category.", "There exists a pretopos $P$ and a conservative, full on subobjects, coherent functor $I: C \\rightarrow P$ such that for any coherent functor $F: C \\rightarrow Q$ where $Q$ is a pretopos, there exists a coherent functor $F^P: P \\rightarrow Q$ making the following diagram commute: $\\begin{tikzcd}P [dr,\"F^P\",dotted] & \\\\C [u,\"I\"] [r,\"F\",swap] & Q.\\end{tikzcd}$" ], [ "Ultracategories", "Ultracategories play a critical role in conceptual completeness.", "In fact, conceptual completeness can be extended to a more powerful duality between classes of ultracategories and pretopoi known as Makkai duality.", "The theory was originally developed in , but the formalism proposed by Lurie is simpler and shown to be strong enough to prove the desired theorems.", "Hence we adopt the formalism found in and encourage the reader to be familiar with its definitions of ultracategories and ultrafunctors.", "Since ultrafunctors are functors, natural transformations between ultrafunctors is a sensible notion of 2-morphism.", "We define $\\mathsf {UltraCat}$ to be the 2-category with objects ultracategories, morphisms ultrafunctors, and 2-morphisms natural transformations.", "For a coherent theory $T$ , we note that the category of models $\\normalfont {\\text{Mod}}(T)$ of $T$ is an ultracategory: the ultrastructure is given by the ultraproduct of models of a theory.The Łos ultraproduct theorem states that this construction yields a model of $T$ .", "Identifying a small pretopos $P$ with its internal logic $P$ , we find that $\\normalfont {\\text{Mod}}(P)$ is equivalent to the category $\\text{Hom}_{\\mathsf {Coh}}(P,{\\mathsf {Set}})$ .", "With this in mind, we define $\\normalfont {\\text{Mod}}(C)$ to be the category $\\text{Hom}_{\\mathsf {Coh}}(C,{\\mathsf {Set}})$ .", "We can now state conceptual completeness and Makkai duality.", "theorem 2.9 (Conceptual Completeness) Let $P$ be a (small) pretopos.", "Then $P$ is equivalent to the category $\\normalfont {\\text{Ult}} (\\normalfont {\\text{Mod}}(P),{\\mathsf {Set}})$ of ultrafunctors from $\\normalfont {\\text{Mod}}(P)$ to ${\\mathsf {Set}}$ .", "In particular, for a coherent functor $F: P_1 \\rightarrow P_2$ between pretopoi, if the induced functor $F^: \\normalfont {\\text{Mod}}(P_2) \\rightarrow \\normalfont {\\text{Mod}}(P_1)$ is an equivalence of categories, then $F$ is also an equivalence of categories.", "theorem 2.10 (Makkai Duality) The map $P \\mapsto \\normalfont {\\text{Mod}}(P)$ is a fully faithful embedding of 2-categories: $\\mathsf {Pretopos}^{\\mathsf {op}}\\subset \\mathsf {UltraCat}.$ Makkai duality is a generalization of Stone duality of Boolean propositional theories and Stone topological spaces.", "This relationship is expounded in , .", "There has also been work in restating and generalizing Makkai duality.", "extends this duality to a fully faithful 2-embedding of coherent topoi into a larger `ultracategorical' category.", "Work by , , based on earlier work by reformulates the underlying syntax (pretopoi)-semantics (ultracategories) duality using classes of topological groupoids." ], [ "Syntactical Preliminaries", "There are two syntactical systems which dominate the field of mathematical logic: classical logic and coherent logic.", "Both of these systems are examples of predicate logic because they include formal sorts (also called types), relation symbols, function symbols, and quantifiers along with the standard logical operations.", "Classical logic can be identified as a special case of coherent logic: coherent logic with negation.", "Coherent logic was pioneered by , though it (and derivatives of coherent logic) pervade any discussion of logic internal to topoi, e.g.", ", .", "On the other hand, our notion of translations between predicate theories is based on , , .", "We will define coherent logic from this latter perspective, nevertheless there is no meaningful difference between the treatments of predicate logic.", "A predicate theory $T$ is defined by a signature $\\Sigma $ and a collection of axioms $\\Delta $ .", "Using the signature, we can construct $\\Sigma $ -formulae (sometimes called $T$ -formulae).", "We formally provide their definitions here.", "Definition 3.1 A coherent signature is a set $\\Sigma $ consisting of the following symbols: Sort symbols $\\sigma $ each with a binary equality relation $=_\\sigma $ .", "A countable list of variables $x_i^\\sigma $ for each sort $\\sigma \\in \\Sigma $ .", "Relation symbols $R \\hookrightarrow \\sigma _1,\\dots ,\\sigma _n$ where $\\sigma _i \\in \\Sigma $ are sorts.", "Moreover, $\\sigma _1,\\dots ,\\sigma _n$ is called the domain of $R$ .", "A relation with empty domain is also called a proposition.", "Function symbols $f: \\sigma _1,\\dots ,\\sigma _n \\rightarrow \\tau $ where $\\sigma _i \\in \\Sigma $ and $\\tau \\in \\Sigma $ are sorts.", "$\\sigma _1,\\dots ,\\sigma _n$ is called the domain of $f$ ; $\\tau $ is the codomain of $f$ , and $n$ is the arity of $f$ .", "A 0-ary function $c: \\emptyset \\rightarrow \\tau $ is also called a constant of type $\\tau $ .", "Logical connectives $\\wedge ,\\vee $ , the existential quantifier $\\exists $ , the tautologous formula $\\top $ , and the contradictory formula $\\bot $ .", "Remark 3.2 Function symbols can be identified with a special class of relations.", "We identify a function with its graph.", "For a function symbol $f: \\sigma \\rightarrow \\tau $ , let $\\Gamma _f \\hookrightarrow \\sigma ,\\tau $ be the formula defined by $\\Gamma _f(x,y) \\equiv \\left( f(x) = y \\right)$ .", "Then for any formula $\\phi \\hookrightarrow \\tau ,\\sigma ^{\\prime }$ , the following sequent is provable: $\\phi (f(t),z) \\dashv \\vdash \\exists y^\\tau \\Gamma _f(t,y) \\wedge \\phi (y,z).$ This result is fundamental to our work.", "Hence we set this logical equivalence true by definition.", "That is, for any function symbol $f$ , the formula $\\phi (f(t),z)$ is an abbreviation for the expression on the right side of the sequent.", "$\\Sigma $ -terms are either variables or formal applications of the form $f(\\vec{t})$ where $f: \\vec{\\sigma } \\rightarrow \\tau $ is a function symbol and $\\vec{t}$ is a list of $\\Sigma $ -terms of type $\\vec{\\sigma }$ .", "Thus, $f(\\vec{t})$ is a $\\Sigma $ -term of type $\\tau $ .", "Moreover, $\\Sigma $ -formulae are formal evaluations $R(\\vec{t})$ where $R$ is a relation whose domain matches the type of $\\vec{t}$ or conjunctions, disjunctions, and quantifications of $\\Sigma $ -formulae.", "Given a formula $\\phi $ , the list of all free variables mentioned in $\\phi $ is called the context of $\\phi $ , and the type of this list is called the domain of $\\phi $ .", "(If the context is empty, we say that $\\phi $ is a sentence.)", "The distinction between terms and lists of terms is often unimportant, so we will use symbols such as $s$ or $t$ to refer to either terms or lists of terms whenever there is no ambiguity.", "We define the substitution class of a formula $\\phi $ to be the $\\alpha $ -equivalence class of $\\phi $ , denoted $[\\phi ]$ .Two formulae $\\phi $ and $\\phi ^{\\prime }$ are $\\alpha $ -equivalent if $\\phi ^{\\prime }$ can be obtained from $\\phi $ by replacing the context of $\\phi $ with a context of the same type.", "Given a context $y$ which matches the context of $\\phi $ , we define $\\phi (y)$ to be the element of $[\\phi ]$ with context $y$ .", "A sequent is a list of $\\Sigma $ -formulae $\\phi _1,\\dots ,\\phi _n \\vdash \\psi $ .", "The union of the contexts of the constituent formulae of a sequent is the context of the sequent.", "A coherent theory $T$ is a pair $(\\Sigma ,\\Delta )$ where $\\Sigma $ is a coherent signature and $\\Delta $ is a collection of $\\Sigma $ -sequents.", "We define $Cn(T)$ to be the set of all sequents provable from $\\Delta $ using the standard rules of deduction.", "The adjustments for classical logic are minimal.", "A classical signature is a coherent signature with two extra symbols $\\lnot $ and $\\forall $ .", "The rules of deduction for classical sequents is adjusted appropriately.", "A classical theory is a theory over a classical signature.", "Two theories $T_1,T_2$ are logically equivalent, written $T_1 =T_2$ , if they have the same signature and $Cn(T_1) = Cn(T_2)$ .", "$T_1$ and $T_2$ are definitionally equivalent if the signature of $T_2$ is logically equivalent to substitution classes of $T_1$ and vice versa.", "We formalize this next.", "Definition 3.3 Let $T = (\\Sigma ,\\Delta )$ be a coherent theory.", "A definitional extension of $T$ is a coherent theory $T^+ = (\\Sigma ^+,\\Delta ^+)$ satisfying the following: $\\Sigma \\subseteq \\Sigma ^+$ and $\\Delta \\subseteq \\Delta ^+$ .", "$\\Sigma $ and $\\Sigma ^+$ have the same sorts up to relabeling.", "For any symbol $S \\in \\Sigma ^+\\setminus \\Sigma $ , there is a $\\Sigma $ -substitution class $[\\phi ]$ with the same domain as $S$ such that $S(x) \\dashv \\vdash \\phi (x)$ is an axiom in $\\Delta ^+$ .", "Such a sequent is called a definition of $S$ in $\\Sigma $ , and we say that $S$ is definable in $\\Sigma $ .", "All sequents in $\\Delta ^+\\setminus \\Delta $ are of the above form.", "Definition 3.4 Two coherent theories $T_1$ and $T_2$ are definitionally equivalent if there exist chains of definitional extensions: $\\begin{array}{c}T_1 \\subset T_1^2 \\subset \\dots \\subset T_1^n, \\\\T_2 \\subset T_2^2 \\subset \\dots \\subset T_2^m\\end{array}$ such that $T_1^n$ is logically equivalent to $T_2^m$ .", "Definitional equivalence has been extensively studied by logicians,For an exposition on definitional equivalence and its history, see .", "but it is restricted to theories with matching sorts.", "Morita extensions, defined in , are the many-sorted analogue to definitional extensions.", "Definition 3.5 Let $T = (\\Sigma ,\\Delta )$ be a coherent theory.", "A Morita extension of $T$ is a coherent theory $T^+ = (\\Sigma ^+,\\Delta ^+)$ satisfying the following conditions: For any sort $\\sigma \\in \\Sigma ^+\\setminus \\Sigma $ , there exists a definition of $\\sigma $ in $\\Sigma $ of one of the forms mentioned below.", "Any non-sort symbol in $\\Sigma ^+\\setminus \\Sigma $ is definable in $\\Sigma $ .", "All sequents in $\\Delta ^+\\setminus \\Delta $ are definitions for $\\Sigma ^+\\setminus \\Sigma $ .", "All appropriate admissibility conditions are provable in $T$ .", "The possible sorts for $\\sigma $ are the following: (Product sort) There are sorts $\\sigma _1,\\dots ,\\sigma _n$ of $\\Sigma $ and function symbols $\\pi _i: \\sigma \\rightarrow \\sigma _i$ of $\\Sigma ^+$ .", "The definition of $\\sigma $ is the sequent: $\\begin{array}{rl}&\\vdash \\exists z^\\sigma \\bigwedge _{i=1}^n \\left( \\pi _i(z) =x_i^{\\sigma _i} \\right),\\\\\\bigwedge _{i=1}^n \\left( \\pi _i(x) = \\pi _i(y) \\right) &\\vdash x = y.\\end{array}$ We usually write $\\sigma $ as $\\sigma _1 \\times \\dots \\times \\sigma _n$ .", "(Terminal sort) As a special 0-ary case of the product sort, we can always define a sort $\\sigma $ corresponding to a singleton.", "It has two defining axioms: $\\begin{array}{rl}& \\vdash \\exists x^\\sigma \\left( x = x \\right), \\\\& \\vdash \\left( x = y \\right).\\end{array}$ (Coproduct sort) There are sorts $\\sigma _1,\\dots ,\\sigma _n$ of $\\Sigma $ and function symbols $\\rho _i: \\sigma _i \\rightarrow \\sigma $ of $\\Sigma ^+$ .", "A definition of $\\sigma $ consists of $n+2$ sequents: $\\begin{array}{rl}& \\vdash \\bigvee _{i=1}^n \\exists x_i^{\\sigma _i} \\left( z^\\sigma =\\rho _i(x_i) \\right), \\\\\\rho _i(x_i^{\\sigma _i}) = \\rho _i(x_k^{\\sigma _i}) &\\vdash x_i = x_k, \\\\\\bigvee _{i\\ne j} \\left( \\rho _i(x_i^{\\sigma _i}) =\\rho _j(x_j^{\\sigma _j}) \\right) &\\vdash \\bot .\\end{array}$ We usually write $\\sigma $ as $\\sigma _1 + \\dots + \\sigma _n$ .", "(Subsort) There is a sort $\\sigma ^{\\prime }$ of $\\Sigma $ , a $\\Sigma $ -substitution class $[\\phi ]$ with domain $\\sigma ^{\\prime }$ , and a function symbol $i: \\sigma \\rightarrow \\sigma ^{\\prime }$ of $\\Sigma ^+$ .", "The definition of $\\sigma $ is the pair of sequents: $\\begin{array}{rl}\\phi (x^{\\sigma ^{\\prime }}) \\dashv \\vdash & \\exists y^\\sigma \\left(i(y) = x \\right), \\\\i(x^{\\sigma ^{\\prime }}) = i(z^{\\sigma ^{\\prime }}) \\vdash & x = z.\\end{array}$ (Quotient sort) There is a sort $\\sigma ^{\\prime }$ of $\\Sigma $ , a $\\Sigma $ -substitution class $[\\phi ] \\hookrightarrow \\sigma ^{\\prime },\\sigma ^{\\prime }$ , and a function symbol $q: \\sigma ^{\\prime } \\rightarrow \\sigma $ of $\\Sigma ^+$ .", "The definition of $\\sigma $ is the pair of sequents: $\\begin{array}{rl}& \\vdash \\exists x^{\\sigma ^{\\prime }} \\left(q(x) = z^\\sigma \\right), \\\\q(x^{\\sigma ^{\\prime }}) = q(y^{\\sigma ^{\\prime }}) & \\vdash \\phi (x,y).\\end{array}$ The admissibility conditions for $\\sigma $ are the sequents ensuring $\\phi $ is an equivalence relation: $\\begin{array}{rl}& \\vdash \\phi (x,x),\\\\\\phi (x,y) & \\vdash \\phi (y,x), \\\\\\phi (x,y) \\wedge \\phi (y,z) & \\vdash \\phi (x,z).\\end{array}$ We sometimes write $\\sigma $ as $\\sigma ^{\\prime }/\\phi $ .", "Definition 3.6 Let $T_1$ and $T_2$ be coherent theories.", "We say that $T_1$ is Morita equivalent to $T_2$ , written $T_1 \\cong T_2$ , if there exist chains of Morita extensions $T_1 \\subset \\dots \\subset T_1^n$ and $T_2 \\subset \\dots \\subset T_2^m$ such that $T_1^n$ and $T_2^m$ are definitionally equivalent." ], [ "The Syntactic Category", "For a coherent theory $T$ , there exists a coherent category $T$ which encodes the logic of $T$ .", "This coherent category is called the syntactic category of $T$ .", "The objects of $T$ are substitution classes of formulae of $T$ .", "The morphisms of $T$ are logical equivalence classes of formulae defining graphs of functional relations between substitution classes.", "Formally, for objects $[\\phi ] \\hookrightarrow \\sigma $ and $[\\psi ]\\hookrightarrow \\tau $ a morphism $\\theta : [\\phi ] \\rightarrow [\\psi ]$ is given by a formula $\\theta \\hookrightarrow \\sigma .\\tau $ such that the following sequents are provable in $T$ : $\\begin{array}{rl}\\phi (x) & \\vdash \\exists y \\theta (x,y), \\\\\\theta (x,y) \\wedge \\theta (x,z) & \\vdash y = z, \\\\\\theta (x,y) & \\vdash \\phi (x) \\wedge \\psi (y),\\end{array}$ where variables and parentheses are dropped whenever contexts are empty.", "The composition of two morphisms $\\theta : [\\phi ,x] \\rightarrow [\\psi ,y]$ and $\\eta :[\\psi ,y] \\rightarrow [\\chi ,z]$ is the formula: $\\left[ \\eta \\circ \\theta \\right](x,z) \\equiv \\exists y( \\theta (x,y) \\wedge \\eta (y,z)).$ Two formulae define the same morphism if they are logically equivalent.", "The seminal work on syntactic categories of coherent theories is , though also explores the theory.", "It is proven in , that $T$ is a coherent category: products are given by conjunctions, unions are disjunctions, the terminal object is $[\\top ]$ , the initial object is $[\\bot ]$ , and the existential quantifier constructs regular epimorphisms.", "Remark 3.7 It is often useful to interpret a formula $\\phi (x)$ in a larger context.", "Let $X = x.z$ where $z$ is a list of $T$ variables.", "We define: $[\\phi ,X] \\equiv [\\phi (x) \\wedge (z = z)].$ The object $[\\top ,x^\\sigma ]$ is common enough that we will abbreviate it to $[\\sigma ]$ .", "A critical result relating the logic of $T$ to the structure of $T$ is proven in : Lemma 3.8 (Subobject Lemma) Let $\\phi (x)$ and $\\psi (y)$ be two formulae in a coherent theory $T$ .", "Then the sequent $\\phi (x) \\vdash \\psi (y)$ is provable in $T$ if and only if $[\\phi ,x.y] \\leqslant [\\psi ,x.y]$ as subobjects in $T$ .", "An interpretation of a coherent signature $\\Sigma $ into a coherent category $C$ is the fundamental unit for categorical semantics.", "We define it next.", "Definition 3.9 An interpretation $M: \\Sigma \\rightarrow C$ is a collection of the rules: For every sort $\\sigma \\in \\Sigma $ , $M\\sigma $ is an object of $C$ .", "For every relation $R \\hookrightarrow \\sigma $ of $\\Sigma $ , $MR$ is a subobject of $M\\sigma $ .", "For every function symbol $f: \\sigma \\rightarrow \\tau $ of $\\Sigma $ , $Mf$ is a morphism $Mf: M\\sigma \\rightarrow M\\tau $ .", "By factoring logical connectives, we note that these rules define a map from $\\Sigma $ -formulae to objects of $C$ .", "Definition 3.10 Let $T$ be a coherent theory with signature $\\Sigma $ .", "A $C$ -model of $T$ is an interpretation $M: \\Sigma \\rightarrow C$ such that: $\\phi (x) \\vdash \\psi (y) \\text{ implies } M[\\phi ,x.y] \\leqslant M[\\psi ,x.y].$ A conservative model is one where this implication is bidirectional.", "A ${\\mathsf {Set}}$ -model is a model of $T$ in the traditional sense.", "Any theory $T$ admits a canonical $T$ interpretation $I: T \\rightarrow T$ defined by sending a formula $\\phi $ to its substitution class $[\\phi ]$ .", "Thus Lemma REF is equivalent to the statement that the canonical interpretation is a conservative model.", ", both prove that a $C$ -model of a theory $T$ is equivalently a coherent functor $M: T \\rightarrow C$ .", "We define $\\normalfont {\\text{Mod}}(T)$ to be the functor category $\\text{Hom}_\\mathsf {Coh}(T,{\\mathsf {Set}})$ .", "If $F: C\\rightarrow D$ is a coherent functor, then there is a corresponding pullback functor $F^: \\normalfont {\\text{Mod}}(D) \\rightarrow \\normalfont {\\text{Mod}}(C)$ defined by $M \\mapsto M \\circ F$ .", "We can endow a subcanonical Grothendieck topology on $T$ by declaring the covering families $f_i: Y_i \\rightarrow X$ to be generated by jointly epimorphic families: those $f_i$ such that the join of the images, $\\bigvee _{i=1}^n\\exists _{f_i} Y_i$ , contains $X$ .", "The corresponding topos of sheaves $Sh(T)$ is the classifying topos for models of $T$ .", "It is proven in that $T_1$ and $T_2$ are Morita equivalent if and only if there is an equivalence of categories $Sh({T_1}) \\cong Sh({T_2})$ .", "This paper can be interpreted as an extension of this result to an equivalence of 2-categories relating morphisms of sites (coherent functors) with translations." ], [ "The Internal Logic", "Let $C$ be a coherent category.", "Lemma REF suggests a relationship between algebra and logic.", "We use this lemma to define a coherent theory associated with $C$ , called the internal logic $C$ of $C$ .", "The signature of $C$ consists of a sort for every object $X$ of $C$ , equality relations $=_X \\hookrightarrow X^T,X^T$ for every object $X$ , and function symbols $f^T: X^T \\rightarrow Y^T$ for every morphism $f: X \\rightarrow Y$ of $C$ .", "If there is no ambiguity, we will omit the superscript $^T$ from symbols of the internal logic.", "The axioms of the internal logic are defined in (see chapter $3.5$ ), but we will state them below.", "For every commutative diagram $fg = h$ for morphisms $f,g,h$ of $C$ , the sequent $\\vdash f (g (x)) = h(x)$ is an axiom of $C$ .", "For every product $X \\times Y$ with projections $\\pi _X,\\pi _Y$ , the following sequents are axioms of $C$ : $\\begin{array}{rl}\\left( \\pi _X(a) = \\pi _X(b) \\right) \\wedge \\left( \\pi _Y(a) = \\pi _Y(b) \\right)& \\vdash a = b, \\\\& \\vdash \\exists a^{X \\times Y} \\left( \\pi _X(a) = x \\wedge \\pi _Y(a) = y \\right).\\end{array}$ For every equalizer: $\\begin{tikzcd}E [r,\"e\"] & X [r,\"f\",shift left] [r,\"g\",shift right,swap] & Y,\\end{tikzcd}$ the following sequents are axioms of $C$ .", "$\\begin{array}{rl}e(x) = e(y) & \\vdash x = y, \\\\& \\vdash fe(x) = ge(x), \\\\f(y) = g(y) & \\vdash \\exists x (e(x) = y).\\end{array}$ For any terminal object $X$ , the sequents $\\vdash x = y$ and $\\vdash \\exists (x = x)$ are axioms of $C$ .", "For any pair of subobjects $i_A: A \\hookrightarrow X$ and $i_B: B\\hookrightarrow X$ with join $j: A \\vee B \\hookrightarrow X$ , the following sequents are axioms of $C$ : $\\begin{array}{rl}\\left( \\exists a^A i_A(a) = x \\vee \\exists b^B i_B(b) = x \\right) & \\vdash \\exists c^{A \\vee B} j(c) = x \\\\\\exists c^{A \\vee B} j(c) = x & \\vdash \\exists a^A (i_A(a) = x) \\vee \\exists b^B \\left( i_B(b) = x \\right).\\end{array}$ For any surjective morphism $f: X \\rightarrow Y$ (i.e.", "one where $Y$ is a subobject of the image $\\exists _f X$ ), the sequent $\\vdash \\exists x^X(f(x) = y)$ is an axiom of $C$ ." ], [ "The 2-Categories $\\mathsf {CTh}_0$ and {{formula:37638033-2c71-40bf-90e6-f126b5ef96d8}}", "The translations we define below have been demonstrated to identify Morita equivalence for certain classes of predicate theories.", "Our goal is to show that these translations are equivalent to coherent functors between syntactic categories.", "We begin by defining a way to map between signatures.", "Definition 4.1 Let $\\Sigma _1$ and $\\Sigma _2$ be two signatures.", "A reconstrual $F: \\Sigma _1 \\rightarrow \\Sigma _2$ is a collection of the following: For every sort $\\sigma \\in \\Sigma _1$ , $F$ identifies a list of sorts $F\\sigma $ of $\\Sigma _2$ .", "This list may be empty.", "For every $\\Sigma _1$ -variable, $x: \\sigma _i$ , $F$ identifies a $\\Sigma _2$ -context $Fx$ with type $F\\sigma $ (if $F\\sigma $ is empty, $Fx$ is the empty context).", "For a list of variables $\\vec{x}$ , $F\\vec{x}$ is the concatenation of each $Fx_i$ for $x_i \\in \\vec{x}$ .", "We also stipulate if $x$ and $y$ are distinct variables, then $Fx$ and $Fy$ are disjoint lists.", "For every sort $\\sigma \\in \\Sigma _1$ , $F$ identifies a $\\Sigma _2$ -substitution class $D_\\sigma \\hookrightarrow F\\sigma $ .", "For a list of sorts $\\vec{\\sigma }$ , we define $D_{\\vec{\\sigma }}$ to be the conjunction: $\\left[ \\bigwedge _{\\sigma _i \\in \\vec{\\sigma }} D_{\\sigma _i}(Fx_i^{\\sigma _i})\\right].$ For every relation $R \\hookrightarrow \\sigma $ of $\\Sigma _1$ , $FR$ is a substitution class with domain $F\\sigma $ .", "For every function $f: \\sigma \\rightarrow \\tau $ of $\\Sigma _1$ , $Ff$ is a substitution class with domain $F\\sigma .", "F\\tau $ .", "Remark 4.2 The map from $\\Sigma _1$ -variables to $\\Sigma _2$ -contexts is a notational crutch.", "The existence of such a map is essentially equivalent to the statement that $\\Sigma _2$ has enough variables to carry out predicate logic.", "This can be guaranteed by assuming a large collection of variables, perhaps a proper class.", "Since the concern of whether a signature $\\Sigma $ has enough variables is mathematically and philosophically uninteresting, we will refrain from specifying the map on $\\Sigma _1$ -variables.", "$F$ extends a map from $\\Sigma _1$ -formulae to $\\Sigma _2$ -formulae by factoring logical connectives (e.g.", "$F(\\phi (x) \\wedge \\psi (y)) = F\\phi (Fx)\\wedge F\\psi (Fy)$ ).", "Furthermore, this map is natural in the sense that $F(\\phi [x/t]) = F\\phi [Fx/Ft]$ where $\\phi [x/t]$ denotes the formula obtained by replacing the context $x$ in $\\phi $ with a matching context $t$ .", "Hence, $F$ also defines a map from $\\Sigma _1$ -substitution classes to $\\Sigma _2$ substitution classes.", "We denote the image of $=_\\sigma $ under $F$ by $E_F^\\sigma $ , though we often abbreviate this to $E_F$ when there is no ambiguity.", "We will also write $F: T_1 \\rightarrow T_2$ if $F$ is a reconstrual from $\\Sigma _1 \\rightarrow \\Sigma _2$ .", "A translation is a reconstrual which preserves provability.", "We provide the formal statement next.", "Definition 4.3 A (weak) translation $F: T_1 \\rightarrow T_2$ between two coherent theories $T_1,T_2$ is a reconstrual $F: \\Sigma _1 \\rightarrow \\Sigma _2$ such that for all sequents $\\phi \\vdash \\psi $ of $T_1$ , we have: $\\phi \\vdash _{T_1} \\psi \\text{ implies } F\\psi \\vdash _{T_2} F\\psi .$ Moreover $F$ is an equality-preserving translation if $E_F^\\sigma (Fx,Fy)$ is logically equivalent to $(Fx = Fy) \\wedge D_\\sigma (Fx) \\wedge D_\\sigma (Fy)$ .", "More specifically, an equality-preserving translation $F: T_1 \\rightarrow T_2$ sends a function symbol $f: \\sigma \\rightarrow \\tau $ to a formula defining a functional relation $Ff: F\\sigma \\rightarrow F\\tau $ .", "Given translations $F: T_1 \\rightarrow T_2$ and $G: T_2 \\rightarrow T_3$ , we can define a composition $GF: T_1 \\rightarrow T_3$ by composing the extended maps between sets of formulae.", "We define $\\mathsf {CTh}_0$ to be the category of coherent theories with morphisms given by translations.This notation was introduced in .", "Similarly, we define $\\mathsf {CThEq}$ to be the category of coherent theories with morphisms given by equality-preserving translations.", "The definition of a translation is also formulable in classical logic.", "To do this, we add the requirement that $F(\\lnot \\phi ) = \\lnot F(\\phi )$ and $F(\\forall x \\phi (x)) = \\forall (Fx) F\\phi (Fx)$ .", "Through Morleyization, we see that classical theories with these translations form a subcategory $\\mathsf {Th}_0$ of $\\mathsf {CTh}_0$ .", "Stipulating that these translations are equality-preserving yields the analogous subcategory $\\mathsf {ThEq}$ .", "In accordance with , a translation $F: T_1 \\rightarrow T_2$ induces a pullback functor $F^: \\normalfont {\\text{Mod}}(T_2) \\rightarrow \\normalfont {\\text{Mod}}(T_1)$ defined by precomposition with $F$ .See Proposition 7.2.3 of .", "We can endow $\\mathsf {CTh}_0$ (and $\\mathsf {Th}_0$ ) with a 2-categorical structure.", "Namely, the 2-morphisms are given by t-maps (originally presented in ).", "Definition 4.4 Let $F,G: T_1 \\rightarrow T_2$ be a pair of translations.", "A t-map $\\chi :F \\Rightarrow G$ is a family of $T_2$ -formulae $\\chi _{\\sigma }\\hookrightarrow F\\sigma ,G\\sigma $ for every sort $\\sigma $ of $T_1$ such that the following sequents are provable in $T_2$ : (x,y) D,F(x) D,G(y) Domain matching Well-defined image EF(x,w) EG(y,z) (w,z) (x,y) D,F(x) y ( D,G(y) (x,y) ) Existence (x,y) (x,z) EG(y,z) Unique image For a list of sorts $\\vec{\\sigma }$ , we define $\\chi _{\\vec{\\sigma }}$ to be the conjunction of each component.", "Additionally, for any $T_1$ -substitution class $[\\phi ]\\hookrightarrow \\vec{\\sigma }$ , $T_2$ must prove: $\\chi _{\\vec{\\sigma }}(x,y) \\wedge F\\phi (x) \\vdash G\\phi (y).$ A t-map is an isomorphism if $T_2$ proves the following sequents as well: D,G(y) x (D,F(x) (x,y)), Onto ((x,y) (w,y)) EF(x,w).", "One-to-one We also require for any $T_1$ -substitution class $\\phi \\hookrightarrow \\vec{\\sigma }$ : $\\chi _{\\vec{\\sigma }}(x,y) \\wedge G\\phi (y) \\vdash F\\phi (x).$ We say that $T_1$ and $T_2$ are weakly intertranslatable (or homotopy equivalent or bi-interpretable) if there are translations $F: T_1 \\rightarrow T_2$ and $G: T_2\\rightarrow T_1$ such that $FG \\simeq 1_{T_2}$ and $GF \\simeq 1_{T_1}$ ." ], [ "Equivalence of $\\mathsf {CThEq}$ and {{formula:6ec1e442-0a16-4a2c-805b-bec26a2fe284}}", "The engine of this paper is the extension of the associations $T \\mapsto T$ and $C \\mapsto C$ to functors $ \\mathsf {CThEq}\\rightarrow \\mathsf {Coh}$ and $ \\mathsf {Coh}\\rightarrow \\mathsf {CThEq}$ .", "We will show that these functors form an equivalence of 2-categories: $ \\mathsf {CThEq}\\leftrightarrows \\mathsf {Coh}.$ This unites the work of with the formalism whose most general form is manifested in the work of , , , .", "This unification has two important consequences.", "Firstly, it shows that translations of predicate theories can be described entirely by a map between signatures.", "Secondly, it creates a bridge with which we can transport key results of categorical logic to a setting more familiar to traditional logicians—thereby explaining the surprising hypotheses necessary to establish the equivalence between intertranslatability and Morita equivalence.", "We begin with the map $T \\mapsto T$ .", "Proposition 5.1 Let $F: T_1 \\rightarrow T_2$ be an equality-preserving translation.", "There exists a coherent functor ${F}: {T_1} \\rightarrow {T_2}$ such that the following square commutes: $\\begin{tikzcd}T_1 [r,\"F\"] [d,\"I\"] & T_2 [d,\"I\"] \\\\{T_1} [r,\"{F}\"] & {T_2}.\\end{tikzcd}$ Moreover $F \\mapsto {F}$ preserves composition, so $\\mathsf {CThEq}\\rightarrow \\mathsf {Coh}$ is a functor of 1-categories.", "[Proof Sketch] Let $[\\phi ]$ be an object of ${T_1}$ with domain $\\sigma $ .", "Define $ {F}[\\phi ] \\equiv \\left[ F\\phi \\wedge D_\\sigma \\right].", "$ For a morphism $\\theta : [\\phi ] \\rightarrow [\\psi ]$ where $[\\psi ]$ has domain $\\tau $ , we define ${F}\\theta $ to be the morphism whose graph is given by the image of $[\\theta ]$ under ${F}$ .", "${F}[\\theta ]$ has a unique image since $F$ is equality-preserving, so this map is well-defined.", "Since $F$ preserves logical connectives, it also preserves composition, so ${F}$ is a functor.", "$F$ preserves conjunctions because ${F}$ is left-exact.", "Since $F$ preserves disjunctions, ${F}$ preserves joins.", "Lastly, $F$ preserves quantification, so ${F}$ preserves images.", "Thus ${F}$ is a coherent functor.", "Recall how ${F}$ is defined on objects: ${G}{F}[\\phi ] = [GF\\phi \\wedge D^{GF}_\\sigma ] = [GF\\phi \\wedge GD^F_\\sigma \\wedge D^G_\\sigma ] = {G}[F\\phi \\wedge D^F_\\sigma ] ={G}({F}[\\phi ]).$ Therefore, $F \\mapsto {F}$ preserves composition.", "This functor can be extended to include the 2-morphisms of $\\mathsf {CThEq}$ .", "We note that the conditions of a t-map $\\chi : F \\Rightarrow G$ describe a family of morphisms $\\chi _\\sigma : [D_\\sigma ^F] \\rightarrow [D_\\sigma ^G]$ .", "Naturality is proven in the following proposition: Proposition 5.2 Let $\\chi : F \\Rightarrow G$ be a t-map of equality-preserving translations $F,G: T_1 \\rightarrow T_2$ .", "Then there exists a natural transformation $\\eta :{F}\\Rightarrow {G}$ , and the association $\\chi \\mapsto \\eta $ preserves composition.", "[Proof Sketch] Define $\\eta : {F}\\Rightarrow {G}$ by setting for each object $\\phi \\hookrightarrow \\vec{\\sigma }$ of ${T_1}$ the morphism $\\eta _\\phi :{F}\\phi \\rightarrow {G}\\phi $ to be $\\chi _{\\vec{\\sigma }}$ .", "We need to prove that the following square commutes for any morphism $\\theta : \\phi \\rightarrow \\psi $ of ${T_1}$ , where $\\phi \\hookrightarrow \\vec{\\sigma }$ and $\\psi \\hookrightarrow \\vec{\\tau }$ : $ \\begin{tikzcd}{F}\\phi [r,\"{F}\\theta \"] [d,\"\\chi _{\\vec{\\sigma }}\"] & {F}\\psi [d,\"\\chi _{\\vec{\\tau }}\"] \\\\{G}\\phi [r,\"{G}\\theta \"] & {G}\\psi .\\end{tikzcd}$ This is true if and only if the following sequent is provable in $T_2$ : $\\exists y^{F\\vec{\\tau }} \\chi _{\\vec{\\tau }}(y,z) \\wedge F\\theta (x,y) \\dashv \\vdash \\exists w^{G\\vec{\\sigma }} G\\theta (w,z) \\wedge \\chi _{\\vec{\\sigma }}(x,w).$ Claim $ \\exists y \\chi _{\\vec{\\tau }} (y,z) \\wedge F\\theta (x,y) \\vdash \\exists wG\\theta (w,z) \\wedge \\chi _{\\vec{\\sigma }}(x,w).$ [Proof of Claim]$ $ 0 y (y,z) F(x,y) Premise (y,z) F(x,y) Elim.", "1 wG G .", "(x,y,w,) Definition (t-map) 2 .", "(x,y,w,) Elim.", "3 (x,w) (y,) Definition (t-map) 4 (y,z) (y,) Intro.", "(2,5) z = Definition (t-map) 6 G(w,z) Definition (t-map) + Equality (4,7) G(w,z) (x,w) Intro.", "(5,8) w G(w,z) (x,w) Intro.", "9 The proof of the converse sequent follows a similar argument.", "We note that the definition of $\\eta $ shows that $\\chi \\mapsto \\eta $ preserves composition.", "Therefore, the functor $ \\mathsf {CThEq}\\rightarrow \\mathsf {Coh}$ extends to a 2-functor.", "We now turn our attention to extending the internal logic map $C \\mapsto C$ to a 2-functor $ \\mathsf {Coh}\\rightarrow \\mathsf {CThEq}$ .", "Proposition 5.3 (Functoriality of Internal Logic) Let ${F}: C \\rightarrow D$ be a coherent functor (between coherent categories).", "There exists an equality-preserving translation $F: C \\rightarrow D$ , and the map ${F}\\mapsto F$ preserves composition.", "We need to define a reconstrual $F: \\Sigma _C \\rightarrow \\Sigma _D$ between the signatures of the internal logics of $C$ and $D$ , respectively.", "For a sort symbol $X^T \\in \\Sigma _C$ , ${F}X$ is an object of $D$ , so define $FX^T \\equiv ({F}X)^T$ .", "For a function $f^T: X^T \\rightarrow Y^T$ , define $Ff^T$ to be the graph of the morphism ${F}f$ .", "Set the domain formulae to be trivial.", "The last piece of the reconstrual to define is the map from $\\Sigma _C$ -variables to lists of $\\Sigma _D$ -variables.", "We claim such a map is essentially equivalent to ensuring each sort of a coherent theory has `enough' variables, see Remark REF .", "To see why $F$ is a translation, we need to show that $F$ sends axioms of $C$ to provable sequents of $D$ .", "We prove a stronger result—namely, we show that the image of any axiom of $C$ is an axiom of $D$ .", "First, ${F}$ is a functor, so the axioms for each commutative diagram in $C$ are preserved.", "Moreover, since ${F}$ is left-exact, the axioms for each product, equalizer, and terminal object are preserved.", "We also have that ${F}$ preserves unions, so any sequent for any join of subobjects is preserved.", "Finally, since ${F}$ preserves essential epimorphisms, any axiom for a surjective morphism is preserved.", "To show that $F$ is equality-preserving, observe that the object $[x =_Xy]$ is the kernel pair of $1_X: X \\rightarrow X$ in $C$ .", "Since ${F}$ is left-exact, it preserves kernel pairs.", "Furthermore, ${F}1_X = 1_{{F}X}$ since ${F}$ is a functor, so the image of $[x =_X y]$ is $[Fx=_{{F}X} Fy]$ .", "Thus, we conclude that $F$ is equality-preserving.", "Similar to the $\\mathsf {CThEq}$ case, $ extends to a 2-functor sending naturaltransformations between coherent functors to t-maps.", "This follows from theearlier argument (preceding Proposition \\ref {t-maps to natural transformations})that a t-map is a collection of morphisms $ XT: [DXTF] [DXTG]$.", "In this case, the domain formulae are trivial, so the t-mapreduces to a collection of morphisms of the form $ XT: F[x =X x] G[x =X x]$.", "In essence this proves the following proposition:$ Proposition 5.4 Let ${F},{G}: C \\rightarrow D$ be coherent functors, and let $\\eta : {F}\\Rightarrow {G}$ be a natural transformation.", "Then there exists a t-map $\\chi : F \\Rightarrow G$ .", "Moreover, $\\eta \\mapsto \\chi $ preserves composition." ], [ "Establishing Equivalence", "We will now show that $ and $ form an equivalence of categories $\\mathsf {CThEq}\\leftrightarrows \\mathsf {Coh}$ .", "To prove this we need to find natural isomorphisms $H:\\Rightarrow 1_{\\mathsf {CThEq}}$ and ${H}: \\Rightarrow 1_{\\mathsf {Coh}}$ .", "We omit the subscripts from the identity functors $1_\\mathsf {CThEq}$ and $1_\\mathsf {Coh}$ for the sake of brevity.", "Proposition 5.5 $\\simeq 1$ .", "Let $T$ be a coherent theory with signature $\\Sigma $ , and let $\\Sigma ^{\\prime }$ be the signature of $(T)$ .", "For a sort $\\sigma \\in \\Sigma $ , define $H_T\\sigma $ to be the sort $[x^\\sigma = x^\\sigma ]^T$ .", "Set the domain formulae to be trivial.", "For a relation $R \\hookrightarrow \\sigma $ in $\\Sigma $ , we have an object $[R]$ of $T$ along with a monomorphism $i_R: R \\rightarrow [x^\\sigma = x^\\sigma ]$ .", "Define $H_T R$ to be the substitution class $[\\exists z^{R^T} i_R^T(z) = x^{\\sigma ^T}]$ .", "For a 0-ary relation (a proposition) $R$ , define $H_T R$ to be the substitution class $[\\exists z^{R^T} i_R^T(z)]$ .", "For a function symbol $f: \\sigma \\rightarrow \\tau $ of $\\Sigma $ , $[f(x) = y]$ is a morphism in $T$ from $[x^\\sigma = x^\\sigma ]$ to $[y^\\tau = y^\\tau ]$ .", "Define $H_T f$ to be the substitution class associated with the graph of this morphism.", "We note that the extension of this map to $\\Sigma $ -formulae sends $\\phi (x)$ to $[\\phi ]^T(x)$ modulo logical equivalence.", "We now show that $H_T$ is an equality-preserving translation.", "For a sequent $\\phi (x) \\vdash \\psi (y)$ in $T$ , we have from Lemma REF that $[\\phi ,x.y] \\leqslant [\\psi ,x.y]$ in $T$ .", "This is equivalent to the statement that $[\\phi ,x.y] \\times _{[\\top ,x.y]}[\\psi ,x.y]$ is isomorphic to $[\\phi ,x.y]$ .", "Applying the axioms for products and the axioms for equalizers of $(T)$ , this implies that $[\\phi ,x.y]^T(x,y) \\vdash [\\psi ,x.y]^T(x,y)$ in $(T)$ .", "Since $[\\phi ,x.y]$ is the object $[\\phi (x) \\wedge y = y]$ , we have that $[\\phi (x),x.y]^T$ is logically equivalent to $[\\phi ]^T(x) \\wedge [y =y]^T(y)$ .", "However, $[y = y]^T(y)$ is logically equivalent to $\\top $ , so $[\\phi (x),x.y]^T$ is logically equivalent to $[\\phi ]^T(x)$ .", "Similarly, $[\\psi ,x.y]^T(x,y)$ is logically equivalent to $[\\psi ]^T(y)$ .", "Thus, $[\\phi ]^T(x) \\vdash [\\psi ]^T(y)$ is provable in $(T)$ , and this is equal to $H_T \\phi (x) \\vdash H_T \\psi (y)$ , so $H_T$ is a translation.", "To show $H_T$ is equality-preserving, note that $[E_{H_T}]$ is a pullback.", "Hence the axioms of products and equalizers in $(T)$ show that $[E_{H_T}]^T$ is the equality relation in $(T)$ , so $H_T$ is equality-preserving.", "We need to show that $H_T$ is one half of a homotopy equivalence.", "We define the second half $\\Theta _T: (T) \\rightarrow T$ first, then we will show that $\\Theta _T$ is a homotopy inverse of $H_T$ .", "Recall that the symbols of $\\Sigma ^{\\prime }$ are given by objects and morphisms in $T$ .", "For a sort $[\\phi ]^T$ of $\\Sigma ^{\\prime }$ , we have a corresponding object $[\\phi ]$ of $T$ with some domain $\\sigma $ .", "Set $\\Theta _T [\\phi ]^T \\equiv \\sigma $ .", "Observe that $[\\phi ]$ is a substitution class, so set the domain formula $D_{[\\phi ]^T} \\equiv [\\phi ]$ .", "For a function symbol $f^T:[\\phi ]^T \\rightarrow [\\psi ]^T$ there is a corresponding morphism $f: [\\phi ] \\rightarrow [\\psi ]$ .", "Define $\\Theta _T f^T$ to be the substitution class $[f]$ .", "Similar to the $H_T$ translation, $\\Theta _T$ is a translation due to Lemma REF .", "We now look at the composition $\\Theta _T H_T: T \\rightarrow T$ .", "For a formula $\\phi \\hookrightarrow \\sigma $ of $T$ , the image $H_T\\phi $ is the formula $\\exists z^{[\\phi ]^T} i_{[\\phi ]^T}(z) = x$ (up to $\\alpha $ -equivalence).", "The morphism $i_\\phi : [\\phi ] \\rightarrow [\\sigma ]$ is just the identity.", "Formally, $i_\\phi (x) = z$ is logically equivalent to $x = z$ .", "$\\Theta _T H_T \\phi (z)$ is the formula $\\exists x^\\sigma \\left( \\phi (x) \\wedge \\left( x = z \\right) \\right)$ , and this is logically equivalent to the formula $\\phi (z)$ .", "Thus, for any formula $\\phi $ , $\\Theta _T H_T \\phi $ is logically equivalent to $\\phi $ , so the choice of t-map is trivial when we define $\\chi _\\sigma (x,y) \\equiv \\left(x =y\\right)$ .", "Therefore $\\chi : \\Theta _T H_T \\Rightarrow 1$ .", "For the composition $H_T \\Theta _T: (T) \\rightarrow (T)$ , note that any formula of $(T)$ is an equality of terms (the only relations of $\\Sigma ^{\\prime }$ are equality relations).", "Thus, by making use of the axioms for commutative diagrams, we know that any atomic formula is logically equivalent to something of the form $f^T(x) = g^T(x)$ , where $f^T:[\\phi ]^T \\rightarrow [\\psi ]^T$ and $g^T: [\\phi ]^T \\rightarrow [\\psi ]^T$ are function symbols.", "The image of this formula under $\\Theta _T$ is $f(x) = g(x)\\wedge \\phi (x)$ .", "The substitution class of this formula is the meet of $[\\phi ]$ with the equalizer $[f(x) = g(x)]$ , so the image of this formula under $H_T$ is logically equivalent to $f^T(x) = g^T(x)$ .", "Hence the trivial t-map $\\chi _{[\\phi ]^T}(x,y) \\equiv \\left(x = y\\right)$ is a 2-isomorphism $\\chi : H_T \\Theta _T \\Rightarrow 1$ .", "The last step is showing $H$ is natural.", "This is straightforward.", "Having demonstrated that $H$ is a natural isomorphism $H: 1 \\Rightarrow $ , we proceed by showing ${H}: 1 \\Rightarrow $ is a natural isomorphism.", "Proposition 5.6 $\\simeq 1$ .", "Since there is a natural correspondence between the sorts of $C$ and the objects of $C$ , along with a natural correspondence between function symbols and morphisms, $$ is identical to 1.", "Thus, ${H}$ is the identity 2-morphism.", "These two propositions establish a 1-categorical equivalence between $\\mathsf {CThEq}$ and $\\mathsf {Coh}$ .", "Recalling the earlier arguments in Proposition REF and Proposition , this extends to the desired equivalence of 2-categories.", "theorem 5.7 The syntactic category functor $ and the internal logic functor $ form an equivalence of 2-categories: $ \\mathsf {CThEq}\\leftrightarrows \\mathsf {Coh}.", "$" ], [ "Extending to $\\mathsf {CTh}_0$ and {{formula:79e37148-457a-4f25-bf2e-91b240637569}}", "Let $F: T_1 \\rightarrow T_2$ be a translation between coherent theories, not necessarily equality-preserving.", "Recall that $E_F$ is the image of the equality relation under $F$ .", "While $E_F$ represents multiple relations, we usually refrain from this level of specificity since keeping track of domains is not difficult.", "Since $=_\\sigma $ is an equivalence relation of $T_1$ , and $F$ is a translation, we know the following sequents are provable in $T_2$ : EF(x,x), Reflexivity EF(x,y) EF(y,x), Symmetry EF(x,y) EF(y,z) EF(x,z).", "Transitvity This shows that $E_F$ is an equivalence relation for contexts of $D_\\sigma $ .", "From the categorical perspective, it follows that $[E_F]$ is a congruence on $[D_\\sigma ]$ .", "Since $F$ is not generally equality–preserving, we cannot naturally map the object $[\\sigma ]$ to $[D_\\sigma ]$ .", "This prevents $F$ from inducing a functor between syntactic categories.", "In order to find a categorical characterization of $\\mathsf {CTh}_0$ , we need to find where to send $[\\sigma ]$ .", "The argument above suggests that general translations induce coherent functors when ${T_2}$ is exact.", "This motivates us to explore the syntactical analogue of an exact category.", "This is known in the literature as elimination of imaginaries.", "Definition 5.8 Let $T$ be a coherent theory.", "$T$ is an exact theory if for any $T$ -formula $\\phi \\hookrightarrow \\sigma ,\\sigma $ such that the sequents $\\begin{array}{rl}&\\vdash \\phi (x,x), \\\\\\phi (x,y) &\\vdash \\phi (y,x),\\\\\\phi (x,y) \\wedge \\phi (y,z) &\\vdash \\phi (x,z)\\end{array}$ are provable in $T$ , there exists a function symbol $p: \\sigma \\rightarrow \\overline{\\sigma }$ such that the following sequents are provable: $\\begin{array}{rl}\\phi (x,y) \\dashv \\vdash & p(x) = p(y), \\\\\\vdash &\\exists y (p(y) = x).\\end{array}$ That is, $T$ is exact if $T$ eliminates imaginaries.", "Definition 5.9 $\\mathsf {ExactCThEq}$ is the sub-2-category of $\\mathsf {CThEq}$ spanned by exact theories.", "Proposition 5.10 $T$ is exact if and only if $T$ is exact.", "A formula $\\phi $ is an equivalence relation if and only if $[\\phi ]$ is a congruence in $T$ .", "The sequents of $\\overline{\\sigma }$ show that $p$ is a coequalizer.", "Corollary 5.11 Theorem REF restricts to an equivalence of 2-categories: $ \\mathsf {ExactCThEq} \\leftrightarrows \\mathsf {ExactCoh}.$ This corollary is close to a characterization of $\\mathsf {CTh}_0$ .", "The next step is showing any theory is homotopy equivalent (in $\\mathsf {CTh}_0$ ) to an exact theory.", "To get there we need two lemmas.", "Lemma 5.12 Let $F: T_1 \\rightarrow T_2$ be a weak translation between exact theories.", "Then $F$ is homotopic to an equality-preserving translation $\\hat{F}: T_1 \\rightarrow T_2$ .", "If $F$ is already equality-preserving, then $\\hat{F}$ is equal to $F$ .", "[Proof Sketch] The key element of this proof is defining $\\hat{F}$ ; the rest of the argument is straightforward.", "Per Theorem REF , it suffices to construct an appropriate functor between syntactic categories.", "Define the functor ${F}: {T_1} \\rightarrow {T_2}$ via ${F}[\\phi ] =\\exists _p [F\\phi ]$ , where $p: [\\sigma ] \\rightarrow [\\sigma _{/{E_F}}]$ is the quotient morphism from the congruence $[E_F]$ .", "Set $\\hat{F}$ to the induced equality-preserving translation from ${F}$ .", "If $F$ is already equality-preserving, then $E_F$ is just the equality relation, so $p$ is the identity.", "Thus, $\\hat{F} = F$ .", "Lemma 5.13 Let $T$ be a coherent theory.", "There exists an exact theory $T^{ex}$ and a homotopy equivalence of weak translations $E_T: T \\rightarrow T^{ex}$ and $R_T:T^{ex}\\rightarrow T$ .", "$T^{ex}$ is called the exact completion of $T$ .", "Assuming Lemma REF , we will first prove the desired equivalence.", "theorem 5.14 $T \\mapsto T^{ex}$ is part of an equivalence of 2-categories: $I \\dashv (-)^{ex}: \\mathsf {CTh}_0 \\leftrightarrows \\mathsf {ExactCThEq}.$ For a translation $F: T_1 \\rightarrow T_2$ in the category $\\mathsf {CTh}_0$ , define $F^{ex}: T_1^{ex}\\rightarrow T_2^{ex}$ as follows.", "Per Lemma , the composition $E_{T_2} \\circ F \\circ R_{T_1}$ is homotopic to an equality-preserving translation.", "Set $F^{ex}$ to this translation.", "Let $I$ be the functor reinterpreting an equality-preserving translation as a weak translation.", "The second part of Lemma REF shows that $(IG)^{ex}= G$ for any equality-preserving translation $G$ between exact theories.", "Therefore, $(-)^{ex}\\circ I \\simeq 1$ , at least as (1)-functors.", "Nevertheless, the action on t-maps is straightforward, so this extends to an equivalence of 2-functors.", "We now need to show that $I \\circ (-)^{ex}\\simeq 1$ .", "This follows from the action of $(-)^{ex}$ on translations along with Lemma .", "That is, consider the commutative diagram: $\\begin{tikzcd}T_1 [r,\"F\"] [d,\"E_{T_1}\",swap] & T_2 [d,\"E_{T_2}\"] \\\\T_1^{ex}[r,\"EFR\"] & T_2^{ex},\\end{tikzcd}$ and recall that $E_T$ and $R_T$ are homotopy inverses for any $T$ .", "Once again the action on t-maps is straightforward, so this establishes $I \\circ (-)^{ex}\\simeq 1$ as 2-functors.", "Remark 5.15 This theorem reveals the paradoxical result that $\\mathsf {CTh}_0$ can be realized as a subcategory of $\\mathsf {CThEq}$ .", "From a philosophical perspective, this shows that no real generality is added by considering weak translations over equality-preserving translations.", "This is because—while adding weak translations adds more morphisms—any theory is weakly intertranslatable with its exact completion.", "We can compose $(-)^{ex}$ with the syntactic category functor $ to yieldthe following corollary:$ Corollary 5.16 There is an equivalence of 2-categories: ${ex}: \\mathsf {CTh}_0 \\leftrightarrows \\mathsf {ExactCoh}.$ Remark 5.17 This last corollary reveals the categorical analogue of a weak translation: a weak translation $F: T_1 \\rightarrow F_2$ is a coherent functor between the exact completions ${T_1}^{ex}\\rightarrow {T_2}^{ex}$ .", "Compare this to an equality-preserving translation, where the exact completion requirement is dropped (Proposition REF ).", "The remainder of this section will be a proof of Lemma REF .", "Recall the exact completion $C^{ex}$ of a coherent category $C$ (Theorem REF ), defined in chapter 8 of .", "There is an inclusion functor $T \\rightarrow T^{ex}$ .", "Define $E_T$ to be the induced translation.", "For a coherent theory $T$ , the internal logic of $T^{ex}$ is constructed from $T$ by appending a sort $\\sigma _{/A}$ and surjective function symbol $p_A: \\sigma \\rightarrow \\sigma _{/A}$ for any equivalence relation $A \\hookrightarrow \\sigma ,\\sigma $ .", "Thus, the internal logic of $T^{ex}$ is an exact theory which is also a Morita extension of $T$ consisting of only quotient sorts.", "The translation $R_T$ follows from the following proposition: Proposition 5.18 Let $T$ be a coherent theory, and let $T^+$ be a Morita extension of $T$ consisting of only quotient sorts.", "Then $T$ and $T^+$ are homotopy equivalent.", "Let $\\Sigma $ be the signature of $T$ , and let $\\Sigma ^+$ be the signature of $T^+$ .", "Based on the observations preceding this proposition, we have an inclusion translation $E_T: T \\rightarrow T^+$ .", "Thus, we just need to find a definition of $R_T: T^+ \\rightarrow T$ .", "We first define the necessary reconstrual: for any symbol $s \\in \\Sigma ^+ \\cap \\Sigma $ , define $R_T s\\equiv s$ , where the right side of this definition is $s$ interpreted in $T$ .", "If $s$ is a sort symbol, let the domain formula $D_s$ be trivial.", "The only symbols in $\\Sigma ^+$ left are the quotient sorts $\\sigma _{/A}$ , their surjections $p_A: \\sigma \\rightarrow \\sigma _{/A}$ , and their equality relations $=_{\\sigma _{/A}} \\hookrightarrow \\sigma _{/A},\\sigma _{/A}$ from the Morita extension (where $A$ is an equivalence relation in $T$ ).", "Define $R_T (\\sigma _{/A}) \\equiv \\sigma $ , and set the domain formula $D_{\\sigma _{/A}}$ to be trivial.", "Define $R_T(=_{\\sigma _{/A}}) \\equiv A$ .", "Lastly, set $R_T(p_A) \\equiv A$ .", "This exhausts all the symbols in $\\Sigma ^+$ , so we have a reconstrual.One may object that the reconstrual is not defined until we describe the map from variables of type $\\sigma _{/A}$ to variables of type $\\sigma $ .", "We will ignore this point; such a map can be specified in multiple ways.", "The next step is verifying $R_T$ is indeed a translation.", "This amounts to checking whether the defining axioms of $\\sigma _{/A}$ are sent to provable sequents in $T$ for any equivalence relation $A$ .", "Consider an arbitrary equivalence relation $A \\hookrightarrow \\sigma ,\\sigma $ .", "Then there are two sequents to check $A(x,y) \\dashv \\vdash p_A(x) = p_A(y)$ and $\\vdash \\exists x^\\sigma \\left( p_A(x) = y \\right)$ .", "Recall $R_T(p_A) \\equiv A$ , so the graph $p_A(x) = y$ is mapped to $A(R_T x,R_T y)$ .", "Thus the first sequent is mapped to $A(x,y) \\dashv \\vdash \\exists u^\\sigma \\exists v^\\sigma A(u,v) \\wedge A(x,u) \\wedge A(y,v)$ , which is provable in $T$ (apply symmetry of $A$ ).", "The second sequent is mapped to $\\vdash \\exists x^\\sigma A(x,y)$ , which is provable in $T$ as well (apply reflexivity of $A$ ).", "Hence $R_T$ is a translation.", "The final step is showing that $R_T$ and $E_T$ are homotopy inverses.", "$E_T$ is an inclusion map, i.e.", "$E_T s \\equiv s$ for any symbol $s \\in \\Sigma $ .", "Moreover, $R_T s \\equiv s$ for any $s \\in \\Sigma ^+ \\cap \\Sigma $ .", "Therefore $R_T E_T$ is identical to the identity translation on $T$ , so we can pick a trivial t-map: $\\chi _\\sigma (x,y) \\equiv (x = y)$ .", "This leaves the reverse direction.", "For sorts $\\sigma \\in \\Sigma ^+ \\cap \\Sigma $ , the choice of t-map is easy: set $\\chi _\\sigma (x,y) \\equiv (x =y)$ .", "This leaves the quotient sorts.", "Set $\\chi _{\\sigma _{/A}}(x,y) \\equiv (p_A(y) = x)$ .", "Proving this is a homotopy is a simple, albeit tedious, argument based on induction on the symbols in $\\Sigma ^+$ , so we will omit it.", "These two t-maps establish $R_T E_T \\simeq 1$ and $E_T R_T \\simeq 1$ , as desired." ], [ "Morita Extensions, Proper Theories, and Pretopos Completions", "Restricting to equality-preserving translations introduces unnecessarily strong restrictions on the kinds of translations between theories we permit.", "There are two reasons for this.", "The first is that a homotopy equivalence in $\\mathsf {CThEq}$ is stronger than weak intertranslatability as defined in , .", "The second reason is that weak intertranslatability is stronger than Morita equivalence.", "Nevertheless, there is a class of theories for which Morita equivalence and weak intertranslatability coincide.", "These are called proper theories, which were originally described in .", "We pick a definition closer to the one in , .", "Definition 6.1 Let $T$ be a coherent theory.", "We say that $T$ is proper if there exist formulae $\\phi ,\\psi \\hookrightarrow \\sigma $ such that $\\phi (x) \\wedge \\psi (x) \\vdash \\bot $ and $\\vdash \\exists x^\\sigma \\phi (x)\\wedge \\exists y^\\sigma \\psi (y)$ are provable in $T$ .", "Moreover, we say that $\\phi $ and $\\psi $ realize $T$ as a proper theory.", "For a classical theory, set $\\psi $ to the formula $\\lnot \\phi $ .", "With this, proved the following theorem: theorem 6.2 (Washington) Two classical proper theories $T_1$ and $T_2$ are weakly intertranslatable if and only if they are Morita equivalent.", "Corollary REF is a generalization of Theorem REF , described using category theory.", "We first make explicit the relationship of that corollary to Washington's theorem.", "theorem 6.3 (Weak intertranslatability is almost Morita equivalence) Let $T_1$ and $T_2$ be coherent theories.", "$T_1$ and $T_2$ are weakly intertranslatable (in $\\mathsf {CTh}_0$ ) if and only if they are Morita equivalent using only products, subsorts, and quotients.", "From Corollary REF , we know that $T_1$ and $T_2$ are weakly intertranslatable if and only if ${T_1}^{ex}$ and ${T_2}^{ex}$ are equivalent categories.", "Since these categories are finitely complete and exact, they are closed under additions of products and quotients.", "Furthermore, subsort extensions do not add any new objects or morphisms.", "Thus the latter condition is the statement that $T_1$ and $T_2$ are Morita equivalent using only products, subsorts, and quotients.", "With this theorem in mind, we will now extend to proper theories and pretopoi.", "This will yield an extension of Theorem REF to coherent logic.", "Recall the following theorem from : theorem 6.4 (Tsementzis) Two coherent theories $T_1$ and $T_2$ are Morita equivalent if and only if ${T_1}$ and ${T_2}$ have equivalent pretopos completions.", "Not all exact coherent categories are pretopoi, so there is additional structure in the internal logic of a coherent category which allows us to leap from an exact category to a pretopos.", "This additional structure is the condition of being a proper theory.", "There are two (essentially equivalent) approaches to proving this.", "For classical logic, showed any coproduct Morita extension of a proper theory is definitionally equivalent to a quotient extension of a product extension (see Chapter 3, Lemma 1 in ).", "Similar work in provides a more categorical perspective, showing that the syntactic category of Shelah's extension of a classical theory is a Boolean pretopos.", "Shelah's extension is equivalent to Makkai and Reyes' exact completion.", "In the notation of this paper, Shelah's extension of $T$ is $T^{ex}$ .", "Both of these approaches generalize to coherent logic without difficulty.", "We state below Harnik's result and provide the necessary step to extend to coherent logic.", "The rest of the proof can be inferred from Theorem 5.3 of .", "theorem 6.5 (Harnik) Let $T$ be a coherent theory.", "If $T$ is proper, then $T^{ex}$ is a pretopos.", "Therefore, $T^{ex}$ is the pretopos completion of $T$ .", "[Proof Sketch] Assume $\\phi ,\\psi \\hookrightarrow \\sigma $ realize $T$ as a proper theory.", "Let $\\tau $ denote the subsort (extension) determined by the class $[\\phi (x) \\vee \\psi (x)]$ .", "For any coproduct $\\sigma _1 + \\sigma _2$ , consider the formula $\\epsilon \\hookrightarrow \\sigma _1 \\times \\sigma _2\\times \\tau , \\sigma _1 \\times \\sigma _2 \\times \\tau $ where $\\epsilon (x_1,y_1,z_1,x_2,y_2,z_2)$ is given by: $\\left( \\phi (z_1) \\wedge \\phi (z_2) \\wedge \\left( x_1 = x_2 \\right) \\right)\\vee \\left( \\psi (z_1) \\wedge \\psi (z_2) \\wedge \\left( y_1 = y_2 \\right)\\right).$ The remainder of the proof is showing that $\\sigma _1 + \\sigma _2$ is definitionally equivalent to $\\left( \\sigma _1 \\times \\sigma _2 \\times \\tau \\right)/\\epsilon $ , which we defer to or .", "The converse, however, was an open question.", "We prove it below.", "theorem 6.6 Let $T$ be a coherent theory.", "If $T^{ex}$ is a pretopos, then $T$ is a proper theory.", "This proof has two parts.", "First, we show the internal logic of $T^{ex}$ is proper.", "The second part is verifying that properness is preserved by weak intertranslatability.", "We conclude the desired result by invoking Corollary REF .", "For the first step, we take advantage of the hypothesis that $T^{ex}$ is a pretopos.", "This implies the object $2 = 1 + 1$ is defined in $T^{ex}$ (where 1 is the terminal object).", "That is, there exist a substitution class $[\\phi ]\\hookrightarrow \\sigma $ and formulae $\\rho _1 \\hookrightarrow \\sigma $ and $\\rho _2\\hookrightarrow \\sigma $ such that $[\\phi ] \\equiv 2$ and $\\rho _1,\\rho _2: 1 \\rightarrow 2$ are the injection morphisms.", "Since $[\\phi ]$ is a coproduct, the sequents $\\rho _1(x) \\wedge \\rho _2(x) \\vdash \\bot $ and $\\vdash \\exists x \\rho _1(x) \\wedge \\exists y \\rho _2(y)$ are provable in the internal logic of $T^{ex}$ .", "Therefore $\\rho _1$ and $\\rho _2$ realize the internal logic as a proper theory.", "For the second step, we can prove a more general result.", "Assume $F: T_1 \\rightarrow T_2$ is a weak translation and $T_1$ is a proper theory.", "We prove that $T_2$ is also a proper theory.", "The more specific result about weak intertranslatability follows immediately from this.", "Suppose $\\phi ,\\psi \\hookrightarrow \\sigma $ realize $T_1$ as a proper theory.", "Since $F$ is a translation, this implies $F\\phi (x) \\wedge F\\psi (x) \\vdash \\bot $ and $\\vdash \\exists x^{F\\sigma } F\\phi (x)\\wedge \\exists y^{F\\sigma } F\\psi (y)$ are provable in $T_2$ .", "Thus, $F\\phi $ and $F\\psi $ realize $T_2$ as a proper theory.", "Corollary 6.7 Let $\\mathsf {Proper}$ denote the 2-category of proper coherent theories with weak translations and t-maps.", "There is an equivalence of 2-categories: ${ex}: \\mathsf {Proper}\\leftrightarrows \\mathsf {Pretopos}.$ Corollary 6.8 Two (coherent) proper theories are Morita equivalent if and only if they are weakly intertranslatable." ], [ "Morita Equivalence Compared to Categorical Equivalence", "The link between translations and pretopoi allows us to introduce Makkai and Reyes' conceptual completeness to the categories of theories.", "theorem 7.1 Let $F: T_1 \\rightarrow T_2$ be a translation of proper theories.", "If $F^:\\normalfont {\\text{Mod}}(T_2) \\rightarrow \\normalfont {\\text{Mod}}(T_1)$ is an equivalence of categories, then $F$ is part of a homotopy equivalence.", "This follows from Corollary REF and Theorem REF .", "In a similar vein to this corollary, we know that translations must preserve the ultracategorical structure of $\\normalfont {\\text{Mod}}(T)$ .", "More precisely, we prove the following: theorem 7.2 Let $T_1$ and $T_2$ be coherent theories such that $F: T_1 \\rightarrow T_2$ is a translation.", "Then $F^: \\normalfont {\\text{Mod}}(T_2) \\rightarrow \\normalfont {\\text{Mod}}(T_1)$ preserves ultraproducts.", "First, note that $F$ induces a unique coherent functor ${F}: {T_1} \\rightarrow {T_2}$ which lifts to a unique coherent functor ${F}^P: P_{T_1} \\rightarrow P_{T_2}$ between pretopos completions.", "The pullback functor $F^$ is given by precomposition of a model $M: {T_2} \\rightarrow {\\mathsf {Set}}$ with ${F}$ .", "Any such $M$ extends to a model $\\tilde{M}: P_{T_2} \\rightarrow {\\mathsf {Set}}$ , and $M\\circ {F}$ is equivalent to $\\tilde{M} \\circ {F}^P$ .", "Since coherent functors between pretopoi preserve ultraproducts (by Makkai duality), precomposition with ${F}^P$ preserves ultraproducts, so $F^$ does as well.", "We can also apply Corollary REF to Makkai duality to obtain a fully faithful embedding: $\\mathsf {Proper}^{\\mathsf {op}}\\rightarrow \\mathsf {UltraCat}.", "$ Furthermore, $\\mathsf {UltraCat}$ is a subcategory of $\\mathsf {Cat}$ , the category of small categories.", "Categorical equivalence of proper theories $T_1$ and $T_2$ is a homotopy equivalence in $\\mathsf {Cat}$ , whereas this embedding implies Morita equivalence is a homotopy equivalence in $\\mathsf {UltraCat}$ .", "This provides an answer to a question originally posed in , namely whether there is a condition $\\mathfrak {P}$ such that Morita equivalence is categorical equivalence combined with $\\mathfrak {P}$ .", "This condition $\\mathfrak {P}$ is the preservation of the ultrastructure of $\\normalfont {\\text{Mod}}(T_1)$ and $\\normalfont {\\text{Mod}}(T_2)$ .", "In other words, we summarize our work in the following theorem: theorem 7.3 Let $T_1$ and $T_2$ be proper theories.", "Then $T_1$ and $T_2$ are Morita equivalent if and only if $T_1$ and $T_2$ are categorically equivalent and such an equivalence lifts to $\\mathsf {UltraCat}$ .", "theorem 7.4 (Makkai Duality of Weak Translations) There is a fully faithful embedding of coherent, proper theories into ultracategories given by the functor $\\normalfont {\\text{Mod}}$ : $\\normalfont {\\text{Mod}}: \\mathsf {Proper}^{\\mathsf {op}}\\rightarrow \\mathsf {UltraCat},$" ], [ "Universal Morita Extensions", "Consider a coherent theory $T$ with a Morita extension $T^+$ consisting of only product sorts and subsorts.", "Since $T$ is finitely complete, there is an equivalence of categories $T \\cong {T^+}$ .", "Porting this equivalence through Theorem REF , we deduce $T$ and $T^+$ are homotopy equivalent in $\\mathsf {CThEq}$ —that is, homotopy equivalent by means of equality-preserving translations.", "Similarly, consider a Morita extension $T^+$ consisting of products, subsorts, and quotients.", "Since $T^{ex}$ is closed under quotients of congruences, we have $T^{ex}\\cong {T^+}^{ex}$ , so by Theorem REF , $T$ and $T^{\\prime }$ are homotopy equivalent in $\\mathsf {CTh}_0$ , i.e.", "homotopy equivalent by means of weak translations.", "These two examples suggest that $_T$ and $_{T}^{ex}$ are universal with respect to Morita extensions.", "We formalize this universal property in the following theorems.", "This framework will show $P̰_T$ is universal with respect to all Morita extensions.", "We say that a translation $F: T_1 \\rightarrow T_2$ is strong if it is equality-preserving and $F[\\phi ]$ has a context of length 1 if and only if $[\\phi ]$ has a context of length 1.", "One can show from this that a strong translation preserves the length of any context.", "Theorems 4.6.17 and 6.6.21 of yield the following useful characterization of strong translations: Lemma 7.5 Two coherent theories $T_1$ and $T_2$ are definitionally equivalent if and only if they are homotopy equivalent via strong translations.", "theorem 7.6 (Morita Completion) Let $T$ be a coherent theory.", "There exists another coherent theory $T^M$ and a strong, conservative translation $I^M_T: T \\rightarrow T^M$ such that the following conditions hold: For any Morita extension $T^+$ of $T$ , ${T^+}^M$ is definitionally equivalent to $T^M$ .", "Any Morita extension ${T^M}^+$ of $T^M$ is definitionally equivalent to $T^M$ .", "Let $J_T$ denote the poset of Morita extensions of $T$ ; then $T^M \\cong \\varinjlim {J_T}$ .", "$T^M$ is called the Morita completion of $T$ .", "Given $T$ , we define the Morita completion $T^M$ to be the internal logic of the pretopos completion $P_T$ .", "From Theorem REF , there is an inclusion functor $T \\rightarrow P_T$ which is conservative.", "From Theorem REF this induces a conservative equality-preserving translation which we will call $I^M: T \\rightarrow T^M$ .", "The derivation of the pretopos completion in shows that $T^M$ is a Morita extension of $T$ .", "In fact, $T^M$ adds all possible sorts obtainable from $T$ by Morita extension.", "Before proving Theorem REF , it will help to elaborate on the structure of $J_T$ .", "Given any Morita extension $T^+$ of $T$ , there is an inclusion reconstrual, denoted $I^+_T: T \\rightarrow T^+$ , given by reinterpreting symbols in $T$ in the larger language of $T^+$ .", "This reconstrual is always a conservative, strong translation (see ).", "Since this is the inclusion translation, it is also induced by the inclusion functor between syntactic categories.", "$J_T$ can be considered a thin category, where the morphisms are these inclusion translations.", "Thus the last point of Theorem REF can be rephrased as the existence of a strong conservative translation $I^M_+: T^+ \\rightarrow T^M$ (for any Morita extension $T^+$ of $T$ ) making the following diagram commute: $\\begin{tikzcd}T [r,\"I^M\"] [dr,\"I^+\",swap] & T^M \\\\& T^+ [u,\"I^M_+\",dotted,swap].\\end{tikzcd}$ [Proof of Theorem REF ] Proof of (1): We first prove the following claim: Claim Let $C$ and $D$ be equivalent coherent categories.", "Then $C$ is definitionally equivalent to $D$ .", "[Proof of Claim] The homotopy equivalence of $C,D$ is given by some pair of coherent functors ${G}, {F}: C \\leftrightarrows D$ .", "The induced equality-preserving translations we will call $F \\equiv $ and $G\\equiv $ .", "Recall the definition of $F$ : $F\\sigma ^T \\equiv ({F}\\sigma )^T$ .", "Since $F$ is also equality-preserving, this shows $F$ is strong.", "The same argument shows $G$ is strong.", "Thus, $T_C$ and $T_D$ are homotopy equivalent via strong translations.", "Lemma REF implies $T_C$ and $T_D$ are definitionally equivalent, proving the claim.", "Equipped with this fact, we note that $P_T \\cong P_{T+}$ , so ${P_T}$ is definitionally equivalent to ${P_{T^+}}$ .", "Proof of (2): Since $T^M$ is the internal logic of a pretopos—which is closed under products, quotients, and coproducts—every sort of $T^{M+}$ is isomorphic to a sort of $T^M$ .", "The inclusion $\\Sigma ^M \\rightarrow \\Sigma ^{M^+}$ induces a reconstrual $T^M \\rightarrow T^{M^+}$ .", "In the other direction there is a reconstrual $\\Sigma ^{M^+} \\rightarrow \\Sigma ^M$ which is the identity on $\\Sigma ^M \\cap \\Sigma ^{M^+}$ and chooses an isomorphic object in $\\Sigma ^M$ for anything in $\\Sigma ^{M^+}\\setminus \\Sigma ^M$ .", "These reconstruals give rise to strong translations that make a homotopy equivalence between $T^M$ and $T^{M+}$ ; as above, this implies definitional equivalence.", "Proof of (3): Since $T^+$ is a Morita extension, ${T^+}$ is the result of adding products, quotients, or coproducts to $T$ .", "Since ${T^M} = P_T$ is a pretopos, any such addition already exists in ${T^M}$ .", "Therefore, we have an inclusion functor ${T^+} \\rightarrow {T^M}$ .", "Let $I_+^M$ be the induced equality-preserving translation.", "This translation is induced by an inclusion functor, so it is strong.", "Since the relevant translations are induced by inclusion functors, the following diagram commutes for any $T^+$ : $\\begin{tikzcd}T [r,\"I^M\"] [dr, \"I^+\", swap] & T^M \\\\& T^+ [u,\"I_+^M\",swap].\\end{tikzcd}$ Furthermore, suppose there exists some $T^{\\prime }$ such that for every $T^+$ there is a map $I_+^{\\prime }: T^+ \\rightarrow T^{\\prime }$ with $I_+^{\\prime } \\circ I^+=I^{\\prime }:T\\rightarrow T^{\\prime }$ for some map $I^{\\prime }$ .", "Then, since $T^M$ is itself a Morita extension of $T$ , there is some $I_M^{\\prime }$ such that $I_M^{\\prime } \\circ I^M=I^{\\prime }$ , so $I_M^{\\prime }\\circ I_+^M \\circ I^+=I^{\\prime }$ .", "Since $I^+$ is monic, this gives us $I_M^{\\prime } \\circ I_+^M=I_+^{\\prime }$ , so $I_+^{\\prime }$ factors through $T^M$ by way of $I_M^{\\prime }$ .", "Since $I_+^M$ is monic, $f \\circ I_+^M=I_+^{\\prime }$ implies $f=I_M^{\\prime }$ .", "Thus we have existence and uniqueness; $T^M$ is indeed the coproduct of $J_T$ .", "If we seek a category of theories in which homotopy equivalence is Morita equivalence—let us call it simply $\\mathsf {CTh}$ , since it is in some sense the most natural analogue to Morita equivalence—this completion yields a candidate category.", "Definition 7.7 Let $\\mathsf {CTh}$ be the 2-category where objects are coherent theories.", "We define $\\normalfont {\\text{Hom}}_\\mathsf {CTh}(T_1,T_2) = \\normalfont {\\text{Hom}}_\\mathsf {CThEq}(T_1^M,T_2^M)$ .", "The 2-morphisms are t-maps.", "Theorem REF shows that $T^M$ has all product sorts (up to definitional equivalence), including the terminal sort.", "Therefore all translations in $\\mathsf {CTh}$ are homotopic to strong translations.", "Let $\\mathsf {CThStrong}$ be the 2-category of coherent theories, strong translations, and t-maps.", "Then these homotopies allow us to equivalently write $\\normalfont {\\text{Hom}}_\\mathsf {CTh}(T_1,T_2) \\cong \\normalfont {\\text{Hom}}_\\mathsf {CThStrong}(T_1^M,T_2^M)$ .The exact completion $T \\mapsto T^{ex}$ is analogous to the Morita completion, restricting to Morita extensions which do not add coproducts.", "A similar argument shows $\\normalfont {\\text{Hom}}_{\\mathsf {CTh}_0}(T_1,T_2) \\cong \\normalfont {\\text{Hom}}_{\\mathsf {CThStrong}}(T_1^{ex},T_2^{ex})$ .", "$T\\mapsto _T$ shares a similar universal property, which we could call lex completion, restricting to Morita extensions which do not add coproducts or quotients.", "This yields $\\normalfont {\\text{Hom}}_{\\mathsf {CThEq}}(T_1,T_2) \\cong \\normalfont {\\text{Hom}}_{\\mathsf {CThStrong}}(_{T_1},_{T_2})$ .", "In either case, we arrive at the following result: theorem 7.8 There is an equivalence of 2-categories: $P: \\mathsf {CTh}\\leftrightarrows \\mathsf {Pretopos}.$" ], [ "Propositional Theories and Weak Intertranslatability", "Consider a propositional theory $T$ .", "We can interpret $T$ as a coherent theory (or a classical theory, depending on the axioms of $T$ ) with no sorts.", "Therefore, we can construct Morita extensions of propositional theories.", "Since propositional theories have far simpler structure than predicate theories, we would like to identify coherent theories which are homotopy equivalent to propositional theories.", "Most arguments and results involving $\\mathsf {Coh}$ and $\\mathsf {CThEq}$ will apply to $\\mathsf {ExactCoh}$ and $\\mathsf {CTh}_0$ .", "We note special considerations where they appear.", "Let $\\mathsf {BDLat}$ be the 2-category with objects bounded distributive lattices and morphisms bounded distributive lattice homomorphisms.", "For morphisms $F,G:L_1\\rightarrow L_2$ , we define a single non-trivial 2-morphism $\\chi : F \\Rightarrow G$ if and only if $Fx\\leqslant Gx$ for all $x\\in L_1$ .", "Lemma 7.9 $\\mathsf {BDLat}$ is equivalent to the full subcategory of $\\mathsf {Coh}$ spanned by thin categories.", "The 1-categorical version of this result is well-known.", "As for natural transformations (2-morphisms), there is at most non-trivial one between any two functors $F,G: L_1\\rightarrow L_2$ by the thinness of $L_2$ .", "In particular, one exists if and only if for every object $x$ of $L_1$ we have a morphism $f_x: F(x) \\rightarrow G(x)$ .", "Since this is true if and only if $F(x)\\leqslant G(x)$ when $F$ and $G$ are considered as lattice homomorphisms, we are done.", "Thus, we state the full inclusion $\\mathsf {BDLat}\\subseteq \\mathsf {Coh}$ .", "Lemma 7.10 $\\mathsf {BDLat}$ is also a full subcategory of $\\mathsf {ExactCoh}$ .", "It suffices to show that every bounded distributive lattice $L$ is an exact category.", "Suppose $K \\rightrightarrows X$ is a congruence in $L$ .", "The reflexivity condition of a congruence tells us we have a chain of morphisms $X \\rightarrow K \\rightarrow X$ , so $X \\leqslant K$ and $K \\leqslant X$ ($L$ is thin).", "$L$ is also a partially ordered set, hence $K = X$ .", "Thus, all congruences are of the form $(X, 1_X, 1_X$ ).", "This is the kernel pair for $1_X : X \\rightarrow X$ , so we conclude that $L$ is exact.", "We proceed by defining the 2-category $\\mathsf {CThProp}$ to be the full subcategory of $\\mathsf {CThEq}$ spanned by theories with no sorts.", "Recall that a 2-morphism is a t-map.", "Ergo, between translations $F$ and $G$ , there exists at most one non-trivial t-map $\\chi : F\\Rightarrow G$ .", "It exists if and only if $F\\phi \\vdash _{T_2} G\\phi $ for all formulae $\\phi $ .", "Since there is no question of equality-preservation when there are no sorts, this is in turn equivalent to the analogous subcategory of $\\mathsf {CTh}_0$ .", "We elide the distinction and simply say $\\mathsf {CThProp}\\subseteq \\mathsf {CThEq}$ .", "theorem 7.11 $ \\mathsf {CThEq}\\leftrightarrows \\mathsf {Coh}$ and ${ex}:\\mathsf {CTh}_0 \\leftrightarrows \\mathsf {ExactCoh}$ restrict to an equivalence $\\mathsf {CThProp}\\leftrightarrows \\mathsf {BDLat}$ .", "Bounded distributive lattices are exact, so by the definition of ${ex}$ it suffices to show that the image of $\\mathsf {CThProp}$ under $ is precisely$ BDLat$.", "Let $ T$ be an object of $ CThProp$.", "For objects $ []$and $ []$ of $ T$, consider morphisms $ : [] []$.", "Since the domains of $ []$ and $ []$ are$$ and $$, the domain of $$ must be$ .=$.", "As such, $ y (x,y) $.", "Thus, the conditions for $$ to be a morphism reduce to:$$\\begin{array}{rl}\\phi & \\vdash \\theta , \\\\\\theta & \\vdash \\phi \\wedge \\psi .\\end{array}$$ This tells us that $$ must imply $$ and be logicallyequivalent to $$.Hence, there exists at most onemorphism from $ []$ to $ []$, andit exists exactly when $$.", "So $ T$ isthin.\\footnote {Specifically, it is the the Lindenbaum lattice L_T,discussed later.", "}$ Now, let $L$ be any member of $\\mathsf {BDLat}$ .", "We define $T_L$ as the theory having one proposition for each element of $L$ , with $P\\vdash Q$ an axiom precisely when $P\\leqslant Q$ in $L$ and no other axioms.", "Thus, ${T_L}$ is a thin coherent category that has a bijection of elements with $L$ , and it has morphisms precisely where $L$ has morphisms.", "This allows us to conclude that ${T_L}\\cong L$ .", "Therefore, the essential image of $ is $ BDLat$, completing the proof.$ This shows that a coherent theory $T$ is homotopy equivalent (in $\\mathsf {CThEq}$ and $\\mathsf {CTh}_0$ ) to a propositional theory exactly when $T$ is thin.", "By Lemma REF (presented later), the syntactic category $T$ is thin if and only if $!_X: X\\rightarrow 1$ is monic for all $X \\in ob(T)$ .", "Since models preserve monics, a morphism in $T$ is monic if and only if its image is monic in all models.", "Hence, $!_X$ is monic if and only if $X$ is always mapped to $\\emptyset $ or $\\lbrace *\\rbrace $ .", "This holds for all $X$ if and only if $\\vdash x^\\sigma = y^\\sigma $ holds for all $\\sigma \\in \\Sigma $ .", "Furthermore, if $!_X: X \\rightarrow 1$ is monic, then $X$ is isomorphic to its image.", "That is, $\\phi \\dashv \\vdash \\exists x \\phi $ is provable for all formulae $\\phi $ of $T$ .", "We remind ourselves that homotopy equivalence in $\\mathsf {CThEq}$ or $\\mathsf {CTh}_0$ is precisely weak intertranslatability (and the question of equality-preservation does not occur with propositional theories).", "Thus, we prove that a theory is intertranslatable with a propositional one if and only if every formula is logically equivalent to a sentence.", "We summarize our results in the following: theorem 7.12 The following are equivalent for a coherent theory $T$ : $T$ is weakly intertranslatable with a propositional theory.", "$T$ is thin.", "$T\\cong Sub_{T}(1)$ .", "$!_X$ is monic for all $X\\in ob(T)$ .", "$\\vdash x^\\sigma =y^\\sigma $ for all sorts $\\sigma \\in \\Sigma $ .", "$\\phi (x)$ is logically equivalent to $\\exists x\\phi (x)$ for all $T$ -formulae $\\phi $ .", "Every $T$ -formula is logically equivalent to a sentence.", "Equipped with $\\mathsf {BDLat}\\cong \\mathsf {CThProp}$ , we are in a better position to inquire about the well-known map $L_{(–)}$ from theories to bounded distributive lattices: the Lindenbaum-Tarski functor.", "For a theory $T$ , the objects of $L_T$ are logical equivalence classes of sentences in $T$ , and $S\\leqslant _{L_T}S^{\\prime }$ if and only if $S\\vdash _T S^{\\prime }$ .", "We show that $L_{(–)}$ is functorial on the source categories $\\mathsf {CThEq}$ and $\\mathsf {CTh}_0$ , and moreover left adjoint to the inclusion (by way of $\\mathsf {BDLat}\\cong \\mathsf {CThProp}$ ) of $\\mathsf {BDLat}$ in both.", "We demonstrate this strict 2-adjunction on the categorial side, i.e.", "$\\mathsf {Coh}\\leftrightarrows \\mathsf {BDLat}$ ; the functor we will call $F$ is really $L_{{(-)}}$ .", "We begin by laying out the functors of the adjunction.", "Let $G$ be the inclusion of $\\mathsf {BDLat}$ in $\\mathsf {Coh}$ .", "It is obviously a full, faithful, and strict 2-functor.", "Lemma 7.13 For a coherent category $C$ , $Sub_C(1)$ is a full, coherent subcategory of $C$ .", "We now define $F$ .", "Given coherent categories $C$ and $D$ , coherent functors $f, g: C \\rightrightarrows D$ , and a natural transformation $\\alpha : f\\Rightarrow g$ , set $F(C)=Sub_C(1)$ , $F(f)={\\left.\\hspace{0.0pt}f \\vphantom{\\big \| } \\right\|_{Sub_1(C)} }:Sub_C(1)\\rightarrow Sub_D(1)$ , and $F(\\alpha )={\\left.\\hspace{0.0pt}\\alpha \\vphantom{\\big \| } \\right\|_{Sub_D(1)} }:{\\left.\\hspace{0.0pt}f \\vphantom{\\big \| } \\right\|_{Sub_C(1)} }\\Rightarrow {\\left.\\hspace{0.0pt}g \\vphantom{\\big \| } \\right\|_{Sub_C(1)} }$ .", "Lemma 7.14 F is a strict 2-functor $\\mathsf {Coh}\\rightarrow \\mathsf {BDLat}$ .", "Coherent functors, being left-exact, preserve terminal objects and monics.", "Hence, $f(Sub_C(1))\\subseteq Sub_D(1)$ , and by restricting the domain and codomainRestricting the codomain is, once we have restricted the domain, purely nominal.", "we have a coherent functor between thin categories.", "This takes the form of a morphism in $\\mathsf {BDLat}$ .", "The functor on the level of 2-morphisms is trivial because $Sub_D(1)$ is thin.", "Since the new functors and natural transformations in the image of $F$ arise from restriction of a functor, composition and identity relationships are respected.", "Hence, $F$ is a strict 2-functor.", "We are now ready to prove the adjunction by means of unit and counit transformations.", "First, we seek a strict 2-natural transformation $\\eta :1_\\mathsf {Coh}\\Rightarrow GF$ .", "In particular, we require a functor $\\eta _C:C\\rightarrow GFC$ for each $C\\in ob(\\mathsf {Coh})$ .", "Now, it is a fact of coherent categories that any morphism $f$ can be factored (uniquely up to isomorphism) into an effective epic $e_f$ and a monic $m_f$ .", "Each object $X$ has a unique morphism $!_X$ to 1, so there is a unique (up to isomorphism) object $Im(!_X)$ such that there is an effective epic $e_{!_X}:X \\rightarrow Im(!_X)$ and monic $m_{!_X}: Im(!_X)\\rightarrow 1$ .", "Clearly, $Im(!_X)\\in ob(Sub_C(1))$ .", "Let $\\eta _C$ be the natural transformation defined by postcomposition with the epic $e_{!_X}$ .", "That is, take every $X\\in ob(C)$ to $Im(!_X)$ and every arrow $f:X\\rightarrow Y$ to the unique morphism $Im(!_X)\\rightarrow Im(!_Y)$ .", "We prove that such a morphism exists by inviting the reader to consider $e_{!_X}\\circ f:X\\rightarrow Im(!_Y)$ , $e_{e_{!_X}\\circ f}: X\\rightarrow Im(e_{!_X}\\circ f)$ and $m_{!_Y}\\circ m_{e_{!_X}\\circ f}: Im(e_{!_X}\\circ f)\\rightarrow 1$ .", "These are effective epic and monic respectively, and they factor $!_X$ , so $Im(e_{!_X}\\circ f) \\cong Im(!_X)$ by some isomorphism $i:Im(!_X)\\rightarrow Im(e_{!_X}\\circ f)$ .", "Thus, we have by construction $m_{e_{!_X}\\circ f} \\circ i: Im(!_X)\\rightarrow Im(!_Y)$ .", "Note that $\\eta _C$ automatically respects identity and composition since its target category is thin, so it is a functor.Here's a cute digression.", "Since $F$ is left adjoint to inclusion (so $\\eta _C(C)\\cong FC$ ), it must be pointwise functorial––namely, $\\eta _C$ .", "What's more, though, $\\eta _C$ is the well-known endofunctor $\\exists $ taking $[\\phi ]$ to $[\\exists x \\phi ]$ , which is itself left adjoint to an inclusion, specifically the inclusion of $Sub_C(1)$ in $C$ .", "This parallelism may or may not have significance beyond aesthetic appeal.", "We must also assign, for every functor $f: C\\rightarrow D$ , a 2-morphism $\\eta _f: FG(f)\\circ \\eta _C \\Rightarrow \\eta _D \\circ f$ .", "However, these two sides are clearly equal, so $\\eta _f$ is the identity for all $f$ , making $\\eta $ a strict 2-natural transformation $1_\\mathsf {Coh}\\Rightarrow GF$ .", "Finally, we seek a strict 2-natural transformation $\\varepsilon : FG\\Rightarrow 1_{\\mathsf {BDLat}}$ .", "We claim that for all objects $A$ in $\\mathsf {BDLat}$ , $FGA=A$ .", "This is because $F$ is defined by restricting to $Sub_A(1)$ .", "Since $A$ is thin, this restriction does nothing.", "As such, every component of $\\varepsilon $ is trivial.", "This gives us $FGF = F$ and $GFG = G$ , so to satisfy the triangle identities we need only confirm that $\\eta _C$ is an isomorphism when its codomain is $C$ .", "This happens exactly when $C \\cong Sub_C(1)$ , and we know that $\\eta _C$ acts as an isomorphism on $Sub_C(1)$ , so the proof is complete.", "We have a strict 2-adjunction.", "Finally, noting further that $G$ is full and faithful (or that $\\varepsilon $ is a natural isomorphism): theorem 7.15 $F\\dashv G: \\mathsf {BDLat}\\leftrightarrows \\mathsf {Coh}$ is a strict, reflective 2-adjunction." ], [ "Propositional Theories and Morita Equivalence", "Since Morita equivalence is generally weaker than weak intertranslatability, the characterization in Theorem REF needs to be extended.", "Recall that $T_1$ and $T_2$ are Morita equivalent if and only if $P_{T_1}$ and $P_{T_2}$ are equivalent pretopoi.", "Therefore, the first to step to identifying Morita equivalence classes of propositional theories is determining the pretopos completion of a bounded distributive lattice.", "Definition 7.16 Let $C$ be a coherent category, and let 1 be a terminal object.", "$C$ is a parapropositional category if for any object $A$ of $C$ , there exists a finite list $\\phi _1,\\dots ,\\phi _n$ of subobjects of 1 such that $A \\cong \\coprod _{i = 1}^n \\phi _i$ .", "theorem 7.17 Let $T$ be a coherent theory.", "$T$ is Morita equivalent to a propositional theory $S$ if and only if $T$ is a parapropositional category.", "Moreover, $S$ is (up to homotopy equivalence) the internal logic of $Sub_{T}(1)$ .", "Assume $T$ is Morita equivalent to a propositional theory $S$ .", "Then by Theorem REF the pretopos completions $P_T$ and $P_S$ are equivalent categories.", "Let $F: P_T \\rightarrow P_S$ and $G: P_S \\rightarrow P_T$ be such an equivalence of categories.", "We have a conservative coherent functor $I: T \\rightarrow P_T$ (see Theorem REF ).", "Let $X$ be an object of $T$ .", "Then $FI(X)$ is an object of $P_S$ .", "Since $S$ is propositional, $S$ is a bounded distributive lattice, so it is an exact category.", "Therefore, any object of $P_S$ is a coproduct of objects of $S = Sub_{P_S}(1)$ .", "In particular $FI(X) \\cong \\coprod _{i=1}^n\\phi _i$ where $\\phi _i$ are objects in $Sub_{P_S}(1)$ .", "$G$ is a coherent functor, so $G\\phi _i$ is a subobject of the terminal object 1 in $P_T$ .", "Therefore $GFI(X) \\cong \\coprod _{i=1}^n G\\phi _i$ .", "Since $F,G$ are an equivalence of categories, this implies $IX \\cong \\coprod _iG\\phi _i$ .", "$I$ is full on subobjects, so this implies $X \\cong \\coprod _iG\\phi _i$ .", "Since $X$ is arbitrary, this shows $T$ is parapropositional.", "Notice $F,G$ preserve the subobject lattices of the terminal object, so $F,G$ restrict to an equivalence of categories $Sub_{T}(1) \\cong Sub_{S}(1) = S$ .", "Thus, the internal logic of $Sub_{T}(1)$ is homotopy equivalent to $S$ .", "Conversely, assume $T$ is parapropositional.", "Then $T$ is the result of appending (some) disjoint coproducts to $Sub_{T}(1)$ , i.e.", "$_T$ is a Morita extension of $Sub_{T}(1)$ ; so $_T$ is Morita equivalent to a propositional theory.", "By Theorem this implies $T$ is Morita equivalent to the same propositional theory.", "Corollary 7.18 Let $T$ be a coherent theory.", "If $T$ has an uncountable model, then $T$ is not Morita equivalent to a propositional theory.", "We prove the contrapositive.", "If $T$ is Morita equivalent to a propositional theory $S$ , then any object $[A]$ of $T$ is a coproduct $\\coprod _{i=1}^n \\phi _i$ where $\\phi _i$ is a subobject of 1.", "Let $X_n$ denote the $n$ -fold coproduct of 1 in the pretopos completion $P_T$ .", "Then $[A]$ is a subobject of $X_n$ .", "Let $M$ be a model of $T$ .", "Then $M$ extends to a model of $P_T$ , hence $M[A] \\subseteq MX_n$ .", "This is true for some $n$ for any $[A]$ .", "Thus, $M = \\bigcup _{i\\in {\\mathbb {N}}} MX_i$ .", "Furthermore, $MX_i$ contains $i$ elements, so $M$ is countable." ], [ "Conclusion", "The previous sections constitute a framework enabling the translation of categorical logic into predicate logic and vice versa, faithfully, as far as bi-interpretability and Morita equivalence are concerned.", "We present a few results demonstrating the promise of porting categorical tools to the study of theoretical equivalence.", "For example, we have refined the relationships among Morita equivalence, bi-interpretability, and other common forms of theoretical equivalence into a chart (Figure REF ), where implication is upward.", "Figure: Comparing standards of theoretical equivalence.The questions marks on this chart indicate a notion of bi-interpretability via translations which account for coproducts like how weak translations account for quotients.", "Such a notion would have to be equivalent to an equality-preserving translation between Morita completions, i.e.", "homotopy equivalence in $\\mathsf {CTh}$ .", "Now that there is a clear definition for a category of theories, $\\mathsf {CThEq}$ or $\\mathsf {CTh}_0$ , we open the gates to new investigations on universal constructions of theories.", "That is, we can now construct e.g.", "(co)limits of coherent theories.", "How these constructions interact with the semantics of coherent logic and with translations is largely unknown.", "However, the presence of such constructions in topos theory hint at a rich analogue for coherent logic." ] ]	2011.14056
[ [ "Finite-volume energy shift of the three-pion ground state" ], [ "Abstract Using the framework of non-relativistic effective field theory, the finite-volume ground-state energy shift is calculated up-to-and-including $O(L^{-6})$ for the system of three pions in the channel with the total isospin $I=1$.", "The relativistic corrections are included perturbatively, up to the same order in the inverse of the box size $L$.", "The obtained explicit expression, together with the known result for the system with maximal isospin $I=3$, can be used for the extraction of two independent effective three-body couplings from the measured ground-state spectrum of three pions." ], [ "Introduction", "The extraction of various hadronic observables in the three particle sector is certainly a challenging task for lattice QCD.", "Such an extraction has become truly realistic only recently, owing to the rapid growth of computational resources available, as well as the progress on the methods and algorithms [1], [2], [3], [4], [5], [6], [7] (see also the closely related papers [8], [9], where the simulations were performed within the scalar $\\varphi ^4$ -theory).", "There is, however, more to this task than merely carrying out larger simulations which require more resources.", "Namely, the extraction of the hadronic parameters, defined in the infinite volume, from the observables, measured on a finite lattices, has proven to be significantly more challenging than in the one- and two-particle sectors.", "The problem has been, in principle, solved by the three-particle quantization condition, which has been derived only recently in three different settings [10], [11], [12], [13], [14], [15].", "In its turn, this development has sparked activities in the field, as seen, e.g., from [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39] (note also earlier work on the related issues [40], [41], [42], [43]).", "The reader is referred to [44] for a recent review on the three-particle problem in a finite volume.", "An alternative approach to the problem focuses on the perturbative shift of the free three-particle energy levels.", "Albeit more restricted from the beginning (e.g., it is clear that the resonances cannot be handled in this manner), the approach is very popular at present.", "The reason for this is that the perturbative approach is very transparent and convenient for the analysis of lattice data.", "In case of identical particles, the expansion parameter is given by $a/L$ , where $a$ denotes the two-body scattering length and $L$ is the spatial size of the cubic box.", "Up to and including $O(L^{-5})$ , only two-body threshold parameters – the scattering length $a$ and the effective range $r$ – enter the expression for the energy shift, and the latter appears at this order only in the excited states.", "At order $L^{-6}$ , a single additional coupling $\\eta _3$ , characterizing the three-body nonderivative interaction, appears in the perturbative energy shift, and hence the measured three-body energies at different values of $L$ can be used to extract the value of this coupling from data.", "Moreover, the perturbative approach can be directly generalized for the systems of four and more particles (which is presently not the case for the exact quantization condition), and the measurements in the many-particle sector can be used to restrict the fit of $\\eta _3$ , since the same coupling appears in all $N$ -particle energies up to and including $O(L^{-6})$ .", "An example of such a fit is given in the recent article [9].", "Application of the perturbation theory to the calculation of the energy shifts in a finite volume has a decades-long history [45], [46], [47], [48].", "This development culminated in Refs.", "[49], [50] where, using the nonrelativistic effective field theory, the $N$ -particle ground-state shift is derived up to and including order $L^{-6}$ and $L^{-7}$ , respectively.", "The relativistic corrections at $O(L^{-6})$ are calculated in the same approach in Ref.", "[51] where, in addition, the inclusion of the electromagnetic interactions is considered.", "Further, carrying out the threshold expansion in the nonrelativistic limit, the expression for the energy shift of the ground and first excited states is derived in Ref.", "[30] at $O(L^{-6})$ .", "On the other hand, the threshold expansion of the relativistic quantization condition leads to the relativistic expression of the ground-state energy at the same order [17].", "Finally, in Ref.", "[9], where the nonrelativistic framework has been used, the relativistic corrections in the ground and first excited states have been calculated up to and including $O(L^{-6})$ .", "All above studies have one thing in common – they are carried out for the system of $N$ identical particles.", "Taking into account that the simplest three-particle system, which can be simulated in lattice QCD, is that of three pions, one may conclude that the expressions, obtained in the above papers, apply to the case with maximal isospin only.", "It is, however, interesting to simulate systems with a different total isospin, in order to extract all low-energy three-pion couplings.", "In addition, studying the three-pion decays on the lattice, one does not encounter, in general, a three-pion system with maximal isospin in the final state.", "A perturbative expression for the energy shift for $I\\ne 3$ at the required accuracy is, however, unavailable in the literature (to the best of our knowledge, a single study [25], which treats different isospin channels, deals with an exact quantization condition and the threshold expansion is not performed there).", "Our present study aims at closing this gap.", "As already mentioned, the results of this study will be certainly useful, for example, in the analysis of lattice data on the three-pion decays, which are expected in the near future.", "The layout of the paper is as follows.", "In Sec.", "we write down the nonrelativistic two- and three-body Lagrangians and perform the matching.", "Further, in Sec.", "we construct the three-pion states with different total isospin $I$ and calculate the potentials – the matrix elements of the interaction Hamiltonian between the three-pion states.", "In Sec.", ", we derive the general expression for the perturbative energy shift up to and including $O(L^{-6})$ in the state with the total isospin $I=1$ .", "Finally, Sec.", "contains our conclusions." ], [ "Non-relativistic effective Lagrangians and matching", "Our calculations closely follow the path of Refs.", "[9], [49], where the identical particles have been considered.", "Throughout this paper, we shall use the nonrelativistic effective field theory framework, including relativistic effects perturbatively.", "We further assume isospin symmetry and denote the common mass of pions by $M$ .", "The Lagrangian, which will be used in the calculations up to and including order $L^{-6}$ , is given by ${\\cal L}=\\sum _{i=\\pm ,0}\\pi _i^\\dagger \\biggl (i\\partial _t-M+\\frac{\\nabla ^2}{2M}+\\frac{\\nabla ^4}{8M^3}\\biggr )\\pi ^{}_i+{\\cal L}_2+{\\cal L}_3\\, .$ Here, $\\pi _\\pm $ and $\\pi ^{}_0$ denote nonrelativistic pion fields, and ${\\cal L}_2$ and ${\\cal L}_3$ are the Lagrangians in the two- and three-particle sectors, respectively.", "Note that only leading relativistic corrections in the kinetic term contribute up to the order $L^{-6}$ .", "Further, at the order we are working, it suffices to retain in ${\\cal L}_2$ the terms up to the two spatial derivatives.", "Then, this Lagrangian takes the form ${\\cal L}_2 &= & \\frac{1}{2}\\, c_1 \\pi _0^\\dagger \\pi _0^\\dagger \\pi ^{}_0 \\pi ^{}_0- \\frac{1}{8}\\, d_1\\biggl ( \\pi _0^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _0^\\dagger \\pi ^{}_0 \\pi ^{}_0+\\mbox{h.c.}\\biggr )\\nonumber \\\\[2mm]&+& 2c_2 \\biggl ( \\pi _+^\\dagger \\pi _0^\\dagger \\pi _+ \\pi ^{}_0 + \\pi _-^\\dagger \\pi _0^\\dagger \\pi ^{}_- \\pi ^{}_0 \\biggr )- \\frac{1}{2}d_2 \\biggl ( \\pi _+^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _0^\\dagger \\pi ^{}_+ \\pi ^{}_0+ \\pi _-^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _0^\\dagger \\pi ^{}_- \\pi ^{}_0 + \\mbox{h.c.} \\biggr )\\nonumber \\\\[2mm]&+& c_3 \\biggl ( \\pi _+^\\dagger \\pi _-^\\dagger \\pi ^{}_0 \\pi ^{}_0 + \\text{h.c.} \\biggr )- \\frac{1}{4}\\,d_3 \\biggl ( \\pi _+^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _-^\\dagger \\pi ^{}_0 \\pi ^{}_0 + \\pi _0^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _0^\\dagger \\pi ^{}_+ \\pi ^{}_- + \\mbox{h.c.} \\biggr )\\nonumber \\\\[2mm]& +& 2c_4 \\pi _+^\\dagger \\pi _-^\\dagger \\pi ^{}_+ \\pi ^{}_-- \\frac{1}{2}\\,d_4\\biggl ( \\pi _+^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _-^\\dagger \\pi ^{}_+ \\pi ^{}_-+\\mbox{h.c.}\\biggr )\\nonumber \\\\[2mm]& +& \\frac{1}{2}\\,c_5 \\biggl ( \\pi _+^\\dagger \\pi _+^\\dagger \\pi ^{}_+ \\pi ^{}_++ \\pi _-^\\dagger \\pi _-^\\dagger \\pi ^{}_- \\pi ^{}_- \\biggr )- \\frac{1}{8}\\,d_5 \\biggl ( \\pi _+^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _+^\\dagger \\pi ^{}_+ \\pi ^{}_++ \\pi _-^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _-^\\dagger \\pi ^{}_- \\pi ^{}_- +\\mbox{h.c.}\\biggr )\\, .$ Here, $\\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}= (\\overset{\\text{\\begin{tikzpicture}[->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}- \\overset{\\text{\\begin{tikzpicture}[<-] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt})/2$ is the Galilean-invariant derivative.", "Note also that there are more terms with two derivatives, which describe the center-of-mass motion of two-particle pairs.", "Since these terms do not contribute to the energy shift of the ground state we are interested in [9], we do not display them here explicitly.", "In addition, it can be checked that the P-wave terms do not contribute to the ground-state energy at this order (albeit they contribute to the excited states).", "These terms are omitted as well.", "In a finite box, the Fourier transform of the field takes the form $\\pi _i({\\bf x},t)=\\frac{1}{L^3}\\,\\sum _{\\bf p}e^{-ip^0t+i{\\bf p}{\\bf x}}a_i({\\bf p})\\, ,$ where the creation and annihilation operators obey the commutation relations: $[a_i({\\bf p}),a_j^\\dagger ({\\bf q})]=L^3\\delta _{ij}\\delta _{{\\bf p}{\\bf q}}\\, .$ The normalized one-particle states are given by $\|\\pi _i({\\bf p})\\rangle =\\frac{1}{L^{3/2}}\\,a^\\dagger _i({\\bf p})\|0\\rangle \\, .$ If dimensional regularization is used, the couplings $c_i$ and $d_i$ can be related to the parameters of the effective-range expansion in the $\\pi \\pi $ scattering phase shift through the matching: $p \\cot \\delta _I(p) = -\\frac{1}{a_I} + \\frac{1}{2} r_I p^2 + \\cdots ,\\quad \\quad I=0,2\\, .$ The matching condition takes the form rClrCl c1 = -4M 13(a0 + 2a2), d1 = -4M 13(a02 r0 + 2a22 r2) , c2 = -4M 12a2, d2 = -4M 12a22 r2 , c3 = -4M 13(a2 - a0), d3 = -4M 13(a22 r2 - a02 r0) , c4 = -4M 16(2a0 + a2), d4 = -4M 16(2a02 r0 + a22 r2) , c5 = -4M a2, d5 = -4M a22 r2 .", "In the above equations, the quantities $\\hat{r}^{}_I$ include relativistic corrections, which stem from the matching to the relativistic amplitude [9]: $\\hat{r}^{}_I=r^{}_I-\\frac{1}{a^{}_IM^2}\\, .$ Further, in the calculations up to and including $O(L^{-6})$ , only the nonderivative Lagrangians in the three-particle sector contribute.", "It is possible to construct two linearly independent terms with this property and, hence, the Lagrangian can be written as ${\\cal L}_3 =\\eta _1 \\biggl ( \\pi _+^\\dagger \\pi ^{}_+ + \\pi _0^\\dagger \\pi ^{}_0 + \\pi _-^\\dagger \\pi ^{}_- \\biggr )^3+ \\eta _2 \\biggl ( \\pi _+^\\dagger \\pi ^{}_+ + \\pi _0^\\dagger \\pi ^{}_0 + \\pi _-^\\dagger \\pi ^{}_- \\biggr )\\biggl ( 2 \\pi _+^\\dagger \\pi _-^\\dagger - \\pi _0^\\dagger \\pi _0^\\dagger \\biggr )\\biggl ( 2 \\pi ^{}_+ \\pi ^{}_- - \\pi ^{}_0\\pi ^{}_0 \\biggr )\\,.$ The low-energy couplings $\\eta _1$ and $\\eta _2$ are ultraviolet-divergent – these divergences cancel the pertinent divergences, which arise in the perturbation theory at higher orders.", "We shall see the examples of such cancellations below.", "Note also that $\\eta _1$ is a counterpart of the coupling $\\eta _3$ , introduced in Ref.", "[49] for the case of three identical particles.", "Namely, one has $\\eta _3=-6\\eta _1$ .", "On the contrary, the coupling $\\eta _2$ does not emerge in the sector with maximal isospin.", "In order to fix this coupling, one has to measure the energy shift in the sector with $I=1$ .", "It is well known that the couplings $\\eta _1$ and $\\eta _2$ , via matching, can be traded for the “divergence free” threshold amplitudes in the three-particle sector.", "Such an amplitude can be defined in different ways.", "In the present context, the definition, introduced in Ref.", "[9], is the most convenient, and we stick to it.", "In order to match two constants, we consider two relativistic on-shell amplitudes $3\\pi ^+\\rightarrow 3\\pi ^+$ and $3\\pi ^0\\rightarrow 3\\pi ^0$ , denoted by ${\\cal M}_+$ and ${\\cal M}_0$ , respectively.", "Next, we have to choose the directions from which the threshold is approached.", "This can be done, e.g., in the following way: ${\\bf p}_1=\\lambda {\\bf e}_y\\, ,\\quad \\quad {\\bf p}_2=\\lambda \\biggl (\\frac{\\sqrt{3}}{2}\\,{\\bf e}_x-\\frac{1}{2}\\,{\\bf e}_y\\biggr )\\, ,\\quad \\quad {\\bf p}_3=-\\lambda \\biggl (\\frac{\\sqrt{3}}{2}\\,{\\bf e}_x+\\frac{1}{2}\\,{\\bf e}_y\\biggr )\\, ,$ and ${\\bf p}_i^{\\prime }=-{\\bf p}_i$ for $i=1,2,3$ .", "Here, ${\\bf p}_i,{\\bf p}^{\\prime }_i$ are the initial and final momenta of particles and ${\\bf e}_{x,y,z}$ denote unit vectors in the direction of pertinent axes.", "The limit $\\lambda \\rightarrow 0$ in the above amplitudes is singular.", "In order to arrive at the regular quantities, one has to subtract the one-particle irreducible piece (pole part) first.", "An exact definition of this quantity can be found in Ref. [9].", "Then, it can be proven that $\\mbox{Re}\\biggl ({\\cal M}_a-{\\cal M}_a^{(pole)}\\biggr )=\\frac{1}{\\lambda }\\,{\\cal M}_a^{(-1)}+\\ln \\frac{\\lambda }{M}\\,{\\cal M}_a^{(l)}+{\\cal M}_a^{(0)}+\\cdots \\, ,\\quad \\quad a=+,0\\, .$ Here, ellipses stand for the terms that vanish, as $\\lambda \\rightarrow 0$ .", "The couplings $\\eta _1$ and $\\eta _2$ can be related to the regular threshold amplitudes ${\\cal M}_a^{(0)}$ .", "This relation can be obtained exactly in the same way as in Ref.", "[9], and we quote here only the final result: $36\\eta _1^r&=&-\\frac{288\\pi ^2}{M}\\,a_2^3\\hat{r}_2-\\frac{576\\pi ^2}{M^3}\\,a_2^2+\\frac{1}{(2M)^3}\\,{\\cal M}_+^{(0)}-\\frac{192\\pi a_2^4}{M}\\,\\bigl (3\\sqrt{3}\\bar{\\delta }^{(d)}-4\\pi \\bar{\\delta }^{(e)}\\bigr )\\, ,\\nonumber \\\\[2mm]36(\\eta _1^r+\\eta _2^r)&=&-\\frac{32\\pi ^2 }{M}\\,(a_0+2a_2)(a_0^2\\hat{r}_0+2a_2^2\\hat{r}_2)-\\frac{64\\pi ^2}{M^3}(a_0+2a_2)^2+\\frac{1}{(2M)^3}\\,{\\cal M}_0^{(0)}\\nonumber \\\\[2mm]&-&\\frac{64\\sqrt{3}\\pi }{3M}\\,(a_0^4+4a_0^3a_2+9a_0^2a_2^2+2a_0a_2^3+11a_2^4)\\bar{\\delta }^{(d)}\\nonumber \\\\[2mm]&+&\\frac{128\\pi ^2}{27M}\\,(2a_0^4+28a_0^3a_2+57a_0^2a_2^2+10a_0a_2^3+65a_2^4)\\bar{\\delta }^{(e)}\\, .$ Here, $\\eta _{1,2}^r$ denote the finite parts of the couplings $\\eta _{1,2}$ (dimensional regularization with the minimal subtraction prescription has been used, and $d$ denotes the number of space dimensions): $\\eta _1&=&\\mu ^{2(d-3)}\\biggl \\lbrace -\\frac{16\\pi a_2^4}{3M}\\,\\frac{1}{d-3}\\,(3\\sqrt{3}-4\\pi )+\\eta _1^r\\biggr \\rbrace \\, ,\\nonumber \\\\[2mm]\\eta _2&=&\\mu ^{2(d-3)}\\biggl \\lbrace \\frac{16\\pi }{243M}\\,\\frac{1}{d-3}\\,(a_0-a_2)\\Big [ 2\\pi \\left(2a_0^3 + 30 a_0^2 a_2 + 87 a_0 a_2^2 +97 a_2^3\\right) - 9\\sqrt{3}\\left(a_0 + 2a_2\\right)\\left(a_0^2 + 3a_0a_2 + 8a_2^2\\right) \\Big ]\\nonumber \\\\[2mm]&+&\\eta _2^r\\biggr \\rbrace \\, .$ Furthermore, the renormalization scale was chosen at $\\mu =M$ in the above equations.", "The quantities $\\bar{\\delta }^{(d)}$ and $\\bar{\\delta }^{(e)}$ , which emerge in Eq.", "(REF ), are equal to $\\bar{\\delta }^{(d,e)}=\\delta ^{(d,e)}-\\Gamma ^{\\prime }(1)-\\ln 4\\pi \\, ,\\quad \\quad \\delta ^{(d)}=-1.090\\ldots \\, ,\\quad \\quad \\delta ^{(e)}=3.926\\ldots \\, .$ where $\\delta ^{(d)}$ and $\\delta ^{(e)}$ are the quantities defined in Appendix A of Ref. [9].", "To summarize, one can express two unknown couplings $\\eta _1^r$ and $\\eta _2^r$ through the regular, real-valued threshold amplitudes ${\\cal M}_+^{(0)}$ and ${\\cal M}_0^{(0)}$ , which can be further extracted from the measured energy levels on the lattice.", "At this stage, we do not see much advantage in working in terms of the amplitudes rather than couplings, since this amounts to the replacing of one set of unknown constants by another one in the fit.", "Note, however, that the amplitudes are observables, whereas the couplings are not.", "For this reason, comparison with other approaches is made easier, if the final result is rewritten in terms of the amplitudes." ], [ "The Fock state and the potentials", "In a finite cubic box with size $L$ , the Fock state of the theory consists of the vectors, containing freely moving particles $\\pi _\\pm $ and $\\pi ^{}_0$ with the discretized momenta ${\\bf p}=2\\pi {\\bf n}/L$ , where ${\\bf n}\\in \\mathbb {Z}^3$ .", "For instance, the closure relation in the three-particle sector with a total charge +1 is written in the following form: $\\mathbb {1}&=&\\frac{1}{2!", "}\\,\\sum _{{\\bf p}_1,{\\bf p}_2,{\\bf p}_3}\|\\pi _+({\\bf p}_1)\\pi _+({\\bf p}_2)\\pi _-({\\bf p}_3)\\rangle \\langle \\pi _+({\\bf p}_1)\\pi _+({\\bf p}_2)\\pi _-({\\bf p}_3)\|\\nonumber \\\\[2mm]&+&\\frac{1}{2!", "}\\,\\sum _{{\\bf p}_1,{\\bf p}_2,{\\bf p}_3}\|\\pi _0({\\bf p}_1)\\pi _0({\\bf p}_2)\\pi _+({\\bf p}_3)\\rangle \\langle \\pi _0({\\bf p}_1)\\pi _0({\\bf p}_2)\\pi _+({\\bf p}_3)\|\\, .$ Note that here the “odd” particle is always labeled by the index 3.", "Instead, one could sum up over three permutations of indices $(123)$ , $(312)$ , $(231)$ , in each of the above sums, dividing everything by $3!$ instead of $2!$ .", "Next, we wish to rewrite this sum as a sum over the three-pion states with a definite total isospin $I$ .", "It is a straightforward exercise to construct such states, using the table of the Clebsh-Gordan coefficients.", "Below, we give maximal vectors $I_z=I$ in each of the representation with $I=0,1,2,3$ : lCl \|3,3= \|+ + + , \|2,2A = 12 ( \|+0+- \|0++) , \|2,2S = 16 ( 2\|++0- \|0++- \|+0+) , \|1,1A = 12 ( \|+00- \|0+0- \|+-++ \|-++) , \|1,1S1 = 13 ( \|+-+- \|00++ \|-++) , \|1,1S2 = 1215 ( 6\|++-+ \|+-++ \|-++- 3\|0+0-3\|+00+ 2\|00+) , \|0,0=16 ( \|+0-- \|+-0- \|0+-+ \|0-++ \|-+0- \|-0+) .", "Here, it is implicitly assumed that the particles in the ket-vectors carry momenta ${\\bf p}_1,{\\bf p}_2,{\\bf p}_3$ , respectively, and the subscripts $S/A$ stand for symmetric/antisymmetric wave functions with respect to $1\\leftrightarrow 2$ .", "Further, the multiplicity of the irreducible representations, corresponding to $I=0,1,2,3$ is $1,3,2,1$ , respectively, i.e., the unperturbed states in the sectors $I=1,2$ are, generally, degenerate, and hence the perturbation theory for the degenerate states should be used in these sectors.", "Note also that this basis is the counterpart of the basis, used in Ref.", "[25], Eqs.", "(2.56) and (2.57) (there, $I_z=0$ for all $I$ has been chosen).", "Let us now focus on the unperturbed ground state with ${\\bf p}_1={\\bf p}_2={\\bf p}_3=0$ .", "From Eq.", "() it is clear that only the vectors with $I=3$ and $I=1$ survive, whereas for $I=0,2$ , the unperturbed ground-state vector vanishes identically – these states will not be seen on the lattice.", "Further, note that the solution for $I=3$ is already known in the literature, whereas the expression for the shift in the $I=1$ state, which will be given below, is new.", "Providing this expression, our formalism that describes the ground state of three pions will become complete.", "At the next step, we wish to rewrite the closure relation in terms of the states with a definite total isospin.", "We shall work with the full basis of states with $I_z=+1$ , where the states with $I=1$ are also contained.", "In the Fock space, there are six basis vectors with $I_z=+1$ : rCrrCrrCr \|v1=\|++- , \|v2=\|+-+ , \|v3=\|-++ , \|v4=\|00+ , \|v5=\|0+0 , \|v6=\|+00 .", "We wish to express these vectors, as linear combinations of the vectors $\|I,I_z\\rangle $ with $I=1,2,3$ and $I_z=1$ .", "The explicit expression of these vectors is given by: $\|f_1\\rangle &=& \\frac{1}{\\sqrt{15}}\\, \\biggl ( 2\|\\pi _+\\pi _+\\pi _-\\rangle +2\|\\pi _+\\pi _-\\pi _+\\rangle +2\|\\pi _-\\pi _+\\pi _+\\rangle - \|\\pi _0\\pi _0\\pi _+\\rangle - \|\\pi _0\\pi _+\\pi _0\\rangle - \|\\pi _+\\pi _0\\pi _0\\rangle \\biggr )\\, ,\\nonumber \\\\[2mm]\|f_2\\rangle &=& \\frac{1}{2\\sqrt{3}}\\,\\biggl ( 2\|\\pi _+\\pi _+\\pi _-\\rangle - \|\\pi _+\\pi _-\\pi _+\\rangle - \|\\pi _-\\pi _+\\pi _+\\rangle + 2\|\\pi _0\\pi _0\\pi _+\\rangle - \|\\pi _0\\pi _+\\pi _0\\rangle -\|\\pi _+\\pi _0\\pi _0\\rangle \\biggr )\\, ,\\nonumber \\\\[2mm]\|f_3\\rangle &=& \\frac{1}{2}\\, \\biggl ( \|\\pi _+\\pi _0\\pi _0\\rangle - \|\\pi _0\\pi _+\\pi _0\\rangle - \|\\pi _+\\pi _-\\pi _+\\rangle + \|\\pi _-\\pi _+\\pi _+\\rangle \\biggr )\\, , \\nonumber \\\\[2mm]\|f_4\\rangle &=&\\frac{1}{2\\sqrt{3}}\\,\\biggl (2\|\\pi _+\\pi _+\\pi _-\\rangle -\|\\pi _+\\pi _-\\pi _+\\rangle -\|\\pi _-\\pi _+\\pi _+\\rangle -2\|\\pi _0\\pi _0\\pi _+\\rangle +\|\\pi _0\\pi _+\\pi _0\\rangle +\|\\pi _+\\pi _0\\pi _0\\rangle \\biggr )\\, ,\\nonumber \\\\[2mm]\|f_5\\rangle &=&\\frac{1}{2}\\,\\biggl (\|\\pi _+\\pi _0\\pi _0\\rangle -\|\\pi _0\\pi _+\\pi _0\\rangle +\|\\pi _+\\pi _-\\pi _+\\rangle -\|\\pi _-\\pi _+\\pi _+\\rangle \\biggr )\\, ,\\nonumber \\\\[2mm]\|f_6\\rangle &=&\\frac{1}{\\sqrt{15}}\\,\\biggl (\|\\pi _+\\pi _+\\pi _-\\rangle +\|\\pi _+\\pi _-\\pi _+\\rangle +\|\\pi _-\\pi _+\\pi _+\\rangle +2\|\\pi _0\\pi _0\\pi _+\\rangle +2\|\\pi _0\\pi _+\\pi _0\\rangle +2\|\\pi _+\\pi _0\\pi _0\\rangle \\biggr )\\, .$ Note that the vectors $\|f_1\\rangle $ and $\|f_2\\rangle $ are linear combinations of the vectors $\|1,1\\rangle _{S_1}$ and $\|1,1\\rangle _{S_2}$ (the same linear combinations, as in Ref.", "[25], Eqs.", "(C.14) and (C.17), again for $I_z=0$ ).", "The reason for such choice will become clear below.", "It is now straightforwardly seen that the basis vectors of one basis can be expressed as linear combinations of the basis vectors of other basis: $\|v_a\\rangle =C_{ab}\|f_b\\rangle $ .", "The orthogonal $6\\times 6$ matrix $C$ is given by: $C=\\begin{pmatrix}\\frac{2}{\\sqrt{15}} & \\frac{1}{\\sqrt{3}} & 0 & \\frac{1}{\\sqrt{3}} & 0 &\\frac{1}{\\sqrt{15}} \\\\[2mm]-\\frac{2}{\\sqrt{15}} & \\frac{1}{2\\sqrt{3}} &\\frac{1}{2} & \\frac{1}{2\\sqrt{3}} &-\\frac{1}{2} & -\\frac{1}{\\sqrt{15}} \\\\[2mm]-\\frac{2}{\\sqrt{15}} & \\frac{1}{2\\sqrt{3}} &-\\frac{1}{2} & \\frac{1}{2\\sqrt{3}} &\\frac{1}{2} & -\\frac{1}{\\sqrt{15}} \\\\[2mm]-\\frac{1}{\\sqrt{15}} & \\frac{1}{\\sqrt{3}} & 0 & -\\frac{1}{\\sqrt{3}} &0 & \\frac{2}{\\sqrt{15}} \\\\[2mm]\\frac{1}{\\sqrt{15}} & \\frac{1}{2\\sqrt{3}} &\\frac{1}{2} & -\\frac{1}{2\\sqrt{3}} &\\frac{1}{2} &-\\frac{2}{\\sqrt{15}} \\\\[2mm]\\frac{1}{\\sqrt{15}} & \\frac{1}{2\\sqrt{3}} &-\\frac{1}{2} & -\\frac{1}{2\\sqrt{3}} &-\\frac{1}{2} &-\\frac{2}{\\sqrt{15}}\\end{pmatrix}\\, .$ Using $CC^T=C^TC=1$ , the closure relation (REF ) can be finally rewritten in the isospin basis: $\\mathbb {1}=\\frac{1}{3!", "}\\sum _{a=1}^6\\sum _{{\\bf p}_1{\\bf p}_2{\\bf p}_3}\|f_a\\rangle \\langle f_a\|\\, .$ The Lippmann-Schwinger equation, which can be obtained in the nonrelativistic effective theory, can be partially diagonalized in this basis.", "In order to derive this equation, we write down the canonical Hamiltonian, obtained from Eq.", "(REF ), in the following form: ${\\bf H}={\\bf H}_0+{\\bf H}_I={\\bf H}_0+{\\bf H}_c+{\\bf H}_d+{\\bf H}_r+{\\bf H}_\\eta \\, .$ Here ${\\bf H}_0$ is the free nonrelativistic Hamiltonian.", "The states, which were constructed above, are the eigenstates of ${\\bf H}_0$ : ${\\bf H}_0\|\\pi _i({\\bf p}_1)\\pi _j({\\bf p}_2)\\pi _k({\\bf p}_3)\\rangle &=&\\frac{1}{2M}\\,({\\bf p}_1^2+{\\bf p}_2^2+{\\bf p}_3^2)\|\\pi _i({\\bf p}_1)\\pi _j({\\bf p}_2)\\pi _k({\\bf p}_3)\\rangle \\nonumber \\\\[2mm]&\\doteq & E_p\|\\pi _i({\\bf p}_1)\\pi _j({\\bf p}_2)\\pi _k({\\bf p}_3)\\rangle \\, ,\\quad \\quad i,j,k=\\pm ,0\\, .$ Further, various terms in the interaction Lagrangian are given by ${\\bf H}_c&=&\\int d^3{\\bf x}\\biggl \\lbrace -\\frac{1}{2}\\, c_1 \\pi _0^\\dagger \\pi _0^\\dagger \\pi ^{}_0 \\pi ^{}_0-2c_2 \\biggl ( \\pi _+^\\dagger \\pi _0^\\dagger \\pi ^{}_+ \\pi ^{}_0+ \\pi _-^\\dagger \\pi _0^\\dagger \\pi ^{}_- \\pi ^{}_0 \\biggr )- c_3 \\biggl ( \\pi _+^\\dagger \\pi _-^\\dagger \\pi ^{}_0 \\pi ^{}_0 + \\text{h.c.} \\biggr )- 2c_4 \\pi _+^\\dagger \\pi _-^\\dagger \\pi ^{}_+ \\pi ^{}_-\\nonumber \\\\[2mm]&-&\\frac{1}{2}\\,c_5 \\biggl ( \\pi _+^\\dagger \\pi _+^\\dagger \\pi ^{}_+ \\pi ^{}_++ \\pi _-^\\dagger \\pi _-^\\dagger \\pi ^{}_- \\pi ^{}_- \\biggr )\\biggr \\rbrace \\, ,\\\\[2mm]{\\bf H}_d&=&\\int d^3{\\bf x}\\biggl \\lbrace \\frac{1}{8}\\, d_1\\biggl ( \\pi _0^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _0^\\dagger \\pi ^{}_0 \\pi ^{}_0+\\mbox{h.c.}\\biggr )+ \\frac{1}{2}d_2 \\biggl ( \\pi _+^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _0^\\dagger \\pi ^{}_+ \\pi ^{}_0+ \\pi _-^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _0^\\dagger \\pi ^{}_- \\pi ^{}_0 + \\mbox{h.c.} \\biggr )\\nonumber \\\\[2mm]&+& \\frac{1}{4}\\,d_3 \\biggl ( \\pi _+^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _-^\\dagger \\pi _0 \\pi ^{}_0 + \\pi _0^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _0^\\dagger \\pi ^{}_+ \\pi ^{}_- + \\mbox{h.c.} \\biggr )+ \\frac{1}{2}\\,d_4\\biggl ( \\pi _+^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _-^\\dagger \\pi ^{}_+ \\pi ^{}_-+\\mbox{h.c.}\\biggr )\\nonumber \\\\[2mm]&+& \\frac{1}{8}\\,d_5 \\biggl ( \\pi _+^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _+^\\dagger \\pi ^{}_+ \\pi ^{}_++ \\pi _-^\\dagger \\overset{\\text{\\begin{tikzpicture}[<->] (0,0) -- (2.ex,0);\\end{tikzpicture}}}{\\nabla }\\hspace{-2.0pt}\\phantom{}^2\\pi _-^\\dagger \\pi ^{}_- \\pi ^{}_- +\\mbox{h.c.}\\biggr )\\biggr \\rbrace \\, ,\\\\[2mm]{\\bf H}_r&=&-\\frac{1}{8M^3}\\sum _{i=\\pm ,0}\\int d^3{\\bf x}\\biggl \\lbrace \\pi _i^\\dagger \\nabla ^4\\pi ^{}_i\\biggr \\rbrace \\, ,\\\\[2mm]{\\bf H}_\\eta &=&\\int d^3{\\bf x}\\biggl \\lbrace -\\eta _1 \\biggl ( \\pi _+^\\dagger \\pi ^{}_+ + \\pi _0^\\dagger \\pi ^{}_0 + \\pi _-^\\dagger \\pi ^{}_- \\biggr )^3- \\eta _2 \\biggl ( \\pi _+^\\dagger \\pi ^{}_+ + \\pi _0^\\dagger \\pi ^{}_0 + \\pi _-^\\dagger \\pi ^{}_- \\biggr )\\biggl ( 2 \\pi _+^\\dagger \\pi _-^\\dagger - \\pi _0^\\dagger \\pi _0^\\dagger \\biggr )\\biggl ( 2 \\pi ^{}_+ \\pi ^{}_- - \\pi ^{}_0\\pi ^{}_0 \\biggr )\\biggr \\rbrace \\,.\\nonumber \\\\$ The integration in the above expressions is carried out in a cubic box with a size $L$ .", "In the operator form, the Lippmann-Schwinger equation can be written as ${\\bf T}(E)={\\bf H}_I+{\\bf H}_I{\\bf G}_0(E){\\bf T}(E)\\, ,\\quad \\quad {\\bf G}_0(E)=\\frac{1}{E-{\\bf H}_0}\\, .$ Let us now sandwich this equation by the three-particle states with the definite total isospin $I$ , use closure relations and take into account the fact that ${\\bf T}$ , ${\\bf H}_I$ and ${\\bf G}_0$ are diagonal in $I$ .", "In particular, we are primarily interested in the sector $I=1$ , where the equation simplifies to $\\langle p\|{\\bf T}_{\\alpha \\beta }(E)\|q\\rangle =\\langle p\|{\\bf V}_{\\alpha \\beta }\|q\\rangle +\\sum _k\\sum _{\\gamma =1}^3\\langle p\|{\\bf V}_{\\alpha \\gamma }\|k\\rangle \\frac{1}{E-E_k}\\,\\langle k\|{\\bf T}_{\\gamma \\beta }(E)\|q\\rangle \\, ,$ where, the symbol $p$ denotes a set of momenta $({\\bf p}_1,{\\bf p}_2,{\\bf p}_3)$ , and so on.", "Further, the matrix elements are defined as $\\langle p\|{\\bf V}_{\\alpha \\beta }\|q\\rangle =\\frac{1}{3!", "}\\,\\langle f_\\alpha ,p\|{\\bf H}_I\|f_\\beta ,q\\rangle \\, ,\\quad \\quad \\langle p\|{\\bf T}_{\\alpha \\beta }(E)\|q\\rangle =\\frac{1}{3!", "}\\,\\langle f_\\alpha ,p\|{\\bf T}(E)\|f_\\beta ,q\\rangle \\, .$ Here, in difference with the indices $a$ and $b$ , the indices $\\alpha ,\\beta ,\\gamma $ run from 1 to 3, picking out the vectors $f_1,f_2,f_3$ from the sector $I=1$ only.", "Thus, the partial diagonalization of the Lippmann-Schwinger equation is achieved in the basis of the isospin vectors (REF ).", "The matrix elements of the interaction Hamiltonian in the above equation can be calculated straightforwardly.", "Below, we merely list different contributions to this matrix element in the sector with $I=1$ : $\\langle f_1,p\|{\\bf H}_c\|f_1,q\\rangle &=& \\frac{4\\pi }{M L^3} \\frac{1}{27} (5a_0 +4a_2)\\biggl [ V_1 + V_2 + V_3 + V^+_1 + V^+_2 + V^+_3\\biggr ]\\, , \\nonumber \\\\[2mm]\\langle f_1,p\|{\\bf H}_c\|f_2,q\\rangle &=& \\frac{4\\pi }{M L^3} \\frac{\\sqrt{5}}{54} (a_0 - a_2)\\biggl [ 2\\left(V_1 + V_2 -2V_3\\right) - 3\\left(V^-_1+V^-_2\\right) - \\left(V^+_1+V^+_2-2V^+_3\\right) \\biggr ]\\, , \\nonumber \\\\[2mm]\\langle f_2,p\|{\\bf H}_c\|f_2,q\\rangle &=& \\frac{4\\pi }{M L^3} \\frac{1}{108} (4a_0 + 5a_2)\\biggl [ \\left(V_1 + V_2 + 4V_3\\right) -\\left(2V^+_1+2V^+_2-V^+_3\\right) \\biggr ]\\, , \\nonumber \\\\[2mm]\\langle f_1,p\|{\\bf H}_c\|f_3,q\\rangle &=& -\\frac{4\\pi }{M L^3} \\frac{\\sqrt{5}}{18\\sqrt{3}} (a_0 - a_2)\\biggl [ 2\\left(V_1-V_2\\right) + \\left(V^-_1-V^-_2-2V^-_3\\right) - \\left(V^+_1 - V^+_2\\right)\\biggr ]\\, , \\nonumber \\\\[2mm]\\langle f_2,p\|{\\bf H}_c\|f_3,q\\rangle &=& -\\frac{4\\pi }{M L^3} \\frac{1}{36\\sqrt{3}} (4a_0 + 5a_2)\\biggl [ \\left(V_1 - V_2\\right) -\\left(V^-_1 - V^-_2+V^-_3\\right) + \\left(V^+_1-V^+_2\\right)\\biggr ] \\, ,\\nonumber \\\\[2mm]\\langle f_3,p\|{\\bf H}_c\|f_3,q\\rangle &=& \\frac{4\\pi }{M L^3} \\frac{1}{36} (4a_0+5a_2)(V_1+V_2-V^+_3)\\, ,$ where $V_1 = \\delta _{\\mathbf {p}_1,\\mathbf {k}_1}\\delta _{\\mathbf {p}_2+\\mathbf {p}_3,\\mathbf {k}_2+\\mathbf {k}_3}, \\quad \\quad V_2 = \\delta _{\\mathbf {p}_2,\\mathbf {k}_2}\\delta _{\\mathbf {p}_1+\\mathbf {p}_3,\\mathbf {k}_1+\\mathbf {k}_3}, \\quad \\quad V_3 = \\delta _{\\mathbf {p}_3,\\mathbf {k}_3}\\delta _{\\mathbf {p}_1+\\mathbf {p}_2,\\mathbf {k}_1+\\mathbf {k}_2},$ and $V^\\pm _1 &=& \\delta _{\\mathbf {p}_2,\\mathbf {k}_3} \\delta _{\\mathbf {p}_1+\\mathbf {p}_3,\\mathbf {k}_1+\\mathbf {k}_2} \\pm \\delta _{\\mathbf {p}_3,\\mathbf {k}_2} \\delta _{\\mathbf {p}_1+\\mathbf {p}_2,\\mathbf {k}_1+\\mathbf {k}_3}\\, , \\nonumber \\\\[2mm]V^\\pm _2&=& \\delta _{\\mathbf {p}_1,\\mathbf {k}_3} \\delta _{\\mathbf {p}_2+\\mathbf {p}_3,\\mathbf {k}_1+\\mathbf {k}_2} \\pm \\delta _{\\mathbf {p}_3,\\mathbf {k}_1} \\delta _{\\mathbf {p}_1+\\mathbf {p}_2,\\mathbf {k}_2+\\mathbf {k}_3}\\, ,\\nonumber \\\\[2mm]V^\\pm _3 &=& \\delta _{\\mathbf {p}_1,\\mathbf {k}_2} \\delta _{\\mathbf {p}_2+\\mathbf {p}_3,\\mathbf {k}_1+\\mathbf {k}_3} \\pm \\delta _{\\mathbf {p}_2,\\mathbf {k}_1} \\delta _{\\mathbf {p}_1+\\mathbf {p}_3,\\mathbf {k}_2+\\mathbf {k}_3}\\, .$ Further, the matrix elements of the operator ${\\bf H}_d$ are also given by Eq.", "(REF ), with the replacement $a_I^{}\\rightarrow a_I^{2}\\hat{r}_I$ and $V^{}_i,V_i^\\pm \\rightarrow \\hat{V}^{}_i,\\hat{V}_i^\\pm $ .", "The latter are again given by Eqs.", "(REF ) and (REF ), with the following replacements done: $\\delta _{\\mathbf {p}_i,\\mathbf {k}_l}\\delta _{\\mathbf {p}_j+\\mathbf {p}_k,\\mathbf {k}_m+\\mathbf {k}_n} \\rightarrow \\frac{1}{16}\\, \\delta _{\\mathbf {p}_i,\\mathbf {k}_l}\\delta _{\\mathbf {p}_j+\\mathbf {p}_k,\\mathbf {k}_m+\\mathbf {k}_n} \\left[ (\\mathbf {p}_j - \\mathbf {p}_k )^2 + (\\mathbf {k}_m - \\mathbf {k}_n )^2 \\right].$ Next, the relativistic correction in the kinetic term has the following matrix elements: $\\langle f_1,p\|{\\bf H}_r\|f_1,q\\rangle &=& -\\frac{1}{96 M^3}\\sum _{i=1}^3 (\\mathbf {k}^4_i + \\mathbf {p}^4_i) \\biggl ( V_{123} + V_{132} + V_{231} + V_{213} + V_{312} + V_{321} \\biggr )\\, , \\nonumber \\\\[2mm]\\langle f_1,p\|{\\bf H}_r\|f_2,q\\rangle &=& 0\\, ,\\nonumber \\\\[2mm]\\langle f_2,p\|{\\bf H}_r\|f_2,q\\rangle &=& -\\frac{1}{192M^3} \\sum _{i=1}^3( \\mathbf {k}^4_i + \\mathbf {p}^4_i) \\biggl ( 2 V_{123} - V_{132} + 2V_{213} - V_{231} - V_{312} - V_{321} \\biggr )\\, , \\nonumber \\\\[2mm]\\langle f_1,p\|{\\bf H}_r\|f_3,q\\rangle &=& 0\\, ,\\nonumber \\\\[2mm]\\langle f_2,p\|{\\bf H}_r\|f_3,q\\rangle &=& \\frac{1}{64\\sqrt{3}M^3} \\sum _{i=1}^3( \\mathbf {k}^4_i + \\mathbf {p}^4_i) \\biggl ( V_{132} - V_{231} + V_{312} - V_{321} \\biggr )\\, , \\nonumber \\\\[2mm]\\langle f_3,p\|{\\bf H}_r\|f_3,q\\rangle &=& -\\frac{1}{192 M^3} \\sum _{i=1}^3( \\mathbf {k}^4_i + \\mathbf {p}^4_i) \\biggl ( 2 V_{123} - 2V_{213} + V_{132} - V_{231} - V_{312} + V_{321} \\biggr )\\, ,$ where $V_{ijk} = \\delta _{\\mathbf {p}_1,\\mathbf {k}_i} \\delta _{\\mathbf {p}_2,\\mathbf {k}_j} \\delta _{\\mathbf {p}_3,\\mathbf {k}_k}\\, .$ Finally, the single non-zero matrix element of the three-particle Hamiltonian is given by: $\\langle f_1,p\|{\\bf H}_\\eta \|f_1,q\\rangle &=&-\\frac{2}{L^6} \\left(3\\eta _1+5\\eta _2\\right)\\, \\delta _{\\mathbf {p}_1+\\mathbf {p}_2+\\mathbf {p}_3,\\mathbf {k}_1+\\mathbf {k}_2+\\mathbf {k}_3}\\, .$" ], [ "Perturbation theory", "The energy level shift in the nonrelativistic effective field theory can be calculated by using the ordinary Rayleigh-Schrödinger perturbation theory.", "A small technical issue might arise here since, as we have seen, there are multiple states with $I=1$ , so, the application of the perturbation theory for the degenerate states might be needed.", "This is, however, not the case for the ground state – as seen from Eq.", "(REF ), only the state $f_1$ has a nonvanishing ground-state wave function.", "Note that this is different in the excited states – in this case, the perturbation theory for the degenerate states should indeed be used.", "In order to derive the perturbative expansion for the ground-state energy, at the first step we single out the contribution from the unperturbed ground-state level ${\\bf k}_1={\\bf k}_2={\\bf k}_3=0$ .", "It can be seen that Eq.", "(REF ) is equivalent to the following system of linear equations: $\\langle p\|{\\bf T}_{\\alpha \\beta }(E)\|q\\rangle &=&\\langle p\|\\Omega _{\\alpha \\beta }(E)\|q\\rangle +\\sum _{\\gamma =1}^3\\langle p\|\\Omega _{\\alpha \\gamma }(E)\|0\\rangle \\frac{1}{E-E_0} \\langle 0\|{\\bf T}_{\\gamma \\beta }(E)\|q\\rangle \\, ,\\\\[2mm]\\langle p\|\\Omega _{\\alpha \\beta }(E)\|q\\rangle &=&\\langle p\|{\\bf V}_{\\alpha \\beta }\|q\\rangle +\\sum _{k\\ne 0}\\sum _{\\gamma =1}^3\\langle p\|{\\bf V}_{\\alpha \\gamma }\|k\\rangle \\frac{1}{E-E_k}\\,\\langle k\|\\Omega _{\\gamma \\beta }(E)\|q\\rangle \\, .$ The shifted energy is given by the position of the pole of the scattering matrix.", "Setting now external momenta to zero, $p=q=0$ , it is seen using Eq.", "(REF ) that the shifted ground-state pole position obeys the secular equation, $E-E_0-\\hat{\\Omega }(E)=0\\, ,$ where $\\hat{\\Omega }(E) =\\Omega _{11}(E)$ and $\\Omega (E)$ is a $3\\times 3$ matrix, whose components coincide with $\\langle 0\|\\Omega _{\\alpha \\beta }(E)\|0\\rangle $ (we remind the reader that all components of this matrix, except $\\alpha =\\beta =1$ , vanish at $p=q=0$ ).", "At the next step, we recall that the quantity $\\hat{\\Omega }(E)$ does not contain the singular denominator $1/(E-E_0)$ , which is excluded in the sum with $k\\ne 0$ .", "Consequently, one may expand $\\hat{\\Omega }(E)$ in Taylor series in the vicinity of the unperturbed ground state, $\\hat{\\Omega }(E)=\\hat{\\Omega }(E_0)+(E-E_0)\\hat{\\Omega }^{\\prime }(E_0)+\\frac{1}{2}\\,(E-E_0)^2\\hat{\\Omega }^{\\prime \\prime }(E_0)+O((E-E_0)^3)\\, .$ The coefficients of this expansion can be further expanded in the inverse powers of $L$ : $\\hat{\\Omega }(E_0) &=& \\frac{\\hat{\\Omega }_{3}(E_0)}{L^3} + \\frac{\\hat{\\Omega }_{4}(E_0)}{L^4} + \\frac{\\hat{\\Omega }_{5}(E_0)}{L^5} + \\frac{\\hat{\\Omega }_{6}(E_0)}{L^6} + O(L^{-7})\\, , \\nonumber \\\\[2mm]\\hat{\\Omega }^{\\prime }(E_0) &=& \\frac{\\hat{\\Omega }^{\\prime }_{2}(E_0)}{L^2} + \\frac{\\hat{\\Omega }^{\\prime }_{3}(E_0)}{L^3} + O(L^{-4})\\, , \\nonumber \\\\[2mm]\\hat{\\Omega }^{\\prime \\prime }(E_0) &=& \\hat{\\Omega }^{\\prime \\prime }_0(E_0) +O(L^{-1})\\, .$ Since $E-E_0=O(L^{-3})$ , the terms, displayed above, are sufficient to calculate the energy shift up to and including order $L^{-6}$ .", "Using iterations, one straightforwardly obtains $E-E_0&=&\\frac{E_3}{L^3}+\\frac{E_4}{L^4}+\\frac{E_5}{L^5}+\\frac{E_6}{L^6}+O(L^{-7})\\, ,\\nonumber \\\\[2mm]E_3&=&\\hat{\\Omega }_3(E_0)\\, ,\\nonumber \\\\[2mm]E_4&=&\\hat{\\Omega }_4(E_0)\\, ,\\nonumber \\\\[2mm]E_5&=&\\hat{\\Omega }_5(E_0)+\\hat{\\Omega }_2^{\\prime }(E_0)\\hat{\\Omega }_3(E_0)\\, ,\\nonumber \\\\[2mm]E_6&=&\\hat{\\Omega }_6(E_0)+\\hat{\\Omega }_3^{\\prime }(E_0)\\hat{\\Omega }_3(E_0)+\\hat{\\Omega }_2^{\\prime }(E_0)\\hat{\\Omega }_4(E_0)+\\frac{1}{2}\\,\\hat{\\Omega }_0^{\\prime \\prime }(E_0)\\hat{\\Omega }_3(E_0)^2\\, .$ The matrix $\\Omega (E)$ can be expanded similarly.", "The coefficients $\\Omega _i(E_0),\\Omega _i^{\\prime }(E_0),\\Omega _i^{\\prime \\prime }(E_0)$ can be calculated by iterating Eq. ().", "Defining a $3\\times 3$ matrix $V_{pq}$ , whose elements are given by $\\langle p\|{\\bf V}_{\\alpha \\beta }\|q\\rangle $ , and attaching superscripts “$c$ ,” “$d$ ,” “$r$ ,” “$\\eta $ ,” to the individual contributions, one gets a compact expression: $\\frac{\\Omega _{3}(E_0)}{L^3} &=& V^c_{00}, \\nonumber \\\\[2mm]\\frac{\\Omega _{4}(E_0)}{L^4} &=& -\\sum _{p \\ne 0} \\frac{V^c_{0p}V^c_{p0}}{E_p}, \\nonumber \\\\[2mm]\\frac{\\Omega _{5}(E_0)}{L^5} &=& \\sum _{p,k \\ne 0} \\frac{V^c_{0p}V^c_{pk}V^c_{k0}}{E_p E_k}, \\nonumber \\\\[2mm]\\frac{\\Omega _{6}(E_0)}{L^6} &=& -\\sum _{p,k,q \\ne 0} \\frac{V^c_{0p}V^c_{pq}V^c_{qk}V^c_{k0}}{E_p E_q E_k} -\\sum _{p \\ne 0} \\frac{V^c_{0p}V^d_{p0}+V^d_{0p}V^c_{p0}}{E_p} + V^\\eta _{00} + \\sum _{p,k \\ne 0} \\frac{V^c_{0p}V^r_{pk}V^c_{k0}}{E_p E_k}, \\\\[2mm]\\phantom{25} \\nonumber \\\\[2mm]\\frac{\\Omega ^{\\prime }_{2}(E_0)}{L^2} &=& -\\sum _{p \\ne 0} \\frac{V^c_{0p}V^c_{p0}}{E_p^2}, \\nonumber \\\\[2mm]\\frac{\\Omega ^{\\prime }_{3}(E_0)}{L^3} &=& 2\\sum _{p,k \\ne 0} \\frac{V^c_{0p}V^c_{pk}V^c_{k0}}{E_p^2 E_k} ,\\\\[2mm]\\phantom{25} \\nonumber \\\\[2mm]\\Omega ^{\\prime \\prime }_{0}(E_0) &=& 2\\sum _{p \\ne 0} \\frac{V^c_{0p}V^c_{p0}}{E_p^3}.$ Taking the component $\\alpha =\\beta =1$ from the above expressions, we finally arrive at the expressions for $\\hat{\\Omega }_i(E_0)$ , $\\hat{\\Omega }^{\\prime }_i(E_0)$ , $\\hat{\\Omega }^{\\prime \\prime }_i(E_0)$ , we are looking for.", "Putting everything together, we obtain the final expression for the energy shift: $E_3&=&\\frac{4 \\pi }{3 M} \\left(5a_0 + 4a_2\\right), \\nonumber \\\\[2mm]E_4&=&-\\frac{4}{3 M} \\left(5a_0^2 + 4a_2^2\\right) I, \\nonumber \\\\[2mm]E_5&=& \\frac{4}{3 M \\pi } \\biggl [ \\left(5a_0^3 + 4a_2^3\\right) I^2- \\frac{1}{9} \\left(55a_0^3 -120 a_0^2 a_2 -60 a_0 a_2^2 + 44 a_2^3\\right) J \\biggr ]\\, , \\nonumber \\\\[2mm]E_6&=&-\\frac{4}{3 M \\pi ^2} \\Big [ \\left(5a_0^4 +4a_2^4\\right) I^3 - \\frac{1}{9} \\left(185a_0^4 -160 a_0^3 a_2 - 80 a_0 a_2^3 + 136 a_2^4\\right) IJ \\nonumber \\\\[2mm]&-& \\frac{1}{27} \\left(5a_0 + 4a_2\\right) \\left(115a_0^3 + 420 a_0^2 a_2 + 300 a_0 a_2^2 + 56 a_2^3\\right) K \\nonumber \\\\[2mm]&+& \\frac{8}{27} \\left(10a_0^4 + 140 a_0^3 a_2 + 285 a_0^2 a_2^2 + 50 a_0 a_2^3 + a_2^4\\right) Q^r \\nonumber \\\\[2mm]&+& \\frac{8}{9} \\left(5a_0^4 + 20 a_0^3 a_2 + 45 a_0^2 a_2^2 + 10 a_0 a_2^3 + a_2^4\\right) R^r \\Big ] \\nonumber \\\\[2mm]&+&\\frac{8\\pi ^2}{3M} \\left(5a_0^3 r_0 + 4a_2^3 r_2\\right) - \\frac{4\\pi ^2}{3M^3} \\left(5a_0^2 + 4 a_2^2\\right) \\nonumber \\\\[2mm]&+& \\frac{64 \\pi }{243 M}\\Big [ 9\\sqrt{3} \\left(5a_0^4 + 20 a_0^3 a_2 + 45 a_0^2 a_2^2 + 10 a_0 a_2^3 + a_2^4\\right) \\nonumber \\\\[2mm]&-& 2\\pi \\left(10a_0^4 + 140 a_0^3 a_2 + 285 a_0^2 a_2^2 + 50 a_0 a_2^3 + a_2^4\\right) \\Big ]\\ln (\\mu L) \\nonumber \\\\[2mm]&-&6 \\eta _1^r - 10 \\eta _2^r \\, .$ Finally, we give a list of all momentum sums that appear in the above equation.", "These are the same quantities that enter the expression in case of the maximal isospin.", "The quantities $I,J,K$ are finite in the dimensional regularization and are given by $I&=&\\sum _{{\\bf n}\\ne 0}\\frac{1}{{\\bf n}^2}=-8.91363291781\\cdots \\, ,\\nonumber \\\\[2mm]J&=&\\sum _{{\\bf n}\\ne 0}\\frac{1}{{\\bf n}^4}=16.532315959\\cdots \\, ,\\nonumber \\\\[2mm]K&=&\\sum _{{\\bf n}\\ne 0}\\frac{1}{{\\bf n}^6}=8.401923974433\\cdots \\, .$ The quantities $Q^r$ and $R^r$ represent the finite part of the double sums over momenta in the minimal subtraction renormalization scheme.", "Introducing a formal notation of a sum in $d$ dimensions,The sums in $d$ dimensions can be interpreted as follows.", "One first performs the Poisson transform and singles out the infinite-volume part, which corresponds to an integral instead of a sum.", "The dimensional regularization and minimal subtraction is applied then to this integral.", "The remainder is ultraviolet-convergent and the limit $d\\rightarrow 3$ can be performed there without a problem.", "we get $\\frac{1}{L^{2d}}\\sum _{{\\bf p},{\\bf q}\\ne 0}\\frac{1}{{\\bf p}^2{\\bf q}^2({\\bf p}^2+{\\bf q}^2+({\\bf p}+{\\bf q})^2)}=\\mu ^{2(d-3)}\\biggl \\lbrace \\frac{1}{48\\pi ^2}\\,\\biggl (\\ln (\\mu L)-\\frac{1}{2(d-3)}\\biggr )+\\frac{1}{(2\\pi )^6}\\,Q^r\\biggr \\rbrace \\, ,\\nonumber \\\\[2mm]\\frac{1}{L^{2d}}\\,\\sum _{{\\bf p}\\ne 0}\\frac{1}{{\\bf p}^4}\\,\\sum _{\\bf q}\\frac{1}{({\\bf p}^2+{\\bf q}^2+({\\bf p}+{\\bf q})^2)}=\\mu ^{2(d-3)}\\biggl \\lbrace -\\frac{\\sqrt{3}}{32\\pi ^3}\\,\\biggl (\\ln (\\mu L)-\\frac{1}{2(d-3)}\\biggr )+\\frac{1}{(2\\pi )^6}\\,R^r\\biggr \\rbrace \\, .$ The finite parts are given by $Q^r&=&-102.1556055\\cdots \\, ,\\nonumber \\\\[2mm]R^r&=&19.186903\\cdots \\, .$ Note that, in the above equations, ${\\bf p}=2\\pi {\\bf n}/L$ , ${\\bf q}=2\\pi {\\bf m}/L$ , and the summation is carried out over ${\\bf n},{\\bf m}\\in \\mathbb {Z}^3$ .", "Equation (REF ) represents our final result.", "It gives the expression for the energy shift of the ground state with $I=1$ up to and including $O(L^{-6})$ .", "Together with the known expression for the case $I=3$ , it provides a full framework for analyzing the lattice data on ground states in the three-pion system.", "Two independent nonderivative couplings $\\eta _{1,2}$ can be separated and extracted from data in a result of this analysis." ], [ "Conclusions", "i) Using the nonrelativistic effective Lagrangian approach, we have evaluated the finite-volume shift of the ground state of three pions in the state with total isospin $I=1$ up to and including $O(L^{-6})$ .", "Since three pions with zero momenta can have only $I=1$ or $I=3$ , the expression, which is derived in the present paper, provides the last remaining missing piece in the description of the three-pion ground state, because the expression in the case of $I=3$ has been known in the literature.", "With this result at hand, one can also study the three-pion decays into all isospin channels.", "ii) An immediate application of the obtained expression could be the extraction of two independent three-body couplings $\\eta _{1,2}$ from the lattice data.", "Moreover, the analysis of the three-particle states might put constraints on the parameters in the two-particle sector – the scattering lengths $a_I$ and the effective radii $r_I$ .", "According to our past experience in the scalar $\\varphi ^4$ -theory [8], [9], these constraints are most effectively implemented when one performs a simultaneous fit of all datasets in the two- and three-particle sectors.", "The input from the three-particle sector might be interesting, because the two-particle scattering parameters (especially those corresponding to the total isospin $I=0$ ) are not very well determined on the lattice at present (for the recent developments in the two-particle sector, we refer the reader, e.g., to [52], [53], [54], [55]).", "iii) Albeit we are working in the nonrelativistic effective theory, our results contain a full set of relativistic corrections up to and including $O(L^{-6})$ .", "In general, such corrections can be systematically included in the nonrelativistic approach up to a desired accuracy.", "iv) The above work is a rather straightforward generalization of the result obtained for the maximal isospin.", "One has, however, to deal with subtle issues, for example, with the degeneracy of levels, which are absent in the former case.", "The level splitting does not occur for the ground state, but one will have to face it, deriving expressions for the shifts of the excited states.", "A framework for the degenerate states can be worked out along the path which is pretty similar to the one considered in the present paper.", "The pertinent explicit formulae are, however, not needed for the study of the ground state, and are therefore not displayed." ], [ "Acknowledgments", "The authors would like to thank U.-G. Meißner, F. Romero-López and C. Urbach for interesting discussions.", "The work of F. M. and A. R. was supported in part by the DFG (CRC 110 “Symmetries and the Emergence of Structure in QCD”, Grant no.", "TRR110).", "A. R., in addition, thanks Volkswagenstiftung (Grant no.", "93562) and the Chinese Academy of Sciences (CAS) President's International Fellowship Initiative (PIFI) (Grant no.", "2021VMB0007) for the partial financial support." ] ]	2011.14178
[ [ "AdaGrasp: Learning an Adaptive Gripper-Aware Grasping Policy" ], [ "Abstract This paper aims to improve robots' versatility and adaptability by allowing them to use a large variety of end-effector tools and quickly adapt to new tools.", "We propose AdaGrasp, a method to learn a single grasping policy that generalizes to novel grippers.", "By training on a large collection of grippers, our algorithm is able to acquire generalizable knowledge of how different grippers should be used in various tasks.", "Given a visual observation of the scene and the gripper, AdaGrasp infers the possible grasp poses and their grasp scores by computing the cross convolution between the shape encodings of the gripper and scene.", "Intuitively, this cross convolution operation can be considered as an efficient way of exhaustively matching the scene geometry with gripper geometry under different grasp poses (i.e., translations and orientations), where a good \"match\" of 3D geometry will lead to a successful grasp.", "We validate our methods in both simulation and real-world environments.", "Our experiment shows that AdaGrasp significantly outperforms the existing multi-gripper grasping policy method, especially when handling cluttered environments and partial observations.", "Video is available at https://youtu.be/kknTYTbORfs" ], [ "Introduction", "In many real-world systems, a robot's end-effector is designed with a specific application in mind, where its specific geometry and kinematic structure often lead to distinct strengths and weaknesses.", "However, the vast majority of robotic research has been limited to single end-effector setups where the learned policy cannot generalize to new gripper hardware without extensive retraining.", "On the other hand, we humans can easily use various tools to accomplish different tasks and quickly adapt to unseen tools.", "Can we allow our robot system to do the same?", "This capability would benefit a robot manipulation system in the following ways: [leftmargin=] Versatility via diversity.", "Since different gripper designs often provide complementary strengths and weaknesses, and by learning to adequately use a diverse set of grippers, the system can effectively improve its versatility on handling a larger variety of objects and tasks.", "Adaptability via generalization.", "Since the learned grasping policy can generalize across different gripper hardware, it can also quickly adapt to new grippers by directly analyzing its geometry and structure.", "It is different from the existing multi-gripper systems [1], [2] that need to collect new training data for any new gripper hardware.", "To achieve this goal, we propose AdaGrasp, a learning-based algorithm that learns a unified policy for different grippers and can generalize to novel gripper designs.", "At its core, AdaGrasp uses cross convolution (CrossConv)[3] operation between the shape encoding of the robot gripper and the scene to infer the grasp score for all possible grasp poses.", "Intuitively, this operation can be considered as an efficient way of exhaustively matching the scene and the gripper geometry under different grasp poses, where a good “match” of their 3D geometry will lead to a successful grasp.", "Figure: Gripper-Aware Grasping Policy.The goal of AdaGrasp is to produce grasping strategies that are conditioned on input gripper description (a,b,c).For example, since the RG2 gripper has a wider fixed opening than WSG 50 (which can control its opening width), it chooses a different grasp pose to avoid double-picking or collision.", "Barrett Hand grasps the big triangle shape, which can be challenging for other two-finger grippers.Figure: Approach Overview.", "At its core, AdaGrasp infers grasp scores for all candidate grasp poses by computing the cross convolution between the gripper encoding produced by the gripper encoder (a) and the scene encoding produced by the scene encoder (b).", "This cross convolution operation (c) matches the scene and gripper encoding under all grasp poses by translating and rotating the gripper kernel, where a good “match” of their encoding results in a high grasp score.", "Different initial opening configuration of a gripper are treated as different grippers and fed to the network in parallel.", "The action associated with the highest grasp score is executed.", "The action for a grasp attempt includes selecting the suitable gripper, deciding its initial joint configuration, and choosing a proper grasp pose.The 3D geometry of a robot gripper and its kinematic structure often inform how it should be used for a given task [4].", "By learning to use a large collection of different grippers, the algorithm should be able to acquire a generalizable knowledge of how different grippers should be used in various tasks.", "For example, a gripper's opening width determines what object shape can fit into the gripper, and the thickness of each finger determines what narrow space the finger can get into without collision.", "Fig.REF illustrates different grasp poses that are suitable for different grippers.", "The primary contribution of this paper is AdaGrasp, a learning-based grasping algorithm that leverages generalized shape matching via cross convolution to produce a grasping policy that works across different gripper hardwares.", "We validate our methods in both simulation and real-world environments.", "Our experiments show that AdaGrasp outperforms the state-of-the-art method for multi-gripper grasping, especially in a cluttered environment and with partial observation." ], [ "Related Work", "Learning-based single-gripper systems.", "Recent data-driven methods have made great progress on learning object-agnostic grasping policies that detect grasps by exploiting visual features, without explicitly using object-specific prior knowledge [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16].", "These algorithms demonstrate the ability to generalize to new objects and scene configurations.", "However, they are often designed and trained with a fixed hardware setup.", "Hence, they cannot adapt to any changes in the gripper hardware without extensive retraining.", "Learning-based multi-gripper systems.", "To take advantage of complementary skills between different grippers, more recent works have started to use multiple end-effectors for grasping.", "For example, both Zeng et al.", "[1], [17] and Mahler et al.", "[18] used a setup with one suction cup gripper and one parallel jaw gripper.", "However, in both the systems, the algorithm learns a separate policy for each gripper, i.e., their policies cannot generalize to new grippers.", "As a result, these algorithms are often limited to a small number of grippers.", "Contact-based grasping policy.", "Many analytical grasping models have been proposed to evaluate grasp quality through contact-point reasoning and force-closure analysis [19], [20], [21], [22], [23].", "The work most related to us is UniGrasp [24], where the algorithm takes in the gripper point cloud and a single object point cloud, samples N points from the object point cloud as contact points for N fingers, and uses inverse kinematics to get gripper joint configuration.", "While a contact-based policy generalizes to new grippers, it also brings in limitations.", "First, since measuring precise contact points in real-world is challenging, the algorithm can only be trained with simulation.", "Moreover, it is trained using static force closure analysis, which does not consider the object dynamics during grasping.", "Second, to reason about force closure, the algorithm assumes a complete object representation as input which relies on a perception algorithm to perfectly detect the target object and provide full 3D geometry.", "Since the algorithm only samples contact points on the object surface, a partial observation of the object will lead to unstable contact point selection and inaccurate force closure evaluation, as we showed in our experiments.", "In contrast, our method's action space won't be limited by partial observation.", "Furthermore, it does not consider the gripper geometry beyond contact points, which increases the likelihood of collision in cluttered environments.", "In contrast, our algorithm does not require any explicit contact point supervision or complete object representation.", "Therefore, it can better handle cluttered environments and partial observation." ], [ "Approach", "The goal of our algorithm is to learn a policy that can produce the optimal grasping strategy for a novel gripper by estimating the probability of grasp success (i.e., grasp score) for all candidate gripper configurations and grasp poses.", "Concretely, taking a visual observation of the scene (RGB-D images) and the gripper design (defined as URDF files) as input, the algorithm infers the possible grasp poses along with their grasp scores that would allow the gripper to successfully grasp a target object.", "The core of our approach is a Grasp Evaluation Network $f_{grasp}(s, g) \\rightarrow a $ that infers the grasp score for all candidate grasp poses $a$ by computing the cross convolution between the gripper encoding $g$ and scene encoding $s$ .", "The grasp pose is parameterized by rotation about the z-axis and 2D translation.", "This cross convolution operation can be considered as an efficient way of exhaustively matching the scene geometry with gripper geometry in all possible grasp poses by translating and rotating the gripper kernel.", "The matching score is finally represented as a dense grasp score map, where a higher value indicates a higher chance of a successful grasp.", "We train the algorithm with a collection of grippers and environment setups and test it with unseen grippers and objects.", "Fig.", "REF shows the network overview, and the following sections provide details of our approach." ], [ "Gripper and Scene Representation", "Gripper encoding.", "The gripper geometry is captured by 10 depth images and encoded as a 3D TSDF volume [25].", "The volume dimension is $64\\times 64\\times 32$ (voxel) with voxel size $v_g=0.004$ (m).", "We compute TSDF volume for the gripper at its initial open state and final closed state and stack them as input $I_g\\in R^{2\\times 64\\times 64\\times 32}$ .", "The gripper encoder network (Fig REF a) starts with two 3D convolution layers with kernel size $3\\times 3\\times 3$ , resulting in a feature $\\in R^{64\\times 32\\times 32\\times 16}$ .", "Then we use one 3D convolution with kernel size $1\\times 1\\times 16$ reducing the z dimension to 1.", "Finally, we use 5 2D convolution layers to produce the gripper features $\\psi (g) \\in R^{16\\times 32\\times 32}$ .", "Scene encoding The input scene is captured with a top-down depth image and encoded as a 3D TSDF volume.", "The workspace dimension is $192\\times 192\\times 64$ (voxel) with a voxel size $v_s=0.002$ (m).", "In multi-object obstacle cases, the obstacle mask is provided as an additional channel.", "This channel will be 0 for other cases.", "The scene volume $I_s\\in R^{2\\times 192\\times 192\\times 64}$ is then fed into the scene encoder network (Fig REF b).", "Similar to the gripper encoder network, it consists of three 3D convolution layers with downsample scale=4, one layer for z-axis reduction, and five 2D convolution layers.", "The output is the scene features $\\phi (s) \\in R^{16\\times 48\\times 48}$ .", "Figure: Training and testing grippers used in our experiments." ], [ "Grasp Evaluation via Shape Matching", "After the encoding network, the scene and gripper geometry are mapped into a query $\\phi (s)$ and key $\\psi (g)$ features.", "We carefully set the number of downsampling size in scene encoder and gripper encoder so that both features share a similar physical receptive field.", "As a result, the spatial alignment is maintained, and shape matching in feature space (via CrossConv) is meaningful.", "The algorithm then computes the cross convolution between the $\\psi (g)$ and $\\phi (s)$ by treating $\\psi (g)$ as the convolution kernel (Fig.", "REF c).", "The output shares the same size as the scene feature $\\phi (s)$ .", "We repeat this step for $r=16$ times [26], each time rotate the scene TSDF volume by $\\theta = 2\\pi /r$ about z-axis.", "Finally, the output of cross convolution is fed into a grasp evaluation network (Fig REF -d) that estimates dense grasp scores for all possible actions $ Q \\in R^{X_s\\times Y_s \\times r} $ , where each grasp score $Q(i,j,k)$ in the $Q$ value map corresponds to one grasp pose.", "The grasp pose is parameterized by its position $(x,y,z)$ and orientation $\\theta = k\\pi /r$ about z-axis, where $ x = x_{min}+v_si$ , $y = y_{min}+v_sj$ , $z=\\mathcal {H}(O(i,j))-0.05$ , $[x_{min}, y_{min}, z_{min}, x_{max}, y_{max}, z_{max}]$ is the workspace bound, $\\mathcal {H}(O(i,j))$ is the height of z-dimention in the scene volume $O$ at location $(i,j)$ .", "During grasp execution, the gripper starts at location $(x,y,z_{max})$ , moves downward along z-axis until having contact with an object or reaching the target position $(x, y, z)$ , and then close its finger.", "The gripper will then move upwards and this execution is considered successful if and only if exactly one target object is lifted $>0.2$ m. Grasping an obstacle or more than one objects is classified as a failure.", "Network training.", "The whole network is trained end-to-end with self-supervised grasping trials, similar to prior work [5], [26].", "Based on the object height after grasping, each grasp trial is labeled with its grasp outcome (1 = success, 0 = failure).", "The network is trained to predict the grasp outcome for all possible actions, and it is supervised by the grasping outcome of the executed action (one action out of $X_s\\times Y_s\\times r$ actions) using softmax loss.", "During training, the network chooses its action using $\\epsilon -$ greedy.", "We use the normalized predicted grasp scores as the probability of choosing each pose.", "At training epoch $e$ , $\\epsilon $ decreases linearly from $\\epsilon _{max}$ to $\\epsilon _{min}$ .", "After $n$ epochs, $\\epsilon =\\epsilon _{min}$ .", "We set $n=2000, \\epsilon _{min}=0.2, \\epsilon _{max}=0.8$ .", "All the grasp trails are stored in a FIFO replay buffer (size=12000).", "At each training step, we sample a batch of examples from the replay buffer with a 1:1 positive to negative ratio.", "We also used data augmentation to overcome overfitting.", "The scene inputs obtained from the replay buffer have a probability of 0.7 to be randomly shifted and rotated.", "We applied the same transformation to the corresponding grasp pose.", "The final model is trained for 5000 epochs, 8 sequences of data collection, and 32 iterations of training per epoch with Adam optimizer and learning rate 0.0005.", "Table: Grasp Success Rate." ], [ "Improving Grasp Quality via Gripper Selection", "To execute the grasp, the algorithm selects the predicted best action from the grasp evaluation network $ a = \\operatornamewithlimits{arg\\,max}_a Q $ .", "However, depending on the input gripper, sometimes even the best action might still not be good enough to achieve a successful grasp (e.g., the input gripper or its initial configuration is too small to enclose the object inside).", "In such cases, the algorithm will compare and select between different input grippers to improve its grasp quality.", "To do so, the network predicts a grasp score for a list of $N$ candidate grippers, then selects the one that produces the highest grasp score.", "Note that the list of candidate grippers can include completely different grippers or the same gripper with different initial joint configurations.", "Since the grasp evaluation network is trained for many grippers, the estimated grasp score for different grippers is naturally comparable, where a higher score indicates a better gripper for the task.", "During testing, we allow the algorithm to choose the best configuration for a given gripper (AdaGrasp-fixGripper in Tab.", "REF ) or choose both the best gripper and its best configuration at the same time (AdaGrasp in Tab.", "REF ).", "Configuration Sampling.", "To sample possible initial configuration for a given gripper, we linearly map the gripper's joint configuration into a scalar value in the range [0,1], where 0 represents the fully closed state, and 1 represents the fully open state.", "Note that the algorithm only needs to choose grippers' initial configuration, since the final configuration is determined – the gripper will always try to close its fingers all the way to its fully closed state.", "During training, each gripper has 4 initial configuration options randomly sampled between 0.4 and 1.0.", "Since two fingers of Barrett Hand have flexible palm joints, we define the following 3 presets: (1) palm joint = 0, two flexible fingers are parallel and next to each other.", "(2) palm joint = $0.1\\pi $ , the angle between two flexible fingers is $0.2\\pi $ .", "(3) palm joint = $0.5\\pi $ and remove the finger with a fixed palm joint.", "This configuration mimics a broken Barrett Hand with two remaining fingers (Barrett Hand-B)." ], [ "Experiments", "We run the following experiments to verify that the proposed AdaGrasp algorithm is able to (1) learn different grasping strategies for different grippers, (2) generalize to new grippers, (3) select a suitable gripper and gripper configuration for a given task.", "We have also provided real-world experiments to validate our approach.", "Scene setup: We use Pybullet [27] as our simulation environment.", "The target objects and obstacles are randomly dropped within a rectangular workspace.", "All objects used in simulation are from Dexnet 2.0 [28] object dataset.", "The training dataset has 801 objects: 400 from the 3DNet subset and 401 from the Kit subset.", "The test dataset has 57 objects: 13 from Adversarial subset and the remaining object from the Kit category that are not used in training.", "For our method, we use a single top-down RGB-D camera to capture the scene.", "For UniGrasp, we use 3 additional cameras to provide a complete 3D point cloud input since it is sensitive to partial observation.", "Tab.", "REF studies both algorithm's performance with respect to scene visibility.", "We tested the following scenarios: [leftmargin=] Single object.", "One random object is dropped into the scene with random position and orientation.", "Multiple objects.", "There are 5 objects in the scene, and the gripper is expected to grasp one object at a time until the scene is empty or a maximum attempt of 7 is reached.", "Multiple objects with obstacles.", "There are 3 targets and 3 obstacles.", "We provide the obstacle mask.", "The algorithm needs to grasp the target object while avoiding obstacles.", "Gripper: We have 7 training grippers and 4 testing grippers as shown in Fig.", "REF .", "One of the testing grippers is Barrett Hand with one finger missing, which is equivalent to a 2 finger gripper.", "During training, grippers are globally scaled by a random factor of $t\\in (0.8,1.2)$ to increase the training gripper diversity.", "During testing, gripper scale is fixed at 1.", "Metric: The algorithm performance is measured by grasp success rate = $\\frac{\\mathrm {\\# successful\\_grasps}}{\\mathrm {\\# total\\_grasp\\_attempts}}$ .", "The grasp success for each attempt is measured by whether the gripper grasps strictly one target.", "For example, in the multi-object setup, grasping two objects simultaneously is considered a failure (double-picking).", "The objects can be grasped in any order.", "We evaluate the algorithms on all grippers separately and use the average performance, except in our final policy, the algorithm has the freedom to select from a set of grippers.", "For each type of scene, the test scene generation is consistent across all algorithms and grippers.", "Algorithm comparisons: [leftmargin=*] UniGrasp [24]: it takes in the gripper point cloud and object point cloud (background removed), samples N (2 or 3) points from the object as contact points for N fingers, respectively, and use inverse kinematics to compute gripper joint configurations for grasp execution.", "We directly test the pre-trained model provided by the authors.", "SceneOnly: a single policy trained using all training grippers (uniformly sampled during training).", "The policy can only access the scene observation without gripper information; hence, it predicts uniformly across all grippers.", "SingleGripper [26]: a learning based grasping method from Zeng et al.", "using only Robotiq 2F-85.", "AdaGrasp-initOnly: the gripper input is the initial gripper state.", "The policy selects the best grasp pose (position and orientation) for a given gripper.", "AdaGrasp-fixConfig: same as AdaGrasp-initOnly, but gripper input has both its initial and final state.", "AdaGrasp-fixGripper: the algorithm linearly samples the gripper configurations and infer grasp score for each configuration.", "Then, the algorithm selects the gripper configuration with the highest grasp score to execute.", "AdaGrasp: On top of the gripper configuration and grasp pose, this algorithm also selects the best gripper with the highest grasp score to use.", "This is our final policy.", "In testing, SceneOnly, AdaGrasp-initOnly, and AdaGrasp-fixConfig uses a random initial configuration sampled from [0.5, 0.625, 0.75, 0.875, 1.0]; AdaGrasp-fixGripper and AdaGrasp will select the configuration from the same list." ], [ "Experimental Results", "Comparison to prior work.", "We compare our approach with state-of-the-art multi-gripper system UniGrasp [24].", "The number of cameras during AdaGrasp's training is randomly chose in {1,2,3,4}.", "Both algorithms are evaluated on test objects and test grippers under a fixed-gripper and fixed-camera setting (i.e., the algorithm can choose the input gripper's initial configuration but cannot switch gripper).", "In the single object case, AdaGrasp-fixGripper achieves better performance (+10%) comparing to UniGrasp.", "The advantage is much more salient in multi-object case, where AdaGrasp-fixGripper is able to outperform UniGrasp by around 60%.", "This result highlights AdaGrasp's ability in handling cluttered environments.", "Fig.", "REF shows qualitative comparisons, where UniGrasp samples contact points on multiple objects or misses potential collisions.", "Another advantage of AdaGrasp is its ability in handling partial observations.", "UniGrasp is very sensitive to the quality and visibility of scene observation since it directly samples contact points from the input point cloud, which is limited to the observed surface (Fig.", "REF -a).", "In contrast, AdaGrasp is able to reason about the object grasp point beyond the visible surfaces using 3D TSDF representation.", "Results in Tab.", "REF demonstrate that when the scene observation is incomplete (i.e., with fewer cameras), UniGrasp's performance decreases significantly, while AdaGrasp has consistent performance.", "Inference time of AdaGrasp is $1.05$ s for each gripper with 5 initial configurations and 16 rotations.", "Figure: Comparisons.", "UniGrasp often fails on incomplete input point clouds since it samples contact points directly from the pointcloud (2nd row with 1 camera).", "It also struggles with cluttered scenes, frequently sampling contact points on multiple objects or failing to account for collision.", "AdaGrasp is able to handle both partial observability and scene clutter.Table: Grasp Succ Rate w.r.t Partial Observation.Can AdaGrasp learn gripper-aware grasping policy?", "To verify AdaGrasp's ability to infer different grasping strategies conditioned on the input gripper, we perform the following experiments.", "All models in Tab.", "REF are trained and tested under single-camera setting.", "First, we compare AdaGrasp-fixConfig with an “SceneOnly” policy, i.e., a single policy trained with all training grippers without the gripper as input.", "Results in Tab.", "REF shows that AdaGrasp-fixConfig's performance is always significantly better than the “SceneOnly”, which demonstrates that AdaGrasp-fixConfig improves the grasp prediction by analyzing the input gripper.", "We visualize the top grasp pose prediction for different grippers given the same scene setup (Fig.", "REF REF ).", "From the visualization, we can see that the algorithm is able to infer diverse grasp poses that are suitable for each input gripper and configuration.", "Figure: Real-world Setup.", "Robot and camera setup (left) and test objects (right).", "Videos of experiments are available in supp.", "video and website.Table: Real-world Grasp Succ Rate on Unseen Grippers and Objects.Figure: Gripper-Aware Grasping Policy.", "Given the same input scene in each row, AdaGrasp predicts a different grasp pose suitable for each gripper.", "Here are example grasps inferred by AdaGrasp for training grippers (left) and testing grippers (right) in multi-object setups (Row 1-2), and multi-object + obstacle setups (Row 3).", "Brown surface: input TSDF.", "Green surface: obstacles input as additional mask.", "More examples available on our website.Figure: Grasp Score Visualization.", "Dense grasp score predictions are shown for 3 out of 16 different grasp orientations.", "The highest grasp score for each orientation is shown at the top left.", "For each gripper, the orientation with the highest score is highlighted in red.In scene (a), the target object is a mug.", "RG2 prefers to grasp the cup's edge or handle, while Barrett Hand prefers to grasp across the whole cup.In scene (b), the target object is surrounded by two obstacles (green).", "We visualize the grasp poses for the WSG 50 gripper under different initial configurations (opening size).", "With a larger opening, the algorithm chooses to grasp vertically (90) to avoid collisions, while with a smaller opening, it chooses to grasp horizontally (22.5) since the object's length is now larger than the gripper width.", "Between these two configurations, the algorithm chooses the wider opening.Can AdaGrasp generalize to new grippers?", "To test the algorithm's adaptability to new gripper hardware, we tested the learned policy with five unseen grippers, including three 2-finger grippers, one 3-finger grippers, and a “damaged” 3-finger gripper (Barrett hand with a missing finger).", "While test grippers are never used during training, AdaGrasp-fixGripper is able to get performance comparable to that on the training grippers.", "In Tab.", "REF AdaGrasp-fixGripper improves the SceneOnly policy performance by 18% to 54%.", "Can AdaGrasp select the right configuration and gripper for a given task?", "To check whether the predicted grasp score is informative for comparing and selecting the gripper's initial configuration, we compare the algorithm performance with and without configuration selection (AdaGrasp-fixConfig v.s.", "AdaGrasp-fixGripper).", "Both algorithms predicts the grasp scores for the same gripper.", "The difference is that AdaGrasp-fixGripper selects the configuration with the highest grasp score while AdaGrasp-fixConfig randomly picks one configuration.", "Compared to AdaGrasp-fixConfig, AdaGrasp-fixGripper performance is better in all cases, improving 5% to 21%.", "This result validates that the predicted grasp score is informative for selecting the best initial configuration.", "Fig.", "REF -b shows an example of configuration selection for WSG 50.", "Similarly, we showed that the grasp score is also comparable across different grippers.", "As a result, the algorithm is able to further improve its grasping performance by choosing the “right tool” (gripper) for a given task at hand (object to grasp).", "Comparing AdaGrasp with AdaGrasp-fixGripper in Tab.", "REF , we can see the 1% to 9% improvement in all scenarios.", "The performance of AdaGrasp is also better than SingerGripper, which only evaluates on Robotiq 2F-85.", "This result indicates that if combined with an automatic tool changing hardware [29], AdaGrasp can improve the grasping performance by allowing the system to properly use a diverse set of grippers.", "Is gripper final state encoding helpful?", "The input gripper encoding includes both gripper's initial and final state.", "It allows the algorithm to reason about the gripper's dynamics during the closing action beyond its static 3D geometry.", "To see the effect of final state encoding, we compare the model without the final-state, which is AdaGrasp-initOnly.", "In almost all test cases, AdaGrasp-fixConfig has a higher success rate, and it is most salient in the multi-object with obstacles setup (up to +23% improvement).", "Moreover, AdaGrasp-fixConfig demonstrates better generalizability when testing on new gripper hardware.", "Real-robot experiment Finally, we validate our method on a real-world robot platform with a UR5 robot and a calibrated RGB-D camera (Intel RealSense D415).", "Fig.", "REF shows the real-world setup and test objects.", "In this experiment, we directly tested AdaGrasp-fixGripper policy trained in simulation on four different physical grippers – WSG 50, RG2, Barrett Hand, and Barrett Hand-B, all of which are unseen during training.", "The test objects used in this experiment include 20 objects from YCB dataset [30] and five 3D printed adversarial objects from DexNet 2.0, all unseen during training.", "For single object tests, we place a single object randomly.", "For multi-object tests, we created 8 scenes each containing 4 randomly chosen objects and made sure that the placement of objects in 8 scenes is consistent across grippers for fair comparison.", "For each multi-object scene, we provide 7 attempts to a gripper for grasping objects.", "The grasp success rates are reported in Tab.", "REF .", "The average success rates for single object and multi-object are 86% and 80.5%, respectively, comparable with the algorithm performance in simulation.", "We noticed that unlike parallel jaw grippers, Barrett Hand and Barrett Hand-B have a curved grasping gait, i.e., fingers take a curved trajectory while closing in.", "Thus, the Barrett Hand cannot create contact at a smaller height and fails to grasp shorter objects like banana and adversarial objects.", "On the other hand, Barrett Hand is good at grasping bigger objects like big triangle or baseball ball, which are challenging for smaller grippers like RG2." ], [ "Conclusion and Future Directions", "We introduced AdaGrasp, a unified policy that generalizes to novel gripper designs.", "Extensive experiments demonstrate that AdaGrasp is able to improve the system's versatility and adaptability, and outperforms the current state-of-the-art multi-gripper grasping method.", "However, since our algorithm focuses on the gripper geometry for mechanical gripper, it does not extend to other gripper types (e.g., suction or deformable) and variable physical parameters (e.g.", "friction).", "It is also limited to top-down grasps due to the reduced action space.", "As future directions, it will be interesting to investigate larger range of gripper types in general dexterous manipulation.", "Grasp Success Rate of each gripper Table: NO_CAPTION" ] ]	2011.14206
[ [ "Unsupervised Spoken Term Discovery on Untranscribed Speech" ], [ "Abstract (Part of the abstract) In this thesis, we investigate the use of unsupervised spoken term discovery in tackling this problem.", "Unsupervised spoken term discovery aims to discover topic-related terminologies in a speech without knowing the phonetic properties of the language and content.", "It can be further divided into two parts: Acoustic segment modelling (ASM) and unsupervised pattern discovery.", "ASM learns the phonetic structures of zero-resource language audio with no phonetic knowledge available, generating self-derived \"phonemes\".", "The audio are labelled with these \"phonemes\" to obtain \"phoneme\" sequences.", "Unsupervised pattern discovery searches for repetitive patterns in the \"phoneme\" sequences.", "The discovered patterns can be grouped to determine the keywords of the audio.", "Multilingual neural network with bottleneck layer is used for feature extraction.", "Experiments show that bottleneck features facilitate the training of ASM compared to conventional features such as MFCC.", "The unsupervised spoken term discovery system is experimented with online lectures covering different topics by different speakers.", "It is shown that the system learns the phonetic information of the language and can discover frequent spoken terms that align with text transcription.", "By using information retrieval technology such as word embedding and TFIDF, it is shown that the discovered keywords can be further used for topic comparison." ], [ "Background", "For a number of decades, great efforts have been put toward developing computing systems that are able to recognize and understand human speech.", "The relevant technology is known as automatic speech recognition (ASR).", "State-of-the-art ASR systems are well developed for most of the major languages in the world.", "It can be arguably said that they are close to human performance in terms of recognition accuracy [1], [2].", "A typical speech recognition system has two key components, namely, acoustic model and language model.", "The acoustic model (AM) maps input speech signals to phonemes or other linguistic units.", "The language model (LM) governs how to derive a word sequence from a phoneme sequence.", "Both the AM and LM are in the form of statistical models or neural network models that are learned from data with properly represented contents.", "For an ASR system to achieve state-of-the-art performance, a large amount of training data are indispensable [3].", "To accomplish effective modeling of a given specific language, the training data must be well defined – with the lexicon information and word-by-word transcriptions being accurately provided.", "When such kinds of data or knowledge resources are not available, which is commonly known as the “low-resource” or “zero-resource” scenario, training a high-performance model remains a great challenge.", "Among the 7,000 languages in the world [4], the top 23 major languages are spoken by more than half of the global population.", "However, half of these 23 languages are still considered as low-resource languages as there are no well developed recognition systems at the moment due to limited data, hence the community has to look into alternative speech technologies which require less data [5], [6].", "Building up the data resource for a new language is not feasible in terms of the time, manpower, and linguistic expertise required.", "In the latest collection of Linguistic Data Consortium (LDC), only 102 languages are covered [7].", "Therefore, there has been increasing research interest in non-traditional speech modelling techniques.", "The Zero Resource Speech Challenges have been organized regularly since 2015 to encourage bench-marking and research exchange on spoken language technology for low-resource languages [8].", "The Low Resource Languages for Emergent Incidents Program (LORELEI) of DARPA in 2015 aimed at language-universal technology that does not rely on huge, manually translated, transcribed or annotated corpora, and is able to efficiently handle practical incidences in low-resource scenarios [9], [10].", "Research on low-resource languages can be categorized according to the following three assumed scenarios: A large amount of un-transcribed data is available with only limited transcribed data are available; Phonetic knowledge about the language is provided, but the available speech data are too little to training a statistical model; Phonetic knowledge about the language is not available, which is referred to as the zero-resource case.", "In the first scenario, a common approach is to locate a small subset of speech data that is informative and representative, e.g., containing typical content, good coverage of phonetic variations, and/or few confusing words.", "These data are then manually transcribed to facilitate so-called active learning [11].", "Another approach is semi-supervised learning, in which a seed model is first trained with a small set of transcribed data to learn the hypothesis of the language, it will then decode the transcription of all unlabelled data [12].", "In the case of limited transcribed data, transfer learning methods can be applied.", "The idea is to transfer linguistic knowledge from a high-resource language in processing the target low-resource language.", "Transfer learning models are trained with transcribed data from one or more high-resource language(s) and refined with the data from the low-resource language.", "When a deep neural network model (DNN) is adopted, the hidden layers are shared, and the softmax outputs represent phonemes of the languages separately.", "By joint training with high-resource data, the ASR system could achieve a better performance than training with limited low-resource data [13].", "In the extreme case that both phonetic knowledge and transcriptions are absent, unsupervised learning is needed to learn the constituting elements and structure of the language completely from audio recordings.", "Specifically, the elements to be learned could be subword units [14], word-like units [15], or phrase-level units [16].", "Automatic discovery of multi-word phrases has been receiving most interest over the past years [5], [17].", "For example, the MIT CSAIL group investigated methods of unsupervised pattern discovery on classroom lectures [16].", "Without requiring any prior linguistic knowledge, unsupervised learning methods can be applied to any low-resource language.", "They are also useful in a wider range of real-world applications that may involve popular spoken languages.", "These applications may involve multi-lingual, code-mixing, and/or accented speech that contain many colloquial terms and non-speech sounds, with unknown and complicated acoustic conditions.", "It is generally impractical and unnecessary to make effort on obtaining formal and accurate transcriptions for such kinds of speech data.", "Numerous studies have been done in this area [16], [18].", "The key technical problem is known as unsupervised acoustic modelling.", "Applications of unsupervised learning can also be applied to non-speech data.", "There are works on audio event detection that search for occurrence of pattern of automatically learnt acoustic units [19], music pattern analysis that learn the structure of music (e.g.", "ABAB) through pattern of music notes and can be applied to different genres such as jazz, classical, etc [20].", "Real world recordings are complex, with environments, acoustic elements and pattern durations that are changing and unpredictable from each of them.", "Traditional modelling methods have not considered enough variation of all the elements, and we do not have complete information yet, unsupervised learning can be considered to be a feasible approach.", "Moreover, considering rapid increasing of information on the Internet, nowadays it is easy to get access to several million terabytes of data [21].", "It is however impossible to apply traditional learning methods on these data as they are mostly unlabelled.", "Exploring the potential of unsupervised learning on these data has a great deal of implications as online data can be fully utilized, which is a better alternative than producing more labelled data.", "Through discovery of repeated data pattern, we can also avoid spending too much time in reading through every single bit of data.", "Summerization on the patterns can provide us useful information that can be represented in much less bytes." ], [ "Thesis objective", "One main interest of this thesis is to explore the use of unsupervised learning on audio that are extracted from Internet, as it is easier to collect data for analysis that are recorded in real world scenario.", "A system is built such that when raw audio recordings without any phonetic and transcribed information are provided as input.", "The system automatically learns the phonetic units of the language, then performs pattern discovery on the phonetic units to obtain repeated word phrases of the recordings.", "The word phrases are then compared across recordings for topic comparison.", "In the system, a bottom-up approach that contains different levels of unsupervised learning are researched and used as shown in Figure REF .", "The hierarchical structure is described as follow: Unsupervised feature extraction that learns to extract representative linguistic representations from zero-resource language.", "Unsupervised units discovery and segmentation that learn the linguistic units and the units boundary information.", "Unsupervised pattern discovery that discovers keyword phrases of each recording through searching for repeated unit sequences (patterns).", "Unsupervised topic comparison on each recording to determine how similar or different the recordings are.", "Figure: Bottom up approach of the fully unsupervised system." ], [ "Organization of Thesis", "The following chapters of the thesis will be presented as follow: Chapter 2 reviews on recent projects related to unsupervised acoustic modelling in the research field and their applications.", "Different approaches related to traditional acoustic speech recognition systems and the use of deep neural networks are also introduced.", "Chapter 3 introduces the first part of the system, unsupervised acoustic modelling, which discovers phonetic unit information of the recording.", "Approaches in extracting language independent features and different clustering methods are discussed.", "Chapter 4 introduces the second part of the system, pattern discovery of the unit sequences generated from the unsupervised acoustic model.", "The proposed metric, algorithm and their effectiveness in discovering patterns are discussed.", "Chapter 5 compares patterns discovered from opencourse lectures.", "Relationship of the patterns and lecture topics is also evaluated.", "Suggestions on potential applications in topic comparison are also given.", "Chapter 6 concludes the whole work and discusses the areas of improvement and future work." ], [ "Background", "This chapter provides the general background of research on spoken pattern discovery and reviews related previous studies.", "We will start by describing conventional automatic speech recognition (ASR) system design, and then focus on acoustic modeling in unsupervised scenario.", "Representative works on spoken term detection are also discussed." ], [ "Fundamentals of ASR", "Automatic speech recognition (ASR) is a technology that enables computers to analyze and convert sound waves of speech into text.", "A typical ASR system is trained to have the ability to map an audio input to a sequence of phonemes or words." ], [ "Probabilistic framework", "Let $O$ be an observed audio signal and $W$ be a word sequence or phoneme sequence.", "$P(W\|O)$ denotes the conditional probability of $W$ given $O$ , indicating how likely $W$ is the cause of $O$ .", "The goal of ASR is to determine the most likely word sequence $W^$ when the observation $O$ is given, i.e., $W^ = {arg\\,max}_W P(W\|O)$ Following the Bayes' Theorem, we have ${arg\\,max}_W P(W\|O) = {arg\\,max}_W \\underbrace{P(O\|W)}_{\\makebox{[}0pt]{\\tiny {acoustic model}}} \\overbrace{P(W)}^{\\makebox{[}0pt]{\\tiny {language model}}},$ in which the maximization is applied on two parts of models: Acoustic model (AM), which describes the mapping between acoustic observation $O$ to linguistic representation $W$ .", "$P(O\|W)$ measures the probability of $O$ being obtained when $W$ is spoken.", "Language model (LM), which represents the language rules/properties governing $W$ .", "$P(W)$ basically measures how likely $W^$ is valid in the language.", "The whole process of ASR is illustrated as in Figure REF .", "The acoustic observation $O$ is typically obtained from the raw audio via a feature extraction process.", "The extracted features $O$ are used as the input of the acoustic model to evaluate $P(O\|W)$ .", "Combining with the language model information, the most likely word sequence $W^$ is determined.", "Figure: The process of ASR." ], [ "Feature extraction", "Feature extraction is the first step of processing raw input speech, aiming to obtain a meaningful representation for subsequent modeling and evaluation.", "A general goal of feature extraction for ASR is to derive a low-dimension feature vector from each short-time frame of speech.", "The features are generally expected to be insensitive against changes of speaker and recording environment and be discriminative to phonemes [22].", "Typically, features at frame level are computed every 10 ms with an analysis window of around 25 ms [23].", "Mel-frequency cepstral coefficients (MFCCs) are by far the best known and most commonly used feature for acoustic modeling of ASR [24].", "Spectral analysis of input speech is applied on the Mel scale, which was inspired by human auditory perception [25].", "The Mel-scaled filter-bank comprises a number of triangular filters as shown in Figure REF .", "MFCC features are computed by taking Discrete Cosine Transform on the log power of these filters' output.", "Figure: Triangle filters that generate MFCC features.Another commonly used feature is known as the perceptual linear predictive (PLP) coefficients [26], which gives an estimate of auditory spectrum based on the concept of hearing psychophysics.", "The Fourier spectrum of speech signal passes through critical-band integration and re-sampling, and the result is multiplied with an equal-loudness curve and compressed with the power law in hearing.", "The inverse Fourier transform is applied to obtain the PLP coefficients.", "MFCC and PLP both provide a representation of smoothed short-term spectrum that is compressed and normalized in the same way as human auditory perception.", "Previous research showed that PLP is more robust to noise than MFCC [27]." ], [ "Acoustic model", "Acoustic model represents the relationship between the audio signal and its corresponding phonemes.", "It learns the relationship with statistical representations of sounds $O$ that make up the word $W$ .", "Given a word $W$ , its pronunciation is formed by sequence of phones $Q = q_1q_2q_3...q_n$ .", "The probability of observed sounds $O$ given $W$ is: $P(O\|W) = \\sum _Q P(O\|Q) P(Q\|W)$ The input audio is not limited to a single word, but can also be a sentence or paragraph.", "We denote the written form of corresponding words as transcription.", "To model the relationship of input audio and transcription, two problems of probability computation are needed: 1) transition probability from phone $q_i$ to $q_{i+1}$ , 2) the output observation probability from $O_t$ to $q_i$ at specific time $t$ .", "In the next section we will explain how the statistical relationships are learnt by a conventional acoustic model that can determine the possible transcription of future incoming audio signal." ], [ "Phone-based GMM-HMM", "As shown in Figure REF ), each phone $q_i$ is represented by a continuous density HMM with states $s_1,s_2...,s_N$ .", "At each time step $t$ , the HMM may make a transition from its current state $i$ to the next connected state $j$ .", "The transition probability from state $s_i$ to $s_j$ is denoted as $a_{ij}$ .", "At the same time, the observation probability $b_j$ is generated at state $j$ with a specific statistical distribution associated with the state.", "The state output distribution can be modelled with a mixture of Gaussians, providing a highly flexible distribution to model speaker, accent and gender difference.", "For the model to work properly, training is needed to adjust model parameters $\\lambda $ of GMM-HMM such that the input audio signal can align with transcription.", "Therefore, exact $O$ and $W$ have to be known for training the model.", "This is called supervised training.", "Expectation-maximisation (EM) can be used to search for suitable model parameters.", "It iterately calculates the likelihood of input-to-output-alignment given certain model parameters, and re-estimates the model parameters accordingly, until the parameters that give the maximum likelihood is reached, i.e.", "${arg\\,max}_\\lambda P(O\|W)$ .", "The trained acoustic model is then capable to process new audio data from the same domain as the training data and recognize the corresponding phones and words." ], [ "DNN based ASR", "The first attempt of using DNN in ASR was the phone recognition system reported in [28], [29].", "They demonstrated clearly better phone accuracy than well-tuned traditional GMM-HMM models.", "Since [30], DNN has been taking over GMM for high-performance acoustic modeling in ASR.", "Many new model structures have been developed, leading to significant and continuous performance improvement." ], [ "Basics of DNN", "The basic form of DNN is a multi-layer feed-forward network built with simple computation units called neurons.", "Each neuron performs computation of a simple function $y_j = f(x_j)$ , where $y_j$ and $x_j$ are the output and input of the neuron, respectively.", "In a feed-forward network, the input to a neuron at an intermediate layer is given as $x_j = b_j + \\sum _i y_iw_{ij},$ where $w_{ij}$ denotes the connection weight from neuron $i$ to $j$ , and $b_j$ is known as the bias of neuron $j$ .", "Training of a DNN refers to the process of determining the values of weights and biases of all neurons in the network.", "This is typically done by minimizing a loss function $L(.", ")$ that quantifies the discrepancy between the actual output of the DNN and the desired output.", "The minimization can be done by stochastic gradient descent, which is an iterative optimization algorithm similar to EM, the derivative of the loss function of each training example is back-propagated, and the values of $w$ are updated and fine-tuned until local minimum of the loss function is met.", "Essentially, the goal of DNN is to approximate non-linear function $f(.", ")$ that can produce $y$ from $x$ in $y = f(x)$ , by learning from the training examples.", "It is commonly used when the function is too hard to be formulated and understood." ], [ "DNN-HMM for ASR", "When the variations in the audio are too large and it is too hard to understand the relationship of the observed audio and states using GMM, DNN can be used to approximate the distribution instead.", "There is a variety of DNN models that have been applied to acoustic modeling.", "A brief review of these models is given below." ], [ "Model structure and training", "The first attempt of using DNN in ASR is deep belief network (DBN) in phone recognition [28], [29], which is a method in training DNN by stacking pre-trained narrow networks (Restricted Boltzmann Machine) together to make it “deep”.", "It replaces traditional GMM from GMM-HMM to DNN-HMM [3] and achieves phone accuracies that are higher than well-tuned traditional models.", "It is now therefore the basic and standard model used in DNN acoustic modelling.", "Compare with GMM, DNN does not require uncorrelated features such as MFCC, therefore other features such as filter bank (fbank) can be used in training DNN-HMM that give better representation to the speech data [31]." ], [ "Convolutional neural network (CNN)", "CNN is widely used in image processing [32].", "Different from normal DNN, activation functions are applied to the nodes that are fully connected.", "CNN also consists of convolution layers at the beginning of the network that convolute the input 2D image to the next layers.", "It also has pooling layers at the latter part that extract the maximum neighbour values to reduce layer resolution.", "In speech recognition, CNN replaces DNN to form a CNN-HMM, with input feature being the 2D spectrogram of the audio signal with frequency and time information.", "The special structure of weight sharing, pooling and local connectivity of CNN enables invariability to slight changes in speech features, making it better in dealing with speaker and environment variations [33]." ], [ "Temporal DNN", "In speech recognition, it is not only important to consider local region information, longer dependencies such as context, referencing from previously appeared words and language structures can also benefit the ASR training.", "This is especially important in time series applications such as speech, audio and video, compare to static data such as image processing.", "Several temporal acoustic models are therefore developed and become more widely used in speech recognition." ], [ "Time delay neural network (TDNN)", "TDNN is first introduced in the application of phone recognition [34].", "Concept of delay is introduced, extra weights representing the input delay are multiplied before computing the total weighted sum to the unit.", "With this design, the network is exposed to sequence of patterns and is able to relate and compare the current input with its past inputs, resulting in more powerful time series data processing ability.", "However, as the structure gets more complicated, the network becomes more complex as well.", "A small TDNN network can consists of several millions parameters and large amount of training data is needed [35]." ], [ "Recurrent neural network (RNN)", "RNN is a DNN with self-connected hidden layers, allowing it to has “memory” on its previous states in processing the input sequence.", "The self-connecting edges have extra weights that determine the importance of the previous unit's states.", "However, when processing long sequential data especially in speech recognition, it faces the vanishing and the exploding gradient problems [36] and therefore is not widely used until the extension to LSTM [37]." ], [ "Long short-term memory (LSTM)", "To solve the vanishing gradient problem, modification to RNN is made by introducing regulating cells that control the flowing of data and error [38].", "Input gate, output gate and forget gate are added to control whether the value should go into the unit, pass to the next unit or reset in the unit respectively.", "In speech processing applications, bidirectional LSTM (BILSTM) is used more often to consider both past and future events into account during sequence training [39], [40].", "Despite higher ability in relating events among the sequence and gives better performance in learning the phoneme sequences, it takes much longer time to train the network." ], [ "End-to-end speech recognition (E2E)", "End-to-end speech recognition trains the whole ASR with one neural network system.", "Training and optimizing acoustic model and language model separately will result in sub-optimal solution of the combined ASR.", "By training the whole ASR as one single system, a better decoding result can be achieved.", "Currently, E2E technique includes connectionist temporal classification (CTC), attention-based encoder decoder and hybrid of the two – attention-based CTC [2]." ], [ "Connectionist temporal classification (CTC)", "Temporal DNN acoustic model only determines the most possible phone of each utterance or frame, which is called framewise classification.", "However, CTC learns the probability of observing the corresponding labels at particular time.", "By multiplying the probability of each label at different time, possible sequence paths with their probabilities corresponding to the observed audio are obtained.", "The best representative phone sequence can be obtained by choosing the sequence path with the highest path probability [41].", "In practice, RNN and LSTM are used in constructing CTC due to their sequence considering property [2], [42].", "It has the same training objective as HMM but outperforms HMM [41].", "CTC represents both acoustic model and language model as one model, and directly searches for model parameters that give the maximum likelihood of the input-output mapping.", "It can be used as end-to-end model that learns the whole ASR to map audio to text without learning intermediate phonemes [43].", "However, since it is a complete end-to-end model, it is hard to interpret intermediate information such as phones." ], [ "Attention-based encoder decoder", "Attention is commonly used in sequence-to-sequence processing such as machine translation and natural language processing, it tells specifically which elements in the sequence should the model places more or less attentions when making decisions [44].", "Encoder-decoder model is used to tackle input and output with variable lengths [45].", "The encoder takes in input speech features and generates intermediate representations, and encoder takes in the representations to output the desired text sequence.", "Content-and-location-awareness is added into the attention mechanism to allow the model to output text with correct word order as the input speech [46], [47].", "While being so flexible, it is difficult to predict proper alignment due to the lack of left-to-right constraints." ], [ "DNN for feature extraction", "Besides acoustic modelling, DNN can be widely applied to other modelling techniques such as feature extraction and language modelling.", "It can also be used in domain adaptation with some modifications to the training process.", "When we do not have enough speech data for a specific language, known as low-resource language, it is hard for traditional DNN acoustic model to achieve satisfying performance when solely trained the low resource language compare to trained with rich resource languages.", "But recently, increasing effort has been put in developing methods to tackle the issues yielded by low resource language as the computation power improves significantly and more advanced technologies are discovered." ], [ "Knowledge transfer", "One method to deal with limited data is knowledge transfer, in which resource-rich data from another domain is used to assist the modelling of resource-limited data.", "In this session, some methods for knowledge transfer are introduced." ], [ "Knowledge distillation", "Another approach to limited data is knowledge distillation, also named as teacher-student model.", "After training classification model (teacher), a new model (student) is trained base on the posterior probability distribution output from the teacher.", "The student is expected to learn the decision boundary information from the teacher and therefore achieves performance as compatible as the teacher with much less model parameters [48].", "This training technique can be applied to model compression (from large model to small model with equal performance), model conversion (from statistical model to neural network model [49]) and domain adaptation (from one high-resource domain to another low-resource domain [50])." ], [ "Multilingual DNN", "Besides training two separate models, another popular method in recognizing low resource language is transfer learning.", "A model is first trained with language with sufficient data, and fine-tune with target language with limited data.", "Transcription is expected to be available.", "It is experimented that training with more languages can significantly improve the result, therefore multilingual training is more preferable nowadays [51].", "In general, this process is also called multi-task learning.", "When training with more than one language(s), the hidden layers of acoustic models of different languages are shared, only the input features and output softmax layers are language specific (Figure REF ).", "It is expected that the rich resource language can help to provide language information for understanding the low resource language better.", "Pre-training with more languages can help the model to learn more generalized language properties and avoid overfitting, and is more suitable to apply to a new language.", "Multi-task learning is a training technique which does not limited to DNN, either statistical models [52] or DNN [53] models including CNN, LSTM, etc.", "can be used." ], [ "DNN as feature extractor", "DNN can also serve as feature extractor.", "The training process is similar to a phone recognizer.", "After training a phone classification network, the speech is feedforward to the network and vectors are extracted from the layer before the softmax, which are the posterior probabilities of the input speech frames corresponding to the phonemes.", "Then they are used as features to train on a GMM-HMM acoustic model.", "It is shown that compared with traditional MFCC, the features generated learn information than can benefit phone classification [54]." ], [ "Bottleneck layer and feature extraction", "Besides learning the posterior probability vectors, when one of the hidden layer is narrowed and with linear activation function, it compresses the data that passes through the layer [55].", "Thus the DNN learns to forward the most important information to the next layers at the same time when solving the classification task without much result degradation [56].", "Features can then be extracted from the linear bottleneck layer by forwarding speech data to the model.", "The features generated are more data driven than in [54].", "Bottleneck layer can be placed at 1) center layer, 2) layer before the softmax output.", "It is better to place at (2) to learn representations that are most sensitive to phoneme variations [57], [58]." ], [ "Multilingual bottleneck network", "Multilingual network with bottleneck layer can extract language independent features for new input audio, preferably when each language is trained on separate softmax output layer [59].", "Figure: Multilingual DNN with bottleneck layerExperiment [60] has been done on investigating the learning effect of bottleneck feature from multilingual neural network, showing that even though the output phoneme labels of different languages are different, phonemes with same pronunciation are still projected to the same IPA symbolIPA (International Phonetic Alphabet) is a standardized representation of the sounds of spoken language, it is independent to any language area when represented as bottleneck feature.", "This concludes that multilingual bottleneck feature is able to learn phonetic information that are generalized and can represent the phonemes of a new untranscribed language.", "Even though the process of training a multilingual bottleneck network is supervised and transcription is required in training, when training with many languages, the network with the bottleneck layer inserted learns the general phonetic properties that can be applied to any language.", "Multilingual bottleneck DNN can then be used as a feature extractor to extract language independent features from unlabelled new language data [61]." ], [ "Acoustic segment modelling (ASM)", "Now we further look into the use of DNN in unsupervised acoustic modelling.", "When there is no linguistic knowledge nor transcription available, supervised acoustic modelling techniques are no longer applicable.", "One of the unsupervised acoustic modelling approach to tackle this scenario is acoustic segment modelling (ASM), it discovers possible phonetic units and build an acoustic model accordingly.", "ASM is first proposed in [62], aiming to learn a self-derived acoustic model for isolated word recognition.", "The trained ASM on word recognition is shown compatible to supervised acoustic model.", "This workflow then becomes standardized for all ASM architecture.", "Typically, ASM consists of three stages (Figure REF ): Initial segmentation that identifies the potential phonetic units and their segmentation information from the input speech.", "Since the phonetic units discovered are not guaranteed to be the real phones of the language, they are called subword units instead.", "Segment clustering and labelling that groups segments into clusters of subword units and labels the speech with corresponding subword units.", "Iterative training of acoustic model with the discovered subword units.", "Figure: Components of Acoustic Segment Modelling (ASM)" ], [ "Segmentation", "The first step for discovering phonemes from the speech is to identify all the potential phonetic unit boundaries, which is called segmentation.", "This process is relatively easier but is also very important.", "After segmentation, segment information such as durations, locations and segment features can be obtained for clustering into subword groups.", "Maximum likelihood segmentation: In [63], number of segments can be determined based on the spectral distortion.", "The segment boundaries can be obtained by minimizing the overall likelihood distortion using dynamic programming based Maximum likelihood (ML) segmentation.", "Dynamic programming algorithm (DP) [64]: Statistical models for the speech data are built to model speaker, channel and speech information similar to an acoustic model.", "Then dynamic programming approach is used to identify the most probable segmentation.", "Maximum-margin segmentation [65]: Given the frame level feature vectors, maximum-margin clustering searches for the boundaries such that the margins between segments are maximized when grouping the frames into segment clusters.", "Bottom-up hierarchical clustering: In [63], time-constrained agglomerative clustering algorithm is used to find the optimal segmentation.", "It begins by treating each frame as an initial cluster and merges these clusters into larger segment clusters until the terminating criteria is met.", "Maximum spectral transition [66]: Phoneme can be analyzed in spectral space, maximum spectral transition can be used to determine the phoneme boundaries with the elimination of those with too short intervals.", "Graph-based observation space: The model in [67] takes in features represented by observation space, such as graph or network, instead of temporal sequence space such as MFCC.", "This provides better information on the change of phonetic properties for segmentation.", "Boundaries are represented as arcs on the graph.", "Nonparametric Bayesian model: Also name as Hidden Markov Model with Dirichlet process priors.", "A Dirichlet process (DP) is a discrete distribution of weighted sum of impulse functions.", "It is often used in Bayesian inference.", "[68] uses nonparametric Bayesian model with Dirichlet process priors to segment the utterances, and uses Gibbs sampler to estimate the segment boundaries.", "In [69], hierarchical Dirichlet processes (HDP) is used, which is HMM with unbounded number of states to segment the utterances.", "Recognizers from other languages: In [70], the untranscribed speech is decoded with language mismatched phoneme recognizers which are trained with high resource languages.", "The segment boundaries produced by different recognizers are merged in form a single set of boundaries.", "The frame-level features in the same segments are averaged to form segment-level features." ], [ "Clustering/Quantization", "With segment-level boundaries available, segmental features can be obtained by combining the frame-level features within the boundaries.", "Segmental features are then grouped into subword clusters that are acoustically similar.", "The subword clusters are then labelled to the speech to generate initial segment sequence for subsequence segment modelling stage.", "Lloyd algorithm [62]: Lloyd algorithm is often used in vector quantization.", "Cluster centroids are computed using a segment codebook.", "The goal of the segment codebook is to generate a set of vectors such that the accumulated segment distortion is minimized.", "Segments are assigned to their nearest cookbook entry to form groups of segments, then the distortion is minimized by updating centroids of the segment groups.", "The process is iterated until converge.", "Gaussian component clustering [71]: A GMM is trained on frame-level features, with the number of Gaussian components set to be the desire subword units.", "Clustering is then performed on the Gaussian components.", "Each cluster is then a small GMM of a subword unit, and the clusters can be used to score the speech segments.", "Segments are labelled with clusters of highest scores to generate the label sequence.", "Segmental Gaussian Mixture Model (SGMM) [72]: Different from GMM, each term in an SGMM is a Gaussian whose mean is a vector trajectory in the cepstral feature space that varies over time to represent time varying characteristics of a sound.", "Each segment is fitted with polynomial trajectory model.", "The pairwise distances between segments are calculated for clustering by binary centroid splitting algorithm.", "The clusters are then used as the basis for generating SGMM.", "SGMM is trained with EM algorithm and then the raw audio is labelled into initial label sequences for the next step.", "Spectral Clustering: With the class-by-segment posterior probabilities generated by recognizers or GMM models, spectral clustering can be used to cluster the speech segments, e.g.", "k-means clustering [73].", "If there are more than one set of segment posterior representations available, multiview spectral embedding can be used to embed the multiple representations into single posterior representation [74], and perform spectral clustering on the embedded vectors." ], [ "Iterative modelling", "After obtaining all the initial segment information and segment clusters.", "ASM is trained iteratively to learn the finalized segment boundaries and clusters.", "Although different models can be used, the training process is consistence, where the speech audio is labelled with initial segment clusters, and trained with the ASM.", "The ASM is then decoded with the same set of audio.", "The decoding result is used as input labels again.", "The training process repeats until the training criteria is met.", "First work in ASM [62] used HMM for iterative modelling, the concept of iterative training is the same as training an acoustic model with the initial segmentation and labels.", "Therefore any statistical model in Section REF and DNN architectures in Section REF can be used for iterative modelling." ], [ "Unsupervised word discovery", "Spoken term detection indexes speech based on the content efficiently.", "It aims at locating spoken terms that appeared in speech, especially if the speech is related to specific topics such as meetings, lectures and conversations.", "Examples of traditional spoken term detection are: spoken term detection using Large Vocabulary Continuous Speech Recognition (LVCSR), acoustic based keyword spotting, and query by example (QBE).", "They require the model to be trained in supervised manner and the target spoken terms are known [75].", "However, in the problem of zero-resource language, there is zero understanding to the language, not to mention knowing the spoken terms we are looking for.Spoken term discovery (STD), is a completely unsupervised method that exploits repeating patterns in the speech signal.", "There are two main approaches in spoken term discovery, one combines the work of developing ASM follow by STD, another one is an integrated STD system.", "In an ASM-STD model, there are 2 main approaches as well: 1) query by example using template matching to discover spoken terms and 2) direct clustering of subword sequences into spoken terms." ], [ "Query by example using template matching", "When both transcription and target keywords of the speech are unavailable, templates are learnt in unsupervised manner for QBE.", "The system first discovers all the possible spoken terms from the speech and saves them as templates.", "The templates are then compared with the speech to discover repeating segment sequences, which are the spoken terms in the recordings." ], [ "Segmental DTW for template matching", "Segmental Dynamic Time Wrapping (DTW) is widely used in template matching, it is similar with frame-level DTW, despite the keywords are compared in segment level to provide more efficient computation [76].", "It scores the similarity of the two sequences.", "One example of using segmental-DTW in QBE with template matching is BBN's work [77].", "They named the subword units discovered using HMM iterative model as self-organized units (SOU) and divided the pattern discovery process into 3 stages: SOU template discovery, template organization and audio segment clustering.", "With the subword units generated by HMM model, SOU sequence can be generated by searching for the 1-best path decoded from the HMM.", "SOU templates can be located by searching for common SOU n-grams from lattices.", "They can be organized by merging similar templates into same groups.", "With the templates, audio segment clustering is done by comparing SOU lattices that match the templates.", "Besides the lattices of SOU, segmental DTW can be applied on different segment representation sequences, such as segments represented by spectrogram [76], posteriorgrams generated from ASM or Gaussian mixture model [71], spectrograms image that capture temporal, frequency and energy information [78]." ], [ "Other approaches", "A sliding window with similar length is used to compute segment features from the sequence.", "The features are then trained with positive and negative examples using SVM [78].", "However, each example requires a SVM classifier and performance decreases with increasing number of keywords." ], [ "Direct clustering of subword sequences", "Instead of discovering the templates for segmental DTW, another approach is to directly cluster all the discovered subword sequences into sequence clusters by grouping similar sequences.", "Each cluster is expected to correspond to a specific spoken term." ], [ "Local alignment with graph clustering", "Alex and James first proposed unsupervised pattern discovery, which applies local alignment follow by graph clustering on subword sequences of recording to discover acoustic patterns [79].", "Local alignment is a modification of segmental DTW, introducing shape constrain and different starting points for comparison.", "Different from segmental DTW which aligns the two complete sequences, local alignment tries to locate matching subsequences within two segment sequences.", "After obtaining the subword subsequences, graph clustering is used to cluster the sequences into clusters.", "The segment positions and their similarities are formulated as graph, the nodes represent the segment locations in time and the edges represent the similarities between the nodes, edges with values larger than the threshold are removed to form clusters of segments.", "The spoken terms are obtained from the finalized clusters obtained by Newman algorithm.", "Clustering techniques are not limited to graph clustering mentioned above.", "Once the subword subsequences are obtained, other clustering techniques such as those introduced in Section REF can be applied." ], [ "Integrated STD", "Besides combining different subsystems to discover spoken terms, there are also models that directly search for the common spoken terms on word-segment-based instead of subword-unit-based.", "Arbitrary-length word segments are embedded to fixed length vectors that facilitates the clustering and topic classification process.", "Bayesian GMM model [80]: A single Bayesian GMM model is used to learn the best segmentation and discover spoken terms through iterative modelling.", "Syllable boundary detection is used to determine all the likely word boundaries.", "Then segmental features are extracted using correspondence autoencoder.", "The features are clustered and trained by the Bayesian GMM model.", "Re-segmentation, feature extraction and re-clustering of segments are done based on the performance of the currently trained model until the configuration that gives the optimal performance is reached.", "K Nearest Neighbour (KNN) clustering [81]: Speech is pre-segmented into possible terms and fix-length term-embedding is applied to produce fix-length vectors of the segments.", "KNN instead of DTW is used to search for common segments.", "Clustering is then applied to group discovered segments into spoken terms.", "Embedded segmental k-means model [82]: There is also work on embedded segmental k-means model, which is very similar with k-mean clustering in learning the vector representations that group acoustic similar segments together." ], [ "Applications of unsupervised spoken term discovery", "Spoken term discovery has been applied to different types of speech, besides corpora that are formally recorded, it can also be applied to real word recordings such as lectures available online [79], child speech [83] and broadcast news [84].", "In terms of applications, it can be used for topic comparison [72], language structure understanding [15] and topic discovery [85]." ], [ "Summary", "In this chapter, we discussed the fundamental of acoustic recognition system, how a conventional acoustic model is trained in supervised manner and how deep neural network technology can be applied.", "However, in the scenario of zero resource language, without the transcript and pre-defined phonetic units for training.", "It is impossible to train an acoustic model with supervised methods.", "The more fundamental challenge to this problem becomes: how can we understand and process an untranscibed speech without any manual interpretation?", "This can be further divided into 2 main sub-problems.", "How can phonetic units be discovered from the speech, such that the process of training an acoustic model based on the discovered units can well represent the untranscribed speech.", "With the speech labelled based on the phonetic units discovered in (1), how can the content of the speech be interpreted?", "How can the topics be discovered and categorized?", "These are what acoustic segment modelling is trying to tackle, it focuses on clustering speech segments into potential phonetic units (subword units), and transcribes the speech according to the phonetic information discovered in unsupervised manner.", "Spoken term discovery then discovers content-related terms from the labelled unit sequences for analyzing.", "In the following chapters, we will discuss how subword units can be discovered using various clustering methods.", "A suitable feature that can well-represent the speech frames for clustering is also investigated.", "After obtaining descent transcription formed by the discovered units, spoken term discovery method is investigated." ], [ "Acoustic Segment Modelling", "Acoustic segment model (ASM) is one of the main approaches to unsupervised acoustic modeling in the absence of speech transcription.", "It involves three sequential steps: initial segmentation, segment clustering and labelling, and iterative model training.", "Since the approach is totally data-driven without requiring any prior knowledge about input speech, effective feature representation plays a vital role in determining the system performance.", "In the present study, a multilingual DNN is trained to serve two purposes.", "On one hand, it is used to perform phone recognition from which an initial segmentation of input utterance can be obtained.", "On the other hand, bottleneck feature (BNF) representations extracted from the DNN are used for segment clustering." ], [ "Multilingual DNN", "As discussed in Chapter 2, a DNN can learn linguistic knowledge from one language and apply to another language.", "This approach could be exploited to achieve knowledge transfer from one or more resource-rich languages to a low-resource target language.", "In this section, two different structures of multilingual DNN are investigated.", "They are namely multilingual DNN with bottleneck layer (Multilingual DNN-BN) and multilingual DNN with stacked bottleneck layer (Multilingual DNN-SBN).", "The DNNs are trained by multi-task learning strategy with a number of existing speech corpora." ], [ "Multilingual DNN with bottleneck layer", "As shown in Figure REF , the Multilingual DNN-BN consists of 4 hidden layers.", "One of the hidden layers has a small dimension of 40, which is a linear transformation layer named the bottleneck layer.", "The other hidden layers all are of dimension 1500.", "The output layer contains 5 blocks, each corresponding to one learning task.", "In this study, 5 existing speech corpora from 4 languages are applied to formulate the 5 learning tasks.", "Details of the speech corpora are given as in Table REF .", "These corpora have similar characteristics, in terms of speaking style, speaker population, and channel condition (e.g., sampling rate of 16 kHz).", "The four languages are selected with considerations on providing a good phonetic coverage and diversity, such that the DNN can learn better features for acoustic modeling of a new language.", "Table: Information of the corpora usedFigure: Architecture of the Multilingual DNN-BN with 5 learning tasksFor each of the 5 learning tasks, a context-dependent GMM-HMM acoustic model is trained in a supervised manner with the respective speech corpora.", "Training of the CD-GMM-HMM follows standard Kaldi recipe, i.e., (1) 13-dimension MFCC features extraction; (2) training a monophone model; (3) training a triphone model with delta features, followed by delta-delta features; (4) triphone HMM model trained with MFCC features transformed with Linear Discriminant Analysis (LDA) and Maximum Likelihood Linear Transform (MLLT); (5) speaker adapted training (SAT), which the model is trained on feature-space Maximum Likelihood Linear Regression (fMLLR) adopted features.", "Supervised training of Multilingual DNN-BN is carried out with the training data from all of the 5 speech corpora and the state-level time alignment produced by the language-specific CD-GMM-HMM.", "Input features to the DNN cover a contextual window of 11 frames and each frame is represented by 23 Mel-scale filter-bank coefficients." ], [ "Multilingual DNN with stacked bottleneck layer", "Previous research shown that bottleneck features extracted from a DNN can be used as a compact and informative speech representation.", "In order to better capture temporal dependency in speech, bottleneck features extracted from a multilingual DNN can be stacked across a certain number of time frames and applied as input features to another DNN [91].", "This architecture, named Multilingual DNN-SBN, is illustrated as in Figure REF .", "The first DNN adopts similar structure and training strategy to Multilingual DNN-BN as described in the last section, except that the dimension of bottleneck layer is changed to 100 to include more information for training the second DNN.", "The input to the second DNN is obtained by stacking the 100-dimension bottleneck layer output of the first DNN over a contextual window of $\\pm 6$ frames.", "The second DNN uses a narrower bottleneck layer of 40 dimensions, and the training procedure is similar to the first DNN.", "It takes in all the 5 corpora for training.", "The 40-dimension bottleneck features from the second DNN are named SBNF (stacked bottleneck features).", "Figure: Multilingual DNN with stacked bottleneck layer." ], [ "Phone recognition with the multilingual DNN", "In this section, the two multilingual DNN models are evaluated in the task of phone recognition.", "They are compared with conventional monolingual acoustic models tested on 2 of the mentioned speech corpora.", "All acoustic models are trained in a supervised manner.", "All 5 corpora are used for training the multilingual DNNs, while only WSJ and CUSENT are used for training monolingual acoustic models.", "The test sets for both monolingual and multilingual DNNs are from WSJ and CUSENT.", "For the monolingual acoustic models, the following model configurations were implemented and evaluated: CD-GMM-HMM The setting is the same as the CD-GMM-HMM described in section REF .", "Subspace Gaussian Mixture Models (SGMMs)-HMM [92] SGMMs represent the parameters of each state GMM as a vector mapping the stacked vectors to a subspace.", "Traing procedure: (1) fMLLR adopted features are used; (2) a trained GMM-HMM is available; (3) the Gaussians are clustered to initialize Universal Background Model (UBM); (4) SGMM is initialized with states’ pdfs equivalent to UBM and phone alignments obtained from GMM-HMM; (5) the SGMM-HMM is trained using EM algorithm.", "DNN/DBN-HMM The DNN is trained using pre-trained DBN layers.", "The DNN has 6 hidden layers, each layer with dimension of 1024.", "The DNN is trained by first pre-training 6 individual DBN layers.", "Then the DBN layers are stacked to form an initial DNN structure.", "The DNN is then fine-tuned iteratively.", "Each of the monolingual acoustic model is trained and tested with one same language only.", "For each model, two ASRs are trained and experimented with two corpora separately, one with WSJ and another one with CUSENT.", "The corpora are split into training sets and test sets, training sets are used to train the ASRs, while test sets are used for evaluation.", "Training procedure of Multilingual DNN-BN and Multilingual DNN-SBN is the same as in Section REF and REF .", "The same 5 corpora are used for training.", "The same test sets from CUSENT and WSJ used in evaluating monolingual ASRs are used to evaluate the multilingual ASRs.", "The performances of ASRs are measured in terms of phone accuracy in decoding the test set.", "The phone error rates attained with the monolingual models and the two multilingual DNNs are compared as in Table REF .", "It is noted that multilingual training really does help improve the performance of acoustic models.", "The monolingual tasks of both WSJ and CUSENT are considered to represent rich-resource training, and the achieved recognition performances are at a fairly high level.", "Nevertheless, by leveraging speech data from other languages, the multilingual DNNs were able to further reduce the phone error rate for WSJ.", "For CUSENT, multilingual DNNs do not perform as good as monolingual DNN, though they are better than all the GMM-HMM based models.", "This may probably due to the loss of certain language specific information, which are learnt in monolingual ASRs trained with resource-rich data.", "It is also seen that the Multilingual DNN-SBN performs better than multilingual DNN-BN, at the cost of significantly longer training time.", "Table: Performances on WSJ and CUSENT corpora using different models" ], [ "Evaluation of multilingual bottleneck features by visualization", "Given a trained multilingual DNN, frame-level bottleneck features could be computed from input speech of any language.", "In this section, we examine the effectiveness of multilingual bottleneck feature in unsupervised segment modelling of a new language and make comparison with conventional MFCC features.", "We trained several DNNs and tested them with a different language to evaluate if the BNF extracted from DNNs are effective in unsupervised phone catagorization.", "Different types of BNFs are extracted from multilingual and monolingual DNNs.", "The DNNs are first trained in supervised manner, BNFs are then obtained by decoding a corpus with language different from training corpora.", "The DNNs experimented and their training procedures are described as follow: A single language DNN-BN With the same architecture as in section REF , expect it is trained on one corpus only: CUSENT (Cantonese).", "Multilingual DNN-BN With the same architecture and training procedure as in Section REF .", "It is trained with all 5 corpora.", "Multilingual DNN-SBN With the same architecture and training procedure as in Section REF .", "It is trained with all 5 corpora.", "After training the DNNs, the DNNs take in MFCC features of the new language.", "Here, telephone conversation corpus named Callhome Spanish [93] is used.", "The corpus has complete new language that the DNNs haven't seen before, it also contains speaking tone, recording channel and sound events (e.g.", "laughter) much different from the training corpora.", "It is expected to evaluate the models' ability in processing new data recorded in different scenario.", "The layers beyond the bottleneck layer are removed, and the bottleneck layer output is extracted as BNFs.", "40 dimension BNFs extracted are compared with 13 dimension MFCC features of the same corpus.", "The features are evaluated in frame-level.", "The phones of the test set are categorized into 6 phonetic groups, represented with different colours shown in Table REF .", "We visualized the frame-level features in 2 dimensional space using PCA [94] follow by t-SNE [95] in Figure REF .", "t-SNE maps high-dimension data to lower dimension in non-linear way.", "t-SNE may have its limitations.", "in which the mapping may be data sensitive and the data sizes and distances cannot be well illustrated.", "However, through visualization, it gives us some insight on whether it is comparatively easier to cluster similar data points together unsupervisedly, such that we can understand which features can facilitate segment clustering more.", "Table: Phone groups and their corresponding colourThe features are analyzed based on phone separability.", "By separability we mean how well data points from same phone groups are projected to same regions, and how well data points from different phone groups are separated.", "It is expected that if it is easier to observe features group by group in t-SNE and there exist clear boundaries to separate frames from different phone groups, the mapping has higher separability and it is more likely to obtain subwords that can well represent the actual phones in segment clustering.", "The visualization is presented in Figure REF .", "Even though BNF obtained from monolingual DNN has better phone separability than MFCCs, frames from same phone groups are still split into different regions.", "It may due to the relatively large extend of phonetic difference between Cantonese and Spanish.", "BNFs obtained from multilingual DNN achieve better separability than monolingual DNN on vowels, fricatives and plosives.", "BNFs obtained from Multilingual DNN-BN have slightly more balanced distribution of phone classes, while BNFs obtained from Multilingual DNN-SBN have slightly better separability.", "Overall, compare with conventional feature, BNFs from Multilingual DNN have better generalization ability on phones and provide better representations that group frames of same phone classes together, making it suitable for unsupervised segment clustering.", "Figure: Stacked BNF from Multilingual DNN" ], [ "Evaluation of multilingual BNFs on minimal-pair ABX task", "The multilingual BNFs are evaluated on the task of unsupervised subword modelling in the ZeroSpeech 2017 Challenge Track 1 [96].", "The challenge covers three languages for development and evaluation, i.e.", "English, French and Mandarin.", "The minimal-pair ABX discriminability metric is used to measure the quality of learned feature representations.", "It involves three stimuli A, B and X.", "A and B are a pair of triphone segments with minimal segmental difference, e.g., “beg” vs.“bag”, , “api”vs.“ati”, and X is a speech segment that contains either A or B.", "In the within-speaker condition, A, B and X are spoken by same speaker.", "In the across-speaker condition, A and B are spoken by the same speaker, while X is from a different speaker.", "BNFs are extracted with the multilingual DNN-BN and the multilingual DNN-SBN as described in Section REF , with the same training data and procedures.", "The performance of multilingual BNFs is compared with the best submitted systems in the ZeroSpeech 2017 Challenge, and in the relevant works of Heck et al.", "[97] and Chen et al.", "[98].", "In [97], the Dirichlet process Gaussian mixture model (DPGMM) was applied to cluster speech feature vectors into subword classes.", "The input features were processed by speaker adaptation in a multi-stage clustering framework.", "In [98], DPGMMs were used to cluster the unlabelled speech into subword units.", "The input MFCC features were processed by vocal tract length normalization (VTLN).", "A multilingual DNN with bottleneck layer was trained on the subword units to extract feature representations.", "Our method is different from the above studies in that out-of-domain transcribed data is employed to facilitate supervised DNN training.", "The similarity between a pair of speech segments is computed as the average frame-level cosine distance averaged along the optimal frame alignment obtained by dynamic time warping.", "The ABX discriminability is given by the error rate, which is the mean over all possible minimal difference triphone pairs in the test set.", "The within-speaker and across-speaker minimal-pair ABX discriminability of different systems are compared in Table REF and REF respectively.", "The baseline system is based on standard MFCC+$\\Delta $ +$\\Delta \\Delta $ features.", "In most cases, BNFs from multilingual DNNs could achieve performance comparable to or better than the best system in the Challenge.", "With similar system structures, the proposed features give better performance than those reported in [98].", "This indicates that, despite he mismatch between training data and evaluation data, increasing the variety of out-of-domain training data is beneficial to improving the quality of BNFs.", "Even though French is a language not involved in the training of multilingual DNNs, the learned features are capable of representing this language well.", "It is also noted that BNFs extracted from multilingual DNN-SBN achieve lower ABX discriminability across speakers, while BNFs extracted from multilingual DNN-BN achieve lower ABX discriminability within speakers.", "The reasons for such incoherent performance need further investigation.", "Table: Results of minimal-pair ABX discriminability across speakersTable: Results of minimal-pair ABX discriminability within speakers" ], [ "Segment clustering", "The multilingual DNNs described in the previous section can be used to decode and segment speech utterances from a new language.", "As shown in Figures REF and REF , the DNNs use 5 blocks of softmax output layers that correspond to different training tasks.", "Given an input utterance, 5 different sets of phone-level time alignment can be obtained.", "By contrasting and combining these multilingual phone boundaries, an initial segmentation of the utterance can be derived by simply merging boundaries that are within an interval of 20 ms.", "Subsequently segment-level feature representations are obtained by averaging frame-level bottleneck features within the same segment according to the boundary information.", "Segment-level features can be computed in different ways besides averaging.", "The next problem is to automatically “discover” a set of segmental units, similar to “phonemes” or “subword units”, by applying clustering algorithms to the initial segments.", "The major challenge in clustering tasks is that the process is highly data sensitive.", "Different clustering algorithms are suitable only for specific data structures and problems [99].", "Considering a 40-dimension BNF obtained from the multilingual DNN, an important process is to understand the structure such that a good clustering method for the problem can be identified.", "Since speech applications typically involve a large amount of data, computational cost and efficiency need to be considered carefully.", "Traditional clustering algorithms can be divided into different categories, based on approaches of partitioning, hierarchical structure, fuzzy theory, distribution, etc.", "Neural network based clustering algorithms are also proposed in recent years [100].", "In this section, a few commonly used clustering algorithms are considered in segment clustering of BNFs.", "These algorithms include: $K$ -means clustering; Hierarchical clustering; Gaussian mixture model; Density-based spatial clustering of applications with noise (DBSCAN)." ], [ "$K$ -means clustering", "$K$ -means clustering [101] aims to partition observation data into $k$ clusters.", "As a result, each data sample is assigned to its closest cluster according to a prescribed distance measure.", "With $n$ observations $\\mathbf {x}_1, ... ,\\mathbf {x}_n$ , k-means clustering partition the samples into $k$ clusters $S = {S_1, S_2, ... , S_k}$ such that the sum of errors between the samples in the same clusters with cluster means $\\mu = \\mu _1,\\mu _2,..., \\mu _k$ are minimized: ${arg\\,min}_{s} \\sum ^k_{i=1} \\sum _{\\mathbf {x} \\in S_i} \|\|\\mathbf {x} - \\mathbf {\\mu _i}\|\|^2$ $K$ -means clustering is suitable for well-separated data classes.", "However there may not have a clear-cut boundary in BNF, especially when there are phones with very similar phonetic properties.", "Also the suitable number of clusters $k$ is not known.", "It is difficult for $k$ -means clustering to find a good cutting boundary and is hard to analysis the quality of clusters obtained without knowing the uncertainty of a sample in a cluster." ], [ "Hierarchical clustering", "Agglomerative hierarchical clustering (AHC) [102] clusters the samples with bottom-up approach.", "It starts by treating each data point as an initial cluster, and subsequently merging pairs of most similar clusters.", "The clustering result can be presented in a dendrogram, which allows visualization of inter-cluster similarity at different stages.", "In AHC, there are two main factors to be considered: a distance metric to determine the similarity between clusters, and a linkage criteria to determine if a pair of clusters should be combined.", "Bottleneck features are high-dimension continuous-valued data.", "Euclidean distance is a commonly used metric.", "Some commonly used linkage criteria are presented in Table REF .", "Table: Some of the linkage criteria used in AHCTo understand the properties of segment-level BNF and determine a suitable linkage criteria, let us inspect and compare the dendrograms produced by three different linkage criteria, namely centroid-linkage, median-linkage and Ward's method, as in Figure REF .", "The input data comprises 1000 frame-level BNFs.", "Centroid and median linkages are very similar as they merge clusters base on the smallest distance between the clusters.", "Ward's method attempts to maximize inter-cluster distance and minimize intra-cluster distance.", "Figure: WardThe dendrograms show that clusters formed by centroid and median linkage tend to be highly unbalanced.", "Some of the clusters contain a very large number of data points while some have very few.", "This is undesirable as we expect that phonetic units in a language should be relatively balanced.", "Ward's method is found to produce better balanced clusters.", "Therefore, Ward's method will be used for segment clustering.", "Although AHC provides very detailed information on the clustering process and distribution in readable form, it requires very large computation time of $O(n^3)$ and memory size of $O(n^2)$ for $n$ samples.", "This makes AHC impossible to apply on large dataset.", "Therefore, a hybrid use of two clustering methods is suggested: AHC can be first applied to a subset of manageable data to determine a decent number of clusters, such that other clustering methods with higher efficiency but less determining information can be used, such as K-means clustering." ], [ "Gaussian mixture model", "Gaussian mixture model is a soft clustering technique, where data samples are not forced to have a unique cluster identity.", "Each data sample is associated with a set of probabilities that indicates its likeliness of belonging to different clusters.", "GMM is a probabilistic model that assumes the data points are generated from a mixture of Gaussian distributions.", "In GMM based clustering, the number of components of GMM $K$ is fixed for clustering.", "It uses the EM algorithm to iteratively search for the parameters of the GMM that maximize the likelihood of data.", "The trained model then assigns each sample to the Gaussian it most probably belongs to." ], [ "Bayesian GMM", "The limitation of GMM is that the number of mixture components is unknown.", "Instead of assigning data samples into pre-specified numbers of clusters and searching for the optimal number through multiple trials, the number of components can be found by maximizing the likelihood of parameters of the GMM model using EM algorithm.", "However, the computation required for maximizing the likelihood is large, especially when number of data samples $N$ is large.", "In nonparametric Bayesian approach [103], [104], dirichlet distribution is used to approximate the posterior distribution of GMM parameters.", "With the upper bound of maximum number of clusters $K$ being defined, the parameters are initialized using k-means, then variational inference (which is an extension of EM algorithm) is used to iteratively update the parameters until convergence.", "The effective number of clusters eventually can be less then K when the weights of unrepresentative clusters become 0 during inference." ], [ "Density-based spatial clustering", "In density-based spatial clustering, data points are grouped based on density.", "Points that lie in high density regions are regarded as clusters, scattered points are regarded as noise.", "Density-based spatial clustering of applications with noise (DBSCAN) is a representative non-parametric method of clustering [105].", "Data samples that are closely packed are grouped as clusters.", "The number of clusters and the size of each cluster are determined by two parameters: radius $r$ and minimum number of points $minPt$ .", "If a data point has $minPt$ or more neighboring points within radius of $r$ , it is called “core point”.", "The points that are reachable from a core point within radius of $r$ are included in the same cluster.", "The searching process continues at these neighbour points, treating them as the centers and searches for their neighbouring points within radius $r$ .", "Searching is iteratively done until all the reachable points are covered.", "Reachable points are assigned to clusters that contain their core points.", "Points that are not reachable are defined as outliers.", "This method is efficient in terms of time and memory complexity.", "One limitation of DBSCAN is that the suitable value of $r$ is not known if we are not familiar with the data.", "The problem is especially harder for high-dimension data.", "Also, DBSCAN is not able to handle data with uneven densities.", "An extension of DBSCAN was proposed in [106], which is known as the hierarchical DBSCAN (HDBSCAN).", "The HDBSCAN operates in a hierarchical manner.", "In order to deal with data with different regional densities, the data is represented by a graph, in which data points are connected by edges.", "The weight of an edge is the reachable distance, i.e.", "the minimum radius such that the point can be assigned to a cluster.", "The radius $r$ increases at each level, from bottom to top, and edges with weight values smaller than $r$ are removed.", "The clustering result can be represented by a dendorgram.", "The problem of our concern is about clustering frame-level features into segments.", "Therefore, instead of keeping the outliers outside the clusters, our goal is to assign every speech frame a corresponding subword label.", "After the HDBCAN is applied, the distances between outliers and core points are computed, and all outliers are assigned to their closest clusters." ], [ "Iterative modelling", "With all speech frames of an audio assigned to clusters, we obtain an initial hypothesis of segment sequence.", "If each segment cluster is treated as a subword unit of speech, the segment sequence could be considered as a pseudo transcription, which could be used for supervised training of acoustic model.", "An acoustic segment model can then be trained using the pseudo transcription.", "This leads to a process of iterative modeling, as described below." ], [ "Training procedure", "DNN-HMM acoustic models [3] are trained to represent the learned subword units, i.e., segment clusters.", "The model training is done in an iterative manner with continuous updating of the pseudo transcriptions.", "The step-by-step procedures are elaborated below: Train an initial set of DNN-HMM acoustic model with the pseudo transcriptions obtained from segment clustering; Decode the training data with the current acoustic model and obtain updated pseudo transcriptions; Train the acoustic model with the updated transcriptions; Repeat Step REF ) and REF ) until convergence.", "The iterative training is carried out with speech data from the target language.", "In this way, acoustic model and pseudo transcriptions are jointly optimized for the target language.", "After terminating the training, the final version of pseudo transcriptions, in the form of subword unit sequences, are used for keyword discovery." ], [ "Models", "Iterative modeling can be applied to all conventional acoustic models, e.g., HMM, GMM, DNN.", "In this study, the most basic DNN architecture is adopted.", "A 6-layer DNN with 1024 nodes per layer is trained.", "The input features are frame-level BNFs.", "The subword units being modeled correspond to the segment clusters obtained as in Section REF ." ], [ "Experiments", "The proposed method of unsupervised acoustic modeling is evaluated on a dataset that was acquired online.", "We use this dataset as a representative task of unsupervised speech modeling of low-resource language, despite that the language being spoken in the dataset is actually not low-resource.", "Our goal is to examine the efficacy of the proposed approach in the context of a real-world application." ], [ "Dataset", "The experimental dataset is built upon unedited video recordings in the MIT OpenCourseWare [107].", "English was used as the primary medium of instruction in these course lectures.", "The lectures are from 5 MIT courses, which are named “Mathematics for Computer Science” (MATH), “Principles of Digital Communication II” (COMM), “Introduction to Computer Science and Programming in Python” (PYTH), “Geometric Folding Algorithms: Linkages, Origami, Polyhedra” (ALGO) and “Discrete Stochastic Processes” (STOC).", "Presumably the audio part of a lecture should contain primarily the voice of the course instructor (professor).", "Due to diverse recording environments and hardware conditions, the recorded lecture may also contain students' voice (e.g., asking or responding to questions), and situational sounds from coughing, laughter, chalk-writing, furniture, etc.", "Table: Information about the course lectures being included in the dataset for experimentsThe recording conditions for lectures in different courses varied greatly.", "Some were recorded in classroom, some in lecture hall; There are different types of microphones: clip-on close-talking mic or built-in mic on video recorder.", "All recordings are in good perceptual quality such that teacher's speech can be heard clearly.", "Each course consists of $12 - 25$ lectures.", "The duration of each lecture is in $45-70$ minutes.", "The course teacher of MATH spoke with French accent.", "The speaking rate in PYTH is relatively fast and that in COMM is slow." ], [ "Clustering results", "Experiments are carried out on two scenarios: 1) training data from lectures of the same course with the same speaker; and 2) training data from lectures of various courses (multiple speaker and varying recording environments).", "The sklearn libraryhttps://scikit-learn.org/stable/modules/clustering.html and [108] in Python are used to implement segment clustering with different types of features and clustering algorithms." ], [ "Single-course training", "We experimented on the course COMM.", "All of the 25 lectures are used in training a course-based ASM.", "Multilingual DNN-BN as described in Section REF is used to extract BNFs from audio input.", "The AHC is used to determine the number of clusters, i.e., 55, and the $k$ -means algorithm is applied to cluster the segmental BNFs into subword units, and producing the initial pseudo transcription.", "The training of DNN-HMM starts with the pseudo transcription, and continue iteratively as described in REF .", "ASMs trained on different acoustic features are compared.", "They include the proposed BNF, conventional MFCC, and filter-bank.", "The number of iterations is 5 for each ASM.", "The degree of convergence is measured by the difference between transcriptions before and after each iteration, which is regarded as subword error rate (SWER) in Table REF .", "Table: Difference in input and output labels on bottleneck and filter bank featureFrom the experimental results, it can be seen that the the ASM trained with BNF could achieve lower SWER and its training converges faster than fbank and MFCC.", "Also the ASM trained with BNF converges at a lower SWER of $2.23\\%$ , compared with those trained with fbank and MFCC with SWER of $4.2\\% - 4.5\\%$ ." ], [ "Multiple-course training", "11 Lectures selected the 5 courses are pooled to train an ASM in the same way as the single-course case.", "A larger number of clusters, i.e., 100, is used in order to model more variations.", "Different clustering algorithms are used to obtain pseudo transcriptions to initialize DNN-HMM training.", "The subword mismatch rate of training an ASM with different sets of pseudo transcription are compared, only the first iteration of DNN-HMM training is used for comparison.", "Table: Subword mismatch rate of the first DNN-HMM iterative training using pseudo transcript from different clustering methodsWhen analysing the performance of k-means clustering, a control experiment is carried out on training ASM from single-course recordings, with cluster number set to be 100.", "The ASM takes in BNFs as input and pseudo transcript generated from the 100 subwords clustered by k-mean, and gives SWER of $36\\%-37\\%$ on its first iteration.", "The subword mismatch for the first iteration (59.10%) in the multi-course case is much higher.", "Apparently k-means clustering does not perform well in clustering BNF under complex speaker and channel conditions.", "Another possible reason is certain clustering algorithms such as k-mean clustering may not be able to well-separate speaker/channel information from phonetic properties in BNF.", "Comparing different clustering methods, it is found that HDBCAN gives the lowest SWER.", "For the clusters generated from GMM and BGMM, there are some clusters that have less than 500 frames (GMM: $\\lbrace 290\\rbrace $ , BGMM: $\\lbrace 365, 295\\rbrace $ ).", "The total number of frames is about $600,000$ .", "Merging these minority clusters into large clusters can be considered." ], [ "Summary", "In the chapter, the proposed procedures in building an ASM is discussed.", "It is shown that bottleneck features produced from Multilingual DNN-BN and Multilingual DNN-SBN can both be suitable features for unsupervised subword modelling, compare with conventional acoustic features.", "However, the training time for Multilingual DNN-SBN is much longer than Multilingual DNN with similar performance.", "Segmental BNF is also suitable for modelling both single speaker and multi-speakers recordings.", "Among different clustering methods, HDBSCAN soft clustering is more preferred with its lower initial subword input/output mismatch, making convergence of training the ASM easier." ], [ "Area of improvement", "As a proof of concept, DNN model used in extracting multilingual bottleneck feature and iterative training in this study is the basic neural network architecture.", "Other advanced architectures such as CNN, BILSTM can be considered in the future.", "However, larger computation is also needed.", "At this moment, theoretical understanding of bottleneck features is still limited.", "Other clustering algorithms and their relationship with bottleneck feature can also be investigated in the future." ], [ "Spoken Term Discovery", "Given a set of input utterances, the ASM produces pseudo transcriptions that characterise its content in terms of sequences of subword-like units.", "For a low-resource language, an interesting and practically useful application is to discover functional spoken terms, e.g., words, from a large collection of speech data in an unsupervised manner.", "This chapter is focused on automatic spoken term discovery from untranscribed lecture recordings.", "Part of the work can be found in [109].", "The proposed method comprises the following procedures: obtain a bag of keyword candidates by identifying repeated sequences of subword units in the pseudo transcriptions; cluster the keyword candidates to discover keywords.", "Figure: Automatic discovery of keywords from pseudo transcription" ], [ "Generating keyword candidates", "Keyword candidates are hypothesized by detecting repetitively occurred sequences of subword units in the pseudo transcriptions.", "The underlying assumption is that if a subword sequence appears multiple times in the transcriptions, it likely corresponds to a keyword or key-phrase.", "It is expected that same word may correspond to different subword sequences, due to pronunciation variation, speaker and environment changes.", "A robust algorithm of candidate sequence identification is needed to cope with these variabilities." ], [ "Sequence alignment", "Identifying similar or closely related sequence segments is an important problem in the area of bio-informatics.", "In [110], an algorithm of inexact matching between a pair of short symbol sequences was described.", "The algorithm finds an alignment between two symbol sequences that gives the best-matching sub-sequence.", "In traditional symbol matching algorithms, all symbols are weighted equally.", "In this study, subword-level acoustic information is used to determine the importance of each symbol.", "The symbol weight is assigned to reflect the likelihood that the symbol represents a speech unit.", "A voice activity detection (VAD) algorithm [111] is first applied to locate speech regions in an utterance.", "The likelihood for a subword unit to represent a phoneme is estimated as the average number of frames with voice activity in each class, with 1 being the highest score and 0 the lowest.", "Expectedly, hypothesized subwords in non-speech regions would have low likelihood values.", "Similarity of two sequences is measured based on local weighted alignment.", "The pseudo-code of this algorithm is given as in Algorithm REF .", "Given 2 subword sequences $A = a_1a_2...a_n$ and $B = b_1b_2...b_m$ .", "A score matrix with size of $n \\times m$ is initiated to store the similarity scores of sub-sequence match {$a_1...a_i, b_1...b_j$ } at each location $i, j$ .", "The scores are calculated based on the accumulative scores of matched and mismatched subwords between $A$ and $B$ .", "The local maximum scores are used to identify termination points of the sub-sequences, and each sub-sequence is obtained by tracing back from the maximum score to its minimum.", "[h] Modified local sequence alignment [1] LocalAlign$A = a_1 a_2 ... a_n, B = b_1 b_2 ... b_m$ $s(a_i,b_j)= {\\left\\lbrace \\begin{array}{ll}+w_{a_i}w_{b_j}&\\text{, if $a_i,b_j$ match}\\\\-w_{a_i}w_{b_j}&\\text{, if $a_i,b_j$ mismatch}\\end{array}\\right.", "}$ Similarity score between sequence elements $a_i$ and $b_j$ Compute an $(n+1) \\times (m+1)$ matrix $\\mathbf {P}$ , where the element $p_{i, j}$ is, $p_{i, j} = {\\left\\lbrace \\begin{array}{ll}0&\\text{, $i$ or $j = 0$}\\\\\\max {\\left\\lbrace \\begin{array}{ll}p_{i-1, j-1} + s(a_i, b_j)\\\\p_{i-1, j}\\\\p_{i, j-1}\\\\0\\end{array}\\right.", "}&\\text{, elsewhere}\\end{array}\\right.", "}$ Traceback from $p_{n,j^}$ ending with an element of $\\mathbf {P}$ equal to 0, where $p_{n,j^}$ are local maxima of $\\lbrace p_{n,j} \| 0 \\le j \\le m \\rbrace $ , to obtain common sub-sequence pairs in $A$ and $B$ .", "Store all obtained sub-sequences with reasonable length ($\\ge 4$ units) into the “bags of sequences”.", "Traditional local alignment traces back from global maximum score and returns only one sub-sequence.", "Whilst the method of using local maximum scores is able to identify multiple matching sub-sequences, regardless of their different order of appearance.", "For example, given the two English sentences: “Today we went hiking” and “We went to a park today”, searching for the global maximum score would result in“we went”, and the proposed method could discover both “we went” and “today”.", "Based on this algorithm, the “bag of sequences” containing only the matching sub-sequences is generated." ], [ "Keyword clustering", "With the “bag of candidate sequences” obtained, the next step is to generate a dictionary from this bag.", "This is a process of sequence pattern clustering.", "Each of the resulted clusters is expected to represent a keyword in the speech database.", "Unlike segment clustering in Chapter 3, the number of keyword clusters is not fixed.", "It depends greatly on the content of the database.", "It is also allowed to have some of the candidate sequences not assigned to any cluster, i.e., outliers." ], [ "Distance metric for subword sequence matching", "Distance metric for symbol string comparison is a critical element in a sequence clustering algorithm.", "It quantifies the similarity between two strings, which in our case are sequences of subword units.", "Levenshtein distance [112] measures the difference between two string sequences by counting the minimum number of character changes required to transform one string to another.", "The types of changes could be insertions, deletions and substitutions, similar to the way of calculating the error rate of an ASR system.", "In terms of Levenshtein distance, the dissimilarity between a pair of string sequences $x$ and $y$ is measured as, $lev_{x,y}(\|x\|,\|y\|)={\\left\\lbrace \\begin{array}{ll}\\max (i,j) & \\text{if $\\min (i,j)=0$,} \\\\[1ex]\\begin{aligned}[b]\\min \\bigl (lev_{x,y}&(i-1,j)+1, \\\\lev_{x,y}&(i,j-1)+1, \\\\lev_{x,y}&(i-1,j-1)+1_{(x_i\\ne y_j)}\\bigr )\\end{aligned} & \\text{otherwise.}\\end{array}\\right.}", "$ where $i$ and $j$ are the subword indexes of sequences $x$ and $y$ respectively, $lev_{x,y}(i,j)$ is the distance between the first $i$ subwords of sequence $x$ (i.e.", "$x_1x_2...x_i$ ) and first $j$ subwords of sequence $y$ (i.e.", "$y_1y_2...y_j$ ) .", "$1_{(x_i\\ne y_j)}$ is a function, which takes the value of 0 if $x_i = y_j$ , and 1 otherwise.", "When using the Levenshtein distance to search for keywords from pseudo transcriptions, a high degree of robustness is required to anticipate that the same keyword could correspond to many variants of subword sequences in natural speech.", "For a short word, deviation of $2-3$ symbols could be very significant, while for a long word we should tolerate a larger number of symbol differences.", "To cope with this issue, the normalized Levenshtein distance $\|\|L(x,y)\|\|$ is defined as, $\|\|L(x,y)\|\| = \\dfrac{L(x,y)}{\\sqrt{\|x\|^2 + \|y\|^2}} .$ Furthermore, subword weights are applied in computing the Levenshtein distance.", "Similar to the modified local alignment in Section REF , the weight of subword is computed as the proportion of frames with speech activity.", "The weighted distance $L_w(x,y)$ is similar with $L(x,y)$ in Equation (REF ), except $1_{(x_i\\ne y_j)}$ is replaced by $w$ , where insertion and deletion cost for subword $x_i$ in $y$ is $w_{x_i}$ , and the substitution cost from ${x_i}$ to $y_j$ is $w_{x_i}w_{y_j}$ .", "Normalized weighted Levenshtein distance is denoted as $\|\|L_w(x,y)\|\|$ , similar to Equation (REF )." ], [ "Leader clustering", "The “bag of subword sequences\" created as in Section REF contains a large number of subword sequences of different lengths.", "These sequences are clustered into groups using the leader clustering algorithm [113], as depicted in Algorithm REF .", "Let $T$ be the radius of each cluster.", "To prevent clusters from overlapping significantly, the minimum distance between a pair of cluster centroids is set to be $aT$ , where $a > 1$ .", "Figure: Illustration of the meaning of parameters TT and aa.Leader clustering is sensitive to initialization of centroids.", "To avoid poor initialization (e.g.", "assigning outliers as centroids), the centroid is updated with the most representative sequence, i.e., the sequence having the least total distance with all intra-cluster members, as shown in Line REF of Algorithm REF .", "The clustering process iterates until the number of clusters does not change further.", "Leader clustering [1] Leaderbag of sequences $S$ Initial a point $i$ to $\\mathrm {centroid}$ each point $p$ in $S$ $\|\|L(i,p)\|\| > aT$ for all $i$ in $\\mathrm {centroid}$ , $a>1$ : Add $p$ to $\\mathrm {centroid}$ Assign each point $p$ in $S$ to its closest cluster $i$ with $\|\|L(i,p)\|\| < T $ .", "each group Update $\\mathrm {centroid}$ with the representative of the cluster (measured by smallest total distance with same group members).", "Repeat steps 2-11 until the number of clusters converge.", "After clustering, sequences in the same cluster that are found to be overlapping in time are removed.", "We then obtain the finalized “keywords” of the recordings." ], [ "Analysis of metric and algorithm ", "Determining suitable parameters for the clustering algorithm is not a straightforward and tractable task.", "In this section, the proposed weighted metric and parameters used in leader clustering are evaluated." ], [ "Weighted distance metric", "The clustering results obtained using the unweighted and the weighted distance metrics are compared.", "In this experiment, all lecture recordings of the course MATH are used for training of the ASM model as well as the generation of subword units.", "Discovery of keywords is performed with the pseudo transcription of only a single lecture, namely “Number Theory I”.", "For leader clustering, we set $a = 1.6$ , $T = 0.325$ .", "As a result, the use of weighted distance metric leads to $36,255$ sub-sequences and 292 keyword clusters, and the unweighted distance metric produces $30,699$ sub-sequences and 264 clusters.", "A few keywords with different sequence lengths are selected for comparison between the two distance metrics.", "Table REF shows the cluster size (number of discovered words in a cluster) and the purity (the proportion of target keyword included in a cluster).", "Table: Clusters containing the target keywords using unweighted and weighted metric in alignment and clustering.For clusters with relatively long sequences ($> 8.5$ symbols), the tendency is that more keywords could be found by using weighted metric, with high purity maintained.", "For partially similar keywords, e.g., “greatest common divisor” and “greatest common divi-”, clustering with unweighted metric tends to merge them in the same cluster, while using weighted metric seems to treat them as being different.", "For clusters of shorter sequences ($< 8$ symbols), the purity decreases more significantly when using weighted metric than unweighted.", "A possible reason is that the radius measured in weighted metric is more loosen than unweighted metric since the respective sequence distance is usually smaller.", "In general, using weighted metric provides more favourable result, i.e., identifying more keywords with good cluster purity, especially for clusters of long sequences." ], [ "Effect of clustering parameters", "The effect of the radius $T$ and the margin factor $a$ is investigated in an experiment on the same lecture as in the previous section.", "The performance of keyword clustering is evaluated with the following parameter values: radius $T = \\lbrace 0.2 , 0.25, 0.3, 0.35, 0.4 \\rbrace $ margin factor $a = \\lbrace 1.2 , 1.4, 1.6 \\rbrace $ A total of $36,255$ sub-sequences are extracted using weighted local sequence alignment.", "With different clustering parameters, the number of clusters generated varies greatly from 82 to $4,644$ , as shown in Figure REF .", "It is noted that larger values of $T$ lead to significantly smaller number of clusters.", "Increasing the value of $a$ reduces the number of clusters, but the effect is not as significant as increasing the radius.", "Figure: Number of discovered keywords corresponding to different parameters TT and aa.A few clusters that correspond to target keywords or key-phrases are analyzed.", "They are “greatest common divisor”, “linear combination”, “number theory”, “algorithm” and “divides”, arranged in the descending order of sequence length.", "The clusters information are presented in Table REF .", "Table: The effect of number of clusters and their purity representing specific keywords.When a small radius $T$ is used, long keywords or key-phrases tend to be split into shorter constituents.", "For example, when radius $T = 0.2$ , $a = 1.6$ , the cluster representing “greatest common divisor” is split into multiple clusters that represent “greatest common divisor”, “-test common divisor”, “greatest common di-” and “the greatest com-”.", "with $T = 0.4$ , all these clusters merge into one.", "The same observation is noticed also for clusters of shorter sequences, “the algorithm” and “this algorithm” are assigned in different clusters when radius $T = 0.2$ , but they are assigned to the same cluster when $T \\ge 0.3$ .", "The purity of clusters increases when $T$ is decreased.", "Reducing the margin factor $a$ tends to encourage to splitting of clusters, though the effect is not as significant as reducing $T$ .", "While using a large radius and large margin help discover long phrases precisely, it may lead to the missing of relatively short phrases (e.g., “number theory” and “algorithm” with $T = 0.4$ , $a =1.6$ ).", "It also causes undesirable grouping of phrases with similar sound sequences, e.g., “the five” is grouped with “divide”." ], [ "Evaluation on ZeroSpeech 2017 spoken term discovery task", "The proposed spoken term discovery system is evaluated with spoken term discovery task in ZeroSpeech 2017 Challenge Track 2 [96].", "Similar as Track 1, the challenge provides three languages for evaluation, namely English, French and Mandarin.", "In the challenge, the spoken term discovery evaluation process is presented as a 3-stages work described as in Figure REF : 1) Matching repeating sequence; 2) Clustering the matching sequences; 3) Token representation of the audio with boundary information.", "Evaluation is done on each of the stage.", "Figure: Evaluation of term discovery in Zerospeech Challenge 2017.Evaluation is done on the discovered clusters with boundary information and cluster members provided by the performers.", "The result is compared with the actual lexicon and word transcription.", "$2-3$ metrics are used in each stage.", "Most of the measures are defined in terms of precision, recall and f-score.", "Precision measures the probability that the discovered element belongs to the actual set (e.g.", "lexicon, word in the transcription).", "Recall measures the probability that the actual element is in the discovered set.", "F-score is the harmonic mean between precision and recall.", "The evaluation in each of the stage are as follow: Matching quality The accuracy of the spoken terms discovered are evaluated using two metrics: NED and Coverage.", "NED measures the normalized edit distance of the word cluster and the word from actual transcription.", "Coverage is corpus that contains the matching pairs discovered.", "Clustering Quality The performance of the system is discovered in terms of lexicon discovery.", "It uses group precision, recall and f-score to evaluate how well the word clusters match on the sequence of phonemes, and type precision, recall and f-score to evaluate how well the cluster match the actual lexicons.", "Parsing Quality The metrics evaluate the performance in terms of word segmentation.", "It measures the word clusters that are aligned with the actual word transcription.", "It uses token precision, recall and f-score to evaluate the number of words that are correctly segmented.", "Boundary precision, recall and F-score is used to evaluate the number of actual word boundaries found.", "To evaluate our proposed spoken term discovery system, we experimented it on the Mandarin data set of $2.5$ hours long.", "Subword units are discovered by clustering on bottleneck features extracted from multilingual DNN-BN presented in Chapter 3.", "For the sequence matching process, we tried two different minimum matching length, which is the minimum matching sequence that will be considered for clustering.", "The experimented matching length are $length = 2$ and $length = 4$ .", "For the word clustering process, we want a straighter matching, with less duplicate word clusters for same keywords, therefore we set radius $T = 0.2$ and margin factor $a = 2.0$ .", "Weighted normalized levenstein distance is used.", "There were not many participants in the spoken term detection track, only 2 systems are presented in the challenge.", "We evaluated our system with the baseline and the 2 participated systems.", "The baseline is a spoken term system proposed by Jansen et al.", "[114].", "It performs spoken term discovery by DTW matching with random projections, follow by connected-component graph clustering.", "One of the performer is Kamper et al.", "[82], which uses k-means to discover repeating acoustic patterns, jointly optimized with exhaustive segmentation.", "The word segments for clustering are represented in fix-dimension.", "Another is Garcıa-Granada et al., which uses supervised Hungarian ASR to decode the speech, followed by sequence matching on the decoded transcription.", "Autoencoder is used to learn the word representation for word filtering using DTW.", "Table: Zerospeech Challange 2017 Track 2 metrics for the spoken term discovery systems on Mandarin dataset.The results of the systems are presented in percentage in Table REF .", "It is observed that our proposed system, when having a small minimum matching length, can achieve reasonable performance compared with other systems in terms of token, type, boundary and NLP quality.", "The word coverage rate $76.6$ is quite high.", "In terms of grouping, our system has reasonable recall rate, but the precision rate is very low, which means although the discovered word clusters maintain a good consistency of subword sequences, the subword sequences in the cluster may not have exact match with actual phonemes.", "It is noted that the evaluation is done only with the cluster boundaries and members information available.", "The discovered subwords are not measured with actual lexicons directly.", "However, it is also understandable since our system perform inexact matching.", "It does not perform any segmentation or further process matching sequences nor word clusters.", "This suggests more work toward word level segmentation can be consider in improving our work.", "Although using a high minimum matching length of 4 gives very low quality in terms of grouping, token, type and boundary, it is interesting to note that even only few words of 80 are discovered, the coverage rate is higher than the baseline.", "Possible reason is that since a straight matching criteria is used, the resulted discovered word clusters contain less non-representing clusters.", "From this evaluation it seems that using a shorter minimum matching length is better, however, the actual parameters used for the algorithm is language and data dependent.", "For example, words in Mardarin usually have a shorter formation of phonemes, while words in English can have various number of phonemes.", "More work on determining the suitable set of parameters for different languages is needed." ], [ "Summary", "In this chapter, local alignment follow by leader clustering is proposed for keyword discovery on the pseudo transcriptions generated from ASM.", "The proposed algorithms are shown to be effective in discovering repeating keywords that is compatible with other available spoken term discovery systems.", "While comparing with the algorithm and metric used, it is observed that subword weighting based on acoustic properties can improve sequence alignment and keyword clustering.", "The selection of clustering parameters also effects the resulting clusters.", "In this set of video lectures, the radius of the weighted normalized Levenshtein distance $T$ is preferred to be around $0.3$ , and the margin factor between clusters $a$ is around $1.4$ ." ], [ "Area of improvement", "Weighting subwords based on voice activity detection improves the clustering performance in general.", "However, there are much more speech characteristics that can be considered.", "Other weighted factors such as 1) weighting based on similarity of the subwords, giving lower penalty when acoustic similar subwords are swapped.", "2) Weighting based on posterior probabilities of the subwords from the ASM, or even weighting each subword unit based on the confusion scores of the ASM decoding result.", "The relationship of algorithm parameters and the experimented languages requires further investigation.", "Improvement can be done on the resulting matching sequences and clusters, such as use of word segmentation." ], [ "Results Analysis and Potential Applications", "In this chapter, we analysis the proposed system with real world data.", "The system combines the ASM in Chapter 3 and keyword discovery in Chapter 4.", "Online lectures are experimented.", "Quality and coverage of the discovered keywords are evaluated in a softer manner.", "The relation between discovered keywords and lecture topics is investigated in further detail.", "Tools for visualizing and analyzing text documents are used.", "They include word embedding and TF-IDF, which reflect the goodness of discovered linguistic information." ], [ "Results of keyword discovery", "The performance of the proposed system is evaluated by comparing and relating the discovered keywords with word-level transcriptions provided at the MIT OpenCourseWare website.", "The same set of course recordings mentioned in Table REF in Section $3.4$ is used.", "The ASM that generates the pseudo transcriptions for keyword discovery is trained on single course only.", "The performance is analyzed in terms of the quality of clusters and the clusters' coverage on lecture content.", "For the quality of discovered clusters, the goal is to have similar subword sequences included in the same cluster, similarity is measured by normalized Levenshtein distance.", "These subword sequences are expected to represent the topic-related keywords or key-phrases frequently spoken in the lecture.", "On the other hand, it is desired that a significant portion of content-related words could be discovered by clustering of subword sequences." ], [ "Quality of clusters", "The clustering results on two selected lectures in the course MATH are examined.", "The parameter values used for leader clustering are $a = 1.6$ , $T = 0.35$ .", "For Lecture 4 (“Number Theory”, 80 min.", "long), the system generates 95 keyword clusters from $34,313$ candidate subword sequences.", "For Lecture 8 (“Graph Theory II: Minimum Spanning Trees”, 83 min.", "long), there are 119 clusters from $25,899$ candidate subword sequences.", "Tables REF and REF list the words corresponding to the 10 longest sequence clusters in the two lectures respectively.", "It is observed that most of these clusters ($\\sim 90\\%$ ) correspond to words that are related to the lecture topic.", "Table: Discovered clusters from lecture “Number Theory I”Table: Discovered clusters from lecture “Graph Theory II: Minimum Spanning Trees”It is noted that clusters with sequence length of 12 subword units or more generally have high purity.", "Each of these clusters provides a valid representation of a specific word or phrase.", "As the sequence length decreases, the cluster's purity tends to decrease.", "Sequences containing less than 5 subword units typically correspond to parts of different words that have similar pronunciations, e.g., “so”, “(al)so”, and “so(lve)”; “in” and “in(teger)”.", "It is noted that the same word may be represented by more than one clusters.", "For example, clusters #116 and #109 of Lecture 8 (Table REF ) both correspond to “vertices”.", "Also, some discovered words may be constituted by shorter sequences of other clusters.", "Some of the clusters represent non-speech sounds, e.g., cluster #77 in Table REF corresponds to “chalk-writing sounds”, which is very common in live recordings of lectures.", "There exists several clusters with “chalk-writing sounds” of various length.", "While listening to the segments in a same cluster, different variations of same word can be observed.", "For example, different tones or different stressed syllables, different paces of speaking a same word, and same word with different recording channels (very clear and distant).", "Even there is influence of background noise to a spoken term such as chalk writing sound, the model is still able to assign it in the same group along with the clean speech of the same keyword.", "Besides speaker and channel variations, phoneme variation is also observed in same clusters, for example, “prime” and “find” , “m divides”, “n divides” and “and divides”, which are not exactly the same but very similar in pronunciation." ], [ "Coverage of discovered words", "For the intended task of keyword discovery, it is desired that a significant portion of the content-related words could be covered by the unsupervisedly generated clusters.", "In this section, we analyzed the automatically discovered keyword clusters with respect to the frequently occurred words and phrases in the ground-truth transcriptions (available at the MIT OpenCourseWare websiteocw.mit.edu/index.htm).", "Word-level trigrams, bigrams and unigrams are computed from the transcription for each lecture session or all lectures in a course, with the function words “is”, “a”, “the”, etc, being discarded.", "For a specific lecture session, the most frequent $N$ -grams are examined one by one, to determine whether the corresponding word(s) can be matched with any of the discovered word clusters.", "There are cases that a cluster may partially match a trigram or bigram.", "If the unmatched part is a function word, e.g.", "“linear combination” versus “linear combination of”, it is regarded as a case of match.", "If the unmatched part is a content word, e.g., “divisor” versus “common divisor”, it is regarded as mismatch." ], [ "Coverage in lecture base", "Firstly, the coverage is measured in lecture base, one lecture from two different courses are examined.", "For Lecture 4 (“Number Theory I”) of the course MATH and Lecture 8 (“Object-Oriented Programming”) of the course PYTH, we analyzed the top 10 trigrams, 20 bigrams and 30 unigrams and matched them with the 10 longest sequence clusters and other highly-populated clusters.", "The details of matching results for both lectures are given as in Table REF and REF .", "The matching rates are found to be $73.3\\%$ and $51.6\\%$ respectively.", "To explain the $20\\%$ difference in the coverage range, we analysis the property of the missing words.", "It is noted that the uncovered unigrams are mostly words with small number of phones, e.g.", "“add”, “car”, “code”, while the covered words are mostly polysyllabic words, e.g., “python\", “coordinate”.", "For the lecture “Number Theory I”, the matching rate for unigrams is higher, due to more complicated phonetic structure of the words." ], [ "Coverage in course base", "Same analysis has also been done for the whole course COMM, which contains 24 lectures of 70 minutes long.", "Keyword discovery is done on the whole course.", "The 100 most frequent trigrams, bigrams and unigrams are examined by comparing with clusters generated from all lectures.", "A high matching rate of $85.8\\%$ is recorded (Table REF ).", "Table: Matching results for course COMM" ], [ "Semantic relation of discovered keywords", "In the field of natural language processing (NLP), word embedding has been commonly adopted to produce vector representations of words or word sequences.", "It has been shown in numerous applications that the learnt word embedding space is able to reflect the linguistic functions and relations of the words.", "For example, vector representations of words with similar meanings are close to each other in terms of common-sense distance measure.", "In this section, the tool of Word2Vec is utilized to evaluate the results of keyword discovery." ], [ "Word2Vec", "Word2Vec [115] is developed to train a neural network by which a high-dimension one-hot word vector is converted into a low-dimension word embedding.", "There are two major approaches, namely continuous bag-of-words model (CBOW) and skip-gram.", "As shown in Figure REF , the two models share similar simple architecture, with an input layer, a projection layer and and an output layer.", "The training task of CBOW is to predict a target word (output) from neighbouring words (input).", "The exact positions of the neighbouring words are not used.", "The skip-gram model aims to classify a word, given other words in context.", "It is found that the skip-gram can model infrequent words better than CBOW.", "After training the model, the embedding space retrieved from the projection layer gives the word vector representations.", "Figure: skip-gram" ], [ "Experimental setup", "Experiment is done with the 24 lectures from course COMM.", "An ASM is trained with these lecture recordings to obtain the pesudo transcriptions.", "Keyword clusters are discovered from the pseudo transcriptions by leader clustering with radius $T = 0.3$ and margin factor $a = 1.4$ .", "A total of $9,605$ clusters are found from $1,371,951$ subword sub-sequences.", "The skip-gram model with 100-dimension embedding is used.", "Illustrated in Figure REF , the input to the model covers 3 left-context word clusters (or phrases) and 3 right-context word clusters, without any overlapping with the target word cluster $w(n)$ .", "Sub-sampling is done with selecting 4 neighbours out of the 6 in training.", "10 negative samples are considered in batch training of 256 clusters at a time.", "Figure: How the neighbours of a word are selected for the skipgram." ], [ "Semantic relation of learned keywords", "We analyze a set of selected clusters, which correspond to identifiable semantic content.", "Each of these clusters is contrasted to its closest neighboring clusters in respect of semantic relation.", "Table REF shows the clusters that contain more than 5 sequences, and Table REF shows those with equal or less than 5 sequences.", "Only those recognizable and unique keywords or key-phrases are shown in the tables, By recognizable we mean that the cluster has a high purity value such that it could be clearly related to a specific word or phrase.", "By unique we mean that if there exist more than one clusters that represent with the same phrase, only one is shown in the tables.", "The corresponding words of the neighbouring clusters are listed in ascending order of their distances from the selected clusters.", "Table: Neighbours of larger clusters (more than 5)Table: Neighbours of smaller clusters (equal or less than 5)It is clear that clusters with larger size are more likely to capture important semantic information and learn meaningful keywords.", "For example, cluster represents the keyword “waveform” has size of 27 .", "Its closest neighboring clusters correspond to closely related words or phrases like “a waveform”, “waveform” and “waveforms”.", "For some of the recognized keywords, the closest neighboring clusters match parts of the words, e.g., cluster referring to “probablity”, with size 62, have closely related clusters that correspond to “proba-” and “-bility”, while for the cluster corresponding to “communication”, “communica-” and “-cation” are its closest neighbors.", "These observations suggest the possibility of merging semantically related keyword clusters.", "They also provide a basis for understanding the structure of the language.", "On the other hand, it is interestingly noted that clusters representing different words/phrases under the same technical topic are closely related in the word embedding space.", "For example, the cluster “minus one” is found to be close to “x1”, “r1”, “square”, “times” and “zero”.", "As for clusters of smaller sizes, relationship between the clusters and their neighbours become less relevant.", "For example, in Table REF , although cluster referring to “probability”, with size 3, has some relevant neighbours “-bility” and “the probability”, they are not its closest neighbours.", "The closest neighbours “study”, “minimize” and “entropy” are less relevant.", "For clusters referring to “separation” and “combination”, their neighbours are totally irrelevant." ], [ "Discussion", "Unlike most applications of Word2Vec, where the words in documents are well defined (linguistically), the automatically discovered word clusters usually do not have 100% purity.", "They typically could be mapped to multiple different phrases.", "The embedded relationship may not in fact reflect the true relationship of the actual phrases.", "Moreover, the clusters may be established based on acoustic-phonetic similarities rather than semantic relation.", "This happens especially on short clusters which represent partial words.", "For example, some clusters referring to “problem” and “proba-(bility)” are found to be close to each other.", "For clusters of short sequences, the meaning of similarity in embedding space could be different from those of long sequences." ], [ "TF-IDF", "Term frequency-inverse document frequency (TF-IDF) [116] makes use of the occurrence count of terms, which could be language units at any level, to characterize, compare and correlate language documents.", "TF-IDF has long been applied successfully to text document retrieval and related tasks.", "Let $t$ denote a term.", "The term frequency of $t$ in document $d$ is given by, $tf(t,d) = freq_{t,d} $ where $d \\in D$ and $D = \\lbrace d_1, ..., d_N\\rbrace $ is a collection of documents concerned.", "The inverse document frequency measures the importance of $t$ by looking into its occurrence counts across all documents, i.e., $idf(t,D) = log{N \\over {\| t \\in d, d \\in D \| }}$ TF-IDF combines $tf$ and $idf$ as $tfidf(t,d) = tf(t,d)*idf(t,D)$ A term with high TF-IDF score for a document means that it is a key term to the document.", "We consider the term likely to be a keyword that can be used to identify the document." ], [ "Experiment on single-course recordings", "In this experiment, the ASM is trained with all lecture recordings of the course COMM.", "Unsupervised word discovery is performed on the pseudo transcriptions generated with the ASM.", "The approach of TF-IDF is applied with each of the discovered word clusters regarded as a term and each lecture in the course COMM is treated as a document.", "The three word clusters with the highest TF-IDF scores in each lecture are shown as in Table REF .", "Words or phrases that do not give information specific to the lecture topic are considered as irrelevant, e.g., “before”, “this particular”, “whatever”, “until here”.", "Among the 72 keywords, only 15 are irrelevant.", "Table: List of lecture topics for COMM and their 3 keywords with highest TF-IDF scoresWith the TF-IDF scores of all discovered keywords and their corresponding lectures obtained, we make an attempt to visualize the score matrix to analyze the relationship of keywords and lecture content.", "Each lecture in COMM is divided into 10-minutes long audio using a sliding window of 5-minutes shift, resulting in $15-20$ sessions for each lecture.", "Each session is treated as one document.", "The TF-IDF score vector of each session is calculated and visualized using t-SNE in Figure REF , with the sessions from same lecture labeled in same shape and colour.", "Figure: t-SNE plot on the TF-IDF score matrix of 24 lectures in COMMEven though the lectures are from same course, most of the sessions from same lecture are projected to close region." ], [ "Finding most similar sessions/lectures", "An experiment on topic comparison is done.", "Given a session, another session that has the most similar content/topic is found using session TF-IDF scores.", "Lectures are cut into 5-minutes long sessions without overlapping.", "Each session is treat as a document and TF-IDF scores of all sessions are calculated.", "The session with TF-IDF score vector that gives the highest cosine similarity with the tested session is selected.", "10 fold cross validation is conducted.", "The average accuracy in finding sessions being in same lectures is 92.73%, with standard deviation of 5.26%.", "This shows that weighting the keywords based on TF-IDF can highlight the distinctive keywords of one lecture from another." ], [ "Experiment on multiple-courses recordings", "To evaluate if topic comparison can also be applied to lectures from various courses, with different speakers and recording environments, we select 11 lectures from 5 different courses in Table REF in Section $3.4$ .", "The topics of the lectures are listed in Table REF .", "Table: Selected courses for the 5 coursesThe same parameter setting of $T=0.3$ and $a=1.4$ is used.", "There are a total of $12,200$ keyword clusters discovered, with each lecture having around $5,640$ clusters on average.", "Around $2,000-3,000$ keywords are shared among different courses.", "Similar with the single-course experiment, each lecture is split into 5-minute sessions.", "The TF-IDF scores are calculated in session-based.", "TF-IDF score vectors of the sessions are projected using t-SNE in Figure REF .", "It is observed that lectures from the same course are in the same region, expect for the course COMM, with lecture 19 being at the same region as STOP.", "Figure: t-SNE plot on the TF-IDF score matrix of 11 lectures from 5 courses" ], [ "Finding most similar lectures", "Lecture retrieval based on TF-IDF score is also performed.", "Given a lecture $D$ , the most similar lecture among the rest of the 10 lectures are obtained by finding the lecture with TF-IDF score vector that gives the highest cosine similarity with $D$ .", "It is found that most of the lectures retrieved are from the same course, expect for COMM lecture 19 “Baseband Detection”, its most similar lecture is “Random Walks and Thresholds” from STOP lecture 22.", "The two lectures' topics do not look similar at first sight, but when we analysis their content, we found that the content is very similar.", "When analysing the 60 most frequent unigrams excluding function words, it is found that $20\\%$ of the words are overlapping, such as “probability” , “likelihood”, “variable”, “function” and “minus”.", "There are also words with very similar meaning, such as “product” in COMM and “times” in STOP, both lectures have mathematics symbols for calculation but with different notations, such as “k”, “t”, “n0” in COMM, “e”, “f”,“h0” in STOP.", "It is shown that even though the two lectures are from different courses, both lectures use very similar methods in solving their problems.", "It would be preferable if the relationship of words with similar meaning, e.g.", "mathematics notation, can be learnt to benefit the topic comparison process.", "Combining the use of Word2Vec can be considered in the future." ], [ "Potential applications", "After analysing the keywords discovered and the result on lecture comparison, the proposed model shows its potential in applications regarding topic comparison and modelling.", "In this section, several potential real world applications are discussed." ], [ "Scenario 1", "There are platforms that provide online courses with different languages available, such as CourseraCoursera is an online learning platform that provides free courses taught by various universities and organizations.", "Link: https://www.coursera.org.", "When the user is watching a lecture and come across some terms that he wants to know more.", "Service can be provide for him to search for other fundamental courses/lectures to learn more about the specific terms.", "Figure: Illustration of the lecture recommendation based on segments system.One application for the proposed system is to allow user to search for similar courses/lectures by clicking and identifying segments he wants to look into.", "The frontend design is illustrate in Figure REF , the keywords discovered are presented as segment transcription, which may or may not be displayed to the user.", "Very similar to interactive transcript, the current segment that is playing will be located and highlighted.", "If the segments are displayed, the user can click on the segment that he is interested to look into.", "If not, he can specify the time interval of the words in the search bar.", "The search engine can look for additional lectures that are related to the segments specified.", "At the back end, the system preprocess all lectures in the database by discovering their spoken terms and compute the TF-IDF scores.", "The information is stored in the database.", "The TF-IDF score of selected session is calculated upon request and is compared with the TF-IDF scores of lectures from all courses.", "The most similar lectures are retrieved and displayed in the result bar.", "The comparison process can also be done by splitting the lectures into sessions, only sessions with highest similarities are displayed." ], [ "Scenario 2", "Another possible application for our system is video suggestion in multimedia sharing platforms such as Youtubehttps://www.youtube.com.", "Traditional video recommendation system uses collaborative filtering approach that suggests related videos based on their popularity and correlation with the current video [117].", "Limitation of this approach is that similarity between videos are measured based on users' activity instead of the content.", "Newly uploaded videos or less popular videos may be underweighted.", "Same situation may happen for videos in low resource languages, which may not have transcription or video tags.", "Figure: Illustration of the video recommendation system.The proposed system can be applied to any languages without the need of pre-labelling the videos with languages or tags.", "Unsupervised spoken term discovery can be done on all videos in the database, follow by calculating the TF-IDF scores of each video based on the discovered keywords.", "The TF-IDF score matrix is stored.", "When a user is watching a video, next suggested videos can be retrieved by searching for the videos whose TF-IDF scores give the highest cosine similarities with the current one.", "The system can also facilitate elders who only speak their own dialects in finding videos they can understand.", "Since ASM training and spoken term discovery process are language independent and they can unsupervisely discover keywords that are acoustically and linguistically similar, the system is able to suggest videos with the same dialects." ], [ "Summary", "In this chapter, the proposed model is experimented with online lectures to evaluate the quality and coverage of keywords discovered.", "The semantic relationship and topic relevance of the discovered keywords are analyzed.", "The proposed model is shown to be effective in discovering keywords that can mostly align with frequent words in actual transcription.", "The discovered keywords are also observed to be context related.", "Word2Vec is used to understand the relationship among the keywords.", "It is found that Word2Vec is able to group similar clusters together.", "Potential improvement to the spoken term discovery result is to merge different clusters corresponding to same phrases, and linking between multiple short clusters and long clusters.", "TF-IDF is used to analysis the relationship of a lecture topic and its discovered keywords.", "It is shown that topic comparison can be done by computing the cosine similarities of the TF-IDF score vectors of the lectures.", "Potential applications on video sharing platform are discussed as well." ], [ "Area of improvement", "The proposed system needs improvement in its ability in identifying keywords of relatively short length.", "In fact, candidate sequences that are shorter than 4 subword units are not included when generating the candidate sequences for clustering (see Algorithm in Section REF ).", "Computation time for clustering process is very long, it may take days to cluster the whole course with 24 lectures.", "Other more robust clustering methods besides leader clustering can be considered, such as Burkhard-Keller Trees [118], KNN-based clustering using k-dimensional tree [119].", "Potential improvement for the application of topic comparison is to combine the use of TF-IDF and Word2Vec, so that topics with similar content but different terminologies can also be discovered easier." ], [ "Conclusion", "Unsupervised acoustic modelling is a critical technology in dealing with zero resource languages.", "Our study has shown that it is possible to develop unsupervised model to represent a language without requiring any linguistic information and transcriptions.", "In acoustic segment modelling, feature extraction is an important part in unit learning and discovery.", "Our study shows that introducing bottleneck layer to DNN is effective method in extracting language independent features that suit any language.", "We also found that bottleneck features can be learnt from training multilingual DNN using resource rich languages.", "The bottleneck features learnt suits the application of subword unit discovery compared to conventional acoustic features by segment clustering.Among different types of clustering methods, density-based clustering can better cluster the bottleneck features proposed.", "By treating untranscribed freestyle lecture as an example of zero resource recordings, an acoustic segment model is built on top of a few courses.", "With reference to bioinformatics symbol searching problems, the subword unit sequences generated from the ASM can be used for spoken pattern discovery.", "It is shown that spoken terms discovered are aligned with real transcription of the lectures, providing evidence that ASM trained using bottleneck feature extracted from multilingual DNN can effectively model untranscribed recordings.", "After experimenting on a set of multi-speakers, multi-topics courses, it is shown that the proposed model is able to perform effectively under both single-speaker-single-channel and multiple-speakers-multiple-channels scenarios.", "The discovered spoken terms can also be further used for lectures comparison, analysing similarity between lectures based on discovered words.", "The relationship between each spoken term can also be understood using Word2Vec." ], [ "Contribution", "This thesis has made the following contributions towards the developing in unsupervised acoustic modelling and spoke term discovery: We proposed using bottleneck feature obtained from multilingual DNN for ASM training.", "The features are language independent and completely data driven, and can effectively represent the training recordings.", "The clustering effectiveness of several clustering algorithms are also investigated.", "Tools that are originated from text and document modelling are used in speech application.", "TF-IDF and Word2Vec are used in understanding relationship between discovered spoken terms and topics of recordings.", "The proposed system can be applied to real word applications such as videos, lectures recommendation in multilingual multimedia platform." ], [ "Limitations", "The evaluation presented in this work requires a more comprehensive analysis.", "In this thesis, the proposed system is only evaluated based on observation of its performance on real data.", "Although metrics from ZeroSpeech Challange are used, the metrics can only evaluate part of the tasks separately and it is difficult to evaluate the whole discovery process.", "Currently, there only exist evaluations on certain tasks, such as feature representations, subword modelling, segmentation.", "However, there are little work regrading the development and evaluation of the complete spoken term discovery system.", "There are also lack of evaluation on the applications of this technology.", "The system can only be concluded as functioning with reasonable performance, but a comprehensive measurement is hard to conduct.", "Also, the proposed system is evaluated on only very few languages.", "Evaluation and extension towards language-independent system still require more work to be done." ], [ "Suggestion for future work", "There are a few directions that can be conducted for future work." ], [ "Better clustering algorithms", "Although it is shown that the clustering algorithms used in segment clustering in Chapter 3 and keywords discovering in Chapter 4 are effective, there may exist other algorithms that can better cluster the data to yield better system performance.", "One possible direction is to look for clustering algorithm that can cluster large amount of data more efficient, so the system can process larger set of recordings.", "In terms of clustering bottleneck features, it is observed that hierarchical clustering (HDBSCAN, AHC) generally have better clustering performance than other clustering algorithms.", "Algorithms that cluster in hierarchical based but are relatively less computation expensive can be investigated.", "Since bottleneck features is being introduced recent years and there are still a lot work to be done in understanding its proprieties.", "More research work in analyzing the data complexity and property of bottleneck features is needed, and to determine suitable clustering algorithms accordingly.", "While clustering subword sequences into keyword clusters, other string clustering algorithms and different metrics can be considered.", "Moreover, alternative linguistic proprieties can be considered in weighting the subword sequences, as summarized in Chapter 4." ], [ "Experiment on low resource language", "In this thesis, experiments are mostly done on opencourse lectures, without a carefully collected and documented recordings as LDC/published corpora.", "Although the recordings used are more daily life related, it is hard to conduct a very detail and structured analysis due to lack of man power in labelling the lectures.", "Tasks specified corpora such as Switchboard, broadcast news for topic classification and keyword search can be experimented for more solid analysis.", "Moreover, the ability of the purposed system in processing zero resource language is not fully utilized.", "Experiment is suggested to conduct on low resource languages to examine the ability in process languages other than English.", "The system and tools discussed in Chapter 5 also be used to discover linguistics knowledge and provide better understanding to unfamiliar languages." ] ]	2011.14060
[ [ "Application of the Kovacic algorithm for the investigation of motion of\n a heavy rigid body with a fixed point in the Hess case" ], [ "Abstract In 1890 German mathematician and physicist W. Hess found new special case of integrability of Euler - Poisson equations of motion of a heavy rigid body with a fixed point.", "In 1892 P. A. Nekrasov proved that the solution of the problem of motion of a heavy rigid body with a fixed point under Hess conditions reduces to integrating the second order linear differential equation.", "In this paper the corresponding linear differential equation is derived and its coefficients are presented in the rational form.", "Using the Kovacic algorithm, we proved that the liouvillian solutions of the corresponding second order linear differential equation exists only in the case, when the moving rigid body is the Lagrange top, or in the case when the constant of the area integral is zero." ], [ "Application of the Kovacic algorithm for the investigation of motion of a heavy rigid body with a fixed point in the Hess case.", "Boris S. Bardin${}^{}$ , Alexander S. Kuleshov${}^{}$ ${}^{}$ Moscow Aviation Institute (national research university), Moscow 125080, Russia E-mail: [email protected] ${}^{*}$ M. V. Lomonosov Moscow State University, Moscow 119234.", "E-mail: [email protected] In 1890 German mathematician and physicist W. Hess [4] found new special case of integrability of Euler – Poisson equations of motion of a heavy rigid body with a fixed point.", "In 1892 P. A. Nekrasov [17], [18] proved that the solution of the problem of motion of a heavy rigid body with a fixed point under Hess conditions reduces to integrating the second order linear differential equation.", "In this paper the corresponding linear differential equation is derived and its coefficients are presented in the rational form.", "Using the Kovacic algorithm [8], we proved that the liouvillian solutions of the corresponding second order linear differential equation exists only in the case, when the moving rigid body is the Lagrange top, or in the case when the constant of the area integral is zero.", "Let us consider the problem of motion of a heavy rigid body with the fixed point $O$ .", "To describe motion of the body we introduce two orthogonal coordinate systems: the fixed system $Oxyz$ and the moving system $Ox_1x_2x_3$ .", "The $Oz$ axis of the fixed system is directed along the upward vertical.", "The $Ox_1x_2x_3$ system is rigidly connected with the moving body and its axes are directed along the principal axes of inertia at $O$ .", "We denote the unit vectors of $Ox_1x_2x_3$ system by ${\\bf e}_1$ , ${\\bf e}_2$ , ${\\bf e}_3$ .", "Let $\\gamma =\\gamma _1{\\bf e}_1+\\gamma _2{\\bf e}_2+\\gamma _3{\\bf e}_3$ be the unit vector of the $Oz$ axis, where $\\gamma _1$ , $\\gamma _2$ , $\\gamma _3$ are projections of this vector onto the $Ox_1x_2x_3$ axes.", "We denote by ${\\bf P}$ the gravity force (directed vertically downward and applied at the center of gravity of the body $G$ ), then we have ${\\bf P}=-Mg{\\gamma }$ , where $M$ is the mass of the body and $g$ is the gravity acceleration.", "We will define the position of the center of mass of the body by the radius – vector ${\\bf r}=\\overrightarrow{OG}=x_1{\\bf e}_1+x_2{\\bf e}_2+x_3{\\bf e}_3,$ where $x_1$ , $x_2$ , $x_3$ are projections of this vector onto $Ox_1x_2x_3$ axes.", "We will apply the principle of angular momentum, taking with respect to the point $O$ , to derive the equations of motion of the body.", "The corresponding vector equation with respect to the fixed coordinate system $Oxyz$ takes the following form: $\\frac{d{\\bf K}}{dt}={\\bf M},$ where ${\\bf M}$ is the total moment of the external forces (the force ${\\bf P}$ in the considered case) about the fixed point $O$ , and ${\\bf K}$ is the angular momentum of the body with respect to the fixed point $O$ .", "This vector can be represented as follows (see.", "[21]): ${\\bf K}=\\mathbb {J}_O\\omega ,$ where $\\omega $ is the angular velocity vector of the body and $\\mathbb {J}_O$ is the tensor of inertia of the body at $O$ .", "It is well known that the rate of change of the vector $K$ – the vector $d{\\bf K}/dt$ , calculated in the fixed coordinate system $Oxyz$ , equals the rate of growth of this vector $\\delta {\\bf K}/{\\delta t}$ (calculated in the moving coordinate system $Ox_1x_2x_3$ ) plus the rate of transport, i.e.", "(see [21]): $\\frac{d{\\bf K}}{dt}=\\frac{\\delta {\\bf K}}{\\delta t}+\\left[\\omega \\times {\\bf K}\\right].$ Taking into account (REF ), we can rewrite equation (REF ) as follows: $\\frac{\\delta {\\bf K}}{\\delta t}+\\left[\\omega \\times {\\bf K}\\right]=\\left[{\\bf r}\\times {\\bf P}\\right].$ In the moving coordinate system $Ox_1x_2x_3$ we will write the vector $\\omega $ , the vector ${\\bf K}$ and the tensor $\\mathbb {J}_O$ as follows: $\\omega =\\omega _1{\\bf e}_1+\\omega _2{\\bf e}_2+\\omega _3{\\bf e}_3,$ $\\mathbb {J}_O=\\left(\\begin{array}{lll}A_1&0&0\\\\ \\\\0&A_2&0\\\\ \\\\0&0&A_3\\end{array}\\right),$ ${\\bf K}=\\mathbb {J}_O\\omega =A_1\\omega _1{\\bf e}_1+A_2\\omega _2{\\bf e}_2+A_3\\omega _3{\\bf e}_3,$ where $A_1$ , $A_2$ , $A_3$ – are principal moments of inertia of the body at $O$ .", "If we take the projections of vector equation (REF ) onto the $Ox_1x_2x_3$ axes we obtain the three scalar equations: $\\begin{array}{l}A_1{\\dot{\\omega }}_1+\\left(A_3-A_2\\right)\\omega _2\\omega _3=Mg\\left(x_3\\gamma _2-x_2\\gamma _3\\right), \\\\ \\\\A_2{\\dot{\\omega }}_2+\\left(A_1-A_3\\right)\\omega _1\\omega _3=Mg\\left(x_1\\gamma _3-x_3\\gamma _1\\right), \\\\ \\\\A_3{\\dot{\\omega }}_3+\\left(A_2-A_1\\right)\\omega _1\\omega _2=Mg\\left(x_2\\gamma _1-x_1\\gamma _2\\right).\\end{array}$ These equations are called the Euler's equations of motion of a rigid body with a fixed point.", "These equations actually involve six parameters $A_1$ , $A_2$ , $A_3$ , $x_1$ , $x_2$ , $x_3$ , characterizing the mass distribution of the body relative to the principal axes of inertia at $O$ , and six unknowns $\\omega _1$ , $\\omega _2$ , $\\omega _3$ , $\\gamma _1$ , $\\gamma _2$ , $\\gamma _3$ to be solved as functions of time $t$ .", "To be able to solve for them we need three more independent equations relating the $\\omega _1$ , $\\omega _2$ , $\\omega _3$ , $\\gamma _1$ , $\\gamma _2$ , $\\gamma _3$ .", "We have, since $\\gamma $ is a space – fixed vector (i.e.", "it is fixed in the $Oxyz$ coordinate system), then $\\frac{\\delta {\\gamma }}{\\delta t}+\\left[\\omega \\times {\\gamma }\\right]=0.$ This vector equation gives the three scalar equations onto the $Ox_1x_2x_3$ axes $\\dot{\\gamma }_1=\\omega _3\\gamma _2-\\omega _2\\gamma _3,\\quad \\dot{\\gamma }_2=\\omega _1\\gamma _3-\\omega _3\\gamma _1,\\quad \\dot{\\gamma }_3=\\omega _2\\gamma _1-\\omega _1\\gamma _2.$ These equations are called the Poisson equations.", "Finally, we obtain the close system of six nonlinear differential equations (REF ), (REF ) with respect to the six unknown functions of time $t$ : $\\omega _1$ , $\\omega _2$ , $\\omega _3$ , $\\gamma _1$ , $\\gamma _2$ , $\\gamma _3$ .", "These equations are called the Euler – Poisson equations.", "It is well known that to solve the Euler – Poisson equations we need to find four independent autonomous first integrals of the system (REF ), (REF ) [2], [14].", "For any values of parameters $A_1$ , $A_2$ , $A_3$ , $x_1$ , $x_2$ , $x_3$ of the body and for any initial conditions we have three first integrals of the Euler – Poisson equations.", "The energy integral $T+V=h$ , where $T$ is the kinetic energy of the body, defined by the formula $T=\\frac{1}{2}\\left(A_1\\omega _1^2+A_2\\omega _2^2+A_3\\omega _3^2\\right),$ and $V$ is the potential energy of the body which has the form: $V=Mg\\left(x_1\\gamma _1+x_2\\gamma _2+x_3\\gamma _3\\right).$ Thus with respect to the axes $Ox_1x_2x_3$ the energy integral takes the form: $\\frac{1}{2}\\left(A_1\\omega _1^2+A_2\\omega _2^2+A_3\\omega _3^2\\right)+Mg\\left(x_1\\gamma _1+x_2\\gamma _2+x_3\\gamma _3\\right)=h.$ The area integral $\\left({\\bf K}\\cdot \\gamma \\right)=C,$ which can be written with respect to the axes $Ox_1x_2x_3$ as follows: $A_1\\omega _1\\gamma _1+A_2\\omega _2\\gamma _2+A_3\\omega _3\\gamma _3=C.$ The geometrical integral $\\gamma _1^2+\\gamma _2^2+\\gamma _3^2=1.$ The energy integral $T+V=h$ , where $T$ is the kinetic energy of the body, defined by the formula $T=\\frac{1}{2}\\left(A_1\\omega _1^2+A_2\\omega _2^2+A_3\\omega _3^2\\right),$ and $V$ is the potential energy of the body which has the form: $V=Mg\\left(x_1\\gamma _1+x_2\\gamma _2+x_3\\gamma _3\\right).$ Thus with respect to the axes $Ox_1x_2x_3$ the energy integral takes the form: $\\frac{1}{2}\\left(A_1\\omega _1^2+A_2\\omega _2^2+A_3\\omega _3^2\\right)+Mg\\left(x_1\\gamma _1+x_2\\gamma _2+x_3\\gamma _3\\right)=h.$ The area integral $\\left({\\bf K}\\cdot \\gamma \\right)=C,$ which can be written with respect to the axes $Ox_1x_2x_3$ as follows: $A_1\\omega _1\\gamma _1+A_2\\omega _2\\gamma _2+A_3\\omega _3\\gamma _3=C.$ The geometrical integral $\\gamma _1^2+\\gamma _2^2+\\gamma _3^2=1.$ Thus for integrability of the Euler – Poisson equations we need to find only one additional autonomous first integral.", "Under appropriate conditions on the parameters $A_1$ , $A_2$ , $A_3$ , $x_1$ , $x_2$ , $x_3$ this integral can exist only in the following three cases [2], [14].", "The Euler case $\\left(x_1=x_2=x_3=0\\right)$ .", "The additional integral has the form: $A_1^2\\omega _1^2+A_2^2\\omega _2^2+A_3^2\\omega _3^2=k^2.$ The Lagrange case $\\left(A_1=A_2, x_1=x_2=0\\right)$ .", "In this case the additional integral has the form: $\\omega _3=\\omega ={\\rm const}.$ The Kovalevskaya case $\\left(A_1=A_2=2A_3, x_2=x_3=0\\right)$ .", "If we denote $\\frac{Mgx_1}{A_3}=n,$ then the additional first integral can be written as follows: $\\left(\\omega _1^2-\\omega _2^2-n\\gamma _1\\right)^2+\\left(2\\omega _1\\omega _2-n\\gamma _2\\right)^2=j.$ The Euler case $\\left(x_1=x_2=x_3=0\\right)$ .", "The additional integral has the form: $A_1^2\\omega _1^2+A_2^2\\omega _2^2+A_3^2\\omega _3^2=k^2.$ The Lagrange case $\\left(A_1=A_2, x_1=x_2=0\\right)$ .", "In this case the additional integral has the form: $\\omega _3=\\omega ={\\rm const}.$ The Kovalevskaya case $\\left(A_1=A_2=2A_3, x_2=x_3=0\\right)$ .", "If we denote $\\frac{Mgx_1}{A_3}=n,$ then the additional first integral can be written as follows: $\\left(\\omega _1^2-\\omega _2^2-n\\gamma _1\\right)^2+\\left(2\\omega _1\\omega _2-n\\gamma _2\\right)^2=j.$ There are no other general cases of integrability of Euler – Poisson equations.", "However, there are several cases, when for the special initial conditions we can find the additional special integral.", "One of these cases is the Hess case which has been firstly described in 1890 by W. Hess [4].", "Let us consider this case in more details.", "We suppose that the parameters of the body $A_1$ , $A_2$ , $A_3$ , $x_1$ , $x_2$ , $x_3$ satisfy the conditions $x_3=0,\\qquad A_2\\left(A_3-A_1\\right)x_2^2=A_1\\left(A_2-A_3\\right)x_1^2,\\qquad A_2\\ge A_3\\ge A_1.$ Let us prove that under conditions (REF ) the Euler – Poisson equations (REF ), (REF ) have an additional special integral (the Hess integral) of the form: $A_1\\omega _1 x_1+A_2\\omega _2 x_2=0.$ To prove the existence of the integral (REF ) we write two first equations of the system (REF ) taking into account conditions (REF ): $\\begin{array}{l}A_1{\\dot{\\omega }}_1+\\left(A_3-A_2\\right)\\omega _2\\omega _3=-Mgx_2\\gamma _3, \\\\ \\\\A_2{\\dot{\\omega }}_2+\\left(A_1-A_3\\right)\\omega _1\\omega _3=Mgx_1\\gamma _3.\\end{array}$ Multiplying the first of equations (REF ) by $x_1$ and the second by $x_2$ and taking their sum we obtain: $\\frac{d}{dt}\\left(A_1\\omega _1 x_1+A_2\\omega _2 x_2\\right)=\\left(A_3-A_1\\right)\\omega _1\\omega _3 x_2+\\left(A_2-A_3\\right)\\omega _2\\omega _3 x_1.$ The right hand part of equation (REF ) can be transformed using (REF ) to the form: $\\left(A_3-A_1\\right)\\omega _1\\omega _3 x_2+\\left(A_2-A_3\\right)\\omega _2\\omega _3x_1=\\frac{\\omega _3\\left(A_2-A_3\\right)x_1}{A_2x_2}\\left(A_1\\omega _1 x_1+A_2\\omega _2 x_2\\right).$ Taking into account equation (REF ) we can conclude from the equation (REF ), that if at initial instant the equation (REF ) is valid then it holds during the whole time of motion of the rigid body.", "Therefore under conditions (REF ) the Euler – Poisson equations (REF ), (REF ) possess the special integral (REF ).", "For the first time this case of special integrability of Euler – Poisson equations was discovered by W. Hess [4].", "Hess found this case of integrability when he searched for singular solutions of his own form of the Euler – Poisson equation [4].", "In 1892 the Hess case was rediscovered by G. G. Appelroth [1] when he analysed the branching of the general solution of Euler – Poisson equations on the complex plane of time using the ideas of S. V. Kovalevskaya [12], [13].", "The detailed analytical investigation of the Hess case has been made by P. A. Nekrasov [17], [18].", "In his papers [17], [18] Nekrasov presented both the Hess conditions and the Hess integral and reduced the solution of the problem to the integration of a second order linear differential equation.", "Nekrasov proved that in the Hess case the solution branches out on the complex plane of time.", "He investigated the analytical properties of the obtained second order linear differential equation and pointed out the basic properties of trajectories on the Poisson sphere.", "He proved also that on the zero level of the area integral the problem is integrable in elliptic functions.", "The Hess integral as well as the reduction to the second order linear differential equation was independently rediscovered in 1895 by Roger Liouville [15].", "The geometrical analysis and the modelling of the Hess top (in particular, on the zero level of the area integral) were given by N E. Zhukovsky [5].", "In the paper [16] B. K. Mlodzeevskii and P. A. Nekrasov proved that under special restrictions solutions of the problem can have asymptotic behaviour.", "The complete analysis of the phase trajectories in a Hess case was made by A. M. Kovalev [9].", "In the paper [11] which substantially uses the results obtained in [9], the authors presented detailed classification of possible hodographs of the angular momentum vector in dependence of constants of the energy integral $h$ and the area integral $C$ .", "In the paper [10] the kinematic description of motion of a rigid body in the Hess case at zero level of the area integral was given by the hodographs method [6], [7].", "In the paper [3] at zero level of the area integral the temporal and spatial evolution of the angular velocity vector and the angular moment vector was investigated.", "The motion of the body with a fixed point in the Hess case was represented by the rolling motion of the ellipsoid of inertia on a fixed plane.", "Since the solution of the problem of motion of a heavy rigid body with a fixed point in the Hess case is reduced to solving the second order linear differential equation it is possible to set up the problem of existence of liouvillian solutions of the corresponding linear differential equation.", "For this purpose it is possible to apply the so-called Kovacic algorithm [8], which allows to find liouvillian solutions of a second order linear differential equation in explicit form.", "If a linear differential equation has no liouvillian solutions, the Kovacic algorithm also allows one to ascertain this fact.", "The necessary condition for the application of the Kovacic algorithm to a second order linear differential equation is that the coefficients of this equation should be rational functions of independent variable.", "Below we derive the second order linear differential equation in the Hess case and reduce its coefficients to the form of rational functions.", "Further we study the problem of existence of liouvillian solutions for the obtained second order linear differential equation using the Kovacic algorithm.", "As the first step to derive the corresponding second order linear differential equation in the Hess case let us write the Euler – Poisson equations in the special coordinate system, proposed by P. V. Kharlamov [6], [7].", "Instead of the principal axes of inertia at $O$ (the $Ox_1x_2x_3$ coordinate system) let us choose the arbitrary trihedral coordinate system $O\\xi _1\\xi _2\\xi _3$ which is rigidly connected with the moving body.", "We denote the unit vectors of the $O\\xi _1\\xi _2\\xi _3$ system by ${\\bf e}_I$ , ${\\bf e}_{II}$ , ${\\bf e}_{III}$ .", "In this coordinate system we can write the angular momentum ${\\bf K}$ as follows: ${\\bf K}=K_1{\\bf e}_I+K_2{\\bf e}_{II}+K_3{\\bf e}_{III}.$ On the other hand, the angular momentum is defined by (REF ) ${\\bf K}=\\mathbb {J}_O\\omega ,$ where $\\omega =\\Omega _1{\\bf e}_I+\\Omega _2{\\bf e}_{II}+\\Omega _3{\\bf e}_{III},$ $\\mathbb {J}_O=\\left(\\begin{array}{ccc}L_{11}&-L_{12}&-L_{13}\\\\ \\\\-L_{21}&L_{22}&-L_{23}\\\\ \\\\-L_{31}&-L_{32}&L_{33}\\end{array}\\right).$ Here the components $L_{ij}$ of inertia tensor $\\mathbb {J}_O$ satisfy the condition $L_{ij}=L_{ji}$ .", "In this tensor components $L_{11}$ , $L_{22}$ , $L_{33}$ are the moments of inertia of the body with respect to $O\\xi _1$ , $O\\xi _2$ , $O\\xi _3$ respectively and $L_{ij}$ , $\\left(i\\ne j\\right)$ are products of inertia.", "Using (REF ), (REF ), (REF ) we can conclude that the angular momentum ${\\bf K}$ has the following components onto $O\\xi _1\\xi _2\\xi _3$ axes: $\\begin{array}{l}K_1=L_{11}\\Omega _1-L_{12}\\Omega _2-L_{13}\\Omega _3, \\\\ \\\\K_2=-L_{21}\\Omega _1+L_{22}\\Omega _2-L_{23}\\Omega _3, \\\\ \\\\K_3=-L_{31}\\Omega _1-L_{32}\\Omega _2+L_{33}\\Omega _3.\\end{array}$ We can write the kinetic energy of the moving body using the angular momentum ${\\bf K}$ by the formula: $T=\\frac{1}{2}\\left({\\bf K}\\cdot {\\omega }\\right).$ Taking into account (REF ), (REF ) the expression for the kinetic energy can be written in the scalar form: $T=\\frac{1}{2}\\left(L_{11}\\Omega _1^2+L_{22}\\Omega _2^2+L_{33}\\Omega _3^2\\right)-L_{23}\\Omega _2\\Omega _3-L_{13}\\Omega _1\\Omega _3-L_{12}\\Omega _1\\Omega _2.$ The inertia tensor (REF ) is a non-singular matrix.", "Therefore equations (REF ) can be represented as the linear nonhomogeneous equations with respect to $\\Omega _1$ , $\\Omega _2$ , $\\Omega _3$ .", "To solve equations (REF ) with respect to $\\Omega _1$ , $\\Omega _2$ , $\\Omega _3$ , we find: $\\begin{array}{l}\\Omega _1=l_{11}K_1+l_{12}K_2+l_{13}K_3, \\\\ \\\\\\Omega _2=l_{21}K_1+l_{22}K_2+l_{23}K_3, \\\\ \\\\\\Omega _3=l_{31}K_1+l_{32}K_2+l_{33}K_3.\\end{array}$ Here we denote: $l_{11}=\\frac{L_{22}L_{33}-L_{23}^2}{\\Delta },\\qquad l_{22}=\\frac{L_{11}L_{33}-L_{13}^2}{\\Delta },\\qquad l_{33}=\\frac{L_{11}L_{22}-L_{12}^2}{\\Delta };$ $l_{23}=l_{32}=\\frac{L_{11}L_{23}+L_{12}L_{13}}{\\Delta },\\qquad l_{13}=l_{31}=\\frac{L_{12}L_{23}+L_{13}L_{22}}{\\Delta },\\qquad l_{12}=l_{21}=\\frac{L_{12}L_{33}+L_{13}L_{23}}{\\Delta };$ $\\Delta =\\det \\left(L_{ij}\\right)=L_{11}L_{22}L_{33}-L_{11}L_{23}^2-L_{33}L_{12}^2-L_{22}L_{13}^2-2L_{12}L_{13}L_{23}.$ Substituting (REF ) in the formula (REF ) for the kinetic energy $T$ , we present it as the function of $K_1$ , $K_2$ , $K_3$ : $T=\\frac{1}{2}\\left(l_{11}K_1^2+l_{22}K_2^2+l_{33}K_3^2\\right)+l_{23}K_2K_3+l_{13}K_1K_3+l_{12}K_1K_2.$ Now we simplify the obtained expression (REF ) for the kinetic energy.", "For this purpose we rotate the coordinate system $O\\xi _1\\xi _2\\xi _3$ about the $O\\xi _1$ axis counter clockwise through an angle $\\alpha $ .", "We denote the obtained coordinate system by $O\\xi _1^\\xi _2^\\xi _3^$ .", "In this case the new components of the vector ${\\bf K}$ , which we denote by $K_1^$ , $K_2^$ , $K_3^$ , can be expressed through the components $K_1$ , $K_2$ , $K_3$ by the following formulae: $K_1=K_1^, \\qquad K_2=K_2^\\cos \\alpha -K_3^\\sin \\alpha ,\\qquad K_3=K_2^\\sin \\alpha +K_3^\\cos \\alpha .$ The expression for the kinetic energy $T$ as a function of components $K_1^$ , $K_2^$ , $K_3^$ of the vector ${\\bf K}$ takes the form: $T=\\frac{1}{2}\\left(l_{11}^\\left(K_1^\\right)^2+l_{22}^\\left(K_2^\\right)^2+l_{33}^\\left(K_3^\\right)^2\\right)+l_{23}^K_2^K_3^+l_{13}^K_1^K_3^+l_{12}^K_1^K_2^.$ In this formula the coefficients $l_{ij}^$ are expressed through the coefficients $l_{ij}$ and the angle $\\alpha $ by the formulae: $l_{11}^=l_{11}, \\quad l_{22}^=l_{22}\\cos ^2\\alpha +l_{33}\\sin ^2\\alpha +l_{23}\\sin 2\\alpha ,$ $l_{33}^=l_{22}\\sin ^2\\alpha +l_{33}\\cos ^2\\alpha -l_{23}\\sin 2\\alpha ,$ $l_{12}^=l_{13}\\sin \\alpha +l_{12}\\cos \\alpha ,\\quad l_{13}^=l_{13}\\cos \\alpha -l_{12}\\sin \\alpha ,$ $l_{23}^=l_{23}\\cos 2\\alpha -\\frac{1}{2}\\left(l_{22}-l_{33}\\right)\\sin 2\\alpha .$ We assume now that the angle $\\alpha $ is defined by the formula: $\\alpha =\\frac{1}{2}\\frac{2l_{23}}{l_{22}-l_{33}}.$ In this case the coefficient $l_{23}^$ in (REF ) becomes zero and the expression for the kinetic energy takes the form: $T=\\frac{1}{2}\\left(l_{11}^\\left(K_1^\\right)^2+l_{22}^\\left(K_2^\\right)^2+l_{33}^\\left(K_3^\\right)^2\\right)+\\left(l_{13}^K_3^+l_{12}^K_2^\\right)K_1^.$ Assuming that the center of mass $G$ of the body does not coincide with the fixed point $O$ we choose the coordinate system $O\\eta _1\\eta _2\\eta _3$ as follows.", "Let the $O\\eta _1$ axis passes through the center of mass $G$ of the body (i.e.", "it is directed along the vector ${\\bf r}$ ) and let the two other axes $O\\eta _2$ and $O\\eta _3$ be directed in such a way that the expression for the kinetic energy $T$ has a form (REF ), i.e.", "it does not contain the product $K_2^K_3^$ .", "This coordinate system was firstly introduced by P. V. Kharlamov [6], [7].", "He called the $O\\eta _1\\eta _2\\eta _3$ system as the special coordinate system.", "Obviously, this coordinate system is uniquely determined, and it will be rigidly connected with a rigid body.", "Using the special coordinate system we will write $a$ , $a_1$ , $a_2$ , $b_1$ , $b_2$ instead of $l_{11}^$ , $l_{22}^$ , $l_{33}^$ , $l_{12}^$ , $l_{13}^$ and we will write $L_1$ , $L_2$ , $L_3$ , $\\omega _I$ , $\\omega _{II}$ , $\\omega _{III}$ instead of $K_1^$ , $K_2^$ , $K_3^$ , $\\Omega _1$ , $\\Omega _2$ , $\\Omega _3$ respectively.", "Finally we have the following expression for the kinetic energy $T$ and components of the angular velocity $\\omega $ of the body onto $O\\eta _1\\eta _2\\eta _3$ axes: $T=\\frac{1}{2}\\left(aL_1^2+a_1L_2^2+a_2L_3^2\\right)+\\left(b_1L_2+b_2L_3\\right)L_1.$ $\\omega _I=aL_1+b_1L_2+b_2L_3,\\qquad \\omega _{II}=a_1L_2+b_1L_1,\\qquad \\omega _{III}=a_2L_3+b_2L_1.$ Using (REF ), (REF ), (REF ), (REF ) we can write the Euler – Poisson equations in the special P. V. Kharlamov coordinate system as follows (see [6], [7]): $\\begin{array}{l}\\dot{L}_1=\\left(a_2-a_1\\right)L_2L_3+\\left(b_2L_2-b_1L_3\\right)L_1,\\\\ \\\\\\dot{L}_2=\\left(a-a_2\\right)L_1L_3+\\left(b_1L_2+b_2L_3\\right)L_3-b_2L_1^2+\\Gamma \\nu _3,\\\\ \\\\\\dot{L}_3=-\\left(a-a_1\\right)L_1L_2-\\left(b_1L_2+b_2L_3\\right)L_2+b_1L_1^2-\\Gamma \\nu _2,\\\\ \\\\\\dot{\\nu }_1=\\left(a_2L_3+b_2L_1\\right)\\nu _2-\\left(a_1L_2+b_1L_1\\right)\\nu _3,\\\\ \\\\\\dot{\\nu }_2=\\left(aL_1+b_1L_2+b_2L_3\\right)\\nu _3-\\left(a_2L_3+b_2L_1\\right)\\nu _1,\\\\ \\\\\\dot{\\nu }_3=-\\left(aL_1+b_1L_2+b_2L_3\\right)\\nu _2+\\left(a_1L_2+b_1L_1\\right)\\nu _1.\\end{array}$ Here $\\nu _1$ , $\\nu _2$ , $\\nu _3$ are projections of the vector $\\gamma $ onto the $O\\eta _1\\eta _2\\eta _3$ axes and $\\Gamma =Mg\\rho $ , where $\\rho =\\sqrt{x_1^2+x_2^2+x_3^2}$ .", "We will use the special coordinate system $O\\eta _1\\eta _2\\eta _3$ for the description of motion of a heavy rigid body with a fixed point in the Hess case.", "In the problem of motion of a heavy rigid body with a fixed point in the Hess case the transformation from the principal axes of inertia $Ox_1x_2x_3$ (the unit vectors ${\\bf e}_1$ , ${\\bf e}_2$ , ${\\bf e}_3$ ) to the special P. V. Kharlamov coordinate system $O\\eta _1\\eta _2\\eta _3$ (the unit vectors ${\\bf e}_I$ , ${\\bf e}_{II}$ , ${\\bf e}_{III}$ ) is defined by the formulae: ${\\bf e}_I={\\bf e}_1\\cos \\alpha +{\\bf e}_2\\sin \\alpha , \\qquad {\\bf e}_{II}=-{\\bf e}_1\\sin \\alpha +{\\bf e}_2\\cos \\alpha ,\\qquad {\\bf e}_{III}={\\bf e}_3,$ where $\\cos \\alpha $ and $\\sin \\alpha $ equal $\\cos \\alpha =\\frac{x_1}{\\sqrt{x_1^2+x_2^2}}, \\qquad \\sin \\alpha =\\frac{x_2}{\\sqrt{x_1^2+x_2^2}}.$ Let us prove now that the unit vectors ${\\bf e}_I$ , ${\\bf e}_{II}$ , ${\\bf e}_{III}$ are indeed the basis vectors of the special P. V. Kharlamov coordinate system.", "We note< first of all, that $x_3=0$ according to (REF ), and therefore the unit vector ${\\bf e}_I$ of the axis $O\\eta _1$ is indeed collinear to the vector $\\overrightarrow{OG}={\\bf r}$ .", "Now let us make sure that the kinetic energy of the body in this case is represented in the form (REF ).", "For this purpose we write the angular momentum as follows: ${\\bf K}=L_1{\\bf e}_I+L_2{\\bf e}_{II}+L_3{\\bf e}_{III}=\\left(L_1\\cos \\alpha -L_2\\sin \\alpha \\right){\\bf e}_1+\\left(L_1\\sin \\alpha +L_2\\cos \\alpha \\right){\\bf e}_2+L_3{\\bf e}_3.$ In the principal axes of inertia the angular momentum has the form ${\\bf K}=K_1{\\bf e}_1+K_2{\\bf e}_2+K_3{\\bf e}_3=A_1\\omega _1{\\bf e}_1+A_2\\omega _2{\\bf e}_2+A_3\\omega _3{\\bf e}_3,$ and the kinetic energy of the body can be written as follows: $T=\\frac{1}{2}\\left(\\frac{K_1^2}{A_1}+\\frac{K_2^2}{A_2}+\\frac{K_3^2}{A_3}\\right).$ It follows from (REF ), (REF ) that the components $L_1$ , $L_2$ , $L_3$ of the angular momentum ${\\bf K}$ with respect to the $O\\eta _1\\eta _2\\eta _3$ coordinate system are connected with the components $K_1$ , $K_2$ , $K_3$ of this vector with respect to the principal axes of inertia by the formulae: $K_1=L_1\\cos \\alpha -L_2\\sin \\alpha ,\\qquad K_2=L_1\\sin \\alpha +L_2\\cos \\alpha ,\\qquad K_3=L_3.$ If we substitute expressions (REF ) to the formula (REF ) for the kinetic energy of the body and take into account the explicit expressions (REF ) for $\\sin \\alpha $ and $\\cos \\alpha $ , we obtain the following expression for the kinetic energy of the body as a function of $L_1$ , $L_2$ , $L_3$ : $T\\!=\\!\\frac{1}{2}\\left(\\frac{x_1^2}{A_1}\\!+\\!\\frac{x_2^2}{A_2}\\right)\\frac{L_1^2}{\\left(x_1^2\\!+\\!x_2^2\\right)}\\!+\\!\\frac{1}{2}\\left(\\frac{x_1^2}{A_2}\\!+\\!\\frac{x_2^2}{A_1}\\right)\\frac{L_2^2}{\\left(x_1^2\\!+\\!x_2^2\\right)}\\!+\\!\\frac{1}{2}\\frac{L_3^2}{A_3}\\!+\\!\\left(\\frac{1}{A_2}\\!-\\!\\frac{1}{A_1}\\right)\\frac{x_1x_2}{\\left(x_1^2\\!+\\!x_2^2\\right)}L_1L_2.$ Note that expression (REF ) for the kinetic energy of the rigid body does not contain the products $L_1L_3$ and $L_2L_3$ .", "This means, that the kinetic energy of the body in the considered case has the form (REF ), i.e.", "the unit vectors ${\\bf e}_I$ , ${\\bf e}_{II}$ , ${\\bf e}_{III}$ are indeed the basis vectors of the special coordinate system by P. V. Kharlamov [6], [7].", "The coefficients of the kinetic energy (REF ) are such, that $b_2=0,\\quad l_{23}^=0.$ Moreover, according to (REF ) we have: $\\frac{A_1x_1^2+A_2x_2^2}{A_1A_2\\left(x_1^2+x_2^2\\right)}=\\frac{1}{A_3},$ and therefore $a_1=a_2.$ Finally expression (REF ) for the kinetic energy of the body with a fixed point in the Hess case can be written as follows: $T=\\frac{1}{2}aL_1^2+\\frac{1}{2}c\\left(L_2^2+L_3^2\\right)+bL_1L_2,$ where we denote $a=\\frac{A_2x_1^2+A_1x_2^2}{A_1A_2\\left(x_1^2+x_2^2\\right)},\\qquad b=\\frac{\\left(A_1-A_2\\right)x_1x_2}{A_1A_2\\left(x_1^2+x_2^2\\right)},\\qquad c=\\frac{1}{A_3}.$ Taking into account expression (REF ) for the kinetic energy we can write the Euler – Poisson equations (REF ) in the form: $\\begin{array}{l}\\dot{L}_1=-bL_1L_3,\\\\ \\\\\\dot{L}_2=\\left(a-c\\right)L_1L_2+bL_2L_3+\\nu _3\\Gamma ,\\\\ \\\\\\dot{L}_3=-\\left(a-c\\right)L_1L_2+bL_1^2-bL_2^2-\\nu _2\\Gamma ,\\\\ \\\\\\dot{\\nu }_1=cL_3\\nu _2-\\left(cL_2+bL_1\\right)\\nu _3,\\\\ \\\\\\dot{\\nu }_2=-cL_3\\nu _1+\\left(aL_1+bL_2\\right)\\nu _3,\\\\ \\\\\\dot{\\nu }_3=\\left(bL_1+cL_2\\right)\\nu _1-\\left(aL_1+bL_2\\right)\\nu _2.\\end{array}$ To find the additional first integral, existing in the Hess case, we consider the first equation of the system (REF ) $\\dot{L}_1=-bL_1L_3.$ In this equation, the right – hand side is equal to the variable $L_1$ itself, multiplied by the coefficient $-bL_3$ bounded in absolute value.", "This means that if at the initial instant of time the quantity $L_1=0 $ , then we have $L_1\\equiv 0.$ The invariant manifold (REF ) (or, in other notations (REF )) together with (REF ) defines the Hess case.", "Under conditions (REF ), (REF ) equations (REF ) are noticeably simplified and take the form $\\begin{array}{c}\\dot{L}_2=bL_2L_3+\\nu _3\\Gamma ,\\quad \\dot{L}_3=-bL_2^2-\\nu _2\\Gamma ,\\\\ \\\\\\dot{\\nu }_1=cL_3\\nu _2-cL_2\\nu _3,\\quad \\dot{\\nu }_2=bL_2\\nu _3-cL_3\\nu _1,\\quad \\dot{\\nu }_3=cL_2\\nu _1-bL_2\\nu _2.\\end{array}$ Equations (REF ) possess the following first integrals: $\\displaystyle \\frac{c}{2}\\left(L_2^2+L_3^2\\right)+\\Gamma \\nu _1=E;\\quad L_2\\nu _2+L_3\\nu _3=k;\\quad \\nu _1^2+\\nu _2^2+\\nu _3^2=1.$ Note that condition $b=0$ corresponds to the Lagrange integrable case in the problem of motion of a heavy rigid body with a fixed point.", "Now let us write the equations (REF ) and the first integrals (REF ) in dimensionless form.", "For this purpose we introduce the dimensionless components of angular momentum $L_2=\\sqrt{\\frac{\\Gamma }{c}}y,\\qquad L_3=\\sqrt{\\frac{\\Gamma }{c}}z,$ and the dimensionless time $\\tau $ : $t=\\frac{\\tau }{\\sqrt{\\Gamma c}}.$ We introduce also the dimensionless parameter $d_1=\\frac{b}{c}.$ and the dimensionless constants of the first integrals $h=\\frac{E}{\\Gamma },\\qquad k_1=k\\sqrt{\\frac{c}{\\Gamma }}.$ Now we can write equations (REF ) in dimensionless form: $\\begin{array}{c}\\displaystyle \\frac{dy}{d\\tau }=d_1yz+\\nu _3,\\qquad \\displaystyle \\frac{dz}{d\\tau }=-d_1y^2-\\nu _2,\\\\ \\\\\\displaystyle \\frac{d{\\nu }_1}{d\\tau }=z\\nu _2-y\\nu _3,\\quad \\displaystyle \\frac{d{\\nu }_2}{d\\tau }=d_1y\\nu _3-z\\nu _1,\\quad \\displaystyle \\frac{d{\\nu }_3}{d\\tau }=y\\nu _1-d_1y\\nu _2.\\end{array}$ System (REF ) possesses the following first integrals: $\\frac{y^2+z^2}{2}+\\nu _1=h,\\quad y\\nu _2+z\\nu _3=k_1,\\quad \\nu _1^2+\\nu _2^2+\\nu _3^2=1.$ From the system (REF ) using (REF ) we will obtain the second order linear differential equation.", "Before we obtain this equation, let us determine the range of parameters $d_1$ , $h$ , $k_1$ .", "It is easy to see that the parameter $k_1$ ranges in the infinite interval $\\left(-\\infty ,\\, +\\infty \\right)$ .", "Since the expression $\\frac{y^2+z^2}{2}$ is nonnegative, then we have for the parameter $h$ the following inequality $h-\\nu _1\\ge 0,\\quad \\mbox{or}\\quad h\\ge \\nu _1.$ The minimal value of the first component $\\nu _1$ of the vector $\\nu $ equals $-1$ .", "Therefore the parameter $h$ ranges in the interval $h\\in \\left[-1,\\;+\\infty \\right).$ The parameter $d_1$ can be represented as follows: $d_1=\\frac{b}{c}=\\frac{\\left(A_1-A_2\\right)x_1x_2}{\\left(A_1x_1^2+A_2x_2^2\\right)}.$ First of all we note that $d_1<0$ according to (REF ).", "We transform the expression for $d_1$ to the form: $d_1=\\frac{\\left(A_1-A_2\\right)x_1x_2}{\\left(A_1x_1^2+A_2x_2^2\\right)}=\\frac{\\left(A_1-A_2\\right)\\frac{x_1}{x_2}}{\\left(A_1\\frac{x_1^2}{x_2^2}+A_2\\right)}.$ From the conditions (REF ) of existence of the Hess integral, we find $\\frac{x_1^2}{x_2^2}=\\frac{A_2\\left(A_3-A_1\\right)}{A_1\\left(A_2-A_3\\right)},\\quad \\mbox{то есть}\\quad \\frac{x_1}{x_2}=\\frac{\\sqrt{A_2\\left(A_3-A_1\\right)}}{\\sqrt{A_1\\left(A_2-A_3\\right)}}.$ Using this equation we can transform expression (REF ) for the parameter $d_1$ as follows: $d_1=-\\sqrt{\\frac{\\left(A_2-A_3\\right)\\left(A_3-A_1\\right)}{A_1A_2}},\\quad \\mbox{or}\\quad d_1^2=\\frac{\\left(A_2-A_3\\right)\\left(A_3-A_1\\right)}{A_1A_2}.$ Since the moments of inertia satisfy the triangle inequality $A_1+A_3>A_2,\\quad \\mbox{i.e.", "}\\quad \\frac{A_2-A_3}{A_1}<1,$ $A_1+A_2>A_3,\\quad \\mbox{i.e.", "}\\quad \\frac{A_3-A_1}{A_2}<1,$ then taking into account these inequalities we have $d_1^2=\\frac{\\left(A_2-A_3\\right)}{A_1}\\cdot \\frac{\\left(A_3-A_1\\right)}{A_2}<1.$ Since $d_1<0$ then we finally obtain that the parameter $d_1$ ranges in the interval $d_1\\in \\left(-1,\\, 0\\right].$ We obtain now the second order linear differential equation from the system (REF ) using (REF ).", "Multiplying the first equation of the system (REF ) by $y$ and the second by $z$ and adding them, we get: $\\frac{d}{d\\tau }\\left(\\frac{y^2+z^2}{2}\\right)=y\\nu _3-z\\nu _2.$ Using the following identity $\\left(y^2+z^2\\right)\\left(\\nu _2^2+\\nu _3^2\\right)=\\left(y\\nu _2+z\\nu _3\\right)^2+\\left(y\\nu _3-z\\nu _2\\right)^2,$ we find from the first integrals (REF ): $\\nu _1=h-\\frac{y^2+z^2}{2}.$ Therefore we have $\\begin{array}{l}\\nu _2^2+\\nu _3^2=1-\\left(h-\\displaystyle \\frac{y^2+z^2}{2}\\right)^2=1-\\left(\\displaystyle \\frac{y^2+z^2}{2}-h\\right)^2,\\\\ \\\\y\\nu _2+z\\nu _3=k_1.\\end{array}$ Finally we obtain $\\left(y\\nu _3-z\\nu _2\\right)^2=\\left(y^2+z^2\\right)\\left(1-\\left(\\frac{y^2+z^2}{2}-h\\right)^2\\right)-k_1^2.$ We will take that $y\\nu _3-z\\nu _2=-\\sqrt{\\left(y^2+z^2\\right)\\left(1-\\left(\\frac{y^2+z^2}{2}-h\\right)^2\\right)-k_1^2}$ (we can choose the arbitrary sign before the square root in (REF )).", "Taking into account (REF ) we can rewrite (REF ) in the form: $\\frac{d}{d\\tau }\\left(\\frac{y^2+z^2}{2}\\right)=-\\sqrt{\\left(y^2+z^2\\right)\\left[1-\\left(\\frac{y^2+z^2}{2}-h\\right)^2\\right]-k_1^2}.$ Now we multiply the first equation of the system (REF ) by $z$ and the second – by $y$ and subtract the first equation from the second equation.", "As a result we obtain: $y\\frac{dz}{d\\tau }-z\\frac{dy}{d\\tau }=-d_1y^3-y\\nu _2-d_1yz^2-z\\nu _3,$ or, taking into account (REF ) $y\\frac{dz}{d\\tau }-z\\frac{dy}{d\\tau }=-d_1y\\left(y^2+z^2\\right)-k_1.$ We pass now from the variables $y$ and $z$ to the polar coordinates $x$ and $\\varphi $ by putting: $y=x\\cos \\varphi ,\\quad z=x\\sin \\varphi .$ Then for the variables $x$ and $\\varphi $ we have the following system of two differential equations: $\\begin{array}{l}x\\displaystyle \\frac{dx}{d\\tau }=-\\sqrt{x^2\\left[1-\\left(\\displaystyle \\frac{x^2}{2}-h\\right)^2\\right]-k_1^2}, \\\\ \\\\x^2\\displaystyle \\frac{d\\varphi }{d\\tau }=-d_1x^3\\cos \\varphi -k_1.\\end{array}$ From this system we obtain the single first order differential equation for the function $\\varphi =\\varphi \\left(x\\right)$ : $\\frac{d\\varphi }{dx}=\\frac{d_1 x^3\\cos \\varphi +k_1}{x\\sqrt{x^2\\left[1-\\left(\\displaystyle \\frac{x^2}{2}-h\\right)^2\\right]-k_1^2}}.$ Note that when we pass from the system (REF ) to the equation (REF ) we exclude the case $x={\\rm const}$ that is, $y^2+z^2={\\rm const}$ or $\\nu _1={\\rm const}$ from consideration.", "Meanwhile for a heavy rigid body with a fixed point in the Hess case there are steady motions for which $\\nu _1=\\nu _1^0={\\rm const}$ (see, for example [19]).", "The substitution $w=\\tan \\frac{\\varphi }{2}$ reduces (REF ) to the Riccati equation: $\\frac{dw}{dx}+\\frac{\\left(d_1x^3-k_1\\right)w^2-d_1x^3-k_1}{2x\\sqrt{x^2\\left[1-\\left(\\displaystyle \\frac{x^2}{2}-h\\right)^2\\right]-k_1^2}}=0.$ or similarly $\\frac{dw}{dx}=-\\frac{d_1 x^3-k_1}{2x\\sqrt{x^2\\left[1-\\left(\\displaystyle \\frac{x^2}{2}-h\\right)^2\\right]-k_1^2}}\\, w^2+\\frac{d_1 x^3+k_1}{2x\\sqrt{x^2\\left[1-\\left(\\displaystyle \\frac{x^2}{2}-h\\right)^2\\right]-k_1^2}}.$ It is well known from the general theory of ordinary differential equations (see, for example [20]), that if the Riccati equation has the form: $\\frac{dw}{dx}=f_2\\left(x\\right)w^2+f_1\\left(x\\right)w+f_0\\left(x\\right),$ then the substitution of the form $u\\left(x\\right)=\\exp \\left(-\\int f_2\\left(x\\right)w\\left(x\\right)dx\\right)$ reduces it to the second order linear differential equation $f_2\\frac{d^2u}{dx^2}-\\left(\\frac{df_2}{dx}+f_1f_2\\right)\\frac{du}{dx}+f_0f_2^2u=0$ or, if we divide this equation by $f_2$ : $\\frac{d^2u}{dx^2}-\\left(\\frac{1}{f_2}\\frac{df_2}{dx}+f_1\\right)\\frac{du}{dx}+f_0f_2u=0.$ In our case $f_2=-\\frac{d_1x^3-k_1}{2x\\sqrt{x^2\\left[1-\\left(\\displaystyle \\frac{x^2}{2}-h\\right)^2\\right]-k_1^2}},\\quad f_1=0,\\quad f_0=\\frac{d_1x^3+k_1}{2x\\sqrt{x^2\\left[1-\\left(\\displaystyle \\frac{x^2}{2}-h\\right)^2\\right]-k_1^2}}.$ Note that the transition from the equation (REF ) to the equation (REF ) is possible only when $f_2\\ne 0$ .", "Taking into account the fact that $x\\ne {\\rm const}$ , the condition $f_2=0$ is equivalent to the simultaneous fulfillment of the conditions $d_1=0,\\quad k_1=0.$ Under the conditions (REF ), the equation (REF ) gives $\\varphi =\\varphi _0={\\rm const}$ .", "It can be shown (see Appendix) that, under the conditions (REF ) a heavy rigid body with a fixed point in the Hess case will perform pendulum nutational oscillations.", "We will assume further that $f_2\\ne 0$ .", "Thus, the problem of motion of a heavy rigid body with a fixed point in the Hess case is reduced to solving the following second order linear differential equation with the rational coefficients: $\\frac{d^2u}{dx^2}+a\\left(x\\right)\\frac{du}{dx}+b\\left(x\\right)u=0,$ $a\\left(x\\right)=\\frac{d_1x^9-4k_1x^6-4d_1\\left(h^2-1\\right)x^5+12k_1hx^4-8k_1^2d_1x^3-8k_1\\left(h^2-1\\right)x^2-4k_1^3}{x\\left(x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2\\right)\\left(d_1x^3-k_1\\right)},$ $b\\left(x\\right)=\\frac{\\left(d_1x^3+k_1\\right)\\left(d_1x^3-k_1\\right)}{x^2\\left(x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2\\right)}.$ Now we can study the problem of existence of liouvillian solutions for the second order linear differential equation (REF ).", "To solve this problem we can use the Kovacic algorithm [8].", "Below we give a brief description of this algorithm.", "Let us consider the differential field $\\mathbb {C}\\left(x\\right)$ of rational functions of one (in general case complex) variable $x$ .", "We accept the standard notations $\\mathbb {Z}$ and $\\mathbb {Q}$ for the sets of integer and rational numbers respectively.", "Our goal is to find a solution of the differential equation $\\frac{d^2z}{dx^2}+a\\left(x\\right)\\frac{dz}{dx}+b\\left(x\\right)z=0,$ where $a\\left(x\\right), b\\left(x\\right)\\in \\mathbb {C}\\left(x\\right)$ .", "In the paper [8] an algorithm has been proposed that allows one to find explicitly the so-called liouvillian solutions of differential equation (REF ), i.e.", "solutions, that can be expressed in terms of liouvillian functions.", "The main advantage of the Kovacic algorithm is precisely that it allows one not only to establish the existence or nonexistence of a solution of differential equation (REF ) expressed in terms of liouvillian functions, but also to present this solution in an explicit form when it exists.", "In turn, liouvillian functions are elements of a liouvillian field, which is defined in the following way.", "Определение 1 Let $F$ be a differential field of functions of one (in general case complex) variable $x$ that contains $\\mathbb {C}\\left(x\\right)$ ; namely $F$ is a field of characteristic zero with a differentiation operator $\\left(\\right)^{\\prime }$ with the following two properties: $\\left(a+b\\right)^{\\prime }=a^{\\prime }+b^{\\prime }$ and $\\left(ab\\right)^{\\prime }=a^{\\prime }b+ab^{\\prime }$ for any $a$ and $b$ in $F$ .", "The field $F$ is liouvillian if there exists a sequence (tower) of differential fields $\\mathbb {C}\\left(x\\right)=F_0\\subseteq F_1\\subseteq \\ldots \\subseteq F_n=F,$ obtained by adjoining one element such that for any $i~=~1,2,\\ldots , n$ we have: $F_i=F_{i-1}\\left(\\alpha \\right),\\;\\mbox{with}\\;\\, \\frac{\\alpha ^{\\prime }}{\\alpha }\\in F_{i-1}$ (i.e.", "$F_i$ is generated by an exponential of an indefinite integral over $F_{i-1}$ ); or $F_i=F_{i-1}\\left(\\alpha \\right),\\;\\mbox{with}\\;\\, \\alpha ^{\\prime }\\in F_{i-1}$ (i.e.", "$F_i$ is generated by an indefinite integral over $F_{i-1}$ ); or $F_i$ is finite algebraic over $F_{i-1}$ (i.e.", "$F_i=F_{i-1}\\left(\\alpha \\right)$ and $\\alpha $ satisfies a polynomial equation of the form $a_0+a_1\\alpha +\\cdots +a_n\\alpha ^n=0,$ where $a_j\\in F_{i-1}$ , $j=0,1,2,\\ldots , n$ and are not all zero).", "Thus, liouvillian functions are built up sequentially from rational functions by using algebraic operations and the operation of indefinite integration and by taking the exponential of a given expression.", "A solution of equation (REF ) that is expressed in terms of liouvillian functions most closely correspond to the notion of a \"close-form solution\"$\\,$ or a \"solution in quadratures\".", "To reduce differential equation (REF ) to a simpler form, we use the following formula $y\\left(x\\right)=z\\left(x\\right)\\exp \\left(\\frac{1}{2}\\int a(x)dx\\right).$ Then equation (REF ) takes the form: $y^{\\prime \\prime }=R\\left(x\\right)y,\\quad R\\left(x\\right)=\\frac{1}{2}a^{\\prime }+\\frac{1}{4}a^2-b,\\quad R\\left(x\\right)\\in \\mathbb {C}\\left(x\\right).$ Hereinafter, it is assumed that that the second order linear differential equation with which the Kovacic algorithm deals is written in the form (REF ).", "The following theorem which has been proved by J. Kovacic [8], determines the structure of a solution of this differential equation.", "Теорема 1 For the differential equation (REF ) only the following four cases are true.", "The differential equation (REF ) has a solution of the form $\\eta =\\exp \\left(\\int \\omega (x)dx\\right)\\quad \\mbox{e}\\quad \\omega \\left(x\\right)\\in \\mathbb {C}\\left(x\\right)$ (liouvillian solution of type 1).", "The differential equation (REF ) has a solution of the form $\\eta =\\exp \\left(\\int \\omega (x)dx\\right),$ where $\\omega \\left(x\\right)$ is an algebraic function of degree 2 over $\\mathbb {C}\\left(x\\right)$ and case 1 does not hold (liouvillian solution of type 2).", "All solutions of differential equation (REF ) are algebraic over $\\mathbb {C}\\left(x\\right)$ and cases 1 and 2 do not hold.", "In this situation a solution of the differential equation (REF ) has the form $\\eta =\\exp \\left(\\int \\omega (x)dx\\right)$ where $\\omega \\left(x\\right)$ is an algebraic function of degree 4, 6 or 12 over $\\mathbb {C}\\left(x\\right)$ (liouvillian solution of type 3).", "Differential equation (REF ) has no liouvillian solutions.", "The differential equation (REF ) has a solution of the form $\\eta =\\exp \\left(\\int \\omega (x)dx\\right)\\quad \\mbox{e}\\quad \\omega \\left(x\\right)\\in \\mathbb {C}\\left(x\\right)$ (liouvillian solution of type 1).", "The differential equation (REF ) has a solution of the form $\\eta =\\exp \\left(\\int \\omega (x)dx\\right),$ where $\\omega \\left(x\\right)$ is an algebraic function of degree 2 over $\\mathbb {C}\\left(x\\right)$ and case 1 does not hold (liouvillian solution of type 2).", "All solutions of differential equation (REF ) are algebraic over $\\mathbb {C}\\left(x\\right)$ and cases 1 and 2 do not hold.", "In this situation a solution of the differential equation (REF ) has the form $\\eta =\\exp \\left(\\int \\omega (x)dx\\right)$ where $\\omega \\left(x\\right)$ is an algebraic function of degree 4, 6 or 12 over $\\mathbb {C}\\left(x\\right)$ (liouvillian solution of type 3).", "Differential equation (REF ) has no liouvillian solutions.", "In order for one of the first three cases listed in Theorem REF to take place the function $R\\left(x\\right)$ in the right hand side of equation (REF ) must satisfy certain conditions.", "These conditions are necessary but not sufficient.", "For example, if the conditions corresponding to Case 1 of Theorem REF are violated, then we must turn to the verification of the conditions corresponding to Cases 2 and 3.", "If these conditions are fulfilled, then we must search for solutions of equation (REF ) exactly in the form, indicated for the corresponding case.", "However, the existence of such a solution is not guaranteed.", "In order to explain the sense of the necessary conditions mentioned, we recall some facts from complex analysis.", "Recall that any analytic function $f$ of a complex variable $z$ can be expanded in a Laurent series in a neighborhood of any point $a$ as follows: $f\\left(z\\right)=a_0+a_1\\left(z-a\\right)+a_2\\left(z-a\\right)^2+\\cdots +\\frac{a_{-1}}{z-a}+\\frac{a_{-2}}{\\left(z-a\\right)^2}+\\cdots .$ The part of this series $a_0+a_1\\left(z-a\\right)+a_2\\left(z-a\\right)^2+\\cdots $ containing nonnegative powers of $z-a$ is called the analytic part of the Laurent series.", "Whereas the other part, namely $\\frac{a_{-1}}{z-a}+\\frac{a_{-2}}{\\left(z-a\\right)^2}+\\cdots $ is called the principal part of the expansion.", "By definition, a point $a$ is called a pole of $f\\left(z\\right)$ of order $n$ if the principal part of the Laurent expansion contains a finite number of terms and the last term has the form $\\frac{a_{-n}}{\\left(z-a\\right)^n}.$ If $f\\left(z\\right)$ is a rational function of $z$ , then a point $a$ is a pole of $f\\left(z\\right)$ of order $n$ if it is a root of the denominator of $f\\left(z\\right)$ of multiplicity $n$ .", "Let $z=\\infty $ be a zero of a function $f\\left(z\\right)$ of order $n$ (i.e., $n$ is the order of the pole at $z=0$ of $f\\left(z\\right)$ ).", "Then we say that $n$ is the order of $f\\left(z\\right)$ at $z=\\infty $ .", "If $f\\left(z\\right)$ is a rational function, then its order at $z=\\infty $ is the difference between the degrees of the denominator and the numerator.", "The following theorem, which has been proved in [8], specifies conditions, that are necessary for one of the first three cases listed in Theorem REF can hold.", "Теорема 2 For the differential equation (REF ) the following conditions are necessary for one of the first cases listed in Theorem REF to hold, i.e.", "for equation (REF ) to have a liouvillian solution of the type specified in description of the corresponding case.", "Each pole of the function $R\\left(x\\right)$ must have even order or else have order 1.", "The order of $R\\left(x\\right)$ at $x=\\infty $ must be even or else be greater than 2.", "The function $R\\left(x\\right)$ must have at least one pole that either has odd order greater than 2 or else has order 2.", "The order of a pole of $R\\left(x\\right)$ cannot exceed 2 and the order of $R\\left(x\\right)$ at $x=\\infty $ must be at least 2.", "If the partial fraction expansion of $R\\left(x\\right)$ has the form $R\\left(x\\right)=\\sum \\limits _i\\frac{\\alpha _i}{\\left(x-c_i\\right)^2}+\\sum \\limits _j\\frac{\\beta _j}{x-d_j},$ then for each $i$ $\\sqrt{1+4\\alpha _i}\\in \\mathbb {Q},\\quad \\sum \\limits _j\\beta _j=0$ and if $\\gamma =\\sum \\limits _i\\alpha _i+\\sum \\limits _j\\beta _j d_j,$ then $\\sqrt{1+4\\gamma }\\in \\mathbb {Q}.$ Each pole of the function $R\\left(x\\right)$ must have even order or else have order 1.", "The order of $R\\left(x\\right)$ at $x=\\infty $ must be even or else be greater than 2.", "The function $R\\left(x\\right)$ must have at least one pole that either has odd order greater than 2 or else has order 2.", "The order of a pole of $R\\left(x\\right)$ cannot exceed 2 and the order of $R\\left(x\\right)$ at $x=\\infty $ must be at least 2.", "If the partial fraction expansion of $R\\left(x\\right)$ has the form $R\\left(x\\right)=\\sum \\limits _i\\frac{\\alpha _i}{\\left(x-c_i\\right)^2}+\\sum \\limits _j\\frac{\\beta _j}{x-d_j},$ then for each $i$ $\\sqrt{1+4\\alpha _i}\\in \\mathbb {Q},\\quad \\sum \\limits _j\\beta _j=0$ and if $\\gamma =\\sum \\limits _i\\alpha _i+\\sum \\limits _j\\beta _j d_j,$ then $\\sqrt{1+4\\gamma }\\in \\mathbb {Q}.$ To find a liouvillian solution of type 1 of the differential equation (REF ), the Kovacic algorithm is stated in the following way (see [8] for details).", "We assume that the necessary conditions for the existence of a solution in case 1 are satisfied and denote the set of finite poles of the function $R\\left(x\\right)$ by $\\Gamma $ .", "Step 1.", "For each $c\\in \\Gamma \\bigcup \\left\\lbrace \\infty \\right\\rbrace $ we define a rational function $\\left[\\sqrt{R}\\right]_c$ and two complex numbers $\\alpha _{c}^{+}$ and $\\alpha _{c}^{-}$ as described below.", "$\\left(c_1\\right)$ If $c\\in \\Gamma $ is a pole of order 1, then $\\left[\\sqrt{R}\\right]_c=0,\\quad \\alpha _{c}^{+}=\\alpha _{c}^{-}=1.$ $\\left(c_2\\right)$ If $c\\in \\Gamma $ is a pole of order 2, then $\\left[\\sqrt{R}\\right]_c=0.$ Let $b$ be the coefficient of $\\displaystyle \\frac{1}{\\left(x-c\\right)^2}$ in the partial fraction expansion of $R\\left(x\\right)$ .", "Then $\\alpha _{c}^{\\pm }=\\frac{1}{2}\\pm \\frac{1}{2}\\sqrt{1+4b}.$ $\\left(c_3\\right)$ If $c\\in \\Gamma $ is a pole of order $2\\nu \\ge 4$ (the order must be even due to the necessary conditions, stated in Theorem REF ), then $\\left[\\sqrt{R}\\right]_c$ is the sum of terms involving $\\displaystyle \\frac{1}{\\left(x-c\\right)^i}$ for $2\\le i\\le \\nu $ in the Laurent expansion of $\\sqrt{R}$ at $c$ .", "There are two possibilities for $\\left[\\sqrt{R}\\right]_c$ that differ by sign; we can choose one of them.", "Thus, $\\left[\\sqrt{R}\\right]_c=\\frac{a}{\\left(x-c\\right)^{\\nu }}+\\cdots +\\frac{d}{\\left(x-c\\right)^2}.$ Let $b$ be the coefficient of $\\displaystyle \\frac{1}{\\left(x-c\\right)^{\\nu +1}}$ in $R-\\left[\\sqrt{R}\\right]^2_c$ .", "Then $\\alpha _{c}^{\\pm }=\\frac{1}{2}\\left(\\pm \\frac{b}{a}+\\nu \\right).$ $\\left(\\infty _1\\right)$ If the order of $R\\left(x\\right)$ at $x=\\infty $ is greater than 2, then $\\left[\\sqrt{R}\\right]_{\\infty }=0,\\quad \\alpha _{\\infty }^{+}=1,\\quad \\alpha _{\\infty }^{-}=0.$ $\\left(\\infty _2\\right)$ If the order of $R\\left(x\\right)$ at $x=\\infty $ is 2, then $\\left[\\sqrt{R}\\right]_{\\infty }=0.$ Let $b$ be the coefficient of $\\displaystyle \\frac{1}{x^2}$ in the Laurent series expansion of $R\\left(x\\right)$ at $x=\\infty $ .", "Then $\\alpha _{\\infty }^{\\pm }=\\frac{1}{2}\\pm \\frac{1}{2}\\sqrt{1+4b}.$ $\\left(\\infty _3\\right)$ If the order $R\\left(x\\right)$ at $x=\\infty $ is $-2\\nu \\le 0$ (it is even due to the necessary conditions stated in Theorem REF ), then the function $\\left[\\sqrt{R}\\right]_{\\infty }$ is the sum of terms involving $x^i$ , $0\\le i\\le \\nu $ of the Laurent expansion of $\\sqrt{R}$ at $x=\\infty $ (one of the two possibilities can be chosen).", "Thus, $\\left[\\sqrt{R}\\right]_{\\infty }=ax^{\\nu }+\\cdots +d.$ Let $b$ be the coefficient of $x^{\\nu -1}$ in $R-\\left(\\left[\\sqrt{R}\\right]_{\\infty }\\right)^2$ .", "Then we have: $\\alpha _{\\infty }^{\\pm }=\\frac{1}{2}\\left(\\pm \\frac{b}{a}-\\nu \\right).$ Step 2.", "For each family $s=\\left(s\\left(c\\right)\\right)_{c\\in \\Gamma \\bigcup \\left\\lbrace \\infty \\right\\rbrace }$ , where $s\\left(c\\right)$ are either $+$ or $-$ let $d=\\alpha _{\\infty }^{s\\left(\\infty \\right)}-\\sum \\limits _{c\\in \\Gamma }\\alpha _c^{s\\left(c\\right)}.$ If $d$ is a non – negative integer, then we introduce the function $\\theta =\\sum \\limits _{c\\in \\Gamma }\\left(s\\left(c\\right)\\left[\\sqrt{R}\\right]_{c}+\\frac{\\alpha _{c}^{s\\left(c\\right)}}{x-c}\\right)+s\\left(\\infty \\right)\\left[\\sqrt{R}\\right]_{\\infty }$ If $d$ is not a non – negative integer, then the family $s$ should be discarded.", "If all tuples $s$ have been rejected, then Case 1 cannot hold.", "Step 3.", "For each family $s$ from Step 2, we search for a monic polynomial $P$ of degree $d$ (the constant $d$ is defined by the formula (REF )), satisfying the differential equation $P^{\\prime \\prime }+2\\theta P^{\\prime }+\\left(\\theta ^{\\prime }+\\theta ^2-R\\right)P=0.$ If such a polynomial exists, then $\\eta =P\\exp \\left(\\int \\theta \\left(x\\right)dx\\right)$ is the solution of the differential equation (REF ).", "If for each tuple $s$ found on Step 2, we cannot find such a polynomial $P$ , then Case 1 cannot hold for the differential equation (REF ).", "For each $c\\in \\Gamma \\bigcup \\left\\lbrace \\infty \\right\\rbrace $ we define a rational function $\\left[\\sqrt{R}\\right]_c$ and two complex numbers $\\alpha _{c}^{+}$ and $\\alpha _{c}^{-}$ as described below.", "$\\left(c_1\\right)$ If $c\\in \\Gamma $ is a pole of order 1, then $\\left[\\sqrt{R}\\right]_c=0,\\quad \\alpha _{c}^{+}=\\alpha _{c}^{-}=1.$ $\\left(c_2\\right)$ If $c\\in \\Gamma $ is a pole of order 2, then $\\left[\\sqrt{R}\\right]_c=0.$ Let $b$ be the coefficient of $\\displaystyle \\frac{1}{\\left(x-c\\right)^2}$ in the partial fraction expansion of $R\\left(x\\right)$ .", "Then $\\alpha _{c}^{\\pm }=\\frac{1}{2}\\pm \\frac{1}{2}\\sqrt{1+4b}.$ $\\left(c_3\\right)$ If $c\\in \\Gamma $ is a pole of order $2\\nu \\ge 4$ (the order must be even due to the necessary conditions, stated in Theorem REF ), then $\\left[\\sqrt{R}\\right]_c$ is the sum of terms involving $\\displaystyle \\frac{1}{\\left(x-c\\right)^i}$ for $2\\le i\\le \\nu $ in the Laurent expansion of $\\sqrt{R}$ at $c$ .", "There are two possibilities for $\\left[\\sqrt{R}\\right]_c$ that differ by sign; we can choose one of them.", "Thus, $\\left[\\sqrt{R}\\right]_c=\\frac{a}{\\left(x-c\\right)^{\\nu }}+\\cdots +\\frac{d}{\\left(x-c\\right)^2}.$ Let $b$ be the coefficient of $\\displaystyle \\frac{1}{\\left(x-c\\right)^{\\nu +1}}$ in $R-\\left[\\sqrt{R}\\right]^2_c$ .", "Then $\\alpha _{c}^{\\pm }=\\frac{1}{2}\\left(\\pm \\frac{b}{a}+\\nu \\right).$ $\\left(\\infty _1\\right)$ If the order of $R\\left(x\\right)$ at $x=\\infty $ is greater than 2, then $\\left[\\sqrt{R}\\right]_{\\infty }=0,\\quad \\alpha _{\\infty }^{+}=1,\\quad \\alpha _{\\infty }^{-}=0.$ $\\left(\\infty _2\\right)$ If the order of $R\\left(x\\right)$ at $x=\\infty $ is 2, then $\\left[\\sqrt{R}\\right]_{\\infty }=0.$ Let $b$ be the coefficient of $\\displaystyle \\frac{1}{x^2}$ in the Laurent series expansion of $R\\left(x\\right)$ at $x=\\infty $ .", "Then $\\alpha _{\\infty }^{\\pm }=\\frac{1}{2}\\pm \\frac{1}{2}\\sqrt{1+4b}.$ $\\left(\\infty _3\\right)$ If the order $R\\left(x\\right)$ at $x=\\infty $ is $-2\\nu \\le 0$ (it is even due to the necessary conditions stated in Theorem REF ), then the function $\\left[\\sqrt{R}\\right]_{\\infty }$ is the sum of terms involving $x^i$ , $0\\le i\\le \\nu $ of the Laurent expansion of $\\sqrt{R}$ at $x=\\infty $ (one of the two possibilities can be chosen).", "Thus, $\\left[\\sqrt{R}\\right]_{\\infty }=ax^{\\nu }+\\cdots +d.$ Let $b$ be the coefficient of $x^{\\nu -1}$ in $R-\\left(\\left[\\sqrt{R}\\right]_{\\infty }\\right)^2$ .", "Then we have: $\\alpha _{\\infty }^{\\pm }=\\frac{1}{2}\\left(\\pm \\frac{b}{a}-\\nu \\right).$ If $c\\in \\Gamma $ is a pole of order 1, then $\\left[\\sqrt{R}\\right]_c=0,\\quad \\alpha _{c}^{+}=\\alpha _{c}^{-}=1.$ If $c\\in \\Gamma $ is a pole of order 2, then $\\left[\\sqrt{R}\\right]_c=0.$ Let $b$ be the coefficient of $\\displaystyle \\frac{1}{\\left(x-c\\right)^2}$ in the partial fraction expansion of $R\\left(x\\right)$ .", "Then $\\alpha _{c}^{\\pm }=\\frac{1}{2}\\pm \\frac{1}{2}\\sqrt{1+4b}.$ If $c\\in \\Gamma $ is a pole of order $2\\nu \\ge 4$ (the order must be even due to the necessary conditions, stated in Theorem REF ), then $\\left[\\sqrt{R}\\right]_c$ is the sum of terms involving $\\displaystyle \\frac{1}{\\left(x-c\\right)^i}$ for $2\\le i\\le \\nu $ in the Laurent expansion of $\\sqrt{R}$ at $c$ .", "There are two possibilities for $\\left[\\sqrt{R}\\right]_c$ that differ by sign; we can choose one of them.", "Thus, $\\left[\\sqrt{R}\\right]_c=\\frac{a}{\\left(x-c\\right)^{\\nu }}+\\cdots +\\frac{d}{\\left(x-c\\right)^2}.$ Let $b$ be the coefficient of $\\displaystyle \\frac{1}{\\left(x-c\\right)^{\\nu +1}}$ in $R-\\left[\\sqrt{R}\\right]^2_c$ .", "Then $\\alpha _{c}^{\\pm }=\\frac{1}{2}\\left(\\pm \\frac{b}{a}+\\nu \\right).$ If the order of $R\\left(x\\right)$ at $x=\\infty $ is greater than 2, then $\\left[\\sqrt{R}\\right]_{\\infty }=0,\\quad \\alpha _{\\infty }^{+}=1,\\quad \\alpha _{\\infty }^{-}=0.$ If the order of $R\\left(x\\right)$ at $x=\\infty $ is 2, then $\\left[\\sqrt{R}\\right]_{\\infty }=0.$ Let $b$ be the coefficient of $\\displaystyle \\frac{1}{x^2}$ in the Laurent series expansion of $R\\left(x\\right)$ at $x=\\infty $ .", "Then $\\alpha _{\\infty }^{\\pm }=\\frac{1}{2}\\pm \\frac{1}{2}\\sqrt{1+4b}.$ If the order $R\\left(x\\right)$ at $x=\\infty $ is $-2\\nu \\le 0$ (it is even due to the necessary conditions stated in Theorem REF ), then the function $\\left[\\sqrt{R}\\right]_{\\infty }$ is the sum of terms involving $x^i$ , $0\\le i\\le \\nu $ of the Laurent expansion of $\\sqrt{R}$ at $x=\\infty $ (one of the two possibilities can be chosen).", "Thus, $\\left[\\sqrt{R}\\right]_{\\infty }=ax^{\\nu }+\\cdots +d.$ Let $b$ be the coefficient of $x^{\\nu -1}$ in $R-\\left(\\left[\\sqrt{R}\\right]_{\\infty }\\right)^2$ .", "Then we have: $\\alpha _{\\infty }^{\\pm }=\\frac{1}{2}\\left(\\pm \\frac{b}{a}-\\nu \\right).$ For each family $s=\\left(s\\left(c\\right)\\right)_{c\\in \\Gamma \\bigcup \\left\\lbrace \\infty \\right\\rbrace }$ , where $s\\left(c\\right)$ are either $+$ or $-$ let $d=\\alpha _{\\infty }^{s\\left(\\infty \\right)}-\\sum \\limits _{c\\in \\Gamma }\\alpha _c^{s\\left(c\\right)}.$ If $d$ is a non – negative integer, then we introduce the function $\\theta =\\sum \\limits _{c\\in \\Gamma }\\left(s\\left(c\\right)\\left[\\sqrt{R}\\right]_{c}+\\frac{\\alpha _{c}^{s\\left(c\\right)}}{x-c}\\right)+s\\left(\\infty \\right)\\left[\\sqrt{R}\\right]_{\\infty }$ If $d$ is not a non – negative integer, then the family $s$ should be discarded.", "If all tuples $s$ have been rejected, then Case 1 cannot hold.", "For each family $s$ from Step 2, we search for a monic polynomial $P$ of degree $d$ (the constant $d$ is defined by the formula (REF )), satisfying the differential equation $P^{\\prime \\prime }+2\\theta P^{\\prime }+\\left(\\theta ^{\\prime }+\\theta ^2-R\\right)P=0.$ If such a polynomial exists, then $\\eta =P\\exp \\left(\\int \\theta \\left(x\\right)dx\\right)$ is the solution of the differential equation (REF ).", "If for each tuple $s$ found on Step 2, we cannot find such a polynomial $P$ , then Case 1 cannot hold for the differential equation (REF ).", "Now we state the Kovacic algorithm to search for a solution of type 2 of differential equation (REF ).", "We denote the set of finite poles of the function $R\\left(x\\right)$ by $\\Gamma $ .", "Step 1.", "For each $c\\in \\Gamma \\bigcup \\left\\lbrace \\infty \\right\\rbrace $ we define the set $E_c$ as follows.", "$\\left(c_1\\right)$ If $c\\in \\Gamma $ is a pole of order 1, then $E_c=\\left\\lbrace 4\\right\\rbrace .$ $\\left(c_2\\right)$ If $c\\in \\Gamma $ is a pole of order 2 and if $b$ is the coefficient of $\\frac{1}{\\left(x-c\\right)^2}$ in the partial fraction expansion of $R\\left(x\\right)$ , then $E_c=\\left\\lbrace \\left(2+k\\sqrt{1+4b}\\right)\\bigcap \\mathbb {Z}\\right\\rbrace ,\\; k=0, \\pm 2.$ $\\left(c_3\\right)$ If $c\\in \\Gamma $ is a pole of order $\\nu >2$ , then $E_c=\\left\\lbrace \\nu \\right\\rbrace .$ $\\left(\\infty _1\\right)$ If $R\\left(x\\right)$ has order $>2$ at $x=\\infty $ , then $E_{\\infty }=\\left\\lbrace 0, 2, 4\\right\\rbrace .$ $\\left(\\infty _2\\right)$ If $R\\left(x\\right)$ has order 2 at $x=\\infty $ and $b$ is the coefficient of $\\frac{1}{x^2}$ in the Laurent series expansion of $R$ at $x=\\infty $ , then $E_{\\infty }=\\left\\lbrace \\left(2+k\\sqrt{1+4b}\\right)\\bigcap \\mathbb {Z}\\right\\rbrace ,\\; k=0, \\pm 2.$ $\\left(\\infty _3\\right)$ If $R\\left(x\\right)$ has order $\\nu <2$ at $x=\\infty $ , then $E_{\\infty }=\\left\\lbrace \\nu \\right\\rbrace .$ Step 2.", "Let us consider the families $s=\\left(e_\\infty , e_c\\right)$ , $c\\in \\Gamma $ , where $e_c\\in E_c$ , $e_\\infty \\in E_\\infty $ and at least one of these numbers is odd.", "Let $d=\\frac{1}{2}\\left(e_{\\infty }-\\sum \\limits _{c\\in \\Gamma } e_c\\right).$ If $d$ is a non – negative integer, the family should be retained, otherwise it should be discarded.", "Step 3.", "For each family retained from Step 2, we form the rational function $\\theta =\\frac{1}{2}\\sum \\limits _{c\\in \\Gamma }\\frac{e_c}{x-c}$ and search for a monic polynomial $P$ of degree $d$ (the constant $d$ is defined by the formula (REF )) such, that $P^{\\prime \\prime \\prime }+3\\theta P^{\\prime \\prime }+\\left(3\\theta ^2+3\\theta ^{\\prime }-4R\\right)P^{\\prime }+\\left(\\theta ^{\\prime \\prime }+3\\theta \\theta ^{\\prime }+\\theta ^3-4R\\theta -2R^{\\prime }\\right)P=0.$ If success is achieved, we set $\\varphi =\\theta +\\frac{P^{\\prime }}{P}$ and let $\\omega $ be a solution of the quadratic equation (algebraic equation of degree 2) of the form: $\\omega ^2-\\varphi \\omega +\\frac{1}{2}\\varphi ^{\\prime }+\\frac{1}{2}\\varphi ^2-R=0.$ Then $\\eta =\\exp \\left(\\int \\omega \\left(x\\right)dx\\right) -$ is a solution of the differential equation (REF ).", "If success is not achieved, then Case 2 cannot hold for the differential equation (REF ).", "For each $c\\in \\Gamma \\bigcup \\left\\lbrace \\infty \\right\\rbrace $ we define the set $E_c$ as follows.", "$\\left(c_1\\right)$ If $c\\in \\Gamma $ is a pole of order 1, then $E_c=\\left\\lbrace 4\\right\\rbrace .$ $\\left(c_2\\right)$ If $c\\in \\Gamma $ is a pole of order 2 and if $b$ is the coefficient of $\\frac{1}{\\left(x-c\\right)^2}$ in the partial fraction expansion of $R\\left(x\\right)$ , then $E_c=\\left\\lbrace \\left(2+k\\sqrt{1+4b}\\right)\\bigcap \\mathbb {Z}\\right\\rbrace ,\\; k=0, \\pm 2.$ $\\left(c_3\\right)$ If $c\\in \\Gamma $ is a pole of order $\\nu >2$ , then $E_c=\\left\\lbrace \\nu \\right\\rbrace .$ $\\left(\\infty _1\\right)$ If $R\\left(x\\right)$ has order $>2$ at $x=\\infty $ , then $E_{\\infty }=\\left\\lbrace 0, 2, 4\\right\\rbrace .$ $\\left(\\infty _2\\right)$ If $R\\left(x\\right)$ has order 2 at $x=\\infty $ and $b$ is the coefficient of $\\frac{1}{x^2}$ in the Laurent series expansion of $R$ at $x=\\infty $ , then $E_{\\infty }=\\left\\lbrace \\left(2+k\\sqrt{1+4b}\\right)\\bigcap \\mathbb {Z}\\right\\rbrace ,\\; k=0, \\pm 2.$ $\\left(\\infty _3\\right)$ If $R\\left(x\\right)$ has order $\\nu <2$ at $x=\\infty $ , then $E_{\\infty }=\\left\\lbrace \\nu \\right\\rbrace .$ If $c\\in \\Gamma $ is a pole of order 1, then $E_c=\\left\\lbrace 4\\right\\rbrace .$ If $c\\in \\Gamma $ is a pole of order 2 and if $b$ is the coefficient of $\\frac{1}{\\left(x-c\\right)^2}$ in the partial fraction expansion of $R\\left(x\\right)$ , then $E_c=\\left\\lbrace \\left(2+k\\sqrt{1+4b}\\right)\\bigcap \\mathbb {Z}\\right\\rbrace ,\\; k=0, \\pm 2.$ If $c\\in \\Gamma $ is a pole of order $\\nu >2$ , then $E_c=\\left\\lbrace \\nu \\right\\rbrace .$ If $R\\left(x\\right)$ has order $>2$ at $x=\\infty $ , then $E_{\\infty }=\\left\\lbrace 0, 2, 4\\right\\rbrace .$ If $R\\left(x\\right)$ has order 2 at $x=\\infty $ and $b$ is the coefficient of $\\frac{1}{x^2}$ in the Laurent series expansion of $R$ at $x=\\infty $ , then $E_{\\infty }=\\left\\lbrace \\left(2+k\\sqrt{1+4b}\\right)\\bigcap \\mathbb {Z}\\right\\rbrace ,\\; k=0, \\pm 2.$ If $R\\left(x\\right)$ has order $\\nu <2$ at $x=\\infty $ , then $E_{\\infty }=\\left\\lbrace \\nu \\right\\rbrace .$ Let us consider the families $s=\\left(e_\\infty , e_c\\right)$ , $c\\in \\Gamma $ , where $e_c\\in E_c$ , $e_\\infty \\in E_\\infty $ and at least one of these numbers is odd.", "Let $d=\\frac{1}{2}\\left(e_{\\infty }-\\sum \\limits _{c\\in \\Gamma } e_c\\right).$ If $d$ is a non – negative integer, the family should be retained, otherwise it should be discarded.", "For each family retained from Step 2, we form the rational function $\\theta =\\frac{1}{2}\\sum \\limits _{c\\in \\Gamma }\\frac{e_c}{x-c}$ and search for a monic polynomial $P$ of degree $d$ (the constant $d$ is defined by the formula (REF )) such, that $P^{\\prime \\prime \\prime }+3\\theta P^{\\prime \\prime }+\\left(3\\theta ^2+3\\theta ^{\\prime }-4R\\right)P^{\\prime }+\\left(\\theta ^{\\prime \\prime }+3\\theta \\theta ^{\\prime }+\\theta ^3-4R\\theta -2R^{\\prime }\\right)P=0.$ If success is achieved, we set $\\varphi =\\theta +\\frac{P^{\\prime }}{P}$ and let $\\omega $ be a solution of the quadratic equation (algebraic equation of degree 2) of the form: $\\omega ^2-\\varphi \\omega +\\frac{1}{2}\\varphi ^{\\prime }+\\frac{1}{2}\\varphi ^2-R=0.$ Then $\\eta =\\exp \\left(\\int \\omega \\left(x\\right)dx\\right) -$ is a solution of the differential equation (REF ).", "If success is not achieved, then Case 2 cannot hold for the differential equation (REF ).", "Similarly the Kovacic algorithm is stated to search for a liouvillian solutions of type 3 of the differential equation (REF ).", "Let us apply now this algorithm to search liouvillian solutions of the second order linear differential equation (REF ).", "So, the differential equation being investigated has the form (REF ).", "In this equation we make a substitution according to (REF ) and reduce it to the form (REF ): $\\frac{d^2y}{dx^2}=R\\left(x\\right)y.$ Here the function $R\\left(x\\right)$ takes the form: $R\\left(x\\right)=\\frac{U\\left(x\\right)}{V\\left(x\\right)},$ $\\begin{array}{l}U\\left(x\\right)\\!=\\!-\\left(1+4d_1^2\\right)d_1^2x^{16}+8\\left(2d_1^2-1\\right)d_1^2hx^{14}+4\\left(2d_1^2+5\\right)d_1k_1x^{13}-8\\left(2d_1^2-7\\right)\\left(h^2-1\\right)d_1^2x^{12}\\!-\\\\ \\\\-8\\left(4d_1^2+19\\right)d_1k_1hx^{11}+8\\left(\\left(1+17d_1^2-2d_1^4\\right)k_1^2-12\\left(h^2-1\\right)d_1^2h\\right)x^{10}+576\\left(h^2-1\\right)k_1^3d_1x^3+\\\\ \\\\+16\\left(2\\left(h^2-1\\right)d_1^2+29h^2-5\\right)d_1k_1x^9+8\\left(6\\left(h^2-1\\right)^2d_1^2-\\left(7+40d_1^2\\right)k_1^2h\\right)x^8-240k_1^4hx^2+\\\\ \\\\+32\\left(\\left(d_1^2-2\\right)k_1^2-21\\left(h^2-1\\right)h\\right)d_1k_1x^7+16\\left(14\\left(h^2-1\\right)d_1^2+8h^2-5\\right)k_1^2x^6+288k_1^5d_1x+\\\\ \\\\+96\\left(4\\left(h^2-1\\right)^2-3k_1^2h\\right)d_1k_1x^5+4\\left(\\left(32d_1^2+35\\right)k_1^2-24\\left(h^2-1\\right)h\\right)k_1^2x^4+48\\left(h^2-1\\right)k_1^4.\\end{array}$ $V\\left(x\\right)=4\\left(d_1x^3-k_1\\right)^2\\left(x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2\\right)^2.$ Thus, it is easy to see, that the function $R\\left(x\\right)$ has nine finite poles of the second order.", "Let us denote the roots of the polynomial $x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2=0$ by $x_1$ , $x_2$ , $x_3$ , $x_4$ , $x_5$ , $x_6$ .", "Note that this polynomial contains only the terms of even degree, therefore its roots satisfy the conditions: $x_2=-x_1,\\qquad x_4=-x_3,\\qquad x_6=-x_5.$ Let us denote the roots of the polynomial $d_1x^3-k_1=0$ by $x_7$ , $x_8$ , $x_9$ .", "Now let us consider the partial fraction expansion of the function $R\\left(x\\right)$ .", "It has the form: $R\\left(x\\right)=-\\frac{3}{16}\\sum \\limits _{i=1}^6\\frac{1}{\\left(x-x_i\\right)^2}+\\sum \\limits _{i=1}^9\\frac{\\gamma _i\\left(x_i\\right)}{x-x_i}+\\frac{3}{4}\\sum \\limits _{i=7}^9\\frac{1}{\\left(x-x_i\\right)^2}.$ The coefficients $\\gamma _i\\left(x_i\\right)$ , $i=1, 2,\\ldots 9$ have a very complicated form and we do not write them explicitly here.", "It is possible to note the following properties of the partial fraction expansion of the function $R\\left(x\\right)$ .", "The coefficients $b_1,\\ldots , b_6$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , $i=1,\\ldots , 6$ are all equal $b_i=-\\frac{3}{16}, \\quad i=1,\\ldots , 6.$ The coefficients $b_7$ , $b_8$ , $b_9$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , $i=7, 8, 9$ are all equal $b_i=\\frac{3}{4}, \\quad i=7, 8, 9.$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ has the form: $R\\left(x\\right)=-\\frac{\\left(1+4d_1^2\\right)}{4x^2}+O\\left(\\frac{1}{x^4}\\right)$ The coefficients $b_1,\\ldots , b_6$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , $i=1,\\ldots , 6$ are all equal $b_i=-\\frac{3}{16}, \\quad i=1,\\ldots , 6.$ The coefficients $b_7$ , $b_8$ , $b_9$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , $i=7, 8, 9$ are all equal $b_i=\\frac{3}{4}, \\quad i=7, 8, 9.$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ has the form: $R\\left(x\\right)=-\\frac{\\left(1+4d_1^2\\right)}{4x^2}+O\\left(\\frac{1}{x^4}\\right)$ Thus, we have $b_{\\infty }=-\\frac{1}{4}-d_1^2,$ and therefore $1+4b_{\\infty }=-4d_1^2$ This means that the numbers $\\alpha ^{\\pm }_{\\infty }$ calculating during the application of the Kovacic algorithm for searching the liouvillian solutions of type 1, are complex numbers if $d_1\\ne 0$ .", "All the remaining numbers $\\alpha ^{\\pm }_c$ are rational.", "They are presented in the following Table.", "Therefore, the number $d$ , calculated by formula (REF ) in the process of searching for liouvillian solutions of type 1, is a complex number for $d_1\\ne 0$ .", "This fact indicates the absence of liouvillian solutions of type 1 for $d_1\\ne 0$ .", "Table: NO_CAPTION Table.", "Numbers $\\alpha ^{\\pm }_c$ for searching liouvillian solutions of type 1.", "Moreover, the coefficient $b_{\\infty }$ coincides with the number $\\gamma $ calculating during the checking of the necessary conditions of existence of liouvillian solutions of type 3 for the differential equation (REF ).", "According to this necessary conditions for existence of liouvillian solutions of type 3 the number $\\sqrt{1+4\\gamma }=\\sqrt{1+4b_{\\infty }}$ should be rational.", "However, when $d_1\\ne 0$ this number is pure imaginary.", "Thus, we can state that for $d_1\\ne 0$ the second order linear differential equation (REF ) (or (REF )) do not have liouvillian solutions of type 3.", "Thus, the following Theorem is valid.", "Теорема 3 If all roots of polynomials (REF ) and (REF ) are distinct and $d_1\\ne 0$ , then the problem of motion of a heavy rigid body with a fixed point in the Hess case has not liouvillian solutions of type 1 and type 3.", "According to Theorem REF , equation (REF ) can have liouvillian solutions of type 1 only when $d_1=0$ , i.e.", "when the moving rigid body with a fixed point is the Lagrange top.", "To search liouvillian solutions of type 1 for the differential equation (REF ) in the Lagrange integrable case we put $d_1=0$ in this equation.", "When $d_1=0$ we can write equation (REF ) as follows: $\\frac{d^2y}{dx^2}=R\\left(x\\right)y=\\frac{U\\left(x\\right)}{V\\left(x\\right)}y,$ $\\begin{array}{l}U\\left(x\\right)=2x^{10}-14hx^8+4\\left(8h^2-5\\right)x^6+\\left(35k_1^2-24h\\left(h^2-1\\right)\\right)x^4-60k_1^2hx^2+12k_1^2\\left(h^2-1\\right),\\end{array}$ $V_1\\left(x\\right)=\\left(x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2\\right)^2.$ We assume that in the equation (REF ) there should be $k_1\\ne 0$ .", "Otherwise, the conditions (REF ) are satisfied and it is impossible to obtain from the equation (REF ) the second-order linear differential equation (REF ) (or (REF )), from which for $d_1=0$ we obtain (REF ).", "It is easy to see that the poles of the function $R\\left(x\\right)$ are the roots of the polynomial (REF ).", "We assume that the polynomial (REF ) has no multiple roots (the possibility of multiple roots for the polynomial (REF ) is considered below, in Section 12).", "Then the function $R\\left(x\\right)$ has six finite poles of the second order.", "We denote these poles by $x_1$ , $x_2$ , $x_3$ , $x_4$ , $x_5$ , $x_6$ .", "Now let us consider the partial fraction expansion of the function $R\\left(x\\right)$ .", "It has the form: $R\\left(x\\right)=-\\frac{3}{16}\\sum \\limits _{i=1}^6\\frac{1}{\\left(x-x_i\\right)^2}+\\sum \\limits _{i=1}^6\\frac{\\gamma _i\\left(x_i\\right)}{x-x_i},$ $\\begin{array}{l}\\gamma _i\\left(x_i\\right)=\\displaystyle \\frac{\\left(48\\left(h^2-1\\right)^3-24k_1^2h\\left(h^2-1\\right)\\left(h^2-13\\right)-\\left(121h^2-177\\right)k_1^4\\right)x_i}{16k_1^2\\left(27k_1^4+8k_1^2h\\left(h^2-9\\right)-16\\left(h^2-1\\right)^2\\right)}+\\\\ \\\\+\\displaystyle \\frac{\\left(32k_1^2\\left(h^4-1\\right)-48h\\left(h^2-1\\right)^2+3k_1^2h\\left(61k_1^2-128h\\right)\\right)x_i^3}{16k_1^2\\left(27k_1^4+8k_1^2h\\left(h^2-9\\right)-16\\left(h^2-1\\right)^2\\right)}+\\\\ \\\\+\\displaystyle \\frac{\\left(48\\left(h^2-1\\right)^2-40k_1^2h\\left(h^2-1\\right)-3\\left(75k_1^2-128h\\right)k_1^2\\right)x_i^5}{64k_1^2\\left(27k_1^4+8k_1^2h\\left(h^2-9\\right)-16\\left(h^2-1\\right)^2\\right)}.\\end{array}$ It is possible to note the following properties of the partial fraction expansion of the function $R\\left(x\\right)$ .", "The coefficients $b_1,\\ldots , b_6$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , $i=1,\\ldots , 6$ are all equal $b_i=-\\frac{3}{16}, \\quad i=1,\\ldots , 6.$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ has the form: $R\\left(x\\right)=\\frac{2}{x^2}+O\\left(\\frac{1}{x^4}\\right).$ The coefficients $b_1,\\ldots , b_6$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , $i=1,\\ldots , 6$ are all equal $b_i=-\\frac{3}{16}, \\quad i=1,\\ldots , 6.$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ has the form: $R\\left(x\\right)=\\frac{2}{x^2}+O\\left(\\frac{1}{x^4}\\right).$ Our goal is to find liouvillian solutions of type 1, type 2 and type 3 of the differential equation (REF ).", "To find liouvillian solutions of (REF ) we will use the Kovacic algorithm.", "First we search liouvillian solutions of type 1.", "According to the Kovacic algorithm, we calculate the numbers $\\alpha ^{\\pm }_c$ .", "For all the roots $x=x_i$ , $i=1,\\ldots , 6$ these numbers are equal $\\alpha _{x_i}^{+}=\\frac{3}{4},\\qquad \\alpha _{x_i}^{-}=\\frac{1}{4}, \\qquad i=1,\\ldots 6.$ The numbers $\\alpha ^{\\pm }_{\\infty }$ are equal $\\alpha ^{+}_{\\infty }=2,\\qquad \\alpha ^{-}_{\\infty }=-1.$ Note that the polynomial (REF ) contains the terms of even degree, therefore its roots can be represented as follows: $x_1=y_1,\\quad x_2=-x_1=-y_1,\\quad x_3=y_2,\\quad x_4=-x_3=-y_2,\\quad x_5=iy_3,\\quad x_6=-x_5=-iy_3.$ Since the numbers $\\alpha ^{\\pm }_{x_i}$ are determined by (REF ) and the numbers $\\alpha ^{\\pm }_{\\infty }$ are determined by (REF ) therefore the number $d$ calculating according to (REF ) can be only zero.", "This takes place for the following sets of signs $+\\,$ and $-:\\,$ $s=\\left(s\\left(\\infty \\right),\\,s\\left(x_1\\right),\\,s\\left(x_2\\right),\\,s\\left(x_3\\right),\\,s\\left(x_4\\right),\\,s\\left(x_5\\right),\\,s\\left(x_6\\right)\\right)$ $\\begin{array}{l}s_1=\\left(+,\\,+,\\,-,\\,-,\\,-,\\,-,\\,-\\right),\\quad s_2=\\left(+,\\,-,\\,+,\\,-,\\,-,\\,-,\\,-\\right), \\quad s_3=\\left(+,\\,-,\\,-,\\,+,\\,-,\\,-,\\,-\\right), \\\\ \\\\s_4=\\left(+,\\,-,\\,-,\\,-,\\,+,\\,-,\\,-\\right),\\quad s_5=\\left(+,\\,-,\\,-,\\,-,\\,-,\\,+,\\,-\\right),\\quad s_6=\\left(+,\\,-,\\,-,\\,-,\\,-,\\,-,\\,+\\right).\\end{array}$ We must check all these sets.", "Let us check now the set $s_1$ .", "For this set of signs let us find the function $\\theta $ according to the formula (REF ).", "This function has the form: $\\theta =\\frac{3}{4\\left(x-y_1\\right)}+\\frac{1}{4\\left(x+y_1\\right)}+\\frac{1}{4\\left(x-y_2\\right)}+\\frac{1}{4\\left(x+y_2\\right)}+\\frac{1}{4\\left(x-iy_3\\right)}+\\frac{1}{4\\left(x+iy_3\\right)}.$ The polynomial $P$ of degree $d=0$ has $P\\equiv 1$ .", "The substitution of the polynomial $P$ and the function $\\theta $ with the differential equation (REF ) reduces it to the form: $\\frac{d\\theta }{dx}+\\theta ^2-R\\left(x\\right)=0.$ The conditions on the parameters $k_1$ and $h$ for which the left hand side of (REF ) becomes zero are the conditions of existence of liouvillian solutions of type 1 for the differential equation (REF ).", "Before we substitute the function $\\theta $ to the equation (REF ), let us simplify it.", "We can rewrite this function in the form: $\\theta =\\frac{1}{2\\left(x-y_1\\right)}+\\frac{1}{4}\\left[\\frac{1}{x-y_1}+\\frac{1}{x+y_1}+\\frac{1}{x-y_2}+\\frac{1}{x+y_2}+\\frac{1}{x-iy_3}+\\frac{1}{x+iy_3}\\right].$ Expression in square brackets can be represented as follows: $\\frac{1}{x-y_1}+\\frac{1}{x+y_1}+\\frac{1}{x-y_2}+\\frac{1}{x+y_2}+\\frac{1}{x-iy_3}+\\frac{1}{x+iy_3}=\\frac{2x\\left(3x^4-8hx^2+4\\left(h^2-1\\right)\\right)}{x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2}.$ Thus, function $\\theta $ has the form: $\\theta =\\frac{1}{2\\left(x-y_1\\right)}+\\frac{x\\left(3x^4-8hx^2+4\\left(h^2-1\\right)\\right)}{2\\left(x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2\\right)}.$ If we substitute this function $\\theta $ to (REF ) we obtain in the left hand side of (REF ) the rational expression.", "The numerator of this expression is the fifth degree polynomial of the form: $P_5\\left(x\\right)=4y_1x^5+\\cdots $ The necessary condition for this polynomial to vanish is the condition $y_1=0$ , that is, the polynomial (REF ) must have a zero root.", "This condition is equivalent to the condition $k_1=0$ .", "However, we previously suggested that $k_1\\ne 0$ .", "Therefore, $y_1\\ne 0$ and we can state that the expression in the left side of the equation (REF ) does not vanish.", "Thus, for the set $s_1$ there are no liouvillian solutions of type 1 in the problem of the motion of a heavy rigid body with a fixed point in the Hess case.", "Similarly, the fact of the absence of liouvillian solutions of type 1 for the sets $s_2, \\ldots , s_6$ is established.", "So, we can state that the following theorem is true.", "Теорема 4 If all roots of polynomial (REF ) are distinct and $d_1=0$ , $k_1\\ne 0$ , then the problem of motion of a heavy rigid body with a fixed point in the Hess case has not liouvillian solutions of type 1.", "Now we continue the study of the existence of liouvillian solutions in the problem of the motion of the Hess top under the additional condition $d_1=0$ , that is, in the problem of the motion of the Lagrange top when the corresponding second-order linear differential equation has the form (REF ).", "Let us study here the problem of the existence of liouvillian solutions of type 2 for the differential equation (REF ).", "According to the Kovacic algorithm for searching the liouvillian solutions of type 2 we should find the sets $E_{x_i}$ , $i=1,\\ldots , 6$ which correspond to finite poles of $R\\left(x\\right)$ and the set $E_{\\infty }$ which corresponds to the pole of $R\\left(x\\right)$ at $x=\\infty $ .", "For the finite poles $x=x_i$ , $i=1,\\ldots , 6$ , which are the roots of the polynomial (REF ), the sets $E_{x_i}$ are all equal $E_{x_i}=\\lbrace 1, 2, 3\\rbrace , \\quad i=1,\\ldots , 6.$ The set $E_{\\infty }$ has the following form: $E_{\\infty }=\\lbrace -4, 2, 8\\rbrace .$ Now we should calculate the constant $d$ using (REF ).", "Note that the minimal value of the sum $\\sum \\limits _{x_i\\in \\Gamma } e_{x_i}$ equals 6.", "Therefore, the maximal value of $d$ , calculated according to (REF ), equals $d=1$ .", "The family $s=\\left(e_{\\infty }, e_1, e_2, e_3, e_4, e_5, e_6\\right)$ corresponding to $d=1$ is $s_1=\\left(8, 1, 1, 1, 1, 1, 1\\right).$ We also have several families, for which we have $d=0$ .", "The following families correspond to $d=0$ : $\\begin{array}{l}s_2=\\left(8, 3, 1, 1, 1, 1, 1\\right),\\quad s_3=\\left(8, 1, 3, 1, 1, 1, 1\\right),\\quad s_4=\\left(8, 1, 1, 3, 1, 1, 1\\right),\\\\ \\\\s_5=\\left(8, 1, 1, 1, 3, 1, 1\\right),\\quad s_6=\\left(8, 1, 1, 1, 1, 3, 1\\right),\\quad s_7=\\left(8, 1, 1, 1, 1, 1, 3\\right),\\\\ \\\\s_8=\\left(8, 2, 2, 1, 1, 1, 1\\right),\\quad s_9=\\left(8, 2, 1, 2, 1, 1, 1\\right),\\quad s_{10}=\\left(8, 2, 1, 1, 2, 1, 1\\right), \\\\ \\\\s_{11}=\\left(8, 2, 1, 1, 1, 2, 1\\right),\\quad s_{12}=\\left(8, 2, 1, 1, 1, 1, 2\\right),\\quad s_{13}=\\left(8, 1, 2, 2, 1, 1, 1\\right),\\\\ \\\\s_{14}=\\left(8, 1, 2, 1, 2, 1, 1\\right),\\quad s_{15}=\\left(8, 1, 2, 1, 1, 2, 1\\right),\\quad s_{16}=\\left(8, 1, 2, 1, 1, 1, 2\\right),\\\\ \\\\s_{17}=\\left(8, 1, 1, 2, 2, 1, 1\\right),\\quad s_{18}=\\left(8, 1, 1, 2, 1, 2, 1\\right),\\quad s_{19}=\\left(8, 1, 1, 2, 1, 1, 2\\right),\\\\ \\\\s_{20}=\\left(8, 1, 1, 1, 2, 2, 1\\right),\\quad s_{21}=\\left(8, 1, 1, 1, 2, 1, 2\\right),\\quad s_{22}=\\left(8, 1, 1, 1, 1, 2, 2\\right).\\end{array}$ We must check all these families.", "We start with the family $s_1$ .", "According to the algorithm let us find the function $\\theta $ by the formula (REF ).", "Since the coefficients at all finite poles are the same, we can write the function $\\theta $ in explicit form.", "For the family $s_1$ this function has the form: $\\theta =\\frac{3x^5-8hx^3+4\\left(h^2-1\\right)x}{x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2}.$ The polynomial $P$ of degree $d=1$ $\\left(P\\equiv x+b\\right)$ should identically satisfy differential equation (REF ).", "After substitution of the polynomial $P$ and the functions $\\theta $ and $R\\left(x\\right)$ in (REF ), we obtain in the left hand side of it the following expression: $\\frac{2Bx\\left(6h-5x^2\\right)}{x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2}.$ This expression becomes identically zero when $B=0$ .", "Therefore, the polynomial $P\\equiv x$ exists for all values $h$ and $k_1$ .", "Thus, we can state the following theorem.", "Теорема 5 In the Lagrange integrable case of motion $d_1=0$ under the Hess conditions (REF ) and also under condition that all the roots of polynomial (REF ) are distinct and $k_1\\ne 0$ , all solutions of a linear differential equation (REF ) are liouvillian solutions of type 2.", "Indeed, it is easy to find the general solution of equation (REF ) for $d_1=0$ .", "In this case the differential equation (REF ) takes the form $\\frac{d^2u}{dx^2}+\\frac{4\\left(x^6-3hx^4+2\\left(h^2-1\\right)x^2+k_1^2\\right)}{x\\left(x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2\\right)}\\frac{du}{dx}-\\frac{k_1^2u}{x^2\\left(x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2\\right)}=0,$ and its general solution can be written as follows: $\\begin{array}{l}u\\left(x\\right)={\\rm C}_1\\exp \\left(\\displaystyle \\int \\displaystyle \\frac{k_1dx}{x\\sqrt{x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2}}\\right)+\\\\ \\\\+{\\rm C}_2\\exp \\left(-\\displaystyle \\int \\displaystyle \\frac{k_1dx}{x\\sqrt{x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2}}\\right).\\end{array}$ If we check the other families $s_2, \\ldots , s_{22}$ we obtain the same results on the existing of liouvillian solutions of type 2 for the differential equation (REF ).", "Now we come back to the investigation of the general case $d_1\\ne 0$ and consider the problem of existence of liouvillian solutions of type 2 for the differential equation (REF ).", "So we are going to study the problem of existence of liouvillian solutions of type 2 for the differential equation (REF ) (or (REF )).", "According to the Kovacic algorithm we firstly define the sets $E_c$ and $E_{\\infty }$ for every pole of the function $R\\left(x\\right)$ .", "For the finite poles $x=x_i$ , $i=1,\\ldots , 6$ , which are roots of the polynomial (REF ), these sets $E_{x_i}$ have the form: $E_{x_i}=\\lbrace 1, 2, 3\\rbrace , \\quad i=1,\\ldots , 6.$ For the finite poles $x=x_i$ , $i=7, 8, 9$ , which are roots of the polynomial (REF ), these sets $E_{x_i}$ have the form: $E_{x_i}=\\lbrace -2, 2, 6\\rbrace , \\quad i=7, 8, 9.$ The set $E_{\\infty }$ contains only one element and this set equals $E_{\\infty }=\\lbrace 2\\rbrace .$ Now we should calculate the constant $d$ using the formula (REF ).", "Note that the minimal values of the sum of the elements of sets corresponding to finite poles is zero.", "Therefore the maximal value of $d$ , calculated according to (REF ), equals $d=1$ .", "The value $d=1$ corresponds to the set $s=\\left(e_{\\infty }, e_1, e_2, e_3, e_4, e_5, e_6, e_7, e_8, e_9\\right)$ , in which the elements $e_{\\infty } $ and $e_i$ , $i=1, 2, \\ldots 9$ are equal $s_1=\\left(2, 1, 1, 1, 1, 1, 1, -2, -2, -2\\right).$ We also have several families for which the constant $d$ , calculated by the formula (REF ), equals to zero.", "The following families correspond to $d=0$ : $\\begin{array}{l}s_2=\\left(2, 3, 1, 1, 1, 1, 1, -2, -2, -2\\right),\\quad s_3=\\left(2, 1, 3, 1, 1, 1, 1, -2, -2, -2\\right),\\\\ \\\\s_4=\\left(2, 1, 1, 3, 1, 1, 1, -2, -2, -2\\right),\\quad s_5=\\left(2, 1, 1, 1, 3, 1, 1, -2, -2, -2\\right),\\\\ \\\\s_6=\\left(2, 1, 1, 1, 1, 3, 1, -2, -2, -2\\right),\\quad s_7=\\left(2, 1, 1, 1, 1, 1, 3, -2, -2, -2\\right),\\\\ \\\\s_8=\\left(2, 2, 2, 1, 1, 1, 1, -2, -2, -2\\right),\\quad s_9=\\left(2, 2, 1, 2, 1, 1, 1, -2, -2, -2\\right),\\\\ \\\\s_{10}=\\left(2, 2, 1, 1, 2, 1, 1, -2, -2, -2\\right),\\quad s_{11}=\\left(2, 2, 1, 1, 1, 2, 1, -2, -2, -2\\right),\\\\ \\\\s_{12}=\\left(2, 2, 1, 1, 1, 1, 2, -2, -2, -2\\right),\\quad s_{13}=\\left(2, 1, 2, 2, 1, 1, 1, -2, -2, -2\\right),\\\\ \\\\s_{14}=\\left(2, 1, 2, 1, 2, 1, 1, -2, -2, -2\\right),\\quad s_{15}=\\left(2, 1, 2, 1, 1, 2, 1, -2, -2, -2\\right),\\\\ \\\\s_{16}=\\left(2, 1, 2, 1, 1, 1, 2, -2, -2, -2\\right),\\quad s_{17}=\\left(2, 1, 1, 2, 2, 1, 1, -2, -2, -2\\right),\\end{array}$ $\\begin{array}{l}s_{18}=\\left(2, 1, 1, 2, 1, 2, 1, -2, -2, -2\\right),\\quad s_{19}=\\left(2, 1, 1, 2, 1, 1, 2, -2, -2, -2\\right),\\\\ \\\\s_{20}=\\left(2, 1, 1, 1, 2, 2, 1, -2, -2, -2\\right),\\quad s_{21}=\\left(2, 1, 1, 1, 2, 1, 2, -2, -2, -2\\right),\\\\ \\\\s_{22}=\\left(2, 1, 1, 1, 1, 2, 2, -2, -2, -2\\right).\\end{array}$ We must check all these families.", "We start with the family $s_1$ .", "According to the algorithm, let us find the function $\\theta $ by the formula (REF ).", "Since the coefficients at all finite poles $x=x_i$ , $i=1, 2, \\ldots , 6$ are the same, and the coefficients at all poles $x=x_i$ , $i=7, 8, 9$ are the same, then we can write the function $\\theta $ in explicit form.", "For the set $s_1$ this function has the form $\\theta =\\frac{3x^5-8hx^3+4\\left(h^2-1\\right)x}{x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2}-\\frac{3d_1x^2}{d_1x^3-k_1}.$ The polynomial $P$ of degree $d=1$ $P=x+B$ should identically satisfy the differential equation (REF ).", "After substitution of $P=x+B$ and the functions $\\theta $ and $R\\left(x\\right)$ to the equation (REF ), we obtain in the left hand side of (REF ) the rational expression.", "The numerator of this expression has a form of the ninth degree polynomial: $P_9=-Bd_1^2\\left(1+4d_1^2\\right)x^9+\\cdots $ Let us put $B=0$ .", "Then the numerator of the rational expression in the left hand side of (REF ) takes the form: $P_7=-12xk_1d_1\\left(d_1x^3-k_1\\right)^2+\\cdots $ This polynomial becomes zero when $k_1=0$ or when $d_1=0$ .", "Therefore we can state the following Theorem based on the verification of the family $s_1$ .", "Теорема 6 For the existence of liouvillian solutions of type 2 in the problem of motion of a heavy rigid body with a fixed point in the Hess case one of the two conditions must be satisfied: $d_1=0\\quad \\mbox{or}\\quad k_1=0.$ In other words, liouvillian solutions of type 2 can exist either in the case, when the moving rigid body is the Lagrange top, or in the Hess case, if the constant of the area integral is zero.", "The fact, that the problem of motion of a heavy rigid body with a fixed point in the Hess case for $k_1=0$ is integrable in elliptic functions (which are liouvillian functions) was firstly discovered by P. A. Nekrasov [17], [18].", "To prove Theorem REF we need to check the families $s_2,\\ldots , s_{22}$ , and to consider various critical cases, which we will discuss further, in Sections 11 – 15.", "We start with families $s_2,\\ldots , s_7$ .", "For the family $s_2$ the function $\\theta $ , calculated by the formula (REF ), has the form: $\\theta =\\frac{1}{2}\\left(\\frac{3}{x-y_1}+\\frac{1}{x+y_1}+\\frac{1}{x-y_2}+\\frac{1}{x+y_2}+\\frac{1}{x-iy_3}+\\frac{1}{x+iy_3}\\right)-\\frac{3d_1x^2}{d_1x^3-k_1}.$ We slightly simplify this expression for the function $\\theta $ .", "We can rewrite it in the form: $\\theta =\\frac{1}{2}\\left[\\frac{1}{x-y_1}+\\frac{1}{x+y_1}+\\frac{1}{x-y_2}+\\frac{1}{x+y_2}+\\frac{1}{x-iy_3}+\\frac{1}{x+iy_3}\\right]+\\frac{1}{x-y_1}-\\frac{3d_1x^2}{d_1x^3-k_1}.$ Expression in square brackets can be represented as follows: $\\frac{1}{x-y_1}+\\frac{1}{x+y_1}+\\frac{1}{x-y_2}+\\frac{1}{x+y_2}+\\frac{1}{x-iy_3}+\\frac{1}{x+iy_3}=\\frac{2x\\left(3x^4-8hx^2+4\\left(h^2-1\\right)\\right)}{x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2}.$ Finally we have the following expression for the function $\\theta $ : $\\theta =\\frac{3x^5-8hx^3+4\\left(h^2-1\\right)x}{x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2}+\\frac{1}{x-y_1}-\\frac{3d_1x^2}{d_1x^3-k_1}.$ The polynomial $P$ of degree $d=0$ has the form $P\\equiv 1$ .", "This polynomial should identically satisfy the differential equation (REF ).", "After substitution of $P=1$ and the functions $\\theta $ and $R\\left(x\\right)$ in this equation, we obtain in the left hand side of (REF ) the rational expression.", "The numerator of this expression has a form of the ninth degree polynomial of $x$ , the leading coefficient of which is equal to $P_{9}=y_1d_1^2\\left(1+4d_1^2\\right)x^9+\\cdots $ If we put $y_1=0$ (this condition is equivalent to the condition $k_1=0$ ), then this polynomial $P_9$ becomes zero.", "Thus, checking of the family $s_2$ gives the same conditions as the checking of the family $s_1$ .", "These conditions have been formulated in Theorem REF .", "Similarly we can check the other families $s_3, \\ldots , s_7$ .", "Let us check now the families $s_8,\\ldots , s_{22}$ .", "For the family $s_8$ the function $\\theta $ , calculated by the formula (REF ), has the form: $\\theta =\\frac{1}{x-y_1}+\\frac{1}{x+y_1}+\\frac{1}{2}\\left(\\frac{1}{x-y_2}+\\frac{1}{x+y_2}+\\frac{1}{x-iy_3}+\\frac{1}{x+iy_3}\\right)-\\frac{3d_1x^2}{d_1x^3-k_1}.$ We rewrite this function as follows: $\\theta =\\frac{1}{2\\left(x\\!-\\!y_1\\right)}+\\frac{1}{2\\left(x\\!+\\!y_1\\right)}+\\frac{1}{2}\\left(\\frac{1}{x\\!-\\!y_1}+\\frac{1}{x\\!+\\!y_1}+\\frac{1}{x\\!-\\!y_2}+\\frac{1}{x\\!+\\!y_2}+\\frac{1}{x\\!-\\!iy_3}+\\frac{1}{x\\!+\\!iy_3}\\right)-\\frac{3d_1x^2}{d_1x^3\\!-\\!k_1}.$ The latter expression can be transformed to the form: $\\theta =\\frac{3x^5-8hx^3+4\\left(h^2-1\\right)x}{x^6-4hx^4+4\\left(h^2-1\\right)x^2+4k_1^2}-\\frac{3d_1x^2}{d_1x^3-k_1}+\\frac{x}{x^2-y_1^2}.$ The polynomial $P$ of degree $d=0$ has the form $P\\equiv 1$ .", "This polynomial should identically satisfy the differential equation (REF ).", "After substituting $P=1$ and the functions $\\theta $ and $R\\left(x\\right)$ in this equation, we obtain in the left hand side of (REF ) the rational expression.", "The numerator of this expression has a form of 13th degree polynomial $P_{13}=4y_1^2d_1^2\\left(1+d_1^2\\right)x^{13}+\\cdots $ If we put $y_1=0$ and $k_1=0$ then this polynomial becomes zero.", "Thus, checking of the family $s_8$ gives the same conditions as checking of the family $s_1$ .", "These conditions have been formulated in Theorem REF .", "Similarly we can check the other families $s_9, \\ldots , s_{22}$ .", "Checking of these families gives us the same conditions as checking of the family $s_1$ .", "Finally we can state that the conditions of existence of liouvillian solutions of type 2 for the differential equation (REF ) are formulated in Theorem REF .", "To confirm the obtained results let us consider the equation (REF ) in the case when $k_1=0$ .", "In the case $k_1=0$ , the function $R\\left(x\\right)$ , defined by (REF ), has the form: $R\\left(x\\right)=\\frac{U\\left(x\\right)}{V\\left(x\\right)},$ $U\\left(x\\right)=-\\left(1+4d_1^2\\right)x^{8}+8h\\left(2d_1^2-1\\right)x^{6}+8\\left(7-2d_1^2\\right)\\left(h^2-1\\right)x^4-96h\\left(h^2-1\\right)x^2+48\\left(h^2-1\\right)^2,$ $V\\left(x\\right)=4x^2\\left(x^2-2h-2\\right)^2\\left(x^2-2h+2\\right)^2.$ Thus, the function $R\\left(x\\right)$ has five finite poles of the second order.", "One of these finite poles is $x_0=0$ and the other four $x_1$ , $x_2$ , $x_3$ , $x_4$ are the roots of two polynomials: $x^2-2h-2=0$ (we denote its roots by $x_1$ , $x_2$ ) and $x^2-2h+2=0$ (we denote its roots by $x_3$ , $x_4$ ).", "Partial fraction expansion of the function $R\\left(x\\right)$ has the form: $R\\left(x\\right)=\\frac{3}{4x^2}-\\sum \\limits _{i=1}^4\\frac{3}{16\\left(x-x_i\\right)^2}-\\sum \\limits _{i=1}^2\\frac{\\left(4d_1^2\\left(h+1\\right)-2h+1\\right)x_i}{32\\left(h+1\\right)\\left(x-x_i\\right)}+\\sum \\limits _{i=3}^4\\frac{\\left(4d_1^2\\left(h-1\\right)-2h-1\\right)x_i}{32\\left(h-1\\right)\\left(x-x_i\\right)}.$ It is possible to note the following properties on the partial fraction expansion of the function $R\\left(x\\right)$ .", "The coefficient $b_0$ of $\\displaystyle \\frac{b_0}{x^2}$ equals $b_0=\\frac{3}{4}.$ The coefficients $b_1,\\ldots , b_4$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , $i=1, 2, 3, 4$ are all equal $b_i=-\\frac{3}{16}, \\quad i=1, 2, 3, 4.$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ has the form: $R\\left(x\\right)=-\\frac{\\left(1+4d_1^2\\right)}{4x^2}+O\\left(\\frac{1}{x^4}\\right).$ The coefficient $b_0$ of $\\displaystyle \\frac{b_0}{x^2}$ equals $b_0=\\frac{3}{4}.$ The coefficients $b_1,\\ldots , b_4$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , $i=1, 2, 3, 4$ are all equal $b_i=-\\frac{3}{16}, \\quad i=1, 2, 3, 4.$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ has the form: $R\\left(x\\right)=-\\frac{\\left(1+4d_1^2\\right)}{4x^2}+O\\left(\\frac{1}{x^4}\\right).$ To find liouvillian solutions of type 2 of the differential equation (REF ) in the case $k_1=0$ we will use the Kovacic algorithm.", "According to the algorithm, let us define the sets $E_{x_i}$ , corresponding to finite poles of the function $R\\left(x\\right)$ and the set $E_{\\infty }$ corresponding to the pole of $R\\left(x\\right)$ at $x=\\infty $ .", "For the pole $x=0$ the corresponding set has the form $E_0=\\lbrace -2, 2, 6\\rbrace .$ For the finite poles $x_i$ , $i=1, 2, 3, 4$ the sets $E_{x_i}$ have the form: $E_{x_i}=\\lbrace 1, 2, 3\\rbrace ,\\quad i=1, 2, 3, 4.$ The set $E_{\\infty }$ contains only one element and this set equals $E_{\\infty }=\\lbrace 2\\rbrace .$ It is easy to see, that the constant $d$ , calculated by the formula (REF ) is a non-negative integer only for the family $s_1=\\lbrace 2, -2, 1, 1, 1, 1\\rbrace ,$ for which we have $d=0$ .", "Using this family, let us find the function $\\theta $ by the formula (REF ).", "This function has a form $\\theta =-\\frac{1}{x}+\\frac{2\\left(x^2-2h\\right)x}{\\left(x^2-2h-2\\right)\\left(x^2-2h+2\\right)}.$ The polynomial $P$ of degree $d=0$ has the form $P\\equiv 1$ .", "This polynomial should identically satisfy the differential equation (REF ).", "After substitution of $P=1$ and the functions $\\theta $ and $R\\left(x\\right)$ to the differential equation (REF ) we find that this equation is satisfied identically for any values of parameters $d_1$ and $h$ .", "Thus, we can state the following theorem.", "Теорема 7 In the case of motion of a heavy rigid body with a fixed point in the Hess case $\\left(d_1\\ne 0\\right)$ , equation (REF ) has liouvillian solutions only in the case $k_1=0$ .", "Indeed, when $k_1=0$ the differential equation (REF ) takes the form: $\\frac{d^2u}{dx^2}+\\frac{x^4-4\\left(h^2-1\\right)}{x\\left(x^2-2h-2\\right)\\left(x^2-2h+2\\right)}\\frac{du}{dx}+\\frac{d_1^2x^2}{\\left(x^2-2h-2\\right)\\left(x^2-2h+2\\right)}u=0.$ The general solution of the equation (REF ) has the form: $\\begin{array}{l}u\\left(x\\right)={\\rm C}_1\\exp \\left(id_1\\displaystyle \\int \\displaystyle \\frac{xdx}{\\sqrt{x^4-4hx^2+4h^2-4}}\\right)+\\\\ \\\\+{\\rm C}_2\\exp \\left(-id_1\\displaystyle \\int \\displaystyle \\frac{xdx}{\\sqrt{x^4-4hx^2+4h^2-4}}\\right).\\end{array}$ It is easy to see, that this function is a liouvillian function of type 2.", "The general solution of the equation (REF ) can be represented also as follows: $\\begin{array}{l}u\\left(x\\right)={\\rm K}_1\\sin \\left(\\displaystyle \\frac{d_1}{2}\\ln \\left(x^2-2h+\\sqrt{x^4-4hx^2+4h^2-4}\\right)\\right)+\\\\ \\\\+{\\rm K}_2\\cos \\left(\\displaystyle \\frac{d_1}{2}\\ln \\left(x^2-2h+\\sqrt{x^4-4hx^2+4h^2-4}\\right)\\right),\\end{array}$ where ${\\rm K}_1$ and ${\\rm K}_2$ are arbitrary constants.", "Thus we can conclude that conditions of existence of liouvillian solutions of type 2 in the problem of motion of a heavy rigid body with a fixed point in a Hess case are formulated in Theorem REF .", "When each of these conditions is fulfilled, liouvillian solutions of type 2 exist regardless of what values the other parameters of the problem take.", "Let us summarize the results.", "In the problem of motion of a heavy rigid body with a fixed point in the Hess case there are no liouvillian solutions of type 1 and type 3.", "Liouvillian solutions of type 2 exist when $d_1=0$ or when $k_1=0$ .", "These results have been obtained in general case, when we suppose that all the nine finite poles of the function $R\\left(x\\right)$ defined by (REF ), are distinct.", "Now we will consider the special cases, when these poles can coincide with each other, i.e.", "the function $R\\left(x\\right)$ can have multiply roots.", "The function $R\\left(x\\right)$ defined by (REF ) has a denominator which is the square of the product of two polynomials: the sixth degree polynomial (REF ) and the cubic polynomial (REF ).", "Let us study now the polynomial (REF ).", "As we already noted, this polynomial includes only even degrees of an independent variable, as a result of which by changing $x^2=z$ we can reduce it to a third-degree polynomial with respect to $z$ : $P_{3z}=z^3-4hz^2+4\\left(h^2-1\\right)z+4k_1^2=0.$ Let us study the type of roots of the polynomial (REF ).", "If we put $z=y+\\frac{4h}{3}$ then the polynomial (REF ) takes the form $y^3-4\\left(1+\\frac{h^2}{3}\\right)y+4\\left(k_1^2+\\frac{4h^3}{27}-\\frac{4h}{3}\\right)=0.$ Thus the third-degree polynomial (REF ) has the form $y^3+3py+2q=0,$ where we denote $p=-\\frac{4}{3}\\left(1+\\frac{h^2}{3}\\right),\\qquad q=2\\left(k_1^2+\\frac{4h^3}{27}-\\frac{4h}{3}\\right).$ The character of the roots of the polynomial (REF ) is determined by the sign of the expression $D=q^2+p^3.$ If $D>0$ then the polynomial (REF ) has one real root and two complex-conjugate roots.", "If $D<0$ then the polynomial (REF ) has three distinct real roots.", "If $D=0$ then the polynomial (REF ) has a multiple root and all of its roots are real.", "In the explicit form expression $D$ can be written as follows: $D=4\\left(k_1^4+\\frac{8}{27}\\left(h^2-9\\right)k_1^2 h-\\frac{16}{27}\\left(h^2-1\\right)^2\\right).$ Thus, if (REF ) is zero, the polynomial (REF ) has multiple roots.", "We consider now the cubic polynomial (REF ).", "The coefficient $d_1$ is such that $d_1\\in \\left(-1,\\, 0\\right]$ .", "Let us denote $k_1=c^3d_1,$ where $c$ is a new parameter.", "Then we can write the polynomial (REF ) as follows: $d_1\\left(x^3-c^3\\right)=0.$ The roots of (REF ) have the following form: $x_1=c, \\quad x_2=\\left(-\\frac{1}{2}+\\frac{i\\sqrt{3}}{2}\\right)c, \\quad x_3=\\left(-\\frac{1}{2}-\\frac{i\\sqrt{3}}{2}\\right)c.$ It is easy to see, that all these roots $x_1$ , $x_2$ , $x_3$ are distinct for $c\\ne 0$ .", "Therefore, since we do not consider the case $k_1=0$ here, the polynomial (REF ) cannot have multiple roots.", "However the situation is possible when one of the roots (REF ) of the polynomial (REF ) is the root of the polynomial (REF ).", "Below we consider in details all possible special cases described in this Section.", "Let us consider the first special case when the polynomial (REF ) has a multiple root.", "First of all we consider the case, when $d_1=0$ , i.e.", "we consider the case of motion of the Lagrange top.", "For $d_1=0$ the cubic polynomial (REF ) becomes a constant.", "Therefore in this case polynomials (REF ) and (REF ) have not the common roots.", "Further in this Section we assume, that $k_1\\ne 0$ .", "The polynomial (REF ) has multiple roots if expression $D$ defined by (REF ), is zero.", "This condition can be represented in the form of a biquadratic equation with respect to $k_1$ : $k_1^4+\\frac{8}{27}\\left(h^2-9\\right)k_1^2 h-\\frac{16}{27}\\left(h^2-1\\right)^2=0.$ From the equation (REF ) we find the value of $k_1^2$ : $k_1^2=\\frac{4}{27}\\left(\\left(h^2+3\\right)\\sqrt{h^2+3}-h\\left(h^2-9\\right)\\right).$ In this case the cubic polynomial (REF ) has three real roots: $z_1=\\frac{4h}{3}-\\frac{4}{3}\\sqrt{h^2+3},$ $z_2=z_3=\\frac{4h}{3}+\\frac{2}{3}\\sqrt{h^2+3}.$ It is easy to see that the roots $z_2$ and $z_3$ are positive.", "This means, that the polynomial (REF ) has one real positive root of multiplicity 2 and one negative root of multiplicity 2.", "In addition, the polynomial (REF ) has two pure imaginary roots, which are square roots of the negative expression $z_1$ .", "Finally in the considered special case the polynomial (REF ) can be represented as follows: $\\left(x^2-z_1\\right)\\left(x^2-z_2\\right)^2.$ Moreover, the expressions $z_1$ and $z_2$ are connected by the relation $z_2=\\frac{z_1}{4}-\\frac{4}{z_1}.$ This relation allows us to essentially simplify the coefficients $a\\left(x\\right)$ and $b\\left(x\\right)$ of the differential equation (REF ).", "Indeed, we choose $z_1$ as a parameter, included in the coefficients $a\\left(x\\right)$ and $b\\left(x\\right)$ .", "The parameters $h$ and $k_1$ are expressed in terms of $z_1$ as follows: $h=\\frac{3}{8}z_1-\\frac{2}{z_1},\\quad k_1=\\left(\\frac{2}{z_1}-\\frac{z_1}{8}\\right)\\sqrt{-z_1}.$ Substitution of expressions (REF ) together with the condition $d_1=0$ to the coefficients $a\\left(x\\right)$ and $b\\left(x\\right)$ of the differential equation (REF ) gives the following expressions for these coefficients: $a\\left(x\\right)=\\frac{16z_1x^4+2\\left(16-7z_1^2\\right)x^2+z_1\\left(z_1^2-16\\right)}{\\left(x^2-z_1\\right)\\left(4z_1x^2+16-z_1^2\\right)x},$ $b\\left(x\\right)=\\frac{z_1\\left(z_1-4\\right)^2\\left(z_1+4\\right)^2}{4\\left(x^2-z_1\\right)\\left(4z_1x^2+16-z_1^2\\right)^2x^2}.$ By the formula (REF ) let us find the function $R\\left(x\\right)$ .", "It is the rational function which denominator has the form: $V\\left(x\\right)=\\left(x^2-z_1\\right)^2\\left(4z_1x^2-z_1^2+16\\right)^2,$ Thus it is easy to see, that the function $R\\left(x\\right)$ has four finite poles of the second order.", "Its partial fraction expansion has a very complicated form and we do not write it explicitly here.", "However it is possible to note the following properties of the partial fraction expansion of the function $R\\left(x\\right)$ .", "The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=1, 2$ are roots of the polynomial $x^2-z_1=0,$ are all equal $b_i=-\\frac{3}{16}, \\quad i=1, 2.$ The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=3, 4$ are roots of the polynomial $4z_1x^2-z_1^2+16=0,$ are all equal $b_i=-\\frac{z_1^2+8}{2\\left(3z_1^2+16\\right)}, \\quad i=3, 4.$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ has the form: $R\\left(x\\right)=\\frac{2}{x^2}+O\\left(\\frac{1}{x^4}\\right).$ The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=1, 2$ are roots of the polynomial $x^2-z_1=0,$ are all equal $b_i=-\\frac{3}{16}, \\quad i=1, 2.$ The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=3, 4$ are roots of the polynomial $4z_1x^2-z_1^2+16=0,$ are all equal $b_i=-\\frac{z_1^2+8}{2\\left(3z_1^2+16\\right)}, \\quad i=3, 4.$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ has the form: $R\\left(x\\right)=\\frac{2}{x^2}+O\\left(\\frac{1}{x^4}\\right).$ Let us find out whether the differential equation (REF ) in the considered critical case can have liouvillian solutions of type 1 (the existence of liouvillian solutions of type 2 for all values of parameters $h$ and $k_1$ was proved above).", "According to the Kovacic algorithm for searching liouvillian solutions of type 1, let us calculate the constants $\\alpha ^{\\pm }_c$ .", "For the finite poles $x=x_i$ , $i=1, 2$ these constants are all equal $\\alpha _{x_i}^+=\\frac{3}{4},\\quad \\alpha _{x_i}^-=\\frac{1}{4}, \\quad i=1, 2.$ The constants $\\alpha _{\\infty }^{\\pm }$ are equal $\\alpha _{\\infty }^+=2,\\quad \\alpha _{\\infty }^-=-1.$ It is easy to see, that the coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x=x_i$ , $i=3, 4$ are the roots of the polynomial (REF ), depend on the parameter $z_1$ .", "Let us estimate these coefficients.", "First of all, we can state, that $b_i<0$ , $i=3, 4$ .", "Let us prove now that the following inequality is valid for these coefficients: $b_i<-\\frac{3}{16},\\quad i=3, 4.$ Indeed, the inequality $-\\frac{z_1^2+8}{2\\left(3z_1^2+16\\right)}<-\\frac{3}{16}$ can be rewritten in the form: $\\frac{z_1^2+8}{3z_1^2+16}-\\frac{3}{8}>0.$ The latter inequality is equivalent to the inequality $\\frac{16-z_1^2}{8\\left(3z_1^2+16\\right)}>0,$ which is valid for all $z_1$ such that $k_1\\ne 0$ .", "Moreover, analysis of the expressions $b_i$ , $i=3, 4$ shows, that these expressions take a minimum for $z_1=0$ .", "This minimum equals $b_i=-\\frac{1}{4}.$ Thus, for the coefficients $b_i$ , $i=3, 4$ we have the following estimate: $-\\frac{1}{4}<b_i<-\\frac{3}{16}.$ Then the constant $\\alpha _{x_i}^+=\\frac{1}{2}+\\frac{1}{2}\\sqrt{1+4b_i}=\\frac{1}{2}-\\frac{z_1}{2\\sqrt{3z_1^2+16}}, \\quad i=3, 4$ satisfies the estimate $\\frac{1}{2}<\\alpha _{x_i}^+<\\frac{3}{4},$ and the constant $\\alpha _{x_i}^-=\\frac{1}{2}-\\frac{1}{2}\\sqrt{1+4b_i}=\\frac{1}{2}+\\frac{z_1}{2\\sqrt{3z_1^2+16}}, \\quad i=3, 4$ satisfies the estimate $\\frac{1}{4}<\\alpha _{x_i}^-<\\frac{1}{2},$ Therefore, in the considered case the constant $d$ , calculated by the formula (REF ), can be only zero.", "The value $d=0$ corresponds to the following sets of signs plus (+) and minus (-):$\\,$ $s=\\left(s\\left(\\infty \\right),\\,s\\left(x_1\\right),\\,s\\left(x_2\\right),\\,s\\left(x_3\\right),\\,s\\left(x_4\\right)\\right)$ $\\begin{array}{l}s_1=\\left(+,\\,+,\\,-,\\,+,\\,-\\right), \\quad s_2=\\left(+,\\,-,\\,+,\\,+,\\,-\\right), \\\\ \\\\s_3=\\left(+,\\,+,\\,-,\\,-,\\,+\\right),\\quad s_4=\\left(+,\\,-,\\,+,\\,-,\\,+\\right).\\end{array}$ We need to check all these sets.", "Let us start with the set $s_1$ .", "For the set $s_1$ we calculate the function $\\theta $ by the formula (REF ).", "For the set $s_1$ this function has the form: $\\theta =\\frac{3}{4\\left(x-x_1\\right)}+\\frac{1}{4\\left(x-x_2\\right)}+\\frac{\\alpha _{x_3}^+}{\\left(x-x_3\\right)}+\\frac{\\alpha _{x_3}^-}{\\left(x-x_4\\right)}.$ The polynomial $P$ of degree $d=0$ has the form $P\\equiv 1$ .", "The substitution of this polynomial to the differential equation (REF ) reduces it to the equation (REF ).", "The substitution of the function $\\theta $ to the differential equation (REF ) gives in the left hand side of this equation the rational expression which vanishes only for $z_1=0$ .", "But we assume that $z_1\\ne 0$ .", "Therefore we can state that the rational expression in the left hand side of (REF ) is not zero.", "Thus, in the considered critical case for the set $s_1$ there are no liouvillian solutions of type 1 in the problem of motion of a heavy rigid body with a fixed point in the Hess case.", "The fact of nonexistence of liouvillian solutions of type 1 for the sets $s_2$ , $s_3$ , $s_4$ in the considered critical case is established similarly.", "Finally, we can state, that there are no liouvillian solutions of type 1 in the problem of motion of a heavy rigid body with a fixed point in the Hess case.", "Now let us consider the case $d_1\\ne 0$ .", "Let us choose $d_1$ and $z_1$ as a parameters, included in the coefficients $a\\left(x\\right)$ and $b\\left(x\\right)$ of the differential equation (REF ).", "Substitution of the expressions (REF ) to the coefficients $a\\left(x\\right)$ and $b\\left(x\\right)$ of the differential equation (REF ) gives the following expressions for these coefficients: $a\\left(x\\right)=\\frac{U_1\\left(x\\right)}{V_1\\left(x\\right)},$ $\\begin{array}{l}U_1\\left(x\\right)=32d_1z_1^2x^7+\\left(z_1^2-16\\right)\\bigl [8d_1z_1x^5+16z_1\\sqrt{-z_1}x^4-16d_1z_1^2x^3-2\\left(7z_1^2-16\\right)\\sqrt{-z_1}x^2+\\\\ \\\\ +z_1\\left(z_1^2-16\\right)\\sqrt{-z_1}\\bigr ],\\end{array}$ $V_1\\left(x\\right)=x\\left(x^2-z_1\\right)\\left(4z_1x^2-z_1^2+16\\right)\\left(8d_1z_1x^3+\\left(z_1^2-16\\right)\\sqrt{-z_1}\\right),$ $b\\left(x\\right)=\\frac{\\left(8d_1z_1x^3+16\\sqrt{-z_1}-z_1^2\\sqrt{-z_1}\\right)\\left(8d_1z_1x^3-16\\sqrt{-z_1}+z_1^2\\sqrt{-z_1}\\right)}{4x^2\\left(x^2-z_1\\right)\\left(4z_1x^2+16-z_1^2\\right)^2}.$ By the formula (REF ) let us find the function $R\\left(x\\right)$ .", "It is the rational function which denominator has the form: $V\\left(x\\right)=\\left(x^2-z_1\\right)^2\\left(4z_1x^2-z_1^2+16\\right)^2\\left(8d_1z_1x^3-16\\sqrt{-z_1}+z_1^2\\sqrt{-z_1}\\right)^2.$ Thus, it is easy to see that the function $R\\left(x\\right)$ has seven finite poles of the second order.", "Its partial fraction expansion has a very complicated form and we do not write it explicitly here.", "However, it is possible to note the following properties of the partial fraction expansion of the function $R\\left(x\\right)$ .", "The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=1, 2, 3$ are roots of the cubic polynomial $8d_1z_1x^3-16\\sqrt{-z_1}+z_1^2\\sqrt{-z_1}=0,$ are all equal $b_i=\\frac{3}{4}, \\quad i=1, 2, 3.$ The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=4, 5$ are roots of the polynomial $x^2-z_1=0,$ are all equal $b_i=-\\frac{3}{16}, \\quad i=4, 5.$ The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=6, 7$ are roots of the polynomial (REF ) are all equal $b_i=\\frac{1}{4}\\left(\\frac{z_1^2\\left(d_1^2-2\\right)-16\\left(d_1^2+1\\right)}{16+3z_1^2}\\right), \\quad i=6, 7.$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ has the form: $R\\left(x\\right)=-\\frac{\\left(1+4d_1^2\\right)}{4x^2}+O\\left(\\frac{1}{x^4}\\right).$ The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=1, 2, 3$ are roots of the cubic polynomial $8d_1z_1x^3-16\\sqrt{-z_1}+z_1^2\\sqrt{-z_1}=0,$ are all equal $b_i=\\frac{3}{4}, \\quad i=1, 2, 3.$ The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=4, 5$ are roots of the polynomial $x^2-z_1=0,$ are all equal $b_i=-\\frac{3}{16}, \\quad i=4, 5.$ The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=6, 7$ are roots of the polynomial (REF ) are all equal $b_i=\\frac{1}{4}\\left(\\frac{z_1^2\\left(d_1^2-2\\right)-16\\left(d_1^2+1\\right)}{16+3z_1^2}\\right), \\quad i=6, 7.$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ has the form: $R\\left(x\\right)=-\\frac{\\left(1+4d_1^2\\right)}{4x^2}+O\\left(\\frac{1}{x^4}\\right).$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ allows us to state that in this special case the differential equation (REF ) can have liouvillian solutions of type 2 only.", "Moreover, it is easy to see that the coefficients $b_i$ of $x=x_i$ , $i=6, 7$ depend on the parameters $d_1$ and $z_1$ .", "Let us estimate these coefficients.", "According to the restrictions on the parameter $d_1$ we can state that $b_i<0$ , $i=6, 7$ .", "Let us prove now that the following inequality is valid for these coefficients: $b_i<-\\frac{3}{16}, \\quad i=6, 7.$ We rewrite the latter inequality in the form: $\\frac{16\\left(d_1^2+1\\right)+\\left(2-d_1^2\\right)z_1^2}{16+3z_1^2}>\\frac{3}{4}.$ The denominator of expression in the left hand side of (REF ) is positive.", "Therefore we can rewrite (REF ) as follows: $16d_1^2+16+2z_1^2-d_1^2z_1^2>12+\\frac{9}{4}z_1^2.$ This inequality is equivalent to the inequality $16\\left(d_1^2+\\frac{1}{4}\\right)>z_1^2\\left(d_1^2+\\frac{1}{4}\\right).$ or $16>z_1^2.$ The latter inequality is valid for all $z_1$ such that $k_1\\ne 0$ .", "To find liouvillian solutions of the differential equation (REF ) in the considered special case we will use the Kovacic algorithm.", "Let us define the sets $E_{x_i}$ , $i=1, \\ldots , 7$ corresponding to finite poles and the set $E_{\\infty }$ corresponding to the pole at $x=\\infty $ .", "For the finite poles $x=x_i$ , $i=1, 2, 3$ the corresponding sets have the form: $E_{x_i}=\\lbrace -2, 2, 6\\rbrace ,\\quad i=1, 2, 3.$ For the finite poles $x_i$ , $i=4, 5$ the corresponding sets $E_{x_i}$ have the form: $E_{x_i}=\\lbrace 1, 2, 3\\rbrace ,\\quad i=4, 5.$ Taking into account estimations made for the coefficients $b_i$ , $i=6, 7$ , we can state that the sets $E_{x_i}$ , corresponding to these poles, contain only one element and have the form $E_{x_i}=\\lbrace 2\\rbrace ,\\quad i=6, 7.$ The set $E_{\\infty }$ corresponding to the pole of $R\\left(x\\right)$ at $x=\\infty $ also contains only one element and has the form $E_{\\infty }=\\lbrace 2\\rbrace .$ Now we should determine the families $s=\\left(e_{\\infty }, e_{x_1}, e_{x_2}, e_{x_3}, e_{x_4}, e_{x_5}, e_{x_6}, e_{x_7}\\right)$ for which the constant $d$ calculating by the formula (REF ), is a non-negative integer.", "It is easy to see, that $d$ is a non-negative integer for three families $s$ only.", "For the family $s_1=\\left(2, -2, -2, -2, 1, 1, 2, 2\\right)$ we have $d=1$ and for the families $s_2=\\left(2, -2, -2, -2, 3, 1, 2, 2\\right),\\quad s_3=\\left(2, -2, -2, -2, 1, 3, 2, 2\\right),$ we have $d=0$ .", "Let us check the family $s_1$ .", "Using the elements of this family we calculate the function $\\theta $ by the formula (REF ).", "For the family $s_1$ this function has the form: $\\theta =-\\frac{24d_1z_1x^2}{8d_1z_1x^3-16\\sqrt{-z_1}+z_1^2\\sqrt{-z_1}}+\\frac{x}{x^2-z_1}+\\frac{8z_1x}{4z_1x^2-z_1^2+16}.$ The polynomial $P$ of degree $d=1$ has the form $P=x+B$ , where $B$ indefinite coefficient.", "This polynomial should identically satisfy differential equation (REF ).", "After substitution of the polynomial $P$ and the functions $\\theta $ and $R\\left(x\\right)$ with this equation we obtain in the left hand side of (REF ) the rational expression.", "The numerator of this expression is the 12th degree polynomial $P_{12}=-8192d_1^3z_1^5\\left(1+4d_1^2\\right)Bx^{12}+\\cdots $ For the leading coefficient of this polynomial to vanish we put $B=0$ .", "Then the numerator of the rational expression in the left hand side of (REF ) takes the form of the 10th degree polynomial $P_{10}=12288z_1^4d_1^4\\sqrt{-z_1}\\left(z_1-4\\right)\\left(z_1+4\\right)x^{10}+\\cdots $ Since $z_1<0$ , then $z_1-4<0$ .", "We do not consider the cases $z_1=0$ (i.e.", "$k_1=0$ ) and $d_1=0$ , because these cases have been already investigated.", "The in case $z_1=-4$ we have $z_2=0$ , i.e.", "$k_1=0$ .", "Therefore the polynomial $P_{10}$ is not identically zero.", "This means that for the for the family $s_1$ the differential equation (REF ) does not have liouvillian solutions of type 2.", "Similarly we can prove that equation (REF ) does not have liouvillian solutions for the families $s_2$ and $s_3$ .", "Finally we can state the following Theorem.", "Теорема 8 In the special case of multiple roots of the polynomial (REF ) the second-order linear differential equation (REF ) (or (REF )) does not have liouvillian solutions.", "Thus, in the first special case we have considered, when the polynomial (REF ) has multiple roots, equation (REF ) does not have liouvillian solutions.", "We turn now to the second special case when the polynomials (REF ) and (REF ) have a common root.", "Let us consider now the case, when the root $x_1=c$ the real root of the polynomial (REF ) from the roots (REF ) is also the root of the polynomial (REF ).", "The parameter $k_1$ is expressed in terms of $x_1=c$ and $d_1$ as follows: $k_1=c^3d_1.$ The condition that $x_1=c$ is a root of the polynomial (REF ) has the form: $\\left(1+4d_1^2\\right)c^4-4hc^2+4h^2-4=0.$ We express $d_1$ from the condition (REF ).", "As a result we obtain: $d_1^2=\\frac{4-\\left(c^2-2h\\right)^2}{4c^4}.$ We will use (REF ) to simplify the coefficients $a\\left(x\\right)$ and $b\\left(x\\right)$ of the linear differential equation (REF ).", "We will consider $c$ and $h$ as parameters included in these coefficients.", "Taking into account (REF ), we can write $a\\left(x\\right)$ and $b\\left(x\\right)$ as follows: $\\begin{array}{l}a\\left(x\\right)=\\displaystyle \\frac{x^2\\left(x^3-3c^3\\right)}{\\left(x-c\\right)\\left(x+c\\right)\\left(x^4+\\left(c^2-4h\\right)x^2+c^4-4c^2h+4h^2-4\\right)}-\\\\ \\\\-\\displaystyle \\frac{4\\left(h^2-1\\right)x^3+\\left(2h-c^2+2\\right)\\left(2h^2-c^2-2\\right)c^3}{x\\left(x-c\\right)\\left(x^2+xc+c^2\\right)\\left(x^4+\\left(c^2-4h\\right)x^2+c^4-4c^2h+4h^2-4\\right)}+\\\\ \\\\+\\displaystyle \\frac{c^2x\\left(12chx+4h^2-4-c^4+4c^2h\\right)}{\\left(x-c\\right)\\left(x+c\\right)\\left(x^2+xc+c^2\\right)\\left(x^4+\\left(c^2-4h\\right)x^2+c^4-4c^2h+4h^2-4\\right)},\\end{array}$ $b\\left(x\\right)=-\\frac{\\left(x^2-xc+c^2\\right)\\left(x^2+xc+c^2\\right)\\left(2h-c^2+2\\right)\\left(2h-c^2-2\\right)}{4c^4x^2\\left(x^4+\\left(c^2-4h\\right)x^2+c^4-4c^2h+4h^2-4\\right)}.$ Now we find the function $R\\left(x\\right)$ by the formula (REF ).", "It is the rational function which denominator has the form: $V\\left(x\\right)=\\left(x-c\\right)^2\\left(x^2+xc+c^2\\right)^2\\left(x+c\\right)^2\\left(x^4+\\left(c^2-4h\\right)x^2+c^4-4c^2h+4h^2-4\\right)^2c^4.$ Thus, the function $R\\left(x\\right)$ has eight finite poles of the second order.", "It has the poles at $x=c$ and $x=-c$ , at $x=x_i$ , $i=1,\\ldots , 4$ , where $x_i$ , $i=1,\\ldots , 4$ are roots of the fourth degree polynomial $x^4+\\left(c^2-4h\\right)x^2+c^4-4c^2h+4h^2-4=0,$ and at $x=x_i$ , $i=5, 6$ , where $x_i$ , $i=5, 6$ are roots of the polynomial $x^2+xc+c^2=0.$ The partial fraction expansion of $R\\left(x\\right)$ has a very complicated form and we do not write it explicitly.", "However it is possible to note the following properties of the partial fraction expansion of the function $R\\left(x\\right)$ .", "The coefficient $b_c$ of $\\displaystyle \\frac{b_c}{\\left(x-c\\right)^2}$ equals $b_c=\\frac{5}{16}.$ The coefficient $b_{-c}$ of $\\displaystyle \\frac{b_{-c}}{\\left(x+c\\right)^2}$ equals $b_{-c}=-\\frac{3}{16}.$ The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=1, 2, 3, 4$ are roots of the polynomial (REF ) are all equal $b_i=-\\frac{3}{16}, \\quad i=1, 2, 3, 4.$ The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=5, 6$ are roots of the polynomial (REF ) are all equal $b_i=\\frac{3}{4}, \\quad i=5, 6.$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ has the form: $R\\left(x\\right)=-\\frac{\\left(1+4d_1^2\\right)}{4x^2}+O\\left(\\frac{1}{x^4}\\right).$ The coefficient $b_c$ of $\\displaystyle \\frac{b_c}{\\left(x-c\\right)^2}$ equals $b_c=\\frac{5}{16}.$ The coefficient $b_{-c}$ of $\\displaystyle \\frac{b_{-c}}{\\left(x+c\\right)^2}$ equals $b_{-c}=-\\frac{3}{16}.$ The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=1, 2, 3, 4$ are roots of the polynomial (REF ) are all equal $b_i=-\\frac{3}{16}, \\quad i=1, 2, 3, 4.$ The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=5, 6$ are roots of the polynomial (REF ) are all equal $b_i=\\frac{3}{4}, \\quad i=5, 6.$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ has the form: $R\\left(x\\right)=-\\frac{\\left(1+4d_1^2\\right)}{4x^2}+O\\left(\\frac{1}{x^4}\\right).$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ allows us to state that the differential equation (REF ) in the considered special case can have liouvillian solutions of type 2 only.", "To find these liouvillian solutions we will use the Kovacic algorithm.", "To find this liouvillian solutions we will use the Kovacic algorithm.", "Let us define the sets $E_{x_i}$ , $i=1, \\ldots , 8$ corresponding to finite poles and the set $E_{\\infty }$ corresponding to the pole of $R\\left(x\\right)$ at $x=\\infty $ .", "For the finite pole $x=c$ the corresponding set has the form: $E_c=\\lbrace -1, 2, 5\\rbrace .$ For the finite pole $x=-c$ the corresponding set has the form: $E_{-c}=\\lbrace 1, 2, 3\\rbrace .$ For the finite poles $x=x_i$ , $i=1, 2, 3, 4$ the corresponding sets $E_{x_i}$ have the form: $E_{x_i}=\\lbrace 1, 2, 3\\rbrace ,\\quad i=1, 2, 3, 4.$ At last, for the finite poles $x=x_i$ , $i=5, 6$ the corresponding sets $E_{x_i}$ have the form: $E_{x_i}=\\lbrace -2, 2, 6\\rbrace ,\\quad i=5, 6.$ The set $E_{\\infty }$ corresponding to the pole of $R\\left(x\\right)$ at $x=\\infty $ contains only one element and has the form: $E_{\\infty }=\\lbrace 2\\rbrace .$ Now we should determine the families $s=\\left(e_{\\infty }, e_{c}, e_{-c}, e_{x_1}, e_{x_2}, e_{x_3}, e_{x_4}, e_{x_5}, e_{x_6}\\right),$ for which the constant $d$ calculated by the formula (REF ), is a non-negative integer.", "It is easy to see, that the minimal value of the sum of the elements of sets, corresponding to finite poles, is zero.", "Therefore, for the family $s_1=\\left(2, -1, 1, 1, 1, 1, 1, -2, -2\\right)$ we have $d=1$ , and for the families $\\begin{array}{l}s_2=\\left(2, -1, 3, 1, 1, 1, 1, -2, -2\\right),\\; s_3=\\left(2, -1, 1, 3, 1, 1, 1, -2, -2\\right),\\\\ \\\\s_4=\\left(2, -1, 1, 1, 3, 1, 1, -2, -2\\right),\\; s_5=\\left(2, -1, 1, 1, 1, 3, 1, -2, -2\\right),\\\\ \\\\s_6=\\left(2, -1, 1, 1, 1, 1, 3, -2, -2\\right),\\; s_7=\\left(2, -1, 2, 2, 1, 1, 1, -2, -2\\right),\\\\ \\\\s_8=\\left(2, -1, 2, 1, 2, 1, 1, -2, -2\\right),\\; s_9=\\left(2, -1, 2, 1, 1, 2, 1, -2, -2\\right),\\\\ \\\\s_{10}=\\left(2, -1, 2, 1, 1, 1, 2, -2, -2\\right),\\; s_{11}=\\left(2, -1, 1, 2, 2, 1, 1, -2, -2\\right),\\\\ \\\\s_{12}=\\left(2, -1, 1, 2, 1, 2, 1, -2, -2\\right),\\; s_{13}=\\left(2, -1, 1, 2, 1, 1, 2, -2, -2\\right),\\\\ \\\\s_{14}=\\left(2, -1, 1, 1, 2, 2, 1, -2, -2\\right),\\; s_{15}=\\left(2, -1, 1, 1, 2, 1, 2, -2, -2\\right),\\\\ \\\\s_{16}=\\left(2, -1, 1, 1, 1, 2, 2, -2, -2\\right)\\end{array}$ we have $d=0$ .", "We must check all these families.", "We start with the family $s_1$ .", "According to the algorithm, let us find the function $\\theta $ by the formula (REF ).", "For the family $s_1$ this function has the form: $\\theta =\\frac{2x^3+\\left(c^2-4h\\right)x}{x^4+\\left(c^2-4h\\right)x^2+c^4-4c^2h+4h^2-4}-\\frac{2x+c}{x^2+xc+c^2}+\\frac{1}{2\\left(x+c\\right)}-\\frac{1}{2\\left(x-c\\right)}.$ The polynomial $P$ of degree $d=1$ equals $P\\equiv x+B$ , where $B$ is indefinite coefficient.", "This polynomial should identically satisfy the differential equation (REF ).", "After substitution of the polynomial $P=x+b$ and the functions $\\theta $ and $R\\left(x\\right)$ to this equation, we obtain in the left hand side of (REF ) the rational expression.", "Its numerator has the form of the 8th degree polynomial $P_8=4\\left(h^2-c^2h-1\\right)Bx^{8}+\\cdots $ If we put $B=0$ then the polynomial $P_8$ takes the form: $P_6=3c^3\\left(2h-c^2+2\\right)\\left(2h-c^2-2\\right)x^6+\\cdots $ The coefficient at $x^6$ of the polynomial $P_6$ vanishes only if $c= 0$ or $d_1=0$ .", "None of these possibilities is considered by us; therefore, the polynomial $P_6$ does not identically vanish.", "We can verify similarly that the sixth degree polynomial obtained from $P_8$ does not identically vanish, if we assume that $h^2-c^2h-1=0.$ Finally this means that equation (REF ) has not liouvillian solutions of type 2 for the family $s_1$ in the considered special case.", "Now let us consider the families $s_2, \\ldots , s_6$ .", "For the family $s_2$ the function $\\theta $ calculated by the formula (REF ), has the form $\\theta =\\frac{2x^3+\\left(c^2-4h\\right)x}{x^4+\\left(c^2-4h\\right)x^2+c^4-4c^2h+4h^2-4}-\\frac{2x+c}{x^2+xc+c^2}+\\frac{3}{2\\left(x+c\\right)}-\\frac{1}{2\\left(x-c\\right)}.$ The polynomial $P$ of degree $d=0$ has the form $P\\equiv 1$ .", "This polynomial should identically satisfy differential equation (REF ).", "After substitution of $P=1$ and the functions $\\theta $ and $R\\left(x\\right)$ to the differential equation (REF ) we obtain in the left hand side of this equation the rational expression.", "Its numerator has the form of 8th degree polynomial $P_8=4\\left(h^2-c^2h-1\\right)x^{8}+\\cdots $ If we assume that condition (REF ) is valid, then the substitution of this condition with $P_8$ reduces it to the polynomial of the 6th degree, which is not zero for any $c$ and $d_1$ such that $c\\ne 0$ and $d_1\\ne 0$ .", "Thus, the differential equation (REF ) has not liouvillian solutions of type 2 for the family $s_2$ .", "Similarly we can check the families $s_3, \\ldots , s_{16}$ .", "For all these families the differential equation (REF ) has not liouvillian solutions of type 2.", "Thus we can state that in the special case when the polynomials (REF ) and (REF ) have a common root, which is not the a multiple root of the polynomial (REF ), the differential equation (REF ) has not liouvillian solutions.", "Now let us consider the case, when the root $x_1=c$ – the real root of the polynomial (REF ) from the roots (REF ) id the multiple root of the polynomial (REF ).", "In this case the cubic polynomial (REF ) obtained from the polynomial (REF ), has the form $P_{3z}=z^3-4hz^2+4\\left(h^2-1\\right)z+4d_1^2c^6=\\left(z-c^2\\right)^2\\left(z+z_1\\right).$ Comparing the coefficients at the same powers of $z$ in two different representations of the polynomial $P_{3z}$ we obtain the following system of equations for the parameters of the problem: $\\begin{array}{l}2c^2-z_1-4h=0,\\\\ \\\\2c^2z_1-c^4+4h^2-4=0,\\\\ \\\\4d_1^2c^6-c^4z_1=0.\\end{array}$ Solving the system (REF ) we find the following expressions for the parameters $h$ , $z_1$ e $d_1$ in terms of the parameter $c$ : $\\begin{array}{l}d_1^2=\\displaystyle \\frac{\\sqrt{c^4+4}}{2c^2}-\\displaystyle \\frac{1}{2}, \\\\ \\\\z_1=4d_1^2c^2=2\\sqrt{c^4+4}-2c^2, \\\\ \\\\h=c^2\\left(\\displaystyle \\frac{1}{2}-d_1^2\\right)=c^2\\left(1-\\displaystyle \\frac{\\sqrt{c^4+4}}{2c^2}\\right)=c^2-\\displaystyle \\frac{\\sqrt{c^4+4}}{2}.\\end{array}$ We will use (REF ) to simplify the coefficients $a\\left(x\\right)$ and $b\\left(x\\right)$ of the differential equation (REF ).", "Taking into account (REF ) we can write the coefficients $a\\left(x\\right)$ and $b\\left(x\\right)$ of the second order linear differential equation (REF ) as follows: $a\\left(x\\right)=\\frac{x^5+2cx^4+4c^2x^3+2c^3x^2+2c^2\\left(2x+c\\right)\\left(\\sqrt{c^4+4}-c^2\\right)}{x\\left(x+c\\right)\\left(x^2+xc+c^2\\right)\\left(x^2+2\\sqrt{c^4+4}-2c^2\\right)},$ $b\\left(x\\right)=\\frac{\\left(x^2+xc+c^2\\right)\\left(x^2-xc+c^2\\right)\\left(\\sqrt{c^4+4}-c^2\\right)}{2c^2x^2\\left(x-c\\right)\\left(x+c\\right)\\left(x^2+2\\sqrt{c^4+4}-2c^2\\right)}.$ Now we find the function $R\\left(x\\right)$ for this special case by the formula (REF ).", "It is the rational function of the rather complicated form which denominator can be written as follows: $V\\left(x\\right)=4c^2\\left(x+c\\right)^2\\left(x^2+xc+c^2\\right)^2\\left(x^2+2\\sqrt{c^4+4}-2c^2\\right)^2\\left(x-c\\right).$ Thus, the function $R\\left(x\\right)$ has six finite poles.", "At $x=c$ function $R\\left(x\\right)$ has the first order pole.", "At $x=-c$ it has the second order pole.", "The function $R\\left(x\\right)$ has also the second order poles at $x=x_i$ , $i=1, 2, 3, 4$ , where $x_1$ and $x_2$ are roots of the polynomial $x^2+xc+c^2=0,$ and $x_3$ and $x_4$ are roots of the polynomial $x^2+2\\sqrt{c^4+4}-2c^2=0.$ The partial fraction expansion of $R\\left(x\\right)$ has a very complicated form and we do not write it here explicitly.", "However it is possible to note the following properties of the partial fraction expansion of the function $R\\left(x\\right)$ .", "The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=1, 2$ are roots of the polynomial (REF ) equal $b_i=\\frac{3}{4}, \\quad i=1, 2.$ The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=3, 4$ are roots of the polynomial (REF ) equal $b_i=-\\frac{3}{16}, \\quad i=3, 4.$ The coefficient $b_{-c}$ of $\\displaystyle \\frac{b_{-c}}{\\left(x+c\\right)^2}$ equals $b_{-c}=-\\frac{1}{4}.$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ has the form: $R\\left(x\\right)=-\\frac{\\left(1+4d_1^2\\right)}{4x^2}+O\\left(\\frac{1}{x^4}\\right),$ where parameter $d_1$ are expressed in terms of $c$ by (REF ).", "The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=1, 2$ are roots of the polynomial (REF ) equal $b_i=\\frac{3}{4}, \\quad i=1, 2.$ The coefficients $b_i$ of $\\displaystyle \\frac{b_i}{\\left(x-x_i\\right)^2}$ , where $x_i$ , $i=3, 4$ are roots of the polynomial (REF ) equal $b_i=-\\frac{3}{16}, \\quad i=3, 4.$ The coefficient $b_{-c}$ of $\\displaystyle \\frac{b_{-c}}{\\left(x+c\\right)^2}$ equals $b_{-c}=-\\frac{1}{4}.$ The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ has the form: $R\\left(x\\right)=-\\frac{\\left(1+4d_1^2\\right)}{4x^2}+O\\left(\\frac{1}{x^4}\\right),$ where parameter $d_1$ are expressed in terms of $c$ by (REF ).", "The Laurent expansion of $R\\left(x\\right)$ at $x=\\infty $ allows us to state that the differential equation (REF ) in the considered special case can have liouvillian solutions of type 2 only.", "To find these liouvillian solutions, we will use the Kovacic algorithm.", "According to the algorithm, let us find the sets $E_{x_i}$ , $i=1, 2,\\ldots , 6$ , corresponding to the finite poles and the set $E_{\\infty }$ corresponding to the pole of $R\\left(x\\right)$ at $x=\\infty $ .", "For the first order finite pole $x=c$ the corresponding set has the form: $E_c=\\lbrace 4\\rbrace .$ For the finite pole of the second order $x=-c$ the corresponding set has the form: $E_{-c}=\\lbrace 2\\rbrace .$ For the finite poles of the second order $x=x_i$ , $i=1, 2$ the corresponding sets $E_{x_i}$ have the form: $E_{x_i}=\\lbrace -2, 2, 6\\rbrace ,\\quad i=1, 2.$ At last, for the finite poles of the second order $x=x_i$ , $i=3, 4$ the corresponding sets $E_{x_i}$ have the form $E_{x_i}=\\lbrace 1, 2, 3\\rbrace ,\\quad i=3, 4.$ The set $E_{\\infty }$ corresponding to the pole of $R\\left(x\\right)$ at $x=\\infty $ contains only one element and has the form: $E_{\\infty }=\\lbrace 2\\rbrace .$ Now we should determine the families $s=\\left(e_{\\infty }, e_{c}, e_{-c}, e_{x_1}, e_{x_2}, e_{x_3}, e_{x_4}, e_{x_5}, e_{x_6}\\right),$ for which the constant $d$ calculated by the formula (REF ), is a non-negative integer.", "It is easy to see, that the minimal value of the sum of elements of sets, corresponding to finite poles equals 4.", "This means that there are no families $s$ , for which the constant $d$ , calculated by (REF ), is a non-negative integer.", "Thus, in the considered special case the differential equation (REF ) has no liouvillian solutions.", "Now let us consider the case when one of the roots $x_2$ or $x_3$ from the roots (REF ) of the cubic polynomial (REF ) is the root of the polynomial (REF ).", "If we substitute any of these roots to (REF ) and take into account that $k_1=c^3d_1$ , we can find, that parameters of the problem should satisfy the system of two conditions: $h^2+hc^2-1=0, \\quad \\left(1+4d_1^2\\right)c^4+2hc^2-2\\left(h^2-1\\right)=0.$ Let us consider the first of these conditions (REF ).", "This is quadratic equation with respect to $h$ .", "Its solutions have the following form: $h_1=\\frac{1}{2}\\sqrt{c^4+4}-\\frac{c^2}{2}, \\quad h_2=-\\frac{1}{2}\\sqrt{c^4+4}-\\frac{c^2}{2}.$ It is easy to see that for $h_2$ the following inequality holds: $-\\frac{1}{2}\\sqrt{c^4+4}-\\frac{c^2}{2}\\le -1,$ Taking into account the range of the parameter $h$ we obtain that $h$ can take only a value $h=h_1$ .", "Thus, we have $h=\\frac{\\sqrt{c^4+4}}{2}-\\frac{c^2}{2}.$ Substituting this expression for $h$ to the second equation of the system (REF ) and solving it with respect to $d_1^2$ , we obtain $d_1^2=-\\frac{2\\sqrt{c^4+4}-c^2}{4c^2}.$ The right hand side of the equation (REF ) is negative for all values of $c\\ne 0$ .", "This means that the condition (REF ) cannot be valid for all values of parameters $c$ and $d_1$ .", "Thus, the polynomials (REF ) and (REF ) cannot have the common non-real root.", "This means, that the considered special case does not hold.", "In this paper we considered the problem of motion of a heavy rigid body with a fixed point in the Hess case.", "The integration of this problem is reduced to solving the second order linear differential equation.", "We reduce this equation to the equation with rational coefficients (REF ).", "Using the Kovacic algorithm we studied the problem of existence of luouvillian solutions of the differential equation (REF ).", "We proved that this equation has liouvillian solutions of type 2 only and only if the moving rigid body is the Lagrange top $\\left(d_1=0\\right)$ or if the constant of the area integral is zero $\\left(k_1=0\\right)$ .", "Thus the problem of existence of liouvillian solutions for the second order linear differential equation (REF ), the integration of which solves the problem of motion of a heavy rigid body with a fixed point in the Hess case, is completely solved.", "Let us find out the motion of the Hess top in the case, when the conditions (REF ) holds in addition to the conditions (REF ).", "Condition $d_1=0$ is equivalent to the condition $\\left(A_1-A_2\\right)x_1x_2=0.$ Further we will assume that $A_1\\ne A_2$ .", "The condition $A_1=A_2$ together with (REF ) corresponds to the case of kinetic symmetry.", "Then (REF ) is equivalent to $x_1=0$ or $x_2=0$ .", "We will consider the case $x_1=0$ (the case $x_2=0$ is studied similarly).", "The condition $x_1=0$ together with (REF ), corresponds to the Lagrange integrable case $x_1=0,\\quad x_3=0,\\quad A_1=A_3,\\quad x_2=a\\ne 0.$ The Hess integral (REF ) takes the form $\\omega _2=0,$ and the area integral can be rewritten as follows: $\\omega _1\\gamma _1+\\omega _3\\gamma _3=0,$ where we taking into account that the constant of this integral is zero.", "The Euler equations (REF ) under conditions (REF ) takes the form: $A_1\\dot{\\omega }_1=-Mga\\gamma _3, \\quad A_1\\dot{\\omega }_3=Mga\\gamma _1.$ If we introduce the standard Euler angles by the formulae $\\gamma _1=\\sin \\theta \\cos \\varphi ,\\quad \\gamma _2=\\cos \\theta ,\\quad \\gamma _3=\\sin \\theta \\sin \\varphi ,$ $\\omega _1=\\dot{\\psi }\\sin \\theta \\cos \\varphi -\\dot{\\theta }\\sin \\varphi ,\\quad \\omega _2=\\dot{\\psi }\\cos \\theta +\\dot{\\varphi },\\quad \\omega _3=\\dot{\\psi }\\sin \\theta \\sin \\varphi +\\dot{\\theta }\\cos \\varphi $ we can rewrite equation (REF ) in the form $\\dot{\\psi }\\cos \\theta +\\dot{\\varphi }=0,$ and the equation (REF ) takes the form: $\\dot{\\psi }\\sin ^2\\theta =0.$ We will consider the case $\\sin \\theta \\ne 0$ (if $\\theta =0$ , then all the components $\\omega _1$ , $\\omega _2$ , and $\\omega _3$ of angular velocity are zero).", "Then, from the condition (REF ) we have $\\dot{\\psi }=0.$ Condition (REF ) gives $\\dot{\\varphi }=0.$ This means that for the considered motions of the Lagrange top we have $\\psi =\\psi _0={\\rm const}, \\quad \\varphi =\\varphi _0={\\rm const}.$ Thus, the components $\\omega _1$ and $\\omega _3$ of angular velocity of the body takes the form: $\\omega _1=-\\dot{\\theta }\\sin \\varphi _0,\\quad \\omega _3=\\dot{\\theta }\\cos \\varphi _0.$ The Euler equations (REF ) are reduced to the differential equation $\\ddot{\\theta }-\\frac{Mga}{A_1}\\sin \\theta =0,$ which describes the pendulum nutational oscillations of the Lagrange top in the vicinity of $\\theta =\\pi $ .", "Thus the motions of a heavy rigid body with a fixed point in the Hess case for which $d_1=0$ and $k_1=0$ correspond to the nutational oscillations of the body.", "This work was supported financially by the Russian Foundation for Basic Researches (grants no.", "19-01-00140, 20-01-00637)." ] ]	2011.14183
[ [ "Two Families of Hypercyclic Non-Convolution Operators" ], [ "Abstract Let $H(\\mathbb{C})$ be the set of all entire functions endowed with the topology of uniform convergence on compact sets.", "Let $\\lambda,b\\in\\mathbb{C}$, let $C_{\\lambda,b}:H(\\mathbb{C})\\to H(\\mathbb{C})$ be the composition operator $C_{\\lambda,b} f(z)=f(\\lambda z+b)$, and let $D$ be the derivative operator.", "We extend results on the hypercyclicity of the non-convolution operators $T_{\\lambda,b}=C_{\\lambda,b} \\circ D$ by showing that whenever $\|\\lambda\|\\geq 1$, the collection of operators \\begin{align} \\{\\psi(T_{\\lambda,b}): \\psi(z)\\in H(\\mathbb{C}), \\psi(0)=0 \\text{ and } \\psi(T_{\\lambda,b}) \\text{ is continuous}\\} \\end{align} forms an algebra under the usual addition and multiplication of operators which consists entirely of hypercyclic operators (i.e., each operator has a dense orbit).", "We also show that the collection of operators \\begin{align} \\{C_{\\lambda,b}\\circ\\varphi(D): \\varphi(z) \\text{ is an entire function of exponential type with } \\varphi(0)=0\\} \\end{align} consists entirely of hypercyclic operators." ], [ "Introduction", "Let $ denote the complex plane and $ H($ be the set of all entire functions endowed with the topology of uniform convergence on compact sets.", "This topology makes $ H($ a separable Fréchet space, which is a locally convex and metrizable topological vector space that is both complete and separable.", "A continuous linear operator $ T$ defined on a Fréchet Space $ F$ is said to be hypercyclic if there exists $ fF$ (called a hypercyclic vector for $ T$) such that the orbit $ {Tn f: nN}$ is dense in $ F$.", "We refer the reader to the books \\cite {GrosseErdmann} and \\cite {BayartMatheron} for a thorough introduction to the study of hypercyclic operators.$ The first example of a hypercyclic operator was given by Birkhoff in 1929, who showed that the translation operator $T:f(z)\\mapsto f(z+1)$ is hypercyclic [3].", "In 1952, MacLane showed that the derivative operator $D:f(z)\\mapsto f^{\\prime }(z)$ is also hypercyclic [10].", "Both of these results were unified in a substantial paper by Godefroy and Shapiro in 1991, who proved that every continuous linear operator $L:H(\\rightarrow H($ which commutes with translations (these operators are called convolution operators) and which is not a scalar multiple of the identity is hypercyclic [6].", "It is then natural to ask for examples of hypercyclic non-convolution operators, that is, operators which are hypercyclic but do not commute with all translations.", "This study was initiated by Aron and Markose in 2004 in [1], where they introduced the non-convolution operators $T_{\\lambda ,b} f(z) = f^{\\prime }(\\lambda z + b)$ , $\\lambda ,b\\in and proved, along with the authors in \\cite {Hallack}, that such operators are hypercyclic when $ \|\|1$.", "Their result was extended by León-Saavedra and Romero de la Rosa to show, among other things, that such operators are not hypercyclic when $ \|\|<1$ in \\cite {LeonRosa}.", "An $ N$-dimensional analogue of these operators was studied in \\cite {Muro}, and other examples of hypercyclic non-convolution operators can be found in \\cite {Vitaly,Petersson}.", "The purpose of this paper is to extend and complement the examples of hypercyclic non-convolution operators given in \\cite {AronMarkose,Hallack,LeonRosa}.$ This paper is organized as follows.", "In Section , we establish some lemmas and prove that if $\\psi (z)$ is an entire function such that $\\psi (0)=0$ and $\\psi (T_{\\lambda ,b})$ is continuous, then the operator $\\psi (T_{\\lambda ,b})$ is hypercyclic if and only if $\|\\lambda \|\\ge 1$ .", "Our other main result is in Section , the description of which requires some established notation.", "Let $D$ be the derivative operator.", "Godefroy and Shapiro showed that an operator $V$ on $H($ is a convolution operator if and only if $V=\\varphi (D)$ , where $\\varphi (z)$ is an entire function of exponential type.", "For $\\lambda ,b\\in , define the composition operator $ C,b:H(H($ by $ C,b f(z)=f(z+b)$.", "If $ (D)$ is a convolution operator, define by $ L,b,$ the operator $ L,b, = C,b (D)$.", "If $ (z)=z$, then $ L,b,=T,b$, so that these operators $ L,b,$ generalize those introduced by Aron and Markose.", "In Section 3, we prove that if $ (0)=0$ and $ \|\|1$, then $ L,b,$ is hypercyclic.$" ], [ "The hypercyclicity of polynomials of $T_{\\lambda ,b}$", "To show that an operator $T$ on $H($ is hypercyclic, we will employ the well-known Hypercyclicity Criterion.", "It states, for our purposes, that a continuous linear operator $T$ is hypercyclic on $H($ if there exists a dense set $\\mathcal {P}\\subset H($ and a sequence of mappings $S_n:\\mathcal {P}\\rightarrow H($ such that $T^n f \\rightarrow 0$ for all $f\\in \\mathcal {P}$ , $S_n f \\rightarrow 0$ for all $f\\in \\mathcal {P}$ , and $T^n S_n f \\rightarrow f$ for all $f\\in \\mathcal {P}$ .", "Actually, there are more general conditions which ensure the hypercyclicity of an operator, but we will not use them.", "We refer the reader to [7] for more details about the Hypercyclicity Criterion.", "As seen in condition (c) above, the mappings $S_n$ act almost as right inverses (as $n\\rightarrow \\infty $ ) for $T^n$ on $\\mathcal {P}$ .", "Borrowing the ideas in [4], we begin with a crucial lemma that will help determine these mappings for a large class of operators.", "Let $I$ be the identity operator on $H($ and $\\mathcal {P}$ be the collection of complex polynomials in $H($ , which is a dense subset of $H($ .", "Lemma 1 Let $\\psi (z)=\\sum _{k=0}^\\infty w_k z^k$ be an entire function such that $\\psi (0)\\ne 0$ .", "Suppose $G:H(\\rightarrow H($ is an operator such that $G(\\mathcal {P})\\subseteq \\mathcal {P}$ , $\\deg Gp < \\deg p$ for all nonzero $p\\in \\mathcal {P}$ , and for all $\\lambda \\in , $ (G) =k=0wk k Gk$ is a continuous linear operator on $ H($.$ Then for each nonzero $\\lambda \\in , there exists a right-inverse mapping $ S(G):PP$ for $ (G)$ such that $ (G)S(G) p = p $ for all $ pP$.", "Moreover, for each non-negative integer $ m$, there exists $ C=C(m)>0$ such that for each positive integer $ n$ there exist constants $ ai,n$, $ 1im$, such that for each nonzero $ and each nonzero polynomial $p\\in \\mathcal {P}$ of degree $m$ , $S_{\\psi (\\lambda G)}^n p$ has the form $ S_{\\psi (\\lambda G)}^n p = w_0^{-n}\\left(I + a_{1,n}G + \\cdots + a_{m,n} G^m\\right) p, $ and the coefficients $a_{i,n}$ satisfy $\|a_{i,n}\|<Cn^m$ for all $i$ and $n$ .", "Let $p$ be a nonzero polynomial, let $m$ be the degree of $p$ , and let $\\lambda \\in be nonzero.", "Since $ Gp < p$, we have that $ (G) p = k=0m wk k Gk p$.$ Let $\\alpha _1,\\alpha _2,\\ldots ,\\alpha _m$ be the zeros of the polynomial $q(z)=w_0+w_1 z + \\cdots + w_m z^m$ , repeated according to multiplicity.", "Since $w_0\\ne 0$ , none of the $ \\alpha _i $ equal zero, and thus $q(z)=a_0(1-z/\\alpha _1)\\cdots (1-z/\\alpha _m)$ .", "Hence we can write $\\psi (\\lambda G) p = w_0\\left(I-\\frac{\\lambda G}{\\alpha _1}\\right)\\left(I-\\frac{\\lambda G}{\\alpha _2}\\right)\\cdots \\left(I-\\frac{\\lambda G}{\\alpha _m}\\right)p.$ We find a right-inverse mapping for each factor $\\left(I-\\frac{\\lambda G}{\\alpha _i}\\right)$ as follows.", "Let $i$ be an integer satisfying $0\\le i\\le m$ .", "Since $\\deg Gp < \\deg p$ , $p &= \\left(I-\\left(\\frac{\\lambda G}{\\alpha _i}\\right)^{m+1}\\right)p = \\left(I-\\frac{\\lambda G}{\\alpha _i}\\right)\\left(I+\\frac{\\lambda G}{\\alpha _i}+ \\frac{\\lambda ^{2} G^{2}}{\\alpha _i^{2}} + \\cdots + \\frac{\\lambda ^{m} G^{m}}{\\alpha _i^{m}}\\right)p.$ Thus define $S_i p = \\left(I+\\frac{\\lambda G}{\\alpha _i}+ \\frac{\\lambda ^{2} G^{2}}{\\alpha _i^{2}} + \\cdots + \\frac{\\lambda ^{m} G^{m}}{\\alpha _i^{m}}\\right)p$ , and observe $S_i p$ is a polynomial of degree $m$ .", "We then define $S_{\\psi (\\lambda G)} p$ as $S_{\\psi (\\lambda G)}p=\\frac{1}{w_0}S_1\\cdots S_m p$ .", "Thus $S_{\\psi (\\lambda G)}p$ has degree $m$ and $\\psi (\\lambda G)S_{\\psi (\\lambda G)}p=p$ .", "By writing the formula for $S_{\\psi (\\lambda G)}p$ as $S_{\\psi (\\lambda G)}p = \\frac{1}{w_0}\\left( \\displaystyle \\prod \\limits _{i=1}^{m} \\left(I+\\frac{\\lambda G}{\\alpha _i}+ \\frac{\\lambda ^{2} G^{2}}{\\alpha _i^{2}} + \\cdots + \\frac{\\lambda ^{m} G^{m}}{\\alpha _i^{m}}\\right) \\right) p,$ we obtain the form for $S_{\\psi (\\lambda G)}^n p$ by multiplying out the above product and keeping only the terms involving $G^i$ for $0\\le i \\le m$ to obtain $S_{\\psi (\\lambda G)}^n p = a_0^{-n}\\left(I + a_{1,n} \\lambda G + \\cdots + a_{m,n} \\lambda ^m G^m\\right) p.$ Let $r=\\max \\lbrace 1,\|\\alpha _1\|^{-1},\|\\alpha _2\|^{-1},\\ldots ,\|\\alpha _m\|^{-1}\\rbrace $ and let $C(mn,i)$ be the coefficient of $y^i$ in the expansion of $(1+y+y^2+y^3+...)^{mn}$ .", "Then $\|a_{i,n}\| \\le r^i C(mn,i)$ .", "Since $(1+y+y^2+\\dots )^{mn} = 1/(1-y)^{mn}$ for $y \\in (-1,1)$ , and the Taylor Series for $1/(1-y)^{mn}$ is $1+mny+\\frac{mn(mn+1)y^2}{2!}+\\frac{mn(mn+1)(mn+2)y^2}{3!", "}+\\dots ,$ we have that $C(mn,i) =\\binom{mn+i-1}{i} = \\left( \\frac{mn}{1}\\right) \\left( \\frac{mn+1}{2}\\right)\\cdots \\left( \\frac{mn+i-1}{i}\\right) \\le (mn)^i,$ which implies that $\|a_{i,n}\|\\le r^i C(mn,i) \\le (rmn)^i \\le (rmn)^m.$ By taking $C=(rm)^m$ , the proof is complete.", "To simplify our presentation, let us recall some notation from above and establish a bit more for the remainder of this paper.", "Let $\\lambda ,b\\in , and recall that we define the operator $ C,b:H(H($ by $ C,b f(z) = f(z+b)$, and the operator $ T,b: H(H($ by $ T, bf(z)=f'(z+b)$.", "With this notation, we can view $ T,b$ as a composition of two operators, namely{\\begin{@align}{1}{-1}T_{\\lambda ,b}=C_{\\lambda ,b} \\circ D.\\end{@align}}More generally, if $ (z)$ is an entire function of exponential type, then we compose $ C,b$ and the convolution operator $ (D)$ to define the operator $ L,b,$ as{\\begin{@align}{1}{-1}L_{\\lambda ,b,\\varphi }=C_{\\lambda ,b} \\circ \\varphi (D).\\end{@align}}Observe that if $ (z)=z$, then $ L,b,=T,b$.", "Our immediate focus will turn to the operators $ L,b,zm=C,bDm$, where $ mN$.$ In this case, one can check that $C_{\\lambda ,b}^n f(z) = f\\left(\\lambda ^n z + \\frac{1-\\lambda ^n}{1-\\lambda }b\\right) f(\\lambda ^n z - r_n)$ , where $r_n=-b\\frac{1-\\lambda ^n}{1-\\lambda }$ .", "A straightforward induction argument then yields that $L_{\\lambda ,b,z^m}^n f(z) = \\lambda ^{\\frac{mn(n-1)}{2}} C_{\\lambda ,b}^n \\circ D^{mn} f(z) = \\lambda ^{\\frac{mn(n-1)}{2}} f^{(mn)}(\\lambda ^n z -r_n).$ We want to use the Hypercyclicity Criterion to show that these operators are hypercyclic when $\|\\lambda \|\\ge 1$ .", "The bulk of the work, as usual, is to determine the right-inverse mappings.", "To motivate what follows, let us describe a derivation of these mappings by closely following the work in [5] to determine a sequence of right-inverse mappings $S_{m,n}$ for $L_{\\lambda ,b,z^m}^n$ defined on the set of complex polynomials $\\mathcal {P}$ .", "Define a “formal operator\" $A_m$ on the set $\\mathcal {D}=\\lbrace 1\\rbrace \\cup \\lbrace d(z+c)^k: k\\in \\mathbb {N}, c,d\\in $ by $A_m(1) = \\frac{(z-b)^m}{m!", "}, \\text{ and} \\\\A_m(d(z+c)^k) = \\frac{k!d(z+c)^{k+m}}{(k+m)!", "}.$ The formal operator $A_m$ acts as an “$m$ th antiderivative\" operator, but it is just a formal tool we use to motivate well-defined right-inverse mappings $S_{m,n}$ .", "With a formal antiderivative operator at hand, it is then natural to try and define $S_{m,1}$ on the basis $\\lbrace 1,z,z^2,\\ldots \\rbrace $ for $\\mathcal {P}$ as $S_{m,1}(z^k)=A_m \\circ C_{\\lambda ,b}^{-1} (z^k),$ and then take $S_{m,n}$ to be $(S_{m,1})^n$ .", "This would yield $S_{m,n}(z^k)=\\frac{k!}{(k+mn)!", "\\lambda ^{kn}\\lambda ^{\\frac{mn(n-1)}{2}}} (z+r_n)^{k+mn},$ which works well with condition (c) of the Hypercyclicity Criterion since $L_{\\lambda ,b,z^m}^n \\circ S_{m,n} = I$ on $\\lbrace 1,z,z^2,\\ldots \\rbrace $ for this choice of $S_{m,n}$ .", "However, one would run into issues checking condition (b), as one would find high powers of $\\lambda $ paired with low powers of $z$ when expanding $(z+r_n)^{k+mn}$ .", "To deal with this issue, we will add a polynomial of degree less than $mn$ to the definition of $S_{m,n}$ in (REF ) to “kill off\" these high powers of $\\lambda $ .", "This new definition of $S_{m,n}$ will then satisfy condition (b) of the Hypercyclicity Criterion, and still satisfy condition (c) because any polynomial of degree less than $mn$ belongs to the kernel of $L_{\\lambda ,b,z^m}^n$ .", "We provide the details in the following lemma.", "Lemma 2 Let $m\\in \\mathbb {N}$ .", "There are linear mappings $S_{m,n}:\\mathcal {P}\\rightarrow \\mathcal {P}$ defined by $S_{m,n}(z^k) = \\frac{k!}{(k+mn)!", "\\lambda ^{kn}\\lambda ^{\\frac{mn(n-1)}{2}}} \\sum _{j=0}^k \\binom{k+mn}{j} z^{k+mn-j} r_n^j,$ such that for all $p\\in \\mathcal {P}$ , $L_{\\lambda ,b,z^m}^n S_{m,n} p=p$ .", "Furthermore, if $\\lbrace \\sigma _n\\rbrace $ is a sequence of complex numbers for which there exists $t>0$ such that $\|\\sigma _n\|\\le t^n$ for all $n\\in \\mathbb {N}$ , then for all $p\\in \\mathcal {P}$ and for all $\\ell \\in \\mathbb {N}$ , $S_{m,\\ell n}(\\sigma _n p)\\rightarrow 0.$ For each monomial $z^k$ , define $S_{m,n}(z^k)$ as above in (REF ), and extend $S_{m,n}$ linearly to $\\mathcal {P}$ .", "Let $\\Delta _{k,n}=\\displaystyle \\frac{k!}{(k+mn)!", "\\lambda ^{kn}\\lambda ^{\\frac{mn(n-1)}{2}}}(z+r_n)^{k+mn}-S_{m,n}(z^k)$ .", "By expanding $(z+r_n)^{k+mn}$ using the Binomial Theorem and cancelling common terms, one can establish that $\\Delta _{k,n}=\\displaystyle \\frac{k!}{(k+mn)!", "\\lambda ^{kn}\\lambda ^{\\frac{mn(n-1)}{2}}} \\sum _{j=k+1}^{k+mn} \\binom{k+mn}{j} z^{k+mn-j} r_n^j.$ Thus the degree of $\\Delta _{k,n}$ is $mn-1$ , and hence it belongs to the kernel of $L_{\\lambda ,b,z^m}^n $ by equation (REF ).", "Using equations (REF ) and (REF ), we compute that $& L_{\\lambda ,b,z^m}^n S_{m,n} (z^k)\\\\& = L_{\\lambda ,b,z^m}^n \\left( S_{m,n}(z^k)+\\Delta _{k,n}\\right)\\\\& = L_{\\lambda ,b,z^m}^n \\left( \\displaystyle \\frac{k!}{(k+mn)!", "\\lambda ^{kn}\\lambda ^{\\frac{mn(n-1)}{2}}}(z+r_n)^{k+mn}\\right)\\\\& = \\frac{k!}{(k+mn)!", "\\lambda ^{kn}\\lambda ^{\\frac{mn(n-1)}{2}}} \\cdot \\frac{\\lambda ^{\\frac{mn(n-1)}{2}}(k+mn)!}{k!}", "(\\lambda ^n z - r_n + r_n)^k \\text{ by (\\ref {L^n eqn})}\\\\& = z^k.$ This verifies that $S_{m,n}$ is a right inverse for $L_{\\lambda ,b,z^m}^n$ on $\\mathcal {P}$ .", "To verify the limit in (REF ), let $p$ be a polynomial and let $\\ell \\in \\mathbb {N}$ .", "By the linearity of $S_{m,\\ell n}$ it suffices to show that $S_{m,\\ell n}(\\sigma _n z^k) \\rightarrow 0$ uniformly on compact subsets of $ for each monomial $ zk$.", "Let $ R>0$, let $ \|z\|R$, and let $ j$ be an integer satisfying $ 0jk$.", "As $ Sm,n(zk)$ is a sum of $ k+1$ terms, to show (\\ref {reduce1}) it suffices to show that each of those terms converges to zero uniformly when $ \|z\|R$.", "That is, it suffices to show{\\begin{@align}{1}{-1}\\left\|\\frac{\\sigma _n k!", "}{(k+m\\ell n)!", "\\lambda ^{k\\ell n}\\lambda ^{\\frac{m\\ell n( \\ell n-1)}{2}}} \\binom{k+m\\ell n}{j} z^{k+m\\ell n-j} r_{\\ell n}^j\\right\|\\rightarrow 0\\end{@align}}uniformly.", "Since $ \|\|1$,{\\begin{@align}{1}{-1}\|r_{\\ell n}\|=\|b\|\\left\|\\sum _{i=0}^{\\ell n-1} \\lambda ^i\\right\|\\le \\ell n\|b\|\\cdot \|\\lambda \|^{\\ell n-1}.\\end{@align}}Thus{\\begin{@align}{1}{-1}& \\left\|\\frac{\\sigma _n k!", "}{(k+m\\ell n)!", "\\lambda ^{k\\ell n}\\lambda ^{\\frac{m\\ell n( \\ell n-1)}{2}}} \\binom{k+m\\ell n}{j} z^{k+m\\ell n-j} r_{\\ell n}^j\\right\|\\\\& \\le \\frac{t^n k!", "R^{k+m\\ell n-j} (\\ell n)^j \|b\|^j \|\\lambda \|^{(\\ell n-1)j}}{(k+m\\ell n)!", "\|\\lambda \|^{k\\ell n}\|\\lambda \|^{\\frac{m\\ell n(\\ell n-1)}{2}}} \\binom{k+m\\ell n}{j} \\text{ by (\\ref {rnbound})}\\\\& = \\frac{t^n k!", "R^{k+m\\ell n-j} (\\ell n)^j \|b\|^j \|\\lambda \|^{(\\ell n-1)j}}{(k+m\\ell n)!", "\|\\lambda \|^{k\\ell n}\|\\lambda \|^{\\frac{m\\ell n(\\ell n-1)}{2}}} \\frac{(k+m\\ell n)!}{j!", "(k+m\\ell n-j)!}", "\\overset{n\\rightarrow \\infty }{\\longrightarrow }0,\\end{@align}}which establishes the limit in (\\ref {reduce2}) and completes the proof.$ With the previous lemmas at hand, we can now use the Hypercyclicity Criterion to show that there are many hypercyclic non-convolution operators that can be generated by $T_{\\lambda ,b}$ .", "Theorem 3 Let $\\lambda ,b\\in with $ \|\|1$, and let $ T,b:H(H($ be the operator defined by $ T,b:f(z)f'(z +b)$.", "If $$ is an entire function such that $ (0)=0$ and $ (T,b)$ is a continuous linear operator, then $ (T,b)$ is hypercyclic.$ Let $\\psi (z)=\\xi (z)z^\\ell $ , where $\\ell \\in \\mathbb {N}$ and $\\xi (z)$ is an entire function with $\\xi (0)\\ne 0$ .", "Then the operator $\\psi (T_{\\lambda ,b})=\\xi (T_{\\lambda ,b})T_{\\lambda ,b}^\\ell $ , and $(\\psi (T_{\\lambda ,b}))^n=(\\xi (T_{\\lambda ,b}))^n T_{\\lambda ,b}^{\\ell n}$ .", "Let $\\mathcal {P}$ be the set of complex polynomials in $H($ , which is a dense subset of $H($ , and let $g\\in \\mathcal {P}$ .", "Since $T_{\\lambda ,b}(g)\\in \\mathcal {P}$ and $\\deg T_{\\lambda ,b}(g) < \\deg g$ , by Lemma REF there is a mapping $S_{\\xi (T_{\\lambda ,b})}:\\mathcal {P}\\rightarrow \\mathcal {P}$ such that $\\xi (T_{\\lambda ,b})^n S_{\\xi (T_{\\lambda ,b})}^n (g)=g$ for all $g\\in \\mathcal {P}$ .", "Since $T_{\\lambda ,b}=L_{\\lambda ,b,z}$ , by Lemma REF there exist linear mappings $S_{1,\\ell n}:\\mathcal {P}\\rightarrow \\mathcal {P}$ such that $T_{\\lambda ,b}^{\\ell n} S_{1,\\ell n}g=g$ for all $g\\in \\mathcal {P}$ .", "Thus for all $g\\in \\mathcal {P}$ , $\\psi (T_{\\lambda ,b})^n S_{1,\\ell n}S_{\\xi (T_{\\lambda ,b})}^n (g) = \\xi (T_{\\lambda ,b})^nT_{\\lambda ,b}^{\\ell n}S_{1,\\ell n}S_{\\xi (T_{\\lambda ,b})}^n (g) = g,$ so the mapping $S_{1,\\ell n}S_{\\xi (T_{\\lambda ,b})}^n$ is a right inverse for $\\psi (T_{\\lambda ,b})^n$ on $\\mathcal {P}$ .", "We check the three conditions of the Hypercyclicity Criterion.", "As just mentioned, the third condition is satisfied.", "Let $p$ be a polynomial of degree $d$ .", "Since $T_{\\lambda ,b}^n p=0$ whenever $n>d$ , and since $\\psi (0)=0$ , we have that $(\\psi (T_{\\lambda ,b}))^n p=0$ whenever $n>d$ .", "Thus the first condition of the Hypercyclicity Criterion is satisfied.", "What remains to check is the second condition.", "Let $\\xi (0)=w_0$ .", "By Lemma REF , there is a constant $C=C(d)>0$ and are constants $a_{i,n}$ for $1\\le i\\le d$ such that $S_{\\xi (T_{\\lambda ,b})}^n p = w_0^{-n}\\left(I + a_{1,n}T_{\\lambda ,b} + \\cdots + a_{d,n} T_{\\lambda ,b}^d\\right) p,$ and $\|a_{i,n}\|<Cn^d$ for all $i$ and $n$ .", "For each positive integer $n$ , let $a_{0,n}=1$ .", "Let $i$ be an integer such that $0\\le i\\le d$ .", "By the linearity of $S_{1,\\ell n}$ and equation (REF ), to show $S_{1,\\ell n}S_{\\xi (T_{\\lambda ,b})}^n p\\overset{n\\rightarrow \\infty }{\\longrightarrow }0$ uniformly on compact subsets of $, it suffices to show that{\\begin{@align}{1}{-1}S_{1,\\ell n} (w_0^{-n}a_{i,n} T_{\\lambda ,b}^i p)\\overset{n\\rightarrow \\infty }{\\longrightarrow }0\\end{@align}}uniformly on compact subsets of $ .", "If $i=0$ , let $\\sigma _n=w_0^{-n}$ and $t=\|w_0\|^{-1}$ .", "Then the limit $(\\ref {Slimit})$ in Lemma REF implies ().", "If $i>0$ , let $\\sigma _n=w_0^{-n}a_{i,n}$ .", "Since $\|w_0^{-n}a_{i,n}\|\\le \|w_0\|^{-n} Cn^d<t^n$ for some $t>0$ , limit (REF ) in Lemma REF again yields (), as desired.", "This shows that the second condition of the Hypercyclicity Criterion is satisfied.", "Hence $\\psi (T_{\\lambda ,b})$ is hypercyclic.", "Theorem REF really provides an algebra of hypercyclic non-convolution operators when $\|\\lambda \|\\ge 1$ (except when $\\lambda =1$ , in which case the operators are indeed convolution operators).", "To see this, let $\\psi _1(z),\\psi _2(z)\\in H($ such that $\\psi _1(0)=0=\\psi _2(0)$ and $\\psi _1(T_{\\lambda ,b})$ , $\\psi _2(T_{\\lambda ,b})$ are continuous.", "Then $\\psi _1(0)+\\psi (0)=0, \\psi _1(0)\\psi _2(0)=0$ , and $\\psi _1(T_{\\lambda ,b})\\psi _2(T_{\\lambda ,b})$ is a continuous operator.", "Hence $\\psi _1(T_{\\lambda ,b})+\\psi _2(T_{\\lambda ,b})$ and $\\psi _1(T_{\\lambda ,b})\\psi _2(T_{\\lambda ,b})$ are hypercyclic by Theorem REF , and any non-zero scalar multiple of them is as well.", "We next show that $\|\\lambda \|\\ge 1$ is necessary for the hypercyclicity of these types of operators.", "Theorem 4 Let $\\lambda ,b\\in with $ \|\|<1$, and let $ T,b:H(H($ be the operator defined by $ T,b:f(z)f'(z +b)$.", "If $ (z)$ is an entire function such that $ (0)=0$ and $ (T,b)$ is a continuous linear operator, then $ (T,b)$ is not hypercyclic.$ We will show that $(\\psi (T_{\\lambda ,b}))^n f\\overset{n\\rightarrow \\infty }{\\longrightarrow }0$ for every $f\\in H($ .", "Since $\\psi (0)=0$ , we may write $\\psi (z)=\\alpha z^\\ell \\xi (z)$ , where $\\alpha \\in \\lbrace 0\\rbrace $ and $\\xi (z)=\\sum _{k=0}^\\infty w_k z^k$ is an entire function such that $\\xi (0)=w_0=1$ .", "Let $c_{j,n}$ be the $j$ th Taylor coefficient of the Taylor series of $(\\xi (z))^n$ centered at zero, so that $(\\xi (z))^n = (1+w_1 z + w_2 z^2 + \\cdots )^n 1+c_{1,n} z + c_{2,n} z^2 +\\cdots ,$ and let $r=\\sup \\lbrace \|w_k\| :k\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace \\rbrace $ .", "Then by the same type of argument used in Lemma REF to obtain the estimate (REF ), we have that $\|c_{j,n}\| \\le r^j\\binom{n+j-1}{j}=r^j \\left( \\frac{n}{1}\\right) \\left( \\frac{n+1}{2}\\right)\\cdots \\left( \\frac{n+j-1}{j}\\right)\\le r^j n^j.$ Now let $f\\in H($ , let $C=\\max \\lbrace \|f(z)\|: \|z\|\\le 1\\rbrace $ , let $R>0$ be given, and let $\|z\|\\le R$ .", "As shown in the proof of [9], there exists an $n_0\\in \\mathbb {N}$ such that for all $n\\ge n_0$ , $\\left\| T_{\\lambda ,b}^n f(z)\\right\| \\le C n!", "2^n \|\\lambda \|^{\\frac{n(n-1)}{2}}.$ Furthermore, since $\\lim _{n\\rightarrow \\infty } 2rn^2\\ell \|\\lambda \|^{\\ell n}=0$ , there exists $n_1\\in \\mathbb {N}$ such that $n\\ge n_1$ implies $2rn^2\\ell \|\\lambda \|^{\\ell n} <1.$ Now let $n\\ge \\max \\lbrace n_0,n_1\\rbrace $ , and let $c_{0,n}=1$ .", "Then $ & \\left\| \\left(\\psi (T_{\\lambda ,b})\\right)^n f(z)\\right\|\\\\ & = \\left\|\\alpha ^n T^{n\\ell } \\left( I + c_{1,n}T + c_{2,n}T^{2} + \\cdots \\right) f(z)\\right\| \\\\ & \\le \|\\alpha \|^n \\sum _{j=0}^\\infty \\left\| c_{j,n}T^{n\\ell +j} f(z) \\right\|\\\\& \\le \|\\alpha \|^n \\sum _{j=0}^\\infty r^j n^j C(n\\ell +j)!2^{n\\ell +j}\|\\lambda \|^{\\frac{(n\\ell +j)(n\\ell +j-1)}{2}} \\text{ by (\\ref {cjnBound}) and (\\ref {LeonRosa})}.$ Let $\\beta _{j,n}$ be the $j$ th term in the sum in the previous line.", "Then (REF ) implies $\\left\|\\frac{\\beta _{j+1,n}}{\\beta _{j,n}}\\right\|=rn(n\\ell +j+1)2\|\\lambda \|^{j+\\ell n}\\le rn(n\\ell (j+2))2\|\\lambda \|^{j+\\ell n}<(j+2)\|\\lambda \|^j.$ We claim that inequality (REF ) implies that $\|\\beta _{j,n}\|< (j+2)!\|\\lambda \|^\\frac{j(j-1)}{2} \|\\beta _{0,n}\|$ for each $j\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace $ .", "To prove this, we proceed by induction.", "The basis for induction follows immediately upon substituting $j=0$ into (REF ).", "For the induction step, we assume that $\|\\beta _{j,n}\|< (j+2)!\|\\lambda \|^\\frac{j(j-1)}{2} \|\\beta _{0,n}\|$ is true for some $j\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace $ .", "Then $\|\\beta _{j+1,n}\| & < (j+3)\|\\lambda \|^j \|\\beta _{j,n}\| \\text{ by (\\ref {ratio})}\\\\& < (j+3)\|\\lambda \|^j (j+2)!\|\\lambda \|^\\frac{j(j-1)}{2}\|\\beta _{0,n}\| \\text{ by the induction hypothesis}\\\\& = (j+3)!\|\\lambda \|^\\frac{j(j+1)}{2}\|\\beta _{0,n}\|,$ which had to be shown.", "This completes the proof of our claim.", "Since $\|\\beta _{j,n}\|< (j+2)!\|\\lambda \|^\\frac{j(j-1)}{2} \|\\beta _{0,n}\|$ for each $j\\in \\mathbb {N}\\cup \\lbrace 0\\rbrace $ , the sum in (REF ) is less than the sum $\|\\beta _{0,n}\|\\sum _{j=0}^\\infty (j+2)!\|\\lambda \|^\\frac{j(j-1)}{2} $ , which converges to zero as $n\\rightarrow \\infty $ since $\|\\beta _{0,n}\|\\overset{n\\rightarrow \\infty }{\\longrightarrow }0$ and $\\sum _{j=0}^\\infty (j+2)!\|\\lambda \|^\\frac{j(j-1)}{2}$ converges by the ratio test.", "This proves that $(\\psi (T_{\\lambda ,b}))^n f\\overset{n\\rightarrow \\infty }{\\longrightarrow }0$ uniformly on compact subsets of $, so $ (T,b)$ cannot by hypercyclic.$ We summarize the previous two theorems in the following characterization.", "Theorem 5 Let $\\lambda ,b\\in and let $ T,b:H(H($ be the operator defined by $ T,b:f(z)f'(z +b)$.", "The algebra of operators{\\begin{@align}{1}{-1}\\lbrace \\psi (T_{\\lambda ,b}): \\psi (z)\\in H(, \\psi (0)=0 \\text{ and } \\psi (T_{\\lambda ,b}) \\text{ is continuous}\\rbrace \\end{@align}}consists entirely of hypercyclic operators if $ \|\|1$ and consists entirely of non-hypercyclic operators if $ \|\|>1$.$ Hypercyclicity of $C_{\\lambda ,b}\\circ \\varphi (D)$ We now look at another generalization of the operators $T_{\\lambda ,b}=C_{\\lambda ,b}\\circ D$ .", "Let $\\varphi (z)$ be an entire function of exponential type, so that the operator $\\varphi (D)$ is a convolution operator.", "We consider in this section the operators $L_{\\lambda ,b,\\varphi }=C_{\\lambda ,b}\\circ \\varphi (D)$ , each of which is a non-convolution operator whenever $\\lambda \\ne 1$ .", "We first prove a type of commutation relation between $C_{\\lambda ,b}$ and $\\varphi (D)$ .", "Lemma 6 Suppose $\\varphi (D)$ is a convolution operator for some non-constant entire function $\\varphi $ of exponential type.", "Let $\\lambda ,b\\in with $ =0$, and let $ C,b: H(H($ be the composition operator $ C,b:f(z)f(z + b)$.", "Then $ C,b (D) = (-1D) C,b $.$ Let $\\varphi (z)=\\sum _{k=0}^\\infty w_k z^k$ and let $f(z)\\in H($ .", "Then $\\varphi (\\lambda ^{-1}D)C_{\\lambda ,b} f(z)=\\varphi (\\lambda ^{-1}D) f(\\lambda z+b) & = \\sum _{k=0}^\\infty w_k \\lambda ^{-k}D^k (f(\\lambda z+b))\\\\&= C_{\\lambda ,b}\\left(\\sum _{k=0}^\\infty w_k D^k f(z) \\right) = C_{\\lambda ,b}\\varphi (D) f(z).$ We now provide yet another family of hypercyclic non-convolution operators.", "Theorem 7 Suppose $\\varphi (D)$ is a convolution operator for some non-constant entire function $\\varphi $ of exponential type with $\\varphi (0) = 0$ .", "Let $\\lambda ,b\\in and let $ C,b: H(H($ be the composition operator $ C,b:f(z)f(z + b)$.", "If $ \|\|1$, then the operator $ L,b,=C,b (D)$ is hypercyclic.$ We first write $\\varphi (z)=z^m \\psi (z)$ , where $\\psi (z)=\\sum _{k=0}^\\infty w_k z^k$ is an entire function of exponential type with $\\psi (0)\\ne 0$ .", "By repeatedly applying Lemma REF , we have that $ L_{\\lambda ,b,\\varphi }^n & = \\underbrace{C_{\\lambda ,b} \\varphi (D) C_{\\lambda ,b} \\varphi (D) \\cdots C_{\\lambda ,b} \\varphi (D)}_{\\text{$n$ times}}\\\\ & = \\varphi \\left(\\lambda ^{-n}D\\right) \\varphi \\left(\\lambda ^{1-n}D\\right) \\cdots \\varphi \\left(\\lambda ^{-1}D\\right) C_{\\lambda ,b}^n\\\\ & = \\frac{D^m}{\\lambda ^{nm}} \\psi \\left(\\lambda ^{-n}D\\right) \\frac{D^m}{\\lambda ^{(n-1)m}} \\psi \\left(\\lambda ^{1-n}D\\right) \\cdots \\frac{D^m}{\\lambda ^{m}}\\psi \\left(\\lambda ^{-1}D\\right) C_{\\lambda ,b}^n\\\\ & = \\psi \\left(\\lambda ^{-n}D\\right) \\psi \\left(\\lambda ^{1-n}D\\right) \\cdots \\psi \\left(\\lambda ^{-1}D\\right) \\frac{1}{\\lambda ^\\frac{nm(n+1)}{2}} D^{nm} C_{\\lambda ,b}^n \\\\ & = \\psi \\left(\\lambda ^{-n}D\\right) \\psi \\left(\\lambda ^{1-n}D\\right) \\cdots \\psi \\left(\\lambda ^{-1}D\\right) \\lambda ^\\frac{nm(n-1)}{2} C_{\\lambda ,b}^n D^{nm}\\\\& = \\psi \\left(\\lambda ^{-n}D\\right) \\psi \\left(\\lambda ^{1-n}D\\right) \\cdots \\psi \\left(\\lambda ^{-1}D\\right) L_{\\lambda ,b,z^m}^n \\text{ by (\\ref {L^n eqn})},$ where $L_{\\lambda ,b,z^m}:H(\\rightarrow H($ is the operator $L_{\\lambda ,b,z^m}:f(z)\\mapsto f^{(m)}(\\lambda z+b)$ considered in Lemma REF .", "Let $\\mathcal {P}$ be the set of complex polynomials in $H($ , which is a dense subset of $H($ .", "Let $p$ be a nonzero polynomial of degree $d$ .", "We define a right-inverse $F_n:\\mathcal {P}\\rightarrow \\mathcal {P}$ for $L_{\\lambda ,b,\\varphi }^n$ on $\\mathcal {P}$ as follows.", "By Lemma REF , there exist $C=C(d)>0$ and constants $a_i\\in 1\\le i \\le d$ , such that for each positive integer $j$ , the mapping $S_{\\psi (\\lambda ^{-j} D)}:\\mathcal {P}\\rightarrow \\mathcal {P}$ defined by $S_{\\psi (\\lambda ^{-j} D)} p =w_0^{-1}\\left(I+a_1 \\lambda ^{-j} D + \\cdots + a_d (\\lambda ^{-j})^d D^d\\right)p$ is a right-inverse for $\\psi (\\lambda ^{-j} D)$ on $\\mathcal {P}$ , and $\|a_i\|<C$ for each $i$ .", "Let $S_{m,n}:\\mathcal {P}\\rightarrow \\mathcal {P}$ be the linear right inverse of $L_{\\lambda ,b,z^m}^n$ as defined in Lemma REF .", "We then define the mapping $F_n:\\mathcal {P}\\rightarrow \\mathcal {P}$ by $F_n p = S_{m,n} S_{\\psi (D)} \\cdots S_{\\psi (\\lambda ^{2-n}D)} S_{\\psi (\\lambda ^{1-n}D)} p,$ which satisfies $L_{\\lambda ,b,\\varphi }^n F_n p =p$ .", "The condition $\\varphi (0)=0$ implies $\\deg L_{\\lambda ,b,\\varphi } p < \\deg p$ , which implies $L_{\\lambda ,b,\\varphi }^n p= 0$ whenever $n>d$ .", "What remains to show for the Hypercyclicity Criterion is that $F_n p\\overset{n\\rightarrow \\infty }{\\longrightarrow }0$ uniformly on compact subsets of $.", "By multiplying out the product $ S(D) S(2-nD) S(1-nD)$ using equation (\\ref {Spsi}), we have that{\\begin{@align}{1}{-1}& S_{\\psi (D)} \\cdots S_{\\psi (\\lambda ^{2-n}D)} S_{\\psi (\\lambda ^{1-n}D)} p\\\\& = w_0^{-n} [I+a_1 D + \\cdots + a_d D^d]\\cdots [I+a_1 \\lambda ^{1-n} D + \\cdots + a_d (\\lambda ^{1-n})^d D^d] p\\\\& = w_0^{-n} [I + c_{1,n} D + \\cdots + c_{d,n} D^d] p,\\end{@align}}where the coefficients $ cj,n$ for $ 1jd$ satisfy{\\begin{@align}{1}{-1}c_{j,n}=\\sum _{j_1+\\cdots + j_n = j} \\frac{a_{j_1}}{(\\lambda ^{j_1})^0} \\frac{ a_{j_2}}{(\\lambda ^{j_2})^1}\\frac{a_{j_3}}{(\\lambda ^{j_3})^2}\\cdots \\frac{a_{j_n}}{(\\lambda ^{j_n})^{n-1}},\\end{@align}}where each $ jk$ is a non-negative integer.$ For each positive integer $n$ , let $c_{0,n}=1$ .", "Since $S_{m,n}$ is linear, to show (REF ) it suffices to show that $S_{m,n}(w_0^{-n}c_{j,n}p)\\overset{n\\rightarrow \\infty }{\\longrightarrow }0.$ uniformly on compact subsets of $ for each integer $ j$ such that $ 0jd$.$ Let $j$ be an integer satisfying $0\\le j\\le d$ .", "The number of terms in the sum () is equal to the number of multinomial coefficients in a multinomial sum, which is $\\displaystyle \\binom{j+n-1}{n-1}$ .", "Let $\\alpha = \\max \\lbrace 1,\|a_1\|,\|a_2\|,\\cdots ,\|a_d\|\\rbrace $ .", "Since $\|\\lambda \|\\ge 1$ , by () we have that $\|c_{j,n}\|\\le \\binom{j+n-1}{n-1}\\alpha ^j \\le n^j\\alpha ^j\\le n^d\\alpha ^d<e^{nd}\\alpha ^d.$ Now let $t=\|w_0\|^{-1}e^d\\alpha ^d$ .", "Then $\|w_0^{-n}c_{j,n}\|\\le t^n$ , and thus $S_{m,n}(w_0^{-n}c_{j,n}p)\\overset{n\\rightarrow \\infty }{\\longrightarrow }0$ uniformly on compact subsets of $ by limit (\\ref {Slimit}) in Lemma \\ref {HallackLemma}, which shows that (\\ref {Smn}) holds and completes the proof.$ Acknowledgements Funding: This work was supported by the St. Olaf College Collaborative Undergraduate Research and Inquiry program." ], [ "Hypercyclicity of $C_{\\lambda ,b}\\circ \\varphi (D)$", "We now look at another generalization of the operators $T_{\\lambda ,b}=C_{\\lambda ,b}\\circ D$ .", "Let $\\varphi (z)$ be an entire function of exponential type, so that the operator $\\varphi (D)$ is a convolution operator.", "We consider in this section the operators $L_{\\lambda ,b,\\varphi }=C_{\\lambda ,b}\\circ \\varphi (D)$ , each of which is a non-convolution operator whenever $\\lambda \\ne 1$ .", "We first prove a type of commutation relation between $C_{\\lambda ,b}$ and $\\varphi (D)$ .", "Lemma 6 Suppose $\\varphi (D)$ is a convolution operator for some non-constant entire function $\\varphi $ of exponential type.", "Let $\\lambda ,b\\in with $ =0$, and let $ C,b: H(H($ be the composition operator $ C,b:f(z)f(z + b)$.", "Then $ C,b (D) = (-1D) C,b $.$ Let $\\varphi (z)=\\sum _{k=0}^\\infty w_k z^k$ and let $f(z)\\in H($ .", "Then $\\varphi (\\lambda ^{-1}D)C_{\\lambda ,b} f(z)=\\varphi (\\lambda ^{-1}D) f(\\lambda z+b) & = \\sum _{k=0}^\\infty w_k \\lambda ^{-k}D^k (f(\\lambda z+b))\\\\&= C_{\\lambda ,b}\\left(\\sum _{k=0}^\\infty w_k D^k f(z) \\right) = C_{\\lambda ,b}\\varphi (D) f(z).$ We now provide yet another family of hypercyclic non-convolution operators.", "Theorem 7 Suppose $\\varphi (D)$ is a convolution operator for some non-constant entire function $\\varphi $ of exponential type with $\\varphi (0) = 0$ .", "Let $\\lambda ,b\\in and let $ C,b: H(H($ be the composition operator $ C,b:f(z)f(z + b)$.", "If $ \|\|1$, then the operator $ L,b,=C,b (D)$ is hypercyclic.$ We first write $\\varphi (z)=z^m \\psi (z)$ , where $\\psi (z)=\\sum _{k=0}^\\infty w_k z^k$ is an entire function of exponential type with $\\psi (0)\\ne 0$ .", "By repeatedly applying Lemma REF , we have that $ L_{\\lambda ,b,\\varphi }^n & = \\underbrace{C_{\\lambda ,b} \\varphi (D) C_{\\lambda ,b} \\varphi (D) \\cdots C_{\\lambda ,b} \\varphi (D)}_{\\text{$n$ times}}\\\\ & = \\varphi \\left(\\lambda ^{-n}D\\right) \\varphi \\left(\\lambda ^{1-n}D\\right) \\cdots \\varphi \\left(\\lambda ^{-1}D\\right) C_{\\lambda ,b}^n\\\\ & = \\frac{D^m}{\\lambda ^{nm}} \\psi \\left(\\lambda ^{-n}D\\right) \\frac{D^m}{\\lambda ^{(n-1)m}} \\psi \\left(\\lambda ^{1-n}D\\right) \\cdots \\frac{D^m}{\\lambda ^{m}}\\psi \\left(\\lambda ^{-1}D\\right) C_{\\lambda ,b}^n\\\\ & = \\psi \\left(\\lambda ^{-n}D\\right) \\psi \\left(\\lambda ^{1-n}D\\right) \\cdots \\psi \\left(\\lambda ^{-1}D\\right) \\frac{1}{\\lambda ^\\frac{nm(n+1)}{2}} D^{nm} C_{\\lambda ,b}^n \\\\ & = \\psi \\left(\\lambda ^{-n}D\\right) \\psi \\left(\\lambda ^{1-n}D\\right) \\cdots \\psi \\left(\\lambda ^{-1}D\\right) \\lambda ^\\frac{nm(n-1)}{2} C_{\\lambda ,b}^n D^{nm}\\\\& = \\psi \\left(\\lambda ^{-n}D\\right) \\psi \\left(\\lambda ^{1-n}D\\right) \\cdots \\psi \\left(\\lambda ^{-1}D\\right) L_{\\lambda ,b,z^m}^n \\text{ by (\\ref {L^n eqn})},$ where $L_{\\lambda ,b,z^m}:H(\\rightarrow H($ is the operator $L_{\\lambda ,b,z^m}:f(z)\\mapsto f^{(m)}(\\lambda z+b)$ considered in Lemma REF .", "Let $\\mathcal {P}$ be the set of complex polynomials in $H($ , which is a dense subset of $H($ .", "Let $p$ be a nonzero polynomial of degree $d$ .", "We define a right-inverse $F_n:\\mathcal {P}\\rightarrow \\mathcal {P}$ for $L_{\\lambda ,b,\\varphi }^n$ on $\\mathcal {P}$ as follows.", "By Lemma REF , there exist $C=C(d)>0$ and constants $a_i\\in 1\\le i \\le d$ , such that for each positive integer $j$ , the mapping $S_{\\psi (\\lambda ^{-j} D)}:\\mathcal {P}\\rightarrow \\mathcal {P}$ defined by $S_{\\psi (\\lambda ^{-j} D)} p =w_0^{-1}\\left(I+a_1 \\lambda ^{-j} D + \\cdots + a_d (\\lambda ^{-j})^d D^d\\right)p$ is a right-inverse for $\\psi (\\lambda ^{-j} D)$ on $\\mathcal {P}$ , and $\|a_i\|<C$ for each $i$ .", "Let $S_{m,n}:\\mathcal {P}\\rightarrow \\mathcal {P}$ be the linear right inverse of $L_{\\lambda ,b,z^m}^n$ as defined in Lemma REF .", "We then define the mapping $F_n:\\mathcal {P}\\rightarrow \\mathcal {P}$ by $F_n p = S_{m,n} S_{\\psi (D)} \\cdots S_{\\psi (\\lambda ^{2-n}D)} S_{\\psi (\\lambda ^{1-n}D)} p,$ which satisfies $L_{\\lambda ,b,\\varphi }^n F_n p =p$ .", "The condition $\\varphi (0)=0$ implies $\\deg L_{\\lambda ,b,\\varphi } p < \\deg p$ , which implies $L_{\\lambda ,b,\\varphi }^n p= 0$ whenever $n>d$ .", "What remains to show for the Hypercyclicity Criterion is that $F_n p\\overset{n\\rightarrow \\infty }{\\longrightarrow }0$ uniformly on compact subsets of $.", "By multiplying out the product $ S(D) S(2-nD) S(1-nD)$ using equation (\\ref {Spsi}), we have that{\\begin{@align}{1}{-1}& S_{\\psi (D)} \\cdots S_{\\psi (\\lambda ^{2-n}D)} S_{\\psi (\\lambda ^{1-n}D)} p\\\\& = w_0^{-n} [I+a_1 D + \\cdots + a_d D^d]\\cdots [I+a_1 \\lambda ^{1-n} D + \\cdots + a_d (\\lambda ^{1-n})^d D^d] p\\\\& = w_0^{-n} [I + c_{1,n} D + \\cdots + c_{d,n} D^d] p,\\end{@align}}where the coefficients $ cj,n$ for $ 1jd$ satisfy{\\begin{@align}{1}{-1}c_{j,n}=\\sum _{j_1+\\cdots + j_n = j} \\frac{a_{j_1}}{(\\lambda ^{j_1})^0} \\frac{ a_{j_2}}{(\\lambda ^{j_2})^1}\\frac{a_{j_3}}{(\\lambda ^{j_3})^2}\\cdots \\frac{a_{j_n}}{(\\lambda ^{j_n})^{n-1}},\\end{@align}}where each $ jk$ is a non-negative integer.$ For each positive integer $n$ , let $c_{0,n}=1$ .", "Since $S_{m,n}$ is linear, to show (REF ) it suffices to show that $S_{m,n}(w_0^{-n}c_{j,n}p)\\overset{n\\rightarrow \\infty }{\\longrightarrow }0.$ uniformly on compact subsets of $ for each integer $ j$ such that $ 0jd$.$ Let $j$ be an integer satisfying $0\\le j\\le d$ .", "The number of terms in the sum () is equal to the number of multinomial coefficients in a multinomial sum, which is $\\displaystyle \\binom{j+n-1}{n-1}$ .", "Let $\\alpha = \\max \\lbrace 1,\|a_1\|,\|a_2\|,\\cdots ,\|a_d\|\\rbrace $ .", "Since $\|\\lambda \|\\ge 1$ , by () we have that $\|c_{j,n}\|\\le \\binom{j+n-1}{n-1}\\alpha ^j \\le n^j\\alpha ^j\\le n^d\\alpha ^d<e^{nd}\\alpha ^d.$ Now let $t=\|w_0\|^{-1}e^d\\alpha ^d$ .", "Then $\|w_0^{-n}c_{j,n}\|\\le t^n$ , and thus $S_{m,n}(w_0^{-n}c_{j,n}p)\\overset{n\\rightarrow \\infty }{\\longrightarrow }0$ uniformly on compact subsets of $ by limit (\\ref {Slimit}) in Lemma \\ref {HallackLemma}, which shows that (\\ref {Smn}) holds and completes the proof.$" ], [ "Acknowledgements", "Funding: This work was supported by the St. Olaf College Collaborative Undergraduate Research and Inquiry program.", "Funding: This work was supported by the St. Olaf College Collaborative Undergraduate Research and Inquiry program." ] ]	2011.14208
[ [ "Defect State Density and Orbital Localization in a-Si:H/c-Si\n Heterojunction and the Role of H" ], [ "Abstract In this paper, we explore the effect of H and its bonding configurations on the defect state density and orbital localization of hydrogenated amorphous Si (a-Si:H)/crystalline Si (c-Si) heterostructures using density functional theory (DFT) studies of model interfaces between amorphous silicon (a- Si)/a-Si:H and c-Si.", "To model the atomic configuration of a-Si on c-Si, melting and quenching simulations were performed using classical molecular dynamics (MD).", "Different hydrogen contents were inserted into the a-Si in different bonding configurations followed by DFT relaxation to create the stable structures of a-Si:H representative of hydrogenated a-Si on crystalline Si surfaces.", "In contrast to crystalline heterojunctions (where the interface density is a maximum at the interface), we find that, in the most energetically stable configurations of H atoms, the defect state density is relatively low at the interface and maximum at the middle of a-Si layer.", "Our structural analysis shows that in these configurations, H atoms do not necessarily bond to dangling bonds or to interface atoms.", "However, they are able to significantly change the atomic structure of the heterostructure and consequently decrease the density of defect states and orbital localization at the a-Si layer and more significantly at the interface of a-Si/c-Si.", "The general form of the modeled defect state distribution demonstrates the passivating role of a-Si:H on c-Si substrates." ], [ "Introduction", "By deposition of a thin layer of a-Si:H on a high quality c-Si substrate, the a-Si:H/c-Si heterojunction solar cells (HIT cells) have achieved the world efficiency record of 26.6% for n-type Si wafers [1].", "Key to the success of HIT devices is using a a-Si:H layer as a passivating and semiconducting film on top of c-Si [2], [3], [4].", "The a-Si:H layer reduces the recombination rate of photo-generated carriers by passiviating defect states at the surface of c-Si and controls the photo-generated carriers transport by creating a proper band bending (band offset) at the interface.", "Engineering these two electronic properties (defect states distribution and band bending) is extremely important to future improve the efficiency of HIT cells.", "H incorporation in different conditions during plasma enhanced chemical vapor deposition of a-Si:H offer a unique way to engineer the mentioned electronic properties [5], [6], [7], [8], [9], [10], [11].", "Due to different fabrication conditions, different H bonding configurations might be generated in a-Si:H layer and at the interface.", "A better understanding of these H bonding configurations on the electronic structure of a-Si:H/c-Si heterojunctions will aim to improve the device performance.", "In the current work, we present our first principle study on the electronic properties of a-Si:H/c-Si(001) heterojunction containing different H concentrations with different bonding configurations.", "Despite the wide range of literature concerning the simulation of the amorphous Si phase, both pure and hydrogenated, either from first principles and by means of semiempirical approaches, we are not aware of many direct simulations of the a-Si:H/c-Si electronic structure, particularly considering different H concentrations and bonding configurations.", "In an early work, Peressi et al.", "have investigated the effect of defects in the electronic properties of a-Si/c-Si [12].", "Tosolini et al.", "[13] studied the atomic structure of the interface between a-Si:H and c-Si using combined tight-bonding molecular dynamics simulations and first principle calculations.", "In another work, Nolan et al.", "[14] investigated the electronic property changes in a-Si:H and c-Si of a-Si:H/c-Si junction which are perturbed by interface formation on three unreconstructed silicon surfaces, namely (100), (110) and (111).", "Recently, by combining magnetic resonance measurement and DFT calculations, George et al.", "[15] studied the atomic structure of interface states of a-Si:H/c-Si heterojunction.", "In these prior works, in general, the authors mainly focused on the atomic structure and electronic properties of a-Si:H/c-Si heterostructures, while the relation between these properties and the H concentration in different bonding configurations was not fully explored.", "The main new contribution of the present work compared to previous studies is threefold: (i) a number of different H configurations with differing concentrations were studied, providing useful insight on the fabrication of high quality a-Si:H/c-Si heterojunctions; (ii) the density of defect states (bandgap states) along the growth direction of the a-Si:H/c-Si heterojunction were calculated, which can be utilized in solar simulation of the electrical behavior of HIT cells, in order to more accurate model their behavior and better engineer of their performance; (iii) orbital localization of the a-Si:H/c-Si heterojunctions were studied, which provides important information that can be used in modeling the recombination rate and effective surface recombination velocity, which in turn strongly affects the efficiency of the HIT solar cells.", "In this paper, we present the results of our first principle calculations performed on a-Si:H/c-Si heterostructure models obtained from Si (100) surfaces.", "These models were initially generated from partially melting and quenching part of the Si on a Si surface using MD simulation and optimized for different H bonding configurations using DFT methods.", "The electronic properties of the simulated structures are calculated to elucidate the sensitivity on H concentration and bonding to the defect state density and orbital localization in the a-Si:H/c-Si heterostructure.", "Our results show that the hydrogen amount and configuration play a key role in the electronic properties.", "Melting and quenching process was simulated on c-Si structures to generate a structural model of a-Si:H/c-Si, as the starting atomic structure for the DFT calculations.", "We used the LAMMPS molecular dynamic code [16] to simulate the melting and quenching process, as we have described previously [17].", "Si atom interactions were described by the Tersoff interatomic potential [18], with cut-offs of 2.7 Å (taper) and 3.0 Å (maximum); this interatomic potential has been successfully applied for modeling Si based amorphous structures, as discussed elsewhere [17], [19].", "The full optimization of all the structures obtained from MD were carried out at the DFT-level as implemented in the Quantum Espresso 5.2.1 software package [20].", "The Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton algorithm, based on the trust radius procedure, was employed as the optimization algorithm for the relaxed structures.", "The Perdew-Burke-Ernzerhof (PBE) [21] exchange-correlation functional was used in both the ionic relaxation and the electronic structure calculations using periodic boundary conditions.", "The core and valence electron interactions were described by the Norm-Conserving Pseudopotential function.", "Unless otherwise stated, an energy cutoff of 12 Ry was employed for the plane-wave basis set and a 4$\\times $ 4$\\times $ 4 k-point mesh is used with the Monkhorst-Pack grid method for the Brillouin-zone sampling in all calculations.", "A Gaussian smearing and fixed method was applied to determine the band occupations and electronic density of states for odd and even number of H atoms in the supercell, respectively." ], [ "Generation of a-Si:H/c-Si Structures", "For modeling a-Si:H/c-Si heterostructures, we used periodically repeated supercells containing one interface for each phase and, therefore, two interfaces.", "We simulated the most common interface, which is that of a-Si:H deposited on a (001) c-Si substrate.", "To model an a-Si/c-Si interface, a tetragonal 001 supercell containing 256 Si atoms was cleaved from a c-Si structure (diamond lattice) with the experimental lattice constant fixed at 5.43 Å ($a_0$ ).", "The lateral dimensions of the supercell were fixed at a=b=2$a_0$ and c=8$a_0$ .", "The longitudinal dimension was chosen according to the experimental density of a-Si:H for given H concentrations [22], [23].", "Once the supercell is determined, the melting and quenching process was performed on the 8 upper layers of the supercell within the constant volume, constant temperature ensemble, while maintaining the c-Si portion in equilibrium at 300 K. In practice, we start with a melting simulation at 3000 K for 10 ps using a time step of 0.1 ps.", "The structure was rapidly quenched close to room temperature (300 K) using a cooling rate of 6$\\times $ 1012 K/s.", "The cooling rate here is consistent with previously reported cooling rates [24], [25], [26], varying between 1011 and 1015 K/s.", "This cooling rate allowed the formation of a disordered a-Si phase interfaced with a crystalline phase.", "After quenching, the complete structure was annealed for 25 ps at 300 K. Finally, the obtained interface was fully optimized by DFT, as discussed above.", "Hydrogenated a-Si structures were obtained by adding H atoms step-by-step to the amorphous constituent of the a-Si/c-Si structure resulting from the MD quenching and DFT optimization above, as described in our previous work [17].", "In each step, one H atom is added to the most stable configuration obtained from the previous step.", "In order to obtain stable configurations at different H concentrations (i.e.", "each step), we sample all possible configurations of adding a H atom to each Si atom in the a-Si phase.", "The number of Si atoms corresponding to the a-Si phase was 128, and, therefore, for each H addition, we generated 128 different configurations by bonding H to a given Si atom.", "After optimizing all the configurations using DFT, the configuration with the lowest energy is taken as the most stable one at a given H concentration, and used for further calculations." ], [ "Atomic and Electronic Structure", "We first analyzed the atomic structure of the supercells generated by the quenching and DFT optimization discussed in the previous section.", "The top panel of Fig.", "1 (a) shows a ball-and-stick representation of the atomic structure of the a-Si/c-Si supercell.", "The supercell contains 256 Si atoms with 128 Si atoms in the a-Si phase.", "After repeating the 3D supercell, we found ten floating bonds and four dangling bonds (Table I) per supercell using a cutoff Si-Si bond length of 2.58 Å, which is 10% longer than experimental Si-Si bond length (2.35 Å).", "Dangling (floating) bonds are missing (extra) chemical bonds of a Si atom from ideally four chemical bonds.", "Figure: Top panel: a) the atomic structure of the simulated a-Si/c-Si supercell.", "Left panel: the layer resolved Si-Si RDF of b) c-Si, c) the interface and d) a-Si.", "Right panel: the projected Density of States (PDOS) plot of e) c-Si, f) the interface and g) a-Si of a-Si/c-Si in comparison with those of a-Si:H/c-Si (5.88 H at%).The structure mostly displays stable 5, 6 or 7 folded rings, and there are no large voids or holes inside.", "The average Si-Si bond length is 2.37 Å with an rms deviation of 0.04 Å while the average Si-Si-Si bond angle is 107.78$^{\\circ }$ with an rms deviation of 12.36$^{\\circ }$ , which is very close to the upper end of experimental angle distribution of 9-11$^{\\circ }$ [27].", "For reference, the Si-Si bond length is 2.35 Å and Si-Si-Si bond angle is 109.47$^{\\circ }$ in the crystalline form of Si.", "The excess energy of the structure is 0.19 eV/atom which is close to the upper end of experimental excess energy range of 0.07-0.15 eV/atom for a-Si [27].", "As mentioned earlier, H atoms were gradually added to a-Si/c-Si supercell in different bonding configurations to obtain a-Si:H/c-Si supercells.", "The average bond length of Si-H bonds in optimized a-Si:H/c-Si structure is 1.5 Å, which is close to the experimental value [28].", "Table I illustrates the number of dangling bonds (DBs) and floating bonds (FBs) per supercell in the most stable a-Si:H/c-Si heterostructures for different atomic percent of H concentration.", "As is clearly seen, the number of DBs and FBs per supercell reduces as the H concentration increases.", "This observation is consistent with the common belief that H incorporation in the growth of a-Si results in passivating DBs and FBs.", "Interestingly, the number of FBs decreases more rapidly than that of DBs by adding H atoms to the supercells, thus indicating that FBs cause more instability compared to DBs.", "All DBs and FBs in the a-Si:H/c-Si heterostructure disappear when the concentration of H reaches 5.88% in the supercell.", "Therefore, for H bonding configurations study, we only focus on DFT results obtained from the structure with a H concentration of 5.88%.", "In order to check the validity of our atomic structure, we compute the region resolved Si-Si radial distribution function (RDF) of the heterostructure along the growth direction z.", "Here, a region is defined as a 3D region with the dimensions of the supercell a=b=c=2$a_0$ centered in a given phase (a-Si, c-Si and interface).", "The RDF gives the probability of finding two atoms in a structure separated by a distance r. Since the RDF is experimentally measured, it provides a strong tool for checking the validity of computationally created structures.", "Table: The number of dangling bonds (Num.DB) and floating bonds (Num.FB) in the a-Si:H/c-Si heterostructure per supercell for different atomic percent of H concentration (H at%).The left panel of Fig.", "1 displays the computed region resolved Si-Si RDF obtained from the modeled supercells.", "We expect to observe sharp peaks in the RDF for crystalline materials due to periodic atomic structure, while broadened and coalesced peaks appear in amorphous materials due to the presence of weak short range order and the lack of long range order.", "Consistent with this argument, we obtained very sharp peaks in the RDF of the crystalline region (Fig.", "1b), and broadened peaks for the RDF of the amorphous region (Fig.", "1d).", "Although the amplitude of the RDF peaks does not give quantitative information because of the statistical noise due to the limited size of the supercell, the RDF peak location can provide valid information for comparison purposes.", "The peak locations in the computed RDFs are consistent with the peak locations in the measured RDFs for both the case of bulk crystalline and amorphous Si [29], [30].", "In addition, the RDF at the interface (Fig.", "1c) gives us a visual indication of the transition between the amorphous and crystalline regions.", "The region resolved RDFs obtained for a-Si:H/c-Si compared to similar ones for a-Si/c-Si show a narrowing of the peaks indicating of increased atomic ordering.", "This atomic order enhancement is particularly clear in the second RDF peak of a-Si:H in comparison with that of a-Si.", "As a result of the atomic structure differences, c-Si and a-Si or a-Si:H have different electronic structure.", "The c-Si electronic structure contains three main characteristic peaks [31], [32] in the valence band and a clear energy gap between valence and conduction bands, with no midgap states in the energy gap.", "In contrast to c-Si, the a-Si or a-Si:H electronic structure contains two main characteristic peaks [33] in the valence band and the well known tail and midgap states in the forbidden gap of its crystalline counterpart.", "These band tail states and midgap states are highly sensitive to the incorporation of hydrogen in the fabrication of a-Si.", "The aforementioned electronic properties are related to c-Si and a-Si(a-Si:H) in the bulk phase, but here we compared them with the computed projected density of states (PDOS) in the crystalline and amorphous region of the modeled supercell in order to further verify the reliability of the generated structures.", "The PDOSs were computed in a middle section of the c-Si and a-Si or a-Si:H supercell, at the maximum distance from the interfaces.", "The right panel of Fig.", "1 shows the computed PDOSs.", "For the c-Si region (Fig.", "1e), the PDOS shows the three characteristic peaks of c-Si bulk and a clear separation between the valence and conduction bands.", "On the other hand, the PDOS of a-Si or a-Si:H (Fig.", "1g) shows a deviation from the c-Si PDOS by converting three to two characteristic peaks and by the appearance of valence and conduction band tails and midgap states due to the presence of undercoordinated Si and strained Si-Si bonds [34].", "As seen, the PDOS in the c-Si regions has the typical form of bulk c-Si and the PDOS in the a-Si regions is typical of bulk a-Si.", "On the other hand, our computed PDOS in the interface region (Fig.", "1f) contains a combination of electronic properties of c-Si and a-Si(a-Si:H).", "While the conduction band of interface PDOS shows three characteristic peaks coming from the c-Si part, the peaks are broadened and collapsed due to the presence of the a-Si part.", "The bandgap of the interface PDOS contains some midgap states, but not as high as the PDOS of a-Si or a-Si:H." ], [ "Defect States and Orbital Localization with Different H Concentration", "After confirming the reliability of our structures in the previous section, we performed electronic structure calculations on a-Si:H/c-Si heterostructure on the most stable H bonding configurations, to understand the role of H in the electronic properties.", "Fig.", "2a shows the total electronic density of states (EDOS) of unhydrogenated a-Si/c-Si in comparison with that of a-Si:H/c-Si in its most stable configuration, for a H concentration of 5.88%.", "We observe that the effect of hydrogenation is small on the EDOS of the overall valence band and conduction band structure of a-Si.", "However, looking in more detail at the EDOS close to the valence and conduction band edge, the effect of adding H to the a-Si structure noticeably changes the electronic structure around the band gap.", "Figure: a) The total EDOS of a-Si/c-Si in comparison to that of a-Si:H/c-Si and b) the EDOS of a-Si:H/c-Si, close to the band gap, for different H concentrations.", "The numbers assigned to each line correspond to the H atomic percent in the a-Si:H layer.Zeroing in on the details near the band edges, Fig.", "2b shows the EDOS of an a-Si:H/c-Si structure with different H concentrations for their most stable configurations.", "It is apparent that, as the H concentration increases, the density of midgap states decreases.", "Interestingly, the rate of reduction of the midgap states is greater for low H concentrations compared to the rate at higher H concentration.", "Structural analysis indicates that this decrease in the most stable configuration is not only due to H bonding to Si dangling bonds, but also due to H bonding to Si under high bond length or strain angle.", "This finding is not unexpected, since Si-Si strained bonds are found to be one of the main sources of midgap and band tail states [17], [34], as well as a main source of DBs and FBs due to a high degree of disorder in the a-Si network.", "Consistent with this argument, we find that in some cases, H atom addition to the Si-Si strained bonds results in non-locally removing DBs and FBs which are considerably further away from the added H location, as we found previously in bulk a-Si [17].", "Looking in more detail at the EDOS for a-Si:H/c-Si compared to a-Si/c-Si (Fig.", "2a), we observe that the reduction in midgap states due to H passivation is associated with an increase in the density of states at the main valence and conduction bands as a consequence of the conservation theorem for the number of eigenstates [34].", "Despite all these changes in the electronic structure of a-Si/c-Si due to H addition, the general form of the electronic structure is preserved for all H concentrations being considered.", "Neither the exchange-correlation functional or the size of the supercell used here are sufficient to precisely determine the density and position of the midgap and tail states, but our calculations still offer a valid model to predict and analyze the basic changes in the electronic structure due to H addition for potential comparison with experiment.", "As mentioned in the introduction, defect induced states in semiconductors are a critical factor affecting the efficiency of optoelectronic devices such as solar cells.", "These states play a key role in carrier recombination, carrier trapping, and carrier transport [35], [36], [37], [38], [39], [40], [41], [42], [43].", "Since the atomic structure changes dramatically going from c-Si to a-Si along the growth direction, it is expected that the density of localized defect states and the orbital localization of the electronic states also change along this direction.", "Therefore, we also investigate how the density of defect states and the orbital localization changes as the phase gradually changes from c-Si to a-Si, and as H is added to the system.", "In several first principle studies, the density of defect states has been calculated by projecting the density of midgap states along simulated supercells [44], [45], [46].", "We use the same approach in order to calculate the total density of defect states.", "The band-gap region is calculated by performing DFT calculations on a c-Si supercell with the same size of the modeled a-Si/c-Si supercell.", "The band gap region extends from 6.27 eV to 6.90 eV, which is less than experimental value due to the well-known bandgap underestimation problem of conventional DFT.", "Fig.", "3 indicates the projected density of states in this band gap range along the z direction of the a-Si:H/c-Si supercell with different stable concentrations of H. As seen in Fig.", "3, for all H concentrations, the density of defect states at the interface is much lower than that at the middle of a-Si:H layer.", "This observation contradicts the often-invoked picture used in the experimental literature of a-Si/c-Si heterostructures of conventional semiconductor heterostructures where the density of states is highest at the interface due to dissimilar bonding, dangling bonds, etc [47].", "In the case of a-Si:H/c-Si, the transition occurs between an ordered form of Si to a less ordered one, where the maximum disorder is in the bulk of the a-Si where the maximum density of localized bandgap states exists, not at the interface between c-Si and a-Si.", "As can be seen from Fig.", "3, the same observed behavior occurs for all H concentrations; however, the overall density of defect states decreases as more H is inserted to the structure.", "Figure: Integrated density of bandgap states along the z-axis of a-Si:H/c-Si supercell with different H concentrations in the most stable configurations.", "The vertical red dashed lines show the location of the interfaces in the periodically repeated supercell.", "The numbers assigned to each line correspond to the H atomic percent in the a-Si:H layer.Because of the presence of structural defects in the atomic structure of an amorphous material, the mobility of holes and electrons in localized states close to the valence and conduction band edges are much lower in comparison to nonlocalized extended states that are above/below the mobility edge of the conduction and valence bands, respectively (due to the predominance of hopping conduction [48]).", "The energy difference between the valence-band mobility edge and the conduction-band mobility edge is the so-called mobility gap.", "Electronic transport in a-Si and a-Si:H is strongly affected by the carriers residing in these localized states within the mobility gap.", "Hence, we investigate the orbital localization in a-Si and a-Si:H and the structural defects that are responsible for orbital localization in the mobility gap of a-Si and H addition effects on this localization.", "The localization of the Kohn-Sham orbitals was studied in detail for the simulated a-Si:H/c-Si structures using the inverse participation ratio (IPR).", "The IPR for an eigenstate $\\Psi _n$ is given as: $IPR_n=\\frac{\\Sigma _{i=1}^I{a_{ni}^4}}{({\\Sigma _{i=1}^I{a_{ni}^2}})^2}$ where $a_{ni}$ is the coefficient of the ith basis set orbital in the nth Kohn-Sham orbital $\\Psi _n$ ($\\Psi _n=\\Sigma _{i=1}^N{a_{ni}\\phi _i}$ ), and N is the total number of basis set orbitals used in the DFT calculations.", "A higher IPR stands for a higher degree of localization.", "Fig.", "4 shows the calculated IPR of all the Kohn-Sham orbitals obtained via DFT versus their energy for a-Si/c-Si and a-Si:H/c-Si for different H concentrations in their most stable configurations.", "We observe a relatively sharp transition between the high and low IPR values (localized and extended electronic states) that suggests the presence of mobility edges in both the valence and conduction bands.", "The general form of the IPR plot is in agreement with previous theoretical and computational studies [49], [26].", "Figure: Orbital localization of a-Si:H/c-Si in different H concentrations: a) 1.56% b) 3.03% c) 4.48% d) 5.88%.From the comparison of the IPR plots shown in Fig.", "4, It is obvious that localization of the Kohn-Sham orbitals significantly decreases as hydrogen is added.", "Since the H atoms add primarily to the strained bond atoms, the decrease in orbital localization is primarily due to removing strained bonds rather than dangling bonds.", "Localized states strongly influence the effective carrier mobility in terms of scattering centers and hopping conduction [50].", "Indeed, in the case of a-Si and a-Si:H, it has been experimentally shown that both the electron and hole mobilities are controlled by traps [51], [52].", "Therefore a lower degree of localization is consistent with the observation of higher electron and hole mobility in a-Si:H compared to a-Si experimentally [53], [54], [55].", "We note that H atoms change the orbital localization inside the mobility gap much more than that outside the mobility gap.", "Therefore, for clearer comparison, we summed all the IPR of the orbitals inside the band gap as an integrated IPR.", "Fig.", "5 shows the integrated IPR of the a-Si:H/c-Si heterostructure as a function of the H concentration.", "As seen, the integrated IPR reduces as H is added to the heterstructure, consistent with what is shown in Fig.", "4.", "The rate of reduction is highest at lower H concentrations and then starts to saturate at higher H concentrations.", "Since the integrated IPR shows the orbital localizations of electronic states in the bandgap, and that localized states in the bandgap are a major source of carrier recombination, this quantity shows that charge recombination rate in a-Si:H/c-Si should decrease with increasing H concentration.", "Figure: Integrated IPR (IIPR) of a-Si:H/c-Si for different H concentration." ], [ "Defect States and Orbital Localization with Different H bonding Configurations", "Experimentally, a-Si:H is typically deposited on a c-Si substrate using plasma enhanced chemical vapor deposition (PECVD) under strongly nonequilibrium conditions, where H bond formation is kinetically rather than thermodynamically limited.", "Hence, different H bonding configurations are expected to occur which are not the energetically lowest state, as considered in the previous section.", "In order to investigate how the exact H bonding configuration energetically affects the electronic structure, we examined the electronic structure of the first three most stable configurations of a-Si:H/c-Si heterostructure for a H concentration of 5.88% (8 H atoms in the amorphous region of the supercell).", "Fig.", "6 shows the EDOS plot close to the band gap of these configurations in comparison with that of unhydrogenated a-Si/c-Si.", "As shown, no matter in which configuration H atoms are added to the a-Si/c-Si structure, they still reduce the density of midgap states.", "However, the amount of this reduction depends on the H bonding configurations, and specifically on the energy, with the number of midgap states increasing with increasing excess energy.", "Structural analysis shows that these two less stable configurations contain DBs and FBs, in contrast to the most stable one, i.e.", "more structural defects.", "Figure: The density of electronic states as a function of energy of a-Si:H/c-Si structures with 5.88% H atomic percent in different configurations in comparison with that of a-Si/c-Si.", "The numbers assigned to each line correspond to the energy of each configuration in eV.Fig.", "7 indicates the projected density of states within the c-Si bandgap along the z (growth) direction of the a-Si:H/c-Si for different H bonding configurations in comparison with that of unhydrogenated a-Si/c-Si.", "Regardless of the H bonding configuration, the density of gap states decreases compared to unhydrogenated case.", "As expected, the density is zero in the crystalline part of heterojunction and increases monotically until it reaches its maximum inside the a-Si:H layer for all configurations.", "In the case of a heterojunction between a crystalline and an amorphous form of the same material, the low defect state density at the interface shows that the a-Si:H layer, at least in its most stable configuration, is an effective passivation layer on the top of the c-Si substrate.", "Thus, there is low defect state density, at the interface that reaches a maximum value in the middle of the a-Si:H layer.", "Since interface states negatively impact the open circuit voltage in solar cells, the low interface state density helps in explaining the high open circuit voltage in HIT cells.", "Interestingly, we find that the low density of defect states at the interface is not due to H bonding to the interface atoms.", "In most of the studied cases, we observe that H atoms preferably bond to atoms in the middle of the a-Si layer.", "This is due to the higher density of defect sites in the middle of a-Si layer in an a-Si/c-Si heterojunction.", "It, thus, results in a defect state density reduction not only in the a-Si layer but also at the interface.", "These results show that H atoms have both local impact in an amorphous network and cause nonlocal changes in the physical and electronic structure.", "This finding suggests that annealing processes in the presence of H atoms can positively impact the quality of a-Si:H/c-Si heterojunction if H can find its most stable configuration, consistent with recent experimental results [56].", "Figure: The local density of defect states for unhydrogenated a-Si/c-Si in comparison with a-Si:H/c-Si with different H bonding configurations of increasing energy.", "The numbers assigned to each line correspond to the energy of each configuration in eV.", "The vertical red dashed lines show the location of the interfaces in the periodically repeated supercell.Fig.", "8 plots the calculated IPR of all the Kohn-Sham orbitals obtained via DFT versus energy for a-Si:H/c-Si in its first three stable configurations.", "It is obvious from the figure that regardless of the H bonding configuration, H addition strongly decreases the localization of the electronic states in the a-Si/c-Si heterostructure.", "Comparing between IPR plots of increasingly energetic configurations of a-Si:H/c-Si heterostructures, the localization of orbitals slowly increases as the energy of the configuration increases.", "According to the Anderson model, the amount of orbital localization in an amorphous material depends on the amount of disorder in the atomic structure.", "Therefore, increase in orbital localization can be attributed to the increase in disorder as the energy of the configuration increases.", "It is worth noting that the distribution of highly localized orbitals also depends on the configuration.", "In the first stable configuration, most localized states are located in conduction band side, whereas the density of localized states increases in the valence side and midgap region in the second and third configuration.", "To obtain a more clear vision about orbital localization versus energy of configuration, integrated IPR for different configurations was calculated as described in the previous section.", "As seen from Fig.", "8e, integrated IPR increases as the energy of the configuration increases.", "Figure: Orbital localization of a-Si:H/c-Si in comparison to that of a-Si:H/c-Si with H concentrations of 5.88% for different H bonding configurations as a function of energy: a) a-Si:H/c-Si with no H, b) 0.14 eV c) 0.50 eV, and d) 0.87 eV energy.", "e) Integrated IPR of a-Si:H/c-Si with 5.88% H atomic percent for different configurations.The main result of this section is that when H atoms bind to Si atoms in less energetically stable configurations, they may still reduce the density of bandgap states and orbital localization in the a-Si/c-Si heterostructure.", "Moreover, by comparing configurations with different excess energy, we find that the electronic properties are strongly affected by the local H bonding configuration depending on its overall energy." ], [ "CONCLUSION", "In conclusion, we employed a combined MD-DFT quenching method to obtain stable configurations of a-Si and a-Si:H. The good agreement between the calculated and experimental RDF and DOS plots, the excess energy and the average bond angle validates the structures used to model the electronic properties of a-Si/c-Si and a-Si:H/c-Si heterostructures.", "The computed DOS plots obtained from DFT calculations performed on different configurations of a-Si:H/c-Si with different H concentrations indicate that H insertion always decreases the density of states in the forbidden gap, but that the amount of this reduction depends on the H concentration and binding configuration.", "In contrast to conventional semiconductor heterostructures, the highest density of bandgap defect derived states does not occur at the heterointerface, but rather increases monotonically from the interface to the bulk of the a-Si, due to transition from ordered to disordered material.", "The interfacial defect density decreases with increasing H concentration which is commensurate with the overall reduction of the defect states throughout the a-Si with H bonding.", "We found that in the most energetically stable configurations, one has the greatest reduction of midgap states." ], [ "ACKNOWLEDGMENTS", "This material is based upon work primarily supported by the Engineering Research Center Program of the National Science Foundation and the Office of Energy Efficiency and Renewable Energy of the Department of Energy under NSF Cooperative Agreement No.EEC-1041895.", "Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect those of the National Science Foundation or Department of Energy." ] ]	2011.14158
[ [ "Text Mining for Processing Interview Data in Computational Social\n Science" ], [ "Abstract We use commercially available text analysis technology to process interview text data from a computational social science study.", "We find that topical clustering and terminological enrichment provide for convenient exploration and quantification of the responses.", "This makes it possible to generate and test hypotheses and to compare textual and non-textual variables, and saves analyst effort.", "We encourage studies in social science to use text analysis, especially for exploratory open-ended studies.", "We discuss how replicability requirements are met by text analysis technology.", "We note that the most recent learning models are not designed with transparency in mind, and that research requires a model to be editable and its decisions to be explainable.", "The tools available today, such as the one used in the present study, are not built for processing interview texts.", "While many of the variables under consideration are quantifiable using lexical statistics, we find that some interesting and potentially valuable features are difficult or impossible to automatise reliably at present.", "We note that there are some potentially interesting applications for traditional natural language processing mechanisms such as named entity recognition and anaphora resolution in this application area.", "We conclude with a suggestion for language technologists to investigate the challenge of processing interview data comprehensively, especially the interplay between question and response, and we encourage social science researchers not to hesitate to use text analysis tools, especially for the exploratory phase of processing interview data.", "?" ], [ "Computational social science and text analysis", "Computational social science, as one of its central methods, gathers and processes text data.", "These data can be obtained through archive studies, media analyses, observational studies, interviews, questionnaires, and many other ways.", "Computational text analysis has a broad palette of tools to create structured knowledge from human language.", "The field of language technology, the basis for text analysis tools, has devoted most of its attention to topical analysis of text, to establish what a text or a segment of text is about.", "This reflects the uses that text analysis has been put to: mostly topical tasks, where relevance and timeliness of the information are the priority.", "The knowledge gathered from texts in computational social science are, in contrast with the general case, only partly topical in nature, and existing text analysis tools and methods are not always well suited to these types of task.", "This paper will examine how text analysis can be of use, and point out some cases where current technology could be improved to be more effective." ], [ "Questionnaires and quantifying text data", "There are multiple reasons to gather human data through interviews or questionnaires where respondents can use their own language to respond to queries rather than e.g.", "multiple choice or graded agreement responses.", "One reason is to make the data gathering situation less formal and more personal and thus encourage the respondent to provide richer data.", "Gathering data through natural human language allows respondents to express what they feel, perceive, believe and value, in a language they are comfortable with which allows the analysis to include the attitude of the respondent to the topic they are responding to.", "In addition, through an analysis of the language used through the entire interview, general observations about the stance and emotional perspective of the respondent can be made.", "Another reason is to allow the respondent more leeway to formulate their responses without too much imposition from a pre-compiled response structure and thus enable the respondent to provide unexpected data or unexpected connections or dependencies between items under consideration [1].", "This is especially true if the study concerns (1) a vague or indefinite subject, (2) treats matter for which there is no established vocabulary and phraseology, or (3) which is sensitive in some way [2].", "An interview situation has the advantage of allowing dynamic follow-up questions which allow specification of aspects that might not otherwise have been detected and will allow the interviewer to probe for bias in the form of tacit assumptions the interviewee might have about concepts that interest the interviewer [3].", "For the above reasons, the design of interview studies and analysis of interview data is an area of methodological debate and research within the social sciences [4], [5], [6].", "While open answers in surveys and questionnaires provide richer data and reduce the effort and difficulty of formulating the questions in exact form before the fact, they move the effort to after-the-fact-processing of the collected data to get useful results.", "Open answers are a challenge for analysts: reporting the collected responses together with more quantitative data elicited from respondents is not obvious.", "Coding procedures—converting open responses into structured form—require time and expertise on the part of the analyst, both of which come at a cost.", "The effort involved in coding open answers is simultaneously intellectually non-trivial and demanding, but still monotonous: analyst fatigue and frustration risks leading to both between-analyst and within-analyst inconsistencies over time in reporting.E.g.", "O’Cathain and Thomas [7] and many others; It takes about 1 minute for a human to categorise an abstract, shown by e.g.", "Macskassy et al.", "[8] when the categories are already given , If the task is to explore a set of responses and define and revise categories or labels as you go it will involve more effort and require more time per item.", "Freeing human effort from routine tasks to more creative challenges motivates the introduction of computational technology in general, and this field is no exception.", "Coding textual responses is an excellent example of human effort expended on tasks which are repetitive, time-consuming, and dreary.", "Processing interview data automatically is an attractive scenario, but is not yet the norm in the social sciences." ], [ "Turning text to quantifiable information", "Language technology has over the last decades developed methods and tools for text analysis, mostly applied to news material, legal documents, technical matters, or other related application domains.", "The methods have been developed with several characteristics in mind.", "They are intended to be efficient and effective for practical application to real life problems.", "They should generalise well from one area to another.", "In addition, their design is often inspired by human communicative behaviour and as a side constraint it is viewed as desirable in many cases that the models conform to known characteristics of humans process language.", "Text analysis methods rely on observable features found in texts: words, multi-word terms, and constructions.", "Deciding which features the methods pay attention to is crucial to the quality of the analysis: are words and multi-word terms sufficiently informative to understand the text?", "Should constructional features, syntactic dependencies, and grammar rules be used?", "In a statistical model, the chosen features also need to be weighted for importance, e.g.", "such that an observation of a frequently occurring feature has less weight than an unusual one or that a combination of several features is more interesting than the individual observations.", "The selected features are then used score texts or text segments quantitatively, to cluster them into coherent subsets, or to classify them into previously established categories.", "All of these three levels of knowledge representation: what features to observe, how they are weighted, and what target categories are of interest, can be learned from text data or explicitly given by a human analyst.", "That distinction illustrates how text analysis in general spans a spectrum from statistical models to knowledge-based methods.", "The former, statistical learning models, are based on the current generation of machine learning methodologies, learn from examples, and tend to be robust in face of linguistic variation; the latter, by explicitly incorporating previous knowledge from theories of language and lexicography, are able to provide more precise and more sophisticated output.", "With recent advances in machine learning, these two main approaches are currently converging.", "Text mining is the general term for the systematic extraction of (somewhat) structured insights from unstructured text material using text analysis methods of various kinds.", "Text mining in various forms predates the use of machine learning, but most recent approaches are based on learning models.", "Text mining in general may use any of a range of methodologies starting from simple keyword spotting to more complex language understanding mechanisms.", "Text mining also refers to a range of applications, from text clustering and categorisation to more specific analyses, such as extracting items of interest from texts, tailored for some application task.", "Text mining as an application area traces its roots to the very first steps of language technology, the Linguistic String Project, and information extraction, which finds and tabulates pre-identified and carefully formulated structural patterns in text [9], [10], [11], [12], [13].", "In recent years, text mining tools have spread to many practical fields and many overviews of text mining can be found: a technical perspective is given e.g.", "by Aggarwal and Zhai [14].", "The most popular text analysis method in computational social science currently is the family of methods known as topic models which is generally understood to refer to the specific strand of probabilistic models originally defined by Blei, [15].", "Topic models assume that each text carries some topic or some number of topics selected from a finite set of possible topics.", "Each topic in play in a text causes some features—terms or constructions, but almost always in practice restricted to single words—to be observable in it.", "Each observable feature is linked to some topic or some topics with some weight.", "As an example, texts which can be observed to contain terms such as helicopter, rotor, airfield, and pilot can be assumed to be about some topic related to helicopters whereas texts which contain the terms cow, milk, cattle, and grazing can be assumed to be about some other topics related to livestock and dairy.", "The connections between terms and topic and the weights of those connections are learned from observing a large number of texts.", "In the approach first formulated by Blei, the topics are also learned from texts, without manual intervention, but this does not necessarily need to be always the case: on a more general level, any procedure which relates texts or text segments to a set of topics based on observed linguistic features can be called a topic model, irrespective of algorithmic details.", "A variant which has been applied to the task of questionnaire processing, structural topic models, allow the analyst to insert previous knowledge of covariation between topics into the analysis [16].", "The result of a topic model analysis for a text collection is a description of what topics appear in the texts and what features characterise those topics: if analyses on different collections are made, a comparison between them can illustrate e.g.", "change over time in topic composition.", "Topic models have lately been used in digital scholarship for mining historical archives and in media monitoring for mining news feeds and the like.", "This introduction of new digital methodology for scholarship has not been uncontroversial [17], [18], [19], [20], [21].", "The debate over how to best use new technologies is lively and goes to the roots of what the ultimate research goals of the humanities and the social sciences are.", "The focus of language technology has been on the analysis of topical and factual content rather than e.g.", "genre, context, author perspective or stance which has effects on how the functionality is evaluated and developed [22].", "The concerns of the humanities and the social sciences are methodologically different and they approach knowledge and what insights can be learned from data differently.", "To some extent, this debate revolves around a technical question: how can technologies that have been developed for some task be transferred to be useful for another?", "What background assumptions does the technology solution bring to the table and what sort of effects on the results and output of the field does it have?", "Computational text analysis has its place in every situation where a collection of text is of a size that would overwhelm a human analyst.", "The collection does not need to be immense for technology to be helpful.", "But introducing computational technology where human effort previously has been the norm will have effects on the output, both positive and negative, both predictable and unexpected.", "For data in social science research, quality criteria go beyond that of productivity gain." ], [ "Text mining for the social sciences", "Text analysis in questionnaire and interview studies involves establishing whether the text verifies or refutes hypotheses of interest, or exploring the data to establish hypotheses for continued study and testing.", "In practice, this means finding if some concepts or entities of interest are mentioned in the text data and to establish how often, in what contexts, and in what way they are mentioned.", "Processing questionnaire and interview texts will entail establishing what general topic is being discussed, but also tracking mentions of e.g.", "persons, organisations, relations between them, and attitudes and feelings with respect to them.", "Potentially, the analysis is able to make use of background variables such as frame of mind, personality type, and outlook of the respondent.", "One of the most valuable features of human language is that it allows its users to vary their way of expressing themselves depending on the social context, the attitude and background of the speaker, and many other factors.", "This is a challenge for analysis methods that rely on observations of term occurrences.", "Most concepts or notions of interest can be referred to with a variety of terms, and most terms may, depending on context and other variables, refer to a multitude of concepts.", "The ambiguity, vagueness, and fluidity over time and across situations of human linguistic expression is often described as a problem.", "This perspective does not do justice to the nature of human communication.", "The adaptability of human language is useful: it allows new terms to be coined, established terms to be recruited into service to fit the momentary needs of some discourse, and various discourses to be associated or contrasted through choices of terms and constructions.", "The challenge for the analyst of our specific use case is in fact exactly the reason why open answers are useful: if the choice of terms and constructions were entirely predictable, the information captured through open answers would be so much less rich and valuable.", "For research tasks, replicability of an analysis is a major quality criterion.", "One must be able to revisit the data and reproduce the analysis results using the methodologies originally used for analysis.", "Furthermore, researchers also wish to generalise from a given study to new populations.", "This is as true for text data as it is for other types of data.", "Text classification and processing is today practically always done by the analyst or researcher in person, or by hired human coders.", "For these scholars to trust computational technology the tools must provide replicable, consistent, and explainable results.", "Given the dynamic and situational characteristics of human language, matching interesting concepts and relevant research variables to observable features in interview responses is a non-trivial challenge.", "Doing this reliably and consistently encompasses every challenge in language understanding.", "It involves terminological, situational, social, dialectal, and individual variation.", "The degrees of freedom are obviously larger for explorative studies where hypotheses are unformed and the concepts are not defined.", "Classifying texts and identifying concepts in responses reliably requires that the classifier has seen large amounts of text in general and has an understanding of the topic being discussed.", "Human analysts have this competence as a matter of course, but text analysis tools need to acquire the competence somehow.", "Recent approaches in text analysis rely on combinations of learning models.", "They are versatile and trainable for application in fields where computational models previously have not been of use.", "Learning models in general are built by providing them with training data which are used to set their internal parameters, weights, or loadings in a way which allows them to repeat their performance on previously unseen data.", "There are numerous learning schemes which rely on various levels of manual intervention and human effort to guide them.", "The practical objective of introducing learning models is to reduce the amount of human effort in configuring a tool to handle the specifics of human language in a new domain.", "Firstly, in recent years, end-to-end training implemented using neural models has become the dominant framework for learning models.", "Neural models are not designed to be configured by human effort, but are instead implemented to learn from examples with as little intervention as possible.", "They are inspired by biological neural architectures and are implemented using a large number of internal statistical parameters which are purposely hidden from the human operator, instead to be set and modified through exposure to judiciously chosen training examples.", "A neural model is trained by being shown a number of training texts together with manually assigned desired category labels.", "It will then use that information to select combinations of observable features in the training texts, for use for future categorisation of previously unseen texts.", "This is a convenient way to build a statistical prediction model with little technical effort, and this combination of approaches has proven useful in many text analysis tasks with impressive performance in recent years.", "The end-to-end training framework corresponds to giving human analysts example results to use as a model for the analysis of new data rather than instructing them by giving them rules for how to proceed.", "Secondly, the background understanding of language is learned through the transfer learning framework, by processing large amounts of previous text and applying that knowledge to numerous later analysis tasks.", "These previous texts do not need to be from the same domain, since the objective is to learn general statistical patterns: which terms and constructions are frequent and which are unusual, e.g.", "By providing a base from which to inspect new data, the transfer learning framework enables learning models to more rapidly establish what is interesting in the data set under analysis.", "This corresponds to requiring human analysts to be well read and erudite.", "Tireless processing at high speed is an obvious benefit of using computational approaches, but answering to the requirement for replicability needs attention before computational models can be reliably applied to texts in social science.", "Statistical models generally and neural models specifically are designed to hide their parameters from the human operator.", "Even if internal parameters were to be inspected, they would be meaningless, taken individually, since neural models encode knowledge as patterns over a multitude of statistical scores rather than isolated in an individual variable.", "This is one of the main advantages of neural models, since it allows the model to take many observable features into account simultaneously, even such features that are weak taken in isolation.", "Such cooccurrence data are impossible to enumerate manually, but can be learned if the training data set is large enough.", "This has the consequence that neural models rely heavily on appropriately chosen training data.", "This is where the transfer learning framework helps by leveraging previously collected data.", "Those data contribute to the analysis in various ways, but what their effect is can only be established if the model comes with a declaration of what data they were trained on.", "These qualities of neural models and the transfer learning framework contribute to making learning text analysis models less transparent than a model which would rely on e.g.", "a set of written explicit categorisation rules.", "This is a known challenge in many applications, and is partially addressed in recent research to make artificial intelligence and learning models explainableE.g.", "in the Explainable Artificial Intelligence (XAI) program launched by DARPA in 2016.. As an additional facet of transparency, for application to research tasks, it is desirable that the model be editable to allow the researcher to manually improve its precision and recall over the data set under consideration for cases where the automatic mechanisms may have not found some correspondence of interest or where they may have overtrained on some singularity in the data set.", "This is in contrast with how end-to-end models typically are understood, but if the aim of a study is not to automatise concept learning but to produce an outcome to be trusted, professional human intervention is an important part of the workflow.", "The structural topic models mentioned above are one way of moving in this direction, since they allow for the explicit introduction of e.g.", "expected covariation between topics [16].", "As a general requirement for using learning models in a research task, models should allow the analyst to inspect the classification or scoring it provides and allow the analyst to adjust internal parameters to fit hypotheses or established knowledge.", "This is not always the case for current learning models." ], [ "Case study", "The case study which motivated us to address these more general methodological questions is an ethnographic exploratory study—\"Intimacy after 60.", "Transition into retirement\"—on women's experiences and current feelings about relationships and intimacy during their transition from working life to retirement.", "The study investigates how mindset and attitude towards relationships with partner, family, friends, and colleagues, with respect to compromises, principles, and conflict resolution relates to how the subjects value three aspects of intimacy: physical, emotional, and intellectual.", "The case study is explorative and is intended to generate patterns which will be used to formulate hypotheses for future testing in representative longitudinal panel studies and the exploratory design was formed with that in mind.", "Recordings of these interviews have been manually transcribed.", "Each interview in the study consists of a number of turns: the interviewer asks a question or prompts the respondent in one turn and the respondent then reacts to the prompt in another turn, ranging in length from terse answers ( to paragraph-length multi-sentence responses (, , ).", "We use turns as the basic unit of analysis.", ".", "I forgot to mention them.", "They're another two that are best friends, yes.", "Name1, and Name2, and I are very close. \"", "\"I think he was not a well person, because he always managed to arrange it so that I would find out.", "Maybe, you know, a piece... \" \"Yes I was\" There is something that I talk about only with one friend, although a couple of friends know about it.", "Yes, but it's funny you should say that–I was recently spending a weekend with friends, Name1 and Name2.", "And Name1 and I are very close.", "Name2 and I are close, but he and I are closer.", "And we were walking on the beach–so tranquilizing, the water–and he said to me, he said, \"You know, it's the stories that they don't tell about ourselves that are the ones that really define us.\"", ".", "\"...\"Oh, well, this is forever, and it'll just be the two of us.\"", "I think we got married so young because we so badly wanted to be together, but in those days it wasn't really very nice to be sexual unless you were married.", "So... \" Not necessarily being taken care of–although my friends always want to be very careful with me and kind–but I think what I want from the relationship is to not be alone, whether it's intellectual, you know, and we spend an hour bashing the president as we did the other night, or, you know, a common interest.", "I have friends that I go birding with, you know?", "I think it's just important not to feel all by myself, not to feel too abandoned.", "Many variables are obviously given directly by study design and respondent selection, through various metadata (e.g.", "demographic and socioeconomic variables).", "As an example, the data collection for the study has been performed in a series of interviews in several cultural areas: North America (NA), Northern Europe (NE), Asia (A), Latin America, Eastern Europe, and the Middle East.", "Countries characterised by recent political upheavals, lack of social peace, or weak rule of law have been grouped under (W).", "The data set currently under consideration consists of some 54 interviews.", "Overview descriptive statistics are given in Table REF .", "While the scale of this study is less than typically is thought to be of interest for automatic processing, even this collection of turns is time-consuming and daunting to process consistently for a human analyst.", "Table: Descriptive statistics for the collected data.Text analysis is used in this study to identify a number of observable textual features which can be quantified into relevant variables for the study.", "The language used by the respondents is not a one on one mapping onto variables.", "Observable features such as term counts need to be combined and sometimes weighted to be aggregated into useful variables.", "Some concepts are mentioned in several turns; some attitudes only emerge across the entire interview; some concepts are only implicitly present and can be inferred by a human reader with some accuracy but are yet beyond the capability of automated methods.", "The tool used in this present case study—Gavagai Explorer—is built primarily for interactive analysis of e.g.", "customer feedback, consumer reviews, or market surveys but has also been used in previous academic research, [23].", "Its functionality is based on term statistics: it uses words and multi-word terms as features to cluster texts.", "Gavagai Explorer is built to be transparent: the models used in the analysis are represented as editable lists of words and terms, which allows the analyst to modify the clustering criteria and recluster interactively.", "Gavagai Explorer uses extensive background knowledge of general language usage based on large amounts of continuously ingested streaming text from published materials, both editorial and social media, and is thus an example of the transfer learning approach discussed above [24].", "Gavagai Explorer splits each text item—in this case, an interview turn—into sentences and clusters those sentences into topically coherent sets by terms that occur in them.", "Each sentence can only be in one cluster, but each turn, since it may contain several sentences and treat several topics, can be in several clusters.", "The clusters and the terms that define them can be inspected and edited by the analyst by adding or deleting terms to enrich and refine them.", "Clusters of no or little interest can be merged with others or discarded entirely.", "Using the background knowledge of language usage, the analyst is given suggestions of suitable synonym or related terms.", "In addition, each turn and each cluster is scored by a number of sentiments which can be defined as sets of terms by the analyst [25]." ], [ "Codable variables", "Some variables of the study are given as direct answers to questions posed by the interviewer.", "With direct answers of the type shown in Examples -, there is a clear and codable datapoint to be found in the response.", "In many cases, the topic is only mentioned in the question such as in Example : responses can be short and abbreviated and refer back to the question through implicit continuation along a topic introduced by the question.", "A question does not reflect the language of the respondent, but text from a question-response pair—and in some cases, from an even longer stretch of discourse—may necessary for the analysis of what the respondent means.", "These responses are straightforward for human coders to extract from the material, but are still fairly tricky for automatic analysis.", "In the present study, such variables have been analysed by hand.", "Automatically processing these kinds of variables would be possible with high precision for many of the cases, but extracting them automatically will entail loss of coverage, using a combination of information extraction and recent advances in sequence tagging with machine learning models.", "Today, the general case is not yet resolved, and the special cases that can be solved will need large amounts of training data to attain any level of reasonable coverage.", "Processing question-response pairs in a longer discourse is today addressed in the context of conversational agents in e.g.", "customer service contexts.", "That methodology shows promise of being applicable for the purpose of processing interviews.", ".", "İNTERVIEWER: Did you work full-time?", "RESPONDENT: I always worked full-time except for when I was in school, yeah.", ".", "İNTERVIEWER: Do you belong to any religious community?", "RESPONDENT: No.", ".", "İNTERVIEWER: On a scale from 1 to 10, how much did your parents encourage you to get an education?", "RESPONDENT: Not so very much.", "I'm going to say 4.", "The variables of greatest interest for this present discussion are those expressed by the respondents in free form in unconstrained discourse.", "Tabulating these can be done through analysis of respondent turns in the text, and the freedom they afford the respondent are the reason to move to open responses in the first place.", "Much of this is fairly simple to detect.", "If someone mentions partner, we know they are talking about someone's partner, and given the interview situation this is almost certainly the respondent's partner.", "By simple tallying of how often some concept is mentioned we get a quantitative score of its importance to the subject in question, and this score can be compared to other respondents' scores.", "This fits well with the goals of general text analysis and especially that of topic modelling discussed earlier.", "Features, especially terms, that are observed to occur frequently are likely to be important in a text, but only if they occur more frequently than expected; especially interesting are features that occur burstily, with local peaks in distribution, indicating that some matter of interest is under treatment [26], [27]: how to compute and assess the importance of term occurrence is central to the theory of information retrieval and related technologies [28].", "This reflects one of the more important aspects of language use: that of referentiality, where language is used to refer to entities, concepts, or notions of interest to discourse participants.", "Referring to the dynamic and flexible qualities of human language discussed above, reference to concepts can be done in many ways, some not entirely predictable before the fact.", "To ensure recall, or coverage, of the analysis, semantically related terms must be included: for the helicopter example given earlier, we might want both synonyms or near synonyms (autogiro, chopper, whirlybird) and other related terms (airfoil, camber, translational lift) to make sure we identify every time the respondent addresses a concept; for the partner example in this case study, we would at a minimum wish to include e.g.", "their husband, their boyfriend, their ex etc, perhaps extending to married, engagement, and so forth.", "This is where transfer learning is of great help: manual enumeration of related terms by the analyst would be difficult to do exhaustively and reliably.", "By using occurrence statistics from general language use it is relatively easy to identify and suggest semantically related and associated terms which yields a model with greater consistency and coverage alike.", "The concept clusters for this present case study include aspects of closeness such as intellectual closeness, emotional closeness, physical closeness, sexual closeness; as well as various interpersonal relations of the respondent such as partner, friends, colleagues, and family.", "These are examined and iteratively refined by editing the terms that define them, resulting in sets of terms of up to a hundred in size.", "Samples of two automatically grouped concept clusters are shown in Examples (clustered around various leisure activities) and (clustered around the concept of friendship).", ".", "İ love theater and movies.", "I travel with my daughter, my cousin, and the last trip I did I did alone.", "Movies I also like and I do knit.", "Netflix is wonderful, we have a glass of wine and peanuts and a movie.", ".", "He did lots of traveling.", "I do yoga two times a week and I dance three times a week and I do gymnastics.", "We do concerts and movies together.", ".", "Ẇith my friends I had more freedom.", "Lately, I let go of friends that don't work out anymore.", "Different groups where I have my best friends.", "I talk about my issues with my best friends.", ".", "Now he has a girlfriend from Place I am intellectually close to my friends and also emotionally.", "I really enjoy talking to my friends about their experience of politics, my friend used to live in Place, he lived there for years during Pinochet.", "Using counts of how often concepts are mentioned, we are able to assess the relative importance of the various types of closeness related to the differential between the various cultural areas, and an overview is given in Table REF .", "The table shows what proportion of the turns bring up the aspect of intimacy in question.", "We see that respondents from Northern Europe do not address physical closeness or intellectual closeness as often as other respondents do, and that respondents from North America discuss sexual closeness more often than others.", "Table: Difference in emphasis on the various aspects of intimacy across cultural areas (percentage of all turns that bring up the facet of intimacy in question)There is an additional referential pattern that has not been captured by the current analysis.", "Coherence in a conversation may be realised using e.g.", "pronominal reference to a previously named person.", "Example shows how the response never mentions \"husband\".", "Interpreting the turn requires the analyst to refer to the question to resolve who is posing the budgetary requests under discussion.", "Similarly, reference by person names is frequent in this material, as in Example , where the respondent uses the first name of a previous husband to refer to him.", ".", "İNTERVIEWER: Was it the same with your first husband?", "Was he supportive of you working full-time when you had kids?", "RESPONDENT: Well, when I had—when we moved to Place in 1989, we didn't have a lot of money, and I would say that he wanted me to do more work, because I wasn't bringing in money.", ".", "İNTERVIEWER: And when you were 23 you lived for how long with ... RESPONDENT: With Name?", "Er, altogether maybe two or three years Pronoun resolution—figuring out who \"he\" or \"him\" refers to locally in discourse such as in Example , is a known and on a theoretical level solved task.", "In this case, we cannot trust such algorithms to resolve the \"he\" to the right referent (are we now discussing the partner, a son-in-law, a co-worker?)", "Similarly, identifying person names is in theory a similarly simple challenge: named entity resolution in general case is a solved task in language technology.", "However, in cases such as in Example , resolving who Name of the various candidate persons mentioned during the interview is involves some knowledge of the limits of the discourse at hand.", "Recently introduced sequence tagging models promise even better coverage than previous knowledge-based models and addressing this challenge promises to be a very fruitful avenue of near future research.", "Other aspects of language use are not as obviously calculable by examining simple term counts, such as those that encode relations between notions that are referred to, those that organize the structure of the discourse, and those that indicate speaker or author attitude, stance, or mood.", "These latter aspects relate directly to variables of interest for computational social sciences: in the case study at hand, e.g.", "tentativeness vs executive capacity; pragmatic vs principled mindset; empathetic vs solipsist outlook and other similar personally or culturally bound concepts.", "There is no obvious and crisp definitional demarcation between features that refer to concepts and features that communicate e.g.", "attitude, but there are some observable differences in distribution.", "Features of the latter kind are sprinkled throughout the text data and cannot be pinpointed to any single utterance.", "In addition, the distribution of attitude features dispersed across texts than features that refer to some concept: skepticism is more frequent in language in general and expressed here and there than terms that refer to e.g.", "helicopters.", "This poses a challenge for computational approaches such as topic modelling which have been optimised to establish what concepts a text refers to.", "Even so, human readers are able with some precision to distinguish many such linguistic factors from reading text, which should be understood to mean that text analysis should be able to establish observable features for them.", "Work on sentiment analysis which focusses on detecting positive vs negative sentiment in text is one example where such variables are detectable [29].", "The palette of human emotion is more complex than so: a text can be written with skepticism, revulsion, anger, or frustration, and what the composition and parameters of variation of human emotion are is under continuous discussion [30], [31], [32], [33], [34], [35], [36], [37] and attendant development of methods to detect sentiment in text.", "In a study such as the present example, some expressions of sentiment are pertinent to the research hypotheses.", "Example gives some samples of items with negative sentiment from the interviews.", "In this study, we defined expressions of feelings and sentiments to suit the hypotheses of the study such as \"Feeling rejected\" or \"Taking initiative\" to be valuable addition to standard positivity and negativity.", "These expressions are defined as lists of terms, which ensures transparency and editability of the model.", "Table: Cultural areas and attitude in text (scores computed based on weighted occurrence of polar terms, attitudinal terms, and hedge terms respectively).", "İ'm so very sorry.", "I didn't expect to be talking about all this horrible stuff.", "Thank you.", "I'm sad hearing that.", "The first one I think was okay, but the second two... Because then you're using sex... Of course they treated her terrible, right?", "I believe that he married her to have access to the children, because he was a monster.", ".", "So I became miserable there.", "But he verbally abused me horribly.", "It was terrible, it was terrible, and nobody knew.", "The model allows us to define variables over all the scores from a respondent, to assess their individual qualities.", "Some of the hypotheses of the current study call for e.g.", "distinguishing pragmatic respondents who put effort into compromise and conflict resolution from respondents who hold their social circles to set standards and whose social actions are bound by explicitly formulated principles; respondents with a positive outlook from those who are more gloomy; respondents who express emotions with intensity from those who are more reticent; and other similar personally or culturally bound concepts.", "In contrast with sentiment, which is here understood as an attitude shown vis-à-vis some mentioned topic, mindset markers are sprinkled throughout responses, cannot be found in any single turn, and need to be aggregated over an entire session.", "If a respondent repeatedly uses terms to indicate bitterness, uncertainty, or exuberance, these can be aggregated.", "Recent interest in authorship profiling has provided some tentative feature sets to explore personality type and various demographic variables from observable features [38]: in this study we have used the attitude scores to compute the difference between number of observed expressions for various positive sentiments and for various negative sentiments as a measure of polarity of emotion for individuals and a sum of squares for both as a measure of intensity of emotion for individuals.", "The variation over aspects of emphasis shown in Table REF , and using attitudinal variables we are able to further analyse it to examine differences in attitude.", "In Table REF we find that hedged or cautious expressions with overt markers of skepticism are more prevalent and intensity of expression is less pronounced in responses from Northern European respondents than in others and that cultural areas where political upheaval has been present show the opposite pattern.", "Across the entire data set, for all individuals, we find e.g.", "that placing high importance on physical closeness is well correlated with placing high importance on intellectual closeness, but that importance of emotional closeness correlates less with the other facets of closeness.", "We also find that those who express themselves with more positive than negative terms more often bring up the concept \"Getting along\" than others." ], [ "Conclusions, Lessons learnt, and Paths Forward", "We have used text analysis technology to process interview text data from a computational social science study.", "The tool we used to process the text data is developed for the analysis of customer feedback and related data: a task similar enough to processing data for the social sciences.", "We found that topical clustering and terminological enrichment provide for convenient exploration and quantification of the responses: tabulating observed concepts allows e.g.", "for correlation analysis and inference of relations between demographic data and attitude towards concepts of interest for the study.", "This makes it possible to simultaneously establish both textual and non-textual variables and to generate hypotheses for further testing.", "Introducing computational tools saves analyst effort even with a text collection on this small scale.", "The tools available today, such as the one used in the present study, are not tailored to the task of processing interview texts.", "We find that some interesting and potentially valuable features are difficult or impossible to automatise reliably at present.", "We especially note as potentially quite useful avenues of further investigation that: Reference to concepts of interest is relatively straightforward to establish whereas attitudinal content in many cases poses greater challenges.", "There is a large and growing body of work on attitude and sentiment analysis in text analysis, but in most cases, the palette of emotions or attitudes will need to be tailored to the needs of the study at hand, which will require more work on the part of the analyst and requires the tool to accommodate such editorial work.", "Variables that are distributed over the entire text of an interview are more challenging to extract: in this case study, e.g., establishing the mindset of the respondent with respect to their social circles, whether they are principled and rule-bound or whether they tend towards compromise and pragmatism.", "Automating such analyses using recent machine learning models is again quite possible, but needs new learning components tailored for this purpose.", "The interplay between questions and responses poses specific requirements for text analysis which are possible to address using today's technology if correspondences of interest are used to train a model.", "This has to our knowledge not yet been attempted.", "Traditional natural language processing mechanisms developed for general application in text analysis such as resolving referents for proper names and pronouns are excellent candidates to improve the analysis, but need to be tailored to the specifics of question-response interplay.", "Using computational text analysis tools for research in social science has great potential to allow for more exploratory open-ended studies with less effort, increase coding consistency, and reduce turnaround time for the analysis of collected data by using tools developed for market purposes.", "We note that any analysis model used for research purposes where previously human effort has been the norm, must provide replicable results.", "To be replicable, the deliberations of a tool must be transparent, to ensure quality, the knowledge it uses must be editable, and to be acceptable for research, classification decisions and scores must be explainable.", "These requirements are necessary both to afford the researcher trust in the analysis results and for review, replication, and reuse by others.", "Many recent methods and tools are not built with these criteria in mind.", "We conclude by urging researchers in language technology to investigate the specific challenges of processing interview data, especially referential issues such as the interplay between questions and responses, and we encourage social science researchers not to hesitate to use text analysis tools, especially for the exploratory phase of processing interview data." ] ]	2011.14037
[ [ "OpenKBP: The open-access knowledge-based planning grand challenge" ], [ "Abstract The purpose of this work is to advance fair and consistent comparisons of dose prediction methods for knowledge-based planning (KBP) in radiation therapy research.", "We hosted OpenKBP, a 2020 AAPM Grand Challenge, and challenged participants to develop the best method for predicting the dose of contoured CT images.", "The models were evaluated according to two separate scores: (1) dose score, which evaluates the full 3D dose distributions, and (2) dose-volume histogram (DVH) score, which evaluates a set DVH metrics.", "Participants were given the data of 340 patients who were treated for head-and-neck cancer with radiation therapy.", "The data was partitioned into training (n=200), validation (n=40), and testing (n=100) datasets.", "All participants performed training and validation with the corresponding datasets during the validation phase of the Challenge, and we ranked the models in the testing phase based on out-of-sample performance.", "The Challenge attracted 195 participants from 28 countries, and 73 of those participants formed 44 teams in the validation phase, which received a total of 1750 submissions.", "The testing phase garnered submissions from 28 teams.", "On average, over the course of the validation phase, participants improved the dose and DVH scores of their models by a factor of 2.7 and 5.7, respectively.", "In the testing phase one model achieved significantly better dose and DVH score than the runner-up models.", "Lastly, many of the top performing teams reported using generalizable techniques (e.g., ensembles) to achieve higher performance than their competition.", "This is the first competition for knowledge-based planning research, and it helped launch the first platform for comparing KBP prediction methods fairly and consistently.", "The OpenKBP datasets are available publicly to help benchmark future KBP research, which has also democratized KBP research by making it accessible to everyone." ], [ "Introduction", "The increasing demand for radiation therapy to treat cancer has led to a growing focus on improving patient flow in clinics.", "[1] Knowledge-based planning (KBP) methods promise to reduce treatment lead time by automatically generating patient-specific treatment plans, thereby streamlining the treatment planning process.", "[2] KBP methods are generally formulated as two-stage pipelines (see Figure REF ).", "In most cases, the first stage is a machine learning (ML) method that predicts the dose distribution that should be delivered to a patient based on contoured CT images, and the second stage is an optimization model that generates a treatment plan based on the predicted dose distribution.", "[3], [4] Figure: Overview of a complete knowledge-based planning pipeline.Research into dose prediction has experienced major growth in the past decade,[5] in part due to the growing sophistication of machine learning and optimization methods in conjunction with advances in computational technology.", "There are two main branches of dose prediction methods: (1) those that predict summary statistics (e.g., dose-volume features)[6], [7], [8], [9] and (2) those that predict entire 3D dose distributions.", "[10], [11], [12], [13], [14], [15] Both branches of dose prediction methods use a wide range of methodologies, e.g., linear regression,[9] principal component analysis,[6], [7] random forest,[10] neural networks.", "[11], [12], [13], [14], [15] All of this KBP research is performed in close collaboration with radiation therapy clinics using private clinical datasets that are generated via local planning protocols.", "[5] Development of KBP models is further challenged by the lack of large open-access datasets and standardized evaluation metrics.", "Existing open-access radiation therapy datasets cater to optimization[16], [17] or classification problems (e.g., segmentation, prognosis).", "[18] Researchers that develop dose prediction models must rely on their own private clinical datasets and different evaluation metrics, which makes it difficult to objectively and rigorously compare the quality of different prediction approaches at a meaningful scale.", "[5] As a result, researchers must recreate published dose prediction models to benchmark their new models via a common dataset and set of evaluation metrics.", "[4], [19] In contrast, open-access datasets and standardized metrics are staples of thriving artificial intelligence-driven fields, as evidenced by the uptake of the CIFAR[20] and ImageNet[21] datasets in the computer vision community over the past decade.", "We launched the Open Knowledge-Based Planning (OpenKBP) Grand Challenge to advance knowledge-based planning by 1) providing a platform to enable fair and consistent comparisons of dose prediction methods and 2) developing the first open-access dataset for KBP.", "Participants of the Challenge used the dataset to train, test, and compare their prediction methods, using a set of standardized evaluation metrics.", "The data and accompanying code-base is freely available at https://github.com/ababier/open-kbp for KBP researchers to use going forward." ], [ "Methods and Materials", "We first describe our process for building and validating the dataset for the Challenge.", "We then describe how the Challenge was organized and delivered.", "Finally, we provide an analysis of the Challenge results.", "This study was approved by the Research Ethics Board at the University of Toronto." ], [ "Data Processing", "Figure REF depicts our data processing approach at a high level, which consisted of four steps: (i) data acquisition, (ii) data cleaning, (iii) plan augmentation, and (iv) data partitioning.", "Figure: Overview of the data processing pipeline.", "nn represents the number of patients at each stage of the pipeline." ], [ "Acquiring the raw data", "We obtained the Digital Imaging and Communications in Medicine (DICOM) files of 217 patients, which we call the raw private data (denoted by $\\mathcal {P}^\\text{raw}$ ), who were treated for oropharyngeal cancer at Princess Margaret Cancer Center.", "Each file included a treatment plan that was delivered from nine approximately equispaced coplanar fields with 6 MV, step-and-shoot, intensity-modulated radiation therapy (IMRT).", "Each patient was prescribed 70 Gy in 35 fractions, with 70 Gy to the high-dose planning target volume (PTV70), 63 Gy to the mid-dose planning target volume (PTV63), and 56 Gy to the low-dose planning target volume (PTV56); a PTV63 was only contoured in 130 of the patients.", "All plans included CT images, contours for regions-of-interest (ROIs), and the dose distributions based on a consistent set of planning protocols.", "We also retrieved clinical DICOM data for 851 patients, which we call the raw public data (denoted by $\\mathcal {O}^\\text{raw}$ ), from four public data sources [22], [23], [24], [25] hosted on The Cancer Imaging Archive (TCIA).", "[18] The data was originally sourced from twelve different institutions between 1999 and 2014.", "Each file contained CT images and contours for the regions of interest (ROIs).", "This collection of files contained several inconsistencies because the data originated from different institutions.", "For example, different institutions may have employed different dose levels, fractionation schemes, ROI naming conventions, languages (English versus French nomenclature), PTV margins (isotropic versus anisotropic margins from CTV), and treatment modalities (3D conformal radiation therapy (3DCRT) versus IMRT)." ], [ "Data cleaning", "In order to standardize and improve the homogeneity of the datasets, we employed a sequence of data cleaning procedures.", "First, we relabeled all of ROIs according to a consistent nomenclature.", "For each patient $p\\in \\mathcal {P}^\\text{raw}\\cup \\mathcal {O}^\\text{raw}$ , we included organ-at-risk (OAR) contours for the brainstem, spinal cord, right parotid, left parotid, larynx, esophagus, and mandible; let $\\mathcal {I}_p$ denote this set of OARs for a patient $p$ .", "All other OAR contours were deleted.", "Also, an OAR was omitted from $\\mathcal {I}_p$ if it was not contoured in the clinical plan (e.g., a patient whose left parotid was not contoured would not have it in the set $\\mathcal {I}_p$ ).", "To construct the set of targets $\\mathcal {T}_p$ , we identified the low-, mid-, and high-dose targets based on their relative dose levels[26] and relabeled them as PTV56, PTV63, and PTV70, respectively.", "Any region with overlapping PTVs was relabeled as a single PTV with a dose-level equal to that of the highest dose-level of those overlapping PTVs.", "Next, we modified target contours in the raw public dataset ($\\mathcal {O}^\\text{raw}$ ) to match the protocols from the private dataset ($\\mathcal {P}^\\text{raw}$ ).", "These modifications helped to fix some of the inconsistencies in contouring (e.g., no PTV margins, anisotropic PTV margins) that were present in the raw public dataset.", "Every PTV was expanded to include the voxels within $5~\\text{mm}$ of its respective clinical target volume (CTV); the PTV was left unchanged in cases where there was no CTV contour associated with the PTV.", "Every PTV was also clipped to be no closer than $5~\\text{mm}$ from the surface of the patient.", "We generated dose influence matrices for each patient in the public dataset $\\mathcal {O}^\\text{raw}$ based on 6 MV step-and-shoot IMRT with nine equispaced coplanar fields at $0^\\circ $ , $40^\\circ $ , $\\hdots $ , $320^\\circ $ .", "Those fields were divided into a set of beamlets $\\mathcal {B}$ that were each $5~\\text{mm}\\times 5~\\text{mm}$ .", "Every patient was also divided into a set of voxels $\\mathcal {V}^p$ that were downsampled to fill axial slices of dimension $128\\times 128$ .", "The relationship between the intensity $w_b$ of beamlet $b$ and dose $d_v$ deposited to voxel $v$ was calculated in MATLAB using the IMRPT library in A Computational Environment for Radiotherapy Research,[27] which we used to form the elements $D_{v, b}$ of each patient's dose influence matrix.", "The dose to a voxel $v$ was calculated as: $d_v= \\sum \\limits _{b \\in \\mathcal {B}} D_{v,b}w_b,\\ \\forall v \\in \\mathcal {V}^p, \\ \\forall p\\in \\mathcal {P}.$ To prepare the patient data for deep learning models, we framed each patient in a $128\\times 128\\times 128$ voxel tensor in two steps.", "First, we calculated the weighted average position of each patient $p$ , using $\\sum \\limits _{b \\in \\mathcal {B}} D_{v,b}$ as the weight for each voxel $v \\in \\mathcal {V}^p$ .", "Second, we applied a bounding box centered on that weighted average position with dimensions of $128\\times 128\\times 128$ voxels.", "We added zero-padding where necessary to ensure consistent tensor volumes.", "Over the course of the data cleaning phase, 390 patients were removed from the public dataset ($\\mathcal {O}^\\text{raw}$ ) for a variety of reasons including missing target contours and issues generating a valid dose influence matrix.", "No patients were removed from our private dataset.", "At the end of the data cleaning step, we had clean private $\\mathcal {P}^\\text{clean}$ and public $\\mathcal {O}^\\text{clean}$ datasets consisting of 217 and 461 patients, respectively." ], [ "Plan augmentation", "Next, we generated synthetic plans for each patient in the clean public dataset and only retained the associated dose distribution.", "These synthetic plans were generated using a variation of a published automated KBP pipeline, [15] which was trained using the cleaned clinical plans from our private dataset $\\mathcal {P}^\\text{clean}$ .", "Figure REF illustrates the plan augmentation process.", "Figure: Overview of the plan augmentation process, which is a two phased approach: (a) the clean private clinical dataset is used to train the dose prediction method, and then (b) the trained method is used in a complete KBP pipeline that intakes the cleaned public data to generate synthetic plans.The dose prediction model in the KBP pipeline was a conditional generative adversarial network (GAN)[28] with the same architecture as the 3D-GAN in Babier et al.", "2020.", "[15] It uses two neural networks: (1) a generator that predicts the dose distribution for a contoured CT image; and (2) a discriminator that predicts whether the input is a predicted or clinical dose distribution.", "We trained the generator to minimize the mean absolute difference between the predicted and clinical dose distributions, which we regularized with the discriminator to encourage the generator to make predictions resembling clinical dose distributions.", "Between each batch update of the generator, we also trained the discriminator to minimize a binary-cross-entropy loss function.", "This GAN model was trained for 200 epochs using the clean private dataset of 217 treatment plans, and it was implemented in Tensorflow 1.12.3 on a Nvidia 1080Ti graphic processing unit (GPU) with $12~\\text{GB}$ of video memory.", "As part of the plan optimization, we added seven optimization structures to each patient in the public dataset to encourage high-quality synthetic plans.", "All of these optimization structures are based on structures that were used to optimize the plans in our private clinical dataset.", "These structures were not included in the final Challenge datasets.", "The first optimization structure was called limPostNeck, which is used to limit dose to the posterior neck.", "The limPostNeck includes all of the non-target voxels between the posterior aspect of a $3~\\text{mm}$ expansion of the spinal cord and the patient posterior; there were 12 cases where no spinal cord was contoured where we extended the brainstem inferiorly to approximate the spinal cord to make the limPostNeck.", "All spinal cord and target voxels were removed from the limPostNeck.", "The other six optimization structures were PTV rings, which we added to encourage high dose gradients around the PTVs.", "We used $2~\\text{mm}$ and $6~\\text{mm}$ rings that include voxels within $0~\\text{mm}$ to $2~\\text{mm}$ and $2~\\text{mm}$ to $6~\\text{mm}$ of the PTV, respectively.", "Any overlap between rings was eliminated by removing voxels in those overlapping regions from the ring of PTV with the lower dose-level.", "All target voxels were also removed from the rings.", "The plan optimization method was a two-stage approach to inverse planning.", "[29] In the first stage, we estimate the objective function weights for a conventional inverse planning model that makes a predicted dose distribution optimal.", "In the second stage, we use the estimated weights and solve the inverse planning model to generate a synthetic treatment plan.", "The objective of the planning model was to minimize the sum of 114 functions: seven per OAR, three per target, and seven per optimization structure.", "The functions for each OAR evaluated the mean dose; maximum dose; and average dose above 0.25, 0.50, 0.75, 0.90, and 0.975 of the maximum predicted dose to that OAR.", "The functions for each target evaluated the mean dose, maximum dose, average dose below the target dose level, and average dose 5% above the target dose level (e.g., average dose above 73.5 Gy in the PTV70).", "The functions for each optimization structure were the same as the OAR functions.", "To ensure that all plans had a similar degree of fluence complexity, all synthetic plans were constrained to a sum-of-positive-gradients (SPG) value of 65.", "[30] Both optimization models were solved in Python 3.6 using Gurobi 9.0.1 (Gurobi Optimization, TX, US) to generate a dose distribution $\\hat{\\textbf {d}}_{p}$ for each patient in the clean public dataset.", "We used Algorithm REF to correct or remove low-quality plans that were generated by our plan augmentation process (i.e., the process in Figure REF ).", "The algorithm curated a set of patients $\\mathcal {O}^\\text{aug}$ with high-quality dose distributions $\\hat{\\textbf {s}}_{p}$ , which were based on the dose distributions $\\hat{\\textbf {d}}_{p}$ for the plans of patients in the clean public dataset $\\mathcal {O}^\\text{clean}$ .", "The algorithm retained any patients that had plans with a high-dose target that received a higher mean dose or 1$^\\text{st}$ percentile dose (${r}$ ) than the mid-dose or low-dose targets (line 3).", "The entire dose was then multiplicatively scaled so that maximum dose to the high-dose target ${\\text{PTV70}}({\\hat{\\textbf {s}}_{p}})$ was no lower than the lowest maximum dose ($\\underline{\\text{D}}_\\text{max}^\\text{\\text{PTV70}}$ ) observed in the plans of the patients from our clean private dataset $\\mathcal {P}^\\text{clean}$ (line 4).", "For each instance where we scaled dose by a constant factor, we also scaled the dose by a random factor $\\varepsilon $ that was sampled from a uniform distribution between 1.00 and 1.05 (i.e., $\\varepsilon \\sim \\mathcal {U}({1.00}, {1.05})$ ) for two reason: (1) we did not want participants to learn a strict cutoff and (2) strict cutoffs are not realistic.", "Next, we reduced the dose $\\hat{\\textbf {s}}_{p}$ so that, for each ROI $r\\in \\mathcal {I}_p\\cup \\mathcal {T}_p$ , it delivered a maximum dose ${r}$ , mean dose ${r}$ , and dose to 99% of voxels ${r}$ that was lower than any plan in our private clinical dataset (line 7).", "We denote the highest value observed in the clinical plans with a bar (e.g., $\\overline{\\text{D}}_\\text{c}^\\text{r}$ for a criteria $c$ and ROI $r$ ).", "Lastly, a patient $p$ was added to $\\mathcal {O}^\\text{aug}$ if that patient's respective dose $\\hat{\\textbf {s}}_{p}$ had a maximum dose to the high-dose target that was between the lower and upper bounds that we observed in our private set of clinical plans (line 8).", "The final size of $\\mathcal {O}^\\text{aug}$ was 340.", "[H] $\\mathcal {O}^\\text{aug}\\leftarrow \\lbrace \\rbrace $ $p\\in \\mathcal {O}^\\text{clean}$ $\\hat{\\textbf {s}}_{p}\\leftarrow \\hat{\\textbf {d}}_{p}$ ${PTV70}({\\hat{\\textbf {s}}_{p}}) \\ge {t}({\\hat{\\textbf {s}}_{p}})$ or ${PTV70}({\\hat{\\textbf {s}}_{p}}) \\ge {t}({\\hat{\\textbf {s}}_{p}}),\\ \\forall t\\in \\mathcal {T}_p$ $\\hat{\\textbf {s}}_{p}\\leftarrow \\hat{\\textbf {s}}_{p}\\times \\text{max}(1,\\ \\underline{\\text{D}}_\\text{max}^\\text{PTV70}/{PTV70}({\\hat{\\textbf {s}}_{p}}) \\times \\varepsilon \\sim \\mathcal {U}({1.00}, {1.05}))$ $r\\in \\mathcal {I}_p\\cup \\mathcal {T}_p$ $c\\in \\lbrace \\text{\\upshape max, mean, 99}\\rbrace $ $\\hat{\\textbf {s}}_{p}\\leftarrow \\hat{\\textbf {s}}_{p}\\times \\text{min}(1,\\ c̥{r}({\\hat{\\textbf {s}}_{p}})/\\overline{\\text{D}}_\\text{c}^\\text{r} \\times \\varepsilon \\sim \\mathcal {U}({0.97}, {1.00}))$ $\\underline{\\text{D}}_\\text{max}^\\text{PTV70} > {PTV70}({\\hat{\\textbf {s}}_{p}}) > \\overline{\\text{D}}_\\text{max}^\\text{PTV70}$ $\\mathcal {O}^\\text{aug}\\leftarrow \\mathcal {O}^\\text{aug}\\cup \\lbrace p\\rbrace $ Improve low-quality plans where possible and construct the set of public patients with high-quality synthetic plan dose distributions $\\hat{\\textbf {s}}_{p}$ ." ], [ "Validation of final competition datasets", "We evaluated the distribution of synthetic dose $\\hat{\\textbf {s}}_{p}$ quality over every patient $p\\in \\mathcal {O}^\\text{aug}$ by comparing it to the distribution of the clinical dose quality over every patient $p\\in \\mathcal {P}^\\text{clean}$ .", "We measured quality using the set of DVH criteria used in the Challenge.", "The distribution of DVH criteria over the population of synthetic doses and clinical doses was visualized with a box plot for each set of criteria.", "For each of the DVH criteria, we used a one-sided Mann-Whitney $U$ test to determine whether the synthetic doses were inferior (null hypothesis) or non-inferior (alternative hypothesis) to the clinical doses, based on an equivalence interval of 2.1 Gy (i.e., 3% of the high-dose level).", "[31] Lower values were better for ${}$ , ${}$ , and ${}$ ; higher values were better for ${}$ and ${}$ .", "For these and all subsequent hypothesis tests, $P < 0.05$ was considered significant.", "The final public dataset $\\mathcal {O}^\\text{aug}$ was randomly split into training $\\mathcal {O}^\\text{train}$ , validation $\\mathcal {O}^\\text{val}$ , and testing $\\mathcal {O}^\\text{test}$ datasets with 200, 40, and 100 patients, respectively.", "Every patient in these datasets had a synthetic dose distribution ($\\hat{\\textbf {s}}_{p}$ ), CT images, structure masks, feasible dose mask (i.e., voxels $v\\in \\mathcal {V}^p$ such that $\\sum \\limits _{b \\in \\mathcal {B}} D_{v,b} > 0$ ), and voxel dimensions.", "This data was released to the participants in phases as described in the next section.", "A detailed description of the data format and files is given in Appendix A." ], [ "Challenge Description", "OpenKBP was hosted as an online competition using CodaLab (Microsoft Research, Redmond, WA).", "Participants could compete in the Challenge as a member of a team or as individuals (i.e., a team of one).", "The Challenge proceeded in two phases.", "In the first (validation) phase, teams developed their models and compared their performance in real time to other teams via a public leaderboard.", "In the second (testing) phase, teams submitted their dose predictions for a new unseen dataset, and we compared their performance to other teams via a hidden leaderboard to determine the final rankings for the Challenge." ], [ "Challenge timeline", "The Challenge took place over four months in 2020.", "Individuals could register to participate in the Challenge anytime after it started on February 21, 2020, which is also when the training and validation data was released to start the first (validation) phase of the Challenge.", "Three months later, on May 22, 2020, the testing data was released to start the second (testing) phase of the Challenge, which ended ten days later on June 1, 2020.", "The final competition rankings (based on testing phase performance) were released four days later on June 5, 2020.", "The Challenge also coincided with the beginning of the COVID-19 pandemic.", "[32] As a result, we extended the validation phase to accommodate for the challenges posed by the pandemic.", "The result was about a one-month delay compared to the originally planned timeline." ], [ "Participants", "OpenKBP was designed with a view towards having a low barrier to entry.", "Registration was free and open to anyone.", "We also offered comprehensive instructions to set up free, high-quality compute resources via Google Colab, for those teams who did not have access to sufficient computational resources otherwise.", "[33] To understand the make-up of the OpenKBP community, we collected demographic information from every participant via a two-part registration survey (see Appendix B).", "The first part of the survey, which was mandatory, collected professional information including their past KBP research experience, primary research area, and academic/industry affiliations.", "The second part of the survey, which was optional, collected equity, diversity, and inclusion (EDI) data including how participants self-identify in terms of gender, race, and disability status, using terminology from the United States Census Bureau." ], [ "Evaluation metrics", "Teams predicted dose distributions or dose-volume histograms for a set of patients and submitted those predictions to our competition on CodaLab.", "For each patient $p$ , we compared the submitted prediction $\\textbf {s}_{p}$ to the corresponding synthetic plan dose distribution $\\hat{\\textbf {s}}_{p}$ via two error measures (1) dose error, $\\alpha _p$ , which measures the mean absolute difference between a submission and its corresponding synthetic plan (i.e., mean absolute voxel-by-voxel difference in dose), and (2) DVH error, $\\beta ^r_{p, c}$ , which measures the absolute difference in DVH criteria between a submission and its corresponding synthetic plan.", "The dose error $\\alpha _p$ was chosen as a general measure of prediction quality that is not radiation therapy specific.", "It was only used to evaluate dose distributions (i.e., not DVH submissions), and it is calculated as $\\alpha _p= \\frac{\\left\|\\left\| \\textbf {s}_{p}- \\hat{\\textbf {s}}_{p}\\right\|\\right\|_1}{\\left\|\\mathcal {V}^p\\right\|},\\ \\forall \\ p\\in \\mathcal {O}^\\text{val}\\cup \\mathcal {O}^\\text{test}.$ The DVH error $\\beta ^r_{p, c}$ was chosen as a clinical measure of prediction quality that is radiation therapy specific.", "It involves a set of DVH criteria $\\mathcal {C}_i$ and $\\mathcal {C}_t$ for each OAR $i\\in \\mathcal {I}_p$ and target $t\\in \\mathcal {T}_p$ , respectively.", "There were two OAR DVH criteria: ${i}$ , which is the mean dose received by OAR $i$ ; and ${i}$ , which is the maximum dose received by 0.1cc of OAR $i$ .", "There were also three target DVH criteria: ${t}$ , ${t}$ , and ${t}$ , which are the doses received by 1% ($99^\\text{th}$ percentile), 95% ($5^\\text{th}$ percentile), and 99% ($1^\\text{st}$ percentile) of voxels in target $t$ , respectively.", "The DVH error was used to evaluate both dose distributions and DVHs, and it is calculated as $\\beta ^r_{p, c}= \\left\| c̥{r}(\\textbf {s}_{p}) - c̥{r}(\\hat{\\textbf {s}}_{p}) \\right\|,\\ \\forall \\ c\\in \\mathcal {C}_r,\\ \\forall \\ r\\in \\mathcal {I}_p\\cup \\mathcal {T}_p,\\ \\forall \\ p\\in \\mathcal {O}^\\text{val}\\cup \\mathcal {O}^\\text{test}.$ We chose to make both error metrics absolute differences to reward models that learn to make realistic dosimetric trade-offs.", "This is critical because prediction models that make unrealistic trade-offs (e.g., predict low dose to an OAR that is unachievable) generally perform worse than models that make realistic trade-offs in full KBP pipelines.", "[9] Building on these error metrics, we scored submissions using dose score $A_h$ and DVH score $B_h$ .", "Both scores are a variation of mean absolute error (MAE).", "The dose score is the mean dose error over all patients in a hold-out set (i.e., $\\mathcal {O}^\\text{val}$ or $\\mathcal {O}^\\text{test}$ ): $A_h= \\frac{1}{\\left\|\\mathcal {O}^h\\right\|}\\sum \\limits _{p\\in \\mathcal {O}^h}\\alpha _p,\\ \\forall \\ h\\in \\lbrace \\text{val, test}\\rbrace .$ The DVH score is the mean DVH error over all criteria from the patients in a hold-out set: $B_h= \\frac{1}{\\sum \\limits _{p\\in \\mathcal {O}^h}\\sum \\limits _{r\\in \\mathcal {I}_p\\cup \\mathcal {T}_p}\|\\mathcal {C}_r\|}\\sum \\limits _{p\\in \\mathcal {O}^h}\\sum \\limits _{r\\in \\mathcal {I}_p\\cup \\mathcal {T}_p}\\sum \\limits _{c\\in \\mathcal {C}_r}\\beta ^r_{p, c},\\ \\forall \\ h\\in \\lbrace \\text{val, test}\\rbrace .$ Using those scores, we ranked all of the submissions to the Challenge in two streams: (1) the dose stream where the team with the lowest (i.e., best) dose score won, and (2) the DVH stream where the team with the lowest (i.e., best) DVH score won." ], [ "Validation phase", "At the start of the validation phase, the full training dataset $\\mathcal {O}^\\text{train}$ of 200 patients was released, and the teams used that data to train their models.", "An out-of-sample validation dataset $\\mathcal {O}^\\text{val}$ , which included data for 40 patients without synthetic plan dose, was also released for teams to validate the out-of-sample performance of their models.", "Predictions made on the validation dataset were submitted directly to our competition on CodaLab where they were scored in the cloud using the held-back synthetic plan dose.", "The resulting scores populated a public leaderboard, but they were not used to determine the winners of the Challenge." ], [ "Testing phase", "The testing dataset $\\mathcal {O}^\\text{test}$ , which included data for 100 patients without synthetic plan dose, was released at the start of the testing phase.", "Teams used the models they developed during the validation phase and made predictions on this new unseen testing dataset.", "Similar to the validation phase, all predictions were submitted to our competition on CodaLab where they were scored in the cloud using the held-back synthetic plan dose.", "However, the resulting dose and DVH scores populated the testing leaderboard that we kept hidden until the competition finished.", "The team that performed best on the testing leaderboard with respect to the dose and DVH score was the winner of the dose and DVH stream, respectively.", "Teams that submitted to the testing leaderboard also responded to a model survey (see Appendix B) to summarize their models." ], [ "Analysis of Challenge Outcomes", "We conducted four analyses.", "First, we summarized the demographics of the participants.", "Second, we evaluated the aggregate improvements made by the teams over the course of the validation phase.", "Third, we compiled and analyzed the final results from the testing phase.", "Fourth, we summarized common modeling techniques that were employed by the participants." ], [ "Participant information", "We examined the registration information of all participants and calculated summary statistics for primary research area, past KBP research experience, country of work/study, and EDI data.", "We compared our aggregated EDI data to comparable data for the population of people who are employed in science and engineering (S&E) in the United States (US)[34] and the general US population.", "[35]" ], [ "Performance over validation phase", "As a retrospective analysis, we evaluated the aggregate improvement of all teams over the validation phase to measure their progress throughout the Challenge.", "We plotted the dose and DVH score against a relative measure of progress towards their final model, which we call the normalized submission count (NSC).", "The NSC is equal to the cumulative number of submissions a team made up to a certain point in time divided by the total number of submissions made in the validation phase.", "For example, if a team made 100 total submissions, the score at NSC = 0.5 represents that team's best recorded performance after their 50$^\\text{th}$ submission.", "For each team, we recorded their best cumulative dose and DVH score in increments of 0.05 NSC.", "At each increment we plotted the average and the 95% confidence interval of those scores over all teams that made more than 20 total submissions." ], [ "Final results in testing phase", "We used a one-sided Wilcoxon signed-rank test to determine whether the set of predictions of the best team in each stream had the same (null hypothesis) or lower (alternative hypothesis) error (i.e., $\\alpha _p$ and $\\beta ^r_{p, c}$ ) than each set of predictions submitted by the other teams.", "To visualize the range of expected error differences, we plotted the difference in dose error over all patients ($n=100$ ) and the difference in DVH error over all DVH criteria ($n=1783$ ) between the winning submission and the runner-up submissions, for the dose stream and DVH stream, respectively.", "As a retrospective sensitivity analysis, we evaluated the submissions according to an alternative scoring function with squared error terms (i.e., $\\alpha _p^2$ and ${\\beta ^r_{p, c}}^2$ ) instead of absolute error terms (i.e., $\\alpha _p$ and $\\beta ^r_{p, c}$ ) to determine if the final competition standings would have changed.", "We refer to the competition and alternative scores as MAE-based and mean squared error (MSE)-based, respectively.", "As a quantitative measure of the alignment between the two ranking methods, we evaluated the rank-order correlation between the rankings for the MAE-based and MSE-based scores via Spearman's rank test." ], [ "Common modeling decisions", "Finally, we present a summary of the model survey information that teams submitted during the testing phase.", "We present common modeling choices (e.g., model architectures), hardware, and software that teams used.", "Lastly, we present a set of techniques that we believe are generalizable to most dose prediction frameworks, based on what teams commonly employed." ], [ "Validation of final competition datasets", "Figure REF compares the quality of the public synthetic dose distributions to the private clinical dose distributions.", "The box plots in the top and bottom row summarize the performance across OAR and target DVH criteria, respectively.", "The public synthetic doses were non-inferior ($P<0.05$ ) to the clinical doses on 19 of the 23 criteria.", "For the remaining four criteria, the synthetic dose was 2.1 Gy worse on average than the clinical dose (3.7% average relative difference).", "While the synthetic doses were not a perfect replication of the clinical doses, they are sufficiently close to representing clinical dose distributions for the purpose of this Challenge and future research that leverages this dataset.", "Figure: The distribution of DVH criteria from the private clinical dose and the public synthetic dose is plotted, and the corresponding PP-values for each criterion are on the right axes.", "The boxes indicate median and interquartile range (IQR).", "Whiskers extend to the minimum of 1.5 times the IQR and the most extreme outlier." ], [ "Participant information", "Table REF summarizes the participation in each phase of the Challenge.", "Overall, 195 people registered to participate, and 73 participants were active during the validation phase.", "A total of 1750 submissions were made to the validation phase, which is an average of 40 submission per team.", "There were 28 unique models submitted in the testing phase.", "Table REF summarizes the participants' past KBP experience and primary area of research.", "Interestingly, $61.5\\%$ of participants had no prior KBP experience and less than half ($42.6\\%$ ) identified medical physics as their primary area of research.", "Machine learning researchers constituted the majority ($50.3\\%$ ) of the participants, and only about one third ($33.3\\%$ ) of those researchers had prior KBP experience.", "Table: Participation throughout each phase of the Challenge.Table: Distribution of participants by primary research area (rows) and whether they have past KBP research experience (columns).Table REF presents the proportion of participants by country of work or study.", "In total, 28 different countries were represented in the Challenge.", "The three countries with the most participants were the United States (32.8%), China (17.4%), and India (11.3%).", "Each of the other 25 counties that were represented had less than 5.1% of the participants.", "Table: The proportion of participants based on country of work or study.In Table REF , we present the aggregate data from our EDI survey.", "Men were overrepresented in OpenKBP (76.9%) compared to the science and engineering population (52.3%) and the general US population (49.2%).", "“Asian American/Asian” was the most common racial or ethnic identity (48.7%) in OpenKBP, much greater than the science and engineering population (13.0%) and the general US population (5.6%).", "On the other hand, individuals who identified as “White” were underrepresented in OpenKBP (21.5%) compared to the science and engineering (68.7%) and US (60.0%) populations.", "Individuals who identified as “African American/Black” (1.0%) and “Hispanic/Latinx” (4.1%) were also underrepresented relative to both baseline populations.", "A relatively large proportion (18.9%) of respondents chose not identify their racial or ethnic identity.", "Lastly, the proportion of OpenKBP participants who identified as having no disability (87.2%) was comparable to both the science and engineering (89.7%) and general US (87.3%) population.", "Fewer respondents identified with having a disability (2.1%) compared to both baselines (10.3% and 12.7%), and the remaining proportion of OpenKBP participants chose not to identify their disability status.", "Table: The equity, diversity, and inclusion data (rows) of three populations of people (columns).", "In order, the columns correspond to the population people who participated in the Challenge (OpenKBP), are employed in the United States in science and engineering (S&E), and live in the United States (US).", "A dash (—) indicates that the data is unavailable." ], [ "Performance over validation phase", "In Figure REF , we plot the distribution of team scores against normalized submission count.", "The plots show that teams generally improved their model throughout the validation phase, however, most teams made the largest improvements early on.", "Overall, the average team improved their dose and DVH score by a factor of 2.7 and 5.7, respectively, over the course of the validation phase.", "Over all of the NSC bins, the best dose and DVH scores were achieved by two and four different teams, respectively.", "There were a total of seven lead changes throughout the validation phase.", "Figure: The distribution of the dose and DVH scores across all teams.", "The solid lines indicate the mean score and the shaded regions indicate the 95% confidence interval.", "A dash lined indicates the best score." ], [ "Final results in testing phase", "Figure REF shows the distribution of error differences between the winning team (i.e., Team 1) and the top 23 runners-up (Teams 2-24).", "Compared to each of the other teams, Team 1 achieved significantly lower dose error over all 100 patients in the testing set ($P<0.05$ ) and significantly lower DVH error over all 1783 DVH criteria ($P<0.05$ ).", "Additionally, when compared to any other team, Team 1 achieved a lower dose and DVH error over at least 75% and 52% of patients and criteria, respectively.", "Figure: The distribution of dose and DVH error differences between the winning team (Team 1) and the top 23 runners-up, ranked by dose score.", "The boxes indicate median and IQR, and a circle indicates the mean.", "Whiskers extend to the minimum of 1.5 times the IQR and the most extreme outlier.Table REF summarizes the relative performance of each team under the MAE-based and MSE-based scores.", "The winner and first runners-up according to MAE-based score would have finished in the same place under the MSE-based score.", "The average absolute rank difference between the two scoring approaches was one.", "The maximum rank difference was four and five for dose score and DVH score, respectively.", "The Spearman's rank-order correlation coefficient for the MAE-based and MSE-based score ranks was 0.983 ($P < 0.001$ ) and 0.981 ($P < 0.001$ ) for the dose and DVH score, respectively.", "Thus, the results of the competition would have been nearly identical had the MSE-based score been used.", "Table: The score and rank that each team achieved in the testing phase according to an MAE-based (i.e., the score used in the Challenge) and MSE-based (i.e., an alternative score) score.", "A positive difference in rank implies that a team performed better on the MSE-based score than on the MAE-based score." ], [ "Common modeling decisions", "According to the model survey, every team in the testing phase trained neural networks to predict dose distributions.", "The majority of those models had architectures based on U-Net,[36] V-Net,[37] and Pix2Pix[28] models.", "All models were built using either a TensorFlow (Google AI, US) or PyTorch (Facebook AI Research, US) framework, but many teams also reported using higher-level libraries like fast.ai[38] to simplify model development.", "To train and develop the models quickly, teams generally used a GPU (e.g., NVIDIA 1080Ti, NVIDIA Titan V); seven teams also reported that they used Google Colab.", "Many teams used generalizable techniques to get better model performance.", "For example, 22 of the 28 teams used some form of data augmentation in their training process, and 15 teams combined two or more augmentation methods.", "Common forms of data augmentation were rotations, flips, crops, and translations.", "Most teams also reported that they normalized dose and CT Hounsfield units.", "Additionally, most teams used a standard loss function, e.g., MAE, MSE, GAN loss.", "There were also some teams that developed radiation therapy specific loss functions (e.g., functions that prioritized regions-of-interest more than the unclassified tissue).", "Lastly, ensemble methods were used by several of the top teams.", "Those methods used multiple neural networks to predict candidate dose distributions that were combined by taking the average prediction." ], [ "Discussion", "There is widespread research interest in knowledge-based planning (KBP) dose prediction methods.", "However, the lack of standardized metrics and datasets make it difficult to measure progress in the field.", "In this paper, we present the first set of standardized metrics and the first open-access dataset for KBP research as part of the OpenKBP Grand Challenge, the first competition for KBP research.", "The Challenge democratizes KBP research by enabling researchers without access to clinical radiation therapy plans to develop state-of-the-art dose prediction methods.", "This spurred the development of 28 unique models and will serve as an important benchmark as field of KBP continues to grow.", "Our open-access dataset contains real patient images that were contoured by clinicians at twelve institutions with different planning protocols.", "There are two major differences in protocol that introduce some variance in how PTVs were drawn.", "First, the raw public clinical data included plans with multiple radiation therapy modalities.", "For example, some of the institutions delivered hybrid-IMRT/3DCRT plans, and those plans had no PTV margins on the lower neck target volumes.", "Second, some of the raw public clinical data is from multiple trials with unconventional contouring in the extent of the target volumes.", "For example, we observed some anisotropic PTV margins that were clipped to omit the OARs.", "These variations are non-existent in the raw private clinical dataset, which contains plans from a single institution where all planning and contouring was done according to a standard process, that was used to create the dose distributions for the competition.", "This variation may have been a factor in the public synthetic dose being non-inferior to the private clinical dose on 19 of 23 dose-volume criteria.", "We proposed two new metrics that quantify the general performance (i.e., dose score) and the clinical performance (i.e., DVH score) of dose prediction methods.", "These two metrics may help measure progress in KBP research, and they will complement other metrics that are typically used in the literature to quantify strengths and weaknesses of a model.", "Other metrics are still important because our scoring metrics are unable to quantify every facet of radiation therapy dose quality.", "For example, the DVH criteria evaluated for the DVH score have varying degrees of clinical importance (e.g., ${\\text{mandible}}$ is much more important than ${\\text{mandible}}$ ).", "We chose to weigh all errors equally because quantifying relative clinical importance is non-trivial and largely dependent on the institution.", "Additionally, since the scores are unweighted it is straightforward to use the scores for all other sites that have OARs and targets (e.g., prostate).", "We aimed to make OpenKBP as accessible as possible in order to build a large and inclusive community.", "By building a large and inclusive community we can ensure that underrepresented populations can contribute to KBP research, which should both accelerate innovation[39] and improve the quality of healthcare.", "[40] As part of this Challenge, we released all competition data in a non-proprietary format (comma-separated value) and a well-documented code repository that helped participants use the data easily and efficiently in Python without costly commercial software.", "The code repository also had instructions to give all participants access to high-quality computational resources at zero cost (i.e., Google Colab).", "In an effort to keep the data manageable for all participants, we also opted to use relatively large voxels (e.g., $3\\text{mm}\\times 3\\text{mm}\\times 2\\text{mm}$ voxels), which ensured that the dose prediction problem was tractable for anyone using Google Colab.", "We conjecture that this manageable data size also helped the teams iterate and improve their models, which is reflected by the number of submissions made by teams in the validation phase (40 submissions on average).", "A limitation of this work is that is uses synthetic dose distributions to augment the real clinical data.", "Those dose distributions were generated by a published KBP pipeline[9] and filtered via Algorithm REF , however, they underwent less scrutiny than clinical plans.", "Extensions of this work should ensure that the top performing models on this dataset also perform well with clinical dose distributions.", "A second limitation is that we can only report commonalities between the top models, which are correlated attributes rather than causal attributes.", "Future work should do ablation testing to isolate exactly what attributes contribute to a good dose prediction model.", "Lastly, all dose predictions were evaluated and ranked based on two scores.", "These scores do not capture all of the strengths and weaknesses of the models submitted to the Challenge." ], [ "Conclusion", "OpenKBP democratizes knowledge-based planning research by making it accessible to everyone.", "It is also the first platform that researchers can use to compare their KBP dose prediction methods in a standardized way.", "The Challenge helps validate our platform and provides a much needed benchmark for the field.", "This new platform should help accelerate the progress in the field of KBP research, much like how ImageNet helped accelerate the progress in the field of computer vision." ], [ "Acknowledgments", "OpenKBP was supported by the American Association of Physicists in Medicine and its Working Group on Grand Challenges.", "The authors also gratefully acknowledge everybody who participated in the Challenge." ], [ "Data Format", "The data for OpenKBP is structured to facilitate the development and validation of dose prediction models.", "In this section, we describe how the data is stored and formatted." ], [ "Summary of data", "The data for each patient is provided as comma-separated values (CSV) files, which are separated into directories with the corresponding patient number.", "The files for each patient include: dose.csv the full 3D dose distribution that was used to treat the patient (in units of Gy).", "ct.csv grey-scale images of the patient prior to treatment (in Hounsfield units).", "There is a mix of 12-bit and 16-bit formats, and we recommend clipping the CT values to be between 0 and 4095 (inclusive) to convert them all to 12-bit number formats, which is the more common convention.", "voxels.csv The dimensions of the patient voxels (in units of mm).", "possible_dose_mask.csv a mask of voxels that can receive dose (i.e., the dose will always be zero where this mask is zero).", "Structure masks a mask that labels any voxel that is contained in the respective structure.", "The tensor for each structure stored as a CSV file under its respective structure name.", "Only structures that were contoured in the patient have CSV files.", "Brainstem.csv mask of brainstem voxels.", "SpinalCord.csv mask of spinal cord voxels.", "RightParotid.csv mask of right parotid voxels.", "LeftParotid.csv mask of left parotid voxels.", "Esophagus.csv mask of esophagus voxels.", "Larynx.csv mask of larynx voxels.", "Mandible.csv mask of mandible voxels.", "PTV56.csv mask of PTV56 voxels.", "PTV63.csv mask of PTV63 voxels.", "PTV70.csv mask of PTV70 voxels." ], [ "Data format", "Other than the file voxels.csv, which contains a list of only three numbers, all of the CSV data in OpenKBP is saved as sparse tensors (i.e., only non-zero values are stored).", "The advantage of this sparse format, compared to dense tensors (i.e., all values are stored), is that the data size is smaller and thus loads into memory faster, which leads to faster model training.", "The disadvantage, is that working with sparse tensors is less intuitive than working with dense tensors.", "In general, we recommend converting the data into dense tensors once it is loaded.", "All of the sparse tensors are stored in CSV files with two columns.", "The first column contains a list of indices.", "The second column contains either a list of values for the corresponding indices, or it contains no values if the tensor is a mask (i.e., where all corresponding values are 1).", "All indices are stored as single numbers that unravel into a 3D (i.e., x-y-z) coordinate system via C-contiguous ordering.", "We provide Python code in our repository to load the data as dense tensors." ], [ "Surveys", "In this section, we present the two mandatory surveys that we released during the Challenge.", "In each survey, respondents answered questions either by writing free-text or by selecting option(s) from a list.", "Mandatory questions are marked with an asterisk ()." ], [ "Registration", "All participants completed the following two part survey to register for OpenKBP." ], [ "Part 1: Professional information", "Please complete this form to be given access to the OpenKBP competition.", "First Name Short-answer text Last Name* Short-answer text E-mail (must be the same address used for your CodaLab account)* Short-answer text Institution name without acronyms (University, Hospital, Company, etc.", ")* Short-answer text Department (Computer Science, Medical Biophysics, Radiation Oncology, Machine Learning, Industrial Engineering, etc.", ")* Short-answer text Primary research area* Medical Physics Machine Learning Optimization Other... Have you done research in knowledge-based planning in the past?", "* Yes No Position* Student Professor Post doctoral fellow Medical physicist Radiation oncologist Industry research Other..." ], [ "Part 2: Equity, diversity, and inclusion", "To help us learn how to support the diversification of researchers in the OpenKBP Grand Challenge, we ask that all applicants complete an equity survey at the time of their registration.", "Equity is one of our competition’s goals.", "We seek to remove barriers to participation for all people including women, LGBTQ individuals, persons with disabilities, Indigenous People and racialized persons and persons of colour.", "Your participation is voluntary and your responses are confidential.", "We hope you will choose to answer these questions to help us bring you an even better competition next time.", "The information we receive from your responses will be used to better understand who has access to the competition, to identify barriers that may exist and areas to develop and/or improve in our rules and procedure to achieve more diversity and equity in the application process.", "All responses will be kept strictly confidential and will be reported only in aggregate so that you cannot be personally identified by your characteristics.", "Do you self-identify as (choose all that apply): Man Women Transgender Prefer not to say Other...", "Please indicate the racial or ethnic groups with which you identify (check all that apply): African American/Black Asian American/Asian Hispanic/Latinx Middle Eastern/North African Native American/Indigenous Native Hawaiian/Other Pacific Islander White Prefer not to say Other... Do you identify as a person with a disability?", "This may mean that either you: (i) have a long-term or recurring condition or health problem which limits the kind or amount of work you can do in the workplace; OR (ii) feel that you may be perceived as limited in the kind or amount of work which you can do because of a physical, mental, sensory, psychiatric, or learning impairment.", "Yes, I identify as a person with a disability No, I do not identify as a person with a disability Prefer not to say" ], [ "Model summary", "Every team that competed in the testing phase of the Challenge also completed the following one part survey to summarize their model." ], [ "Model survey", "Please describe your final dose prediction model using this survey.", "We will consider your submission complete only if this survey is submitted.", "Any submission made on CodaLab that is not associated with a survey response will be considered void, and it will not be ranked in the final leaderboard.", "We may also reach out to you for more information.", "This survey includes 5 long answer questions, and we expect the cumulative word count of your responses to be about 350 words.", "We provide an estimated word count for your response to each question.", "These estimates are only a guide and you may provide more detail where you see fit.", "Please reach out if you have any questions or need clarification.", "For teams, only one team member should submit this form.", "Username on CodaLab* Short-answer text Team Name on CodaLab (enter N/A if you have no team)* Short-answer text Broadly speaking, how would you describe your model?", "* Linear regression Random forest Neural network Gradient boosted trees Support vector machines Other...", "Briefly describe your approach.", "($\\mathbf {\\sim 150}$ words)* Long-answer text What would you say is the biggest contributing factor(s) to your models efficacy?", "($\\mathbf {\\sim 100}$ words)* Long-answer text Did you use transfer learning?", "* Yes No Please describe any data augmentation methods that you used (e.g., rotations, private clinical dataset)?", "($\\mathbf {\\sim 50}$ words)* Long-answer text Briefly describe your loss function.", "Did you use radiation therapy specific metrics in your loss function (e.g., max dose to PTV)?", "($\\mathbf {\\sim 50}$ words)* Long-answer text Briefly describe the hardware (e.g., GPU model, CPU model) or cloud resources (e.g., Google Colab) that you used.", "($\\mathbf {\\sim 25}$ words)* Long-answer text Please leave any other comments about your process here.", "Long-answer text Provide a link to the code repository that will recreate your model.", "We will include links to all the provided repositories from our existing OpenKBP Github to enable new users to build on a library of existing models.", "You may also provide a repository link at a later date, but it is not required.", "Short-answer text" ] ]	2011.14076
[ [ "Thermodynamic Consistent Neural Networks for Learning Material\n Interfacial Mechanics" ], [ "Abstract For multilayer materials in thin substrate systems, interfacial failure is one of the most challenges.", "The traction-separation relations (TSR) quantitatively describe the mechanical behavior of a material interface undergoing openings, which is critical to understand and predict interfacial failures under complex loadings.", "However, existing theoretical models have limitations on enough complexity and flexibility to well learn the real-world TSR from experimental observations.", "A neural network can fit well along with the loading paths but often fails to obey the laws of physics, due to a lack of experimental data and understanding of the hidden physical mechanism.", "In this paper, we propose a thermodynamic consistent neural network (TCNN) approach to build a data-driven model of the TSR with sparse experimental data.", "The TCNN leverages recent advances in physics-informed neural networks (PINN) that encode prior physical information into the loss function and efficiently train the neural networks using automatic differentiation.", "We investigate three thermodynamic consistent principles, i.e., positive energy dissipation, steepest energy dissipation gradient, and energy conservative loading path.", "All of them are mathematically formulated and embedded into a neural network model with a novel defined loss function.", "A real-world experiment demonstrates the superior performance of TCNN, and we find that TCNN provides an accurate prediction of the whole TSR surface and significantly reduces the violated prediction against the laws of physics." ], [ "Introduction", "Traction-separation relations play a key role in understanding the mechanical behavior of a material interface undergoing openings and predicting interfacial failures under complex loading conditions [8], [14], [16], [15].", "However, the entire process is tedious, complex, and unreliable due to three main problems: (1) there has not been a universal and robust approach that can extract the TSR from the cohesive zone using far-field measurements; (2) there has not been a unified approach to model TSR directly from the experimental data; and (3) the experimental design is typically pre-designed, which cannot provide efficient coverage for the testing space [8].", "Therefore, it is urgent to develop a data-driven approach to model interfacial TSR, which allows us to effectively learn from sparse experimental data and comply with thermodynamic consistency [13].", "Deep learning has achieved remarkable success in diverse applications [3], [5] including computer vision and natural language processing, but its use in real-world engineering fields with small data is limited.", "For the TSR problem, a neural network can fit well along the loading paths but often fails to obey physical laws, due to a lack of experimental data and understanding of the inherent mechanism.", "To this end, we seek a physics-informed approach that enables us to encode physical laws as prior information into deep learning models, which can mitigate the issue caused by a lack of data [21], [20].", "Recent advances in physics-informed neural networks (PINN) [9], [10], [11] that have been used in a wide range of engineering applications including fluid mechanics [7], [12], bio-medical engineering [4], nanophotonics [1], [18] and computational materials science[19], [6], may bring an opportunity to address this challenge.", "PINN aims at solving supervised learning tasks while respecting any given law of physics described by general nonlinear PDE.", "The trained neural networks represent a class of data-efficient approximators that naturally encode underlying physical laws as prior information.", "This important feature of PINN enables solving inverse problems with limited data observations [17].", "However, it is a non-trivial task to simply use the PINN for the TSR problem because there are three challenges: (1) the law of physics hidden in TSR is complicated and can not be explicitly described by PDE governing equations; (2) the thermodynamic consistency in TSR is more abstract and difficult to be extracted than typical PDE-based governing equations, such as boundary conditions in PINN; and (3) the prior information from thermodynamic consistency needs to be formulated and implemented into neural network models in a rigorous and data-driven approach.", "To address these challenges, our core contributions in this paper can be summarized as follows: [leftmargin=10pt] We propose a novel thermodynamic consistent neural network (TCNN) approach to model the material interface mechanics with sparse experimental data We extract three thermodynamic consistency principles, i.e., positive energy dissipation, steepest energy dissipation gradient, and energy conservative loading path, from complex TSR problems We formulate the physical knowledge mathematically, encode the prior information as physics constraints that are then embedded into a neural network model with a new loss function" ], [ "Data-driven modeling of TSR using TCNN", "Interfacial traction separation relations (TSR) For a 2-layer structure undergoing interfacial fracture process as shown in Figure REF (a), a cohesive layer in between these two evolved substrates provides tractions.", "This cohesive layer is assumed to be homogeneous, which enables us to consider it as an assembly of identical “springs” connecting the two layers.", "The interfacial fracture process is then reproduced with the elongation and failure process of each “spring” along the interface.", "For a stretched spring connecting the upper and lower layer, the normal and tangential tractions ($\\sigma _n, \\sigma _t)$ change correspondingly with the normal and tangential separations ($\\delta _n, \\delta _t$ ), which are defined by the change of the relative distances between the end points of the “spring”.", "Depending on the loading and boundary conditions, the ratio between the tangential and normal separations (i.e., the mode-mix) varies.", "To quantitatively describe this relation, the vectorial separation is defined as the Euclidean norm of normal and tangential separation components, shown as $\|\\delta \|$ in Figure REF .", "The mode mix is represented by the phase angle, which is defined as the arctangent of the ratio between normal and tangential separations, shown as $\\phi $ in Figure REF .", "Similarly, the vectorial traction is defined as $\|\\sigma \|$ in Figure REF .", "With these definitions, the relation between tractions ($\\sigma _n, \\sigma _t$ ) and separations ($\\delta _n, \\delta _t$ ) is defined as traction separation relation (TSR) which constitutes the interfacial mechanical property.", "Figure: Illustration of TSR problem: (a) Schematics of a double cantilever beam under end loaded split experiment, (b) traction and separations for one spring and (c) a typical TSR curve.Thermodynamic consistent principles We extract three principles and discuss them as follows: [leftmargin=10pt] $\\textup {TC1}$ : Positive energy dissipation.", "Damage mechanics at the interface is the foundation for cohesive modeling and the cohesive zone can be represented as the partial fracture at the interface.", "The total interfacial fracture resistant energy can then be described as a damaging manner $\\Gamma = (1-d_n)\\Gamma _n + (1-d_t) \\Gamma _t$ where $\\Gamma $ is the total interfacial toughness, $\\Gamma _n$ and $\\Gamma _t$ are the normal and shear toughness for the intact interface, $d_n$ and $d_t$ are the damage parameters ranging from $[0,1]$ , and can be defined by $d_n(\\delta _n) = 1 - {J_n(\\delta _n)}/{\\Gamma _n}, \\quad d_t(\\delta _t) = 1 - {J_t(\\delta _t)}/{\\Gamma _t}$ where $\\Gamma _n = \\max (J_n)$ and $\\Gamma _t = \\max (J_t)$ are normal and tangential toughness respectively.", "$J_n$ and $J_t$ are normal and tangential $J$ -integrals which are defined as $J_n(\\delta _n) = \\int _{0}^{\\delta _n} \\sigma _n(x)dx, \\quad J_t(\\delta _t) = \\int _{0}^{\\delta _t} \\sigma _t(x)dx $ For a monotonic loading during the experiment, the energy dissipation for the interfacial delamination should be positive $D = \\Gamma _n\\dot{d}_n + \\Gamma _t\\dot{d}_t \\ge 0.$ This implies that the rate of the damage parameters should be positive for the monotonic loading, $\\frac{\\partial {d_n}}{\\partial {\\delta _n}} \\ge 0, \\quad \\frac{\\partial {d_t}}{\\partial {\\delta _t}} \\ge 0$ $\\textup {TC2}$ : Steepest energy dissipation gradient.", "In addition to the energy dissipation, the dissipation rate should reach the local maximum when the interfacial separation reaches local maximum, $\\left\| \\frac{\\partial d}{\\partial \\delta _{\\max }}\\right\|_{\\phi = \\phi _0} = \\max \\left\|\\frac{\\partial d}{\\partial \\mathbf {\\delta }} \\right\|$ where $\\delta _{\\max }$ is the vecorial direction towards the largest increment of interfacial separation given a mode-mix phase angle $\\phi = \\arctan (\\delta _t/\\delta _n)$ .", "This ensures the fastest energy dissipation follows the fixed loading path.", "This can be illustrated by damage parameters versus the total separation and the phase angle in Figure REF .", "The steepest descend will be guaranteed when the projected total separation for each monotonic loading step is the largest, which is given by $\\left(\\frac{\\partial d_n}{\\partial \\delta } \\right)_{\\phi =\\phi _0, \\delta = \|\\delta \|} < \\left(\\frac{\\partial d_n}{\\partial \\delta } \\right)_{\\phi \\ne \\phi _0, \\delta \\ne \|\\delta \|}, \\quad \\left(\\frac{\\partial d_t}{\\partial \\delta } \\right)_{\\phi =\\phi _0, \\delta = \|\\delta \|} < \\left(\\frac{\\partial d_t}{\\partial \\delta } \\right)_{\\phi \\ne \\phi _0, \\delta \\ne \|\\delta \|}$ $\\textup {TC3}$ : Energy conservative loading path.", "The last physics constrain is the fulfillment of energy conservation law, which is the energy dissipation along the vectorial path should equal to the sum of energy dissipation in both normal and tangential direction, as shown in Figure REF .", "This requires the ratio between the normal tangential stress should be equal to those of separations, i.e., $\\mathbf {J}_{\\textup {total}} (\\sigma _n, \\sigma _t, \\delta _n, \\delta _t) = \\mathbf {J}_n(\\sigma _n, \\delta _n) + \\mathbf {J}_{t} (\\sigma _t, \\delta _t)$ Thus we have $\\frac{d \\sigma _t}{d \\sigma _n} = \\frac{d \\delta _t}{d \\delta _n} = \\tan \\phi $ In review of all three thermodynamics consistent conditions, TC1 and TC2 are based on the irreversible nature of the energy dissipation as stated in the second law of thermodynamics, thus are the stronger constrains.", "TC3 condition is a relatively weaker condition than TC1 and TC2, because it is based on the assumption that the delamination occurs at the interface with no frictional energy dissipation, which may not be true based on some of the experiment observations [15].", "Figure: Three thermodynamic consistent principles for modeling TSRThermodynamic consistent neural network (TCNN) We propose a TCNN framework by embedding the thermodynamic consistency principles as physical constraints into deep neural networks.", "Figure REF presents the schematic of a TCNN for solving TSR prediction with three thermodynamic consistent constraints.", "The inputs of the DNN model are separation norm $\|\\delta \|$ and phase angle $\\phi $ as defined in Figure REF , and the outputs are normal and tangential $J$ -integrals ($J_n, J_t$ ), as defined in Eq.", "(REF ).", "The goal is to predict the $J$ -integral surfaces using trained DNN model with collected experimental data.", "[leftmargin=10pt] $\\mathcal {L}_{\\textup {MSE}}$ : a regular expected MSE loss to measure the mismatch with the given data observations $Y^=(J_n^, J_t^)$ , which is defined as $\\mathcal {L}_{\\textup {MSE}} = \\frac{1}{z} \\sum _{i=1}^z \\left(\\left\\Vert Y_i - Y_i^{}\\right\\Vert _2^2 \\right)$ where $i=1,..,z$ is the number of data.", "$\\mathcal {L}_{\\textup {TC}_1}$ : TC1 is imposed on constraint paths separately on normal and tangential output $J$ -integral surfaces.", "We use max function to compare the values so that only the positive part is added to the loss, $\\mathcal {L}_{\\textup {TC}_1} = \\frac{1}{2m} \\sum _{j=1}^m \\left( \\max _{i \\in [1, z-1]}\\left\\lbrace \\frac{d_n^{(i+1,j)} - d_n^{(i,j)}}{\\delta _n^{(i+1,j)} - \\delta _n^{(i,j)}}, \\ 0\\right\\rbrace + \\max _{i \\in [1, z-1]}\\left\\lbrace \\frac{d_t^{(i+1,j)} - d_t^{(i,j)}}{\\delta _t^{(i+1,j)} - \\delta _t^{(i,j)}}, \\ 0\\right\\rbrace \\right)$ $\\mathcal {L}_{\\textup {TC}_2}$ : TC2 constraints the gradient vector direction along the fixed phase angle loading paths $\\mathcal {L}_{\\textup {TC}_2} = \\frac{1}{2mz} \\left(\\sum _{i=1}^{z-1} \\sum _{j=1}^{m-1} \\max \\left\\lbrace \\frac{\\partial d_n^{(i,j)}}{\\partial \|\\delta \|} - \\frac{\\partial d_n^{(i,j)}}{\\partial \\phi } , \\ 0 \\right\\rbrace + \\max \\left\\lbrace \\frac{\\partial d_t^{(i,j)}}{\\partial \|\\delta \|} - \\frac{\\partial d_t^{(i,j)}}{\\partial \\phi } , \\ 0 \\right\\rbrace \\right)$ $\\mathcal {L}_{\\textup {TC}_3}$ : Different from TC1 and TC2 constraints, TC3 condition is an equality constraint, $\\mathcal {L}_{\\textup {TC}_3} = \\frac{1}{mz}\\sum _{j=1}^m \\sum _{i=1}^z \\left\| \\frac{\\sigma _t^{(i,j)}}{\\sigma _n^{(i,j)}} -\\tan (\\phi ^{(j)}) \\right\|$ $\\mathcal {L}$ : the total loss is defined by the weighted summation of each loss functions $\\mathcal {L} = \\lambda _0 \\mathcal {L}_{\\textup {MSE}} + \\lambda _1 \\mathcal {L}_{\\textup {TC}_1} + \\lambda _2 \\mathcal {L}_{\\textup {TC}_2} + \\lambda _3 \\mathcal {L}_{\\textup {TC}_3}$ where $\\lambda _i, i=0,1,2,3$ refer to the weights of the loss functions and satisfies $\\sum _{i=0}^{3} \\lambda _i=1$ .", "Figure: Schematic of a thermodynamic consistent neural network (TCNN) for solving TSR problems.", "The total loss consists of the loss from data observations and thermodynamic consistency." ], [ "Experiments and results", "We collected data from real-world experimental measurements where the normal and shear TSR for a silicon/epoxy interface were determined over normal mode-mixes range from -53° to 87.5° using non-symmetric end-loaded split (ELS) and end-notched flexure (ENF) specimens [15].", "A total 10 loading paths with 236 TSR data points are feed to the TCNN model in which a neural network model is constructed with two layers and there are 60 neurons for each layer, and we use tanh as the activation function and Adam as the optimizer.", "The hyperparameters are tuned by trust-region Bayesian optimization [2].", "Figure REF and Figure REF shows the performance of TCNN on TSR surface using sparse experimental data.", "Note that the learned TSR model can capture the complex surface even only limited data is given and the thermodynamic violations of interfacial fracture toughness is around 5% that is much lower than the cases without encoding thermodynamic constraints (typically 20% - 40%).", "Figure: TCNN performance on TSR data: predicted TSR surface using experimental dataFigure: Thermodynamics violations of predicted interfacial fracture toughness.", "Note that the red color represents the thermodynamic violations, blue color means no violations." ], [ "Conclusion", "In this paper, we propose a thermodynamic consistent neural network method for solving the traction separation relation problems in material interactial mechanics.", "Three thermodynamic consistency principles are formulated as physics constraints and mathematically imposed to a deep neural network model with a novel loss function.", "By introducing these prior physical knowledge into the deep learning model, we demonstrate that the TCNN enables to well predict the entire traction surface, and significantly reduce the violated prediction against the laws of thermodynamics." ] ]	2011.14172
[ [ "Close Latency--Security Trade-off for the Nakamoto Consensus" ], [ "Abstract Bitcoin is a peer-to-peer electronic cash system invented by Nakamoto in 2008.", "While it has attracted much research interest, its exact latency and security properties remain open.", "Existing analyses provide security and latency (or confirmation time) guarantees that are too loose for practical use.", "In fact the best known upper bounds are several orders of magnitude larger than a lower bound due to a well-known private-mining attack.", "This paper describes a continuous-time model for blockchains and develops a rigorous analysis that yields close upper and lower bounds for the latency--security trade-off.", "For example, when the adversary controls 10\\% of the total mining power and the block propagation delays are within 10 seconds, a Bitcoin block is secured with less than $10^{-3}$ error probability if it is confirmed after four hours, or with less than $10^{-9}$ error probability if confirmed after ten hours.", "These confirmation times are about two hours away from their corresponding lower bounds.", "To establish such close bounds, the blockchain security question is reduced to a race between the Poisson adversarial mining process and a renewal process formed by a certain species of honest blocks.", "The moment generation functions of relevant renewal times are derived in closed form.", "The general formulas from the analysis are then applied to study the latency--security trade-off of several well-known proof-of-work longest-chain cryptocurrencies.", "Guidance is also provided on how to set parameters for different purposes." ], [ "Introduction", "Bitcoin was invented by Nakamoto [22] in 2008 as a peer-to-peer electronic cash system.", "It builds a distributed ledger commonly referred to as a blockchain.", "The Bitcoin blockchain is a sequence of transaction-recording blocks which begins with a genesis block, and chains every subsequent block to a parent block using a cryptographic hash.", "Producing a new block requires proof-of-work (PoW) mining: a nonce must be included such that the block's hash value satisfies a difficulty requirement.", "An honest miner follows the longest-chain rule, i.e., it always tries to mine a block at the maximum height.", "Miners form a peer-to-peer network to inform each other of newly mined or received blocks.", "A block (and the transactions within) cannot be immediately confirmed upon its inclusion in the blockchain due to a phenomenon called forking.", "Different blocks may be mined and published at around the same time, so different honest miners may extend different blockchains.", "Forking may also occur as a result of adversarial miners deviating from the longest-chain rule.", "Forking presents opportunities for double spending, which may happen if a transaction buried in a longest fork at one time is not included in another fork that later overtakes the former fork.", "In fact, unlike classic Byzantine fault tolerant protocols, the Bitcoin protocol only admits probabilistic guarantees.", "The latency (or confirmation time) of a block in Nakamoto-style consensus protocols (Nakamoto consensus for short) depends on the desired security level.", "The goal of this work is to obtain close estimates of the fundamental latency–security trade-off for the Nakamoto consensus.", "While the consensus protocol is simple and elegant, a rigorous analysis of its security is very challenging.", "The original Bitcoin white paper [22] only analyzes a single specific attack, called the private attack, which is to mine an adversarial fork in private.", "Nakamoto showed that the probability the adversary's private fork overtakes the main blockchain vanishes exponentially with the latency (see also [27]).", "It is not until six years later that Garay et al.", "[10] provided the first proof that the Nakamoto consensus is secure against all possible attacks.", "One major limitation of [10] is that their round-based lock-step synchrony model essentially abstracts away block propagation delays.", "Several follow-up works [24], [25], [16], [1] have extended the analysis to the $\\Delta $ -synchrony model in which the rounds in which different honest miners observe the same block may differ by up to a known upper bound $\\Delta $ .", "So far, most existing analyses against all possible attacks [10], [11], [24], [25], [16], [1] (including a few concurrent and follow-up works , [23], [6], [12]) focus on establishing asymptotic trends using the big $O(\\cdot )$ or big $\\Omega (\\cdot )$ notation.", "These results are not concrete bounds for any given security level because the latency implied could be hours, days, or even years depending on the unknown constants.", "The constants in some of these asymptotic results have been worked out in [20] to reveal that the latency upper bounds are several orders of magnitude higher than the best known lower bounds.", "Thus, despite their theoretical value, existing analyses of the Nakamoto consensus provide little guidance on the actual confirmation time, security guarantees, or parameter selection in practice.", "In this paper, we explicitly and closely characterize the trade-off between latency and security for Nakamoto-style PoW consensus protocols.", "For Bitcoin, the latency results we prove are typically within a few hours to lower bounds from the private attack.", "The gap remains nearly constant at different security levels, and is hence proportionally insignificant for high security levels.", "For example, with a 10% adversarial mining power, a mining rate of one block every 10 minutes, and a maximum block propagation delay of 10 seconds, a block in the Nakamoto consensus is secured with $10^{-3}$ error probability if it is confirmed four hours after it is mined, or with $10^{-9}$ error probability if it is confirmed ten hours after it is mined.", "As a reference, one must wait for at least two hours or six hours and forty minutes before confirming for $10^{-3}$ and$10^{-9}$ security levels, respectively.", "In contrast, the best prior bounds under essentially the same setting put the latency guarantees at thousands of hours or more [1], [20], [19].", "Since Bitcoin's rise to fame, numerous altcoins and Bitcoin hard forks have adopted the Nakamoto consensus protocol with very different parameters.", "Those parameters are mostly determined in an ad-hoc or empirical manner.", "This paper provides theoretical and quantitative tools to reason about the effects and trade-offs of these parameters on different metrics in the Nakamoto consensus including confirmation time, throughput, and fault tolerance.", "We use these new tools to analyze and compare various altcoins and offer new insights and recommendations for setting parameters.", "Some new techniques developed in this paper may be of independent interests.", "Assuming all block propagation delays are under $\\Delta $ units of time, we show the arrivals of several species of honest blocks form (ordinary) renewal processes [2].", "That is, the inter-arrival times of such a process are independent and identically distributed (i.i.d.).", "We show that the adversary must match the so-called double-laggers in order to succeed in any attack.", "We derive the moment generating function (MGF) of their inter-arrival times in closed form.", "This allows us to calculate quite accurately the probability that more double-laggers are mined than adversarial blocks over any confirmation time, which leads to a close bound of the fundamental latency–security trade-off.", "Our main contributions include: 1) The blockchain security question is reduced to a race between a renewal process and a Poisson process.", "2) We provide an explicit formula for the security guarantee as a function of the latency.", "This is equivalent to an upper bound on the latency that guarantees any desired security level.", "3) By means of numerical analysis, the latency upper bound is shown to be close to a lower bound due to the private attack.", "4) We quantify how the block propagation delay bound, mining rates, and other parameters affect the latency–security trade-off.", "5) We compare the performance and security of several prominent PoW longest-chain cryptocurrencies.", "We note that several existing proofs in the literature are flawed (see Section REF and the end of Section REF ).", "In this paper, we carefully circumvent some common mistakes and develop a fully rigorous analysis.", "The remainder of this paper is organized as follows.", "Section reviews the Nakamoto consensus and describes a model for it.", "Section presents the main theorems.", "Section presents a simple analysis of the zero-delay case.", "Section develops a technical proof for the latency–security guarantee with delay.", "Section calculates a bound due to the private attack.", "Section is devoted to numerical results and discussions.", "Section concludes the paper.", "The Nakamoto consensus centers around the PoW mechanism and the “longest-chain-win” rule.", "The gist of the protocol can be described succinctly: At any point in time, every honest miner attempts to mine a new block that extends the longest blockchain in that miner's view; once a new block is mined or received, the honest miner publishes or forwards it through the peer-to-peer mining network.", "If an honest miner mines a block at time $t$ , the block must extend the miner's longest blockchain immediately before $t$ .", "The honest miner will also immediately publish the block through a peer-to-peer network.", "Under the $\\Delta $ -synchrony model, where $\\Delta $ is a known upper bound on block propagation delays, all other miners will receive the block by time $t+\\Delta $ .", "Note that $\\Delta $ upper bounds the end-to-end delay between every pair of miners regardless of the number of hops between them.", "In contrast, if an adversarial miner mines a block at time $t$ , the block may extend any blockchain mined by time $t$ and may be presented to individual honest miners at any time after $t$ ." ], [ "Formal Model for the Nakamoto Consensus", "In this subsection we present a very lean model for blockchain and mining processes which capture the essence of the Nakamoto consensus for our purposes.", "[Block and mining] A genesis block, also referred to as block 0, is mined at time 0.", "Subsequent blocks are referred to as block 1, block 2, and so on, in the order they are mined.", "Let $T_b$ denote the time when block $b$ is mined.A block in a practical blockchain system is a data structure that contains a unique identifier, a reference to its parent block, and some application-level data.", "The block number and mining time are tools in our analysis and are not necessarily included in the block.", "The probability that two blocks are mined at the same time is zero in the continuous-time model.", "Nevertheless, for mathematical rigor, we can break ties in some deterministic manner even if they occur (with no impact on the security–latency trade-off).", "[Blockchain and height] Every block has a unique parent block that is mined strictly before it.", "We use $b̑ \\in \\lbrace 0,1,...,b-1\\rbrace $ to denote block $b$ 's parent block number.", "The sequence $b_{0},b_{1}, \\dots , b_{n}$ defines a blockchain if $b_{0}=0$ and $=b_{i-1}$ for $i =1,\\dots ,n$ .", "It is also referred to as blockchain $b_n$ since $b_n$ uniquely identifies it.", "The height of both block $b_i$ and blockchain $b_i$ is said to be $i$ .", "Throughout this paper, “by time $t$ ” means “during $(0, t]$ ”.", "[A miner's view] A miner's view at time $t$ is a subset of all blocks mined by time $t$ .", "A miner's view can only increase over time.", "A block is in its own miner's view from the time it is mined.", "[A miner's longest blockchain] A blockchain is in a miner's view at time $t$ if all blocks of the blockchain are in the miner's view at time $t$ .", "A miner's longest blockchain at time $t$ is a blockchain with the maximum height in the miner's view at time $t$ .", "Ties are broken in an arbitrary manner.The Bitcoin protocol favors the earliest to enter the view.", "How ties are broken has essentially no impact on the security–latency trade-off.", "[Honest and adversarial miners] Each miner is either honest or adversarial.", "A block is said to be honest (resp.", "adversarial) if it is mined by an honest (resp.", "adversarial) miner.", "An honest block mined at time $t$ must extend its miner's longest blockchain immediately before $t$ .", "We assume all block propagation delays are upper bounded by $\\Delta $ units of time in the following sense.", "[Block propagation delay bound $\\Delta $ ] If a block is in any honest miner's view by time $t$ , then it is in all miners' views by time $t+\\Delta $ .", "The adversary may use an arbitrary strategy subject to (only) the preceding constraints.", "Specifically, an adversary can choose to extend any existing blockchain.", "Once an adversarial block is mined, its miner can determine when it enters each individual honest miner's view subject to the delay bound $\\Delta $ (Definition REF ).", "This treatment cannot be fully rigorous without a well-defined probability space.", "At first it appears to be intricate to fully described blockchains and an all-encompassing probability space.", "One option (adopted in [6]) is to define blockchains as branches of a random tree that depend on the adversary's strategies as well as the network topology and delays.", "Other authors include in their probability space the random hashing outcomes as well as adversarial strategies (e.g., [10]).", "For our purposes it is sufficient (and most convenient) to include no more than the mining times of the honest and adversarial blocks in the probability space.", "We show that barring a certain “bad event” in this probability space, blockchain consistency is guaranteed under all adversarial strategies and network schedules (under $\\Delta $ -synchrony).", "Thus, the adversary's strategies and the network schedules do not have to be included in the probability space.", "[Mining processes] Let $H_t$ (resp.", "$A_t$ ) denote the total number of honest (resp.", "adversarial) blocks mined during $(0, t]$ .", "We assume $(H_t, t\\ge 0)$ and ($A_t, t\\ge 0)$ to be independent homogeneous Poisson point processes with rate $\\alpha $ and $\\beta $ , respectively.", "In lieu of specifying the number of honest and adversarial miners, this model is only concerned with their respective aggregate (constant) mining rates.", "This is in the same spirit as the permissionless nature of the Nakamoto consensus.", "In fact the model and techniques apply to both centralized and decentralized mining.", "Table REF illustrates frequently used notations in this paper.", "Table: Some frequently used notations." ], [ "Main Results", "Throughout this section, the time unit is arbitrary and fixed.", "Let $\\alpha >0$ denote the total honest mining rate and let $\\beta >0$ denote the total adversarial mining rate, both in blocks per unit of time.", "[Achievable security latency function] A blockchain protocol is said to achieve the security latency function $\\overline{\\epsilon }(\\cdot )$ if, for all $s>0$ and $t>0$ , barring an event with probability no greater than $\\overline{\\epsilon }(t)$ , all blocks that are mined by time $s$ and included in some honest miner's longest blockchain at time $s+t$ are also included in all honest miners' longest blockchains at all later times.", "[Unachievable security level for given latency] We say a blockchain system cannot achieve the security level $\\underline{\\epsilon }(\\cdot )$ for a given latency $t>0$ , if for some $s>0$ , there exists an attack such that some block that is mined by time $s$ and is included in a longest blockchain in some honest view at time $s+t$ is excluded from a longest blockchain in some honest view at some point in time after $s+t$ with probability $\\underline{\\epsilon }(t)$ .", "The view adopted in Definitions and is that an attack on the interval $[s,s+t]$ is successful if any honest miner “commits” a block by time $s+t$ and then commits a different block at the same height at some time $r\\ge s+t$ .", "We allow the adversary to mount an attack tailored for $[s,s+t]$ from as early as time 0.", "However, the interval of interest $[s,s+t]$ is exogenous, i.e., the adversary is not allowed to adapt $s$ to their advantage.", "The Nakamoto consensus described in Section satisfies the following security–latency trade-offs.", "Suppose $\\beta <\\alpha $ and $\\Delta =0$ .", "For every $t>0$ , the Nakamoto consensus achieves the following security–latency function: 0(t) = (1+)2 ( - ( - )2 t ) .", "For every latency $t>0$ , the Nakamoto consensus cannot achieve the security level 0(t) = k=0p(k-1; t, t) ( )k ( 1 + k ( 1 - ) ) where $p(\\cdot ;\\cdot ,\\cdot )$ denotes the probability mass function (pmf) of the Skellam distribution [29].", "Theorems and have their counterparts in the case of non-zero propagation delays: [Achievable security latency function] Suppose $\\Delta >0$ and < e-2 .", "Let (v) = v - v2 v2 - v - v e(v-) + 2 e2(v-) and let $\\theta $ denote the smallest positive zero of the denominator in ().", "The Nakamoto consensus achieves the following security–latency function: (t) = v (0, ) c2(v) e- (v-(v)) t where $c(v)&= e^{\\eta (v)\\beta \\Delta }\\left(1-\\frac{\\beta }{\\alpha }e^{2\\alpha \\Delta }\\right)\\frac{\\eta (v)}{\\frac{1}{1+\\eta (\\eta (v)\\beta )}-\\frac{1}{1+\\eta (v)} } .$ As $c(v)$ and $\\eta (v)$ are infinitely differentiable functions, the optimization in () is numerically easy.", "While the variable $v$ is optimized in () for each latency $t$ of interest, by picking a maximum universal exponent we obtain a slightly weaker exponential security–latency function: '(t) = c2(u) e-(u-(u))t where $u$ is a maximizer of the exponent, i.e., $u-\\eta (u)\\beta \\ge v-\\eta (v)\\beta $ for all $v\\in (0,\\theta )$ .", "For many practical parameters and typical security levels of interest, $\\overline{\\epsilon }(t)$ in () is very close to an exponential function in $t$ .", "Hence () provides an excellent approximation.", "Theorems and guarantee for all honest miners that all blocks in their longest blockchains that are received $t$ units of time earlier (hence, mined at least that much earlier) will always remain in all honest miners' longest blockchains in the future, except for a probability that is (essentially) exponentially small in $t$ .", "Unlike existing asymptotic results, these results provide a concrete and close upper bound on the probability of consistency violation under for a given confirmation time.", "Equivalently, these results also upper bound the required confirmation time for every desired security level.", "[Unachievable security lower for given latency] Let () = (1-)(- - ) - e(1-) (+-) and q(n) = {ll (0) + '(0) if n=0 (n+1)(0)/(n+1)!", "if n=1,2,.... For every latency $t>0$ , the Nakamoto consensus cannot achieve the following security level: $\\underline{\\epsilon }(t)&=\\left(1-\\frac{\\beta }{\\alpha }\\right)e^{(\\alpha -\\beta )t}\\sum _{n=0}^\\infty \\sum _{{k=0\\\\k+n>0}}^\\infty q(n) \\left(\\frac{\\beta }{\\alpha }\\right)^k \\\\& \\qquad \\times F_1(k;\\alpha t) \\, \\overline{F}_2(t-(n+k)\\Delta ;n+k,\\alpha )$ where $F_1(\\cdot ;\\lambda )$ denote the cumulative distribution function (cdf) of a Poisson distribution with mean $\\lambda $ and $\\overline{F}_2(\\cdot ;n,\\alpha )$ denotes the complementary cdf of the Erlang distribution with shape parameter $n$ and rate parameter $\\alpha $ .", "Evidently, if the infinite sums in () and () are replaced by partial sums for numerical evaluation, the resulting (tighter) security level remains unachievable." ], [ "Remarks", "Theorems and assume the delay $\\Delta >0$ .", "The bounds therein still apply if we set $\\Delta =0$ , but are slightly looser than the bounds in Theorems and for the zero-delay case.", "It is important to include the time of interest $s$ in Definitions and .", "The “bad events” for security breach depend on $s$ as well as the latency $t$ .", "These well-defined events are concerned with block mining times, not how blocks form blockchains.To be rigorous, we do not make claims such as “the blockchain/protocol/system satisfies consistency or liveness properties with probability ...” because those properties themselves are not events in the probability space defined here.", "We note that a number of previous analyses on the Nakamoto consensus assume a finite lifespan of the protocol [10], [1], that is, a maximum round number is defined, at which round the protocol terminates.", "The probability of consistency depends on the maximum round number.", "In contrast, this paper does not assume a finite lifespan.", "Theorem states that, barring a small probability event, confirmed blocks remain permanently in all miners' longest blockchains into the arbitrary future.", "Even though we provide the same security guarantee for every blockchain after the confirmation latency $t$ , no one can simultaneously guarantee the same for all blocks that will ever be confirmed.", "This is a simple consequence of Murphy's Law: If an adversary keeps trying new episodes of attacks, with probability 1 a bad event will eventually occur to revert some confirmed honest blocks.", "For technical convenience, we regard a block in a miner's longest blockchain to be confirmed after a certain amount of time elapses since the block is mined or enters the miner's view.", "Nakamoto [22] originally proposed confirming a block after it is sufficiently deep in an honest miner's longest blockchain.", "We believe both confirmation rules are easy to use in practice.", "And the two confirmation rules imply each other in probability (see Appendix for further discussion).", "Figure: Bitcoin's latency–security trade-off with α+β=1/600\\alpha +\\beta =1/600 blocks per second and Δ=10\\Delta =10 seconds." ], [ "Numerical Examples", "The latency–security trade-off under several different sets of parameters is plotted in Figure REF .", "The mining rate is set to Bitcoin's one block per 600 seconds, or $\\alpha +\\beta =1/600$ blocks/second.", "The propagation delay bound is assumed to be $\\Delta =10$ seconds.", "The latency upper and lower bounds are computed using Theorems and , respectively.", "In Figure REF , all bounds appear to be exponential for all but very small latency and high error probabilities.", "This implies the exponential bound () is a good approximation of () in Theorem for the typical range of parameters of interest here.", "It is instructive to examine concrete data points in Figure REF : If the adversarial share of the total network mining rate is 10% ($\\alpha :\\beta =9:1$ ), then a confirmation time of four hours is sufficient to achieve $10^{-3}$ security level, and a ten-hour confirmation achieves $10^{-9}$ security level.", "These results are about two hours away from the corresponding lower bounds.", "Also, for every additional hour of latency, the security improves by a factor of approximately 20.", "If the adversarial share of the mining rate increases to 25% ($\\alpha :\\beta =3:1$ ), then 10 hours 40 minutes and 28 hours 45 minutes of confirmation times achieve $10^{-3}$ and $10^{-9}$ security levels, respectively, and the gap between the upper and lower bounds is between five and seven hours.", "In general, the gap is proportionally insignificant at high security levels but can be otherwise at low security levels.", "For given mining rates, the gaps are similar at different security levels.", "This indicates the lower bound () is also approximately exponential with a slightly steeper exponent than that of the upper bound." ], [ "The Mining Race", "A key proof technique in this paper is to reduce the blockchain security question to a race between the Poisson adversarial mining process and a counting process formed by a certain species of honest blocks.", "In the zero-delay case, the latter is exactly the Poisson honest mining process.", "The race is analyzed in Section .", "In the bounded-delay case, we identify a species of honest blocks that are necessary for the adversary to match block-by-block.", "We then upper bound the probability that a matching can occur to yield an achievable security latency function (Theorem ) in Section .", "Conversely, we identify another species of honest blocks which are sufficient for the adversary to match to succeed with a specific attack.", "We then lower bound the probability of such a match to yield the unachievability result (Theorem ) in Section ." ], [ "The Zero-delay Case: The Race between Two Poisson Processes", "In this section we analyze the fundamental security–latency trade-off of the Nakamoto consensus in the special case of zero block propagation delays.", "In this case, a private attack is known to be the worst among all possible attacks [6].", "While the gist of the security–latency trade-off is known, the thorough treatment here including both “pre-mining” and “post-mining” gains demonstrates some key techniques we shall use later in the much more complicated case with positive delay bound." ], [ "Achievable Security Latency Function", "A sufficient condition for blocks mined by time $s$ and confirmed at time $r=s+t$ to remain permanent is that more honest than adversarial blocks are mined during every interval $(c,d]$ that covers $(s,r]$ .", "We let $F_{s,r}$ denote the “bad event” that Hc,dAc,d for some $c\\in [0,s]$ and $d\\in [r,\\infty )$ .", "Note that (REF ) is equivalent to ( Hc,s - Ac,s ) + ( Hs,r - As,r ) + ( Hr,d - Ar,d ) 0 .", "For convenience, let M- = c[0,s] { Hc,s - Ac,s } M+ = d[r,) { Hr,d - Ar,d } where the adversary attains the maximum pre-mining gain if the attack began at the minimizer $c$ , and the minimizer $d$ is the point of time where the adversary attains the maximum post-mining gain.", "It is easy to see that the bad event can be expressed as Fs,r = { M- + Hs,r - As,r + M+ 0 } .", "Because the $H$ and $A$ processes are independent and memoryless, the variables $M_-, H_{s,r}, A_{s,r}, M_+$ are mutually independent.", "Note that $M_+$ is equal to the minimum of an asymmetric random walk where each step is $+1$ with probability $\\alpha /(\\alpha +\\beta )$ and $-1$ with probability $\\beta /(\\alpha +\\beta )$ .", "It is well known that $-M_+$ is a geometric random variable with parameter $\\beta /\\alpha $ [15]: P(-M+=k) = ( 1- ) ( )k, k=0,1,....", "Similarly, $M_-$ is equal to the minimum of the same random walk but with a finite duration $s$ .", "The distribution of $M_-$ converges to (REF ) as $s\\rightarrow \\infty $ .", "Evidently, $M_-$ is lower bounded by a random variable that is identically distributed as $M_+$ .", "We use the Chernoff's to upper bound the error probability: For every $\\rho \\in (0,\\alpha /\\beta )$ , P(Fs,r) E[ -(M- + Hs,r - As,r + M+) ] ( E[ -M+ ] )2 E[ As,r ] E[ -Hs,r ] = ( - - )2 e(r-s)(-1) e(r-s)(-1-1) where we have used the following Laplace-Stieltjes transforms: Let $X$ be a Poisson random variable with parameter $q$ .", "For every $\\rho >0$ , we have E[ X ] = e(-1) .", "Let $L$ be a geometric random variable with pmf $(1-q)q^k$ , $k=0,1,\\dots $ .", "For every $\\rho >0$ , we have E[ L ] = 1-q1-q .", "To obtain the tightest bound, we may optimize $\\rho $ by setting the derivative with respect to $\\rho $ to zero.", "It is straightforward to show that the best $\\rho $ is the unique real-valued root of a third order polynomial.", "For simplicity, however, we use a specific choice = which is asymptotically the best choice with $r-s\\rightarrow \\infty $ (it minimizes $\\beta \\rho +\\alpha /\\rho $ ).", "The upper bound (REF ) then becomes () in Theorem ." ], [ "Unachievable Security for Given Latency", "To examine how tight the achievable functions are, we shall compare them to unachievable security levels due to the following well-known attack: [Private attack with pre-mining and post-mining] Let $s>0$ and $t>0$ be given.", "The adversary mines in private and publishes no blocks until it attains a longest blockchain in the entire system at or after time $s+t$ , at which point it publishes this blockchain to conclude a successful attack.", "Specifically, until time $s$ , the adversary always mines off the highest of all blocks in the system except for the highest honest block (which it tries to undermine).", "From time $s$ onward, the adversary always mines off the tip of its longest private blockchain.", "Until time $s$ , as soon as the adversary's private blockchain is more than one height behind, it starts afresh to mine off the parent of the highest honest block.", "Let $L_r$ , $0\\le r\\le s$ , denote the pre-mining gain at time $r$ , which is defined as the adversarial advantage relative to the parent of the highest honest block by time $r$ .", "It is not difficult to see that $(L_r, 0\\le r\\le s)$ is a birth–death process [28] with constant birth rate $\\beta $ and death rate $\\alpha $ .", "For all practical purposes $s$ is sufficiently large, so we regard the process to be in its steady state.", "Hence the pmf of the pre-mining gain at time $s$ (denoted as $L$ ) is said to be the geometric distribution with parameter $\\beta /\\alpha $ (the same as (REF )).", "Following similar arguments as in Section REF , the adversary wins the race if the honest mining advantage during $(s,s+t]$ fails to counter the adversary's pre-mining and post-mining gains: 1 + Hs,s+t - As,s+t L - M+ where the “1” is due to the highest honest block mined by $s$ .", "The difference of the two independent Poisson random variables takes the Skellam distribution [29].", "Also, the complementary cdf of the sum of i.i.d.", "geometric random variables $L$ and $-M_+$ takes the form of ( )n ( 1 + n ( 1 - ) ) .", "Hence the probability of the event (REF ) can be rewritten as () in Theorem ." ], [ "Achievable Security with Bounded Delay (Proof of Theorem 3.5)", "As in the zero-delay case studied in Section REF , we establish the achievable security latency function by studying a race between two mining processes.", "Because delays cause forking, we only count a special species of honest blocks called loners.", "A loner is an honest block that is not mined within $\\Delta $ units of time of other honest blocks.", "Let $b$ be a block mined by time $s$ and included in some honest miner's longest blockchain at time $s+t$ .", "Section REF proves that block $b$ must be “permanent” if for all ${c} \\le s$ and ${d} \\ge s+t$ , more loners than adversarial blocks are mined during $({c,d}]$ .", "The race between the loner counting process and the adversarial mining process is difficult to analyze directly.", "To make progress, we identify a renewal process as a surrogate of the loner process in Section REF .", "In Appendix we develop a general formula to upper bound the probability that a renewal process loses the race against an independent Poisson process (this formula is useful in its own right).", "In Section REF we invoke a special case of the formula to prove Theorem .", "For convenience and better intuition, we specifically choose the time unit to be equal to the block propagation delay bound in this proof.", "Hence $\\Delta $ units of time in Theorem becomes one (new) unit of time here.", "This obviously normalizes the block propagation delay bound to 1 new unit of time.", "Consequently, the mining rate, aka the expected number of blocks mined per new unit of time, is equal to the expected number of blocks mined per maximum allowable delay.", "With slight abuse of notion, we still use $\\alpha $ and $\\beta $ as the mining rates under the new time unit.", "At the end of the analysis we will recover Theorem with an arbitrary time unit." ], [ "Consistency Barring Loner Deficiency", "[Publication] A block is said to be published by time $t$ if it is included in at least one honest miner's view by time $t$ .", "A blockchain is said to be published by time $t$ if all of its blocks are published by time $t$ .", "[$t$ -credibility] A blockchain is said to be $t$ -credible if it is published by time $t$ and its height is no less than the height of any blockchain published by time $t-1$ .", "If $t$ is unspecified, the blockchain is simply said to be credible (in context).", "There can be multiple $t$ -credible blockchains, which may or may not be of the same height.", "Once a block is published, it takes no more than 1 unit of time to propagate to all miners.", "Hence at time $t$ , an honest miner must have seen all blockchains published by $t-1$ .", "It follows that every honest miner's longest blockchain must be $t$ -credible.", "As we shall see, it is unnecessary to keep tabs of individual miner's views as far as the fundamental security is concerned.", "Focusing on credible blockchains allows us to develop a simple rigorous proof with minimal notation.", "[Lagger] An honest block mined at time $t$ is called a lagger if it is the only honest block mined during $[t-1, t]$ .", "By convention, the genesis block is honest and is regarded as the 0-th lagger.", "[Loner] An honest block mined at time $t$ is called a loner if it is the only honest block mined during ${[t-1, t+1]}$ .", "A loner is the only honest block at its height.", "Suppose block $b$ mined at time $t$ is a loner.", "By definition, no other honest block is mined during $[t-1, t+1]$ .", "By Definitions REF and REF , block $b$ is in all honest miners' views by time $t+1$ .", "Thus, the height of every honest block mined after $t+1$ is at least ${b̉+1}$ .", "Similarly, if an honest block is mined before $t-1$ , its height must be smaller than $b̉$ ; otherwise, block $b$ 's height would be at least $b̉+1$ .", "Hence, no other honest block has height $h(b)$ .", "Suppose $0\\le s < r$ .", "Let $A_{s,r}=A_r-A_s$ denote the total number of adversarial blocks mined during $(s,r]$ .", "Let $H_{s,r}= H_r-H_s$ denote the total number of honest blocks mined during time interval $(s,r]$ .", "Let $V_{s,r}$ denote the total number of laggers mined during $(s,r]$ .", "Let $Y_{s,r}$ denote the total number of loners mined during $(s,r]$ .", "By convention, for every counting process $X$ , we let $X_{s,r} = 0$ for all $s \\ge r$ .", "Suppose $t \\le r$ .", "Let $s$ denote the mining time of the highest honest block shared by a $t$ -credible blockchain and an $r$ -credible blockchain.", "Then Ys+1,t-1 As,r.", "Let block $e$ denote the highest honest block shared by $r$ -credible blockchain $d$ and $t$ -credible blockchain $d^{\\prime }$ with $T_e = s$ .", "Let block $b$ denote the highest block shared by blockchains $d$ and $d^{\\prime }$ .", "Blocks $b$ and $e$ may or may not be the same block.", "An illustration of the relationship between these blocks is as follows: $&\\fbox{\\color {white}a} -\\cdots -\\cdots -\\fbox{d} \\;\\; \\\\&\\,\\;\|\\quad \\; \\qquad \\qquad \\qquad \\text{time } r \\\\\\fbox{e}-\\cdots -&\\fbox{b}-\\cdots -\\fbox{d'} \\qquad \\qquad \\\\\\text{time } T_e=s \\qquad & \\qquad \\qquad \\qquad \\text{time } t$ If $t-s\\le {2}$ or no loner is mined during $(s+1, t-1]$ , obviously $ Y_{{s}+{ 1},t-{ 1}} = 0 \\le A_{{s},r}$ .", "Otherwise, consider loner $c$ mined during $({s}+{ 1}, t-{ 1}]$ .", "We next show that $c$ can be paired with an adversary block mined during $(s,r]$ .", "Since blockchain $e$ is $s$ -credible and block $c$ is mined after time $s+1$ , we have $c̉ > ẻ$ .", "Since blockchain $d$ is $r$ -credible and blockchain $d^{\\prime }$ is $t$ -credible, we have $c̉ \\le \\min \\lbrace d̉, \\rbrace $ .", "Consider the following two only possible cases: If $ẻ < c̉ \\le b̉$ , there exists at least one adversarial block at height $h(c)$ because all blocks in blockchain $d$ between block $e$ (exclusive) and block $b$ (inclusive) are adversarial by definition.", "If $b̉ < c̉ \\le \\min \\lbrace d̉, \\rbrace $ , there is at least one adversarial block at height $h(c)$ , because two divergent blockchains exist but loner $c$ is the only honest block at its height by Lemma REF .", "Thus, for every loner mined during $({s}+1, {t}-{1}]$ , at least one adversarial block must be mined during $({s},r]$ at the same height.", "In particular, the adversarial block must be mined before $r$ because it is published by $r$ .", "Hence (REF ) must hold.", "We define a “bad event”: For all $s,r \\ge 0$ and $\\epsilon \\in (0,1)$ , let Fs,r = a[0,s], b [r,) {Ya+1,b-1- Aa,b} .", "Let $\\epsilon \\in (0,1)$ be fixed for now.", "We will eventually send $\\epsilon \\rightarrow 0$ .", "Suppose block $b$ is mined by time $s$ and is included in a $t$ -credible blockchain.", "Then, barring event $F_{s,t}$ , block $b$ is included in all $r$ -credible blockchains for all $r\\ge t$ .", "We first establish the result for $r\\in [t,t+\\epsilon ]$ and then prove the lemma by induction.", "Fix arbitrary $r\\in [t,t+\\epsilon ]$ .", "Let block $e$ denote the highest honest block shared by an $r$ -credible blockchain and a $t$ -credible blockchain that includes block $b$ .", "We have YTe+1,r-1- YTe+1,t-1 ATe,r where (REF ) is due to Lemma REF .", "Barring $F_{s,t}$ , Ya+1,r-1- > Aa,r holds for all $a\\in [0,s]$ .", "Hence for (REF ) to hold, we must have $T_e>s$ .", "Since $s\\ge T_b$ by assumption, block $b$ must be included in blockchain $e$ , which implies that block $b$ must also be included in the $r$ -credible blockchain.", "Suppose the lemma holds for $r\\in [t,t+n\\epsilon ]$ for some positive integer $n$ .", "We show the lemma also holds for $r\\in [r,t+(n+1)\\epsilon ]$ as follows: Let $t^{\\prime }=t+n\\epsilon $ .", "If $F_{s,t}$ does not occur, then $F_{s,t^{\\prime }}$ does not occur either.", "Because block $b$ is included in a $t^{\\prime }$ -credible blockchain, a repetition of the $r\\in [t,t+\\epsilon ]$ case with $t$ replaced by $t^{\\prime }$ implies that block $b$ is included in all $r$ -credible blockchains with $r\\in [t^{\\prime },t^{\\prime }+\\epsilon ]$ .", "Hence lemma holds also for $r\\in [t,t+(n+1)\\epsilon ]$ .", "The lemma is then established by induction on $n$ .", "Lemma REF guarantees that a block with some confirmation time is permanent/irreversible barring the bad event $F_{s,r}$ .", "It remains to upper bound $P(F_{s,r})$ as a function of the confirmation time $r-s$ .", "Throughout this paper, the arbitrary interval of interest $[s,r]$ is exogenous, so that both the honest and the adversarial mining processes are Poisson during the interval.", "We note that a recurrent mistake in the literature is to define a time interval according to some miners' views, actions or other observed outcomes.", "The boundaries of such an interval are in fact complicated random variables in an all-encompassing probability space.", "Several prior analyses (see, e.g., [26], [23], [19]) fail to account for the (subtle but crucial) fact that the mining processes are no longer Poisson when conditioned on those random variables." ], [ "Some Moment Generating Functions", "In this subsection we derive the MGF of several types of inter-arrival times.", "These will be useful in the analysis of $P(F_{s,r})$ .", "Let $Z$ be an exponential random variable with mean $1/\\alpha $ .", "Let the MGF of $Z$ conditioned on $Z\\le 1$ be denoted as $\\phi _0(u)$ .", "Then 0(u) = {ll (1 - eu-) (1-e-) (-u) if u 1-e- if u = .", ".", "The probability density function (pdf) of $Z$ conditioned on $Z\\le 1$ is simply 1-e- e-z 1{0<z<1} where $1_{\\lbrace \\cdot \\rbrace }$ represents the indicator function which takes the values of 1 or 0 depending on whether the condition in the braces holds or not.", "The conditional MGF is thus 0(u) = E[ eu Z \| Z1 ] = 01 1-e- e-zeuz d z which becomes (REF ).", "Let $Z$ be an exponential random variable with mean $1/\\alpha $ .", "Let the MGF of $Z$ conditioned on $Z>1$ be denoted as $\\phi _1(u)$ .", "Then 1(u) = eu-u where the region of convergence is $u\\in (-\\infty ,\\alpha )$ .", "Conditioned on $Z>1$ , the pdf of $Z$ is given by e(-z+1) 1{z>1}.", "The conditional MGF is thus: 1(u) = E[ euZ \| Z>1 ] = 1+ e(-z+1)euz d z.", "The integral converges if and only if $u<\\alpha $ , where the result is given by (REF ).", "Recall the genesis block is the 0-th lagger.", "For $i=1,2,\\dots $ , let $X_i$ denote the time elapsed between the mining times of the $(i-1)$ -st and the $i$ -th lagger.", "Let $K_i$ denote the number of honest blocks mined between the $(i-1)$ -st lagger (excluded) and the $i$ -th lagger (included).", "$(X_1,K_1), (X_2,K_2), \\dots $ are i.i.d.", "Lemma REF is proved in Appendix .", "The lemma implies that laggers form an ordinary renewal process.", "Double-laggers.", "The loner process is not easy to characterize since whether a block mined at time $t$ is a loner depends not only on the past but also on future blocks (in $(t,t+1]$ ).", "In order to count loners, we examine a tightly-related species of honest blocks defined as follows.", "[Double-lagger] The first honest block mined after a loner is called a double-lagger.", "Note that a loner is also a lagger.", "So whenever two laggers are mined in a row, the former one is a loner and the latter one is a double-lagger.", "As such, there is a one-to-one correspondence between loners and double-laggers.", "We shall prove the independence of inter-double-lagger times and derive their MGFs, thus establishing the arrivals of double-laggers as a renewal process.", "Let $s̭{r}$ denote the total number of double-laggers mined during $(s,r]$ .", "Let $I_{1}$ denote the time the first double-lagger arrives.", "Let $J_1$ be the number of laggers after the genesis block until the first double-lagger (included).", "For $i>1$ , let $I_{i}$ denote the time elapsed between the $(i-1)$ -st and the $i$ -th double-lagger.", "Let $J_i$ be the number of laggers between the $(i-1)$ -st double-lagger to the $i$ -th double-lagger.", "For all $0 \\le s \\le r$ , Ys,r s̭r-1.", "Because loners and double-laggers appear in consecutive pairs, every double-lagger mined during $(s,r]$ but the first corresponds to a distinct loner mined during $(s,r]$ .", "$(I_{1},J_1), (I_{2},J_2),\\dots $ are i.i.d.", "Lemma REF is proved in Appendix .", "The time from a lagger to the next double-lagger follows the same distribution as an inter-double-lagger time.", "Let blocks $b$ , $c$ , $d$ be consecutive honest blocks.", "Evidently, $T_c-T_b$ and $T_d-T_c$ are i.i.d.", "exponential random variables.", "Let $Q$ be the time elapsed from block $d$ to the next double-lagger.", "If $d$ is a lagger, then $Q$ does not depend on whether $c$ is a lagger.", "Thus, for all $x$ , P(Qx \| Td-Tc>1) = P(Qx \| Td-Tc>1, Tc-Tb>1) .", "The left hand side is the cdf of the time between a lagger and the next double-lagger; the right hand side is the cdf of an inter-double-lagger time.", "Hence the proof.", "For convenience we define the following function $g_\\alpha (u)$ : g(u) = u2 - u - u eu- + 2 e2(u-).", "Evidently $g_\\alpha (u)>0$ if $u\\le 0$ and $g_\\alpha (\\alpha )=0$ .", "Also, $g_\\alpha (u)$ is differentiable with bounded derivative on $[0,\\alpha ]$ .", "From now on, let $u_0$ denote the smallest zero of $g_\\alpha (\\cdot )$ , i.e., $g_\\alpha (u_0)=0$ , and $g_\\alpha (u)\\ne 0$ for all $u\\in [0,u_0)$ .", "We must have $0<u_0\\le \\alpha $ .", "The MGF of the inter-double-lagger time is (u) = 1 + u - u2 u2 - u - u e(u-) + 2 e2(u-) where the region of convergence is $(-\\infty ,u_0)$ .", "Lemma REF is proved in Appendix .", "The key is to study a Markov process: The initial state is a lagger.", "With a known probability a double-lagger follows immediately to terminate the process.", "With the remaining probability we visit a non-lagger state a geometric number of times until we return to the initial lagger state.", "This allows us to write a recursion for the MGF of the inter-double-lagger time (aka the time till the double-lagger terminal state), the solution of which is (REF )." ], [ "Race between the Double-Lagger Process and a Poisson Process", "By Lemma REF , it suffices to study the race between the double-lagger process and the adversarial mining process.", "In Appendix , we study the race between a general renewal process and a Poisson process to obtain the following result: Let $(W_t)_{t\\ge 0}$ denote a renewal process where the mean of the renewal time is denoted by $m$ .", "Let the MGF of the renewal time be denoted by $\\phi (u)$ where the region of convergence is $(-\\infty ,{\\overline{u}})$ , where ${\\overline{u}}$ may be $+\\infty $ .", "Let $(A_t)_{t\\ge 0}$ denote a Poisson process with rate $\\beta $ .", "Let $\\mu ,\\nu \\ge 0$ be fixed.", "For $s,t>0$ , the probability that there exists $c\\in [0,s]$ and $d\\in [s+t,\\infty )$ such that the number of renewals in $(c,d]$ is at most the number of Poisson arrivals in $(c-\\mu ,d+\\nu )$ plus $n$ is upper bounded: $& P( \\exists \\, c\\in [0,s], \\, d\\in [s+t,\\infty ) \\text{ such that }W_{c,d} \\le A_{c-\\mu ,d+\\nu } + n ) \\\\&\\le \\exp ((\\phi (u)-1)\\beta (\\mu +\\nu ))\\phi ^{n+1}(u) \\mathcal {L}^2( \\phi (u) )\\exp ( - \\psi (u) t )$ for all $u\\in (0,{\\overline{u}})$ where (u) = u + - (u) and L(r) = (r-1) ( 1- m ) r ( ((r-1)) )-1 - 1 .", "Theorem REF applies to arbitrary renewal processes.", "In this section, the renewal process of interest is the double-lagger process, where the mean and MGF of the inter-arrival time given by $m=e^{-2\\alpha }/\\alpha $ and $\\phi (u)$ in (REF ), respectively.", "Define $\\psi (u)$ using (REF ) and (REF ).", "If $\\beta < \\alpha e^{-2\\alpha }$ , then there exists $u^\\in (0,u_0)$ such that $\\psi (u) > 0$ for all $u\\in (0, u^]$ .", "Evidently, $\\psi (0) = 0$ .", "It is not difficult to calculate the derivative '(0) = 1-'(0) = 1-e-2.", "If $\\beta < \\alpha e^{-2\\alpha }$ , we have $\\psi ^{\\prime }(0)> 0$ .", "By continuity, there must exist a $u^<u_0$ such that $\\psi ^{\\prime }(u)>0$ and $\\psi (u)>0$ for all $u\\in (0,u^]$ .", "Let $u_1>0$ be the smallest positive number such that $\\psi (u) = 0$ .", "Note that as $u\\rightarrow u_0$ , we have $\\phi (u)\\rightarrow \\infty $ and $\\psi (u) \\rightarrow -\\infty $ .", "By Lemma REF , $u_1$ exists and $u^* < u_1<u_0$ .", "Indeed, Lemma REF guarantees that the optimized exponent $\\psi (u)$ is positive if $\\beta <\\alpha e^{-2\\alpha }$ .", "Theorem is basically a special case of Theorem REF where the renewal process is the double-lagger process.", "Let $\\mu =1$ and $\\nu =1+\\epsilon $ .", "Due to Lemma REF , we given $n=1$ extra count to the Poisson process.", "For convenience we let (u)=(u)-1.", "The right hand side of (REF ) becomes $e^{(2+\\epsilon )\\zeta (u)\\beta }\\left(1-\\frac{\\beta }{\\alpha }e^{2\\alpha }\\right)^2\\left( \\frac{\\zeta (u)}{\\frac{1}{1+\\zeta (\\zeta (u)\\beta )}-\\frac{1}{1+\\zeta (u)} } \\right)^2e^{- (u-\\zeta (u)\\beta ) t}.$ It is interesting to note that the squared terms are the product of two identical multiplicative factors, one due to pre-mining and the other due to post-mining.", "Lastly, we recover Theorem for the original arbitrary time unit.", "In particular, the block propagation delays are bounded by $\\Delta $ time units.", "To reintroduce $\\Delta $ into the result, we let $\\tau =r\\Delta $ , $\\sigma =s\\Delta $ , $\\alpha ^{\\prime }=\\alpha /\\Delta $ , $\\beta ^{\\prime }=\\beta /\\Delta $ , and $v=u/\\Delta $ .", "These new variables and parameters are then defined under the original time unit.", "We define (v) = (v) .", "Plugging in (REF ), we have (v) = 'v - v2 v2 - ' v - ' v e(v-') + '2 e2(v-') .", "Suppose a block is mined by time $\\sigma $ and is included in a $\\tau $ -credible blockchain.", "Applying Lemma REF , the block is included in all future credible blockchains barring the event $F_{\\frac{\\sigma }{\\Delta },\\frac{\\tau }{\\Delta }}$ .", "Using Theorem REF with $\\mu =1$ , $\\nu =1+\\epsilon $ , and $n=1$ with the time unit conversion and then letting $\\epsilon \\rightarrow 0$ , an upper bound of $P(F_{\\frac{\\sigma }{\\Delta },\\frac{\\tau }{\\Delta }})$ is obtained as Theorem , where $v_1$ is the smallest positive number such that $\\psi (v_1)=0$ .", "It is easy to see that all blocks mined by $\\sigma $ and included in a $\\tau $ -credible blockchain must be included in all credible blockchains thereafter barring the event $F_{\\frac{\\sigma }{\\Delta },\\frac{\\tau }{\\Delta }}$ , whose probability is upper bounded by () with the optimized parameter $v$ .", "This conclusion is equivalent to the main theorem (with minor abuse of notation we still use $\\alpha $ and $\\beta $ to replace $\\alpha ^{\\prime }$ and $\\beta ^{\\prime }$ , respectively, to follow some convention).", "Thus, Theorem is proved." ], [ "Liveness and Private Attack with Bounded Delay", "In this section, we study liveness properties and generalize the analysis in Section REF to the case where block propagation delays are upper bounded.", "Following Section , we let the time units be such that delay bound is 1 unit of time." ], [ "Blockchain Growth and Liveness", "[Jumper] We say the genesis block is the 0-th jumper.", "After the genesis block, every jumper is the first honest block mined at least one maximum delay after the previous jumper.", "For $i=1,2,\\dots $ , let $M_i$ denote the time elapsed between the $(i-1)$ -st jumper and the $i$ -th jumper.", "The following result is evident: The inter-jumper times $M_1,M_2,\\dots $ are i.i.d.", "and $M_i-1$ follows the exponential distribution with mean $1/\\alpha $ .", "Because $P(M_i>1)=1$ for all $i$ , all jumpers have different heights (almost surely).", "Let $J_{s,r}$ denote the number of jumpers mined during time interval $(s,r]$ .", "For simplicity, we also assume that individual honest miners have infinitesimal mining power, so that almost surely no individual honest miner mines two blocks within one maximum delay.This is the worst case for honest miners.", "Using Lemma REF , it is straightforward to establish the following result concerning the height of the longest (or credible) blockchains.", "[Blockchain growth] For all $s,t\\ge 0$ , every honest miner's longest blockchain at time $s+t$ must be at least $n$ higher than every honest miner's longest blockchain at time $s$ with probability no less than F2(t-1-n; n, ) where $F_2(\\cdot ;n,\\alpha )$ denotes the Erlang cdf with shape parameter $n$ and rate $\\alpha $ .", "If $t\\le 1$ then $F_2(t-1-n;n,\\alpha )=0$ , so the lemma holds trivially.", "We assume $t>1$ .", "All jumpers mined during $(s,s+t-1]$ must be in every honest miner's views by $s+t$ , where the first jumper is higher than the miner's longest blockchain at time $s$ and the last jumper is no higher than the miner's longest blockchain at time $s+t$ .", "Hence the probability of interest is no less than P( Js,s+t-1 n ) P(J0,t-1n) where the inequality is easily justified using a special case of Lemma .", "The event that $n$ or more jumpers are mined during $(0,t-1]$ is the same as that $n$ inter-jumper times can fit in a duration of $t-1$ .", "Hence P( Js,s+t-1 n ) P(M1 + ...+ Mn t-1) = P( (M1-1) + ...+ (Mn-1) t-1-n ) = F2( t-1-n; n, ) where (REF ) is because $M_1-1, \\dots , M_n-1$ are i.i.d.", "exponential random variables whose sum has the Erlang distribution.", "With the preceding techniques, it is also straightforward to establish the following result for blockchain quality or liveness.", "[Blockchain liveness] Let $s>0$ and $t>1$ .", "In every honest miner's longest blockchain at time $s+t$ , the probability that $n$ or more of those blocks are honest blocks mined during $(s,s+t]$ is lower bounded by i=0e-t(t)ii!", "F2(t-i-n-1;i+n,) .", "Since $J_{s,s+t-1}$ jumpers are mined during $(s,s+t-1]$ (with different heights), and at most $A_{s,s+t}$ of them are matched by adversarial blocks, the number of surplus jumper blocks lower bounds the number of honest blocks in any honest miner's longest blockchain that are mined during $(s,s+t]$ .", "The said probability is thus lower bounded by P( Js,s+t-1 - As,s+t n ) = i=0 P( As,s+t = i ) P(Js,s+t-1 i+n) = i=0e-t(t)ii!", "P(M1+...+Mi+n t-1) which is equal to (REF ) by (REF )." ], [ "Private Attack", "In addition to following the private attack of Definition REF , the adversary can also manipulate the block propagation delays in the following manner: 1) Until time $s$ , it presents all honest blocks to all honest miners with no delay, so that there is a unique blockchain consisting of honest blocks only.", "(This is not the most dangerous attack, but it is easy to analyze and provides a bound on the unachievable security function).", "2) From time $s$ onward, the adversary delays all honest blocks by the maximum allowed time $\\Delta $ .", "Since the adversary makes all delays zero before time $s$ , the pre-mining gain $L$ has the geometric pmf (REF ).", "After time $s$ , the first higher honest block is mined in exponential time, and the following honest blocks of increasing heights are mined with inter-jumper times.", "The adversary wins the race if there exists $d\\ge t$ such that the honest blockchain's growth during $(s,d-1]$ is no greater than $A_{s,d}+L-1$ .", "The growth of the honest blockchain during $(s,d-1]$ is identically distributed as the number of jumpers mined in $(0,d-s)$ .", "Hence the probability that the adversary wins is equal to the probability of Es,t = d[s+t,) { J0,d-s A0,d-s + L - 1 } = d[t,) { J0,t + Jt,d A0,t + At,d + L - 1 } .", "The jumper counts $J_{0,t}$ and $J_{t,d}$ are in general dependent.", "Using () in Lemma , we can replace $J_{t,d}$ by an identically distributed $J^{\\prime }_{0,d-t}$ which is independent of $J_{0,t}$ with a penalty of 1 count to yield P(Es,t) P( {J0,t A0,t + L - 1 } d(t,) { J0,t + J'0,d-t A0,t + At,d + L - 2 } ) = P( J0,t A0,t + N + L - 1 ) where N = { 0, -1- r[0,) { J'0,r - A'0,r } } lower bounds the post-mining gain, i.e., the maximum surplus the adversary can gain via private mining after time $s+t$ .", "We have a closed form expression for the Laplace-Stieltjes transform of $\\min _{r\\in [0,\\infty )} \\lbrace J^{\\prime }_{0,r} - A^{\\prime }_{0,r} \\rbrace $ [17].", "That is, letting $\\phi $ be $\\phi _1$ given by (REF ) and $m=1+\\frac{1}{\\alpha }$ in (REF ), we can express the transform in the form of (), the derivatives of which can be used to recover the pmf of $N$ as ().", "It is not difficult to show that the sum $A_{0,t}+L$ has the following pmf: P( A0,t+L = k ) = ki=0 (1-) ()k-i e-t(t)ii!", "= (1-) ()k e(-) t F1(k;t) where $F_1(k;\\alpha t)$ stands for the cdf of a Poisson distribution with mean $\\alpha t$ .", "From (REF ), we have P( Es,t ) n=0 k=0P(N=n, A0,t+L=k) P( J0,t n + k - 1 ) = n=0P(N=n) k=0P(A0,t+L=k) ( 1 - F2(t-n-k,n+k,) ) .", "Plugging (REF ) into (REF ) proves Theorem , in which we also return the time units to the original ones." ], [ "Methodology", "Metrics.", "The performance metrics of a Nakamoto-style protocol include latency for a given security level, throughput, and fault tolerance (the upper limit of the fraction of adversarial mining in a secure system).", "In this section, we numerically compute the trade-off between different performance metrics of popular Nakamoto-style cryptocurrencies (Bitcoin Cash, Ethereum, etc.)", "and discuss their parameter selections.", "Block propagation delay.", "The above metrics crucially depend on the block generation rate (or the total mining rate), maximum block size, and block propagation delay.", "The former two are explicitly specified in the protocol.", "The block propagation delay, however, depends on network conditions.", "Block propagation delays in the Bitcoin network have been measured in [5], [3], [13].", "Such measurements are in general lacking for other cryptocurrencies.", "It is observed in [5] that there is a linear relationship between propagation delays and block size.", "In this section, we assume the block propagation delay upper bound is determined by the block size $S$ (in KB) according to the following formula: = aS + b .", "We determine the coefficients $a$ and $b$ using propagation delay data from Bitcoin and Ethereum monitoring websites.", "In Bitcoin, the block size is about 1 MB.", "The propagation delay of Bitcoin blocks fluctuates over the years with an overall decreasing trend [21]; the 90th percentile of block propagation is 4 seconds on average as of May 2021.", "Since $\\Delta $ in our model needs to be an upper bound on propagation delay, we assume $\\Delta =10$ seconds for a 1 MB Bitcoin block.", "According to [8], the 90th percentile of Ethereum block propagation is between 1.5 and 1.75 seconds for an average block size of 25 KB.", "We round it up to 2 seconds for an upper bound.", "Using these data points, we estimate $a = 0.0098$ and $b = 0.208$ in formula (REF ).", "Figure: Latency–security trade-off with Δ=10\\Delta =10 seconds and the percentage of adversarial mining power is 25%." ], [ "Confirmation time", "The latency–security trade-off is shown in Figure REF in Section for the Bitcoin parameters.", "Figure REF illustrates how the trade-off changes if the block generation rate increases by 10 folds (to 1 block per minute), with everything else held the same.", "The latency is much shorter under the higher block generation rate in this case.", "Also note that as the block generation rate further increases, the latency is longer.", "Figure: Latency required for different propagation delays.", "The percentage of adversarial mining power is 25%.Figure REF illustrates sufficient confirmation times due to Theorem for different security levels, block generation rates, and block propagation delays.", "The case of $\\Delta =0$ is plotted using () in Theorem .", "As expected, the latency is larger with longer block propagation delay and/or stronger security level requirement.", "Interestingly, increasing the block generation rate first reduces latency but eventually causes the latency to rise rapidly (due to excessive forking).", "From the graph, with the Bitcoin block propagation delay around 5 to 15 seconds, a sweet spot for block generation rate is between 50 and 200 blocks per hour in terms of optimizing latency.", "Figure: Achievable latency for different throughput with 10 -9 10^{-9} security level and 25% adversarial mining power.Table: Parameters and performances of Nakamoto-style Protocols.", "The percentage of adversarial mining power is 25%.", "In formula (), a = 0.0098 and b = 0.208.We recall that Theorem requires condition () to hold, i.e., the honest-to-adversarial mining ratio is bounded by $\\beta /\\alpha < e^{-2\\alpha \\Delta }$ , where $\\alpha e^{-2\\alpha \\Delta }$ is the exact rate that loners are mined.", "The actual sufficient and necessary condition for the consistency of the Nakamoto consensus in the infinite-latency limit is given in [6], [12]: < 11+ .", "The product $\\alpha \\Delta $ is equal to the average number of honest blocks mined per maximum delay period, which is usually set to a very small number to curb forking.", "If $\\alpha \\Delta \\ll 1$ , as is the case in Bitcoin, the condition () is almost identical to (REF )." ], [ "The latency–throughput trade-off", "A larger block size may benefit throughput by carrying more transactions.", "On the other hand, the larger block size increases the propagation delay, which causes longer latency.", "A protocol designer may want to find a sweet spot that leads to the most desirable latency and throughput.", "The throughput metric can be defined in a few different ways, ranging from the “best-case” throughput where the adversarial miners follow the protocol, to the “worst-case” throughput where the adversarial miners not only mine empty blocks but also use a selfish mining type of attack [9], [10] to displace honest blocks.", "In this paper, we choose to focus on the “best-case” throughput, which is the throughput under normal operation and is perhaps what protocol designers have in mind when setting parameters.", "Figure REF illustrates the minimum latency required for a given throughput according to Theorem (it is actually a latency upper bound).", "For several target throughput numbers (20, 30, 40, 50, and 60 KB per second), we also mark the corresponding latency bound.", "We see that we can achieve a latency less of around one hour with a throughput about 30 KB per second.", "This is achieved by increasing the block generation rate and decreasing the block size." ], [ "Case Studies in the Current Ecosystem", "Some PoW protocols attempt to better Bitcoin by increasing the block size (Bitcoin Cash) or the block generation rate (Litecoin).", "This subsection discusses the performance of these Bitcoin-like protocols.", "Table REF describes the parameters, estimated propagation delay, and performances of the aforementioned protocols.", "Bitcoin Cash (BCH) is a hard fork of Bitcoin from 2017.", "BCH aims to increase the throughput by increasing the maximum block size to 8 MB while keeping the same block generation rate as Bitcoin [18].", "As a result, the latency is increased from around 30 hours to 43 hours for $10^{-9}$ security level.", "Had BCH increase the block generation rate (instead of the block size) by eight times, it would have obtained the same eight fold throughput improvement while at the same time shortened the latency by a factor of eight or so.", "Litecoin is also a fork of the Bitcoin Core client that dates back to 2011.", "Litecoin decreases the block generation time from 10 minutes to 2.5 minutes per block [14].", "For Litecoin, the latency is 8 hours 39 minutes, and the current throughput is 6.7 KB per second (for $10^{-9}$ security level and 25% percentage of adversarial mining power).", "Dogecoin is a cryptocurrency developed in 2013 as a joke, making fun of the wild speculation in cryptocurrencies.", "It is a fork of Litecoin blockchain with higher block generation rate (from 2.5 minute per block to one minute per block).", "The throughput of Dogecoin blockchain increases to 16.6 KB/second (compared with 6.7 KB/second and 1.7 KB/second for Litecoin and Bitcoin, respectively).", "Meanwhile, the $10^{-9}$ security-level confirmation time decreases to 5 hours (compared with 8.6 hours and 29.6 hours for Litecoin and Bitcoin, respectively).", "Zcash.", "Proposed in 2016, Zcash is aims to provide enhanced privacy features.", "In 2017, Zcash doubled the maximum block size from 1 MB to 2 MB [4].", "Zcash also decreased the block interval from 10 minutes to 1.25 minutes [32].", "Similar to Litecoin, ZCash can be improved by increasing the block generation rate (higher throughput) and/or decreasing block size (shorter latency).", "Ethereum.", "The second largest cryptocurrency platform Ethereum has the block generation rate of 15 seconds per block [30], [7].", "The maximum gas consumption for each Ethereum block is 12.5$\\times 10^{6}$ .", "Given that 21000 gas must be paid for each transaction and 68 gas must be paid for each non-zero byte of the transaction[31], we estimate the maximum block size of an Ethereum block is 183 KB.", "Ethereum increases the block generation rate and decrease the block size.", "From Figure REF , for the throughput of around 12 KB per second, the latency bound is around 1 hour for $10^{-9}$ security level, which is close to the current confirmation time of 1 hour and 6 minutes.", "The Ethereum parameters appear to be well-chosen.", "Summary.", "In general, most of Nakamoto-style cryptocurrencies start with Bitcoin as the baseline and aim to improve its throughput.", "Since Bitcoin has a very low block generation rate, the best option according to a principled method is to increase its block generation rate.", "Additional improvements can be obtained by decreasing the block size and further increasing the block generation rate.", "This will not only increases throughput but also shortens the latency.", "Unfortunately, almost all of the cryptocurrencies we looked at either kept the block size the same as Bitcoin or went in the opposite direction to increase the block size, partly due to a lack of principled methodology.", "The only exception is Ethereum; Ethereum's parameters are very close to the optimal ones recommended in Figure REF ." ], [ "Conclusion and Future Directions", "We have developed concrete bounds for the latency–security trade-off for the Nakamoto consensus under the assumption that the block propagation delays are upper bounded by a constant.", "If the total block generation rate (normalized by the block propagation delay bound) is not very high and the adversarial mining power is well below that of honest mining, the upper and lower bounds are quite close.", "We have also applied the new formulas to analyze existing PoW longest-chain cryptocurrencies.", "Recent works [6], [12] have established the exact fault tolerance under high mining rate, but concrete bounds on latency beyond the lower tolerance studied in this paper remain open.", "Also, only asymptotic bounds are known for the Nakamoto consensus with dynamic participation and/or difficulty adjustment [11], .", "It would be interesting to establish concrete latency–security trade-offs in that more general setting.", "We would like to thank David Tse and Sreeram Kannan for stimulating discussions about this work." ], [ "Converting Confirmation Depth to Confirmation Time", "Let (,) = { k: e- i=k ii! }", ".", "Then for every $\\tau >0$ , the probability that $\\kappa ( (\\alpha +\\beta )\\tau , \\epsilon )$ or more blocks are mined in $\\tau $ units of time is no greater than $\\epsilon $ .", "For example, we learn from Figure REF that 4.35 hours of latency guarantees a security level of 0.0005.", "Using (), we obtain that $\\kappa (4.35\\times 6,0.0005)=45$ .", "Hence if one counts 45 confirmation blocks, it implies that at least 4.35 hours have elapsed with error probability 0.0005.", "In all, 45 confirmation blocks guarantees $10^{-3}$ security level, assuming that at most 10% of the total mining power is adversarial.", "At 10% adversarial mining power, Nakamoto [22] estimated that confirming after six blocks beats private attack at least 99.9% of the time.", "In contrast, 45 confirmation blocks guarantees the same security level regardless of what attack the adversary chooses to employ.", "We also note that while on average six blocks take only one hour to mine, with probability $10^{-3}$ it takes 2.75 hours or more to mine." ], [ "Proof of Lemma ", "For $i = 1, 2, \\dots $ , let $Z_i$ denote the inter-arrival time between the $i$ -th honest block and the $(i-1)$ -st honest block throughout the appendices.", "They are i.i.d.", "exponential random variables with mean $1/\\alpha $ .", "For convenience, we introduce the following shorthand notation within the bounds of this proof: Ln = K1+...+Kn ln = k1+...+kn for $n=1,2,\\dots $ .", "By convention, we let $L_0=l_0=0$ .", "It is easy to see that Xi = ZLi-1+1 + ...+ ZLi holds for $i=1,2,\\dots $ .", "Also, the event that $K_i=k_i$ is equivalent to the event that ZLi-1+1 1, ..., ZLi-1+ki-1 1, ZLi-1+ki > 1 .", "Given $K_1=k_1, \\dots , K_i = k_i$ , the event $X_i\\le x$ is equivalent to the event that Zli-1+1 + ...+ Zlix.", "For all positive integers $n, k_1, k_2, \\dots , k_n$ and real numbers $x_1, x_2, \\dots , x_n$ , we have P(X1x1, K1 = k1, ..., Xnxn, Kn = kn) = P(Z1++Zl1x1, Z11, ..., Zl1-11, Zl1>1, , Zln-1+1++Zlnxn, Zln-1+11, ..., Zln-11, Zln>1) = P(Z1++Zk1x1, Z11, ..., Zk1-11, Zk1>1 ) P( Zln-1+1++Zlnxn, Zln-1+11, ..., Zln-11, Zln>1) which is a product of $n$ probabilities, where () is because $Z_1,Z_2,\\dots $ are i.i.d.", "Let us define a two-variable function f( x, k ) = P( X1 x, K1=k ) for all $x\\in (-\\infty ,\\infty )$ and $k\\in \\lbrace 0,1,\\dots \\rbrace $ .", "The $i$ -th probability on the right hand side of () can be reduced as follows: P(Zli-1+1++Zlixi, Zli-1+11, ..., Zli-11, Zli>1) = P(Z1++Zkixi, Z11, ..., Zki-11, Zki>1) = f( X1 xi, K1=ki ) for all $i= 1,\\dots ,n$ , where () is because $Z_1,Z_2,\\dots $ is stationary.", "Applying () to () yields P( X1x1, K1=k1, ..., Xnxn, Kn = kn) = P(X1x1, K1=k1) P(X1xn, K1=kn) = f( x1, k1 ) f(xn, kn) .", "Hence the joint probability distribution of $(X_i,K_i)^n_{i=1}$ decomposes and each term takes exactly the same form.", "Thus Lemma REF is established." ], [ "Proof of Lemma ", "This proof takes the same form as the proof of Lemma REF .", "For convenience, we introduce the following shorthand within the bounds of this proof: Mn = J1+...+Jn mn = j1+...+jn for $n=1,2,\\dots $ .", "By convention, we let $M_0=m_0=0$ .", "It is easy to see that Ii = XMi-1+1 + + XMi holds for $i=1,2,\\dots $ .", "Also, the event $J_i=j_i$ is equivalent to the event that KMi-1+1>1, ..., KMi-1+ji-1>1, KMi-1+ji=1 .", "Given $J_1=j_1,\\dots ,J_i=j_i$ , the event $I_{i}\\le w$ is equivalent to Xmi-1+1+...+Xmiw.", "For all positive integers $n, j_1,\\dots , j_n$ and real numbers $w_1,\\dots , w_n$ , we have P(I1w1, J1=j1, ..., Inwn, Jn = jn) = P(X1+...+Xj1w1, K1>1, ..., Kj1-1>1, Kj1=1, ..., Xmn-1+1 + ...+ Xmnwn, Kmn-1+1>1,..., Kmn-1>1, Kmn=1 ) = P(X1+...+Xj1w1, K1>1, ..., Kj1-1>1, Kj1=1 ) P( Xmn-1+1 + ...+ Xmnwn, Kmn-1+1>1,..., Kmn-1>1,Kmn=1) which is the product of $n$ probabilities, where () is due to Lemma REF , i.e., $(X_1,K_1),(X_2,K_2),\\dots $ are i.i.d.", "Moreover, the $i$ -th probability on the right hand side of () can be reduced as: $& P(X_{m_{i-1}+1} + \\dots + X_{m_{i}}\\le w, \\\\& \\quad \\qquad K_{m_{i-1}+1}>1, \\dots ,K_{m_{i}-1}>1,K_{m_i}=1)\\\\=~&P(X_{1}+\\dots +X_{j_i}\\le w, \\\\& \\quad \\qquad K_1>1, \\dots , K_{j_i-1}>1,K_{j_i}=1)$ for all $i= 1,\\dots ,n$ .", "Applying (REF ) to () yields $& {\\color {white}=}P(I_{1}\\le w_1, J_1=j_1, \\dots , I_{n}\\le w_n, J_n = j_n) \\\\& =P(I_{1}\\le w_1, J_1=j_1)\\cdots P(I_{1}\\le w_n, J_1 = j_n).", "$ Hence the joint probability distribution of $(I_{i},J_i)_{i=1}^n$ decomposes and each term takes exactly the same function form.", "Thus Lemma REF is established." ], [ "Proof of Lemma ", "By Lemma REF , it suffices to consider $I_{1}$ , the arrival time of the first double-lagger starting from time 0.", "Let $K$ denote the number of honest blocks until (including) the first lagger and let $b_1,\\dots ,b_K$ denote that sequence of blocks.", "Then blocks $b_1, \\dots , b_{K-1}$ are non-laggers, and block $b_K$ is a lagger (it may or may not be a double-lagger).", "With probability $e^{-\\alpha }$ , $Z_1>1$ .", "In this case, block $b_1$ is a double-lagger since the genesis block is a lagger.", "We know $K=1$ and $I_{1} = Z_1$ .", "With probability $1-e^{-\\alpha }$ , $Z_1 \\le 1$ .", "Then block $b_1$ is not a lagger.", "We have $Z_1\\le 1, \\dots , Z_{K-1}\\le 1, Z_K>1$ .", "Let $I^{\\prime }$ denote the time from lagger $b_K$ to the next double-lagger.", "Then we can write I1 = {ll Z1 if Z1 > 1 Z1 + ...+ ZK + I' if Z1 1 .", "where $I^{\\prime }$ follows the same distribution as $I_{1}$ by Lemma REF .", "Thus the MGF of $I_{1}$ can be calculated as E[ euI1 ] = (1-e-) E[ eu(Z1+...+ZK+I') \| Z1 1 ] + e-E[ euZ1 \| Z1 > 1 ] = (1-e-) E[ eu(Z1+...+ZK) \| Z11 ] E[ euI' ] + e-E[ euZ1 \| Z1 > 1 ] = (1-e-) E[ eu(Z1+...+ZK) \| Z11 ] E[ euI1 ] + e-E[ euZ1 \| Z1>1 ] where () is because $I^{\\prime }$ and $I_i$ s are independent, and the fixed-point equation () is because $I^{\\prime }$ is identically distributed as $I_{1}$ .", "If (1-e-) E[ eu(Z1+...+ZK) \| Z11 ] <1 rearranging () yields E[ euI1 ] = e-E[euZ1 \| Z1>1] 1 - (1-e-) E[eu(Z1+...+ZK) \| Z11] .", "We shall revisit the condition () shortly.", "Note that P(K = k\|Z11) = (1-e-)k-2 e-, k = 2,3,....", "Hence E[eu(Z1+...+ZK) \| Z11] = k=2 P(K = k\|Z11) E[eu(Z1+...+Zk) \| K=k, Z11] = k=2(1-e-)k-2e- E[eu(Z1+...+Zk) \| Z11, ..., Zk-11, Zk > 1] = k=2 (1-e-)k-2e-E[euZ1 \| Z11] E[euZk-1 \| Zk-11] E[euZk \| Zk>1] = k=2(1-e-)k-2e- k-10(u)1(u) = e-0(u)1(u) k=0(1-e-)kk0(u) where () is due to mutual independence of inter-arrival times and () is due to Lemmas REF and REF .", "If (1-e-)0(u)<1, then the series sum converges to yield E[eu(Z1+...+ZK) \| Z11] = e-0(u)1(u)1-(1-e-)0(u) .", "Let us examine the conditions () and ().", "Note that (1-e-)0(u) = {ll 1-eue1-u if u , if u = .", ".", "It is clear that () is less than 1 for $u\\le 0$ .", "For $u>0$ , because $e^u/u$ is monotone decreasing on $(0,1)$ and monotone increasing on $(1,+\\infty )$ , there must exist $u\\le \\alpha $ that satisfies $(1-e^{-\\alpha })\\phi _0(u)=1$ .", "Hence the region of convergence must be a subset of $(-\\infty ,\\alpha )$ .", "Let $g_\\alpha (u)$ be defined as in (REF ).", "Using (), it is straightforward to show that (1-e-) E[ eu(Z1+...+ZK) \| Z11 ] = 1 is equivalent to $g_\\alpha (u)=0$ .", "Let $u_0>0$ be the smallest number that satisfies $g_\\alpha (u_0)=0$ .", "Then $u_0$ exists and $0<u_0\\le \\alpha $ according to the discussion in Section REF .", "Also, $u<u_0$ implies ().", "Therefore, the region of convergence for the MGF is $(-\\infty ,u_0)$ .", "For $u<u_0$ , we have by () and Lemma REF : E[ euI1 ] = e-1(u)(1-(1-e-)0(u))1-(1-e-)0(u) - e-(1-e-)0(u)1(u) which becomes (REF )." ], [ "Race Between a Renewal Process and a Poisson Process", "In this appendix we consider the race between a renewal process and a Poisson process.", "Let $(W_t)_{t\\ge 0}$ denote the renewal process whose i.i.d.", "renewal times are denoted as $I_1,I_2,\\dots $ .", "Let $(A_t)_{t\\ge 0}$ denote the Poisson process with rate $\\beta $ .", "Let $[s,r]$ denote the interval of interest.", "We give the Poisson process an early start by $\\mu >0$ units of time and a late finish of $\\nu \\ge 0$ units of time.", "We also give the Poisson process an extra advantage of $n\\ge 0$ .", "We say the Poisson process wins the race if there exists $c\\in (0,s]$ and $d\\in (t,\\infty )$ such that Wc,d Ac-,d+ + n ; otherwise the renewal process $W$ wins the race.", "The losing event is thus exactly: Fs,r = c[0,s], d[r,) { Wc,d Ac-,d+ + n } .", "We divide the race to three segments: before $s$ , from $s$ to $r$ , and after $r$ .", "While the Poisson process is memoryless, the renewal process has memory, so the number of renewals in those three segments are dependent.", "For convenience let us introduce two processes $(W^{\\prime }_t)_{t\\ge 0}$ and $(W^{\\prime \\prime }_t)_{t\\ge 0}$ which are i.i.d.", "as $(W_t)_{t\\ge 0}$ .", "Using the following result, we bound $P(F_{s,r})$ by studying a race with three independent segments.", "Let $d\\ge r\\ge s\\ge 0$ .", "The random variable $W_{s,d}=W_{s,r}+W_{r,d}$ is stochastically bounded: P( Ws,r + W'0,d-r x-1 ) P( Ws,r + Wr,d x ) P( Ws,r + W'0,d-r x ) P( W0,r-s + W'0,d-r x ) for every real number $x$ .", "We relegate the proof of Lemma to Appendix .", "Using () and () yields P( Fs,t ) P( c[0,s], d[t,) { W'0,s-c + W0,r-s + W”0,d-r Ac-,d+ + n } ) where we also use the fact that the adversarial process $A$ is independent of $W$ , $W^{\\prime }$ and $W^{\\prime \\prime }$ .", "The inequality on the right hand side of () is equivalent to ( W'0,s-c - Ac-,s- ) + ( W0,r-s - As-,r+ - n ) + ( W”0,d-r - Ar+,d+ ) 0 .", "Let us define M- = c[0,s] { W'0,s-c - Ac-,s- } M+ = d[r,) { W”0,d-r - Ar+,d+ } .", "Using (), it is straightforward to check that the union event on the right hand side of () is identical to M- + ( W0,r-s - As-,r+ - n ) + M+ 0 .", "It is important to note that $W_{s,r}$ , $A_{s-\\mu ,r+\\nu }$ , $M_-$ , and $M_+$ are mutually independent.", "We note that $M_-$ stochastically dominates $M_+$ as it is seeking the minimum gap between two processes over a shorter period of time.", "For our convenience we introduce $M^{\\prime }_+$ to be i.i.d.", "with $M_+$ .", "Then, P(Fs,r) = P( M- + ( W0,r-s - As-,r+ - n ) + M+ 0 ) P( W0,r-s As-,r+ + n - M+ - M'+ ).", "Let the moment generating function of $W$ 's renewal time be denoted as (u) = E[ euI1 ] .", "We then have that for every $u\\ge 0$ and positive integer $l$ , P( W0,r-s l-1 ) P( I1 + ...+ Il r-s ) E[ ( u (I1 + ...+ Il - r+s) ) ] = ( (u) )l e-u(r-s) .", "The Laplace-Stieltjes transform of $M_+$ , defined as L() = E[ -M+ ] is derived in [17] as (REF ) where $m=\\mathbb {E}\\left[I_1\\right]$ therein.", "Plugging () into (), we have P(Fs,r) E[ ((u)) As-,r+ + n - M'+ - M+ + 1 ] = E[ ((u)) As-,r+ ] ((u)) n + 1 ( E[ ((u)) M+ ] )2 (((u)--u)(r-s)) (((u)-1)(+)) ( (u) )n+1 ( L( (u) ) )2 where we have used the Laplace-Stieltjes transforms of $M_+$ as well as $A_{s-\\mu ,r+\\nu }$ which has a Poisson distribution with mean $\\beta (r-s+\\mu +\\nu )$ .", "Hence Theorem REF is established." ], [ "Proof of Lemma E.1", "In this proof we make some simple intuitions precise.", "Let $(W_t)_{t\\ge 0}$ and $(W^{\\prime }_t)_{t\\ge 0}$ denote two i.i.d.", "renewal processes.", "Let $I_1,I_2,\\dots $ and $I^{\\prime }_1,I^{\\prime }_2,\\dots $ denote their respective i.i.d.", "renewal times.", "We write P( Ws,t + Wt,d x ) = m=0 n=0 P( Wt,d x-n, W0,s=m, Ws,t=n ).", "The idea here is to divide the event into exclusive events corresponding to $m$ renewals in $(0,s]$ and $n$ renewals in $(s,t]$ .", "With $W_{0,s}=m$ and $W_{s,t}=n$ , the first arrival after $t$ is the $(m+n+1)$ -st renewal.", "Regardless of the time from $t$ to this arrival, as long as $t+I_{m+n+2}+\\dots +I_{m+x+1} > d$ , we know that the $(m+x+1)$ -st arrival is after $d$ , so that at most $(m+x)-(m+n)=x-n$ renewals can occur during $(t,d]$ , i.e., $W_{t,d}\\le x-n$ .", "Hence () leads to P( Ws,t + Wt,d x ) m=0 n=0 P( Im+n+2+...+Im+x+1 > d-t, W0,s=m, Ws,t=n ) .", "Because $I_{m+n+2},\\dots ,I_{m+x+1}$ are renewal times after $t$ , they are independent of $W_{0,s}$ and $W_{s,t}$ .", "Therefore, () is equivalent to P( Ws,t + Wt,d x ) m=0 n=0 P( I'1+...+I'x-n > d-t, W0,s=m, Ws,t=n ) = m=0 n=0 P( W'0,d-tx-n-1, W0,s=m, Ws,t=n ) = P( Ws,t + W'0,d-t x-1 ) .", "Hence the proof of ().", "To prove (), we begin with () and write P( Ws,t + Wt,d x ) = m=0 n=0 P( t+Y+Im+n+2+...+Im+x+1 > d, W0,s=m, Ws,t=n ) = m=0 n=0 P( t+Y+I'2+...+I'x-n+1 > d, W0,s=m, Ws,t=n ) where $Y$ denote the time between $t$ and the first renewal after $t$ and we have used the same aforementioned independence argument to arrive at ().", "Conditioned on $W_{0,s}=m$ and $W_{s,t}=n$ , the variable $Y$ is equal to part of the renewal time $I_{m+n+1}$ .", "Hence P(Yy\|W0,s=m,Ws,t=n) P(Im+n+1 y\|W0,s=m,Ws,t=n) = P(I'1 y) i.e., $Y$ is (conditionally) stochastically dominated by the renewal time.", "In fact the intuition is very simple: If one has waited for some time, then the time to the next arrival should be shorter than the renewal time.", "Using () and (), we have P( Ws,t + Wt,d x ) m=0 n=0 P( t+I'1+I'2+...+I'x-n+1 > d, W0,s=m, Ws,t=n ) = m=0 n=0 P( W'0,d-t x-n, W0,s=m, Ws,t=n ) = P( Ws,t + W'0,d-t x ) ." ] ]	2011.14051
[ [ "Incorporating Hidden Layer representation into Adversarial Attacks and\n Defences" ], [ "Abstract In this paper, we propose a defence strategy to improve adversarial robustness by incorporating hidden layer representation.", "The key of this defence strategy aims to compress or filter input information including adversarial perturbation.", "And this defence strategy can be regarded as an activation function which can be applied to any kind of neural network.", "We also prove theoretically the effectiveness of this defense strategy under certain conditions.", "Besides, incorporating hidden layer representation we propose three types of adversarial attacks to generate three types of adversarial examples, respectively.", "The experiments show that our defence method can significantly improve the adversarial robustness of deep neural networks which achieves the state-of-the-art performance even though we do not adopt adversarial training." ], [ "Introduction", "Recent deep neural networks (DNNs) make breakthroughs in many areas such as computer vision, speech recognition and so on.", "The power of DNNs brings people infinite reverie.", "Many significant and interesting works on DNNs bursting out.", "There are many kinds of DNNs as such convolution neural network (e.g.", "LeNet [1], AlexNet [2], ResNet [3]), recurrent neural network (e.g.", "LSTM [4]).", "As the rapid development of DNNs, more and more people focus on the security of DNNs.", "In particular, creating adversarial examples and defending adversarial attack are crucial techniques in the security of DNNs.", "The vulnerability [5] of DNNs receives great attention since it has been found.", "Great researches are focusing on this topic in literature.", "These researches can be roughly classified into two categories, such as adversarial attacks [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] and adversarial defences [5], [16], [17], [18], [19], [20], [21], [22].", "The methods of adversarial attack always want to create the adversarial examples which can fool the DNNs.", "Furthermore, the adversarial examples are usually imperceptible to a human.", "Inversely, adversarial defences make DNNs more robustness to adversarial examples.", "The adversarial attacks and adversarial defences just like a zero-sum game.", "Many works follow such definition of adversarial example that adversarial example is very close to original natural images but fool the DNNs [5].", "In the context of this definition, an adversarial example is imperceptible to human (We call them imperceptible adversarial examples).", "However, there are some other forms of adversarial examples.", "For example, some researches add some patches to the original image for fooling the DNNs [23], [24], [25], [26].", "These examples are called adversarial patches which can fool the DNNs though they are perceptive to human.", "Moreover, the adversarial noise images manipulated willfully by humans can fool the DNNs [27].", "Though these noise images are no meaning for human, DNNs still classify them as a certain class with high confidence (we call them as unrecognizable adversarial examples).", "Therefore, the definition of adversarial examples, that is imperceptible to human and fool DNNs is not very suitable.", "To solve this problem, this paper represents a novel definition of adversarial examples which is more generalized than that [5].", "We want to look for more generalized definition which can adaptive current most of the adversarial examples.", "Then, based on our generalized definition, we proposed three types of attacks and a defensive mechanism.", "In this paper, we mainly focus on defensive mechanism.", "Firstly, we present that the adversarial example cannot change the objective reality since all machine tasks are in the service of humanity, and humans understand the world according to the objective reality.", "For example, as shown in ”B” part of Fig.", "REF , the left subfigure is an English word, called ”F”.", "We assume it is an objective reality (e.g.", "human and machine all called it as ”F”).", "Then we get medial subfigure by adding little pixel in the left subfigure.", "These perturbation does not bother humans recognition in pixel space, but the machine may be confused with other words in some feature space.", "We consider it an adversarial example if the model classifies it wrongly.", "Furthermore, the right subfigure also made by adding little pixel in the left subfigure.", "We find that in pixel space it becomes another word which is called ”E”.", "Secondly, it is obvious that not all adversarial examples are imperceptible to humans such as Adversarial patches and Unrecognizable adversarial examples.", "We point out that it is significantly different between human vision and DNNs to recognize the images.", "As shown in ”A” part of Fig.", "REF , we find that when humans identify clean or disturbing images, the answer is always unique, whereas machines cannot give a definite answer.", "If the machine misidentified the disturbing image, such a sample is known as an adversarial example.", "Besides, if an adversarial example is imperceptible to human, the ”imperceptibility” is defined in pixel space.", "Therefore, we emphasize that the human recognizes an image in pixel space while the machine does that in a special space which is a machine ” vision” space (”C” part of Fig.", "REF ).", "We call as Feature space.", "Therefore, we draw two rules about generalized adversarial examples as followed: The adversarial example cannot change the objective reality.", "The adversarial example is imperceptible to the machine in Feature space rather than to human vision.", "According to these two rules, we assume that existing a function which can map the original pixel space to the feature space.", "We describe adversarial examples as those that are imperceptible to human in the feature space and can fool the DNN.", "Then, we provided three types of attacks included most of the current attack methods based on our generalized form of adversarial examples.", "Finally, based on the generalized adversarial examples, we propose a defence mechanism to improve the adversarial robustness of the model.", "Our defence strategy will compress the information of input and filter many of that, including the adversarial perturbation.", "And we theoretically certify the effectiveness of the defence under a simple case.", "Besides, we provide a certain defensive function and empirically verify the effectiveness of our defence mechanism.", "It is noteworthy that our defensive model achieves state-of-the-art performance.", "The contributions are listed as below: We draw a novel and more generalized definition of adversarial examples by keeping the properties of imperception and fooling in a feature space rather than original space or pixel space since the cognition mechanism of human is remarkably different with the machine.", "Due to this manipulation, we give the attacker more freedom to modify the original image (e.g.", "adversary can add small imperceptible perturbation or adversarial patches or even using an adversarial noise image to instead original image).", "Based on our generalized form of adversarial examples, we provide three types of adversarial attack to generate imperceptible adversarial examples, unrecognizable adversarial examples and adversarial patch, which are shown in Fig.", "REF .", "Most of the existed attack methods can be derived from our definition of adversarial examples.", "According to the generalized form of adversarial examples, we propose a defensive mechanism against adversarial attacks.", "We point out that a good defensive strategy needs to have two properties to prevent the perturbation from pixel space.", "Our defence method is simple and significantly improve the adversarial robustness of the DNNs.", "The attack effectivity of our attack method of generating imperceptible adversarial examples is better than most of attacks method such as FGSM [6], L2BIA [7], LBFGS [5] and so on.", "It drops the accuracy from $98.49\\%$ to $59.59\\%$ , which is better than CW [10] on MNIST dataset.", "Moreover, it drops the accuracy respectively from $88.92\\%$ to $46.69\\%$ on CIFAR10 and from $69.60\\%$ to $49.07\\%$ on ImageNet.", "Our performance is very close to or even better than that of PGD [16] and CW [10], which are state-of-the-art attack method.", "And even though we do not use adversarial training, our defending method achieves state-of-the-art performance which defeat all existed methods.", "In the rest of this paper, Section introduce the related work in literature.", "In Section , we define generalized adversarial examples.", "In Section , we derive three types of adversarial attacks from the definition of generalized adversarial examples.", "In Section , we introduce our adversarial defence which is state-of-the-art without adversarial training.", "In Section , we conduct large experiments and confirm that our methods are effective.", "Finally, Section concludes the paper.", "Recently DNNs are powerful function and have made the advanced achievements in many domains such as Image Classify [2], [28], NLP [29], [30], Object Detection [31], [23], [32], Semantic classification [33], [31], [34] and many more.", "With the burst spread of DNNs, more and more people pay attention to the security of DNNs.", "We may want to (or not want to) apply the DNNs depends on the extent to which the adversary generates the adversarial examples.", "Natural language processing (NLP) make great progress in recent years.", "The model based on DNNs achieves state-of-the-art performance in various fields such as language modelling [35], [36], syntactic parsing [37], machine translation [38], [29] and many more.", "Then, the attention has transformed the risk of the DNNs in NLP.", "Recent work has shown [39] it is possible to generate adversarial examples by adding noises into texts, which fool the DNNs.", "This work firstly finds adversarial examples in texts, leading an arm begins between attack and defence in texts.", "In the context of object detection, recent works find various interesting adversarial examples [31].", "For example, C. Xie et al.", "propose an algorithm called Dense Adversary Generation (DAG) to generate imperceptible adversarial examples.", "To decrease computing time, They utilize regional proposal network to produce the possible targets and then sum up the loss from these targets.", "Another form of adversarial examples in object detection is an adversarial patch [23] which generate a small image to stick it on the original image.", "Even these adversarial examples can attack the physical world (e.g.", "attacking Yolo [32]).", "In the semantic classification domain, the adversary is allowed manipulating fewer pixels than other domain since each perturbation is responsible for at least one-pixel segmentation [33], [31], [34].", "These attacks include non-targeted attacks and targeted attacks.", "However, they generate adversarial examples which are imperceptible to humans.", "This paper focuses on image classification.", "The model based on DNNs makes excellent progress in image classification since Olga Russakovsky et al.", "created AlexNet [40] which achieved champion in ILSVRC 2010.", "After that, many excellent DNNs (such as GoogleNet [41], VGG [42], ResNet [3] etc.)", "are proposed in this domain.", "However, when people indulge in the feast of DNNs, it is found that the DNNs are incredibly vulnerable to adversarial examples [5].", "The study of adversarial examples becomes significant in this domain.", "Usually, the successful rate of adversarial attack is related to how much distortion adversary can manipulate in original images.", "Szegedy et al.", "[5] use L2-norm to quantify the difference between adversarial images and original images.", "However, this metric is not necessarily applicable in other domain such as NLP.", "According to the knowledge of adversary, adversarial attacks can be classified as white-box attacks and black-box attacks.", "In this paper, We suppose the adversary can access the detail of DNNs, including parameters and framework.", "With this strong assumption, we can construct aggressive adversarial examples and then also utilize them to black-box attacks since previous work [17] find the adversarial examples have a property called transferability that perturbations crafted on an undefended model often transfer to an adversarially trained one." ], [ "Adversarial example", "Szegedy et al.", "[5] firstly find the vulnerability of the DNNs.", "They show that original image added small perturbation, called adversarial example, can fool DNNs.", "They firstly describe searching adversarial examples as a box-constrained optimization problem: $\\begin{aligned}\\min \\quad &\\Vert r\\Vert _2 \\\\s.t.", "\\quad &f(x+r) = l \\\\&x+r\\in [0,1]^m\\end{aligned}$ $f$ means a classifier (the DNN) mapping pixel space to a discrete category set.", "$x$ is the raw image and $r$ is the perturbation which is limited in $[0,1]$ .", "Moreover, $l$ is a targeted label.", "Solving this formula, we can construct the adversarial example which the model classifies $x+r$ as $l$ .", "This intriguing property rises significant attention on the security of the DNNs.", "Then, a significant number of works [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] try to generate all kinds of adversarial examples which can fool the DNNs such as Fast Gradient Sign Method [6], Basic Iterative Method [7], Jacobian-based Saliency Map Attack [8], DeepFool [9], CW [10], PGD [16], One Pixel Attack [13] and so on.", "Goodfellow et al.", "[6] firstly propose based-gradient attack, which updates along the direction of the signal function of the pixel gradient to obtain the adversarial examples.", "It is a fast method since it updates one step.", "Based on FGSM, Kurakin et al.", "[7] propose a more powerful attack method which is multi-update to generate adversarial examples.", "Also using multiple iterations, Deepfool [9] utilize a linear approximation method to produce adversarial examples by searching the minimum distance from a clean example to an adversarial example.", "Looking for more powerful and non-gradient based attack method, Carlini et al.", "[10] propose various objective functions and distance metrics to generate adversarial examples which can disable the Distillation Defense [18].", "PGD [16] is the first-order adversary which is the most potent attack method among the first-order attack methods.", "Madry et al.", "[16] study the adversarial robustness of DNNs and utilize projected gradient descent to search for more aggressive adversarial examples.", "Their method can significantly improve resistance to most of the adversarial attacks.", "However, most of the researches search the adversarial examples follow the definition of adversarial examples [5] which are imperceptible to humans but can fool DNNs.", "Those researches focus on studying the small perturbation.", "Some particular adversarial examples have been found as the development of adversarial examples.", "Brown et al.", "[23] firstly introduce adversarial patch attacks for image classifiers.", "They design a small patch than the size of the original image and put the small patch into the original image.", "Those adversarial examples are called adversarial patches those that are applied in object detection [24], [25], [43], [44].", "Moreover, many works find that adversarial patches exist in the physical world [45], [26], [32], [46].", "The adversarial patch is not imperceptible to humans.", "And the perturbation usually does not confuse humans judge.", "However, it can fool the DNNs.", "Anh et al.", "[27] find that some unrecognizable for humans can be classified as a class with high confidence by DNNs.", "Based evolutionary algorithms algorithm, they proposed a new algorithm called the multi-dimensional archive of phenotypic elites MAP-Elites [47], enable them to evolve the population better.", "Sara et al.", "[48] show that the representation of the raw image in a DNN can be operated to approximate those of other natural images by adding a minor, imperceptible perturbation to the original image.", "They focus on the internal layers of DNN representations.", "They create an adversarial example which approximated the original image, but its internal representation appears remarkably different from the original image.", "These researches are of great significance and raise some questions about DNN representations, as well as the vulnerability of DNN." ], [ "Adversarial Defense", "Adversarial defence [5], [16], [17], [18], [19], [20], [21], [22] makes DNNs more robust to adversarial examples.", "Papernot et al.", "[18] propose Defensive Distillation to defence DNN against adversarial examples.", "By controlling the temperature in the distillation network, they train a DNN with hard-label and train other DNN with soft-label.", "And the two networks have the same framework.", "However, this defence will fail under CW [10] attack method.", "Adversarial training is one of the most effective defence methods.", "Szegedy et al.", "[5] and Madry et al.", "[16] use their attack method to generate adversarial examples.", "Then combining clean examples and adversarial examples, they utilize them to train a DNN, which can significant improve adversarial robustness of the DNN.", "However, such defence method will fail under more powerful attack methods and is time-consuming.", "Tramer et al.", "[17] generate adversarial examples by various adversarial attacks and propose adversarial ensemble training which augments training data with perturbations transferred from other models.", "However, recent work [49] points out that adversarial training would cause obfuscated gradients that lead to a false sense of security in defences against adversarial examples.", "They propose three types of obfuscated gradients and design adversarial attacks that successfully attack all defensive papers in ICLR 2018 except that for CW [10]." ], [ "Notation", "In this subsetion we will show some mathematical symbols used in this paper.", "The details are shown in Table REF .", "Table: mathematical symbols" ], [ "Generalized Adversarial Examples", "In this section, to begin, we introduce three existed types of adversarial attack.", "These three types of adversarial attack include most of the current adversarial attack methods.", "By analyzing the characters of these attacks, we present a more generalized definition of adversarial examples which can explain the existent of adversarial examples in DNNs.", "Then, we introduce that the generalized adversarial examples can be degenerate currently existed adversarial examples by adding some restricted condition.", "Many works follow the definition of adversarial examples drawn by Szedegy et al.", "[5].", "We refer to this type of adversarial examples as Imperceptible Adversarial Examples which is very close to the original image but can fool the DNN.", "Moreover, Nguyen et al.", "[27] find a new type of adversarial examples which are unrecognizable to human.", "We refer to it as Unrecognizable Adversarial Examples that are hard to understand for human but is classified by a DNN with high confidence.", "Another type of adversarial examples is Adversarial Patches which are created by adding some small patches to original images [23].", "These adversarial examples have a familiar character that can fool the DNN, but their original images are classified correctly by DNNs.", "The difference between these adversarial examples is the magnitude of change in input.", "The details are shown below: Imperceptible Adversarial Examples: allows the adversary to change the whole image limiting in bounded-norm.", "Unrecognizable Adversarial Examples: allows the adversary to change the whole image without limiting, but the adversarial examples after adding perturbation should be unrecognizable to human.", "Adversarial Patches: allows the adversary to change the image in a confined region.", "Therefore, in this paper, we want to explore the unknown type of adversarial examples and give a definition of adversarial examples that will help us understand adversarial examples.", "Let us begin with two questions: A) If an example of $A$ class become an example of $B$ class after adding perturbation and the DNNs misclassify it, is it an adversarial example?", "A) Are adversarial examples imperceptible to humans?", "For the former one, we argue that the adversary cannot change the objective reality since all machine tasks are in the service of humanity, and the humans understand the world according to the objective reality.", "In practice, an adversarial example usually consists of a clean sample with a slight perturbation, e.g., imperceptible adversarial sample.", "It is this imperceptibility that makes it difficult for the perturbation to change the objective reality of the original example.", "However, the Adversarial patch does not take into account this imperceptibility, which may cause the sample to become another type of sample after adding disturbance.", "For example, as shown in B part if Fig.", "REF , the left subfigure is an English word, called ”F”.", "We assume it is an objective reality (e.g.", "human and machine all called it as ”F”).", "Then we get medial subfigure by adding little pixel in the left subfigure.", "These perturbation does not bother humans recognition in pixel space, but the machine may be confused with other words in some feature space.", "We consider it adversarial example if the model classifies it wrongly.", "Moreover, the right subfigure also made by adding little pixel in the left subfigure.", "We find that in pixel space it becomes another word which is called ”E”.", "It is meaningless that we call a sample as an adversarial example in this case if the model classifies it wrongly.", "For the after one, it is obvious that we can perceive the perturbation of adversarial patches and unrecognizable adversarial examples.", "Furthermore, for Imperceptible Adversarial Examples, if the limitation of perturbation is relaxed enough, we also can perceive the small perturbation.", "We point out that it is significantly different between human vision and DNNs to recognize the images.", "It is a common phenomenon that the perturbation that hard to recognize by human but may be easy to recognize for a DNN.", "Inversely, the perturbation such as adversarial patches that are easy to recognize by human but maybe are hard to recognize for a DNN.", "As shown in ”A” part of Fig.", "REF , we find that when humans identify clean or disturbing images, the answer is always unique, whereas DNN cannot give a definite answer.", "If the machine misidentified the disturbing image, such a sample is known as an adversarial example.", "Besides, if an adversarial example is imperceptible to human, the ”imperceptibility” is defined in pixel space.", "Therefore, we emphasize that the human recognizes an image in pixel space while the machine does that in a particular space which is a machine ”vision” space (”C” part of Fig.", "REF ).", "We call as Feature space.", "We now draw the definition of adversarial examples which refer to it as Generalized Adversarial Examples which include previous three types of adversarial examples.", "To begin, we draw two rules about adversarial examples as below: The adversarial example can’t change the objective reality.", "The adversarial example is imperceptible to machine in feature space rather than to human vision.", "According to these two rules, we assume that existing a function which can map the original pixel space to the feature space.", "Then, we define generalized adversarial examples.", "Define 1 The generalized adversarial examples are those that do not change original objective reality and are imperceptible to the machine in feature space and can fool the machine.", "We try to describe it formally.", "Let $I$ represents the original natural image and $y$ represents a class of $I$ .", "$T$ is a function which maps the pixel space to the feature space.", "Moreover, $f$ maps the feature space to the category space.", "Therefore, image recognition can be described as below $f(T(I))=f(F)=y$ Where $T(I)=F$ .", "Then, the generalized adversarial examples are described as: If $T$ is an identity function, Adversarial Tiny Perturbation is the degenerate case of Generalized Adversarial Examples.", "If $\\Delta $ is limited in the shape of patches, then Adversarial Patches is the degenerate case of Generalized Adversarial Examples.", "If $I+\\Delta $ is unrecognizable to human, in that case, Generalized Adversarial Examples degenerate to Adversarial Noise Perturbation." ], [ "Generalized attack mechanism", "According to the last section (Generalized adversarial examples), we get the formal definition of Generalized Adversarial Examples.", "In this paper, we mainly focus on adversarial defence.", "Therefore, adversarial attacks are briefly introduced.", "Now we formulate the problem of searching a generalized adversarial example for an image $I$ as follows: $\\begin{aligned}\\min \\quad &\\Vert T(I) - T(I+\\Delta )\\Vert _p \\\\s.t.", "\\quad &f(T(I+\\Delta )) \\ne y\\end{aligned}$ From the perspective of generalized adversarial example, we provided three versions about searching for generalized adversarial examples.", "The key idea of these attacks is searching adversarial examples by maximizing the gap between original images and adversarial examples.", "The first one is tiny perturbation version described as below: $\\begin{aligned}\\min \\quad &\\Vert \\Delta \\Vert _p - \\alpha \\cdot \\Vert T(I) - T(I+\\Delta )\\Vert _p \\\\s.t.", "\\quad &f(T(I+\\Delta )) \\ne y \\\\&\\Vert \\Delta \\Vert _p \\le \\delta \\end{aligned}$ where $\\alpha $ is the hyperparameter and $\\delta $ is the small value for humans perception.", "Eq.", "REF try to find an adversarial example which is imperceptible to human by minimizing $\\delta $ and maximize $\\Vert T(I)-T(I+\\Delta )\\Vert $ .", "The pixel space for human vision is similar to the feature space for DNNs.", "It is sensitive for DNNs to the perturbation in feature space.", "Applying this type of attack, the adversarial examples closed to the original image are created which are imperceptible to humans.", "However, the features are extracted from the adversarial examples by DNNs are different from that of original images.", "That is why the adversarial examples can fool the DNNs.", "The second one is called confidence attack, which makes DNN classifies a natural unrecognized image for human or a natural image from other domain with high confidence.", "The detailed formula is described as below: $\\begin{aligned}\\min \\quad &\\Vert T(I_{in}) - T(I_{out}+\\Delta )\\Vert _p \\\\s.t.", "\\quad &f(T(I+\\Delta )) \\ne y\\end{aligned}$ $I_{in}$ represents a natural image which comes from the original domain.", "And $I_{out}$ represents an image which comes from some domain or original domain.", "If $I_{out}$ is unrecognizable to human, then we can create an adversarial example which is unrecognizable for human but is classified by a DNN with high confidence.", "In that case, the attack is similar to the work [4] that produces images that are unrecognizable for humans, but a DNN believes to be recognizable objects with high confidence.", "If $I_{out}$ came from original domain, then this type attack is similar to the work [1] which shows that the representation of a raw image in a DNN can be operated to approximate those of other natural images by adding a minor, imperceptible perturbation to the original image.", "Referring to [1], $I_{in}$ is guided image, and $I_{out}$ is the source image.", "The third is patch version described as below: $\\begin{aligned}\\min \\quad &-\\Vert T(I) - T(I+P_{\\Delta })\\Vert _p \\\\s.t.", "\\quad &f(T(I+P_{\\Delta })) \\ne y \\\\&P_{\\Delta } is\\ a\\ small\\ patch\\end{aligned}$ In Eq.", "REF , $P_{\\Delta }$ is a smaller patch than original image.", "This attack allows adversary to manipulate pixels in confined region (e.g.", "$P_{\\Delta }$ ) but is no limit to choose the values yielding the max or min value.", "And $P_{\\Delta }$ have a specific shape such as square, circle and so on.", "In practical, when optimizing the Eq.", "REF , we need to know where $P_{\\Delta }$ locate in and what size that is.", "Usually, the position and size of $P_{\\Delta }$ can be randomly initialized as long as the patch does not change the objective reality of original image.", "Therefore, the size of $P_{\\Delta }$ always is much smaller than that of original image.", "In Eq.", "REF , $P_{\\Delta }$ is a smaller patch than the original image.", "This attack allows an adversary to manipulate pixels in the confined region (e.g.", "$P_{\\Delta }$ ) but is no limit to choose the values yielding the max or min value.", "Moreover, $P_{\\Delta }$ has a specific shape such as square, circle and so on.", "In practical, when optimizing the Eq.", "REF , we need to know where $P_{\\Delta }$ locate in and what size that is.", "Usually, the position and size of $P_{\\Delta }$ can be randomly initialized as long as the patch does not change the objective reality of the original image.", "Therefore, the size of $P_{\\Delta }$ always is much smaller than that of the original image." ], [ "Generalized defense mechanism", "In this section, we introduce a defence mechanism against adversarial examples based on generalized adversarial examples.", "In feature space, we define a function $D$ which can smooth the perturbation from pixel space.", "Recalled that $T$ is a function which maps pixels value to feature value and $f$ classifies the feature to a class, we construct the defence strategy as below: $\\begin{aligned}\\min \\quad \\mathop {E}_{(I,y) \\backsim \\mathcal {D}} [L(f\\cdot D \\cdot T(I), y)]\\end{aligned}$ where $\\mathcal {D}=\\lbrace (I^{(i)}, y^{(i)})\\rbrace _{i=1}^N$ is a dataset with $N$ samples.", "And $f \\cdot D \\cdot T(I)=f(D(T(I)))$ .", "$L$ is a loss function.", "According to the definition of generalized adversarial examples, the perturbations made by the adversary in pixel space are not strictly limited.", "The goal of the adversary is to manipulate perturbations which can cause significant discrepancy than original that when they transferred from pixel space to feature space.", "If we can prevent the disturbance from propagating backwards, then the model will naturally be adversarial robust.", "The simple idea is to design a filter like function and then deploy it between the pixel space and the feature space.", "The purpose of this function is to filter and compress input information.", "Because DNNs usually have multiple convolution kernels, the extracted features are redundant, which provides feasible conditions for this design scheme.", "When the filter compresses the input information, it loses some of the original information, including adversarial perturbation.", "However, because the input information is redundant, this filter does not have much impact on the performance of the model.", "Secondly, the filter should be differentiable or segmented differentiable, which facilitates its combination with DNN.", "Therefore, we propose that function $D$ have some property from the inverse perspective of an adversary.", "$D$ has sufficient smoothness.", "It is uniform continuity or uniform continuity in segments.", "This property ensures that the small perturbations in $T$ are limited in a fixed range after they are transferring.", "The range of $D$ is a denumerable finite set.", "The critical problem of adversarial defence is how to prevent the small perturbations slowly amplifying in a DNN.", "If the interval of the domain of $D$ contains the interval of the range, the perturbations will be compressed.", "This compression naturally leads to the loss of information.", "However, it is this compression effect that makes the impact of small perturbations on the results even smaller.", "Considering a single network $f(x)=Wx$ where $W$ is a $m\\times n$ vector in feature space.", "$m$ is the number of categories.", "And $n$ is the number of features.", "Let us denote $\\mathcal {I}$ as pixel space and denote $\\mathcal {F}$ as feature space.", "$\\forall I \\in \\mathcal {I}$ , we have $T(I)=F \\in \\mathcal {F}$ .", "$D$ is a defence strategy.", "Then, we have the following theorem: Theorem 1 Suppose $D$ satisfies the Lipschitz continuity condition, then $f\\cdot D$ also satisfies the Lipschitz continuity condition.", "Existing a constant $K$ , we have $\\Vert f(D(F+\\Delta )) - f(D(F))\\Vert _2 \\le K\\cdot \\Vert \\Delta \\Vert _2$ Proof 1 Since $D$ satisfies the Lipschitz continuity condition, then existing a constant $c$ , we have $\\Vert D(F+\\Delta ) - D(F)\\Vert _2 \\le c \\Vert \\Delta \\Vert $ Then, $\\begin{split}&\\Vert f(D(F+\\Delta )) - f(D(F))\\Vert _2 \\\\&\\le \\Vert W\\cdot D(F+\\Delta ) + b - (W\\cdot F + b) \\Vert _2 \\\\&= \\Vert W\\cdot (D(F+\\Delta ) - D(F)) \\Vert _2 \\\\&\\le \\Vert W\\Vert _2 \\cdot \\Vert D(F+\\Delta ) - D(F)\\Vert _2 \\\\&\\le c\\Vert W\\Vert _2 \\cdot \\Vert \\Delta \\Vert _2\\end{split}$ Let us denote $K = c\\Vert W\\Vert _2$ , then $(\\ref {eq:theorem:1}) \\le K \\cdot \\Vert \\Delta \\Vert _2$ Proof done.", "In Theorem REF , we give a strong supposition that $D$ satisfies the Lipschitz continuity condition, leading $f\\cdot D \\cdot T$ also satisfies the Lipschitz continuity condition.", "With this conclusion, the effect of perturbation can be under-control.", "Next, we will show how to design a reasonable function $D$ .", "For satisfying the above two property, and we propose a specific function: $D(F) = sign(F) = {\\left\\lbrace \\begin{array}{ll}1, &\\quad F_i \\ge 0 \\\\-1, &\\quad F_i < 0\\end{array}\\right.", "}$ Since Eq.", "REF is bounded, $D$ satisfies Lipschitz continuity condition.", "And the range of $D$ is $\\lbrace 1,-1\\rbrace $ .", "In section , we will show that applying Eq.", "REF can significantly improve the adversarial robustness of the model at the same time, keeping accuracy on clean examples.", "By observing Eq.", "REF , it is found that the equation is a piecewise differentiable function whose non-differentiable point is the origin.", "The defence function $D$ is too simplistic and extreme.", "Because the input is all integers or all negative Numbers, the model loses all available information through this function, resulting in network convergence failure.", "In the experiment section , we will introduce that the deeper the network layer where function $D$ (Eq.", "REF ) is located, the worse the performance of the network under clean samples.", "Therefore, based on equation Eq.", "REF , we propose a more general defence function as following: $D(F) = sign(F) = {\\left\\lbrace \\begin{array}{ll}b_u, &\\quad F_i \\in [a_u, +\\infty ] \\\\..., &\\quad F_i \\in [...,...] \\\\b_l, &\\quad F_i \\in [a_l, -\\infty ]\\end{array}\\right.", "}$ where $b_u$ is the upper bound, $b_l$ is the lower bound, and $\\lbrace a_u,..,a_l\\rbrace $ are the non-differentiable points.", "[] Defense Algorithm [1] An dataset $\\mathcal {D}=\\lbrace (I^{(i)}, y^{(i)})\\rbrace _{i=1}^N$ ; A trained network $G=f\\cdot D \\cdot T=(I;\\theta )$ Initialize hyperparameter $k$ Initialize epoches i=1 to epoches $k$ steps Sample $m$ samples $d=\\lbrace I^{(1)},I^{(2)},\\cdots , I^{(m)}\\rbrace $ Update the parameters by descending its stochastic gradient: $\\bigtriangledown \\frac{1}{m}\\sum _{i=1}^{m} {y^{(i)} \\cdot log(G(I^{(i)}))}$ Output $\\theta $" ], [ "Experiments", "In this section, we use Python 3.6 and Jupyter Notebook to conduct three types of adversarial attack method.", "Using toolbox Advertorch [50], we implement several adversarial attack methods as a baseline.", "For imperceptible adversarial examples, we compared our method with existed attack methods on three datasets such as MNIST [51], CIFAR10 [52] and IMAGENET [40].", "Secondly, we use our method to generate various types of unrecognizable adversarial examples.", "Finally, we test our method, which generates the adversarial patches.", "In the setting in which the patch size is same, we compared our method with existed methods.", "MNIST dataset consists of 60,000 training samples and 10,000 samples, each of which is a 28x28 pixel handwriting digital image.", "CIFAR10 dataset is composed of 60,000 32x32 colour image, 50,000 for training and 10,000 for testing.", "ImageNet is large colour image dataset which consists of more than 1,400,000 natural images and 1000 classes.", "For any sample from MNIST, CIFAR10 and ImageNet, we make a preprocess on it, e.g., normalization described as below: $I = \\frac{I-\\mu }{\\sigma }$ where $\\mu $ is the mean pixel value of the whole dataset and $\\sigma $ is the standard deviation of that.", "The detail is showed in Table REF .", "Table: THE MEAN VALUE AND STANDARD DEVIATION IN MNIST, CIFAR10 AND IMAGENET" ], [ "Imperceptible Adversarial Examples", "In this subsection, we test the first attack method (e.g.", "Eq.", "REF ) on MNIST, CIFAR10 and ImageNet.", "Our tiny attack method is a novel adversarial attack method.", "We compare it with existed methods such as FGSM [6], PGD [16], SinglePixel [13], CW [10], LBFGS [5] and L2BIA [7].", "We utilize the LeNet [1], ResNet18 [3] and pre-trained VGG19 [42] to test MNIST, CIFAR10 and ImageNet, respectively.", "We set $\\epsilon $ as 0.3 when attacking on MNIST and 0.03 on ImageNet.", "Moreover, we set respectively $\\alpha $ as 0.3 and 0.03 when attacking CIFAR10.", "For multi-step attacks, let the number of iterations uniformly to 100 steps.", "In the best case, our attack method can drop the accuracy down to $59.95\\%$ (Table REF ) with $\\epsilon $ =0.3 on MNIST.", "Our attack method is second only PGD.", "For more challenging cases, such as imperceptible adversarial examples on CIFAR10, our method drop the accuracy down to $3.03\\%$ , with $\\epsilon $ =0.3 (Table REF ) which is second only PGD, and $46.69\\%$ with $\\epsilon $ =0.03 (Table REF ) which is second only PGD and CW.", "Moreover, we test our tiny attack method on ImageNet with $\\epsilon $ =0.03.", "The adversary drops the accuracy down to $49.07\\%$ (Table REF ), which is second only PGD and CW.", "To validate the effectivity of our attack method, we repeat experiments by adjusting the $\\alpha $ value of Eq.", "REF .", "Fig.", "REF shows the effectivity of attack with different $\\alpha $ .", "It can be seen that the closer $\\alpha $ is to zeros, the worse the performance of attack.", "Moreover, we find that when locating in [0,1], the attack achieves the best performance.", "As $\\alpha $ gradually increases, the success of the attack first rises rapidly and then becomes stable.", "Figure: A trend of attack's power with various α\\alpha values.", "We find that closer α\\alpha to zero, worse performance of attack." ], [ "Unrecognizable Adversarial Examples", "In this subsection, we utilize Eq.", "REF to create unrecognizable adversarial examples to fool VGG19 neural network, which is trained with ImageNet.", "We regard the images which came from ImageNet as an image from the domain (labelled as $I_{in}$ ).", "And any images which do not belong to $I_{in}$ are out of the domain (labelled as $I_{out}$ ).", "Our goal is creating some noise images and regular images which are classified by DNNs as a class with high confidence." ], [ "Irregular image", "Firstly, we randomly generate a noise image.", "Then randomly choosing an image in the domain to guide the noise to close each other in feature space.", "We find that it is easy to create such a noise image.", "And the DNN classifies it as a class with high confidence (Fig.", "REF )." ], [ "regular image", "Once again, we utilize Eq.", "REF to generate unrecognizable adversarial examples.", "However, those are regular images rather than noise images.", "To find suitable images from out of domain, we utilize the Colored Brodatz Texture (CBT) database and the Multiband Texture (MBT) database [53].", "CBT is a coloured version of the original 112 Brodatz grayscale texture images.", "And MBT is a collection of 154 colour images.", "The colour of the MBT images is mainly the result of inter-band and intra-band texture variation.", "Based on those images, we search unrecognizable adversarial examples in the out domain by minimizing the gay between the image from domain and the image from out domain in feature space.", "Though those images are very different from that from ImageNet, the DNN still mostly believes that they are from the domain (Fig.", "REF ).", "Figure: There are three types of adversarial examples (e.g.", "imperceptible adversarial examples, unrecognizable adversarial examples and adversarial patch).", "(a) Some imperceptible adversarial examples are generated from ImageNet by the attack method of the first version (Eq.", ").", "(b) Some unrecognizable adversarial examples are generated by the attack method of the second version (Eq.", ").", "We utilize the noisy image and the image from the Colored Brodatz Texture (CBT) and the Multiband Texture (MBT) dataset to produce the unrecognizable adversarial example which is classified a class with high confidence.", "(c) Some adversarial patch examples are generated by the attack method of the third version (Eq.", ")." ], [ "Adversarial Patch", "In this subsection, we try to generate adversarial patches.", "Adopting the setting in [23], we set the patch size as $q\\times q$ where $q=\\lfloor ((Length\\ of\\ image)^2c)^{0.5}\\rfloor $ .", "And $c$ is a constant.", "Because the image size in MNIST and CIFR10 datasets are relatively small, the generated adversarial patches are easy to modify the objective reality, so we only generate the adversarial patches to attack VGG19 network on ImageNet.", "The pixel of the patch can be manipulated freely.", "And the position of the patch is randomly located.", "Some examples are shown in Fig.", "REF .", "We show that our method can successfully generate adversarial patches.", "Since limiting facility, we randomly choose 1,000 images from validation dataset of ImageNet to generate their adversarial patches.", "Moreover, we test the accuracy in a different setting.", "The details are described in Fig.", "REF .", "We find that our attack can effectively generate adversarial patches and is better than the Baseline [23] in most of the setting.", "Figure: A trend of attack's power with various sizes of patch.", "We find that our attack can effectively generate adversarial patches and is better than the Baseline in most of setting.We use the part of convolution layers as function $T$ and use the part of linear layers as classifier $f$ .", "We set epochs as 30.", "The learning rate is $0.01$ at the beginning, and then half every 10 epochs.", "The momentum is $0.95$ .", "The batch size is 64.", "The results of the experiment are shown in Tabel REF .", "We find that our defence strategy almost invalidates all adversaries.", "At the same setting, our defending method can effectively defence gradient-based attacks (such as PGD) or non-gradient-based attacks (such as CW).", "Table: Defense on MNIST: Performance of our defending strategy against different adversaries for ϵ=0.3\\epsilon =0.3" ], [ "CIFAR10", "Considering a more complicated case, we use ResNet18 as our basic neural network to train on CIFAR10.", "We set epochs as 30.", "The learning rate is $0.01$ at the beginning, and then half every 10 epochs.", "The momentum is $0.95$ .", "The batch size is 64.", "We try to insert function $D$ into ResNet18.", "The part of ResNet18 before $D$ is considered as $T$ .", "Moreover, the part of ResNet18 after $D$ is considered as $f$ .", "The results of the experiment are shown in Tabel REF .", "As Table REF showed, all adversaries fail to attack our model.", "Table: Defense on CIFAR10: Performance of our defending strategy against different adversaries for ϵ=0.03\\epsilon =0.03" ], [ "ImageNet", "In this part, we apply our defence method in ImageNet dataset.", "We adopt ResNet50 as our model and set epochs as 90.", "Then learning rate is $0.01$ at the beginning and then is adjusted according to this formulation $lr=0.01 0.1^{\\lfloor epoch/30\\rfloor }$ .", "The momentum is $0.95$ .", "The batch size is 256.", "The defensive function is deployed in the back of the first convolutional layer.", "The detail results of the experiments are shown in Table REF .", "As a result, all adversaries fail to attack our defensive model.", "It is noted that the accuracy under different attacks is almost the same with that on clean samples.", "It demonstrates that our defence method can successfully extend more realistic and complicated case.", "Table: Defense on ImageNet: Performance of our defending strategy against different adversaries for ϵ=0.03\\epsilon =0.03" ], [ "Analysis", "We find that our defending method will decrease the accuracy of the model on clean samples.", "And all adversaries fail to attack our model.", "Therefore, if we can prevent the drop in the accuracy of the model on clean samples, the adversarial robustness of the model can be further improved.", "To search the best position where the function $D$ is, we do a simple test in the network layers of LeNet and ResNet18.", "As shown in Fig.", "REF , we find that our defence strategy is best deployed at the front of the network.", "The further function $D$ is deployed, the lower the recognition rate of the model on clean samples, and the ability of the model to resist the attack decreases accordingly.", "Moreover, the recognition rate of our defence model for clean samples is very similar to that of the model for adversarial samples.", "This indicates that the adversaries have failed to attack the defence model.", "We present that function $D$ acts to compress the information.", "After the feature passes through the function $D$ , much information is lost, including adversarial perturbation.", "In the latter part of the neural network, information is extracted step by step.", "If the function $D$ is placed in the back half of the neural network, a large amount of useful information will be lost, making the performance of the model worse.", "Figure: The effect of setting location of function D on network performance.", "(a) Lenet have five layers include two convolutions and three connective layers.", "(b) ResNet18 have a convolution layer, a fully connective layer and four blocks.Moreover, we analyze why function $D$ can effectively play a defensive role by visualizing the changes in the input found after convolution.", "We use the trained ResNet50 network to visualize the input, the output of the first layer of convolution, and the output after function $D$ .", "Function $D$ is Eq.", "REF .", "We compare the degree of contamination of inputs under different disturbances.", "As Fig.", "REF shown, when function $D$ filters the information, it filters the adversarial perturbations to some extent.", "When $\\epsilon \\le 0.06$ , function $D$ filters out adversarial perturbations almost entirely (Columns 2 and 3).", "In the case without function $D$ , the input is heavily contaminated (column 1 and column 4).", "Figure: We use the trained ResNet50 network to visualize the input, the output of the first layer of convolution and the output after function DD.", "Function DD is Eq.", ".", "We compare the degree of contamination of inputs under different disturbances." ], [ "Conclusion", "We show that, according to our definition of adversarial examples, we can generate imperceptible adversarial examples, unrecognizable adversarial examples and adversarial patches and construct defending strategy.", "Our attack methods can effectively generate adversarial examples whose performances are better than the most attack methods (except for PGD and CW in some setting).", "Furthermore, our defending strategy achieve state-of-the-art performance.", "As far as we know, our defence is the best performance than existed methods such as adversarial training.", "Since we do not use adversarial training, our defence is less time-consuming than adversarial training.", "In this paper, we define generalized adversarial examples to include three types of adversarial examples.", "Furthermore, our attack methods are focus on a single hidden layer.", "Starting from our definition, we propose three attack methods and one defence strategy.", "We present that we can construct more powerful attack methods by manipulating multi-hidden layers or adaptively manipulating multi-hidden layers.", "That is what we want to improve in the future.", "Our defence strategy achieves excellent performance.", "However, even if our defence method can defend against the attack, the improper setting will lead to the model’s accuracy decreased in the clean sample.", "Therefore, there is room for improvement in our defence in the future." ], [ "Acknowledgment", "This work was supported in part by Key Project of Natural Science Foundation of China (Grant 61732011), in part by the National Natural Science Foundation of China (Grants 61976141, 61772344 and 61732011), in part by the Natural Science Foundation of SZU (827-000230), and in part by the Interdisciplinary Innovation Team of Shenzhen University." ] ]	2011.14045
[ [ "Time-frequency representation of nonstationary signals: the IMFogram" ], [ "Abstract Iterative filtering methods were introduced around 2010 to improve definitions and measurements of structural features in signal processing.", "Like many applied techniques, they present considerable challenges for mathematicians to theorize their effectiveness and limitations in commercial and scientific usages.", "In this paper we recast iterative filtering methods in a mathematical abstraction more conducive to their understanding and applications.", "We also introduce a new visualization of simultaneous local frequencies and amplitudes.", "By combining a theoretical and practical exposition, we hope to stimulate efforts to understand better these methods.", "Our approach acknowledges the influence of Ciprian Foias, who was passionate about pure, applied, and applications of mathematics." ], [ "Introduction ", "A common way to analyze a complex signal is to decompose it as a superposition of simpler ones.", "Many methods have been devised, from Fourier series to the more modern wavelets.", "These methods are extremely successful in dealing with stationary signals.", "However they are less adapted to non-stationary ones because the decomposition is done in the frequency domain, and when transformed back to the time or spatial domain, yields stationary components.", "In contrast, the iterative filtering approach discussed in this paper offers a compromise between separating frequencies and retaining non-stationarity." ], [ " Overview.", "The traditional approach to time-frequency analysis is rooted in Fourier analysis and was later developed into wavelets and other frame decompositions [7], [10], [15], [17], [27], [1], [11], [12], [2], [13].", "These techniques, overviewed in [42], rely on predetermined bases or frames that are not data-driven.", "Consequently, some results may depend on the chosen basis and may not reflect intrinsic properties of the signal [15].", "These methods often assume that the signal is stationary, or near-stationary.", "Localization with distribution kernel models tries to remove the stationarity assumption.", "It attempts to balance bias, forcing to localize more, with stability, forcing to localize less and average more.", "These traditional decompositions are linear, that is why the corresponding time-frequency representation are defined of linear-type.", "In the last decades some method have been proposed to improve the linear-type time-frequency representations, such as the reassignment method [2] and the synchrosqueezing transform [12].", "Bilinear-type techniques have been also proposed, like the Wigner-Ville distribution and the Cohen's or affine class.", "However these techniques proved to be limited in producing a clean and sharp time-frequency representation [15], [42].", "For all these reasons, more data-driven decompositions have been designed.", "In particular, the empirical mode decomposition [23], which divides the signal into several simple oscillatory components.", "The components are obtained inductively, removing from the signal the average between its upper and lower envelopes, these envelopes being defined by interpolation between the local extremum (maximum for the upper envelope, minimum for the lower envelope).", "Each component, called an intrinsic mode in [23], may then be analyzed separately in the time-frequency domain [22], [25].", "Informally, intrinsic modes try to equate the number of extrema and zero-crossing, making them nicely oscillating, and the average between their upper and lower envelopes is near 0, making them well balanced.", "Decomposition in empirical modes is successful in a wide range of applications [9], [31], [46], [24], [38], [37].", "However, the algorithm may be unstable and may not separate well nearby frequencies, creating the so-called mode mixing problem [33].", "As a consequence, several variants have been proposed [43], [45], [39], [40].", "These methods received considerable attentions from the scientific community, as indicated by Huang and collaborators' papers receiving so far more than 30,000 citations on Scopus.", "Intrinsic modes, produced using the aforementioned techniques, allow to overcome artificial spectrum spread caused by sudden changes in local frequencies, formalized in the Heisenberg-Gabor uncertainty principle [15].", "Regrading the actual computation of the frequency and amplitude content of each intrinsic mode, different approaches have been proposed in the literature, like in [4], [25], [34].", "Hybrid methods optimizing a decomposition on a fixed basis or frame with data-driven amplitude or phase modulation have also been proposed [19], [20], [47], [8], [18], [14], [35], [28], [32]." ], [ "Paper outline. ", "The work of [4] highlights the vast literature on the subject, and from there one can easily get a better view of the progress that has been made during the last 20 years.", "In this paper we consider the iterative filtering method and its variation using the fast Fourier transform, the fast iterative filtering, FIF algorithm, that are inspired by the empirical mode decomposition [23].", "Twenty years ago, Ciprian Foiaş, Mike Jolly, and Wing Suet Li [16] published in the area of numerical analysis.", "Ciprian joked that the method was communist: one step forward and two steps backward!", "Here we are making a capitalist approach: after two steps forward (iterative filtering and its fast version), we take one step backward, for a broader view of the methodology!", "The basic idea is to convolve the signal with a filter determined by features of the signal, and to decompose the signal into finitely many components, called intrinsic mode functions (IMFs).", "Since the filter is determined by the signal itself, each IMF keeps a significant amount of non-stationarity and local frequency characteristics from the signal.", "This makes them easier to interpret by practitioners.", "Our approach is not definitive: we have a theoretical framework, some interesting examples, a new visualization, but complete mathematical justifications remain elusive.", "It is easy to explain the basic principle of iterative filtering method for continuous signals.", "However, in practice, one will most likely encounter discrete finite time series.", "We may also deal with spatial signals, like air pressure on earth, moisture in a field, possibly discretized.", "Some signals, such as those relating gyroscopic data and other sensors may be indexed by the sphere.", "The unifying framework of groups allows us to develop a method that covers many applications.", "Some of our developments may be carried out in homogeneous spaces, a setting that includes the most intriguing examples, but key ideas become hidden in a more complex formalism.", "We present the theoretical framework in section 2.", "Section 3 specializes it to time series.", "A new visualization scheme, the IMFogram (pronounced like “infogram”), is introduced in section 4.", "We present some numerical examples in section 5." ], [ "The theoretical framework ", "Signals are viewed as real valued functions defined on a group such as $\\mathbb {R}$ or $\\mathbb {Z}$ for time series, $\\mathbb {R}^2$ or $\\mathbb {Z}^2$ for spatial signals, the circle or some discretization of it for periodic ones.", "Because we need convolutions and Fourier transforms, we restrict $G$ to be a locally compact abelian group written additively, with a Haar measure, denoted by $\\lambda $ .", "If $G$ is finite, $\\lambda $ is the counting measure up to normalization.", "The following recalls basic abstract Fourier analysis, see [26].", "The convolution of two functions $u$ and $v$ defined on $G$ is $u\\star v(g) = \\int _G u(g-h) v(h) \\,{\\rm d}\\lambda (h) \\, ,\\quad g\\in G.$ The characters of $G$ are all the continuous homomorphisms from $G$ to the complex unit circle.", "The set of all characters, $\\widehat{G}$ , is an abelian group under the pointwise multiplication, called the dual group of $G$ .", "The Fourier transform ${\\mathcal {F}}$ maps linearly a function $u$ defined on $G$ to the function defined on the dual group by ${\\mathcal {F}}u(\\chi ) = \\int u(g)\\overline{\\chi }(g)\\,{\\rm d}\\lambda (g), \\quad \\chi \\in \\widehat{G}\\, .$ It satisfies ${\\mathcal {F}}(u\\star v) = {\\mathcal {F}}u\\,{\\mathcal {F}}v$ .", "It is possible to find a Haar measure on $\\widehat{G}$ , $\\,{\\rm d}\\chi $ , such that the inverse of the Fourier transform is given by ${\\mathcal {F}}^{-1}u(g) = \\int _{\\widehat{G}} u\\bigl (\\chi (g)\\bigr ) \\,{\\rm d}\\chi \\, , \\qquad g\\in G\\,.$ We continue our discussion by defining the central player of the FIF method, the filter.", "Recall that a function $w$ on $G$ is even if $w(x) = w(-x)$ for every $x$ in $G$ .", "Definition 1 Let $G$ be a locally compact abelian group, written additively, with Haar measure $\\lambda $ .", "(i) A function $w$ on $G$ is a filter if it is nonnegative, even, bounded, and $\\displaystyle {\\int w \\,{\\rm d}\\lambda = 1}$ .", "(ii) A double convolution filter $w$ is the self-convolution of a filter $\\tilde{w}$ , that is, $w = \\tilde{w} \\star \\tilde{w}$ .", "(iv) The size, or the length, of a filter $w$ is the Haar measure of its support, $\\ell (w) = \\lambda \\lbrace \\, w>0\\,\\rbrace $ .", "This definition implies that convolutions of filters are filters.", "The range of the Fourier transform of a filter is in $[\\,-1,1\\,]$ , and that of a double convolution filter is in $[\\,0,1\\,]$ .", "Consider a possibly non-stationary signal $s$ in $L^2(G)$ .", "Its moving average with respect to a filter $w$ is defined as the convolution $\\nonumber \\mathcal {C}_w s = w \\star s \\,.$ We subtract the moving average from the signal and obtain the variation of the signal around its $w$ -moving average, $\\nonumber \\mathcal {V}_ws = s - \\mathcal {C}_ws.$ Iterating $p$ -times the linear operator $\\mathcal {V}_w$ we obtain the linear IMF operator, $\\nonumber \\mathcal {I}_{w,p}=\\mathcal {V}_w^p .$ We will discuss how to choose the filter $w$ and this $p$ later.", "Since filters are bounded, IMF operators are endomorphisms on both $L_1(G)$ and $L_2(G)$ .", "We can rewrite these operators on the dual group using the Fourier transform.", "Representing the pointwise multiplication of functions by a dot ($\\cdot $ ), $\\nonumber \\mathcal {F}\\,\\mathcal {I}_{w,p}s = (1 - \\mathcal {F}w)^p \\cdot \\mathcal {F}s$ Taking the inverse Fourier transform, $\\nonumber \\mathcal {I}_{w,p}s = \\mathcal {F}^{-1}\\bigl ((1 - \\mathcal {F}w)^p \\cdot \\mathcal {F}s\\bigr ) \\, .$ The construction of IMFs is iterative.", "Starting with a signal $s$ , we obtain the first IMF as some ${\\mathcal {I}}_{w_1,p_1}s$ .", "We then consider the remainder, $s_1 = s - {\\mathcal {I}}_{w_1,p_1}s$ and construct a second IMF as some ${\\mathcal {I}}_{w_2,p_2}s_1$ .", "More generally, given a reminder $s_{n-1}$ , we define the $n$ -th IMF as ${\\mathcal {I}}_{w_n,p_n}s_{n-1}$ for some $w_n$ and $p_n$ , and the next reminder as $s_{n} = s_{n-1} - {\\mathcal {I}}_{w_n,p_n}s_{n-1}$ .", "Whenever $G$ and its dual are finite, all the linear operators involved so far may be written as finite matrices.", "We agree to list the elements of $G$ and its character group $\\widehat{G}$ in some chosen specific order.", "A function $f$ on $G$ is then a vector $\\bigl (f(g)\\bigr )_{g\\in G}$ and the Fourier transform $\\mathcal {F}$ is the matrix ${\\mathcal {F}} = \\bigl (\\overline{\\chi }(g)\\bigr )_{\\chi \\in \\widehat{G},\\,g\\in G}$ , with its rows indexed by the character group $\\widehat{G}$ and its columns by the group $G$ .", "Then $\\nonumber \\mathcal {V}_{w} = \\mathcal {F}^{-1} {\\rm diag}(1 - \\mathcal {F}w) \\mathcal {F} \\, ,$ $\\mathcal {I}_{w,p} = \\mathcal {V}_{w}^p= \\mathcal {F}^{-1}{\\rm diag}(1 - \\mathcal {F}w)^p \\mathcal {F} \\, .$ The $n$ -th IMF is obtained by applying a linear operator of the following form to $s$ : $\\nonumber \\mathcal {F}^{-1} {\\rm diag}(1 - \\mathcal {F}w_n)^{p_n} \\prod _{k=1}^{n-1} \\bigl ( 1 - {\\rm diag}(1 - \\mathcal {F}w_k)^{p_k} \\bigr ) \\mathcal {F}\\, .$ However, the $n$ -th IMF is not a linear function of $s$ since the $w_k$ and $p_k$ are determined by $s$ .", "Assume that $w$ is a double convolution filter, ensuring that its Fourier transform is in $[\\,0,1\\,]$ .", "Then $\\nonumber \\lim _{p\\rightarrow \\infty } {\\rm diag}(1 - \\mathcal {F}w)^p = {\\rm diag}\\bigl ({1}\\lbrace \\mathcal {F}w(\\chi )=0\\rbrace \\bigr )_{\\chi \\in \\widehat{G}} \\, .$ This limit is the orthogonal projection on the subspace of functions in $L^2(\\widehat{G})$ that vanish on the support of $\\mathcal {F}w$ .", "It is a bandpass filter when $G$ is $\\mathbb {Z}_{N}$ , and it preserves no non-stationarity information from the original signal.", "In practice, the choice of $p$ reflects the desire that the IMFs focus on a section of the spectrum, that is $ {\\rm diag}(1 - \\mathcal {F}w)^p $ is fairly concentrated, yet not the characteristic function of a set.", "This allows the IMF to retain some non-stationarity features." ], [ "Application to time dependent signals", "Consider a signal $\\sigma $ on the time interval $[\\,0,L\\,]$ sampled at rate $B$ per time unit.", "This signal is represented as a vector of size $N=BL$ .", "As it is customary in the signal processing literature, we wrap this signal on a discrete circle: consider the quotient group $\\mathbb {Z}_N=\\mathbb {Z}/N\\mathbb {Z}$ and set $t_i=i/B$ , $i\\in \\mathbb {Z}_N$ .", "Our discretized and wrapped signal is $s=(\\sigma (t_i))_{i\\in \\mathbb {Z}_N}$ .", "The discrete Fourier transform $\\mathcal {F}s$ is calculated on the points $k/L$ , $k\\in \\mathbb {Z}_N$ , in the interval $[\\,0,B\\,]$ .", "Let $\\omega $ be $e^{-2\\pi \\imath /N}$ , the $n$ -th root of unity.", "The Fourier transform operator $\\mathcal {F}$ is the $N\\times N$ matrix with entries ${\\mathcal {F}}_{i,j}= \\omega ^{i-j}/\\sqrt{N}$ , with $i,j = 0, 1, \\dots , N-1$ .", "Next we discuss three key ingredients of the FIF algorithm: the choice of the filters $w_j$ , the powers $p_j$ , and how to ensure that the algorithm stops in finite time.", "We only give a brief discussion that suffices to convey the essential ideas, and refer the reader to [6], [3] for details." ], [ "Choosing the filters", "As indicated at the end of last section, in view of (REF ), we are more interested in the shape of $\\mathcal {F}w$ , the Fourier transform of the filter than the filter itself.", "Since $w$ is compactly supported, its Fourier transform has unbounded support.", "Definition REF (ii) implies that $\\mathcal {F}{w}(0)=1$ .", "For the particular filter that we pick, $\\mathcal {F}{w}$ decreases from $\\mathcal {F}{w}(0)$ until it reaches zero.", "After this smallest positive zero $\\xi $ , it oscillates with smaller and smaller amplitude.", "By requiring $\\mathcal {F}w$ to be smaller than some desired quantity to the right of $\\xi $ we create a damping effect in the frequency domain.", "By damping and not annihilating frequencies, the IMF retains the non-stationarity of the signal, and yet concentrates the frequency range of the IMF above $\\xi $ .", "We observe that the zero set of $\\mathcal {F}w$ is preserved by self convolution of $w$ since $\\mathcal {F}w^{\\star k}=(\\mathcal {F}w)^k$ .", "We use only double convolution filters.", "This ensures that $\\mathcal {F}w$ is in $[\\,0,1\\,]$ and $\\mathcal {F}w^{\\star k}$ decreases with $k$ except at the origin.", "Since $\\ell (w^{\\star k})=\\min \\lbrace k\\ell (w), N\\rbrace $ , the filter length of $w^{\\star k}$ increases with $k$ .", "A long filter length makes moving averaging less local, and therefore increases stationarity.", "In practice we choose a double convolution filter $w_0$ with support $[\\,-1/2, 1/2\\,]$ and take a dilation $w(\\cdot )=w_0(\\cdot /\\ell )/\\ell $ .", "To choose the filter length and control the frequency focus, we compute the median of the distances between two consecutive local extrema.", "This median is an estimate of the half period of the highest observed frequency.", "We take the filter length $\\ell $ to be $\\nu $ times this median, where $\\nu $ is a tuning parameter that accounts for the concentration of the filter.", "We follow the iterative construction of the IMFs described in the previous section.", "The $j$ -th filter $w_j$ is constructed as a dilation of $w_0$ as indicated above from the local extrema of the signal $s_{j-1}$ .", "Each filter focuses on sections of the spectrum of the signal.", "The first one, $w_1$ , focuses on a group of highest frequencies, the second one, $w_2$ , the next group of frequencies, etc.", "If $\\xi _{j}$ is the first positive zero for the $j$ -th filter, the first $\\text{IMF}$ focuses on frequencies around and higher than $\\xi _{1}$ , the second $\\text{IMF}$ focuses on frequencies around $\\xi _{2}$ , etc.", "Because of the damping effect, there is a small residue of frequencies that are higher than $\\xi _{j}$ in the $j$ -th IMF.", "Numerically we have observed from the periodogram of the $j$ -th IMF that its energy is concentrated around the frequency $\\xi _{j}$ , and very little beyond $\\xi _{j-1}$ .", "Put differently, IMFs tend to keep together nearby frequencies on the important parts of the periodogram." ], [ "Deciding the power $\\mathbf {p_j}$ ", "Equation (REF ) gives us an estimate for the norm of the matrix $\\Vert \\mathcal {V}_{w} ^{p+1} - \\mathcal {V}_{w}^{p} \\Vert $ .", "Indeed, write $D$ for ${\\rm diag}(1 - \\mathcal {F}w)$ .", "The function $(1-x)^px$ being bounded by $1/(ep)$ on $[\\,0,1\\,]$ , each diagonal element of $(1-D)^pD$ is between 0 and $1/(ep)$ .", "Since $\\mathcal {F}$ is an isometry, we then have $\\nonumber \\Vert \\mathcal {V}_{w}^{p+1} - \\mathcal {V}_{w}^{p} \\Vert = \\Vert \\mathcal {F}^{-1}(1-D)^pD\\mathcal {F}\\Vert \\le \\Vert (1-D)^{p}D\\Vert \\le 1/(ep).$ In practice, we set a threshold $\\delta $ , typically $10^{-3}$ or $10^{-4}$ , and for a signal $s$ , we choose $p$ to be the smallest integer such that $\\Vert \\mathcal {I}_{w,p+1}s - \\mathcal {I}_{w,p} s \\Vert _2\\le \\delta \\Vert s\\Vert _2$ .", "The actual number $p$ is much smaller than that from the theoretical estimate." ], [ "Ensuring the algorithm terminates", "Because IMFs are not bandpass filters but instead just focus on sections of the spectrum, the remainder signal $s_{j}$ may still contains some high frequencies.", "This makes possible for the filter length chosen as in section REF to decrease.", "To prevent such a decrease we force the filter length to increase by $10\\%$ or some appropriate amount of the previous filter length when this happens.", "It is also possible that an IMF is insignificant, that is, $\\Vert \\mathcal {I}_{w, 1} s_{j}\\Vert _2\\le \\delta \\Vert s_{j}\\Vert _2$ .", "In this case, we redo the iteration by forcing the filter length to increase by $10\\%$ .", "Since the signal is on the discrete circle, its frequency domain contains finitely many points.", "This $10\\%$ increase guarantees that the algorithm stops.", "In real life applications that we encountered, there are sufficiently many significant IMFs, and only a trend with at most one extrema is left as the last remainder." ], [ "IMFogram", "The periodogram and its localized version, the spectrogram, are plots adapted to Fourier decomposition.", "Because each Fourier component has a specific frequency and these components are mutually orthogonal, the total energy of the signal is the sum of each component's energy.", "In contrast, each IMF does not have a fix frequency and the IMFs are not mutually orthogonal.", "The IMFogram is a simple analog of the spectrogram that can be quickly computed on IMF decompositions.", "It is defined as follows: Let $\\eta $ be a parameter — 5 to 20 in practice.", "Consider an IMF, denoted by $f$ , which was produced with a filter of length $\\ell $ .", "Its local energy at $t$ is approximated by $\\nonumber E_f(t) =\\frac{1}{2\\eta \\ell }\\int _{t-\\eta \\ell }^{t+\\eta \\ell } f(\\tau )^2 \\,{\\rm d}\\tau .$ Because an IMF focuses on a narrow frequency section, the local frequency of $f$ may be approximated by $\\nonumber \\Omega _f(t) = \\frac{1}{4\\eta \\ell } \\times \\text{number of $0$-crossings of $f$ over the interval $[\\,t-\\eta \\ell , t+\\eta \\ell \\,]$}.$ There are many other possible approximations discussed comprehensively in [21].", "We obtain time, frequency, energy triples $(t_i, \\Omega _{f}(t_i), E_{f}(t_i))_{i\\in \\mathbb {Z}_N}$ for $f$ .", "Consider a discretized signal $s$ as in section .", "Its time domain in $\\lbrace i/B: i\\in \\mathbb {Z}_N\\rbrace $ and its frequency domain is $(\\mathbb {Z}_N/L)\\cap [\\,0,B/2\\,]$ .", "We partition the time-frequency domain in rectangles.", "For a rectangle $R$ , let $\\Pi _{t}R$ be its projection onto the time coordinate and $\\Pi _{\\omega }R$ its projection onto the frequency coordinate.", "Since the time domain is discrete, the cardinality $\\# \\Pi _t R$ is finite.", "Let $s_1, s_2, \\dots , s_k$ be the IMF decomposition of $s$ .", "To a rectangle $R$ we associate an energy defined as a sum of the average local energies of each IMF when the local frequency lies in $\\Pi _{\\omega }R$ , that is $\\nonumber E_s(R) = \\sum _{1\\le j \\le k} \\frac{1}{\\#\\Pi _{t}R}\\sum _{\\tau \\in \\Pi _{t}R} E_{s_j}(\\tau ){1}\\Bigl \\lbrace \\Omega _f(\\tau ) \\in \\Pi _{\\omega }R \\Bigr \\rbrace $ We recommend to choose the length of $\\Pi _{t}R$ comparable to the smallest filter length $\\ell (w_1)$ so that changes in high frequencies are well represented.", "Depending on the application, one may take rectangles of different sizes to cover the time and frequency domain.", "The IMFogram of $s$ is the plot of the step function that equals to $E_s(R)$ on each rectangle $R$ .", "Sometimes one may want to discard small IMFs when plotting the IMFogram for a signal." ], [ "Numerical Examples", "In this section we present a few examples of time frequency analysis of both synthetic and real world signals, and the ability of FIF to handle higher dimensional data.", "The double convolution filter $w_0$ that we use for these examples is obtained by taking $\\tilde{w}$ to be the stationary solution from a Fokker-Plank equation [4].", "This approach shares similarities with [41] where filters are solutions of a Fokker-Plank equation, and filter length is in one-to-one correspondence with the time in the evolution equation.", "Our approach results in a faster algorithm because the filters are dilation of a fixed $w_0$ ." ], [ "Synthetic signal", "Consider a synthetic signal mixing a nonlinear chirp (rapid change in frequency) and a bandpass time-varying noise, Figure REF .", "This example was studied in other previous works, like for instance in [44].", "Figure: Time frequency representation of one chirp in time-varying noise.", "Left: spectrogram.", "Right: IMFogram.Figure REF compares the classical short time window spectrogram and IMFogram.", "As we see, the IMFogram is more concentrated on the time-frequencies used to generate the signal." ], [ "Piano recording", " Figure: NO_CAPTION We consider the time frequency analysis of the first four notes of A vous dirais-je Maman.", "The four notes from the right hand are Do Do Sol Sol in quarter notes and the left hand are Do Sol in half notes.", "Figure REF shows the original recording, top row, and the 4-th, 5-th, 6-th, and 7-th IMFs, and their periodograms.", "We see that the 4-th IMF pick up the upper Sol, the next one the Do beat opening the piece.", "The left hand is captured by the 6-th and 7-th IMFs.", "We also see some harmonics of the notes, which reflects the fact that piano notes are complex sounds, especially on the lower register.", "Compare with spectrogram, we see in Figure REF that the IMFogram gives a sharper view of the notes.", "This example suggests that FIF may be of value for automatic scoring, which is useful for music transcription and its applications.", "Figure: Time frequency representation of the first four quarter notes of the A vous dirais-je Maman.", "Left: Spectrogram.", "Right: IMFogram.Figure: Top row: original piano recording and its periodogram.", "Next four rows: 4-th to 7-th IMFs and their periodograms." ], [ "Global temperature", "To show the ability of FIF to handle 2-dimensional data.", "The Earth's air temperature measurements are made available through the NCEP/NCAR Reanalysis project.", "The original NCEP Reanalysis data have been provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA [29] and have been edited by the Climatic Research Unit, University of East Anglia [30].", "They consists of measurements taken on a global grid at an altitude of 2 meters from the surface in Kelvin.", "We choose the data from January 1, 2014, for a quarter-spherical window between $0^{\\circ } $ N – $75^{\\circ } $ N and $90^{\\circ }$ W – $90^{\\circ } $ E. This covers eastern North America, most of the north Atlantic Ocean, Europe, Africa above equator, western and continental Asia, as illustrated in the top right panel of Figure REF .", "Some of the IMFs produced by a two dimensional FIF method [36], [5] are exhibited on the second and third rows of Figure REF show.", "The bottom right panel shows a broad trend: temperatures are about constant and warm on the equator, decreasing on the more continental land masses of the northern hemisphere, with the Atlantic ocean warmer than those northern land masses.", "The bottom left panel shows broad fluctuations around this trend.", "The continental climates of northern USA and Canada, similar to that of Russia, are markedly colder than other regions.", "On the middle right panel, the depression near $90^{\\circ }$ E – $31^{\\circ }$ N corresponds to the Himalaya region, while that around $85^{\\circ }$ W – $50^{\\circ }$ E corresponds to the north of the Great Lakes area.", "The middle left panel gives other fluctuations that may have other interpretations.", "Figure: Top left: Earth's air temperature (in K) on January 1, 2014, in a quarter-spherical window covering Eastern America and West/Central Eurasia.", "Top right: land and sea map corresponding to the data.", "Middle and bottom rows: some of the IMFs components generated by the FIF2 algorithm." ], [ "Conclusion", "The examples show the value of the IMFogram and confirm that of iterative filtering methods.", "The IMFogram is developed as a visualization tool to “see” the local frequency and amplitude information of a signal simultaneously.", "By varying the partition of frequency-amplitude space, it provides the visualization in multiple resolutions.", "Not only the IMFogram can be used together with the FIF and EMD algorithms as demonstrated in this paper, but also with other decomposition strategies for non-stationary signals.", "By bringing sharper frequency capability than the spectrogram, the IMFogram seems a good candidate to improve automated frequency identification techniques.", "The similarities and differences between spectrogram and IMFogram shown in the examples presented in this work suggest that there is a mathematical connection between these two time-frequency representations.", "We plan to explore this direction of research in a future work.", "The group theoretic framework that we introduced shows that time indexed and spaced indexed methodologies can be unified and that iterative filtering methods can be applied to a wider range of problems than previously considered.", "This framework also raises interesting questions on discretization and sampling on homogeneous spaces and Lie groups which we hope to tackle in a future work." ] ]	2011.14209
[ [ "On Parameterized Complexity of Liquid Democracy" ], [ "Abstract In liquid democracy, each voter either votes herself or delegates her vote to some other voter.", "This gives rise to what is called a delegation graph.", "To decide the voters who eventually votes along with the subset of voters whose votes they give, we need to resolve the cycles in the delegation graph.", "This gives rise to the Resolve Delegation problem where we need to find an acyclic sub-graph of the delegation graph such that the number of voters whose votes they give is bounded above by some integer {\\lambda}.", "Putting a cap on the number of voters whose votes a voter gives enable the system designer restrict the power of any individual voter.", "The Resolve Delegation problem is already known to be NP-hard.", "In this paper we study the parameterized complexity of this problem.", "We show that Resolve Delegation is para-NP-hard with respect to parameters {\\lambda}, number of sink nodes and the maximum degree of the delegation graph.", "We also show that Resolve Delegation is W[1]-hard even with respect to the treewidth of the delegation graph.", "We complement our negative results by exhibiting FPT algorithms with respect to some other parameters.", "We finally show that a related problem, which we call Resolve Fractional Delegation, is polynomial time solvable." ], [ "Introduction", "In a direct democracy, agents vote for a candidate by themselves.", "In liquid democracy, the voters can delegate their votes to other agents who can vote on their behalf.", "Suppose voter 1 delegates her vote to voter 2 and voters 2 and 3 delegate their votes to voter 4.", "Then voter 4 has a voting power equivalent to 4 individual votes.", "That is delegations are transitive.", "This particular feature can make liquid democracy a disruptive approach to democratic voting system.", "This happens because such a voting system can lead to what we call a super-voter who has a lot of voting power.", "So now the candidates instead of trying to appease the general public can do behind the closed door dealings with the super-voters and try to win the election in an unfair manner.", "In order to deal with this issue, a central mechanism ensures that no super-voter has a lot of voting power.", "Formally we do it as follows.", "We create a delegation graph where the set of vertices is the set of voters and we have a directed edge from vertex $i$ to vertex $j$ if voter $i$ delegates her vote to voter $j$ .", "We observe that delegation graph may contain cycles.", "Every voter is also allowed to delegate her vote to more than one other voters and let the system decide her final delegation.", "We use a central mechanism to find a acyclic sub-graph of the delegation graph such that no super-voter (the vertices having out-degree 0) has a lot of voting power.", "We call this problem Resolve Delegation.", "An empirical investigation of the existence and influence of super-voters was done by [9].", "They showed that the super-voters can be powerful although they seem to act in a responsible manner according to their results.", "There have been a few theoretical work in this area by [7],[4] and [8].", "A detailed theoretical work especially on the approximation algorithms in this setting was done by [6].", "Some other important work in Liquid democracy includes [2] and [3]." ], [ "Our Contribution", "We study parameterized complexity of the Resolve Delegation problem with respect to various natural parameters.", "In particular, we consider the number of sink vertices ($t$ ), maximum allowed weight $\\lambda $ of any sink in the final delegation graph, maximum degree ($\\Delta $ ), tree-width, number of edges deleted in optimal solution ($e_{rem}$ ), number of non-sink vertices ($\|\\mathcal {V} \\backslash \\mathcal {T} \|$ ).", "The number of sink vertices corresponds to the number of influential voters which is often a small number in practice.", "This makes the number of sink vertices an important parameter to study.", "Similarly, the parameter $\\lambda $ corresponds to the “power” of a voter.", "Since the input to the problem is a graph, it is natural to study parameters, for example, tree-width (by ignoring the directions of the edges) and the number of edges that one needs to delete in an optimal solution.", "We summarize our results in tbl:summary.", "We finally show that Resolve Delegation is polynomial time solvable if we allow fractional delegations [thm:fractionaldelegation].", "Table: Summary of results.A directed graph $\\mathcal {G} $ is a tuple $(\\mathcal {V},\\mathcal {E})$ where $\\mathcal {E} \\subseteq \\lbrace (x,y): x,y\\in \\mathcal {V},x\\ne y\\rbrace $ .", "For a graph $\\mathcal {G}$ , we denote its set of vertices by $\\mathcal {V} [\\mathcal {G} ]$ , its set of edges by $\\mathcal {E} [\\mathcal {G} ]$ , the number of vertices by $n$ , and the number of edges by $m$ .", "Given a graph $\\mathcal {G} =(\\mathcal {V},\\mathcal {E})$ , a sub-graph $\\mathcal {H} =(\\mathcal {V} ^\\prime ,\\mathcal {E} ^\\prime )$ is a graph such that (i) $\\mathcal {V} ^\\prime \\subseteq \\mathcal {V} $ , (ii) $\\mathcal {E} ^\\prime \\subseteq \\mathcal {E} $ , and (iii) for every $(x,y)\\in \\mathcal {E} ^\\prime $ , we have $x,y\\in \\mathcal {V} ^\\prime $ .", "A sub-graph $\\mathcal {H}$ of a graph $\\mathcal {G}$ is called a spanning sub-graph if $\\mathcal {V} [\\mathcal {H} ]=\\mathcal {V} [\\mathcal {G} ]$ and induced sub-graph if $\\mathcal {E} [\\mathcal {H} ]=\\lbrace (x,y)\\in \\mathcal {E} [\\mathcal {G} ]: x,y\\in \\mathcal {V} [\\mathcal {H} ]\\rbrace $ .", "Given an induced path $P$ of a graph, we define end vertex as vertex with 0 outdegree in $P$ and start vertex as a vertex with 0 indegree in $P$ ." ], [ "Problem Definition", "We now define our problem formally.", "[Resolve Delegation ] Given a directed graph $\\mathcal {G} =(\\mathcal {V},\\mathcal {E})$ (also known as delegation graph) with the set $\\mathcal {T} \\subseteq \\mathcal {V} $ as its set of sink vertices and an integer $\\lambda $ , decide if there exists a spanning sub-graph $\\mathcal {H} \\subseteq \\mathcal {G} $ such that The out-degree of every vertex in $\\mathcal {V} \\setminus \\mathcal {T} $ is exactly 1 For every sink vertex $t\\in \\mathcal {T} $ , the number of vertices (including $t$ ) in $\\mathcal {V}$ which has a path to $t$ in the sub-graph $\\mathcal {H}$ is at most $\\lambda $ We denote an arbitrary instance of Resolve Delegation by $(\\mathcal {G},\\lambda )$ .", "In the spanning sub-graph $\\mathcal {H} \\subseteq \\mathcal {G} $ , if there is a path from $u$ to $v$ in $\\mathcal {H}$ such that all the vertices on this path except $v$ has out-degree 1, then we say that vertex $u$ delegates to vertex $v$ .", "In any spanning sub-graph $\\mathcal {H} \\subseteq \\mathcal {G} $ with the out-degree of every vertex in $\\mathcal {V} \\setminus \\mathcal {T} $ is exactly 1 (we call sub-graph $\\mathcal {H}$ a feasible solution), weight of a tree rooted at the sink vertex $u$ is the number of vertices (including $u$ ) that have a directed path to $u$ .", "We study parameterized complexity of Resolve Delegation with respect to $t$ , $\\lambda $ , and the maximum degree $\\Delta $ of the input graph as our parameters.", "In the optimization version of Resolve Delegation, we aim to minimize $\\lambda $ ." ], [ "Results: Algorithmic Hardness", "Our first result shows that Resolve Delegation is $\\mathsf {NP}$ -complete even if we have only 3 sink vertices.", "For that, we exhibit reduction from the Two Vertex Disjoint Paths problem.", "[Two Vertex Disjoint Paths ] Given a directed graph $\\mathcal {G} =(\\mathcal {V},\\mathcal {E})$ , two pairs $(s_1,t_1)$ and $(s_2,t_2)$ of vertices which are all different from each other, compute if there exists two vertex disjoint paths $\\mathcal {P} _1$ and $\\mathcal {P} _2$ where $\\mathcal {P} _i$ is a path from $s_i$ to $t_i$ for $i\\in [2]$ .", "We denote an arbitrary instance of it by $(\\mathcal {G},s_1,t_1,s_2,t_2)$ .", "We know that Two Vertex Disjoint Paths is $\\mathsf {NP}$ -complete [5].", "The idea is to add paths containing large number of nodes in the instance of Resolve Delegation which we are creating using the instance of Two Vertex Disjoint Paths.", "This key idea will make both the instances equivalent.", "The Resolve Delegation problem is $\\mathsf {NP}$ -complete even if we have only 3 sink vertices.", "In particular, Resolve Delegation is para-$\\mathsf {NP}\\text{-hard}$ with respect to the parameter $t$ .", "The Resolve Delegation problem clearly belongs to $\\mathsf {NP}$ .", "To show its $\\mathsf {NP}$ -hardness, we reduce from Two Vertex Disjoint Paths.", "Let $(\\mathcal {G} =(\\mathcal {V},\\mathcal {E}),s_1,t_1,s_2,t_2)$ be an arbitrary instance of Two Vertex Disjoint Paths.", "Let $n=\|\\mathcal {V} \|$ .", "We consider the following instance $(\\mathcal {G} ^\\prime =(\\mathcal {V} ^\\prime ,\\mathcal {E} ^\\prime ),\\lambda )$ .", "$\\mathcal {V} ^\\prime &= \\lbrace a_v: v\\in \\mathcal {V} \\rbrace \\cup \\mathcal {D} _1 \\cup \\mathcal {D} _1^\\prime \\cup \\mathcal {D} _2\\cup \\mathcal {D} _2^\\prime \\cup \\mathcal {D} _3 \\text{ where}\\\\& \|\\mathcal {D} _1\|=\|\\mathcal {D} _2^\\prime \|=10n, \|\\mathcal {D} _1^\\prime \|=\|\\mathcal {D} _2\|=5n,\|\\mathcal {D} _3\|=15n\\\\\\mathcal {E} ^\\prime &= \\lbrace (a_u, a_v): (u,v)\\in \\mathcal {E} \\rbrace \\cup \\mathcal {F} $ We now describe the edges in $\\mathcal {F}$ .", "Each $\\mathcal {D} _1, \\mathcal {D} _1^\\prime , \\mathcal {D} _2,\\mathcal {D} _2^\\prime $ and $\\mathcal {D} _3$ induces a path in $\\mathcal {G} ^\\prime $ and thus the edges in these paths are part of $\\mathcal {F}$ .", "The end vertices of the path induced on $\\mathcal {D} _1$ and $\\mathcal {D} _2$ be respectively $d_1$ and $d_2$ .", "The start vertices of the path induced on $\\mathcal {D} _1^\\prime $ and $\\mathcal {D} _2^\\prime $ be respectively $d_1^\\prime $ and $d_2^\\prime $ .", "The end vertices of the path induced on $\\mathcal {D} _1^\\prime ,\\mathcal {D} _2^\\prime $ and $\\mathcal {D} _3$ be $t_1^\\prime ,t_2^\\prime $ and $t_3^\\prime $ respectively.", "The set $\\mathcal {F}$ also contains the edges in $\\lbrace (d_1,a_{s_1}), (d_2,a_{s_2}), (a_{t_1},d_1^\\prime ), (a_{t_2},d_2^\\prime )\\rbrace $ .", "$\\mathcal {F}$ also contains edge $(a_v, t_3^\\prime )$ $\\forall v \\in \\mathcal {V} $ .", "This finishes the description of $\\mathcal {F}$ and thus the description of $\\mathcal {G} ^\\prime $ .", "We observe that $\\mathcal {G} ^\\prime $ has exactly 3 sink vertices, namely $t_1^\\prime ,t_2^\\prime $ and $t_3^\\prime $ .", "Finally we define $\\lambda =17n$ .", "We claim that the two instances are equivalent.", "In one direction, let us assume that the Two Vertex Disjoint Paths instance is a yes instance.", "For all $i\\in [2]$ , let $\\mathcal {P} _i$ be a path from $s_i$ to $t_i$ in $\\mathcal {G}$ such that $\\mathcal {P} _1$ and $\\mathcal {P} _2$ are vertex disjoint.", "We build the solution $\\mathcal {H}$ for Resolve Delegation by first adding the set of edges $\\lbrace (u,v)\|\\text{outdegree of $u$ is 1}\\rbrace $ .", "Then we add the paths $\\mathcal {P} _1$ and $\\mathcal {P} _2$ .", "Then we add the edges $(a_{t_1},d_1^\\prime ), (a_{t_2},d_2^\\prime )$ .", "Then for each vertex $u$ in the set $\\mathcal {V} ^r$ $=\\lbrace a_v \| v \\in \\mathcal {V} \\rbrace \\backslash \\mathcal {V} [\\mathcal {P} _1 \\cup \\mathcal {P} _2] $ , add the edge $(u,t_3^\\prime )$ to $\\mathcal {H}$ .", "We observe that the out degree of every vertex is exactly 1 in $\\mathcal {H} $ except the sink vertices in $\\mathcal {G} ^\\prime $ (which are $t_1^\\prime ,t_2^\\prime $ and $t_3^\\prime $ ).", "Also since $\\mathcal {H}$ contains the path $\\mathcal {P} _i$ , every vertex in $\\mathcal {D} _i$ has a path to $t_i^\\prime $ for $i\\in [2]$ .", "Of course, every vertex in $\\mathcal {D} _i^\\prime $ has a path to $t_i^\\prime $ for $i\\in [2]$ and every vertex in $\\mathcal {D} _3$ delegates to $t_3^\\prime $ .", "Hence $\\forall i\\in [3]$ , the number of vertices which has a path to $t_i$ in $\\mathcal {H} ^\\prime $ is at most $16n$ which is less than $\\lambda $ .", "Hence the Resolve Delegation instance is a yes instance.", "In the other direction, let us assume that the Resolve Delegation instance is a yes instance.", "Let $\\mathcal {H} ^\\prime =(\\mathcal {V} ^\\prime ,\\mathcal {E} ^{\\prime \\prime })\\subseteq \\mathcal {G} ^\\prime $ be a spanning sub-graph of $\\mathcal {G} ^\\prime $ such that (i) the out degree of every vertex which is not a sink is exactly 1, (ii) there are at most $\\lambda \\;(=17n)$ vertices (including the sink nodes) in $\\mathcal {H} ^\\prime $ which has a path to $t_i^\\prime $ for $i\\in [3]$ .", "Note that $a_{s_1}$ must have a path $\\mathcal {P} _1^\\prime $ to $a_{t_1}$ in $\\mathcal {H} ^\\prime $ otherwise at least $20n$ vertices have path to either $t_{2}^\\prime $ or $t_3^\\prime $ in $\\mathcal {H} ^\\prime $ which is a contradiction (since $\\lambda =17n$ ).", "Similarly $a_{s_2}$ must have a path $\\mathcal {P} _2^\\prime $ to $a_{t_2}$ in $\\mathcal {H} ^\\prime $ otherwise at least $20n$ vertices have path to either $t_{1}^\\prime $ or $t_3^\\prime $ in $\\mathcal {H} ^\\prime $ which is a contradiction (since $\\lambda =17n$ ).", "Since, for $i\\in [2]$ , we have a path $\\mathcal {P} _i^\\prime $ from $a_{s_i}$ to $a_{t_i}$ in $\\mathcal {H} ^\\prime $ and the out-degree of every vertex in $\\mathcal {H} ^\\prime $ except $t_1^\\prime ,t_2^\\prime $ and $t_3^\\prime $ is 1, the paths $\\mathcal {P} _1^\\prime $ and $\\mathcal {P} _2^\\prime $ are vertex disjoint.", "We define path $\\mathcal {P} _i=\\lbrace (u,v): (a_u,a_v)\\in \\mathcal {P} _i^\\prime \\rbrace $ in $\\mathcal {G}$ for $i\\in [2]$ .", "Since $\\mathcal {P} _1^\\prime $ and $\\mathcal {P} _2^\\prime $ are vertex disjoint, it follows that $\\mathcal {P} _1$ and $\\mathcal {P} _2$ are also vertex disjoint.", "Thus the Resolve Delegation instance is a yes instance.", "$\\Box $ We next show that Resolve Delegation is $\\mathsf {NP}$ -complete even if we have $\\lambda =3$ and $\\Delta =3$ .", "For that we exhibit a reduction from $(3,\\text{B}2)$ -SAT which is known to be $\\mathsf {NP}$ -complete [1].", "[$(3,\\text{B}2)$ -SAT ] Given a set $\\mathcal {X} =\\lbrace x_i: i\\in [n]\\rbrace $ of $n$ variables and a set $\\mathcal {C} =\\lbrace C_j: j\\in [m]\\rbrace $ of $m$ 3-CNF clauses on $\\mathcal {X}$ such that, for every $i\\in [n]$ , $x_i$ and $\\bar{x}_i$ each appear in exactly 2 clauses, compute if there exists any Boolean assignment to the variables which satisfy all the $m$ clauses simultaneously.", "We denote an arbitrary instance of $(3,\\text{B}2)$ -SAT by $(\\mathcal {X},\\mathcal {C})$ .", "For each literal and clause in $(3,\\text{B}2)$ -SAT we add a node in the instance of Resolve Delegation and we add some special set of edges and nodes so that $\\lambda =3$ and both the out-degree and in-degree of every vertex is at most 3 The Resolve Delegation problem is $\\mathsf {NP}$ -complete even if we have $\\lambda =3$ and both the out-degree and in-degree of every vertex is at most 3.", "In particular, Resolve Delegation is para-$\\mathsf {NP}\\text{-hard}$ with respect to the parameter $(\\lambda ,\\Delta )$ .", "The Resolve Delegation problem clearly belongs to $\\mathsf {NP}$ .", "To show its $\\mathsf {NP}$ -hardness, we reduce from $(3,\\text{B}2)$ -SAT.", "Let $(\\mathcal {X} =\\lbrace x_i:{i\\in [n]}\\rbrace , \\mathcal {C} =\\lbrace C_j: j\\in [m]\\rbrace )$ be an arbitrary instance of $(3,\\text{B}2)$ -SAT.", "We define a function $f:\\lbrace x_i,\\bar{x}_i: i\\in [n]\\rbrace \\longrightarrow \\lbrace a_i,\\bar{a}_i: i\\in [n]\\rbrace $ as $f(x_i)=a_i$ and $f(\\bar{x}_i)=\\bar{a}_i$ for $i\\in [n]$ .", "We consider the following instance $(\\mathcal {G} =(\\mathcal {V},\\mathcal {E}),\\lambda )$ .", "$\\mathcal {V} &= \\lbrace a_i, \\bar{a}_i, d_{i,1}, d_{i,2}: i\\in [n]\\rbrace \\cup \\lbrace y_j: j\\in [m]\\rbrace \\\\\\mathcal {E} &= \\lbrace (y_j,f(l_1^j)), (y_j,f(l_2^j)), (y_j,f(l_3^j)): C_j=(l_1^j\\vee l_2^j\\vee l_3^j), j\\in [m]\\rbrace \\\\&\\cup \\lbrace (d_{i,2}, d_{i,1}), (d_{i,1}, a_i), (d_{i,1},\\bar{a}_i): i\\in [n]\\rbrace \\\\\\lambda &= 3$ We observe that both the in-degree and out-degree of every vertex in $\\mathcal {G}$ is at most 3.", "Also $\\Delta =3$ .", "We now claim that the two instances are equivalent.", "Suppose the $(3,\\text{B}2)$ -SAT instance is a yes instance.", "Let $g:\\lbrace x_i: i\\in [n]\\rbrace \\longrightarrow \\lbrace \\text{{\\sc true}}, \\text{{\\sc false}} \\rbrace $ be a satisfying assignment of the $(3,\\text{B}2)$ -SAT instance.", "We define another function $h(g,j)=f(l), j\\in [m],$ for some literal $l$ which appears in the clause $C_j$ and $g$ sets it to true.", "We consider the following sub-graph $\\mathcal {H} \\subseteq \\mathcal {G} $ $\\mathcal {E} [\\mathcal {H} ] &= \\lbrace (d_{i,2}, d_{i,1}): i\\in [n]\\rbrace \\\\&\\cup \\lbrace (d_{i,1},a_i): i\\in [n], g(x_i)=\\text{{\\sc false}} \\rbrace \\\\&\\cup \\lbrace (d_{i,1},\\bar{a}_i ): i\\in [n], g(x_i)=\\text{{\\sc true}} \\rbrace \\\\&\\cup \\lbrace (y_j, h(g,j)): j\\in [m]\\rbrace $ We observe that $\\mathcal {H}$ is a spanning sub-graph of $\\mathcal {G}$ such that (i) every non-sink vertices in $\\mathcal {G}$ has exactly one outgoing edge in $\\mathcal {H}$ and (ii) for each sink vertex in $\\mathcal {G}$ , there are at most 3 vertices (including the sink itself) which has a path to it.", "Hence the Resolve Delegation instance is a yes instance.", "In the other direction, let the Resolve Delegation instance is a yes instance.", "Let $\\mathcal {H} \\subseteq \\mathcal {G} $ be a sub-graph of $\\mathcal {G}$ such that (i) every non-sink vertices in $\\mathcal {G}$ has exactly one outgoing edge in $\\mathcal {H}$ and (ii) for each sink vertex in $\\mathcal {G}$ , there are at most 3 vertices (including the sink itself) which has a path to it.", "We define an assignment $g:\\lbrace x_i: i\\in [n]\\rbrace \\longrightarrow \\lbrace \\text{{\\sc true}}, \\text{{\\sc false}} \\rbrace $ as $g(x_i)=\\text{{\\sc false}} $ if $(d_{i,1},a_i)\\in \\mathcal {E} [\\mathcal {H} ]$ and true otherwise.", "We claim that $g$ is a satisfying assignment for the $(3,\\text{B}2)$ -SAT instance.", "Suppose not, then there exists a clause $C_j=(l_1^j\\vee l_2^j\\vee l_3^j)$ for some $j\\in [m]$ whom $g$ does not satisfy.", "We define functions $f_1,f_2:\\lbrace x_i,\\bar{x}_i: i\\in [n]\\rbrace \\longrightarrow \\lbrace d_{i,1},d_{i,2}: i\\in [n]\\rbrace $ as $f_1(x_i)=f_1(\\bar{x}_i)=d_{i,1}$ and $f_2(x_i)=f_2(\\bar{x}_i)=d_{i,2}$ .", "We observe that the sink vertex $f(l_i^j)$ is reachable from both $f_1(l_i^j)$ and $f_2(l_i^j)$ in $\\mathcal {H}$ for every $i\\in [3]$ .", "Since $\\lambda =3$ , we do not have a path from $y_j$ to any of $f(l_i), i\\in [3]$ which is a contradiction since the non-sink vertex $y_j$ must have out-degree 1 in $\\mathcal {H}$ .", "Hence $g$ is a satisfying assignment for the $(3,\\text{B}2)$ -SAT instance and thus the instance is a yes instance.", "$\\Box $ Given that the input graph is both bipartite and directed acyclic graph, the Resolve Delegation problem is $\\mathsf {NP}$ -complete even if we have $\\lambda =3$ and both the out-degree and in-degree of every vertex is at most 3 which concludes that Resolve Delegation is para-$\\mathsf {NP}\\text{-hard}$ with respect to the parameter $(\\lambda ,\\Delta )$ .", "The corollary follows as the resulting graph $\\mathcal {G}$ from reduction of $(3,\\text{B}2)$ -SAT instance in Theorem is bipartite as $\\mathcal {V} $ can be partitioned into 2 independent sets $\\mathcal {V} _1=\\lbrace y_j:j\\in [m]\\rbrace \\cup \\lbrace d_{i,1}:i\\in [n]\\rbrace $ and $\\mathcal {V} _2=\\lbrace a_i,\\bar{a}_i,d_{i,2}:i\\in [n]\\rbrace .$ Also $\\mathcal {G}$ is Directed Acyclic graph as it doesn't have directed cycles.", "$\\Box $ A (positive integral) $edge$ $weighting$ of a graph $G$ is a mapping $w$ that assigns to each edge of $G$ a positive integer.", "An $orientation$ of $G$ is a mapping $\\Lambda :E(G)\\rightarrow V(G)\\times V(G)$ with $\\Lambda ((u,v))\\in \\lbrace (u,v),(v,u)\\rbrace $ .", "The $weighted$ $outdegree$ of a vertex $v\\in V(G)$ w.r.t an edge weighting $w$ and an orientation $\\Lambda $ is defined as $d^{+}_{G,w,\\Lambda }(v)=\\sum _{(v,u)\\in E(G)\\text{ with } \\Lambda ((v,u))=(v,u)}w((v,u))$ .", "(Minimum Maximum Outdegree).", "Given a graph $G$ , an edge weighting $w$ of $G$ in unary and a positive integer $r$ , is there an orientation $\\Lambda $ of $G$ such that $d^{+}_{G,w,\\Lambda }(v)\\leqslant r$ for each $v\\in V(G)$ ?", "[10] Minimum Maximum Outdegree is W[1]-hard when parameterized by the treewidth of the instance graph We now show that Resolve Delegation is W[1]-hard when parameterized by the treewidth of the instance graph.", "We reduce from Minimum Maximum Outdegree with instance graph $G$ to Resolve Delegation by first creating a replica of the $G$ and then taking an edge $(u,v)$ with weight $w$ and replacing it with a path of $w$ nodes with the end vertex having edges to $u$ and $v$ .", "Resolve Delegation is W[1]-hard when parameterized by the treewidth of the instance graph To prove W[1]-Hardness we reduce from Minimum Maximum Outdegree to Resolve Delegation.", "Let a graph $G(V,E)$ with an edge weighting $w$ in unary and a positive integer $r$ be an arbitrary instance of Minimum Maximum Outdegree.", "Minimum Maximum Outdegree is considered to be a YES instance if the weighted outdegree of every vertex is upper bounded by $r$ .", "Now using the instance of Minimum Maximum Outdegree we create an instance $(\\mathcal {H},r+1)$ of Resolve Delegation.", "Let us construct a graph $\\mathcal {H=(V,E)}$ where $\\mathcal {V}=V_1\\cup V_2$ .", "$V_1=\\lbrace b_u:u\\in V\\rbrace $ .", "$\\forall (u,v)\\in E$ add the set of vertices $\\lbrace a_{uv_1},a_{uv_2},\\ldots ,a_{uv_{w(u,v)}}\\rbrace $ to $V_2$ .", "$\\forall (u,v) \\in E$ , $(a_{uv_1},b_u)\\in \\mathcal {E}$ , $(a_{uv_1},b_v)\\in \\mathcal {E}$ and $\\forall i \\in [w(u,v)]\\setminus \\lbrace 1\\rbrace $ , $(a_{uv_i},a_{uv_{i-1}})\\in \\mathcal {E}$ .", "This completes the construction of $\\mathcal {H}$ with $V_1$ as the sink nodes.", "It is trivial to observe the fact that $tw(\\mathcal {H})\\leqslant tw(G)+2$ .", "We now prove that the Minimum Maximum Outdegree is an YES instance iff the Resolve Delegation is an YES instance Let Minimum Maximum Outdegree be a YES instance.", "Let $\\Lambda $ be the orientation of $G$ which makes Minimum Maximum Outdegree an YES instance.", "We consider the following sub-graph $\\mathcal {H}^{\\prime }\\subseteq \\mathcal {H} $ $\\mathcal {E} [\\mathcal {H}^{\\prime }] &= \\lbrace (a_{uv_i},a_{uv_{i-1}}): i\\in [w(u,v)]\\setminus \\lbrace 1\\rbrace ,(u,v)\\in E\\rbrace \\\\&\\cup \\lbrace (a_{uv_1},b_u): (u,v)\\in E, \\Lambda ((u,v))=(u,v)\\rbrace $ We observe that $\\mathcal {H}$ ' is a spanning sub-graph of $\\mathcal {H}$ such that (i) every non-sink vertices in $\\mathcal {H}$ has exactly one outgoing edge in $\\mathcal {H}$ ' and (ii) for each sink vertex in $\\mathcal {H}$ , there are at most $r+1$ vertices (including the sink itself) which has a path to it.", "Hence the Resolve Delegation instance is a yes instance.", "Let Resolve Delegation be a YES instance.", "Let $\\mathcal {H}$ ' be the spanning sub-graph of $\\mathcal {H}$ which make Resolve Delegation a YES instance.", "Let the edges in $\\mathcal {H}$ ' be denoted by $\\mathcal {E}$ '.", "We consider the following orientation $\\Lambda $ of $G$ $ \\Lambda ((u,v))= \\left\\lbrace \\begin{array}{rcl}(u,v) & \\mbox{if} & (a_{uv_1},b_u)\\in \\mathcal {E}^{\\prime } \\\\(v,u) & \\mbox{otherwise}\\end{array}\\right.$ Clearly weighted outdegree of every vertex in $G$ is atmost $r$ .", "Therefore Minimum Maximum Outdegree is an YES instance.", "This concludes the proof of this theorem $\\Box $ Resolve Delegation is W[1]-hard when parameterized by the treewidth even when the input graph is both Bipartite and Directed Acyclic Graph.", "In the instance of Resolve Delegation created in Theorem , graph $\\mathcal {H}$ is Bipartite as there is no odd cycle in the underlying undirected graph.", "Also graph $\\mathcal {H}$ is Directed Acyclic Graph (DAG) as there is no directed cycle.", "$\\Box $" ], [ "$\\mathsf {FPT}$ Algorithms", "We now prresent our $\\mathsf {FPT}$ algorithms.", "Observation 1 There is a kernel for Resolve Delegation consisting of at most $\\lambda t$ vertices.", "In particular, there is an $\\mathsf {FPT}$ algorithm for the Resolve Delegation problem parameterized by $(\\lambda ,t)$ .", "If the number $n$ of vertices in the input graph is more than $\\lambda t$ , then the instance is clearly a no instance.", "Hence, we have $n\\leqslant \\lambda t$ .", "$\\Box $ In this section we define the notion of weights for the nodes in the subgraph $\\mathcal {H}$ of the delegation graph $\\mathcal {G}$ .", "We define weight of all nodes $u$ in $\\mathcal {G}$ to be 1.", "To get a notion of weight of a vertex $u$ in a subgraph $\\mathcal {H}$ , it can be considered as a number which is one more than the number of nodes who have delegated their vote to $u$ and then have been removed from the graph $\\mathcal {G}$ during the construction of $\\mathcal {H}$ .", "If $\\mathcal {H}$ is a forest such that every non-sink node has an outdegree 1, then clearly the weight of the tree rooted at a sink node say $t$ is sum of the weights of the nodes in the tree.", "We now show Resolve Delegation is FPT w.r.t number of non-sink nodes by using the technique of bounded search tree by the branching on set of vertices satisfying some key properties.", "The Resolve Delegation problem has a FPT with respect to the parameter $k$ which is the number of non-sink nodes in $\\mathcal {G}$ (delegation graph).", "Let us denote the problem instance by $(\\mathcal {G},\\lambda ,k)$ .", "Now we present the following reduction and branching rules.", "Reduction RD.1.", "If there is a vertex $v$ in $\\mathcal {V} $ with only one outgoing edge to a vertex $u$ ($u,v$ are distinct), delete $v$ from graph and increase weight of $u$ by the weight of $v$ .", "The incoming edges which were incident on $v$ (except the self loops if any) are now incident on $u$ .", "Safeness of Reduction RD.1.", "is trivial as a node $v$ with single outgoing edge can only delegate the votes it has got (this includes $v$ 's own vote and the votes of other nodes who have delegated to $v$ so far) to the only neighbor $u$ it has got.", "Reduction RD.2 Remove self loops if any.", "Safeness of Reduction RD.2.", "follows from the fact that no non-sink node can delegate to itself Reduction RD.3.", "If $\\mathcal {G}$ contains a non-sink node $v$ with outdegree more than $2(k-1)$ and indegree 0, delete $v$ from $\\mathcal {G}$ .", "The new instance is $(\\mathcal {G}-v,\\lambda ,k-1)$ Safeness of Reduction RD.3.", "is due to the fact that if we have a vertex $v$ with outdegree greater than $2(k-1)$ , it implies that it has an outgoing edge to at least $k$ sink nodes.", "Let us denote these sink nodes by set $S$ .", "So, irrespective of the delegations made by other vertices, there will exist one sink node $t^{\\prime }\\in S$ such that none of the other $k-1$ non-sink nodes have delegated to $t^{\\prime }$ and hence we can delegate $v$ to $t^{\\prime }$ and still not increase the maximum weight of the sink node.", "Branching B.1.", "Pick a vertex $v$ such that the outdegree is more than $2(k-1)$ and indegree is $k^{\\prime }>0$ .", "Note that $k^{\\prime }\\leqslant k-1$ .", "Each of $k^{\\prime }$ nodes having an outgoing edge to $v$ can either delegate to $v$ or not delegate it.", "So we have $2^{k^{\\prime }}$ possibilities and hence we can create $2^{k^{\\prime }}$ subproblems.", "In each possibility if a node $u_1$ is delegating to $v$ then we delete all the outgoing edges of $u_1$ expect $(u_1,v)$ and if we have a node $u_2$ which doesn't delegate to $v$ then we delete the outgoing edge from $u_2$ to $v$ .", "In each of the $2^{k^{\\prime }}$ instances of graph created first apply R.D.1, then R.D.2, and then finally R.D.3.", "Now solve the problem recursively for each of the $2^{k^{\\prime }}$ instances created by considering each of them as a subproblem.", "If a non-sink node $u$ has delegated to $v$ then $u$ gets deleted due to R.D.1 and if none of the non-sink nodes delegate to $v$ then $v$ gets deleted to R.D.3.", "So therefore, the new parameter (number of non-sink nodes) for the smaller subproblems gets reduced by at least 1.", "Given a directed delegation graph $\\mathcal {G} $ , the algorithm works as follows.", "It first applies Reductions RD.1., RD.2.,RD.3.", "and Branching Rule B.1 exhaustively and in the same order.", "The parameter (number of non-sink nodes) decreases by at least 1 for each of the subproblems as explained earlier.", "If we can't apply the branching rule B.1 to a given subproblem it implies that there is no non-sink node such that the outdegree is more than $2(k-1)$ and indegree is greater than 0.", "Also due to R.D.3 we don't have any non-sink node with outdegree more than $2(k-1)$ and indegree equal to 0.", "So we can do a brute force by considering every possible delegations and solve this instance in $O(k^{k}\\cdot n^{O(1)})$ running time.", "Note that our algorithm will only look at the feasible solutions of Resolve Delegation while brute forcing for a subproblem.", "Also since every node of bounded search tree splits into at most $2^{k-1}$ subproblems and height of the tree is $O(k)$ , we get $f(k)$ leaves (where $f(k)$ is a function of $k$ only).", "Clearly the time taken at every node is bounded by $g(k)\\cdot n^{O(1)}$ where $g(k)$ is a function of $k$ only.", "Thus, the total time used by the algorithm is at-most $O(f(k)\\cdot g(k)\\cdot n^{O(1)})$ which gives us an FPT for Resolve Delegation.", "$\\Box $ We now show Resolve Delegation is FPT w.r.t number of edges to be deleted from delegation graph by using the technique of bounded search tree by the branching on set of edges satisfying some key properties.", "The Resolve Delegation problem has a FPT with respect to the parameter $k$ which is the number of edges to be deleted from delegation graph.", "The parameter $k$ is the number of edges to be deleted.", "Given any instance $\\mathcal {G} $ of problem , every feasible solution graph $\\mathcal {G} _{\\mathcal {T}}$ is a forest with trees with set of roots as set of all sink nodes $\\mathcal {T} $ .", "Clearly then $k=\|\\mathcal {E} \|-\|\\mathcal {V} \|+\|\\mathcal {T} \|$ .", "Let us denote the problem instance by $(\\mathcal {G},\\lambda ,k)$ .", "Observation 2 If $k>0$ and only the sink nodes have outdegree 0, then there is a non-sink node with outdegree atleast 2.", "Sum of outdegree of all the non-sink nodes is greater than $\|\\mathcal {V} \|-\|\\mathcal {T} \|$ .", "Hence the observation follows from pigeon hole principle.", "Branching B.1.", "Let $k>0$ .", "Consider the vertex with maximum outdegree.", "If $l$ is the outdegree of one such vertex $v$ , delete one of the two groups of edges $\\lbrace 1,\\ldots ,\\lfloor l/2 \\rfloor \\rbrace $ and $\\lbrace \\lfloor l/2 \\rfloor +1, \\ldots , l\\rbrace $ outgoing from $v$ .", "Then solve the problem recursively for two new subproblems with new parameter $k^{^{\\prime }}\\leqslant k-1$ .", "Now we describe why the Branching B.1 is safe.", "Note that the Branching B.1 is triggered only when $k>0$ .", "It follows from Observation REF that outdegree of $v$ is at least 2.", "Consider the degree of $v$ to be $l$ and the corresponding outgoing edges from $v$ to be $\\lbrace 1,\\ldots ,l\\rbrace $ .", "Since $v$ can delegate only to exactly one of its neighbours connected by $\\lbrace 1,\\ldots ,l\\rbrace $ , other $l-1$ edges need to be deleted from delegation graph as they can not be a part of feasible solution.", "If we partition the set of edges into two disjoint sets $\\lbrace 1,\\ldots ,\\lfloor l/2 \\rfloor \\rbrace $ and $\\lbrace \\lfloor l/2 \\rfloor +1, \\ldots , l\\rbrace $ , only one out of the two groups can be a part of feasible solution.", "This allows us to delete the other half set say $\\lbrace \\lfloor l/2 \\rfloor +1, \\ldots , l\\rbrace $ .", "As we know that $\|l\|\\geqslant 2$ which comes from the fact that outdegree of vertex $v$ is at least 2.", "The problem now reduces to a smaller instance $\\mathcal {G} ^{^{\\prime }}$ with edges $\\mathcal {E} ^{^{\\prime }}[\\mathcal {G} ^\\prime ]$ $=$ $\\mathcal {E} [\\mathcal {G} ] \\backslash \\lbrace \\lfloor l/2 \\rfloor +1, \\ldots , l\\rbrace $ and parameter number of edges to be deleted as $k^{^{\\prime }} \\leqslant k-1$ .", "Thus way we get a bounded search tree with only constant number of subproblems at each branch such that at each recursive step the height of search tree reduces by at least one.", "Given a directed delegation graph $\\mathcal {G} $ , the algorithm works as follows.", "As long as $k>0$ , Branching Rule B.1 is applied exhaustively in the bounded search tree.", "Note that Branching Rule B.1 brings down the parameter $k$ in every call by at least 1.", "Whenever the parameter $k$ becomes 0, we have a feasible solution as the non-sink nodes have the outdegree of 1.", "Now we can easily check in polynomial time whether the feasible solution is a YES instance or a NO instance.", "At every recursive call we decrease the parameter by at least 1 and thus the height of the tree is at most $k$ .", "Also since every node of bounded search tree splits into two, we get $O(2^{k})$ leaves.", "Clearly the time taken at every node is bounded by $n^{O(1)}$ .", "Thus if $f(k) = O(2^{k})$ be the number of nodes in the bounded search tree, the total time used by the algorithm is at most $O(2^{k}n^{O(1)})$ which gives us an FPT for Resolve Delegation.", "$\\Box $" ], [ "Structural Results", "There exists a linear programming formulation for the optimization version of Resolve Delegation where fractional delegation of votes is allowed.", "Thus the fractional variant is solvable in polynomial time.", "We consider the fractional variant of Liquid Democracy Delegation Problem where it is allowed to fractionally delegate votes of a source (delegator) to multiple nodes such that total number of votes being delegated is conserved at the delegator.", "We formally define conservation while formulating the LP for the problem.", "LP formulation follows similar to the LP formulation of flow-problems (e.g.", "Max-FLow-MinCut etc).", "We assign $x_{u,v}$ as weight to every edge $(u,v)\\in \\mathcal {E} [\\mathcal {G} ]$ which corresponds to the fractional weight of votes delegated from vertex $u$ to $v$ ( for all $u,v \\in \\mathcal {V} [\\mathcal {G} ]$ .", "For all other $x_{u,v}$ where $(u,v)$ pair doesn't correspond to an edge of delegation graph we assign value 0.", "It immediately follows that for all sink nodes $t\\in \\mathcal {T} [\\mathcal {G} ]$ , total weight of fractional votes being delegated to each sink-node $t$ (including that of the sink node $t$ ) is $\\sum \\limits _{v\\in \\mathcal {V} \\backslash \\mathcal {T}} x_{v,t}+1$ $\\forall t \\in \\mathcal {T} $ .", "For all other non-sink nodes $s \\in \\mathcal {V} \\backslash \\mathcal {T} $ , node $s$ obeys conservation as follows : $\\sum \\limits _{u\\in \\mathcal {V} \\backslash \\mathcal {T}} x_{u,s}$ + 1 = $\\sum \\limits _{v\\in \\mathcal {V}} x_{s,v}$ , $\\forall s \\in \\mathcal {V} \\backslash \\mathcal {T} $ Our aim is to minimize the maximum weight of votes delegated to any sink node (including that of the sink node).", "The corresponding LP formulation is: $\\text{minimize }z$ $z\\geqslant \\sum \\limits _{v\\in \\mathcal {V} \\backslash \\mathcal {T}} x_{v,t}+1$ , $\\forall t\\in \\mathcal {T} $ $\\sum \\limits _{u\\in \\mathcal {V} \\backslash \\mathcal {T}} x_{u,s}$ + 1 = $\\sum \\limits _{v\\in \\mathcal {V}} x_{s,v}$ , $\\forall s \\in \\mathcal {V} \\backslash \\mathcal {T} $ [Follows from conservation] $x_{u,v} \\geqslant 0$ ,$\\forall (u,v) \\in \\mathcal {E} [\\mathcal {G} ] $ $x_{u,v} = 0$ ,$\\forall (u,v) \\notin \\mathcal {E} [\\mathcal {G} ] $ $\\Box $" ], [ "Conclusion and Future Direction", "We have studied the parameterized complexity of a fundamental problem in liquid democracy, namely Resolve Delegation.", "We considered various natural parameters for the problem including the number of sink vertices, maximum allowed weight of any sink in the final delegation graph, maximum degree of any vertex, tree-width, the number of edges that one deletes in an optimal solution, number of non-sink vertices.", "We also show that a related problem which we call Resolve Fractional Delegation is polynomial time solvable.", "An important future work is to resolve the complexity of Resolve Delegation if the input graph is already acyclic or tree.", "We know that there exists a $\\Omega (\\log n)$ lower bound on the approximation factor of optimizing the maximum allowed weight of any sink [6].", "It would be interesting to see if there exsits $\\mathsf {FPT}$ algorithms achieving a approximation factor of $o(\\log n)$ ." ] ]	2011.14192
[ [ "A note on the Grover walk and the generalized Ihara zeta function of the\n one-dimensional integer lattice" ], [ "Abstract Chinta, Jorgenson and Karlsson introduced a generalized version of the determinant formula for the Ihara zeta function associated to finite or infinite regular graphs.", "On the other hand, Konno and Sato obtained a formula of the characteristic polynomial of the Grover matrix by using the determinant expression for the second weighted zeta function of a finite graph.", "In this paper, we focus on a relationship between the Grover walk and the generalized Ihara zeta function.", "That is to say, we treat the generalized Ihara zeta function of the one-dimensional integer lattice as a limit of the Ihara zeta function of the cycle graph." ], [ "Introduction", "Ihara [9] introduced the Ihara zeta functions of graphs, and showed that the reciprocal of the Ihara zeta function of a regular graph is an explicit polynomial.", "Afterwards, the Ihara zeta function of a finite graph was studied in [12], [13], [14], [8], [1], [4].", "Furthermore, the Ihara zeta function of a finite graph was extended to an infinite graph in [1], [3], [5], [6], [7], and its determinant expressions were presented.", "Chinta, Jorgenson and Karlsson [2] gave a generalized version of the determinant formula for the Ihara zeta function associated to finite or infinite regular graphs.", "A discrete-time quantum walk is a quantum analog of the classical random walk on a graph whose state vector is governed by a matrix called the time evolution matrix.", "The time evolution matrix of a discrete-time quantum walk in a graph is closely related to the Ihara zeta function of a graph.", "Ren et al.", "[11] gave a relationship between the discrete-time quantum walk and the Ihara zeta function of a graph.", "Konno and Sato [10] obtained a formula of the characteristic polynomial of the Grover matrix by using the determinant expression for the second weighted zeta function of a graph.", "In this paper, we consider the relation between the Grover walk and the generalized Ihara zeta function, and present the generalized Ihara zeta function of the one-dimensional integer lattice as a limit of the Ihara zeta function of the cycle graph.", "In Section 2, we state a review for the Ihara zeta function of a finite graph and the generalized Ihara zeta function of a finite or infinite vertex transitive graph.", "In Section 3, we deal with the Grover walk on a graph as a discrete-time quantum walk on a graph.", "In Section 4, we treat the generalized Ihara zeta function of $ \\mathbb {Z} $ as a limit of the Ihara zeta function of the cycle graph $C_n $ with $n$ vertices." ], [ "The Ihara zeta function of a graph", "All graphs in this paper are assumed to be simple.", "Let $G=(V(G),E(G))$ be a connected graph (without multiple edges and loops) with the set $V(G)$ of vertices and the set $E(G)$ of unoriented edges $uv$ joining two vertices $u$ and $v$ .", "For $uv \\in E(G)$ , an arc $(u,v)$ is the oriented edge from $u$ to $v$ .", "Let $D_G$ be the symmetric digraph corresponding to $G$ .", "Set $D(G)= \\lbrace (u,v),(v,u) \\mid uv \\in E(G) \\rbrace $ .", "For $e=(u,v) \\in D(G)$ , set $u=o(e)$ and $v=t(e)$ .", "Furthermore, let $e^{-1}=(v,u)$ be the inverse of $e=(u,v)$ .", "For $v \\in V(G)$ , the degree $\\deg {}_G v = \\deg v = d_v $ of $v$ is the number of vertices adjacent to $v$ in $G$ .", "A path $P$ of length $n$ in $G$ is a sequence $P=(e_1, \\cdots ,e_n )$ of $n$ arcs such that $e_i \\in D(G)$ , $t( e_i )=o( e_{i+1} )(1 \\le i \\le n-1)$ .", "If $e_i =( v_{i-1} , v_i )$ for $i=1, \\cdots , n$ , then we write $P=(v_0, v_1, \\cdots ,v_{n-1}, v_n )$ .", "Set $ \\mid P \\mid =n$ , $o(P)=o( e_1 )$ and $t(P)=t( e_n )$ .", "Also, $P$ is called an $(o(P),t(P))$ -path.", "We say that a path $P=( e_1 , \\cdots , e_n )$ has a backtracking if $ e^{-1}_{i+1} =e_i $ for some $i(1 \\le i \\le n-1)$ .", "A $(v, w)$ -path is called a $v$ -cycle (or $v$ -closed path) if $v=w$ .", "A cycle $C=( e_1 , \\ldots , e_r )$ has a tail if $e_r = e^{-1}_1 $ .", "A cycle $C$ is reduced if $C$ has a neither a backtracking nor a tail.", "For a natural number $k \\in \\mathbb {N} $ , let $N_k $ be the number of reduced cycles of length $k$ in $G$ .", "The Ihara zeta function of a graph $G$ is a function of a complex variable $u$ with $\|u\|$ sufficiently small, defined by ${\\bf Z} (G, u)= {\\bf Z}_G (u)= \\exp ( \\sum ^{\\infty }_{k=1} \\frac{N_k }{k} u^k ) .$ Let $G$ be a connected graph with $n$ vertices $v_1, \\cdots ,v_n $ .", "The adjacency matrix ${\\bf A}= {\\bf A} (G)=(a_{ij} )$ is the square matrix such that $a_{ij} =1$ if $v_i$ and $v_j$ are adjacent, and $a_{ij} =0$ otherwise.", "The degree of a vertex $v_i$ of $G$ is defined by $ \\deg v_i = \\deg {}_G v_i = \\mid \\lbrace v_j \\mid v_i v_j \\in E(G) \\rbrace \\mid $ .", "If $ \\deg {}_G v=k$ (constant) for each $v \\in V(G)$ , then $G$ is called $k$ -regular.", "Theorem 1 (Ihara; Bass) Let $G$ be a connected graph.", "Then the reciprocal of the Ihara zeta function of $G$ is given by ${\\bf Z} (G,u )^{-1} =(1- u^2 )^{r-1}\\det ( {\\bf I} -u {\\bf A} (G)+ u^2 ( {\\bf D} - {\\bf I} )) ,$ where $r$ is the Betti number of $G$ , and ${\\bf D} =( d_{ij} )$ is the diagonal matrix with $d_{ii} = \\deg v_i$ and $d_{ij} =0, i \\ne j ,(V(G)= \\lbrace v_1 , \\cdots , v_n \\rbrace )$ .", "Let $G=(V(G),E(G))$ be a vertex transitive $(q+1)$ -regular graph and $ x_0 \\in V(G)$ a fixed vertex.", "Then the generalized Ihara zeta function $\\zeta {}_G (u)$ of $G$ is defined by $\\zeta {}_G (u)= \\exp ( \\sum ^{\\infty }_{m=1} \\frac{N^0_m }{m} u^m ) ,$ where $N^0_m $ is the number of reduced $x_0$ -cycles of length $m$ in $G$ .", "Note that, for a finite graph, the classical Ihara zeta function is just the above Ihara zeta function raised to the power equaling the number of vertices.", "Furthermore, the Laplacian of $G$ is given by $\\Delta = \\Delta (G) = {\\bf D} - {\\bf A} (G).$ A formula for the generalized Ihara zeta function of a vertex transitive graph is given as follows: Theorem 2 (Chinta, Jorgenson and Karlsson) Let $G$ be a vertex transitive $(q+1)$ -regular graph with spectral measure $\\mu $ for the Laplacian.", "Then $\\zeta {}_G (u)^{-1} =(1-u^2 )^{(q-1)/2} \\exp ( \\int \\log (1-(q+1- \\lambda )u+q u^2 ) d \\mu ( \\lambda )) .$" ], [ "The Grover walk on a graph", "Let $G$ be a connected graph with $n$ vertices and $m$ edges.", "Set $V(G)= \\lbrace v_1 , \\ldots , v_n \\rbrace $ and $d_j = d_{v_j} = \\deg v_j , \\ j=1, \\ldots , n$ .", "For $u \\in V(G)$ , let $D(u)= \\lbrace e \\in D(G) \\mid t(e)=u \\rbrace $ .", "Furthermore, let $\\alpha {}_u , \\ u \\in V(G)$ be a unit vector with respect to $D(u)$ , that is, $\\alpha {}_u (e) =\\left\\lbrace \\begin{array}{ll}non \\ zero \\ complex \\ number \\ & \\mbox{if $e \\in D(u)$, } \\\\0 & \\mbox{otherwise, }\\end{array}\\right.$ where $ \\alpha {}_u (e) $ is the entry of $\\alpha {}_u $ corresponding to the arc $e \\in D(G)$ .", "Now, a $2m \\times 2m$ matrix ${\\bf C} $ is given as follows: ${\\bf C} =2 \\sum _{u \\in V(G)} \| \\alpha {}_u \\rangle \\langle \\alpha {}_u \| - {\\bf I}_{2m } .$ The matrix C is the coin operator of the considered quantum walk.", "Note that ${\\bf C} $ is unitary.", "Then the time evolution matrix ${\\bf U} $ is defined by ${\\bf U} = {\\bf S} {\\bf C} ,$ where ${\\bf S} =( S_{ef} )_{e,f \\in D(G)} $ is given by $S_{ef} =\\left\\lbrace \\begin{array}{ll}1 & \\mbox{if $f= e^{-1} $, } \\\\0 & \\mbox{otherwise.", "}\\end{array}\\right.$ The matrix ${\\bf S} $ is called the shift operator.", "The time evolution of a quantum walk on $G$ through ${\\bf U} $ is given by $\\psi {}_{t+1} = {\\bf U} \\psi {}_t .$ Here, $\\psi {}_{t+1}, \\psi {}_t $ are the states.", "Note that the state $\\psi {}_t $ is written with respect to the initial state $\\psi {}_0 $ as follows: $\\psi {}_{t} = {\\bf U}^t \\psi {}_0 .$ A quantum walk on $G$ with ${\\bf U} $ as a time evolution matrix is called a coined quantum walk on $G$ .", "If $\\alpha {}_u (e)= \\frac{1}{\\sqrt{d_u}} $ for $e \\in D(u)$ , then the time evolution matrix ${\\bf U} $ is called the Grover matrix of $G$ , and a quantum walk on $G$ with the Grover matrix as a time evolution matrix is called a Grover walk on $G$ .", "Thus, the Grover matrix ${\\bf U} ={\\bf U} (G)=( U_{ef} )_{e,f \\in D(G)} $ of $G$ is defined by $U_{ef} =\\left\\lbrace \\begin{array}{ll}2/d_{t(f)} (=2/d_{o(e)} ) & \\mbox{if $t(f)=o(e)$ and $f \\ne e^{-1} $, } \\\\2/d_{t(f)} -1 & \\mbox{if $f= e^{-1} $, } \\\\0 & \\mbox{otherwise}\\end{array}\\right.$ Let $G$ be a connected graph with $n$ vertices and $m$ edges.", "Then the $n \\times n$ matrix ${\\bf T}_n (G)=( T_{uv} )_{u,v \\in V(G)}$ is given as follows: $T_{uv} =\\left\\lbrace \\begin{array}{ll}1/( \\deg {}_G u) & \\mbox{if $(u,v) \\in D(G)$, } \\\\0 & \\mbox{otherwise.", "}\\end{array}\\right.$ Note that the matrix ${\\bf T} (G)$ is the transition probability matrix of the simple random walk on $G$ .", "Theorem 3 (Konno and Sato) Let $G$ be a connected graph with $n$ vertices $v_1 , \\ldots , v_n $ and $m$ edges.", "Then the characteristic polynomial for the Grover matrix ${\\bf U}$ of $G$ is given by $\\begin{array}{rcl}\\det ( \\lambda {\\bf I}_{2m} - {\\bf U} ) & = & ( \\lambda {}^2 -1)^{m-n} \\det (( \\lambda {}^2 +1) {\\bf I}_n -2 \\lambda {\\bf T} (G)) \\\\\\ & & \\\\\\ & = & \\frac{( \\lambda {}^2 -1)^{m-n} \\det (( \\lambda {}^2 +1) {\\bf D} -2 \\lambda {\\bf A} (G))}{d_{v_1} \\cdots d_{v_n }} .\\end{array}$ From Theorem 3, the following equation for the Grover matrix on a graph is obtained.", "Corollary 1 Let $G$ be a connected graph with $n$ vertices $v_1 , \\ldots , v_n $ and $m$ edges.", "Then the characteristic polynomial for the Grover matrix ${\\bf U}$ of $G$ is given by $\\begin{array}{rcl}\\det ( {\\bf I}_{2m} -u {\\bf U} ) & = & (1- u^2 )^{m-n} \\det ((1+ u^2 ) {\\bf I}_n -2u {\\bf T} (G)) \\\\\\ & & \\\\\\ & = & \\frac{(1- u^2 )^{m-n} \\det ((1+ u^2 ) {\\bf D} -2u {\\bf A} (G))}{d_{v_1} \\cdots d_{v_n }} .\\end{array}$" ], [ "The generalized Ihara zeta function of $\\mathbb {Z}$", "Let the cycle graph $C_n $ be the connected 2-regular graph with $n$ vertcies.", "If $n \\rightarrow \\infty $ , then the limit of $C_n $ is the one-dimensional integer lattice $\\mathbb {Z} $ .", "Then we consider the Grover walk on $\\mathbb {Z} $ .", "This quantum walk is a $\\left[\\begin{array}{cc}1 & 0 \\\\0 & 1\\end{array}\\right]$ free quantum walk.", "Let ${\\bf U}^{(s)}_n $ be the Grover matrix on $C_n $ and ${\\bf P}^{(s)}_n $ the transition probability matrix of the simple random walk on $C_n $ .", "Then we have ${\\ P}^{(s)}_n = \\frac{1}{2} {\\bf A} (C_n ) .$ By Corollary 1, we have $\\det ({\\bf I}_{2n} -u {\\bf U}^{(s)}_n ) =(1- u^2 )^{n-n} \\det ((1+u^2 ) {\\bf I}_n -2u {\\bf P}^{(s)}_n )=(1- u^2 )^{n-n} \\det ( {\\bf I}_n -u {\\bf A} (C_n )+ u^2 {\\bf I}_n) .$ That is, ${\\bf Z} ( C_n , u)^{-1} = \\det ({\\bf I}_{2n} -u {\\bf U}^{(s)}_n ) .$ By the fact that, for a finite graph, the classical Ihara zeta function is just the above Ihara zeta function raised to the power equaling the number of vertices, we have $\\begin{array}{rcl}\\zeta {}_{C_n } (u)^{-1} & = & {\\bf Z} (C_n , u)^{-1/n} = \\det ( {\\bf I}_{2m} -u {\\bf U}^{(s)}_n )^{1/n} \\\\\\ & & \\\\\\ & = & \\lbrace \\det ((1+u^2 ) {\\bf I}_n -2u {\\bf P}^{(s)}_n ) \\rbrace {}^{1/n } \\\\\\ & & \\\\\\ & = & \\lbrace \\prod _{ \\lambda \\in {\\rm Spec}( {\\bf P}^{(s)}_n )} ((1+u^2 )-2u \\lambda ) \\rbrace {}^{1/n} \\\\\\ & & \\\\\\ & = & \\exp [ \\log \\lbrace ( \\prod _{ \\lambda \\in {\\rm Spec}( {\\bf P}^{(s)}_n )} ((1+u^2 )-2u \\lambda ))^{1/n} \\rbrace ] \\\\\\ & & \\\\\\ & = & \\exp [ \\frac{1}{n} \\sum _{\\lambda \\in {\\rm Spec} ( {\\bf P}^{(s)}_n )} \\log ((1+u^2 )-2u \\lambda ) ] .\\end{array}$ Now, since $\\det ({\\bf I}_{2n} -u {\\bf U}^{(s)}_n )= {\\bf Z} (C_n , u)^{-1} = (1- u^n)^2 ,$ we have $\\det ( \\lambda {\\bf I}_{2n} - {\\bf U}^{(s)}_n )=( \\lambda {}^n -1)^2 .$ Thus, ${\\rm Spec} ( {\\bf U}^{(s)}_n )= \\lbrace [ e^{i \\theta {}_0 } ]^2 , [ e^{i \\theta {}_1 } ]^2 , \\ldots , [ e^{i \\theta {}_{n-1} } ]^2 \\rbrace ,$ where ${\\rm Spec} ({\\bf F})$ is the spectra of a square matrix ${\\bf F}$ , and $\\theta {}_k = \\frac{2 \\pi k}{n} \\ (k=0,1, \\ldots , n-1) .$ Furthermore, by Theorem 3 (Konno-Sato Theorem), we obtain the following spectral mapping theorem: $\\det ( \\lambda {\\bf I}_{2n} - {\\bf U}^{(s)}_n )=(2 \\lambda )^n ( \\lambda {}^2 -1)^0 \\det ( \\frac{ \\lambda +1}{2 \\lambda } {\\bf I}_n - {\\bf P}^{(s)}_n )=(2 \\lambda )^n \\det ( \\frac{ \\lambda + \\overline{ \\lambda } }{2} {\\bf I}_n - {\\bf P}^{(s)}_n ) .$ If $ \\lambda =e^{i \\theta {}_k } ( k=0,1, \\ldots , n-1)$ , then we obtain $\\frac{ \\lambda + \\overline{ \\lambda } }{2} = \\cos \\theta {}_k .$ That is, ${\\rm Spec} ( {\\bf P}^{(s)}_n )= \\lbrace [ \\cos \\theta {}_k ]^1 \\mid k=0,1, \\ldots , n-1\\rbrace .$ Thus, we have $\\zeta {}_{C_n } (u)^{-1} = \\exp \\Bigl [ \\sum ^{n-1}_{k=0} \\log \\Bigl ( (1+u^2 )-2u \\cos \\Bigl ( \\frac{2 \\pi k}{n} \\Bigr ) \\Bigr ) \\frac{1}{2 \\pi }\\times \\frac{2 \\pi }{n} \\Bigr ] .$ When $n \\rightarrow \\infty $ , then we have $ \\displaystyle \\begin{array}{rcl}\\displaystyle \\lim {}_{n \\rightarrow \\infty } \\zeta {}_{C_n } (u)^{-1} & = & \\displaystyle \\exp \\Bigl [ \\int ^{2 \\pi }_0 \\log \\Bigl ( (1+u^2 )-2u \\cos x \\Bigr ) \\frac{dx}{2 \\pi } \\Bigr ] \\\\\\ & & \\\\\\ & = & \\displaystyle \\frac{u^2 +1+\| u^2 -1\|}{2} = \\left\\lbrace \\begin{array}{ll}1 & \\mbox{if $\|u\| <1$, } \\\\u^2 & \\mbox{if $\|u\| \\ge 1$.", "}\\end{array}\\right.\\end{array}$ On the other hand, we see that $\\zeta {}_{\\mathbb {Z}} (u)=1 ,$ since $\\mathbb {Z} $ has no reduced cycle.", "Therefore we obtain the following result.", "Theorem 4 $\\lim {}_{n \\rightarrow \\infty } \\zeta {}_{C_n } (u)=\\zeta {}_{\\mathbb {Z}} (u) \\ for \\ \|u\| <1.$ From now on we consider a relation between Theorem 1.3 given by Chinta et al.", "[2] and Theorem 3.", "Their result for $ \\mathbb {Z} $ case gives $ \\displaystyle \\begin{array}{rcl}\\displaystyle \\zeta {}_{\\mathbb {Z}} (u)^{-1} & = & \\displaystyle (1-u^2 )^{(1-1)/2} \\exp ( \\int \\log (1-2u+ u^2 +u \\lambda ) d \\mu ( \\lambda )) \\\\\\ & & \\\\\\ & = & \\displaystyle \\exp ( \\int \\log (1-2u +u^2 + \\lambda u ) d \\mu ( \\lambda )) \\ \\ \\ \\ \\ \\ (1) .\\end{array}$ Noting that $\\Delta (C_n )= {\\bf D} - {\\bf A} (C_n )=2( {\\bf I}_n - {\\bf P}^{(s)}_n ) $ , we have $\\begin{array}{rcl}\\zeta {}_{C_n } (u)^{-1} & = & {\\bf Z} (C_n , u)^{-1/n} = \\det ( {\\bf I}_{2m} -u {\\bf U}^{(s)}_n )^{1/n} \\\\\\ & & \\\\\\ & = & \\lbrace \\det ((1-2u+u^2 ) {\\bf I}_n +u \\Delta (C_n )) \\rbrace {}^{1/n } .\\end{array}$ Thus a similar argument in the proof of Theorem implies $\\lim {}_{n \\rightarrow \\infty } \\zeta {}_{C_n } (u)^{-1} =\\exp \\Bigl [ \\int ^{2 \\pi }_0 \\log ((1-2u+u^2 )+2u (1-\\cos x )) \\frac{dx}{2 \\pi } \\Bigr ] .$ Remark that the right-hand side of this equality is noting but that of Eq.", "(1).", "Then we have Corollary 2 $\\displaystyle \\lim {}_{n \\rightarrow \\infty } \\zeta {}_{C_n } (u)^{-1} = \\zeta {}_{\\mathbb {Z}} (u)^{-1}= \\exp \\Bigl ( \\int \\log (1-2u +u^2 + \\lambda u ) d \\mu ( \\lambda ) \\Bigr ) ,$ where $\\lambda d \\mu ( \\lambda ) \\sim 2(1 -\\cos x) \\frac{dx}{2 \\pi } \\ \\ \\ \\ \\ on \\ [0, 2 \\pi ) .$" ] ]	2011.14162
[ [ "Construction of Rank $2$ Indecomposable Modules in Grassmannian Cluster\n Categories" ], [ "Abstract The category ${\\rm CM}(B_{k,n}) $ of Cohen-Macaulay modules over a quotient $B_{k,n}$ of a preprojective algebra provides a categorification of the cluster algebra structure on the coordinate ring of the Grassmannian variety of $k$-dimensional subspaces in $\\mathbb C^n$, \\cite{JKS16}.", "Among the indecomposable modules in this category are the rank $1$ modules which are in bijection with $k$-subsets of $\\{1,2,\\dots,n\\}$, and their explicit construction has been given by Jensen, King and Su.", "These are the building blocks of the category as any module in ${\\rm CM}(B_{k,n}) $ can be filtered by them.", "In this paper we give an explicit construction of rank 2 modules.", "With this, we give all indecomposable rank 2 modules in the cases when $k=3$ and $k=4$.", "In particular, we cover the tame cases and go beyond them.", "We also characterise the modules among them which are uniquely determined by their filtrations.", "For $k\\ge 4$, we exhibit infinite families of non-isomorphic rank 2 modules having the same filtration." ], [ "Introduction", "One of the key initial examples of Fomin and Zelevinsky's theory of cluster algebras [6] was the homogeneous coordinate ring $\\operatorname{\\mathbb {C}}\\nolimits [\\operatorname{Gr}\\nolimits (2,n)]$ of the Grassmannian of 2-dimensional subspaces of $\\operatorname{\\mathbb {C}}\\nolimits ^n$ .", "Scott proved in [16] that this cluster structure can be generalized to the coordinate ring $\\operatorname{\\mathbb {C}}\\nolimits [\\operatorname{Gr}\\nolimits (k,n)]$ .", "This has sparked a lot of research activities in cluster theory, e.g.", "[18], [9], [12], [11], [15], [4], [13], [14], [7], [17].", "An aditive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian variety of $k$ -dimensional subspaces in $\\mathbb {C}^n$ has been given by Geiss, Leclerc, and Schroer [8], [10] in terms of a subcategory of the category of finite dimensional modules over the preprojective algebra of type $A_{n-1}$ .", "Jensen, King, and Su [13] gave a new additive categorification of this cluster structure using the maximal Cohen-Macaulay modules over the completion of an algebra $B_{k,n}$ which is a quotient of the preprojective algebra of type $A_{n-1}$ .", "In the category ${\\rm CM}(B_{k,n}) $ of Cohen-Macaulay modules over $B_{k,n}$ , which is called the Grassmannian cluster category, among the indecomposable modules are the rank 1 modules which are known to be in bijection with $k$ -subsets of $\\lbrace 1,2,\\dots ,n\\rbrace $ , and their explicit construction has been given in [13].", "For a given $k$ -subset $I$ , the corresponding rank 1 module is denoted by $L_I$ .", "Also, we refer to $k$ -subsets as rims, because of the way we use them to visualize rank 1 modules (see Section 2).", "Rank 1 modules are the building blocks of the category as any module in ${\\rm CM}(B_{k,n}) $ can be filtered by rank 1 modules (the filtration is noted in the profile of a module, [13]).", "The number of rank 1 modules appearing in the filtration of a given module is called the rank of that module.", "The aim of this paper is to explicitly construct rank 2 indecomposable Cohen-Macaulay $B_{k,n}$ modules in the cases when $k=3$ and $k=4$ .", "In particular, we construct all indecomposable rank 2 modules in the tame cases $(3,9)$ and $(4,8)$ , and more generally, for an arbitrary $k$ , we construct all indecomposable modules of rank 2 whose rank 1 filtration layers $L_I$ and $L_J$ satisfy the condition $\|I\\cap J\|\\ge k-4$ .", "Once we have the construction, we investigate the question of uniqueness.", "Here, the central notions are that of $r$ -interlacing (Definition REF ) and of the poset of a given rank 2 module (Section ).", "If $I$ and $J$ are $k$ -subsets of $\\lbrace 1,\\ldots , n\\rbrace $ , then $I$ and $J$ are said to be $r$ -interlacing if there exist subsets $\\lbrace i_1,i_3,\\dots ,i_{2r-1}\\rbrace \\subset I\\setminus J$ and $\\lbrace i_2,i_4,\\dots , i_{2r}\\rbrace \\subset J\\setminus I$ such that $i_1<i_2<i_3<\\dots <i_{2r}<i_1$ (cyclically) and if there exist no larger subsets of $I$ and of $J$ with this property.", "The filtration layers of a module $M$ give a poset structure.", "In rank 2, if $I$ and $J$ are $r$ -interlacing, the sets $I$ and $J$ form a number $r_1\\le r$ of boxes in the so-called lattice diagram of $M$ (see Section for details on how we picture $M$ with its filtration layers).", "The associated poset is $1^{r_1}\\mid 2$ ; the poset consists of a tree with one vertex of degree $r_1$ and $r_1$ leaves, it has dimension 1 at the leaves and dimension 2 at the central vertex.", "See Figure REF .", "Figure: The profile of a module with 4-interlacing layers forming 3 boxes with poset 1 3 ∣21^3\\mid 2.The dashed line shows the rim of L I L_I with arrows x i x_i, i∈Ii\\in I indicated.The solid line below is the rim of L J L_J, with arrows x i x_i, i∈Ji\\in J indicated.A partial answer to the question of indecomposability of a rank 2 module in terms of its poset is given in the following proposition.", "Proposition 1.1 ([2], Remark 3.2) Let $M\\in {\\rm CM}(B_{k,n})$ be an indecomposable module with profile $I\\mid J$ .", "Then $I$ and $J$ are $r$ -interlacing and their poset is $1^{r_1}\\mid 2$ , where $r\\ge r_1\\ge 3$ .", "This proposition tells us that when dealing with rank 2 indecomposable modules, we can assume that the poset of such a module is of the form $1^{r_1}\\mid 2$ , for $r_1\\ge 3$ , and that its layers are $r$ -interlacing, where $r\\ge r_1\\ge 3$ .", "We say that $I$ and $J$ are almost tightly 3-interlacing if $I\\mid J$ has poset $1^3\\mid 2$ , and $I\\setminus J=\\lbrace a_1\\rbrace \\cup \\lbrace a_2\\rbrace \\cup \\lbrace a_3, \\dots , a_{3+r}\\rbrace $ , $J\\setminus I=\\lbrace b_1\\rbrace \\cup \\lbrace b_2\\rbrace \\cup \\lbrace b_3, \\dots , b_{3+r}\\rbrace $ , $r\\ge 0$ , and $b_1<a_1<b_2<a_2<b_3< \\dots <b_{3+r}<a_3< \\dots < a_{3+r}.$ Our main results are the following two theorems.", "Theorem 1.2 (Theorem REF , Theorem REF ) An indecomposable rank 2 module $M\\in {\\rm CM}(B_{k,n})$ is uniquely determined by its profile if and only if its poset is $1^3\\mid 2$ and its layers are almost tightly 3-interlacing.", "More precisely, in the case of $r$ -interlacing rank 1 layers with poset $1^{r_1}\\mid 2$ , where $r\\ge r_1\\ge 4$ , we show that there are infinitely many non-isomorphic rank 2 modules with the same profile, e.g.", "there are infinitely many non-isomorphic indecomposable rank 2 modules with filtration $\\lbrace 1,3,5,7\\rbrace \\mid \\lbrace 2,4,6,8\\rbrace $ in the tame case $(4,8)$ .", "Theorem 1.3 (Theorem REF ) Let $M$ be an indecomposable rank 2 module with profile $I\\mid J$ , where $I$ and $J$ are $r$ -interlacing with poset $1^{r_1}\\mid 2$ , where $r\\ge r_1\\ge 4$ .", "Then there are infinitely many non-isomorphic rank 2 indecomposable modules from ${\\rm CM}(B_{k,n})$ with profile $I \\mid J$ .", "In the case $r=r_1=4$ , we show that this infinite family of indecomposable modules with the profile $I\\mid J$ is parameterized by the set $\\mathbb {C}\\setminus {\\lbrace 0,1,-1\\rbrace }$ where two points from this set are identified if their sum is 0.", "For the filtration layers $I$ and $J$ of an indecomposable module with profile $I\\mid J$ , we construct all decomposable rank 2 modules that are extensions of these rank 1 modules, i.e.", "we construct all decomposable modules that appear as middle terms in short exact sequences with $I$ and $J$ as end terms.", "The paper is organized as follows.", "In Section , we recall the definitions and key results about Grassmannian cluster categories.", "In Section , we give the construction of rank 2 modules in the case when the layers are tightly 3-interlacing.", "This covers in particular the tame case $(3,9)$ and almost all rank 2 modules in the tame case $(4,8)$ .", "Section is devoted to the cases of non-tightly 3-interlacing layers.", "Section is devoted to the case of tightly 4-interlacing layers, which completes the case $(4,8)$ .", "In the last section, we deal with the general case of $r$ -interlacing, when $r\\ge 4$ , and we show that there are infinitely many non-isomorphic rank 2 indecomposable modules with the same filtration." ], [ "Acknowledgments", "We thank Matthew Pressland and Alastair King for numerous helpful conversations.", "K. B. was supported by a Royal Society Wolfson Fellowship.", "She is currently on leave from the University of Graz.", "D.B.", "was supported by the Austrian Science Fund Project Number P29807-35.", "J.-R.L.", "was supported by the Austrian Science Fund (FWF): M 2633-N32 Lise Meitner Program." ], [ "Preliminaries", "We follow the exposition from [13], [1], [2] in order to introduce notation and background results.", "Let $\\Gamma _n$ be the quiver of the boundary algebra, with vertices $1,2,\\dots , n$ on a cycle and arrows $x_i: i-1\\rightarrow i$ , $y_i:i\\rightarrow i-1$ .", "We write ${\\rm CM}(B_{k,n})$ for the category of maximal Cohen-Macaulay modules for the completed path algebra $B_{k,n}$ of $\\Gamma _n$ , with relations $xy-yx$ and $x^k-y^{n-k}$ (at every vertex).", "The centre of $B_{k,n}$ is $Z:=\\operatorname{\\mathbb {C}}\\nolimits [\|t\|]$ , where $t=\\sum _ix_iy_i$ .", "For example, when $n=5$ we have the quiver [scale=1] jin 1,...,5 (90-72j:1.5cm) node[black] (wj) $\\bullet $ ; (162-72j:1.5cm) node[black] (vj) ; [->,>=latex] (vj) edge[black,bend left=30,thick] node[black,auto] $x_{\\j }$ (wj); [->,>=latex] (wj) edge[black,bend left=30,thick] node[black,auto] $y_{\\j }$ (vj); (90:1.5cm) node[above=3pt] 5; (162:1.5cm) node[above left] 4; (234:1.5cm) node[below left] 3; (306:1.5cm) node[below right] 2; (18:1.5cm) node[above right] 1; The algebra $B_{k,n}$ coincides with the quotient of the completed path algebra of the graph $C$ (a circular graph with vertices $C_0=\\mathbb {Z}_n$ set clockwise around a circle, and with the set of edges, $C_1$ , also labeled by $\\mathbb {Z}_n$ , with edge $i$ joining vertices $i-1$ and $i$ ), i.e.", "the doubled quiver as above, by the closure of the ideal generated by the relations above (we view the completed path algebra of the graph $C$ as a topological algebra via the $m$ -adic topology, where $m$ is the two-sided ideal generated by the arrows of the quiver, see [5]).", "The algebra $B_{k,n}$ , that we will often denote by $B$ when there is no ambiguity, was introduced in [13].", "Observe that $B_{k,n}$ is isomorphic to $B_{n-k,n}$ , so we will always take $k\\le \\frac{n}{2}$ .", "The (maximal) Cohen-Macaulay $B$ -modules are precisely those which are free as $Z$ -modules.", "Such a module $M$ is given by a representation $\\lbrace M_i\\,:\\,i\\in C_0\\rbrace $ of the quiver with each $M_i$ a free $Z$ -module of the same rank (which is the rank of $M$ ).", "Definition 2.1 ([13], Definition 3.5) For any $B_{k,n}$ -module $M$ and $K$ the field of fractions of $Z$ , the rank of $M$ , denoted by ${\\rm rk}(M)$ , is defined to be ${\\rm rk}(M) = {\\rm len}(M \\otimes _Z K)$ .", "Note that $B\\otimes _Z K\\cong M_n ( K)$ , which is a simple algebra.", "It is easy to check that the rank is additive on short exact sequences, that ${\\rm rk} (M) = 0$ for any finite-dimensional $B$ -module (because these are torsion over $Z$ ) and that, for any Cohen-Macaulay $B$ -module $M$ and every idempotent $e_j$ , $1\\le j\\le n$ , ${\\rm rk}_Z(e_j M) = {\\rm rk}(M)$ , so that, in particular, ${\\rm rk}_Z(M) = n {\\rm rk}(M)$ .", "Definition 2.2 ([13], Definition 5.1) For any $k$ -subset $I$ of $C_1$ , we define a rank 1 $B$ -module $L_I = (U_i,\\ i\\in C_0 \\,;\\, x_i,y_i,\\, i\\in C_1)$ as follows.", "For each vertex $i\\in C_0$ , set $U_i=\\mathbb {C}[[t]]$ and, for each edge $i\\in C_1$ , set $x_i\\colon U_{i-1}\\rightarrow U_{i}$ to be multiplication by 1 if $i\\in I$ , and by $t$ if $i\\notin I$ , $y_i\\colon U_{i}\\rightarrow U_{i-1}$ to be multiplication by $t$ if $i\\in I$ , and by 1 if $i\\notin I$ .", "The module $L_I$ can be represented by a lattice diagram $\\mathcal {L}_I$ in which $U_0,U_1,U_2,\\ldots , U_n$ are represented by columns of vertices (dots) from left to right (with $U_0$ and $U_n$ to be identified), going down infinitely.", "The vertices in each column correspond to the natural monomial $\\mathbb {C}$ -basis of $\\mathbb {C}[t]$ .", "The column corresponding to $U_{i+1}$ is displaced half a step vertically downwards (respectively, upwards) in relation to $U_i$ if $i+1\\in I$ (respectively, $i+1\\notin I$ ), and the actions of $x_i$ and $y_i$ are shown as diagonal arrows.", "Note that the $k$ -subset $I$ can then be read off as the set of labels on the arrows pointing down to the right which are exposed to the top of the diagram.", "For example, the lattice diagram $\\mathcal {L}_{\\lbrace 1,4,5\\rbrace }$ in the case $k=3$ , $n=8$ , is shown in the following picture Figure: Lattice diagram of the module L {1,4,5} L_{\\lbrace 1,4,5\\rbrace }We see from Figure REF that the module $L_I$ is determined by its upper boundary, denoted by the thick lines, which we refer to as the rim of the module $L_I$ (this is why we call the $k$ -subset $I$ the rim of $L_I$ ).", "Throughout this paper we will identify a rank 1 module $L_I$ with its rim.", "Moreover, most of the time we will omit the arrows in the rim of $L_I$ and represent it as an undirected graph.", "We say that $i$ is a peak of the rim $I$ if $i\\notin I$ and $i+1\\in I$ .", "In the above example, the peaks of $I=\\lbrace 1,4,5\\rbrace $ are 3 and 8.", "We say that $i$ is a valley of the rim $I$ if $i\\in I$ and $i+1\\notin I$ .", "In the above example, the valleys of $I=\\lbrace 1,4,5\\rbrace $ are 1 and 5.", "Proposition 2.3 ([13], Proposition 5.2) Every rank 1 Cohen-Macaulay $B$ -module is isomorphic to $L_I$ for some unique $k$ -subset $I$ of $C_1$ .", "Every $B$ -module has a canonical endomorphism given by multiplication by $t\\in Z$ .", "For ${L}_I$ this corresponds to shifting $\\mathcal {L}_I$ one step downwards.", "Since $Z$ is central, ${\\rm Hom}_B(M,N)$ is a $Z$ -module for arbitrary $B$ -modules $M$ and $N$ .", "If $M,N$ are free $Z$ -modules, then so is ${\\rm Hom}_B(M,N)$ .", "In particular, for any two rank 1 Cohen-Macaulay $B$ -modules $L_I$ and $L_J$ , ${\\rm Hom}_B(L_I,L_J)$ is a free module of rank 1 over $Z=\\mathbb {C}[[t]]$ , generated by the canonical map given by placing the lattice of $L_I$ inside the lattice of $L_J$ as far up as possible so that no part of the rim of $L_I$ is strictly above the rim of $L_J$ [13].", "Definition 2.4 ($r$ -interlacing) Let $I$ and $J$ be two $k$ -subsets of $\\lbrace 1,\\dots ,n\\rbrace $ .", "The sets $I$ and $J$ are said to be $r$ -interlacing if there exist subsets $\\lbrace i_1,i_3,\\dots ,i_{2r-1}\\rbrace \\subset I\\setminus J$ and $\\lbrace i_2,i_4,\\dots , i_{2r}\\rbrace \\subset J\\setminus I$ such that $i_1<i_2<i_3<\\dots <i_{2r}<i_1$ (cyclically) and if there exist no larger subsets of $I$ and of $J$ with this property.", "We say that $I$ and $J$ are tightly $r$ -interlacing if they are $r$ -interlacing and $\|I\\cap J\|=k-r.$ Definition 2.5 A $B$ -module is rigid if ${\\rm Ext}^1_B (M,M)=0$ .", "If $I$ and $J$ are $r$ -interlacing $k$ -subsets, where $r<2$ , then ${\\rm Ext}_{B}^1(L_I,L_J)=0$ , in particular, rank 1 modules are rigid (see [13]).", "Every rigid indecomposable $M$ of rank $n$ in ${\\rm CM}(B)$ has a filtration having factors $L_{I_1},L_{I_{2}},\\dots , L_{I_n}$ of rank 1.", "This filtration is noted in its profile, ${\\rm pr} (M) = I_1 \\mid I_2\\mid \\ldots \\mid I_n$ , [13].", "In the case of a rank 2 module $M$ with filtration $L_I\\mid L_J$ (i.e.", "with profile $I\\mid J$ ), we picture this module by drawing the rim $J$ below the rim $I$ , in such a way that $J$ is placed as far up as possible so that no part of the rim $J$ is strictly above the rim $I$ .", "We refer to this picture of $M$ as its lattice diagram.", "Note that there is at least one point where the rims $I$ and $J$ meet (see Figure REF for an example).", "Remark 2.6 Suppose that the two $k$ -subsets $I$ and $J$ are $r$ -interlacing and that $M$ is a rank 2 module with profile $I\\mid J$ .", "Then the two rims in the lattice diagram of $M$ form a number of regions between the points where the two rims meet but differ in direction before and/or after meeting.", "We call these regions the boxes formed by the rims or by the profile.", "The term box is a combinatorial tool which will be very useful in finding conditions for indecomposability.", "Let us point out, however, that the module $M$ might be a direct sum in which case the lattice diagram is really a pair of lattice diagrams of rank 1 modules.", "We still view the corresponding diagram as forming boxes.", "If $I$ and $J$ are $r$ -interlacing, then they form exactly $r$ -boxes if and only if they are tightly $r$ -interlacing.", "(If we consider the lattice diagram as an infinite branched graph in the plane, the boxes are the closures of the finite regions in the complement.)", "A lattice diagram with three boxes is shown in Figure REF .", "If $M$ is a rank 2 module with $r_1$ boxes, with $r_1\\le r$ , the poset structure associated with $M$ is $1^{r_1}\\mid 2$ , see Figure REF .", "For background on the poset associated with an indecomposable module or with its profile, we refer the reader to [13] and to [3].", "Consider the tame cases $(k,n)=(3,9)$ or $(k,n)=(4,8)$ and let $M$ be a rigid indecomposable rank 2 module of ${\\rm CM}(B_{k,n})$ .", "Then $M\\cong L_I\\mid L_J$ where $I$ and $J$ are 3-interlacing, [2].", "Furthermore, we also know that if $I$ and $J$ are tightly 3-interlacing and if $M\\cong L_I\\mid L_J$ , then $M$ is indecomposable, [2].", "We therefore start studying pairs of tightly 3-interlacing $k$ -subsets in order to construct indecomposable rank 2 modules and will later consider higher interlacing.", "Throughout the paper, our strategy to prove a module is indecomposable is to show that its endomorphism ring does not have non-trivial idempotent elements.", "When we deal with a decomposable rank 2 module, in order to determine the summands of this module, we construct a non-trivial idempotent in its endomorphism ring, and then find corresponding eigenvectors at each vertex of the quiver and check the action of the morphisms $x_i$ on these eigenvectors." ], [ "Tight 3-interlacing", "In this section we give a construction of rank 2 indecomposable modules with the profile $I\\mid J$ in the case when $I$ and $J$ are tightly 3-interlacing $k$ -subsets, i.e.", "when $\|I\\setminus J\|=\|J\\setminus I\|=3$ and non-common elements of $I$ and $J$ interlace.", "This covers all indecomposable rank 2 modules in the tame case $(3,9)$ and almost all indecomposable rank 2 modules in the tame case $(4,8)$ .", "We want to define a rank 2 module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ with filtration $L_I\\mid L_J$ in a similar way as rank 1 modules are defined in ${\\rm CM}(B_{k,n})$ .", "Let $V_i:=\\operatorname{\\mathbb {C}}\\nolimits [\|t\|]\\oplus \\operatorname{\\mathbb {C}}\\nolimits [\|t\|]$ , $i=1,\\dots , n$ .", "The module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ has $V_i$ at each vertex $1,2,\\dots , n$ of $\\Gamma _n$ .", "In order to have a module structure for $B_{k,n}$ , for every $i$ we need to define $x_i\\colon V_{i-1}\\rightarrow V_{i}$ and $y_i\\colon V_{i}\\rightarrow V_{i-1}$ in such a way that $x_iy_i=t\\cdot \\operatorname{id}\\nolimits $ and $x^k=y^{n-k}$ .", "Since $L_J$ is a submodule of a rank 2 module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ , and $L_I$ is the quotient, if we extend the basis of $L_J$ to the basis of the module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ , then with respect to that basis all the matrices $x_i$ , $y_i$ must be upper triangular with diagonal entries from the set $\\lbrace 1,t\\rbrace $ .", "More precisely, the diagonal of $x_i$ (resp.", "$y_i$ ) is $(1,t)$ (resp.", "$(t,1)$ ) if $i\\in J\\setminus I$ , it is $(t,1)$ (resp.", "$(1,t)$ ) if $i\\in I\\setminus J$ , $(t,t)$ (resp.", "$(1,1)$ ) if $i\\in I^c\\cap J^c$ , and $(1,1)$ (resp.", "$(t,t)$ ) if $i\\in I\\cap J$ .", "The only entries in all these matrices that are left to be determined are the ones in the upper right corner.", "Let us assume that $n=6$ , $I=\\lbrace 1,3,5\\rbrace $ , and $J=\\lbrace 2,4,6\\rbrace $ .", "In the general case, the arguments are the same.", "Denote by $b_i$ the upper right corner element of $x_i$ .", "From $x_iy_i=t\\cdot \\operatorname{id}\\nolimits $ , we have that the upper right corner element of $y_i$ is $-b_i$ .", "From the relation $x^k=y^{n-k}$ it follows that $\\sum _{i=1}^6b_i=0$ .", "Thus, our module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is 0@<.8ex>[rr] t b1 0 1 1 @<.8ex>[rr] 1 b2 0 t @<.8ex>[ll] 1 -b1 0 t 2 @<.8ex>[rr] t b3 0 1 @<.8ex>[ll] t -b2 0 1 3 @<.8ex>[rr] 1 b4 0 t @<.8ex>[ll] 1 -b3 0 t 4 @<.8ex>[rr] t b5 0 1 @<.8ex>[ll] t -b4 0 1 5 @<.8ex>[rr] 1 b6 0 t @<.8ex>[ll] 1 -b5 0 t 6 @<.8ex>[ll] t -b6 0 1 with $\\sum b_i=0$ .", "We say that $\\mathbb {M}(I,J)$ is determined by the 6-tuple $(b_1,b_2,b_3, b_4,b_5,b_6)$ ." ], [ "Divisibility conditions for (in)decomposability", "Let $I,J$ be two $k$ -subsets and $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ be given by the tuple $(b_1,b_2,b_3, b_4,b_5,b_6)$ with $\\sum b_i=0$ .", "The question is how to determine the $b_i$ 's so that the module $\\mathbb {M}(I,J)$ is indecomposable.", "Assume first that $\\mathbb {M}(I,J)$ is decomposable and that $L_J$ is a direct summand of $\\mathbb {M}(I,J)$ .", "Then there exists a retraction $\\mu =(\\mu _i)_{i=1}^6$ such that $\\mu _i\\circ \\theta _i=\\operatorname{id}\\nolimits $ , where $(\\theta _i)_{i=1}^6$ is the natural injection of $L_J$ into $\\mathbb {M}(I,J)$ .", "Using the same basis as before, we can assume that $\\mu _i=[1\\,\\, \\alpha _i ]$ for some $\\alpha _i\\in \\operatorname{\\mathbb {C}}\\nolimits [[t]]$ .", "From the commutativity relations we have $\\operatorname{id}\\nolimits \\circ \\mu _i=\\mu _{i+1}\\circ x_{i+1}$ for $i$ odd, and $t\\cdot \\operatorname{id}\\nolimits \\circ \\mu _i=\\mu _{i+1}\\circ x_{i+1}$ for $i$ even.", "It follows that $\\alpha _i=b_{i+1}+t\\alpha _{i+1}$ for $i$ odd, and $t\\alpha _i=b_{i+1}+\\alpha _{i+1}$ for $i$ even.", "From this we have $t(\\alpha _2-\\alpha _4)&=b_3+b_4,\\\\t(\\alpha _4-\\alpha _6)&=b_5+b_6,\\\\t(\\alpha _6-\\alpha _2)&=b_1+b_2.$ Thus, if $L_J$ is a direct summand of $\\mathbb {M}(I,J)$ , then $t\| b_3+b_4$ , $t\|b_5+b_6$ , and $t\| b_1+b_2$ (and we can easily find elements $\\alpha _i$ , $i=1,\\dots ,6$ , satisfying previous equations).", "If only one of these conditions is not met, then $L_J$ is not a direct summand of $M$ .", "For example, if we choose $b_2=-b_3=0$ , $b_4=-b_5=1$ , and $b_6=-b_1=2$ in the construction of the module $\\mathbb {M}(I,J)$ , then $L_J$ is not a direct summand of $\\mathbb {M}(I,J)$ .", "Our aim is to study the structure of the module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ in terms of the divisibility conditions the coefficients $b_i$ satisfy.", "Remark 3.1 If $L_J$ is not a summand of $M$ , it does not mean that $M$ is indecomposable (cf.", "Theorem 3.12 in [2]).", "Let us now consider the general case, i.e.", "let $\\mathbb {M}(I,J)$ be the module as defined above, but in general terms when $I$ and $J$ are tightly 3-interlacing.", "Write $I\\setminus J$ as $\\lbrace i_1,i_2,i_3\\rbrace $ and $J\\setminus I=\\lbrace j_1,j_2,j_3\\rbrace $ so that $1\\le i_1<j_1<i_2<j_2<i_3<j_3\\le n$ .", "Define $&&&&&&x_{i_{r}}&=\\begin{pmatrix} t& b_{2r-1} \\\\ 0 & 1 \\end{pmatrix},& x_{j_{r}}&=\\begin{pmatrix} 1& b_{2r} \\\\ 0 & t \\end{pmatrix}, &&&&&\\\\&&&&&&y_{i_{r}}&=\\begin{pmatrix} 1& -b_{2r-1} \\\\ 0 & t \\end{pmatrix}, &y_{j_{r}}&=\\begin{pmatrix} t& -b_{2r} \\\\ 0 & 1 \\end{pmatrix},&&&&&$ for $r=1,2,3$ (see the previous figure for $n=6$ ).", "For $i\\in I^c \\cap J^c$ , we set $x_i=\\begin{pmatrix} t & 0 \\\\ 0 & t \\end{pmatrix}$ and $y_i=\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}$ .", "For $i\\in I\\cap J$ , we set $x_i=\\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}$ and $y_i=\\begin{pmatrix} t & 0 \\\\ 0 & t \\end{pmatrix}$ .", "Also, we assume that $\\sum _{r=1}^6b_r=0$ .", "Note that for $i\\in (I^c \\cap J^c)\\cup (I\\cap J)$ we define the matrices $x_i$ and $y_i$ to be diagonal, i.e.", "we assume that the upper right corner of $x_i$ and $y_i$ is 0 if $i\\in (I^c \\cap J^c)\\cup (I\\cap J)$ .", "We can achieve this under a suitable base change of $V_i$ .", "By construction it holds that $xy=yx$ and $x^k=y^{n-k}$ at all vertices, and that $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is free over the centre of the boundary algebra.", "Hence, the following proposition holds.", "Proposition 3.2 The above-constructed module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is in ${\\rm CM}(B_{k,n})$ .", "For the remainder of the paper, if $w=t^av$ , for some positive integer $a$ , then $t^{-a}w$ stands for $v$ .", "Proposition 3.3 Let $I,J$ be tightly 3-interlacing, $n\\ge 6$ arbitrary, $I\\setminus J=\\lbrace i_1,i_2,i_3\\rbrace $ , and $J\\setminus I=\\lbrace j_1,j_2,j_3\\rbrace $ , where $1\\le i_1<j_1<i_2<j_2<i_3<j_3\\le n$ .", "If $\\varphi =( \\varphi _i)_{i=1}^n\\in $ Hom$(\\operatorname{\\mathbb {M}}\\nolimits (I,J),\\operatorname{\\mathbb {M}}\\nolimits (I,J))$ , then $\\varphi _{j_3}&=\\begin{pmatrix}a & b \\\\ c & d \\end{pmatrix},\\\\\\varphi _{i_1}&= \\begin{pmatrix}a+b_1t^{-1}c & tb+(d-a)b_1-b_1^2t^{-1}c \\\\ t^{-1}c& d-b_1t^{-1}c \\end{pmatrix},\\\\\\varphi _{j_1}&=\\begin{pmatrix}a+(b_1+b_2)t^{-1}c & b+t^{-1}((d-a)(b_1+b_2)-(b_1+b_2)^2t^{-1}c) \\\\ c& d-(b_1+b_2)t^{-1}c \\end{pmatrix},\\\\\\varphi _{i_2}&=\\begin{pmatrix}a+(b_1+b_2+b_3)t^{-1}c & tb+(d-a)(b_1+b_2+b_3)-(b_1+b_2+b_3)^2t^{-1}c \\\\ t^{-1}c& d-(b_1+b_2+b_3)t^{-1}c \\end{pmatrix},\\\\\\varphi _{j_2}&=\\begin{pmatrix}a+(b_1+b_2+b_3+b_4)t^{-1}c & b+t^{-1}((d-a)(b_1+b_2+b_3+b_4)-(b_1+b_2+b_3+b_4)^2t^{-1}c) \\\\ c& d-(b_1+b_2+b_3+b_4)t^{-1}c \\end{pmatrix},\\\\\\varphi _{i_3}&=\\begin{pmatrix}a-b_6t^{-1}c & tb-(d-a)b_6-b_6^2t^{-1}c \\\\ t^{-1}c& d+b_6t^{-1}c \\end{pmatrix},\\\\\\varphi _{i}&=\\varphi _{i-1}, \\,\\, \\text{for } i\\in (I^c \\cap J^c)\\cup (I\\cap J),$ with $a,b,c,d\\in \\operatorname{\\mathbb {C}}\\nolimits [\|t\|]$ .", "Furthermore, $t\\mid c$ , $t\\mid (d-a)(b_1+b_2)-(b_1+b_2)^2t^{-1}c$ , and $t\\mid (d-a)(b_1+b_2+b_3+b_4)-(b_1+b_2+b_3+b_4)^2t^{-1}c$ .", "First we consider the case $n=6$ .", "Let $\\varphi =(\\varphi _1,\\dots , \\varphi _6)$ be an endomorphism of $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ , where each $\\varphi _i$ is an element of $M_2(\\operatorname{\\mathbb {C}}\\nolimits [\|t\|])$ (matrices over the centre).", "6@<.8ex>[rr] t b1 0 1 @<.1ex>[ddd]6 1 @<.8ex>[rr] 1 b2 0 t @<.8ex>[ll] 1 -b1 0 t @<.1ex>[ddd]1 2 @<.8ex>[rr] t b3 0 1 @<.8ex>[ll] t -b2 0 1 @<.1ex>[ddd]2 3 @<.8ex>[rr] 1 b4 0 t @<.8ex>[ll] 1 -b3 0 t @<.1ex>[ddd]3 4 @<.8ex>[rr] t b5 0 1 @<.8ex>[ll] t -b4 0 1 @<.1ex>[ddd]4 5 @<.8ex>[rr] 1 b6 0 t @<.8ex>[ll] 1 -b5 0 t @<.1ex>[ddd]5 6 @<.8ex>[ll] t -b6 0 1 @<.1ex>[ddd]6 6@<.8ex>[rr] t b1 0 1 1 @<.8ex>[rr] 1 b2 0 t @<.8ex>[ll] 1 -b1 0 t 2 @<.8ex>[rr] t b3 0 1 @<.8ex>[ll] t -b2 0 1 3 @<.8ex>[rr] 1 b4 0 t @<.8ex>[ll] 1 -b3 0 t 4 @<.8ex>[rr] t b5 0 1 @<.8ex>[ll] t -b4 0 1 5 @<.8ex>[rr] 1 b6 0 t @<.8ex>[ll] 1 -b5 0 t 6 @<.8ex>[ll] t -b6 0 1 We use commutativity relations $x_{i+1}\\varphi _i = \\varphi _{i+1}x_{i+1}$ , i.e.", "we check the relations: $\\begin{array}{clccl}(i) & x_2\\varphi _1 = \\varphi _2x_2, & \\quad & (ii) & x_3\\varphi _2 = \\varphi _3x_3, \\\\(iii) & x_4\\varphi _3 = \\varphi _4x_4, & & (iv) & x_5\\varphi _4 = \\varphi _5x_5, \\\\(v) & x_6\\varphi _5 = \\varphi _6x_6, & & (vi) & x_1\\varphi _6 = \\varphi _1 x_1.\\end{array}$ Let $\\varphi _0=\\begin{pmatrix} a & b \\\\ c & d \\end{pmatrix}$ and $\\varphi _1=\\begin{pmatrix} e & f \\\\ g & h \\end{pmatrix}$ .", "From (vi), we get $\\begin{pmatrix} at+b_1c & bt+b_1d \\\\ c & d\\end{pmatrix} = \\begin{pmatrix} et & eb_1+f \\\\ gt & h+gb_1 \\end{pmatrix},$ and $at+b_1c=et$ , $bt+b_1d=eb_1+f$ , $c=gt$ , and $d=gb_1+h$ .", "It follows that $t\\mid c$ , and that $\\varphi _1=\\begin{pmatrix}a+b_1t^{-1}c & tb+(d-a)b_1-b_1^2t^{-1}c \\\\ t^{-1}c& d-b_1t^{-1}c \\end{pmatrix}.$ Similarly, if $\\varphi _2=\\begin{pmatrix} e & f \\\\ g & h \\end{pmatrix}$ , then equality $x_2\\varphi _1 = \\varphi _2x_2$ yields $\\varphi _2=\\begin{pmatrix}a+(b_1+b_2)t^{-1}c & b+t^{-1}((d-a)(b_1+b_2)-(b_1+b_2)^2t^{-1}c) \\\\ c& d-(b_1+b_2)t^{-1}c \\end{pmatrix},$ and $t\\mid (d-a)(b_1+b_2)-(b_1+b_2)^2t^{-1}c$ .", "The rest of the proof for $\\varphi _3$ , $\\varphi _4$ , and $\\varphi _5$ is analogous.", "We omit the details of elementary, but tedious computation.", "In the general case of arbitrary $n$ , the proof is almost the same as for $n=6$ .", "The only thing left to note is that if $i\\in (I^c \\cap J^c)\\cup (I\\cap J)$ , then $x_i$ is a scalar matrix (either identity or $t$ times identity), so from $x_{i}\\varphi _{i-1}=\\varphi _{i}x_{i}$ , it follows immediately that $\\varphi _{i-1}=\\varphi _{i}$ .", "Remark 3.4 Take $\\varphi $ as in Proposition REF .", "The morphism $\\varphi $ also satisfies the other six relations $\\varphi _i y_{i+1}=y_{i+1}\\varphi _{i+1}$ .", "Indeed, if $x_{i+1}\\varphi _i=\\varphi _{i+1}x_{i+1}$ , then if we multiply this equality by $y_{i+1}$ both from the left and right, we obtain $t \\cdot \\varphi _i y_{i+1}=t\\cdot y_{i+1}\\varphi _{i+1} $ .", "Since $t$ is a regular element in $\\mathbb {C}[[t]]$ , after cancellation by $t$ we obtain $\\varphi _i y_{i+1}=y_{i+1}\\varphi _{i+1}$ .", "Also, note that in order to prove that $\\varphi $ is idempotent we only need to make sure that only one $\\varphi _i$ is idempotent.", "If $\\varphi _i$ is idempotent, then from $x_{i+1}\\varphi _i=\\varphi _{i+1}x_{i+1}$ when we multiply by $\\varphi _{i+1}$ from the left, we have $\\varphi _{i+1}x_{i+1}\\varphi _i=\\varphi ^2_{i+1}x_{i+1}$ , and multiplying by $\\varphi _i$ from the right we get $\\varphi _{i+1}x_{i+1}\\varphi _i=x_{i+1}\\varphi _i^2$ .", "Then $x_{i+1}\\varphi ^2_i=\\varphi ^2_{i+1}x_{i+1}$ and $x_{i+1}\\varphi _i=\\varphi ^2_{i+1}x_{i+1}$ as $\\varphi _i$ is idempotent, and subsequently, $\\varphi _{i+1}x_{i+1}=\\varphi ^2_{i+1}x_{i+1}$ , yielding $\\varphi _{i+1}=\\varphi ^2_{i+1}$ after multiplication by $y_{i+1}$ from the right and cancellation by $t$ .", "We now give necessary and sufficient conditions for the module $\\mathbb {M}(I,J)$ to be indecomposable.", "Theorem 3.5 Let $I$ and $J$ be tightly 3-interlacing.", "The module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is indecomposable if and only if $t\\nmid b_{1}+b_{2}$ , $t\\nmid b_{3}+b_{4}$ , and $t\\nmid b_{5}+b_{6}$ .", "As before, it is sufficient to consider the case $n=6$ , so we can assume that $I=\\lbrace 1,3,5\\rbrace $ and $J=\\lbrace 2,4,6\\rbrace $ .", "Let $\\varphi =( \\varphi _i)_{i=1}^6\\in $ Hom$(\\operatorname{\\mathbb {M}}\\nolimits (I,J),\\operatorname{\\mathbb {M}}\\nolimits (I,J))$ be an idempotent homomorphism and assume that $\\varphi _{0}=\\begin{pmatrix}a & b \\\\ c & d \\end{pmatrix}$ .", "From the previous proposition we know that $t&\\mid c, \\\\t&\\mid (d-a)(b_1+b_2)-(b_1+b_2)^2t^{-1}c, \\\\t&\\mid (d-a)(b_1+b_2+b_3+b_4)-(b_1+b_2+b_3+b_4)^2t^{-1}c. $ Assume that $t\\nmid b_1+b_2$ , $t\\nmid b_3+b_4$ , and $t\\nmid b_5+b_6$ .", "Since $t\\nmid b_1+b_2$ , it follows from relation () that $t\\mid d-a-(b_1+b_2)t^{-1}c.$ Similarly, since $t\\nmid b_5+b_6=-(b_1+b_2+b_3+b_4)$ , from relation () follows that $t\\mid d-a-(b_1+b_2+b_3+b_4)t^{-1}c.$ Thus, it must hold that $t\\mid (b_3+b_4)t^{-1}c,$ and since $t\\nmid b_3+b_4$ , it must be that $t\\mid t^{-1}c$ , and subsequently that $t\\mid d-a$ .", "From the fact that $\\varphi _0$ is idempotent and $t\\mid c$ it follows that $t\\mid a-a^2$ and $t\\mid d-d^2$ .", "Also, from $\\varphi _0^2=\\varphi _0$ it follows that either $a=d$ or $a+d=1$ .", "If $a=d$ , then $b=c=0$ (otherwise $a=d=\\frac{1}{2}$ and $\\frac{1}{4}=bc$ , which is not possible as $c$ is divisible by $t$ ), and $a=d=1$ or $a=d=0$ giving us the trivial idempotents.", "If $a+d=1$ , then $t\\mid a$ or $t\\mid d$ .", "Taking into account that $t\\mid d-a$ , we conclude that $t\\mid a$ and $t\\mid d$ .", "This implies that $1=a+d$ is divisible by $t$ , which is not true.", "Thus, the only idempotent homomorphisms of $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ are the trivial ones.", "Hence, $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is indecomposable.", "Assume that $t$ divides at least one of the elements $b_1+b_2$ , $b_3+b_4$ , $b_5+b_6$ .", "If $t$ divides all three of them, we have seen before that $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is the direct sum of $L_I$ and $L_J$ , hence it is a decomposable module.", "The remaining case is when one of $b_1+b_2$ , $b_3+b_4$ , $b_5+b_6$ is divisible by $t$ and the other two are not divisible by $t$ .", "Note that it is not possible that two of them are divisible by $t$ , and one is not, because they sum up to zero.", "Let $t\\mid b_5+b_6$ , $t\\nmid b_1+b_2$ , and $t\\nmid b_3+b_4$ .", "In order to find a non-trivial idempotent $\\varphi $ , note that the relation () holds because $t\\mid b_5+b_6=-(b_1+b_2+b_3+b_4)$ .", "Hence, we only need to find elements $a$ , $b$ , $c$ , and $d$ in such a way that $t\\mid c$ and $t\\mid d-a-(b_1+b_2)t^{-1}c$ .", "Recall that if $a=d$ , then we only obtain the trivial idempotents because $t\\mid c$ .", "So it must be $a+d=1$ if we want to find a non-trivial idempotent.", "If we choose $a=1$ , $d=0$ , then $t\\mid 1+(b_1+b_2)t^{-1}c$ .", "Thus $(b_1+b_2)t^{-1}c=-1+tg,$ for some $g$ .", "We can take $g=0$ , i.e.", "$c=t(b_1+b_2)^{-1}$ (recall that $b_1+b_2$ is invertible because $t\\nmid b_1+b_2$ ) giving us the idempotent ($b=0$ since $a-a^2=bc$ and $c\\ne 0$ ) $\\varphi _0=\\begin{pmatrix}1& 0\\\\-t(b_1+b_2)^{-1}&0\\end{pmatrix}.$ Its orthogonal complement is the idempotent $\\begin{pmatrix}0& 0\\\\t(b_1+b_2)^{-1}&1\\end{pmatrix}.$ Since these are non-trivial idempotents, it follows that the module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is decomposable, what we needed to prove.", "Example 3.6 Let $k=3$ , $n=6$ , $I=\\lbrace 1,3,5\\rbrace $ and $J=\\lbrace 2,4,6\\rbrace $ .", "If $b_1=-2$ , $b_2=0$ , $b_3=0$ , $b_4=1$ , $b_5=-1$ , and $b_6=2$ , then it is easily checked that $t\\nmid b_i+b_{i+1}$ , $i=1,3,5$ , and that $\\sum _{i=1}^6b_i=0$ , thus giving us an indecomposable $B_{k,n}$ -module: 6@<.8ex>[rr] t -2 0 1 1 @<.8ex>[rr] 1 0 0 t @<.8ex>[ll] 1 2 0 t 2 @<.8ex>[rr] t 0 0 1 @<.8ex>[ll] t 0 0 1 3 @<.8ex>[rr] 1 1 0 t @<.8ex>[ll] 1 0 0 t 4 @<.8ex>[rr] t -1 0 1 @<.8ex>[ll] t -1 0 1 5 @<.8ex>[rr] 1 2 0 t @<.8ex>[ll] 1 1 0 t 6 @<.8ex>[ll] t -2 0 1 The lattice diagram (showing only the rims) of $M=L_{135}\\mid L_{246}$ is $\\includegraphics [width=6cm]{135-246}$ Proposition 3.7 If $t\\nmid b_{1}+b_{2}$ , $t\\nmid b_{3}+b_{4}$ , and $t\\mid b_{5}+b_{6}$ , then $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is isomorphic to $L_{\\lbrace i_1,j_1,i_3\\rbrace \\cup (I\\cap J)}\\oplus L_{\\lbrace i_2,j_2, j_3\\rbrace \\cup (I\\cap J)}.$ In the proof of the previous theorem we constructed a non-trivial idempotent endomorphism $\\varphi _0=\\begin{pmatrix}1& 0\\\\-t(b_1+b_2)^{-1}&0\\end{pmatrix}.$ It remains to show that $\\operatorname{\\mathbb {M}}\\nolimits (I,J)\\cong L_{\\lbrace i_1,j_1,i_3\\rbrace }\\oplus L_{\\lbrace i_2,j_2, j_3\\rbrace }.$ We know that $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is the direct sum of rank 1 modules $L_X$ and $L_Y$ for some $X$ and $Y$ .", "Let us determine $X$ and $Y$ .", "For this, we take, at vertex $i$ , eigenvectors $v_i$ and $w_i$ corresponding to the eigenvalue 1 of the idempotents $\\varphi _i$ and $1-\\varphi _i$ respectively.", "For example, $v_0=[1\\,\\, ,\\,\\, -t(b_1+b_2)^{-1}]^t$ , $w_0=[0\\,\\, ,\\,\\, 1]^t$ , $v_1= [ 1-b_1(b_1+b_2)^{-1} \\,\\, ,\\,\\, -(b_1+b_2)^{-1} ]^t$ and $w_1= [b_1 \\,\\, , \\,\\, 1 ]^t$ , and so on.", "A basis for $L_X$ is $\\lbrace v_i \\mid i=0,\\ldots , 5\\rbrace $ , and a basis for $L_Y$ is $\\lbrace w_i \\mid i=0,\\ldots , 5\\rbrace $ .", "Direct computation gives us that $x_1v_0=tv_1$ , $x_2v_1=tv_2$ , $x_3v_2=v_3$ , $x_4v_3=v_4$ , $x_5v_4=tv_5$ , and $x_6v_5=v_0$ .", "Thus, $X=\\lbrace 3,4,6\\rbrace $ .", "Analogously, $Y=\\lbrace 1,2,5\\rbrace .$ In the general case, it is easily seen that this means that $X=\\lbrace i_2,j_2,j_3\\rbrace \\cup (I\\cap J)$ and $Y=\\lbrace i_1,j_1,i_3\\rbrace \\cup (I\\cap J)$ because $x_i$ is a scalar matrix for $i\\in (I\\cap J)\\cup (I^c\\cap J^c)$ .", "Remark 3.8 If $n=6$ , in the case when $t\\nmid b_1+b_2$ , $t\\nmid b_3+b_4$ , and $t\\mid b_5+b_6$ , if we just rename the vertices of the quiver by adding 2 (modulo $n$ ) to every vertex or by adding 4 to every vertex, we obtain two modules that correspond to the cases when $t\\mid b_1+b_2$ , $t\\nmid b_3+b_4$ , $t\\nmid b_5+b_6$ and $t\\nmid b_1+b_2$ , $t\\mid b_3+b_4$ , $t\\nmid b_5+b_6$ .", "In the general case, these two modules are direct sums $L_{\\lbrace i_1, i_2,j_2\\rbrace \\cup (I\\cap J)}\\oplus L_{\\lbrace j_1, i_3,j_3\\rbrace \\cup (I\\cap J)}$ and $L_{\\lbrace i_2, i_3,j_3\\rbrace \\cup (I\\cap J)}\\oplus L_{\\lbrace i_1,j_1, j_2\\rbrace \\cup (I\\cap J)}.$ Example 3.9 When $n=6$ , $I=\\lbrace 1,3,5\\rbrace $ , and $J=\\lbrace 2,4,6\\rbrace $ , an indecomposable module which has $L_J$ as a submodule and $L_I$ as a quotient module is given in Example REF .", "Also, there are four different decomposable modules appearing as the middle term in a short exact sequence that has $L_I$ (as a quotient) and $L_J$ (as a submodule) as end terms: $0\\longrightarrow L_J\\longrightarrow L_{\\lbrace 1, 3, 5\\rbrace }\\oplus L_{\\lbrace 2,4, 6\\rbrace } \\longrightarrow L_I \\longrightarrow 0,\\\\0\\longrightarrow L_J\\longrightarrow L_{\\lbrace 1, 2, 5\\rbrace }\\oplus L_{\\lbrace 3,4, 6\\rbrace } \\longrightarrow L_I \\longrightarrow 0,\\\\0\\longrightarrow L_J\\longrightarrow L_{\\lbrace 1, 3, 4\\rbrace }\\oplus L_{\\lbrace 2,5, 6\\rbrace } \\longrightarrow L_I \\longrightarrow 0,\\\\0\\longrightarrow L_J\\longrightarrow L_{\\lbrace 1, 2, 4\\rbrace }\\oplus L_{\\lbrace 3,5, 6\\rbrace } \\longrightarrow L_I \\longrightarrow 0.$ The profiles of the four modules that appear in the middle in these short exact sequences are illustrated in Figure REF .", "Note that the two rims now stand for two rank 1 modules which are direct summands of the module.", "The pictures each show two lattice diagrams which are overlaid so we can compare the positions of the peaks.", "In particular, the two lattice diagrams in (a) look like the lattice diagram of the indecomposable extension described in Example REF .", "Figure: The pairs of lattice diagrams of decomposable extensions between L {1,3,5} L_{\\lbrace 1, 3, 5\\rbrace } and L {2,4,6} L_{\\lbrace 2,4, 6\\rbrace }.Remark 3.10 Note that for $I=\\lbrace 1,3,5\\rbrace $ and $J=\\lbrace 2,4,6\\rbrace $ there is a non-trivial extension ${0@{->}[r]&L_J@{->}[rr]^{[\\operatorname{id}\\nolimits , -f_1]^t\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,}&& L_{\\lbrace 1, 3, 5\\rbrace }\\oplus L_{\\lbrace 2,4, 6\\rbrace } @{->}[rr]^{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,[f_2, f_3]} && L_I @{->}[r]& 0,}$ where $f_i$ , $i=1,2,3$ , is the canonical map between rank 1 modules (see Section , this is a homomorphism of minimal codimension).", "The middle term is equal to the direct sum of the end terms, but the maps make this short exact sequence not isomorphic to the trivial sequence.", "Now we turn our attention to the question of uniqueness of the constructed indecomposable module.", "If we choose a different set of values for $b_i$ , i.e.", "we choose a 6-tuple different from $(-2,0,0,1,-1,2)$ , so that the conditions $t\\nmid b_3+b_4$ , $t\\nmid b_5+b_6$ , and $t\\nmid b_1+b_2$ are fulfilled, then we obtain a module which is not the same as the above constructed module $\\mathbb {M}(I,J)$ in Example REF .", "In the next theorem, we will show directly that for different choices of 6-tuples giving us indecomposable modules with the same filtration $L_I \\mid L_J$ we obtain isomorphic modules.", "Thus, there is a unique indecomposable module with filtration $L_I \\mid L_J$ .", "Theorem 3.11 Let $(b_{1}, b_{2}, b_{3}, b_{4}, b_{5}, b_{6} )$ and $(c_{1}, c_{2}, c_{3}, c_{4}, c_{5}, c_{6} )$ be different 6-tuples corresponding to indecomposable modules $M_1$ and $M_2$ , respectively, as constructed in Theorem REF .", "Then the modules $M_1$ and $M_2$ are isomorphic.", "As before, it is sufficient to consider the $n=6$ case.", "We will explicitly construct an isomorphism $\\varphi =(\\varphi _i)_{i=0}^5$ between the two modules, where $\\varphi _i: V_i\\longrightarrow W_i$ , and $V_i$ and $W_i$ are the vector spaces at vertex $i$ of the modules $M_1$ and $M_2$ respectively.", "Also, each $\\varphi _i$ is invertible.", "Recall that $b_i$ (resp.", "$c_i$ ) is the right upper corner element of $x_i$ for the module $M_1$ (resp.", "$M_2$ ): 0@<.8ex>[rr] t b1 0 1 @<.1ex>[ddd]0 1 @<.8ex>[rr] 1 b2 0 t @<.8ex>[ll] 1 -b1 0 t @<.1ex>[ddd]1 2 @<.8ex>[rr] t b3 0 1 @<.8ex>[ll] t -b2 0 1 @<.1ex>[ddd]2 3 @<.8ex>[rr] 1 b4 0 t @<.8ex>[ll] 1 -b3 0 t @<.1ex>[ddd]3 4 @<.8ex>[rr] t b5 0 1 @<.8ex>[ll] t -b4 0 1 @<.1ex>[ddd]4 5 @<.8ex>[rr] 1 b6 0 t @<.8ex>[ll] 1 -b5 0 t @<.1ex>[ddd]5 6 @<.8ex>[ll] t -b6 0 1 @<.1ex>[ddd]0 0@<.8ex>[rr] t c1 0 1 1 @<.8ex>[rr] 1 c2 0 t @<.8ex>[ll] 1 -c1 0 t 2 @<.8ex>[rr] t c3 0 1 @<.8ex>[ll] t -c2 0 1 3 @<.8ex>[rr] 1 c4 0 t @<.8ex>[ll] 1 -c3 0 t 4 @<.8ex>[rr] t c5 0 1 @<.8ex>[ll] t -c4 0 1 5 @<.8ex>[rr] 1 c6 0 t @<.8ex>[ll] 1 -c5 0 t 0 @<.8ex>[ll] t -c6 0 1 Let us assume that $\\varphi _i=\\begin{pmatrix}\\alpha _i & \\beta _i \\\\ \\gamma _i & \\delta _i \\end{pmatrix}$ , for $i=0,\\ldots , 5$ .", "Then from $\\varphi _1 \\begin{pmatrix} t & b_1 \\\\ 0 & 1 \\end{pmatrix}=\\begin{pmatrix}t & c_1 \\\\ 0 & 1 \\end{pmatrix} \\varphi _0$ , we obtain $t\\mid \\gamma _0$ , $t \\gamma _1= \\gamma _0$ , $\\alpha _1=\\alpha _0+c_1t^{-1}\\gamma _0$ , $\\beta _1=\\beta _0t-\\alpha _0b_1+c_1\\delta _0-b_1c_1t^{-1}\\gamma _0$ , and $\\delta _1=\\delta _0-b_1t^{-1}\\gamma _0$ .", "Hence, $\\varphi _1=\\begin{pmatrix} \\alpha _0+c_1t^{-1}\\gamma _0 & \\beta _0t-\\alpha _0b_1+c_1\\delta _0-b_1c_1t^{-1}\\gamma _0 \\\\ t^{-1}\\gamma _0 & \\delta _0-b_1t^{-1}\\gamma _0 \\end{pmatrix}.$ Since $t\\mid \\gamma _0$ and we would like $\\varphi _0$ to be invertible, then it must be that $t\\nmid \\alpha _0$ and $t\\nmid \\delta _0$ .", "Then the inverse of $\\varphi _0$ is $\\frac{1}{\\alpha _0\\delta _0-\\beta _0\\gamma _0}\\begin{pmatrix} \\delta _0&-\\beta _0 \\\\ -\\gamma _0 & \\alpha _0 \\end{pmatrix}.$ From $\\varphi _2 \\begin{pmatrix} 1 & b_2 \\\\ 0 & t \\end{pmatrix}=\\begin{pmatrix}1 & c_2 \\\\ 0 & t \\end{pmatrix} \\varphi _1$ , we obtain that $\\varphi _2=\\begin{pmatrix} \\alpha _0+(c_1+c_2)t^{-1}\\gamma _0 & \\beta _0+t^{-1}(-\\alpha _0(b_1+b_2)+(c_1+c_2)\\delta _0 -(b_1+b_2)(c_1+c_2)t^{-1}\\gamma _0) \\\\ \\gamma _0 & \\delta _0-(b_1+b_2)t^{-1}\\gamma _0 \\end{pmatrix},$ where $t\\mid -\\alpha _0(b_1+b_2)+(c_1+c_2)\\delta _0 -(b_1+b_2)(c_1+c_2)t^{-1}\\gamma _0.$ Analogously, it is easily computed that $\\varphi _3&=\\begin{pmatrix} \\alpha _0+(c_1+c_2+c_3)t^{-1}\\gamma _0 & \\beta _0t-\\alpha _0(\\sum _{i=1}^3b_i)+(\\sum _{i=1}^3c_i)\\delta _0-(\\sum _{i=1}^3b_i)(\\sum _{i=1}^3c_i)t^{-1}\\gamma _0 \\\\ t^{-1}\\gamma _0 & \\delta _0-(b_1+b_2+b_3)t^{-1}\\gamma _0 \\end{pmatrix},\\\\\\varphi _4&=\\begin{pmatrix} \\alpha _0-(c_5+c_6)t^{-1}\\gamma _0 & \\beta _0+t^{-1}(\\alpha _0(b_5+b_6)-(c_5+c_6)\\delta _0 -(b_5+b_6)(c_5+c_6)t^{-1}\\gamma _0) \\\\ \\gamma _0 & \\delta _0+(b_5+b_6)t^{-1}\\gamma _0 \\end{pmatrix},\\\\\\varphi _5&=\\begin{pmatrix} \\alpha _0-c_6t^{-1}\\gamma _0 & \\beta _0t+\\alpha _0b_6-c_6\\delta _0-b_6c_6t^{-1}\\gamma _0 \\\\ t^{-1}\\gamma _0 & \\delta _0+b_6t^{-1}\\gamma _0 \\end{pmatrix},$ where $t\\mid \\alpha _0(b_5+b_6)-(c_5+c_6)\\delta _0 -(b_5+b_6)(c_5+c_6)t^{-1}\\gamma _0.$ In order to find an isomorphism $\\varphi $ , we must determine $\\alpha _0$ , $\\beta _0$ , $\\gamma _0$ , and $\\delta _0$ satisfying the following conditions: $t\\mid \\gamma _0$ , $t\\nmid \\alpha _0$ , $t\\nmid \\delta _0$ , and $t&\\mid -\\alpha _0(b_1+b_2)+(c_1+c_2)\\delta _0 -(b_1+b_2)(c_1+c_2)t^{-1}\\gamma _0,\\\\t&\\mid \\alpha _0(b_5+b_6)-(c_5+c_6)\\delta _0 -(b_5+b_6)(c_5+c_6)t^{-1}\\gamma _0.$ Note that there are no conditions attached to $\\beta _0$ so we set it to be 0.", "If we set $-\\alpha _0(b_1+b_2)+(c_1+c_2)\\delta _0 -(b_1+b_2)(c_1+c_2)t^{-1}\\gamma _0&=0,\\\\\\alpha _0(b_5+b_6)-(c_5+c_6)\\delta _0 -(b_5+b_6)(c_5+c_6)t^{-1}\\gamma _0&=0,$ then we get $\\alpha _0(b_5+b_6)\\left[1+\\frac{c_5+c_6}{c_1+c_2} \\right]-\\delta _0(c_5+c_6)\\left[1+\\frac{b_5+b_6}{b_1+b_2} \\right]=0.$ If $t\\mid 1+\\frac{c_5+c_6}{c_1+c_2}$ , then from $\\sum _{i=1}^6c_i=0$ , we get $t\\mid c_3+c_4$ , which is not true.", "The same holds for $1+\\frac{b_5+b_6}{b_1+b_2}$ , so both of these elements are invertible.", "Thus, if we set $\\delta _0=1$ , then we get $\\alpha _0=\\frac{(c_1+c_2)(b_3+b_4)(c_5+c_6)}{(b_1+b_2)(c_3+c_4)(b_5+b_6)},$ and $\\gamma _0=t \\frac{(c_3+c_4)(b_5+b_6)-(b_3+b_4)(c_5+c_6)}{(b_1+b_2)(c_3+c_4)(b_5+b_6)}.$ Hence, $\\varphi _0=\\begin{pmatrix} \\displaystyle \\frac{(c_1+c_2)(b_3+b_4)(c_5+c_6)}{(b_1+b_2)(c_3+c_4)(b_5+b_6)} & \\,\\,\\,\\,\\,\\,\\,\\, 0 \\\\ t \\displaystyle \\frac{(c_3+c_4)(b_5+b_6)-(b_3+b_4)(c_5+c_6)}{(b_1+b_2)(c_3+c_4)(b_5+b_6)} & \\,\\,\\,\\,\\,\\,\\,\\,1 \\end{pmatrix}.$ The other invertible matrices $\\varphi _i$ are now determined from the above equalities.", "Note that all of them are invertible because their determinant is equal to $\\alpha _0\\delta _0-\\beta _0\\gamma _0$ which is an invertible element.", "Also, $\\beta _0=\\beta _2=\\beta _4=0,$ and $\\gamma _0=\\gamma _2=\\gamma _4=t\\gamma _1=t\\gamma _3=t\\gamma _5.$ Example 3.12 Assume that $n=6$ .", "We use the previous theorem to construct an isomorphism between the modules corresponding to the 6-tuples $(b_1, b_2, b_3, b_4, b_5, b_6 )=(-2,0,0,1,-1,2)$ and $(c_1, c_2, c_3, c_4, c_5, c_6 )=(0,1,-1,2,-2,0)$ .", "From the previous theorem we get that $\\varphi _0=\\begin{pmatrix} 1 & 0 \\\\ -\\frac{3}{2}t & 1 \\end{pmatrix}$ , $\\varphi _1=\\varphi _3=\\varphi _5=\\begin{pmatrix} 1 & 2 \\\\ -\\frac{3}{2} & -2\\end{pmatrix}$ , $\\varphi _2=\\begin{pmatrix} -\\frac{1}{2} & 0 \\\\ -\\frac{3}{2} t & -2 \\end{pmatrix}$ , $\\varphi _4=\\begin{pmatrix} -2 & 0 \\\\ -\\frac{3}{2}t & -\\frac{1}{2} \\end{pmatrix}$ .", "Remark 3.13 In the case when $t\\nmid b_{1}+b_{2}$ , $t\\nmid b_{3}+b_{4}$ , and $t\\mid b_{5}+b_{6}$ , we have seen that the module in question is isomorphic to the direct sum $L_{\\lbrace i_1,j_1,i_3\\rbrace \\cup (I\\cap J)}\\oplus L_{\\lbrace i_2,j_2, j_3\\rbrace \\cup (I\\cap J)}$ .", "This means that regardless of the choice of the elements $b_i$ that satisfy these conditions, we get a module that is isomorphic to the same direct sum of rank 1 modules.", "Obviously, the same holds when $t\\mid b_{1}+b_{2}$ , $t\\mid b_{3}+b_{4}$ , and $t\\mid b_{5}+b_{6}$ , in which case we get the direct sum $L_I\\oplus L_J$ .", "Thus, once we know which divisibility conditions our coefficients fulfil, we immediately know which module we are dealing with." ], [ "Almost tight 3-interlacing", "In the tame case $(4,8)$ , besides the indecomposable modules of rank 2 that we have already constructed, i.e.", "the modules whose layers $I$ and $J$ are 3-interlacing and satisfy the condition $\|I\\cap J\|=k-3$ , there are also the cases of non-tightly 3-interlacing layers with poset $1^3\\mid 2$ and of 4-interlacing layers (with poset $1^4\\mid 2$ ).", "In this section, we deal with the former case.", "Recall that the two rims form three boxes in this case (Remark REF ).", "This happens exactly for pairs of subsets $J=\\lbrace i,i+2,i+4,i+5\\rbrace $ , $I=\\lbrace i+1,i+3,i+6,i+7\\rbrace $ , e.g.", "for the profile $2478\\mid 1356$ .", "We prove that there is a unique indecomposable rank 2 module with filtration $I\\mid J$ .", "We will work in a more general setup which we explain now.", "So we assume for now, for general $k$ and $n$ , that $I$ and $J$ are $r$ -interlacing for some $r\\ge 3$ and that $I\\mid J$ forms three boxes with poset $1^3\\mid 2$ .", "Figure: The profile of a module with 3-interlacing layers, with poset 1 3 ∣21^3\\mid 2, and with all boxes squares.We would like to construct an indecomposable rank 2 module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ with $L_J$ as submodule and $L_I$ as quotient.", "As before, we define $x_{i}=\\begin{pmatrix} t& b_{i} \\\\ 0 & 1 \\end{pmatrix}$ and $y_{i}=\\begin{pmatrix} 1& -b_{i} \\\\ 0 & t \\end{pmatrix}$ , if $i\\in I\\setminus J$ , and $x_{i}=\\begin{pmatrix} 1& b_{i} \\\\ 0 & t \\end{pmatrix}$ and $y_{i}=\\begin{pmatrix} t& -b_{i} \\\\ 0 & 1 \\end{pmatrix}$ if $i\\in J\\setminus I$ .", "For any other $i$ we define $x_i$ to be the identity matrix and $y_i$ to be $t\\cdot \\operatorname{id}\\nolimits $ if $i\\in I\\cap J$ , and $x_i=t\\cdot \\operatorname{id}\\nolimits $ , $y_i=\\operatorname{id}\\nolimits $ if $i\\in I^c\\cap J^c$ .", "This gives an element of ${\\rm CM}(B_{k,n})$ , Proposition REF , as we will explain now.", "In order to have a representation for $B_{k,n}$ , our matrices have to satisfy the relation $x^k=y^{n-k}$ .", "After multiplication by $x^{n-k}$ from the left, we conclude that this is equivalent to $x^n=t^{n-k}\\cdot \\operatorname{id}\\nolimits $ because $x_iy_i=t$ .", "When we compute such a product $x_nx_{n-1}\\cdots x_1$ we get the matrix $\\begin{pmatrix} t^{n-k}& z \\\\ 0 & t^{n-k} \\end{pmatrix}$ , where $z$ is a linear combination of the coefficients $b_i$ over the centre $Z$ .", "This linear combination must be zero if we want to have a $B_{k,n}$ -module structure.", "Remark 4.1 Note that for all $i\\in (I\\cap J)\\cup (I^c\\cap J^c)$ , $x_i$ is equal to $\\operatorname{id}\\nolimits $ or $t\\operatorname{id}\\nolimits $ , so in the product $x^n$ , any such $x_i$ does not contribute to $z$ , or more precisely, it cancels out in $x^n=t^{n-k}\\cdot \\operatorname{id}\\nolimits $ .", "Therefore, in finding conditions for $z=0$ , we will assume $(I\\cap J)\\cup (I^c\\cap J^c)=\\emptyset $ , i.e.", "that the rims of $L_I$ and of $L_J$ have no parallel segments and that $(k,n)$ are modified accordingly to $(k^{\\prime },n^{\\prime })$ for some $k^{\\prime }\\le k$ , $n^{\\prime }\\le n$ .", "This implies in particular that the boxes are symmetric around horizontal axes and that $n^{\\prime }=2k^{\\prime }$ .", "We consider the product of all $x_i$ appearing in a single box.", "Since we have removed all common elements of $I$ and $J$ and of $I^c$ and $J^c$ , the boxes are separated by the three points where the rims meet.", "We call them the three branching points of $I\\mid J$.", "In other words, let $\\lbrace i_1,i_2,i_3\\rbrace \\subset I$ be the positions where the arrow $x_{i_m}$ in the rim of $L_I$ ends at the position/height (in the lattice diagram) where the arrow $y_{i_m}$ starts in the rim of $L_J$ , $m=1,2,3$ .", "And let $\\lbrace j_1,j_2,j_3\\rbrace \\subset J$ be defined as $j_m=i_m+1$ .", "We call the $i_m$ 's the branching points for $I$ and the $j_m$ 's the branching points for $J$ .", "(We will give the definition of branching points in the general setting later, cf.", "Definition REF .)", "The profile $I \\mid J$ of a rank 2 module with 6-interlacing layers, with poset $1^3\\mid 2$ , and with $I\\cap J=I^c\\cap J^c=\\emptyset $ is given in Figure REF , for $(k,n)=(8,16)$ .", "Figure: The profile of a module with 3 branching points and 3 junctions.Note that all the boxes now fit between consecutive branching points $i_m,i_{m+1}\\in I$ , but that there might be some points, along the boundary of a box where the rims deviate from forming a square: considering the rim of $L_I$ , these points are precisely the valleys of $I$ (and by symmetry, the peaks of $J$ ), i.e.", "points $i\\in I$ (and thus $i\\notin J$ ) such that $i+1\\notin I$ (and thus $i+1\\in J$ ).", "We call such a point a junction of $I$ (of $J$ , by the symmetry).", "In Figure REF , the first box has two junctions at 4 and at 6 in $I$ .", "By definition, branching points are not junctions.", "For the remainder of this section we assume that $I\\mid J$ is a profile whose boxes are squares, i.e., that there are no junctions.", "Figure: A profile with squared boxes and no parallel lines.Consider the first box in the reduced setting (no parallel segments, only squared boxes), i.e.", "the squared box with starting/ending points $i_1,i_2$ .", "The size of the set $I\\cap (i_1,i_2]$ is the same as the size of the set $J\\cap (i_1,i_2]$ .", "We call this number the size of the box, and denote it by $l_1:=\\frac{1}{2}\|\\lbrace i_1+1,\\dots , i_2\\rbrace \|$ , $l_2$ and $l_3$ are defined accordingly.", "Then the product of the matrices $x_{i_2}x_{i_2-1}\\cdots x_{i_1+l_1+1}$ (see Figure REF ) on the second half of the first box is $\\begin{pmatrix} t^{l_1}& b_{i_2}+b_{i_2-1}t+\\dots + b_{i_1+l_1+1}t^{l_1-1} \\\\ 0 & 1 \\end{pmatrix}.$ Note that the peaks of $I$ are $i_1+l_1,$ $i_2+l_2$ , and $i_3+l_3$ .", "In the linear combination $b_{i_2}+b_{i_2-1}t+\\dots + b_{i_1+l_1+1}t^{l_1-1}$ there is only one term that is potentially not divisible by $t$ , the one corresponding to the branching point $i_2$ .", "Denote the sum $b_{i_2}+b_{i_2-1}t+\\dots + b_{i_1+l_1+1}t^{l_1-1}$ by $B_{i_2}$ .", "On the first half of the box, the product of matrices $x_{i_1+l_1}x_{i_1+l_1-1}\\cdots x_{i_1+1}$ is $\\begin{pmatrix} 1& b_{j_1}+b_{i_1+1}t+\\dots + b_{i_1+l_1}t^{l_1-1} \\\\ 0 & t^{l_1} \\end{pmatrix}.$ Note that $b_{j_1}$ is the only term potentially not divisible by $t$ in the sum $b_{j_1}+b_{j_1+1}t+\\dots + b_{i_1+l_1}t^{l_1-1}$ , which we denote by $B_{j_1}$ .", "The product $x_{i_2}x_{i_2-1}\\cdots x_{j_1}$ is thus equal to $\\begin{pmatrix} t^{l_1}& t^{l_1}(B_{j_1}+B_{i_2}) \\\\ 0 & t^{l_1} \\end{pmatrix}.$ Similarly, for the remaining two boxes we get that the corresponding products are (with the obvious analogous notation): $\\begin{pmatrix} t^{l_2}& t^{l_2}(B_{j_2}+B_{i_3}) \\\\ 0 & t^{l_2} \\end{pmatrix}$ and $\\begin{pmatrix} t^{l_3}& t^{l_3} (B_{j_3}+B_{i_1}) \\\\ 0 & t^{l_3} \\end{pmatrix}.$ Therefore, the whole product $x^n$ is $\\begin{pmatrix} t^{n-k}& t^{n-k} (B_{j_3}+B_{i_1}+B_{j_2}+B_{i_3}+B_{j_1}+B_{i_2} )\\\\ 0 & t^{n-k} \\end{pmatrix}.$ The following proposition is now obvious.", "Proposition 4.2 If $B_{i_1}+ B_{i_2}+B_{i_3}+B_{j_1}+B_{j_2}+ B_{j_3} =0$ , then $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is a rank 2 Cohen-Macaulay module.", "From now on we assume that $B_{i_1}+ B_{i_2}+B_{i_3}+B_{j_1}+B_{j_2}+ B_{j_3} =0$ .", "We first determine necessary and sufficient conditions for the module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ to be isomorphic to the direct sum $L_I\\oplus L_J$ .", "Assume first that $\\mathbb {M}(I,J)$ is decomposable and that $L_J$ is a direct summand of $\\mathbb {M}(I,J)$ .", "Then there exists a retraction $\\mu =(\\mu _i)_{i=1}^n$ such that $\\mu _i\\circ \\theta _i=\\operatorname{id}\\nolimits $ , where $(\\theta _i)_{i=1}^n$ is the natural injection of $L_J$ into $\\mathbb {M}(I,J)$ .", "Using the same basis as before, we can assume that $\\mu _i=[1\\,\\, \\alpha _i ]$ for some $\\alpha _i\\in \\operatorname{\\mathbb {C}}\\nolimits [[t]]$ .", "From the commutativity relations we have $\\operatorname{id}\\nolimits \\circ \\mu _i=\\mu _{i+1}\\circ x_{i+1}$ for $i+1\\in J\\setminus I$ , and $t\\cdot \\operatorname{id}\\nolimits \\circ \\mu _i=\\mu _{i+1}\\circ x_{i+1}$ for $i+1\\in I\\setminus J$ .", "It follows that $\\alpha _i=b_{i+1}+t\\alpha _{i+1}$ for $i+1\\in J\\setminus I$ , and $t\\alpha _i=b_{i+1}+\\alpha _{i+1}$ for $i+1\\in I\\setminus J$ .", "From this we have $t^{l_3} \\alpha _{i_3+l_3}-t^{l_1}\\alpha _{i_1+l_1}&=B_{i_1}+B_{j_1},\\\\t^{l_2} \\alpha _{i_2+l_2}-t^{l_3}\\alpha _{i_3+l_3}&=B_{i_3}+B_{j_3},\\\\t^{l_1} \\alpha _{i_1+l_1}-t^{l_2}\\alpha _{i_2+l_2}&=B_{i_2}+B_{j_2}.$ It follows that $t^{\\min \\lbrace {l_{g-1}, l_g}\\rbrace }\\mid B_{i_g}+B_{j_g}$ , for $g=1,2,3$ .", "In the opposite direction, if $t^{\\min \\lbrace {l_{g-1}, l_g}\\rbrace }\\mid B_{i_g}+B_{j_g}$ , for $g=1,2,3$ , then we can easily find the coefficients $\\alpha _{i_g+l_g}$ , $g=1,2,3,$ so that the above equalities hold.", "Assume, without loss of generality, that $l_3\\ge l_2\\ge l_1.$ Then by setting $\\alpha _{i_3+l_3}=0$ we obtain $\\alpha _{i_1+l_1}=-t^{-l_1}(B_{i_1}+B_{j_1})$ and $\\alpha _{i_2+l_2}=t^{-l_2}(B_{i_3}+B_{j_3})$ .", "The coefficient $\\alpha _{i+1}$ , where $i+1$ is not a peak of $I$ , is determined directly from $\\alpha _i=b_{i+1}+t\\alpha _{i+1}$ , when $i+1\\in J\\setminus I$ , or $t\\alpha _i=b_{i+1}+\\alpha _{i+1}$ , when $i+1\\in I\\setminus J$ .", "This defines a retraction $\\mu $ , and thus proves the following proposition.", "Proposition 4.3 The module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is isomorphic to $L_I\\oplus L_J$ if and only if $t^{\\min \\lbrace {l_{g-1}, l_g}\\rbrace }\\mid B_{i_g}+B_{j_g}$ , for $g=1,2,3$ .", "We give necessary and sufficient conditions for the module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ to be indecomposable.", "Theorem 4.4 The module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is indecomposable if and only if $t^{\\min \\lbrace l_3,l_1\\rbrace }\\nmid B_{i_1}+B_{j_1}$ , $t^{\\min \\lbrace l_1,l_2\\rbrace }\\nmid B_{i_2}+B_{j_2}$ , and $t^{\\min \\lbrace l_2,l_3\\rbrace }\\nmid B_{i_3}+B_{j_3}$ .", "Let $\\varphi =(\\varphi _{i_1})_{i=1}^n$ be an idempotent endomorphism of $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ and let $\\varphi _{i_3+l_3}= \\begin{pmatrix} a& b \\\\ c & d \\end{pmatrix}$ (see Figure REF ).", "From $x_{i_1+l_1}x_{i_1+l_1-1}\\cdots x_{j_1}\\cdot x_{i_1}\\cdots x_{i_3+l_3+1}\\varphi _{i_3+l_3}=\\varphi _{i_1+l_1} x_{i_1+l_1}x_{i_1+l_1-1}\\cdots x_{j_1}\\cdot x_{i_1}\\cdots x_{i_3+l_3+1}$ follows that $\\varphi _{i_1+l_1}= \\begin{pmatrix} a+t^{-l_3}(B_{i_1}+B_{j_1})c& t^{-l_1}[ b t^{l_3}+(d-a)(B_{i_1}+B_{j_1})-t^{-l_3}(B_{i_1}+B_{j_1})^2c] \\\\ t^{l_1-l_3}c & d-t^{-l_3}(B_{i_1}+B_{j_1})c \\end{pmatrix}, $ where $t^{l_3-l_1}\\mid c$ , $t^{l_1}\\mid (d-a)(B_{i_1}+B_{j_1})-t^{-l_3}(B_{i_1}+B_{j_1})^2c$ , and $t^{l_3}\\mid (B_{i_1}+B_{j_1})c$ .", "Here, we assume, without loss of generality, that $l_1\\le l_2\\le l_3$ .", "From $x_{i_3+l_3}\\cdots x_{j_3}\\cdot x_{i_3}\\cdots x_{i_2+l_2+1}\\varphi _{i_2+l_2}=\\varphi _{i_3+l_3} x_{i_3+l_3}\\cdots x_{j_3}\\cdot x_{i_3}\\cdots x_{i_2+l_2+1}$ follows that $\\varphi _{i_2+l_2}= \\begin{pmatrix} a-t^{-l_3}(B_{i_3}+B_{j_3})c& t^{-l_2}[b t^{l_3}-(d-a)(B_{i_3}+B_{j_3})-t^{-l_3}(B_{i_3}+B_{j_3})^2c] \\\\ t^{l_2-l_3}c & d+t^{-l_3}(B_{i_3}+B_{j_3})c \\end{pmatrix}, $ where $t^{l_3-l_2}\\mid c$ , $t^{l_2}\\mid (d-a)(B_{i_3}+B_{j_3})+t^{-l_3}(B_{i_3}+B_{j_3})^2c$ , and $t^{l_3}\\mid (B_{i_3}+B_{j_3})c$ .", "Assume that $t^{\\min \\lbrace l_3,l_1\\rbrace }\\nmid B_{i_1}+B_{j_1}$ , $t^{\\min \\lbrace l_1,l_2\\rbrace }\\nmid B_{i_2}+B_{j_2}$ , and $t^{\\min \\lbrace l_2,l_3\\rbrace }\\nmid B_{i_3}+B_{j_3}$ .", "Then for $g=1,2,3,$ there exists $s_g$ such that $0\\le s_g<\\min \\lbrace l_{g-1},l_g\\rbrace $ , $t^{s_g}\\mid B_{i_g}+B_{j_g}$ , and $t^{s_g+1}\\nmid B_{i_g}+B_{j_g}$ .", "Note that since $B_{i_1}+ B_{i_2}+B_{i_3}+B_{j_1}+B_{j_2}+ B_{j_3} =0$ it is not possible that one of the $s_g$ 's is strictly less than the other two, i.e.", "the smallest two have to be equal.", "For the coefficients of $\\varphi _{i_3+j_3}$ it holds that $t^{l_1-s_1}&\\mid (d-a)t^{-s_1}(B_{i_1}+B_{j_1})-t^{-s_1}(B_{i_1}+B_{j_1})t^{-l_3}(B_{i_1}+B_{j_1})c,\\\\t^{l_2-s_3}&\\mid (d-a)t^{-s_3}(B_{i_3}+B_{j_3})+t^{-s_3}(B_{i_3}+B_{j_3})t^{-l_3}(B_{i_3}+B_{j_3})c.$ It follows that $t^{l_1-s_1}\\mid (d-a)-t^{-l_3}(B_{i_1}+B_{j_1})c$ and $t^{l_2-s_3}\\mid (d-a)+t^{-l_3}(B_{i_3}+B_{j_3})c$ .", "Then $t^{\\min \\lbrace l_1-s_1, l_2-s_3\\rbrace }\\mid t^{-l_3}(B_{i_2}+B_{j_2})c$ and subsequently $t^{\\min \\lbrace l_1-s_1,l_2-s_3\\rbrace }\\mid t^{-l_3}t^{s_2}c$ .", "If $\\min \\lbrace l_1-s_1,l_2-s_3\\rbrace =l_2-s_3$ , then $t^{l_2-s_3}\\mid t^{-l_3}t^{s_2}t^{-s_3}(B_{i_3}+B_{j_3})c$ and $t^{l_2}\\mid t^{-l_3}t^{s_2}(B_{i_3}+B_{j_3})c$ .", "Since $l_2>s_2$ , we have $t\\mid t^{-l_3}(B_{i_3}+B_{j_3})c$ and $t\\mid d-a$ .", "Assume now that $\\min \\lbrace l_1-s_1,l_2-s_3\\rbrace =l_1-s_1$ .", "From $t^{l_1-s_1}\\mid t^{-l_3}t^{s_2}t^{-s_1}(B_{i_1}+B_{j_1})c$ we conclude directly that $t^{l_1-s_2}\\mid t^{-l_3}(B_{i_1}+B_{j_1})c$ .", "Since $l_1>s_2$ , we have that $t\\mid t^{-l_3}(B_{i_1}+B_{j_1})c$ and $t\\mid d-a$ .", "From $t\\mid d-a$ follows that $a=d$ and $b=c=0$ , giving us the trivial idempotents.", "Hence, the module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is indecomposable.", "Assume that at least one of the conditions $t^{\\min \\lbrace l_3,l_1\\rbrace }\\nmid B_{i_1}+B_{j_1}$ , $t^{\\min \\lbrace l_1,l_2\\rbrace }\\nmid B_{i_2}+B_{j_2}$ , and $t^{\\min \\lbrace l_2,l_3\\rbrace }\\nmid B_{i_3}+B_{j_3}$ does not hold.", "If $t^{\\min \\lbrace l_2,l_3\\rbrace }\\mid B_{i_3}+B_{j_3}$ , $t^{\\min \\lbrace l_3,l_1\\rbrace }\\mid B_{i_1}+B_{j_1}$ , and $t^{\\min \\lbrace l_1,l_2\\rbrace }\\mid B_{i_2}+B_{j_2}$ , then, by Proposition REF , $\\operatorname{\\mathbb {M}}\\nolimits (I,J)\\cong L_I\\oplus L_J.$ If $t^{\\min \\lbrace l_2,l_3\\rbrace }\\mid B_{i_3}+B_{j_3}$ , and $t^{\\min \\lbrace l_3,l_1\\rbrace }\\nmid B_{i_1}+B_{j_1}$ or $t^{\\min \\lbrace l_1,l_2\\rbrace }\\nmid B_{i_2}+B_{j_2}$ , then we repeat the same procedure as in the previous section to construct a non-trivial idempotent.", "Assume that $t^{\\min \\lbrace l_3,l_1\\rbrace }\\nmid B_{i_1}+B_{j_1}$ (analogous arguments are used if we were to assume that $t^{\\min \\lbrace l_1,l_2\\rbrace }\\nmid B_{i_2}+B_{j_2}$ ).", "The only divisibility conditions that the elements of the matrix $\\varphi _{i_3+l_3}= \\begin{pmatrix} a& b \\\\ c & d \\end{pmatrix}$ have to fulfill are $t^{l_3-s_1}\\mid c$ and $t^{l_1}\\mid (d-a)(B_{i_1}+B_{j_1})-t^{-l_3}(B_{i_1}+B_{j_1})^2c$ .", "The latter condition is equivalent to the condition $t^{l_1-s_1}\\mid (d-a)-t^{-l_3}(B_{i_1}+B_{j_1})c$ .", "Recall that if $a=d$ , then we only obtain the trivial idempotents because $t\\mid c$ .", "So it must be $a+d=1$ .", "If we set $a=1$ , $d=0$ , $c=-t^{l_3-s_1}(t^{-s_1}(B_{i_1}+B_{j_1}))^{-1}$ , and $b=0$ , we get the idempotent $\\varphi _{i_3+l_3}=\\begin{pmatrix}1& 0\\\\-t^{l_3-s_1}(t^{-s_1}(B_{i_1}+B_{j_1}))^{-1}&0\\end{pmatrix}.$ Its orthogonal complement is the idempotent $\\begin{pmatrix}0& 0\\\\t^{l_3-s_1}(t^{-s_1}(B_{i_1}+B_{j_1}))^{-1}&1\\end{pmatrix}.$ From $x_i\\varphi _{i-1}=\\varphi _ix_i$ , we easily determine idempotents $\\varphi _i$ , for all $i$ .", "Since these are non-trivial idempotents, it follows that the module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is decomposable, what we needed to prove.", "The case when $t^{\\min \\lbrace l_3,l_1\\rbrace }\\mid B_{i_1}+B_{j_1}$ , and $t^{\\min \\lbrace l_2,l_3\\rbrace }\\nmid B_{i_3}+B_{j_3}$ or $t^{\\min \\lbrace l_1,l_2\\rbrace }\\nmid B_{i_2}+B_{j_2}$ , and the case when $t^{\\min \\lbrace l_1,l_2\\rbrace }\\mid B_{i_2}+B_{j_2}$ , and $t^{\\min \\lbrace l_3,l_1\\rbrace }\\nmid B_{i_1}+B_{j_1}$ or $t^{\\min \\lbrace l_2,l_3\\rbrace }\\nmid B_{i_3}+B_{j_3}$ are treated similarly, so we omit the details.", "Remark 4.5 All arguments from the previous theorem hold more generally, for example, when the boxes are rectangular.", "By Remark REF , the matrices $x_i$ are all scalar for the parts that are parallel, so they can be ignored in all computations.", "So the arguments hold for $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ where the profiles can be reduced to a profile satisfying the conditions of the theorem by removing indices in $(I\\cap J)\\cup (I^c\\cap J^c)$ from the $k$ -subsets.", "Remark 4.6 As in Proposition REF , it is possible to give an explicit combinatorial description of the summands of $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ when this module is decomposable.", "We do not give this combinatorial description here, but we note that the coefficients $s_1,s_2,$ and $s_3$ from the proof of the previous theorem play a crucial role in determining the summands.", "For example, if $t^{\\min \\lbrace l_2,l_3\\rbrace }\\mid B_{i_3}+B_{j_3}$ , $t^{\\min \\lbrace l_3,l_1\\rbrace }\\nmid B_{i_1}+B_{j_1}$ , $t^{\\min \\lbrace l_1,l_2\\rbrace }\\nmid B_{i_2}+B_{j_2}$ , and $s_1=s_2=0$ , then $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is isomorphic to $L_X \\oplus L_Y$ where $X=(J\\cup (I\\cap (i_3,i_1]))\\setminus (J\\cap (i_3,i_1]))$ and $Y=(I\\cup (J\\cap (i_3,i_1])))\\setminus ( I\\cap (i_3,i_1]).$ The result of Proposition REF is a special case of this result.", "Corollary 4.7 In the case $(4,8)$ , if $I=\\lbrace 2,4,7,8\\rbrace $ and $J=\\lbrace 1,3,5,6\\rbrace $ , then $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is indecomposable if and only if $t\\nmid b_2+b_3$ , $t\\nmid b_4+b_5$ , and $t\\nmid b_8+b_1$ .", "Furthermore, if $t\\nmid b_2+b_3$ , $t\\mid b_4+b_5$ , $t\\nmid b_8+b_1$ , then $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is isomorphic to $L_{\\lbrace 2,3,5,6\\rbrace }\\oplus L_{\\lbrace 1,4,7,8\\rbrace }$ .", "If $t\\mid b_2+b_3$ , $t\\nmid b_4+b_5$ , $t\\nmid b_8+b_1$ , then $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is isomorphic to $L_{\\lbrace 2,4,5,6\\rbrace }\\oplus L_{\\lbrace 1,3,7,8\\rbrace }$ .", "If $t\\nmid b_2+b_3$ , $t\\nmid b_4+b_5$ , $t\\mid b_8+b_1$ , then $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is isomorphic to $L_{\\lbrace 2,3,7,8\\rbrace }\\oplus L_{\\lbrace 1,4,5,6\\rbrace }$ .", "If $t\\mid b_2+b_3$ , $t\\mid b_4+b_5$ , $t\\mid b_8+b_1$ , then $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is isomorphic to $L_I\\oplus L_J$ .", "We now consider the question of uniqueness of the rank 2 modules we constructed.", "In case we have a decomposable rank 2 module, the module is completely determined by the divisibility conditions its coefficients $b_i$ , $i=1,\\ldots ,n$ , satisfy.", "Thus, two sets of coefficients $b_i$ define isomorphic decomposable modules if and only if they satisfy the same divisibility conditions.", "What remains to study is the case when we have an indecomposable module.", "The next theorem tells us that in this case too, there is a unique rank 2 indecomposable module with $L_{\\lbrace 1,3,5,6\\rbrace }$ as a submodule and $L_{\\lbrace 2,4,7,8\\rbrace }$ as a quotient.", "Let $(b_i)$ and $(c_i)$ be different $n$ -tuples corresponding to indecomposable modules $\\operatorname{\\mathbb {M}}\\nolimits _1$ and $\\operatorname{\\mathbb {M}}\\nolimits _2$ with filtration $L_I\\mid L_J$ from Theorem REF , satisfying the indecomposability conditions of Theorem REF .", "Thus, assume that $t^{\\min \\lbrace l_{g-1},l_{g}\\rbrace }\\nmid B_{i_g}+B_{j_g}$ , $t^{\\min \\lbrace l_{g-1},l_{g}\\rbrace }\\nmid C_{i_g}+C_{j_g}$ , for $g=1,2,3.$ Then for each $g$ , there exists $s^b_g$ such that $0\\le s^b_g<\\min \\lbrace l_{g-1},l_g\\rbrace $ , $t^{s^b_g}\\mid B_{i_g}+B_{j_g}$ , and $t^{s^b_g+1}\\nmid B_{i_g}+B_{j_g}$ , and there exists $s^c_g$ such that $0\\le s^c_g<\\min \\lbrace l_{g-1},l_g\\rbrace $ , $t^{s^c_g}\\mid C_{i_g}+C_{j_g}$ , and $t^{s^c_g+1}\\nmid C_{i_g}+C_{j_g}$ .", "Recall from the proof of Theorem REF that since $B_{i_1}+ B_{i_2}+B_{i_3}+B_{j_1}+B_{j_2}+ B_{j_3} =0$ it is not possible that one of the $s^b_g$ 's is strictly less than the other two, i.e.", "the smallest two have to be equal.", "The same holds for $s^c_g$ 's.", "Theorem 4.8 Let $(b_i)$ and $(c_i)$ be different $n$ -tuples corresponding to indecomposable modules $\\operatorname{\\mathbb {M}}\\nolimits _1$ and $\\operatorname{\\mathbb {M}}\\nolimits _2$ with filtration $L_I\\mid L_J$ satisfying the indecomposability conditions of Theorem REF .", "Then the modules $\\operatorname{\\mathbb {M}}\\nolimits _1$ and $\\operatorname{\\mathbb {M}}\\nolimits _2$ are isomorphic if and only if $s_g^c=s_g^b$ for $g=1,2,3$ .", "Let us assume that there is an isomorphism $\\varphi =(\\varphi _i)_{i=1}^n$ between $\\operatorname{\\mathbb {M}}\\nolimits _1$ and $\\operatorname{\\mathbb {M}}\\nolimits _2$ .", "If $\\varphi _{i_3+l_3}=\\begin{pmatrix}\\alpha & \\beta \\\\ \\gamma & \\delta \\end{pmatrix}$ , then from $x_{i_1+l_1}x_{i_1+l_1-1}\\cdots x_{j_1}\\cdot x_{i_1}\\cdots x_{i_3+l_3+1}\\varphi _{i_3+l_3}=\\varphi _{i_1+l_1} x_{i_1+l_1}x_{i_1+l_1-1}\\cdots x_{j_1}\\cdot x_{i_1}\\cdots x_{i_3+l_3+1}$ follows that $\\varphi _{i_1+l_1}$ is equal to $\\begin{pmatrix} \\alpha +t^{-l_3}(C_{i_1}+C_{j_1})\\gamma & t^{-l_1}[t^{l_3}\\beta -\\alpha (B_{i_1}+B_{j_1}) + (C_{i_1}+C_{j_1})\\delta -(B_{i_1}+B_{j_1})t^{-l_3}(C_{i_1}+C_{j_1})\\gamma ]\\\\ t^{l_1-l_3}\\gamma & \\delta -t^{-l_3}(B_{i_1}+B_{j_1})\\gamma \\end{pmatrix},$ and $\\varphi _{i_2+l_2}$ is equal to $\\begin{pmatrix} \\alpha -t^{-l_3}(C_{i_3}+C_{j_3})\\gamma & t^{-l_2}[t^{l_3}\\beta +\\alpha (B_{i_3}+B_{j_3}) -(C_{i_3}+C_{j_3})\\delta -(B_{i_3}+B_{j_3})t^{-l_3}(C_{i_3}+C_{j_3})\\gamma ]\\\\ t^{l_2-l_3}\\gamma & \\delta +t^{-l_3}(B_{i_3}+B_{j_3})\\gamma \\end{pmatrix},$ where $t^{l_3-l_2}\\mid \\gamma $ , $t^{l_3-l_1}\\mid \\gamma $ , $t^{l_3}\\mid (B_{i_1}+B_{j_1})\\gamma $ , $t^{l_3} \\mid (C_{i_1}+C_{j_1})\\gamma $ , $t^{l_1}\\mid -\\alpha (B_{i_1}+B_{j_1}) + (C_{i_1}+C_{j_1})\\delta -(B_{i_1}+B_{j_1})t^{-l_3}(C_{i_1}+C_{j_1})\\gamma $ , and $t^{l_2}\\mid \\alpha (B_{i_3}+B_{j_3}) -(C_{i_3}+C_{j_3})\\delta -(B_{i_3}+B_{j_3})t^{-l_3}(C_{i_3}+C_{j_3})\\gamma $ .", "Here, without loss of generality we assume that $l_3$ is the largest amongst $l_1,l_2,l_3$ .", "The coefficients $\\alpha $ and $\\delta $ are invertible because $t\\mid \\gamma $ and $\\varphi _{i_3+l_3}$ is invertible.", "If $s_1^b>s^c_1$ , then $t^{-s_1^c}(B_{i_1}+B_{j_1})$ is divisible by $t$ .", "From $t^{l_1}\\mid -\\alpha (B_{i_1}+B_{j_1}) + (C_{i_1}+C_{j_1})\\delta -(B_{i_1}+B_{j_1})t^{-l_3}(C_{i_1}+C_{j_1})\\gamma $ it follows that $t^{l_1-s_1^c}\\mid -\\alpha t^{-s_1^c}(B_{i_1}+B_{j_1}) + t^{-s_1^c}(C_{i_1}+C_{j_1})\\delta -t^{-s_1^c}(B_{i_1}+B_{j_1})t^{-l_3}(C_{i_1}+C_{j_1})\\gamma .$ This implies that $t\\mid t^{-s_1^c}(C_{i_1}+C_{j_1})\\delta $ , but neither $\\delta $ nor $t^{-s_1^c}(C_{i_1}+C_{j_1})$ is divisible by $t$ .", "This is a contradiction.", "The proof is analogous if $s_2^b>s^c_2$ or $s_3^b>s^c_3$ .", "Thus if $s_g^c\\ne s_g^b$ for at least one $g$ , the modules $\\operatorname{\\mathbb {M}}\\nolimits _1$ and $\\operatorname{\\mathbb {M}}\\nolimits _2$ are not isomorphic.", "Assume that $s_g^c=s_g^b$ for $g=1,2,3$ .", "From $t^{l_1}\\mid -\\alpha (B_{i_1}+B_{j_1}) + (C_{i_1}+C_{j_1})\\delta -(B_{i_1}+B_{j_1})t^{-l_3}(C_{i_1}+C_{j_1})\\gamma $ and $t^{l_2}\\mid \\alpha (B_{i_3}+B_{j_3}) -(C_{i_3}+C_{j_3})\\delta -(B_{i_3}+B_{j_3})t^{-l_3}(C_{i_3}+C_{j_3})\\gamma $ we have $t^{\\min \\lbrace l_1,l_2\\rbrace }\\mid \\alpha (B_{i_1}+B_{j_1}) (C_{i_2}+C_{j_2}) (B_{i_3}+B_{j_3}) - \\delta (C_{i_1}+C_{j_1}) (B_{i_2}+B_{j_2}) (C_{i_3}+C_{j_3}).$ If we set $\\alpha (B_{i_1}+B_{j_1}) (C_{i_2}+C_{j_2}) (B_{i_3}+B_{j_3}) - \\delta (C_{i_1}+C_{j_1}) (B_{i_2}+B_{j_2}) (C_{i_3}+C_{j_3})=0$ , then by defining $\\delta =1$ , we get $\\alpha =\\displaystyle \\frac{ [t^{-s_1^c}(C_{i_1}+C_{j_1})][t^{-s_2^b}(B_{i_2}+B_{j_2})][t^{-s_3^c}(C_{i_3}+C_{j_3})]}{[t^{-s_1^b}(B_{i_1}+B_{j_1})][t^{-s_2^c}(C_{i_2}+C_{j_2})][t^{-s_3^b}(B_{i_3}+B_{j_3})]}.", "$ By setting $-\\alpha (B_{i_1}+B_{j_1}) + (C_{i_1}+C_{j_1})\\delta -(B_{i_1}+B_{j_1})t^{-l_3}(C_{i_1}+C_{j_1})\\gamma =0$ , we obtain $\\gamma =t^{l_3} \\displaystyle \\frac{[t^{-s_2^c}(C_{i_2}+C_{j_2})][t^{-s_3^b}(B_{i_3}+B_{j_3})]-[t^{-s_2^b}(B_{i_2}+B_{j_2})][t^{-s_3^c}(C_{i_3}+C_{j_3})]}{[t^{-s_1^b}(B_{i_1}+B_{j_1})][t^{-s_2^c}(C_{i_2}+C_{j_2})][t^{-s_3^b}(B_{i_3}+B_{j_3})]}.$ The coefficients $\\alpha , \\beta $ , and $\\gamma $ of $\\varphi _{i_3+l_3}$ satisfy the necessary divisibility conditions, it is left to set $\\beta =0$ to obtain the isomorphism $\\varphi _{i_3+l_3}=\\begin{pmatrix} \\displaystyle \\frac{ [t^{-s_1^c}(C_{i_1}+C_{j_1})][t^{-s_2^b}(B_{i_2}+B_{j_2})][t^{-s_3^c}(C_{i_3}+C_{j_3})]}{[t^{-s_1^b}(B_{i_1}+B_{j_1})][t^{-s_2^c}(C_{i_2}+C_{j_2})][t^{-s_3^b}(B_{i_3}+B_{j_3})]}& \\,\\,\\,\\,\\,\\,\\,\\, 0 \\\\ t^{l_3} \\displaystyle \\frac{[t^{-s_2^c}(C_{i_2}+C_{j_2})][t^{-s_3^b}(B_{i_3}+B_{j_3})]-[t^{-s_2^b}(B_{i_2}+B_{j_2})][t^{-s_3^c}(C_{i_3}+C_{j_3})]}{[t^{-s_1^b}(B_{i_1}+B_{j_1})][t^{-s_2^c}(C_{i_2}+C_{j_2})][t^{-s_3^b}(B_{i_3}+B_{j_3})]} & \\,\\,\\,\\,\\,\\,\\,\\,1 \\end{pmatrix}.$ Once we know $\\varphi _{i_3+l_3}$ , $\\varphi _{i_2+l_2}$ , and $\\varphi _{i_1+l_1}$ , we easily compute $\\varphi _i$ , for all $i$ , again by using relations $x_i\\varphi _{i-1}=\\varphi _ix_i.$ Corollary 4.9 In the tame case $(4,8)$ , up to isomorphism, there is a unique rank 2 indecomposable module with filtration layers $2478\|1356$ .", "In this case, $s_g=0$ , for $g=1,2,3$ , and the statement follows from the previous theorem.", "Example 4.10 Observe the case $(5,10)$ .", "Let $I=\\lbrace 1,2,5,6,8\\rbrace $ and $J=\\lbrace 3,4,7,9,10\\rbrace $ .", "For this profile, there are three boxes of sizes $l_1=1,$ $l_2=2$ , and $l_3=2$ .", "Define $b_1=0$ , $b_2=t$ , $b_3=b_4=b_5=0$ , $b_6=1$ , $b_7=-t$ , $b_8=-1$ , and $b_9=b_{10}=0$ .", "It holds that $\\sum _{i=1}^{10}b_i=0$ , $t\\nmid b_7+b_6+tb_5$ , $t\\nmid b_8+b_9+tb_{10}$ , $t^2\\nmid b_2+tb_1+b_3+tb_4$ , and $t\\mid b_2+tb_1+b_3+tb_4$ .", "Thus, $s^b_1=s^b_2=0$ and $s^b_3=1$ .", "Denote the corresponding module by $\\operatorname{\\mathbb {M}}\\nolimits _1$ .", "This module is indecomposable by Theorem REF .", "Define $c_1=0$ , $c_2=1$ , $c_3=c_4=c_5=0$ , $c_6=1$ , $c_7=0$ , $c_8=-2$ , and $c_9=c_{10}=0$ .", "It holds that $\\sum _{i=1}^{10}c_i=0$ , $t\\nmid c_7+c_6+tc_5$ , $t\\nmid c_8+c_9+tc_{10}$ , $t^2\\nmid c_2+tc_1+c_3+tc_4$ , and $t\\nmid c_2+tc_1+c_3+tc_4$ .", "Thus, $s^c_1=s^c_2=s^c_3=0$ .", "Denote the corresponding module by $\\operatorname{\\mathbb {M}}\\nolimits _2$ .", "This module is indecomposable by Theorem REF .", "By Theorem REF , the modules $\\operatorname{\\mathbb {M}}\\nolimits _1$ and $\\operatorname{\\mathbb {M}}\\nolimits _2$ are not isomorphic.", "Corollary 4.11 Let $I\\mid J$ be the filtration from Theorem REF and assume that $l_1\\le l_2\\le l_3$ .", "Up to isomorphism, there are $l_1(\\frac{l_1-1}{2}+l_2)$ indecomposable modules with filtration $I\\mid J$ .", "In order to count the non-isomorphic modules, we have to count the number of different triples $s_1,s_2,s_3$ .", "Recall that the smallest two among $s_1,s_2$ and $s_3$ have to be equal because $B_{i_1}+ B_{i_2}+B_{i_3}+B_{j_1}+B_{j_2}+ B_{j_3} =0$ .", "Also, recall that $s_i<\\min \\lbrace s_{i-1},s_i\\rbrace $ .", "Assume that $s_1=i$ .", "If $s_2$ or $s_3$ is less than $s_1$ , then $s_3=s_2$ and there are $i$ different choices for $s_3=s_2$ .", "If $s_1$ is less than $s_2$ , then $s_1$ has to be equal to $s_3$ and vice versa, if $s_1$ is less than $s_3$ , then $s_1$ has to be equal to $s_2$ .", "In total there are $l_1-i-1$ options in the former case, and $l_2-i-1$ in the latter case.", "We also have to count the case when $s_1=s_2=s_3=i$ .", "Thus, there are $\\sum _{i=0}^{l_1-1}(i+1+l_1-i-1+l_2-i-1)=\\sum _{i=0}^{l_1-1}(l_1+l_2-i-1)=l_1(\\frac{l_1-1}{2}+l_2).$ The expression $l_1(\\frac{l_1-1}{2}+l_2)$ is equal to 1 if and only if $l_1=l_2=1$ .", "In all other cases it is strictly greater than 1.", "Thus, there is a unique indecomposable module for a given filtration only when $l_1=l_2=1$ , i.e.", "only when two of the boxes of the profile are of size 1.", "We give this special type of a profile a name.", "We say that $I$ and $J$ are almost tightly 3-interlacing if $I\\setminus J=\\lbrace a_1\\rbrace \\cup \\lbrace a_2\\rbrace \\cup \\lbrace a_3, \\dots , a_{3+r}\\rbrace $ and $J\\setminus I=\\lbrace b_1\\rbrace \\cup \\lbrace b_2\\rbrace \\cup \\lbrace b_3, \\dots , b_{3+r}\\rbrace $ , $r\\ge 0$ , and $b_1<a_1<b_2<a_2<b_3< \\dots <b_{3+r}<a_3< \\dots < a_{3+r}.$ Combinatorially, $I$ and $J$ are almost tightly 3-interlacing if the profile $I\\mid J$ has three squared boxes, at least two of them of size 1.", "Here, we assume that the potential parallel lines of the rims have been removed in order to simplify notation.", "Note that, by definition, tightly 3-interlacing layers are also almost tightly 3-interlacing.", "Corollary 4.12 Let $I\\mid J$ be the filtration from Theorem REF .", "There is a unique rank 2 indecomposable module with filtration $I\\mid J$ if and only if $I$ and $J$ are almost tightly 3-interlacing.", "Remark 4.13 In the general case of layers with profile $1^3\\mid 2$ , i.e.", "in the case when $I$ and $J$ form three boxes that are not necessarily squares or rectangles (meaning that there are junctions), one can construct multiple non-isomorphic indecomposable modules with the given filtration.", "To see this, when constructing indecomposable modules we deal with junction points in the following way.", "Observe Figure REF .", "For the junction points 4 and 6 of the leftmost box, we define $b_4=b_5=b_6=b_7=0$ .", "Similarly, for the rightmost box we set $b_{14}=b_{15}=0$ .", "In this way we practically ignore the junctions because for each junction point $i$ it holds that $x_{i+1}x_i=t\\cdot {\\rm id}$ so it does not interfere with our computation, and we can behave as if we were in the case when all boxes are squares.", "If there are at least two boxes with junction points, then as in the case when all boxes are squares it follows that there are multiple non-isomorphic indecomposable modules.", "In the case when there is only one box with junction points, then we use the arguments from the last section (by treating one of the junctions as if it were a branching point) to construct multiple non-isomorphic indecomposable modules.", "Example 4.14 In the tame case $(4,8)$ , there is only one type of a profile that is almost tightly 3-interlacing.", "Such a profile is $2478\\mid 1356$ (and all profiles obtained from this profile by adding $a$ to each element of both 4-subsets, for $a=1,\\dots , 7$ ).", "To construct modules with this profile we define $x_i=\\begin{pmatrix} t& b_{i} \\\\ 0 & 1 \\end{pmatrix}$ , $y_i=\\begin{pmatrix} 1& -b_{i} \\\\ 0 & t \\end{pmatrix},$ for $i=2,4,7,8$ and $x_{i}=\\begin{pmatrix} 1& b_{i} \\\\ 0 & t \\end{pmatrix},$ $y_i=\\begin{pmatrix} t& -b_{i} \\\\ 0 & 1 \\end{pmatrix},$ for $i=1,3,5,6$ .", "Note that the $x_i$ 's are almost the same as for a module with the profile $2468\\mid 1357$ constructed in the next section, with only the ones at vertices 6 and 7 changing places.", "In order for this to be a module we assume that $b_1+b_2+b_3+b_4+b_5+b_8+t(b_6+b_7)=0.$ Denote this module again by $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ .", "As for the module $135\\mid 246$ , it is easily seen that $L_J$ is a summand of $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ if and only if $t\\mid b_8+b_1$ , $t\\mid b_2+b_3$ , and $t\\mid b_4+b_5$ .", "The module $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is indecomposable if and only if $t\\nmid b_2+b_3$ , $t\\nmid b_4+b_5$ , $t\\nmid b_8+b_1$ .", "If $t\\nmid b_2+b_3$ , $t\\mid b_4+b_5$ , $t\\nmid b_8+b_1$ , then $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is isomorphic to $L_{\\lbrace 2,3,5,6\\rbrace }\\oplus L_{\\lbrace 1,4,7,8\\rbrace }$ .", "If $t\\mid b_2+b_3$ , $t\\nmid b_4+b_5$ , $t\\nmid b_8+b_1$ , then $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is isomorphic to $L_{\\lbrace 2,4,5,6\\rbrace }\\oplus L_{\\lbrace 1,3,7,8\\rbrace }$ .", "If $t\\nmid b_2+b_3$ , $t\\nmid b_4+b_5$ , $t\\mid b_8+b_1$ , then $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ is isomorphic to $L_{\\lbrace 2,3,7,8\\rbrace }\\oplus L_{\\lbrace 1,4,5,6\\rbrace }$ .", "There are four different decomposable modules appearing as the middle term in a short exact sequence that has $L_I$ (as a quotient) and $L_J$ (as a submodule) as end terms: $0\\longrightarrow L_J\\longrightarrow L_{\\lbrace 1, 3, 5,6\\rbrace }\\oplus L_{\\lbrace 2,4, 7,8\\rbrace } \\longrightarrow L_I \\longrightarrow 0,\\\\0\\longrightarrow L_J\\longrightarrow L_{\\lbrace 1, 4, 7,8 \\rbrace }\\oplus L_{\\lbrace 2,3,5, 6\\rbrace } \\longrightarrow L_I \\longrightarrow 0,\\\\0\\longrightarrow L_J\\longrightarrow L_{\\lbrace 1, 3, 7,8\\rbrace }\\oplus L_{\\lbrace 2,4,5, 6\\rbrace } \\longrightarrow L_I \\longrightarrow 0,\\\\0\\longrightarrow L_J\\longrightarrow L_{\\lbrace 2,3,7,8\\rbrace } \\oplus L_{\\lbrace 1, 4, 5,6 \\rbrace }\\longrightarrow L_I \\longrightarrow 0.$ The pairs of profiles of the four modules that appear in the middle in these short exact sequences can be pictured as follows (similarly as in Example REF ).", "Figure: The pairs of profiles of decomposable extensions between L {1,3,5,6} L_{\\lbrace 1, 3, 5,6\\rbrace } and L {2,4,7,8} L_{\\lbrace 2,4, 7,8\\rbrace }." ], [ "Tight 4-interlacing", "In the tame case $(4,8)$ , there is only one type of configuration of layers with 4-interlacing, $1357\\mid 2468$ (and the one obtained by adding 1 to each element of the two 4-subsets).", "We study this now.", "Let $I=\\lbrace 1,3,5,7\\rbrace $ and $J=\\lbrace 2,4,6,8\\rbrace .$ The construction is the same as for the module $135\\mid 246$ , we just have two more vertices.", "So assume that $x_i=\\begin{pmatrix} t& b_{i} \\\\ 0 & 1 \\end{pmatrix}$ for odd $i$ and $x_{i}=\\begin{pmatrix} 1& b_{i} \\\\ 0 & t \\end{pmatrix}$ for even $i$ .", "From $x^k=y^{n-k}$ it follows that $\\sum _1^8b_i=0$ .", "We denote the constructed module by $\\operatorname{\\mathbb {M}}\\nolimits $ and study the structure of this module with respect to the divisibility conditions of the coefficients $b_i$ .", "Just as in the case of the module $135\\mid 246$ (Section REF ), one can argue that $\\operatorname{\\mathbb {M}}\\nolimits =L_I\\oplus L_J$ if and only if $t\\mid b_1+b_2$ , $t\\mid b_3+b_4$ , $t\\mid b_5+b_6$ , $t\\mid b_7+b_8$ .", "In order to determine the structure of the module $\\operatorname{\\mathbb {M}}\\nolimits $ when these four divisibility conditions are not fulfilled, first we determine the structure of an endomorphism of this module.", "If $\\varphi =(\\varphi _i)_{i=0}^7$ is an endomorphism of $\\operatorname{\\mathbb {M}}\\nolimits $ and $\\varphi _0=\\begin{pmatrix}a & b \\\\ c & d \\end{pmatrix}$ , then $\\varphi _{2i+1}&=\\begin{pmatrix}a+(b_1+\\dots +b_{2i+1})t^{-1}c & tb+(d-a)(b_1+\\dots +b_{2i+1})-(b_1+\\dots +b_{2i+1})^2t^{-1}c \\\\ t^{-1}c& d-(b_1+\\dots +b_{2i+1})t^{-1}c \\end{pmatrix},\\\\\\varphi _{2i}&=\\begin{pmatrix}a+(b_1+\\dots +b_{2i})t^{-1}c & b+t^{-1}((d-a)(b_1+\\dots +b_{2i})-(b_1+\\dots +b_{2i})^2t^{-1}c) \\\\ c& d-(b_1+\\dots +b_{2i})t^{-1}c \\end{pmatrix},$ where $t\\mid c$ , and $ t&\\mid (d-a)(b_1+b_2)-(b_1+b_2)^2t^{-1}c,\\\\ \\nonumber t&\\mid (d-a)(b_1+b_2+b_3+b_4)-(b_1+b_2+b_3+b_4)^2t^{-1}c,\\\\ \\nonumber t&\\mid (d-a)(b_1+b_2+b_3+b_4+b_5+b_6)-(b_1+b_2+b_3+b_4+b_5+b_6)^2t^{-1}c. \\nonumber $ We distinguish between different cases depending on whether the sums $b_1+b_2$ , $b_3+b_4$ , $b_5+b_6$ , and $b_7+b_8$ are divisible by $t$ or not.", "We will call these the four divisibility conditions $t\\mid b_1+b_2$ , $t\\mid b_3+b_4$ , $t\\mid b_5+b_6$ and $t\\mid b_7+b_8$ , and write (div) to abbreviate.", "There are three base cases: one of the sums is divisible by $t$ and three are not, two are divisible by $t$ and two are not, and none of the sums is divisible by $t$ .", "We will see that $\\operatorname{\\mathbb {M}}\\nolimits $ is indecomposable in the first case and partly in the third case.", "We will explain how the module decomposes in the other cases.", "Furthermore, we will also show that there are infinitely many non-isomorphic modules with the same filtration for the indecomposable case when none of the sums is divisible by $t$ ." ], [ "Only one of the sums is divisible by $t$ (Case 1)", "We first assume that $t\\nmid b_1+b_2$ , $t\\nmid b_3+b_4$ , $t\\nmid b_5+b_6$ , and $t\\mid b_7+b_8$ .", "Theorem 5.1 The above defined module $\\operatorname{\\mathbb {M}}\\nolimits $ is indecomposable if $t\\nmid b_1+b_2$ , $t\\nmid b_3+b_4$ , $t\\nmid b_5+b_6$ , and $t\\mid b_7+b_8$ .", "As in the proof of Theorem REF for the module $135\\mid 246$ , we repeat the same arguments using the divisibility conditions $t&\\mid (d-a)(b_1+b_2)-(b_1+b_2)^2t^{-1}c,\\\\ t&\\mid (d-a)(b_1+b_2+b_3+b_4)-(b_1+b_2+b_3+b_4)^2t^{-1}c, $ to conclude that the only possible idempotent endomorphisms of $\\operatorname{\\mathbb {M}}\\nolimits $ are the trivial ones.", "We only note that $t\\nmid b_1+b_2+b_3+b_4$ because if it were not so, then $t\\mid b_5+b_6$ which is not true.", "In the previous theorem it suffices to choose $b_1=0$ , $b_2=1$ , $b_3=2$ , $b_4=0$ , $b_5=0$ , $b_6=-3$ , $b_7=-1$ , and $b_8=1$ in order to fulfil the conditions of the theorem and to have an indecomposable module.", "In the next theorem we show that this module only depends on the divisibility conditions of the coefficients $b_i$ , so if we have two different 8-tuples satisfying the same divisibility conditions, then they give rise to isomorphic modules.", "Theorem 5.2 Let $(c_1, c_2, c_3, c_4, c_5, c_6,c_7,c_8 )$ be an 8-tuple such that $t\\nmid c_1+c_2$ , $t\\nmid c_3+c_4$ , $t\\nmid c_5+c_6$ , and $t\\mid c_7+c_8$ .", "If $\\operatorname{\\mathbb {M}}\\nolimits ^{\\prime }$ is the module determined by this 8-tuple, then the modules $\\operatorname{\\mathbb {M}}\\nolimits ^{\\prime }$ and $\\operatorname{\\mathbb {M}}\\nolimits $ are isomorphic.", "As in the proof of Theorem REF for the module $135\\mid 246$ , we explicitly construct an isomorphism $\\varphi =(\\varphi _i)_{i=0}^7$ between the two modules, where $\\varphi _i: V_i\\longrightarrow W_i$ , and $V_i$ and $W_i$ are the vector spaces at vertex $i$ of the modules $\\operatorname{\\mathbb {M}}\\nolimits $ and $\\operatorname{\\mathbb {M}}\\nolimits ^{\\prime }$ respectively.", "Let us assume that $\\varphi _0=\\begin{pmatrix}\\alpha & \\beta \\\\ \\gamma & \\delta \\end{pmatrix}$ .", "Then by repeating the same calculations as for the module $135\\mid 246$ we get $\\varphi _{2i+1}&= \\begin{pmatrix} \\alpha +(c_1+\\dots +c_{2i+1})t^{-1}\\gamma & \\beta t-\\alpha \\sum _{j=1}^{2i+1} b_j+\\delta \\sum _{j=1}^{2i+1} c_j -(\\sum _{j=1}^{2i+1} b_j)(\\sum _{j=1}^{2i+1} c_j)t^{-1}\\gamma \\\\ t^{-1}\\gamma & \\delta -(b_1+\\dots b_{2i+1})t^{-1}\\gamma _0 \\end{pmatrix} ,\\\\\\varphi _{2i}&=\\begin{pmatrix} \\alpha +(c_1+\\dots +c_{2i})t^{-1}\\gamma & \\beta +t^{-1}(-\\alpha \\sum _{j=1}^{2i} b_j+\\delta \\sum _{j=1}^{2i} c_j -t^{-1}\\gamma \\sum _{j=1}^{2i} b_j \\sum _{j=1}^{2i} c_j) \\\\ \\gamma & \\delta -(b_1+\\dots +b_{2i})t^{-1}\\gamma \\end{pmatrix},$ where $t\\mid \\gamma $ and $t&\\mid -\\alpha (b_1+b_2)+(c_1+c_2)\\delta -(b_1+b_2)(c_1+c_2)t^{-1}\\gamma ,\\\\t&\\mid -\\alpha (b_1+b_2+b_3+b_4)+(c_1+c_2+c_3+c_4)\\delta -(b_1+b_2+b_3+b_4)(c_1+c_2+c_3+c_4)t^{-1}\\gamma ,\\\\t&\\mid -\\alpha (b_1+b_2+b_3+b_4+b_5+b_6)+(c_1+c_2+c_3+c_4+c_5+c_6)\\delta -t^{-1}\\gamma \\sum _{i=1}^6b_i \\sum _{i=1}^6c_i.$ Since $t\\mid \\gamma $ and we would like $\\varphi $ to be invertible, then it must be that $t\\nmid \\alpha $ and $t\\nmid \\delta $ .", "Then the inverse of $\\varphi _0$ is $\\frac{1}{\\alpha \\delta -\\beta \\gamma }\\begin{pmatrix} \\delta &-\\beta \\\\ -\\gamma & \\alpha \\end{pmatrix}.$ In order to find an isomorphism $\\varphi $ , note that the last divisibility condition is fulfilled because $t\\mid b_7+b_8$ and $t\\mid c_7+c_8$ .", "Also note that the condition $t\\mid -\\alpha (b_1+b_2+b_3+b_4)+(c_1+c_2+c_3+c_4)\\delta -(b_1+b_2+b_3+b_4)(c_1+c_2+c_3+c_4)t^{-1}\\gamma $ is equivalent to the condition $t\\mid \\alpha (b_5+b_6)-(c_5+c_6)\\delta -(b_5+b_6)(c_5+c_6)t^{-1}\\gamma .$ Now we repeat the same calculations as for the module $135\\mid 246$ .", "Note that there are no conditions attached to $\\beta $ so we set it to be 0.", "If we set $-\\alpha (b_1+b_2)+(c_1+c_2)\\delta -(b_1+b_2)(c_1+c_2)t^{-1}\\gamma &=0,\\\\\\alpha (b_5+b_6)-(c_5+c_6)\\delta -(b_5+b_6)(c_5+c_6)t^{-1}\\gamma &=0,$ then we get $\\alpha (b_5+b_6)\\left[1+\\frac{c_5+c_6}{c_1+c_2} \\right]-\\delta (c_5+c_6)\\left[1+\\frac{b_5+b_6}{b_1+b_2} \\right]=0.$ If $t\\mid 1+\\frac{c_5+c_6}{c_1+c_2}$ , then from $\\sum _{i=1}^8c_i=0$ , we get $t\\mid c_3+c_4$ , which is not true.", "The same holds for $1+\\frac{b_5+b_6}{b_1+b_2}$ , so both of these elements are invertible.", "Thus, if we set $\\delta =1$ , then we get $\\alpha =\\frac{(c_1+c_2)(b_3+b_4)(c_5+c_6)}{(b_1+b_2)(c_3+c_4)(b_5+b_6)},$ and $\\gamma =t \\frac{(c_3+c_4)(b_5+b_6)-(b_3+b_4)(c_5+c_6)}{(b_1+b_2)(c_3+c_4)(b_5+b_6)}.$ Hence, $\\varphi _0=\\begin{pmatrix} \\displaystyle \\frac{(c_1+c_2)(b_3+b_4)(c_5+c_6)}{(b_1+b_2)(c_3+c_4)(b_5+b_6)} & \\,\\,\\,\\,\\,\\,\\,\\, 0 \\\\ t \\displaystyle \\frac{(c_3+c_4)(b_5+b_6)-(b_3+b_4)(c_5+c_6)}{(b_1+b_2)(c_3+c_4)(b_5+b_6)} & \\,\\,\\,\\,\\,\\,\\,\\,1 \\end{pmatrix}.$ The other invertible matrices $\\varphi _i$ are now determined from the above equalities.", "Note that all of them are invertible because their determinant is equal to $\\alpha \\delta -\\beta \\gamma $ which is an invertible element.", "We denote the unique module (up to isomorphism) from Theorem REF by $\\operatorname{\\mathbb {M}}\\nolimits _{7,8}$ .", "It is obvious, due to the symmetry of the arguments, that there are also modules $\\operatorname{\\mathbb {M}}\\nolimits _{1,2}$ , $\\operatorname{\\mathbb {M}}\\nolimits _{3,4}$ and $\\operatorname{\\mathbb {M}}\\nolimits _{5,6}$ that correspond to the remaining three possible divisibility conditions for Case 1, e.g.", "$\\operatorname{\\mathbb {M}}\\nolimits _{1,2}$ corresponds to the case when $t\\mid b_1+b_2$ , $t\\nmid b_3+b_4$ , $t\\nmid b_5+b_6$ , and $t\\nmid b_7+b_8$ .", "In the next statement we prove that no two of these modules are isomorphic to each other.", "Proposition 5.3 There are no isomorphic modules amongst $\\operatorname{\\mathbb {M}}\\nolimits _{1,2}$ , $\\operatorname{\\mathbb {M}}\\nolimits _{3,4}$ , $\\operatorname{\\mathbb {M}}\\nolimits _{5,6}$ and $\\operatorname{\\mathbb {M}}\\nolimits _{7,8}$ .", "Due to the symmetry of the arguments, we only show that $\\operatorname{\\mathbb {M}}\\nolimits _{7,8}$ is not isomorphic to any of the other modules.", "Assume otherwise, that $\\operatorname{\\mathbb {M}}\\nolimits _{7,8}$ is isomorphic to $\\operatorname{\\mathbb {M}}\\nolimits _{i,i+1}$ , where $i$ is 1, 3 or 5.", "Then there is an isomorphism between these two modules.", "Keeping the same notation from the proof of the Theorem REF , we have that this isomorphism has to satisfy the following divisibility conditions: $t&\\mid -\\alpha (b_1+b_2)+(c_1+c_2)\\delta -(b_1+b_2)(c_1+c_2)t^{-1}\\gamma ,\\\\t&\\mid -\\alpha (b_1+b_2+b_3+b_4)+(c_1+c_2+c_3+c_4)\\delta -(b_1+b_2+b_3+b_4)(c_1+c_2+c_3+c_4)t^{-1}\\gamma ,\\\\t&\\mid -\\alpha (b_1+b_2+b_3+b_4+b_5+b_6)+(c_1+c_2+c_3+c_4+c_5+c_6)\\delta -t^{-1}\\gamma \\sum _{i=0}^6 b_i\\sum _{i=0}^6 c_i.$ Here, the coefficients $b_j$ correspond to $\\operatorname{\\mathbb {M}}\\nolimits _{7,8}$ and $c_j$ correspond to $\\operatorname{\\mathbb {M}}\\nolimits _{i,i+1}$ .", "Since $t\\mid b_1+b_2+b_3+b_4+b_5+b_6=-(b_7+b_8)$ , from the last condition it follows that $t\\mid \\delta (c_1+c_2+c_3+c_4+c_5+c_6)$ .", "Since $t\\nmid \\delta $ , it must be $t\\mid (c_1+c_2+c_3+c_4+c_5+c_6)$ .", "Then $t\\mid c_7+c_8=-(c_1+c_2+c_3+c_4+c_5+c_6)$ which is in contradiction with our assumption that $t\\nmid c_7+c_8$ .", "It follows from the previous proposition that we have now constructed four non-isomorphic rank 2 modules whose filtration is $L_{1357}\\mid L_{2468}$ .", "Before we show that in fact there are infinitely many, we consider the other two cases for the divisibility conditions." ], [ "Exactly two of the sums are divisible by $t$ (Case 2)", "There are two subcases.", "The first subcase is when the divisible sums are consecutive, e.g.", "when $t\\mid b_1+b_2$ , $t\\mid b_3+b_4$ , $t\\nmid b_5+b_6$ and $t\\nmid b_7+b_8$ .", "The second subcase is when the divisible sums are not consecutive, e.g.", "when $t\\mid b_1+b_2$ , $t\\mid b_5+b_6$ , $t\\nmid b_3+b_4$ and $t\\nmid b_7+b_8$ .", "Assume first that $t\\mid b_1+b_2$ , $t\\mid b_3+b_4$ , $t\\nmid b_5+b_6$ , and $t\\nmid b_7+b_8$ .", "Theorem 5.4 If $t\\mid b_1+b_2$ , $t\\mid b_3+b_4$ , $t\\nmid b_5+b_6$ , and $t\\nmid b_7+b_8$ , then the module $\\operatorname{\\mathbb {M}}\\nolimits $ is isomorphic to $L_{\\lbrace 1,3,5,6\\rbrace }\\oplus L_{\\lbrace 2,4,7,8\\rbrace }$ .", "We show that $\\operatorname{\\mathbb {M}}\\nolimits $ is decomposable by constructing a non-trivial idempotent endomorphism of $\\operatorname{\\mathbb {M}}\\nolimits $ .", "Recall that an endomorphism $\\varphi =(\\varphi _i)_{i=0}^7$ of $\\operatorname{\\mathbb {M}}\\nolimits $ , where $\\varphi _0=\\begin{pmatrix}a & b \\\\ c & d \\end{pmatrix}$ , satisfies the divisibility conditions (REF ).", "Since $t\\mid b_1+b_2$ , $t\\mid b_3+b_4$ , these conditions reduce to a single condition $t\\mid (d-a)(b_5+b_6)-(b_5+b_6)^2t^{-1}c$ .", "From $t\\nmid b_5+b_6$ we conclude that $t\\mid (d-a)-(b_5+b_6)t^{-1}c$ .", "To construct a non-trivial idempotent homomorphism, as in the case $n=6$ , we set $\\alpha =1$ , $\\delta =0=\\beta $ , and $\\gamma =-t(b_5+b_6)^{-1}$ .", "Thus, $\\varphi _0=\\begin{pmatrix}1& 0\\\\-t(b_5+b_6)^{-1}&0\\end{pmatrix}.$ Its orthogonal complement is the idempotent $\\begin{pmatrix}0& 0\\\\t(b_5+b_6)^{-1}&1\\end{pmatrix}.$ Since these are non-trivial idempotents, it follows that the module $\\operatorname{\\mathbb {M}}\\nolimits $ is decomposable.", "It remains to show that $\\operatorname{\\mathbb {M}}\\nolimits \\cong L_{\\lbrace 2,4,7,8\\rbrace }\\oplus L_{\\lbrace 1,3,5,6\\rbrace }.$ We know that $\\operatorname{\\mathbb {M}}\\nolimits $ is the direct sum of rank 1 modules $L_X$ and $L_Y$ for some $X$ and $Y$ .", "Let us determine $X$ and $Y$ .", "For this, we take, at vertex $i$ , eigenvectors $v_i$ and $w_i$ corresponding to the eigenvalue 1 of the idempotents $\\varphi _i$ and $1-\\varphi _i$ respectively.", "For example, $v_0=[1\\,\\, ,\\,\\, -t(b_5+b_6)^{-1}]^t$ , $w_0=[0\\,\\, ,\\,\\, 1]^t$ , $v_1= [ 1-b_1(b_5+b_6)^{-1} \\,\\, ,\\,\\, -(b_5+b_6)^{-1} ]^t$ and $w_1= [b_1 \\,\\, , \\,\\, 1 ]^t$ , and so on.", "A basis for $L_X$ is $\\lbrace v_i \\mid i=0,\\ldots , 7\\rbrace $ , and a basis for $L_Y$ is $\\lbrace w_i \\mid i=0,\\ldots , 7\\rbrace $ .", "Direct computation gives us that $x_1v_0=tv_1$ , $x_2v_1=v_2$ , $x_3v_2=tv_3$ , $x_4v_3=v_4$ , $x_5v_4=tv_5$ , $x_6v_5=tv_6$ , $x_7v_6=v_7$ , $x_8v_7=v_0$ .", "Thus, $X=\\lbrace 2,4,7,8\\rbrace $ .", "Analogously, $Y=\\lbrace 1,3,5,6\\rbrace .$ Remark 5.5 In the case when $t\\nmid b_1+b_2$ , $t\\mid b_3+b_4$ , $t\\mid b_5+b_6$ , and $t\\nmid b_7+b_8$ , $\\operatorname{\\mathbb {M}}\\nolimits $ is the direct sum $L_{\\lbrace 3,5,7,8\\rbrace }\\oplus L_{\\lbrace 1,2,4,6\\rbrace }$ .", "Similarly, by suitable renaming of the vertices of the quiver, we obtain two more direct sums $L_{\\lbrace 1,2,5,7\\rbrace }\\oplus L_{\\lbrace 3,4,6,8\\rbrace }$ and $L_{\\lbrace 1,3,4,7\\rbrace }\\oplus L_{\\lbrace 2,5,6,8\\rbrace }$ that have $L_J$ as a submodule and $L_I$ as a quotient module, and there are short exact sequences with $L_I$ and $L_J$ as end terms: $(a) \\quad 0\\longrightarrow L_J\\longrightarrow L_{\\lbrace 1, 3, 5, 6\\rbrace }\\oplus L_{\\lbrace 2,4, 7, 8 \\rbrace } \\longrightarrow L_I \\longrightarrow 0,\\\\(b) \\quad 0\\longrightarrow L_J\\longrightarrow L_{\\lbrace 3,5,7,8\\rbrace } \\oplus L_{\\lbrace 1,2,4,6\\rbrace } \\longrightarrow L_I \\longrightarrow 0,\\\\(c) \\quad 0\\longrightarrow L_J\\longrightarrow L_{\\lbrace 1,2,5,7\\rbrace }\\oplus L_{\\lbrace 3,4,6,8\\rbrace } \\longrightarrow L_I \\longrightarrow 0,\\\\(d) \\quad 0\\longrightarrow L_J\\longrightarrow L_{\\lbrace 1,3,4,7\\rbrace }\\oplus L_{\\lbrace 2,5,6,8\\rbrace } \\longrightarrow L_I \\longrightarrow 0.$ Here, (a) is the case where $t\\mid (b_1+b_2)$ and $t\\mid (b_3+b_4)$ , (b) the case where $t\\mid (b_3+b_4)$ and $t\\mid (b_5+b_6)$ , (c) the case where $t\\mid (b_5+b_6)$ and $t\\mid (b_7+b_8)$ , and (d) the case $t\\mid (b_7+b_8)$ and $t\\mid (b_1+b_2)$ .", "The pairs of profiles of the four decomposable modules that appear in the middle in these short exact sequences can be pictured as follows: Figure: The pairs of profiles of decomposable extensions between L {1,3,5,7} L_{\\lbrace 1, 3, 5,7\\rbrace } and L {2,4,6,8} L_{\\lbrace 2,4, 6,8\\rbrace }.Assume now that $t\\mid b_1+b_2$ , $t\\mid b_5+b_6$ , $t\\nmid b_3+b_4$ , and $t\\nmid b_7+b_8$ .", "Theorem 5.6 If $t\\mid b_1+b_2$ , $t\\mid b_5+b_6$ , $t\\nmid b_3+b_4$ , and $t\\nmid b_7+b_8$ , then the module $\\operatorname{\\mathbb {M}}\\nolimits $ is isomorphic to $L_{\\lbrace 1,3,4,6\\rbrace }\\oplus L_{\\lbrace 2,5,7,8\\rbrace }$ .", "The only difference from the proof of the previous statement is that the divisibility conditions are now reduced to the condition $t\\mid (d-a)-(b_3+b_4)t^{-1}c$ .", "To construct a non-trivial idempotent homomorphism, we set $\\alpha =1$ , $\\delta =0=\\beta $ , and $\\gamma =-t(b_3+b_4)^{-1}$ .", "Thus, $\\varphi _0=\\begin{pmatrix}1& 0\\\\-t(b_3+b_4)^{-1}&0\\end{pmatrix},$ and the rest of proof is analogous to the proof of the previous statement.", "Remark 5.7 In the case when $t\\nmid b_1+b_2$ , $t\\mid b_3+b_4$ , $t\\nmid b_5+b_6$ , and $t\\mid b_7+b_8$ , $\\operatorname{\\mathbb {M}}\\nolimits $ is the direct sum $L_{\\lbrace 1,2,4,7\\rbrace }\\oplus L_{\\lbrace 3,5,6,8\\rbrace }$ .", "The pairs of profiles of these two modules can be pictured as follows: Figure: The pairs of profiles of decomposable extensions between L {1,3,5,7} L_{\\lbrace 1, 3, 5,7\\rbrace } and L {2,4,6,8} L_{\\lbrace 2,4, 6,8\\rbrace }." ], [ "None of the four sums is divisible by $t$ (Case 3)", "There are two subcases we have to consider.", "The first subcase is when all sums $b_i+b_{i+1}+b_{i+2}+b_{i+3}$ , for $i=1,3,5,7$ , are divisible by $t$ , the second subcase is when at least one of these sums is not divisible by $t$ .", "In the latter case, we get infinitely many non-isomorphic indecomposable modules as we will show.", "In this subsection, we always assume that none of the four divisibility conditions $t\\mid b_i+b_{i+1}$ , $i$ odd, which we continue to abbreviate as (div), is satisfied.", "We first consider the case where all sums $b_i+b_{i+1}+b_{i+2}+b_{i+3}$ are divisible by $t$ .", "Theorem 5.8 Assume that the $(b_i)_i$ satisfy none of the four divisibility conditions (div) but that $t\\mid b_i+b_{i+1}+b_{i+2}+b_{i+3}$ , for $i=1,3,5,7$ .", "Then $\\operatorname{\\mathbb {M}}\\nolimits \\cong L_{\\lbrace 1,2,5,6\\rbrace }\\oplus L_{\\lbrace 3,4,7,8\\rbrace } .$ As before, we will construct a non-trivial idempotent endomorphism of $\\operatorname{\\mathbb {M}}\\nolimits $ to prove that it is a decomposable module.", "Also, as before, our endomorphism has to satisfy the conditions (REF ).", "From $t\\mid b_1+b_{2}+b_{3}+b_{4}$ it follows that the conditions (REF ) reduce to a single condition $t\\mid (d-a)-(b_1+b_2)t^{-1}c.$ Note that this condition is equivalent to the condition $t\\mid (d-a)-(b_5+b_6)t^{-1}c.$ As before, we obtain a non-trivial idempotent $\\varphi _0=\\begin{pmatrix}1& 0\\\\-t(b_1+b_2)^{-1}&0\\end{pmatrix}.$ The rest of the proof is analogous to the proof of the previous two statements.", "Remark 5.9 The pairs of profiles of $L_{\\lbrace 1,2,5,6\\rbrace }\\oplus L_{\\lbrace 3,4,7,8\\rbrace }$ can be pictured as follows: Figure: The pairs of profile of a decomposable extension between L {1,3,5,7} L_{\\lbrace 1, 3, 5,7\\rbrace } and L {2,4,6,8} L_{\\lbrace 2,4, 6,8\\rbrace }.Assume now that for the tuple $(b_i)_i$ one of the consecutive sums of four entries is not divisible by $t$ .", "Proposition 5.10 If none of the four divisibility conditions (div) holds for $(b_i)_i$ and if there exists an $i\\in \\lbrace 1,3,5,7\\rbrace $ such that $t\\nmid b_i+b_{i+1}+b_{i+2}+b_{i+3}$ , then $\\operatorname{\\mathbb {M}}\\nolimits $ is indecomposable.", "Assume, without loss of generality, that $t\\nmid b_1+b_2+b_3+b_4$ .", "As in the proof of indecomposability of the module $135\\mid 246$ the endomorphism of $\\operatorname{\\mathbb {M}}\\nolimits $ has to satisfy the conditions $t&\\mid (d-a)(b_1+b_2)-(b_1+b_2)^2t^{-1}c,\\\\t&\\mid (d-a)(b_1+b_2+b_3+b_4)-(b_1+b_2+b_3+b_4)^2t^{-1}c,\\\\t&\\mid (d-a)(b_1+b_2+b_3+b_4+b_5+b_6)-(b_1+b_2+b_3+b_4+b_5+b_6)^2t^{-1}c.$ Now we repeat the same arguments as for the module $135\\mid 246$ .", "From the first two conditions we get that $t\\mid (b_3+b_4)t^{-1}c$ .", "Since $t\\nmid b_3+b_4$ , it follows that $t\\mid t^{-1}c$ and that $t\\mid d-a$ .", "If $a+d=1$ , then, because $t\\mid a$ or $t\\mid 1-a$ , it must be that $t\\mid 1$ , which is not true.", "Thus, it must be $a=d$ and $c=b=0$ , giving us only trivial idempotents.", "Now that we know that whenever $t\\nmid b_i+b_{i+1}+b_{i+2}+b_{i+3}$ , for at least one $i$ , the module $\\operatorname{\\mathbb {M}}\\nolimits $ with the given $(b_i)_i$ is indecomposable, we would like to know if the constructed modules are isomorphic.", "Since $\\sum b_i=0$ , we have $t\\nmid b_i+b_{i+1}+b_{i+2}+b_{i+3}$ for some $i$ if and only if $t\\nmid b_{i+4}+b_{i+5} + b_{i_6} + b_{i_7}$ .", "Therefore, these conditions come in pairs of “complementary” sums.", "Thus we have to distinguish between the cases when two of the sums $b_i+b_{i+1}+b_{i+2}+b_{i+3}$ are divisible by $t$ and two are not, and when none of these sums of four consecutive $b_i$ 's is divisible by $t$ .", "We will see that in the first case, we only get two indecomposable modules (up to isomorphism) while in the latter case, we get infinitely many.", "Let us assume that $(c_i)_1^8$ is another 8-tuple satisfying none of the divisibility conditions (div) and such that $t\\nmid c_i+c_{i+1}+c_{i+2}+c_{i+3}$ , for some $i\\in \\lbrace 1,3,5,7\\rbrace $ .", "Denote the module given by these $(c_i)_i$ by $\\operatorname{\\mathbb {M}}\\nolimits ^{\\prime }$ .", "In the following two propositions we consider the case when both $b_i$ 's and $c_i$ 's have two of the above mentioned sums divisible by $t$ , and two sums not divisible by $t$ .", "Proposition 5.11 If $t\\nmid b_1+b_2+b_3+b_4$ , $t\\mid b_3+b_4+b_5+b_6$ , $t\\nmid c_1+c_2+c_3+c_4$ , and $t\\mid c_3+c_4+c_5+c_6$ , then $\\operatorname{\\mathbb {M}}\\nolimits $ and $\\operatorname{\\mathbb {M}}\\nolimits ^{\\prime }$ are isomorphic.", "Keeping the same notation as before when constructing isomorphisms, it must hold: $t&\\mid -\\alpha (b_1+b_2)+(c_1+c_2)\\delta -(b_1+b_2)(c_1+c_2)t^{-1}\\gamma ,\\\\t&\\mid -\\alpha (b_1+b_2+b_3+b_4)+(c_1+c_2+c_3+c_4)\\delta -(b_1+b_2+b_3+b_4)(c_1+c_2+c_3+c_4)t^{-1}\\gamma ,\\\\t&\\mid -\\alpha (b_1+b_2+b_3+b_4+b_5+b_6)+(c_1+c_2+c_3+c_4+c_5+c_6)\\delta -t^{-1}\\gamma \\sum _{i=1}^6b_i\\sum _{i=1}^6c_i.$ Since $t\\mid b_3+b_4+b_5+b_6$ and $t\\mid c_3+c_4+c_5+c_6$ , the above conditions reduce to the first two conditions.", "Now, we set $-\\alpha (b_1+b_2)+(c_1+c_2)\\delta -(b_1+b_2)(c_1+c_2)t^{-1}\\gamma =0$ and $-\\alpha (b_1+b_2+b_3+b_4)+(c_1+c_2+c_3+c_4)\\delta -(b_1+b_2+b_3+b_4)(c_1+c_2+c_3+c_4)t^{-1}\\gamma =0.$ Subsequently, $\\alpha (b_1+b_2+b_3+b_4)(c_3+c_4)(c_1+c_2)^{-1}-\\delta (c_1+c_2+c_3+c_4) (b_3+b_4)(b_1+b_2)^{-1} =0$ .", "By setting $\\alpha =1$ , $\\beta =0$ , we get that $\\delta =[(b_1+b_2)(b_1+b_2+b_3+b_4)(c_3+c_4)] [(c_1+c_2)(c_1+c_2+c_3+c_4) (b_3+b_4)]^{-1}$ , and $\\gamma =-(c_1+c_2)^{-1}+\\delta (b_1+b_2)^{-1}$ , giving us an isomorphism between $\\operatorname{\\mathbb {M}}\\nolimits $ and $\\operatorname{\\mathbb {M}}\\nolimits ^{\\prime }$ .", "Proposition 5.12 If $t\\nmid b_1+b_2+b_3+b_4$ , $t\\mid b_3+b_4+b_5+b_6$ , $t\\mid c_1+c_2+c_3+c_4$ , and $t\\nmid c_3+c_4+c_5+c_6$ , then $\\operatorname{\\mathbb {M}}\\nolimits $ and $\\operatorname{\\mathbb {M}}\\nolimits ^{\\prime }$ are not isomorphic.", "If there were an isomorphism between $\\operatorname{\\mathbb {M}}\\nolimits $ and $\\operatorname{\\mathbb {M}}\\nolimits ^{\\prime }$ , then its coefficients would satisfy $t&\\mid -\\alpha (b_1+b_2)+(c_1+c_2)\\delta -(b_1+b_2)(c_1+c_2)t^{-1}\\gamma ,\\\\t&\\mid -\\alpha (b_1+b_2+b_3+b_4)+(c_1+c_2+c_3+c_4)\\delta -(b_1+b_2+b_3+b_4)(c_1+c_2+c_3+c_4)t^{-1}\\gamma ,\\\\t&\\mid -\\alpha (b_1+b_2+b_3+b_4+b_5+b_6)+(c_1+c_2+c_3+c_4+c_5+c_6)\\delta -t^{-1}\\gamma \\sum _{i=1}^6b_i\\sum _{i=1}^6c_i.$ From the second condition we obtain $t\\mid \\alpha (b_1+b_2+b_3+b_4)$ .", "But $t\\nmid b_1+b_2+b_3+b_4$ and $t\\nmid \\alpha $ , which is a contradiction.", "Remark 5.13 The same arguments used in the proof of the previous proposition tell us that the two modules $\\operatorname{\\mathbb {M}}\\nolimits $ and $\\operatorname{\\mathbb {M}}\\nolimits ^{\\prime }$ from Proposition REF are two new non-isomorphic indecomposable modules which are not isomorphic to any of the modules $\\operatorname{\\mathbb {M}}\\nolimits _{1,2}$ , $\\operatorname{\\mathbb {M}}\\nolimits _{3,4}$ , $\\operatorname{\\mathbb {M}}\\nolimits _{5,6}$ and $\\operatorname{\\mathbb {M}}\\nolimits _{7,8}$ constructed before.", "For example, if the $b_i$ 's correspond to the module $\\operatorname{\\mathbb {M}}\\nolimits _{7,8}$ and the $c_i$ 's correspond to the module $\\operatorname{\\mathbb {M}}\\nolimits $ , then from the third relation in the proof of Proposition REF we obtain that $t\\mid \\delta (c_1+c_2+c_3+c_4+c_5+c_6)$ , yielding $t\\mid \\delta (c_1+c_2)$ , which is not true since $t\\nmid \\delta $ and $t\\nmid (c_1+c_2)$ .", "We are only left to examine if, in the case when none of the sums $b_i+b_{i+1}+b_{i+2}+b_{i+3}$ is divisible by $t$ , for two different tuples we obtain isomorphic modules.", "In the following theorem we assume that $b_i$ 's correspond to the module $\\operatorname{\\mathbb {M}}\\nolimits $ and $c_i$ 's correspond to the module $\\operatorname{\\mathbb {M}}\\nolimits ^{\\prime }$ .", "Also, we assume that $t\\nmid b_i+b_{i+1}$ $t\\nmid c_i+c_{i+1}$ for odd $i$ .", "Theorem 5.14 If $t\\nmid b_i+b_{i+1}+b_{i+2}+b_{i+3}$ and $t\\nmid c_i+c_{i+1}+c_{i+2}+c_{i+3}$ , for $i=1,3,5,7,$ then the modules $\\operatorname{\\mathbb {M}}\\nolimits $ and $\\operatorname{\\mathbb {M}}\\nolimits ^{\\prime }$ are isomorphic if and only if $t\\mid (b_1+ b_2) (c_3+ c_4) (b_5+ b_6) (c_7+ c_8) -(c_1+ c_2) (b_3+ b_4) (c_5+ c_6) (b_7+ b_8).$ As before, if there were an isomorphism between $\\operatorname{\\mathbb {M}}\\nolimits $ and $\\operatorname{\\mathbb {M}}\\nolimits ^{\\prime }$ , its coefficients would have to satisfy the following conditions: $t&\\mid -\\alpha (b_1+b_2)+(c_1+c_2)\\delta -(b_1+b_2)(c_1+c_2)t^{-1}\\gamma ,\\\\t&\\mid -\\alpha (b_1+b_2+b_3+b_4)+(c_1+c_2+c_3+c_4)\\delta -(b_1+b_2+b_3+b_4)(c_1+c_2+c_3+c_4)t^{-1}\\gamma ,\\\\t&\\mid \\alpha (b_7+b_8)-(c_7+c_8)\\delta -(b_7+b_8)(c_7+c_8)t^{-1}\\gamma .$ From these we get that $t&\\mid \\alpha (c_3+ c_4)[(c_1+ c_2)(c_1+ c_2+c_3+ c_4)]^{-1}-\\delta (b_3+ b_4)[(b_1+ b_2)(b_1+ b_2+b_3+ b_4)]^{-1},\\\\t&\\mid \\alpha (c_5+ c_6)[(c_7+ c_8)(c_1+ c_2+c_3+ c_4)]^{-1}-\\delta (b_5+ b_6)[(b_7+ b_8)(b_1+ b_2+b_3+ b_4)]^{-1}.$ Finally, from the last two relations we get $t\\mid \\alpha [(b_1+ b_2) (c_3+ c_4) (b_5+ b_6) (c_7+ c_8) -(c_1+ c_2) (b_3+ b_4) (c_5+ c_6) (b_7+ b_8)].$ If $t\\nmid (b_1+ b_2) (c_3+ c_4) (b_5+ b_6) (c_7+ c_8) -(c_1+ c_2) (b_3+ b_4) (c_5+ c_6) (b_7+ b_8)$ , then there is no isomorphism between $\\operatorname{\\mathbb {M}}\\nolimits ^{\\prime }$ and $\\operatorname{\\mathbb {M}}\\nolimits $ .", "If $t\\mid (b_1+ b_2) (c_3+ c_4) (b_5+ b_6) (c_7+ c_8) -(c_1+ c_2) (b_3+ b_4) (c_5+ c_6) (b_7+ b_8), $ then we simply set $\\alpha =1$ , and compute $\\delta $ and $\\gamma $ from the above relations (as before, we set $\\beta =0$ ).", "Remark 5.15 It is easily shown that none of the indecomposable modules from Theorem REF is isomorphic to any of the indecomposable modules from the previous cases.", "To prove this, we use the same arguments as in Remark REF .", "We will now parametrize the non-isomorphic indecomposable modules from Theorem REF .", "Let $\\beta \\in \\mathbb {C}\\setminus {\\lbrace 0,1,-1\\rbrace }$ .", "Keeping the notation from the theorem, choose the parameters $b_i$ in the following way: $b_1+b_2=-(b_5+b_6)=1$ , $b_3+b_4=-(b_7+b_8)=\\beta $ .", "Then, $t\\nmid b_i+b_{i+1}$ and $t\\nmid b_i+b_{i+1}+b_{i+2}+b_{i+3}$ , for odd $i$ .", "Denote the indecomposable module that corresponds to these coefficients by $\\operatorname{\\mathbb {M}}\\nolimits _{\\beta }$ .", "Corollary 5.16 There are infinitely many non-isomorphic rank 2 indecomposable modules in ${\\rm CM}(B_{4,8})$ with profile $1357\\mid 2468$ .", "Let $\\alpha \\in \\mathbb {C}\\setminus {\\lbrace 0,1,-1\\rbrace }$ , with $\\beta \\ne \\pm \\alpha $ , and $\\operatorname{\\mathbb {M}}\\nolimits _{\\alpha }$ be the corresponding indecomposable module.", "Then $\\operatorname{\\mathbb {M}}\\nolimits _{\\alpha }$ and $\\operatorname{\\mathbb {M}}\\nolimits _{\\beta }$ are not isomorphic.", "Indeed, assuming that the coefficients $c_i$ correspond to $\\operatorname{\\mathbb {M}}\\nolimits _{\\alpha }$ , we have that $t\\nmid (b_1+ b_2) (c_3+ c_4) (b_5+ b_6) (c_7+ c_8) -(c_1+ c_2) (b_3+ b_4) (c_5+ c_6) (b_7+ b_8)=\\alpha ^{2}-\\beta ^{2},$ since $\\alpha \\ne \\pm \\beta $ , so by the previous theorem the corresponding modules are not isomorphic.", "From the proof of the previous corollary it follows that the two modules $\\operatorname{\\mathbb {M}}\\nolimits _{\\alpha }$ and $\\operatorname{\\mathbb {M}}\\nolimits _{\\beta }$ are isomorphic if and only if $\\alpha =\\pm \\beta $ .", "Thus, the non-isomorphic indecomposable modules of this form are parameterized by $\\mathbb {C}\\setminus {\\lbrace 0,1,-1\\rbrace }$ , where two points in this set are identified if they sum up to 0.", "In the next proposition we show that every indecomposable module as in Theorem REF is isomorphic to $\\operatorname{\\mathbb {M}}\\nolimits _{\\beta }$ for some $\\beta $ .", "Proposition 5.17 Let $\\operatorname{\\mathbb {M}}\\nolimits $ be a rank 2 indecomposable module with the corresponding coefficients $c_i$ satisfying $t\\nmid c_i+c_{i+1}$ , $t\\nmid c_i+c_{i+1}+c_{i+2}+c_{i+3}$ , for odd $i$ .", "Then there exists $\\beta \\in \\mathbb {C}\\setminus {\\lbrace 0,1,-1\\rbrace }$ such that $\\operatorname{\\mathbb {M}}\\nolimits $ is isomorphic to $\\operatorname{\\mathbb {M}}\\nolimits _{\\beta }$ .", "Let $c_i+c_{i+1}=C_i$ , for $i=1,3,5,7.$ Since the coefficients of $\\operatorname{\\mathbb {M}}\\nolimits _{\\beta }$ satisfy $b_1+b_2=-(b_5+b_6)=1$ , $b_3+b_4=-(b_7+b_8)=\\beta $ , it follows from Theorem REF that we need to find $\\beta $ satisfying $t\\mid \\beta ^2C_1C_5-C_3C_7$ .", "If $\\gamma _i$ is the constant term of $C_i$ , then we choose $\\beta $ to be a square root of $(\\gamma _3\\gamma _7)(\\gamma _1\\gamma _5)^{-1}$ .", "Note that by the divisibility conditions (div), $\\gamma _1\\gamma _5\\ne 0$ and $\\gamma _3\\gamma _7\\ne 0$ .", "In particular, $\\beta \\ne 0$ .", "If $\\beta =\\pm 1$ , then $\\gamma _1\\gamma _5=\\gamma _3\\gamma _7$ .", "From $C_1+C_3+C_5+C_7=0$ , we get after multiplying by $C_5$ that $(\\gamma _3+\\gamma _5)(\\gamma _5+\\gamma _7)=0$ which is not possible.", "Hence, $\\beta \\ne \\pm 1.$" ], [ "The general case: $r$ -interlacing, {{formula:9202fdb8-7aa8-4b92-b734-703a9d975e92}}", "In this section we generalize the results from the previous two sections.", "We deal with the case of $r$ -interlacing, where $r\\ge 4$ and at least four boxes (i.e.", "$r$ -interlacing rims where $r\\ge r_1>3$ ).", "We prove that if $I$ and $J$ are $r$ -interlacing with poset $1^{r_1}\\mid 2$ , where $r\\ge 4$ and $r\\ge r_1> 3$ , then there exist non-isomorphic indecomposable modules with the given filtration $L_I\\mid L_J$ .", "It follows that in this case, as in Section , the profile of a module does not uniquely determine the module.", "Let $I$ and $J$ be $r$ -interlacing and form $r_1$ boxes.", "In general, $I\\cap J$ and $I^c\\cap J^c$ are non-empty.", "We have to modify our definition of branching points $i_m$ and associated points $j_m$ for the general setting.", "Definition 6.1 Let $I$ and $J$ be two $k$ -subsets such that their lattice diagram forms $r_1$ boxes.", "The branching points of the lattice diagram $I \\mid J$ are defined to be the points where the boxes end, i.e.", "$i\\in I\\setminus J$ is a branching point if $i+1\\notin I$ and the two rims meet at $i$ .", "We denote them by $\\lbrace i_1,i_2,\\dots , i_{r_1}\\rbrace $ .", "In addition, we define the points $\\lbrace j_1,j_2,\\dots ,j_{r_1}\\rbrace $ at the beginning of the boxes as the set of $j\\in J\\setminus I$ such that $j-1\\notin J$ and such that $j_m$ is minimal in $\\lbrace i_m+1,\\dots , i_{m+1}\\rbrace $ (cyclically) with this property.", "The size of the box ending at $i_m$ is defined to be the number of elements of $I\\setminus J$ for that box.", "Let $r\\ge 4$ .", "If $k$ is arbitrary and $I$ , $J$ are such that $I$ and $J$ are $r$ -interlacing and $I\\mid J$ has poset of the form $1^{r_1}\\mid 2$ , where $r\\ge r_1>3$ (see Figure REF ), then we are able to construct more than one non-isomorphic indecomposable rank 2 module which has $L_J$ as a submodule and $L_I$ as the quotient as follows.", "Denote this module by $\\operatorname{\\mathbb {M}}\\nolimits (I,J)$ .", "We mimic the same procedure as for the module $1357\\mid 2468$ .", "Figure: The profile of a module with 5-interlacing layers and with poset 1 4 ∣21^4\\mid 2, for (k,n)=(6,12)(k,n)=(6,12).The two rims form $r_1$ boxes.", "Denote the $r_1$ branching points from $I$ where the boxes of the two rims end by $i_1,\\dots , i_{r_1}$ and their counterparts in $J$ by $j_1,\\dots , j_{r_1}$ , as in Definition REF .", "For these branching points, we set $x_{i_l}=\\begin{pmatrix} t& b_{i_l} \\\\ 0 & 1 \\end{pmatrix}$ and $x_{j_l}=\\begin{pmatrix} 1& b_{j_l} \\\\ 0 & t \\end{pmatrix}$ for $l=1,\\dots , r_1$ .", "For all other vertices $i$ we define $x_i$ (resp.", "$y_i$ ) to be diagonal matrices as follows: the diagonal of $x_i$ (resp.", "$y_i$ ) is $(1,t)$ (resp.", "$(t,1)$ ) if $i\\in J\\setminus I$ , it is $(t,1)$ (resp.", "$(1,t)$ ) if $i\\in I\\setminus J$ , $(t,t)$ (resp.", "$(1,1)$ ) if $i\\in I^c\\cap J^c$ , and $(1,1)$ (resp.", "$(t,t)$ ) if $i\\in I\\cap J$ .", "We also assume that $\\sum _1^{r_1}(b_{i_l}+b_{j_l})=0$ so that we have a module structure.", "Now we assume that the following divisibility conditions hold for the $b_i$ 's at the first three branching points, as in Theorem REF (recalling from its proof that $t\\nmid b_{1}+b_{2}+b_{3}+b_{4}$ ) for the module $1357\\mid 2468$ : $t\\nmid b_{i_1}+b_{j_1}$ , $t\\nmid b_{i_2}+b_{j_2}$ , $t\\nmid b_{i_3}+b_{j_3}$ , with $t\\nmid b_{i_1}+b_{j_1}+b_{i_2}+b_{j_2}$ , and $t\\mid b_{i_l}+b_{j_l}$ for $l\\ge 4$ .", "As in the previous sections, it is now easy to prove that this module is indecomposable by invoking the same divisibility arguments as before.", "Obviously, we could start at any branching point in order to obtain additional $r_1-1$ indecomposable modules.", "As in Proposition REF , one can easily prove that no two of these $r_1$ indecomposable rank 2 modules are isomorphic.", "Therefore, we have the following proposition.", "Proposition 6.2 If $I$ and $J$ are $r$ -interlacing and $I\\mid J$ has the poset $1^{r_1}\\mid 2$ , where $r\\ge r_1> 3$ , then there are more than one indecomposable rank 2 modules with the profile $I\\mid J$ .", "Furthermore, it is easy to adapt the proof of Theorem REF and Corollary REF to the general case when $r\\ge 4$ in order to obtain the following theorem (we omit the proof).", "Theorem 6.3 Let $I$ and $J$ be $r$ -interlacing with poset $1^{r_1}\\mid 2$ , where $r\\ge r_1> 3$ .", "There are infinitely many non-isomorphic rank 2 indecomposable modules in ${\\rm CM}(B_{k,n})$ with profile $I \\mid J$ .", "For given $r$ -interlacing $k$ -subsets $I$ and $J$ with poset $1^{r_1}\\mid 2$ , where $r\\ge r_1> 3$ , we note that as the number $r_1$ increases the parameterization of non-isomorphic indecomposable rank 2 modules with filtration $L_I\\mid L_J$ becomes more complicated.", "In the case $r=r_1=4$ , we have seen that the family of non-isomorphic indecomposable modules with filtration $L_I\\mid L_J$ is parameterized by the set $\\mathbb {C}\\setminus {\\lbrace 0,1,-1\\rbrace }$ up to sign (if $\\alpha =-\\beta $ , then $\\operatorname{\\mathbb {M}}\\nolimits _{\\alpha }\\cong \\operatorname{\\mathbb {M}}\\nolimits _{\\beta }$ ).", "Here, we do not pursue the classification of these non-isomorphic indecomposable modules, but it would be nice to have this sort of classification for general $r\\ge r_1$ ." ] ]	2011.14176
[ [ "Rules of the Road: Safety and Liveness Guarantees for Autonomous\n Vehicles" ], [ "Abstract The ability to guarantee safety and progress for all vehicles is vital to the success of the autonomous vehicle industry.", "We present a framework for designing autonomous vehicle behavior in a way that is safe and guarantees progress for all agents.", "In this paper, we first introduce a new game paradigm which we term the quasi-simultaneous game.", "We then define an agent protocol that all agents must use to make decisions in this quasi-simultaneous game setting.", "According to the protocol, agents first select an intended action using a behavioral profile.", "Then, the protocol defines whether an agent has precedence to take its intended action or must take a sub-optimal action.", "The protocol ensures safety under all traffic conditions and liveness for all agents under `sparse' traffic conditions.", "We provide proofs of correctness of the protocol and validate our results in simulation." ], [ "Introduction", "A prerequisite for introducing autonomous vehicles into our society is a compelling proof of their safety and efficacy.", "Unfortunately, designing agent strategies in interactive multi-agent settings is extremely difficult since agent behavior is highly coupled and the computational complexity grows exponentially when reasoning about joint action spaces.", "Most approaches for designing agent behavior focus on designing an individual agent's strategy while modeling interactions with other agents using some interactive behavioral model.", "Minimum violation motion-planning has been proposed to help the vehicle choose the trajectory that minimizes violation of a set of ordered rules [26], [30].", "Rulebooks are a way to set priorities among possibly conflicting sets of specifications [5].", "The game-theoretic approach has been to model agent decision-making as interacting partially-observable Markov Decision Processes (POMDPs) [3], [10].", "These methods often capture the reactivity of agents by modeling a reward function defined on a joint action space but suffers from the curse of dimensionality.", "Data-driven methods are used to learn interactive models between agents and design an optimal strategy for an individual agent based on this learned model [21], [20].", "When designing an individual agent strategy, how other agents are assumed to be behaving is not explicitly defined—thereby preventing the ability to make complete safety guarantees.", "Instead of reasoning about safety on the individual agent level, the authors in [24] introduce the idea of reasoning about safety as a property of the collective of agents.", "In particular, they introduce the idea of social laws, which are a set of rules imposed upon all agents in a multiagent system to ensure some desirable global behaviors like safety or progress [24], [27].", "The design of social laws is intended to achieve the desirable global behavioral properties in a minimally-restrictive way [24].", "The problem of automatically synthesizing useful social laws for a set of agents for a general state space, however, has been shown to be NP-complete [24].", "Model checking tools have also been designed to verify correctness of agent protocols for multi-agent systems, but these do not solve the protocol synthesis problem [14], [27].", "The Responsibility-Sensitive-Safety (RSS) framework [23] adopts a similar top-down philosophy for guaranteeing safety by providing a set of rules like maintaining distance, yielding, etc, but does not provide guarantees of agent progress.", "Similarly, the Assume-Guarantee framework for autonomous vehicles introduced in [18] dictates all agents must abide by some behavioral contract where agents make decisions according to a behavioral profile.", "With all agents operating according to the behavioral profile, the interactions are not necessarily coordinated.", "In particular, there might be multiple agents with conflicting goals.", "The process for resolving multiple conflicting processes in a local, decentralized manner is addressed in the Drinking Philosopher problem, which provides a mechanism for resolving conflicts by defining a local, decentralized algorithm for assigning precedence among agents [6].", "We introduce an agent protocol that is an adaptation of the Drinking Philosopher problem.", "The agent protocol is defined so agents use a behavioral profile to select an intended action.", "Additional constraints specified in the profile, determine when an agent has precedence in taking its intended action.", "Unlike [22], our framework leverages the structure of the driving road network and takes into account the inertial properties of agents.", "The main contributions of this paper are as follows: 1) The introduction of a new game paradigm, which we term the quasi-simultaneous discrete-time multi-agent game, 2) the definition of an agent protocol that defines local rules agents must use to select their actions, 3) safety and liveness proofs when all agents operate according to these local rules and 4) simulations as proof of concept of the safety and liveness guarantees." ], [ "Quasi-Simultaneous Discrete-Time Game", "We propose a quasi-simultaneous discrete-time game paradigm, which we motivate by looking at the shortcomings of more traditional game paradigms.", "In synchronous games, all agents in the game are making decisions simultaneously.", "Since agents are making decisions in the absence of other agent behaviors, it does not capture the sequential nature of real-life decision making.", "Turn-based games offer potential for capturing sequential decision-making, but the turns are often assigned arbitrarily.", "The quasi-simultaneous discrete-time game offers a way to assign turns, but in a turn order based on the agent states defined with respect to the road network.", "A state associated with a set of variables is an assignment of values to those variables.", "A game evolves by a sequence of state changes.", "A quasi-simultaneous game has the following two properties regarding state changes: 1) each agent will get to take a turn in each time-step of the game and 2) each agent must make their turn in an order that emerges from a locally-defined precedence assignment algorithm.", "We define a quasi-simultaneous game where all agents act in a local, decentralized manner as follows $\\mathfrak {G} = \\langle \\mathfrak {A}, \\mathcal {Y}, Act_{[\\cdot ]}, P \\rangle $ , where $\\mathfrak {A}$ is the set of all agents in the game, $\\mathcal {Y}$ is the set of all variables in the game, $Act_{\\text{Ag}}$ be the set of all possible actions $\\text{Ag}$ can take.", "Finally, $P: \\mathcal {Y} \\rightarrow \\text{PolyForest}(\\mathfrak {A})$ , is the precedence assignment function where $\\text{PolyForest}$ is an operator that maps a set to a polyforest graph object.", "The polyforest, with its nodes and directed edges, defines the global turn order (of precedence) of the set of all agents based on the agent states." ], [ "Specific Agent Class", "In order to make global guarantees on safety and progress, we first only consider a single specific class of agents whose attributes, dynamics, motion-planner, and perception capabilities are described in more detail in the following section.", "Although assuming a single class of agents seems very restrictive, the work can be easily extended to accommodate additional variants of the agent class.", "These extensions, however, are beyond the scope of this work." ], [ "Agent Attributes", "Each agent $\\text{Ag}$ is characterized by a set of variables $\\mathcal {V}_{\\text{Ag}} \\subseteq \\mathcal {Y}$ .", "We define $\\lbrace \\texttt {Id}_{\\text{Ag}}, \\texttt {Tc}_{\\text{Ag}}, \\texttt {Goal}_{\\text{Ag}} \\rbrace ~\\subseteq ~{\\mathcal {V}}_{\\text{Ag}}$ where $\\texttt {Id}_{\\text{Ag}}$ , $\\texttt {Tc}_{\\text{Ag}}$ , and $\\texttt {Goal}_{\\text{Ag}}$ are the agent's ID number, token count and goal respectively.", "The token count and ID are defined in greater depth in Section REF .", "Agents are assumed to have the capability of querying the token counts of neighboring agents.", "In this paper, we only consider car agents such that if $\\text{Ag} \\in \\mathfrak {A}$ , then $\\mathcal {V}_{\\text{Ag}}$ includes $x_{\\text{Ag}}$ , $y_{\\text{Ag}}$ , $\\theta _{\\text{Ag}}$ , $v_{\\text{Ag}}$ , namely its absolute coordinates, heading and velocity.", "We let $S_{\\text{Ag}}$ denote the set that contains all possible states of these variables in $\\mathcal {V}_{\\text{Ag}}$ .", "$\\mathcal {V}_{\\text{Ag}}$ also has parameters: $ {a_{\\text{min}}}_{\\text{Ag}} \\in \\mathbb {Z}, {a_{\\text{max}}}_{\\text{Ag}}\\in \\mathbb {Z}, {v_{\\text{min}}}_{\\text{Ag}}\\in \\mathbb {Z} \\text{ and } {v_{\\text{max}}}_{\\text{Ag}}\\in \\mathbb {Z}$ which define the minimum and maximum accelerations and velocities respectively.", "The agent control actions are defined by two parameters: 1) an acceleration value $\\text{acc}_{\\text{Ag}}$ between ${a_{\\text{min}}}_{\\text{Ag}}$ and ${a_{\\text{max}}}_{\\text{Ag}}$ and 2) a steer maneuver $\\gamma _{\\text{Ag}} \\in $ {left-turn, right-turn, left-lane change, right-lane change, straight}.", "The discrete agent dynamics works as follows.", "At a given state $s \\in S_{\\text{Ag}}$ at time $t$ , for a given control action $(\\text{acc}_{Ag}, \\gamma _{Ag})$ , the agent first applies the acceleration to update its velocity $s.v_{Ag,t+1} = s.v_{Ag,t} + \\text{acc}_{\\text{Ag}}$ .", "Once the velocity is applied, the steer maneuver (if at the proper velocity) is taken and the agent occupies a set of grid-points, specified in Fig.", "REF , while taking its maneuver.", "The agent state-transition function $\\tau _{\\text{Ag}}: S_{\\text{Ag}} \\times Act_{\\text{Ag}} \\rightarrow S_{\\text{Ag}}$ defines the state an agent will transition to by taking an action $a$ at a given state $s_{\\text{Ag}}$ and the state precondition $\\rho _{\\text{Ag}}:S_{\\text{Ag}} \\rightarrow 2^{Act_{\\text{Ag}}}$ functions defines the set of allowable actions at a given state.", "Figure: Shows different grid point occupancy associated with different discrete agent maneuvers.", "Note the grid point occupancy represents a conservative space in which the agent may occupy when taking the associated maneuver.During an agent state transition, an agent may, depending on the maneuver, occupy a set of grid points.", "Before and after the state transition, the agent is assumed to occupy only a single grid point.", "Fig.", "REF shows the grid point occupancy for different agent maneuvers.", "The concept of grid point occupancy is defined as follows: Definition 3.1 (Grid Point Occupancy) The notion of grid point occupancy is captured by the definitions of the following maps for each $\\text{Ag} \\in \\mathfrak {A}$ .", "To define the grid point an agent is occupying at a given time we use the map: $\\mathcal {G}_{\\text{Ag},t}: S_{\\text{Ag}} \\rightarrow 2^{G}$ , mapping each agent to the single grid point the agent occupies.", "By a slight abuse of notation, we let $\\mathcal {G}_{Ag,t}: S_{\\text{Ag}} \\times {Act}_{\\text{Ag}} \\rightarrow 2^{G}$ be a function that maps each $s \\in S_{\\text{Ag}}$ and $a \\in \\rho _{\\text{Ag}}(s)$ to denote the set of all grid points that are occupied by the agent $\\text{Ag}$ when it takes an allowable action $a$ from state $s$ at the time-step $t$ .", "Here we assume that any graph-based planning algorithm can be used to specify an agent's motion plan, where the motion plan is a set of critical points along the graph that the agent must reach in order to get to its destination." ], [ "Agent Backup Plan Action", "A backup plan is a reserved set of actions an agent is entitled to execute at any time while being immune to being at fault for a collision if one occurs.", "In other words, an agent will always be able to safely take its backup plan action.", "We show if each agent can maintain the ability to safely execute its own backup plan (i.e.", "keep a far enough distance behind a lead agent), the safety of the collective system safety is guaranteed.", "The default backup plan, which we refer to as $a_{\\text{bp}}$ adopted here is that of applying maximal deceleration until a complete stop is achieved.", "Note, it may take multiple time-steps for an agent to come to a complete stop because of the inertial dynamics of the agent." ], [ "Limits on Agent Perception", "In real-life, agents make decisions based on local information.", "We model this locality by defining a region of grid points around which agents have access to the full state and intentions of the other agents.", "We assume agents have different perception capabilities in different contexts of the road network.", "For road segments, the region around which agents make decisions cannot be arbitrarily defined.", "In fact, an agent's bubble must depend on its state, and the agent attributes and dynamics of all agents in the game.", "In particular, the bubble can be defined as follows: Definition 3.2 (Bubble) Let $\\text{Ag}$ with state $s_0 \\in S_{\\text{Ag}}$ .", "Then the bubble of $\\text{Ag}$ with respect to agents of the same type is written as $\\mathcal {B}_{\\text{Ag}}(s_0)$ .", "The bubble is the minimal region of space (set of grid points) agents need to have full information over to guarantee they can make a decision that will preserve safety under the defined protocol.", "The details for the construction of the bubble for an agent with a particular set of attributes and dynamics can be found in the Appendix.", "At intersections, agents are assumed to be able to see across the intersection when making decisions about crossing the intersection.", "More precisely, any $\\text{Ag}$ must be able to know about any $\\text{Ag}^{\\prime } \\in \\mathfrak {A}$ that is in the lanes of oncoming traffic.", "The computation of the exact region of perception necessary depends on the agent dynamics." ], [ "Road Network Environment", "Here we introduce the structure of the road network environment that agents are assumed to be operating on.", "The road network is a grid world with additional structure (e.g.", "lanes, bundles, road segments, intersections, etc.).", "The road network is formalized as follows: Definition 4.1 (Road Network) A road network $\\mathfrak {R}$ is a graph $\\mathfrak {R} = (G, E)$ where $G$ is the set of grid points and $E$ is the set of edges that represent immediate adjacency in the Cartesian space among grid points.", "Note that each grid point $g\\in G$ has a set of associated properties $\\mathcal {P}$ , where $\\mathcal {P} = \\lbrace p, d, \\texttt {lo} \\rbrace $ which denote the Cartesian coordinate, drivability of the grid point and the set of legal orientations allowed on the grid point respectively.", "Note, $p\\in \\mathbb {Z}^2$ , $d \\in \\lbrace 0, 1\\rbrace $ and $\\texttt {lo} \\in \\lbrace \\texttt {north}, \\texttt {east}, \\texttt {south}, \\texttt {west}\\rbrace $ .", "$\\mathcal {S}_{\\text{sources}}$ ($\\mathcal {S}_{\\text{sinks}}$ ) are the set of grid points agents can enter or leave the road network from.", "Each intersection of the road network is governed by traffic lights.", "The road network is hierarchically decomposed into lanes, bundles and road segments, where a lane $La(g)$ defines a set of grid points that contains $g$ and all grid points that form a line going through $g$ and a bundle $Bu(g)$ is a set of grid points that make up a set of lanes that are adjacent or equal to the lane containing $g$ and have the same legal orientation.", "Each bundle can be decomposed into a set of road segments $RS$ , where the intersections are used to partition each bundle into a set of road segments.", "These road components can be seen in Fig.REF .", "We introduce the following graph definition since it will be used in the liveness proof.", "Definition 4.2 (Road Network Dependency Graph) The road network dependency graph is a graph $G_{\\text{dep}} = (RS, E)$ where nodes are road segments and a directed edge $(rs_1, rs_2)$ denotes that agents on $rs_1$ depends on the clearance of agents in $rs_2$ to make forward progress." ], [ "the Agent Protocol", "The protocol is the set of rules agents use to select which action to ultimately take at a given time step.", "According to the protocol, agents first select an intended action using a profile.", "The protocol then defines additional rules that an agent uses to determine whether it has priority to take its intended action, and if not, which alternative, less-optimal actions it is allowed to take.", "The protocol is defined in a way that 1) scales well in the number of agents 2) is interpretable so there is a consistent and transparent way agents make their decisions 3) ensures safety and progress of all agents.", "In this section, we introduce the components that form the agent protocol that make it such that all these properties are satisfied." ], [ "Agent Precedence Assignment", "The definition of the quasi-simultaneous game requires agents to locally assign precedence, i.e.", "have a set of rules to define how to establish which agents have higher, lower, equal or incomparable precedence to it.", "Thus, the first element of the agent protocol is defining the agents' local precedence assignment algorithm so each agent knows its turn order relative to neighboring agents.", "Our precedence assignment algorithm is motivated by capturing how precedence among agents is generally established in real-life scenarios on a road network.", "In particular, since agents are designed to move in the forward direction, we aim to capture the natural inclination of agents to react to the actions of agents visibly ahead of it.", "Before presenting the precedence assignment rules, we must introduce a few definitions.", "Let us define: $\\text{proj}^{B}_{\\text{long}}: \\mathfrak {A} \\rightarrow \\mathbb {Z}$ , which is restricted to only be defined on the bundle $B$ .", "In other words, $\\text{proj}^{B}_{\\text{long}}(Ag)$ is the mapping from an agent (and its state) to its scalar projection onto the longitudinal axis of the bundle $B$ the agent $\\text{Ag}$ is in.", "If $\\text{proj}^B_{\\text{long}}(\\text{Ag}^{\\prime }) < \\text{proj}^B_{\\text{long}}(\\text{Ag})$ , then the agent $\\text{Ag}^{\\prime }$ is behind $\\text{Ag}$ in $B$ .", "The following rules can be used to define the precedence relation among agents $Ag$ and $Ag^{\\prime }$ .", "If $\\text{proj}^B_{\\text{long}}(\\text{Ag}^{\\prime }) < \\text{proj}^B_{\\text{long}}(\\text{Ag})$ and $Bu(\\text{Ag}^{\\prime })= Bu(\\text{Ag})$ , then $\\text{Ag}^{\\prime } \\prec \\text{Ag}$ , i.e.", "if agents are in the same bundle and $\\text{Ag}$ is longitudinally ahead of $\\text{Ag}^{\\prime }$ , $\\text{Ag}$ has higher precedence than $\\text{Ag}^{\\prime }$ .", "If $\\text{proj}^B_{\\text{long}}(\\text{Ag}^{\\prime }) = \\text{proj}^B_{\\text{long}}(\\text{Ag})$ and $Bu(\\text{Ag}^{\\prime })= Bu(\\text{Ag})$ , then $\\text{Ag} \\sim Ag^{\\prime }$ and we say $\\text{Ag}$ and $\\text{Ag}^{\\prime }$ are equivalent in precedence.", "If $\\text{Ag}^{\\prime }$ and $\\text{Ag}$ are not in the same bundle, then the two agents are incomparable.", "Each agent $\\text{Ag}\\in \\mathfrak {A}$ only assigns precedence according to the above rules locally to agents within its local region.", "Thus, we must show if all agents locally assign precedence according to these rules, a globally-consistent turn precedence among all agents is established.", "The linear ordering induced by these local rules are used to prove this.", "The reader is referred to the Appendix for the full proof.", "Figure: Rules for precedence assignment.Even when this turn-order is established, there is still some ambiguity as to which agents have precedence.", "The ambiguity is resolved through the conflict-cluster resolution, introduced in Section REF ." ], [ "Behavioral Profile", "The way in which agents select actions is the fundamental role of the agent protocol.", "The behavioral profile serves the purpose of defining which action an agent intends to take at a given time-step $t$ .", "We define a specific assume-guarantee profile with the mathematical properties defined in [18].", "In particular, we define a set of ten different specifications (rules) and place a hierarchy of importance (ordering) on these rules.", "Figure: Assume-guarantee profile that shows ordering of specifications, where specifications on the same tier are incomparable to one another and Tier 1 has highest priority.Each of the specifications is associated with an oracle that evaluates whether or not an agent taking an action $a$ satisfies the specification.", "The reader is referred to the Appendix for the precise oracle definitions.", "The consistent-evaluating function, defined on this agent profile, will evaluate actions based on which subset of specifications they satisfy–giving priority to actions that satisfy the highest number of highest-valued specifications, as described in [18].", "The action with the highest value is then selected as the action the agent intends to takes.", "For this work, the agent profile defined in Fig.", "REF is used to define both the agent's intended action $a_i$ and best straight action $a_{\\text{st}}$ defined in Definition REF .", "Since an agent would never propose a lane-change action if $O_{\\text{Ag},t,\\text{dynamic safety}}(s,a,u)$ were included in the profile, it is not included in the selection of the intended action $a_i$ , but rather evaluated later downstream in the protocol." ], [ "Conflict-Cluster Resolution", "At every time-step $t$ , each agent will know when to take its turn based on its local precedence assignment algorithm.", "Before taking its turn, the agent will have selected an intended action $a_i$ using the Agent profile.", "When it is the agent's turn to select an action, it must choose whether or not to take it's intended action $a_i$ .", "When the intended actions of multiple agents conflict, the conflict-cluster resolution is a token-based querying method used to help agents determine which agent has priority in taking its action.", "Under the assumption agents have access to the intentions of other agents within a local region as defined in Section REF , agents can use the following criteria to define when it conflicts with another agent.", "Definition 5.1 (Agent-Action Conflict) Let us consider an agent $\\text{Ag}$ is currently at state $s \\in S_{\\text{Ag}}$ and wants to take action $a$ and an agent $\\text{Ag}^{\\prime }$ at state $s^{\\prime } \\in S_{\\text{Ag}^{\\prime }}$ wants to take action $a^{\\prime }$ .", "We write an agent-action conflict exists $(\\text{Ag}, s, a) \\dagger (\\text{Ag}^{\\prime }, s^{\\prime }, a^{\\prime })$ , if each of the agents taking their respective actions will cause them to overlap in occupancy grid points or end up in a configuration where the agent behind does not have a valid safe backup plan action.", "In the case that an agent's action is in conflict with another agents' action, the agent must send a conflict request that ultimately serves as a bid the agent is making to take its intended action.", "It cannot, however, send requests to just any agent (e.g.", "agents in front of it).", "The following criteria are used to determine the properties that must hold in order for an agent $\\text{Ag}$ to send a conflict request to agent $\\text{Ag}^{\\prime }$ : 1) $\\text{Ag}$ 's intended action $a_i$ is a lane-change action, 2) $\\text{Ag}^{\\prime } \\in \\mathcal {B}_{Ag}(s)$ , i.e.", "$\\text{Ag}^{\\prime }$ is in agent $\\text{Ag}$ 's bubble, 3) $\\text{Ag}^{\\prime } \\precsim Ag$ , i.e.", "$\\text{Ag}$ has equivalent or higher precedence than $\\text{Ag}^{\\prime }$ , 4) $Ag$ and $Ag^{\\prime }$ have the same heading, 5) $(Ag, a_i) \\dagger (Ag^{\\prime },a_i^{\\prime })$ : agents intended actions are in conflict with one another, and 6) $\\mathcal {F}_{\\text{Ag}}(u,a_i) = \\texttt {F}$ , where $\\mathcal {F}_{\\text{Ag}}(u,a_i)$ is the max-yielding-not enough flag and is defined below.", "Definition 5.2 (maximum-yielding-not-enough flag) The maximum-yielding-not-enough flag $\\mathcal {F}_{\\text{Ag}}:\\mathcal {U} \\times Act_{\\text{Ag}} \\rightarrow \\mathbb {B}$ is set to $\\texttt {T}$ when $\\text{Ag}$ is in a configuration where if $\\text{Ag}$ did a lane-change, $Ag$ would still violate the safety of $\\text{Ag}^{\\prime }$ 's backup plan action even if $\\text{Ag'}$ applied its own backup plan action.", "We note that if $\\mathcal {F}_{\\text{Ag}}(u, a_i)$ is set, $\\text{Ag}$ cannot send a conflict request by the last condition.", "Even though $\\text{Ag}$ does not send a request, it must use the information that the flag has been set in the agent's Action Selection Strategy defined in Section REF .", "After a complete exchange of conflict requests, each agent will be a part of a cluster of agents that define the set of agents it is ultimately bidding for its priority (to take its intended action) over.", "These clusters of agents are defined as follows: Definition 5.3 (Conflict Cluster) A conflict cluster for an agent $\\text{Ag}$ is defined as $\\mathcal {C}_{Ag} = \\lbrace Ag^{\\prime } \\in \\mathfrak {A} \\mid Ag \\texttt { send } \\text{Ag}^{\\prime } \\text{ or } Ag^{\\prime } \\texttt { send } Ag\\rbrace $ , where $\\text{Ag} \\texttt { send } Ag^{\\prime }$ implies $\\text{Ag}$ has sent a conflict request to $\\text{Ag}^{\\prime }$ .", "An agents' conflict cluster defines the set of agents in its bubble that an agent is in conflict with.", "Fig.", "REF shows an example scenario and each agents' conflict clusters.", "Once the conflict requests have been sent and an agent can thereby identify the other agents in its conflict cluster, it needs to establish whether or not the conflict resolution has resolved in it's favor.", "Figure: An example scenario with agents in a given configuration of agents, their intended actions and their respective conflict clusters.Once an agent has determined which agents are in its conflict cluster, it must determine whether or not it has the priority to take its intended action.", "The token resolution scheme is the way in which agents determine whether they have precedence.", "The token resolution strategy must be designed to be fair, meaning each agent will always eventually wins their conflict resolution.", "The resolution is therefore based on the agents' token counts $\\texttt {Tc}$ , which is updated by agents to represent how many times an agent has been unable to take a forward progress action thus far.", "The token count updates according to the agent's chosen action.", "In particular, if $\\text{Ag}$ selects action $a$ : if $O_{\\text{forward progress}}(s,a,u) = \\texttt {T}$ , the the token count resets to 0, otherwise it increases by 1.", "Then, a fair strategy would be to make it so that the agent with the highest amount of tokens wins in its own conflict cluster.", "Thus, we define a token resolution indicator variable for each $\\text{Ag}$ as $\\mathcal {W}_{\\text{Ag}} \\in \\mathbb {B}$ , indicating whether or not the agent has won in its conflict cluster.", "The conflict cluster resolution indicator variable $\\mathcal {W}_{\\text{Ag}}$ evaluates to $\\texttt {T}$ if $Ag$ has the highest amount of tokens in its conflict cluster, where ties are broken via agent ID comparison." ], [ "Action Selection Strategy", "The Action Selection Strategy is a decision tree that defines whether or not an agent is allowed to take its intended action $a_i$ and if it is not, which alternative action it should take.", "In the case where an agent is not allowed to take $a_i$ , the agent is restricted to take either: the best straight action $a_{st}$ , which is defined in Definition REF , or its backup plan action $a_{bp}$ , where the best straight action is defined as follows: Definition 5.4 (Best Straight Action) Let us consider $\\text{Ag}$ and its associated action set $\\rho _{\\text{Ag}}(s)$ .", "The best straight action is the action $a \\in \\rho _{\\text{Ag}(s)}$ that is the highest-ranked action (according to the profile defined in Section REF ), among the set of all actions for which $\\gamma _{\\text{Ag}} = \\text{straight}$ .", "The decision tree branches are defined based on the following five conditions: 1) $a_i$ , the agent's and other agents' (in its bubble) intended actions 2) $\\text{Ag}$ 's role in conflict request cluster being a) a conflict request sender, b) a conflict request receiver, c) both a sender and a receiver, d) neither sender or receiver, 3) the agent's conflict cluster resolution $\\mathcal {W}_{\\text{Ag}}$ , 4) evaluation of $O_{\\text{Ag}, t, \\text{dynamic safety}}(s,a_i,u)$ and 5) $\\mathcal {F}_{\\text{Ag}}(u, a_i)$ for $\\text{Ag}$ is raised, where $\\mathcal {F}_{\\text{Ag}}(u, a_i)$ is the maximal-yielding-not-enough flag defined in Section REF .", "If an agent receives a conflict cluster request and loses their conflict cluster resolution, according to the action selection strategy, the agent must take its backup plan action $a_{\\text{bp}}$ .", "An agent is only allowed to take a lane-change action when the agent is a winner of its conflict cluster resolution, $\\mathcal {F}_{\\text{Ag}} = \\texttt {F}$ and the dynamic safety oracle evaluates to true (i.e.", "$O_{\\text{Ag}, t, \\text{dynamic safety}}(s,a_i,u) = \\texttt {T}$ ).", "Finally, an agent that loses in its conflict cluster but did not send requests must take $a_{\\text{st}}$ .", "A figure showing the full decision-tree logic for selecting actions can be found in the Appendix.", "The agent protocol, as described in the above sections, has been designed in a way such that if all agents are selecting actions via the protocol, we can provide formal guarantees on safety and liveness.", "Theses safety and liveness proofs are given in the following sections." ], [ "Formal Guarantees", "Before introducing the formal guarantees of safety and liveness and their respective proofs, we first make explicit the assumptions that must hold on agents and the road network.", "Each $\\text{Ag} \\in \\mathfrak {A}$ has access to the traffic light states.", "There is no communication error in the conflict requests, token count queries and the agent intention signals.", "All intersections in the road network $R$ are governed by traffic lights.", "The traffic lights are designed to coordinate traffic such that if agents respect the traffic light rules, they will not collide.", "Agents follow the agent dynamics defined in Section REF .", "For $t=0$ , $\\forall \\text{Ag} \\in \\mathfrak {A}$ in the quasi-simultaneous game is initialized to be located on a distinct grid point on the road network and have a safe backup plan action $a_{bp}$ such that $S_{\\text{Ag}, bp}(s, u) = \\texttt {T}$ .", "The traffic lights are red a window of time $\\Delta t_{\\text{tl}}$ such that $t_{\\text{min}}<\\Delta t_{\\text{tl}}<\\infty $ , where $t_{\\text{min}}$ is defined so agents are slowed down long enough so agents that have been waiting can take a lane-change action.", "More details can be found in the Appendix.", "The static obstacles are not on any grid point $g$ where $g.d = 1$ .", "Each $\\text{Ag}$ treats its respective goal $\\text{Ag}.\\texttt {g}$ as a static obstacle.", "Bundles in the road network $\\mathfrak {R}$ have no more than 2 lanes.", "All intersections in the road network $\\mathfrak {R}$ are governed by traffic lights." ], [ "Safety Guarantee", "Safety is guaranteed when agents do not collide with one another.", "An agent causes collision when it takes an action that satisfies the following condition.", "Definition 6.1 (Collision) An agent $\\text{Ag}$ that takes an action $a \\in Act_{\\text{Ag}}$ will cause collision if the grid point occupancy of $Ag$ ever overlaps with the grid point occupancy of another agent $Ag^{\\prime }$ or a static obstacle $O_{\\text{st}}$ .", "A strategy where agents simply take actions that avoid collision in the current time-step is insufficient for guaranteeing safety because of the inertial properties of the agent dynamics.", "The agent protocol has therefore been defined so an agent also avoids violating the safety of its own and any other agent's backup plan action $a_{bp}$ defined in Section REF .", "An agent's backup plan action $a_{bp}$ is evaluated to be safe when the following conditions hold: Definition 6.2 [Safety of a Backup Plan Action] Let us define the safety of an agent's backup plan action $S_{\\text{Ag}, bp}:\\mathcal {U} = \\mathbb {B}$ , where $\\mathbb {B} = \\lbrace \\texttt {T}, \\texttt {F} \\rbrace $ is an indicator variable that determines whether an agent's backup plan action is safe or not.", "It is defined as: $S_{Ag, bp}(u) = \\wedge _{o \\in O} o(s, a_{bp}, u)$ where the set $O$ is the set of all oracles in the top three tiers of the agent profile defined in Section REF .", "An agent $\\text{Ag}$ takes an action $a \\in Act_{\\text{Ag}}$ that violates the safety backup plan action of another agent $\\text{Ag}^{\\prime }$ when the following conditions hold: Definition 6.3 (Safety Backup Plan Violation Action) Let us consider an agent $\\text{Ag}$ that is taking an action $a \\in Act_{\\text{Ag}}$ , and another agent $\\text{Ag}^{\\prime }$ .", "The action $(\\text{Ag},a) \\bot \\text{Ag}^{\\prime }$ , i.e.", "agent $\\text{Ag}$ violates the safety backup plan of an agent $\\text{Ag}^{\\prime }$ when by taking an action $a$ , then $S_{\\text{Ag}^{\\prime }, bp}(u^{\\prime }) = \\texttt {F}$ , where $u^{\\prime }$ is the state of the game after $\\text{Ag}$ has taken its action.", "In other words, by taking the action, the agent has ended in a state such that it violates the safety of its own or another agents' backup plan action.", "The safety proof is based on the premise that all agents only take actions that do not collide with other agents and maintain the invariance of the safety of their own and other agents' safety backup plan actions.", "The safety theorem statement and the proof sketch are as follows.", "We can treat the quasi-simultaneous game as a program, where each of the agents are separate concurrent processes.", "A safety property for a program has the form $P \\Rightarrow \\square Q$ , where $P$ and $Q$ are immediate assertions.", "This means if the program starts with $P$ true, then $Q$ is always true throughout its execution [15].", "Theorem 6.1 (Safety Guarantee) Given all agents $\\text{Ag} \\in \\mathfrak {A}$ in the quasi-simultaneous game select actions in accordance to the Agent Protocol specified in Section , then we can show the safety property $P \\Rightarrow \\square Q$ , where the assertion $P$ is an assertion that the state of the game is such that $\\forall Ag, S_{\\text{Ag}, bp}(s,u) = \\texttt {T}$ , i.e.", "each agent has a backup plan action that is safe, as defined in Section REF .", "We denote $P_t$ as the assertion over the state of the game at the beginning of the time-step $t$ , before agents take their respective actions.", "$Q_t$ is the assertion that the agents never occupy the same grid point when taking their respective action at time step $t$ .", "The following is a proof sketch.", "To prove an assertion of this form, we need to find an invariant assertion $I$ for which i) $P \\Rightarrow I$ ii) $I \\Rightarrow \\square I$ and iii) $I \\Rightarrow Q$ hold.", "We define $I$ to be the assertion that holds on the actions that agents select to take at a time-step.", "We denote $I_t$ to be the assertion on the actions agents take at time $t$ such that $\\forall Ag$ , $\\text{Ag}$ takes $a\\in Act_{\\text{Ag}}$ where 1) it does not collide with other agents and 2) it does not violate the safety of other agents' back up plan actions (i.e.", "$\\forall \\text{Ag}, S_{\\text{Ag}, bp}(u^{\\prime }) = \\texttt {T}$ where $s^{\\prime } = \\tau _{\\text{Ag}}(s, a)$ , and $u^{\\prime }$ is the corresponding global state of the game after each $\\text{Ag}$ has taken its respective action $a$ ).", "We can prove $P \\Rightarrow \\square Q$ by showing the following: $P_t \\Rightarrow I_t$ .", "This is equivalent to showing that if all agents are in a state where $P$ is satisfied at time $t$ , then all agents will take actions at time $t$ where the $I$ holds.", "This can be proven by showing agents will take actions that satisfy the conditions of $I$ as long as they are begin a state where all agents have a safe backup action and they select actions according to the protocol.", "$I \\Rightarrow \\square I$ .", "If agents take actions such that at time $t$ such that the assertion $I_t$ holds, then by the definition of the assertion $I$ , agents will end up in a state where at time t+1, assertion $P$ holds, meaning $I_t \\Rightarrow P_{t+1}$ .", "Since $P_{t+1} \\Rightarrow I_{t+1}$ from REF , we get $I \\Rightarrow \\square I$ .", "$I \\Rightarrow Q$ .", "If all agents take actions according to the assertions in $I$ , then collisions will not occur.", "This follows from the definition of $I$ .", "The reader is referred to the Appendix for a full proof.", "Proof of safety alone is not sufficient reason to argue for the effectiveness of the protocol, as all agents could simply stop for all time and safety would be guaranteed.", "A liveness guarantee, i.e.", "proof that all agents will eventually make it to their final destination, is critical.", "In the following section, we present liveness guarantees." ], [ "Liveness Guarantees", "A liveness property asserts that program execution eventually reaches some desirable state [15].", "In this paper, we describe the eventual desirable state for each agent is to reach their respective final destinations.", "Unfortunately, deadlock occurs when agents indefinitely wait for resources held by other agents [19].", "Since the Manhattan grid road network has loops, agents can enter a configuration in which each agent in the loop is indefinitely waiting for a resource held by another agent.", "When the density of agents in the road network is high enough, deadlocks along these loops will occur.", "We can therefore guarantee liveness only when certain assumptions hold on the density of the road network.", "Definition 6.4 (Sparse Traffic Conditions) Let $M$ denote the number of grid points in the smallest loop (defined by legal orientation) of the road network, not including grid points $g\\in \\mathcal {S}_{\\text{intersections}}$ .", "The sparsity condition must be such that $N < M-1$ , where $N$ is the number of agents in the road network.", "The number of agents has to be such that the smallest loop does not become completely saturated, in which deadlock would occur.", "Note, these sparsity conditions are conservative because it is a bound defined by the worst possible assignment of agents and their destinations.", "Now, we introduce the liveness guarantees under these sparse traffic conditions.", "The proof of liveness is based on the fact that 1) agent profile include progress specifications and 2) conflict precedence is resolved by giving priority to the agent that has waited the longest time (a quantity that is reflected by token counts).", "Theorem 6.2 (Liveness Under Sparse Traffic Conditions) Under the Sparse Traffic Assumption given by Definition REF and given all agents $\\text{Ag} \\in \\mathfrak {A}$ in the quasi-simultaneous game select actions in accordance to the Agent Protocol specified in Section , liveness is guaranteed, i.e.", "all $\\text{Ag} \\in \\mathfrak {A}$ will always eventually reach their respective goals.", "The following is a proof sketch.", "The invariance of a no-deadlock state follows from the sparsity assumption and the invariance of safety (no collision) follows from the safety proof.", "Inductive arguments related to control flow are used to show that all $\\text{Ag}$ will always eventually take $a \\in Act_{\\text{Ag}}$ where $O_{\\text{forward progress}}(s,a,u) = \\texttt {T}$ .", "Let us consider a road segment $r \\in RS$ that contains grid point(s) $g \\in \\mathcal {S}_{\\text{sinks}}$ meaning that the road segment contains grid points with sink nodes.", "Inductive arguments based on the agents' longitudinal distance to destination grid points are used to show every $\\text{Ag} \\in r$ will be able to always eventually take $a \\in Act_{\\text{Ag}}$ for which the forward progress oracle $O_{\\text{forward progress}}(s,a,u) = \\texttt {T}$ .", "Let us consider a road segment $rs \\in RS$ .", "Let us assume $\\forall rs \\in RS, \\exists (rs, rs^{\\prime }) \\in G_{\\text{dep}}$ meaning that the clearance of $rs$ depends on the clearance of all $rs^{\\prime }$ .", "Inductive arguments based on agents' longitudinal distance to the front of the intersection show any $\\text{Ag}$ on $rs$ will always eventually take $a \\in Act_{\\text{Ag}}$ where the forward progress oracle $O_{\\text{forward progress}}(s,a,u) = \\texttt {T}$ .", "For any $\\mathfrak {R}$ where the dependency graph $G_{\\text{dep}}$ (as defined in Definition REF ) is a directed-acyclic-graph (DAG), inductive arguments based on the linear ordering of road segments $rs \\in G_{\\text{dep}}$ , combined with the arguments REF -REF , can be used to prove all $\\text{Ag} \\in \\mathfrak {A}$ will always eventually take $a\\in Act_{\\text{Ag}}$ for which the forward progress oracle $O_{\\text{forward progress}}(s,a,u)=\\texttt {T}$ .", "When the graph $G_{\\text{dep}}$ is cyclic, the Sparsity Assumption REF allows for similar induction arguments in REF to apply.", "By the above inductive arguments and the definition of $O_{\\text{forward progress}}(s,a,u)$ , all $\\text{Ag}$ will always eventually take actions that allow them to make progress towards their respective destinations.", "The reader is referred to the Appendix for a full proof." ], [ "Simulation Results", "In order to streamline discrete-time multi-agent simulations, we have built a traffic game simulation platform called Road Scenario Emulator (RoSE).", "We use RoSE to generate different game scenarios and simulate how agents will all behave if they each follow the agent strategy protocol introduced in this paper.", "Figure: City blocks map environment.We simulate the game with randomized initialization of spawning agents at the source nodes for three different road network environments: 1) the straight road segment, 2) small city blocks grid and 3) large city blocks grid.", "A snapshot of a small city blocks grid simulation is shown in Fig.", "REF .", "The agent attributes in this simulation are as follows: $v_{\\text{min}}=0$ , $v_{\\text{max}}=3$ , $a_{\\text{min}}=-1$ , and $a_{\\text{max}}=1$ .", "For each road network environment, we simulate the game 100 times for $t=250$ time-steps.", "During each time-step, agents will spontaneously spawn with some defined probability $p$ at the source nodes and are randomly assigned a sink node as their destination.", "Agents that make it to their destinations exit the map.", "For all game simulation trials, collision does not occur.", "Although liveness is only guaranteed in sparse traffic conditions, we simulate for a number of agents $N > M-1$ specified in the sparsity condition and agents do not enter a deadlock state.", "In particular, over the 100 trials for each of the maps (straight, small and large city blocks), on average $77\\%$ , $36\\%$ and $43\\%$ made it to their respective destinations on the respective maps by the end of the 250 time-steps." ], [ "Conclusion and Future Works", "In this paper, we have proposed a novel paradigm for designing safety-critical decision-making modules for agents whose behavior is extremely complex and highly-coupled with other agents.", "The main distinction of our proposed architecture from the existing literature, is the shift from thinking of each agents as separate, individual entities, to agents as a collective where all all agents adopt a common local, decentralized protocol.", "The protocol defines the agent attributes, the region it must reason over (i.e.", "the bubble), how the agent chooses its intended agent, and how it ultimately selects which action to take.", "With this protocol, we are able to formally guarantee specifications safety and liveness (under sparse traffic conditions) for all agents.", "We validate the safety and liveness guarantees in a randomized simulation environment.", "The current work still lacks 1) liveness guarantees in all scenarios, 2) robustness to imperfect sensory information and 3) does not account for other agent types like pedestrians and cyclists.", "Future work on modifying the agent strategy architecture to prevent the occurrence of the loop deadlock introduced in Section REF from occurring.", "Additionally, the architecture must be modified in a way to effectively accommodate impartial and imperfect information.", "We also hope to accommodate a diverse, heterogenous set of car agents and also other agent types like pedestrians and cyclists.", "Although the work needs to be extended to make more applicable to real-life systems, we believe this work is a first step towards defining a comprehensive method for guaranteeing safety and liveness for all agents in an extremely dynamic and complex environment." ], [ "Acknowledgments", "We would like to acknowledge K. Mani Chandy who provided valuable input and to Giovanna Amorim for her contributions to the simulation code.", "K.X.C., R.M.M., and T.P-M. jointly conceived the conceptual framework.", "K.X.C.", "and T.P-M. jointly developed the problem formulation and theoretical approach.", "K.X.C.", "worked out the main proofs with input from T.P-M. K.X.C.", "drafted the manuscript and figures with input from T.P-M. S-J.C. and R.M.M.", "provided guidance on the overall approach and provided feedback on the final manuscript." ], [ "Road Network", "The following defines the set of properties that grid points can have.", "The set of properties $\\mathcal {P} = \\lbrace p, d,$ lo$\\rbrace $ of each grid point $g \\in G$ .", "$p \\in \\mathbb {Z}^2$ denotes the Cartesian coordinate of the grid point, $d \\in \\lbrace 0, 1\\rbrace $ , which is an indicator variale that defines whether or not the grid point is drivable, lo is the legal orientation, where the legal orientation is an element of the set $ \\lbrace \\texttt {north}, \\texttt {east}, \\texttt {south}, \\texttt {west}\\rbrace $ .", "The set lo may be empty when the grid point is not drivable.", "Figure: Road network decomposition where each box represents a grid point.The following are sets of labeled grid points on the road network map.", "$\\mathcal {S}_{\\text{intersection}}$ : A set of grid points that contains all grid points with more than one legal orientation.", "$\\mathcal {S}_{\\text{traffic light}}$ : A set of grid points that represent the traffic light states in the vertical or horizontal direction via its color (for every intersection).", "The road network is hierarchically decomposed into lanes and bundles, which are defined informally as follows: Lanes: Let lane $La(g)$ denote a set of grid points that contains all grid points that are in the same `lane' as $g$ .", "$La(g) = \\lbrace g^{\\prime } \| \\text{proj}_x(g^{\\prime }.p) = \\text{proj}_x(g.p)$ or $\\text{proj}_y(g.p) = \\text{proj}_y(g.p),$ $g^{\\prime }.\\phi _l = g.\\phi _l, g.\\texttt {drivable} = g^{\\prime }.\\text{drivable} = 1 \\rbrace $ .", "Bundles: First, we define the set of adjacent lanes to lane $La(g)$ as $\\texttt {adj}(La(g)) = \\lbrace La(g^{\\prime }) \\mid \\exists e = (\\hat{g}, \\hat{g}^{\\prime }) \\in \\mathfrak {R} \\text{ s.t. }", "(\\hat{g} \\in La(g), \\hat{g^{\\prime }} \\in La(g^{\\prime })) \\text{ and } \\hat{g}.\\phi _l = \\hat{g^{\\prime }}.\\phi _l\\rbrace $ .", "This represents the set of lanes $La(g)$ in the same direction that the lane is adjacent to.", "Let $N(g)$ = adj(La(g)).", "Let bundle $Bu(g)$ denote a set of lanes that are all connected to one another and is defined recursively as follows: $Bu(g) ={\\left\\lbrace \\begin{array}{ll}La(g) \\cup N(g) \\text{ if } N(g) \\ne \\emptyset \\\\La(g) & \\text{otherwise}.\\end{array}\\right.", "}$ For clarity of the road network decomposition, refer to Fig.", "REF .", "With slight abuse of notation, we let $La(\\text{Ag})$ refer to the lane ID associated with the grid point $(s.x_{\\text{Ag}}, s.y_{\\text{Ag}})$ , and $Bu(\\text{Ag})$ mean the bundle ID associated with the lane $La(\\text{Ag})$ ." ], [ "Agent Backup Plan Action", "Definition 8.1 (Backup Plan Action) The backup plan action $a_{bp}$ is a control action where $a = a_{\\text{min}}$ and when applying $a_{\\text{min}}$ causes the agent's velocity to go below 0, $a=\\text{max}(a_{\\text{min}}, -s.v_{Ag})$ and $\\gamma _{\\text{Ag}} = \\texttt {straight}$ ." ], [ "Bubble Construction", "In order to define the bubble for the agent dynamics specified in Section REF , we present some preliminary definitions.", "We first introduce the backup plan node set (which is defined recursively) as follows: Definition 8.2 (Backup Plan Node Set) Let $\\text{Ag} \\in \\mathfrak {A}$ and $s_0 \\in S_{\\text{Ag}}$ .", "The backup plan grid point set $BP_{\\text{Ag}}(s_0)$ is all the grid points agent $\\text{Ag}$ occupies as it applies maximum deceleration to come to a complete stop.", "$BP_{\\text{Ag}}(s_0) ={\\left\\lbrace \\begin{array}{ll}\\mathcal {G}_{\\text{Ag}}(s_0, a_{{\\text{bp}}}) \\cup BP_{\\text{Ag}}(\\tau _{\\text{Ag}}(s_0,a_{{\\text{bp}}})) & \\text{if $\\tau _{\\text{Ag}}(s_0, a_{{\\text{bp}}}).v \\ne 0$} \\\\\\mathcal {G}_{\\text{Ag}}(\\tau (s_0, a_{{\\text{bp}}})) & \\text{otherwise}.\\end{array}\\right.", "}$ where $a_{\\text{min}}$ is the agent's action of applying maximal deceleration while keeping the steering wheel at the neutral position.", "Definition 8.3 (Forward/Backward Reachable States) The (1-step) forward reachable state set of agent $\\text{Ag}$ denoted $\\mathcal {R}_{\\text{Ag}}(s_0)$ represents the set of all states reachable by $\\text{Ag}$ from the state $s_0$ .", "The forward reachable set is defined as $\\mathcal {R}_{\\text{Ag}}(s_0) \\triangleq \\lbrace s \\in S_{\\text{Ag}} \\mid \\exists a \\in \\rho _{\\text{Ag}}(s_0).s = \\tau (s_0, a)\\rbrace $ .", "Similarly, we define the (1-step) backward reachable state set $\\mathcal {R}^{-1}_{\\text{Ag}}(s_0)$ as the set of all states from which the state $s_0$ can be reached by $\\text{Ag}$ .", "Formally, $\\mathcal {R}^{-1}_{\\text{Ag}}(s_0) \\triangleq \\lbrace s \\in S_{\\text{Ag}} \\mid \\exists s \\in S_{\\text{Ag}}.", "\\exists a \\in \\rho _{\\text{Ag}}(s).", "s_0 = \\tau (s, a) \\rbrace .$ Definition 8.4 (Forward Reachable Nodes) We denote by $\\mathcal {G}^{\\mathcal {R}}_{\\text{Ag}}(s_0)$ the forward reachable node set, namely, the set of all grid points that can be occupied upon taking the actions that brings the agent $\\text{Ag}$ from its current state $s_0$ to a state in $\\mathcal {R}_{\\text{Ag}}(s_0)$ .", "Specifically, $\\mathcal {G}^{\\mathcal {R}}_{\\text{Ag}}(s_0) \\triangleq \\bigcup _{a \\in \\rho _{Ag}(s_0)} \\mathcal {G}_{\\text{Ag}}(s_0, a)$ This set represents all the possible grid points that can be occupied by an agent in the next time step.", "Definition 8.5 (Occupancy Preimage) For $n \\in G$ , where $G$ are the nodes in the road network graph $\\mathfrak {R}$ , the occupancy preimage $\\mathcal {G}^{\\mathcal {R}^{-1}}_{\\text{Ag}}(n) $ is the set of states of agent $\\text{Ag}$ from which there is an action that causes $n$ to be occupied in the next time step.", "Formally, $\\mathcal {G}^{\\mathcal {R}^{-1}}_{\\text{Ag}}(n) = \\lbrace s \\in S_{\\text{Ag}} \\mid \\exists a \\in \\rho _{\\text{Ag}}(s).", "n \\in \\mathcal {G}_{\\text{Ag}}(s, a)\\rbrace $ In the next section, we define several different sets of grid points that are defined to represent the locations where two agents may possibly interfere with one another, which are shown in Fig.", "REF .", "The bubble is defined to be the union of these sets of grid points.", "Figure: Bubble if all Ag∈𝔄\\text{Ag}\\in \\mathfrak {A} have the Agent Dynamics specified in Section .", "Construction of this set defined in the Appendix.We begin by considering the ego agent whose bubble we are defining.", "In particular, let us again consider an agent $\\text{Ag}$ at state $s_0 \\in S_{Ag}$ .", "The corresponding grid point set $\\mathcal {G}_{\\text{Ag}}^{\\mathcal {R}}(s_0)$ is shown in the left-most figure in Fig.", "REF .", "The grid points an agent occupies when executing its backup plan from a state in the agent's forward reachable set $\\mathcal {R}_{\\text{Ag}}(s_0)$ is given by: $\\mathcal {G}^{\\mathcal {R},BP}_{\\text{Ag}}(s_0) \\triangleq \\bigcup _{s \\in \\mathcal {R}_{\\text{Ag}}(s_0)} BP_{\\text{Ag}}(s)$ These grid points are shown in the second from the left sub-figure in Fig.", "REF .", "The set-valued map $\\mathcal {Z}_{\\text{Ag}}(s_0) \\triangleq \\mathcal {G}^{\\mathcal {R}}_{\\text{Ag}}(s_0) \\cup \\mathcal {G}^{\\mathcal {R},BP}_{\\text{Ag}}(s_0).$ represents all the grid points an agent can possibly reach in the next state or in the following time step were it to execute its backup plan.", "Let $\\text{Ag}^{\\prime } \\in \\mathfrak {A}$ and $\\text{Ag}^{\\prime } \\ne \\text{Ag}$ .", "The set: $\\mathcal {S}_{\\text{Ag}^{\\prime }}^{\\mathcal {R}}(\\text{Ag}, s_0) \\triangleq \\bigcup _{n \\in \\mathcal {Z}_{\\text{Ag}}(s_0)}\\mathcal {G}^{\\mathcal {R}^{-1}}_{\\text{Ag}^{\\prime }}(n)$ defines the set of all states in which another agent $\\text{Ag}^{\\prime }$ can reach any grid point in the other agents' forward reachable grid points $\\mathcal {Z}_{\\text{Ag}}(s_0)$ .", "Let us define the grid point projection of these states as $\\mathcal {G}_{\\text{Ag}^{\\prime }}^{\\mathcal {R}}(\\text{Ag}, s_0) \\triangleq \\lbrace \\mathcal {G}_{\\text{Ag}^{\\prime }}(s) \\mid s \\in \\mathcal {S}_{\\text{Ag}^{\\prime }}^{\\mathcal {R}}(\\text{Ag}, s_0) \\rbrace .$ These grid points are defined in the third from the left subfigure in Fig.", "REF .", "The bubble also needs to include any state where an agent $\\text{Ag}^{\\prime }$ where the agent has so much momentum it cannot stop fast enough to avoid collision with the agent $\\text{Ag}$ .", "To define the set of states from which this might occur, let us define the set: $\\mathcal {S}^{BP}_{Ag^{\\prime }}(Ag, s_0) = \\lbrace s \\in S_{Ag^{\\prime }} \\mid BP_{Ag^{\\prime }}(s) \\cap \\mathcal {Z}_{Ag}(s_0) \\ne \\emptyset \\rbrace .$ If another agent $\\text{Ag}^{\\prime }$ occupies a state in this set, then execution of that agent's backup plan will cause it to intersect with the set of grid points that are in agents set $\\mathcal {Z}_{\\text{Ag}}(s_0)$ .", "Let $\\mathcal {S}^{\\mathcal {R}, BP}_{\\text{Ag}^{\\prime }}(\\text{Ag}, s_0) = \\bigcup _{s \\in \\mathcal {S}^{BP}_{\\text{Ag}^{\\prime }}(\\text{Ag})} \\mathcal {R}_{\\text{Ag}^{\\prime }}^{-1}(s).$ This is the set of all states backward reachable to the states in $\\mathcal {S}^{BP}_{\\text{Ag}^{\\prime }}(\\text{Ag}, s_0)$ .", "If an agent $\\text{Ag}^{\\prime }$ occupies any of these states, it will end up in a state where its backup plan will intersect with agent $\\text{Ag}$ 's potential grid points that are defined in $\\mathcal {Z}_{\\text{Ag}}$ .", "We project this set of states to a set of grid points as $\\mathcal {G}^{\\mathcal {R}, BP}_{\\text{Ag}^{\\prime }}(\\text{Ag}, s_0) = \\lbrace \\mathcal {G}_{Ag^{\\prime }}(s) \\mid s \\in \\mathcal {S}^{BP}_{Ag^{\\prime }}(Ag, s_0)\\rbrace .$ Note, this set of grid points is shown in the right-most subfigure in Fig.", "REF .", "The bubble is then defined as the union of all the sets of grid points specified above.", "Definition 8.6 (Bubble) Let us consider an agent $\\text{Ag}$ with state $s_0 \\in S_{\\text{Ag}}$ and agent $\\text{Ag}^{\\prime }$ be another agent.", "Then the bubble of $\\text{Ag}$ with respect to agents of the same type as $\\text{Ag}^{\\prime }$ is given by $\\mathcal {B}_{\\text{Ag}/\\text{Ag}^{\\prime }}(s_0) \\triangleq \\mathcal {Z}_{\\text{Ag}}(s_0) \\cup \\mathcal {G}_{\\text{Ag}^{\\prime }}^{\\mathcal {R}}(\\text{Ag}, s_0) \\cup \\mathcal {G}^{\\mathcal {R}, BP}_{\\text{Ag}^{\\prime }}(\\text{Ag}, s_0).$ Note that under almost all circumstances, we should have $\\mathcal {Z}_{\\text{Ag}}(s_0) \\subseteq \\mathcal {G}^{\\mathcal {R}}_{\\text{Ag}^{\\prime }}(Ag, s_0) \\subseteq \\mathcal {G}^{\\mathcal {R}, BP}_{\\text{Ag}^{\\prime }}(\\text{Ag}, s_0)$ so $\\mathcal {B}_{\\text{Ag}}(s_0)$ is simply equal to $\\mathcal {G}^{\\mathcal {R}, BP}_{\\text{Ag}^{\\prime }}(\\text{Ag}, s_0)$ .", "This holds true for the abstract dynamics we consider in this paper.", "This means the bubble contains any grid points in which another agent $\\text{Ag}^{\\prime }$ occupying those grid points can interfere (via its own forward reachable states or the backup plan it would use in any of its forward reachable states) with at least one of agent $\\text{Ag}$ 's next possible actions and the backup plan it would use if it were to take any one of those next actions." ], [ "Global Precedence Consistency", "Lemma 8.1 If all agents assign precedence according to the local precedence assignment rules to agents in their respective bubbles, then the precedence relations will induce a polyforest on $\\mathfrak {A}/\\sim $ , where $S/\\sim $ defines the quotient set of a set $S$ .", "Suppose there is a cycle $C$ in $\\mathfrak {A}/\\sim $ .", "For each of the equivalent classes in $C$ ($C$ must have at least 2 to be a cycle), choose a representative from $\\mathfrak {A}$ to form a set $R_C$ .", "Let $\\text{Ag} \\in R_C$ be one of these representatives.", "Applying the second local precedence assignment rule inductively, we can see that all agents in $R_C$ must be from $\\text{Ag}$ 's bundle.", "By the first local precedence assignment rule, any $C$ edge must be from an agent with lower projected value to one with a higher projected value in this bundle.", "Since these values are totally ordered (being integers), they must be the same.", "This implies that $C$ only has one equivalence class, a contradiction.", "The acyclicity of the polyforest structure implies the consistency of local agent precedence assignments.", "Note, the local precedence assignment algorithm establishes the order in which agents are taking turns." ], [ "Oracle Definitions", " $O_{\\text{Ag},t,\\text{unprotected left-turn safety}}(s, a, u)$ returns T when the action $a$ from the state $s$ will result in the complete execution of a safe, unprotected left-turn (invariant to agent precedence).", "Note, an unprotected left turn spans over multiple time-steps.", "The oracle will return T if $\\text{Ag}$ has been waiting to take left-turn (while traffic light is green), traffic light turns red, and no agents in oncoming lanes.", "$O_{\\text{static safety}}(s, a, u)$ returns T when the action $a$ from state $s$ will not cause the agent to collide with a static obstacle or end up in a state where the agent's safety backup plan $a_{bp}$ with respect to the static obstacle is no longer safe.", "$O_{\\text{traffic light law}}(s, a, u)$ returns T if the action $a$ from the state $s$ satisfies the traffic light laws (not crossing into intersection when red.", "It also requires that $\\text{Ag}$ be able to take $a_{bp}$ from $s^{\\prime } = \\tau _{\\text{Ag}}(s,a)$ and not violate the traffic-light law.", "$O_{\\text{traffic orientation law}}(s, a, u)$ returns T if the action $a$ from the state $s$ follows the legal road orientation.", "$O_{\\text{traffic intersection clearance law}}(s, a, u)$ returns T if the action causes the agent to enter the intersection and not leave it when the traffic light turns red.", "Returns T if the action causes the agent to end in a state where its backup plan action will cause the agent to enter the intersection and not be able to leave it when the traffic light turns red.", "$O_{\\text{traffic intersection lane change law}}(s, a, u)$ returns T if the action is such that $\\gamma _{Ag} = \\lbrace \\texttt {left-lane change}, \\texttt {right-lane change} \\rbrace $ and the agent either begins in an intersection or ends up in the intersection after taking the action.", "$O_{\\text{maintains progress}}(s, a, u)$ returns T if the action $a$ from the state $s$ stays the same distance to its goal." ], [ "Safety Lemmas", "In the following lemma, we show that an agent cannot send (or receive) a conflict request to (from) an agent outside its bubble.", "Lemma 8.2 Let us consider agent $\\text{Ag}$ with state $s$ and agent $\\text{Ag}^{\\prime }$ at state $s^{\\prime }$ .", "$\\text{Ag} \\texttt { send } \\text{Ag}^{\\prime } \\Rightarrow \\text{Ag} \\in \\mathcal {B}_{\\text{Ag}^{\\prime }}(s^{\\prime })$ .", "If $A \\texttt { send } B$ this means that all of the conditions specified in Section REF , particularly that $(A, a_i) \\dagger (B, a_i^{\\prime })$ .", "This condition is only valid if $\\text{proj}_G s \\in \\mathcal {G}_{F, B}(B,A)$ or $\\text{proj}_G s \\in \\mathcal {G}_{F, BP}(B,A)$ .", "Membership of Agent A's state in either of these sets implies $A \\in \\mathcal {B}(B)$ .", "The following lemma follows from the lemma above.", "Lemma 8.3 At most one agent will win in each agent's conflict cluster.", "W.l.o.g.", "let us consider an agent $\\text{Ag}$ and its respective conflict cluster $\\mathcal {C}(\\text{Ag})$ .", "It follows from Lemma REF that $\\forall \\text{Ag}^{\\prime }$ , s.t.", "$ \\text{Ag} \\texttt { send } \\text{Ag}^{\\prime } \\text{Ag}^{\\prime } \\in \\mathcal {B}_{Ag}(s)$ and $\\text{Ag} \\in \\mathcal {B}_{Ag^{\\prime }}(s^{\\prime })$ .", "It also follows that $\\forall \\text{Ag}^{\\prime } \\text{ s.t.", "}, \\text{Ag} \\texttt { send } \\text{Ag}^{\\prime }, \\text{Ag} \\in \\mathcal {B}_{\\text{Ag}^{\\prime }}(s^{\\prime })$ and $\\text{Ag}^{\\prime } \\in \\mathcal {B}_{\\text{Ag}}(s)$ .", "This means an agent has access to all token counts and IDs of all agents in its conflict cluster, and all agents in its conflict cluster have access to the agent's token count and ID.", "The conflict resolution implies that all agent edges are incident to the winning agent, where edges point to the agent they cede to.", "This implies that at most one agent can be the winner of each cluster.", "Less than one winner (per conflict cluster) will occur when an agent that is in the intersection of more than one conflict cluster wins.", "The following lemma states that if all $\\text{Ag} \\in \\mathfrak {A}$ are following the Agent Protocol, an agent $\\text{Ag}$ will not take an action that will cause it to 1) collide with or 2) violate the safety backup plan of another agent outside its bubble $\\mathcal {B}_{\\text{Ag}}(s)$ .", "Lemma 8.4 If $\\text{Ag}$ is following the Agent Protocol, and $S_{\\text{Ag}, bp}(u) = \\texttt {T}$ , $\\text{Ag}$ will only choose an action $a \\in Act_{\\text{Ag}}$ for which the following two conditions hold: 1) $\\mathcal {G}_{\\text{Ag}}(s, a) \\cap (\\cup _{\\text{Ag}^{\\prime }\\in S}\\mathcal {G}_{\\text{Ag}^{\\prime }}(s^{\\prime }, a^{\\prime })) = \\emptyset $ and 2) $\\forall \\text{Ag}^{\\prime }\\in S$ , $\\lnot ((\\text{Ag}, a)\\bot \\text{Ag}^{\\prime })$ , where the set $S \\triangleq \\lbrace \\text{Ag}^{\\prime }\|\\text{Ag}^{\\prime } \\notin \\mathcal {B}_{\\text{Ag}}(s) \\wedge ((\\text{Ag}^{\\prime } \\sim \\text{Ag}) \\vee (\\text{Ag}^{\\prime } \\prec \\text{Ag}) \\vee (\\text{Ag} \\prec \\text{Ag'})) \\rbrace $ .", "This follows from the definition of the agent bubble, whose construction is defined in REF .", "The following lemma states that an agent $\\text{Ag}$ following the Agent Protocol will not take an action for which it violates the safety of its own backup plan.", "Lemma 8.5 If $\\text{Ag}$ is following the Agent Protocol, and $S_{\\text{Ag}, bp}(u) = \\texttt {T}$ , $\\text{Ag}$ will only choose an action $a \\in Act_{\\text{Ag}}$ for which the following condition holds: $\\forall \\text{Ag}^{\\prime }\\in S$ , $\\lnot ((\\text{Ag}, a)\\bot Ag^{\\prime })$ , where $S = \\lbrace \\text{Ag}\\rbrace $ .", "We prove this by using specific definition of elements in the Agent Protocol.", "Let us first show that any action $a\\in Act_{\\text{Ag}}$ that $\\text{Ag}$ takes will satisfy the oracles in the top two tiers (safety and traffic rules) of $\\text{Ag}$ 's profile defined in Section.", "REF .", "According to the Action Selection Strategy defined in Section REF , $\\text{Ag}$ will choose one of three actions: the agent's intended action $a_i$ , the best straight action $a_{st}$ , or its backup plan action $a_{bp}$ .", "Let us consider the actions $a_i$ and $a_{st}$ .", "Both $a_i$ and $a_{st}$ are selected via the Agent Profile and consistent-function evaluator defined in Section REF .", "Since $S_{\\text{Ag}, bp}(u) = \\texttt {T}$ , the agent will have at least one action ($a_{bp}$ ) for which the top two tiers of specifications are satisfied.", "By definition of the Agent Profile and the consistent evaluator function, if $S_{\\text{Ag}, bp}(u) = \\texttt {T}$ , the safety backup plan action $a_{bp}$ will always be chosen over an action where any of the specifications in the top two tiers of the profile are not satisfied.", "By REF and REF , $\\text{Ag}$ will have $a \\in Act_{Ag}$ and will choose an action for which the top two tiers of the Agent Profile are satisfied and thus $a_{i}$ and $a_{st}$ are actions where all oracles in the top two tiers of the profile are satisfied.", "Let us consider the action $a_{bp}$ .", "This follows from the assumption that $S_{\\text{Ag}, bp}(u) = \\texttt {T}$ and the definition of $S_{\\text{Ag}, bp}(u)$ .", "If the oracles in the top two tiers are satisfied by an action $a$ , by the definition of the oracles in Section REF , this implies that the action $a$ will take $\\text{Ag}$ to a state $s^{\\prime }$ and the system will be in a new global state $u^{\\prime }$ where $S_{Ag, bp}(u^{\\prime })= \\texttt {T}$ .", "$S_{Ag, bp}(u^{\\prime })= T$ means $\\text{Ag}$ will end up in a state where $a_{bp}$ will be an action that satisfies traffic rules, avoids inevitable collision with static obstacles, and $\\lnot ((Ag,a_i) \\bot \\text{Ag})$ .", "The following lemma states that if all $\\text{Ag} \\in \\mathfrak {A}$ are following the Agent Protocol, any agent $\\text{Ag}$ will not take an action for which it collides with or violates the safety backup plan of any agent with higher precedence.", "Lemma 8.6 If $\\text{Ag}$ is following the Agent Protocol, and $S_{\\text{Ag}, bp}(u) = \\texttt {T}$ , $\\text{Ag}$ will only choose an action $a \\in Act_{\\text{Ag}}$ for which the following two conditions hold: 1) $\\mathcal {G}_{Ag}(s, a) \\cap (\\cup _{Ag^{\\prime }\\in S}\\mathcal {G}_{\\text{Ag}^{\\prime }}(s^{\\prime }, a^{\\prime })) = \\emptyset $ and 2) $\\forall \\text{Ag}^{\\prime }\\in S$ , $\\lnot ((Ag, a)\\bot Ag^{\\prime })$ , where the set $S \\triangleq \\lbrace \\text{Ag}^{\\prime }\|\\text{Ag} \\prec \\text{Ag}^{\\prime }\\rbrace $ , i.e.", "agents with higher precedence than $\\text{Ag}$ .", "We prove this by using arguments based on the definition of precedence, the Agent Protocol, and Agent Dynamics.", "Let us first consider all $\\text{Ag}^{\\prime }$ where $\\text{Ag} \\prec \\text{Ag}^{\\prime }$ and $Ag^{\\prime } \\notin \\mathcal {B}_{\\text{Ag}}(s)$ .", "Proof by Lemma REF .", "Now, let us consider all $\\text{Ag}^{\\prime }$ where $\\text{Ag} \\prec \\text{Ag}^{\\prime }$ and $\\text{Ag}^{\\prime } \\in \\mathcal {B}_{\\text{Ag}}(s)$ .", "According to Lemma REF , $\\text{Ag}$ will only take an action that satisfies all oracles in the top two tiers, including $O_{\\text{dynamic safety}}(s, a, u)$ .", "Since $a$ is such that $O_{\\text{dynamic safety}}(s, a, u) = \\texttt {T}$ , by definition of the oracle, $Ag$ will not cause collision with any $Ag^{\\prime } \\in \\mathcal {B}_{\\text{Ag}}(s)$ .", "For any $\\text{Ag} \\prec \\text{Ag}^{\\prime }$ , where $\\text{Ag}^{\\prime }$ has higher precedence than $\\text{Ag}$ , then $\\text{proj}_{\\text{long}}(\\text{Ag}) < \\text{proj}_{\\text{long}} (\\text{Ag}^{\\prime })$ , i.e.", "$\\text{Ag}^{\\prime }$ is longitudinally ahead of $\\text{Ag}$ .", "In order for $(\\text{Ag}, a) \\bot \\text{Ag}^{\\prime }$ , the action $a$ would have to be such that $s_f = \\tau _{Ag}(s,a)$ , and $ La(s_f) = La(s^{\\prime })$ and $\\text{proj}_{\\text{long}}(\\text{Ag}) > \\text{proj}_{\\text{long}} (\\text{Ag}^{\\prime })$ , where $\\text{Ag}$ is directly in front of $\\text{Ag}^{\\prime }$ .", "Because of the agent dynamics defined in Section REF , any $a$ such that $(\\text{Ag}, a) \\bot \\text{Ag}^{\\prime }$ will require $\\mathcal {G}(\\text{Ag},a) \\cap \\mathcal {G}(\\text{Ag}^{\\prime }) \\ne \\emptyset $ .", "Thus, any such action $a$ will not satisfy the oracle $O_{\\text{dynamic safety}}(s, a, u)$ .", "Since $S_{\\text{Ag}, bp}(u) = \\texttt {T}$ , by Assumption 6 in Section , the agent will have at least one action $a_{bp}$ for which $O_{\\text{dynamic safety}}(s, a, u) = \\texttt {T}$ .", "Since the agent will only choose an action for which $O_{\\text{dynamic safety}}(s,a,u) = \\texttt {T}$ and it always has at least one action $a_{bp}$ that satisfies the oracle, the agent will always choose an action for which $O_{\\text{dynamic safety}}(s,a,u) = \\texttt {T}$ and thus will take an action such that $\\lnot ((\\text{Ag}, a) \\bot \\text{Ag}^{\\prime })$ .", "The following lemma states that if all $\\text{Ag} \\in \\mathfrak {A}$ are following the Agent Protocol, any agent $\\text{Ag}$ will not take an action for which it collides with or violates the safety backup plan of any agent with lower precedence.", "Lemma 8.7 If $\\text{Ag}$ is following the Agent Protocol, and $S_{\\text{Ag}, bp}(u) = \\texttt {T}$ , $\\text{Ag}$ will only choose an action $a \\in Act_{\\text{Ag}}$ for which the following two conditions hold: 1) $\\mathcal {G}_{Ag}(s, a) \\cap (\\cup _{Ag^{\\prime }\\in S}\\mathcal {G}_{\\text{Ag}^{\\prime }}(s^{\\prime }, a^{\\prime })) = \\emptyset $ and 2) $\\forall Ag^{\\prime }\\in S$ , $\\lnot ((Ag, a)\\bot Ag^{\\prime })$ , where the set $S \\triangleq \\lbrace \\text{Ag}^{\\prime } \| \\text{Ag}^{\\prime } \\prec \\text{Ag} \\rbrace $ , i.e.", "agents with lower precedence than $\\text{Ag}$ .", "We prove this by using arguments based on the definition of precedence, the Agent Protocol, and Agent Dynamics.", "Let us first consider all $\\text{Ag}^{\\prime }$ where $\\text{Ag} \\prec \\text{Ag}^{\\prime }$ and $\\text{Ag}^{\\prime } \\notin \\mathcal {B}_{\\text{Ag}}(s)$ .", "Proof by Lemma REF .", "Now, let us consider all $\\text{Ag}^{\\prime }$ where $\\text{Ag} \\prec \\text{Ag}^{\\prime }$ and $\\text{Ag}^{\\prime } \\in \\mathcal {B}_{\\text{Ag}}(s)$ .", "According to REF , $\\text{Ag}$ will only take an action that satisfies all oracles in the top two tiers, including $O_{\\text{dynamic safety}}(s,a,u)$ .", "Since $a$ is such that $O_{\\text{dynamic safety}}(s, a, u) = \\texttt {T}$ , by definition of the oracle, $Ag$ will not cause collision with any $Ag^{\\prime } \\in \\mathcal {B}_{\\text{Ag}}(s)$ .", "According to the Action Selection Strategy defined in Section REF , $\\text{Ag}$ will choose one of three actions: the agent's intended action $a_i$ , the best straight action $a_{st}$ , or its backup plan action $a_{bp}$ .", "Let us consider the backup plan action $a_{bp}$ .", "By violation of safety backup plan, $((\\text{Ag}, a_{bp}) \\bot \\text{Ag}^{\\prime })$ only if $La(\\text{Ag}) = La(\\text{Ag}^{\\prime })$ .", "W.l.o.g., let us consider $\\text{Ag}^{\\prime }$ that is directly behind $\\text{Ag}$ .", "Since $S_{\\text{Ag}^{\\prime }, bp}(s, u) = \\texttt {T}$ , by Assumption 6 in Section , $O_{\\text{dynamic safety}}(s, a_{bp},u) = \\texttt {T}$ , meaning $\\text{Ag}^{\\prime }$ will be far enough behind $\\text{Ag}$ so that if $\\text{Ag}$ executes its backup plan action $a_{bp}$ , $\\text{Ag}^{\\prime }$ can safely execute its own backup plan action.", "Thus, by Definition REF , $\\lnot ((\\text{Ag}, a_{bp}) \\bot \\text{Ag}^{\\prime })$ .", "Let us consider the best straight action $a_{st}$ .", "This follows from the arguments made in REF , since $a_{st}$ is a less severe action than $a_{bp}$ .", "Let us consider the intended action $a_i$ .", "Let us consider when $\\gamma _{\\text{Ag}} = \\lbrace \\texttt {straight}\\rbrace $ .", "This follows from REF .", "Let us consider when $\\gamma _{\\text{Ag}} \\in \\lbrace \\texttt {right-turn}, \\texttt {left-turn} \\rbrace $ .", "If $\\text{Ag}$ takes such an action, $\\text{Ag}$ will end up in a state where $Bu(\\text{Ag}^{\\prime }) \\ne Bu(\\text{Ag})$ and from Definition REF , agents in different bundles cannot violate each others' backup plans.", "Let us consider when $\\gamma _{\\text{Ag}} \\in \\lbrace \\texttt {right-lane change}$ $\\texttt {left-lane change} \\rbrace $ .", "$(\\text{Ag}, a_i) \\bot \\text{Ag}^{\\prime }$ when $a_i$ is a lane change and the agents $\\text{Ag}$ and $\\text{Ag}^{\\prime }$ are at a state such that $s_f = \\tau (s, a_i)$ and $s_f^{\\prime } = \\tau (s^{\\prime }, a_{bp})$ , respectively, where $d(s_f, s_f^{\\prime }) < gap_{req}$ , where $d(s_f, s_f^{\\prime })$ is the $l_2$ distance between $s_f$ and $s_f^{\\prime }$ .", "When this condition holds, the agent's max-yielding-not-enough flag $\\mathcal {F}_{\\text{Ag}}(u, a_i)$ defined in Section REF will be set.", "According to the action-selection strategy, $\\text{Ag}$ will only take $a_i$ when $\\mathcal {F}_{\\text{Ag}}(u, a_i) = \\texttt {F}$ .", "Thus, $\\text{Ag}$ will only take $a_i$ when $\\lnot ((Ag, a_{i}) \\bot Ag^{\\prime })$ .", "The following lemma states that if all $\\text{Ag} \\in \\mathfrak {A}$ are following the Agent Protocol, any agent $\\text{Ag}$ will not take an action for which it collides with or violates the safety backup plan of any agent with equal precedence.", "Lemma 8.8 If $\\text{Ag}$ is following the Agent Protocol, and $S_{\\text{Ag}, bp}(u) = \\texttt {T}$ , $\\text{Ag}$ will only choose an action $a \\in Act_{\\text{Ag}}$ for which the following two conditions hold: 1) $\\mathcal {G}_{\\text{Ag}}(s, a) \\cap (\\cup _{\\text{Ag}^{\\prime }\\in S}\\mathcal {G}_{\\text{Ag}^{\\prime }}(s^{\\prime }, a^{\\prime })) = \\emptyset $ and 2) $\\forall \\text{Ag}^{\\prime }\\in S$ , $\\lnot ((\\text{Ag}, a)\\bot \\text{Ag}^{\\prime })$ , where the set $S \\triangleq \\lbrace \\text{Ag}^{\\prime } \| \\text{Ag}^{\\prime } \\sim \\text{Ag} \\rbrace $ , i.e.", "agents with equivalent precedence as the agent.", "We prove this by using arguments based on the definition of precedence, Agent Dynamics, and the Agent Protocol.", "Let us first consider all $\\text{Ag}^{\\prime }$ where $\\text{Ag} \\prec \\text{Ag}^{\\prime }$ and $\\text{Ag}^{\\prime } \\notin \\mathcal {B}_{\\text{Ag}}(s)$ .", "Proof by Lemma REF .", "Now, let us consider all $\\text{Ag}^{\\prime }$ where $\\text{Ag} \\prec \\text{Ag}^{\\prime }$ and $\\text{Ag}^{\\prime } \\in \\mathcal {B}_{\\text{Ag}}(s)$ .", "Let us first consider the agent itself, since an agent has equivalent precedence to itself.", "This is true by Lemma REF .", "This can be proven for any other agents of equivalent precedence that is not the agent itself as follows.", "Agents with equal precedence take actions simultaneously so $O_{\\text{dynamic safety}}(s,a,u)$ does not guarantee no collision.", "According to the Action Selection Strategy defined in Section REF , $\\text{Ag}$ will choose one of three actions: the agent's intended action $a_i$ , the best straight action $a_{st}$ , or its backup plan action $a_{bp}$ .", "By definition of precedence assignment, any $\\text{Ag}^{\\prime }$ for which $\\text{Ag}^{\\prime } \\sim \\text{Ag}$ will be such that $La(\\text{Ag}) \\ne La(\\text{Ag}^{\\prime })$ .", "Let us show if $\\text{Ag}$ selects $a_{bp}$ , it will 1) not collide with any $\\text{Ag}^{\\prime }\\in S$ and 2) $\\lnot ((\\text{Ag}, a_{bp}) \\bot \\text{Ag}^{\\prime })$ .", "W.l.o.g., let us consider $\\text{Ag}^{\\prime }$ where $\\text{Ag}^{\\prime } \\sim Ag$ .", "The flag $\\mathcal {F}_{\\text{Ag'}}(u, a_i) = \\texttt {T}$ if $\\text{Ag}^{\\prime }s$ intended action $a_i$ causes collision with $\\text{Ag}$ or $(Ag^{\\prime }, a_{i}) \\bot Ag$ , i.e.", "it collides with or violates the safety of $\\text{Ag}$ 's backup plan action.", "By the action-selection-strategy, $\\text{Ag}^{\\prime }$ will not take the action $a_i$ when $\\mathcal {F}_{\\text{Ag'}}(u,a_i) = \\texttt {T}$ , so this guarantees $\\text{Ag}$ will not collide with $\\text{Ag}^{\\prime }$ when $\\text{Ag}$ takes $a_{bp}$ .", "By the Agent Dynamics, $\\text{Ag}$ 's backup plan action cannot cause $\\text{Ag}$ to end up in a position where it can violate $\\text{Ag}^{\\prime }$ 's backup plan without colliding with it–for which $\\text{Ag}^{\\prime }$ 's flag $\\mathcal {F}_{\\text{Ag}}(u, a_i)$ would be set.", "Let us show that $\\text{Ag}$ will only choose an $a_{st}$ if it will 1) not collide with $\\text{Ag}^{\\prime }\\in S$ and 2) $\\lnot ((\\text{Ag}, a_{st}) \\bot \\text{Ag}^{\\prime })$ .", "When $a_{st} = a_{bp}$ , then the arguments in REF hold.", "$\\text{Ag}$ selects an $a_{st}$ that is not $a_{bp}$ only when 1) its conflict cluster is empty (i.e.", "$C_{\\text{Ag}} = \\emptyset $ ) or 2) when it has received a conflict request from another agent and it has won its conflict cluster resolution (i.e.", "$W_{\\text{Ag}} = \\texttt {T}$ ).", "If $C_{\\text{Ag}} = \\emptyset $ , by definition of how conflict clusters are defined in Section REF , the agent's action $a_{st}$ will not cause $\\text{Ag}$ to collide with any $\\text{Ag}^{\\prime } \\in S$ , and $\\forall \\text{Ag}^{\\prime } \\in S, \\lnot ((\\text{Ag}, a_{st}) \\bot \\text{Ag}^{\\prime })$ .", "In the case $\\text{Ag}$ has received a conflict request and has won $W_{\\text{Ag}}$ , by Lemma REF , if $W_{\\text{Ag}} = \\texttt {T}$ , it will be the only agent in its conflict cluster that has won.", "By definition of the conflict cluster, any $\\text{\\text{Ag}}^{\\prime } \\in C_{\\text{Ag}}$ where $\\text{Ag} \\sim \\text{Ag}^{\\prime }$ will take a straight action.", "Since agents of equivalent precedence are initially in separate lanes by REF and any $\\text{Ag}^{\\prime } \\in S$ will take a straight action, then $La(s_{\\text{Ag}, t+1}) \\ne La(s_{\\text{Ag'}, t+1})$ when $\\text{Ag}$ takes $a_{st}$ .", "Thus, by definition of agent dynamics and Definition REF , the action will not cause $\\text{Ag}$ to collide with any $\\text{Ag}^{\\prime } \\in S$ , and $\\forall \\text{Ag}^{\\prime } \\in S, \\lnot ((\\text{Ag}, a_{st}) \\bot \\text{Ag}^{\\prime })$ .", "Let us show that $\\text{Ag}$ will only choose an $a_{i}$ if it will 1) not collide with any $\\text{Ag}^{\\prime } \\in S$ and 2) $\\lnot ((\\text{Ag}, a_{i}) \\bot \\text{Ag}^{\\prime })$ .", "Let us consider when $\\gamma _{\\text{Ag}} = \\texttt {straight}$ for $a_i$ .", "This follows from the same arguments presented in REF .", "Let us consider when $\\gamma _{\\text{Ag}} \\in \\lbrace \\texttt {right-turn, left-turn}\\rbrace $ for $a_i$ .", "This follows from the fact that all other agents are following the Agent Protocol and will not take a lane-change action in the intersection, and because of the definition of the Agent Dynamics and Road Network.", "Let us consider when $\\gamma _{\\text{Ag}} \\in \\lbrace \\texttt {right-lane change}$ , $\\texttt {left-lane change} \\rbrace $ .", "$\\text{Ag}$ will only take its intended action $a_i$ if the flag $\\mathcal {F}_{\\text{Ag}}(u, a_i) = \\texttt {F}$ , and in the case that it is part of a conflict cluster, it is the winner of the conflict cluster resolution, i.e.", "$\\mathcal {W}_{\\text{Ag}} = \\texttt {T}$ .", "By definition of $\\mathcal {F}_{\\text{Ag}}(u, a_i)$ , the agent will not take $a_i$ when $a_i$ causes $\\text{Ag}$ to collide with any agent $\\text{Ag}^{\\prime } \\in S$ or when it causes $Ag$ to violate the safety of the back up plan of another agent $Ag^{\\prime }$ , i.e.", "$ \\exists Ag^{\\prime }$ s.t.", "$(\\text{Ag}, a_i) \\bot \\text{Ag}^{\\prime }$ .", "In the case the agent has received a conflict request and has won $\\mathcal {W}_{\\text{Ag}}$ , by Lemma REF , if $\\mathcal {W}_{\\text{Ag}} = \\texttt {T}$ , it will be the only agent in its conflict cluster that has won.", "By definition of the conflict cluster, any $\\text{Ag}^{\\prime } \\in C_{\\text{Ag}}$ where $\\text{Ag} \\sim \\text{Ag}^{\\prime }$ will take its backup plan action $a_{bp}$ , and thus $s_f = \\tau (s, a_{st})$ , and $s_f^{\\prime } = \\tau (s, a_{bp})$ , where $ d(s_f, s_f^{\\prime }) \\ge gap_{\\text{req}} $ .", "Thus, $a_i$ will only be selected when $a_i$ does not cause $\\text{Ag}$ to collide with any $\\text{Ag}^{\\prime } \\in S$ and $\\forall \\text{Ag}^{\\prime } \\in S, \\lnot ((\\text{Ag}, a_i) \\bot \\text{Ag}^{\\prime })$ .", "The following lemma states that if all $\\text{Ag} \\in \\mathfrak {A}$ are following the Agent Protocol, any agent $\\text{Ag}$ will not take an action for which it collides with or violates the safety backup plan of any agent with incomparable precedence to it.", "Lemma 8.9 If $\\text{Ag}$ is following the Agent Protocol, and $S_{\\text{Ag}, bp}(u) = \\texttt {T}$ , $\\text{Ag}$ will only choose an action $a \\in Act_{\\text{Ag}}$ for which the following two conditions hold: 1) $\\mathcal {G}_{\\text{Ag}}(s, a) \\cap (\\cup _{\\text{Ag}^{\\prime }\\in S}\\mathcal {G}_{\\text{Ag}^{\\prime }}(s^{\\prime }, a^{\\prime })) = \\emptyset $ and 2) $\\forall \\text{Ag}^{\\prime }\\in S$ , $\\lnot ((\\text{Ag}, a)\\bot \\text{Ag}^{\\prime })$ , where the set $S \\triangleq \\lbrace \\text{Ag}^{\\prime } \| \\text{Ag}^{\\prime } \\lnot \\sim \\text{Ag} \\rbrace $ , i.e.", "agents with precedence incomparable to the agent.", "We prove this by using arguments based on the definition of precedence, Agent Dynamics, and the Agent Protocol.", "Let us show when $\\text{Ag}$ chooses $a_{bp}$ , it will 1) not collide with any $\\text{Ag}^{\\prime } \\in S$ and 2) $\\lnot ((\\text{Ag}, a_{bp}) \\bot \\text{Ag}^{\\prime })$ .", "Since $S_{\\text{Ag}, bp}(u) = \\texttt {T}$ , the agent will have at least one action ($a_{bp}$ ) for which the top two tiers of specifications are satisfied.", "By REF , the action $a_{bp}$ will only take $\\text{Ag}$ into the intersection if traffic light is green.", "By Assumption 4, all traffic lights are coordinated so if agents respect traffic light rules, they will not collide.", "By the assumption that all other $\\text{Ag}^{\\prime } \\in \\mathfrak {G}$ are obeying the same protocol, each agent will only take actions that satisfy the top two tiers of their profile.", "Any $\\text{Ag}^{\\prime }$ in a perpendicular bundle will not enter the intersection since they have a red light.", "Thus, $\\text{Ag}$ cannot collide or violate the backup plan of agents in perpendicular bundles.", "Any $\\text{Ag}^{\\prime }$ in an oncoming traffic bundle must only take an unprotected left-turn when it satisfies $O_{\\text{unprotected left-turn}}(s,a,u)$ .", "Thus $\\text{Ag}$ will not collide or violate the backup plan of agents in bundles of oncoming traffic.", "Let us show that when $\\text{Ag}$ chooses $a_{st}$ , it will 1) not collide with any $\\text{Ag}^{\\prime } \\in S$ and 2) $\\lnot ((\\text{Ag}, a_{st}) \\bot \\text{Ag}^{\\prime })$ .", "Since $a_{st}$ is chosen according to the Agent Profile, it will only be a straight action that is not $a_{bp}$ as long as it satisfies the top-two tiers of the profile and more.", "Thus, $a_{st}$ will only take $\\text{Ag}$ into intersection if traffic light is green.", "By the same arguments in REF , this holds.", "Let us show that when $\\text{Ag}$ chooses $a_{i}$ , it will 1) not collide with any $\\text{Ag}^{\\prime } \\in S$ and 2) $\\lnot ((\\text{Ag}, a_{i}) \\bot \\text{Ag}^{\\prime })$ .", "Let us consider when $a_i$ is such that $\\gamma _{Ag} = \\texttt {straight}$ .", "This follows from the same arguments presented in REF .", "Let us consider when $a_i$ is such that $\\gamma _{Ag} \\in $ $\\lbrace \\texttt {left-lane change$ , $right-lane change}\\rbrace $ .", "$\\text{Ag}$ will never select such an action at an intersection since $O_{\\text{intersection lane-change}}(s,a,u)$ will evaluate to $\\texttt {F}$ .", "Let us consider when $a_i$ is such that $\\gamma _{Ag} \\in \\\\ \\lbrace \\texttt {left-turn, right-turn}\\rbrace $ .", "By the assumption that all other agents are following the Agent Protocol, all $\\text{Ag}^{\\prime }$ that are in bundle perpendicular to $Bu(Ag)$ will not be in the intersection and will not collide with $\\text{Ag}$ .", "Further, the traffic light oracle $O_{\\text{traffic light}}(s,a,u) = \\texttt {T}$ only when $\\lnot ((\\text{Ag}, a_i) \\bot \\text{Ag}^{\\prime })$ when $\\gamma _{\\text{Ag}} = \\texttt {right-turn}$ .", "Thus, when $\\gamma _{\\text{Ag}} = \\texttt {right-turn}$ proof by REF and REF .", "For an action $a_i$ where $\\gamma _{\\text{Ag}} = \\texttt {left-turn}, \\text{Ag}$ will only take $a_i$ if $O_{\\text{traffic-light}}(s,a,u) = \\texttt {T}$ and $O_{\\text{unprotected left-turn}}(s,a,u) = \\texttt {T}$ .", "Since all agents are following the law based on Proof REF , $O_{\\text{traffic light}}(s,a,u)=\\texttt {T}$ means action will not cause the agent to collide with or violate the safety of the backup plan in perpendicular bundles.", "By the definition of the unprotected-left-turn oracle, $Ag$ will only take the left-turn action when it does not violate the safety of the backup plan of agents in oncoming traffic." ], [ "Safety Proof", "Theorem 8.10 Given all agents $\\text{Ag} \\in \\mathfrak {A}$ in the quasi-simultaneous game select actions in accordance to the Agent Protocol specified in Section , we can show the safety property $P \\Rightarrow \\square Q$ , where the assertion $P$ is an assertion that the state of the game is such that $\\forall Ag, S_{\\text{Ag}, bp}(s,u) = \\texttt {T}$ , i.e.", "each agent has a backup plan action that is safe, as defined in REF .", "We denote $P_t$ as the assertion over the state of the game at the beginning of the time-step $t$ , before agents take their respective actions.", "$Q$ is the assertion that the agents never occupy the same grid point in the same time-step (e.g.", "collision never occurs when agents take their respective actions during that time-step).", "We denote $Q_t$ as the assertion for the agent states/actions taken at time-step $t$ .", "To prove an assertion of this form, we need to find an invariant assertion $I$ for which i) $P \\Rightarrow I$ , ii) $I \\Rightarrow \\square I$ , and iii) $I \\Rightarrow Q$ hold.", "We define $I$ to be the assertion that holds on the actions that agents select to take at a time-step.", "We denote $I_t$ to be the assertion on the actions agents take at time $t$ such that $\\forall Ag$ , $\\text{Ag}$ takes $a\\in Act_{\\text{Ag}}$ where 1) it does not collide with other agents and 2) $\\forall \\text{Ag}, S_{\\text{Ag}, bp}(u^{\\prime }) = \\texttt {T}$ where $s^{\\prime } = \\tau _{\\text{Ag}}(s, a)$ , and $u^{\\prime }$ is the corresponding global state of the game after $\\text{Ag}$ has taken its action $a$ .", "It suffices to assume: Each $\\text{Ag} \\in \\mathfrak {A}$ has access to the traffic light states.", "There is no communication error in the conflict requests, token count queries, and the agent intention signals.", "All intersections in the road network $R$ are governed by traffic lights.", "The traffic lights are designed to coordinate traffic such that if agents respect the traffic light rules, they will not collide.", "Agents follow the agent dynamics defined in Section REF .", "For $t=0$ , $\\forall \\text{Ag} \\in \\mathfrak {A}$ in the quasi-simultaneous game is initialized to: Be located on a distinct grid point on the road network.", "Have a safe backup plan action $a_{bp}$ such that $S_{\\text{Ag}, bp}(s, u) = \\texttt {T}$ .", "We can prove $P \\Rightarrow \\square Q$ by showing the following: $P_t \\Rightarrow I_t$ .", "This is equivalent to showing that if all agents are in a state where $P$ is satisfied at time $t$ , then all agents will take actions at time $t$ where the $I$ holds.", "In the case that the assertion $P_t$ holds, let us show that $\\text{Ag}$ will only choose an action $a \\in Act_{Ag}$ for which the following two conditions hold: 1) $\\mathcal {G}_{Ag}(s, a) \\cap (\\cup _{Ag^{\\prime }\\in S}\\mathcal {G}_{\\text{Ag}^{\\prime }}(s^{\\prime }, a^{\\prime })) = \\emptyset $ and 2) $\\forall Ag^{\\prime }\\in S$ , $\\lnot ((\\text{Ag}, a)\\bot \\text{Ag}^{\\prime })$ , where the set $S$ is: The set $S \\triangleq \\lbrace \\text{Ag}^{\\prime }\|\\text{Ag} \\prec \\text{Ag}^{\\prime }\\rbrace $ , i.e.", "agents with higher precedence than $\\text{Ag}$ .", "Proof by Lemma REF .", "$S \\triangleq \\lbrace \\text{Ag}^{\\prime } \| \\text{Ag}^{\\prime } \\prec \\text{Ag} \\rbrace $ , i.e.", "agents with lower precedence than $\\text{Ag}$ .", "Proof by Lemma REF .", "$S \\triangleq \\lbrace \\text{Ag}^{\\prime } \| \\text{Ag}^{\\prime } \\sim \\text{Ag} \\rbrace $ , i.e.", "agents with equal precedence than the agent.", "Proof by Lemma REF .", "$S \\triangleq \\lbrace \\text{Ag}^{\\prime } \| \\text{Ag}^{\\prime } \\lnot \\sim \\text{Ag} \\rbrace $ , i.e.", "agents with precedence incomparable to the agent.", "Proof by Lemma REF .", "The set of all agents, agents with lower precedence, higher precedence, equal precedence, and incomparable precedence, is complete and includes all agents.", "By REF -REF and REF , an agent will not take an action that will cause collision with any other agents (including itself) or violate the safety of the safety backup plan of all other agents, and thus any action taken by any agent will be such that following the action, the assertion $P$ still holds.", "$P_t \\Rightarrow I_t$ .", "This is equivalent to showing that if all agents are in a state where $P$ is satisfied at time $t$ , then all agents will take actions at time $t$ where the $I$ holds.", "This can be proven using arguments based on the design of the Agent Protocol.", "More details can be found in Lemmas A.REF -A.REF in the Appendix.", "$I \\Rightarrow \\square I$ .", "If agents take actions at time $t$ such that the assertion $I_t$ holds, then by the definition of the assertion $I$ , agents will end up in a state where at time t+1, assertion $P$ holds, meaning $I_t \\Rightarrow P_{t+1}$ .", "Since $P_{t+1} \\Rightarrow I_{t+1}$ , from REF , we get $I \\Rightarrow \\square I$ .", "$I \\Rightarrow Q$ .", "This is equivalent to showing that if all agents take actions according to the assertions in $I$ , then collisions will not occur.", "This follows from the invariant assertion that agents are taking actions that do not cause collision, and the fact that all $\\text{Ag}$ have a safe backup plan action $a_{bp}$ to choose from, and thus will always be able to (and will) take an action from which it can avoid collision in future time steps." ], [ "Liveness Lemmas", "Lemma 8.11 If the only $a \\in Act_{Ag}$ for an agent $\\text{Ag}$ for which $O_{\\text{destination reachability}}(s,a,u) = \\texttt {T}$ and $O_{\\text{forward progress}}(s,a,u) = \\texttt {T}$ is an action such that: $\\gamma _{\\text{Ag}} \\in \\lbrace \\texttt {right-turn, left-turn}\\rbrace $ and the grid-point $s_f = \\tau _{\\text{Ag}}(s, a)$ is unoccupied (for a left-turn, where $a$ is the final action of the left-turn maneuver), $\\text{Ag}$ will always eventually take $a$ .", "W.l.o.g., let us consider agent $\\text{Ag} \\in \\mathfrak {A}$ in the quasi-simultaneous game $\\mathfrak {G}$ .", "We prove this by showing that all criteria required by the Agent Protocol are always eventually satisfied, thereby allowing $\\text{Ag}$ to take action $a$ .", "By the definition of $\\mathfrak {R}$ and the agent dynamics, when $\\text{Ag}$ is in a position where only $\\gamma _{\\text{Ag}} \\in \\lbrace \\texttt {right-turn}, \\texttt {left-turn}\\rbrace $ , it will neither send nor receive requests from other agents and $\\mathcal {F}_{\\text{Ag}}(u, a_i)$ will never be set to T. In accordance with the Action Selection Strategy, for $\\text{Ag}$ to take action $a$ , all the oracles in the Agent Profile must be simultaneously satisfied (so it will be selected over any other $a^{\\prime }\\in Act_{\\text{Ag}}$ ).", "Thus, we show: The following oracle evaluations will always hold when $\\text{Ag}$ is in this state: $O_{\\text{traffic intersection lane-change}}(s,a,u)=\\texttt {T}$ ,$O_{\\text{legal orientation}}(s,a,u)=\\texttt {T}$ , $O_{\\text{static safety}}(s,a,u) = \\texttt {T}$ and $ O_{\\text{traffic intersection clearance}}(s,a,u) = \\texttt {T}$ .", "The first oracle is true vacuously and the following are true by the road network constraints and agent dynamics, Assumption 8, and the assumption in the lemma statement that $s_f = \\tau (s,a)$ is unoccupied respectively.", "To show that the following oracles will always eventually simultaneously hold true, let us first consider when $\\gamma = \\lbrace \\texttt {right-turn}\\rbrace $ .", "By the assumption, the traffic light is red for a finite time, and when the traffic light is green, $O_{\\text{traffic light}}(s,a,u) = \\texttt {T}$ .", "$O_{\\text{unprotected left-turn}}(s,a,u)$ is vacuously true for a right-turn action.", "Since $O_{\\text{traffic intersection clearance}}(s,a,u) = \\texttt {T}$ and by the safety proof REF , all $\\text{Ag}$ are only taking actions in accordance with traffic laws so there will never be any $\\text{Ag}^{\\prime }\\in \\mathfrak {A}$ blocking the intersection, making $O_{\\text{dynamic safety}}(s,a,u) = \\texttt {T}$ .", "Thus, all oracles are always eventually simultaneously satisfied and $\\text{Ag}$ can take $a$ where $\\gamma = \\lbrace \\texttt {right-turn}\\rbrace $ Let us consider when $\\gamma _{\\text{Ag}} = \\lbrace \\texttt {left-turn} \\rbrace $ .", "By Assumption 7, traffic lights are green for a finite time.", "By the safety proof REF , all $\\text{Ag}$ are only taking actions in accordance with traffic laws so there will never be any $\\text{Ag}^{\\prime }\\in \\mathfrak {A}$ blocking the intersection.", "When $\\gamma _{\\text{Ag}} = \\texttt {left-turn}$ , by definition of the unprotected left-turn oracle, $\\square \\lozenge O_{\\text{unprotected left-turn}}(s,a,u)$ , specifically when the traffic light switches from green to red and $\\text{Ag}$ has been waiting at the traffic light.", "Thus, $\\square \\lozenge O_{\\text{unprotected left-turn}}(s,a,u)$ after the light turns from green to red.", "Further, $O_{\\text{unprotected left-turn}}(s,a,u) = \\texttt {T}$ combined with $O_{\\text{traffic intersection clearance}}(s,a,u) = \\texttt {T}$ implies $O_{\\text{dynamic safety}}(s,a,u) = \\texttt {T}$ .", "Thus, all oracles are always eventually simultaneously satisfied and $\\text{Ag}$ can take $a$ where $\\gamma = \\lbrace \\texttt {left-turn}\\rbrace $ .", "Thus, we have shown all oracles in the Agent Profile will always eventually be satisfied, and $\\text{Ag}$ will take $a$ such that $O_{\\text{destination reachability}}(s,a,u)= \\texttt {T}$ and $O_{\\text{forward progress}}(s,a,u)= \\texttt {T}$ .", "Lemma 8.12 If the only $a \\in Act_{\\text{Ag}}$ for which $O_{\\text{destination reachability}}(s,a,u)= \\texttt {T}$ and $O_{\\text{forward progress}}(s,a,u)= \\texttt {T}$ is when $a$ has $\\gamma _{\\text{Ag}} \\in \\lbrace \\texttt {right-lane change, left-lane change}\\rbrace $ and the grid-point(s) $\\mathcal {G}(s,a)$ is (are) either unoccupied or agents that occupy these grid points will always eventually clear these grid points, $\\text{Ag}$ will always eventually take this action $a$ .", "W.l.o.g., let us consider agent $\\text{Ag} \\in \\mathfrak {A}$ in the quasi-simultaneous game $\\mathfrak {G}$ .", "We prove this by showing that all criteria required by the Agent Protocol are always eventually satisfied, thereby allowing $\\text{Ag}$ to take its action $a$ .", "Let us consider Case A, when $a$ is such that $s_f = \\tau _{\\text{Ag}}(s,a) = \\texttt {Goal}_{\\text{Ag}}$ , i.e.", "the action takes the agent to its goal, and let us show that $\\text{Ag}$ will always eventually be able to take $a$ .", "In accordance with the Action Selection Strategy, for $\\text{Ag}$ to take $a$ is that 1) all the oracles in the agent profile must be simultaneously satisfied (so the action $a$ is chosen over any other $a^{\\prime }\\in Act_{\\text{Ag}}$ , 2) $\\mathcal {F}_{\\text{Ag}}(u, a_i)=0$ , and 3) $W_{\\text{Ag}} = \\texttt {T}$ .", "We first show all the oracles for $\\text{Ag}$ will always be simultaneously satisfied: When $\\text{Ag}$ is in this state, the following oracle evaluations always hold: $O_{\\text{traffic light}}(s,a,u) = \\texttt {T}$ , $O_{\\text{traffic intersection lane-change}}(s,a,u)=\\texttt {T}$ , $O_{\\text{unprotected left turn}}(s,a,u)=\\texttt {T}$ , $\\square \\lozenge O_{\\text{traffic intersection clearance}}(s,a,u)$ , $O_{\\text{static safety}}(s,a,u) = \\texttt {T}$ , $O_{\\text{traffic orientation}}(s,a,u)=\\texttt {T}$ .", "The first four hold vacuously, the others hold by Assumption 8, and the last holds by Agent dynamics and the Road Network.", "$O_{\\text{dynamic safety}}(s,a,u)=\\texttt {T}$ .", "By the definition Road Network $\\mathfrak {R}$ , agent dynamics in Section REF , and the condition that $\\forall \\text{Ag} \\in \\mathfrak {A}$ will leave $\\mathfrak {R}$ (i.e.", "$\\text{Ag}$ does not occupy any grid point on $\\mathfrak {R}$ when it reaches its respective goal $\\texttt {Goal}_{\\text{Ag}}$ ).", "Thus, $O_{\\text{dynamic safety}}(s,a,u)=\\texttt {T}$ whenever an agent is in this state.", "In accordance with the action selection strategy, for $\\text{Ag}$ to take $a$ , it must be that $\\mathcal {F}_{\\text{Ag}}(u, a_i)=0$ , i.e.", "the max-yielding-flag-not-enough must not be set.", "Let us show that this is always true.", "The only $\\text{Ag}^{\\prime }$ that can cause the $\\mathcal {F}_{\\text{Ag}}(u, a_i)=1$ of $\\text{Ag}$ is when an agent $\\text{Ag}^{\\prime }$ is in a state where $La(Ag^{\\prime }) = \\texttt {Goal}_{\\text{Ag}}$ .", "W.l.o.g.", "let us consider such an $\\text{Ag}^{\\prime }$ .", "By liveness Assumption 9, upon approaching the goal, the agent $\\text{Ag}^{\\prime }$ must be in a state where $\\text{Ag}^{\\prime }$ backup plan action $a_{bp}$ will allow it to a complete stop before reaching its goal.", "By REF , $\\text{Ag}^{\\prime }$ will always be in a state for which the max-yielding-not-enough flag for $\\text{Ag}$ is $\\mathcal {F}_{\\text{Ag}}(u, a_i)=0$ .", "In order for $\\text{Ag}$ to take $a$ , it must be that $W_{Ag}=1$ .", "Let us show that this is always eventually true.", "In the case that $\\text{Ag}$ has the maximum number of tokens, $\\mathcal {W}_{Ag} = 1$ and $\\text{Ag}$ will be able to take its forward action since all criteria are satisfied.", "Any $\\text{Ag}^{\\prime } \\in \\mathcal {C}_{Ag}$ will be of equal or lower precedence than $\\text{Ag}$ .", "Any $\\text{Ag}^{\\prime }$ with the maximum number of tokens will move to its goal since $\\mathcal {W}_{\\text{Ag}}=1$ and all the other criteria required for that agent to take its action will be true.", "By definition of the Action Selection Strategy in Section REF , any agent $\\hat{\\text{Ag}}$ that replaces $\\text{Ag}^{\\prime }$ will have taken a forward progress action and its respective token count will reset to 0.", "Thus, any $\\text{Ag}^{\\prime }$ will be allowed to take its action before $\\text{Ag}$ , but $\\text{Ag}$ 's token count $\\texttt {Tc}_{\\text{Ag}}$ will increase by one for every time-step this occurs.", "Thus, by REF and by REF , $\\text{Ag}$ will always eventually have the highest token count in its conflict cluster such that $W_{\\text{Ag}}=1$ .", "Since conditions REF and REF are always true, and REF is always eventually true, then all conditions will simultaneously always eventually be true and the $\\text{Ag}$ will always eventually take the action $a$ .", "Let us consider Case B, when $a$ is the final action to take for an agent to reach its sub-goal (i.e.", "a critical left-turn or right-turn tile), and let us show $\\text{Ag}$ will always eventually be able to take a forward progress action where $\\gamma _{\\text{Ag}} \\in \\lbrace \\texttt {left-lane change}, \\texttt {right-lane change}\\rbrace $ .", "In accordance with the Action Selection Strategy, for $\\text{Ag}$ to take $a$ is that 1) $W_{\\text{Ag}}=1$ , 2) $\\mathcal {F}_{\\text{Ag}}(u, a_i)=0$ , i.e.", "the max-yielding-flag-not-enough must not be set and 3) all the oracles in the Agent Profile must be simultaneously satisfied.", "Let us first consider when $W_{\\text{Ag}}=1$ , then $\\square W_{\\text{Ag}}$ until $\\text{Ag}$ takes its forward progress action $a$ because by definition of $W_{\\text{Ag}}$ , $\\text{Ag}$ has the highest token count in its conflict cluster, $\\text{Ag}.\\texttt {tc} = \\text{Ag}.\\texttt {tc}+1$ , while $\\text{Ag}$ does not select $a$ (and thus does not make forward progress) and any $\\text{Ag}$ that newly enters $\\text{Ag}$ 's conflict cluster will have a token count of 0.", "All the oracles are either vacuously or trivially satisfied by the assumptions except for $O_{\\text{dynamic safety}}(s,a,u)$ .", "By the Assumption 7, the traffic light will always cycle through red-to-green and green-to-red at the intersection $\\text{Ag}$ is located at.", "By the Assumption on the minimum duration of the red traffic light, all $\\text{Ag}^{\\prime }$ will be in a state such that $\\mathcal {F}_{\\text{Ag}}(u, a_i) = 0$ .", "By the lemma assumption that all $Ag^{\\prime }$ occupying grid points will always eventually take their respective forward progress actions, $\\square \\lozenge O_{\\text{dynamic safety}}(s,a,u)$ .", "Thus, all criteria for which $\\text{Ag}$ can take its forward progress action $a$ will be simultaneously satisfied.", "When $W_{\\text{Ag}}$ = 0, we must show $\\square \\lozenge W_{\\text{Ag}}$ .", "For $\\text{Ag}$ , all agents in its conflict cluster have equal or lower precedence and are not in the same lane as $\\text{Ag}$ .", "For any such $\\text{Ag}^{\\prime }$ with equal precedence, $\\text{Ag}^{\\prime }$ will always eventually take its forward progress action by the arguments in REF -REF if $\\text{Ag}^{\\prime }$ intends to make a lane-change.", "By the lemma assumption, any agents $Ag^{\\prime }$ occupying the grid points that $Ag$ needs to take its action will always eventually take its forward progress action so $\\square \\lozenge O_{\\text{dynamic safety}}(s,a,u)$ .", "Any $\\hat{\\text{Ag}}$ with lower precedence and higher token count that $\\text{Ag}$ will take $\\text{Ag}^{\\prime }$ 's position and in doing so will have a token count of 0 and any $\\text{Ag}$ that replaces any agents with higher token count than $\\text{Ag}$ and is in $\\text{Ag}$ 's conflict cluster will have token count 0.", "Thus $\\square \\lozenge W_{\\text{Ag}}$ .", "Lemma 8.13 Let us consider a road segment $rs \\in RS$ where there exist grid points $g \\in \\mathcal {S}_{\\text{sinks}}$ .", "Every $\\text{Ag} \\in rs$ will always eventually be able to take $a \\in Act_{\\text{Ag}}$ for which $O_{\\text{forward progress}}(s,a,u) = \\texttt {T}$ .", "We prove this by induction.", "W.l.o.g, let us consider $\\text{Ag} \\in \\mathfrak {A}$ .", "Let $m_{\\text{Ag}}=\\text{proj}_{\\text{long}}(\\texttt {Goal}_{\\text{Ag}}) -\\text{proj}_{\\text{long}}(Ag.s)$ .", "Base Case: $m_{\\text{Ag}}=1$ , i.e.", "$\\text{Ag}$ only requires a single action $a$ to reach its goal $\\texttt {Goal}_{\\text{Ag}}$ .", "If $a$ is such that $\\gamma _{Ag} \\in \\lbrace \\texttt {left-lane change, right-lane change} \\rbrace $ , then $\\text{Ag}$ will take always eventually this action by Lemma REF .", "If $a$ is such that $\\gamma _{\\text{Ag}} = \\texttt {straight}$ : In accordance with the Action Selection Strategy, for $\\text{Ag}$ to take $a$ is that 1) all the oracles in the agent profile must be simultaneously satisfied (so the action $a$ is chosen over any other $a^{\\prime }\\in Act_{\\text{Ag}}$ , and 2) $W_{\\text{Ag}}$ = 1.", "First, we show that all oracles in the agent profile will always be simultaneously satisfied.", "These all follow from the same arguments presented when $\\gamma _{Ag} = \\lbrace \\text{right-lane change}, \\text{left-lane change}\\rbrace $ in Case A in Lemma REF .", "In accordance with the Action Selection Strategy, we must show that $\\square \\lozenge W_{\\text{Ag}}$ .", "This is vacuously true since no $\\text{Ag}$ will be in the agent's conflict cluster when an agent is in this state.", "Case $m=N$ : Let us assume that any $\\forall \\text{Ag}$ where $m_{\\text{Ag}} = N$ always eventually take $a \\in Act_{\\text{Ag}}$ for which $O_{\\text{forward progress}}(s,a,u)=\\texttt {T}$ .", "Case $m=N+1$ : Let us show $\\forall \\text{Ag}$ where $m_{\\text{Ag}}=N+1$ always eventually take $a$ for which $O_{\\text{forward progress}} = \\texttt {T}$ .", "Any $\\text{Ag}$ for which $m_{\\text{Ag}} > 1$ will always have an $a$ where $\\gamma _{\\text{Ag}} = \\texttt {straight}$ such that $O_{\\text{forward progress}}(s,a,u) = \\texttt {T}$ .", "Thus, we show that $\\text{Ag}$ always eventually will take $\\gamma _{\\text{Ag}} = \\texttt {straight}$ such that $O_{\\text{forward progress}}(s,a,u) = \\texttt {T}$ .", "W.l.o.g., let us consider $\\text{Ag}$ for which $m_{\\text{Ag}} = N+1$ .", "In accordance with the Action Selection Strategy, for $\\text{Ag}$ to take $a$ is 1) $W_{\\text{Ag}}$ = 1 and 2) all the oracles in the agent profile must be simultaneously satisfied (so the action $a$ is chosen over any other $a^{\\prime }\\in Act_{\\text{Ag}}$ ).", "In accordance with the Action Selection Strategy, we must show $\\square \\lozenge W_{\\text{Ag}}$ .", "Any $\\text{Ag}^{\\prime } \\in \\mathcal {C}_{\\text{Ag}}$ will be an agent of equal or higher precedence and in separate lane.", "Any such agent with higher token count than $\\text{Ag}$ that is in its conflict cluster will always eventually be able to go by the inductive assumption in REF .", "After all such agents take a forward progress action, they will no longer be in $\\text{Ag}$ 's conflict cluster and $\\text{Ag}$ will have the highest token count since all $Ag$ that newly enter the conflict cluster will have token count of 0.", "After the assignment $W_{\\text{Ag}} = 1$ , $\\square W_{\\text{Ag}}$ until $\\text{Ag}$ selects $a$ .", "This is true because by definition of $W_{\\text{Ag}}$ , $\\text{Ag}$ has the highest token count in its conflict cluster, $\\text{Ag}.\\texttt {tc} = Ag.\\texttt {tc}+1$ , while $\\text{Ag}$ does not select $a$ , and any $\\text{Ag}$ that enters $\\text{Ag}$ 's conflict cluster will have a token count of 0.", "Let us show that the oracles in the Agent Profile will always evaluate to T. The same arguments hold here as in Lemma REF .REF for all oracles except for $O_{\\text{dynamic safety}}(s,a,u)$ , where $\\square \\lozenge O_{\\text{dynamic safety}}(s,a,u) = \\texttt {T}$ by the inductive Assumption REF .", "Lemma 8.14 Let $\\text{Ag}$ be on a road segment $rs \\in RS$ , where $RS$ is the set of nodes in the dependency road network dependency graph $\\mathcal {G}_{\\text{dep}}$ .", "Let $rs$ be a road segment for which $\\forall rs^{\\prime } \\in RS s.t.", "\\exists e: (rs^{\\prime }, rs)$ .", "Each road segment $rs^{\\prime }$ has vacancies in the grid points where $\\text{Ag} \\in rs$ would occupy if it crossed the intersection (i.e.", "$s_f = \\tau _{\\text{Ag}}(s,a)$ ), and we show that $\\text{Ag}$ will always eventually take an action $a \\in Act_{\\text{Ag}}$ where $O_{\\text{progress oracle}}(s,a,u) = \\texttt {T}$ .", "We prove this with induction.", "W.l.o.g., let us consider $\\text{Ag} \\in \\mathfrak {A}$ .", "Let $m_{\\text{Ag}}=\\text{proj}_{\\text{long}}(g_{\\text{front of rs}}) -\\text{proj}_{\\text{long}}(\\text{Ag}.s)$ , where $g_{\\text{front of intersection}}$ represents a grid point at the front of the road segment.", "Base Case $m_{\\text{Ag}} = 0$ : Let us consider an $\\text{Ag}$ whose next action will take will bring $\\text{Ag}$ to cross into the intersection and show that $\\text{Ag}$ will always eventually take $a$ for which $O_{\\text{forward progress}}(s,a,u) = \\texttt {T}$ .", "If the only $a$ where $O_{\\text{forward progress}} = \\texttt {T}$ is such that $\\gamma _{\\text{Ag}} \\in \\lbrace \\texttt {left-turn}, \\texttt { right-turn}$ }, proof by Lemma REF .", "If the only $a$ where $O_{\\text{forward progress}}(s,a,u) = \\texttt {T}$ is such that $\\gamma _{\\text{Ag}} = \\texttt {straight}$ .", "In accordance with the Action Selection Strategy, for $\\text{Ag}$ to take $a$ is that 1) all the oracles in the Agent Profile must be simultaneously satisfied (so the action $a$ is chosen over any other $a^{\\prime }\\in Act_{\\text{Ag}}$ , 2) $W_{\\text{Ag}}$ = 1.", "$O_{\\text{unprotected left-turn}}(s,a,u)= \\texttt {T}$ , $O_{\\text{traffic intersection lane-change}}(s,a,u)= \\texttt {T}$ , $O_{\\text{static safety}}(s,a,u)= \\texttt {T}$ , $O_{\\text{traffic intersection clearance}}(s,a,u) = \\texttt {T}$ $O_{\\text{legal orientation}}(s,a,u)= \\texttt {T}$ .", "The first two oracles are true vacuously, followed by Assumption 8, and by agent dynamics and the road network $\\mathfrak {R}$ definition, respectively, and by the assumption in the lemma statement.", "$\\square \\lozenge O_{\\text{traffic light}}(s,a,u)$ by Assumption 7.", "$O_{\\text{dynamic obstacle}}(s,a,u) = \\texttt {T}$ because by the safety proof, all $\\text{Ag}$ take $a\\in Act_{\\text{Ag}}$ that satisfy the first top tiers of the agent profile so there will be no $\\text{Ag}^{\\prime }\\in \\mathfrak {A}$ that are in the intersection when the traffic light for $\\text{Ag}$ is green.", "Thus, whenever $O_{\\text{traffic light}}(s,a,u) = \\texttt {T}$ , then it $O_{\\text{dynamic obstacle}}(s,a,u) = \\texttt {T}$ as well.", "$W_{Ag}=1$ vacuously since neither $\\text{Ag}$ or any $\\text{Ag}^{\\prime } \\in \\mathfrak {A}$ will send a conflict request at the front of the intersection since all $a_i$ must satisfy $O_{\\text{traffic intersection lane-change}}(s,a,u)$ according to the Safety Proof in Section AREF .", "By the safety proof in REF , $\\text{Ag}$ will only take $a\\in Act_{\\text{Ag}}$ that satisfy the top two tiers of the Agent Profile, so $\\text{Ag}$ will not take an $a$ where $\\gamma _{\\text{Ag}} \\in \\lbrace \\texttt {left-lane change}, \\texttt {right-lane change}$ } into an intersection.", "Case $m_{Ag} = N$ : Let us assume that $\\text{Ag}$ with $m_{\\text{Ag}} = N$ will always eventually take $a \\in Act_{\\text{Ag}}$ for which $O_{\\text{forward progress}}(s,a,u)=\\texttt {T}$ .", "Case $m_{Ag} = N+1$ : Let us show that any $\\text{Ag}$ that is at a longitudinal distance of $N+1$ from the destination will always eventually take $a$ for which $O_{\\text{forward progress}}(s,a,u) = \\texttt {T}$ .", "Let us consider when $\\text{Ag}$ 's only $a$ such that $O_{\\text{forward progress}}(s,a,u) = \\texttt {T}$ is $\\gamma _{\\text{Ag}} \\in \\lbrace \\texttt {right-lane change}, \\texttt {left-lane change}$ }.", "Although $Ag$ may not have priority (since it does not have max tokens in its conflict cluster), any $Ag$ that occupies grid points $\\mathcal {G}(s,a,u)$ will always eventually make forward progress by Argument REF .", "Further, Once these agents have made forward progress, any $\\hat{Ag}$ that replace $\\text{Ag}^{\\prime }$ will have a $\\texttt {Tc}_{Ag}=0$ and since $Ag$ is always increasing its token counts as it cannot make forward progress, it will always eventually have the max tokens and thus have priority over those grid points.", "Thus, this can be proven by using Case B in Lemma REF .", "For all other $a\\in Act_{\\text{Ag}}$ are actions for which $\\gamma _{\\text{Ag}} = \\texttt {straight}$ , and the same arguments as in the proof of straight actions for $rs$ with $g \\in \\mathcal {S}_{\\text{sinks}}$ in REF hold." ], [ "Liveness Proof", "Theorem 8.15 (Liveness Under Sparse Traffic Conditions) Under the Sparse Traffic Assumption given by REF and given all agents $\\text{Ag} \\in \\mathfrak {A}$ in the quasi-simultaneous game select actions in accordance with the agent protocol specified in Section , liveness is guaranteed, i.e.", "all $\\text{Ag} \\in \\mathfrak {A}$ will always eventually reach their respective goals.", "It suffices to assume: $\\forall \\text{Ag} \\in \\mathfrak {A}$ , $\\forall \\text{Ag}^{\\prime } \\in \\mathbb {B}_{\\text{Ag}}$ , $\\text{Ag}$ knows $\\text{Ag}^{\\prime }.s, Ag^{\\prime }.i$ , i.e.", "the other agent's state $\\text{Ag}.s$ and intended action $a_i$ and all $\\text{Ag}$ within a region around the intersection defined in the Appendix.", "Each $\\text{Ag} \\in \\mathfrak {A}$ has access to the traffic light states.", "There is no communication error in the conflict requests, token count queries, and the agent intention signals.", "For $t=0$ , $\\forall \\text{Ag} \\in \\mathfrak {A}$ in the quasi-simultaneous game is initialized to: Be located on a distinct grid point on the road network.", "Have a safe backup plan action $a_{bp}$ such that $S_{\\text{Ag}, bp}(u) = \\texttt {T}$ .", "The traffic lights are red for some time window $\\Delta t_{\\text{tl}}$ such that $t_{\\text{min}}<\\Delta t_{\\text{tl}}<\\infty $ , where $t_{\\text{min}}$ is defined in the Appendix in Section REF .", "The static obstacles are not on any grid point $g$ where $g.d= 1$ .", "Each $\\text{Ag}$ treats its respective goal $\\text{Ag}.\\texttt {g}$ as a static obstacle.", "Bundles in the road network $\\mathfrak {R}$ have no more than 2 lanes.", "The road network $R$ is such that all intersections are governed by traffic lights.", "and prove: Let us consider a road segment $r \\in RS$ that contains grid point(s) $g \\in \\mathcal {S}_{\\text{sinks}}$ .", "Every $\\text{Ag} \\in r$ will be able to always eventually take $a \\in Act_{Ag}$ for which $O_{\\text{forward progress}}(s,a,u) = \\texttt {T}$ .", "Let us consider a road segment $rs \\in RS$ .", "Let us assume $\\forall rs \\in RS, \\exists (rs, rs^{\\prime }) \\in G_{\\text{dep}}$ , i.e.", "the clearance of $rs$ depends on the clearance of all $rs^{\\prime }$ .", "We use inductive reasoning to show that any $\\text{Ag}$ on $rs$ will always eventually take an $a \\in Act_{\\text{Ag}}$ where $O_{\\text{forward progress}}(s,a,u) = \\texttt {T}$ .", "For any $\\mathfrak {R}$ where the dependency graph $G_{\\text{dep}}$ (as defined in REF ) is a directed-acylcic-graph (DAG), we prove all $\\text{Ag} \\in \\mathfrak {A}$ will always eventually take $a\\in Act_{Ag}$ for which $O_{\\text{forward progress}}(s,a,u)=\\texttt {T}$ inductively as follows.", "A topological sorting of a directed acyclic graph G = (V, E) is a linear ordering of vertices V such that $(u, v) \\in E \\rightarrow u$ appears before $v$ in ordering.", "If and only if a graph $G$ is a DAG, then $G$ has a topological sorting.", "Since $G_{\\text{dep}}$ is a $DAG$ , it has a topological sorting.", "We can then use an argument by induction on the linear ordering provided by the topological sorting to show that all $\\text{Ag}$ always eventually take $a\\in Act_{Ag}$ for which $O_{\\text{forward progress}}(s,a,u)=\\texttt {T}$ .", "Let $l$ denote the linear order associated with the road network dependency graph $G_{\\text{dep}}$ , where an ordering of $l=0$ denotes a road segment with source nodes.", "Base Case $l=0$ .", "This can be proven true by Lemma REF .", "Let us assume this is true for any road segment where $l=N$ .", "Under the Inductive Assumption REF , there will be clearance in any road segment that agent $\\text{Ag}$ depends on for $\\text{Ag}$ to make forward progress to its destination.", "Since all $\\text{Ag}$ are following the traffic laws by the Safety proof in REF , the clearance spots will be given precedence to $\\text{Ag} \\in rs$ for a positive, finite time, and thus the assumptions required in Lemma REF and REF used to prove Lemma REF will hold.", "Thus, the Lemma REF to show that all $\\text{Ag}$ for which $l=N+1$ always eventually take an action for which $O_{\\text{forward progress}}(s,a,u)=\\texttt {T}$ .", "When the graph $G_{\\text{dep}}$ is cyclic, the Sparsity Assumption REF can be used to prove all agents always eventually take an action for which $O_{\\text{forward progress}}(s,a,u)=\\texttt {T}$ .", "The sparsity assumption REF ensures that there is at least one vacancy in any map loop.", "Let us consider $Ag$ inside a map loop.", "Let us consider $Ag$ in the loop for which the vacancy is directly ahead of $Ag$ .", "If the vacancy is directly ahead of $Ag$ , then if the only forward progress action $a$ keeps $Ag$ in the loop, $Ag$ will always eventually take its action by Lemmas REF , REF and the arguments in Lemma REF REF .", "If the only forward progress action $a$ makes $Ag$ leave the loop, $Ag$ will always eventually take its action by the sparsity assumption REF and the inductive arguments in REF .", "By REF , it can then be inductively shown that any $Ag$ in the loop will always eventually have a vacancy for which it can take a forward progress action.", "Let us consider $Ag$ on a road segment that is not part of a map loop.", "Let us consider an action $a$ that takes $Ag$ into a map loop.", "If the grid point required by $Ag$ to make forward progress is occupied, by REF , it will always eventually be unoccupied.", "If the only action $Ag$ can take is such that $\\gamma _{Ag}=\\lbrace \\texttt {lane-change}\\rbrace $ since all $Ag^{\\prime }$ in the loop are reset when they take forward progress action, $Ag$ will always eventually have the max token count.", "Thus, the same arguments in Lemma REF hold.", "If the only action $Ag$ can take is such that $Ag$ crosses into an intersection, the traffic light rules ensure that $Ag$ has precedence over any $Ag$ in the loop.", "Thus, $Ag$ will always eventually take a forward progress action by Lemma REF and Lemma REF REF .", "For any action $a$ that does not take $Ag$ into a map loop, $Ag$ can take a forward action because of the sparsity assumptions REF and the inductive arguments in REF .", "By the induction arguments and by definition of the forward progress oracle $O_{\\text{forward progress}}(s,a,u)$ , all $\\text{Ag}$ will always eventually take actions that allow them to make progress to their respective destinations, and liveness is guaranteed." ], [ "Traffic Light Assumptions", "A traffic light grid point contains three states $g.s=\\lbrace \\texttt {red}, \\texttt {yellow}, \\texttt {green} \\rbrace $ .", "The traffic lights at each intersection are coordinated so that if all agents obey the traffic signals, collision will not occur (i.e.", "the lights for the same intersection will never be simultaneously green) and the lights are both red for long enough such that $\\text{Ag}$ that entered the intersection when the light was $\\texttt {yellow}$ will be able to make it across the intersection before the other traffic light turns $\\texttt {green}$ ." ], [ "Traffic Light Minimum Time", "In order to guarantee that agents will always eventually be able to make a lane-change to a critical tile, the traffic light has to be red for sufficiently long such that any $Ag^{\\prime }$ that may cause $\\mathcal {F}_{\\text{Ag}}(u, a_i) = \\texttt {T}$ is slowed down for long enough such that $Ag$ can take its lane-change action.", "This can be computed simply once given the dynamics of $\\text{Ag}$ .", "Normally a simple heuristic can be used instead of computing this specific lower-bound." ], [ "Simulation Environment Features", "A road network environment, complete with legal lane orientations, intersections, and traffic lights, can be specified via a CSV file.", "The specified (by the user) road network environment forms a map data structure graph, which decomposes the roads into bundles, mentioned in REF .", "The map will automatically parse the boundaries and lane directions of the road network to define where agents can either spawn from or exit the road network.", "In each game scenario, agents will randomly spawn according to a specified spawn rate.", "Each agent has the following attributes in our simulation: parameters like min and max velocity and accelerations, dynamics specified by agent actions and their corresponding occupancy grids, goal location, agent color, ID, token count.", "Note, these attributes can be modified depending on what the user wants to include.", "For each agent, a graph-planning algorithm is used to compute a high-level motion plan on the map graph to get the agent to its goal.", "Each game scenario is comprised of the road network graph and a set of agents (constantly changing over time as new agents spawn and old agents reach their goals and leave).", "The game is simulated forward for a specified number of time steps and the traces from the simulation are saved.", "The animation module in RoSE animates the traces from the simulated game.", "RoSE also offers a collection of debugging tools to help reconstruct scenarios that occurred during a simulated game.", "If the user would like to regenerate the same initialization, the simulation has a feature where users can specify a specific randomization seed.", "There is a configuration tool that allows users to prescribe the states of a set of agents and their respective goals.", "A final debugging tool outputs the variables of the agent that were relevant to the decision-making process." ] ]	2011.14148
[ [ "Multidimensional Persistence Module Classification via Lattice-Theoretic\n Convolutions" ], [ "Abstract Multiparameter persistent homology has been largely neglected as an input to machine learning algorithms.", "We consider the use of lattice-based convolutional neural network layers as a tool for the analysis of features arising from multiparameter persistence modules.", "We find that these show promise as an alternative to convolutions for the classification of multidimensional persistence modules." ], [ "Introduction", "Persistent homology has the ability to discern both the global topology [10] and local geometry [5] of finite metric spaces (e.g.", "embedded weighted graphs, point clouds in $^d$ ) making it a befitting feature for the purposes of training a neural network.", "Single-dimensional homological persistence has drawn recent attention in deep learning [12], [21], [3].", "This is, in part, due to a wide range of efficient software libraries [19], [11], [2] for computing persistent homology, as well as a growing cookbook of recipes for featurizing barcodes from single dimensional persistence, including persistence images [1], persistence landscapes [4], and more exotic methods [13].", "Multidimensional persistence generalizes single-dimensional persistent homology in order to tackle filtrations parameterized in multiple dimensions.", "Unfortunately, there is no complete compact barcode-like characterization of multidimensional persistence modules [7].", "We must make do with incomplete invariants.", "There are various algebraic invariants; in this paper we will use the Hilbert function and the multi-graded Betti numbersCaveat lector: the multi-graded (algebraic) Betti numbers need not be confused with the topological Betti numbers, i.e.", "the rank of homology., both of which are $\\mathbb {N}$ -valued functions on the parameter space.", "The Hilbert function is nothing more than the pointwise (topological) Betti numbers, and the multi-graded Betti numbers have a geometric interpretation in terms of births and deaths [14].", "The rank invariant, another invariant, is shown to be complete in the case of single dimensional persistence [7]; due to its difficulty to compute it is not considered here.", "Multidimensional persistence suffers two hindrances to its usefulness in machine learning.", "First, software for computing multidimensional persistence is scarce.", "To the authors' knowledge, RIVET is the only available software for computing multidimensional persistence [16]; RIVET specializes to 2-dimensional persistence and is focused on interactive visualization rather than machine-interpretable output.", "Second, there has been only very recent and preliminary activity [23], [8] focusing on extraction of suitable features for multidimensional persistent homology.", "A full-fledged deep learning pipeline for multidimensional persistence waits in the wings.", "We hope to ignite further interest in filling both of these gaps.", "In this paper, we propose a naive featurization of multidimensional persistence modules based on the aforementioned invariants, and design an architecture for classifying these persistence modules.", "This architecture employs a lattice-theoretic notion of convolution, thereby respecting the order relation of the parameters of the persistence module.", "We implement our model and compare the performance of our proposed lattice-convolutional architecture with a (simplified) standard convolutional architecture." ], [ "Related work", "Very recent advances have been made to featurize multidimensional persistence modules via landscapes [23] and images [8].", "Our naive approach at featurizing 2-dimensional persistence modules has the advantage of being readily computable with existing software [16].", "Persistence modules supported on lattices are a popular object of study as of late.", "By leveraging Möbius inversion, these persistence modules are shown to generate a stable persistence diagram [17] as well as factor through a convenient category with pseudo-inverses [15]." ], [ "Backgound", "Space constraints require that this section be laconic.", "For a primer on persistent homology, see [10], [6]; for multiparameter persistent homology, see [16].", "An introduction to lattices may be found in [9]." ], [ "Rips complexes and persistent homology", "Let $(\\mathcal {M},d)$ be a finite metric space.", "The Vietoris-Rips complex of $\\mathcal {M}$ at scale $r$ is the abstract simplicial complex $\\text{Rips}_r(\\mathcal {M})$ whose simplices are subsets of $\\mathcal {M}$ of diameter at most $r$ .", "There is a natural inclusion $\\text{Rips}_r(\\mathcal {M})\\rightarrow \\text{Rips}_{r^{\\prime }}(\\mathcal {M})$ for $r \\le r^{\\prime }$ .", "Applying the simplicial homology functor (with coefficients in a field $k$ ) $H_i$ to $\\text{Rips}_r(\\mathcal {M})$ produces a sequence of vector spaces $PH_i(r)$ .", "The inclusions $\\text{Rips}_r(\\mathcal {M}) \\rightarrow \\text{Rips}_{r^{\\prime }}(\\mathcal {M})$ induce maps $PH_i(r) \\rightarrow PH_i(r^{\\prime })$ , producing the data of a persistence module.", "This structure can be compactly described as a functor from $$ , viewed as a category via its standard order structure, to the category $_k$ of vector spaces over $k$ .", "The simplicity of the category $$ gives these persistence modules simple structure: they decompose as direct sums of interval modules $I_{[a,b)}$ , which have $I_{[a,b)}(r) = k$ for $a \\le r < b$ and zero otherwise.", "The maps are the identity where possible and the zero map otherwise." ], [ "Multiparameter persistence", "The Rips construction produces a filtration of simplicial complexes from a finite metric space; it is natural to consider the behavior of the homology functor over a pair of coherent filtrations.", "Consider a finite metric space $(\\mathcal {M}, d)$ and a filtration function $\\rho :\\mathcal {M} \\rightarrow $ .", "This data specifies a bifiltration of simplicial complexes given by $\\mathbb {X}_{r,t} = \\text{Rips}_r\\lbrace x \\in \\mathcal {M} \\mid \\rho (x) \\le t\\rbrace .$ There is a natural inclusion $\\mathbb {X}_{r,t} \\hookrightarrow \\mathbb {X}_{r^{\\prime },t^{\\prime }}$ whenever $(r,t) \\le (r^{\\prime },t^{\\prime })$ in the lattice $\\times $ .", "Composing with the homology functor produces a 2-parameter persistence module $PH_{i}: _+ \\times \\rightarrow ; \\quad (r,t) \\mapsto H_i(X_{r,t}).$ While there does not exist a complete discrete invariant for $PH_i$ , we can extract meaningful features.", "Two particularly informative types of features are the Hilbert function $\\text{Hilb}: _+ \\times \\rightarrow _+;\\quad (r,t) \\mapsto \\dim (PH_i(r,t)),$ and the multi-graded Betti numbers $\\xi _j: _+ \\times \\rightarrow _+$ , for $j =0,1,2$ .", "For $PH_i$ as above, the Hilbert function counts the number of connected components ($i=0$ ), cycles ($i=1$ ), or higher dimensional voids ($i >1$ ) of the complex $\\mathbb {X}_{r,t}$ at each $(r,t) \\in _+\\times $ .", "The multi-graded Betti numbers, on the other hand, capture information about births and deaths of homology classes.$\\xi _j(r,t)$ counts the rank of the $j$ th term of the free resolution of $PH_i$ at grading $(r,t)$ [16]." ], [ "Lattice-theoretic signal processing", "Classical signal processing proceeds by constructing filters for space- or time-indexed signals using convolutional operators.", "These implicitly rely on the algebraic properties of the underlying space, in particular the existence of well-behaved translation operators.", "This fact is exploited in the general framework of algebraic signal processing [22] and extended to more general domains by the theory of graph signal processing [18].", "We here describe a similar extension to signals on a finite lattice proposed in [20].", "A lattice $L$ is a partially ordered set in which every pair of elements $x,y$ has a greatest lower bound (the meet $x \\wedge y$ ) and a least upper bound (the join $x \\vee y$ ).", "These operations and their properties produce an algebraic characterization of lattices; the ordering can be recovered from the algebra and vice versa.", "The key insight of [20] is that the meet and join operations on a lattice define two “shift operators” that can be exploited to define convolutional filters for signals on a lattice.", "That is, for two signals $f,g: L \\rightarrow $ , where $L$ is a lattice, we define $(f _\\wedge g)(x) = \\sum _{a \\in L}f(x\\wedge a)g(a) \\;\\;\\text{and}\\;\\; (f _\\vee g)(x) = \\sum _{a \\in L}f(x\\vee a)g(a).$ A nice class of examples of lattices are given by the sets $^n$ viewed as partially ordered sets, with the ordering $(x_1,\\ldots ,x_n) \\le (y_1,\\ldots ,y_n)$ whenever $x_i \\le y_i$ for all $1 \\le i \\le n$ .", "These are, of course, the indexing sets for persistent homology, suggesting that lattice convolutions over $^n$ or its finite sublattices may be useful in processing data coming from multiparameter persistence computations." ], [ "Lattice Convolutional Neural Networks", "Convolutions over $^2$ (with its abelian group structure) have served as an easily parameterized and efficient set of linear operations adapted to the structure of images.", "Their extreme utility in computer vision problems is owed to the translation equivariance properties of images: humans naturally recognize an image translated via an additive reparameterization as equivalent to the original.", "The data of a multidimensional persistence module is also indexed by $^n$ or a regular finite subset thereof, but its natural algebraic structure is not that of an abelian group.", "Rather, with its partial order structure, the indexing set is a lattice.", "In processing signals associated with a persistence module, it may be useful to take this structure into account rather than imposing the abelian group structure implied by standard convolutions.", "To this end, we construct a lattice convolution-based neural network layer suitable for use with features originating from multidimensional persistence modules.", "To the authors' knowledge, such an architecture has not previously been described, although a special case (where the underlying lattice is a power set) has been implemented in [24].", "We specialize the convolutions described in Section REF to the particular case of regular finite sublattices of $^2$ .", "These may be represented (up to isomorphism) as $L = [m]\\times [n]$ , where $[n]$ is the ordered set $\\lbrace 0,1,\\dots ,n\\rbrace $ .", "The meet and join operations are easily computed elementwise: $(r,t) \\wedge (r^{\\prime },t^{\\prime }) = (\\min (r,r^{\\prime }),\\min (t,t^{\\prime }));\\quad (r,t)\\vee (r^{\\prime },t^{\\prime }) =(\\max (r,r^{\\prime }),\\max (t,t^{\\prime })).$ A lattice convolution layer takes as input an $N_{\\text{in}}$ -dimensional signal $f : [m] \\times [n] \\rightarrow ^{N_{\\text{in}}}$ and outputs an $N_{\\text{out}}$ -dimensional signal $ [m] \\times [n] \\rightarrow ^{N_{\\text{out}}}$ .", "The layer's parameters are given by a function $g: [m]\\times [n] \\rightarrow ^{N_{\\text{out}}\\times N_{\\text{in}}}$ .", "If we label the entries of $f(x,y)$ by $f_i$ and the entries of $g(x,y)$ by $g^i_j$ , the layer then acts by $\\text{MeetConv}(f)(x,y)^j = \\sum _{i} (f_i \\ast _{\\wedge } g^i_j)(x,y) = \\sum _i\\sum _{(a,b) \\in [m]\\times [n]} f_i(x \\wedge a, y \\wedge b)g^i_j(a,b)$ in the case of convolution with respect to the meet operation, and $\\text{JoinConv}(f)(x,y)^j = \\sum _{i} (f_i \\ast _{\\vee } g^i_j)(x,y) = \\sum _i\\sum _{(a,b) \\in [m]\\times [n]} f_i(x \\vee a, y \\vee b)g^i_j(a,b)$ in the case of convolution with respect to the join operation." ], [ "Remark", "Traditional convolutional neural networks are useful in part because the convolution kernels (here the functions $g$ ) can have very small support, reducing the number of parameters that must be learned.", "In the standard convolutional setting, these kernels are implicitly supported in a neighborhood of the origin, but the location of the kernel is not usually explicitly specified.", "In the lattice setting, we do need to specify where the kernel resides.", "In the abelian group case, the kernel is supported near the identity, and similarly, when we treat our domain as a lattice, the kernel should be supported near the neutral element of the operation.", "That is, for a meet convolution, $g$ should be supported at the maximum $(m,n)$ , and for a join convolution, $g$ should be supported at the minimum $(0,0)$ .", "This ensures that the convolution operators are capable of preserving information at every point of the space.", "For instance, if $g(0,0) \\ne 0$ , then $(f \\ast _{\\vee } g)(x,y)$ is a sum of terms including $f(x,y)g(0,0)$ , so all information from one layer can be passed to the next layer.", "A bit more is necessary to avoid degenerate convolutions.", "A kernel supported in a geometrically small neighborhood of the neutral element results in trivial receptive fields for most neurons.", "For example, if $(x,y)$ is greater than the maximal element of the support of $g$ , $(f \\ast _{\\vee } g)(x,y)$ is simply a scalar multiple of $f(x,y)$ .", "As a result, we hypothesize that the appropriate support sets for lattice-convolutional kernels are evenly-spaced sublattices including both $(0,0)$ and $(m,n)$ ." ], [ "Experiments", "We use a portion of the Princeton ModelNet dataset [25] as a source of finite metric spaces.", "This dataset consists of hundreds of 3-dimensional CAD models representing objects from 10 classes.", "We sample points from the 3D models to produce finite metric spaces embedded in $^3$ .", "We then compute the corresponding multidimensional persistence modules, from which we produce features used as an input to a convolutional neural net classifier.", "The pipline thus begins with a 3D polyhedral model, of which 3000 vertices are sampled to produce a point cloud in $^3$ .", "This point cloud then produces a bifiltered simplicial complex, whose degree-0 persistent homology we calculate using RIVET [16], sampled at a discrete grid of $40\\times 40$ points, producing lattice-indexed signals given by the Hilbert function and the multi-graded Betti numbers $\\xi _0, \\xi _1, \\xi _2$ ; four features in total.", "These are then passed to the classifier, which produces a class prediction, in this case one of 10 possible household objects.", "As the filter function on these data sets, we use the codensity function $\\rho _{\\text{codense}}(x;k) = \\left( \\frac{1}{k} \\sum _{y \\in N_k(x)} d(x,y) \\right)^{-1},$ where $N_k(x)$ is the set of the $k$ nearest neighbors to $x$ ; we select $k=100$ ." ], [ "Remark", "The name codensity is appropriate because the points in the densest regions of $\\mathcal {M}$ appear earlier in the filtration.", "A folk theorem is that the 2-parameter persistent homology of a Rips/codensity bifiltration is stable (w.r.t.", "interleaving distance) under non-Hausdorff noise (e.g.", "adding or removing a small number of point samples).", "We compare the performance of two convolutional networks on this classification task.", "One uses the lattice-convolution based layers described in Section , and the other uses standard convolutional layers.", "Each has three convolutional layers followed by two fully connected layers.", "The lattice-based convolution layers are of the form $\\alpha \\cdot \\text{MeetConv}(x) +(1-\\alpha )\\cdot \\text{JoinConv}(x)$ for a hyperparameter $\\alpha \\in [0,1]$ .", "We set $\\alpha = \\frac{1}{2}$ .", "All convolution kernels have dimension $4 \\times 4$ , hidden convolution layers have 16 features, and the final convolution layer has 8 features.", "The first two convolution layers are followed by max-pooling layers with a $2 \\times 2$ kernel.", "For the lattice convolutional layers, the support of the kernel lay in an evenly spaced $4 \\times 4$ grid of points in $[m]\\times [n]$ , while the standard convolutional layers had a traditional contiguously supported kernel.", "The inner fully connected layer has 32 features.", "We use a cross entropy loss function with a softmax in the final layer.", "The lattice-based convolutional architecture is summarized in Figure REF .", "The networks are trained using the Adam gradient algorithm with learning rate $2 \\times 10^{-4}$ for 300 epochs.", "We reserve 10% of the data for testing.", "Results are shown in Figure REF .", "The lattice convolutional network slightly underperforms the standard convolutional classifier.", "This is somewhat disappointing, and perhaps indicates that the relevant information encoded in degree-0 persistent homology is best captured with a geometric notion of locality in $^2$ .", "This work has by no means exhausted the possibilities for lattice convolutions in multidimensional persistence, but does suggest circumspection in evaluating their future use." ], [ "Discussion", "Our proposed featurization for persistence modules is rather naive, but well adapted to the use of lattice convolutions as a data processing method.", "The lattice convolutional neural network shows promise as a method for classifying features arising from a multiparameter persistence module.", "The algebraic perspective on partially ordered sets, exemplified by lattices, may also offer approaches to featurizing more complex invariants of persistence modules.", "In particular, the incidence algebra may offer a natural way to represent the rank invariant [7] in a way amenable to convolution-like operations.", "We hope with these brief experiments to inspire further work on featurizing multidimensional persistence for use in machine learning algorithms.", "Figure: The architecture of our lattice-based convolutional neural network.Figure: Comparison learning curves for the lattice neural network andstandard convolutional neural network." ], [ "Acknowledgments", "The authors would like to thank Chris Wendler for replicating our original experiment and pointing out a coding error in our pytorch implementation.", "We would like to thank the reviewers for providing helpful feedback.", "The author [HR] is partially supported by Office of Naval Research (Grant No.", "N00014-1442-16-1-2010), and the author [JH] is supported by the National Science Foundation (NSF-DMS #1547357)." ] ]	2011.14057
[ [ "Conjecture $\\mathcal{O}$ holds for some Horospherical Varieties of\n Picard Rank 1" ], [ "Abstract Property $\\mathcal{O}$ for an arbitrary complex, Fano manifold $X$, is a statement about the eigenvalues of the linear operator obtained from the quantum multiplication of the anticanonical class of $X$.", "Conjecture $\\mathcal{O}$ is a conjecture that Property $\\mathcal{O}$ holds for any Fano variety.", "Pasquier listed the smooth non-homogeneous horospherical varieties of Picard rank 1 into five classes.", "Conjecture $\\mathcal{O}$ has already been shown to hold for the odd symplectic Grassmannians which is one of these classes.", "We will show that Conjecture $\\mathcal{O}$ holds for two more classes and an example in a third class of Pasquier's list.", "The theory of Perron-Frobenius reduces our proofs to be graph-theoretic in nature." ], [ "arrows decorations.markings vertex=[ circle, draw, inner sep=0pt, minimum size=2pt,]" ] ]	2011.14154
[ [ "Feedback Effects in Repeat-Use Criminal Risk Assessments" ], [ "Abstract In the criminal legal context, risk assessment algorithms are touted as data-driven, well-tested tools.", "Studies known as validation tests are typically cited by practitioners to show that a particular risk assessment algorithm has predictive accuracy, establishes legitimate differences between risk groups, and maintains some measure of group fairness in treatment.", "To establish these important goals, most tests use a one-shot, single-point measurement.", "Using a Polya Urn model, we explore the implication of feedback effects in sequential scoring-decision processes.", "We show through simulation that risk can propagate over sequential decisions in ways that are not captured by one-shot tests.", "For example, even a very small or undetectable level of bias in risk allocation can amplify over sequential risk-based decisions, leading to observable group differences after a number of decision iterations.", "Risk assessment tools operate in a highly complex and path-dependent process, fraught with historical inequity.", "We conclude from this study that these tools do not properly account for compounding effects, and require new approaches to development and auditing." ], [ "Introduction", "As machine learning techniques have developed to replicate human decision-making, their use has forced a reconciliation with existing decision policies: can statistics do better?", "Are the statistics unfair, and are they more unfair than people?", "A number of influential papers in 2015 [11], [12] suggested that accuracy in statistical forecasting methods can and should be used in ‘important’ contexts, where people’s freedom or health or finances are on the line, since these algorithms come with demonstrable accuracy levels.", "These contexts include sentencing and pre-trial decisions, credit scoring, medical testing and selective education access.", "Since then, the release of a ProPublica investigation of a common bail algorithm [1] and retorts from the Criminology field [6], [10] have forced a reckoning among theorists and practitioners about what fairness goals can and cannot be achieved.", "Researchers have emphasized shifting focus from predictions to treatment effects, acknowledging that many of these high-impact decisions are, indeed, highly impactful on individual life-courses [2].", "This revelation introduces the relatively new and under-analyzed topic of fairness in relation to repeated decision processes.", "Individual studies have demonstrated that ‘predictive feedback loops’ can lead to disproportionate over-policing in certain neighborhoods [16], and that these loops can be modeled and simulated to demonstrate sub-optimal allocation in policing and compliance contexts [8], [4].", "The sequential-decision context is truly the norm, rather than the outlier.", "In virtually all high-impact scoring or testing systems, these processes occur (or may occur) numerous times throughout individual life-courses and each are both highly dependent on the past and highly impactful on individuals' futures.", "In light of sequential dependence in high-impact algorithms, this paper analyzes current methods for validating scoring systems as accurate and fair.", "In the criminal legal context, new risk assessment algorithms are touted as data-driven, well-tested tools and often cite one or multiple validation studies that demonstrate a tool’s predictive accuracy and predictive parity between defendants of differing protected classes.", "Virtually all use a single-point-in-time, batch setting to analyze fairness and accountability concerns, with the exception of a few studies about how change in scores over time can better predict future scores [18], [13], [19], [14].", "We show that these tests are not catered to the criminal legal domains, where decisions often occur sequentially at multiple times through a defendant's life.", "We take a close look at the statistical methods used by these studies, and show using simulation experiments that risk assessment tests can fail at meeting a number of fairness definitions even while passing instantial validity tests." ], [ "Validation and One-Shot Testing", "Risk assessment algorithms are developed and then tested for `validity’.", "These experiments, formerly only concerned with predictive validity, now test various potential biases that algorithms may exhibit in new populations.", "Validation experiments have therefore become an important aspect of the risk-assessment development process, and validity is seen as a necessary requisite for any risk assessment algorithm in use.", "What does validity mean?", "While there has been some controversy over the way in which risk assessment tools get developed,In Philadelphia, for example, recidivism was being measured as re-arrest rate, and because of public opposition the sentencing commission began measuring it as subsequent conviction rate.", "remarkably little analysis has been conducted of the best practices for validation in risk assessment.", "As a result, many validation experiments resemble one another.", "Typically, the studies measure a tool’s predictive capacity by analyzing post-conviction arrest rates over a short time-frame.", "They take a group of defendants released from the same jurisdiction in a given time-frame, and determine the average re-arrest rate of defendants with different risk scores over a typical period of one or two years.", "For example, Lowenkamp et al.", "conducted a validation experiment in which they tested the LSI-R and the LSI-Screening Version, which screens defendants to decide whether to administer the more in-depth LSI-R assessment [15].", "Using a look-ahead period of 1.5 years, the study measured re-arrest rate and re-conviction rate, and found that a higher LSI-R score is positively correlated with future incarceration.", "Interestingly, algorithmic risk assessments tend to find disparate validity levels when the same algorithm is used on racially distinct populations.", "Fass et al.", "in 2008 published validation data on the Level of Service Inventory - Revised (LSI-R) algorithm, as well as COMPAS [9].", "Using a dataset of 975 offenders released into the community between 1999-2002 from New Jersey, the measurement period was 12 months.", "The purpose of the study was to see whether these algorithms, trained on mostly white populations, are invalid for a population like New Jersey, which has has “substantial minority” representation in incarceration.", "The study finds “inconsistent validity when tested on ethnic/racial populations” [9], meaning the predictive validity may suffer as the result of differences between the training cohort used to develop the algorithm and the actual demographic breakdown of a jurisdiction.", "Demichele et al.", "in “The Public Safety Assessment: A Re-Validation” use data from Kentucky provided by the Laurence and John Arnold Foundation, which developed the PSA.", "The study measured actual failure-to-appear, new criminal activity, and new violent criminal activity before a trial.", "They found that the PSA exhibited broad validity, but found a discrepancy based on race [5].", "Beyond recidivism, a few studies have focused on the relationship between risk assessment-driven decisions and other life outcomes, including earnings and family life.", "Bruce Western and Sara McLanahan in 2000 published a study entitled “Fathers Behind Bars” that finds alarming impacts of incarceration on family life.", "A sentence to incarceration was found to lower the odds of parents living together by 50-70% [20].", "Dobbie et al.", "published a study that demonstrated that pre-trial detention in Philadelphia on increased conviction rates, decreased future income projects and decreased the probability that defendants would receive government welfare benefits later in life [7].", "The Prison Policy Initiative reports an unemployment rate above 27% for formerly incarcerated people, and find a particularly pronounced effects of incarceration on employment prospects for women of color [3].", "Given the deeply impactful nature of risk-based decisions, validation experiments are surprisingly limited in scope.", "The outcome variable - typically rearrests in a one or two-year window - fail to capture the many ways that a risk-assessment can impact an individual’s family, employment, income, and attitudes - all of which may be relevant in considering recidivism.", "Perhaps more importantly, the various aspects of life impacted by detention are precisely the risk factors that may get picked up by a subsequent judicial decision.", "By treating risk assessment as instantial and analyzing longitudinal effects of a single assignment of risk, validation experiments are only observing part of the picture.", "When we consider the tangible impacts of judicial decisions and relate these impacts to future decisions, we see that there are possible feedback effects in the criminal system.", "The dependence of subsequent judicial decisions on prior judicial decisions is rampant.", "Sentencing guidelines suggest (and often require) judges to give longer sentences to repeat offenders, for example.", "The very notion of responsivity in criminal treatment requires periodic assessments that determine the `progress’ or treatment effect over time for a given defender, and shape punishment accordingly.", "However, treatment of sequential risk-assessments and the possible harms of feedback is missing from a literature that has so exhaustively debated whether incarceration has a criminogenic effect.", "This paper explores how compounding in criminal justice impacts defendants.", "The treatment of risk assessment as innocuous, objective, statistical prediction has clouded rigorous theoretical exploration of lifetime compounding in criminal punishment.", "Using data from Philadelphia, we find that higher confinement sentences significantly increase cumulative future incarceration sentences for defendants.", "Synthesizing data from Philadelphia with a theoretical understanding of feedback in algorithmic risk assessment, we will discuss implications for judges and defendants." ], [ "Contributions", "This paper is meant to critically evaluate the current vetting and auditing process for high-stakes, repeated-use risk assessment algorithms that are deployed in the U.S. criminal legal system.", "First, we develop a generalized sequential scoring-decision model, which can be used in simulation experiments to test for possible compounding effects in group fairness, uncertainty, and punishment.", "Then, using simulation experiments, we demonstrate that a risk assessment can pass validity tests and still exhibit problems with predictive accuracy, group-fairness, and risk-group-difference.", "The broader argument put forward by this paper is that current validation tests do not consider sequential feedback, and are therefore insufficient to approve criminal risk assessments for use.", "Algorithms used in the criminal legal system, credit system, and in other high-impact domains should test for unintended impacts when used repeatedly." ], [ "Model Problem Setting", "We offer a model of repeated high-impact decisions that will help us simulate the purpose and pitfalls of validation tests.", "We use a binary observation-decision system that allows each decision to impact the underlying propensity for a failed observation.", "We can imagine this context as being a repeated parole decision, where an officer uses a risk score at each meeting to decide whether to impose a more restrictive policy on a parolee (e.g.", "curfew), thus limiting employment opportunities and increasing the probability of unlawful behavior.", "Each periodic parole meeting there is some observation of whether the rules were broken, a re-assessment of risk, and a new binary treatment decision.", "The context also has parallels in credit decisions, regulatory compliance checks, ad clicks, and more." ], [ "General Modelling Assumptions", "We begin with a simple model of risk-needs driven decisions.", "Given that existing risk assessment services emphasize their wide applicability, some algorithms are adopted at numerous stages in criminal proceedings.", "Other jurisdictions may use different assessments for policing, bail, sentencing and parole.", "Starting simple, we model risk assessments as instantaneous binary decisions that are separated in time.", "Each decision occurs sequentially, and the outcome is either “high risk” or “low risk”, as visualized in Figure 1.", "Figure: Sequential decision context diagramWe assume here that risk assessments are conducted $T$ times throughout a person’s life, and that the assessment $r_t$ measures some underlying probability of future criminality $p_t \\in [0,1]$ .", "The risk assessment $r$ fully dictates a decision $X_i$ , which denotes some choice of high-risk or low-risk treatment (e.g.", "increased surveillance, or prison security level): $X_i \\in \\bigg {\\lbrace } \\begin{array}{c}\\ 1, \\ \\ \\ \\ \\ \\textit {if defendant is classified high-risk} \\\\0,\\ \\ \\ \\ \\ \\textit {if defendant is classified low-risk}\\end{array}$ We model each assessment using the current state of the world before decision $i$ , denoted $S_{i-1}$ .", "The assessment is a random variable and not deterministic because risk assessment algorithms do not solely determine defendant outcomes - the ultimate decision is still up to a judge, who references the risk assessment score as part of the broader pre-trial policy decision.", "We wish to explore the possibility that outcomes of assessments may impact and alter future assessments.", "As such, our model must enable us to analyze cases where the outcome variable $X_i$ may impact the probability of high-risk classification for $X_{i+1}, X_{i+2}, ... , X_N$ .", "The probability of a high-risk classification at decision $i$ can thus be thought of as a function of some defendant information $D_i$ (gender, race, age) and the history prior decisions, $H_i$ .", "We write the current state of beliefs at $i$ as $S_i = \\lbrace D_i, H_i\\rbrace $ .", "We more accurately portray this dependence on the history of decisions as a branching process, rather than a sequence of decisions, in Figure 2.", "Figure: Branching and Path Dependence in a Binary Risk Classification ScorerEvery major risk assessment algorithm uses information about criminal history to assess risk.", "PSA, for example, measures a defendant's number of prior misdemeanors, felonies, convictions, and violent convictions.", "These numbers add various point values to a risk assessment score, and a threshold value may determine pre-trial detention or cash bail amounts.", "Therefore, the PSA and most (if not all) other algorithms have a reinforcement effect.", "After an individual is convicted with a felony charge, every subsequent risk assessment for the rest of his life will use his criminal history to increase his risk score.", "Thus, initial assessments of risk can hold more `weight' in determining lifetime treatment than later assessments.", "If a person is identified as high-risk in their first encounter with the criminal system, known effects on future crime rates, employment, family life, taxes, and other features will increase the likelihood of subsequent encounters.", "This property of reinforcement is key to modeling our system.", "The process is not Markovian: history matters, and our state of beliefs changes over time.", "Instead, we understand the changing effects of sequential risk-assessments as an Urn process, derived from the classic Pólya Urn model in mathematics [17]." ], [ "Dependence and Reinforcement", "Let's say each risk assessment decision affects subsequent decisions as follows: If $X_{i-1}$ is the risk-assessment outcome for decision $i-1$ , the subsequent probability of a high-risk decision $p_{i}$ is a weighted average between $p_{i-1}$ , the prior probability, and $X_{i-1}$ , the most recent classification: $p_{i} = p_{i-1}\\left[\\gamma _{i}\\right] + X_{i-1}\\left[1-\\gamma _{i}\\right], \\ \\ \\ \\ i \\in \\lbrace 2,...,N\\rbrace ,\\ \\ \\gamma _i \\in [0,1]$ This means that we model updates in risk score by averaging the prior assumed risk and the outcome of a new assessment.", "The $X_{i-1}$ term can be thought of as the marginal effect of a new classification on defendant risk.", "To model reinforcement, we allow $\\gamma _i$ to increase as $i$ increases, letting prior risk score $p_{i-1}$ hold more importance as a defendant is older and has more history.", "This should make intuitive sense - if a defendant has lived out most of his life with a certain propensity for criminal activity (`risk'), the effect of a new assessment should carry less weight.", "Using the above intuition, we'll start by assuming the following relationship between $\\gamma _i$ and $i$ (the number of encounters with the criminal justice system): $\\gamma _i = \\frac{i}{i+1}$ To understand the equation above, let's consider the value of $\\gamma _i$ for varying $i$ .", "In a first encounter with criminal courts where $i=1$ , we'd have $\\gamma _1 = \\frac{1}{2}$ .", "Risk assessment outcome $X_1$ would thus have a very strong impact on future risk assessments.", "When $i$ is high, however, $\\gamma _i$ approaches 1 and new assessments would diminish in weight.", "This is the reinforcement property we're seeking - the more decisions that go by, the less weighty they are in determining a person's lifetime experience with the state's criminal system.", "Thus, our formula for $P(X_{i} \| D,H_i)$ is: $ P(X_{i} \| p_{i-1}, X_{i-1}) = p_{i-1}\\left[\\frac{i}{i+1}\\right] + X_{i-1}\\left[\\frac{1}{i+1}\\right], \\ \\ \\ \\ i \\in \\lbrace 2,...,N\\rbrace $ Let's assume temporarily that every defendant starts off with a probability of high-risk classification $p_1=\\frac{1}{2}$ .", "We model the effect of sequential risk-assessments for different defendants by implementing our iterative equation.", "Below are sample paths for 5 defendants who are subject to ten periodic, evenly spaced assessments over time: Figure: NO_CAPTION In the plot above, each color represents an individual who encounters criminal risk assessments throughout their life.", "Notice that this plot behaves in accordance with the reinforcement effect - initial assessments have large effects on $p_i$ , and later assessments only marginally change the course of the risk level.", "Indeed, the for very large $i$ the risk level approaches a straight-line, meaning that the system reaches a stable propensity for criminal activity.", "Below are the paths of the same five defendants, this time over a total of 100 assessments (so 90 additional assessments): Figure: NO_CAPTION While it is unrealistic that a single person would have one hundred exactly evenly spaced and identical assessments throughout their lives, the behavior of our model seems to cohere with our knowledge of risk-assessments - their output impacts future assessments in a way that reinforces their classification.", "In other words, people detained after being identified as high-risk are more likely to re-offend, spend time in jail, have financial trouble, lose employment, or receive a guilty charge - all of which will affect their level of `risk'." ], [ "Pòlya's Urn Generalization", "The model derived above is an Urn process.", "Borrowing a few theorems from probability theory, we can begin to understand the large-scale, long-term effects that might come about when algorithms are used consecutively throughout a person's life.", "Pòlya's Urn can be used to model path-dependent branching processes that are 'exchangeable', meaning the order of prior events does not matter.This is an assumption that may not hold true for our case, because many algorithms care about how recent a historical event took place.", "PSA, for example, cares about prior failures to appear in court in the past two years.", "However, for the most part, algorithms consider the aggregate number of historical events - number of prior felonies, misdemeanors, convictions, etc.", "These indicators are all exchangeable in the sense that it doesn't matter when in the defendant's life they occurred.", "The model asks what the long-term distribution of blue balls will be in the following random process: An urn contains $R_t$ red balls and $B_t$ blue balls.", "Start at $t=0$ , with an initial mix of $R_0$ and $B_0$ balls.", "for iteration $t \\in \\lbrace 1,...,T\\rbrace $ : Pick a ball randomly from the urn.", "For the ball picked, return it and $k$ additional balls of the same color to the urn." ], [ "Urn Equivalence to a Risk Assessment Model", "We can model reinforcement in algorithmic decision-making as an urn process.", "Our basic defendant model replicates exactly the basic Pòlya process with $R_0 = 1$ , $B_0 = 1$ , and $k=1$ .", "We derive the equivalence in the two processes below.", "Denote the color of the ball selected by pick $i \\in \\lbrace 1,2,...,N\\rbrace $ as: $\\tilde{X}_i \\in \\bigg {\\lbrace } \\begin{array}{c}\\ 1, \\ \\ \\ \\ \\ \\textit {if blue ball is picked} \\\\0,\\ \\ \\ \\ \\ \\textit {if red ball is picked}\\end{array}$ Assuming each ball is picked with equal probability, the probability of picking blue in is given by: $P(\\tilde{X}_i = 1) = \\frac{B_{i-1}}{B_{i-1} + R_{i-1}}$ The total number of ball in the urn is $n_i = R_i + B_i$ .", "The probability of picking blue given all prior picks is denoted as $\\tilde{p}_i$ .", "We can always find $\\tilde{p}_i$ by dividing the number of blue balls in the urn by the total number of balls.", "We've shown that $p_i = \\frac{B_{i-1}}{n_{i-1}}$ .", "After the $i^{th}$ pick, what will be the probability of picking blue?", "We inevitably add $k$ balls into the urn, so $n_{i} = n_{i-1} + k$ .", "In the event that our pick is red, we still have $B_{i-1}$ blue balls, so the probability of picking blue decreases to $\\frac{B_{i-1}}{n_{i-1} + k}$ .", "If we do pick blue, however, the probability increases to $\\frac{B_{i-1} + k}{n_{i-1} + k}$ .", "Thus, the probability of picking blue on the $(i+1)^{th}$ pick, given $B_0, n_0$ and $\\tilde{X}_1$ , is: $\\tilde{p}_{i+1} = \\frac{B_{i-1} + \\tilde{X}_ik}{n_{i-1} + k} $ With a bit of algebra, we can define this probability in terms of the probability for the prior pick: $\\tilde{p}_{i+1} = \\frac{B_{i-1}}{n_{i-1} + k} + \\tilde{X}_i\\frac{k}{n_{i-1} + k} = \\left[\\frac{B_{i-1}}{n_{i-1}}\\right]\\frac{n_{i-1}}{n_{i-1}+k} + \\tilde{X}_i\\frac{k}{n_{i-1} + k}$ $\\therefore \\ \\tilde{p}_{i+1} = \\tilde{p}_{i}\\frac{n_{i-1}}{n_{i-1}+k} + \\tilde{X}_i\\frac{k}{n_{i-1} + k}$ When $k=1$ and $R_0 = B_0 = 1$ , how does $n_i$ behave?", "It starts at $n_0 = 2$ , and after each pick it increments by $k=1$ .", "Thus, $n_i = 2+i$ .", "Equivalently, $n_{i-1} = 1+i$ , and $n_{i-2} = i$ .", "Using the relationship derived above, a shift in index yields the probability of picking blue $\\tilde{p}_i$ for $i \\in \\lbrace 2,...,N\\rbrace $ : $ \\tilde{p}_{i} = \\tilde{p}_{i-1}\\frac{n_{i-2}}{n_{i-2}+k} + \\tilde{X}_{i-1}\\frac{k}{n_{i-2} + k} = \\tilde{p}_{i-1}\\left[\\frac{i}{i+1}\\right] + \\tilde{X}_{i-1}\\left[\\frac{1}{i+1}\\right]$ Notice the equivalence to equation REF .", "We've shown the probability for picking blue at each iteration of the classic Pólya Urn process exactly equals the probability of a high-risk classification in our simple model of sequential risk assessments, where $\\tilde{p}_i = p_i$ and $ \\tilde{X}_i = X_i$ ." ], [ "Long Run Behavior", "When we say that a sequence of random decisions might exhibit reinforcement, we now know that this means something deeper mathematically.", "Random processes with reinforcement behave in certain ways that might be problematic in the context of criminal policy.", "We have a general sense that algorithmic decisions in criminal justice impact defendants profoundly, and likely impact future encounters with law enforcement.", "Leveraging insights from probability theory, we can begin to understand the danger of policies that have compounding effects.", "To start, we analyze the long-term treatment of individuals that are subject to sequential risk-based decisions.", "In Robin Pemantle's “A Survey of Random Processes with Reinforcement\" (2006), the following theorem is reported about Pòlya's Urn process: Theorem 2.1: The random variable $p_i = \\frac{B_i}{B_i + R_i}$ converges almost surely for large $i$ to a limit $P$ .", "The distribution of $P$ is: $P \\sim \\beta (a,b)$ where $a = \\frac{B_0}{k}$ and $b = \\frac{R_0}{k}$ .", "In the case where $a=b=1$ , the limit variable $P$ is uniform on $[0,1]$ .", "[17] Theorem 2.1 lays out how we can expect our modeled risk assessments to behave over many iterations.", "If one person undergoes risk assessments numerous times throughout their life, they may end up in radically different places depending on the risk-assessment outcome.", "They may be able to steer clear of subsequent confinement and re-arrest, or they may be continuously surveiled and repeatedly penalized by the state.", "For a preliminary understanding of how inter-dependence in repeated risk assessments can impact a population, we use our initial modeling assumption that $p_1 = 0.5$ (so $B_0 = R_0$ and $a = b$ ), and imagine varying the parameter that determines the bearing of prior assessments on updated assessments, $k$ (which defines $\\gamma $ ).", "If we decrease $k$ to $0.1$ so that $a = b = \\frac{B_0}{k} = 10$ , we have the following long-term distribution for defendant risk.", "See Figures 3 and 4.", "Figure: Urn Model Plot, p i p_i versus ii for 30 defendants over 15 consecutive risk assessments, k=0.1k=0.1When decisions have little impact on people's lives (and potential subsequent risk assessments), we see consistency in long-term outcomes.", "Everyone starts with a risk score of $0.5$ , and all end up somewhere near there even after many assessments.", "However, if algorithmic-driven decisions are more sensitive to the effect of prior decisions with $a = b = \\frac{B_0}{k} = 0.1$ , then we can see very problematic behavior in the long term.", "See Figures 5 and 6.", "Figure: Urn Model Plot, p i p_i versus ii for 30 defendants over 15 consecutive risk assessments, k=10k=10In this second case, we begin with defendants that are identical in attributes, with an initial probability of high-risk classification $p_1 = 0.5$ .", "However, simply because of the effect of risk-based decision making, defendants end up with radically different risk levels, and are highly likely to be pushed to an extreme (no criminal risk, 0, and extreme criminal risk, 1).", "Of course, these results are purely theoretical and do not come from real observed processes.", "But they motivate the importance of scrutinizing how algorithms are used in practice.", "Algorithms may be validated to ensure that biases are mitigated to a certain confidence threshold.", "But even tiny disparities in the system described by the second plot above can profoundly impact outcomes." ], [ "Discussion", "Understanding that sequential feedback-effects exist in criminal legal decisions forces us to re-evaluate the ways that validations are currently used.", "The effect of prison time and similar decisions on future encounters with criminal punishment implies that algorithmic risk-assessment tools cannot be assessed using instantial experiments at one time in a defendant’s life.", "If larger sentences are associated with greater prison time, it is likely that longer sentences hold bearing on future risk assessment.", "A more severe sentence may lead parole officers to have more discretion over parolees.", "It may increase a defendant’s association with other criminals.", "This kind of dependence between decisions is clear from sentencing tables and three-strikes rules, which recommend that judges give exaggerated sentences to repeat-offenders.", "Since judicial decisions appear to feed into one another sequentially over a defendant’s life time, it is important to consider models that encompass compounding effects.", "Risk assessment algorithms and validation experiments fail to adequately address the potential of feedback effects over time.", "Rigorously considering the impacts of dependent, sequential decisions will be necessary for deploying any high-stakes algorithm." ], [ "Broader Impact", "My hope is that this inquiry exposes some of the shortcomings of auditing in high-impact ML domains.", "The discussion and analysis were specifically about the criminal legal space; however, many of the findings are relevant to the use of high-impact ML algorithms in many fields.", "In credit and medicine, for instance, risk determinations are premised on historical access to resources (e.g.", "capital or medical attention), so when future triage decisions are made, risk-based decisions will always exhibit the effects of historical decisions.", "None of these systems should treat risk as exogenous or innate and should instead have the goal of minimizing harm.", "I'd like to acknowledge Miklos Racz, my undergraduate research advisor who has been helping me pursue and build on my research after graduating.", "I'd like to acknowledge my friends, family, colleagues, and role models who have provided me with all the skills and access necessary to submit to a venue like this one." ] ]	2011.14075
[ [ "Using Multiple Subwords to Improve English-Esperanto Automated Literary\n Translation Quality" ], [ "Abstract Building Machine Translation (MT) systems for low-resource languages remains challenging.", "For many language pairs, parallel data are not widely available, and in such cases MT models do not achieve results comparable to those seen with high-resource languages.", "When data are scarce, it is of paramount importance to make optimal use of the limited material available.", "To that end, in this paper we propose employing the same parallel sentences multiple times, only changing the way the words are split each time.", "For this purpose we use several Byte Pair Encoding models, with various merge operations used in their configuration.", "In our experiments, we use this technique to expand the available data and improve an MT system involving a low-resource language pair, namely English-Esperanto.", "As an additional contribution, we made available a set of English-Esperanto parallel data in the literary domain." ], [ "Introduction", "In this paper, we use the constructed language Esperanto to illustrate potential improvements in the automatic translation of material from low-resource languages.", "Languages are considered low-resource when there is little textual material available in the form of electronically stored corpora.", "They pose significant challenges in the field of Machine Translation (MT), since it is difficult to build models that perform adequately using small amounts of data.", "Multiple techniques have been developed to improve MT in conditions of data scarcity.", "A popular approach is to translate indirectly via a pivot language [30], [6], [14], [19].", "Moreover, indirect translation can be used for creating additional training data.", "A further useful technique for expanding the dataset is back-translation [24].", "This procedure consists of automatically translating a monolingual text from the target language into the selected source language, and then using the resulting parallel set as training data so the model benefits from this additional information.", "Although the quality of these sentence pairs is not as high as that of human-translated sentences (the source side contains mistakes produced by the MT system), the pairs are still useful when used as training data, because they do often improve the models [18].", "Nonetheless, for some languages, the available data are in such short supply that MT models used for generating back-translated sentences may produce a high proportion of noisy sentences.", "The use of noisy sentences for building MT models could ultimately have a negative impact on the quality of the MT system's outputs [8], and therefore they are often removed [9], [27], [22].", "We propose employing another technique to augment datasets: using the same set of sentences multiple times, but in slightly altered form each time.", "Specifically, we modify the sentences by using different Byte Pair Encoding (BPE) [25] merge operations.", "We perform a fine-grained analysis, exploring the use of different splitting options on the source side, on the target side, and on both sides." ], [ "Previous work", "This research is inspired by techniques for augmenting the training set artificially.", "One of these techniques is back-translation [24], which involves creating artificial source-side sentences by translating a monolingual set in the target language.", "Similar techniques include the use of several models to generate sentences [21], [26], or the use of synthetic data on the target side [3], [13].", "A technique that involves multiple segmentation is subword regularization [12], in which candidate sentences with different splits are sampled, either probabilistically or using a language model for training.", "In the work of [20], different splits are used to build an English-Thai MT model.", "As the Thai language does not use whitespace separation between words, different splits can be applied, to address the fact that all the words and sub-words are joined together in the final output.", "More recently, [23] introduced BPE-dropout, an improvement on standard BPE consisting of randomly dropping merges when training the model, such that a single word can have several segmentations." ], [ "The Esperanto language", "This article is concerned with improving MT models for Esperanto, the most successful constructed international language [1].", "It was created in the late nineteenth century, and is said to be currently spoken by over 2 million people, spread across more than 100 countries [5].", "During its first century of development, Esperanto was principally maintained by means of membership-based organisations.", "Currently, internet applications such as Duolingo are supporting the wider spread of the language among new enthusiasts.", "While many Esperanto speakers have sought to develop the language through translation, the body of work available - particularly in digital formats - remains relatively small, making Esperanto a clear example of a low-resource language.", "Esperanto loosely derives its lexicon from several Indo-European languages, and shares some typological characteristics with, among others, Russian, English, and French [16].", "In contrast to most natural languages, Esperanto's most distinctive characteristic is its regularity.", "The grammar consists of a very limited set of operations, to which there are, in principle, no exceptions.", "Furthermore, the language is agglutinative, and its suffixes are independently meaningful and invariable.", "For instance, virino, the word for woman, consists of the compound parts vir [adult human], in [female], and o [entity] (as the 'o' ending is used for all nouns).", "The word for mother, patrino, largely refers to the same semantic categories, and is therefore structurally highly similar.", "As a consequence of this internal consistency, Esperanto learners can quickly expand their vocabulary by learning to segment words into their various parts, which can then be used to construct new words by morphological analogy.", "Because of its affinity with many other languages, and because of the thoroughly logical composition of its vocabulary, Esperanto has historically been central to several experiments in MT, most notably regarding its potential function as a pivot language between European languages [7].", "In this study, however, we focus on automatic translation into Esperanto for its own sake." ], [ "Research Questions", "We propose building MT models using training data composed of a dataset split into multiple variants with a different configuration of BPE, as presented in Figure REF .", "At the top of the figure, one can see that the same parallel set has been processed using BPE with 89,500, 50,000 and 10,000 operations (trained separately for each language).", "The MT model represented on the left has been built using the same dataset replicated three times, the only difference being that on the target side, different splits were implemented.", "Similarly, the MT model in the centre is built with different splits on the source side.", "The last model, represented on the right, combines different splits both on the source and the target side.", "In order to evaluate the models, we use a test set that is split with a single BPE strategy (i.e.", "using 89,500 merge operations, the default proposed in the work of [25]).", "Therefore, using different merge operations on the source side of the training data may not have as big an impact as when they are applied to the target side (not all the words will match those in the test set).", "However, the addition of other BPE configurations could in principle still be useful to improve modeling for the source language.", "In Section we describe the settings of the MT and the data used for training.", "In Section we analyze the results achieved by the baseline system.", "This paper's experiments are divided into three sections.", "Each of these sections describes and also provides the evaluation of a model.", "The sections are the following: Combination of dataset with different merge operations on the target side (Section REF ).", "Combination of dataset with different merge operations on the source side (Section REF ).", "Combination of dataset with different merge operations on both the source and target side (Section REF ).", "In Section , we compare translation examples from the different models and analyze the different outcomes.", "Finally, in Section we conclude and propose how these experiments could be expanded in future work.", "Figure: Diagram with the experiments" ], [ "Experimental Settings", "The NMT systems we build are Transformer [31] models, based on OpenNMT [10].", "Models are trained for a maximum of 30K steps using the recommended parameters.https://opennmt.net/OpenNMT-py/FAQ.html We have selected the model with the lowest perplexity on the development set." ], [ "Dataset", "For training the models we use the Tatoeba, GlobalVoices and bible-uedin [4] datasets from OPUS project.http://opus.nlpl.eu/ Our dataset thus contains material from the Bible, from news sources, and from less domain-specific multilingual translation examples.", "The sentences are randomly shuffled, after which 302,768 sentences are used as a training set and the other 1,000 as our dev set.", "All the sentences are tokenized and truecased.", "BPE is applied using several merge operations.", "We use 89,500 operations as a starting point and explore other splits that produce smaller subword units (by using a lower number of merge operations).", "In our experiments we work with 50,000, 20,000 and 10,000 operations.", "We also concatenate the dev set using the same configuration of BPE." ], [ "Test Set", "In order to evaluate the quality of the models, two test sets are translated.", "The test sets are the same for all models.", "In addition to tokenization and truecase, we also use BPE with 89,500 merge operations.", "We do not use (or combine) other BPE configurations.", "The translations are evaluated using the BLEU [15] metric.", "The first test set is taken from the OPUS (Books) dataset [28] (1562 sentences).", "Specifically, the test set consists of material from two texts available in English and in Esperanto translation, namely Carroll's Alice's Adventures in Wonderland [2] and Poe's The Fall of the House of Usher [17].https://farkastranslations.com/bilingual_books.php The second test set (which contains 1256 sentences) consists of an English and an Esperanto version of Oscar Wilde's Salomé [32], https://en.wikisource.org/wiki/Salom%C3%A9 and http://www.gutenberg.org/ebooks/63064 a play originally written in French.", "As an additional contribution to this paper, we have made a set of aligned sentences from the texts available via OPUS.http://opus.nlpl.eu/Salome-v1.php Both test sets are in the literary domain, which is especially challenging [29] for MT models.", "Not only do the test sets contain numerous personal names and uncommon vocabulary, they are also highly creative and, at times, experimental.", "For instance, in Alice's Adventures in Wonderland, grammatical and lexical principles are often challenged on purpose to portray a character's individual traits (i.e.", "the Mock Turtle sings of Beau–ootiful soo–oop!.", "In Salome, characters regularly produce complex similes and metaphors to describe one another.", "The text is a variation on a religious theme, and heavily draws on Biblical imagery.", "While such material is highly challenging, the inclusion of Biblical matter in the training data may have a positive impact on the overall results." ], [ "Baseline MT", "In Table REF we present the models trained with the training data using different merge operations on the target side.", "The rows of the table correspond to the evaluation of the model, using the same data.", "The only difference is the number of BPE merge operations that have been used on the target side.", "As the test set is split using 89,500 merge operations, it would not be beneficial to apply BPE with merge operations other than 89,500 on the source side.", "In fact, when using BPE with 50,000, 20,000 and 10,000 operations on the source side, the BLEU score for the translation of the Books data is only 5.75, 5.70 and 5.76, respectively, and 14.30, 13.24, and 14.53 for Salome Table REF shows that the four models achieve similar results.", "As mentioned before, the Books set contains complex grammatical and lexical constructions, which makes it more difficult to translate.", "This is also evidenced in the table as BLEU scores of the Books set are lower than those of the Salome set.", "Moreover, there is no correlation between the number of merge operations and the performance.", "For example, we observe a small drop in the performance when decreasing the number of merge operations from 89,500 to 50,000, but the performance improves slightly when the number of operations is further decreased to 20,000." ], [ "Different Merge Operations on the Target Side", "In the first set of experiments we explore the models when the sentences in the parallel set are replicated by changing only the number of BPE merge operations used on the target side.", "We perform two sets of experiments: one where we keep the duplicates (sentences that remain the same after being split with different BPE configurations), and another where duplicates are removed.", "Table: Model performance using different merge operations on the target side.In Table REF we present the results of the models when trained with a different concatenation of datasets.", "The first column specifies the datasets used in the training.", "For example, the row TRG89500 & TRG50000 indicates that the training set used for building the MT model consists of sentences split using 89,500 and 50,000 merge operations, respectively We mark in bold those scores that exceed 6.89 BLEU points, i.e.", "the maximum score achieved by the baseline models presented in Table REF .", "The scores receive an asterisk when the improvements are statistically significant at p=0.01.", "Statistical significance has been computed using Bootstrap Resampling [11].", "In the table, we find that scores tend to be higher when duplicate sentence pairs are removed.", "By doing this the dataset is reduced by between 30% and 45%.", "In the second subtable, all the BLEU scores indicate improvements over the baseline, whereas in the first subtable some models, such as TRG89500 & TRG50000, have a lower score.", "The best performance is seen when for the multiple settings used, the number of merge operations differs greatly.", "For example, the highest scores are achieved when mixing 89,500 and 10,000 operations (i.e.", "the TRG89500 & TRG10000 rows in both subtables), the uppermost and the lowermost number of operations used in the experiments.", "The same principle holds true for those models built by combining three or four datasets." ], [ "Different Merge Operations on the Source Side", "The next set of experiments explores the use of several merge operations on the source side.", "In this case, when combining the datasets, we ensure that the SRC89500 set is used, as the test set has been processed using 89,500 operations.", "We present the results in Table REF .", "Those scores that are higher than the baselines of Table REF are marked in bold.", "Our observations are similar to those obtained in Section REF .", "The best results are observed when the duplicate sentences are removed (between 25% and 40% of the sentences are removed) and the merge operation settings are the furthest apart (89,500 and 10,000).", "Most of the models using several BPE configurations on the source side perform better than the baseline models.", "However, when compared to the experiments in the previous section (Table REF ), the performance is lower." ], [ "Different Merge Operations on both Source and Target Side", "The last set of experiments consists of building a model with data created using different splits both on the source and on the target side.", "We perform experiments based on the outcomes observed in the previous section.", "Thus, two models are built.", "One combines the datasets split using BPE with 89,500 and 10,000 merge operations (both source and target side) and the other model, All, combines the dataset with all the splits (i.e.", "89,500, 50,000, 20,000 and 10,000).Note that we use all possible combinations.", "For example, the training set of the All model is built combining $4*4=16$ datasets.", "The duplicates are removed, as this approach showed the best results.", "We present the translation quality of the test set using these models in Table REF .", "We see that the use of different splits on both the source and target sides tends to achieve the best results when compared both to baselines and to the experiments in the previous sectionsWe observed that the output tends to be more similar to the splits following the TRG89500 configuration.." ], [ "Comparison of Outputs", "In Table REF , we show some translation examples of the models that, as discussed in the previous sections, achieved the best performance.", "We mark in bold some important differences across the translations.", "The first example, drawn from Alice in Wonderland, contains a joke.", "Alice, who is collecting her thoughts, aims to voice her opinion, and starts out by saying I don't think.... Before she can finish her sentence, however, the Mad Hatter interrupts her by stating that in that case, she should not speak.", "The human Esperanto translation makes this joke very explicit by repeating the emphasis on 'not thinking', whereas in English the transition is more subtle.", "Two of the systems, while differing in exact word order, succeed in reproducing the joke (TRG89500 and TRG89500 & TRG10000).", "In the other two models, either the crucial element do [so], which realises the inference, is omitted, or the meaning is mistakenly changed to a positive imperative: vi devus diri [you should say].", "It can further be observed in the sentences that none of the systems translates the Hatter's name meaningfully.", "Either the name remains the same, or it is slightly altered from the original, in a seemingly random manner.", "Interestingly, Alice's name is adapted to Alico, which conforms to the rule that all Esperanto names end in -o (or, in some cases -a), but the adaptation does not equal the human choice for Alicio.", "The second example, also taken from Alice's Adventures in Wonderland, is concerned with a particular fixed expression in the English language: venture to say.", "The baseline system does not translate this mark of politeness, while the other models do provide varying translations (i.e.", "decidis, sukcesis and entrepenis, which correspond to the past tenses of the verb to decide, to succeed and to undertake).", "While none of them is completely correct (when compared to the human translation), all of them are fairly transparent in context, and foreground different aspects of meaning contained in the English venture.", "With reference to the Salome test set, we find in the entire translated text numerous small and relatively inconsequential vocabulary differences across systems (e.g.", "veston or mantelon for referring to a piece of clothing), as well as varying preferences for orthographically similar verb tenses (e.g lacigis or lacigas, past and present tense of the verb to tire or wear out).", "At times, the systems differ in their translation of multi-word units such as sacred person, which is translated either as the literal sankta homo or as the more interpretative sanktulo [saint].", "Overall, the systems perform well when translating the play's dense symbolism, as illustrated in Table REF .", "The examples in the table are similes, which start with the explicit comparative phrase it is like.", "In the first example, the baseline system does not manage to reproduce the reference to serpentoj [snakes], although the mention of turmentoj [afflictions] does offer an interesting metaphorical perspective.", "The system SRC89500 & SRC10000 does not produce a correct translation, but those systems trained with different splits on the target side (i.e.", "the SRC10000 & SRC89500 and All systems) provide a remarkably good translation of the source.", "Similarly, in the last example included in the table, the baseline system fails to reproduce the meaning of the original (the knife falls apart instead of cutting the fruit), whereas all systems with multiple segmentation are successful in conveying a variant of the poetic image presented in the source text.", "In short, the examples in Table REF indicate that a combination of different merge operations may improve results for translation into Esperanto, a language for which limited resources are available.", "In a number of cases, the systems succeed in translating highly uncommon constructions in the context of humorous and poetic literary discourse." ], [ "Conclusion and Future Work", "In this work, we have aimed to improve an English-Esperanto MT system by using multiple instances of the same sentence pair, split with different configurations of BPE.", "In our experiments, the best performance tends to be achieved when splitting strategies are applied both on the source and target side, duplicate parallel sentences are removed, and the number of merge operations used are very different from each other.", "In our experiments, the best results are achieved when all the split-combinations are used on both sides.", "Although the goal of these experiments is to find a technique to improve the MT models when the available data are very limited, this technique could also be applied in scenarios where data are abundant.", "It should be noted that Esperanto is perhaps a particularly suitable candidate for word-split methods, as the language's vocabulary consists of fixed chunks that are combined to form transparent compounds.", "However, the techniques applied here are in principle language-independent.", "Finally, although we demonstrated that combining sentences with different merge operations improves the model, in this paper we could not determine the best configuration to use.", "Similarly, the test set that we used was processed using 89,500 merge operations.", "If the test set had been processed with a different BPE configuration the performance could have been different, especially when using models with different split configurations on the source side.", "Extensions of this work could involve finding an optimal configuration for achieving the best results, or testing the performance when combined with other word-splitting techniques." ], [ "Acknowledgments", "This research has been supported by the ADAPT Centre for Digital Content Technology which is funded under the SFI Research Centres Programme (Grant 13/RC/2106).", "The QuantiQual Project, generously funded by the Irish Research Council’s COALESCE scheme (COALESCE/2019/117)." ] ]	2011.14190
[ [ "A Backward SDE Method for Uncertainty Quantification in Deep Learning" ], [ "Abstract We develop a probabilistic machine learning method, which formulates a class of stochastic neural networks by a stochastic optimal control problem.", "An efficient stochastic gradient descent algorithm is introduced under the stochastic maximum principle framework.", "Numerical experiments for applications of stochastic neural networks are carried out to validate the effectiveness of our methodology." ], [ "Introduction", "In this paper, we introduce an efficient computational framework to quantify the uncertainty of a class of deep neural networks (DNNs).", "The DNN has emerged from machine learning and becomes one of the most extensively studied research topics in scientific computing and data science.", "As a type of artificial neural network with multiple layers, the DNN is capable to model complex systems and its applications cover wide range among various scientific and engineering disciplines.", "However, despite phenomenal success, the deterministic output of DNNs can not produce probabilistic predictions for the uncertain nature of scientific research.", "Therefore, it's difficult to apply DNNs to solve real-world scientific problems.", "To address the challenge of uncertainty quantification, probabilistic machine learning approaches are developed based on probabilistic modeling .", "The state-of-the-art probabilistic learning approach is the Bayesian neural network (BNN) , .", "The main idea of the BNN is to treat parameters in a DNN as random variables, and then approximate distributions of these random parameters.", "Instead of searching for the optimal parameters by deterministic optimization, the BNN utilizes Bayesian optimization to derive parameter distributions.", "As a result, the estimated random parameters generate random output, which characterize the uncertainty of the target model.", "Although the BNN approach provides a neat conceptual framework to quantify the uncertainty in probabilistic machine learning, carrying out Bayesian optimization to estimate a large number of parameters in the BNN poses several computational challenges such like high dimensional integration, high dimensional optimization, and high dimensional approximation.", "In recent studies, the Markov Chain Monte Carlo (MCMC) sampling method is introduced to address the challenge of high dimensional integration in Bayesian inference, and a class of gradient based methods, such like Langevin MCMC and Hamiltonian MCMC , are applied to improve the efficiency of high dimensional optimization.", "However, the primary computational challenge of approximating high dimensional distributions still remains.", "Thus training BNNs is still a computationally extensive task.", "In this work, we focus on an alternative probabilistic learning approach, which allows us to avoid Bayesian optimization.", "The neural network architecture in our approach is an extension of the so-called “Neural ODE”, which formulates the evolution of hidden layers in the DNN as a discretized ordinary differential equation (ODE) system .", "The Neural ODE formulation of deep learning provides a mathematical description for the residual neural network, which is an important neural network structure in machine learning.", "To incorporate uncertainties in the neural network, instead of treating parameters as random variables in the Bayesian approach, we add Gaussian-type noises to the hidden layers and construct a stochastic ordinary differential equation (SDE) formulation of DNN, which will be called the stochastic neural network (SNN) in this paper.", "Such a stochastic construction of neural network has been extensively studied recently , , .", "The major difference between the SNN and the BNN is that the uncertainty of SNN is generated by the diffusion term of SDE and the coefficient of diffusion controls the probabilistic output of SNN.", "Therefore, the point estimation for diffusion coefficients in the SNN approach could achieve the goal of uncertainty quantification for probabilistic learning.", "While the construction of SDE type neural networks and justification for SNNs are well accepted, the training process is also challenging for SNNs.", "In the deterministic DNN, the optimal parameters are determined by (stochastic) gradient descent optimization, and the gradient is calculated by the chain rule.", "However, when trying to find the gradient with respect to parameters in an SDE, the standard chain rule is not applicable due to the stochastic integrals, and Itô calculus is needed, which makes the derivation of the gradient complicated.", "To derive a mathematical expression for the gradient, we formulate the SNN model as a stochastic optimal control problem.", "Specifically, we consider the state of neurons in SNN as controlled state, and the SNN parameters are considered as control terms that govern the controlled state to meet the optimal cost condition, which is chosen to be the minimum loss for the SNN output.", "In this way, the training process for the SNN, which seeks the optimal parameters, is equivalent to the solving process for the stochastic optimal control problem, which determines the optimal control.", "The theoretical framework that we adopt to solve the SNN version of stochastic optimal control problem is the stochastic maximum principle (SMP) due to its advantage in solving high dimensional problems — compared with its alternative approach, i.e.", "the dynamic programming principle , .", "Therefore, our method can also be considered as an “SMP approach for SNNs”.", "The central idea of SMP is that a stochastic optimal control problem must satisfy an optimality condition of a function called the Hamiltonian, which consists of solutions of an adjoint backward SDE (BSDE).", "In this way, the gradient with respect to the optimal control is expressed by solutions of the adjoint BSDE, and hence solving the stochastic optimal control problem through SMP requires obtaining solutions of the BSDE at each gradient descent iteration step.", "Although several successful numerical methods are developed to solve BSDEs , , , , , approximating solutions of BSDEs in the high dimensional (controlled) state space at each iteration step is still extremely challenging.", "The main theme of our computational method to implement the SMP approach for SNNs is to solve for the optimal control, which is equivalent to the optimal parameters in SNNs, under the SMP framework without solving the adjoint BSDE in the high dimensional state space.", "The key to the success of our method is the fact that solutions of the adjoint BSDE are used to formulate the gradient, which will be used in the optimization procedure for the optimal control, and obtaining solutions of BSDEs in the entire state space is not the goal of optimization.", "In other words, the adjoint BSDE provides a mechanism for the “backpropogation” in the training process of SNN, and the optimal parameters of the SNN are determined by gradient-based optimization, where the gradient consists of solutions of the adjoint BSDE.", "In data driven optimization problems, gradient-based optimization is often carried out by stochastic gradient descent (SGD), which utilizes one data sample to represent the entire data set.", "A novel concept that we shall introduce in this work is that the random samples in the state space that characterize solutions of the adjoint BSDE can be considered as “pseudo-data” in the optimization procedure, and we extend the application of SGD by treating the state samples of the adjoint BSDE as a source of data.", "In this connection, we apply SGD to both the real training data and the pseudo state sample data in the optimization procedure, and we select one training data and one sample data each time to represent their corresponding data sets.", "As a result, at each SGD iteration step, we only pick one sample-path in the state space and solve the BSDE along the chosen sample-path.", "In this way, we avoid solving the BSDE in the entire high dimensional state space repeatedly at each gradient descent iteration step, which makes our SGD optimization an efficient numerical method to implement the SMP approach for SNNs.", "The rest of this paper is organized as following.", "In Section , we introduce the mathematical foundation of our SMP approach for SNNs.", "In Section , we derive an SGD optimization algorithm to solve the stochastic optimal control problem under the SMP framework, which is equivalent to the training algorithm that determines the desired optimal parameters in the SNN.", "Numerical experiments will be presented in Section to examine the performance of our algorithm in three applications of SNNs, and we will give some concluding remarks and the plan of future work in Section ." ], [ "Stochastic maximum principle approach for stochastic neural networks", "In this section, we introduce the mathematical framework of our stochastic maximum principle approach for stochastic neural networks (SNNs).", "The main theme of our approach is to formulate a class of SNNs by stochastic differential equations, which describe stochastic forward propagation of deep neural networks (DNNs), and then treat the training process as a stochastic optimal control problem, which will be solved by stochastic maximum principle." ], [ "Stochastic neural networks", "The SNN structure that we consider in this work is given by the following model $X_{n+1} = X_n + h F(X_n, \\theta _n ) + \\sqrt{h} \\sigma _n \\omega _n, \\qquad n = 0, 1, 2, \\cdots , N-1,$ where $X_n : = [x^1, x^2, \\cdots x^L] \\in \\mathbb {R}^{L \\times d}$ denotes the vector that contains all the $L$ neurons at the $n$ -th layer in a DNN; $F$ is the pre-chosen activation function, which is commonly picked among the sigmoid function, the hyperbolic tangent function, and the rectified linear unit function (ReLu); $h$ is a fixed positive constant that stabilizes the network; $\\theta _n$ represents neural network parameters (such like weights and biases), which determine the output of the neural network and will be learned through the “training process”; $\\lbrace \\omega _n\\rbrace _n : = \\lbrace \\omega _n\\rbrace _{n=0}^{N-1}$ is a sequence of i.i.d.", "standard Gaussian random variables, together with the coefficient matrices $\\lbrace \\sigma _n\\rbrace _n$ , the stochastic terms $\\lbrace \\sigma _n \\omega _n\\rbrace _n$ generate artificial noises that bring uncertainties to a deterministic DNN, which allows the SNN to produce random output to reflect stochastic behaviors of the target model.", "The initial state $X_0 \\in \\mathbb {R}^d$ in (REF ) is the input variable in a neural network and $X_N$ is the output.", "Different from the Bayesian neural network (BNN) formulation of probabilistic machine learning, which let the parameter $\\theta $ in a standard DNN be a random variable and apply Bayesian inference to generate the empirical distribution of the parameter from training data, we introduce uncertainties to the SNN model by adding a random noise term.", "When $\\sigma _n = 0$ , the above SNN model reduces to a standard DNN, which can be formulated by the so-called “neural ordinary differential equation” (Neural ODE).", "The main concept of Neural ODE is to describe a DNN as a discretized ODE system, where standard analysis for ODEs can be applied, and the training process for a DNN is equivalent to solving a deterministic optimal control problem .", "Following the argument in Neural ODE, we consider the SNN model in the form of the stochastic sequence (REF ) as a discretized stochastic (ordinary) differential equation (SDE), where the activation function $F$ defines the drift coefficient and $\\sigma $ is the diffusion coefficient, and the propagation of the SDE system implements the forward propagation in the SNN.", "In connection with the optimal control perspective on DNN, the training process for SNN is equivalent to a stochastic optimal control problem.", "In what follows, we introduce our stochastic optimal control formulation of SNN." ], [ "Stochastic optimal control formulation of stochastic neural networks", "We consider the following continuous form of the SNN model (REF ) in a complete filtered probability space $(\\Omega , \\mathcal {F}, \\mathbb {F}^W, \\mathbb {P})$ $X_{t} = X_0 + \\int _{0}^{T} F(X_s, \\theta _s )ds + \\int _0^T \\sigma _s dW_s,$ where $W:= \\lbrace W_t\\rbrace _{0 \\le t \\le T}$ is a standard Brownian motion (corresponding to the Gaussian noise $\\lbrace \\omega _n\\rbrace _n$ ) and $\\sigma $ is the diffusion coefficient.", "The stochastic integral $\\int _0^T \\sigma _s dW_s$ on the right hand side of the equation (REF ) is an Itô type integral, which brings uncertainties to the forward propagation in the SNN.", "The temporal index $T>0$ is a given positive constant that represents the output layer of the neural network.", "When a specific temporal discretization on the interval $[0, T]$ is chosen, the depth of the SNN is determined.", "In this work, instead of treating $\\theta $ in the activation function $F$ as a parameter to be optimized in the learning process, we consider $\\theta $ and $\\sigma $ as control terms in a stochastic optimal control problem.", "Since the uncertainties in probabilistic learning and stochastic behaviors of the SNN model are described by the stochastic integral in (REF ), we let both $\\theta $ and $\\sigma $ be deterministic control processes in this work for efficiency of optimization.", "For convenience of presentation, we let $u := [\\theta , \\sigma ]$ be the control vector in an admissible control set denoted by $\\mathcal {U}[0, T]$ , and we rewrite the continuous SNN model (REF ) as the following controlled process in its differential form $dX_{t} = f(X_t, u_t )dt + g(u_t) dW_t, \\qquad 0 \\le t \\le T,$ where $f(X_t, u_t) = F(X_t, \\theta _t)$ , $g(u_t) = \\sigma _t$ , and $X_t$ is usually called the “state process”.", "When the stochastic optimal control framework is used to formulate a machine learning problem, the control $u$ aims to minimize the discrepancy between the SNN output and the data.", "Therefore, we define the cost function in our stochastic optimal control problem as $J(u) := {\\mathbb {E}} [\\Phi (X_T, \\Gamma )],$ where $\\Gamma $ is a random variable that generates training data in machine learning, which also depends on $X_0$ , and $\\Phi (X_T, \\Gamma ): = \\Vert X_T - \\Gamma \\Vert _{Loss}$ is a loss function with its corresponding error norm $\\Vert \\cdot \\Vert _{Loss}$ .", "The goal of stochastic optimal control problems is to find an optimal control $\\bar{u}$ that minimizes the cost $J$ , i.e.", "find $\\bar{u}$ such that $J(\\bar{u}) = \\inf _{u \\in \\mathcal {U}[0, T]} J(u).$ In this way, the optimal control $\\bar{u}$ also minimizes the difference between $X_T$ and $\\Gamma $ in the loss function, and hence $\\bar{u}$ is equivalent to the desired optimal parameters in the SNN.", "There are two well-known approaches to solve the stochastic optimal control problem – the dynamic programing and the stochastic maximum principle.", "The main theme of the dynamic programming approach is to solve the stochastic optimal control problem through numerical solutions for the Hamilton-Jacobi-Bellman (HJB) equation, which is a nonlinear partial differential equation (PDE) .", "On the other hand, the stochastic maximum principle approach aims to seek the optimal control that satisfies an optimality condition of a function called the (stochastic) Hamiltonian, and it's typically achieved by gradient descent type optimization methods .", "In most machine learning applications, neural networks contain large number of neurons, and the number of neurons corresponds to the dimension of HJB equations.", "Therefore, it is very difficult to apply the dynamic programming approach to solve stochastic optimal control problems that we use to formulate the SNN model — due to the “curse of dimensionality” of solving high dimensional PDEs, and we adopt the stochastic maximum principle approach in this work." ], [ "Stochastic maximum principle approach for the stochastic optimal control", "In stochastic maximum principle (SMP), we assume that the optimal control $\\bar{u}$ is in the interior of $\\mathcal {U}[0, T]$ .", "Then, by applying the Gâteaux derivative of $J$ with respect to $u$ , one can derive that the gradient process of the control function $J$ with respect to the control process on the interval $[0, T]$ has the following expression (see ) $J^{\\prime }_u(t, u_t) = {\\mathbb {E}} \\big [f^{\\prime }_u(X_t, u_t)^T Y_t + g_u^{\\prime }(u_t)^T Z_t\\big ],$ where $X$ is the controlled state process introduced in (REF ), $Y$ and $Z$ satisfy the following adjoint backward SDE, which is corresponding to the (forward) state process $X_t$ , $dY_t = - f^{\\prime }_x(X_t, \\bar{u}_t)^T Y_t dt + Z_t dW_t, \\qquad Y_T = \\Phi ^{\\prime }_x(X_T, \\ \\Gamma ).$ The pair $(Y, Z)$ is called the adapted solutions of the above adjoint backward SDE (BSDE), where $Y$ propagates backwards from $T$ to 0 with initial condition $Y_T = \\Phi ^{\\prime }_x(X_T, \\ \\Gamma )$ , and $Z$ is the martingale representation of $Y$ with respect to $W$ .", "An important property of the BSDE (REF ) for the continuous SNN model (REF ) is that the solutions $Y_t$ and $Z_t$ are functions of $X_t$ given the state $X_t$ .", "With the gradient process $J^{\\prime }_u$ , the standard approach to find the optimal control $\\bar{u}$ is the gradient descent optimization.", "Specifically, for a pre-chosen initial guess $u^0$ , we carry out the following gradient descent iteration $u_t^{k+1} = u_t^k - \\eta _k J^{\\prime }_u(t, u_t^k), \\qquad k = 0, 1, 2, \\ \\cdots , \\quad 0 \\le t \\le T,$ where $\\eta _k>0$ is the step-size of the gradient.", "Since the stochastic optimal control problem (REF ) - (REF ) is designed to formulate the SNN model, the optimization procedure to find the optimal control $\\bar{u}$ is equivalent to the training process to determine the “optimal parameter” that fits the training data, and the gradient descent step-size $\\eta _k$ is also the “learning rate” in machine learning.", "We want to mention that in supervised learning, there's an essential procedure called “backpropagation”, which computes the gradient of the loss function with respect to network parameters.", "However, due to the stochastic nature of SNN, mathematical tools (such like the chain rule) that derive the gradient in the deterministic backpropagation are not applicable.", "In this way, the BSDE (REF ) in the SMP approach can be considered as a stochastic version of backpropagation in machine learning that generates the gradient process." ], [ "An efficient stochastic gradient descent algorithm for stochastic maximum principle", "Our numerical schemes are constructed on discrete points over the interval $[0, T]$ with a discretization defined by $\\Pi ^N : = \\Big \\lbrace t_n \| 0=t_0 < t_1 < t_2 < \\cdots t_n < t_{n+1} < \\cdots < t_N=T \\Big \\rbrace ,$ where $N$ is the partition number, which is equivalent to the depth of SNNs.", "When $\\Pi ^N$ is a uniform partition, we have $h = \\frac{T}{N}$ .", "In most DNN models, the step-size $h$ is chosen to be equal to 1 — although some analysis outcomes in Neural ODE show that appropriately designed step-sizes may stabilize DNNs , .", "To develop our stochastic gradient descent algorithm for searching the optimal control, we shall first provide numerical solutions for BSDEs." ], [ "Numerical solutions for backward stochastic differential equations", "Our numerical schemes for BSDEs consist of numerical simulation for the controlled state process (REF ) and numerical solutions for the adjoint BSDE (REF ).", "The derivation of numerical schemes is based on approximation for integrals on sub-intervals $[t_n, t_{n+1}]$ , $0 \\le n \\le N-1$ .", "Since the controlled state process is a standard forward SDE (with the control term $u$ ), we apply the Euler-Maruyama scheme and derive the following approximation equation for $X$ on the sub-interval $[t_n, t_{n+1}]$ , $X_{t_{n+1}} = X_{t_n} + h f(X_{t_n}, u_{t_n}) + g(u_{t_n}) \\Delta W_{t_n} + R_X^n, \\qquad 0 \\le n \\le N-1,$ where $\\Delta W_{t_n}:= W_{t_{n+1}} - W_{t_n}$ and $R_X^n : = \\int _{t_n}^{t_{n+1}} f(X_s, u_s) ds - h f(X_{t_n}, u_{t_n}) + \\int _{t_n}^{t_{n+1}} g(u_s) d W_s - g(u_{t_n}) \\Delta W_{t_n}$ is the approximation error term.", "We can see that the approximation scheme (REF ) (without the approximation error term $R_X^n$ ) is equivalent to the discrete SNN model (REF ) given the definitions of $f$ and $g$ in the continuous controlled process (REF ), and the control term $u_{t_n}$ plays the role of parameters $\\theta _n$ and $\\sigma _n$ .", "Therefore, the forward simulation of the state process $X$ in our stochastic optimal control problem coincides the forward propagation of the SNN model.", "To derive numerical schemes for the adjoint BSDE, we consider the equation (REF ) on the sub-interval $[t_n, t_{n+1}]$ , $0 \\le n \\le N-1$ , i.e.", "$Y_{t_n} = Y_{t_{n+1}} + \\int _{t_n}^{t_{n+1}} f_x^{\\prime }(X_s, u_s) Y_s ds - \\int _{t_n}^{t_{n+1}} Z_s dW_s.$ We take the conditional expectation $ {\\mathbb {E}} _{n}^X[\\cdot ] : = {\\mathbb {E}} [\\cdot \| X_{t_n}]$ on both sides of the above equation and approximate the deterministic integral by using the right-point formula to get $Y_{t_n} = {\\mathbb {E}} _{n}^X\\big [Y_{t_{n+1}} \\big ] + h {\\mathbb {E}} _{n}^X\\big [ f_x^{\\prime }(X_{t_{n+1}}, u_{t_{n+1}}) Y_{t_{n+1}} \\big ] + R_Y^n,$ where we have used the fact $ Y_{t_n} = {\\mathbb {E}} _{n}^X\\big [ Y_{t_n}\\big ]$ due to the adaptedness of the solution $Y$ with respect to $W$ , and the stochastic integral becomes 0 under the expectation $ {\\mathbb {E}} _{n}^X[\\cdot ]$ , i.e.", "$ {\\mathbb {E}} _{n}^X[\\int _{t_n}^{t_{n+1}} Z_s dW_s] = 0$ .", "The error term $R_Y^n : = {\\mathbb {E}} _n^X\\Big [ \\int _{t_n}^{t_{n+1}} f_x^{\\prime }(X_s, u_s) ds - f_x^{\\prime }(X_{t_{n+1}}, u_{t_{n+1}}) Y_{t_{n+1}}h \\Big ]$ on the right hand side of (REF ) contains the approximation error for the deterministic integral, and we can derive a numerical scheme to solve for $Y_{t_n}$ by dropping the error term $R_Y^n$ .", "From the above discussion, we can see that the stochastic integral term contains the solution $Z$ and it's eliminated by the expectation.", "To maintain the stochastic integral and get a numerical scheme for $Z$ , we use the left-point formula to approximate both the deterministic integral and the stochastic integral in (REF ).", "As a result, we obtain $Y_{t_n} = Y_{t_{n+1}} + h f_x^{\\prime }(X_{t_n}, u_{t_n}) Y_{t_n} - Z_{t_{n}} \\Delta W_{t_n} + R_{Z}^n,$ where $R_{Z}^n = \\int _{t_n}^{t_{n+1}} f_x^{\\prime }(X_s, u_s) ds - f_x^{\\prime }(X_{t_{n}}, u_{t_{n}}) Y_{t_{n}}h - \\int _{t_n}^{t_{n+1}} Z_s dW_s + Z_{t_{n}} \\Delta W_{t_n}$ is the approximation error term.", "Then we multiply $\\Delta W_{t_n}$ on both sides of (REF ) and take conditional expectation $ {\\mathbb {E}} _{n}^X[\\cdot ]$ .", "Since $X$ and $Y$ are adapted to $W$ , we know that $ {\\mathbb {E}} _n^X[ Y_{t_{n}} \\Delta W_{t_n}] = 0$ and $ {\\mathbb {E}} _n^X[ f_x^{\\prime }(X_{t_{n}}, u_{t_{n}}) Y_{t_{n}}h \\Delta W_{t_n}] = 0$ .", "Therefore, the equation (REF ) becomes $ {\\mathbb {E}} _n^X\\big [ Z_{t_{n}}\\big ] h = {\\mathbb {E}} _n^X[ Y_{t_{n+1}} \\Delta W_{t_n}] + {\\mathbb {E}} _n^X[R_{Z}^n \\Delta W_{t_n}],$ where the left hand side of the above equation is obtained by the fact that $Z$ is also $W$ adapted, which gives us $ {\\mathbb {E}} _n^X\\big [ Z_{t_{n}} (\\Delta W_{t_n})^2\\big ] = {\\mathbb {E}} _n^X\\big [ Z_{t_{n}}\\big ] h. $ By dropping the approximation error terms $R_x^n$ and $R_{Y}^n$ in (REF ) and (REF ), respectively, and dropping the error $ {\\mathbb {E}} _n^X[R_{Z}^n \\Delta W_{t_n}]$ in (REF ), we obtain our numerical method to solve the BSDE (REF ): For $n = N-1, N-2, \\cdots , 2, 1, 0$ , we solve the BSDE (REF ) with the following schemes $Y_n = {\\mathbb {E}} _n^X[ Y_{n+1}] + h {\\mathbb {E}} _n^X\\big [ f_x^{\\prime }(X_{n+1}, u_{t_{n+1}}) Y_{n+1} \\big ], \\qquad Z_n = \\frac{ {\\mathbb {E}} _n^X[ Y_{n+1} \\Delta W_{t_n}]}{h},$ where $X_{n+1}$ is the approximation for the forward controlled state $X_{t_{n+1}}$ , which is obtained by the following discrete scheme for the state process, i.e.", "$X_{n+1} = X_{n} + f(X_n, u_{t_n}) h + g(u_{t_n}) \\Delta W_{t_n}, $ and $Y_n$ , $Z_n$ are numerical approximations for $Y_{t_n}$ , $Z_{t_n}$ , respectively.", "In practical simulations, we let $\\Delta W_{t_n} = \\sqrt{h} \\omega _n$ , where $\\omega _n$ is the Gaussian random variable in the SNN model (REF ).", "In our SMP approach for SNNs, the scheme (REF ) has the same formulation of the $N$ -layer SNN model, and the simulation of $X_n$ from $n = 0$ to $n = N$ is equivalent to the forward propagation of SNNs, where the input layer of the SNN is $X_0$ and the output layer of the SNN is $X_N$ .", "At the same time, the numerical schemes (REF ) for the adjoint BSDE provide a mechanism for the backpropagation of SNNs in the training process since they compose the gradient with respect to the control, which is equivalent to SNN parameters.", "In this way, numerical schemes (REF )-(REF ) give us a computational framework to train the SNN model introduced in (REF ).", "With numerical solutions $\\lbrace Y_n \\rbrace _n$ and $\\lbrace Z_n\\rbrace _n$ obtained in the schemes (REF ), we have the approximated gradient with respect to control as following $\\bar{J}^{\\prime }_u(n, u_{t_n}) = {\\mathbb {E}} \\big [f^{\\prime }_u(X_{n}, u_{t_n})^T Y_n + g_u^{\\prime }(u_{t_n})^T Z_n\\big ], \\qquad n = 0, 1, 2, \\cdots , N-1,$ where $\\bar{J}^{\\prime }_u(n, u_{t_n})$ is an approximation for $J^{\\prime }_u(t_n, u_{t_n})$ by using approximated solutions $X_n$ , $Y_n$ and $Z_n$ .", "Then, the gradient descent iteration on the discretization $\\Pi ^N$ can be carried out as following $\\begin{aligned}u_n^{k+1} = & u_n^k - \\eta _k \\bar{J}^{\\prime }_u(n, u_n^k) \\\\= & u_n^k - \\eta _k {\\mathbb {E}} \\big [f^{\\prime }_u(X^k_{n}, u_{n}^k)^T Y^k_n + g_u^{\\prime }(u_{n}^k)^T Z^k_n\\big ], \\quad n = 0,1, \\cdots , N-1\\end{aligned}$ where $X_n^k$ , $Y_n^k$ and $Z_n^k$ are obtained under the control $\\lbrace u_n^k\\rbrace _n$ , i.e.", "choose $u_{{t_n+1}} = u_{n+1}^k$ and $u_{{t_n}} = u_{n}^k$ in schemes (REF ) and (REF ) respectively.", "Then, for a pre-chosen integer $K$ as our stopping criteria, we let $u_n^K$ be our estimate for the optimal control $\\bar{u}_{t_n}$ .", "In order to implement the schemes (REF )-(REF ), one needs to approximate the (conditional) expectations.", "Well-known numerical methods for approximating expectations include numerical integrations and Monte Carlo simulations.", "Since our computational framework for stochastic optimal control problems is introduced to solve the SNN problem, the controlled state $X$ , which is used to model stacked neurons, is typically a high dimensional random variable.", "Therefore, numerical integration methods are usually not feasible.", "In what follows, we introduce the Monte Carlo method to approximate the expectations as a fully calculated approach for the gradient descent iteration (REF ).", "To proceed, we first use the scheme (REF ) to generate simulations of $X$ as Monte Carlo samples corresponding to the controlled state process, i.e.", "$X_{n+1}^{m, k} = X_{n}^{m, k} + f(X_n^{m, k}, u_{n}^k) h + g(u_{n}^k) \\sqrt{h} \\omega _n^m, \\qquad 0 \\le n \\le N-1, \\quad m = 1, 2, \\cdots , M,$ where $M$ is the number of Monte Carlo samples that we use to describe the state $X$ , and $\\omega _n^m$ represents the $m$ -th realization of the Gaussian random variable $\\omega _n$ .", "Then, we approximate conditional expectations in (REF ) by Monte Carlo averages, i.e.", "$\\begin{aligned} {\\mathbb {E}} ^X_n[Y_{n+1}^k] \\approx \\sum _{m=1}^M\\frac{ Y^k_{n+1}(X_{n+1}^{m, k})}{M}, \\quad & {\\mathbb {E}} ^X_n[f_x^{\\prime }(X_{n+1}, u_{n+1}^k) Y_{n+1}^k] \\approx \\sum _{m=1}^M\\frac{ f_x^{\\prime }(X_{n+1}^{m, k}, u_{n+1}^k) Y^k_{n+1}(X_{n+1}^{m, k})}{M}, \\\\ {\\mathbb {E}} ^X_n[Y_{n+1}^k \\Delta W_{t_n}] \\approx & \\sum _{m=1}^M\\frac{ Y^k_{n+1}(X_{n+1}^{m, k}) \\sqrt{h} \\omega _n^m }{M}.\\end{aligned}$ It's worthy to point out that the side condition of the adjoint BSDE (REF ), i.e.", "$Y_T = \\Phi ^{\\prime }_x(X_T, \\Gamma )$ , contains the random variable $\\Gamma $ , which describes data.", "Hence the numerical schemes that we introduced to solve the adjoint BSDE need to be carried out on each data sample $\\gamma \\sim \\Gamma $ , and an extra Monte Carlo average is needed for the data variable $\\Gamma $ when calculating the expectation in the gradient descent iteration (REF ).", "As a result, we use two layers of Monte Carlo simulation (for the state process $X$ and for the data variable $\\Gamma $ ) to approximate the expectation and rewrite the gradient descent iteration as $u_n^{k+1} = u_n^k - \\eta _k \\sum _{q=1}^{Q}\\sum _{m=1}^M \\frac{f^{\\prime }_u(X_{n}^{m, k}, u_{n}^k)^T Y^{k}_n\|_{\\gamma _q}(X_{n}^{m, k}) + g_u^{\\prime }(u_{n}^k)^T Z^k_n\|_{\\gamma _q}(X_{n}^{m, k})}{M Q},$ where $Y^{k}_n\|_{\\gamma _q}$ and $Z^{k}_n\|_{\\gamma _q}$ denote approximated solutions $Y^k_n$ and $Z^k_n$ corresponding to the data sample $\\gamma _q \\sim \\Gamma $ in the side condition, and $Q$ is the total number of available training data, which can be considered as Monte Carlo samples that describe the data variable $\\Gamma $ .", "Although the Monte Carlo approach is the state-of-the-art method to approximate expectations, when the dimension of the controlled state $X$ is high, which is common in SNN applications, and when the discretization number $N$ is large, i.e.", "the SNN is deep, the Monte Carlo sampling number $M$ needs to be very large, which makes the computational cost to implement the gradient descent optimization (REF ) very high.", "Moreover, it's important to recall that the values of $Y$ and $Z$ correspond to the state of $X$ .", "Therefore we also need to derive mappings from $X$ to solutions $Y$ and $Z$ , which can be considered as spatial approximation for solutions of BSDEs.", "In most existing numerical methods, this is accomplished by function approximation.", "Specifically, a set of spatial points that describe the random variable $X$ are chosen, and people typically use approximated solution values on those pre-chosen spatial points to construct interpolatory approximations for solutions in the state space.", "In this way, the function values $Y_{n}^k\|_{\\gamma _q}(X_{n}^{m, k})$ and $Z_{n}^k\|_{\\gamma _q}(X_{n}^{m, k})$ in the scheme (REF ) are calculated by numerical interpolation methods, such like polynomial interpolations, sparse grid interpolations, or meshfree approximations.", "Recently, several machine learning type approximation methods are developed, which give us effective global meshfree approximators to simulate functions , .", "In spite of the fact that function approximation is a well-established field and conventional numerical methods are very successful, obtaining a complete approximation for a high dimensional function is still an extremely challenging task.", "Especially, in the gradient descent optimization procedure (REF ), we need to solve the BSDE system (REF ) for every data $\\gamma _q$ that we pick as the side condition $Y_T= \\Phi ^{\\prime }_x(X_T, \\gamma _q)$ to get one update for the estimated optimal control.", "Thus complete numerical implementations for the iteration scheme (REF ) is computationally expensive.", "To address the aforementioned computational challenges in Monte Carlo simulation and high dimensional function approximation, in the next subsection, we introduce an efficient stochastic gradient descent algorithm to carry out the optimization procedure for the optimal control." ], [ "Stochastic gradient descent optimization", "In our stochastic optimal control formulation of the SNN model, the cost function $J$ (defined in (REF )) is the expectation of the loss function $\\Phi (X_T, \\Gamma )$ , where $\\Gamma $ is the random variable corresponding to training data.", "The main theme of the stochastic gradient descent (SGD) optimization is that instead of using the average of all the data samples as a Monte Carlo approximation for the expectation, we can randomly select one data sample from the data set to approximate the expectation in each gradient descent iteration step.", "Following this methodology, the gradient descent iteration (REF ) that we use to search for the optimal control can be achieved through the following scheme $u_n^{k+1} = u_n^k - \\eta _k {\\mathbb {E}} \\big [f^{\\prime }_u(X_{n}, u_{n}^k)^T Y^k_n + g_u^{\\prime }(u_{n}^k)^T Z^k_n\\big ]\\big \|_{\\gamma ^k \\in \\lbrace \\gamma _q\\rbrace _{q=1}^Q}, \\quad n = 0,1, \\cdots , N-1,$ where $\\gamma ^k$ is a data randomly selected from the entire training data set $\\lbrace \\gamma _q\\rbrace _{q=1}^Q$ at each iteration step $k$ , and the random selection $\\gamma ^k$ appears in the side condition of the adjoint BSDE (REF ), i.e.", "$Y_T = \\Phi ^{\\prime }_x(X_T, \\ \\gamma ^k)$ .", "On the other hand, although we can replace the data variable $\\Gamma $ by its random representation $\\gamma ^k$ in the SGD optimization, we still need to keep the expectation in (REF ) since the system also contains randomness caused by the Brownian motion $W$ in the state process $X$ , which is used to model the artificial noises in the SNN model.", "In this way, the fully calculated gradient descent iteration (REF ) can be simplified as $u_n^{k+1} = u_n^k - \\eta _k \\sum _{m=1}^M \\frac{f^{\\prime }_u(X_{n}^{m, k}, u_{n}^k)^T Y^{k}_n\|_{\\gamma ^k}(X_{n}^{m, k}) + g_u^{\\prime }(u_{n}^k)^T Z^k_n\|_{\\gamma ^k}(X_{n}^{m, k})}{M }.$ We can see from the above SGD iteration scheme that we only need to solve the BSDE (REF ) once to get an update for the estimated optimal control — instead of solving the entire BSDE system $Q$ -times (corresponding to different selection of data $\\gamma _q$ ) in each iteration step to get the complete Monte Carlo approximation for the expectation.", "However, the major computational challenges of large-number Monte Carlo simulations and high dimensional function approximation for solutions $Y$ and $Z$ still remain.", "A novel concept that we want to introduce in this work is that the noises in the SNN model can also be considered as a source of data.", "In other words, we treat the simulated sample paths for the controlled state process $X$ as “pseudo-data” that we use to calculate expectations in the Monte Carlo approximation.", "Therefore, the methodology of single-sample representation for random variables in SGD optimization can be extended and applied to our computational framework to describe the gradient process, which contains both the data variable $\\Gamma $ and the state process $X$ .", "Following this methodology, we generate a simulated sample-path for the state process at each iteration step $k$ as $X_{n+1}^k = X_{n}^k + f(X_n^k, u_{n}^k) h + g(u_{n}^k) \\sqrt{h} \\omega _n^k, \\qquad 0 \\le n \\le N-1.$ Corresponding to the sample-path $\\lbrace X_n^k\\rbrace _n$ and the randomly selected data $\\gamma ^k$ from the training data set, we also introduce the following path-wise numerical schemes to implement the schemes (REF ) for the BSDE $\\hat{Y}_n^k = \\hat{Y}_{n+1}^k + h f_x^{\\prime }(X_{n+1}^k, u_{n+1}^k) \\hat{Y}_{n+1}^k, \\qquad \\hat{Z}_n^k = \\frac{\\hat{Y}_{n+1}^k \\sqrt{h} \\omega _n^k}{h},\\vspace{-3.00003pt}$ where expectations in the schemes (REF ) are represented by single-realization of samples indexed by the iteration step $k$ , and $\\hat{Y}_n^k$ and $\\hat{Z}_n^k$ are approximated solutions for $Y$ and $Z$ corresponding to the state sample $X_{n}^k$ , i.e.", "$\\hat{Y}_n^k \\approx Y_{t_n}(X_n^k)$ and $\\hat{Z}_n^k \\approx Z_{t_n}(X_n^k)$ .", "In this way, at each gradient descent iteration step, we generate a sample-path $\\lbrace X_n^k\\rbrace _n$ for the state process and then solve the BSDE along this sample path with the schemes (REF ) to get a pair of simulated paths $\\lbrace ( \\hat{Y}_n^k, \\ \\hat{Z}_n^k)\\rbrace _n$ .", "Then, with simulated sample paths obtained by (REF )-(REF ), we introduce the following SGD optimization scheme $u_n^{k+1} = u_n^{k} - \\eta _k \\Big ( f^{\\prime }_u(X_{n}^k, u_{n}^k)^T \\hat{Y}^{k}_n+ g_u^{\\prime }(u_{n}^k)^T \\hat{Z}^k_n \\Big ),$ where $f^{\\prime }_u(X_{n}^k, u_{n}^k)^T \\hat{Y}^{k}_n+ g_u^{\\prime }(u_{n}^k)^T \\hat{Z}^k_n$ is the single-sample representation for the gradient, which is under expectation.", "As a result, we obtain our estimated optimal control $\\lbrace \\hat{u}_n\\rbrace _n := \\lbrace u_{n}^{K} \\rbrace _n$ .", "We notice that in the schemes (REF ), we only use one sample of each random variable to approximate conditional expectations instead of using the Monte Carlo average of a set of samples, and the simulated paths $\\lbrace \\hat{Y}_n^k\\rbrace _n$ and $\\lbrace \\hat{Z}_n^k\\rbrace _n$ only describe solutions of the adjoint BSDE (REF ) corresponding to a given simulated state sample path $\\lbrace X_n^k\\rbrace _n$ , which are not complete numerical approximations for $Y$ and $Z$ in the state space.", "However, we need to point out that the solutions $Y$ and $Z$ of the adjoint BSDE are used to formulate the gradient process, and hence the conditional expectations in the numerical schemes (REF ) are also used to compose the approximated gradient process.", "In this connection, the justification for using one sample to represent a random variable in the SGD optimization can also be applied to explain the single-sample representation for the gradient process in (REF ).", "Although we are aware that a complete description for the state variable $X$ is necessary to accurately characterize the solutions $Y$ and $Z$ since they are both functions of $X$ , it is important to emphasize that the purpose of solving the optimal control problem (REF ) - (REF ) is to find the optimal control $\\bar{u}$ , which is equivalent to the optimal parameters in the SNN model (REF ), and it's not necessary to obtain accurate numerical solutions for $Y$ and $Z$ ." ], [ "Summary of the algorithm", "We summarize the numerical algorithm of our SMP approach for SNNs in Algorithm REF .", "Table: Conclusion and future work" ] ]	2011.14145
[ [ "Batch Normalization with Enhanced Linear Transformation" ], [ "Abstract Batch normalization (BN) is a fundamental unit in modern deep networks, in which a linear transformation module was designed for improving BN's flexibility of fitting complex data distributions.", "In this paper, we demonstrate properly enhancing this linear transformation module can effectively improve the ability of BN.", "Specifically, rather than using a single neuron, we propose to additionally consider each neuron's neighborhood for calculating the outputs of the linear transformation.", "Our method, named BNET, can be implemented with 2-3 lines of code in most deep learning libraries.", "Despite the simplicity, BNET brings consistent performance gains over a wide range of backbones and visual benchmarks.", "Moreover, we verify that BNET accelerates the convergence of network training and enhances spatial information by assigning the important neurons with larger weights accordingly.", "The code is available at https://github.com/yuhuixu1993/BNET." ], [ "Introduction", "Deep learning has reshaped the computer vision community in recent years [23].", "Most vision problems, including classification [41], [16], detection [36], [26], and segmentation [7], [50], fall into the pipeline that starts with extracting image features using deep networks.", "When training very deep networks, batch normalization (BN) [21] is a standard tool to regularize the distribution of neural responses so as to improve the numerical stability of optimization.", "There are many variants of BN, differing from each other mostly in the ways of partitioning input data, including by instances [44], channels [32], groups [47], positions [24], and image domains [51].", "These variants generally shared the same module, often referred to as linear transformation, which applies a pair of learnable coefficients to restore the representations of the normalized neural responses.", "As a result, the output is no longer constrained within a zero-mean, unit-variance distribution, and the model has a stronger ability in fitting the real data distribution.", "Table: Compared to BN, BNET-3 helps ResNet-50 gain consistent improvements on four different visual tasks, including classification, detection, segmentation and action recognition.", "The corresponding evaluation metrics are accuracy, AP, mIOU, and accuracy, respectively.Note that the extra computational costs added by BNET are negligible.The above analysis implies that the learning ability of the linear transformation module in BN affects the flexibility of deep networks.", "However, we notice that existing methods have mostly assumed the linear transformation module takes a single neuron as the input and outputs the corresponding response—the referable information generally is much fewer than other operations like convolution and pooling.", "Such a design of the linear transformation module potentially limit the ability of BN in fitting much more complex data distributions.", "To this end, we present a straightforward method to enhance the standard BN—rather than just use a single neuron, we allow the linear transformation module to calculate the output based on a set of neighboring neurons in the same channel.", "Our method is easy to implement, , as shown in Algorithm REF , PyTorch [33] instantiation can be as simple as switching off the affine option in BN and appending a channel-wise convolution [9] afterwards.", "This logic is easily transplanted to other deep learning libraries and can be implemented within 2–3 lines of code.", "We name this modified BN as BNET, short for BN with Enhanced Transformation, and use BNET-$k$ to indicate a $k\\times k$ channel-wise convolution being used for linear transformation.", "We demonstrate the effectiveness of the proposed BNET on a wide range of network backbones in several visual benchmarks.", "As shown in Table REF , by simply replacing BN with BNET, the standard ResNet-50 backbone achieves a $76.8\\%$ (+$0.5\\%$ ) top-1 accuracy on ImageNet classification, a $39.5\\%$ (+$2.0\\%$ ) AP on MS-COCO detection, and a $77.4\\%$ (+$1.1\\%$ ) mIOU on Cityscapes semantic segmentation.", "BNET works better if a large kernel size is adopted, , compared to BNET-3, BNET-7 can further improve the detection AP by $1.2\\%$ on MS-COCO at marginally increased costs of $$$$ 2%$ more FLOPs.", "More results are provided in the experimental section.$ Besides these quantitative results, we also show that BNET enjoys faster convergence in network training.", "This property saves computational costs especially when strong regularizations (, AutoAugment [11]) are added to the training framework—by equipping networks with BNET, the demands of requiring extra training epochs is much alleviated.", "Additionally, by performing linear regression on the input-output pairs of BNET, an interesting observation is that the neurons that are related to important objects or contexts will be significantly enhanced.", "This observation offers a conjecture that the faster convergence property of BNET is partly due to its stronger ability in capturing spatial cues.", "In summary, the main contribution of this paper lies in a simple enhancement of BN that consistently improves recognition models.", "Our study delivers an important messages to the community that batch normalization needs a tradeoff between normalization and flexibility.", "We look forward to more products along this research direction." ], [ "Related Works", "paragraph4 .5em plus1ex minus.2ex-.5emNormalization methods.", "Layer Response Normalization (LRN) was an early normalization method that computed the statistics in a small neighborhood of each pixel and was applied in early models [17].", "Batch Normalization (BN) [21] accelerated training and improved generalization by computing the mean and variance more globally along the batch dimension.", "By contrast, Layer Normalization (LN) [2] computed the statistics of all channels in a layer and was proven to help the training of recurrent neural networks.", "Instance Normalization (IN) [44] performed normalization along each individual channel and was commonly used in image generation [20].", "A pair of parameters $\\gamma $ and $\\beta $ were learned to scale and shift the normalized features along the channels.", "The above three normalization methods can also be applied jointly, $e.g.$ , IBN-Net [32] integrated IN and BN to eliminate the appearance in DNNs and Switchable Normalization (SN) [30] combines all the three normalizers using a differentiable method to learn the ratio of each one.", "Instead of normalizing features, Weight Normalization (WN) [37] and Weight Standardization (WS) [35] proposed to normalize the weights.", "Decorrelated Batch Normalization (DBN) [19] extended BN by decorrelating features using the covariance matrix computed over a mini-batch.", "However, most of the proposed normalization methods focused “where” and “how” to normalize and ignore the linear recovery part of these normalization methods.", "paragraph4 .5em plus1ex minus.2ex-.5emUnderstanding the normalization in DNNs.", "Many efforts have been made to understand and analyze the effectiveness of BN.", "Cai [5] proved the convergence of gradient descent with BN (BNGD) for arbitrary learning rates for the weights.", "Santurkar [39] found that BN made the optimization landscape significantly smoother.", "Bjorck [3] empirically showed that BN enabled training with large gradients steps which may result in diverging loss and activations growing uncontrollably with network depth.", "Li [25] found a way which can jointly use dropout [43] and BN to boost the performance.", "In addition to directly analyzing the effectiveness of BN, some works try to train DNNs without BN to better understand the normalization layers.", "Zhang [49] introduced a new initialization method to solve the exploding and vanishing gradient problem without BN.", "Qi [34] successfully trained deep vanilla ConvNets without normalization nor skip connections by enforcing the convolution kernels to be near isometric during initialization and training.", "Shao [40] proposed RescaleNet which can be trained without normalization layers and without performance degradation.", "In this paper, we find that the recovery parameters are important for the training of DNNs and improve it by incorporating contextual information." ], [ "Background and Motivations", "Batch Normalization is proposed to normalize each channel of input features into zero mean and unit variance.", "Considering the input of a layer in DNNs over a mini-batch: the inputs share the same channel index of $N$ samples are normalized together.", "$\\hat{\\mathbf {x}}^{(n)}_c=\\frac{1}{\\sigma _c}(\\mathbf {x}^{(n)}_c-\\mu _c),$ where $c$ and $n$ are the channel index and sample index, respectively.", "$\\mu $ and $\\sigma $ in Equation (REF ) are the mean and standard deviation computed as follows: $\\mu _c=\\frac{1}{N}\\sum _{n=1}^N \\mathbf {x}^{(n)}_c, \\sigma _c = \\sqrt{\\frac{1}{N}\\sum _{n=1}^N (\\mathbf {x}^{(n)}_c-\\mu _c)^2+\\epsilon },$ where $\\epsilon $ is a small constant.", "In order to recover the representation ability of the normalized feature, a pair of per-channel parameters $\\gamma $ and $\\beta $ is learned.", "$\\mathbf {y}^{(n)}_c=\\gamma \\hat{\\mathbf {x}}^{(n)}_c+\\beta $ Figure: The framework of BN and BNET-kk.", "We consider the per-channel parameters {γ,β\\lbrace \\gamma ,\\beta } as an equivalent 1×11\\times 1 depth-wise convolution to help better understanding the extended BNET-kk.Contextual information has been verified important in many visual benchmarks.", "Contextual conditioned activations [14], [31] were proposed to improve the performance of visual recognition.", "SE-Net utilized attention modules to provide contextual information.", "Non-local networks [46] captured the long-range dependencies and facilitate many tasks.", "Relation networks [18] modeled the relation between objects by means of their appearance feature and geometry to improve object detection.", "One important role of BN is fitting the distribution of the inputs by recovering the statistics of the inputs.", "The previous linear recovery assumes that the input pixels are spatial-wise independent which is the opposite of the case.", "For the input with a complex environment and multiple objects, it is hard to capture its distribution with the independence assumptions.", "In such cases, contextual information is very important.", "In this paper, we mainly focus on how to embed contextual information in BN.", "For a neighborhood $\\mathcal {S}_{\\hat{x}_i}$ of the normalized input $\\hat{x}_i$ , we give a generalized formulation of the recovery step in BN which considers the contextual information and local dependencies: $y_i=\\sum _{j\\in \\mathcal {S}_{\\hat{x}_i}}f_j(\\hat{x}_j)+\\beta ,$ where $j$ is the index of the neighbors and $f_j$ is the recover functions.", "BN is a special case of Equation (REF ) when only $\\hat{x}_i$ itself is included in ${S}_{\\hat{x}_i}$ ." ], [ "The Formulation of BNET", "Different from previous works that mainly focus on studying different ways to calculate normalization statistics, we hereby investigate the effects of the linear recovery transformations of BN.", "The primitive target of linear transformation parameters $\\gamma $ and $\\beta $ is to fit the distribution of the input and recover representation ability, however, two simple parameters of each channel can hardly accomplish this job without contextual information provided especially when the input contains complex scenes.", "Here, based on the general formulation in Equation (REF ), we adopt a parameterized linear transformation $M{(\\cdot ;\\theta )}$ for simplicity.", "Therefore, the recovered features can be computed as follows: $\\mathbf {y}^{(n)}_c=M{(\\hat{\\mathbf {x}}^{(n)}_c;\\theta )}+\\beta ,$ where $\\theta $ denotes the learned parameters.", "To capture contextual information, a simple instantiation of $M{(\\cdot ;\\theta )}$ is by using $(k\\times k)$ depth-wise convolution [9].", "We name this method as BNET-$k$ , short for Batch Normalization with Enhanced Linear Transformation, where $k$ denotes the kernel size of depth-wise convolution.", "An illustration is provided in Figure REF .", "BNET-$k$ can be easily implemented by a few lines of code based on the original implementation of BN in PyTorch [33] and TensorFlow [1].", "Algorithm REF provides the code of BNET in PyTorch.", "[t] BNET code in PyTorch-like style whitefullflexiblebcodeblue # width: number of input channels # k: the kernel size of the transformation class BNET2d(nn.BatchNorm2d): def __init__(self, width, args, k=3, kwargs): super(BNET2d, self).__init__(width, args, affine=False, **kwargs) self.bnconv = nn.Conv2d(width, width, k, padding=(k-1) // 2, groups=width, bias=True) def forward(self, x): return self.bnconv(super(BNET2d, self).forward(x))" ], [ "Advantages over Previous Mechanisms", "BNET offers a plug-and-play option to improve BN, a standard module in most modern deep networks.", "Before continuing with extensive experiments, we briefly analyze its advantages over some past works.", "To the best of our knowledge, this is the first work that improves BN by enhancing the linear transformation.", "It is complementary to prior BN variants that mainly contributed to partitioning the input data into different groups [2], [44], [47].", "BNET requires additional parameters to enhance the linear transformation, but the increased amount is often much smaller than that of the convolutional parameters.", "In other words, we find an objective that has been undervalued and verify that small extra costs lead to big gains.", "Another work that relates to BNET is FReLU [31], which introduced context information to the activation function (e.g., ReLU).", "We point out that BNET enjoys a stronger ability in fitting different distributions, and BNET can be combined with FReLU for better recognition performance.", "In ImageNet classification on ResNet-50, adding BNET-3 upon FReLU improves the top-1 accuracy from 77.5% to 77.9%, and the benefit transfers to MS-COCO object detection, claiming an AP gain of 0.4% (from 39.8% to 40.2%)." ], [ "Experiments", "In this section, we will present the experimental results, including image classification on ImageNet [12], object detection and instance segmentation on MS-COCO [28], video recognition on UCF-101 [42] and semantic segmentation on Cityscapes [10]." ], [ "Image Classification", "To better evaluate the effectiveness of our proposed BNET, we conduct image classification experiments on ImageNet [12].", "It contains $1\\rm {,}000$ object categories, and $1.3\\mathrm {M}$ training images and $50\\mathrm {K}$ validation images, all of which are high-resolution and roughly equally distributed over all classes.", "Various architectures are adopted.", "First, we apply BNET on ResNet [16] and ResNeXt [48].", "Then, experiments on efficient models (MobileNetV2 [38]) and low-precision models (Bi-real Net [29]) are presented.", "Experiments of training with Auto-Augment [11] are discussed, afterwards.", "In these experiments, the input image size is fixed to be $224\\times 224$ .", "For a fair comparison, we train all the models in the same code base with the same settings.", "The learning rate is set to $0.1$ initially, and is multiplied by $0.1$ after every 30 epochs.", "We use SGD to train the models for a total of 100 epochs, where the weight decay is set to 0.0001 and the momentum is set to 0.9.", "For ResNet-18 and ResNet-50, the training batch is set to 256 for 4 GPUs (Nvidia 1080Ti).", "We use 8 GPUs (Nvidia 1080Ti) to train ResNet-101 and ResNeXt-50 with a batch-size 256.", "Table REF shows the major experimental results on ImageNet.", "The proposed BNET consistently outperforms BN on ResNet with different depths and ResNetXt-50 with negligible computation cost.", "For example, ResNet-101 with BNET-3 has a remarkable increase of 0.9% on Top-1 and 0.4% on Top-5.", "Figure REF and Figure REF depict the loss and accuracy curves of ResNet-50 (BN) and ResNet-50 (BN-3).", "We find that model using BNET converges faster than the model using BN.", "Table: Image classification results on ImageNet using ResNet and ResNeXt .Figure: The training curves of ResNet-50 using BN and BNET-3 on ImageNet.", "We consider two situations: Figure and Figure are the loss and accuracy curves without using Auto-Augment while Figure and Figure are the loss and accuracy curves using Auto-Augment." ], [ "Comparison on Lightweight Models", "paragraph4 .5em plus1ex minus.2ex-.5emMobileNetV2.", "We apply BNET on an efficient architecture, i.e., MobilenetV2 [38].", "The Models are optimized by momentum SGD, with an initial learning rate of $0.05$ (annealed down to zero following a cosine schedule without restart), a momentum of 0.9, and a weight decay of $4\\times 10^{-5}$ .", "They are trained by 8 GPUs with a batch size of 256 for a total of 150 epochs.", "Table REF shows that the Top-1 and Top-5 accuracies of MobileNetV2 with different widths using BN and BNET.", "We observe that BNET increases the Top-1 and Top-5 accuracies of MobileNetV2 (1.0) about 0.5% and 0.6%, respectively.", "Furthermore, the increase of the Top-1/Top-5 accuracy of MobileNetV2 becomes more significant as the width of the model is narrowed down.", "Table: Image classification results on ImageNet using MobileNetV2 and Bi-real Net paragraph4 .5em plus1ex minus.2ex-.5emBi-real Net.", "Next we evaluate BNET on low-precision models.", "We adopt Bi-real-net [29] as our basic model for its competitive results in binary models (both weights and features are binarized).", "We use the same training settings as the official PyTorch implementation$\\endcsname $https://github.com/liuzechun/Bi-Real-net .", "The Adam optimizer [22] is adopted with an initial learning rate 0.001 which linearly decays to 0 after 256 epochs.", "The batch size is set to 512 for 8 GPUs.", "Table REF reports the results of Bi-real Net using BNET-3 and BN.", "With the help of BNET, the Top-1 and Top-5 accuracies of Bi-real net are significantly increased by 1.1% and 0.3%, respectively.", "It indicates that besides the full-precision models in Section REF , BNET also benefits the training of low-precision models." ], [ "Training with Auto-Augment", "To further demonstrate the out-standing fitting abilities of BNET, in this section, experiments of training with augmentations such as Auto-Augment [11] are presented.", "The training settings are the same as the settings in Section REF except when the total epochs are 270.", "When the total epochs are set as 270, the learning rate is set to $0.1$ initially, and is multiplied by $0.1$ after every 80 epochs.", "We report the results of ResNet-50 trained with Auto-Augment using BN and BNET in Table REF .", "Two training epoch settings (100 and 270) are provided for better comparisons.", "ResNet-50 (BN) is unable to fit the strong augmentation of Auto-augment with even a 0.2% Top-1 accuracy drop compared to the results in Table REF when only trained 100 epochs.", "On the contrary, ResNet-50 (BNET-3) reveals better fitting abilities towards Auto-Augment with 0.4% Top-1 accuracy gain over the model without Auto-Augment.", "Besides, ResNet-50 (BNET-3) outperforms ResNet-50 (BN) by a Top-1 error of 0.3% when trained 270 epochs.", "Figure REF and Figure REF illustrate the loss and accuracy curves of ResNet-50 which is trained 100 epochs with Auto-Augment.", "With the benefit of BNET, the loss drops more sharply compared with BN.", "Table: Image classification results on ImageNet using ResNet .", "AA denotes the Auto-Augment." ], [ "Detection and Segmentation on MS-COCO", "We conduct object detection and instance segmentation experiments on MS-COCO [28] to evaluate the generalization performance of BNET on different tasks by using the models pre-trained on ImageNet as the backbones.", "The MS-COCO dataset has 80 object categories.", "We train the entire models on the trainval dataset, which is obtained by a standard pipeline that excludes $5\\mathrm {K}$ images from the val set, merges the rest data into the $80\\mathrm {K}$ train set and uses the minival set for testing.", "All the experiments on MS-COCO are implemented on the PyTorch [33] based MMDetection [6].", "Figure: The training curves of Faster R-CNN on MS-COCO using ResNet-50 as the backbone." ], [ "Object Detection", "We use the standard configuration of Faster R-CNN [36] with FPN [26] and ResNet/ResNeXt as the backbone architectures.", "The input image size is 1333$\\times $ 800.", "We train the models on 8 GPUs with 2 images per GPU (effective mini-batch size of 16).", "The backbones of all models are pre-trained on ImageNet classification (Table REF ) with the statistics of BN frozen.", "Following the 1$\\times $ schedule in MMDetection, all models are trained for 12 epochs using synchronized SGD with a weight decay of 0.0001 and a momentum of 0.9.", "The learning rate is initialized to 0.02 and is decayed by a factor of 10 at the 9th and 11th epochs.", "Results of the 3$\\times $ schedule are also provided.", "Table REF provides the results in terms of Average Precision (AP) of Faster R-CNN trained with BN and BNET.", "There are several observations.", "First, BNET consistently outperforms BN on different backbone architectures.", "For example, ResNet-50 (BNET-3) has an increase of 2.0 AP comparing to the ResNet-50 (BN).", "Second, increasing the kernel size of BNET would further boost the detection scores as a bigger kernel size can bring more contextual information.", "Particularly, ResNet-101 (BNET-7) attains a 1.2% gain in terms of AP over ResNet-101 (BNET-3).", "Third, in addition to precision, we also compare the computational cost including GFLOPs and FPS.", "The increased computational cost is negligible on all of the different backbone architectures.", "Detailed error analysis is presented in Appendix C and the results of RetinaNet [27] are shown in Appendix B.", "The training curves of Faster-RCNN using ResNet-50 as the backbone are drawn in Figure REF .", "The training loss of the model with BNET declines sharply and the AP score of BNET benefited model is consistently higher than the baseline model in each training epoch." ], [ "Instance Segmentation", "For the instance segmentation experiments, we use Mask R-CNN [15] with FPN [26] as the basic framework and ResNet [16] as the backbone architecture.", "We follow the 1$\\times $ schedule of MMDetection, which is the same as that in the detection experiments.", "Comparison results of BNET and the baseline methods on instance segmentation are reported in Table REF .", "Similar to the detection results of Faster R-CNN, we can also observe a great improvement of BNET compared with the original BN baseline model.", "The improvements become more significant when the kernel size of BNET is increased." ], [ "Semantic Segmentation on Cityscapes", "Experiment results of semantic segmentation on Cityscapes [10] are further presented to validate the effectiveness of BNET.", "The Cityscapes dataset contains 19 categories which includes 5,000 finely annotated images, 2,975 for training, 500 for validation, and 1525 for testing.", "We use the PSPNet [50] as the segmentation framework and the input image size is 512$\\times $ 1024.", "For the training settings, we use the poly learning rate policy where the base learning rate is 0.01 and the power is 0.9.", "We use a weight decay of 0.0005, and 8 GPUs with a batch size of 2 on each GPU to train $40\\mathrm {K}$ iterations.", "Table REF summarizes the results of PSPNet with BNET and BN on the Cityscapes val set.", "With the similar model size and computational cost, BNET-3 achieves better performance, i.e., a 1.1% gain over BN.", "Table: Semantic segmentation results on Cityscapes using PSPNet with ResNet-50 as backbone." ], [ "Action Recognition on UCF-101", "We further show the general applicability of BNET on the task of action recognition.", "Specifically, the experiments are conducted on the UCF-101 [42] dataset.", "It contains $13\\mathrm {K}$ videos which are annotated into 101 action classes.", "We compare BN and BNET on the first split of UCF-101.", "We select TSN [45] as the base framework and use ResNet-50 as the backbone which is pre-trained on ImageNet (Section REF ).", "For simplicity, we only use the RGB input.", "We utilize the SGD to learn the network parameters on 8 GPUs, where the batch size is set to 256 and momentum set to 0.9.", "The learning rate is initialized to 0.001, and decayed by a factor of 10 after every 30 epochs.", "Table REF provides the results of TSN with different normalization methods.", "BNET-3 brings remarkable gains of 4.3% and 1.5% on Top-1 and Top-5 accuracies, respectively.", "However, BNET-5 is not able to give more improvements but more computational cost.", "Table: Action recognition results of TSN on UCF-101 (split 1) using ResNet as backbone." ], [ "Ablation Study", "paragraph4 .5em plus1ex minus.2ex-.5emComparisons of different linear transformations.", "We first conduct experiments on different choices of linear transformation which may have different reception fields and embed diverse context information.", "We compare the performance of $\\lbrace 1\\times 1, 3\\times 3, 5\\times 5, 3\\times 3\\ (dilation=2)\\rbrace $ using ResNet-50 as the base model on ImageNet.", "Table REF provides the results.", "From the results in the table, we can conclude that $3\\times 3$ depth-wise convolution is the best choice for BNET on ImageNet dataset.", "However, the performance of object detection and classification is not strictly consistent.", "As shown in Table REF and Table REF , $5\\times 5$ depth-wise convolution obtains better performance on the MS-COCO even though the pre-trained performance is worse.", "This phenomenon indicates that a task like Object Detection prefers a larger kernel size $k$ as a larger kernel-size $k$ in BNET-$k$ can produce more contextual information and a broader reception field.", "paragraph4 .5em plus1ex minus.2ex-.5emDiscussions about the plug-in position.", "We conduct experiments about where to add BNET to offer a better trade-off between accuracy and computational cost.", "There exist three BN layers in a residual bottleneck block which is denoted as A, B and C, respectively.", "A and C are the BN layers after the $1\\times 1$ convolution and B is the BN after the $3\\times 3$ .", "Experiment results are shown in Table REF .", "We conclude that replacing the BN layer in C is the best choice this may because C has wider input channels than A and C. Further, we consider the case that replacing all the BN layers in A, B and C by BNET.", "However, it can not provide extra improvement which means that configuration C has provided enough contextual information.", "Table: The positions of BNET in the residual BottleneckTable: Compared with adding a convolution layer by using ResNet-50 on ImageNet.Table: Applications on different normalization layers by using ResNet-50 on ImageNet.Figure: The visualization of enhancement heat-maps of BNET-3.", "The first row is the input image from MS-COCO with bounding box and {ℌ 2 ,ℌ 3 ,ℌ 4 ,ℌ 5 }\\lbrace \\mathfrak {H}_2,\\mathfrak {H}_3,\\mathfrak {H}_4,\\mathfrak {H}_5\\rbrace are enhancement heat-maps of BNET in each stage’s last residual block which corresponds to the responses fed into the feature pyramid network.paragraph4 .5em plus1ex minus.2ex-.5emComparison to adding a convolution layer.", "At first glance, the proposed BNET looks very simple.", "An immediate concern arises: what is the difference with simply adding a convolution layer in the residual block?", "To answer this question, we perform additional experiments and compare the results in Table REF .", "We add a “ReLU\" layer followed by a “$3\\times 3$ depth-wise convolution\" layer in the residual bottleneck.", "The results show that simply adding a convolution layer in the bottleneck is unable to boost performance, on the contrary, the Top-1 and Top-5 accuracy even drops by 0.6% and 0.4%, respectively.", "paragraph4 .5em plus1ex minus.2ex-.5emAdding contextual information to other normalization layers.", "In addition to BN, many normalization methods use the same linear recovery as BN, such as IN [44], SN [30] and GN [47].", "Can contextual information boost their performance?", "To answer this question, we enhance the linear transformations of GN and SN which is similar to BNET-3.", "Table REF provides the experiment results of ResNet-50 on ImageNet.", "In addition to BN, both of GN and SN can benefit from the contextual information." ], [ "What Was Enhanced by BNET?", "To answer the question, we delve deep into BNET by observing the behavior of the linear transformation.", "We use the Faster R-CNN model (equipped with BNET-3) trained for MS-COCO object detection.", "We choose four layers, denoted by $\\mathfrak {H}_2$ –$\\mathfrak {H}_5$ , that correspond to the responses fed into the feature pyramid network.", "For each layer, we investigate the input-output pairs (before and after the linear transformation of BNET).", "Note that in the vanilla BN, these pairs follow a simple linear correlation, but using BNET, we can regress a linear function upon these pairs and observe which outputs are larger than the regressed value.", "We record the corresponding positions for the enhanced outputs and sum up them across all the channels, and obtain the enhancement heatmap.", "We show these heatmaps in Figure REF .", "One can observe that mostly, the enhanced positions correspond to meaningful contents, which can be interesting textures in low-level layers (e.g., $\\mathfrak {H}_2$ ) and objects in high-level layers (e.g., $\\mathfrak {H}_5$ ).", "These results indicate that BNET has the ability to enhance important regions.", "We conjecture that this property contributes to the improved recognition accuracy as well as the faster convergence of BNET." ], [ "Conclusion", "In this paper, we propose BNET, a variant of BN that enhances the linear transformation module by referring to the neighboring responses of each neuron.", "BNET is easily implemented, requires little extra computational costs, and achieves consistent accuracy gain in various visual recognition tasks.", "We demonstrate that designing a more sophisticated module to balance normalization and flexibility leads to better performance, and believe that this research topic has higher potentials in the future.", "Table: Detection results on MS-COCO using RetinaNet and FPN with BN and BNET as normalization methods.†\\dagger FPS is measured on a single Nvidia 1080Ti GPU and the batch size is set as 1.Table: Error analysis on MS-COCO using Faster R-CNN and FPN with BN and BNET as normalization methods.", "The error analysis is performed by means of TIDE .Table: Semantic segmentation results of DeeplabV3 using ResNet101 as the backbone architecture on PASCAL VOC 2012" ], [ "Visualizations of the ResNet-50 (BNET)", "Figure REF depicts the attention maps of ResNet-50 (BN) and ResNet-50 (BNET).", "$g_0,g_1,g_2,g_3$ are the attention maps of the last residual block in each stage of ResNet-50.", "One can observe that, the attention maps of ResNet (BNET) are more focused on the target object (e.g., $g_3$ ) when compared with the attention maps of ResNet-50 (BN).", "Figure: Attention maps of different stages in ResNet-50 (BNET-3).", "Six input images from ImageNet are fed into ResNet50 (BN) and ResNet (BNET-3).", "In each sub-figure, the first row is the attentions maps of ResNet (BNET-3) and the second row belong to ResNet50 (BN)." ], [ "RetinaNet Results on MS-COCO", "We use RetinaNet [27] as the basic framework using ResNet-101 as the backbone architecture.", "Experiments are performed using MMdetection [6].", "The backbones are pre-trained on ImageNet classification with the statistics of BN frozen.", "Following the 1$\\times $ schedule in MMDetection, all models are trained for 12 epochs using synchronized SGD with a weight decay of 0.0001 and a momentum of 0.9.", "The learning rate is initialized to 0.02 and is decayed by a factor of 10 at the 9th and 11th epochs.", "Table REF provides the results of RetinaNet with BNET and BN as the normalization methods.", "The model using BNET obtains a gain of 2.2% over the model using BN." ], [ "Error Analysis on MS-COCO", "We provide the analysis using the tool TIDE [4] on MS-COCO.", "We adopt Faster R-CNN and use four kinds of ResNet-50 backbone architectures, i.e., ResNet-50 (BN), ResNet-50 (BNET-3), ResNet-50 (BNET-5), and ResNet-50 (BNET-7).", "Table REF provides the error analysis of BNET and BN based models on MS-COCO.", "Follow [4], we calculate four kinds of sub-classified errors (e.g., Classification Error $(E_{cls)}$ , Localization Error $(E_{loc)}$ , Background Error $(E_{bkg)}$ , and Missed GT Error $(E_{miss)}$ ), False Positive Error and False Negative Error.", "All the errors are calculated by the $\\Delta AP$ proposed in [4].", "All the three models with BNET decrease the classification error greatly.", "The localization errors of BNET-3 and BNET-7 have also decreased.", "We also plot the relative contribution of each error in Figure REF .", "Figure: The relative contributions of different errors in Faster R-CNN with FPN using ResNet-50 (BN), ResNet-50 (BNET-3), ResNet-50 (BNET-5), and ResNet-50 (BNET-7) as the backbone architectures." ], [ "Segmentation Results on VOC2012", "In this section, we provide experiment results of semantic segmentation on PASCAL VOC 2012 [13] to compare BNET with BN baseline.", "The PPASCAL VOC dataset contains 20 foreground object classes and one background class.", "We augment the original dataset with the extra annotations which contains 10,582 (train aug) training images.", "We use the DeeplabV3 [7] as the segmentation framework.", "Table REF summarizes the results of DeeplabV3 with BNET and BN on the PASCAL VOC 2012 set.", "With similar model size and computational cost, BNET-3 achieves better performance: a 1.0% gain over BN." ] ]	2011.14150
[ [ "A $\\dbar$-steepest descent method for oscillatory Riemann-Hilbert\n problems" ], [ "Abstract We study the asymptotic behavior of Riemann-Hilbert problems (RHP) arising in the AKNS hierarchy of integrable equations.", "Our analysis is based on the $\\dbar$-steepest descent method.", "We consider RHPs arising from the inverse scattering transform of the AKNS hierarchy with $H^{1,1}(\\R)$ initial data.", "The analysis will be divided into three regions: fast decay region, oscillating region and self-similarity region (the Painlev\\'e region).", "The resulting formulas can be directly applied to study the long-time asymptotic of the solutions of integrable equations such as NLS, mKdV and their higher-order generalizations." ], [ "Introduction", "The long-time behavior of solutions of the initial-value problem for nonlinear evolution integrable PDEs has been studied extensively.", "It is well-known that the long-time asymptotic analysis for the integrable PDEs can be, via inverse scattering, formulated as a problem of finding asymptotics of certain oscillatory RHPs.", "A countless number of papers (see e.g.", "[19], [32] and the references therein) have been devoted to studying the asymptotic behavior of a certain type of oscillatory 2 by 2 matrix RHPs, which is also the main subject of the current study.", "The most influential is the nonlinear steepest descent method (or the Deift-Zhou method), which was published in Annals of mathematics [14] in 1993.", "Before the Deift-Zhou work, A.R.Its [22] proposed a direct method, via an isomonodromic deformation, to study the asymptotics of a RHP arising in studying the long-time behavior of the nonlinear Schrödinger (NLS) equation.", "Ten years after the Deift-Zhou method was published, Deift and Zhou extended their method to study the long-time behavior of the defocusing NLS equation on some weight Sobolev space.", "Between 1993 and 2003, the Deift-Zhou method had been applied to not only the long-time behavior of integrable systems, but also equilibrium measure for logarithmic potentials [12], the strong asymptotics of orthogonal polynomials [13] and many other other fields in mathematical physics.", "Shortly after the Deift-Zhou 2003 paper, McLaughlin and Miller [29] proposed another generalization to the Deift-Zhou method: the so-called $\\bar{\\partial }$ -steepest descent.", "This method was first applied to studying the long-time asymptotics of the defocusing NLS equation in 2008 [17], see also its extension version [18].", "Comparing to the Deift-Zhou method, the $\\bar{\\partial }$ -steepest descent method provides a more elementary way and more tractable way of analyzing the error terms.", "Since then, the $\\bar{\\partial }$ -steepest descent method has been applied to many long-time asymptotic studies for nonlinear integrable PDEs, such as the focusing NLS equation [5], the KdV equation [21], the mKdV equation[7], the sine-Gordon equation[8], the fifth order mKdV equation [23] and many others.", "It is worth mentioning that the mKdV and fifth-order mKdV equations belong to the mKdV hierarchy we will consider in the current work.", "In fact, by carefully checking [7] and [23], we find there are many similar analyses which motivate us to study the whole mKdV hierarchy at once.", "In the current paper, we will study an oscillatory 2 by 2 matrix RHP arising in studying the long-time asymptotics of the mKdV hierarchy.", "We will discuss the defocusing case (i.e., without solitons).", "The focusing case will be treated somewhere else in the future.", "The main analysis is based on the idea of $\\bar{\\partial }$ -steepest descent [17], [29].", "In the study of Cauchy initial-value problems of integrable systems by means of inverse scattering, the following RHP appears: Riemann-Hilbert problem 1.1 Looking for a 2 by 2 matrix-valued function $m(z)$ such that $m$ is analytic off the real line $\\mathbb {R}$ ; for $z\\in \\mathbb {R}$ , we have $m_+=m_-v_{\\theta }(z),\\quad z\\in \\mathbb {R},$ where $m_{\\pm }(z)=\\lim _{\\epsilon \\rightarrow 0^{+}}m(z\\pm i\\epsilon ),z\\in \\mathbb {R}$ , and the jump reads $v_{\\theta }(z)=\\begin{pmatrix}1-\|R(z)\|^2 & -\\bar{R}(z)e^{-2it\\theta }\\\\R(z)e^{2it\\theta } & 1\\end{pmatrix},$ where $R(z)$ is the reflection coefficient in performing inverse scattering with given initial data, see, e.g., (REF ), and $\\theta =\\theta (z;x/t)$ is a polynomial of $z$ with coefficients depends on $x/t$ ; $m(z)=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty .$ In this paper, we consider the following defocusing mKdV type reduction of the AKNS hierarchy (shortly, we call it the mKdV hierarchy): Fixing $n$ as an positive odd integer, we consider $\\psi _x(x,t;z)&=\\left(-iz\\sigma _3+\\begin{pmatrix}0&q(x,t)\\\\q(x,t)&0\\end{pmatrix}\\right)\\psi (x,t;z),\\\\\\psi _t(x,t;z)&=\\left(\\sum _{k=0}^nQ_k(x,t)z^k\\right)\\psi (x,t;z),$ where $\\sigma _3=\\text{diag}(1,-1)$ , $q(x,t)$ is the potential which solves a certain $1+1$ dimensional integrable equation, and $Q_k$ is determined by certain recursion relation (for details, see [1]).", "In the case of the mKdV hierarchy, $Q_n(x,t)$ is a constant with respect to $x,t$ .", "The corresponding nonlinear integrable PDE is worked out by the the zero curvature condition, which is also equivalent to $\\psi _{xt}=\\psi _{tx}$ .", "In this paper, we will study the Cauchy initial-value problem for integrable PDEs generated from the defocusing mKdV hierarchy, with the initial data belonging to $H^{1,1}(\\mathbb {R})=\\lbrace f\\in L^2\|f^{\\prime }\\in L^2, xf\\in L^2\\rbrace $ .", "Due to Zhou's result [33], after direct scattering, the reflection coefficient $R(z)$ also belongs to $H^{1,1}(\\mathbb {R})$ .", "By performing the time evolution, we arrive at the RHP REF .", "The first part (oscillating region) of the analysis is slightly more general than the one in the AKNS hierarchy, by making the following assumptions on the phase function: $\\theta $ is a real polynomial of degree $n$ with respect to $z$ , with coefficients depends on $x/t$ ; $\\theta ^{\\prime }(z_j)=0,\\theta ^{\\prime \\prime }(z_j)\\ne 0$ for $j=1,\\cdots , l$ , where $l$ denotes the number of real stationary phase points.", "Remark 1.2 For the defocusing mKdV hierarchy case, $n$ in the first assumption corresponds to the $\\frac{n-1}{2}$ th member of the hierarchy.", "Since in mKdV hierarchy, $n$ is an odd number, say $n=2k-1,k\\in \\mathbb {Z}_+$ , we will only need to study the phase function of the type: $c_1z+c_2z^{2k+1},\\ k\\in \\mathbb {Z}_+,$ and $c_1,c_2$ are some constants.", "The purpose of the second assumption includes the case of linear combination of several members in the mKdV hierarchy, which is again integrable.", "In such situation, we will see a generic polynomial of $z$ with coefficients depends on $x/t$ .", "Before we establish the main results, we first introduce some notations.", "Let's denote the weighted Sobolev space by $H^{k,j}(\\mathbb {R}):=\\lbrace f(x)\\in L^2(\\mathbb {R}):\\partial _x^sf\\in L^2(\\mathbb {R}),\\ s=1,\\cdots ,k,\\ x^jf(x)\\in L^2(\\mathbb {R})\\rbrace ,$ with norm $\\Vert f\\Vert _{H^{k,j}}:=\\left(\\Vert f\\Vert ^2_{L^2}+\\sum _{l=1}^k\\Vert \\partial _x^lf\\Vert ^2_{L^2}+\\sum _{m=1}^j\\Vert x^mf\\Vert ^2_{L^2}\\right)^{1/2}.$ Next we define the meaning of the long-time behavior in the three regions we are concerned with as follows.", "Long-time behavior of the potential in the oscillating region means taking the $t\\rightarrow \\infty $ limit of $q(x,t)$ along the ray $x=-ct,c>0,t\\rightarrow \\infty $ .", "Long-time behavior of the potential in the fast decaying region means taking the $t\\rightarrow \\infty $ limit of $q(x,t)$ along the ray $x=ct,c>0,t\\rightarrow \\infty $ .", "Long-time behavior of the potential in the Painlevé region means taking the $t\\rightarrow \\infty $ limit of $q(x,t)$ along the curve $x=c(nt)^{1/n},c\\ne 0,\\, t\\rightarrow \\infty $ , where $n$ is the degree of the polynomial phase function $\\theta $ .", "Theorem 1.3 In the oscillating region, provided that the initial dataDue to Zhou's theorem [33], $R(z)$ belongs to $H^{1,n-1}(dz)$ , then the time evolving reflection coefficient $R(z)e^{\\pm 2it\\theta }$ will stay in $H^{1,1}(dz)$ since the degree of $\\theta $ is $n$ .", "$q(x,0)\\in H^{n-1,1}(\\mathbb {R},dx)$ , the long-time behavior for the potentials $q(x,t)$ reads $q(x,t)&=q_{as}(x,t)+\\mathcal {O}(t^{-3/4}),t\\rightarrow \\infty ,$ where $q_{as}(x,t)&=-2i\\sum _{j=1}^{l}\\frac{\|\\eta (z_j)\|^{1/2}}{\\sqrt{2t\\theta ^{\\prime \\prime }(z_j)}}e^{i\\varphi (t)},\\\\\\varphi (t)&=\\frac{\\pi }{4}-\\arg \\Gamma (-i\\eta (z_j))\\\\&-2t\\theta (z_j)-\\frac{\\eta (z_j)}{2}\\log \|2t\\theta ^{\\prime \\prime }(z_j)\|+2\\arg (\\delta _j)+\\arg (R_j),$ and the phase function $\\theta $ will depend only on $z$ along any ray in the oscillating region, $\\lbrace z_j\\rbrace _{j=1}^l$ are the real stationary phase points of the phase function, and $\\delta (z)&=\\exp {\\left(\\frac{1}{2\\pi i}\\int _{D_-}\\frac{\\log (1-\|R(s)\|^2)}{s-z}\\text{ds}\\right)},z\\in D_-, \\\\D_-&=\\lbrace z\\in \\mathbb {R}:\\theta ^{\\prime }(z)<0\\rbrace ,\\\\\\eta (z)&=-\\frac{1}{2\\pi }\\log (1-\|R(z)\|^2),\\\\R_j&=R(z_j),\\quad j=1,..,l,\\\\\\delta _j&=\\lim _{\\begin{array}{c}z=z_j+\\rho e^{i\\phi },\\\\ \\rho \\rightarrow 0,\\\\ \\text{fix } \\phi \\in (0,\\pi /2)\\end{array}}\\delta (z)(z-z_j)^{i\\eta (z_j)}.$ Here $R(z)$ is the reflection coefficient generated from the standard inverse scattering procedure, see equation (REF ).", "Corollary 1.4 For the case of the AKNS hierarchy, in the oscillating region, the phase function $\\theta (z)=\\frac{x}{t}z+cz^{n},c>0$ and has just two real stationary phase points: $z_{\\pm }=\\pm \\left\|-\\frac{x}{nct}\\right\|^{\\frac{1}{n-1}}$ , and then the long-time asymptotics for the potentials in the AKNS hierarchy are merely a special case of Theorem REF .", "Theorem 1.5 In the fast decay region, the long-time behavior for the potential reads $q(x,t)=\\mathcal {O}(t^{-1}), \\quad t\\rightarrow \\infty .$ Theorem 1.6 In the Painlevé region, the long-time behavior for the potential reads $q(x,t)=(nt)^{-\\frac{1}{n}}u_n(x(nt)^{-\\frac{1}{n}})+\\mathcal {O}(t^{-\\frac{3}{2n}}),\\quad t\\rightarrow \\infty ,$ where $u_n$ solves the $n^{th}$ member of the Painlevé II hierarchy." ], [ "Outline", "In section , we simply review the inverse scattering for the AKNS hierarchy.", "In section , we summarize the idea of the $\\bar{\\partial }$ -steepest descent method following [18], [7].", "In the following sections we first discuss the long-time behavior of the potential in the oscillating region.", "The general workflow is shown in Fig.REF .", "The first step (see section ) is so-called conjugation by which one can simultaneously factorize the jump matrix to lower/upper triangle and upper/lower triangle.", "The next step (see section ) is so called lenses-opening.", "In each interval where $\\theta $ is monotonic, we can deform those intervals into new contours which are off the real line and the exponential terms will decay as $t$ goes to infinity on the new contours.", "The core idea of this step of the $\\bar{\\partial }$ -steepest descent is to use Stokes' theorem to transfer contour integrals to double integrals, while in the original Deift-Zhou's method, this step is done by first performing rational approximation then analytic continuation.", "After lenses-opening, we will end up at a mixed $\\bar{\\partial }$ -RHP.", "Next, from section and section , we will first approximate the pure RHP.", "Three main steps of approximating the pure RHP are so-called localization, phase reduction and contribution separation, which lead to an exact solvable model RHP (also called Its' isomonodromy problem).", "Due to the exact solvability of the model RHP and the small norm theory, one can establish the existence and uniqueness of the pure RHP part of the mixed $\\bar{\\partial }$ -RHP.", "The last step (see section ) is to estimate the errors by analyze the pure $\\bar{\\partial }$ -problem which dominates the errors generated by approximating the pure RHP.", "Undo all the steps, we will eventually prove Theorem REF .", "Then, in section , we will study the long-time behavior of the potential in the fast decaying region.", "Following similar analysis, we end up proving Theorem REF .", "The final section (see section ) is devoted to proving Theorem REF .", "In that section, we first give an algorithm to generate the Painlevé II hierarchy.", "Following the method of $\\bar{\\partial }$ -steepest descent, we represent the long-time behavior of the potential by the solution to a member of the Painlevé II hierarchy.", "In this section, we simply review the inverse scattering transform for the AKNS hierarchy in a certain weighted $L^2$ Sobolev space.", "For more details, we direct readers to Zhou's paper [33].", "The AKNS hierarchy is the integrable hierarchy associated with the following spectral problem: $\\psi _x(x,t;z)=(-iz\\sigma _3+Q(x,t))\\psi (x,t;z),$ where $\\sigma _3=\\begin{pmatrix}1 &0\\\\ 0&-1 \\end{pmatrix}$ and $Q(x,t)=\\begin{pmatrix}0&q(x,t)\\\\r(x,t)&0 \\end{pmatrix}$ .", "In the current paper, we only consider the defocusing type reduction: $r(x,t)=q(x,t)\\in \\mathbb {R},$ and we assume $q(x,t=0)\\in H^{n-1,1}(\\mathbb {R},dx)$This guarantees the time evolving of the initial data will stay in $H^{1,1}$ .", "Roughly speaking, from Zhou's work, we know $q(x,0)\\in H^{n-1,1}\\subset H^{1,1}$ is mapped to $R(z)\\in H^{1,n-1}$ .", "Time evolution of the reflection coefficient gives $R(z)e^{itz^n}$ , which belongs to $H^{1,1}$ due to the fact that $R(z)\\in H^{1,n-1}$ , and then the inverse scattering leads to $q(x,t)\\in H^{1,1}$ .. For $t=0$ , we are looking for solutions (so-called Jost solutions) of equation (REF ) in $H^{1,1}(\\mathbb {R},dx)$ , which satisfy the following boundary conditions at infinity: $\\psi _{\\pm }=e^{ixz\\sigma _3}+o(1),\\quad x\\rightarrow \\pm \\infty .$ The scattering matrix $S(z)$ is then defined as $S(z):=\\psi _+\\psi _-^{-1}.$ It is well known that $S$ enjoys the following properties: for $z\\in \\mathbb {R}$ , $S(z)=\\begin{pmatrix}a(z) & \\bar{b}(z)\\\\b(z) & \\bar{a}(z)\\end{pmatrix},$ where $a,b$ can be represented in terms of the initial data and the eigenfunctions $\\psi $ .", "To find such representations, we consider $\\mu ^{(\\pm )}=\\psi _\\pm e^{-izx\\sigma _3}.$ Then the spectral problem (REF ) becomes: $(\\mu ^{(\\pm )})_x=iz[\\sigma _3,\\mu ^{(\\pm )}]+Q\\mu ^{(\\pm )}.$ Then the representations of $a,b$ read $a(z)&=\\mu ^{(+)}_{11}(x\\rightarrow -\\infty )=1-\\int _{\\mathbb {R}}q(y)\\mu _{21}^{(+)}(y,z)dy,\\\\b(z)&=\\mu ^{(+)}_{12}(x\\rightarrow -\\infty )=-\\int _{\\mathbb {R}}e^{2iyz}q(y)\\mu _{22}^{(+)}(y,z)dy.$ From the above representations, it is straightforward to show that $a(z)=1+\\mathcal {O}(1/z)$ and $b(z)=\\mathcal {O}(1/z)$ as $z\\rightarrow \\infty $ , and $a(z)$ can be analytically extended to the upper half plane.", "Now, setting $m_+(x,z)&=(\\mu _1^{(+)}(x,z)/a(z),\\mu _2^{(-)}(x,z)),\\ \\operatorname{Im}{z}\\ge 0,\\\\m_-(x,z)&=(\\mu _1^{(-)}(x,z),\\mu _2^{(+)}(x,z)/\\bar{a}(z)),\\ \\operatorname{Im}{z}\\le 0,$ we can then define the jump matrix on the real line by $v(z)=e^{-izx\\text{ ad}\\sigma _3}(m_-^{-1}m_+).$ A direct computation shows $v(z)=\\begin{pmatrix}1-\|R(z)\|^2 & -\\bar{R}(z)\\\\R(z) & 1\\end{pmatrix},$ where $R(z)=\\frac{b(z)}{a(z)}.$ The deformation of the spectral problem (REF ) with respect to $t$ is governed by the following equation: $\\psi _t(x,t;z)&=\\left(\\sum _{k=0}^nQ_k(x,t)z^k\\right)\\psi (x,t;z).$ To generate isospectral flow, $\\psi $ need to satisfy the compatiblity condition, i.e., $\\psi _{xt}=\\psi _{tx}$ .", "By this condition, one can uniquely determine $Q_k$ if the integration constants are assumed to be all zeros.", "One can systematically determines the $Q_k$ 's via associated Lie algebra techniques, see for example [24].", "Through the Lie algebra, one can show the AKNS hierarchy is integrable, i.e, there are infinite many conservation laws.", "Moreover, using the powerful trace identity [31], one can easily show the bi-Hamiltonian structure of the AKNS hierarchy.", "Moreover, under the same framework, one can show that any linear combinations of the time-evolution problem are also integrable.", "The compatibility condition of (REF ) and (REF ) generates integrable PDEs, including the defocusing nonlinear Schrödinger equation, the modified KdV equation, the fifth-order modified KdV equation.", "Due to the decaying of the potential $Q$ , it is easy to show the time evolution of the jump matrix $v$ is trivial.", "Formally speaking, since $S\\psi _-=\\psi _+$ , taking derivatives with respect to $t$ on both sides leads to $S_t\\psi _-+S\\psi _{-,t}=\\psi _{+,t},$ then by the time evolution equation on $\\psi $ , we have $S_t\\psi _-+S\\left(\\sum _{k=0}^nQ_k(x,t)z^k\\right)\\psi _-&=\\left(\\sum _{k=0}^nQ_k(x,t)z^k\\right)\\psi _+\\\\&=\\left(\\sum _{k=0}^nQ_k(x,t)z^k\\right)S\\psi _-,$ letting $x\\rightarrow -\\infty $ , and since for the case of AKNS flows, all coefficients of $z^k,k=0,\\cdots ,n-1,$ will vanish, we arrive at (see, e.g., [25], [28]): $S_t=[Q_nz^n,S].$ It is of our current interest that $Q_n=-icz^n\\sigma _3$ for some positive constant $c$ .", "Therefore, we have the time evolution for the scattering matrix $S(z;t)=e^{-icz^nt\\text{ ad }{\\sigma _3}}S(z),$ where $e^{\\text{ ad }{\\sigma _3}}(\\cdot ):=e^{\\sigma _3}(\\cdot )e^{-\\sigma _3}.$ This implies the time evolution of the jump matrix $v(z)$ (see (REF )), and we have $v_\\theta (z):=e^{-it\\theta (z;x,t)\\text{ ad }{\\sigma _3}}v(z),$ where (in the case of the AKNS hierarchy) $\\theta (z;x,t)=\\frac{x}{t}z+cz^n$ for some positive constant $c$ .", "Finally, we formulate the direct scattering problem as a Riemann-Hilbert problem as follows: Riemann-Hilbert problem 2.1 Looking for a 2 by 2 matrix-valued functions $m(z;x,t)$ such that $m(z)$ is analytic in $\\mathbb {R}$ ; $m_+=m_-v_\\theta ,\\quad z\\in \\mathbb {R}$ ; $m=I+m_1(x,t)/z+\\mathcal {O}(1/z^2)$ , $z\\rightarrow \\infty $ ; where $v_\\theta $ is defined in equation (REF ) and $m_{\\pm }=\\lim _{\\epsilon \\downarrow 0}m(z\\pm i\\epsilon )$ .", "From the equation (REF ), and the definition of the jump matrix $v$ , we can recover the potential by $\\begin{split}q(x,t)&=-2i\\lim _{z\\rightarrow \\infty }[z(m-I)]_{12}\\\\ &=-2i (m_1(x,t))_{12}.\\end{split}$ In the following sections, we will perform the $\\bar{\\partial }$ -steepest descent method and study the asymptotic behavior for $t$ being sufficiently large." ], [ "Overview of the strategies", "In this section, we will simply review the idea of Deift-Zhou's nonlinear steepest descent method and its variation, the $\\bar{\\partial }$ -steepest descent method.", "In general, the key step in both methods is to deform the RHP.", "After the deformation, the new RHP is expected to be approximable locally as $t$ goes to $\\infty $ .", "Next, we will explain the main ideas of both methods.", "The notations in this section are used in this section only.", "Let us consider the following RHP on $\\mathbb {R}_+$ : $M_+(z)&=M_-(z)e^{-it\\theta (z)\\text{ ad }{\\sigma _3}}V(z),\\quad z\\in \\mathbb {R}_+,\\\\M(z)&\\rightarrow I,\\quad z\\rightarrow \\infty .$ The main ideaA good summary of this method can be found in [16].", "of Deift-Zhou's method is to find a factorization of $V(z)$ , say, $V(z)=V_-(z)V_+(z)$ , such that $V_{\\pm }(z)$ can be approximated by $\\tilde{V}_{\\pm }(z)$ which are analytic in the sectors $\\Omega _+$ and $\\Omega _-$ respectively, see Fig.REF .", "Figure: Contour deformationBy introducing a new analytic function $\\tilde{M}$ as follows: $\\tilde{M}\|_{\\Omega }&=M,\\\\\\tilde{M}\|_{\\Omega _+}&=M\\tilde{V}_+^{-1},\\\\\\tilde{M}\|_{\\Omega _-}&=M\\tilde{V}_-,$ we arrive at a new RHP: $\\tilde{M}_+&=\\tilde{M}_-\\tilde{V},\\quad z\\in \\mathbb {R}_+\\cup \\Sigma _1\\cup \\Sigma _2,\\\\\\tilde{V}&={\\left\\lbrace \\begin{array}{ll}\\tilde{V}_+,\\quad z\\in \\Sigma _1,\\\\\\tilde{V}_-,\\quad z\\in \\Sigma _2,\\\\\\tilde{V}_-^{-1}V_-V_+\\tilde{V}_+^{-1},\\quad z\\in \\mathbb {R}_+.\\end{array}\\right.", "}$ Also, we want to guarantee that based on the signatures of the phase function $\\operatorname{Re}{(i\\theta (z))}$ in each sector, the new jumps converge rapidly to the identity away from $O$ as $t\\rightarrow \\infty $ .", "Usually, one needs to deform the RHP several times.", "Eventually, the initial RHP can be approximated locally by the following fairly simple model RHP: $M^{\\sharp }_+&=M^{\\sharp }_-e^{-it\\tilde{\\theta }(z)\\text{ ad }{\\sigma _3}}V(0),\\quad z\\in \\mathbb {R}_+,\\\\M^{\\sharp }&\\rightarrow I,\\quad z\\rightarrow \\infty ,$ where $\\tilde{\\theta }(z)$ is a certain rational approximation to $\\theta (z)$ near $z=0$ .", "This model RHP can be solved explicitly and by undoing all deformations, one can track all errors in the middle steps.", "The Deift-Zhou method of analyzing errors is heavily based on the harmonic analysis for the Cauchy operators on contours, however, the $\\bar{\\partial }$ -steepest descent method transfers the error estimation to some fairly simple estimations of certain double integrals.", "A natural way of connecting the contour integrals to the double integrals is to use Stokes' theorem (or the Cauchy-Green theorem): for any $C^1(\\mathbb {R}^2\\rightarrow $ function $f(z):=f(z,\\bar{z})$ , we have $\\int _{\\partial \\Omega }f(z)dz=2i\\int _{\\Omega }\\frac{\\partial f(z)}{\\partial \\bar{z}}dxdy,$ where $z=x+iy$ .", "So in the $\\bar{\\partial }$ -steepest descent theory, we try to find an interpolation, say $E(z)$ , between the old contour and the new one.", "Such an $E$ satisfies $E(z)={\\left\\lbrace \\begin{array}{ll}V_+(0),\\quad z\\in \\Sigma _1^-,\\\\V_-(0),\\quad z\\in \\Sigma _2^+,\\\\V_+(z),\\quad z\\in \\mathbb {R}_+^+,\\\\V_-(z),\\quad z\\in \\mathbb {R}_+^-,\\\\I,\\quad z\\in \\Sigma _1^+\\cup \\Sigma _2^-,\\end{array}\\right.", "}$ where all contours are orientated from $O$ to $\\infty $ and $\\Gamma ^{\\pm }$ mean the limit from left/right, and it is $C^1$ in $\\overline{\\Omega _+\\cup \\Omega _-}$ .", "Also, we want $e^{it\\tilde{\\theta }(z)\\text{ ad }{\\sigma _3}}V_{\\pm }(0)$ go to 0 as $t\\rightarrow \\infty $ .", "Now, let us set $\\hat{M}=M(z)E(z)$ , we obtain the so-called $\\bar{\\partial }$ -RHP: (The RHP) $\\hat{M}_+=\\hat{M}_-e^{-it{\\theta }(z)\\text{ ad }{\\sigma _3}}\\hat{V}(z),\\quad z\\in \\Sigma _1\\cup \\Sigma _2$ , where $\\hat{V}(z)={\\left\\lbrace \\begin{array}{ll}e^{-it{\\theta }(z)\\text{ ad }{\\sigma _3}}V_+(0),\\quad z\\in \\Sigma _1,\\\\e^{-it{\\theta }(z)\\text{ ad }{\\sigma _3}}V_-(0),\\quad z\\in \\Sigma _2.\\end{array}\\right.", "}$ (The $\\bar{\\partial }$ -problem) For any $z\\in , we have{\\begin{@align}{1}{-1}\\bar{\\partial }\\hat{M}=\\hat{M}E^{-1}\\bar{\\partial }{E}.\\end{@align}}$ The deformation of the RHP follows from Deift-Zhou's method, but the error estimations here are transferred to a dbar problem, which turns out to be equivalent to some singular integral equation with respect to the area measure.", "Then through some fairly simple estimates on the double integrals, one will obtain the same error estimates as the Deift-Zhou method.", "In the following sections, we will apply the $\\bar{\\partial }$ -steepest descent to the defocusing mKdV type reduction of the AKNS hierarchy.", "Conjugation In this section, we will factorize the jump matrix (as defined by equation (REF ) ) in a way that it can be used for deforming the RHP.", "It is easy to see that the jump matrix enjoys the following two kinds of factorization: $v_\\theta (z)={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & -\\bar{R}(z)e^{-2it\\theta }\\\\0& 1\\end{pmatrix}\\begin{pmatrix}1 & 0\\\\R(z)e^{2it\\theta } & 1\\end{pmatrix},\\\\\\begin{pmatrix}1 & 0\\\\\\frac{R(z)}{1-\|R(z)\|^2}e^{2it\\theta }& 1\\end{pmatrix}(1-\|R\|^2)^{\\sigma _3}\\begin{pmatrix}1 & -\\frac{\\bar{R}(z)}{1-\|R\|^2}e^{-2it\\theta }\\\\0 & 1\\end{pmatrix}.\\end{array}\\right.", "}$ In the light of the main ideas we described in the last section, we want to remove the middle term in the second factorization.", "By doing so, we can eventually find the proper factorization based on the signatures of the $\\operatorname{Re}(i\\theta )$ .", "Due to our assumptions on $\\theta $ , near a stationary phase point (say $\|z-z_j\|<\\epsilon $ , for some small positive $\\epsilon $ ), $\\theta =\\theta (z_j)+\\frac{\\theta ^{\\prime \\prime }(z_j)}{2}(z-z_j)^2+\\mathcal {O}(\|z-z_j\|^3)$ .", "If $\\theta ^{\\prime \\prime }(z_j)>0$ , then $\\operatorname{Re}(i\\theta (z))$ is negative in the line (I): $\\lbrace z=z_j+re^{i\\alpha }, r\\in (-\\epsilon ,\\epsilon ) \\text{ with fixed } \\alpha \\in (0,\\pi /2)\\rbrace $ , and it is positive in the line (II): $\\lbrace z=z_j+re^{i\\alpha }, r\\in (-\\epsilon ,\\epsilon ) \\text{ with fixed }\\alpha \\in (-\\pi /2,0)\\rbrace $ .", "On the line (I), notice that $e^{2it\\theta }$ decays to 0 as $t\\rightarrow \\infty $ , we can deform the jump on the contour right to the stationary phase point using the first factorization.", "With the same argument on the line (II), we can deform the jump on the contour left to the stationary phase point using the second factorization.", "If $\\theta ^{\\prime \\prime }(z_j)<0$ , notice now $e^{2it\\theta }$ decays to 0 as $t\\rightarrow \\infty $ on the line (II), and thus we need the second factorization for the jump on the contour right to the stationary phase point and the first factorization for the jump on the contour left to the stationary phase point.", "Motivated by the above arguments, we denote $D_\\pm =\\lbrace z\\in \\mathbb {R}: \\pm \\theta ^{\\prime }(z)>0\\rbrace $ To eliminate the diagonal matrix in the second factorization, we introduce a scalar RHP: $\\begin{split}\\delta _+&=\\delta _-[(1-\|R\|^2)\\chi _{D_-}+\\chi _{D_+}],\\quad z\\in \\mathbb {R},\\\\\\delta (z)&=1+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty .\\end{split}$ Then by conjugating the initial RHP, we arrive at a new RHP: Riemann-Hilbert problem 4.1 Looking for a 2 by 2 matrix-valued function $m^{[1]}(z;x,t)$ such that $m^{[1]}_+=m^{[1]}_-\\delta _-^{\\sigma _3}v_\\theta \\delta _+^{-\\sigma _3},\\quad z\\in \\mathbb {R};$ $m^{[1]}=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty .$ By denoting $v_\\theta ^{[1]}:=\\delta _-^{\\sigma _3}v_\\theta \\delta _+^{-\\sigma _3}$ , the new jump matrix reads $v_\\theta ^{[1]}(z)={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & -\\bar{R}(z)\\delta ^2(z)e^{-2it\\theta }\\\\0 & 1\\end{pmatrix}\\begin{pmatrix}1 & 0\\\\R(z)\\delta ^{-2}e^{2it\\theta } & 1\\end{pmatrix},\\quad z\\in D_+,\\\\\\begin{pmatrix}1 & 0\\\\\\frac{R(z)\\delta ^{-2}_-e^{2it\\theta }}{1-\|R\|^2} & 1\\end{pmatrix}\\begin{pmatrix}1 & -\\frac{\\bar{R}(z)\\delta _+^2(z)e^{-2it\\theta }}{1-\|R\|^2}\\\\0 & 1\\end{pmatrix},\\quad z\\in D_-.\\end{array}\\right.", "}$ The scalar RHP (REF ) has been carefully studied in the literature (see for example [4] Lemma 23.3, [15], [32] and [19]).", "Here we just list some of the properties.", "First, the solution to the RHP (REF ) can be represented as follows: $\\log {(\\delta (z))}=(C_{D_-}(\\log (1-\|R\|^2)))(z),z\\in D_-,$ where the Cauchy operator $(C_{D_-}f)(z)=\\frac{1}{2\\pi i}\\int _{D_-}\\frac{f(s)}{s-z}ds$ .", "Since we assume $R(z)\\in H^{1,1}_1(\\mathbb {R},dz)=H^{1,1}\\cap \\lbrace f:\|f\|<1\\rbrace $ , one can show $\\log (1-\|R\|^2)$ is in $H^{1,0}$ , and then by the Sobolev embedding, we know it is also Hölder continuous with index $1/2$ .", "Then, by the Privalov-Plemelj theorem, which says that Cauchy operator perseveres Hölder continuity with index less than 1, one can show $\\log (\\delta (z))$ is Hölder continuous with index $1/2$ except for the end points.", "Next we study the behavior near those points.", "Let us denote $\\eta (z)=-\\frac{1}{2\\pi }\\log (1-\|R(z)\|^2), \\quad z\\in \\mathbb {R}.$ We will prove the following proposition.", "First we define a tent function supported on the interval $[-\\epsilon ,\\epsilon ]$ , $s_\\epsilon (z)={\\left\\lbrace \\begin{array}{ll}0,\\quad \|z\|\\ge \\epsilon \\\\-\\frac{1}{\\epsilon }z+1,\\quad 0< z<\\epsilon ,\\\\\\frac{1}{\\epsilon }z+1,\\quad -\\epsilon < z\\le 0.\\end{array}\\right.", "}$ Proposition 4.2 For each $\\epsilon >0$ , and $\\epsilon \\le \\frac{1}{3}\\min _{j\\ne k}\|z_j-z_k\|, $ there exists a neighborhood $I=I(\\epsilon )$ such that the identity $\\log (\\delta (z))&=i\\int _{D_-\\backslash I}\\frac{\\eta (s)}{s-z}ds+i\\sum _{j=1}^l\\left[\\eta (z_j)(1+\\log (z-z_j))\\right]\\varepsilon _j\\\\&+i\\sum _{j=1}^l{\\int _{I\\cap D_-}\\frac{\\eta (s)-\\eta _j(s)}{s-z}ds}\\\\&+i\\sum _{j=1}^l\\frac{1}{\\epsilon }\\eta (z_j)[(z-z_j)\\log (z-z_j)-(z-z_j+\\varepsilon _j\\epsilon )\\log (z-z_j+\\varepsilon _j\\epsilon )]$ is true, where $\\varepsilon _j=\\text{sgn}(\\theta ^{\\prime \\prime }(z_j))$ , $\\eta _j(z)=\\eta (z_j)s_\\epsilon (z-z_j)$ and see (REF ) for the definition of $\\eta $ .", "As for the logarithm function, the branch is chosen such that $\\text{argument}\\in (-\\pi ,\\pi )$ .", "Let $I=\\cup _{j=1}^l(I_{j+}\\cup I_{j-})$ , where $I_{j\\pm }=\\lbrace z:0<\\pm (z-z_j)<\\epsilon \\rbrace $ .", "Now we have $\\log (\\delta (z))&=i\\int _{D_-\\backslash I}\\frac{\\eta (s)}{s-z}ds\\\\&+i\\sum _{j=1}^l(\\int _{I_{j-}\\cap D_-}+\\int _{I_{j+}\\cap D_-}\\frac{\\eta (s)}{s-z}ds).$ For each $j$ , we have $\\int _{I_{j-}}\\frac{\\eta (s)}{s-z}ds=\\int _{I_{j-}}\\frac{\\eta (s)-\\eta _j(s)}{s-z}ds+\\int _{I_{j-}}\\frac{\\eta _j(s)}{s-z}ds.$ The first integral on the right hand side is the non-tangential limit as $z\\rightarrow z_j$ and the second one generates a logarithm singularity near $z_j$ .", "In fact, direct computation shows $\\int _{I_{j-}}\\frac{\\eta _j(s)}{s-z}ds&=\\eta (z_j)+\\frac{1}{\\epsilon }[(z-z_j)\\log (z-z_j)-(z-z_j+\\epsilon )\\log (z-z_j+\\epsilon )]\\eta (z_j)\\\\&+\\eta (z_j)\\log (z-z_j).$ Similarly, for $I_{j+}$ , $\\int _{I_{j+}}\\frac{\\eta _j(s)}{s-z}ds&=-\\eta (z_j)+\\frac{1}{\\epsilon }[(z-z_j)\\log (z-z_j)-(z-z_j-\\epsilon )\\log (z-z_j-\\epsilon )]\\\\&-\\eta (z_j)\\log (z-z_j).$ And note that only one of the $I_{j\\pm }\\cap D_-$ is nonempty, which depends on the sign of the second derivative of the phase function $\\theta $ .", "By assembling all together, the proof is done.", "Remark 4.3 The proposition tells us how the function $\\delta (z)$ behavior near the saddle points.", "Near the saddle points $z_j$ , $\\delta (z)$ has a mild singularity $(z-z_j)^{i\\eta (z_j)}$ .", "Fortunately, those singularities are bounded along any ray off $\\mathbb {R}$ and hence in some sense they do not affect asymptotics much.", "It is worth mentioning that one can ignore the mild singularity by introducing an auxiliary function, see Lemma 3.1 in [18].", "Lenses opening The purpose of lens-opening is to deform the RHP on the real line to a new RHP on new contours such that jumps on the new contours will rapidly decay to $I$ as $t\\rightarrow \\infty $ .", "We first study the signature of $\\operatorname{Im}{\\theta }$ near the saddle point $z_j$ .", "Figure: Notations for studying signatures of Im(θ(z))\\operatorname{Im}(\\theta (z)) near z j z_jLet us denote $I_{j+}=[z_j,\\frac{z_j+z_{j+1}}{2}]$ and $I_{j-}=[\\frac{z_j+z_{j-1}}{2},z_j]$ .", "Two cases need to be discussed.", "The first case is $\\theta ^{\\prime \\prime }(z_j)>0$ , and so we have $I_{j\\pm }\\subset D_{\\pm }$ .", "The second case is $\\theta ^{\\prime \\prime }(z_j)<0$ , and then we have $I_{j\\pm }\\subset D_{\\mp }$ .", "Recall the factorization of the conjugated jump matrix $v^{[1]}_\\theta $ , to deform it from $I_{j+}$ to $\\Sigma _{j,1}$ , we need make sure the exponential term $e^{2it\\theta (z)}$ decays rapidly to $I$ on $\\Sigma _{j,1}$ , and thus we need to discuss $\\operatorname{Re}(i\\theta )$ on $\\Sigma _{j,1}$ .", "Considering a Taylor approximation of $\\theta (z)$ near $z_j$ , we have $\\theta (z)=\\theta (z_j)+\\varepsilon _jA_j(z-z_j)^2+O(\|z-z_j\|^3)$ , where $A_j=\\left\|\\frac{\\theta ^{\\prime \\prime }(z_j)}{2}\\right\|$ and $\\varepsilon _j=\\text{sgn}\\lbrace \\theta ^{\\prime \\prime }(z_j)\\rbrace $ .", "Let $z=z_j+u+iv=z_j+\\rho e^{i\\phi }$ .", "Then $\\operatorname{Im}(\\theta (z))=\\varepsilon _j A_j\\rho ^2 \\sin (2\\phi )+O(\\rho ^3)$ , where $\\phi \\in (0,\\alpha ]$ is fixed.", "Now we define the regions $\\Omega _{j,n},\\ n=1,\\cdots ,6,$ as follows: $\\begin{split}\\Omega _{j,1}&=\\left\\lbrace z=z_j+\\rho e^{i\\phi },\\phi \\in (0,\\alpha ],\\rho \\in (0,\\frac{\|z_j-z_{j+1}\|}{2\\cos {\\alpha }}),\\operatorname{Re}z\\in I^{\\varepsilon _j}_{j+}\\right\\rbrace ,\\\\\\Omega _{j,3}&=\\left\\lbrace z=z_j+\\rho e^{i\\phi },\\phi \\in [\\pi -\\alpha ,\\pi ),\\rho \\in (0,\\frac{\|z_j-z_{j-1}\|}{2\\cos {\\alpha }}),\\operatorname{Re}z\\in I^{\\varepsilon _j}_{j-}\\right\\rbrace ,\\\\\\Omega _{j,2}&=+\\backslash (\\Omega _{j,1}\\cup \\Omega _{j,3}),\\\\\\Omega _{j,4}&=\\left\\lbrace z=z_j+\\rho e^{i\\phi },\\phi \\in (\\pi ,\\pi +\\alpha ],\\rho \\in (0,\\frac{\|z_j-z_{j-1}\|}{2\\cos {\\alpha }}),\\operatorname{Re}z\\in I^{\\varepsilon _j}_{j-}\\right\\rbrace ,\\\\\\Omega _{j,6}&=\\left\\lbrace z=z_j+\\rho e^{i\\phi },\\phi \\in [-\\alpha ,0),\\rho \\in (0,\\frac{\|z_j-z_{j+1}\|}{2\\cos {\\alpha }}),\\operatorname{Re}z\\in I^{\\varepsilon _j}_{j+}\\right\\rbrace ,\\\\\\Omega _{j,5}&=-\\backslash (\\Omega _{j,4}\\cup \\Omega _{j,6}),\\end{split}$ where $I^{\\varepsilon _j}_{j\\pm }={\\left\\lbrace \\begin{array}{ll}I_{j\\pm },\\quad \\varepsilon _j=1,\\\\I_{j\\mp },\\quad \\varepsilon _j=-1.\\end{array}\\right.", "}$ Since the number of real saddle points is finite, we can always choose a sufficiently small $\\alpha $ , such that for each $j$ , $e^{2it\\theta }$ decays to 0 in $\\Omega _{j,1}\\cup \\Omega _{j,4}$ and $e^{-2it\\theta }$ decays to 0 in $\\Omega _{j,3}\\cup \\Omega _{j,6}$ .", "Now we are in the position to open the lenses.", "First we introduce a bounded smooth function $\\mathcal {K}$ defined on $[0,2\\pi ]$ such that $\\begin{split}\\mathcal {K}(0)=1,\\\\\\mathcal {K}(\\alpha )=0,\\\\\\text{Period of }\\mathcal {K} \\text{ is $\\pi $},\\\\\\mathcal {K} \\text{ is even function}.\\end{split}$ Consider $\\varepsilon _j=1$ first.", "Then the $\\bar{\\partial }$ extension functions are as follows.", "Let $z-z_j=u+iv=\\rho e^{i\\phi }$ ,and for the case $\\varepsilon _j=1$ , we set $\\begin{split}E_{j,1}(z)&=\\mathcal {K}(\\phi )R(u+z_j)\\delta ^{-2}(z)\\\\&+[1-\\mathcal {K}(\\phi )]R(z_j)\\delta _j^{-2}(z-z_j)^{-2i\\varepsilon _j\\eta (z_j)},\\quad z\\in \\Omega _{j,1},\\\\E_{j,3}(z)&=\\mathcal {K}(\\pi -\\phi )(-\\frac{\\bar{R}(u+z_j)}{1-\|R(u+z_j)\|^2}\\delta _+^2(z))\\\\&+[1-\\mathcal {K}(\\pi -\\phi )](-\\frac{\\bar{R}(z_j)}{1-\|R(z_j)\|^2}\\delta _{j}^2(z-z_j)^{2i\\varepsilon _j\\eta (z_j)}),\\quad z\\in \\Omega _{j,3},\\\\E_{j,4}(z)&=\\mathcal {K}(\\pi +\\phi )(\\frac{R(z_j+u)}{1-\|R(z_j+u)\|^2}\\delta _-^{-2}(z))\\\\&+[1-\\mathcal {K}(\\pi +\\phi )](\\frac{R(z_j)}{1-\|R(z_j)\|^2}\\delta _j^{-2}(z-z_j)^{-2i\\varepsilon _j\\eta (z_j)}),\\quad z\\in \\Omega _{j,4},\\\\E_{j,6}(z)&=\\mathcal {K}(-\\phi )(-\\bar{R}(z_j+u)\\delta ^2(z))\\\\&+[1-\\mathcal {K}(-\\phi )](-\\bar{R}(z_j)\\delta _j^2(z-z_j)^{2i\\varepsilon _j\\eta (z_j)}),\\quad z\\in \\Omega _{j,6},\\\\\\end{split}$ where $\\delta _j=\\lim _{\\begin{array}{c}z=z_j+\\rho e^{i\\phi },\\\\ \\rho \\rightarrow 0,\\\\ \\phi \\in (0,\\pi /2)\\end{array}}\\delta (z)(z-z_j)^{i\\eta (z_j)}.$ For the case $\\varepsilon _j=-1$ , one only needs to switch the index 1 with 3 and 4 with 6.", "For the sake of simplicity, in what follows, we focus just on the case $\\varepsilon _j=1$ .", "The extension functions can be considered as interpolations between jumps on the old and new contours.", "Using the extension functions $E_{j,k}, k=1,3,4,6$ , we can construct the lens-opening matrices $O(z)$ as follows: $O(z)={\\left\\lbrace \\begin{array}{ll}O_{j,n}(z)=\\begin{pmatrix}1 & 0\\\\(-1)^nE_{j,n}e^{2it\\theta (z)}& 1\\end{pmatrix},\\quad z\\in \\Omega _{j,n},\\quad n=1,4,\\\\O_{j,m}(z)=\\begin{pmatrix}1 & (-1)^{m}E_{j,m}e^{-2it\\theta (z)}\\\\0 & 1\\end{pmatrix},\\quad z\\in \\Omega _{j,m},\\quad m=3,6,\\\\O_{j,k}(z)=I,\\quad z\\in \\Omega _{j,k},\\quad k=2,5.\\end{array}\\right.", "}$ Then lens-opening is performed by multiplying $O(z)$ to the right of the matrix $m^{[1]}$ .", "Let us denote $m^{[2]}(z)=m^{[1]}(z)O(z),z\\in \\mathbb {R}$ .", "Due to the lacking of analyticity of $O(z)$ (in fact, since we only assume $R(z)\\in C^1(\\mathbb {R})$ , $O(z)$ is also just in $C^1(\\mathbb {R}^2)$Here, $R(z)\\in C^1(\\mathbb {R})$ means $R(z)$ is a function defined on the real line with continuous first order derivative.", "While since $O(z)$ is a matrix-valued function defined on the complex plan, so $O(z)\\in C^1(\\mathbb {R}^2)$ means all the entries have continuous first-order derivatives with respect to $z$ and $\\bar{z}$ .", "), we arrive at the following mixed $\\bar{\\partial }$ -Riemann-Hilbert problem($\\bar{\\partial }$ -RHP): Mixed $\\bar{\\partial }$ -Riemann-Hilbert problem 5.1 Looking for a 2 by 2 matrix-valued function $m^{[2]}$ such that The RHP $m^{[2]}(z)\\in C^1(\\mathbb {R}^2\\Sigma )$ ; $m^{[2]}_+=m^{[2]}_-v^{[2]}_\\theta ,\\quad z\\in \\cup _{j=1,...,l,k=1,2,3,4}\\Sigma _{j,k},$ where the jump matrices read $v^{[2]}_\\theta ={\\left\\lbrace \\begin{array}{ll}O_{j,1}^{-1}, \\quad z\\in \\Sigma _{j,1},\\\\O_{j,3}^{-1}, \\quad z\\in \\Sigma _{j,2},\\\\O_{j,4}, \\quad z\\in \\Sigma _{j,3},\\\\O_{j,6}, \\quad z\\in \\Sigma _{j,4};\\end{array}\\right.", "}$ $m^{[2]}(z)=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty $ .", "The $\\bar{\\partial }$ -problem For $z\\in , we have{\\begin{@align}{1}{-1}\\bar{\\partial }m^{[2]}(z)=m^{[2]}(z)\\bar{\\partial }O(z).\\end{@align}}$ To close this section, we state a bound estimate for $\\bar{\\partial }E_{j,k}$ , which will be used in later sections.", "Lemma 5.2 For $j=1\\cdots l,\\, k=1,2,3,4$ , and $z\\in \\Omega _{j,k},u=\\operatorname{Re}(z-z_j)$ , $\|\\bar{\\partial }E_{j,k}(z)\|\\le c(\|z-z_j\|^{-1/2}+\|R^{\\prime }(u+z_j)\|).$ In the polar coordinates, $\\bar{\\partial }=\\frac{e^{i\\phi }}{2}(\\partial _\\rho +i\\rho ^{-1}\\partial _{\\phi })$ .", "For $z$ in any ray starting from $z_j$ and off the real line, we have $\\bar{\\partial }E_{j,1}(z)&=\\frac{ie^{i\\phi }\\mathcal {K}^{\\prime }(\\phi )}{2\\rho }[R(u+z_j)\\delta ^{-2}(z)-R(z_j)\\delta ^{-2}_j(z-z_j)^{-2i\\eta (z_j)}]\\\\&+\\mathcal {K}(\\phi )R^{\\prime }(u+z_j)\\delta ^{-2}(z).$ From Proposition REF , we know $\|\\delta (z)-\\delta _j(z-z_j)^{i\\eta (z_j)}\|\\le c\|z-z_j\|^{1/2}$ .", "Also since $\\delta (z)^{-1}=e^{-C_{D_-}(\\log {(1-\|R\|^2)})},$ it is evident that $\\delta (z)^{-1}$ is bounded.", "Therefore $\|\\delta ^{-2}(z)-\\delta ^{-2}_j(z-z_j)^{-2i\\eta (z_j)}\|\\le c\|z-z_j\|^{1/2}.$ And we haveIn the middle steps, $c$ means a generic positive constant.", "$\| R(u+z_j)&\\delta ^{-2}(z)-R(z_j)\\delta ^{-2}_j(z-z_j)^{-2i\\eta (z_j)}\|\\\\&\\le \|R(u+z_j)-R(z_j)\|\|\\delta ^{-2}(z)\|\\\\&+\|\\delta ^{-2}(z)-\\delta ^{-2}_j(z-z_j)^{-2i\\eta (z_j)}\|\|R(z_j)\|\\\\&\\le c\|\\int _{z_j}^{u+z_j}R^{\\prime }(s)ds\|+c\|z-z_j\|^{1/2}\\\\& \\text{by Cauchy-Schwartz inequality}\\\\& \\le c\\Vert R^{\\prime }\\Vert _{L^2}\|z-z_j\|^{1/2}+c\|z-z_j\|^{1/2}\\\\& \\le c\|z-z_j\|^{1/2}.$ Therefore $\\begin{split}\|\\bar{\\partial }E_{j,1}(z)\|&\\le c\\rho ^{-1}\|z-z_j\|^{1/2}+c\|R^{\\prime }(u+z_j)\|\\\\&\\le c(\|z-z_j\|^{-1/2}+\|R^{\\prime }(u+z_j)\|).\\end{split}$ Here we have use the fact that $u\\le \\rho $ , which implies $\|z-z_j\|^{1/2}/\\rho =u^{1/2}/\\rho \\le u^{-1/2}$ .", "Noting also that $\\sup \|R\|<1$ , we have $\\frac{R}{1-\|R\|^2}\\le \\frac{R}{1-\\sup \|R\|}$ , and thus all the estimates for $E_{j,1}$ can be smoothly moved to $E_{j,k},k=3,4,6$ .", "Separate contributions and phase reduction The RHP and the mixed $\\bar{\\partial }$ -RHP we have discussed above are global.", "In this section, we shall approximate the global RHP by performing two steps: (1) separate contributions from each stationary phase point, (2) phase reduction.", "Before that, let us first consider two saddle points $z_j,z_{j+1}$ , and discuss $\\varepsilon _j=1=-\\varepsilon _{j+1}$ for example.", "We will first remove the vertical segments, see Fig.", "(REF ): $\\Sigma _{j+\\frac{1}{2}}:=\\Omega _{j,1}\\cap \\Omega _{j+1,3} \\cup \\Omega _{j,6}\\cap \\Omega _{j+1,4}\\backslash \\mathbb {R},$ where $\\Omega _{j,\\cdot }$ 's are defined in REF .", "Figure: Jumps in a small triangular region.Recall the constructions of $E_{j,1}$ and $E_{j+1,3}$ (see (REF )), the boundary value of $m^{[2]}(z)$ on $\\Sigma _{j+\\frac{1}{2}}$ from $\\Omega _{j,1}$ is $m^{[1]}(z_{j+1/2}+iv)O_{j,1}(z_{j+1/2}+iv),$ while from $\\Omega _{j+1,3}$ it is $m^{[1]}(z_{j+1/2}+iv)O_{j+1,3}(z_{j+1/2}+iv).$ Both correspond to locally increasing parts of the phase function, and thus correspond to an upper/lower factorization.", "So the jump on the new contour $\\Sigma _{j+1/2}$ is $O_{j+1,3}O^{-1}_{j,1}(z)$ , $z=z_{j+\\frac{1}{2}}+iv$ , where the nontrivial entry is (regarding the property of $\\mathcal {K}$ and definitions of those matrix $O_{j,k}$ , see (REF ) and (REF ) ): $(1-\\mathcal {K}(\\phi ))&[R(z_j)\\delta _j^{-2}(z_{j+1/2}-z_j+iv)^{-2i\\eta (z_j)}\\\\&-R(z_{j+1})\\delta _{j+1}^{-2}(z_{j+1/2}-z_{j+1}+iv)^{-2i\\eta (z_{j+1})}]e^{2it\\theta (z_{j+1/2}+iv)},$ with $v\\in (0,(z_{j+1/2}-z_j)\\tan (\\alpha ))$ and $\\phi =\\arg {(z-z_j)}$ .", "Note that $\|(z_{j+1/2}-z_j+iv)^{-2i\\eta (z_j)}\|&=e^{2\\eta (z_j)\\phi }\\le e^{2\\eta (z_j)\\alpha }.$ and $\|e^{2it\\theta (z_{j+1/2}+iv)}\|\\le ce^{-2tdv},\\quad d=(z_{j+1}-z_{j})/2.$ Thus we have, for any $z\\in \\Sigma _{j+\\frac{1}{2}}$ , $O_{j+1,3}O^{-1}_{j,1}-I=\\mathcal {O}(e^{-ct}),\\quad t\\rightarrow \\infty ,$ where $c$ is some generic positive constant.", "Since the jump is close to $I$ , by a small norm theory, the solution will also be close to $I$ .", "In fact, we have the following estimate for the potential $\|\\lim _{z\\rightarrow \\infty }z&(m^{[2]}\|_{\\Sigma _{j+\\frac{1}{2}}}-I)\|\\\\& \\le \\frac{1}{2\\pi }\\int _0^{d\\tan {(\\alpha )}}\\left\|m^{[2]}_-(z_{j+1/2}+is)\\right\|\\left\|O_{j+1,3}O_{j,1}^{-1}(z_{j+1/2}+is)-I\\right\|\\text{ds}\\\\&\\le \\frac{1}{2\\pi }\\int _0^{d\\tan {(\\alpha )}}\\left\|m^{[2]}_-(z_{j+1/2}+is)\\right\|e^{-2tsd}\\text{ds}\\\\&\\le \\frac{1}{2\\pi }\\Vert m^{[1]}\|_{\\Sigma _{j+\\frac{1}{2} ,3}}\\Vert _\\infty \\Vert O_{j+1,3}\\Vert _\\infty \\int _{0}^{d\\tan {(\\alpha )}}e^{-2tsd}\\text{ds}\\\\&=\\mathcal {O}(t^{-1}),$ where we assume $m^{[1]}$ , as a solution to the conjugated RHP, existsThe existence and uniqueness will be discussed later..", "So it is analytic in a neighborhood of $\\Sigma _{j+\\frac{1}{2}}$ and hence it is bounded on $\\Sigma _{j+\\frac{1}{2}}$ .", "By the definition (see (REF )) of $O_{j+1,3}$ , it is continuous in $\\Sigma _{j+\\frac{1}{2}}$ and does not blow up at the endpoints of $\\Sigma _{j+\\frac{1}{2}}$ .", "So $\\Vert O_{j+1,3}\\Vert _\\infty $ is also finiteHere the $L^\\infty (\\Sigma )$ norm $\\Vert f(z)\\Vert _{\\infty }$ means $\\sup _{z\\in \\Sigma }\|f(z)\|$ , where $\|f(z)\|=\\max _{i,j=1,2,z\\in \\Sigma }\|f_{i,j}(z)\|$ ..", "Therefore, we can remove all those vertical segments by paying a price of error $\\mathcal {O}(t^{-1})$ , which will be dominated by the error generated by the $\\bar{\\partial }$ -problem (it is $\\mathcal {O}(t^{-3/4})$ , we will show it in a moment.)", "Let us denote the new RHP by $\\tilde{m}^{[2]}$ .", "To make it clear, we note that the jumps for $\\tilde{m}^{[2]}$ are $\\tilde{v}^{[2]}(z)={\\left\\lbrace \\begin{array}{ll}v^{[2]}(z),\\quad z\\in \\cup _{j=1,..,l,k=1,2,3,4}\\Sigma _{j,k},\\\\I,\\quad z\\in \\cup _{j=1,..,l}\\Sigma _{j+\\frac{1}{2}}\\cup \\mathbb {R}.\\end{array}\\right.", "}$ Next, we will show that the RHP for $\\tilde{m}^{[2]}$ can be localized to each saddle point.", "For example, near $z_j$ , along the segment $\\Sigma _{j,1}:z=z_j+u+iv, \\arg {z}=\\alpha $ , we have $\|E_{j,1}e^{2it\\theta }\|\\le ce^{-2t\\tan (\\alpha )u^2}$ It is well-known [14], [19] that the $\|E_{j,1}e^{2it\\theta }\|\\le ce^{-2t\\tan (\\alpha )u^2}$ , where let $u\\ge u_0>0$ , and then the jump matrix will go to $I$ with decaying rate at $\\mathcal {O}(e^{-ct}),c>0$ , as $t\\rightarrow \\infty $ .", "The RHP is localized in the small neighborhoods of those stationary phase points.", "Note that near each $z_j$ , we have $\\theta (z)=\\theta (z_0)+\\frac{\\theta ^{\\prime \\prime }(z_0)(z-z_0)^2}{2}+\\mathcal {O}{(\|z-z_j\|^3)}.$ By a similar argument of Lemma 3.35 in [16] or subsection 8.2 in [19] for the phase reduction, the error generated by reducing the phase function $\\theta $ to $\\theta (z_0)+\\frac{\\theta ^{\\prime \\prime }(z_0)(z-z_0)^2}{2}$ will be bounded by $\\mathcal {O}{(t^{-1})}$ .", "Both analysis of the mentioned references are based on the analysis of the so-called Beals-Coifman operator [3].", "Now we shall simply describe it here.", "For the sake of simplicity, we only consider the RHP on the contour $\\Sigma _{j,1}$ (for more details, we direct the interested reader to [15]): Riemann-Hilbert problem 6.1 Looking for 2 by 2 matrix-valued function $\\tilde{m}^{[2]}$ such that $\\tilde{m}(z)$ is analytic off $\\Sigma _{j,1}$ ; $\\tilde{m}_+=\\tilde{m}_-v^{[2]},\\quad z\\in \\Sigma _{j,1};$ $\\tilde{m}=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty $ .", "Since $E_{j,1}\|_{\\Sigma _{j,1}}$ is analytic near $\\Sigma _{j,1}$ for $z$ away from $z_j$ , and enjoys a factorization$(w^-,w^+)$ will be called the factorization data for the jump matrix.", ": $(I-w^-)^{-1}(I+w^+),$ where $w^-&=I-(v^{[2]})^{-1}=(v^{[2]})-I,\\\\w^+&=0,$ and the superscribes $\\pm $ indicate the analyticitiy in the left/right neighborhood of the the contour.", "Following the definition in [3], we define the Beals-Coifman operator, for any $f\\in L^2(\\Sigma _{j,1})$ , as follows: $C_w(f)=C_+(fw^-)+C_-(fw^+),$ where $C$ means the usual Cauchy operator, i.e., $Cf(z)=\\frac{1}{2\\pi i}\\int _{\\Sigma _{j,1}}\\frac{f(s)}{s-z}\\text{ds},$ and $C_{\\pm }$ means the non-tangential limits from left/right side.", "The following proposition, which plays a fundamental role in Deift-Zhou's method, is well-known.", "Proposition 6.2 (see also proposition 2.11 in [15]) If $\\mu \\in I+L^2$ solves the singular integral equation: $\\mu =I+C_w(\\mu ).$ Then the (unique) solution to the RHP for $\\tilde{m}$ readsHere $w=w^++w^-$ .", "$\\tilde{m}=I+C(\\mu w).$ Then follow the localization principle in [14], [19], [32], and the simple argument on the vertical segments, we arrive at a new RHP on the new contours: fixing $\\rho _0>0$ small, define $\\Sigma _{j,1}^o&:=\\lbrace z:z=z_j+\\rho e^{i\\alpha },\\rho \\in (0,\\rho _0)\\rbrace ,\\\\\\Sigma _{j,2}^o&:=\\lbrace z:z=z_j+\\rho e^{i(\\pi -\\alpha )},\\rho \\in (0,\\rho _0)\\rbrace ,\\\\\\Sigma _{j,3}^o&:=\\lbrace z:z=z_j-\\rho e^{i\\alpha },\\rho \\in (0,\\rho _0)\\rbrace ,\\\\\\Sigma _{j,4}^o&:=\\lbrace z:z=z_j+\\rho e^{i(\\alpha -\\pi )},\\rho \\in (0,\\rho _0)\\rbrace .$ Then with the new contour (see Fig.REF ) $\\Sigma ^o=\\cup _{j=1,..,l,k=1,2,3,4}\\Sigma _{j,k}^o$ , the new RHP reads as follows: Riemann-Hilbert problem 6.3 Looking for a 2 by 2 matrix-valued function $\\hat{m}^{[2]}$ such that $\\hat{m}^{[2]}_+=\\hat{m}^{[2]}_-\\hat{v}^{[2]},\\quad z\\in \\Sigma ^o,$ with $\\hat{v}^{[2]}=\\tilde{v}^{[2]}\|_{\\Sigma ^0};$ $\\hat{m}^{[2]}=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty .$ Figure: New contours, dashed line segments are those deleted parts.Moreover, since the potential of the mKdV hierarchy can be recovered by the formula (REF ), which can also be written as the Beals-Coifman solution: $q_{RHP}(x,t)=-\\frac{1}{2\\pi i}\\int _{\\Sigma }((I-C_w)^{-1}I)w(s)\\text{ds}.$ Then, by localization, we have $\\int _{\\Sigma }((I-C_w)^{-1}I)w(s)\\text{ds}=\\int _{\\Sigma ^o}((I-C_w)^{-1}I)w(s)\\text{ds}+\\mathcal {O}(t^{-1}), \\quad t\\rightarrow \\infty ,$ where $\\Sigma $ is the the contour before localization and $w$ can be easily defined in each cross since the jumps are all triangle matrices and all entries in the diagonal are one.", "Let us denote $q^o_{RHP}(x,t)=-\\frac{1}{2\\pi i}\\int _{\\Sigma ^o}((I-C_w)^{-1}I)w(s)\\text{ds}.$ Then from the localization principal, we have $q_{RHP}(x,t)=q^o_{RHP}(x,t)+\\mathcal {O}(t^{-1}),\\quad t\\rightarrow \\infty .$ Moreover, we define the RHP ($m^{[3]}$ ) which corresponds to the local Beals-Coifman solution (i.e.", "$q^o_{RHP}$ ) as follows: Riemann-Hilbert problem 6.4 Looking for a 2 by 2 matrix-valued function $m^{[3]}$ such that $m^{[3]}_+=m^{[3]}_-v^{[3]}(z),\\quad z\\in \\Sigma ^o$ , with jump matrix reads $v^{[3]}=\\hat{v}^{[2]}\\upharpoonright _{\\Sigma ^o};$ $m^{[3]}=I+\\mathcal {O}{(z^{-1})},\\quad z\\rightarrow \\infty $ .", "However, the integral $\\int _{\\Sigma ^o}((I-C_w)^{-1}I)w(s)\\text{ds}$ is still hard to compute, and following the Deift-Zhou method, we need to separate the contributions from each stationary phase point.", "Thus, we need the following important lemma.", "Lemma 6.5 (see equation (3.64) or proposition 3.66 in [14]) As $t\\rightarrow \\infty $ , $\\int _{\\Sigma ^o}(\\left(1-C_w\\right)^{-1}I)w=\\sum _{j=1}^{l}\\int _{\\Sigma ^o_j}((1-C_{w_j})^{-1}I)w_j+\\mathcal {O}(t^{-1}),$ where $w_j$ is the factorization data supported on $\\Sigma ^o_j=\\cup _{k=1}^4\\Sigma ^o_{j,k}$ , $w=\\sum _{j=1}^lw_j$ and $\\Sigma ^o=\\cup _{j}^l\\Sigma ^o_j$ .", "First, recall the following observation by Varzugin [32], $(1-C_w)(1+\\sum _jC_{w_j}(1-C_{w_j})^{-1})=1-\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1}.$ With the hints from this observation, we need to estimate the norms of $C_{w_j}C_{w_k}$ from $L^\\infty $ to $L^2$ and from $L^2$ to $L^2$ .", "Also from next section (with a small norm argument), we know $(1-C_{w_j})^{-1}$ are uniformly bounded in $L^2$ sense.", "Now let us focus on the contour $\\Sigma ^o_{j,1}$ , and $\\varepsilon =1$ .", "Then the nontrivial entry of the factorization data is $E_{j,1}(z)e^{-2it\\theta (z)},z\\in \\Sigma ^o_{j,1}$ , and thus we have $\|w_{j}\\upharpoonright _{\\Sigma ^o_{j,1}}\|\\le ce^{-2t\\tan (\\alpha )u^2},$ which implies that $\\Vert w_{j}\\upharpoonright _{\\Sigma ^o_{j,1}}\\Vert _{L^1}=\\mathcal {O}(t^{-1/2})$ and $\\Vert w_{j}\\upharpoonright _{\\Sigma ^o_{j,1}}\\Vert _{L^2}=\\mathcal {O}(t^{-1/4})$ .", "Then following exactly the same steps in the proof of [14], Lemma 3.5, we have for $j\\ne k$ $\\Vert C_{w_j}C_{w_k}\\Vert _{L^2(\\Sigma ^o)}&=\\mathcal {O}(t^{-1/2}),\\\\\\Vert C_{w_j}C_{w_k}\\Vert _{L^\\infty \\rightarrow L^2(\\Sigma ^o)}&=\\mathcal {O}(t^{-3/4}).$ Then use the resolvent identities and the Cauchy-Schwartz inequality, $((1-C_w)^{-1}I)&=I+\\sum _{j=1}^{l}C_{w_j}(1-C_{w_j})^{-1}I\\\\&+[1+\\sum _{j=1}^{l}C_{w_j}(1-C_{w_j})^{-1}][1-\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1}]^{-1}\\\\&(\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1})I\\\\&=I+\\sum _{j=1}^{l}C_{w_j}(1-C_{w_j})^{-1}I+ABDI,$ where $A&:=1+\\sum _{j=1}^{l}C_{w_j}(1-C_{w_j})^{-1},\\\\B&:=[1-\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1}]^{-1},\\\\D&:=\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1},$ and thus $\|\\int _{\\Sigma ^o}ABDIw\|&\\le \\Vert A\\Vert _{L^2}\\Vert B\\Vert _{L^2}\\Vert D\\Vert _{L^\\infty \\rightarrow L^2}\\Vert w\\Vert _{L^2}\\\\&\\le c t^{-3/4}t^{-1/4}=\\mathcal {O}{(t^{-1})}.$ Then applying the restriction lemma ([14], Lemma 2.56), we have $\\int _{\\Sigma ^o}(I+C_{w_j}(1-C_{w_j})^{-1}I)w\\upharpoonright _{\\Sigma ^o_j}&=\\int _{\\Sigma ^o_j}(I+C_{w_j}(1-C_{w_j})^{-1}I)w\\\\&=\\int _{\\Sigma ^o_j}((1-C_{w_j})^{-1}I)w_j.$ Therefore, the proof is done.", "A model Riemann-Hilbert problem In the previous section, we have reduce the global RHP to $l$ local RHPs near each stationary phase point due to Lemma REF .", "In fact, near each stationary phase point, we need to compute the integral $\\int _{\\Sigma _j^o}((1-C_{w_j})^{-1}I)w_j$ , which is equivalent to a local RHP.", "In this section, we will approximate the local RHPs by a model RHP which can be solved explicitly by solving a parabolic-cylinder equation.", "Consider the following RHP: Riemann-Hilbert problem 7.1 Looking for a 2 by 2 matrix-valued function $P(\\xi ;R)$ such that $P_+(\\xi ;R)=P_-(\\xi ;R)J(\\xi ),\\xi \\in \\mathbb {R}$ , where $J(\\xi )=\\begin{pmatrix}1-\|R\|^2&-\\bar{R}\\\\R&1\\end{pmatrix}$ is a constant matrix with respect to $\\xi $ and the constant $R$ satisfies $\|R\|<1$ ; $P(\\xi ;R)=\\xi ^{i\\eta \\sigma _3}e^{-i\\frac{\\xi ^2}{4}\\sigma _3}(I+P_1\\xi ^{-1}+\\mathcal {O}(\\xi ^{-2})),\\quad \\xi \\rightarrow \\infty $ , where $P_1=\\begin{pmatrix}0&\\beta \\\\\\bar{\\beta }&0\\end{pmatrix}$ .", "Then by Liouville's argument, $P^{\\prime }P^{-1}$ is analytic and thus $P^{\\prime }(\\xi )=(-\\frac{i\\xi }{2}\\sigma _3-\\frac{i}{2}[\\sigma _3,P_1])P(\\xi ),$ which can be solved in terms of the parabolic-cylinder equation, and apply the asymptotics formulas we can eventually determine that $\\beta =\\frac{\\sqrt{2\\pi }e^{i\\pi /4}e^{-\\pi \\eta /2}}{R\\Gamma (-a)},$ where $a=i\\eta ,$ with $\\eta =-\\frac{1}{2\\pi }\\log {(1-\|R\|^2)}$ .", "The above result has been presented in the literatureThe first description of this model RHP was presented by A. R. Its [22].", "Later examples of the model can be find in [14], [15], [17], [19], [32], [26], [27].", "in many ways.", "Here we follows the representations in [14].", "Next, we will connect this model RHP to the original RHP.", "Recall, near stationary phase point $z_j$ , we need to estimate integral $\\int _{\\Sigma _j^o}((1-C_{w_j})^{-1}I)(w_{j+}+w_{j-})$ , which is equivalent to solve the following RHP ($m^{[3,j]},\\quad j=1,\\cdots ,l$ ): $m^{[3,j]}_+(z)=m^{[3,j]}_-(z)v^{[3,j]}(z),z\\in \\Sigma _j^o$ .", "The jump matrix reads $v^{[3,j]}(z)={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1&0\\\\R_j^\\sharp (z-z_j)^{-2i\\eta (z_j)}e^{-it\\theta ^{\\prime \\prime }(z_j)(z-z_j)^2}&1\\end{pmatrix},z\\in \\Sigma _{j,1}^o,\\\\\\begin{pmatrix}1&-\\frac{\\bar{R}_j^\\sharp }{1-\|R^\\sharp _j\|^2}(z-z_j)^{2i\\eta (z_j)}e^{it\\theta ^{\\prime \\prime }(z_j)(z-z_j)^2}\\\\0&1\\end{pmatrix},z\\in \\Sigma _{j,2}^o,\\\\\\begin{pmatrix}1&0\\\\\\frac{R_j^\\sharp }{1-\|R^\\sharp _j\|^2}(z-z_j)^{2i\\eta (z_j)}e^{-it\\theta ^{\\prime \\prime }(z_j)(z-z_j)^2}&1\\end{pmatrix},z\\in \\Sigma _{j,3}^o,\\\\\\begin{pmatrix}1&-\\bar{R}_j^\\sharp (z-z_j)^{-2i\\eta (z_j)}e^{it\\theta ^{\\prime \\prime }(z_j)(z-z_j)^2}\\\\0&1\\end{pmatrix},z\\in \\Sigma _{j,4}^o,\\end{array}\\right.", "}$ where $R_j^\\sharp =R_j\\delta ^{-2}_je^{-2it\\theta (z_j)}$ ; $m^{[3,j]}=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty .$ Set $\\xi =(2t\\theta ^{\\prime \\prime }(z_j))^{1/2}(z-z_j)$ and by closing lenses, we arrive at an equivalent RHP on the real line: $m^{[4,j]}(\\xi )_+=m^{[4]}_-v^{[4,j]}(\\xi ),\\xi \\in \\Sigma _{j}^p$ .", "The new jump is $v^{[4,j]}(\\xi )=(2\\theta ^{\\prime \\prime }(z_j)t)^{-\\frac{i\\eta (z_j)}{2}\\text{ ad }\\sigma _3}\\xi ^{i\\eta (z_j)\\text{ ad }\\sigma _3}e^{-\\frac{i\\xi ^2}{4}\\text{ ad }\\sigma _3}\\begin{pmatrix}1-\|R_j^\\sharp \|^2&-\\bar{R}_j^\\sharp \\\\R_j^\\sharp &1\\end{pmatrix};$ $m^{[4,j]}=I+\\mathcal {O}(\\xi ^{-1}),\\xi \\rightarrow \\infty .$ Comparing with the model RHP, we observe that $m^{[4,j]}(\\xi )(2\\theta ^{\\prime \\prime }(z_j)t)^{-\\frac{i\\eta (z_j)}{2}\\sigma _3}\\xi ^{i\\eta (z_j)\\sigma _3}e^{-\\frac{i\\xi ^2}{4}\\sigma _3}$ solves the model RHP, which leads to $m^{[4]}_{1,12}&=\\frac{\\sqrt{2\\pi }e^{i\\pi /4}e^{-\\pi \\eta (z_j)/2}}{R^\\sharp _j\\Gamma (-i\\eta (z_j))},\\\\m^{[4]}_{1,21}&=\\frac{-\\sqrt{2\\pi }e^{-i\\pi /4}e^{-\\pi \\eta (z_j)/2}}{\\bar{R}^\\sharp _j\\Gamma (i\\eta (z_j))}.$ Changing the variable $\\xi $ back to $z$ , we have $m^{[3,j]}_{1,12}(t)&= (2t\\theta ^{\\prime \\prime }(z_j))^{-\\frac{1}{2}-\\frac{i\\eta (z_j)}{2}}\\frac{\\sqrt{2\\pi }e^{i\\pi /4}e^{-\\pi \\eta (z_j)/2}}{R_j^\\sharp \\Gamma (-i\\eta (z_j))},\\\\m^{[3,j]}_{1,21}(t)&=- (2t\\theta ^{\\prime \\prime }(z_j))^{-\\frac{1}{2}+\\frac{i\\eta (z_j)}{2}}\\frac{\\sqrt{2\\pi }e^{-i\\pi /4}e^{-\\pi \\eta (z_j)/2}}{\\bar{R}_j^\\sharp \\Gamma (i\\eta (z_j))}.$ Noting that $R^\\sharp _j=R_j\\delta ^{-2}_je^{2it\\theta (z_j)}$ , one can rewrite in a neat way: $m^{[3,j]}_{1,12}(t)=\\frac{\|\\eta (z_j)\|^{1/2}}{\\sqrt{2t\\theta ^{\\prime \\prime }(z_j)}}e^{i\\varphi (t)},\\\\m^{[3,j]}_{1,21}(t)=\\frac{\|\\eta (z_j)\|^{1/2}}{\\sqrt{2t\\theta ^{\\prime \\prime }(z_j)}}e^{-i\\varphi (t)},$ where the phase is $\\varphi (t)=\\frac{\\pi }{4}-\\arg \\Gamma (-i\\eta (z_j))-2t\\theta (z_j)-\\frac{\\eta (z_j)}{2}\\log \|2t\\theta ^{\\prime \\prime }(z_j)\|+2\\arg (\\delta _j)+\\arg (R_j).$ Here we have used the fact that $\|\\beta \|^2=\\eta $ .", "Denoting $q_{as}(x,t)=-2i\\sum _{j=1}^{l}\\frac{\|\\eta (z_j)\|^{1/2}}{\\sqrt{2t\\theta ^{\\prime \\prime }(z_j)}}e^{i\\varphi (t)},$ then the connection formula (REF ) and Lemma REF lead to $q^o_{RHP}(x,t)=q_{as}(x,t)+\\mathcal {O}(t^{-1}),\\quad t\\rightarrow \\infty .$ Errors from the pure $\\bar{\\partial }$ -problem In this section, we will discuss the error generated from the pure $\\bar{\\partial }$ -problem of $m^{[2]}$ .", "Let us denote $E(z)=m^{[2]}({m}^{[2]}_{RHP})^{-1},$ where $m^{[2]}_{RHP}$ denotes the solution to the pure RHP part of $m^{[2]}$ .", "Assuming the existence (which we will be provided in the next section), and by the normalization condition, we have $E(z)=1+(m_1^{[2]}-{m}^{[2]}_{RHP,1})z^{-1}+\\mathcal {O}(z^{-2}),\\quad z\\rightarrow \\infty .$ Due to the procedure of localization and separation of the contributions, we can approximate $m^{[2]}_{RHP}$ by $\\hat{m}^{[2]}$ , and the error of approximating the potential is of $\\mathcal {O}(t^{-1})$ as $t\\rightarrow \\infty $ .", "Thus, by the equation(REF ), $q(x,t)=q_{RHP}(x,t)+\\mathcal {O}(t^{-1})+\\lim _{z\\rightarrow \\infty }z(E-I),\\quad t\\rightarrow \\infty .$ Moreover, from this construction (equation (REF )), there is no jump on the contours $\\Sigma _{j,k},k=1,2,3,4,$ but only a pure $\\bar{\\partial }$ -problem is left due to the non-analyticity.", "The $\\bar{\\partial }$ -problem reads $\\bar{\\partial }E=EW,$ where $W(z)=m^{[2]}_{RHP}\\bar{\\partial }O(z)(m^{[2]}_{RHP})^{-1}.$ From the normalization condition of $m^{[2]}_{RHP}$ , we see it is uniformly bounded by $\\frac{c}{1-\\sup {R}}$ .", "And to estimate the errors of recovering the potential, one actually needs to estimate $\\lim _{z\\rightarrow \\infty }z(E-I)$ , where the limit can be chosen along any rays that are not parallel to $\\mathbb {R}$ .", "For simplicity, we will take the imaginary axis.", "The $\\bar{\\partial }$ -problem is equivalent to the following Fredholm integral equation by a simple application of the generalized Cauchy integral formula: $E(z)=I-\\frac{1}{\\pi }\\int _{\\frac{E(s)W(s)}{s-z}dA(s).", "}In the following, we will show for each fixed z\\in , \\mathcal {K}_W(E)(z):=\\int _{\\frac{E(s)W(s)}{s-z}dA(s) is bounded and then by the dominated convergence theorem, we will show \\lim _{z\\rightarrow \\infty }z(E-I)=\\mathcal {O}(t^{-3/4}).", "First of all, since m^{[3]} is uniformly bounded, upon settng z=z_j+u+iv, we have{\\begin{@align}{1}{-1}\\Vert W\\Vert _{\\infty }\\lesssim {\\left\\lbrace \\begin{array}{ll}\|\\bar{\\partial }E_{j,k}\|e^{-2t\\theta ^{\\prime \\prime }(z_j)uv},z\\in \\Omega _{j,k},k=1,4,\\\\\|\\bar{\\partial }E_{j,k}\|e^{2t\\theta ^{\\prime \\prime }(z_j)uv},z\\in \\Omega _{j,k},k=3,6,\\\\\\end{array}\\right.", "},\\end{@align}}where 0\\le a\\lesssim b means there exists C>0 such that a\\le Cb.", "Then we have{\\begin{@align}{1}{-1}\\mathcal {K}_W(E)\\le \\Vert E\\Vert _\\infty \\int _{\\Vert W(s)\\Vert _\\infty }{\|s-z\|}dA(s).\\end{@align}}We claim the following lemma:\\begin{lemma}Let \\Omega =\\lbrace s:s=\\rho e^{i\\phi }, \\rho \\ge 0,\\phi \\in [0,\\pi /4]\\rbrace , and z\\in \\Omega .", "Then{\\begin{@align}{1}{-1}\\int _\\Omega \\frac{\|u^2+v^2\|^{-1/4}e^{-tuv}}{\|u+iv-z\|}dudv=\\mathcal {O}(t^{-1/4}).\\end{@align}}\\end{lemma}\\begin{proof}Since there are two singularities of the integrand at z and (0,0).", "In the first case, set z\\ne 0, and let d=dist(z,0).", "We split \\Omega into three parts: \\Omega _1\\cup \\Omega _2\\cup \\Omega _3, where \\Omega _1=\\lbrace s:\|s\|<d/3\\rbrace \\cap \\Omega , \\Omega _2=\\lbrace s:\|s-z\|<d/3\\rbrace \\cap \\Omega and \\Omega _3=\\Omega \\backslash (\\Omega _1\\cup \\Omega _2).", "In the region \\Omega _1, \|s-z\|\\ge 2d/3, and thus\\begin{equation}\\begin{split}\|\\int _{\\Omega _1}\\frac{\|u^2+v^2\|^{-1/4}e^{-tuv}}{\|u+iv-z\|}dudv\|&\\le \\frac{3}{2d}\\int _{0}^\\infty \\int _0^u\\frac{e^{-tuv}}{(u^2+v^2)^{1/4}}dvdu\\\\& \\text{ substituted $v=wu$}\\\\&\\le \\frac{3}{2d}\\int _0^\\infty \\int _0^1\\frac{e^{-tu^2w}}{(1+w^2)^{1/4}}u^{1/2}dwdu\\\\&\\le \\frac{3}{2d}\\int _0^\\infty \\int _0^1 e^{-tu^2w}u^{1/2}dwdu\\\\&=\\frac{3}{2d}\\int _0^\\infty \\frac{1-e^{-tu^2}}{tu^{3/2}}du\\\\&=\\frac{3}{2d}\\frac{1}{2}t^{-3/4}\\int _0^\\infty \\frac{1-e^{-u}}{u^{5/4}}du\\\\&=\\frac{3}{d}\\Gamma (3/4)t^{-3/4}.\\end{split}\\end{equation}In the region \\Omega _2, \|s\|^{-1/2}\\le (2d/3)^{-1/2}, we have{\\begin{@align}{1}{-1}\|\\int _{\\Omega _1}\\frac{\|u^2+v^2\|^{-1/4}e^{-tuv}}{\|u+iv-z\|}dudv\|&\\le \\sqrt{\\frac{3}{2d}}\\int _{\\Omega _2}\\frac{e^{-tuv}}{((u-x)^2+(v-y)^2)^{1/2}}dvdu\\\\&\\le \\sqrt{\\frac{3}{2d}}\\int _{0}^{d/3}\\int _0^{2\\pi }e^{-t(x+\\rho \\cos (\\theta ))(y+\\rho \\sin (\\theta ))}d\\theta d\\rho \\\\&\\le \\frac{2\\pi }{3}\\sqrt{\\frac{3d}{2}}e^{-txy}.\\end{@align}}While in the region \\Omega _3,{\\begin{@align}{1}{-1}\|\\int _{\\Omega _3}\\frac{\|u^2+v^2\|^{-1/4}e^{-tuv}}{\|u+iv-z\|}dudv\|&\\le \\int _{0}^{\\infty }\\int _0^u e^{-tuv}dvdu=\\mathcal {O}(t^{-1}).\\end{@align}}Now consider z=0.", "We have{\\begin{@align}{1}{-1}\|\\int _{\\Omega }\\frac{e^{-tuv}}{(u^2+v^2)^{3/4}}dA(u,v)\|&=\\int _{0}^\\infty \\int _0^u\\frac{e^{-tuv}}{(u^2+v^2)^{3/4}}dvdu\\\\&=\\int _0^\\infty \\int _0^1\\frac{e^{-tu^2w}}{(1+w^2)^{3/4}u^{1/2}}dwdu\\\\&\\le \\int _0^\\infty \\int _0^1\\frac{e^{-tu^2w}}{u^{1/2}}dwdu\\\\&=\\int _0^\\infty \\frac{1-e^{-tu^2}}{tu^{5/2}}du\\\\&=\\int _0^\\infty \\frac{1-e^{-u}}{tt^{-5/4}u^{5/4}}t^{-1/2}\\frac{1}{2}u^{-\\frac{1}{2}}du\\\\&=\\frac{1}{2}t^{-1/4}\\int _0^\\infty \\frac{1-e^{-u}}{u^{7/4}}du\\\\&=\\frac{3}{8}t^{-1/4}\\Gamma (1/4).\\end{@align}}By assembling all together, the proof is done.\\end{proof}\\begin{remark}The essential fact that makes the above true is the rapid decay of the exponential factor in the region.", "And the lemma also tells us that those mild singularities, which have rational order growth, can be absorbed by the exponential factor.", "Back to our situation, after some elementary transformations (translation and rotation), the estimation of \\int _{\\Vert W(s)\\Vert _\\infty }{\|s-z\|}dA(s) will eventually reduce to a similar situation discussed in the above lemma.\\end{remark}Based on Lemma \\ref {estimates of W}, we know that when t is sufficiently large, \\Vert \\mathcal {K}_W\\Vert <1 and thus the resolvent is uniformly bounded, and we obtain the following estimate by taking a standard Neumann series, for some sufficiently large t_0,{\\begin{@align}{1}{-1}\\Vert E-I\\Vert _\\infty =\\Vert \\mathcal {K}_W(1-\\mathcal {K}_W)^{-1} I\\Vert _\\infty \\le \\frac{ct^{-1/4}}{1-ct^{-1/4}}\\le ct^{-1/4}, \\quad t>t_0.\\end{@align}}Now since for each z\\in \\Omega _{j,k}, we have \|\\bar{\\partial }E_{j,k}(z)\|\\le c(\|z-z_j\|^{-1/2}+\|R^{\\prime }(u+z_j)\|), and apply the dominated convergence theorem, we have{\\begin{@align}{1}{-1}\\lim _{z\\rightarrow \\infty }\|z(E-I)\|&\\le \\frac{1}{\\pi }\\sum _{j=1}^{l}\\sum _{k=1}^4\\Vert E\\Vert _{L^\\infty }\\int _{\\Omega _{j,k}}\\Vert W\\Vert _\\infty ds,\\end{@align}}and use the Lemma \\ref {estimates of W} again, we will eventually have:{\\begin{@align}{1}{-1}E_1=\\lim _{z\\rightarrow \\infty }\|z(E-I)\|=\\mathcal {O}(t^{-3/4}).\\end{@align}}}\\section {Asymptotics representation}First, we summarize all the steps as following (see Fig.\\ref {fig:steps}):\\begin{enumerate}[label={(0)}]\\item Initial RHP m^{[0]}=m, see RHP \\ref {RHP m 0}.\\item Conjugate initial RHP to obtain m^{[1]}=m^{[0]}\\delta ^{\\sigma _3}, see RHP \\ref {RHP m 1}.\\item Open lenses to obtain a mixed \\bar{\\partial }-RHP \\ref {dbar RHP}.\\item Approximate the RHP part m^{[2]}_{RHP} of m^{[2]} by removing \\Sigma _{j+\\frac{1}{2}} (see RHP \\ref {RHP m tilde}), localization (see RHP \\ref {RHP m hat}), reducing the phase function and separating the contributions (see RHP \\ref {RHP m 3}).", "The error term is \\mathcal {O}(t^{-1}).", "Note those exponential decaying errors are absorbed by \\mathcal {O}(t^{-1}).\\item Comparing m^{[2]} and m^{[2]}_{RHP} and computing the error by analysis a pure \\bar{\\partial }-problem.", "The error term is \\mathcal {O}(t^{-3/4}).\\end{enumerate}$ Figure: Steps of the ∂ ¯\\bar{\\partial }-steepest method.Now by undoing all the steps, we arrive at: $m^{[0]}(z)=E(z)m^{[2]}_{RHP}(z)O^{-1}(z)\\delta ^{-\\sigma _3}.$ Since $O(z)$ uniformly converges to $I$ as $z\\rightarrow \\infty $ , and $\\delta ^{-\\sigma _3}$ is a diagonal matrix, they do not affect the recovering of the potential.", "Thus we obtain $q(x,t)&=-2i(m^{[2]}_{RHP,1,12}+E_{1,12})\\\\&=q_{RHP}(x,t)-2iE_{1,12}\\\\& \\text{ by (\\ref {q rhp to q o rhp}),(\\ref {q o rhp to q as})}\\\\&=q_{as}(x,t)+\\mathcal {O}(t^{-1})-2iE_{1,12}\\\\& \\text{ by (\\ref {dbar error})}\\\\&=q_{as}(x,t)+\\mathcal {O}(t^{-1})+\\mathcal {O}(t^{-3/4})\\\\&=q_{as}(x,t)+\\mathcal {O}(t^{-3/4}),$ where $q_{as}(x,t)$ is given by equation (REF ).", "Remark 8.1 Note that due to the analysis in the section 7, according to Proposition 2.6 and Proposition 2.11 of [15], together with the small norm theory, the existence and uniqueness of the model RHP implies, via the estimates of the corresponding Beals-Coifman operators, the existence and uniqueness of RHP REF .", "Similarly, we obtain the existence and uniqueness of $\\tilde{m}^{[2]}$ , $\\hat{m}^{[2]}$ and eventually $m^{[2]}_{RHP}$ .", "Remark 8.2 From equation (REF ), we know $q_{as}$ is $\\mathcal {O}(t^{-1/2})$ as $t\\rightarrow \\infty $ in the region $x<0$ and consider the limit along the ray $x=-ct$ for some positive constant $c$ .", "Fast decaying region In this section and the next section, we will focus only on the case of the defocusing mKdV flow.", "In this case, the phase function reads $\\theta (z;x,t)=\\frac{x}{t}z+cz^n,\\quad n \\text{ is an odd positive integer.", "}$ In the previous sections, we have derived the asymptotic solutions to the defocusing mKdV flow in the oscillating region, namely, along the ray $x=-\\nu t,\\ \\nu >0,\\ t\\rightarrow \\infty $ .", "In this section, we consider the long-time behavior along the ray $x=\\nu t,\\ \\nu >0,\\ t\\rightarrow \\infty $ , which we call it the fast decaying region as we will soon prove in this region, the solution decay like $\\mathcal {O}(t^{-1})$ , which is faster than the leading term in the oscillating region, i.e., $\\mathcal {O}(t^{-1/2})$ , as $t\\rightarrow \\infty $ .", "In the fast decaying region, the phase function enjoys the following properties: There exits $\\epsilon =\\epsilon (n,\\nu )>0$ such that $\\pm \\operatorname{Im}(\\theta )>0$ in the strips $\\lbrace z:\\pm \\operatorname{Im}(z)\\in (0,\\epsilon )\\rbrace $ , respectively.", "There exits $M\\in (0,1/\\epsilon )$ such that $\\operatorname{Im}(\\theta ) \\ge nvu^{n-1}$ for $\|u\|\\ge M\\epsilon $ and $\\operatorname{Im}(\\theta )\\ge v(1-(M\\epsilon )^2)$ for $\|u\|\\le M\\epsilon $ .", "Here $z=u+iv$ .", "First we will formulate the RHP as follows: Riemann-Hilbert problem 9.1 Given $R(z)\\in H^{1,1}(\\mathbb {R})$ , looking for a 2 by 2 matrix-value function $m$ such that $m_+=m_-e^{-it\\theta (z)\\text{ ad }{\\sigma _3}}v(z),z\\in \\mathbb {R},$ where the jump matrix is given by $v(z)=\\begin{pmatrix}1-\|R\|^2& -\\bar{R}\\\\R & 1\\\\\\end{pmatrix}=\\begin{pmatrix}1 & -\\bar{R}\\\\0& 1\\\\\\end{pmatrix}\\begin{pmatrix}1 & 0\\\\R & 1\\\\\\end{pmatrix};$ $m=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty $ .", "Theorem 9.2 For the above RHP, the solution $m$ enjoys the following asymptotics as $t\\rightarrow \\infty $ : $m_1(t)=\\mathcal {O}(t^{-1}).$ where $m=I+m_1(t)/z+\\mathcal {O}(z^{-2}),\\ z\\rightarrow \\infty $ .", "Figure: ∂ ¯\\bar{\\partial }-extension for the case of the fast decaying region.", "Here we only draw the case when n=5n=5.", "For generic odd nn, there are n-1 2\\frac{n-1}{2} curves of Imθ=0\\operatorname{Im}{\\theta }=0 in the upper and in the lower half plane.In the light of $\\bar{\\partial }$ -steepest descent, to open the lens, we multiple a smooth function $O(z)$ to $m$ , where $O(z)$ is given by $O(z)={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & 0\\\\\\frac{{-R}(\\operatorname{Re}{z})e^{2it\\theta (z)}}{1+(\\operatorname{Im}{z})^2} & 1\\\\\\end{pmatrix}, \\quad z\\in {\\Omega }_1,\\\\\\begin{pmatrix}1 & \\frac{\\bar{R}(\\operatorname{Re}{z})e^{-2it\\theta (z)}}{1+(\\operatorname{Im}{z})^2}\\\\0 &1\\\\\\end{pmatrix},\\quad z\\in \\Omega ^_{1},\\\\I,\\quad z \\in (\\Omega _1\\cup \\Omega ^_1),\\end{array}\\right.", "}$ where (see Fig.REF ) $\\Omega _1&=\\lbrace z:\\operatorname{Im}{z}\\in (0,\\epsilon ) \\rbrace ,\\\\\\Omega ^_1&=\\lbrace z:\\operatorname{Im}{z}\\in (-\\epsilon ,0) \\rbrace .$ Let us denote $\\Sigma _1=\\lbrace z:\\operatorname{Im}{z}=\\epsilon \\rbrace $ , see Fig.REF , and let $\\tilde{m}=mO,\\quad z\\in $ Now as usual, we obtain a $\\bar{\\partial }$ -RHP, due to the exponential decaying of the off-diagonal term, and the jump matrix of the RHP part will approach $I$ .", "Hence by a small norm argument, we know the solution will close to $I$ as $z\\rightarrow \\infty $ .", "Denote the solution to the pure RHP by $m^\\sharp $ , and small norm theory leads to $m^\\sharp =I+\\mathcal {O}(e^{-c(\\epsilon )t}),c(\\epsilon )>0,z\\rightarrow \\infty $ .", "Next, consider $E=\\tilde{m}(m^\\sharp )^{-1}.$ By direct computation one can show $E$ doesn't have any jump on $\\Sigma _1$ and it satisfies a pure $\\bar{\\partial }$ -problem: $\\bar{\\partial }E=EW,$ where $W={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}0 & m^{\\sharp }e^{-2it\\theta (z)}\\bar{\\partial }(\\frac{\\bar{R}(\\operatorname{Re}{z})}{1+(\\operatorname{Im}{z})^2}) (m^\\sharp )^{-1} \\\\0 & 0\\end{pmatrix},\\quad z\\in \\Omega _1,\\\\\\begin{pmatrix}0 & 0\\\\m^{\\sharp }e^{2it\\theta (z)}\\bar{\\partial }(\\frac{-{R}(\\operatorname{Re}{z})}{1+(\\operatorname{Im}{z})^2}) (m^\\sharp )^{-1} & 0\\end{pmatrix},\\quad z\\in \\Omega _1^,\\\\0,\\quad z\\in (\\Omega _1\\cup \\Omega _1^),\\end{array}\\right.", "}$ where $\\bar{\\partial }=\\frac{1}{2}(\\partial _{\\operatorname{Re}{z}}+i\\partial _{\\operatorname{Im}{z}})$ .", "Since $R,\\bar{R}\\in H^{1,1}$ , $\\bar{\\partial }(\\frac{\\bar{R}(\\operatorname{Re}{z})}{1+(\\operatorname{Im}{z})^2}),\\bar{\\partial }(\\frac{-{R}(\\operatorname{Re}{z})}{1+(\\operatorname{Im}{z})^2})$ are uniformly bounded by some non-negative $L^2(\\mathbb {R})$ function $f(\\operatorname{Re}{z})$ .", "Note that $m^\\sharp $ is uniformly close to $I$ , and setting $z=u+iv$ , and considering $z\\in \\Omega _1$ first, we have $\\Vert W\\Vert _\\infty \\le f(u)e^{-t\\operatorname{Im}{\\theta (u,v)}}, \\forall u\\in \\mathbb {R}, v\\in (0,\\epsilon ).$ By the same procure as the one in section 8, the error of approximating $m$ by the identity matrix is given by the following integral (since there is only one non-trivial entry of $W$ ): $\\Delta :=\\int _0^\\epsilon \\int _\\mathbb {R}f(u)e^{-t\\operatorname{Im}{\\theta }}dudv.$ Split the $u$ into two regions: (1) $\|u\|\\le M\\epsilon $ , (2) $\|u\|\\ge M\\epsilon $ .", "And denote them by $\\Delta _1$ , $\\Delta _2$ respectively.", "Then $\\Delta =\\Delta _1+\\Delta _2$ .", "And $\\Delta _1&\\le \\int _0^\\epsilon \\int _{-M\\epsilon }^{M\\epsilon }f(u)e^{-tv(1-M^2\\epsilon ^2)}dudv\\\\&\\text{by Cauchy-Schwartz }\\\\&\\le \\Vert f\\Vert _{L^2(\\mathbb {R})}(2M\\epsilon )^{1/2}\\frac{1-e^{-t\\epsilon (1-M^2\\epsilon ^2)}}{t(1-M^2\\epsilon ^2)}\\\\&=\\mathcal {O}(t^{-1}).$ On the other hand, $\\Delta _2&\\le \\int _0^\\epsilon \\int _{\|u\|\\ge M\\epsilon }f(u)e^{-ntvu^{n-1}}dudv\\\\&=\\int _{\|u\|\\ge M\\epsilon } f(u)\\int _0^\\epsilon e^{-ntvu^{n-1}}dvdu\\\\&\\le t^{-1}\\Vert f\\Vert _{L^2}(\\int _{\|u\|\\ge M\\epsilon }(\\frac{1-e^{-ntvu^{n-1}}}{nu^{n-1}})^2du)^{1/2}\\\\&\\le t^{-1}\\Vert f\\Vert _{L^2} \\frac{n}{n-2}(M\\epsilon )^{-(n-2)}\\\\&=\\mathcal {O}(t^{-1}).$ Similarly, we can prove that for $z\\in \\Omega _1^$ , we also have the error estimate $\\mathcal {O}(t^{-1})$ .", "Assembling all together, we conclude that the error term is $\\mathcal {O}(t^{-1})$ , and $m_1=\\mathcal {O}(t^{-1})$ , as $ t\\rightarrow \\infty .$ Painlevé region In this section, we first derive the Painlevé II hierarchy based on some RHP.", "Then, we will connect the long-time behavior of the mKdV hierarchy in the so-called Painlevé region to solutions of the Painlevé II hierarchy.", "Painlevé II hierarchy As mentioned in [2], the mKdV equation is can be transferred to the Painlevé II equation.", "The authors in [2] also suggest the connection between integrable PDEs with Painlevé equations.", "In [10], the authors explicitly derived the Painlevé II hierarchy from self-symmetry reduction of the mKdV hierarchy (see page 59 of [10].", "And also [11]).", "In this section, we will provide a slight different (as comparing to [11]) algorithm based on Riemann-Hilbert problems to generate the Painlevé II hierarchy.", "Let's denote $\\Theta (x,z)=xz+\\frac{c}{n}z^n$ , and suppose $Y$ solves the following RHP: $Y_+&=Y_-e^{i\\Theta \\sigma _3}v_0e^{-i\\Theta \\sigma _3},\\quad z\\in \\Sigma _n,\\\\Y&=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty .$ where the contour $\\Sigma _n$ consists of all stokes lines $\\lbrace z:\\operatorname{Im}{\\Theta (z)}=0\\rbrace $ and $v_0$ is a constant 2 by 2 matrix that is independent of $x,z$ .", "Now let $\\tilde{Y}=Ye^{i\\Theta \\sigma _3}$ , and we arrive at a new RHP: $\\tilde{Y}_+&=\\tilde{Y}_-v_0,\\quad z\\in \\Sigma _n,\\\\\\tilde{Y}&=(I+\\mathcal {O}(z^{-1}))e^{i\\Theta \\sigma _3},\\quad z\\rightarrow \\infty .$ Since $v_0$ is constant, it is easily to check, by Louisville's argument, that both $\\partial _z\\tilde{Y}\\tilde{Y}^{-1}$ and $\\partial _x\\tilde{Y}\\tilde{Y}^{-1}$ are polynomial of $z$ .", "Hence we obtain the following two differential equations: $\\partial _x\\tilde{Y}\\tilde{Y}^{-1}&=A(x,z),\\\\\\partial _z\\tilde{Y}\\tilde{Y}^{-1}&=B(x,z).$ If we assume $Y&=I+\\sum _{j=1}^{n-1}{Y_j(x)z^{-j}}+\\mathcal {O}(z^{-n}),\\quad z\\rightarrow \\infty ,\\\\\\underline{Y}&=Y^{-1}=I+\\sum _{j=1}^{n-1}{\\underline{Y}_j(x)z^{-j}}+\\mathcal {O}(z^{-n}),\\quad z\\rightarrow \\infty ,$ then a direct computation shows $A&=i[Y_1,\\sigma _3]+iz\\sigma _3,\\\\B&=ix\\sigma _3+icz^{n-1}\\sigma _3+icz^{n-2}[Y_1,\\sigma _3]\\\\&+\\sum _{k=2}^{n-1}icz^{n-1-k}(Y_k\\sigma _3+\\sigma _3\\underline{Y}_k+\\sum _{j=1}^{k-1}Y_{k-j}\\sigma _3\\underline{Y}_j).$ Since $Y_{x,z}=Y_{z,x}$ , we have $A_z-B_x+[A,B]=0.$ Set $Y_j=\\begin{pmatrix}p_j(x) & u_j(x)\\\\v_j(x)& q_j(x)\\end{pmatrix},\\quad j=1,..,n-2,$ where $p_j,q_j,u_j,v_j$ are smooth functions of $x$ .", "To guarantee (REF ), all the coefficients of $z$ must vanish.", "Those equations can be solved recursively.", "Eventually, by eliminating $u_j,v_j, j=2,..,n-2$ , and let $v_1=u_1$ , we will arrive at a nonlinear ODE of $u_1$Surprisingly, the dependence on $p_j,q_j$ will disappear., which turns out to be a member of the hierarchy of Painlevé II equations .", "We list the first few of them: $n=3&: -8cu^3+cu_{xx}-4xu=0,\\\\n=5&:-24cu^5+10cu^2u_{xx}+10cuu_x^2-\\frac{c}{4}u_{xxxx}-4xu=0,\\\\n=7&: -80cu^7+70cu^4u_{xx}+140cu^3u_x^2-\\frac{7cu^2u_{xxxx}}{2}\\\\\\quad \\quad & +(-\\frac{21}{2}cu^2_{xx}-14cu_xu_{xxx}-4x)u+\\frac{c}{16}u_{xxxxxx}-\\frac{35}{2}cu_x^2u_{xx}=0.$ In the current article, we focus only on the odd members.", "In fact, $n=3$ corresponds to the mKdV equation, $n=5$ corresponds to the 5th order mKdV, and so on.", "In the following subsection, we will show how to connect the long-time asymptotics behavior of the mKdV hierarchy to the solutions to the Painlevé II hierarchy.", "Painlevé Region Recall the phase functions of the AKNS hierarchy of mKdV type equations are $\\theta (z;x,t)=xz+ctz^n, \\quad n\\, \\text{is odd}.$ By the Painlevé region we mean a collection of all the curves $x=s(nt)^{1/n},s\\ne 0$ , by rescaling $z\\rightarrow (nt)^{-\\frac{1}{n}}\\xi $ , we have $\\Theta (\\xi )=s\\xi +\\frac{c}{n}\\xi ^n.$ Now the modulus of the stationary phase points of (REF ) is $\|z_0\|=\\left\|-\\frac{x}{ct}\\right\|^{\\frac{1}{n-1}}=\\mathcal {O}(t^{-\\frac{1}{n}}),$ and however, after scaling, the modulus of the stationary phase points of $\\Theta (\\xi )$ is $\\xi _0=\|z_0\|t^{\\frac{1}{n}},$ which is fixed as $t\\rightarrow \\infty $ .", "A direct computation shows for any odd $n$ , one can always perform lens-opening to the rays $\\lbrace z\\in \\mathbb {R}:\|z\|>\|\\xi _0\|\\rbrace $ , due to the signature of $\\operatorname{Re}(i\\theta )$ , see Fig.REF .", "Figure: Signature of Re(iθ)\\operatorname{Re}(i\\theta ).", "The green region: Re(iθ)>0\\operatorname{Re}(i\\theta )>0 when x<0x<0; The red region: Re(iθ)>0\\operatorname{Re}(i\\theta )>0 when x>0x>0; The yellow region: the overlapping region of red and green; The white region: Re(iθ)<0\\operatorname{Re}(i\\theta )<0.", "Here we only plot the signatures of Re(iθ)\\operatorname{Re}{(i\\theta )} for n=9n=9.", "Other odd nn, the region plot looks very similar.Note that $e^{-i\\theta (z)\\text{ ad }\\sigma _3}v(z)&=e^{-i\\Theta (\\xi )\\text{ ad }\\sigma _3}v(\\xi )\\\\&=\\begin{pmatrix}1-\|R\|^2 & -\\bar{R}e^{-2i\\Theta }\\\\Re^{2i\\Theta } & 1\\end{pmatrix}\\\\&=\\begin{pmatrix}1 & -\\bar{R}e^{-2i\\Theta }\\\\0 & 1\\end{pmatrix}\\begin{pmatrix}1 & 0\\\\Re^{2i\\Theta } & 1\\end{pmatrix}.$ We can deform the the contour $\\lbrace z\\in \\mathbb {R}:\|z\|>\|\\xi _0\|\\rbrace $ as before and get the deformed contour as follows (see Fig.REF ): Fix a positive constant $\\alpha <\\frac{\\pi }{n}$Such a choice of $\\alpha $ guarantees that the new contours will stay within the regions where the corresponding exponential term will decay (considering Fig.REF )., $\\Sigma _0&=\\lbrace z\\in \\mathbb {R}:-\\xi _0\\le z\\le \\xi _0\\rbrace ,\\\\\\Sigma _1&=\\lbrace z:z=\\xi _0+\\rho e^{i\\alpha },\\rho \\in (0,\\infty )\\rbrace ,\\\\\\Sigma _2&=\\lbrace z:z=-\\xi _0+\\rho e^{-i\\alpha },\\rho \\in (-\\infty ,0)\\rbrace ,\\\\\\Sigma _3&=\\lbrace z:z=-\\xi _0+\\rho e^{i\\alpha },\\rho \\in (-\\infty ,0)\\rbrace ,\\\\\\Sigma _4&=\\lbrace z:z=\\xi _0+\\rho e^{-i\\alpha },\\rho \\in (0,\\infty )\\rbrace ,$ and we define the regions as follows: $\\Omega _1&=\\lbrace z:z=\\xi _0+\\rho e^{i\\phi },\\rho \\in (0,\\infty ),\\phi \\in (0,\\alpha )\\rbrace ,\\\\\\Omega _2&=+\\backslash (\\Omega _1\\cup \\Omega _3),\\\\\\Omega _3&=\\lbrace z:z=-\\xi _0+\\rho e^{-i\\phi },\\rho \\in (-\\infty ,0),\\phi \\in (-\\alpha ,0)\\rbrace ,\\\\\\Omega _4&=\\lbrace z:z=-\\xi _0+\\rho e^{i\\phi },\\rho \\in (-\\infty ,0),\\phi \\in (0,\\alpha )\\rbrace ,\\\\\\Omega _5&={-}\\backslash (\\Omega _4\\cup \\Omega _6),\\\\\\Omega _6&=\\lbrace z:z=\\xi _0+\\rho e^{i\\phi },\\rho \\in (0,\\infty ),\\phi \\in (-\\alpha ,0)\\rbrace .$ Figure: Contour for ∂ ¯\\bar{\\partial }-RHP.As before, set the original RHP as $m^{[1]}$ with jump $e^{-i\\theta (z)\\text{ ad }\\sigma _3}v(z)$ .", "After re-scaling and $\\bar{\\partial }$ -lenses opening, we set $m^{[2]}(\\xi )=m^{[1]}O(\\gamma )$ , where the lenses opening matrix is $O(\\gamma )={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & 0\\\\-E_+e^{2i\\Theta (\\gamma )} & 1\\end{pmatrix},\\quad \\gamma \\in \\Omega _1\\cup \\Omega _3,\\\\\\begin{pmatrix}1 & -E_-e^{-2i\\Theta (\\gamma )}\\\\0 & 1\\end{pmatrix},\\quad \\gamma \\in \\Omega _4\\cup \\Omega _6,\\\\I,\\quad \\gamma \\in \\Omega _2\\cup \\Omega _5,\\end{array}\\right.", "}$ where $E_+(\\gamma )&=\\mathcal {K}(\\phi )R\\left((nt)^{-\\frac{1}{n}}\\xi \\right)+(1-\\mathcal {K}(\\phi ))R(\\tilde{\\xi }_0(nt)^{-\\frac{1}{n}}),\\\\E_-&(\\gamma )=\\overline{E_+(\\gamma )},\\\\\\gamma &={\\left\\lbrace \\begin{array}{ll}\\xi _0+\\rho e^{i\\phi },\\quad \\text{if }\\gamma \\in \\Omega _1\\cup \\Omega _6,\\\\-\\xi _0+\\rho e^{i\\phi },\\quad \\text{if }\\gamma \\in \\Omega _3\\cup \\Omega _4,\\end{array}\\right.}", "\\\\\\xi &=\\operatorname{Re}(\\gamma ),\\\\\\tilde{\\xi }_0&={\\left\\lbrace \\begin{array}{ll}\\xi _0,\\quad \\text{if }\\gamma \\in \\Omega _1\\cup \\Omega _6,\\\\-\\xi _0,\\quad \\text{if }\\gamma \\in \\Omega _3\\cup \\Omega _4.\\end{array}\\right.", "}$ Now we arrive at the following $\\bar{\\partial }$ -RHP: Mixed $\\bar{\\partial }$ -Riemann-Hilbert problem 10.1 Looking for a 2 by 2 matrix-valued function $m^{[2]}$ such that The RHP: (1.a) $m^{[2]}(\\gamma )\\in C^1(\\mathbb {R}^2\\backslash \\Sigma )$ and $m^{[2]}(z)=I+\\mathcal {O}(\\gamma ^{-1}),\\gamma \\rightarrow \\infty $ ; (1.b) the jumps on $\\Sigma _1$ and $\\Sigma _2$ are $e^{-i\\Theta (\\xi )\\text{ ad }\\sigma _3}v_+$ , and the jumps on $\\Sigma _3$ and $\\Sigma _4$ are $e^{-i\\Theta (\\xi )\\text{ ad }\\sigma _3}v_-$ , where $v_=\\begin{pmatrix}1 & \\bar{R}\\\\0 & 1\\end{pmatrix},\\quad v_+=\\begin{pmatrix}1 & 0\\\\R & 1\\end{pmatrix}.$ The jump on $\\Sigma _0$ is $e^{-i\\Theta \\text{ ad }\\sigma _3}v((nt)^{-\\frac{1}{n}}\\xi ),$ , and the jumps on $\\lbrace z\\in \\mathbb {R}:\|z\|>\|\\xi _0\|\\rbrace $ is $I$ .", "The $\\bar{\\partial }$ -problem: For $z\\in , we have{\\begin{@align}{1}{-1}\\bar{\\partial }m^{[2]}(\\xi )=m^{[2]}(\\xi )\\bar{\\partial }O(\\xi ).\\end{@align}}$ Again, we will need the following lemma in order to estimate errors from the $\\bar{\\partial }$ -problem.", "Lemma 10.2 For $\\gamma \\in \\Omega _{1,3,4,6}$ , $\\xi =\\operatorname{Re}\\gamma $ , $\|\\bar{\\partial }E_{\\pm }(\\gamma )\|\\le (nt)^{-\\frac{1}{n}}\|(nt)^{-\\frac{1}{n}}(\\xi -\\xi _0)\|^{-\\frac{1}{2}}\\Vert R\\Vert _{H^{1,0}}+(nt)^{-\\frac{1}{n}}\|R^{\\prime }((nt)^{-\\frac{1}{n}}\\xi )\|.$ For brevity, we only prove for the region $\\Omega _1$ .", "Using the polar coordinates, we have $\|\\bar{\\partial }E_+(\\gamma )\|&=\\left\|\\frac{ie^{i\\phi }}{2\\rho }\\mathcal {K}^{\\prime }(\\phi )\\left[R\\left((nt)^{-\\frac{1}{n}}\\xi \\right)-R(\\xi _0(nt)^{-\\frac{1}{n}})\\right]+\\mathcal {K}(\\phi )R^{\\prime }\\left((nt)^{-\\frac{1}{n}}\\xi \\right)(nt)^{-\\frac{1}{n}}\\right\| \\\\&\\text{by Cauchy-Schwartz inequality}\\\\&\\le \\left\|\\frac{\\Vert R\\Vert _{H^{1,0}}\|(nt)^{-\\frac{1}{n}}\\xi -\\xi _0(nt)^{-\\frac{1}{n}}\|^{1/2}}{\\gamma -\\xi _0}\\right\|+(nt)^{-\\frac{1}{n}}\\left\|R^{\\prime }\\left((nt)^{-\\frac{1}{n}}\\xi \\right)\\right\|\\\\&\\le (nt)^{-\\frac{1}{n}}\|(nt)^{-\\frac{1}{n}}(\\xi -\\xi _0)\|^{-\\frac{1}{2}}\\Vert R\\Vert _{H^{1,0}}+(nt)^{-\\frac{1}{n}}\|R^{\\prime }((nt)^{-\\frac{1}{n}}\\xi )\|.$ Similarly, we can prove for other regions.", "Next, consider a pure RHP $m^{[3]}$ which satisfies exactly the RHP part of $\\bar{\\partial }$ -RHP($m^{[2]}$ ).", "$m^{[3]}$ can be approximated by the RHP corresponding to a special solution of the Painlevé II hierarchyAs for the existence of the RHP $m^{[3]}$ , which is not completely trivial due to the fact that solutions to the Painlevé II equations have poles, we refer the readers to the book[20] for the details.", "Since for $\\gamma \\in \\Omega _1$ , $&\\left\|\\left(R(\\xi (nt)^{-\\frac{1}{n}})-R(0)\\right)e^{2i\\Theta (\\gamma )}\\right\|\\\\&\\le \|\\xi (nt)^{-\\frac{2}{n}}\|^{\\frac{1}{2}}\\Vert R\\Vert _{H^{1,0}}e^{2\\operatorname{Re}{i\\Theta (\\gamma )}}\\\\&\\le (nt)^{-\\frac{1}{n}}\|\\operatorname{Re}\\gamma \|^{\\frac{1}{2}}\\Vert R\\Vert _{H^{1,0}}e^{2\\operatorname{Re}{i\\Theta (\\gamma )}},$ it is evident that $\\Vert Re^{2i\\Theta }-R(0)e^{2i\\Theta }\\Vert _{L^{\\infty }\\cap L^1\\cap L^2}\\le c(nt)^{-\\frac{1}{n}}.$ Let $m^{[4]}$ solves the RHP formed by replacing $R(\\pm \\xi _0(nt)^{-1/n})$ and its complex conjugate in the jumps of $m^{[3]}$ along $\\Sigma _k,k=1,2,3,4$ by $R(0)$ and $\\bar{R}(0)$ respectively.", "Then, by the small norm theory, the errors between the corresponding potential is given by $error_{3,4}&=\\lim _{\\gamma \\rightarrow \\infty }\|\\gamma (m^{[4]}_{12}-m^{[3]}_{12})\|\\\\&\\le c \\int _{\\Sigma } \|(R(\\operatorname{Re}(s) (nt)^{-\\frac{1}{n}})-R(0))e^{2i\\Theta (s)}\|ds\\\\&\\le c(nt)^{-\\frac{1}{n}}.$ Then since now the jumps are all analytic, we can perform an analytic deformation and arrive at the green contours as show in Fig.REF .", "Let's denote the new RHP by $m^{[5]}(\\gamma )$ , and we arrive at the following RHP: Riemann-Hilbert problem 10.3 Looking for a 2 by 2 matrix-valued function $m^{[5]}$ such that $m^{[5]}$ is analytic off the contours $\\cup _{k=1,2,3,4}\\Sigma _{k}^{[5]}$ ; $m^{[5]}_+=m^{[5]}_-v^{[5]},\\quad z\\in \\cup _{k=1,2,3,4}\\Sigma _{k}^{[5]},$ where $v^{[5]}={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & 0\\\\R(0)e^{2i\\Theta (\\gamma )}&1\\end{pmatrix},\\quad \\gamma \\in \\Sigma _{1}^{[5]}\\cup \\Sigma _2^{[5]},\\\\\\begin{pmatrix}1 &\\bar{R}(0)e^{-2i\\Theta (\\gamma )}\\\\0&1\\end{pmatrix},\\quad \\gamma \\in \\Sigma _{3}^{[5]}\\cup \\Sigma _4^{[5]}.\\end{array}\\right.", "}$ Here the new contours (see Fig.REF ) are $\\Sigma _1^{[5]}&=\\lbrace z:z=\\rho e^{i\\alpha },\\rho \\in (0,\\infty )\\rbrace ,\\\\\\Sigma _2^{[5]}&=\\lbrace z:z=\\rho e^{-i\\alpha },\\rho \\in (-\\infty ,0)\\rbrace ,\\\\\\Sigma _3^{[5]}&=\\lbrace z:z=\\rho e^{i\\alpha },\\rho \\in (-\\infty ,0)\\rbrace ,\\\\\\Sigma _4^{[5]}&=\\lbrace z:z=\\rho e^{-i\\alpha },\\rho \\in (0,\\infty )\\rbrace .$ Figure: Contour for m [4] m^{[4]}(Green part).Then according to the previous subsection, the $(1,2)$ entry of the solution $m^{[5]}$ , similarly the solution $m^{[4]}$ , is the solution to the Painlevé II hierarchy, i.e., $m^{[4]}_{12}(\\gamma )=m^{[5]}_{12}(\\gamma ),$ Hence we have $P^{II}_k(s)=\\lim _{\\gamma \\rightarrow \\infty }\\gamma m^{[5]}_{12}$ where $P^{II}_k$ solves the $k^{th}$ equation in the Painlevé II hierarchy, where $k=\\frac{n-1}{2}$ .", "Now let's consider the error generated from the $\\bar{\\partial }$ -extension.", "Recall that the error $E$ satisfies a pure $\\bar{\\partial }$ -problem: $\\bar{\\partial }E&=EW,\\\\W&=m^{[3]}\\bar{\\partial }O(m^{[3]})^{-1}.$ As before, the $\\bar{\\partial }$ -equation is equivalent to an integral equation which reads $E(z)=I+\\frac{1}{\\pi }\\int _{\\frac{E(s)W(s)}{z-s}\\text{d}A(s)=I+\\mathcal {K}(E).", "}As before, we can show that the resolvent always exists for large t. So we only need to estimate the true error which is: \\lim _{z\\rightarrow \\infty }z(E-I).", "In fact, we have{\\begin{@align}{1}{-1}\\lim _{z\\rightarrow \\infty }\|z(E-I)\|&=\|\\int _\\mathbb {C}EWds\|\\\\&\\le c\\Vert E\\Vert _\\infty \\int _{\\Omega }\|\\bar{\\partial }O\|ds.\\end{@align}}For the sake of simplicity, we only estimate the integral on the right hand side in the region of the top right corner.", "Note there is only one entry which is nonzero in \\bar{\\partial }O, which is one of the E_{\\pm } and we split the integral into two parts in the obvious way, i.e.,{\\begin{@align}{1}{-1}\\int _\\Omega \|\\bar{\\partial }O\|ds&\\le I_1+I_2\\\\&=\\int _\\Omega (nt)^{-\\frac{1}{2n}}\|\\operatorname{Re}{s}-\\xi _0\|\\Vert R\\Vert _{H^{1,0}}e^{2\\operatorname{Re}{i\\Theta (s)}}ds\\\\&+\\int _{\\Omega }(nt)^{-\\frac{1}{nt}}\|R^{\\prime }((nt)^{-\\frac{1}{n}}s)\|e^{2\\operatorname{Re}{i\\Theta (s)}}ds.\\end{@align}}As we know from previous sections, e^{\\operatorname{Re}{2i\\Theta (s)}}\\le ce^{-2\|\\Theta ^{\\prime \\prime }(\\xi _0)\|uv} in the region \\lbrace z=u+iv:u>\\xi _0,0<v<\\alpha u\\rbrace for some small \\alpha , where s=u+iv+\\xi _0.", "Then we have{\\begin{@align}{1}{-1}I_1&\\le (nt)^{-\\frac{1}{2n}}\\int _{\\Omega }\|\\operatorname{Re}{s}-\\xi _0\|^{-1/2}e^{-cuv}dudv\\\\&\\le (nt)^{-\\frac{1}{2n}}\\int _0^\\infty \\int _0^{\\alpha u}u^{-1/2}e^{-cuv}dudv\\\\&\\le C(nt)^{-\\frac{1}{2n}}\\int _0^\\infty \\frac{1-e^{-2\\alpha \|\\Theta ^{\\prime \\prime }(\\xi _0)\|} }{u^{3/2}}du\\\\&=\\mathcal {O}\\left((nt)^{-\\frac{1}{2n}}\\right),\\end{@align}}and{\\begin{@align}{1}{-1}I_2&\\le (nt)^{-\\frac{1}{n}}\\int \|R^{\\prime }((nt)^{-\\frac{1}{2n}}\\operatorname{Re}{s})\|e^{-cuv}dudv\\\\&\\text{ by Cauchy-Schwartz inequality}\\\\&\\le (nt)^{-\\frac{1}{n}}\\Vert R\\Vert _{H^{1,0}}\\int _0^\\infty (\\int _{\\alpha v}^\\infty e^{-2cuv}du)^{1/2}dv\\\\&\\le (nt)^{-\\frac{1}{n}}\\Vert R\\Vert _{H^{1,0}}\\int _0^\\infty \\frac{e^{-c\\alpha v^2}}{\\sqrt{2\\alpha cv}}dc\\\\&=\\mathcal {O}((nt)^{-\\frac{1}{n}}).\\end{@align}}Thus, we arrive at{\\begin{@align}{1}{-1}\\bar{\\partial }\\text{Error} = \\mathcal {O}((nt)^{-\\frac{1}{2n}}).\\end{@align}}And we undo all the deformations, we obtain{\\begin{@align}{1}{-1}m^{[1]}((nt)^{-\\frac{1}{n}}\\gamma )&=m^{[2]}(\\gamma )O^{-1}(\\gamma )\\\\&=(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })m^{[3]}(\\gamma )O^{-1}(\\gamma )\\\\&=(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })m^{[4]}(\\gamma )O^{-1}(\\gamma )\\\\&=(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })m^{[5]}(\\gamma )O^{-1}(\\gamma ).\\end{@align}}It can also be rewritten in terms of the variable z:{\\begin{@align}{1}{-1}m^{[1]}(z)=\\left(1+\\frac{\\mathcal {O}{(t^{-1/(2n)}})}{z(nt)^{1/n}}\\right)m^{[5]}((nt)^{1/n}z)+\\mathcal {O}{(z^{-2})},\\quad z\\rightarrow \\infty .\\end{@align}}Since m^{[5]} corresponds to the RHP for the Painlevé II hierarchy, we have{\\begin{@align}{1}{-1}m^{[5]}(\\gamma )=I+\\frac{m_1^{[5]}(s)}{\\gamma }+\\mathcal {O}(\\gamma ^{-1}),\\end{@align}}where \\gamma =z(nt)^{1/n}.$ Thus, $m^{[1]}(z)&=\\left(1+\\frac{\\mathcal {O}{(t^{-\\frac{1}{2n}})}}{z(nt)^{1/n}}\\right)\\left(1+\\frac{m_1^{[5]}(s)}{z(nt)^{1/n}}+\\mathcal {O}(z^{-2})\\right)\\\\&=I+\\frac{m_1^{[5]}(s)}{z(nt)^{1/n}}+\\frac{\\mathcal {O}{(t^{-\\frac{1}{2n}})}}{z(nt)^{1/n}}+\\mathcal {O}(z^{-2}).$ Since $m_1^{[5]}(s)$ is connected to solutions of the Painlevé II hierarchy, we conclude that $q(x,t)&=\\lim _{z\\rightarrow \\infty }z(m^{[1]}-I)\\\\&=(nt)^{-\\frac{1}{n}}u_n(x(nt)^{-\\frac{1}{n}})+\\mathcal {O}{(t^{-\\frac{3}{2n}})},$ where $u_n$ solves the $\\frac{n-1}{2}$ th equation of the Painlevé II hierarchy.", "The odd integer $n$ corresponds to the $\\frac{n-1}{2}$ th member in the mKdV hierarchy.", "Remark 10.4 As for the asymptotics for the Painlevé II equation, we refer the readers to the classical book [20].", "There are also some recent works related to Painlevé II hierarchy, see for example [30],[9],[6]." ], [ "Conjugation", "In this section, we will factorize the jump matrix (as defined by equation (REF ) ) in a way that it can be used for deforming the RHP.", "It is easy to see that the jump matrix enjoys the following two kinds of factorization: $v_\\theta (z)={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & -\\bar{R}(z)e^{-2it\\theta }\\\\0& 1\\end{pmatrix}\\begin{pmatrix}1 & 0\\\\R(z)e^{2it\\theta } & 1\\end{pmatrix},\\\\\\begin{pmatrix}1 & 0\\\\\\frac{R(z)}{1-\|R(z)\|^2}e^{2it\\theta }& 1\\end{pmatrix}(1-\|R\|^2)^{\\sigma _3}\\begin{pmatrix}1 & -\\frac{\\bar{R}(z)}{1-\|R\|^2}e^{-2it\\theta }\\\\0 & 1\\end{pmatrix}.\\end{array}\\right.", "}$ In the light of the main ideas we described in the last section, we want to remove the middle term in the second factorization.", "By doing so, we can eventually find the proper factorization based on the signatures of the $\\operatorname{Re}(i\\theta )$ .", "Due to our assumptions on $\\theta $ , near a stationary phase point (say $\|z-z_j\|<\\epsilon $ , for some small positive $\\epsilon $ ), $\\theta =\\theta (z_j)+\\frac{\\theta ^{\\prime \\prime }(z_j)}{2}(z-z_j)^2+\\mathcal {O}(\|z-z_j\|^3)$ .", "If $\\theta ^{\\prime \\prime }(z_j)>0$ , then $\\operatorname{Re}(i\\theta (z))$ is negative in the line (I): $\\lbrace z=z_j+re^{i\\alpha }, r\\in (-\\epsilon ,\\epsilon ) \\text{ with fixed } \\alpha \\in (0,\\pi /2)\\rbrace $ , and it is positive in the line (II): $\\lbrace z=z_j+re^{i\\alpha }, r\\in (-\\epsilon ,\\epsilon ) \\text{ with fixed }\\alpha \\in (-\\pi /2,0)\\rbrace $ .", "On the line (I), notice that $e^{2it\\theta }$ decays to 0 as $t\\rightarrow \\infty $ , we can deform the jump on the contour right to the stationary phase point using the first factorization.", "With the same argument on the line (II), we can deform the jump on the contour left to the stationary phase point using the second factorization.", "If $\\theta ^{\\prime \\prime }(z_j)<0$ , notice now $e^{2it\\theta }$ decays to 0 as $t\\rightarrow \\infty $ on the line (II), and thus we need the second factorization for the jump on the contour right to the stationary phase point and the first factorization for the jump on the contour left to the stationary phase point.", "Motivated by the above arguments, we denote $D_\\pm =\\lbrace z\\in \\mathbb {R}: \\pm \\theta ^{\\prime }(z)>0\\rbrace $ To eliminate the diagonal matrix in the second factorization, we introduce a scalar RHP: $\\begin{split}\\delta _+&=\\delta _-[(1-\|R\|^2)\\chi _{D_-}+\\chi _{D_+}],\\quad z\\in \\mathbb {R},\\\\\\delta (z)&=1+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty .\\end{split}$ Then by conjugating the initial RHP, we arrive at a new RHP: Riemann-Hilbert problem 4.1 Looking for a 2 by 2 matrix-valued function $m^{[1]}(z;x,t)$ such that $m^{[1]}_+=m^{[1]}_-\\delta _-^{\\sigma _3}v_\\theta \\delta _+^{-\\sigma _3},\\quad z\\in \\mathbb {R};$ $m^{[1]}=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty .$ By denoting $v_\\theta ^{[1]}:=\\delta _-^{\\sigma _3}v_\\theta \\delta _+^{-\\sigma _3}$ , the new jump matrix reads $v_\\theta ^{[1]}(z)={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & -\\bar{R}(z)\\delta ^2(z)e^{-2it\\theta }\\\\0 & 1\\end{pmatrix}\\begin{pmatrix}1 & 0\\\\R(z)\\delta ^{-2}e^{2it\\theta } & 1\\end{pmatrix},\\quad z\\in D_+,\\\\\\begin{pmatrix}1 & 0\\\\\\frac{R(z)\\delta ^{-2}_-e^{2it\\theta }}{1-\|R\|^2} & 1\\end{pmatrix}\\begin{pmatrix}1 & -\\frac{\\bar{R}(z)\\delta _+^2(z)e^{-2it\\theta }}{1-\|R\|^2}\\\\0 & 1\\end{pmatrix},\\quad z\\in D_-.\\end{array}\\right.", "}$ The scalar RHP (REF ) has been carefully studied in the literature (see for example [4] Lemma 23.3, [15], [32] and [19]).", "Here we just list some of the properties.", "First, the solution to the RHP (REF ) can be represented as follows: $\\log {(\\delta (z))}=(C_{D_-}(\\log (1-\|R\|^2)))(z),z\\in D_-,$ where the Cauchy operator $(C_{D_-}f)(z)=\\frac{1}{2\\pi i}\\int _{D_-}\\frac{f(s)}{s-z}ds$ .", "Since we assume $R(z)\\in H^{1,1}_1(\\mathbb {R},dz)=H^{1,1}\\cap \\lbrace f:\|f\|<1\\rbrace $ , one can show $\\log (1-\|R\|^2)$ is in $H^{1,0}$ , and then by the Sobolev embedding, we know it is also Hölder continuous with index $1/2$ .", "Then, by the Privalov-Plemelj theorem, which says that Cauchy operator perseveres Hölder continuity with index less than 1, one can show $\\log (\\delta (z))$ is Hölder continuous with index $1/2$ except for the end points.", "Next we study the behavior near those points.", "Let us denote $\\eta (z)=-\\frac{1}{2\\pi }\\log (1-\|R(z)\|^2), \\quad z\\in \\mathbb {R}.$ We will prove the following proposition.", "First we define a tent function supported on the interval $[-\\epsilon ,\\epsilon ]$ , $s_\\epsilon (z)={\\left\\lbrace \\begin{array}{ll}0,\\quad \|z\|\\ge \\epsilon \\\\-\\frac{1}{\\epsilon }z+1,\\quad 0< z<\\epsilon ,\\\\\\frac{1}{\\epsilon }z+1,\\quad -\\epsilon < z\\le 0.\\end{array}\\right.", "}$ Proposition 4.2 For each $\\epsilon >0$ , and $\\epsilon \\le \\frac{1}{3}\\min _{j\\ne k}\|z_j-z_k\|, $ there exists a neighborhood $I=I(\\epsilon )$ such that the identity $\\log (\\delta (z))&=i\\int _{D_-\\backslash I}\\frac{\\eta (s)}{s-z}ds+i\\sum _{j=1}^l\\left[\\eta (z_j)(1+\\log (z-z_j))\\right]\\varepsilon _j\\\\&+i\\sum _{j=1}^l{\\int _{I\\cap D_-}\\frac{\\eta (s)-\\eta _j(s)}{s-z}ds}\\\\&+i\\sum _{j=1}^l\\frac{1}{\\epsilon }\\eta (z_j)[(z-z_j)\\log (z-z_j)-(z-z_j+\\varepsilon _j\\epsilon )\\log (z-z_j+\\varepsilon _j\\epsilon )]$ is true, where $\\varepsilon _j=\\text{sgn}(\\theta ^{\\prime \\prime }(z_j))$ , $\\eta _j(z)=\\eta (z_j)s_\\epsilon (z-z_j)$ and see (REF ) for the definition of $\\eta $ .", "As for the logarithm function, the branch is chosen such that $\\text{argument}\\in (-\\pi ,\\pi )$ .", "Let $I=\\cup _{j=1}^l(I_{j+}\\cup I_{j-})$ , where $I_{j\\pm }=\\lbrace z:0<\\pm (z-z_j)<\\epsilon \\rbrace $ .", "Now we have $\\log (\\delta (z))&=i\\int _{D_-\\backslash I}\\frac{\\eta (s)}{s-z}ds\\\\&+i\\sum _{j=1}^l(\\int _{I_{j-}\\cap D_-}+\\int _{I_{j+}\\cap D_-}\\frac{\\eta (s)}{s-z}ds).$ For each $j$ , we have $\\int _{I_{j-}}\\frac{\\eta (s)}{s-z}ds=\\int _{I_{j-}}\\frac{\\eta (s)-\\eta _j(s)}{s-z}ds+\\int _{I_{j-}}\\frac{\\eta _j(s)}{s-z}ds.$ The first integral on the right hand side is the non-tangential limit as $z\\rightarrow z_j$ and the second one generates a logarithm singularity near $z_j$ .", "In fact, direct computation shows $\\int _{I_{j-}}\\frac{\\eta _j(s)}{s-z}ds&=\\eta (z_j)+\\frac{1}{\\epsilon }[(z-z_j)\\log (z-z_j)-(z-z_j+\\epsilon )\\log (z-z_j+\\epsilon )]\\eta (z_j)\\\\&+\\eta (z_j)\\log (z-z_j).$ Similarly, for $I_{j+}$ , $\\int _{I_{j+}}\\frac{\\eta _j(s)}{s-z}ds&=-\\eta (z_j)+\\frac{1}{\\epsilon }[(z-z_j)\\log (z-z_j)-(z-z_j-\\epsilon )\\log (z-z_j-\\epsilon )]\\\\&-\\eta (z_j)\\log (z-z_j).$ And note that only one of the $I_{j\\pm }\\cap D_-$ is nonempty, which depends on the sign of the second derivative of the phase function $\\theta $ .", "By assembling all together, the proof is done.", "Remark 4.3 The proposition tells us how the function $\\delta (z)$ behavior near the saddle points.", "Near the saddle points $z_j$ , $\\delta (z)$ has a mild singularity $(z-z_j)^{i\\eta (z_j)}$ .", "Fortunately, those singularities are bounded along any ray off $\\mathbb {R}$ and hence in some sense they do not affect asymptotics much.", "It is worth mentioning that one can ignore the mild singularity by introducing an auxiliary function, see Lemma 3.1 in [18]." ], [ "Lenses opening", "The purpose of lens-opening is to deform the RHP on the real line to a new RHP on new contours such that jumps on the new contours will rapidly decay to $I$ as $t\\rightarrow \\infty $ .", "We first study the signature of $\\operatorname{Im}{\\theta }$ near the saddle point $z_j$ .", "Figure: Notations for studying signatures of Im(θ(z))\\operatorname{Im}(\\theta (z)) near z j z_jLet us denote $I_{j+}=[z_j,\\frac{z_j+z_{j+1}}{2}]$ and $I_{j-}=[\\frac{z_j+z_{j-1}}{2},z_j]$ .", "Two cases need to be discussed.", "The first case is $\\theta ^{\\prime \\prime }(z_j)>0$ , and so we have $I_{j\\pm }\\subset D_{\\pm }$ .", "The second case is $\\theta ^{\\prime \\prime }(z_j)<0$ , and then we have $I_{j\\pm }\\subset D_{\\mp }$ .", "Recall the factorization of the conjugated jump matrix $v^{[1]}_\\theta $ , to deform it from $I_{j+}$ to $\\Sigma _{j,1}$ , we need make sure the exponential term $e^{2it\\theta (z)}$ decays rapidly to $I$ on $\\Sigma _{j,1}$ , and thus we need to discuss $\\operatorname{Re}(i\\theta )$ on $\\Sigma _{j,1}$ .", "Considering a Taylor approximation of $\\theta (z)$ near $z_j$ , we have $\\theta (z)=\\theta (z_j)+\\varepsilon _jA_j(z-z_j)^2+O(\|z-z_j\|^3)$ , where $A_j=\\left\|\\frac{\\theta ^{\\prime \\prime }(z_j)}{2}\\right\|$ and $\\varepsilon _j=\\text{sgn}\\lbrace \\theta ^{\\prime \\prime }(z_j)\\rbrace $ .", "Let $z=z_j+u+iv=z_j+\\rho e^{i\\phi }$ .", "Then $\\operatorname{Im}(\\theta (z))=\\varepsilon _j A_j\\rho ^2 \\sin (2\\phi )+O(\\rho ^3)$ , where $\\phi \\in (0,\\alpha ]$ is fixed.", "Now we define the regions $\\Omega _{j,n},\\ n=1,\\cdots ,6,$ as follows: $\\begin{split}\\Omega _{j,1}&=\\left\\lbrace z=z_j+\\rho e^{i\\phi },\\phi \\in (0,\\alpha ],\\rho \\in (0,\\frac{\|z_j-z_{j+1}\|}{2\\cos {\\alpha }}),\\operatorname{Re}z\\in I^{\\varepsilon _j}_{j+}\\right\\rbrace ,\\\\\\Omega _{j,3}&=\\left\\lbrace z=z_j+\\rho e^{i\\phi },\\phi \\in [\\pi -\\alpha ,\\pi ),\\rho \\in (0,\\frac{\|z_j-z_{j-1}\|}{2\\cos {\\alpha }}),\\operatorname{Re}z\\in I^{\\varepsilon _j}_{j-}\\right\\rbrace ,\\\\\\Omega _{j,2}&=+\\backslash (\\Omega _{j,1}\\cup \\Omega _{j,3}),\\\\\\Omega _{j,4}&=\\left\\lbrace z=z_j+\\rho e^{i\\phi },\\phi \\in (\\pi ,\\pi +\\alpha ],\\rho \\in (0,\\frac{\|z_j-z_{j-1}\|}{2\\cos {\\alpha }}),\\operatorname{Re}z\\in I^{\\varepsilon _j}_{j-}\\right\\rbrace ,\\\\\\Omega _{j,6}&=\\left\\lbrace z=z_j+\\rho e^{i\\phi },\\phi \\in [-\\alpha ,0),\\rho \\in (0,\\frac{\|z_j-z_{j+1}\|}{2\\cos {\\alpha }}),\\operatorname{Re}z\\in I^{\\varepsilon _j}_{j+}\\right\\rbrace ,\\\\\\Omega _{j,5}&=-\\backslash (\\Omega _{j,4}\\cup \\Omega _{j,6}),\\end{split}$ where $I^{\\varepsilon _j}_{j\\pm }={\\left\\lbrace \\begin{array}{ll}I_{j\\pm },\\quad \\varepsilon _j=1,\\\\I_{j\\mp },\\quad \\varepsilon _j=-1.\\end{array}\\right.", "}$ Since the number of real saddle points is finite, we can always choose a sufficiently small $\\alpha $ , such that for each $j$ , $e^{2it\\theta }$ decays to 0 in $\\Omega _{j,1}\\cup \\Omega _{j,4}$ and $e^{-2it\\theta }$ decays to 0 in $\\Omega _{j,3}\\cup \\Omega _{j,6}$ .", "Now we are in the position to open the lenses.", "First we introduce a bounded smooth function $\\mathcal {K}$ defined on $[0,2\\pi ]$ such that $\\begin{split}\\mathcal {K}(0)=1,\\\\\\mathcal {K}(\\alpha )=0,\\\\\\text{Period of }\\mathcal {K} \\text{ is $\\pi $},\\\\\\mathcal {K} \\text{ is even function}.\\end{split}$ Consider $\\varepsilon _j=1$ first.", "Then the $\\bar{\\partial }$ extension functions are as follows.", "Let $z-z_j=u+iv=\\rho e^{i\\phi }$ ,and for the case $\\varepsilon _j=1$ , we set $\\begin{split}E_{j,1}(z)&=\\mathcal {K}(\\phi )R(u+z_j)\\delta ^{-2}(z)\\\\&+[1-\\mathcal {K}(\\phi )]R(z_j)\\delta _j^{-2}(z-z_j)^{-2i\\varepsilon _j\\eta (z_j)},\\quad z\\in \\Omega _{j,1},\\\\E_{j,3}(z)&=\\mathcal {K}(\\pi -\\phi )(-\\frac{\\bar{R}(u+z_j)}{1-\|R(u+z_j)\|^2}\\delta _+^2(z))\\\\&+[1-\\mathcal {K}(\\pi -\\phi )](-\\frac{\\bar{R}(z_j)}{1-\|R(z_j)\|^2}\\delta _{j}^2(z-z_j)^{2i\\varepsilon _j\\eta (z_j)}),\\quad z\\in \\Omega _{j,3},\\\\E_{j,4}(z)&=\\mathcal {K}(\\pi +\\phi )(\\frac{R(z_j+u)}{1-\|R(z_j+u)\|^2}\\delta _-^{-2}(z))\\\\&+[1-\\mathcal {K}(\\pi +\\phi )](\\frac{R(z_j)}{1-\|R(z_j)\|^2}\\delta _j^{-2}(z-z_j)^{-2i\\varepsilon _j\\eta (z_j)}),\\quad z\\in \\Omega _{j,4},\\\\E_{j,6}(z)&=\\mathcal {K}(-\\phi )(-\\bar{R}(z_j+u)\\delta ^2(z))\\\\&+[1-\\mathcal {K}(-\\phi )](-\\bar{R}(z_j)\\delta _j^2(z-z_j)^{2i\\varepsilon _j\\eta (z_j)}),\\quad z\\in \\Omega _{j,6},\\\\\\end{split}$ where $\\delta _j=\\lim _{\\begin{array}{c}z=z_j+\\rho e^{i\\phi },\\\\ \\rho \\rightarrow 0,\\\\ \\phi \\in (0,\\pi /2)\\end{array}}\\delta (z)(z-z_j)^{i\\eta (z_j)}.$ For the case $\\varepsilon _j=-1$ , one only needs to switch the index 1 with 3 and 4 with 6.", "For the sake of simplicity, in what follows, we focus just on the case $\\varepsilon _j=1$ .", "The extension functions can be considered as interpolations between jumps on the old and new contours.", "Using the extension functions $E_{j,k}, k=1,3,4,6$ , we can construct the lens-opening matrices $O(z)$ as follows: $O(z)={\\left\\lbrace \\begin{array}{ll}O_{j,n}(z)=\\begin{pmatrix}1 & 0\\\\(-1)^nE_{j,n}e^{2it\\theta (z)}& 1\\end{pmatrix},\\quad z\\in \\Omega _{j,n},\\quad n=1,4,\\\\O_{j,m}(z)=\\begin{pmatrix}1 & (-1)^{m}E_{j,m}e^{-2it\\theta (z)}\\\\0 & 1\\end{pmatrix},\\quad z\\in \\Omega _{j,m},\\quad m=3,6,\\\\O_{j,k}(z)=I,\\quad z\\in \\Omega _{j,k},\\quad k=2,5.\\end{array}\\right.", "}$ Then lens-opening is performed by multiplying $O(z)$ to the right of the matrix $m^{[1]}$ .", "Let us denote $m^{[2]}(z)=m^{[1]}(z)O(z),z\\in \\mathbb {R}$ .", "Due to the lacking of analyticity of $O(z)$ (in fact, since we only assume $R(z)\\in C^1(\\mathbb {R})$ , $O(z)$ is also just in $C^1(\\mathbb {R}^2)$Here, $R(z)\\in C^1(\\mathbb {R})$ means $R(z)$ is a function defined on the real line with continuous first order derivative.", "While since $O(z)$ is a matrix-valued function defined on the complex plan, so $O(z)\\in C^1(\\mathbb {R}^2)$ means all the entries have continuous first-order derivatives with respect to $z$ and $\\bar{z}$ .", "), we arrive at the following mixed $\\bar{\\partial }$ -Riemann-Hilbert problem($\\bar{\\partial }$ -RHP): Mixed $\\bar{\\partial }$ -Riemann-Hilbert problem 5.1 Looking for a 2 by 2 matrix-valued function $m^{[2]}$ such that The RHP $m^{[2]}(z)\\in C^1(\\mathbb {R}^2\\Sigma )$ ; $m^{[2]}_+=m^{[2]}_-v^{[2]}_\\theta ,\\quad z\\in \\cup _{j=1,...,l,k=1,2,3,4}\\Sigma _{j,k},$ where the jump matrices read $v^{[2]}_\\theta ={\\left\\lbrace \\begin{array}{ll}O_{j,1}^{-1}, \\quad z\\in \\Sigma _{j,1},\\\\O_{j,3}^{-1}, \\quad z\\in \\Sigma _{j,2},\\\\O_{j,4}, \\quad z\\in \\Sigma _{j,3},\\\\O_{j,6}, \\quad z\\in \\Sigma _{j,4};\\end{array}\\right.", "}$ $m^{[2]}(z)=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty $ .", "The $\\bar{\\partial }$ -problem For $z\\in , we have{\\begin{@align}{1}{-1}\\bar{\\partial }m^{[2]}(z)=m^{[2]}(z)\\bar{\\partial }O(z).\\end{@align}}$ To close this section, we state a bound estimate for $\\bar{\\partial }E_{j,k}$ , which will be used in later sections.", "Lemma 5.2 For $j=1\\cdots l,\\, k=1,2,3,4$ , and $z\\in \\Omega _{j,k},u=\\operatorname{Re}(z-z_j)$ , $\|\\bar{\\partial }E_{j,k}(z)\|\\le c(\|z-z_j\|^{-1/2}+\|R^{\\prime }(u+z_j)\|).$ In the polar coordinates, $\\bar{\\partial }=\\frac{e^{i\\phi }}{2}(\\partial _\\rho +i\\rho ^{-1}\\partial _{\\phi })$ .", "For $z$ in any ray starting from $z_j$ and off the real line, we have $\\bar{\\partial }E_{j,1}(z)&=\\frac{ie^{i\\phi }\\mathcal {K}^{\\prime }(\\phi )}{2\\rho }[R(u+z_j)\\delta ^{-2}(z)-R(z_j)\\delta ^{-2}_j(z-z_j)^{-2i\\eta (z_j)}]\\\\&+\\mathcal {K}(\\phi )R^{\\prime }(u+z_j)\\delta ^{-2}(z).$ From Proposition REF , we know $\|\\delta (z)-\\delta _j(z-z_j)^{i\\eta (z_j)}\|\\le c\|z-z_j\|^{1/2}$ .", "Also since $\\delta (z)^{-1}=e^{-C_{D_-}(\\log {(1-\|R\|^2)})},$ it is evident that $\\delta (z)^{-1}$ is bounded.", "Therefore $\|\\delta ^{-2}(z)-\\delta ^{-2}_j(z-z_j)^{-2i\\eta (z_j)}\|\\le c\|z-z_j\|^{1/2}.$ And we haveIn the middle steps, $c$ means a generic positive constant.", "$\| R(u+z_j)&\\delta ^{-2}(z)-R(z_j)\\delta ^{-2}_j(z-z_j)^{-2i\\eta (z_j)}\|\\\\&\\le \|R(u+z_j)-R(z_j)\|\|\\delta ^{-2}(z)\|\\\\&+\|\\delta ^{-2}(z)-\\delta ^{-2}_j(z-z_j)^{-2i\\eta (z_j)}\|\|R(z_j)\|\\\\&\\le c\|\\int _{z_j}^{u+z_j}R^{\\prime }(s)ds\|+c\|z-z_j\|^{1/2}\\\\& \\text{by Cauchy-Schwartz inequality}\\\\& \\le c\\Vert R^{\\prime }\\Vert _{L^2}\|z-z_j\|^{1/2}+c\|z-z_j\|^{1/2}\\\\& \\le c\|z-z_j\|^{1/2}.$ Therefore $\\begin{split}\|\\bar{\\partial }E_{j,1}(z)\|&\\le c\\rho ^{-1}\|z-z_j\|^{1/2}+c\|R^{\\prime }(u+z_j)\|\\\\&\\le c(\|z-z_j\|^{-1/2}+\|R^{\\prime }(u+z_j)\|).\\end{split}$ Here we have use the fact that $u\\le \\rho $ , which implies $\|z-z_j\|^{1/2}/\\rho =u^{1/2}/\\rho \\le u^{-1/2}$ .", "Noting also that $\\sup \|R\|<1$ , we have $\\frac{R}{1-\|R\|^2}\\le \\frac{R}{1-\\sup \|R\|}$ , and thus all the estimates for $E_{j,1}$ can be smoothly moved to $E_{j,k},k=3,4,6$ .", "Separate contributions and phase reduction The RHP and the mixed $\\bar{\\partial }$ -RHP we have discussed above are global.", "In this section, we shall approximate the global RHP by performing two steps: (1) separate contributions from each stationary phase point, (2) phase reduction.", "Before that, let us first consider two saddle points $z_j,z_{j+1}$ , and discuss $\\varepsilon _j=1=-\\varepsilon _{j+1}$ for example.", "We will first remove the vertical segments, see Fig.", "(REF ): $\\Sigma _{j+\\frac{1}{2}}:=\\Omega _{j,1}\\cap \\Omega _{j+1,3} \\cup \\Omega _{j,6}\\cap \\Omega _{j+1,4}\\backslash \\mathbb {R},$ where $\\Omega _{j,\\cdot }$ 's are defined in REF .", "Figure: Jumps in a small triangular region.Recall the constructions of $E_{j,1}$ and $E_{j+1,3}$ (see (REF )), the boundary value of $m^{[2]}(z)$ on $\\Sigma _{j+\\frac{1}{2}}$ from $\\Omega _{j,1}$ is $m^{[1]}(z_{j+1/2}+iv)O_{j,1}(z_{j+1/2}+iv),$ while from $\\Omega _{j+1,3}$ it is $m^{[1]}(z_{j+1/2}+iv)O_{j+1,3}(z_{j+1/2}+iv).$ Both correspond to locally increasing parts of the phase function, and thus correspond to an upper/lower factorization.", "So the jump on the new contour $\\Sigma _{j+1/2}$ is $O_{j+1,3}O^{-1}_{j,1}(z)$ , $z=z_{j+\\frac{1}{2}}+iv$ , where the nontrivial entry is (regarding the property of $\\mathcal {K}$ and definitions of those matrix $O_{j,k}$ , see (REF ) and (REF ) ): $(1-\\mathcal {K}(\\phi ))&[R(z_j)\\delta _j^{-2}(z_{j+1/2}-z_j+iv)^{-2i\\eta (z_j)}\\\\&-R(z_{j+1})\\delta _{j+1}^{-2}(z_{j+1/2}-z_{j+1}+iv)^{-2i\\eta (z_{j+1})}]e^{2it\\theta (z_{j+1/2}+iv)},$ with $v\\in (0,(z_{j+1/2}-z_j)\\tan (\\alpha ))$ and $\\phi =\\arg {(z-z_j)}$ .", "Note that $\|(z_{j+1/2}-z_j+iv)^{-2i\\eta (z_j)}\|&=e^{2\\eta (z_j)\\phi }\\le e^{2\\eta (z_j)\\alpha }.$ and $\|e^{2it\\theta (z_{j+1/2}+iv)}\|\\le ce^{-2tdv},\\quad d=(z_{j+1}-z_{j})/2.$ Thus we have, for any $z\\in \\Sigma _{j+\\frac{1}{2}}$ , $O_{j+1,3}O^{-1}_{j,1}-I=\\mathcal {O}(e^{-ct}),\\quad t\\rightarrow \\infty ,$ where $c$ is some generic positive constant.", "Since the jump is close to $I$ , by a small norm theory, the solution will also be close to $I$ .", "In fact, we have the following estimate for the potential $\|\\lim _{z\\rightarrow \\infty }z&(m^{[2]}\|_{\\Sigma _{j+\\frac{1}{2}}}-I)\|\\\\& \\le \\frac{1}{2\\pi }\\int _0^{d\\tan {(\\alpha )}}\\left\|m^{[2]}_-(z_{j+1/2}+is)\\right\|\\left\|O_{j+1,3}O_{j,1}^{-1}(z_{j+1/2}+is)-I\\right\|\\text{ds}\\\\&\\le \\frac{1}{2\\pi }\\int _0^{d\\tan {(\\alpha )}}\\left\|m^{[2]}_-(z_{j+1/2}+is)\\right\|e^{-2tsd}\\text{ds}\\\\&\\le \\frac{1}{2\\pi }\\Vert m^{[1]}\|_{\\Sigma _{j+\\frac{1}{2} ,3}}\\Vert _\\infty \\Vert O_{j+1,3}\\Vert _\\infty \\int _{0}^{d\\tan {(\\alpha )}}e^{-2tsd}\\text{ds}\\\\&=\\mathcal {O}(t^{-1}),$ where we assume $m^{[1]}$ , as a solution to the conjugated RHP, existsThe existence and uniqueness will be discussed later..", "So it is analytic in a neighborhood of $\\Sigma _{j+\\frac{1}{2}}$ and hence it is bounded on $\\Sigma _{j+\\frac{1}{2}}$ .", "By the definition (see (REF )) of $O_{j+1,3}$ , it is continuous in $\\Sigma _{j+\\frac{1}{2}}$ and does not blow up at the endpoints of $\\Sigma _{j+\\frac{1}{2}}$ .", "So $\\Vert O_{j+1,3}\\Vert _\\infty $ is also finiteHere the $L^\\infty (\\Sigma )$ norm $\\Vert f(z)\\Vert _{\\infty }$ means $\\sup _{z\\in \\Sigma }\|f(z)\|$ , where $\|f(z)\|=\\max _{i,j=1,2,z\\in \\Sigma }\|f_{i,j}(z)\|$ ..", "Therefore, we can remove all those vertical segments by paying a price of error $\\mathcal {O}(t^{-1})$ , which will be dominated by the error generated by the $\\bar{\\partial }$ -problem (it is $\\mathcal {O}(t^{-3/4})$ , we will show it in a moment.)", "Let us denote the new RHP by $\\tilde{m}^{[2]}$ .", "To make it clear, we note that the jumps for $\\tilde{m}^{[2]}$ are $\\tilde{v}^{[2]}(z)={\\left\\lbrace \\begin{array}{ll}v^{[2]}(z),\\quad z\\in \\cup _{j=1,..,l,k=1,2,3,4}\\Sigma _{j,k},\\\\I,\\quad z\\in \\cup _{j=1,..,l}\\Sigma _{j+\\frac{1}{2}}\\cup \\mathbb {R}.\\end{array}\\right.", "}$ Next, we will show that the RHP for $\\tilde{m}^{[2]}$ can be localized to each saddle point.", "For example, near $z_j$ , along the segment $\\Sigma _{j,1}:z=z_j+u+iv, \\arg {z}=\\alpha $ , we have $\|E_{j,1}e^{2it\\theta }\|\\le ce^{-2t\\tan (\\alpha )u^2}$ It is well-known [14], [19] that the $\|E_{j,1}e^{2it\\theta }\|\\le ce^{-2t\\tan (\\alpha )u^2}$ , where let $u\\ge u_0>0$ , and then the jump matrix will go to $I$ with decaying rate at $\\mathcal {O}(e^{-ct}),c>0$ , as $t\\rightarrow \\infty $ .", "The RHP is localized in the small neighborhoods of those stationary phase points.", "Note that near each $z_j$ , we have $\\theta (z)=\\theta (z_0)+\\frac{\\theta ^{\\prime \\prime }(z_0)(z-z_0)^2}{2}+\\mathcal {O}{(\|z-z_j\|^3)}.$ By a similar argument of Lemma 3.35 in [16] or subsection 8.2 in [19] for the phase reduction, the error generated by reducing the phase function $\\theta $ to $\\theta (z_0)+\\frac{\\theta ^{\\prime \\prime }(z_0)(z-z_0)^2}{2}$ will be bounded by $\\mathcal {O}{(t^{-1})}$ .", "Both analysis of the mentioned references are based on the analysis of the so-called Beals-Coifman operator [3].", "Now we shall simply describe it here.", "For the sake of simplicity, we only consider the RHP on the contour $\\Sigma _{j,1}$ (for more details, we direct the interested reader to [15]): Riemann-Hilbert problem 6.1 Looking for 2 by 2 matrix-valued function $\\tilde{m}^{[2]}$ such that $\\tilde{m}(z)$ is analytic off $\\Sigma _{j,1}$ ; $\\tilde{m}_+=\\tilde{m}_-v^{[2]},\\quad z\\in \\Sigma _{j,1};$ $\\tilde{m}=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty $ .", "Since $E_{j,1}\|_{\\Sigma _{j,1}}$ is analytic near $\\Sigma _{j,1}$ for $z$ away from $z_j$ , and enjoys a factorization$(w^-,w^+)$ will be called the factorization data for the jump matrix.", ": $(I-w^-)^{-1}(I+w^+),$ where $w^-&=I-(v^{[2]})^{-1}=(v^{[2]})-I,\\\\w^+&=0,$ and the superscribes $\\pm $ indicate the analyticitiy in the left/right neighborhood of the the contour.", "Following the definition in [3], we define the Beals-Coifman operator, for any $f\\in L^2(\\Sigma _{j,1})$ , as follows: $C_w(f)=C_+(fw^-)+C_-(fw^+),$ where $C$ means the usual Cauchy operator, i.e., $Cf(z)=\\frac{1}{2\\pi i}\\int _{\\Sigma _{j,1}}\\frac{f(s)}{s-z}\\text{ds},$ and $C_{\\pm }$ means the non-tangential limits from left/right side.", "The following proposition, which plays a fundamental role in Deift-Zhou's method, is well-known.", "Proposition 6.2 (see also proposition 2.11 in [15]) If $\\mu \\in I+L^2$ solves the singular integral equation: $\\mu =I+C_w(\\mu ).$ Then the (unique) solution to the RHP for $\\tilde{m}$ readsHere $w=w^++w^-$ .", "$\\tilde{m}=I+C(\\mu w).$ Then follow the localization principle in [14], [19], [32], and the simple argument on the vertical segments, we arrive at a new RHP on the new contours: fixing $\\rho _0>0$ small, define $\\Sigma _{j,1}^o&:=\\lbrace z:z=z_j+\\rho e^{i\\alpha },\\rho \\in (0,\\rho _0)\\rbrace ,\\\\\\Sigma _{j,2}^o&:=\\lbrace z:z=z_j+\\rho e^{i(\\pi -\\alpha )},\\rho \\in (0,\\rho _0)\\rbrace ,\\\\\\Sigma _{j,3}^o&:=\\lbrace z:z=z_j-\\rho e^{i\\alpha },\\rho \\in (0,\\rho _0)\\rbrace ,\\\\\\Sigma _{j,4}^o&:=\\lbrace z:z=z_j+\\rho e^{i(\\alpha -\\pi )},\\rho \\in (0,\\rho _0)\\rbrace .$ Then with the new contour (see Fig.REF ) $\\Sigma ^o=\\cup _{j=1,..,l,k=1,2,3,4}\\Sigma _{j,k}^o$ , the new RHP reads as follows: Riemann-Hilbert problem 6.3 Looking for a 2 by 2 matrix-valued function $\\hat{m}^{[2]}$ such that $\\hat{m}^{[2]}_+=\\hat{m}^{[2]}_-\\hat{v}^{[2]},\\quad z\\in \\Sigma ^o,$ with $\\hat{v}^{[2]}=\\tilde{v}^{[2]}\|_{\\Sigma ^0};$ $\\hat{m}^{[2]}=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty .$ Figure: New contours, dashed line segments are those deleted parts.Moreover, since the potential of the mKdV hierarchy can be recovered by the formula (REF ), which can also be written as the Beals-Coifman solution: $q_{RHP}(x,t)=-\\frac{1}{2\\pi i}\\int _{\\Sigma }((I-C_w)^{-1}I)w(s)\\text{ds}.$ Then, by localization, we have $\\int _{\\Sigma }((I-C_w)^{-1}I)w(s)\\text{ds}=\\int _{\\Sigma ^o}((I-C_w)^{-1}I)w(s)\\text{ds}+\\mathcal {O}(t^{-1}), \\quad t\\rightarrow \\infty ,$ where $\\Sigma $ is the the contour before localization and $w$ can be easily defined in each cross since the jumps are all triangle matrices and all entries in the diagonal are one.", "Let us denote $q^o_{RHP}(x,t)=-\\frac{1}{2\\pi i}\\int _{\\Sigma ^o}((I-C_w)^{-1}I)w(s)\\text{ds}.$ Then from the localization principal, we have $q_{RHP}(x,t)=q^o_{RHP}(x,t)+\\mathcal {O}(t^{-1}),\\quad t\\rightarrow \\infty .$ Moreover, we define the RHP ($m^{[3]}$ ) which corresponds to the local Beals-Coifman solution (i.e.", "$q^o_{RHP}$ ) as follows: Riemann-Hilbert problem 6.4 Looking for a 2 by 2 matrix-valued function $m^{[3]}$ such that $m^{[3]}_+=m^{[3]}_-v^{[3]}(z),\\quad z\\in \\Sigma ^o$ , with jump matrix reads $v^{[3]}=\\hat{v}^{[2]}\\upharpoonright _{\\Sigma ^o};$ $m^{[3]}=I+\\mathcal {O}{(z^{-1})},\\quad z\\rightarrow \\infty $ .", "However, the integral $\\int _{\\Sigma ^o}((I-C_w)^{-1}I)w(s)\\text{ds}$ is still hard to compute, and following the Deift-Zhou method, we need to separate the contributions from each stationary phase point.", "Thus, we need the following important lemma.", "Lemma 6.5 (see equation (3.64) or proposition 3.66 in [14]) As $t\\rightarrow \\infty $ , $\\int _{\\Sigma ^o}(\\left(1-C_w\\right)^{-1}I)w=\\sum _{j=1}^{l}\\int _{\\Sigma ^o_j}((1-C_{w_j})^{-1}I)w_j+\\mathcal {O}(t^{-1}),$ where $w_j$ is the factorization data supported on $\\Sigma ^o_j=\\cup _{k=1}^4\\Sigma ^o_{j,k}$ , $w=\\sum _{j=1}^lw_j$ and $\\Sigma ^o=\\cup _{j}^l\\Sigma ^o_j$ .", "First, recall the following observation by Varzugin [32], $(1-C_w)(1+\\sum _jC_{w_j}(1-C_{w_j})^{-1})=1-\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1}.$ With the hints from this observation, we need to estimate the norms of $C_{w_j}C_{w_k}$ from $L^\\infty $ to $L^2$ and from $L^2$ to $L^2$ .", "Also from next section (with a small norm argument), we know $(1-C_{w_j})^{-1}$ are uniformly bounded in $L^2$ sense.", "Now let us focus on the contour $\\Sigma ^o_{j,1}$ , and $\\varepsilon =1$ .", "Then the nontrivial entry of the factorization data is $E_{j,1}(z)e^{-2it\\theta (z)},z\\in \\Sigma ^o_{j,1}$ , and thus we have $\|w_{j}\\upharpoonright _{\\Sigma ^o_{j,1}}\|\\le ce^{-2t\\tan (\\alpha )u^2},$ which implies that $\\Vert w_{j}\\upharpoonright _{\\Sigma ^o_{j,1}}\\Vert _{L^1}=\\mathcal {O}(t^{-1/2})$ and $\\Vert w_{j}\\upharpoonright _{\\Sigma ^o_{j,1}}\\Vert _{L^2}=\\mathcal {O}(t^{-1/4})$ .", "Then following exactly the same steps in the proof of [14], Lemma 3.5, we have for $j\\ne k$ $\\Vert C_{w_j}C_{w_k}\\Vert _{L^2(\\Sigma ^o)}&=\\mathcal {O}(t^{-1/2}),\\\\\\Vert C_{w_j}C_{w_k}\\Vert _{L^\\infty \\rightarrow L^2(\\Sigma ^o)}&=\\mathcal {O}(t^{-3/4}).$ Then use the resolvent identities and the Cauchy-Schwartz inequality, $((1-C_w)^{-1}I)&=I+\\sum _{j=1}^{l}C_{w_j}(1-C_{w_j})^{-1}I\\\\&+[1+\\sum _{j=1}^{l}C_{w_j}(1-C_{w_j})^{-1}][1-\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1}]^{-1}\\\\&(\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1})I\\\\&=I+\\sum _{j=1}^{l}C_{w_j}(1-C_{w_j})^{-1}I+ABDI,$ where $A&:=1+\\sum _{j=1}^{l}C_{w_j}(1-C_{w_j})^{-1},\\\\B&:=[1-\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1}]^{-1},\\\\D&:=\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1},$ and thus $\|\\int _{\\Sigma ^o}ABDIw\|&\\le \\Vert A\\Vert _{L^2}\\Vert B\\Vert _{L^2}\\Vert D\\Vert _{L^\\infty \\rightarrow L^2}\\Vert w\\Vert _{L^2}\\\\&\\le c t^{-3/4}t^{-1/4}=\\mathcal {O}{(t^{-1})}.$ Then applying the restriction lemma ([14], Lemma 2.56), we have $\\int _{\\Sigma ^o}(I+C_{w_j}(1-C_{w_j})^{-1}I)w\\upharpoonright _{\\Sigma ^o_j}&=\\int _{\\Sigma ^o_j}(I+C_{w_j}(1-C_{w_j})^{-1}I)w\\\\&=\\int _{\\Sigma ^o_j}((1-C_{w_j})^{-1}I)w_j.$ Therefore, the proof is done.", "A model Riemann-Hilbert problem In the previous section, we have reduce the global RHP to $l$ local RHPs near each stationary phase point due to Lemma REF .", "In fact, near each stationary phase point, we need to compute the integral $\\int _{\\Sigma _j^o}((1-C_{w_j})^{-1}I)w_j$ , which is equivalent to a local RHP.", "In this section, we will approximate the local RHPs by a model RHP which can be solved explicitly by solving a parabolic-cylinder equation.", "Consider the following RHP: Riemann-Hilbert problem 7.1 Looking for a 2 by 2 matrix-valued function $P(\\xi ;R)$ such that $P_+(\\xi ;R)=P_-(\\xi ;R)J(\\xi ),\\xi \\in \\mathbb {R}$ , where $J(\\xi )=\\begin{pmatrix}1-\|R\|^2&-\\bar{R}\\\\R&1\\end{pmatrix}$ is a constant matrix with respect to $\\xi $ and the constant $R$ satisfies $\|R\|<1$ ; $P(\\xi ;R)=\\xi ^{i\\eta \\sigma _3}e^{-i\\frac{\\xi ^2}{4}\\sigma _3}(I+P_1\\xi ^{-1}+\\mathcal {O}(\\xi ^{-2})),\\quad \\xi \\rightarrow \\infty $ , where $P_1=\\begin{pmatrix}0&\\beta \\\\\\bar{\\beta }&0\\end{pmatrix}$ .", "Then by Liouville's argument, $P^{\\prime }P^{-1}$ is analytic and thus $P^{\\prime }(\\xi )=(-\\frac{i\\xi }{2}\\sigma _3-\\frac{i}{2}[\\sigma _3,P_1])P(\\xi ),$ which can be solved in terms of the parabolic-cylinder equation, and apply the asymptotics formulas we can eventually determine that $\\beta =\\frac{\\sqrt{2\\pi }e^{i\\pi /4}e^{-\\pi \\eta /2}}{R\\Gamma (-a)},$ where $a=i\\eta ,$ with $\\eta =-\\frac{1}{2\\pi }\\log {(1-\|R\|^2)}$ .", "The above result has been presented in the literatureThe first description of this model RHP was presented by A. R. Its [22].", "Later examples of the model can be find in [14], [15], [17], [19], [32], [26], [27].", "in many ways.", "Here we follows the representations in [14].", "Next, we will connect this model RHP to the original RHP.", "Recall, near stationary phase point $z_j$ , we need to estimate integral $\\int _{\\Sigma _j^o}((1-C_{w_j})^{-1}I)(w_{j+}+w_{j-})$ , which is equivalent to solve the following RHP ($m^{[3,j]},\\quad j=1,\\cdots ,l$ ): $m^{[3,j]}_+(z)=m^{[3,j]}_-(z)v^{[3,j]}(z),z\\in \\Sigma _j^o$ .", "The jump matrix reads $v^{[3,j]}(z)={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1&0\\\\R_j^\\sharp (z-z_j)^{-2i\\eta (z_j)}e^{-it\\theta ^{\\prime \\prime }(z_j)(z-z_j)^2}&1\\end{pmatrix},z\\in \\Sigma _{j,1}^o,\\\\\\begin{pmatrix}1&-\\frac{\\bar{R}_j^\\sharp }{1-\|R^\\sharp _j\|^2}(z-z_j)^{2i\\eta (z_j)}e^{it\\theta ^{\\prime \\prime }(z_j)(z-z_j)^2}\\\\0&1\\end{pmatrix},z\\in \\Sigma _{j,2}^o,\\\\\\begin{pmatrix}1&0\\\\\\frac{R_j^\\sharp }{1-\|R^\\sharp _j\|^2}(z-z_j)^{2i\\eta (z_j)}e^{-it\\theta ^{\\prime \\prime }(z_j)(z-z_j)^2}&1\\end{pmatrix},z\\in \\Sigma _{j,3}^o,\\\\\\begin{pmatrix}1&-\\bar{R}_j^\\sharp (z-z_j)^{-2i\\eta (z_j)}e^{it\\theta ^{\\prime \\prime }(z_j)(z-z_j)^2}\\\\0&1\\end{pmatrix},z\\in \\Sigma _{j,4}^o,\\end{array}\\right.", "}$ where $R_j^\\sharp =R_j\\delta ^{-2}_je^{-2it\\theta (z_j)}$ ; $m^{[3,j]}=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty .$ Set $\\xi =(2t\\theta ^{\\prime \\prime }(z_j))^{1/2}(z-z_j)$ and by closing lenses, we arrive at an equivalent RHP on the real line: $m^{[4,j]}(\\xi )_+=m^{[4]}_-v^{[4,j]}(\\xi ),\\xi \\in \\Sigma _{j}^p$ .", "The new jump is $v^{[4,j]}(\\xi )=(2\\theta ^{\\prime \\prime }(z_j)t)^{-\\frac{i\\eta (z_j)}{2}\\text{ ad }\\sigma _3}\\xi ^{i\\eta (z_j)\\text{ ad }\\sigma _3}e^{-\\frac{i\\xi ^2}{4}\\text{ ad }\\sigma _3}\\begin{pmatrix}1-\|R_j^\\sharp \|^2&-\\bar{R}_j^\\sharp \\\\R_j^\\sharp &1\\end{pmatrix};$ $m^{[4,j]}=I+\\mathcal {O}(\\xi ^{-1}),\\xi \\rightarrow \\infty .$ Comparing with the model RHP, we observe that $m^{[4,j]}(\\xi )(2\\theta ^{\\prime \\prime }(z_j)t)^{-\\frac{i\\eta (z_j)}{2}\\sigma _3}\\xi ^{i\\eta (z_j)\\sigma _3}e^{-\\frac{i\\xi ^2}{4}\\sigma _3}$ solves the model RHP, which leads to $m^{[4]}_{1,12}&=\\frac{\\sqrt{2\\pi }e^{i\\pi /4}e^{-\\pi \\eta (z_j)/2}}{R^\\sharp _j\\Gamma (-i\\eta (z_j))},\\\\m^{[4]}_{1,21}&=\\frac{-\\sqrt{2\\pi }e^{-i\\pi /4}e^{-\\pi \\eta (z_j)/2}}{\\bar{R}^\\sharp _j\\Gamma (i\\eta (z_j))}.$ Changing the variable $\\xi $ back to $z$ , we have $m^{[3,j]}_{1,12}(t)&= (2t\\theta ^{\\prime \\prime }(z_j))^{-\\frac{1}{2}-\\frac{i\\eta (z_j)}{2}}\\frac{\\sqrt{2\\pi }e^{i\\pi /4}e^{-\\pi \\eta (z_j)/2}}{R_j^\\sharp \\Gamma (-i\\eta (z_j))},\\\\m^{[3,j]}_{1,21}(t)&=- (2t\\theta ^{\\prime \\prime }(z_j))^{-\\frac{1}{2}+\\frac{i\\eta (z_j)}{2}}\\frac{\\sqrt{2\\pi }e^{-i\\pi /4}e^{-\\pi \\eta (z_j)/2}}{\\bar{R}_j^\\sharp \\Gamma (i\\eta (z_j))}.$ Noting that $R^\\sharp _j=R_j\\delta ^{-2}_je^{2it\\theta (z_j)}$ , one can rewrite in a neat way: $m^{[3,j]}_{1,12}(t)=\\frac{\|\\eta (z_j)\|^{1/2}}{\\sqrt{2t\\theta ^{\\prime \\prime }(z_j)}}e^{i\\varphi (t)},\\\\m^{[3,j]}_{1,21}(t)=\\frac{\|\\eta (z_j)\|^{1/2}}{\\sqrt{2t\\theta ^{\\prime \\prime }(z_j)}}e^{-i\\varphi (t)},$ where the phase is $\\varphi (t)=\\frac{\\pi }{4}-\\arg \\Gamma (-i\\eta (z_j))-2t\\theta (z_j)-\\frac{\\eta (z_j)}{2}\\log \|2t\\theta ^{\\prime \\prime }(z_j)\|+2\\arg (\\delta _j)+\\arg (R_j).$ Here we have used the fact that $\|\\beta \|^2=\\eta $ .", "Denoting $q_{as}(x,t)=-2i\\sum _{j=1}^{l}\\frac{\|\\eta (z_j)\|^{1/2}}{\\sqrt{2t\\theta ^{\\prime \\prime }(z_j)}}e^{i\\varphi (t)},$ then the connection formula (REF ) and Lemma REF lead to $q^o_{RHP}(x,t)=q_{as}(x,t)+\\mathcal {O}(t^{-1}),\\quad t\\rightarrow \\infty .$ Errors from the pure $\\bar{\\partial }$ -problem In this section, we will discuss the error generated from the pure $\\bar{\\partial }$ -problem of $m^{[2]}$ .", "Let us denote $E(z)=m^{[2]}({m}^{[2]}_{RHP})^{-1},$ where $m^{[2]}_{RHP}$ denotes the solution to the pure RHP part of $m^{[2]}$ .", "Assuming the existence (which we will be provided in the next section), and by the normalization condition, we have $E(z)=1+(m_1^{[2]}-{m}^{[2]}_{RHP,1})z^{-1}+\\mathcal {O}(z^{-2}),\\quad z\\rightarrow \\infty .$ Due to the procedure of localization and separation of the contributions, we can approximate $m^{[2]}_{RHP}$ by $\\hat{m}^{[2]}$ , and the error of approximating the potential is of $\\mathcal {O}(t^{-1})$ as $t\\rightarrow \\infty $ .", "Thus, by the equation(REF ), $q(x,t)=q_{RHP}(x,t)+\\mathcal {O}(t^{-1})+\\lim _{z\\rightarrow \\infty }z(E-I),\\quad t\\rightarrow \\infty .$ Moreover, from this construction (equation (REF )), there is no jump on the contours $\\Sigma _{j,k},k=1,2,3,4,$ but only a pure $\\bar{\\partial }$ -problem is left due to the non-analyticity.", "The $\\bar{\\partial }$ -problem reads $\\bar{\\partial }E=EW,$ where $W(z)=m^{[2]}_{RHP}\\bar{\\partial }O(z)(m^{[2]}_{RHP})^{-1}.$ From the normalization condition of $m^{[2]}_{RHP}$ , we see it is uniformly bounded by $\\frac{c}{1-\\sup {R}}$ .", "And to estimate the errors of recovering the potential, one actually needs to estimate $\\lim _{z\\rightarrow \\infty }z(E-I)$ , where the limit can be chosen along any rays that are not parallel to $\\mathbb {R}$ .", "For simplicity, we will take the imaginary axis.", "The $\\bar{\\partial }$ -problem is equivalent to the following Fredholm integral equation by a simple application of the generalized Cauchy integral formula: $E(z)=I-\\frac{1}{\\pi }\\int _{\\frac{E(s)W(s)}{s-z}dA(s).", "}In the following, we will show for each fixed z\\in , \\mathcal {K}_W(E)(z):=\\int _{\\frac{E(s)W(s)}{s-z}dA(s) is bounded and then by the dominated convergence theorem, we will show \\lim _{z\\rightarrow \\infty }z(E-I)=\\mathcal {O}(t^{-3/4}).", "First of all, since m^{[3]} is uniformly bounded, upon settng z=z_j+u+iv, we have{\\begin{@align}{1}{-1}\\Vert W\\Vert _{\\infty }\\lesssim {\\left\\lbrace \\begin{array}{ll}\|\\bar{\\partial }E_{j,k}\|e^{-2t\\theta ^{\\prime \\prime }(z_j)uv},z\\in \\Omega _{j,k},k=1,4,\\\\\|\\bar{\\partial }E_{j,k}\|e^{2t\\theta ^{\\prime \\prime }(z_j)uv},z\\in \\Omega _{j,k},k=3,6,\\\\\\end{array}\\right.", "},\\end{@align}}where 0\\le a\\lesssim b means there exists C>0 such that a\\le Cb.", "Then we have{\\begin{@align}{1}{-1}\\mathcal {K}_W(E)\\le \\Vert E\\Vert _\\infty \\int _{\\Vert W(s)\\Vert _\\infty }{\|s-z\|}dA(s).\\end{@align}}We claim the following lemma:\\begin{lemma}Let \\Omega =\\lbrace s:s=\\rho e^{i\\phi }, \\rho \\ge 0,\\phi \\in [0,\\pi /4]\\rbrace , and z\\in \\Omega .", "Then{\\begin{@align}{1}{-1}\\int _\\Omega \\frac{\|u^2+v^2\|^{-1/4}e^{-tuv}}{\|u+iv-z\|}dudv=\\mathcal {O}(t^{-1/4}).\\end{@align}}\\end{lemma}\\begin{proof}Since there are two singularities of the integrand at z and (0,0).", "In the first case, set z\\ne 0, and let d=dist(z,0).", "We split \\Omega into three parts: \\Omega _1\\cup \\Omega _2\\cup \\Omega _3, where \\Omega _1=\\lbrace s:\|s\|<d/3\\rbrace \\cap \\Omega , \\Omega _2=\\lbrace s:\|s-z\|<d/3\\rbrace \\cap \\Omega and \\Omega _3=\\Omega \\backslash (\\Omega _1\\cup \\Omega _2).", "In the region \\Omega _1, \|s-z\|\\ge 2d/3, and thus\\begin{equation}\\begin{split}\|\\int _{\\Omega _1}\\frac{\|u^2+v^2\|^{-1/4}e^{-tuv}}{\|u+iv-z\|}dudv\|&\\le \\frac{3}{2d}\\int _{0}^\\infty \\int _0^u\\frac{e^{-tuv}}{(u^2+v^2)^{1/4}}dvdu\\\\& \\text{ substituted $v=wu$}\\\\&\\le \\frac{3}{2d}\\int _0^\\infty \\int _0^1\\frac{e^{-tu^2w}}{(1+w^2)^{1/4}}u^{1/2}dwdu\\\\&\\le \\frac{3}{2d}\\int _0^\\infty \\int _0^1 e^{-tu^2w}u^{1/2}dwdu\\\\&=\\frac{3}{2d}\\int _0^\\infty \\frac{1-e^{-tu^2}}{tu^{3/2}}du\\\\&=\\frac{3}{2d}\\frac{1}{2}t^{-3/4}\\int _0^\\infty \\frac{1-e^{-u}}{u^{5/4}}du\\\\&=\\frac{3}{d}\\Gamma (3/4)t^{-3/4}.\\end{split}\\end{equation}In the region \\Omega _2, \|s\|^{-1/2}\\le (2d/3)^{-1/2}, we have{\\begin{@align}{1}{-1}\|\\int _{\\Omega _1}\\frac{\|u^2+v^2\|^{-1/4}e^{-tuv}}{\|u+iv-z\|}dudv\|&\\le \\sqrt{\\frac{3}{2d}}\\int _{\\Omega _2}\\frac{e^{-tuv}}{((u-x)^2+(v-y)^2)^{1/2}}dvdu\\\\&\\le \\sqrt{\\frac{3}{2d}}\\int _{0}^{d/3}\\int _0^{2\\pi }e^{-t(x+\\rho \\cos (\\theta ))(y+\\rho \\sin (\\theta ))}d\\theta d\\rho \\\\&\\le \\frac{2\\pi }{3}\\sqrt{\\frac{3d}{2}}e^{-txy}.\\end{@align}}While in the region \\Omega _3,{\\begin{@align}{1}{-1}\|\\int _{\\Omega _3}\\frac{\|u^2+v^2\|^{-1/4}e^{-tuv}}{\|u+iv-z\|}dudv\|&\\le \\int _{0}^{\\infty }\\int _0^u e^{-tuv}dvdu=\\mathcal {O}(t^{-1}).\\end{@align}}Now consider z=0.", "We have{\\begin{@align}{1}{-1}\|\\int _{\\Omega }\\frac{e^{-tuv}}{(u^2+v^2)^{3/4}}dA(u,v)\|&=\\int _{0}^\\infty \\int _0^u\\frac{e^{-tuv}}{(u^2+v^2)^{3/4}}dvdu\\\\&=\\int _0^\\infty \\int _0^1\\frac{e^{-tu^2w}}{(1+w^2)^{3/4}u^{1/2}}dwdu\\\\&\\le \\int _0^\\infty \\int _0^1\\frac{e^{-tu^2w}}{u^{1/2}}dwdu\\\\&=\\int _0^\\infty \\frac{1-e^{-tu^2}}{tu^{5/2}}du\\\\&=\\int _0^\\infty \\frac{1-e^{-u}}{tt^{-5/4}u^{5/4}}t^{-1/2}\\frac{1}{2}u^{-\\frac{1}{2}}du\\\\&=\\frac{1}{2}t^{-1/4}\\int _0^\\infty \\frac{1-e^{-u}}{u^{7/4}}du\\\\&=\\frac{3}{8}t^{-1/4}\\Gamma (1/4).\\end{@align}}By assembling all together, the proof is done.\\end{proof}\\begin{remark}The essential fact that makes the above true is the rapid decay of the exponential factor in the region.", "And the lemma also tells us that those mild singularities, which have rational order growth, can be absorbed by the exponential factor.", "Back to our situation, after some elementary transformations (translation and rotation), the estimation of \\int _{\\Vert W(s)\\Vert _\\infty }{\|s-z\|}dA(s) will eventually reduce to a similar situation discussed in the above lemma.\\end{remark}Based on Lemma \\ref {estimates of W}, we know that when t is sufficiently large, \\Vert \\mathcal {K}_W\\Vert <1 and thus the resolvent is uniformly bounded, and we obtain the following estimate by taking a standard Neumann series, for some sufficiently large t_0,{\\begin{@align}{1}{-1}\\Vert E-I\\Vert _\\infty =\\Vert \\mathcal {K}_W(1-\\mathcal {K}_W)^{-1} I\\Vert _\\infty \\le \\frac{ct^{-1/4}}{1-ct^{-1/4}}\\le ct^{-1/4}, \\quad t>t_0.\\end{@align}}Now since for each z\\in \\Omega _{j,k}, we have \|\\bar{\\partial }E_{j,k}(z)\|\\le c(\|z-z_j\|^{-1/2}+\|R^{\\prime }(u+z_j)\|), and apply the dominated convergence theorem, we have{\\begin{@align}{1}{-1}\\lim _{z\\rightarrow \\infty }\|z(E-I)\|&\\le \\frac{1}{\\pi }\\sum _{j=1}^{l}\\sum _{k=1}^4\\Vert E\\Vert _{L^\\infty }\\int _{\\Omega _{j,k}}\\Vert W\\Vert _\\infty ds,\\end{@align}}and use the Lemma \\ref {estimates of W} again, we will eventually have:{\\begin{@align}{1}{-1}E_1=\\lim _{z\\rightarrow \\infty }\|z(E-I)\|=\\mathcal {O}(t^{-3/4}).\\end{@align}}}\\section {Asymptotics representation}First, we summarize all the steps as following (see Fig.\\ref {fig:steps}):\\begin{enumerate}[label={(0)}]\\item Initial RHP m^{[0]}=m, see RHP \\ref {RHP m 0}.\\item Conjugate initial RHP to obtain m^{[1]}=m^{[0]}\\delta ^{\\sigma _3}, see RHP \\ref {RHP m 1}.\\item Open lenses to obtain a mixed \\bar{\\partial }-RHP \\ref {dbar RHP}.\\item Approximate the RHP part m^{[2]}_{RHP} of m^{[2]} by removing \\Sigma _{j+\\frac{1}{2}} (see RHP \\ref {RHP m tilde}), localization (see RHP \\ref {RHP m hat}), reducing the phase function and separating the contributions (see RHP \\ref {RHP m 3}).", "The error term is \\mathcal {O}(t^{-1}).", "Note those exponential decaying errors are absorbed by \\mathcal {O}(t^{-1}).\\item Comparing m^{[2]} and m^{[2]}_{RHP} and computing the error by analysis a pure \\bar{\\partial }-problem.", "The error term is \\mathcal {O}(t^{-3/4}).\\end{enumerate}$ Figure: Steps of the ∂ ¯\\bar{\\partial }-steepest method.Now by undoing all the steps, we arrive at: $m^{[0]}(z)=E(z)m^{[2]}_{RHP}(z)O^{-1}(z)\\delta ^{-\\sigma _3}.$ Since $O(z)$ uniformly converges to $I$ as $z\\rightarrow \\infty $ , and $\\delta ^{-\\sigma _3}$ is a diagonal matrix, they do not affect the recovering of the potential.", "Thus we obtain $q(x,t)&=-2i(m^{[2]}_{RHP,1,12}+E_{1,12})\\\\&=q_{RHP}(x,t)-2iE_{1,12}\\\\& \\text{ by (\\ref {q rhp to q o rhp}),(\\ref {q o rhp to q as})}\\\\&=q_{as}(x,t)+\\mathcal {O}(t^{-1})-2iE_{1,12}\\\\& \\text{ by (\\ref {dbar error})}\\\\&=q_{as}(x,t)+\\mathcal {O}(t^{-1})+\\mathcal {O}(t^{-3/4})\\\\&=q_{as}(x,t)+\\mathcal {O}(t^{-3/4}),$ where $q_{as}(x,t)$ is given by equation (REF ).", "Remark 8.1 Note that due to the analysis in the section 7, according to Proposition 2.6 and Proposition 2.11 of [15], together with the small norm theory, the existence and uniqueness of the model RHP implies, via the estimates of the corresponding Beals-Coifman operators, the existence and uniqueness of RHP REF .", "Similarly, we obtain the existence and uniqueness of $\\tilde{m}^{[2]}$ , $\\hat{m}^{[2]}$ and eventually $m^{[2]}_{RHP}$ .", "Remark 8.2 From equation (REF ), we know $q_{as}$ is $\\mathcal {O}(t^{-1/2})$ as $t\\rightarrow \\infty $ in the region $x<0$ and consider the limit along the ray $x=-ct$ for some positive constant $c$ .", "Fast decaying region In this section and the next section, we will focus only on the case of the defocusing mKdV flow.", "In this case, the phase function reads $\\theta (z;x,t)=\\frac{x}{t}z+cz^n,\\quad n \\text{ is an odd positive integer.", "}$ In the previous sections, we have derived the asymptotic solutions to the defocusing mKdV flow in the oscillating region, namely, along the ray $x=-\\nu t,\\ \\nu >0,\\ t\\rightarrow \\infty $ .", "In this section, we consider the long-time behavior along the ray $x=\\nu t,\\ \\nu >0,\\ t\\rightarrow \\infty $ , which we call it the fast decaying region as we will soon prove in this region, the solution decay like $\\mathcal {O}(t^{-1})$ , which is faster than the leading term in the oscillating region, i.e., $\\mathcal {O}(t^{-1/2})$ , as $t\\rightarrow \\infty $ .", "In the fast decaying region, the phase function enjoys the following properties: There exits $\\epsilon =\\epsilon (n,\\nu )>0$ such that $\\pm \\operatorname{Im}(\\theta )>0$ in the strips $\\lbrace z:\\pm \\operatorname{Im}(z)\\in (0,\\epsilon )\\rbrace $ , respectively.", "There exits $M\\in (0,1/\\epsilon )$ such that $\\operatorname{Im}(\\theta ) \\ge nvu^{n-1}$ for $\|u\|\\ge M\\epsilon $ and $\\operatorname{Im}(\\theta )\\ge v(1-(M\\epsilon )^2)$ for $\|u\|\\le M\\epsilon $ .", "Here $z=u+iv$ .", "First we will formulate the RHP as follows: Riemann-Hilbert problem 9.1 Given $R(z)\\in H^{1,1}(\\mathbb {R})$ , looking for a 2 by 2 matrix-value function $m$ such that $m_+=m_-e^{-it\\theta (z)\\text{ ad }{\\sigma _3}}v(z),z\\in \\mathbb {R},$ where the jump matrix is given by $v(z)=\\begin{pmatrix}1-\|R\|^2& -\\bar{R}\\\\R & 1\\\\\\end{pmatrix}=\\begin{pmatrix}1 & -\\bar{R}\\\\0& 1\\\\\\end{pmatrix}\\begin{pmatrix}1 & 0\\\\R & 1\\\\\\end{pmatrix};$ $m=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty $ .", "Theorem 9.2 For the above RHP, the solution $m$ enjoys the following asymptotics as $t\\rightarrow \\infty $ : $m_1(t)=\\mathcal {O}(t^{-1}).$ where $m=I+m_1(t)/z+\\mathcal {O}(z^{-2}),\\ z\\rightarrow \\infty $ .", "Figure: ∂ ¯\\bar{\\partial }-extension for the case of the fast decaying region.", "Here we only draw the case when n=5n=5.", "For generic odd nn, there are n-1 2\\frac{n-1}{2} curves of Imθ=0\\operatorname{Im}{\\theta }=0 in the upper and in the lower half plane.In the light of $\\bar{\\partial }$ -steepest descent, to open the lens, we multiple a smooth function $O(z)$ to $m$ , where $O(z)$ is given by $O(z)={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & 0\\\\\\frac{{-R}(\\operatorname{Re}{z})e^{2it\\theta (z)}}{1+(\\operatorname{Im}{z})^2} & 1\\\\\\end{pmatrix}, \\quad z\\in {\\Omega }_1,\\\\\\begin{pmatrix}1 & \\frac{\\bar{R}(\\operatorname{Re}{z})e^{-2it\\theta (z)}}{1+(\\operatorname{Im}{z})^2}\\\\0 &1\\\\\\end{pmatrix},\\quad z\\in \\Omega ^_{1},\\\\I,\\quad z \\in (\\Omega _1\\cup \\Omega ^_1),\\end{array}\\right.", "}$ where (see Fig.REF ) $\\Omega _1&=\\lbrace z:\\operatorname{Im}{z}\\in (0,\\epsilon ) \\rbrace ,\\\\\\Omega ^_1&=\\lbrace z:\\operatorname{Im}{z}\\in (-\\epsilon ,0) \\rbrace .$ Let us denote $\\Sigma _1=\\lbrace z:\\operatorname{Im}{z}=\\epsilon \\rbrace $ , see Fig.REF , and let $\\tilde{m}=mO,\\quad z\\in $ Now as usual, we obtain a $\\bar{\\partial }$ -RHP, due to the exponential decaying of the off-diagonal term, and the jump matrix of the RHP part will approach $I$ .", "Hence by a small norm argument, we know the solution will close to $I$ as $z\\rightarrow \\infty $ .", "Denote the solution to the pure RHP by $m^\\sharp $ , and small norm theory leads to $m^\\sharp =I+\\mathcal {O}(e^{-c(\\epsilon )t}),c(\\epsilon )>0,z\\rightarrow \\infty $ .", "Next, consider $E=\\tilde{m}(m^\\sharp )^{-1}.$ By direct computation one can show $E$ doesn't have any jump on $\\Sigma _1$ and it satisfies a pure $\\bar{\\partial }$ -problem: $\\bar{\\partial }E=EW,$ where $W={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}0 & m^{\\sharp }e^{-2it\\theta (z)}\\bar{\\partial }(\\frac{\\bar{R}(\\operatorname{Re}{z})}{1+(\\operatorname{Im}{z})^2}) (m^\\sharp )^{-1} \\\\0 & 0\\end{pmatrix},\\quad z\\in \\Omega _1,\\\\\\begin{pmatrix}0 & 0\\\\m^{\\sharp }e^{2it\\theta (z)}\\bar{\\partial }(\\frac{-{R}(\\operatorname{Re}{z})}{1+(\\operatorname{Im}{z})^2}) (m^\\sharp )^{-1} & 0\\end{pmatrix},\\quad z\\in \\Omega _1^,\\\\0,\\quad z\\in (\\Omega _1\\cup \\Omega _1^),\\end{array}\\right.", "}$ where $\\bar{\\partial }=\\frac{1}{2}(\\partial _{\\operatorname{Re}{z}}+i\\partial _{\\operatorname{Im}{z}})$ .", "Since $R,\\bar{R}\\in H^{1,1}$ , $\\bar{\\partial }(\\frac{\\bar{R}(\\operatorname{Re}{z})}{1+(\\operatorname{Im}{z})^2}),\\bar{\\partial }(\\frac{-{R}(\\operatorname{Re}{z})}{1+(\\operatorname{Im}{z})^2})$ are uniformly bounded by some non-negative $L^2(\\mathbb {R})$ function $f(\\operatorname{Re}{z})$ .", "Note that $m^\\sharp $ is uniformly close to $I$ , and setting $z=u+iv$ , and considering $z\\in \\Omega _1$ first, we have $\\Vert W\\Vert _\\infty \\le f(u)e^{-t\\operatorname{Im}{\\theta (u,v)}}, \\forall u\\in \\mathbb {R}, v\\in (0,\\epsilon ).$ By the same procure as the one in section 8, the error of approximating $m$ by the identity matrix is given by the following integral (since there is only one non-trivial entry of $W$ ): $\\Delta :=\\int _0^\\epsilon \\int _\\mathbb {R}f(u)e^{-t\\operatorname{Im}{\\theta }}dudv.$ Split the $u$ into two regions: (1) $\|u\|\\le M\\epsilon $ , (2) $\|u\|\\ge M\\epsilon $ .", "And denote them by $\\Delta _1$ , $\\Delta _2$ respectively.", "Then $\\Delta =\\Delta _1+\\Delta _2$ .", "And $\\Delta _1&\\le \\int _0^\\epsilon \\int _{-M\\epsilon }^{M\\epsilon }f(u)e^{-tv(1-M^2\\epsilon ^2)}dudv\\\\&\\text{by Cauchy-Schwartz }\\\\&\\le \\Vert f\\Vert _{L^2(\\mathbb {R})}(2M\\epsilon )^{1/2}\\frac{1-e^{-t\\epsilon (1-M^2\\epsilon ^2)}}{t(1-M^2\\epsilon ^2)}\\\\&=\\mathcal {O}(t^{-1}).$ On the other hand, $\\Delta _2&\\le \\int _0^\\epsilon \\int _{\|u\|\\ge M\\epsilon }f(u)e^{-ntvu^{n-1}}dudv\\\\&=\\int _{\|u\|\\ge M\\epsilon } f(u)\\int _0^\\epsilon e^{-ntvu^{n-1}}dvdu\\\\&\\le t^{-1}\\Vert f\\Vert _{L^2}(\\int _{\|u\|\\ge M\\epsilon }(\\frac{1-e^{-ntvu^{n-1}}}{nu^{n-1}})^2du)^{1/2}\\\\&\\le t^{-1}\\Vert f\\Vert _{L^2} \\frac{n}{n-2}(M\\epsilon )^{-(n-2)}\\\\&=\\mathcal {O}(t^{-1}).$ Similarly, we can prove that for $z\\in \\Omega _1^$ , we also have the error estimate $\\mathcal {O}(t^{-1})$ .", "Assembling all together, we conclude that the error term is $\\mathcal {O}(t^{-1})$ , and $m_1=\\mathcal {O}(t^{-1})$ , as $ t\\rightarrow \\infty .$ Painlevé region In this section, we first derive the Painlevé II hierarchy based on some RHP.", "Then, we will connect the long-time behavior of the mKdV hierarchy in the so-called Painlevé region to solutions of the Painlevé II hierarchy.", "Painlevé II hierarchy As mentioned in [2], the mKdV equation is can be transferred to the Painlevé II equation.", "The authors in [2] also suggest the connection between integrable PDEs with Painlevé equations.", "In [10], the authors explicitly derived the Painlevé II hierarchy from self-symmetry reduction of the mKdV hierarchy (see page 59 of [10].", "And also [11]).", "In this section, we will provide a slight different (as comparing to [11]) algorithm based on Riemann-Hilbert problems to generate the Painlevé II hierarchy.", "Let's denote $\\Theta (x,z)=xz+\\frac{c}{n}z^n$ , and suppose $Y$ solves the following RHP: $Y_+&=Y_-e^{i\\Theta \\sigma _3}v_0e^{-i\\Theta \\sigma _3},\\quad z\\in \\Sigma _n,\\\\Y&=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty .$ where the contour $\\Sigma _n$ consists of all stokes lines $\\lbrace z:\\operatorname{Im}{\\Theta (z)}=0\\rbrace $ and $v_0$ is a constant 2 by 2 matrix that is independent of $x,z$ .", "Now let $\\tilde{Y}=Ye^{i\\Theta \\sigma _3}$ , and we arrive at a new RHP: $\\tilde{Y}_+&=\\tilde{Y}_-v_0,\\quad z\\in \\Sigma _n,\\\\\\tilde{Y}&=(I+\\mathcal {O}(z^{-1}))e^{i\\Theta \\sigma _3},\\quad z\\rightarrow \\infty .$ Since $v_0$ is constant, it is easily to check, by Louisville's argument, that both $\\partial _z\\tilde{Y}\\tilde{Y}^{-1}$ and $\\partial _x\\tilde{Y}\\tilde{Y}^{-1}$ are polynomial of $z$ .", "Hence we obtain the following two differential equations: $\\partial _x\\tilde{Y}\\tilde{Y}^{-1}&=A(x,z),\\\\\\partial _z\\tilde{Y}\\tilde{Y}^{-1}&=B(x,z).$ If we assume $Y&=I+\\sum _{j=1}^{n-1}{Y_j(x)z^{-j}}+\\mathcal {O}(z^{-n}),\\quad z\\rightarrow \\infty ,\\\\\\underline{Y}&=Y^{-1}=I+\\sum _{j=1}^{n-1}{\\underline{Y}_j(x)z^{-j}}+\\mathcal {O}(z^{-n}),\\quad z\\rightarrow \\infty ,$ then a direct computation shows $A&=i[Y_1,\\sigma _3]+iz\\sigma _3,\\\\B&=ix\\sigma _3+icz^{n-1}\\sigma _3+icz^{n-2}[Y_1,\\sigma _3]\\\\&+\\sum _{k=2}^{n-1}icz^{n-1-k}(Y_k\\sigma _3+\\sigma _3\\underline{Y}_k+\\sum _{j=1}^{k-1}Y_{k-j}\\sigma _3\\underline{Y}_j).$ Since $Y_{x,z}=Y_{z,x}$ , we have $A_z-B_x+[A,B]=0.$ Set $Y_j=\\begin{pmatrix}p_j(x) & u_j(x)\\\\v_j(x)& q_j(x)\\end{pmatrix},\\quad j=1,..,n-2,$ where $p_j,q_j,u_j,v_j$ are smooth functions of $x$ .", "To guarantee (REF ), all the coefficients of $z$ must vanish.", "Those equations can be solved recursively.", "Eventually, by eliminating $u_j,v_j, j=2,..,n-2$ , and let $v_1=u_1$ , we will arrive at a nonlinear ODE of $u_1$Surprisingly, the dependence on $p_j,q_j$ will disappear., which turns out to be a member of the hierarchy of Painlevé II equations .", "We list the first few of them: $n=3&: -8cu^3+cu_{xx}-4xu=0,\\\\n=5&:-24cu^5+10cu^2u_{xx}+10cuu_x^2-\\frac{c}{4}u_{xxxx}-4xu=0,\\\\n=7&: -80cu^7+70cu^4u_{xx}+140cu^3u_x^2-\\frac{7cu^2u_{xxxx}}{2}\\\\\\quad \\quad & +(-\\frac{21}{2}cu^2_{xx}-14cu_xu_{xxx}-4x)u+\\frac{c}{16}u_{xxxxxx}-\\frac{35}{2}cu_x^2u_{xx}=0.$ In the current article, we focus only on the odd members.", "In fact, $n=3$ corresponds to the mKdV equation, $n=5$ corresponds to the 5th order mKdV, and so on.", "In the following subsection, we will show how to connect the long-time asymptotics behavior of the mKdV hierarchy to the solutions to the Painlevé II hierarchy.", "Painlevé Region Recall the phase functions of the AKNS hierarchy of mKdV type equations are $\\theta (z;x,t)=xz+ctz^n, \\quad n\\, \\text{is odd}.$ By the Painlevé region we mean a collection of all the curves $x=s(nt)^{1/n},s\\ne 0$ , by rescaling $z\\rightarrow (nt)^{-\\frac{1}{n}}\\xi $ , we have $\\Theta (\\xi )=s\\xi +\\frac{c}{n}\\xi ^n.$ Now the modulus of the stationary phase points of (REF ) is $\|z_0\|=\\left\|-\\frac{x}{ct}\\right\|^{\\frac{1}{n-1}}=\\mathcal {O}(t^{-\\frac{1}{n}}),$ and however, after scaling, the modulus of the stationary phase points of $\\Theta (\\xi )$ is $\\xi _0=\|z_0\|t^{\\frac{1}{n}},$ which is fixed as $t\\rightarrow \\infty $ .", "A direct computation shows for any odd $n$ , one can always perform lens-opening to the rays $\\lbrace z\\in \\mathbb {R}:\|z\|>\|\\xi _0\|\\rbrace $ , due to the signature of $\\operatorname{Re}(i\\theta )$ , see Fig.REF .", "Figure: Signature of Re(iθ)\\operatorname{Re}(i\\theta ).", "The green region: Re(iθ)>0\\operatorname{Re}(i\\theta )>0 when x<0x<0; The red region: Re(iθ)>0\\operatorname{Re}(i\\theta )>0 when x>0x>0; The yellow region: the overlapping region of red and green; The white region: Re(iθ)<0\\operatorname{Re}(i\\theta )<0.", "Here we only plot the signatures of Re(iθ)\\operatorname{Re}{(i\\theta )} for n=9n=9.", "Other odd nn, the region plot looks very similar.Note that $e^{-i\\theta (z)\\text{ ad }\\sigma _3}v(z)&=e^{-i\\Theta (\\xi )\\text{ ad }\\sigma _3}v(\\xi )\\\\&=\\begin{pmatrix}1-\|R\|^2 & -\\bar{R}e^{-2i\\Theta }\\\\Re^{2i\\Theta } & 1\\end{pmatrix}\\\\&=\\begin{pmatrix}1 & -\\bar{R}e^{-2i\\Theta }\\\\0 & 1\\end{pmatrix}\\begin{pmatrix}1 & 0\\\\Re^{2i\\Theta } & 1\\end{pmatrix}.$ We can deform the the contour $\\lbrace z\\in \\mathbb {R}:\|z\|>\|\\xi _0\|\\rbrace $ as before and get the deformed contour as follows (see Fig.REF ): Fix a positive constant $\\alpha <\\frac{\\pi }{n}$Such a choice of $\\alpha $ guarantees that the new contours will stay within the regions where the corresponding exponential term will decay (considering Fig.REF )., $\\Sigma _0&=\\lbrace z\\in \\mathbb {R}:-\\xi _0\\le z\\le \\xi _0\\rbrace ,\\\\\\Sigma _1&=\\lbrace z:z=\\xi _0+\\rho e^{i\\alpha },\\rho \\in (0,\\infty )\\rbrace ,\\\\\\Sigma _2&=\\lbrace z:z=-\\xi _0+\\rho e^{-i\\alpha },\\rho \\in (-\\infty ,0)\\rbrace ,\\\\\\Sigma _3&=\\lbrace z:z=-\\xi _0+\\rho e^{i\\alpha },\\rho \\in (-\\infty ,0)\\rbrace ,\\\\\\Sigma _4&=\\lbrace z:z=\\xi _0+\\rho e^{-i\\alpha },\\rho \\in (0,\\infty )\\rbrace ,$ and we define the regions as follows: $\\Omega _1&=\\lbrace z:z=\\xi _0+\\rho e^{i\\phi },\\rho \\in (0,\\infty ),\\phi \\in (0,\\alpha )\\rbrace ,\\\\\\Omega _2&=+\\backslash (\\Omega _1\\cup \\Omega _3),\\\\\\Omega _3&=\\lbrace z:z=-\\xi _0+\\rho e^{-i\\phi },\\rho \\in (-\\infty ,0),\\phi \\in (-\\alpha ,0)\\rbrace ,\\\\\\Omega _4&=\\lbrace z:z=-\\xi _0+\\rho e^{i\\phi },\\rho \\in (-\\infty ,0),\\phi \\in (0,\\alpha )\\rbrace ,\\\\\\Omega _5&={-}\\backslash (\\Omega _4\\cup \\Omega _6),\\\\\\Omega _6&=\\lbrace z:z=\\xi _0+\\rho e^{i\\phi },\\rho \\in (0,\\infty ),\\phi \\in (-\\alpha ,0)\\rbrace .$ Figure: Contour for ∂ ¯\\bar{\\partial }-RHP.As before, set the original RHP as $m^{[1]}$ with jump $e^{-i\\theta (z)\\text{ ad }\\sigma _3}v(z)$ .", "After re-scaling and $\\bar{\\partial }$ -lenses opening, we set $m^{[2]}(\\xi )=m^{[1]}O(\\gamma )$ , where the lenses opening matrix is $O(\\gamma )={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & 0\\\\-E_+e^{2i\\Theta (\\gamma )} & 1\\end{pmatrix},\\quad \\gamma \\in \\Omega _1\\cup \\Omega _3,\\\\\\begin{pmatrix}1 & -E_-e^{-2i\\Theta (\\gamma )}\\\\0 & 1\\end{pmatrix},\\quad \\gamma \\in \\Omega _4\\cup \\Omega _6,\\\\I,\\quad \\gamma \\in \\Omega _2\\cup \\Omega _5,\\end{array}\\right.", "}$ where $E_+(\\gamma )&=\\mathcal {K}(\\phi )R\\left((nt)^{-\\frac{1}{n}}\\xi \\right)+(1-\\mathcal {K}(\\phi ))R(\\tilde{\\xi }_0(nt)^{-\\frac{1}{n}}),\\\\E_-&(\\gamma )=\\overline{E_+(\\gamma )},\\\\\\gamma &={\\left\\lbrace \\begin{array}{ll}\\xi _0+\\rho e^{i\\phi },\\quad \\text{if }\\gamma \\in \\Omega _1\\cup \\Omega _6,\\\\-\\xi _0+\\rho e^{i\\phi },\\quad \\text{if }\\gamma \\in \\Omega _3\\cup \\Omega _4,\\end{array}\\right.}", "\\\\\\xi &=\\operatorname{Re}(\\gamma ),\\\\\\tilde{\\xi }_0&={\\left\\lbrace \\begin{array}{ll}\\xi _0,\\quad \\text{if }\\gamma \\in \\Omega _1\\cup \\Omega _6,\\\\-\\xi _0,\\quad \\text{if }\\gamma \\in \\Omega _3\\cup \\Omega _4.\\end{array}\\right.", "}$ Now we arrive at the following $\\bar{\\partial }$ -RHP: Mixed $\\bar{\\partial }$ -Riemann-Hilbert problem 10.1 Looking for a 2 by 2 matrix-valued function $m^{[2]}$ such that The RHP: (1.a) $m^{[2]}(\\gamma )\\in C^1(\\mathbb {R}^2\\backslash \\Sigma )$ and $m^{[2]}(z)=I+\\mathcal {O}(\\gamma ^{-1}),\\gamma \\rightarrow \\infty $ ; (1.b) the jumps on $\\Sigma _1$ and $\\Sigma _2$ are $e^{-i\\Theta (\\xi )\\text{ ad }\\sigma _3}v_+$ , and the jumps on $\\Sigma _3$ and $\\Sigma _4$ are $e^{-i\\Theta (\\xi )\\text{ ad }\\sigma _3}v_-$ , where $v_=\\begin{pmatrix}1 & \\bar{R}\\\\0 & 1\\end{pmatrix},\\quad v_+=\\begin{pmatrix}1 & 0\\\\R & 1\\end{pmatrix}.$ The jump on $\\Sigma _0$ is $e^{-i\\Theta \\text{ ad }\\sigma _3}v((nt)^{-\\frac{1}{n}}\\xi ),$ , and the jumps on $\\lbrace z\\in \\mathbb {R}:\|z\|>\|\\xi _0\|\\rbrace $ is $I$ .", "The $\\bar{\\partial }$ -problem: For $z\\in , we have{\\begin{@align}{1}{-1}\\bar{\\partial }m^{[2]}(\\xi )=m^{[2]}(\\xi )\\bar{\\partial }O(\\xi ).\\end{@align}}$ Again, we will need the following lemma in order to estimate errors from the $\\bar{\\partial }$ -problem.", "Lemma 10.2 For $\\gamma \\in \\Omega _{1,3,4,6}$ , $\\xi =\\operatorname{Re}\\gamma $ , $\|\\bar{\\partial }E_{\\pm }(\\gamma )\|\\le (nt)^{-\\frac{1}{n}}\|(nt)^{-\\frac{1}{n}}(\\xi -\\xi _0)\|^{-\\frac{1}{2}}\\Vert R\\Vert _{H^{1,0}}+(nt)^{-\\frac{1}{n}}\|R^{\\prime }((nt)^{-\\frac{1}{n}}\\xi )\|.$ For brevity, we only prove for the region $\\Omega _1$ .", "Using the polar coordinates, we have $\|\\bar{\\partial }E_+(\\gamma )\|&=\\left\|\\frac{ie^{i\\phi }}{2\\rho }\\mathcal {K}^{\\prime }(\\phi )\\left[R\\left((nt)^{-\\frac{1}{n}}\\xi \\right)-R(\\xi _0(nt)^{-\\frac{1}{n}})\\right]+\\mathcal {K}(\\phi )R^{\\prime }\\left((nt)^{-\\frac{1}{n}}\\xi \\right)(nt)^{-\\frac{1}{n}}\\right\| \\\\&\\text{by Cauchy-Schwartz inequality}\\\\&\\le \\left\|\\frac{\\Vert R\\Vert _{H^{1,0}}\|(nt)^{-\\frac{1}{n}}\\xi -\\xi _0(nt)^{-\\frac{1}{n}}\|^{1/2}}{\\gamma -\\xi _0}\\right\|+(nt)^{-\\frac{1}{n}}\\left\|R^{\\prime }\\left((nt)^{-\\frac{1}{n}}\\xi \\right)\\right\|\\\\&\\le (nt)^{-\\frac{1}{n}}\|(nt)^{-\\frac{1}{n}}(\\xi -\\xi _0)\|^{-\\frac{1}{2}}\\Vert R\\Vert _{H^{1,0}}+(nt)^{-\\frac{1}{n}}\|R^{\\prime }((nt)^{-\\frac{1}{n}}\\xi )\|.$ Similarly, we can prove for other regions.", "Next, consider a pure RHP $m^{[3]}$ which satisfies exactly the RHP part of $\\bar{\\partial }$ -RHP($m^{[2]}$ ).", "$m^{[3]}$ can be approximated by the RHP corresponding to a special solution of the Painlevé II hierarchyAs for the existence of the RHP $m^{[3]}$ , which is not completely trivial due to the fact that solutions to the Painlevé II equations have poles, we refer the readers to the book[20] for the details.", "Since for $\\gamma \\in \\Omega _1$ , $&\\left\|\\left(R(\\xi (nt)^{-\\frac{1}{n}})-R(0)\\right)e^{2i\\Theta (\\gamma )}\\right\|\\\\&\\le \|\\xi (nt)^{-\\frac{2}{n}}\|^{\\frac{1}{2}}\\Vert R\\Vert _{H^{1,0}}e^{2\\operatorname{Re}{i\\Theta (\\gamma )}}\\\\&\\le (nt)^{-\\frac{1}{n}}\|\\operatorname{Re}\\gamma \|^{\\frac{1}{2}}\\Vert R\\Vert _{H^{1,0}}e^{2\\operatorname{Re}{i\\Theta (\\gamma )}},$ it is evident that $\\Vert Re^{2i\\Theta }-R(0)e^{2i\\Theta }\\Vert _{L^{\\infty }\\cap L^1\\cap L^2}\\le c(nt)^{-\\frac{1}{n}}.$ Let $m^{[4]}$ solves the RHP formed by replacing $R(\\pm \\xi _0(nt)^{-1/n})$ and its complex conjugate in the jumps of $m^{[3]}$ along $\\Sigma _k,k=1,2,3,4$ by $R(0)$ and $\\bar{R}(0)$ respectively.", "Then, by the small norm theory, the errors between the corresponding potential is given by $error_{3,4}&=\\lim _{\\gamma \\rightarrow \\infty }\|\\gamma (m^{[4]}_{12}-m^{[3]}_{12})\|\\\\&\\le c \\int _{\\Sigma } \|(R(\\operatorname{Re}(s) (nt)^{-\\frac{1}{n}})-R(0))e^{2i\\Theta (s)}\|ds\\\\&\\le c(nt)^{-\\frac{1}{n}}.$ Then since now the jumps are all analytic, we can perform an analytic deformation and arrive at the green contours as show in Fig.REF .", "Let's denote the new RHP by $m^{[5]}(\\gamma )$ , and we arrive at the following RHP: Riemann-Hilbert problem 10.3 Looking for a 2 by 2 matrix-valued function $m^{[5]}$ such that $m^{[5]}$ is analytic off the contours $\\cup _{k=1,2,3,4}\\Sigma _{k}^{[5]}$ ; $m^{[5]}_+=m^{[5]}_-v^{[5]},\\quad z\\in \\cup _{k=1,2,3,4}\\Sigma _{k}^{[5]},$ where $v^{[5]}={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & 0\\\\R(0)e^{2i\\Theta (\\gamma )}&1\\end{pmatrix},\\quad \\gamma \\in \\Sigma _{1}^{[5]}\\cup \\Sigma _2^{[5]},\\\\\\begin{pmatrix}1 &\\bar{R}(0)e^{-2i\\Theta (\\gamma )}\\\\0&1\\end{pmatrix},\\quad \\gamma \\in \\Sigma _{3}^{[5]}\\cup \\Sigma _4^{[5]}.\\end{array}\\right.", "}$ Here the new contours (see Fig.REF ) are $\\Sigma _1^{[5]}&=\\lbrace z:z=\\rho e^{i\\alpha },\\rho \\in (0,\\infty )\\rbrace ,\\\\\\Sigma _2^{[5]}&=\\lbrace z:z=\\rho e^{-i\\alpha },\\rho \\in (-\\infty ,0)\\rbrace ,\\\\\\Sigma _3^{[5]}&=\\lbrace z:z=\\rho e^{i\\alpha },\\rho \\in (-\\infty ,0)\\rbrace ,\\\\\\Sigma _4^{[5]}&=\\lbrace z:z=\\rho e^{-i\\alpha },\\rho \\in (0,\\infty )\\rbrace .$ Figure: Contour for m [4] m^{[4]}(Green part).Then according to the previous subsection, the $(1,2)$ entry of the solution $m^{[5]}$ , similarly the solution $m^{[4]}$ , is the solution to the Painlevé II hierarchy, i.e., $m^{[4]}_{12}(\\gamma )=m^{[5]}_{12}(\\gamma ),$ Hence we have $P^{II}_k(s)=\\lim _{\\gamma \\rightarrow \\infty }\\gamma m^{[5]}_{12}$ where $P^{II}_k$ solves the $k^{th}$ equation in the Painlevé II hierarchy, where $k=\\frac{n-1}{2}$ .", "Now let's consider the error generated from the $\\bar{\\partial }$ -extension.", "Recall that the error $E$ satisfies a pure $\\bar{\\partial }$ -problem: $\\bar{\\partial }E&=EW,\\\\W&=m^{[3]}\\bar{\\partial }O(m^{[3]})^{-1}.$ As before, the $\\bar{\\partial }$ -equation is equivalent to an integral equation which reads $E(z)=I+\\frac{1}{\\pi }\\int _{\\frac{E(s)W(s)}{z-s}\\text{d}A(s)=I+\\mathcal {K}(E).", "}As before, we can show that the resolvent always exists for large t. So we only need to estimate the true error which is: \\lim _{z\\rightarrow \\infty }z(E-I).", "In fact, we have{\\begin{@align}{1}{-1}\\lim _{z\\rightarrow \\infty }\|z(E-I)\|&=\|\\int _\\mathbb {C}EWds\|\\\\&\\le c\\Vert E\\Vert _\\infty \\int _{\\Omega }\|\\bar{\\partial }O\|ds.\\end{@align}}For the sake of simplicity, we only estimate the integral on the right hand side in the region of the top right corner.", "Note there is only one entry which is nonzero in \\bar{\\partial }O, which is one of the E_{\\pm } and we split the integral into two parts in the obvious way, i.e.,{\\begin{@align}{1}{-1}\\int _\\Omega \|\\bar{\\partial }O\|ds&\\le I_1+I_2\\\\&=\\int _\\Omega (nt)^{-\\frac{1}{2n}}\|\\operatorname{Re}{s}-\\xi _0\|\\Vert R\\Vert _{H^{1,0}}e^{2\\operatorname{Re}{i\\Theta (s)}}ds\\\\&+\\int _{\\Omega }(nt)^{-\\frac{1}{nt}}\|R^{\\prime }((nt)^{-\\frac{1}{n}}s)\|e^{2\\operatorname{Re}{i\\Theta (s)}}ds.\\end{@align}}As we know from previous sections, e^{\\operatorname{Re}{2i\\Theta (s)}}\\le ce^{-2\|\\Theta ^{\\prime \\prime }(\\xi _0)\|uv} in the region \\lbrace z=u+iv:u>\\xi _0,0<v<\\alpha u\\rbrace for some small \\alpha , where s=u+iv+\\xi _0.", "Then we have{\\begin{@align}{1}{-1}I_1&\\le (nt)^{-\\frac{1}{2n}}\\int _{\\Omega }\|\\operatorname{Re}{s}-\\xi _0\|^{-1/2}e^{-cuv}dudv\\\\&\\le (nt)^{-\\frac{1}{2n}}\\int _0^\\infty \\int _0^{\\alpha u}u^{-1/2}e^{-cuv}dudv\\\\&\\le C(nt)^{-\\frac{1}{2n}}\\int _0^\\infty \\frac{1-e^{-2\\alpha \|\\Theta ^{\\prime \\prime }(\\xi _0)\|} }{u^{3/2}}du\\\\&=\\mathcal {O}\\left((nt)^{-\\frac{1}{2n}}\\right),\\end{@align}}and{\\begin{@align}{1}{-1}I_2&\\le (nt)^{-\\frac{1}{n}}\\int \|R^{\\prime }((nt)^{-\\frac{1}{2n}}\\operatorname{Re}{s})\|e^{-cuv}dudv\\\\&\\text{ by Cauchy-Schwartz inequality}\\\\&\\le (nt)^{-\\frac{1}{n}}\\Vert R\\Vert _{H^{1,0}}\\int _0^\\infty (\\int _{\\alpha v}^\\infty e^{-2cuv}du)^{1/2}dv\\\\&\\le (nt)^{-\\frac{1}{n}}\\Vert R\\Vert _{H^{1,0}}\\int _0^\\infty \\frac{e^{-c\\alpha v^2}}{\\sqrt{2\\alpha cv}}dc\\\\&=\\mathcal {O}((nt)^{-\\frac{1}{n}}).\\end{@align}}Thus, we arrive at{\\begin{@align}{1}{-1}\\bar{\\partial }\\text{Error} = \\mathcal {O}((nt)^{-\\frac{1}{2n}}).\\end{@align}}And we undo all the deformations, we obtain{\\begin{@align}{1}{-1}m^{[1]}((nt)^{-\\frac{1}{n}}\\gamma )&=m^{[2]}(\\gamma )O^{-1}(\\gamma )\\\\&=(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })m^{[3]}(\\gamma )O^{-1}(\\gamma )\\\\&=(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })m^{[4]}(\\gamma )O^{-1}(\\gamma )\\\\&=(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })m^{[5]}(\\gamma )O^{-1}(\\gamma ).\\end{@align}}It can also be rewritten in terms of the variable z:{\\begin{@align}{1}{-1}m^{[1]}(z)=\\left(1+\\frac{\\mathcal {O}{(t^{-1/(2n)}})}{z(nt)^{1/n}}\\right)m^{[5]}((nt)^{1/n}z)+\\mathcal {O}{(z^{-2})},\\quad z\\rightarrow \\infty .\\end{@align}}Since m^{[5]} corresponds to the RHP for the Painlevé II hierarchy, we have{\\begin{@align}{1}{-1}m^{[5]}(\\gamma )=I+\\frac{m_1^{[5]}(s)}{\\gamma }+\\mathcal {O}(\\gamma ^{-1}),\\end{@align}}where \\gamma =z(nt)^{1/n}.$ Thus, $m^{[1]}(z)&=\\left(1+\\frac{\\mathcal {O}{(t^{-\\frac{1}{2n}})}}{z(nt)^{1/n}}\\right)\\left(1+\\frac{m_1^{[5]}(s)}{z(nt)^{1/n}}+\\mathcal {O}(z^{-2})\\right)\\\\&=I+\\frac{m_1^{[5]}(s)}{z(nt)^{1/n}}+\\frac{\\mathcal {O}{(t^{-\\frac{1}{2n}})}}{z(nt)^{1/n}}+\\mathcal {O}(z^{-2}).$ Since $m_1^{[5]}(s)$ is connected to solutions of the Painlevé II hierarchy, we conclude that $q(x,t)&=\\lim _{z\\rightarrow \\infty }z(m^{[1]}-I)\\\\&=(nt)^{-\\frac{1}{n}}u_n(x(nt)^{-\\frac{1}{n}})+\\mathcal {O}{(t^{-\\frac{3}{2n}})},$ where $u_n$ solves the $\\frac{n-1}{2}$ th equation of the Painlevé II hierarchy.", "The odd integer $n$ corresponds to the $\\frac{n-1}{2}$ th member in the mKdV hierarchy.", "Remark 10.4 As for the asymptotics for the Painlevé II equation, we refer the readers to the classical book [20].", "There are also some recent works related to Painlevé II hierarchy, see for example [30],[9],[6]." ], [ "Separate contributions and phase reduction", "The RHP and the mixed $\\bar{\\partial }$ -RHP we have discussed above are global.", "In this section, we shall approximate the global RHP by performing two steps: (1) separate contributions from each stationary phase point, (2) phase reduction.", "Before that, let us first consider two saddle points $z_j,z_{j+1}$ , and discuss $\\varepsilon _j=1=-\\varepsilon _{j+1}$ for example.", "We will first remove the vertical segments, see Fig.", "(REF ): $\\Sigma _{j+\\frac{1}{2}}:=\\Omega _{j,1}\\cap \\Omega _{j+1,3} \\cup \\Omega _{j,6}\\cap \\Omega _{j+1,4}\\backslash \\mathbb {R},$ where $\\Omega _{j,\\cdot }$ 's are defined in REF .", "Figure: Jumps in a small triangular region.Recall the constructions of $E_{j,1}$ and $E_{j+1,3}$ (see (REF )), the boundary value of $m^{[2]}(z)$ on $\\Sigma _{j+\\frac{1}{2}}$ from $\\Omega _{j,1}$ is $m^{[1]}(z_{j+1/2}+iv)O_{j,1}(z_{j+1/2}+iv),$ while from $\\Omega _{j+1,3}$ it is $m^{[1]}(z_{j+1/2}+iv)O_{j+1,3}(z_{j+1/2}+iv).$ Both correspond to locally increasing parts of the phase function, and thus correspond to an upper/lower factorization.", "So the jump on the new contour $\\Sigma _{j+1/2}$ is $O_{j+1,3}O^{-1}_{j,1}(z)$ , $z=z_{j+\\frac{1}{2}}+iv$ , where the nontrivial entry is (regarding the property of $\\mathcal {K}$ and definitions of those matrix $O_{j,k}$ , see (REF ) and (REF ) ): $(1-\\mathcal {K}(\\phi ))&[R(z_j)\\delta _j^{-2}(z_{j+1/2}-z_j+iv)^{-2i\\eta (z_j)}\\\\&-R(z_{j+1})\\delta _{j+1}^{-2}(z_{j+1/2}-z_{j+1}+iv)^{-2i\\eta (z_{j+1})}]e^{2it\\theta (z_{j+1/2}+iv)},$ with $v\\in (0,(z_{j+1/2}-z_j)\\tan (\\alpha ))$ and $\\phi =\\arg {(z-z_j)}$ .", "Note that $\|(z_{j+1/2}-z_j+iv)^{-2i\\eta (z_j)}\|&=e^{2\\eta (z_j)\\phi }\\le e^{2\\eta (z_j)\\alpha }.$ and $\|e^{2it\\theta (z_{j+1/2}+iv)}\|\\le ce^{-2tdv},\\quad d=(z_{j+1}-z_{j})/2.$ Thus we have, for any $z\\in \\Sigma _{j+\\frac{1}{2}}$ , $O_{j+1,3}O^{-1}_{j,1}-I=\\mathcal {O}(e^{-ct}),\\quad t\\rightarrow \\infty ,$ where $c$ is some generic positive constant.", "Since the jump is close to $I$ , by a small norm theory, the solution will also be close to $I$ .", "In fact, we have the following estimate for the potential $\|\\lim _{z\\rightarrow \\infty }z&(m^{[2]}\|_{\\Sigma _{j+\\frac{1}{2}}}-I)\|\\\\& \\le \\frac{1}{2\\pi }\\int _0^{d\\tan {(\\alpha )}}\\left\|m^{[2]}_-(z_{j+1/2}+is)\\right\|\\left\|O_{j+1,3}O_{j,1}^{-1}(z_{j+1/2}+is)-I\\right\|\\text{ds}\\\\&\\le \\frac{1}{2\\pi }\\int _0^{d\\tan {(\\alpha )}}\\left\|m^{[2]}_-(z_{j+1/2}+is)\\right\|e^{-2tsd}\\text{ds}\\\\&\\le \\frac{1}{2\\pi }\\Vert m^{[1]}\|_{\\Sigma _{j+\\frac{1}{2} ,3}}\\Vert _\\infty \\Vert O_{j+1,3}\\Vert _\\infty \\int _{0}^{d\\tan {(\\alpha )}}e^{-2tsd}\\text{ds}\\\\&=\\mathcal {O}(t^{-1}),$ where we assume $m^{[1]}$ , as a solution to the conjugated RHP, existsThe existence and uniqueness will be discussed later..", "So it is analytic in a neighborhood of $\\Sigma _{j+\\frac{1}{2}}$ and hence it is bounded on $\\Sigma _{j+\\frac{1}{2}}$ .", "By the definition (see (REF )) of $O_{j+1,3}$ , it is continuous in $\\Sigma _{j+\\frac{1}{2}}$ and does not blow up at the endpoints of $\\Sigma _{j+\\frac{1}{2}}$ .", "So $\\Vert O_{j+1,3}\\Vert _\\infty $ is also finiteHere the $L^\\infty (\\Sigma )$ norm $\\Vert f(z)\\Vert _{\\infty }$ means $\\sup _{z\\in \\Sigma }\|f(z)\|$ , where $\|f(z)\|=\\max _{i,j=1,2,z\\in \\Sigma }\|f_{i,j}(z)\|$ ..", "Therefore, we can remove all those vertical segments by paying a price of error $\\mathcal {O}(t^{-1})$ , which will be dominated by the error generated by the $\\bar{\\partial }$ -problem (it is $\\mathcal {O}(t^{-3/4})$ , we will show it in a moment.)", "Let us denote the new RHP by $\\tilde{m}^{[2]}$ .", "To make it clear, we note that the jumps for $\\tilde{m}^{[2]}$ are $\\tilde{v}^{[2]}(z)={\\left\\lbrace \\begin{array}{ll}v^{[2]}(z),\\quad z\\in \\cup _{j=1,..,l,k=1,2,3,4}\\Sigma _{j,k},\\\\I,\\quad z\\in \\cup _{j=1,..,l}\\Sigma _{j+\\frac{1}{2}}\\cup \\mathbb {R}.\\end{array}\\right.", "}$ Next, we will show that the RHP for $\\tilde{m}^{[2]}$ can be localized to each saddle point.", "For example, near $z_j$ , along the segment $\\Sigma _{j,1}:z=z_j+u+iv, \\arg {z}=\\alpha $ , we have $\|E_{j,1}e^{2it\\theta }\|\\le ce^{-2t\\tan (\\alpha )u^2}$ It is well-known [14], [19] that the $\|E_{j,1}e^{2it\\theta }\|\\le ce^{-2t\\tan (\\alpha )u^2}$ , where let $u\\ge u_0>0$ , and then the jump matrix will go to $I$ with decaying rate at $\\mathcal {O}(e^{-ct}),c>0$ , as $t\\rightarrow \\infty $ .", "The RHP is localized in the small neighborhoods of those stationary phase points.", "Note that near each $z_j$ , we have $\\theta (z)=\\theta (z_0)+\\frac{\\theta ^{\\prime \\prime }(z_0)(z-z_0)^2}{2}+\\mathcal {O}{(\|z-z_j\|^3)}.$ By a similar argument of Lemma 3.35 in [16] or subsection 8.2 in [19] for the phase reduction, the error generated by reducing the phase function $\\theta $ to $\\theta (z_0)+\\frac{\\theta ^{\\prime \\prime }(z_0)(z-z_0)^2}{2}$ will be bounded by $\\mathcal {O}{(t^{-1})}$ .", "Both analysis of the mentioned references are based on the analysis of the so-called Beals-Coifman operator [3].", "Now we shall simply describe it here.", "For the sake of simplicity, we only consider the RHP on the contour $\\Sigma _{j,1}$ (for more details, we direct the interested reader to [15]): Riemann-Hilbert problem 6.1 Looking for 2 by 2 matrix-valued function $\\tilde{m}^{[2]}$ such that $\\tilde{m}(z)$ is analytic off $\\Sigma _{j,1}$ ; $\\tilde{m}_+=\\tilde{m}_-v^{[2]},\\quad z\\in \\Sigma _{j,1};$ $\\tilde{m}=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty $ .", "Since $E_{j,1}\|_{\\Sigma _{j,1}}$ is analytic near $\\Sigma _{j,1}$ for $z$ away from $z_j$ , and enjoys a factorization$(w^-,w^+)$ will be called the factorization data for the jump matrix.", ": $(I-w^-)^{-1}(I+w^+),$ where $w^-&=I-(v^{[2]})^{-1}=(v^{[2]})-I,\\\\w^+&=0,$ and the superscribes $\\pm $ indicate the analyticitiy in the left/right neighborhood of the the contour.", "Following the definition in [3], we define the Beals-Coifman operator, for any $f\\in L^2(\\Sigma _{j,1})$ , as follows: $C_w(f)=C_+(fw^-)+C_-(fw^+),$ where $C$ means the usual Cauchy operator, i.e., $Cf(z)=\\frac{1}{2\\pi i}\\int _{\\Sigma _{j,1}}\\frac{f(s)}{s-z}\\text{ds},$ and $C_{\\pm }$ means the non-tangential limits from left/right side.", "The following proposition, which plays a fundamental role in Deift-Zhou's method, is well-known.", "Proposition 6.2 (see also proposition 2.11 in [15]) If $\\mu \\in I+L^2$ solves the singular integral equation: $\\mu =I+C_w(\\mu ).$ Then the (unique) solution to the RHP for $\\tilde{m}$ readsHere $w=w^++w^-$ .", "$\\tilde{m}=I+C(\\mu w).$ Then follow the localization principle in [14], [19], [32], and the simple argument on the vertical segments, we arrive at a new RHP on the new contours: fixing $\\rho _0>0$ small, define $\\Sigma _{j,1}^o&:=\\lbrace z:z=z_j+\\rho e^{i\\alpha },\\rho \\in (0,\\rho _0)\\rbrace ,\\\\\\Sigma _{j,2}^o&:=\\lbrace z:z=z_j+\\rho e^{i(\\pi -\\alpha )},\\rho \\in (0,\\rho _0)\\rbrace ,\\\\\\Sigma _{j,3}^o&:=\\lbrace z:z=z_j-\\rho e^{i\\alpha },\\rho \\in (0,\\rho _0)\\rbrace ,\\\\\\Sigma _{j,4}^o&:=\\lbrace z:z=z_j+\\rho e^{i(\\alpha -\\pi )},\\rho \\in (0,\\rho _0)\\rbrace .$ Then with the new contour (see Fig.REF ) $\\Sigma ^o=\\cup _{j=1,..,l,k=1,2,3,4}\\Sigma _{j,k}^o$ , the new RHP reads as follows: Riemann-Hilbert problem 6.3 Looking for a 2 by 2 matrix-valued function $\\hat{m}^{[2]}$ such that $\\hat{m}^{[2]}_+=\\hat{m}^{[2]}_-\\hat{v}^{[2]},\\quad z\\in \\Sigma ^o,$ with $\\hat{v}^{[2]}=\\tilde{v}^{[2]}\|_{\\Sigma ^0};$ $\\hat{m}^{[2]}=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty .$ Figure: New contours, dashed line segments are those deleted parts.Moreover, since the potential of the mKdV hierarchy can be recovered by the formula (REF ), which can also be written as the Beals-Coifman solution: $q_{RHP}(x,t)=-\\frac{1}{2\\pi i}\\int _{\\Sigma }((I-C_w)^{-1}I)w(s)\\text{ds}.$ Then, by localization, we have $\\int _{\\Sigma }((I-C_w)^{-1}I)w(s)\\text{ds}=\\int _{\\Sigma ^o}((I-C_w)^{-1}I)w(s)\\text{ds}+\\mathcal {O}(t^{-1}), \\quad t\\rightarrow \\infty ,$ where $\\Sigma $ is the the contour before localization and $w$ can be easily defined in each cross since the jumps are all triangle matrices and all entries in the diagonal are one.", "Let us denote $q^o_{RHP}(x,t)=-\\frac{1}{2\\pi i}\\int _{\\Sigma ^o}((I-C_w)^{-1}I)w(s)\\text{ds}.$ Then from the localization principal, we have $q_{RHP}(x,t)=q^o_{RHP}(x,t)+\\mathcal {O}(t^{-1}),\\quad t\\rightarrow \\infty .$ Moreover, we define the RHP ($m^{[3]}$ ) which corresponds to the local Beals-Coifman solution (i.e.", "$q^o_{RHP}$ ) as follows: Riemann-Hilbert problem 6.4 Looking for a 2 by 2 matrix-valued function $m^{[3]}$ such that $m^{[3]}_+=m^{[3]}_-v^{[3]}(z),\\quad z\\in \\Sigma ^o$ , with jump matrix reads $v^{[3]}=\\hat{v}^{[2]}\\upharpoonright _{\\Sigma ^o};$ $m^{[3]}=I+\\mathcal {O}{(z^{-1})},\\quad z\\rightarrow \\infty $ .", "However, the integral $\\int _{\\Sigma ^o}((I-C_w)^{-1}I)w(s)\\text{ds}$ is still hard to compute, and following the Deift-Zhou method, we need to separate the contributions from each stationary phase point.", "Thus, we need the following important lemma.", "Lemma 6.5 (see equation (3.64) or proposition 3.66 in [14]) As $t\\rightarrow \\infty $ , $\\int _{\\Sigma ^o}(\\left(1-C_w\\right)^{-1}I)w=\\sum _{j=1}^{l}\\int _{\\Sigma ^o_j}((1-C_{w_j})^{-1}I)w_j+\\mathcal {O}(t^{-1}),$ where $w_j$ is the factorization data supported on $\\Sigma ^o_j=\\cup _{k=1}^4\\Sigma ^o_{j,k}$ , $w=\\sum _{j=1}^lw_j$ and $\\Sigma ^o=\\cup _{j}^l\\Sigma ^o_j$ .", "First, recall the following observation by Varzugin [32], $(1-C_w)(1+\\sum _jC_{w_j}(1-C_{w_j})^{-1})=1-\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1}.$ With the hints from this observation, we need to estimate the norms of $C_{w_j}C_{w_k}$ from $L^\\infty $ to $L^2$ and from $L^2$ to $L^2$ .", "Also from next section (with a small norm argument), we know $(1-C_{w_j})^{-1}$ are uniformly bounded in $L^2$ sense.", "Now let us focus on the contour $\\Sigma ^o_{j,1}$ , and $\\varepsilon =1$ .", "Then the nontrivial entry of the factorization data is $E_{j,1}(z)e^{-2it\\theta (z)},z\\in \\Sigma ^o_{j,1}$ , and thus we have $\|w_{j}\\upharpoonright _{\\Sigma ^o_{j,1}}\|\\le ce^{-2t\\tan (\\alpha )u^2},$ which implies that $\\Vert w_{j}\\upharpoonright _{\\Sigma ^o_{j,1}}\\Vert _{L^1}=\\mathcal {O}(t^{-1/2})$ and $\\Vert w_{j}\\upharpoonright _{\\Sigma ^o_{j,1}}\\Vert _{L^2}=\\mathcal {O}(t^{-1/4})$ .", "Then following exactly the same steps in the proof of [14], Lemma 3.5, we have for $j\\ne k$ $\\Vert C_{w_j}C_{w_k}\\Vert _{L^2(\\Sigma ^o)}&=\\mathcal {O}(t^{-1/2}),\\\\\\Vert C_{w_j}C_{w_k}\\Vert _{L^\\infty \\rightarrow L^2(\\Sigma ^o)}&=\\mathcal {O}(t^{-3/4}).$ Then use the resolvent identities and the Cauchy-Schwartz inequality, $((1-C_w)^{-1}I)&=I+\\sum _{j=1}^{l}C_{w_j}(1-C_{w_j})^{-1}I\\\\&+[1+\\sum _{j=1}^{l}C_{w_j}(1-C_{w_j})^{-1}][1-\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1}]^{-1}\\\\&(\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1})I\\\\&=I+\\sum _{j=1}^{l}C_{w_j}(1-C_{w_j})^{-1}I+ABDI,$ where $A&:=1+\\sum _{j=1}^{l}C_{w_j}(1-C_{w_j})^{-1},\\\\B&:=[1-\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1}]^{-1},\\\\D&:=\\sum _{j\\ne k}C_{w_j}C_{w_k}(1-C_{w_k})^{-1},$ and thus $\|\\int _{\\Sigma ^o}ABDIw\|&\\le \\Vert A\\Vert _{L^2}\\Vert B\\Vert _{L^2}\\Vert D\\Vert _{L^\\infty \\rightarrow L^2}\\Vert w\\Vert _{L^2}\\\\&\\le c t^{-3/4}t^{-1/4}=\\mathcal {O}{(t^{-1})}.$ Then applying the restriction lemma ([14], Lemma 2.56), we have $\\int _{\\Sigma ^o}(I+C_{w_j}(1-C_{w_j})^{-1}I)w\\upharpoonright _{\\Sigma ^o_j}&=\\int _{\\Sigma ^o_j}(I+C_{w_j}(1-C_{w_j})^{-1}I)w\\\\&=\\int _{\\Sigma ^o_j}((1-C_{w_j})^{-1}I)w_j.$ Therefore, the proof is done." ], [ "A model Riemann-Hilbert problem", "In the previous section, we have reduce the global RHP to $l$ local RHPs near each stationary phase point due to Lemma REF .", "In fact, near each stationary phase point, we need to compute the integral $\\int _{\\Sigma _j^o}((1-C_{w_j})^{-1}I)w_j$ , which is equivalent to a local RHP.", "In this section, we will approximate the local RHPs by a model RHP which can be solved explicitly by solving a parabolic-cylinder equation.", "Consider the following RHP: Riemann-Hilbert problem 7.1 Looking for a 2 by 2 matrix-valued function $P(\\xi ;R)$ such that $P_+(\\xi ;R)=P_-(\\xi ;R)J(\\xi ),\\xi \\in \\mathbb {R}$ , where $J(\\xi )=\\begin{pmatrix}1-\|R\|^2&-\\bar{R}\\\\R&1\\end{pmatrix}$ is a constant matrix with respect to $\\xi $ and the constant $R$ satisfies $\|R\|<1$ ; $P(\\xi ;R)=\\xi ^{i\\eta \\sigma _3}e^{-i\\frac{\\xi ^2}{4}\\sigma _3}(I+P_1\\xi ^{-1}+\\mathcal {O}(\\xi ^{-2})),\\quad \\xi \\rightarrow \\infty $ , where $P_1=\\begin{pmatrix}0&\\beta \\\\\\bar{\\beta }&0\\end{pmatrix}$ .", "Then by Liouville's argument, $P^{\\prime }P^{-1}$ is analytic and thus $P^{\\prime }(\\xi )=(-\\frac{i\\xi }{2}\\sigma _3-\\frac{i}{2}[\\sigma _3,P_1])P(\\xi ),$ which can be solved in terms of the parabolic-cylinder equation, and apply the asymptotics formulas we can eventually determine that $\\beta =\\frac{\\sqrt{2\\pi }e^{i\\pi /4}e^{-\\pi \\eta /2}}{R\\Gamma (-a)},$ where $a=i\\eta ,$ with $\\eta =-\\frac{1}{2\\pi }\\log {(1-\|R\|^2)}$ .", "The above result has been presented in the literatureThe first description of this model RHP was presented by A. R. Its [22].", "Later examples of the model can be find in [14], [15], [17], [19], [32], [26], [27].", "in many ways.", "Here we follows the representations in [14].", "Next, we will connect this model RHP to the original RHP.", "Recall, near stationary phase point $z_j$ , we need to estimate integral $\\int _{\\Sigma _j^o}((1-C_{w_j})^{-1}I)(w_{j+}+w_{j-})$ , which is equivalent to solve the following RHP ($m^{[3,j]},\\quad j=1,\\cdots ,l$ ): $m^{[3,j]}_+(z)=m^{[3,j]}_-(z)v^{[3,j]}(z),z\\in \\Sigma _j^o$ .", "The jump matrix reads $v^{[3,j]}(z)={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1&0\\\\R_j^\\sharp (z-z_j)^{-2i\\eta (z_j)}e^{-it\\theta ^{\\prime \\prime }(z_j)(z-z_j)^2}&1\\end{pmatrix},z\\in \\Sigma _{j,1}^o,\\\\\\begin{pmatrix}1&-\\frac{\\bar{R}_j^\\sharp }{1-\|R^\\sharp _j\|^2}(z-z_j)^{2i\\eta (z_j)}e^{it\\theta ^{\\prime \\prime }(z_j)(z-z_j)^2}\\\\0&1\\end{pmatrix},z\\in \\Sigma _{j,2}^o,\\\\\\begin{pmatrix}1&0\\\\\\frac{R_j^\\sharp }{1-\|R^\\sharp _j\|^2}(z-z_j)^{2i\\eta (z_j)}e^{-it\\theta ^{\\prime \\prime }(z_j)(z-z_j)^2}&1\\end{pmatrix},z\\in \\Sigma _{j,3}^o,\\\\\\begin{pmatrix}1&-\\bar{R}_j^\\sharp (z-z_j)^{-2i\\eta (z_j)}e^{it\\theta ^{\\prime \\prime }(z_j)(z-z_j)^2}\\\\0&1\\end{pmatrix},z\\in \\Sigma _{j,4}^o,\\end{array}\\right.", "}$ where $R_j^\\sharp =R_j\\delta ^{-2}_je^{-2it\\theta (z_j)}$ ; $m^{[3,j]}=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty .$ Set $\\xi =(2t\\theta ^{\\prime \\prime }(z_j))^{1/2}(z-z_j)$ and by closing lenses, we arrive at an equivalent RHP on the real line: $m^{[4,j]}(\\xi )_+=m^{[4]}_-v^{[4,j]}(\\xi ),\\xi \\in \\Sigma _{j}^p$ .", "The new jump is $v^{[4,j]}(\\xi )=(2\\theta ^{\\prime \\prime }(z_j)t)^{-\\frac{i\\eta (z_j)}{2}\\text{ ad }\\sigma _3}\\xi ^{i\\eta (z_j)\\text{ ad }\\sigma _3}e^{-\\frac{i\\xi ^2}{4}\\text{ ad }\\sigma _3}\\begin{pmatrix}1-\|R_j^\\sharp \|^2&-\\bar{R}_j^\\sharp \\\\R_j^\\sharp &1\\end{pmatrix};$ $m^{[4,j]}=I+\\mathcal {O}(\\xi ^{-1}),\\xi \\rightarrow \\infty .$ Comparing with the model RHP, we observe that $m^{[4,j]}(\\xi )(2\\theta ^{\\prime \\prime }(z_j)t)^{-\\frac{i\\eta (z_j)}{2}\\sigma _3}\\xi ^{i\\eta (z_j)\\sigma _3}e^{-\\frac{i\\xi ^2}{4}\\sigma _3}$ solves the model RHP, which leads to $m^{[4]}_{1,12}&=\\frac{\\sqrt{2\\pi }e^{i\\pi /4}e^{-\\pi \\eta (z_j)/2}}{R^\\sharp _j\\Gamma (-i\\eta (z_j))},\\\\m^{[4]}_{1,21}&=\\frac{-\\sqrt{2\\pi }e^{-i\\pi /4}e^{-\\pi \\eta (z_j)/2}}{\\bar{R}^\\sharp _j\\Gamma (i\\eta (z_j))}.$ Changing the variable $\\xi $ back to $z$ , we have $m^{[3,j]}_{1,12}(t)&= (2t\\theta ^{\\prime \\prime }(z_j))^{-\\frac{1}{2}-\\frac{i\\eta (z_j)}{2}}\\frac{\\sqrt{2\\pi }e^{i\\pi /4}e^{-\\pi \\eta (z_j)/2}}{R_j^\\sharp \\Gamma (-i\\eta (z_j))},\\\\m^{[3,j]}_{1,21}(t)&=- (2t\\theta ^{\\prime \\prime }(z_j))^{-\\frac{1}{2}+\\frac{i\\eta (z_j)}{2}}\\frac{\\sqrt{2\\pi }e^{-i\\pi /4}e^{-\\pi \\eta (z_j)/2}}{\\bar{R}_j^\\sharp \\Gamma (i\\eta (z_j))}.$ Noting that $R^\\sharp _j=R_j\\delta ^{-2}_je^{2it\\theta (z_j)}$ , one can rewrite in a neat way: $m^{[3,j]}_{1,12}(t)=\\frac{\|\\eta (z_j)\|^{1/2}}{\\sqrt{2t\\theta ^{\\prime \\prime }(z_j)}}e^{i\\varphi (t)},\\\\m^{[3,j]}_{1,21}(t)=\\frac{\|\\eta (z_j)\|^{1/2}}{\\sqrt{2t\\theta ^{\\prime \\prime }(z_j)}}e^{-i\\varphi (t)},$ where the phase is $\\varphi (t)=\\frac{\\pi }{4}-\\arg \\Gamma (-i\\eta (z_j))-2t\\theta (z_j)-\\frac{\\eta (z_j)}{2}\\log \|2t\\theta ^{\\prime \\prime }(z_j)\|+2\\arg (\\delta _j)+\\arg (R_j).$ Here we have used the fact that $\|\\beta \|^2=\\eta $ .", "Denoting $q_{as}(x,t)=-2i\\sum _{j=1}^{l}\\frac{\|\\eta (z_j)\|^{1/2}}{\\sqrt{2t\\theta ^{\\prime \\prime }(z_j)}}e^{i\\varphi (t)},$ then the connection formula (REF ) and Lemma REF lead to $q^o_{RHP}(x,t)=q_{as}(x,t)+\\mathcal {O}(t^{-1}),\\quad t\\rightarrow \\infty .$" ], [ "Errors from the pure $\\bar{\\partial }$ -problem", "In this section, we will discuss the error generated from the pure $\\bar{\\partial }$ -problem of $m^{[2]}$ .", "Let us denote $E(z)=m^{[2]}({m}^{[2]}_{RHP})^{-1},$ where $m^{[2]}_{RHP}$ denotes the solution to the pure RHP part of $m^{[2]}$ .", "Assuming the existence (which we will be provided in the next section), and by the normalization condition, we have $E(z)=1+(m_1^{[2]}-{m}^{[2]}_{RHP,1})z^{-1}+\\mathcal {O}(z^{-2}),\\quad z\\rightarrow \\infty .$ Due to the procedure of localization and separation of the contributions, we can approximate $m^{[2]}_{RHP}$ by $\\hat{m}^{[2]}$ , and the error of approximating the potential is of $\\mathcal {O}(t^{-1})$ as $t\\rightarrow \\infty $ .", "Thus, by the equation(REF ), $q(x,t)=q_{RHP}(x,t)+\\mathcal {O}(t^{-1})+\\lim _{z\\rightarrow \\infty }z(E-I),\\quad t\\rightarrow \\infty .$ Moreover, from this construction (equation (REF )), there is no jump on the contours $\\Sigma _{j,k},k=1,2,3,4,$ but only a pure $\\bar{\\partial }$ -problem is left due to the non-analyticity.", "The $\\bar{\\partial }$ -problem reads $\\bar{\\partial }E=EW,$ where $W(z)=m^{[2]}_{RHP}\\bar{\\partial }O(z)(m^{[2]}_{RHP})^{-1}.$ From the normalization condition of $m^{[2]}_{RHP}$ , we see it is uniformly bounded by $\\frac{c}{1-\\sup {R}}$ .", "And to estimate the errors of recovering the potential, one actually needs to estimate $\\lim _{z\\rightarrow \\infty }z(E-I)$ , where the limit can be chosen along any rays that are not parallel to $\\mathbb {R}$ .", "For simplicity, we will take the imaginary axis.", "The $\\bar{\\partial }$ -problem is equivalent to the following Fredholm integral equation by a simple application of the generalized Cauchy integral formula: $E(z)=I-\\frac{1}{\\pi }\\int _{\\frac{E(s)W(s)}{s-z}dA(s).", "}In the following, we will show for each fixed z\\in , \\mathcal {K}_W(E)(z):=\\int _{\\frac{E(s)W(s)}{s-z}dA(s) is bounded and then by the dominated convergence theorem, we will show \\lim _{z\\rightarrow \\infty }z(E-I)=\\mathcal {O}(t^{-3/4}).", "First of all, since m^{[3]} is uniformly bounded, upon settng z=z_j+u+iv, we have{\\begin{@align}{1}{-1}\\Vert W\\Vert _{\\infty }\\lesssim {\\left\\lbrace \\begin{array}{ll}\|\\bar{\\partial }E_{j,k}\|e^{-2t\\theta ^{\\prime \\prime }(z_j)uv},z\\in \\Omega _{j,k},k=1,4,\\\\\|\\bar{\\partial }E_{j,k}\|e^{2t\\theta ^{\\prime \\prime }(z_j)uv},z\\in \\Omega _{j,k},k=3,6,\\\\\\end{array}\\right.", "},\\end{@align}}where 0\\le a\\lesssim b means there exists C>0 such that a\\le Cb.", "Then we have{\\begin{@align}{1}{-1}\\mathcal {K}_W(E)\\le \\Vert E\\Vert _\\infty \\int _{\\Vert W(s)\\Vert _\\infty }{\|s-z\|}dA(s).\\end{@align}}We claim the following lemma:\\begin{lemma}Let \\Omega =\\lbrace s:s=\\rho e^{i\\phi }, \\rho \\ge 0,\\phi \\in [0,\\pi /4]\\rbrace , and z\\in \\Omega .", "Then{\\begin{@align}{1}{-1}\\int _\\Omega \\frac{\|u^2+v^2\|^{-1/4}e^{-tuv}}{\|u+iv-z\|}dudv=\\mathcal {O}(t^{-1/4}).\\end{@align}}\\end{lemma}\\begin{proof}Since there are two singularities of the integrand at z and (0,0).", "In the first case, set z\\ne 0, and let d=dist(z,0).", "We split \\Omega into three parts: \\Omega _1\\cup \\Omega _2\\cup \\Omega _3, where \\Omega _1=\\lbrace s:\|s\|<d/3\\rbrace \\cap \\Omega , \\Omega _2=\\lbrace s:\|s-z\|<d/3\\rbrace \\cap \\Omega and \\Omega _3=\\Omega \\backslash (\\Omega _1\\cup \\Omega _2).", "In the region \\Omega _1, \|s-z\|\\ge 2d/3, and thus\\begin{equation}\\begin{split}\|\\int _{\\Omega _1}\\frac{\|u^2+v^2\|^{-1/4}e^{-tuv}}{\|u+iv-z\|}dudv\|&\\le \\frac{3}{2d}\\int _{0}^\\infty \\int _0^u\\frac{e^{-tuv}}{(u^2+v^2)^{1/4}}dvdu\\\\& \\text{ substituted $v=wu$}\\\\&\\le \\frac{3}{2d}\\int _0^\\infty \\int _0^1\\frac{e^{-tu^2w}}{(1+w^2)^{1/4}}u^{1/2}dwdu\\\\&\\le \\frac{3}{2d}\\int _0^\\infty \\int _0^1 e^{-tu^2w}u^{1/2}dwdu\\\\&=\\frac{3}{2d}\\int _0^\\infty \\frac{1-e^{-tu^2}}{tu^{3/2}}du\\\\&=\\frac{3}{2d}\\frac{1}{2}t^{-3/4}\\int _0^\\infty \\frac{1-e^{-u}}{u^{5/4}}du\\\\&=\\frac{3}{d}\\Gamma (3/4)t^{-3/4}.\\end{split}\\end{equation}In the region \\Omega _2, \|s\|^{-1/2}\\le (2d/3)^{-1/2}, we have{\\begin{@align}{1}{-1}\|\\int _{\\Omega _1}\\frac{\|u^2+v^2\|^{-1/4}e^{-tuv}}{\|u+iv-z\|}dudv\|&\\le \\sqrt{\\frac{3}{2d}}\\int _{\\Omega _2}\\frac{e^{-tuv}}{((u-x)^2+(v-y)^2)^{1/2}}dvdu\\\\&\\le \\sqrt{\\frac{3}{2d}}\\int _{0}^{d/3}\\int _0^{2\\pi }e^{-t(x+\\rho \\cos (\\theta ))(y+\\rho \\sin (\\theta ))}d\\theta d\\rho \\\\&\\le \\frac{2\\pi }{3}\\sqrt{\\frac{3d}{2}}e^{-txy}.\\end{@align}}While in the region \\Omega _3,{\\begin{@align}{1}{-1}\|\\int _{\\Omega _3}\\frac{\|u^2+v^2\|^{-1/4}e^{-tuv}}{\|u+iv-z\|}dudv\|&\\le \\int _{0}^{\\infty }\\int _0^u e^{-tuv}dvdu=\\mathcal {O}(t^{-1}).\\end{@align}}Now consider z=0.", "We have{\\begin{@align}{1}{-1}\|\\int _{\\Omega }\\frac{e^{-tuv}}{(u^2+v^2)^{3/4}}dA(u,v)\|&=\\int _{0}^\\infty \\int _0^u\\frac{e^{-tuv}}{(u^2+v^2)^{3/4}}dvdu\\\\&=\\int _0^\\infty \\int _0^1\\frac{e^{-tu^2w}}{(1+w^2)^{3/4}u^{1/2}}dwdu\\\\&\\le \\int _0^\\infty \\int _0^1\\frac{e^{-tu^2w}}{u^{1/2}}dwdu\\\\&=\\int _0^\\infty \\frac{1-e^{-tu^2}}{tu^{5/2}}du\\\\&=\\int _0^\\infty \\frac{1-e^{-u}}{tt^{-5/4}u^{5/4}}t^{-1/2}\\frac{1}{2}u^{-\\frac{1}{2}}du\\\\&=\\frac{1}{2}t^{-1/4}\\int _0^\\infty \\frac{1-e^{-u}}{u^{7/4}}du\\\\&=\\frac{3}{8}t^{-1/4}\\Gamma (1/4).\\end{@align}}By assembling all together, the proof is done.\\end{proof}\\begin{remark}The essential fact that makes the above true is the rapid decay of the exponential factor in the region.", "And the lemma also tells us that those mild singularities, which have rational order growth, can be absorbed by the exponential factor.", "Back to our situation, after some elementary transformations (translation and rotation), the estimation of \\int _{\\Vert W(s)\\Vert _\\infty }{\|s-z\|}dA(s) will eventually reduce to a similar situation discussed in the above lemma.\\end{remark}Based on Lemma \\ref {estimates of W}, we know that when t is sufficiently large, \\Vert \\mathcal {K}_W\\Vert <1 and thus the resolvent is uniformly bounded, and we obtain the following estimate by taking a standard Neumann series, for some sufficiently large t_0,{\\begin{@align}{1}{-1}\\Vert E-I\\Vert _\\infty =\\Vert \\mathcal {K}_W(1-\\mathcal {K}_W)^{-1} I\\Vert _\\infty \\le \\frac{ct^{-1/4}}{1-ct^{-1/4}}\\le ct^{-1/4}, \\quad t>t_0.\\end{@align}}Now since for each z\\in \\Omega _{j,k}, we have \|\\bar{\\partial }E_{j,k}(z)\|\\le c(\|z-z_j\|^{-1/2}+\|R^{\\prime }(u+z_j)\|), and apply the dominated convergence theorem, we have{\\begin{@align}{1}{-1}\\lim _{z\\rightarrow \\infty }\|z(E-I)\|&\\le \\frac{1}{\\pi }\\sum _{j=1}^{l}\\sum _{k=1}^4\\Vert E\\Vert _{L^\\infty }\\int _{\\Omega _{j,k}}\\Vert W\\Vert _\\infty ds,\\end{@align}}and use the Lemma \\ref {estimates of W} again, we will eventually have:{\\begin{@align}{1}{-1}E_1=\\lim _{z\\rightarrow \\infty }\|z(E-I)\|=\\mathcal {O}(t^{-3/4}).\\end{@align}}}\\section {Asymptotics representation}First, we summarize all the steps as following (see Fig.\\ref {fig:steps}):\\begin{enumerate}[label={(0)}]\\item Initial RHP m^{[0]}=m, see RHP \\ref {RHP m 0}.\\item Conjugate initial RHP to obtain m^{[1]}=m^{[0]}\\delta ^{\\sigma _3}, see RHP \\ref {RHP m 1}.\\item Open lenses to obtain a mixed \\bar{\\partial }-RHP \\ref {dbar RHP}.\\item Approximate the RHP part m^{[2]}_{RHP} of m^{[2]} by removing \\Sigma _{j+\\frac{1}{2}} (see RHP \\ref {RHP m tilde}), localization (see RHP \\ref {RHP m hat}), reducing the phase function and separating the contributions (see RHP \\ref {RHP m 3}).", "The error term is \\mathcal {O}(t^{-1}).", "Note those exponential decaying errors are absorbed by \\mathcal {O}(t^{-1}).\\item Comparing m^{[2]} and m^{[2]}_{RHP} and computing the error by analysis a pure \\bar{\\partial }-problem.", "The error term is \\mathcal {O}(t^{-3/4}).\\end{enumerate}$ Figure: Steps of the ∂ ¯\\bar{\\partial }-steepest method.Now by undoing all the steps, we arrive at: $m^{[0]}(z)=E(z)m^{[2]}_{RHP}(z)O^{-1}(z)\\delta ^{-\\sigma _3}.$ Since $O(z)$ uniformly converges to $I$ as $z\\rightarrow \\infty $ , and $\\delta ^{-\\sigma _3}$ is a diagonal matrix, they do not affect the recovering of the potential.", "Thus we obtain $q(x,t)&=-2i(m^{[2]}_{RHP,1,12}+E_{1,12})\\\\&=q_{RHP}(x,t)-2iE_{1,12}\\\\& \\text{ by (\\ref {q rhp to q o rhp}),(\\ref {q o rhp to q as})}\\\\&=q_{as}(x,t)+\\mathcal {O}(t^{-1})-2iE_{1,12}\\\\& \\text{ by (\\ref {dbar error})}\\\\&=q_{as}(x,t)+\\mathcal {O}(t^{-1})+\\mathcal {O}(t^{-3/4})\\\\&=q_{as}(x,t)+\\mathcal {O}(t^{-3/4}),$ where $q_{as}(x,t)$ is given by equation (REF ).", "Remark 8.1 Note that due to the analysis in the section 7, according to Proposition 2.6 and Proposition 2.11 of [15], together with the small norm theory, the existence and uniqueness of the model RHP implies, via the estimates of the corresponding Beals-Coifman operators, the existence and uniqueness of RHP REF .", "Similarly, we obtain the existence and uniqueness of $\\tilde{m}^{[2]}$ , $\\hat{m}^{[2]}$ and eventually $m^{[2]}_{RHP}$ .", "Remark 8.2 From equation (REF ), we know $q_{as}$ is $\\mathcal {O}(t^{-1/2})$ as $t\\rightarrow \\infty $ in the region $x<0$ and consider the limit along the ray $x=-ct$ for some positive constant $c$ ." ], [ "Fast decaying region", "In this section and the next section, we will focus only on the case of the defocusing mKdV flow.", "In this case, the phase function reads $\\theta (z;x,t)=\\frac{x}{t}z+cz^n,\\quad n \\text{ is an odd positive integer.", "}$ In the previous sections, we have derived the asymptotic solutions to the defocusing mKdV flow in the oscillating region, namely, along the ray $x=-\\nu t,\\ \\nu >0,\\ t\\rightarrow \\infty $ .", "In this section, we consider the long-time behavior along the ray $x=\\nu t,\\ \\nu >0,\\ t\\rightarrow \\infty $ , which we call it the fast decaying region as we will soon prove in this region, the solution decay like $\\mathcal {O}(t^{-1})$ , which is faster than the leading term in the oscillating region, i.e., $\\mathcal {O}(t^{-1/2})$ , as $t\\rightarrow \\infty $ .", "In the fast decaying region, the phase function enjoys the following properties: There exits $\\epsilon =\\epsilon (n,\\nu )>0$ such that $\\pm \\operatorname{Im}(\\theta )>0$ in the strips $\\lbrace z:\\pm \\operatorname{Im}(z)\\in (0,\\epsilon )\\rbrace $ , respectively.", "There exits $M\\in (0,1/\\epsilon )$ such that $\\operatorname{Im}(\\theta ) \\ge nvu^{n-1}$ for $\|u\|\\ge M\\epsilon $ and $\\operatorname{Im}(\\theta )\\ge v(1-(M\\epsilon )^2)$ for $\|u\|\\le M\\epsilon $ .", "Here $z=u+iv$ .", "First we will formulate the RHP as follows: Riemann-Hilbert problem 9.1 Given $R(z)\\in H^{1,1}(\\mathbb {R})$ , looking for a 2 by 2 matrix-value function $m$ such that $m_+=m_-e^{-it\\theta (z)\\text{ ad }{\\sigma _3}}v(z),z\\in \\mathbb {R},$ where the jump matrix is given by $v(z)=\\begin{pmatrix}1-\|R\|^2& -\\bar{R}\\\\R & 1\\\\\\end{pmatrix}=\\begin{pmatrix}1 & -\\bar{R}\\\\0& 1\\\\\\end{pmatrix}\\begin{pmatrix}1 & 0\\\\R & 1\\\\\\end{pmatrix};$ $m=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty $ .", "Theorem 9.2 For the above RHP, the solution $m$ enjoys the following asymptotics as $t\\rightarrow \\infty $ : $m_1(t)=\\mathcal {O}(t^{-1}).$ where $m=I+m_1(t)/z+\\mathcal {O}(z^{-2}),\\ z\\rightarrow \\infty $ .", "Figure: ∂ ¯\\bar{\\partial }-extension for the case of the fast decaying region.", "Here we only draw the case when n=5n=5.", "For generic odd nn, there are n-1 2\\frac{n-1}{2} curves of Imθ=0\\operatorname{Im}{\\theta }=0 in the upper and in the lower half plane.In the light of $\\bar{\\partial }$ -steepest descent, to open the lens, we multiple a smooth function $O(z)$ to $m$ , where $O(z)$ is given by $O(z)={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & 0\\\\\\frac{{-R}(\\operatorname{Re}{z})e^{2it\\theta (z)}}{1+(\\operatorname{Im}{z})^2} & 1\\\\\\end{pmatrix}, \\quad z\\in {\\Omega }_1,\\\\\\begin{pmatrix}1 & \\frac{\\bar{R}(\\operatorname{Re}{z})e^{-2it\\theta (z)}}{1+(\\operatorname{Im}{z})^2}\\\\0 &1\\\\\\end{pmatrix},\\quad z\\in \\Omega ^_{1},\\\\I,\\quad z \\in (\\Omega _1\\cup \\Omega ^_1),\\end{array}\\right.", "}$ where (see Fig.REF ) $\\Omega _1&=\\lbrace z:\\operatorname{Im}{z}\\in (0,\\epsilon ) \\rbrace ,\\\\\\Omega ^_1&=\\lbrace z:\\operatorname{Im}{z}\\in (-\\epsilon ,0) \\rbrace .$ Let us denote $\\Sigma _1=\\lbrace z:\\operatorname{Im}{z}=\\epsilon \\rbrace $ , see Fig.REF , and let $\\tilde{m}=mO,\\quad z\\in $ Now as usual, we obtain a $\\bar{\\partial }$ -RHP, due to the exponential decaying of the off-diagonal term, and the jump matrix of the RHP part will approach $I$ .", "Hence by a small norm argument, we know the solution will close to $I$ as $z\\rightarrow \\infty $ .", "Denote the solution to the pure RHP by $m^\\sharp $ , and small norm theory leads to $m^\\sharp =I+\\mathcal {O}(e^{-c(\\epsilon )t}),c(\\epsilon )>0,z\\rightarrow \\infty $ .", "Next, consider $E=\\tilde{m}(m^\\sharp )^{-1}.$ By direct computation one can show $E$ doesn't have any jump on $\\Sigma _1$ and it satisfies a pure $\\bar{\\partial }$ -problem: $\\bar{\\partial }E=EW,$ where $W={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}0 & m^{\\sharp }e^{-2it\\theta (z)}\\bar{\\partial }(\\frac{\\bar{R}(\\operatorname{Re}{z})}{1+(\\operatorname{Im}{z})^2}) (m^\\sharp )^{-1} \\\\0 & 0\\end{pmatrix},\\quad z\\in \\Omega _1,\\\\\\begin{pmatrix}0 & 0\\\\m^{\\sharp }e^{2it\\theta (z)}\\bar{\\partial }(\\frac{-{R}(\\operatorname{Re}{z})}{1+(\\operatorname{Im}{z})^2}) (m^\\sharp )^{-1} & 0\\end{pmatrix},\\quad z\\in \\Omega _1^,\\\\0,\\quad z\\in (\\Omega _1\\cup \\Omega _1^),\\end{array}\\right.", "}$ where $\\bar{\\partial }=\\frac{1}{2}(\\partial _{\\operatorname{Re}{z}}+i\\partial _{\\operatorname{Im}{z}})$ .", "Since $R,\\bar{R}\\in H^{1,1}$ , $\\bar{\\partial }(\\frac{\\bar{R}(\\operatorname{Re}{z})}{1+(\\operatorname{Im}{z})^2}),\\bar{\\partial }(\\frac{-{R}(\\operatorname{Re}{z})}{1+(\\operatorname{Im}{z})^2})$ are uniformly bounded by some non-negative $L^2(\\mathbb {R})$ function $f(\\operatorname{Re}{z})$ .", "Note that $m^\\sharp $ is uniformly close to $I$ , and setting $z=u+iv$ , and considering $z\\in \\Omega _1$ first, we have $\\Vert W\\Vert _\\infty \\le f(u)e^{-t\\operatorname{Im}{\\theta (u,v)}}, \\forall u\\in \\mathbb {R}, v\\in (0,\\epsilon ).$ By the same procure as the one in section 8, the error of approximating $m$ by the identity matrix is given by the following integral (since there is only one non-trivial entry of $W$ ): $\\Delta :=\\int _0^\\epsilon \\int _\\mathbb {R}f(u)e^{-t\\operatorname{Im}{\\theta }}dudv.$ Split the $u$ into two regions: (1) $\|u\|\\le M\\epsilon $ , (2) $\|u\|\\ge M\\epsilon $ .", "And denote them by $\\Delta _1$ , $\\Delta _2$ respectively.", "Then $\\Delta =\\Delta _1+\\Delta _2$ .", "And $\\Delta _1&\\le \\int _0^\\epsilon \\int _{-M\\epsilon }^{M\\epsilon }f(u)e^{-tv(1-M^2\\epsilon ^2)}dudv\\\\&\\text{by Cauchy-Schwartz }\\\\&\\le \\Vert f\\Vert _{L^2(\\mathbb {R})}(2M\\epsilon )^{1/2}\\frac{1-e^{-t\\epsilon (1-M^2\\epsilon ^2)}}{t(1-M^2\\epsilon ^2)}\\\\&=\\mathcal {O}(t^{-1}).$ On the other hand, $\\Delta _2&\\le \\int _0^\\epsilon \\int _{\|u\|\\ge M\\epsilon }f(u)e^{-ntvu^{n-1}}dudv\\\\&=\\int _{\|u\|\\ge M\\epsilon } f(u)\\int _0^\\epsilon e^{-ntvu^{n-1}}dvdu\\\\&\\le t^{-1}\\Vert f\\Vert _{L^2}(\\int _{\|u\|\\ge M\\epsilon }(\\frac{1-e^{-ntvu^{n-1}}}{nu^{n-1}})^2du)^{1/2}\\\\&\\le t^{-1}\\Vert f\\Vert _{L^2} \\frac{n}{n-2}(M\\epsilon )^{-(n-2)}\\\\&=\\mathcal {O}(t^{-1}).$ Similarly, we can prove that for $z\\in \\Omega _1^$ , we also have the error estimate $\\mathcal {O}(t^{-1})$ .", "Assembling all together, we conclude that the error term is $\\mathcal {O}(t^{-1})$ , and $m_1=\\mathcal {O}(t^{-1})$ , as $ t\\rightarrow \\infty .$" ], [ "Painlevé region", "In this section, we first derive the Painlevé II hierarchy based on some RHP.", "Then, we will connect the long-time behavior of the mKdV hierarchy in the so-called Painlevé region to solutions of the Painlevé II hierarchy." ], [ "Painlevé II hierarchy", "As mentioned in [2], the mKdV equation is can be transferred to the Painlevé II equation.", "The authors in [2] also suggest the connection between integrable PDEs with Painlevé equations.", "In [10], the authors explicitly derived the Painlevé II hierarchy from self-symmetry reduction of the mKdV hierarchy (see page 59 of [10].", "And also [11]).", "In this section, we will provide a slight different (as comparing to [11]) algorithm based on Riemann-Hilbert problems to generate the Painlevé II hierarchy.", "Let's denote $\\Theta (x,z)=xz+\\frac{c}{n}z^n$ , and suppose $Y$ solves the following RHP: $Y_+&=Y_-e^{i\\Theta \\sigma _3}v_0e^{-i\\Theta \\sigma _3},\\quad z\\in \\Sigma _n,\\\\Y&=I+\\mathcal {O}(z^{-1}),\\quad z\\rightarrow \\infty .$ where the contour $\\Sigma _n$ consists of all stokes lines $\\lbrace z:\\operatorname{Im}{\\Theta (z)}=0\\rbrace $ and $v_0$ is a constant 2 by 2 matrix that is independent of $x,z$ .", "Now let $\\tilde{Y}=Ye^{i\\Theta \\sigma _3}$ , and we arrive at a new RHP: $\\tilde{Y}_+&=\\tilde{Y}_-v_0,\\quad z\\in \\Sigma _n,\\\\\\tilde{Y}&=(I+\\mathcal {O}(z^{-1}))e^{i\\Theta \\sigma _3},\\quad z\\rightarrow \\infty .$ Since $v_0$ is constant, it is easily to check, by Louisville's argument, that both $\\partial _z\\tilde{Y}\\tilde{Y}^{-1}$ and $\\partial _x\\tilde{Y}\\tilde{Y}^{-1}$ are polynomial of $z$ .", "Hence we obtain the following two differential equations: $\\partial _x\\tilde{Y}\\tilde{Y}^{-1}&=A(x,z),\\\\\\partial _z\\tilde{Y}\\tilde{Y}^{-1}&=B(x,z).$ If we assume $Y&=I+\\sum _{j=1}^{n-1}{Y_j(x)z^{-j}}+\\mathcal {O}(z^{-n}),\\quad z\\rightarrow \\infty ,\\\\\\underline{Y}&=Y^{-1}=I+\\sum _{j=1}^{n-1}{\\underline{Y}_j(x)z^{-j}}+\\mathcal {O}(z^{-n}),\\quad z\\rightarrow \\infty ,$ then a direct computation shows $A&=i[Y_1,\\sigma _3]+iz\\sigma _3,\\\\B&=ix\\sigma _3+icz^{n-1}\\sigma _3+icz^{n-2}[Y_1,\\sigma _3]\\\\&+\\sum _{k=2}^{n-1}icz^{n-1-k}(Y_k\\sigma _3+\\sigma _3\\underline{Y}_k+\\sum _{j=1}^{k-1}Y_{k-j}\\sigma _3\\underline{Y}_j).$ Since $Y_{x,z}=Y_{z,x}$ , we have $A_z-B_x+[A,B]=0.$ Set $Y_j=\\begin{pmatrix}p_j(x) & u_j(x)\\\\v_j(x)& q_j(x)\\end{pmatrix},\\quad j=1,..,n-2,$ where $p_j,q_j,u_j,v_j$ are smooth functions of $x$ .", "To guarantee (REF ), all the coefficients of $z$ must vanish.", "Those equations can be solved recursively.", "Eventually, by eliminating $u_j,v_j, j=2,..,n-2$ , and let $v_1=u_1$ , we will arrive at a nonlinear ODE of $u_1$Surprisingly, the dependence on $p_j,q_j$ will disappear., which turns out to be a member of the hierarchy of Painlevé II equations .", "We list the first few of them: $n=3&: -8cu^3+cu_{xx}-4xu=0,\\\\n=5&:-24cu^5+10cu^2u_{xx}+10cuu_x^2-\\frac{c}{4}u_{xxxx}-4xu=0,\\\\n=7&: -80cu^7+70cu^4u_{xx}+140cu^3u_x^2-\\frac{7cu^2u_{xxxx}}{2}\\\\\\quad \\quad & +(-\\frac{21}{2}cu^2_{xx}-14cu_xu_{xxx}-4x)u+\\frac{c}{16}u_{xxxxxx}-\\frac{35}{2}cu_x^2u_{xx}=0.$ In the current article, we focus only on the odd members.", "In fact, $n=3$ corresponds to the mKdV equation, $n=5$ corresponds to the 5th order mKdV, and so on.", "In the following subsection, we will show how to connect the long-time asymptotics behavior of the mKdV hierarchy to the solutions to the Painlevé II hierarchy." ], [ "Painlevé Region", "Recall the phase functions of the AKNS hierarchy of mKdV type equations are $\\theta (z;x,t)=xz+ctz^n, \\quad n\\, \\text{is odd}.$ By the Painlevé region we mean a collection of all the curves $x=s(nt)^{1/n},s\\ne 0$ , by rescaling $z\\rightarrow (nt)^{-\\frac{1}{n}}\\xi $ , we have $\\Theta (\\xi )=s\\xi +\\frac{c}{n}\\xi ^n.$ Now the modulus of the stationary phase points of (REF ) is $\|z_0\|=\\left\|-\\frac{x}{ct}\\right\|^{\\frac{1}{n-1}}=\\mathcal {O}(t^{-\\frac{1}{n}}),$ and however, after scaling, the modulus of the stationary phase points of $\\Theta (\\xi )$ is $\\xi _0=\|z_0\|t^{\\frac{1}{n}},$ which is fixed as $t\\rightarrow \\infty $ .", "A direct computation shows for any odd $n$ , one can always perform lens-opening to the rays $\\lbrace z\\in \\mathbb {R}:\|z\|>\|\\xi _0\|\\rbrace $ , due to the signature of $\\operatorname{Re}(i\\theta )$ , see Fig.REF .", "Figure: Signature of Re(iθ)\\operatorname{Re}(i\\theta ).", "The green region: Re(iθ)>0\\operatorname{Re}(i\\theta )>0 when x<0x<0; The red region: Re(iθ)>0\\operatorname{Re}(i\\theta )>0 when x>0x>0; The yellow region: the overlapping region of red and green; The white region: Re(iθ)<0\\operatorname{Re}(i\\theta )<0.", "Here we only plot the signatures of Re(iθ)\\operatorname{Re}{(i\\theta )} for n=9n=9.", "Other odd nn, the region plot looks very similar.Note that $e^{-i\\theta (z)\\text{ ad }\\sigma _3}v(z)&=e^{-i\\Theta (\\xi )\\text{ ad }\\sigma _3}v(\\xi )\\\\&=\\begin{pmatrix}1-\|R\|^2 & -\\bar{R}e^{-2i\\Theta }\\\\Re^{2i\\Theta } & 1\\end{pmatrix}\\\\&=\\begin{pmatrix}1 & -\\bar{R}e^{-2i\\Theta }\\\\0 & 1\\end{pmatrix}\\begin{pmatrix}1 & 0\\\\Re^{2i\\Theta } & 1\\end{pmatrix}.$ We can deform the the contour $\\lbrace z\\in \\mathbb {R}:\|z\|>\|\\xi _0\|\\rbrace $ as before and get the deformed contour as follows (see Fig.REF ): Fix a positive constant $\\alpha <\\frac{\\pi }{n}$Such a choice of $\\alpha $ guarantees that the new contours will stay within the regions where the corresponding exponential term will decay (considering Fig.REF )., $\\Sigma _0&=\\lbrace z\\in \\mathbb {R}:-\\xi _0\\le z\\le \\xi _0\\rbrace ,\\\\\\Sigma _1&=\\lbrace z:z=\\xi _0+\\rho e^{i\\alpha },\\rho \\in (0,\\infty )\\rbrace ,\\\\\\Sigma _2&=\\lbrace z:z=-\\xi _0+\\rho e^{-i\\alpha },\\rho \\in (-\\infty ,0)\\rbrace ,\\\\\\Sigma _3&=\\lbrace z:z=-\\xi _0+\\rho e^{i\\alpha },\\rho \\in (-\\infty ,0)\\rbrace ,\\\\\\Sigma _4&=\\lbrace z:z=\\xi _0+\\rho e^{-i\\alpha },\\rho \\in (0,\\infty )\\rbrace ,$ and we define the regions as follows: $\\Omega _1&=\\lbrace z:z=\\xi _0+\\rho e^{i\\phi },\\rho \\in (0,\\infty ),\\phi \\in (0,\\alpha )\\rbrace ,\\\\\\Omega _2&=+\\backslash (\\Omega _1\\cup \\Omega _3),\\\\\\Omega _3&=\\lbrace z:z=-\\xi _0+\\rho e^{-i\\phi },\\rho \\in (-\\infty ,0),\\phi \\in (-\\alpha ,0)\\rbrace ,\\\\\\Omega _4&=\\lbrace z:z=-\\xi _0+\\rho e^{i\\phi },\\rho \\in (-\\infty ,0),\\phi \\in (0,\\alpha )\\rbrace ,\\\\\\Omega _5&={-}\\backslash (\\Omega _4\\cup \\Omega _6),\\\\\\Omega _6&=\\lbrace z:z=\\xi _0+\\rho e^{i\\phi },\\rho \\in (0,\\infty ),\\phi \\in (-\\alpha ,0)\\rbrace .$ Figure: Contour for ∂ ¯\\bar{\\partial }-RHP.As before, set the original RHP as $m^{[1]}$ with jump $e^{-i\\theta (z)\\text{ ad }\\sigma _3}v(z)$ .", "After re-scaling and $\\bar{\\partial }$ -lenses opening, we set $m^{[2]}(\\xi )=m^{[1]}O(\\gamma )$ , where the lenses opening matrix is $O(\\gamma )={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & 0\\\\-E_+e^{2i\\Theta (\\gamma )} & 1\\end{pmatrix},\\quad \\gamma \\in \\Omega _1\\cup \\Omega _3,\\\\\\begin{pmatrix}1 & -E_-e^{-2i\\Theta (\\gamma )}\\\\0 & 1\\end{pmatrix},\\quad \\gamma \\in \\Omega _4\\cup \\Omega _6,\\\\I,\\quad \\gamma \\in \\Omega _2\\cup \\Omega _5,\\end{array}\\right.", "}$ where $E_+(\\gamma )&=\\mathcal {K}(\\phi )R\\left((nt)^{-\\frac{1}{n}}\\xi \\right)+(1-\\mathcal {K}(\\phi ))R(\\tilde{\\xi }_0(nt)^{-\\frac{1}{n}}),\\\\E_-&(\\gamma )=\\overline{E_+(\\gamma )},\\\\\\gamma &={\\left\\lbrace \\begin{array}{ll}\\xi _0+\\rho e^{i\\phi },\\quad \\text{if }\\gamma \\in \\Omega _1\\cup \\Omega _6,\\\\-\\xi _0+\\rho e^{i\\phi },\\quad \\text{if }\\gamma \\in \\Omega _3\\cup \\Omega _4,\\end{array}\\right.}", "\\\\\\xi &=\\operatorname{Re}(\\gamma ),\\\\\\tilde{\\xi }_0&={\\left\\lbrace \\begin{array}{ll}\\xi _0,\\quad \\text{if }\\gamma \\in \\Omega _1\\cup \\Omega _6,\\\\-\\xi _0,\\quad \\text{if }\\gamma \\in \\Omega _3\\cup \\Omega _4.\\end{array}\\right.", "}$ Now we arrive at the following $\\bar{\\partial }$ -RHP: Mixed $\\bar{\\partial }$ -Riemann-Hilbert problem 10.1 Looking for a 2 by 2 matrix-valued function $m^{[2]}$ such that The RHP: (1.a) $m^{[2]}(\\gamma )\\in C^1(\\mathbb {R}^2\\backslash \\Sigma )$ and $m^{[2]}(z)=I+\\mathcal {O}(\\gamma ^{-1}),\\gamma \\rightarrow \\infty $ ; (1.b) the jumps on $\\Sigma _1$ and $\\Sigma _2$ are $e^{-i\\Theta (\\xi )\\text{ ad }\\sigma _3}v_+$ , and the jumps on $\\Sigma _3$ and $\\Sigma _4$ are $e^{-i\\Theta (\\xi )\\text{ ad }\\sigma _3}v_-$ , where $v_=\\begin{pmatrix}1 & \\bar{R}\\\\0 & 1\\end{pmatrix},\\quad v_+=\\begin{pmatrix}1 & 0\\\\R & 1\\end{pmatrix}.$ The jump on $\\Sigma _0$ is $e^{-i\\Theta \\text{ ad }\\sigma _3}v((nt)^{-\\frac{1}{n}}\\xi ),$ , and the jumps on $\\lbrace z\\in \\mathbb {R}:\|z\|>\|\\xi _0\|\\rbrace $ is $I$ .", "The $\\bar{\\partial }$ -problem: For $z\\in , we have{\\begin{@align}{1}{-1}\\bar{\\partial }m^{[2]}(\\xi )=m^{[2]}(\\xi )\\bar{\\partial }O(\\xi ).\\end{@align}}$ Again, we will need the following lemma in order to estimate errors from the $\\bar{\\partial }$ -problem.", "Lemma 10.2 For $\\gamma \\in \\Omega _{1,3,4,6}$ , $\\xi =\\operatorname{Re}\\gamma $ , $\|\\bar{\\partial }E_{\\pm }(\\gamma )\|\\le (nt)^{-\\frac{1}{n}}\|(nt)^{-\\frac{1}{n}}(\\xi -\\xi _0)\|^{-\\frac{1}{2}}\\Vert R\\Vert _{H^{1,0}}+(nt)^{-\\frac{1}{n}}\|R^{\\prime }((nt)^{-\\frac{1}{n}}\\xi )\|.$ For brevity, we only prove for the region $\\Omega _1$ .", "Using the polar coordinates, we have $\|\\bar{\\partial }E_+(\\gamma )\|&=\\left\|\\frac{ie^{i\\phi }}{2\\rho }\\mathcal {K}^{\\prime }(\\phi )\\left[R\\left((nt)^{-\\frac{1}{n}}\\xi \\right)-R(\\xi _0(nt)^{-\\frac{1}{n}})\\right]+\\mathcal {K}(\\phi )R^{\\prime }\\left((nt)^{-\\frac{1}{n}}\\xi \\right)(nt)^{-\\frac{1}{n}}\\right\| \\\\&\\text{by Cauchy-Schwartz inequality}\\\\&\\le \\left\|\\frac{\\Vert R\\Vert _{H^{1,0}}\|(nt)^{-\\frac{1}{n}}\\xi -\\xi _0(nt)^{-\\frac{1}{n}}\|^{1/2}}{\\gamma -\\xi _0}\\right\|+(nt)^{-\\frac{1}{n}}\\left\|R^{\\prime }\\left((nt)^{-\\frac{1}{n}}\\xi \\right)\\right\|\\\\&\\le (nt)^{-\\frac{1}{n}}\|(nt)^{-\\frac{1}{n}}(\\xi -\\xi _0)\|^{-\\frac{1}{2}}\\Vert R\\Vert _{H^{1,0}}+(nt)^{-\\frac{1}{n}}\|R^{\\prime }((nt)^{-\\frac{1}{n}}\\xi )\|.$ Similarly, we can prove for other regions.", "Next, consider a pure RHP $m^{[3]}$ which satisfies exactly the RHP part of $\\bar{\\partial }$ -RHP($m^{[2]}$ ).", "$m^{[3]}$ can be approximated by the RHP corresponding to a special solution of the Painlevé II hierarchyAs for the existence of the RHP $m^{[3]}$ , which is not completely trivial due to the fact that solutions to the Painlevé II equations have poles, we refer the readers to the book[20] for the details.", "Since for $\\gamma \\in \\Omega _1$ , $&\\left\|\\left(R(\\xi (nt)^{-\\frac{1}{n}})-R(0)\\right)e^{2i\\Theta (\\gamma )}\\right\|\\\\&\\le \|\\xi (nt)^{-\\frac{2}{n}}\|^{\\frac{1}{2}}\\Vert R\\Vert _{H^{1,0}}e^{2\\operatorname{Re}{i\\Theta (\\gamma )}}\\\\&\\le (nt)^{-\\frac{1}{n}}\|\\operatorname{Re}\\gamma \|^{\\frac{1}{2}}\\Vert R\\Vert _{H^{1,0}}e^{2\\operatorname{Re}{i\\Theta (\\gamma )}},$ it is evident that $\\Vert Re^{2i\\Theta }-R(0)e^{2i\\Theta }\\Vert _{L^{\\infty }\\cap L^1\\cap L^2}\\le c(nt)^{-\\frac{1}{n}}.$ Let $m^{[4]}$ solves the RHP formed by replacing $R(\\pm \\xi _0(nt)^{-1/n})$ and its complex conjugate in the jumps of $m^{[3]}$ along $\\Sigma _k,k=1,2,3,4$ by $R(0)$ and $\\bar{R}(0)$ respectively.", "Then, by the small norm theory, the errors between the corresponding potential is given by $error_{3,4}&=\\lim _{\\gamma \\rightarrow \\infty }\|\\gamma (m^{[4]}_{12}-m^{[3]}_{12})\|\\\\&\\le c \\int _{\\Sigma } \|(R(\\operatorname{Re}(s) (nt)^{-\\frac{1}{n}})-R(0))e^{2i\\Theta (s)}\|ds\\\\&\\le c(nt)^{-\\frac{1}{n}}.$ Then since now the jumps are all analytic, we can perform an analytic deformation and arrive at the green contours as show in Fig.REF .", "Let's denote the new RHP by $m^{[5]}(\\gamma )$ , and we arrive at the following RHP: Riemann-Hilbert problem 10.3 Looking for a 2 by 2 matrix-valued function $m^{[5]}$ such that $m^{[5]}$ is analytic off the contours $\\cup _{k=1,2,3,4}\\Sigma _{k}^{[5]}$ ; $m^{[5]}_+=m^{[5]}_-v^{[5]},\\quad z\\in \\cup _{k=1,2,3,4}\\Sigma _{k}^{[5]},$ where $v^{[5]}={\\left\\lbrace \\begin{array}{ll}\\begin{pmatrix}1 & 0\\\\R(0)e^{2i\\Theta (\\gamma )}&1\\end{pmatrix},\\quad \\gamma \\in \\Sigma _{1}^{[5]}\\cup \\Sigma _2^{[5]},\\\\\\begin{pmatrix}1 &\\bar{R}(0)e^{-2i\\Theta (\\gamma )}\\\\0&1\\end{pmatrix},\\quad \\gamma \\in \\Sigma _{3}^{[5]}\\cup \\Sigma _4^{[5]}.\\end{array}\\right.", "}$ Here the new contours (see Fig.REF ) are $\\Sigma _1^{[5]}&=\\lbrace z:z=\\rho e^{i\\alpha },\\rho \\in (0,\\infty )\\rbrace ,\\\\\\Sigma _2^{[5]}&=\\lbrace z:z=\\rho e^{-i\\alpha },\\rho \\in (-\\infty ,0)\\rbrace ,\\\\\\Sigma _3^{[5]}&=\\lbrace z:z=\\rho e^{i\\alpha },\\rho \\in (-\\infty ,0)\\rbrace ,\\\\\\Sigma _4^{[5]}&=\\lbrace z:z=\\rho e^{-i\\alpha },\\rho \\in (0,\\infty )\\rbrace .$ Figure: Contour for m [4] m^{[4]}(Green part).Then according to the previous subsection, the $(1,2)$ entry of the solution $m^{[5]}$ , similarly the solution $m^{[4]}$ , is the solution to the Painlevé II hierarchy, i.e., $m^{[4]}_{12}(\\gamma )=m^{[5]}_{12}(\\gamma ),$ Hence we have $P^{II}_k(s)=\\lim _{\\gamma \\rightarrow \\infty }\\gamma m^{[5]}_{12}$ where $P^{II}_k$ solves the $k^{th}$ equation in the Painlevé II hierarchy, where $k=\\frac{n-1}{2}$ .", "Now let's consider the error generated from the $\\bar{\\partial }$ -extension.", "Recall that the error $E$ satisfies a pure $\\bar{\\partial }$ -problem: $\\bar{\\partial }E&=EW,\\\\W&=m^{[3]}\\bar{\\partial }O(m^{[3]})^{-1}.$ As before, the $\\bar{\\partial }$ -equation is equivalent to an integral equation which reads $E(z)=I+\\frac{1}{\\pi }\\int _{\\frac{E(s)W(s)}{z-s}\\text{d}A(s)=I+\\mathcal {K}(E).", "}As before, we can show that the resolvent always exists for large t. So we only need to estimate the true error which is: \\lim _{z\\rightarrow \\infty }z(E-I).", "In fact, we have{\\begin{@align}{1}{-1}\\lim _{z\\rightarrow \\infty }\|z(E-I)\|&=\|\\int _\\mathbb {C}EWds\|\\\\&\\le c\\Vert E\\Vert _\\infty \\int _{\\Omega }\|\\bar{\\partial }O\|ds.\\end{@align}}For the sake of simplicity, we only estimate the integral on the right hand side in the region of the top right corner.", "Note there is only one entry which is nonzero in \\bar{\\partial }O, which is one of the E_{\\pm } and we split the integral into two parts in the obvious way, i.e.,{\\begin{@align}{1}{-1}\\int _\\Omega \|\\bar{\\partial }O\|ds&\\le I_1+I_2\\\\&=\\int _\\Omega (nt)^{-\\frac{1}{2n}}\|\\operatorname{Re}{s}-\\xi _0\|\\Vert R\\Vert _{H^{1,0}}e^{2\\operatorname{Re}{i\\Theta (s)}}ds\\\\&+\\int _{\\Omega }(nt)^{-\\frac{1}{nt}}\|R^{\\prime }((nt)^{-\\frac{1}{n}}s)\|e^{2\\operatorname{Re}{i\\Theta (s)}}ds.\\end{@align}}As we know from previous sections, e^{\\operatorname{Re}{2i\\Theta (s)}}\\le ce^{-2\|\\Theta ^{\\prime \\prime }(\\xi _0)\|uv} in the region \\lbrace z=u+iv:u>\\xi _0,0<v<\\alpha u\\rbrace for some small \\alpha , where s=u+iv+\\xi _0.", "Then we have{\\begin{@align}{1}{-1}I_1&\\le (nt)^{-\\frac{1}{2n}}\\int _{\\Omega }\|\\operatorname{Re}{s}-\\xi _0\|^{-1/2}e^{-cuv}dudv\\\\&\\le (nt)^{-\\frac{1}{2n}}\\int _0^\\infty \\int _0^{\\alpha u}u^{-1/2}e^{-cuv}dudv\\\\&\\le C(nt)^{-\\frac{1}{2n}}\\int _0^\\infty \\frac{1-e^{-2\\alpha \|\\Theta ^{\\prime \\prime }(\\xi _0)\|} }{u^{3/2}}du\\\\&=\\mathcal {O}\\left((nt)^{-\\frac{1}{2n}}\\right),\\end{@align}}and{\\begin{@align}{1}{-1}I_2&\\le (nt)^{-\\frac{1}{n}}\\int \|R^{\\prime }((nt)^{-\\frac{1}{2n}}\\operatorname{Re}{s})\|e^{-cuv}dudv\\\\&\\text{ by Cauchy-Schwartz inequality}\\\\&\\le (nt)^{-\\frac{1}{n}}\\Vert R\\Vert _{H^{1,0}}\\int _0^\\infty (\\int _{\\alpha v}^\\infty e^{-2cuv}du)^{1/2}dv\\\\&\\le (nt)^{-\\frac{1}{n}}\\Vert R\\Vert _{H^{1,0}}\\int _0^\\infty \\frac{e^{-c\\alpha v^2}}{\\sqrt{2\\alpha cv}}dc\\\\&=\\mathcal {O}((nt)^{-\\frac{1}{n}}).\\end{@align}}Thus, we arrive at{\\begin{@align}{1}{-1}\\bar{\\partial }\\text{Error} = \\mathcal {O}((nt)^{-\\frac{1}{2n}}).\\end{@align}}And we undo all the deformations, we obtain{\\begin{@align}{1}{-1}m^{[1]}((nt)^{-\\frac{1}{n}}\\gamma )&=m^{[2]}(\\gamma )O^{-1}(\\gamma )\\\\&=(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })m^{[3]}(\\gamma )O^{-1}(\\gamma )\\\\&=(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })m^{[4]}(\\gamma )O^{-1}(\\gamma )\\\\&=(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })(1+\\frac{\\mathcal {O}{(t^{\\frac{1}{2n}})}}{\\gamma })m^{[5]}(\\gamma )O^{-1}(\\gamma ).\\end{@align}}It can also be rewritten in terms of the variable z:{\\begin{@align}{1}{-1}m^{[1]}(z)=\\left(1+\\frac{\\mathcal {O}{(t^{-1/(2n)}})}{z(nt)^{1/n}}\\right)m^{[5]}((nt)^{1/n}z)+\\mathcal {O}{(z^{-2})},\\quad z\\rightarrow \\infty .\\end{@align}}Since m^{[5]} corresponds to the RHP for the Painlevé II hierarchy, we have{\\begin{@align}{1}{-1}m^{[5]}(\\gamma )=I+\\frac{m_1^{[5]}(s)}{\\gamma }+\\mathcal {O}(\\gamma ^{-1}),\\end{@align}}where \\gamma =z(nt)^{1/n}.$ Thus, $m^{[1]}(z)&=\\left(1+\\frac{\\mathcal {O}{(t^{-\\frac{1}{2n}})}}{z(nt)^{1/n}}\\right)\\left(1+\\frac{m_1^{[5]}(s)}{z(nt)^{1/n}}+\\mathcal {O}(z^{-2})\\right)\\\\&=I+\\frac{m_1^{[5]}(s)}{z(nt)^{1/n}}+\\frac{\\mathcal {O}{(t^{-\\frac{1}{2n}})}}{z(nt)^{1/n}}+\\mathcal {O}(z^{-2}).$ Since $m_1^{[5]}(s)$ is connected to solutions of the Painlevé II hierarchy, we conclude that $q(x,t)&=\\lim _{z\\rightarrow \\infty }z(m^{[1]}-I)\\\\&=(nt)^{-\\frac{1}{n}}u_n(x(nt)^{-\\frac{1}{n}})+\\mathcal {O}{(t^{-\\frac{3}{2n}})},$ where $u_n$ solves the $\\frac{n-1}{2}$ th equation of the Painlevé II hierarchy.", "The odd integer $n$ corresponds to the $\\frac{n-1}{2}$ th member in the mKdV hierarchy.", "Remark 10.4 As for the asymptotics for the Painlevé II equation, we refer the readers to the classical book [20].", "There are also some recent works related to Painlevé II hierarchy, see for example [30],[9],[6]." ] ]	2011.14205
[ [ "Demography of galactic technosignatures" ], [ "Abstract Probabilistic arguments about the existence of technological life beyond Earth traditionally refer to the Drake equation to draw possible estimates of the number of technologically advanced civilizations releasing, either intentionally or not, electromagnetic emissions in the Milky Way.", "Here, we introduce other indicators than Drake's number $N_D$ to develop a demography of artificial emissions populating the Galaxy.", "We focus on three main categories of statistically independent signals (isotropic, narrow beams, and rotating beacons) to calculate the average number $N_G$ of emission processes present in the Galaxy and the average number of them crossing Earth, $\\bar{k}$, which is a quantity amenable to statistical estimation from direct observations.", "We show that $\\bar{k}$ coincides with $N_D$ only for isotropic emissions, while $\\bar{k}$ can be orders of magnitude smaller than $N_D$ in the case of highly directional signals.", "We further show that while $N_D$ gives the number of emissions being released at the present time, $N_G$ considers also the signals from no longer active emitters but whose emissions still occupy the Galaxy.", "We find that as long as the average longevity of the emissions is shorter than about $10^5$ yr, $N_G$ is fully determined by the rate of emissions alone, in contrast to $N_D$ and $\\bar{k}$ which depend also on the emission longevity.", "Finally, using analytic formulas of $N_G$, $N_D$, and $\\bar{k}$ determined for each type of emission processes here considered, we provide a comprehensive overview of the values these quantities can possibly achieve as functions of the emission birthrates, longevities, and directionality." ], [ "Introduction", "As new extrasolar worlds are being routinely discovered, there is an ever mounting evidence that a significant fraction of exoplanets may have environmental conditions suitable for developing life [8], [27], [37].", "In the hunt for signs of life beyond the solar system, the search for biosignatures from the atmosphere and surface of extrasolar Earth-like planets is moving its first steps and will likely dominate the exoplanet science in the next decades [25].", "The prospects of life elsewhere in the Galaxy have also reinvigorated the longstanding search for putative signs of technologically savvy life from beyond Earth.", "The search for such technosignatures is going indeed through a phase of intense activity, boosted primarily by large scale private initiatives such as the Breakthrough Listen program [10], [23], [30] and benefiting of significant advances in detector technologies.", "The searches for bio- and technosignatures are two complementary strategies in the general quest of finding life elsewhere as they probe different remotely detectable byproducts of life.", "Future searches for spectroscopic biosignatures will however target exoplanets up to only a few tens of light years from Earth [31], while current telescopes are potentially capable of detecting, for example, radio emissions from artificial sources of technological level comparable to our own located well beyond 100 ly from the Earth [18].", "Moreover, searches for remotely detectable technosignatures probe a parameter search space so large that the absence of detection to date does not warrant any firm conclusion about the existence of potentially detectable exocivilizations [34], [32].", "This point has been recently emphasized by [35] who compare the tiny fraction of the parameter space explored so far to the ratio of the volume of a small swimming pool to that of the Earth's oceans.", "On the theoretical side, the prospects of technological life existing elsewhere and the probability of its detection are undetermined as well.", "In this context, the famous Drake equation has traditionally inspired the search for electromagnetic (EM) technosignatures [6], [7].", "In its compact form, the Drake equation equates the mean number of active emitters, $N_D$ , with the product between the average rate of emergence of communicating civilizations, $\\Gamma $ , and the average longevity, $\\bar{L}$ , of the emission processes: $N_D=\\Gamma \\bar{L}.$ In the original formulation of Eq.", "(REF ), $\\Gamma $ was expressed as a product of probability factors, later grouped in astrophysical and biological/evolutionary probability events [28], though to be conducive to the emergence of technological life capable of releasing EM emissions.", "A vast literature has been devoted to the analysis of (REF ) [4], [26], [15], [16], [2] and to the discussion of each entry of the original Drake equation thought, nowadays, only the astrophysical contributions to $\\Gamma $ are known with some confidence [1], [29].", "Not surprisingly, lacking any empirical knowledge about the rate of abio- and technogenesis beyond Earth or the size of $\\bar{L}$ , estimates of $N_D$ span several orders of magnitude, even for the Milky Way galaxy alone [12].", "Although Eq.", "(REF ) is the most celebrated equation in the field of technosignatures, it is not the only vehicle to study their statistical properties.", "For example, of particular importance for assessing the probability of detection are the emission processes that cross our planet, as only those can be potentially detected [19].", "Their average number, denoted $\\bar{k}$ , is thus a quantity that can be empirically estimated from observations, at least in principle.", "In practice, however, the limited sensitivity of current telescopes and the aforementioned vastness of the parameter search space permits only a probabilistic inference of the range of possible values of $\\bar{k}$ compatible with observations [21], [11].", "Even if $\\bar{k}$ is sometimes confused with the Drake number, it actually coincides with $N_D$ only by assuming a constant birthrate of emissions that are either entirely isotropic, that is, radiating in all directions, or otherwise all directed towards our planet.", "In more general scenarios that contemplate anisotropic EM emissions, such as randomly directed beam-like signals or beacons sweeping across space, only the fraction of emissions directed towards the Earth can be potentially detected, implying $N_D\\ge \\bar{k}$ .", "Another quantity of interest discussed here is $N_G$ , the average total number of emission processes present in the Galaxy.", "According to this definition, $N_G$ contains both the processes generated by emitters that are still transmitting, whose average number is $N_D$ , and those that come from emitters that are no longer active, but whose emissions still occupy the Galaxy (regardless of whether or not they intersect the Earth).", "In full generality, $N_G\\ge N_D$ , where the equality sign holds true either if there are no galactic emissions ($N_G=0$ and $N_D=0$ ) or if the only emissions present in the Galaxy come from emitters currently radiating.", "The three quantities $N_G$ , $N_D$ , and $\\bar{k}$ are the main statistical parameters characterizing the demography of technosignatures in the Milky Way, from which other quantities and properties of interest can be derived.", "For example $\\bar{k}/N_G$ gives the fraction of galactic emissions intersecting Earth, while it can be demonstrated that $1-\\exp (-\\bar{k})$ yields the fraction of the galactic volume occupied by the emissions [19].", "Furthermore, we see from the discussion above that the sequence of nested inequalities, $N_G\\ge N_D\\ge \\bar{k},$ holds true for all types of EM emissions (isotropic, anisotropic) and for any combination of them, implying that $\\bar{k}$ , the quantity that could be possibly estimated from observations, sets a lower limit to the population ($N_G$ ) of emissions filling the Galaxy.", "Here, we present a detailed study of $N_G$ , $N_D$ , and $\\bar{k}$ to ascertain their dependence on the birthrate, the longevity, and the geometry of the emission processes.", "We base our analysis on the presumption that the rate of technogenesis in the Milky Way has not changed significantly during the recent history of the Galaxy (a few million years) and that the population of artificial EM sources can be described by a collection of statistically independent emitters.", "Table: Legend of symbols used in the text and their meaning.", "The subscript ii refers to the different types of emissions considered here:isotropic signals (i=i= iso), random beams (i=i= rb), and rotating lighhouses (i=i= lh)." ], [ "Emission processes", "We start by defining our model of emission processes generated by artificial emitters in the Milky Way galaxy.", "We focus on the thin disk component of the Galaxy containing roughly $10^{10}$ potentially habitable planets within a radius $R_G\\simeq 60$ kly from the galactic centre, located at the origin of a cartesian reference frame with axes $x$ , $y$ , and $z$ .", "We approximate the thin disk by an effectively two-dimensional disk of radius $R_G$ on the $x$ -$y$ plane.", "We make the assumption that the artificial emitters are located at random sites $\\mathbf {r}=(x,y)$ relative to the galactic centre and that they are statistically independent of each other, meaning that their birthrates and longevities are random variables uncorrelated with $\\mathbf {r}$ .", "In the following, we shall employ the generic term “emission process\" to indicate an artificial EM radiation of any wavelength and power spectrum that is emitted either continuously or not during a time duration $L$ .", "We shall however distinguish the emission processes according to their isotropy/anisotropy and to the geometry of the volume occupied by their radiations by assigning to them a distinct type or class labelled by the index $i$ .", "In particular, here we shall focus on three prototypical types of emissions: isotropic radiations ($i=$ iso), randomly directed narrow beams (denoted as “random beams\", $i=$ rb), and narrow beams emitted by rotating lighthouses (denoted simply as “lighthouses\", $i=$ lh).", "Furthermore, we shall assume that the radiations propagate unperturbed throughout the Galaxy at the speed of light $c$ .", "In so doing, we are neglecting scattering and absorption processes by the interstellar medium, which is a highly idealized setup meant to illustrate more clearly the effects of the emission geometries.", "We allow the possibility that the different types of emission processes can have correspondingly different birthrates and longevities.", "For example, a continuous isotropic emission in the infrared could be resulting from the waste heat produced by a civilization exploiting the energy of its sun [9], as for type II civilizations of the Kardashev scale [24].", "The corresponding emission birthrate would be presumably lower, and the emission longevity longer, than that of a less technologically developed (or less energy harvesting) civilization targeting other planets with radio signals to just advertise its existence.", "We therefore introduce the rate of appearance per unit area for emitters of type $i$ , $\\gamma _i(\\mathbf {r},t)$ , defined so that $\\gamma _i(\\mathbf {r},t)d\\mathbf {r}dt$ gives the expected number of $i$ -emitters within an area element $d\\mathbf {r}$ about $\\mathbf {r}$ that started emitting within a time interval $dt$ centered at a time $t$ before present.", "Likewise, we associate to all processes of type $i$ a common probability distribution function (PDF) of the longevity, denoted $\\rho _i(L)$ , such that $\\int _0^\\infty \\!dL\\,\\rho _i(L)=1$ for each $i$ .", "The emitter rate of emergence vanishes for distances on the galactic disk larger than $R_\\textrm {G}$ , so that integrating $\\gamma _i(\\mathbf {r},t)$ over $\\mathbf {r}$ gives the birthrate of the emission processes of type $i$ in the entire Galaxy: $\\Gamma _i(t)=\\int \\!d\\mathbf {r}\\,\\gamma _i(\\mathbf {r},t).$ Finally, owing to the statistical independence of the emission processes, the sum of the birthrates of each type of emission gives the total rate of appearance of all emission processes: $\\Gamma (t)=\\sum _i \\Gamma _i(t).$" ], [ "Average number of active emitters (Drake's $N_D$ )", "The Drake equation can be directly derived from considering the number of emitters that are currently transmitting [20].", "To see this, we note that for any galactic emitter that started an emission process at a time $t$ in the past, the necessary condition that at present time the emitter is still active is that the time elapsed since its birth is shorter than the emission longevity, that is, $t\\le L$ .", "The time integral of $\\Gamma _i(t)$ from $t=0$ to $t=L$ gives therefore the expected number of emitters of type $i$ and longevity $L$ that are still emitting.", "The average number of active $i$ -emitters is obtained by marginalizing $L$ with respect to the PDF associated to the processes of type $i$ : $N_D^i=\\int _0^\\infty \\!dL\\, \\rho _i(L)\\int _0^L\\!", "dt\\,\\Gamma _i(t).$ We take the time-scale over which the rate $\\Gamma _i(t)$ is expected to show appreciable variations to be much larger than $L$ , even for longevity values distributed over several million years.", "In so doing, we are assuming that the emitter birthrates did not change significantly during the recent history of the Galaxy and can be taken constant in Eq.", "(REF ), leading to: $N_D^i=\\Gamma _i\\bar{L}_i,$ where $\\bar{L}_i=\\int _0^\\infty \\!dL\\,\\rho _i(L)L$ is the mean longevity of the emission processes of type $i$ (see Table REF for a list of symbols used in this paper and their meaning).", "Equation (REF ) is the Drake equation relative to signals of type $i$ under the steady-state hypothesis.", "By setting $x_i=\\Gamma _i/\\Gamma $ with $\\sum _i x_i=1$ , and defining $\\bar{L}=\\sum _i x_i\\bar{L}_i$ as the longevity averaged over all types of emission processes, the sum of $N_D^i$ over all $i$ 's leads to the usual Drake equation in the compact form: $N_D=\\Gamma \\bar{L}.$ It is worth stressing that the steady-state hypothesis upon which the derivation of Eq.", "(REF ) rests would be less justifiable if we were considering active emitters from a region extending over Giga light-years, as in this case the temporal dependence of the emission birthrates should be taken into account." ], [ "Average number of emission processes at Earth ($\\bar{k}$ )", "While in deriving $N_D$ we only needed to count the number of active emitters without specifying the characteristics of their emission processes, to calculate the mean number of emission processes intersecting Earth, $\\bar{k}$ , we have to specify the conditions under which such intersections occur.", "First, we note that for an emission process that started at a time $t$ in the past, the emitted EM radiations, traveling through space at the speed of light $c$ , fill at the present time a more or less extended region of space that depends on the longevity and directionality of the emission process.", "This region can be continuous, as in the case of an emission process (either isotropic or anisotropic) lasting a time $L$ without interruptions, or discontinuous as for an emitter having sent during $L$ a sequence of intermittent signals.", "In the latter case, if we assume that the train of signals is crossing Earth, there is a finite probability of the Earth not being illuminated at a given instant of time [18], which may lead us to overlook this process in the calculation of $\\bar{k}$ .", "Similar considerations apply also to intrinsically continuous emission processes that appear discontinuous or intermittent from the Earth's viewpoint, as it is the case of a rotating beacon whose beamed signal crosses Earth periodically.", "Also, the signal intermittency may be due to variations in the emitted power with minimum flux at the receiver below the detection threshold [18] or to scintillation effects due to the interstellar medium [5].", "To avoid ambiguities in determining $\\bar{k}$ , we shall treat any intermittent (as seen from the Earth) emission of total longevity $L$ as an effectively continuous process lasting the same amount of duration time.", "Operatively, we could think of a periodic signal of period $T$ and duty cycle $w$ impinging upon the Earth during an observational time interval $\\Delta t$ .", "The condition $\\Delta t/T\\ge 1-w$ ensures that the “on\" phase of the emission crosses Earth at least once during $\\Delta t$ , so that the process is “detectable\" with probability one and can be added to the list of processes crossing Earth.", "We further note that among the requisites an intermittent signal should have to be recognized as a bona fide technosignature, the recurrence of detection is one of the most important [14]." ], [ "Isotropic emissions", "Let us consider an emitter located at $\\mathbf {r}$ that started emitting an isotropic process at a time $t$ in the past and for a duration $L$ .", "If the process is intrinsically continuous, at the present time the region of space filled by the EM waves is a spherical concentric shell of outer radius $ct$ and thickness $cL$ , centered on the emitter position $\\mathbf {r}$ .", "In the case of a intermittent isotropic process of period $T$ and duty cycle $w$ , this region encompasses a sequence of nested concentric spherical shells, each of thickness $cwT$ , of consecutive outer radii differing by $cT$ .", "Conforming to the above prescription for intermittent emissions, we ignore the internal structure of the encompassing shell by treating it as an effectively continuous spherical shell of width $cL$ .", "The condition that the shell intersects the Earth is fulfilled by the requirement [19], [2] $ct-cL\\le \\vert \\mathbf {r}-\\mathbf {r}_\\textrm {E}\\vert \\le ct,$ where $\\mathbf {r}_\\textrm {E}$ is the vector position of the Earth.", "If $\\gamma _\\textrm {iso}(\\mathbf {r},t)$ (where the subscript “iso\" stands for isotropic) is the process birthrate per unit volume and $\\rho _\\textrm {iso}(L)$ is the PDF of the longevity, the average number of spherical shells at Earth is obtained by marginalizing the condition (REF ) over $\\mathbf {r}$ , $L$ , and $t$ : $\\bar{k}_\\textrm {iso}=\\int \\!", "dL\\,\\rho _\\textrm {iso}(L)\\int \\!", "d\\mathbf {r}\\int _{\\vert \\mathbf {r}-\\mathbf {r}_\\textrm {E}\\vert /c}^{\\vert \\mathbf {r}-\\mathbf {r}_\\textrm {E}\\vert /c+L}\\!", "\\!\\!", "dt\\,\\gamma _\\textrm {iso}(\\mathbf {r},t).$ As done in Sec.", "REF , we neglect the temporal dependence of the birthrate, $\\gamma _\\textrm {iso}(\\mathbf {r},t)\\simeq \\gamma _\\textrm {iso}(\\mathbf {r})$ , so that Eq.", "(REF ) reduces to: $\\bar{k}_\\textrm {iso}=\\int \\!", "dL\\,\\rho _\\textrm {iso}(L) L\\int \\!", "d\\mathbf {r}\\,\\gamma _\\textrm {iso}(\\mathbf {r})=\\Gamma _\\textrm {iso}\\bar{L}_\\textrm {iso},$ which, as anticipated in Sec.", "REF , coincides with the Drake number $N_D^\\textrm {iso}$ relative to isotropic emission processes.", "Note that since we have taken a time independent birthrate, $\\mathbf {r}_E$ has dropped off Eq.", "(REF ), meaning that $\\bar{k}_\\textrm {iso}$ actually gives the mean number of emissions crossing any given point in the Galaxy.", "This holds true also for other types of emission processes as long as the corresponding birthrates do not depend on $t$ ." ], [ "Anisotropic emissions: random beams and lighthouses", "In the case of anisotropic signals, the region of space filled by the EM radiation does not cover all directions and, therefore, only the fraction of signals that are directed towards the Earth can contribute to $\\bar{k}$ .", "For example, a prototypical anisotropic signal often discussed in the literature is that of a conical beam of opening angle $\\alpha _0$ pointing to a given direction over the full lifetime of the emission process.", "As shown in Appendix REF , if such beamed signals are generated with a constant birthrate and their orientation is distributed uniformly over the unit sphere (3D case), the average number of random beams (rb) intersecting Earth will be proportional to the solid angle subtended by the beams, that is: $\\begin{array}{lll}\\bar{k}_\\textrm {rb}=\\dfrac{\\langle \\alpha _0^2\\rangle }{16}\\Gamma _\\textrm {rb}\\bar{L}_\\textrm {rb}, & \\textrm {(3D random beams)}\\end{array}$ where $\\langle \\cdots \\rangle $ denotes an average over the beam apertures (assumed to be narrow), $\\bar{L}_\\textrm {rb}$ is the average longevity of the beams, and $\\Gamma _\\textrm {rb}$ is their birthrate.", "We see therefore that contrary to the case of isotropic emission processes, the mean number of beams crossing Earth can be many orders of magnitude smaller than the corresponding Drake's number $N_D^\\textrm {rb}=\\Gamma _\\textrm {rb}\\bar{L}_\\textrm {rb}$ .", "Taking for example beam apertures comparable to that of the Arecibo radar ($\\sim 2\\simeq 6\\times 10^{-4}$ rad) Eq.", "REF yields $\\bar{k}_\\textrm {rb}/N_D^\\textrm {rb}\\sim 2\\times 10^{-8}$ , and even smaller values of $\\bar{k}_\\textrm {rb}/N_D^\\textrm {rb}$ are obtained by assuming optical or infrared laser emissions of apertures under an arcsecond [22], [33].", "Instead of pointing towards random directions in space, another hypothetical scenario is that in which the emitters generate beams directed preferably along the galactic plane in order to enhance the probability of being detected by other civilizations.", "In this two dimensional (2D) case, the mean number of beams illuminating the Earth becomes (see REF ): $\\begin{array}{ll}\\bar{k}_\\textrm {rb}=\\dfrac{\\langle \\alpha _0\\rangle }{2\\pi }\\Gamma _\\textrm {rb}\\bar{L}_\\textrm {rb}, & \\textrm {(2D random beams)}\\end{array}$ so that $\\bar{k}_\\textrm {rb}/N_D^\\textrm {rb}\\sim 10^{-4}$ for $\\langle \\alpha _0\\rangle $ comparable to that of the Arecibo radar.", "Another type of anisotropic signal is that of a narrow beam emitted by a rotating source, like a lighthouse rotating with constant angular velocity.", "This kind of process generates a continuous, radiation-filled spiralling beam revolving around the emitter and expanding at the speed of light.", "The spiral cross section increases with the beam aperture $\\alpha _0$ and the distance from the source.", "Even if this kind of process is intrinsically continuous, an expanding spiral impinging upon Earth will be perceived as a periodic sequence of signals.", "For example, if the spin axis is perpendicular to the galactic plane, the spiral generated by a rotating conical beam of angle aperture $\\alpha _0$ will periodically cross Earth's line-of-sight with duty cycle $\\alpha _0/2\\pi $ .", "In a manner similar to what we have done for the case of intrinsically discontinuous signals, we discard the signal intermittency perceived at Earth by introducing an effective volume encompassing the spiralling beam, whose construction is detailed in the Appendix REF .", "As long as the rotation axes are oriented uniformly over the unit sphere (3D case), we find that the mean number of lighthouse (lh) signals crossing Earth reduces to: $\\begin{array}{cc}\\bar{k}_\\textrm {lh}=\\dfrac{\\displaystyle \\langle \\alpha _0\\rangle }{\\displaystyle 2}\\Gamma _\\textrm {lh}\\bar{L}_\\textrm {lh}, &\\textrm {(3D lighthouses)}\\end{array}$ while when the spin axes are perpendicular to the galactic plane (2D case), $\\bar{k}_\\textrm {lh}$ becomes: $\\begin{array}{lr}\\bar{k}_\\textrm {lh}=\\Gamma _\\textrm {lh}\\bar{L}_\\textrm {lh}, & \\textrm {(2D lighthouses)}.\\end{array}$ The signals generated by lighthouses have therefore much larger values of $\\bar{k}$ than those of random beams with comparable birthrates and longevities, as shown in Fig.", "REF .", "In particular, from Eqs.", "(REF )-(REF ) we see that $\\bar{k}_\\textrm {rb}/\\bar{k}_\\textrm {lh}$ scales as $\\sim \\langle \\alpha _0\\rangle N_D^\\textrm {rb}/N_D^\\textrm {lh}$ for 2D or 3D anisotropic signals of similar apertures, meaning for example that over $\\sim 10^3$ active Arecibo-like beams must thus be added to each active lighthouse of comparable $\\langle \\alpha _0\\rangle $ to have analogous values of $\\bar{k}$ .", "Furthermore, $\\bar{k}_\\textrm {lh}$ for the 2D case turns out to be independent of $\\alpha _0$ , as for isotropic processes.", "This is an interesting result, implying that an observer on Earth has the same chances of being illuminated by a 2D lighthouse as by a (pulsed) isotropic signal if the two have equal Drake's numbers, Fig.", "REF .", "Figure: Average number of emissions of type ii crossing Earth, k ¯ i \\bar{k}_i, over the corresponding Drake number N D i N_D^i plotted as a functionof the beam aperture α 0 \\alpha _0 for the different types ii of processes considered in this work." ], [ "Average number of emission processes intersecting the Galaxy ($N_G$ )", "So far, the only temporal variable required to calculating $N_D^i$ and $\\bar{k}_i$ has been the signal longevity $L$ .", "In deriving $N_G$ , we shall introduces an additional time-scale, $t_G=\\frac{2R_G}{c},$ defined as the time required by a photon to travel, unperturbed, across two opposite edges of the Milky Way ($\\approx 10^5$ yr).", "Contrary to $L$ and $\\Gamma _i$ , $t_G$ is an astrophysical quantity, specific to our Galaxy, that is independent of any assumption about the existence and/or the properties of the artificial emissions.", "Figure: Top view of the galactic disk of radius R G R_G with superimposed areas covered by emissions of different geometries originated by a sourcelocated at (r,0)(r,0).", "(a): the intersection with the plane xx-yy of the spherical shell generated by an isotropic emission forms an annulus of inner radius c(t-L)c(t-L),which intersects the galactic disk if c(t-L)≤R G +rc(t-L)\\le R_G+r." ], [ "Isotropic emissions", "The necessary condition for an isotropic signal intersecting the Galaxy is that there is a non-null intersection between the spherical shell and the galactic disk.", "As shown in Fig.", "REF , this is fulfilled by requiring the inner radius of the shell to be smaller than the maximum distance of the emitter from the edge of the galactic disk, that is, $c(t-L)\\le r+R_G$ .", "The number of isotropic emissions intersecting the Galaxy is therefore: $N_G^\\textrm {iso}&=\\int \\!", "dL\\rho _\\textrm {iso}(L)\\int \\!", "d\\mathbf {r}\\!\\int _0^{(r+R_G)/c+L}\\!", "dt\\,\\gamma _\\textrm {iso}(\\mathbf {r})\\nonumber \\\\&=\\Gamma _\\textrm {iso}\\bar{L}_\\textrm {iso}+\\frac{1}{c}\\int \\!", "d\\mathbf {r}\\gamma _\\textrm {iso}(\\mathbf {r})(r+R_G).$ Using a birthrate that is uniform over the entire galactic disk, $\\gamma _\\textrm {iso}(\\mathbf {r})=\\theta (R_G-r)\\Gamma _\\textrm {iso}/\\pi R_G^2$ , the above expression reduces to: $N_G^\\textrm {iso}=\\Gamma _\\textrm {iso}\\left(\\bar{L}_\\textrm {iso}+\\frac{5}{6}t_G\\right),$ where $t_G$ is the time-scale given in Eq.", "(REF ).", "Other functional forms of $\\gamma _\\textrm {iso}(\\mathbf {r})$ affects only the prefactor of $t_G$ .", "For example, taking $\\gamma _\\textrm {iso}(\\mathbf {r})\\propto \\theta (R_G-r)\\exp (-r/r_s)$ with $r_s=8.15$ kly [21], the numerical factor $5/6$ ($\\simeq 0.833$ ) in (REF ) becomes $\\simeq 0.646$ .", "An interesting feature of Eq.", "(REF ) is that using (REF ) we can replace $\\Gamma _\\textrm {iso}$ by $\\bar{k}_\\textrm {iso}/\\bar{L}_\\textrm {iso}$ , yielding: $N_G^\\textrm {iso}=\\bar{k}_\\textrm {iso}\\left(1+\\frac{5}{6}\\frac{t_G}{\\bar{L}_\\textrm {iso}}\\right),$ so that for $\\bar{L}_\\textrm {iso}\\ll t_G\\approx 10^5$ yr the expected number of the emissions intersecting the Galaxy can be much larger than that of the emissions crossing Earth.", "For example, even if $\\bar{k}_\\textrm {iso}$ is only $\\approx 0.1$ and $\\bar{L}_\\textrm {iso}\\approx 10^2$ yr, $N_G^\\textrm {iso}$ is nevertheless of the order $10^2$ .", "As we shall see below, the directionality of the signals can amplify even more the difference between $\\bar{k}_i$ and $N_G^i$ ." ], [ "Anisotropic emissions: random beams and lighthouses", "The calculations of the average number of random beams present in the Galaxy, $N_G^\\textrm {rb}$ , and the one relative to the lighthouse signals, $N_G^\\textrm {lh}$ , are detailed in the Appendixes REF and REF , respectively.", "Here we report only the final expressions obtained under the assumption of small opening angles and a spatially uniform birthrate of the emitters: $N_G^\\textrm {rb}=\\left\\lbrace \\begin{array}{lr}\\Gamma _\\textrm {rb}\\left(\\bar{L}_\\textrm {rb}+\\dfrac{\\langle \\alpha _0\\rangle }{2}\\dfrac{4}{3\\pi }t_G\\right), & \\textrm {3D random beams} \\\\[8pt]\\Gamma _\\textrm {rb}\\left(\\bar{L}_\\textrm {rb}+\\dfrac{4}{3\\pi }t_G\\right), & \\textrm {2D random beams},\\end{array}\\right.$ $N_G^\\textrm {lh}=\\left\\lbrace \\begin{array}{lr}\\Gamma _\\textrm {lh}\\left(\\bar{L}_\\textrm {lh}+\\dfrac{2}{\\pi }t_G\\right), & \\textrm {3D lighthouses} \\\\[8pt]\\Gamma _\\textrm {lh}\\left(\\bar{L}_\\textrm {lh}+\\dfrac{5}{6}t_G\\right), & \\textrm {2D lighthouses}.\\end{array}\\right.$ The relevant result of these calculations is that for all but one case (that is, the 3D random beams) the mean number of anisotropic emissions intersecting the galactic plane is independent of the angular aperture, and is therefore comparable to that obtained for the case of isotropic emissions with similar birthrates and longevities." ], [ "Discussion", "Table REF summarizes the analytic expressions of $N_G^i$ and $\\bar{k}_i$ derived in the previous section.", "For each type of emission process, the Drake number $N_D^i=\\Gamma _i\\bar{L}_i$ is the only quantity that does not depend on the geometry of the emission process and we shall therefore focus our discussion primarily on $\\bar{k}_i$ and $N_G^i$ .", "Figures REF -REF show $\\bar{k}_i$ as a function of the mean signal longevity $\\bar{L}_i$ and the population of signals in the Galaxy $N_G^i$ , with $i=\\textrm {iso}$ (isotropic, Fig.REF ), $i=\\textrm {rb}$ (random beams,Fig.REF ), and $i=\\textrm {lh}$ (lighthouses, Fig.REF ).", "The results have been obtained by taking $R_G=60$ kly for the galactic radius, corresponding to $t_G=2R_G/c=1.2\\times 10^{5}$ yr.", "The red solid lines demarcate the boundary between $\\bar{k}_i >1$ (red colour scale) and $\\bar{k}_i <1$ (blue colour scale), while the black solid lines indicate $N_G^i$ calculated for constant values of the emission birthrate $\\Gamma _i$ .", "The results shown in Figs.", "REF and REF have been obtained assuming $\\langle \\alpha _0^2\\rangle \\simeq \\langle \\alpha _0\\rangle ^2$ and an average beam aperture of 2, corresponding to $\\langle \\alpha _0\\rangle \\simeq 6\\times 10^{-4}$ rad.", "Results for different beam apertures can be easily obtained using the expressions in Table REF .", "Figure: Expected number of isotropic emission processes at Earth, k ¯ iso \\bar{k}_\\textrm {iso}, as a function of the mean signal longevityL ¯ iso \\bar{L}_\\textrm {iso} and the average number N G iso N_G^\\textrm {iso} of isotropic emissions intersecting the Galaxy.", "The red solid lineindicates the special value k ¯ iso =1\\bar{k}_\\textrm {iso}=1, while the black solid lines show N G iso N_G^\\textrm {iso} calculated for different valuesof the emission birthrate Γ iso \\Gamma _\\textrm {iso}.A first interesting feature is the behaviour of the galactic population of technosignatures, $N_G^i$ , as a function of $\\bar{L}_i$ for fixed $\\Gamma _i$ (black solid lines).", "While $N_G^i$ increases proportionally to the signal longevity for $\\bar{L}_i\\gtrsim t_G$ , reaching asymptotically the corresponding Drake's number $N_D^i=\\bar{L}_i\\Gamma _i$ , for $\\bar{L}_i\\lesssim t_G\\sim 10^5$ yr it reduces to $N_G^i\\simeq \\Gamma _i t_G\\simeq \\Gamma _i\\times (10^5 \\textrm {yr}),$ for all types of emission processes with the exception of random beams in 3D.", "In this case $N_G^\\textrm {rb}$ scales as $\\langle \\alpha _0\\rangle \\Gamma _\\textrm {rb} t_G$ for $\\bar{L}_\\textrm {rb}\\lesssim \\langle \\alpha _0\\rangle t_G$ .", "Figure: Expected number of random narrow beams crossing Earth, k ¯ rb \\bar{k}_\\textrm {rb}, as a function of their mean longevityL ¯ rb \\bar{L}_\\textrm {rb} and the average number N G rb N_G^\\textrm {rb} of beams intersecting the Galaxy.", "In panel (a) the orientations of the beamaxes are distributed uniformly along the galactic disk (2D case), while in panel (b) they are distributed uniformly over the 3D space.", "Thebeam aperture is fixed at 2, corresponding to α 0 =6×10 -4 \\alpha _0=6\\times 10^{-4} rad.The red solid lines indicate the special value k ¯ rb =1\\bar{k}_\\textrm {rb}=1, while the black solid lines show N G rb N_G^\\textrm {rb} calculated fordifferent values of the emission birthrate Γ rb \\Gamma _\\textrm {rb}.Figure: Expected number of emissions from rotating beacons crossing Earth, k ¯ lh \\bar{k}_\\textrm {lh}, as a function of their mean longevityL ¯ lh \\bar{L}_\\textrm {lh} and the average galactic population N G lh N_G^\\textrm {lh}.", "In panel (a) the orientations of the spinaxes are kept perpendicular to the galactic plane (2D case), while in panel (b) they are distributed uniformly over the 3D space.The red solid lines indicate the special value k ¯ lh =1\\bar{k}_\\textrm {lh}=1, while the black solid lines show N G lh N_G^\\textrm {lh} calculated fordifferent values of the emission birthrate Γ lh \\Gamma _\\textrm {lh}.Equation (REF ) is remarkable because it prescripts the galactic population of emission processes to be proportional to only the birthrate $\\Gamma _i$ , regardless of the signal longevity as long as it is assumed $\\bar{L}_i$ to be less than $\\sim 10^5$ yr.", "This is an advantage compared to the Drake's number $N_D^i$ , where in addition to $\\Gamma _i$ the longevity of the signals is a further object of speculations.", "For a wide range of $\\bar{L}_i$ values, we can thus conjecture about the size of $N_G^i$ by reasoning only in terms of the signal birthrate.", "To this end it is instructive to compare $\\Gamma _i$ with the rate of formation of habitable planets in the Milky Way, $\\Gamma _\\textrm {P}$ , whose estimates place it in the range $0.01$ -$0.1$ planet per year [3], [36], [17], [1].", "Let us first make the hypothesis that each habitable planet can be the potential source of no more than one artificial emission.", "This would correspond to $\\Gamma _\\textrm {P}$ being a theoretical upper limit of $\\Gamma _i$ .", "Under this assumption, the resulting galactic population of both isotropic emissions (Fig.", "REF ) and rotating beacons (Fig.", "REF ) would be bounded from above by $\\max (N_G^i)\\sim 10^3$ -$10^4$ , or somewhat less for 2D random beams of Fig.", "REF (a), which is essentially the number of habitable planets being formed during a timespan of order $t_G\\sim 10^5$ yr.", "Such an upper limit of $N_G$ entails a corresponding lower bound on the average distance $a_E$ between the emitters.", "Indeed, since within our working assumption $N_G$ corresponds to the number of emitters releasing the emissions, their number density can be expressed as $\\rho _E=N_G/\\pi R_G^2$ .", "This allows us to find from $\\pi a_E^2\\rho _E\\sim 1$ that $a_E\\sim R_G/\\sqrt{N_G}$ , thereby implying that the lower bound on $a_E$ is of the order $10^3$ ly for $\\bar{L}\\lesssim t_G$ .", "Following the same reasoning, we see that the typical relative distance between emitters estimated by the Drake equation, $a_D\\sim R_G/\\sqrt{N_D}$ , scales for $\\bar{L}\\lesssim t_G$ as $a_D\\sim a_E\\sqrt{t_G/\\bar{L}}$ .", "The difference between $a_G$ and $a_D$ stems from the fact that the Drake's number gives the average population of active emitters, which are only a fraction of all $N_G$ emitters whose signals are present in the Galaxy.", "For example, while assuming $\\Gamma _i\\sim \\Gamma _P$ and $\\bar{L}\\lesssim 10$ yr gives $N_G\\sim 10^3$ and $a_E\\sim 10^3$ ly, the Drake equation yields $N_D\\lesssim 1$ and a value of $a_D$ comparable to or larger than the diameter of the Galaxy, meaning that in this case out of $\\sim 10^3$ galactic emissions essentially none comes from currently active emitters.", "As we have seen in the previous section, in addition to $N_D$ the longevity $\\bar{L}$ constraints also the number of the emission processes crossing our planet, which is further affected by the directionality of the signals (Fig.", "REF ).", "Assuming therefore a large number of galactic emissions does not automatically imply similarly large values of $\\bar{k}_i$ .", "For example, even taking $\\Gamma _i\\sim 0.1$ yr$^{-1}$ (that is, $N_G^i\\simeq 10^4$ for $\\bar{L}_i<t_G$ ), the expected maximum value of $\\bar{k}_i$ ranges from $\\sim 10^3$ for $\\bar{L}_i=10^4$ yr down to $\\sim 0.1$ for $\\bar{L}_i=1$ yr in the case of isotropic emissions (Fig.", "REF ) and 2D lighthouses [Fig.", "REF (a)].", "Within the same range of signal longevities, the upper bound on $\\bar{k}_i$ of beams in 2D and rotating beacons in 3D drops to only $\\sim 10^{-5}$ -$10^{-1}$ , Figs.", "REF (a) and REF (b).", "A special situation is represented by a collection of beamed signals with axis orientations distributed in the 3D space [Fig.", "REF (b)].", "In this case, $N_G^\\textrm {rb}$ becomes independent of the signal longevity only when $\\bar{L}_\\textrm {rb}\\lesssim \\langle \\alpha _0\\rangle t_G$ , which for $\\langle \\alpha _0\\rangle \\sim 2$ represents lifetimes smaller than $\\sim 10$ yr.", "In this limit, $N_G^\\textrm {rb}\\lesssim 1$ for $\\Gamma _\\textrm {rb}\\sim 0.1$ yr$^{-1}$ and the resulting $\\bar{k}_\\textrm {rb}$ is upper bounded by a negligible $\\sim 5\\times 10^{-8}$ .", "Values of $N_G$ of the order of $10^5$ can nevertheless be reached for 3D beams lasting at least 1 Myr, but even in this case $\\bar{k}_\\textrm {rb}$ is only $\\sim 2\\times 10^{-3}$ .", "Let us pause one moment to consider the implications of assuming each habitable planet being the potential source of at most one emission process.", "As shown above, this hypothesis entails an upper bound of $N_G^i$ of the order $10^3$ -$10^4$ , implying therefore the possibility of technogenesis arising on each habitable planet during the last $\\sim 10^5$ years.", "This exceeds by far the most optimistic stances, as such an assumption would imply not only a non-zero probability that abiogenesis is ubiquitous in the Milky Way, but also that intelligence and technology are inevitable outcomes of the evolutionary path of life on each inhabited planet.", "As long as a one-to-one correspondence between emission processes and planets is maintained, an upper bound on $\\Gamma _i$ (and so on $N_G^i$ ) should be more reasonably placed to much lower values than $\\Gamma _P$ , leading to $\\max (N_G^i)\\ll 10^3$ -$10^4$ and to correspondingly small values of $\\bar{k}_i$ .", "Our model, however, does not distinguish whether the emission events have occurred once or multiple times within $t_G$ on a given planet, nor does it rule out the possibility of the emitters far outnumbering the planets in which technology arose, as for example self-replicating robotized lighthouses swarming in the free space.", "Within such scenarios, $\\Gamma _i$ could thus be larger than the rate of emergence of technological civilizations capable of releasing technosignatures and perhaps even comparable to, or in excess of, $\\Gamma _P$ .", "The plausibility of a galactic population of $\\sim 10^4$ short lived (i.e., $\\bar{L}_i\\lesssim t_G$ ) emissions should however be weighed against the requirement that all these emissions must have been released during the last $\\sim 10^5$ years in order to fill the galaxy.", "Of course, it is still possible to have significantly large values of $N_G^i$ and $\\bar{k}_i$ even for relatively low birthrates if the mean longevity is so long to prevail over the small values of $\\Gamma _i$ .", "For example, signals emitted from isotropic sources or 2D lighthouses with a rate as small as $\\sim 10^{-5}$ yr$^{-1}$ would bring values of $N_G^i$ and $\\bar{k}$ larger $\\sim 100$ if their longevities exceeded $\\sim 10$ Myr.", "Similar values of $N_G^i$ are obtained for 2D beams and 3D lighthouses with $\\Gamma _i\\sim 10^{-5}$ yr$^{-1}$ and $\\bar{L}_i\\sim 10$ Myr, but the reduced solid angle for $\\langle \\alpha _0\\rangle \\sim 2$ makes $\\bar{k}_i$ as small as $\\sim 10^{-2}$ (which drops to $\\sim 10^{-8}$ in the case of 3D random beams).", "So far, we have discussed each type of emission processes separately to study the effect of $\\Gamma _i$ , $\\bar{L}_i$ and of the signal directionality on $N_G^i$ and $\\bar{k}_i$ .", "However, in the most general case, different types of processes may be present simultaneously in the Galaxy and the contribution of each $i$ -process to the total $N_G$ and $\\bar{k}$ depends on the respective occurrence frequency.", "To see this, we note that the different expressions of $N_G^i$ and $\\bar{k}_i$ given in Table REF have the form $N_G^i=\\Gamma _i(\\bar{L}_i+u_i t_G)$ and $\\bar{k}_i=v_i\\Gamma _i\\bar{L}_i$ , where $u_i$ and $v_i$ are the dimensionless factors taking account the geometry and the directionality of the signals.", "Owing to the assumed statistical independence of the emitters, the quantities $N_D$ , $N_G$ , and $\\bar{k}$ are simple linear combinations of the different types of processes.", "We can thus write: $N_G&=\\sum _i \\Gamma _i(\\bar{L}_i+u_it_G)=\\Gamma \\bar{L}+\\Gamma t_G\\sum _i x_i u_i, \\\\\\bar{k}&=\\sum _i v_i\\Gamma _i\\bar{L}_i=\\Gamma \\sum _ix_iv_i\\bar{L}_i,$ where, as done in Sec.REF , $x_i=\\Gamma _i/\\Gamma $ and $\\bar{L}=\\sum _i x_i\\bar{L}_i$ .", "Considerations similar to those discussed in the previous section apply therefore also to the more general case.", "In particular, as seen from Eq.", "(REF ), as long as the total signal longevity $\\bar{L}$ is smaller than $\\sim t_G\\sim 10^5$ yr, the total number of processes intersecting the Galaxy results to be proportional to $\\Gamma t_G\\sim (10^5\\,\\textrm {yr})\\times \\Gamma $ , regardless of $\\bar{L}$ .", "Speculations about the abundance of short-lived ($\\bar{L}\\lesssim 10^5$ yr) emissions in the Galaxy can thus be framed in terms of possible upper bounds on the total birthrate $\\Gamma $ .", "From Eq.", "() we see that the contribution of each type of emission to the total number of processes crossing Earth strongly depends on the relative abundance of signal types and their longevities.", "As shown in Figs.", "REF -REF , the contribution to $\\bar{k}$ of isotropic processes and lighthouses in 2D would likely dominate over other types of emissions of similar birthrates.", "For example, assuming that the fraction of rotating beacons sweeping the galactic plane is comparable to that of 3D beamed emissions, $x_\\textrm {lh}\\sim x_\\textrm {rb}$ , the two would contribute equally to $\\bar{k}$ only if the mean longevity of the 3D beams is about $16/\\langle \\alpha _0^2\\rangle $ times larger than that of the rotating beacons.", "For beam apertures of 2 this corresponds to a factor $\\sim 10^7$ , so for a given fraction of 2D lighthouses lasting in average 10 years an equal amount of 3D beams requires a longevity of $\\sim 100$ Myr to contribute equally to $\\bar{k}$ .", "As a last consideration, we note that the total birthrate $\\Gamma $ in Eqs.", "(REF ) and () can be eliminated using the Drake number $N_D=\\Gamma \\bar{L}$ , yielding: $N_G&=N_D\\left(1+\\dfrac{t_G}{\\bar{L}}\\sum _i x_i u_i\\right), \\\\\\bar{k}&=N_D\\sum _i x_iv_i\\dfrac{\\bar{L}_i}{\\bar{L}},$ allowing us to translate in terms of $N_G$ and $\\bar{k}$ the rich literature devoted to the Drake equation.", "By further eliminating $N_D$ from (REF ) and () we get $\\frac{\\bar{k}}{N_G}=\\dfrac{\\sum _i x_iv_i\\bar{L}_i}{\\bar{L}+t_G\\sum _i x_i u_i},$ which expresses the fraction of galactic signals crossing Earth in terms of the remaining unknown temporal variables: the longevities.", "We note that Eq.", "REF generalizes a similar formula derived for the case of isotropic signals in [20] and [21].", "The two formulas are however not fully equivalent because in those works $\\bar{k}$ was put in relation to the number of emission processes released during the last $t_G$ years rather than using the number $N_G$ of emissions physically intersecting the Galaxy." ], [ "Conclusions", "In this paper, we have introduced other statistical quantities than the Drake number $N_D$ to characterize the population of EM technosignatures in the Milky Way.", "We have considered the average number of EM emissions present in the Galaxy, $N_G$ , and the average number $\\bar{k}$ of emissions intersecting the Earth (or any other site in the Galaxy).", "Unlike $N_D$ , $\\bar{k}$ and $N_G$ provide measures of the number of emission processes that are not necessarily released by currently active emitters, but that can be potentially detected on Earth ($\\bar{k}$ ) or that still occupy physically the Galaxy ($N_G$ ).", "In order to study how these indicators are affected by the signal directionality we have considered in addition to the case of isotropic emission processes also strongly anisotropic ones like narrow beams pointing in random directions and rotating beacons.", "Under the assumption that the emission birthrates did not change during the recent history of the Galaxy, we have shown that $\\bar{k}=N_D$ only for isotropic processes and for emissions originating from rotating beacons sweeping the galactic disk.", "In all the other cases considered (beamed signals directed randomly and lighthouses with tilted rotation axis) $\\bar{k}$ can be orders of magnitudes smaller than the Drake number, showing that $N_D$ may largely overestimate the possible occurrence of signals that can be remotely detected.", "We have further discussed at length $N_G$ as the proper indicator of the galactic abundance of technosignatures.", "We have shown that $N_G$ , leaving aside the special case of narrow beams directed uniformly in 3D space, is only marginally affected by the signal directionality.", "A central result of the present study is that $N_G$ becomes independent of the signal longevity if this is shorter than about $10^5$ ly, yielding therefore a measure of the abundance of galactic technosignatures that depends only on the emission birthrate." ], [ "Acknowledgements", "The author thanks Amedeo Balbi and Geoffrey W. Marcy for fruitful discussions.", "The data underlying this article are available in the article." ], [ "$\\bar{k}$ for random beams", "Let us consider an emitter located at $\\mathbf {r}$ transmitting since a time $t$ before present a conical beam of aperture $\\alpha _0$ [13], [19].", "During the entire lifetime $L$ of the emission, the beam axis is held oriented along the direction of a unit vector $\\hat{n}$ .", "The region of space filled by the radiation is the intersection between a cone of apex at $\\mathbf {r}$ and a spherical shell centred on the cone apex with outer radius $ct$ and thickness $cL$ .", "As done for the isotropic case, we neglect the internal structure of this region arising in the case of an intermittent beam.", "The angular sector formed by the overlap of the conical beam with the galactic plane (grey region in Fig.", "REF ) subtends the angle $\\beta =\\left\\lbrace \\begin{array}{ll}2\\arccos \\!\\left[\\dfrac{\\cos (\\alpha _0/2)}{\\sin (\\theta )}\\right], & \\vert \\theta -\\pi /2\\vert \\le \\alpha _0/2 \\\\ [8pt]0, & \\textrm {otherwise}\\end{array}\\right.$ where $\\theta \\in [0, \\pi ]$ is the angle formed by $\\hat{n}$ with the $z$ -axis.", "From this construction, we se that the beam will cross the Earth if $\\mathbf {r}_\\textrm {E}$ is located within the angular sector, that is, if Eq.", "(REF ) is satisfied and $\\vert \\phi \\vert \\le \\beta /2$ , where $\\phi $ is the angle formed by $\\mathbf {r}_\\textrm {E}-\\mathbf {r}$ and the projection of $\\hat{n}$ on the $x$ -$y$ plane, Fig.", "REF .", "Figure: Schematic illustration of a beamed emission of duration LL and aperture α 0 \\alpha _0.The Earth and the emitter positions are denoted respectively by the vectors 𝐫\\mathbf {r} and𝐫 E \\mathbf {r}_\\textrm {E}.", "The beam axis of the lighthouse is oriented along the direction n ^\\hat{n} and forms an angle θ\\theta withthe zz-axis.", "φ\\phi is the angle formed by the direction of 𝐫-𝐫 E \\mathbf {r}-\\mathbf {r}_\\textrm {E} and the projectionof n ^\\hat{n} over the galactic disk.The grey region denotes the overlap area between the galactic disk and the beam.By adopting a constant birthrate of beamed signals with random orientations of $\\hat{n}$ , the integration over $t$ under the condition (REF ) yields the factor $L$ , as in Eq.", "(REF ).", "Introducing the random beam (rb) emission rate $\\Gamma _\\textrm {rb}$ and the corresponding average longevity $\\bar{L}_\\textrm {rb}$ , the mean number of beamed signals crossing Earth reduces therefore to: $\\bar{k}_\\textrm {rb}(\\alpha _0)=\\Gamma _\\textrm {rb}\\bar{L}_\\textrm {rb}\\int \\!d\\hat{n}\\,g(\\hat{n})\\Theta (\\beta /2-\\vert \\phi \\vert ),$ where $d\\hat{n}=d\\phi d\\theta \\sin \\theta $ , $g(\\hat{n})$ is the PDF of the direction of $\\hat{n}$ , and $\\Theta (x)=1$ for $x\\ge 0$ and $\\Theta (x)=0$ or $x<0$ is the unit step function.", "In the case in which the beams are oriented uniformly in three dimensions (3D), the PDF of $\\hat{n}$ is $g(\\hat{n})=1/4\\pi $ and using Eq.", "(REF ) the integration over $\\hat{n}$ yields $\\frac{1}{2}[1-\\cos (\\alpha _0/2)]$ , which is simply the fractional solid angle subtended by the beam [19].", "For beam directions distributed uniformly over the two-dimensional (2D) galactic plane, $g(\\hat{n})$ is a Dirac-delta function peaked at $\\theta =\\pi /2$ , $g(\\hat{n})=\\delta (\\theta -\\pi /2)/2\\pi $ , and the orientational average reduces simply to $\\alpha _0/(2\\pi )$ .", "For random 3D and 2D beam orientations we obtain therefore: $\\bar{k}_\\textrm {rb}(\\alpha _0)=\\left\\lbrace \\begin{array}{ll}\\dfrac{1-\\cos (\\alpha _0/2)}{2}\\Gamma _\\textrm {rb}\\bar{L}_\\textrm {rb}, & \\textrm {3D random beams}, \\\\[8pt]\\dfrac{\\alpha _0}{ 2\\pi }\\Gamma _\\textrm {rb}\\bar{L}_\\textrm {rb}, & \\textrm {2D randombeams}.\\end{array}\\right.$ Under the assumption that the beams have angular apertures distributed over small values of $\\alpha _0$ , Eq.", "(REF ) reduces to: $\\bar{k}_\\textrm {rb}=\\langle \\bar{k}_\\textrm {rb}(\\alpha _0)\\rangle \\simeq \\left\\lbrace \\begin{array}{ll}\\dfrac{\\langle \\alpha _0^2\\rangle }{16}\\Gamma _\\textrm {rb}\\bar{L}_\\textrm {rb}, & \\textrm {3D random beams}, \\\\[8pt]\\dfrac{\\langle \\alpha _0\\rangle }{ 2\\pi }\\Gamma _\\textrm {rb}\\bar{L}_\\textrm {rb}, & \\textrm {2D random beams},\\end{array}\\right.$ where $\\langle \\cdots \\rangle $ denotes an average over the $\\alpha _0$ values." ], [ "$\\bar{k}$ for lighthouses", "We take a lighthouse (lh) located at $\\mathbf {r}$ that started transmitting at a time $t$ in the past and for a duration $L$ a conical beam of angular aperture $\\alpha _0$ .", "The effective volume encompassing the region physically filled by the EM radiation is formed by the overlap between the volume swept by a cone (of aperture $\\alpha _0$ ) rotating about the spin axis $\\hat{n}$ with apex at $\\mathbf {r}$ and a spherical shell concentric to $\\mathbf {r}$ of outer radius $ct$ and thickness $cL$ (see Fig.", "REF ).", "The overlap of the effective volume with the galactic plane forms two annular sectors (shown by grey areas in Fig.", "REF ), symmetric with respect to the line of nodes formed by the intersection of the rotation plane with the galactic disk, and subtending the angle $\\alpha $ given by $\\alpha =\\left\\lbrace \\begin{array}{ll}\\pi , & 0\\le \\theta \\le \\alpha _0/2, \\\\ [8pt]2\\arcsin \\!\\left[\\dfrac{\\sin (\\alpha _0/2)}{\\sin (\\theta )}\\right], & \\alpha _0/2<\\theta \\le \\pi /2,\\end{array}\\right.$ where now $\\theta \\in [0, \\pi /2]$ is the angle formed by $\\hat{n}$ with the $z$ direction.", "As done for the case of random beamed signals, the intersection of the angular sector with the Earth is given by the condition (REF ) and $\\vert \\phi \\vert \\le \\alpha /2$ , where $\\phi $ is the angle formed by $\\mathbf {r}_\\textrm {E}-\\mathbf {r}$ and the line of nodes (see Fig.", "REF ).", "For a constant birthrate of the lighthouses, the average number of randomly distributed spiralling beams crossing Earth is thus given by: $\\bar{k}_\\textrm {lh}(\\alpha _0)=\\Gamma _\\textrm {lh}\\bar{L}_\\textrm {lh}\\int \\!d\\hat{n}\\,g(\\hat{n})\\Theta (\\alpha /2-\\vert \\phi \\vert ),$ If $\\hat{n}$ is distributed uniformly over the unit sphere (3D case), using Eq.", "(REF ) and $g(\\hat{n})=1/4\\pi $ , the integral over $\\theta $ reduces exactly to $\\sin (\\alpha _0/2)$ for $\\alpha _0\\le \\pi $ and 1 otherwise.", "In the case the spin axis is perpendicular to the galactic plane (2D case), $g(\\theta )$ is a Dirac-delta peak at $\\theta =0$ , so that the angular average in Eq.", "(REF ) yields 1.", "Expanding Eq.", "(REF ) for small beam apertures, $\\bar{k}_\\textrm {lh}=\\langle \\bar{k}_\\textrm {lh}(\\alpha _0)\\rangle $ reads: $\\bar{k}_\\textrm {lh}=\\left\\lbrace \\begin{array}{ll}\\dfrac{\\displaystyle \\langle \\alpha _0\\rangle }{\\displaystyle 2}\\Gamma _\\textrm {lh}\\bar{L}_\\textrm {lh}, & \\textrm {3D lighthouses}, \\\\\\Gamma _\\textrm {lh}\\bar{L}_\\textrm {lh}, & \\textrm {2D lighthouses}.\\end{array}\\right.$ Figure: Top view of the galactic disk of radius R G R_G with superimposed areas covered by emissions of different geometries originated by a sourcelocated at (r,0)(r,0).", "(a): The intersection of a beam with the xx-yy plane forms an annular sector (orange region) subtendingan angle β\\beta , Eq.", "(), and an inner arc at distance c(t-L)c(t-L) from the emitter.", "ℓ(r,ϕ)\\ell (r,\\varphi ) is the distance from the emitter to the edge of the Galaxyalong the direction of the beam axis projected on the xx-yy plane.", "A narrow beam intersects the galactic disk when c(t-L)≤ℓ(r,ϕ)c(t-L)\\le \\ell (r,\\varphi ).", "(b): The two annular sectors of angle α\\alpha ,Eq.", "(), denote the overlap area between the xx-yy plane and the effective volume spanned by a rotating lighthouse with spin axis tilted with respectto the zz-axis (see Fig. ).", "The intersection of the galactic disk with the rotation plane forms two segments of length ℓ(r,ϕ)\\ell (r,\\varphi ) andℓ(r,ϕ+π)\\ell (r,\\varphi +\\pi ).", "The longest of them must be larger than c(t-L)c(t-L) in order for the effective volume to intersect the galactic disk." ], [ "$N_G$ for random beams", "In deriving the average number of random beams intersecting the Galaxy, $N_G^\\textrm {rb}$ , we shall retain only the contributions at the lowest order in the beam aperture $\\alpha _0$ , which simplifies considerably the calculation.", "We take an emitter to be located along the $x$ -axis, $\\mathbf {r}=(r,0)$ , with the beam axis directed along $\\hat{n}$ forming an azimuthal angle $\\varphi $ with $\\mathbf {r}$ , Fig.", "REF (a).", "The projection of the beam axis on the $x$ -$y$ plane defines the distance $\\ell (r,\\varphi )=\\sqrt{R_G^2-r^2\\sin (\\varphi )^2}-r\\cos (\\varphi )$ measured from the emitter position to the edge of the Galaxy (i.e.", "the circle of radius $R_G$ ).", "As seen from Eq.", "(REF ), the intersection of the beam with the $x$ -$y$ plane is non-null only if the polar angle of $\\hat{n}$ is such that $\\vert \\theta -\\pi /2\\vert \\le \\alpha _0/2$ .", "Since $\\alpha _0\\ll 1$ , $\\hat{n}$ lies approximately on the $x$ -$y$ plane and a beam emitted at time $t$ for a duration $L$ will intersect the galactic disk only when $c(t-L)$ is smaller than $\\ell (r,\\varphi )$ , as shown in Fig.", "REF (a).", "Conversely, if $\\vert \\theta -\\pi /2\\vert > \\alpha _0/2$ the only beams that intersect the galactic disk are those that are still being transmitted at the present time, that is, those such that $t<L$ .", "After integration over $t$ , $N_G^\\textrm {rb}$ at the lowest order in $\\alpha _0$ is therefore given by: $N_G^\\textrm {rb}&=\\int \\!dL\\rho _\\textrm {rb}(L)\\int \\!", "d\\mathbf {r}\\gamma _\\textrm {rb}(\\mathbf {r})\\int \\!d\\hat{n}\\, g(\\hat{n})\\nonumber \\\\&\\times \\left[L+\\Theta \\left(\\frac{\\alpha _0}{2}-\\left\|\\theta -\\frac{\\pi }{2}\\right\|\\right)\\frac{\\ell (r,\\varphi )}{c}\\right]\\nonumber \\\\&=\\Gamma _\\textrm {rb}\\bar{L}_\\textrm {rb}+t_G\\frac{\\eta }{\\pi }\\int \\!d\\mathbf {r}\\gamma _\\textrm {rb}(\\mathbf {r})E(r/R_G),$ where $\\eta =\\alpha _0/2$ or $\\eta =1$ if the direction of $\\hat{n}$ is distributed uniformly in 3D space or in the $x$ -$y$ plane, and $E(x)=\\int _0^{\\pi /2}\\!d\\varphi \\,\\sqrt{1-x^2\\sin (\\varphi )^2}$ is the complete elliptic integral of the second kind.", "Using a birthrate that is constant over the galactic disk, the integration over $\\mathbf {r}$ yields: $N_G^\\textrm {rb}=\\left\\lbrace \\begin{array}{ll}\\Gamma _\\textrm {rb}\\left(\\bar{L}_\\textrm {rb}+\\dfrac{\\langle \\alpha _0\\rangle }{2}\\dfrac{4}{3\\pi }t_G\\right), & \\textrm {3D random beams} \\\\[8pt]\\Gamma _\\textrm {rb}\\left(\\bar{L}_\\textrm {rb}+\\dfrac{4}{3\\pi }t_G\\right), & \\textrm {2D random beams},\\end{array}\\right.$ from which we see that $N_G^\\textrm {rb}$ of narrow beams depends on the angular aperture only in the 3D case." ], [ "$N_G$ for lighthouses", "In the case the spin axis of a rotating beacon is parallel to the $z$ -axis, the effective volume encompassing the radiation (Fig.", "REF ) intersects the Galaxy as long as $c(t-L)$ is smaller than the maximum distance of the emitter from the galactic edge, in full equivalence with the isotropic case.", "Assuming a spatially uniform birthrate, the number of rotating beacon signals intersecting the Milky Way is thus $N_G^{lh}=\\Gamma _\\textrm {lh}(\\bar{L}_\\textrm {lh}+\\frac{5}{6}t_G)$ , as in Eq.", "(REF ).", "In the more general case in which $\\hat{n}$ forms an angle $\\theta $ with the $z$ -direction, at the lowest order in $\\alpha _0$ it suffices to calculate $N_G^{lh}$ by considering the intersection of the rotation plane with the galactic disk, which forms an angle $\\varphi $ with the $x$ -axis.", "As shown in Fig.", "REF (b), the emitter, located at $(r,0)$ , cuts the intersection line in two segments, generally of different lengths.", "The longest of these segments has length $\\ell _\\textrm {max}(r,\\varphi )=\\ell (r,\\varphi +\\pi )$ for $0\\le \\varphi \\le \\pi /2$ and $\\ell _\\textrm {max}(r,\\varphi )=\\ell (r,\\varphi )$ for $\\pi /2\\le \\varphi \\le \\pi $ , where $\\ell (r,\\varphi )$ is given in Eq.", "(REF ).", "Since a non-null intersection with the galactic disk is obtained by requiring the inner edge of the effective volume, $c(t-L)$ , to be smaller than $\\ell _\\textrm {max}(r,\\varphi )$ , we obtain: $N_G^\\textrm {lh}&=\\!\\int \\!\\!dL\\rho _\\textrm {lh}(L)\\!\\int \\!\\!", "d\\mathbf {r}\\gamma _\\textrm {lh}(\\mathbf {r})\\!\\int \\!d\\hat{n}\\, g(\\hat{n})\\!\\left[L+\\frac{\\ell _\\textrm {max}(r,\\varphi )}{c}\\right]\\nonumber \\\\&=\\Gamma _\\textrm {lh}\\bar{L}_\\textrm {lh}+\\frac{t_G}{\\pi }\\int \\!d\\mathbf {r}\\gamma _\\textrm {lh}(\\mathbf {r})[E(r/R_G)+r/R_G],$ where we have assumed that the orientation of $\\hat{n}$ is distributed uniformly over 3D.", "For a spatially uniform $\\gamma _\\textrm {lh}(\\mathbf {r})$ the integration over $\\mathbf {r}$ yields $2\\Gamma _\\textrm {lh}$ , so that for the two cases examined ($\\hat{n}$ random and $\\hat{n}\\parallel z$ ) $N_G^\\textrm {lh}$ reduces to: $N_G^\\textrm {lh}=\\left\\lbrace \\begin{array}{ll}\\Gamma _\\textrm {lh}\\left(\\bar{L}_\\textrm {lh}+\\dfrac{2}{\\pi }t_G\\right), & \\textrm {3D lighthouses} \\\\[8pt]\\Gamma _\\textrm {lh}\\left(\\bar{L}_\\textrm {lh}+\\dfrac{5}{6}t_G\\right), & \\textrm {2D lighthouses}.\\end{array}\\right.$" ] ]	2011.14147
[ [ "Inter-slice Context Residual Learning for 3D Medical Image Segmentation" ], [ "Abstract Automated and accurate 3D medical image segmentation plays an essential role in assisting medical professionals to evaluate disease progresses and make fast therapeutic schedules.", "Although deep convolutional neural networks (DCNNs) have widely applied to this task, the accuracy of these models still need to be further improved mainly due to their limited ability to 3D context perception.", "In this paper, we propose the 3D context residual network (ConResNet) for the accurate segmentation of 3D medical images.", "This model consists of an encoder, a segmentation decoder, and a context residual decoder.", "We design the context residual module and use it to bridge both decoders at each scale.", "Each context residual module contains both context residual mapping and context attention mapping, the formal aims to explicitly learn the inter-slice context information and the latter uses such context as a kind of attention to boost the segmentation accuracy.", "We evaluated this model on the MICCAI 2018 Brain Tumor Segmentation (BraTS) dataset and NIH Pancreas Segmentation (Pancreas-CT) dataset.", "Our results not only demonstrate the effectiveness of the proposed 3D context residual learning scheme but also indicate that the proposed ConResNet is more accurate than six top-ranking methods in brain tumor segmentation and seven top-ranking methods in pancreas segmentation.", "Code is available at https://git.io/ConResNet" ], [ "Introduction", "Medical images provide visual representations of the anatomy or function of living bodies, which are essential for clinical analysis and medical intervention.", "Segmentation of 3D medical images aims to predict the semantic category (i.e.", "a specific organ or lesion) of each voxel, and is a fundamental and critical task in computer aided diagnosis (CAD), since accurate segmentation of organs or tumors is of value not only in facilitating the diagnosis but also in assessing the severity and prognosis of the disease.", "This segmentation task, however, is extremely challenging due to the low soft tissue contrast and the heterogeneity of organs and tumors in shape, size, and location.", "Moreover, since the acquisition and annotation of medical data are expensive, there is usually a lack of sufficient annotated data to train segmentation models, which renders an even more challenging task for 3D medical image segmentation[1], [2], [3].", "Recent years have witnessed the amazing success of deep convolutional neural networks (DCNNs) in image segmentation.", "Many attempts have been made to strengthen the ability of DCNNs to medical image segmentation.", "For instance, the encoder-decoder architecture has been improved in various ways to keep the low-level detailed information and obtain sharp object boundaries [4], [5], [6], the spatial pyramid pooling has been used to exploit the multi-scale information [6], the atrous convolution has been incorporated into segmentation models to expand the receptive field efficiently [7], and several attention learning mechanisms have been introduced to segmentation models, enabling them to focus more on certain locations and / or channels [8], [9].", "On 3D medical image segmentation tasks, 3D DCNNs have demonstrated striking improvements over their 2D counterparts [10], [11], [12], [13], since they are able to explore the contextual information contained across slices, which contributes significantly to better segmentation performance.", "We therefore advocate that the accuracy of 3D medical image segmentation can be further improved by capturing and using the inter-slice contextual information more effectively.", "However, it is difficult to capture the inter-slice context information in volumetric images with complex anatomical structures.", "Take a brain tumor in a magnetic resonance (MR) sequence for example.", "In Fig.", "REF , the first row shows four adjacent slices sampled from the sequence in which there is a brain tumor, the second row gives the ground truth tumor region in each slice, and the voxel-wise difference of the tumor region between any two adjacent slices, including extending outward regions or contracting inward regions, is displayed in the third row.", "We define such difference as the inter-slice context residual, which appears on or near the tumor surface and contains the essential and intriguing morphological information of the tumor, since we can use it, together with the tumor region in any slice, to reconstruct the shape of the 3D tumor.", "Intuitively, exploring the inter-slice context residual in a segmentation process must be beneficial to improving the accuracy.", "Unfortunately, since the tumor regions in two adjacent slices have the similar shape and size, the inter-slice context residual is usually tiny, and hence has never been characterized directly.", "In this paper, we propose the 3D context residual network (ConResNet) for accurate segmentation of 3D medical images.", "ConResNet has an encoder-decoder architecture, containing an encoder for feature extraction and two decoders for the generation of segmentation masks and inter-slice context residuals, respectively.", "The context residual (ConRes) decoder takes the residual feature maps of adjacent slices produced by the segmentation decoder as its input, and also provides feedback to the segmentation decoder as a kind of attention guidance, aiming to boost the ability of the segmentation model to perceive and use the inter-slice context information effectively (see Fig.", "REF ).", "The design of ConRes decoder is conceptually generic and compatible with any existing 3D DCNN-based medical image segmentation model.", "We evaluate the proposed ConResNet model on the MICCAI 2018 Brain Tumor Segmentation (BraTS) dataset and NIH pancreas segmentation dataset and achieve the state-of-the-art performance on both segmentation tasks.", "The main contributions of this paper are summarized as follows: We suggest adding the ConRes decoder to an encoder-decoder structure to explicitly boost the model's ability to 3D context perception and thus improve the segmentation accuracy.", "We design the context residual module, which is used between the segmentation decoder and ConRes decoder at each scale, for simultaneous context residual mapping and context attention mapping.", "We propose an accurate 3D medical image segmentation model called ConResNet, which achieves improved performance over state-of-the-art methods on both brain tumor segmentation task and pancreas segmentation task.", "Figure: (a) Diagram of the proposed ConResNet model, which has an encoder-decoder architecture, including a shared encoder (orange), a segmentation decoder (blue), and a ConRes decoder (green).", "(b) Context residual module.The context of an object provides the information of its surroundings, and hence is essential for semantic segmentation.", "Many strategies have been developed for context learning, which can be roughly grouped into three categories.", "First, to explore the context information at multiple scales, Zhao et al.", "[14] employed the pyramid spatial pooling strategy at different scales to aggregate multi-scale global information.", "Chen et al.", "[6] introduced the atrous spatial pyramid pooling with different dilated rates to parallel branches, which perform multi-scale representation aggregation.", "Second, to expand the receptive field, Yu et al.", "[7] replaced the conventional convolution by the atrous convolution with an adjustable dilated rate, which shows superior performance on many computer vision tasks, like segmentation [15], detection[16], and super-resolution[17].", "Peng et al.", "[18] utilized large kernels to capture rich global context information, which is beneficial for dense per-pixel prediction tasks.", "Third, many attention based methods were proposed to filter out the extraneous information.", "Hu et al.", "[19] recalibrated adaptively channel-wise feature responses by explicitly modeling the channel-wise interdependencies of convolutional features.", "Wang et al.", "[20] designed an attention module with a bottom-up and top-down feed-forward structure to learn soft attention weights.", "In our previous work [21], we exploited the intrinsic self-attention ability of deep networks by using the high-level features to generate the attention map for low-level layers.", "These strategies have demonstrated their effectiveness in many 2D scenarios, and some of them have been extended to 3D cases.", "Wang et al.", "[22] utilized a 3D fully convolutional networks (FCN) with group dilated convolutions to segment the prostate in MR sequences.", "Schlemper et al.", "explored a 3D attention gated network which highlights task-related salient regions by embedding the attention mechanism into a U-like segmentation network for 3D organ segmentation.", "We also attempted to use the 3D atrous spatial pyramid pooling to capture multi-scale information for liver tumor segmentation in CT volumes [23].", "Although improving the performance, these 2D and 3D strategies fail to characterize directly the inter-slice context residual information, which is particularly important for 3D medical image segmentation." ], [ "Residual learning", "Residual learning can be traced back to the seminal work by He et al.", "[24], Since then, it has been successfully applied to many computer vision tasks, including classification, segmentation, detection, and tracking.", "The idea of residual learning is to learn the residual between the input and output feature maps using the skip connection that directly jumps from input to output.", "With the skip connection, residual learning eliminates the degradation problem, and hence makes it possible to train very deep networks with up to hundreds, even thousands, of layers.", "In this work, the proposed ConRes decoder targets at perceiving the feature residual between two adjacent slices, and thus strengthens the model's ability to use the 3D context information for segmentation." ], [ "Medical image segmentation", "Recently, both 2D and 3D DCNNs have become increasingly prevalent tools for medical image segmentation [25], [26], [27], [23], [28], [29].", "Ronneberger et al.", "[4] proposed a U-like architecture (UNet) that consists of an encoder path to capture segmentation-related high-level semantics and a symmetric decoder with skip connections from the encoder to generate segmentation results, and achieved excellent performance on several 2D medical image segmentation tasks.", "Fu et al.", "[30] designed a 2D M-Net, which combines a multi-scale U-like network with a side-output layer, and boosted the performance of optic disc and cup segmentation.", "Chen et al.", "[31] utilized a 2D deep contour-aware network to harness the multi-level contextual feature representation in an end-to-end way for effective gland segmentation.", "When applied to 3D medical images, these 2D models perform the segmentation task in a slice-by-slice manner, and hence cannot capture inter-slice context information, leading to limited segmentation accuracy [32].", "With 3D convolutions, 3D DCNNs can directly process volumetric data and thus have distinct advantages over 2D DCNNs in 3D medical image segmentation.", "Dou et al.", "[11] designed a 3D fully convolutional network to generate high-quality score maps for automated 3D liver segmentation.", "Li et al.", "[33] incorporated a multi-scale context module into a 3D segmentation network to use the multi-scale contextual information for inter-vertebral discs segmentation.", "Meanwhile, Chen et al.", "[34] integrated the contour length and region constraints into the loss function to enforce the smoothness of segmented regions.", "Karimi and Salcudean [35] introduced a Hausdorff distance based loss function to minimise the maximum deviation between the prediction and ground truth surfaces of targets.", "Previously, we developed a light-weight hybrid convolutional network which replaces the 3D convolutions at bottom of network with low-cost 2D convolutions to reduce the model parameters and improve the segmentation performance with only the limited training data [23].", "Instead of designing a new segmentation model, we propose to improve the accuracy of existing 3D DCNNs via adding the ConRes decoder to them, which is able to capture and use the inter-slice context residual information.", "Let a 3D medical image be denoted by $\\textbf {X} \\in \\textbf {R}^{S\\times H \\times W}$ , where $S$ is the number of slices, and $H$ and $W$ are height and width of each slice, respectively.", "Its ground-truth segmentation mask is denoted by $\\textbf {Y}_{seg} \\in \\textbf {R}^{S\\times H \\times W}$ , in which the semantic label of each voxel is $\\textbf {Y}_{seg}^{s, h, w} = \\lbrace 0: background, 1: object\\rbrace $ .", "Its ground-truth context residual mask is denoted by $\\textbf {Y}_{res} \\in \\textbf {R}^{S\\times H \\times W}$ , in which each element is calculated as $\\textbf {Y}_{res}^{s+1, h, w} = \\left\| \\textbf {Y}_{seg}^{s+1, h, w} - \\textbf {Y}_{seg}^{s, h, w} \\right\|$ where $\\textbf {Y}_{res}^{s+1, h, w}=0$ means that the voxels at $(h,w)$ on two adjacent ($s^{th}$ and $s+1^{th}$ ) slices belong to the same semantic categories, i.e., both of them are background or foreground, and $\\textbf {Y}_{res}^{s+1, h, w}=1$ means that one voxel is background and the other is foreground.", "The proposed ConResNet aims to predict the segmentation mask $\\textbf {Y}_{seg}$ and residual mask $\\textbf {Y}_{res}$ simultaneously, formally shown as follows $\\textbf {P}_{seg}, \\textbf {P}_{res} = f(\\textbf {X};\\mathbf {\\theta })$ where $\\textbf {P}_{seg} \\in \\textbf {R}^{S\\times H \\times W}$ is the segmentation prediction of $\\textbf {X}$ , $\\textbf {P}_{res} \\in \\textbf {R}^{S\\times H \\times W}$ is the residual prediction of $\\textbf {X}$ , and $\\mathbf {\\theta }$ represents the parameters of ConResNet.", "Accordingly, ConResNet has a shared encoder for feature extraction and two decoders for the prediction of $\\textbf {Y}_{seg}$ and $\\textbf {Y}_{res}$ , respectively.", "The context residual module is designed to bridge both decoders, which is embedded in each pair of layers of dual decoders.", "The semantic features from the segmentation decoder are transformed to the context residual features through a new defined operation, called context residual mapping, and input to the ConRes decoder for the refinement.", "Besides, the ConRes decoder, in turn, provides an attention guidance via the context attention mapping to strengthen the 3D context perceiving ability of the segmentation decoder.", "The diagram of ConResNet is illustrated in Fig.", "REF .", "We now delve into the details of this model." ], [ "Shared Encoder", "In the proposed ConResNet, the shared encoder consists of nine residual blocks, each consisting of two $3\\times 3\\times 3$ convolutional layers and a skip connection from input to output.", "Due to the limited GPU memory, we have to train the network with very small batch size, like one sample per GPU.", "Therefore, to speed up the training process, we use the group normalization [36] with a group number of eight, which is insensitive to the scale of batch size, to adjust and scale the activation of each layer.", "Besides, we use the weight standardization algorithm [37] to accelerate micro-batch training by normalizing the weights of convolutional layers.", "As shown in Fig.", "REF , the encoding process can be divided into five stages.", "In the first stage, the input is processed by a convolutional layer with 32 kernels and a residual block.", "In each of next three stages, the data is processed by a convolutional layer with doubled kernels and a stride of 2 and two residual blocks.", "Thus, we gradually down-sample the feature maps to 1/8 of the input size and, simultaneously, increase the number of channels from 32 to 256, leading to the expansion of receptive field and reduction of computation.", "In the last stage, we employ two residual blocks, which use the atrous convolution with a dilated rate of 2, to further expand the receptive field while keeping the feature resolution for more details of the shape and edge." ], [ "Dual Decoders", "Our ConResNet contains two decoders, i.e., a segmentation decoder and a ConRes decoder.", "The generation of a segmentation masks by the segmentation decoder or a residual mask by the ConRes decoder consists of three stages.", "In each stage, we propose a context residual module to bridge the segmentation decoder and ConRes decoder.", "Specifically, the segmentation decoder first uses the trilinear interpolation to up-sample the previous feature maps, and then fuses them with the low-level features passing from the encoder using element-wise summation.", "The context residual of fused features are transmitted to the ConRes decoder.", "The ConRes decoder fuses the context residual features passing from the segmentation decoder and the features passing from the previous layer, and then refine them to predict the residual mask.", "Moreover, the inter-slice context residual information generated by the ConRes decoder is transmitted back to the segmentation decoder as an attention guidance to boost the ability of the segmentation decoder to perceive 3D context.", "To match the number of channels in the encoder, we keep halving the channels after each up-sampling operation.", "As a result, we obtain a segmentation mask and the corresponding inter-slice context residual mask for each input volume.", "Figure: Illustration of the context residual in images or feature maps.", "(a) Context residual of brain tumors (green) existing in two adjacent MR slices.", "(b) Context residual existing in the feature maps (extracted from output of encoder), where the left column gives the channel-wise average of the two adjacent feature maps along the inter-slice dimension.", "Note that each intensity value in the residual map indicates the level of dissimilarity between two voxels located at the same position in adjacent slices." ], [ "Context Residual Module", "Each object in a 3D medical image, such as an organ or a tumor, gradually changes cross-sectional areas on 2D slices.", "As highlighted by those white pixels in Fig.", "REF (a), the context residual indicates the discrepancy between the cross-sectional areas on two adjacent slices, which is usually tiny.", "We suspect that such discrepancy can also be found in the feature maps of adjacent slices produced the segmentation decoder.", "To verify this, we visualize the channel-wise average of the feature maps of two adjacent slices and their difference in Fig.", "REF (b)).", "It shows that those two average feature maps look similar in most regions and the dissimilarity only appear in a small fraction of the slice, which is similar to the pattern of context residual shown in Fig.", "REF (a).", "Therefore, we design a context residual module to capture the context residual based on the feature maps produced the segmentation decoder.", "Fig.", "REF (b) shows the diagram of the context residual module.", "It is composed of two main paths, i.e., a segmentation path (top) and a context residual path (down).", "In the segmentation path, the input is the element-wise summation of the segmentation features $\\textbf {I}_{seg}$ coming from the previous layer and the low-level features $\\textbf {I}_{skip}$ passing from the same-scale layer in the encoder, and the output is the segmentation feature $\\textbf {O}_{seg}$ .", "In the context residual path, the input is the context residual features $\\textbf {I}_{res}$ coming from the previous layer and the output is the context residual features $\\textbf {O}_{res}$ .", "We design two strategies, i.e., context residual mapping and context attention mapping, to connect two paths for better context perceiving and semantic segmentation.", "Context residual mapping: We use a weighted layer to refine the element-wise summation of $\\textbf {I}_{seg}$ and $\\textbf {I}_{skip}$ , formally shown as $\\textbf {F} = {\\rm Conv}( \\textbf {I}_{seg} \\oplus \\textbf {I}_{skip}),$ where $\\oplus $ represents the element-wise summation, ${\\rm Conv}$ represents the weighted layer, including convolution, group normalization, and ReLU activation.", "The obtained feature map $\\textbf {F}$ has a size of $S^{\\prime } \\times H^{\\prime } \\times W^{\\prime }$ , where $S^{\\prime }, H^{\\prime }, W^{\\prime }$ represents the inter-slice depth, spatial height and width, respectively.", "The context residual $\\textbf {G}$ is calculated as the position-wise absolute difference between each pair of adjacent features along the inter-slice dimension $S$ , shown as follows $\\textbf {G}^{s+1, h,w} = \|\\sigma (\\textbf {F}^{s+1, h,w}) - \\sigma (\\textbf {F}^{s, h,w})\|,$ where $\\sigma (\\cdot )$ is the sigmoid function.", "Note that $\\textbf {G}$ has a size of $(S^{\\prime }-1)\\times H^{\\prime } \\times W^{\\prime }$ .", "For the convenience of subsequent processing, we pad the first-slice of $\\textbf {G}$ to make it the same size of $\\textbf {F}$ .", "Then, we combine the generated context residual feature map $\\textbf {G}$ with $\\textbf {I}_{res}$ from the previous context residual layer as the output of context residual path, shown as follows $\\textbf {O}_{res} = {\\rm Conv}({\\rm Conv}(\\textbf {G}) \\oplus \\textbf {I}_{res}).$ Context attention mapping: The context residual mapping generates the inter-slice context residual feature map and feeds it to the residual path for the context residual learning.", "In the meantime, the output of context residual path can be used as a kind of attention to boost the context perceiving ability of the segmentation path.", "To this end, we introduce the simple but effective context attention mapping.", "We apply the sigmoid function to the context residual feature map $\\textbf {O}_{res} $ and obtain the context attention weights, which is employed as an attention gate to activate the context residual regions and prompt the sensitivity of 3D context variance.", "Since the context attention highlights only the residual region (see Fig.", "REF ), directly applying the context attention weights to the feature maps may attenuate both the background and foreground.", "Hence, we define the context attention weighted output of the segmentation path as $\\textbf {O}_{seg} = \\textbf {F} \\otimes (1+\\sigma (\\textbf {O}_{res}))$ where $\\otimes $ means the element-wise multiplication." ], [ "Network Optimization", "For the segmentation decoder, the loss function is defined as the following combination of the cross entropy loss and Dice loss $\\small L_{seg} = \\sum _{s,h,w} l_{bce}(\\textbf {P}_{seg}^{s,h,w}, \\textbf {Y}_{seg}^{s,h,w}) - \\frac{2 \\sum _{s,h,w} \\textbf {P}_{seg}^{s,h,w} \\textbf {Y}_{seg}^{s,h,w}}{\\sum _{s,h,w}(\\textbf {P}_{seg}^{s,h,w} + \\textbf {Y}_{seg}^{s,h,w}) + \\epsilon }$ where $l_{bce}$ is the binary cross entropy loss, $\\textbf {P}_{seg}^{shw}$ and $\\textbf {Y}_{seg}^{shw}$ are the predicted and ground-truth segmentation label at $(s, h, w)$ , respectively, and $\\epsilon $ is a smooth factor in the Dice loss.", "Training the ConRes decoder is supervised by the ground-truth context residual mask $\\textbf {Y}^{res}$ .", "The deep supervision technique [11] (see Fig.", "REF ) is adopted to accelerate the convergence.", "Note that an additional convolutional layer is employed to predict the residual masks for each deep supervision operation.", "Since the foreground and background voxels are highly imbalanced in each context residual mask, we set a weight for foreground voxels in the binary cross entropy loss function, shown as follows $L_{res}^{(i)} = \\sum _{s,h,w} w_k l_{bce}({\\textbf {P}_{res}^{s,h,w}}^{(i)}, \\textbf {Y}_{res}^{s,h,w})$ where $i$ ($=0, 1, 2$ ) represents $i$ -th context residual prediction, the class weight $w_k = \\log \\frac{V}{V_k}$ , and $V_k$ is the number of voxels belonging to class $k$ .", "The overall loss for context residual supervision is defined as $L_{res} = L_{res}^{0} + \\lambda (L_{res}^{1} + L_{res}^{2})$ where the trade-off parameter $\\lambda $ controls the influence of the loss in front layers and is empirically set to 0.5.", "Hence, the proposed ConResNet can be jointly optimized by the core segmentation and auxiliary context residual supervisions via minimizing the joint loss $L = L_{seg} + L_{res} $ in an end-to-end manner." ], [ "Datasets", "Two publicly available datasets were used for this study.", "BraTS Dataset: The BraTS dataset [51] was collected and shared by the MICCAI 2018 Brain Tumor Segmentation Challenge [51].", "The aim of this challenge is to develop automated segmentation algorithms to delineate intrinsically heterogeneous brain tumors, i.e., (1) enhancing tumor (ET), (2) tumor core (TC) that consists of ET, necrotic and non-enhancing tumor core, and (3) whole tumor (WT) that contains TC and the peritumoral edema.", "The BraTS dataset has 285 cases for training and 66 cases for online testing.", "Each case contains four MR sequences, including the T1, T1c, T2, and FLAIR.", "All sequences were registered to the same anatomical template and interpolated to the same size of $155\\times 240\\times 240$ voxels and the same voxel size of $1.0\\times 1.0\\times 1.0$ $mm^3$ .", "The voxel-wise segmentation ground truth of training cases are publicly available, but the ground truth of validation cases are withheld for online evaluation.", "Pancreas-CT Dataset: Pancreas-CThttps://wiki.cancerimagingarchive.net/display/Public/Pancreas-CT#82af2dca8f2443b1bef1e85ac73acd44 is the most authoritative open source dataset for pancreas segmentation, which is collected by the National Institutes of Health Clinical Center [52].", "This dataset contains 82 contrasted-enhanced abdominal 3D CT scans from 53 male and 27 female subjects.", "The size of each scan is $512\\times 512\\times (181\\sim 466)$ voxels and their voxel spacing varies from 0.5mm to 1.0mm.", "Following [45], [46], [47], [48], [49], [50], we adopted the 4-fold cross-validation and split randomly the dataset into four fixed and roughly equal folds.", "The average segmentation performance over all 82 cases was reported." ], [ "Implementation Details", "We implemented our ConResNet using PyTorch and performed all experiments on a workstation with two NVIDIA 2080Ti GPUs.", "During the training, we optimized ConResNet using the Adam algorithm with a batch size of 2 and weight decay of 0.0005.", "We set the initial learning rate to 0.0001 and decayed it according to a polynomial schedule $lr^{t}=lr^{0}\\times (1-t/T)^{0.9}$ where $t$ is the index of current iteration, and $T$ is the total number of iterations.", "To alleviate the overfitting, we employed simple online data augmentation techniques, including randomly scaling and flipping along three dimensions.", "On the BraTS dataset, we first normalized the voxel values in each MR sequence to the standard normal distribution, and then concatenated four available modalities into a multimodal volume with 4 channels [42].", "In the training stage, we randomly cropped sub-volumes of size $80\\times 160\\times 160$ as training samples.", "We treated this multi-class segmentation problem as three binary segmentation tasks.", "Accordingly, the last decoding layer of our ConResNet has three output channels, which use the sigmoid activation to produce the segmentation results of ET, WT, and TC, respectively.", "In this way, we optimized three sub-regions (i.e., ET, WT, and TC) directly.", "On the Pancreas-CT dataset, we truncated the HU values of all voxels to the range [-100, +240] to remove irrelevant information [48], and then linearly mapped HU values to [0, 1].", "We randomly cropped $64\\times 120\\times 120$ patches as training samples.", "Evaluation Metrics: For this study, we adopted the Dice coefficient and Hausdorff distance to measure the performance of all segmentation methods on the BraTS dataset [38], [41], [42], [43], [44], and adopted the mean Dice score, max Dice score, and min Dice score as golden performance indicators on the Pancreas-CT dataset [45], [46], [47], [48], [49], [50].", "The Dice coefficient is a statistic used to gauge the overlapping between a segmentation prediction $\\textbf {P}_{seg}$ and the corresponding ground truth $\\textbf {Y}_{seg}$ .", "Specifically, the Dice score is defined as $Dice = 2 \\times \\frac{\|\\textbf {P}_{seg} \\cap \\textbf {Y}_{seg}\|}{\|\\textbf {P}_{seg}\| + \|\\textbf {Y}_{seg} \| + \\epsilon }$ The Hausdorff distance evaluates the quality of segmentation boundaries by computing the maximum distance between the prediction and its ground truth, defined as follows $\\begin{aligned}HD = \\max \\lbrace &\\sup _{p\\in \\partial \\textbf {P}_{seg}} \\inf _{y\\in \\partial \\textbf {Y}_{seg}} \\left\\Vert p-y \\right\\Vert _2 , \\\\&\\sup _{y\\in \\partial \\textbf {Y}_{seg}} \\inf _{p\\in \\partial \\textbf {P}_{seg}} \\left\\Vert p-y \\right\\Vert _2 \\rbrace \\end{aligned}$ where $\\partial \\textbf {P}_{seg}$ and $\\partial \\textbf {Y}_{seg}$ represents the surface point sets of $\\textbf {P}_{seg}$ and $\\textbf {Y}_{seg}$ , respectively.", "A large Dice coefficient or a small Hausdorff distance indicates a more accurate segmentation result." ], [ "Results", "Comparison on the BraTS dataset: On the BraTS dataset, we compared the proposed ConResNet to six segmentation models top-ranked in the BraTS 2018 Challenge, including CascadeNet [38], DMFNet [39], OM-Net [40], DeepSCAN [41], VAE-Seg [42], EnsembleNets [43], nnUnet [44].", "CascadeNet [38] is a triple cascaded network, which hierarchically segments WT, TC and ET.", "OM-Net [40] separates multi-class segmentation tasks into one deep model and only requires one-pass computation for brain tumor segmentation.", "DMFNet [39] is built upon a multi-fiber unit, embedding the weighted 3D dilated convolution to gain multi-scale image representation for improved segmentation.", "DeepSCAN [41] is a densely connected segmentation model that uses dilated convolutions to increase the receptive field.", "VAE-seg [42] contains a variational auto-encoder to reconstruct the input image itself in order to regularize the segmentation tasks.", "EnsembleNets [43] is an ensemble of multiple segmentation models, each being specially designed to learn contextual and attentive information.", "nnUnet [44] incorporates with additional tricks into UNet to boost the segmentation performance.", "The Dice scores and Hausdorff distance obtained by using these models to segment three sub-regions, i.e., ET, WT, and TC, were given in TABLE REF .", "Note that the Dice scores and Hausdorff distances of compared methods were reported in the original papers.", "It shows that our ConResNet achieves the largest Dice score for ET segmentation and smallest Hausdorff distance for WT segmentation, whereas nnUnet achieves the largest Dice score for WT segmentation and smallest Hausdorff distance for ET segmentation.", "Since no model performs consistently better than others when measured by both metrics on three segmentation tasks, we also give the rank (immediately after the metric score) of each model in terms of each metric on each task, where `1' means performing best and `7' means performing worst.", "To assess the overall performance of each model, we shows the summation of the ranks over all metrics in the last column, which is treated as the golden performance indicators for brain tumor segmentation.", "It reveals that our ConResNet outperforms other six state-of-the-art methods on this 3D medical image segmentation problem, as indicated by the smallest Sum-Score.", "It should be noted that, although achieving an Sum-Score close to ours, nnUnet uses additional data for co-training.", "Note that all competing methods use the ensemble strategy.", "nnUnet and VAE-Seg improve the segmentation performance partly due to the ensemble of five and 10 models, respectively.", "EnsembleNets averages the probabilities predicted by multiple models, including MC-Net, OM-Net and their variants [43].", "Thanks to the context residual learning strategy, we achieve competitive results using the ensemble of three ConResNets, which were trained independently.", "Table: Performance of our ConResNet with / without the context residual mapping and context attention mapping on the BraTS validation dataset.Figure: Segmentation results obtained by applying our ConResNet with / without the context residual mapping and context attention mapping to six cases on the BraTS validation dataset.", "From left to right: (a) 2D ground truth overlaid on FLAIR slices, (e) ground truth of 3D tumor, and the 2D and 3D segmentation results of (b,f) the baseline model, (c,g) the baseline with context residual, and (d,h) the baseline with both context residual and context attention (ConResNet).", "ET: Yellow; TC: Yellow + Red; WT: Yellow + Red + Green.Comparison on the Pancreas-CT dataset: On the Pancreas-CT dataset, we compared the proposed ConResNet to HNN [45], Attention-UNet [8], CNN-RNN [46], RSTN [47], ResDSN [48], Bayesian [49], and Coarse2Fine [50], which are top-ranked segmentation methods on this task.", "HNN [45] is a holistically-nested DCNN approach to pancreas localization and segmentation, exploiting multi-view spatial pooling and combining interior and boundary mid-level cues.", "As an improved version of UNet, Attention-UNet [8] incorporates a novel attention gate module into the model to force it to focus on target structures of varying shapes and sizes.", "CNN-RNN [46] employs the recurrent neural networks to address the problem of spatial non-smoothness of inter-slice pancreas segmentation along adjacent slices.", "RSTN [47] repeatedly transforms the segmentation probability map from previous iterations into the spatial prior and use it in the current iteration to relate the coarse and fine stages.", "ResDSN [48] is composed of a coarse model and a fine model, the former obtains the rough location of the pancreas and the latter refines the segmentation based on the coarse result.", "Bayesian [49] leverages the Bayesian model to incorporate the rich shape priors learned from statistical shape models into the deep neural network and thus improves the performance in pancreas segmentation.", "Coarse2Fine [50] is also a two stage framework for pancreas segmentation, which contains the coarse segmentation for candidate region generation and the fine segmentation of smaller regions-of-interest.", "The mean Dice, max Dice, and min Dice of our ConResNet and these methods were listed in TABLE REF .", "Note that the scores of competing methods were adopted from the original papers.", "It shows ConResNet achieves the highest mean Dice of 86.06%, highest max Dice of 92.00%, and highest min Dice of 73.40%, outperforming the baseline and other competitive methods in terms of all metrics.", "Besides, we visualize the segmentation results of the baseline model that does not contain the ConRes decoder and our ConResNet model in Fig.", "REF .", "Note that the baseline model has the same encoder and segmentation decoder as ConResNet, including number of channels, network depth, and training strategies.", "It shows that our ConResNet generates better segmentation results, which are more similar to the ground truth than the results of the baseline model.", "Visualization of the learned context residual attention maps: In Fig.", "REF , we visualized the learned context residual attention maps and compared the feature maps obtained using or without using the context attention mapping.", "It reveals that the learned context residual attention maps highlight the context residual positions of brain tumor sub-regions.", "With the context attention mapping, the obtained feature maps have an improved ability to highlight the boundaries of those sub-regions, which is beneficial for better segmentation.", "Figure: Visualization of the learned context residual attention maps: (a) input FLAIR MR slices, (b) feature maps obtained without the attention, (c) learned context residual attention maps, and (d) feature maps obtained with the attention.", "Note that the feature maps shown here are the channel-wise summation of the output of the last context residual module.Figure: Visualization of feature maps produced by different models.", "All the feature maps are extracted from the output of first stage in the decoder.", "From top to down: (a) T1-weighted brain MR slices as input, (b) ground truth, and feature maps produced by (c) the baseline model, (d) baseline with context residual, and (e) baseline with both context residual and context attention (ConResNet).", "Note that the feature maps shown here are the channel-wise summation." ], [ "Ablation Analysis", "The major contribution of this work is to add the ConRes decoder and context residual models, which enable both context residual mapping and context attention mapping, to an encoder-decoder structure for improved segmentation performance.", "To verify the effectiveness of this design, we performed ablation experiments on the BraTS dataset.", "For the convenience of quantitative evaluation, we randomly selected 35 cases from training set to form a local validation set.", "Thus, we have 250 training cases when evaluating on the validation set (35 cases) and have 285 cases when evaluating on the testing set (66 cases).", "The performance our ConResNet with / without the context residual mapping and context attention mapping on the local validation set was listed in Table REF .", "Note that we kept other settings such as the network width, depth, and training strategies the same to ensure a fair comparison.", "It reveals that using the context residual mapping helps the segmentation model achieve larger Dice scores and smaller Hausdorff distances on ET, WT, and TC, and the performance improvement is substantial in all metrics.", "Meanwhile, incorporating the context attention mapping into the model with the context residual mapping can further improve the segmentation performance in all metrics.", "Consequently, comparing to the baseline model that uses neither the context residual mapping nor the context attention mapping, our ConResNet improves the Dice scores by 1.66%, 0.83%, and 2.27% and decreases the Hausdorff distance by 1.76, 0.41, and 1.44 for the segmentation of ET, WT, and TC, respectively.", "The segmentation results overlaid on FLAIR slices and the 3D visualization of segmented brain tumors were displayed in Fig.", "REF .", "It shows that the results produced by our ConResNet are more similar to the ground truth.", "Both the quantitative evaluation in Table REF and qualitative comparison in Fig.", "REF demonstrate the effectiveness of the proposed ConResNet for 3D medical image segmentation.", "In Fig.", "REF , we also visualize the channel-wise summation of the feature maps obtained by the baseline model, the model with only context residual mapping, and our ConResNet that uses both context residual mapping and context attention mapping.", "It shows that the feature maps produced by our model focus more on the target region than the feature maps produced by other models, which is beneficial for the segmentation." ], [ "Deep Supervision", "The proposed ConResNet also uses deep supervision, which is controlled by the trade-off parameter $\\lambda $ (see Eq.(8)).", "To validate the contribution made by deep supervision, we trained the model on the BraTS dataset four times when setting $\\lambda $ to 0, 0.1, 0.5, and 1, respectively.", "Note that $\\lambda =0$ means no deep supervision.", "In Fig.", "REF , we observe the lowest training loss and the best performance on the local validation set when $\\lambda = 0.5$ .", "Therefore, we empirically set $\\lambda $ to 0.5 in our experiments.", "Figure: Plot of training loss of our ConResNet versus iterations when setting λ\\lambda to different values.", "The table reports the validation performance under the different setting of λ\\lambda ." ], [ "Learning Context Residual along Other Views", "Our ConResNet is not limited by the inter-slice residual information along the axial view.", "It is easy to employ the proposed context residual learning to sagittal and coronal views.", "Taking the sagittal view for example, we compute the position-wise absolute difference between each pair of adjacent features along longitudinal axis, and train our ConResNet to predict the sagittal-view context residual mask.", "As shown in TABLE REF , we compared the performance of our ConResNet model on the local validation set, considering the residual information along the axial view, sagittal view, and coronal view, respectively.", "It shows that comparable performance was obtained when considering each of three views.", "It also reveals that ConResNet with any view is superior to the baseline model in terms of all evaluation metrics.", "These results demonstrate the effectiveness of the proposed context residual learning, which is not limited to the axial view.", "Table: Performance of ConResNet when learning context residual along axial, sagittal, and coronal views." ], [ "Applying to Multi-class Problems", "Let us consider a multi-class segmentation task with $C$ categories, where the ground-truth segmentation mask is denoted by $\\textbf {Y}_{seg} \\in \\textbf {R}^{(C+1)\\times S\\times H \\times W}$ .", "For each category $c\\in \\lbrace 1,2,...,C\\rbrace $ , we specifically compute its context residual mask as $\\textbf {Y}_{res}^{c, s+1, h, w} = \\left\| \\textbf {Y}_{seg}^{c, s+1, h, w} - \\textbf {Y}_{seg}^{c, s, h, w} \\right\|$ In the multi-class mode, ConResNet predicts the segmentation mask after the $softmax$ activation and predicts the residual mask of each category after the $sigmoid$ activation.", "Besides, we ignore the residual mask of background and just consider the error back-propagation of C categories during the training process.", "We compared the multi-class ConResNet to its binary counterpart on the BraTS dataset.", "The brain tumor segmentation task is formulated either into three binary segmentation problems, including ET vs. others, WT vs. others, and TC vs. others, or into a four-class segmentation problem, where the four classes include: 0 - background, 1 - necrotic and non-enhancing tumor core, 2 - peritumoral edema, and 3 - enhancing tumor.", "We still evaluated the segmentation performance on three sub-regions as done in the BraTS Challenge.", "TABLE REF shows the performance of the baseline model and our ConResNet in either the binary or multi-class mode on the local validation set.", "It reveals that the binary mode leads to the better performance than the multi-class mode, which may owe to the direct optimization of three sub-regions.", "However, our ConResNet achieves better performance than the baseline model in both binary and multi-class settings.", "Table: Comparison of ConResNet and the baseline model in binary or multi-class mode.", "Mode: B - “Binary\"; M - “Multi-class\"." ], [ "Comparing to Boundary Loss-based Methods", "The methods reported in [35] and [53] use newly designed boundary-wise loss functions to force the model to pay attention to boundary pixels on the object surface.", "Different intrinsically from them, the proposed ConResNet aims to learn the inter-slice context residual, which contains the essential and intriguing morphological information of the tumor, and use the context residual to boost the segmentation performance via simultaneous context residual mapping and context attention mapping.", "Although context residual voxels appear on or near to the tumor surface, using the proposed context residual learning is different from defining a boundary loss.", "In fact, the boundary loss can be incorporated into our model to possibly further improve its performance.", "In this case, our ConResNet has a compound loss function $L_{compound} = \\alpha \\times L + (1-\\alpha ) \\times L_{BD}$ where $\\alpha = 1- t \\cdot \\frac{1-0.01}{T}$ is a weighting factor, and $L_{BD}$ is the boundary loss proposed in [53].", "We compared the baseline model and our ConResNet model with and without using the boundary loss on the local validation set.", "TABLE REF show that using the boundary loss can further improve the performance of the baseline model and our ConResNet.", "Nevertheless, it also shows that, although the boundary loss is beneficial for this segmentation task, our ConResNet even outperforms the baseline model with the boundary loss.", "In general, adding a boundary loss or using the context residual learning is able to effectively help the model pay more attention to boundaries so as to improve the segmentation performance, particularly in localizing the surface of each target volume.", "Interestingly, both approaches address the discrepancies in the boundaries from different perspectives.", "Jointly using the context residual learning and a boundary loss is able to produce a mutual promotion.", "Table: Comparison of baseline and ConResNet with and without boundary loss (BD)." ], [ "Efficiency Analysis", "Our ConResNet has dual decoders, which, inevitably, raises the number of parameters and computations.", "TABLE REF lists the number of parameters, model size, Giga Floating-point Operations Per Second (GFLOPs) of VAE-Seg, baseline and our ConResNet.", "Note that the GFLOPs was calculated when the input is a 3D volume of size $80\\times 160\\times 160$ .", "It shows that our ConResNet suffers from slightly more parameters (an increase of 2.66%), more computations (an increase of 18.64%), and a larger mode size (an increase of 2.53%) than the baseline.", "However, considering the performance improvement shown in TABLE REF and Fig.", "REF , we believe that such a moderate increase of the complexity is acceptable.", "Moreover, although it is somewhat slower than the baseline model, our ConResNet can perform brain tumor segmentation almost in real-time, and hence has the potential to be used in clinical practice.", "Table: Number of parameters, model size and GFLOPs of different models.In this paper, we propose the ConResNet with explicit 3D context learning to boost the ability of DCNNs to perceive inter-slice context for accurate segmentation of volumetric medical images.", "We have evaluated this model on the BraTS dataset and Pancreas-CT dataset.", "Our results indicate that the proposed ConResNet outperforms state-of-the-art method on both brain tumor segmentation and pancreas segmentation tasks.", "Our ablation study also demonstrates the effectiveness of the proposed context learning, including the context residual mapping and context attention mapping.", "In our future work, we plan to treat the predicted residual mask as a prior that highlights the error-prone regions and to concatenate it with the input image as the input of another segmentation network for refined result.", "Besides, we will investigate how to combine the proposed 3D context learning with self-supervised learning, and thus extend this work to semi-supervised segmentation problems." ] ]	2011.14155
[ [ "Online Search with Maximum Clearance" ], [ "Abstract We study the setting in which a mobile agent must locate a hidden target in a bounded or unbounded environment, with no information about the hider's position.", "In particular, we consider online search, in which the performance of the search strategy is evaluated by its worst case competitive ratio.", "We introduce a multi-criteria search problem in which the searcher has a budget on its allotted search time, and the objective is to design strategies that are competitively efficient, respect the budget, and maximize the total searched ground.", "We give analytically optimal strategies for the line and the star environments, and efficient heuristics for general networks." ], [ "Introduction", "We study a general search problem, in which a mobile agent with unit speed seeks to locate a target that hides in some unknown position of the environment.", "Specifically, we are given an environment which may be bounded or unbounded, with a point $O$ designated as its root.", "There is an immobile target (or hider) $H$ that is hiding in some unknown point in the environment, whereas the searcher is initially placed at the root $O$ .", "The searcher has no information concerning the hider's position.", "A search strategy $S$ determines the precise way in which the searcher explores the environment, and we assume deterministic strategies.", "The cost of $S$ given hider $H$ , denoted by $d(S,H)$ , is the total distance traversed by the searcher the first time it reaches the location of $H$ , or equivalently the total search time.", "There is a natural way to evaluate the performance of the search strategy that goes back to [8] and [7]: we can compare the cost paid by the searcher in a worst-case scenario to the cost paid in the ideal situation where the searcher knows the hider's position.", "We define the competitive ratio of strategy $S$ as ${\\tt {cr}} (S)=\\sup _H \\frac{d(S,H)}{d(H)},$ with $d(H)$ the distance of $H$ from $O$ in the environment.", "Competitive analysis allows to evaluate a search strategy under a status of complete uncertainty, and provides strict, worst-case guarantees.", "Competitive analysis has been applied to several search problems in robotics, for example [32], [31], [33] [22].", "See also the survey [19].", "In this work we will study the following classes of environments: First, we consider the problem of searching on the line, informally known as the cow path problem [25], in which the environment is the unbounded, infinite line.", "Next, we consider a generalization of linear search, in which the environment consists of $m$ unbounded rays, concurrent at $O$ ; this problem is known as the $m$ -ray search or star search problem.", "This environment can model much broader settings in which we seek an intelligent allocation of resources to tasks under uncertainty.", "Thus, it is a very useful paradigm that arises often in applications such as the design of interruptible systems based on contract algorithms [9], [1], [28], or pipeline filter ordering [11].", "Last, we consider general undirected, edge-weighted graph networks, and a target that can hide anywhere over an edge or a vertex of this graph.", "In some previous work, online search may refer to the setting in which the searcher has no information about the environment or the position of the target.", "In this work we assume that the searcher knows the environment, but not the precise position of the target.", "This is in line with some foundational work on competitive analysis of online search algorithms, e.g.", "[27]." ], [ "Searching with a budget", "Most previous work on competitive analysis of searching has assumed that a target is indeed present, and so the searcher will eventually locate it.", "Thus, the only consideration is minimizing the competitive ratio.", "However, this assumption does not reflect realistic settings.", "Consider the example of Search-And-Rescue (SAR) operations: first, it is possible that the search mission may fail to locate the missing person, in which case searching should resume from its starting point instead of continuing fruitlessly for an exorbitant amount of time.", "Second, and more importantly, SAR operations come with logistical constraints, notably in terms of the time alloted to the mission.", "To account for such situations, in this work we study online search in the setting where the searcher has a certain budget $T$ , which reflects the total amount of search time that it can afford, and a desired competitive ratio $R$ that the search must attain.", "If the target is found within this budget, the search is successful, otherwise it is deemed unsuccessful.", "We impose two optimization constraints on the search.", "First, it must be competitively efficient, i.e., its competitive ratio, as expressed by (REF ) is at most $R$ , whether it succeeds or not.", "Second, if the search is unsuccessful, the search has maximized the total clearance by time $T$ .", "In the case of the environments we study in this work, the clearance is the measure of the part of the environment that the searcher has explored by time $T$ .", "We call this problem the Maximum Clearance problem with budget $T$ and competitive ratio $R$, and we denote it by MaxClear(R,T).", "It should be clear that the competitive ratio and the clearance are in a trade-off relation with respect to any given budget $T$ : by reducing the competitive efficiency, one can improve the clearance, and vice versa.", "Hence, our goal is to find strategies that attain the optimal tradeoff, in a Pareto sense, between these two objectives." ], [ "Contributions", "We study Maximum Clearance in three environments: the unbounded line, the unbounded star, and a fixed network.", "We begin with the line: here we show how to use a linear programming formulation to obtain a Pareto-optimal solution.", "We also show that the Pareto-optimal strategy has a natural interpretation as the best among two simple strategies.", "We then move to the $m$ -ray star, which generalizes the line, and which is more challenging.", "Here, we argue that the intuitive strategies that are optimal for the line are not optimal for the star.", "We thus need to exploit the structure of the LP formulation, so as to give a Pareto-optimal strategy.", "We do not require an LP solver, instead, we show how to compute the theoretically optimal strategy efficiently, in time $O(m\\log m \\log T+ m \\log T \\log \\log T)$ .", "Experimental evaluations confirm the superiority of this optimal strategy over other candidate solutions to the problem.", "Finally, we consider the setting in which the environment consists of a network.", "Here, there is a complication: we do not known the optimal competitive ratio as, for example, in the star (the problem is NP-hard if the target hides on vertices), and only $O(1)$ approximations of the optimal competitive ratio are known [4].", "Hence, in this context, we define MaxClear(R,T) with $R\\ge 1$ , as the problem of maximizing clearance given budget $T$ , while guaranteeing that the strategy is an $R$ -approximation of the optimal competitive ratio.", "Previous approaches to competitive searching in networks typically involve a combination of a solution to the Chinese Postman Problem (CPP) [14] with iterative doubling of the search radius.", "For our problem, we strengthen this heuristic using the Rural Postman Problem (RPP) [15], in which only a subset of the network edges need to be traversed.", "While RPP has been applied to the problem of online coverage in robotics [34], [13], to the best of our knowledge, no previous work on competitive search has addressed its benefits.", "Although there is no gain on the theoretical competitive ratio, our experimental analysis shows that it has significant benefits over the CPP-based approach.", "We demonstrate this with experiments using real-world data from the library Transportation Network Test Problems [6], which model big cities.", "We conclude with some extensions and applications.", "We first explain how our techniques can be applied to a problem “dual” to Maximum Clearance, which we call Earliest Clearance.", "We also show some implications of our work for contract scheduling problems.", "In particular, we explain how our results extend those of [3] for contract scheduling with end guarantees." ], [ "Other related work", "It has long been known that linear search has optimal competitive ratio 9 [7], which is achieved by a simple strategy based on iterative doubling.", "Star search on $m$ rays also has a long history of research, going back to [17] who showed that the optimal competitive ratio is $R^_m=1+2\\rho _m^, \\ \\textrm {where } \\rho _m^=\\frac{m^m}{(m-1)^{m-1}},$ a result that was later rediscovered by computer scientists [5].", "Star search has been studied from the algorithmic point of view in several settings, such as randomized strategies [26]; multi-searcher strategies [30]; searching with an upper bound on the target distance [21], [10]; fault-tolerant search [28]; and probabilistic search [24], [25].", "For general, edge-weighted networks only $O(1)$ -approximation strategies are known [27], [4]." ], [ "Preliminaries", "For the $m$ -ray star, we assume the rays are numbered $0, \\ldots ,m-1$ .", "A search strategy for the star is defined as $\\lbrace (x_i,r_i)\\rbrace _{i\\ge 1}$ , with the semantics that in the $i$ -th step, the searcher starts from $O$ , visits ray $r_i$ to length $x_i$ , then returns to $O$ .", "A cyclic strategy is a strategy for which $r_i =i \\bmod m$ ; we will thus often omit the $r_i$ 's for such strategies, since they are implied.", "We make the standing assumption that the target is hiding at least at unit distance from the root, otherwise there is no strategy of bounded competitive ratio.", "A geometric strategy is a cyclic strategy in which $x_i=b^i$ , for some $b>1$ , which we call the base.", "Geometric strategies are important since they often give optimally competitive solutions to search problems on a star.", "For instance, the optimal competitive ratio $R_m^$ is achieved by a geometric strategy with base $b=\\frac{m}{m-1}$ [17].", "In general, the competitive ratio of a cyclic strategy with base $b$ is equal to $1+2\\frac{b^m}{b-1}$ [16].", "By applying standard calculus, it follows that, for any given $R = 1+2\\rho \\ge R_m^$ , the geometric strategy with base $b$ is $R$ -competitive if and only if $b \\in [\\zeta _1, \\zeta _2]$ , where $\\zeta _i$ are the positive roots of the characteristic polynomial $p(t) = t^m-\\rho t + \\rho $ .", "A less known family of strategies for the $m$ -ray star is the set of strategies which maximize the searched length at the $i$ -th step.", "Formally, we want $x_i$ to be as large as possible, so that the strategy $X=(x_i)$ has competitive ratio $R=1+2\\rho $ .", "It turns out that this problem has indeed a solution, and as shown in [24], the resulting strategy $Z=(z_i)$ is one in which the search lengths are defined by the linear recurrence relation $z_{m+i}=\\rho (z_{i+1}-z_i)$ .", "[24] give a solution to the recurrence for $\\rho =\\rho _m^$ .", "We can show that $Z$ is in fact uniquely defined for all values of $R\\ge R^_m$ , and give a closed-form expression for $z_i$ , as a function of $\\zeta _1$ and $\\zeta _2$ , defined above (Appendix).", "Following the terminology of [2] we call $Z$ the aggressive strategy of competitive ratio $R$, or simply the aggressive strategy when $R$ is implied.", "For the star we will use a family of linear inequalities involving the search lengths $x_i$ to model the requirement that the search is $R$ -competitive.", "Such inequalities are often used in competitive search, see e.g.", "[29], [21].", "Each inequality comes from an adversarial position of the target: for a search strategy of the form $X=\\lbrace (x_i,r_i)\\rbrace $ in the star, the placements of the target which maximize the competitive ratio are on ray $r_j$ and at distance $x_j+\\epsilon $ , for all $j$ and for infinitesimally small $\\epsilon $ (i.e., the searcher barely misses the target at step $j$ ).", "There is, however, a subtlety in enforcing competitiveness in our problem.", "In particular, we need to filter out some strategies that can be $R$ -competitive up to time $T$ , but are artificial.", "To illustrate this, consider the case of the line, and a strategy $S$ that walks only to the right of $O$ up to time $T$ (it helps to think of $T$ as very large).", "This strategy is 1-competitive in the time interval $[0,T]$ , and obviously maximizes clearance, but intuitively is not a realistic solution.", "The reason for this is that $S$ discards the entire left side with respect to $R$ -competitiveness.", "Specifically, for a point at distance 1 to the left of $O$ , any extension $S^{\\prime }$ of $S$ will incur a competitive ratio of at least $2T+1$ , which can be enormous.", "We thus need to enforce a property that intuitively states that a feasible strategy $S$ to our problem should be extendable to an $R$ -competitive strategy $S^{\\prime }$ that can detect targets hiding infinitesimally beyond the boundary that has been explored by time $T$ in $S$ .", "We call this property extendability of an $R$ -competitive strategy.", "We give a formal definition in the Appendix concerning our environments, although this intuitive description will suffice for the purposes of modeling and analysis.", "Our experimental evaluation shows that the optimal extendable strategy on the star performs significantly better than other candidate strategies, which further justifies the use of this notion." ], [ "A warm-up: Maximum Clearance on the line", "We begin with the simplest environment: an unbounded line with root $O$ .", "Fix a competitive ratio $R=1+2\\rho $ , for some $\\rho \\ge \\rho _2^=4$ .", "Without loss of generality, we assume cyclic strategies $X=(x_i)$ such that $x_{i+2} >x_{i}$ , for all $i$ .", "Let ${\\cal S}_k$ denote the set of all strategies $X=(x_1,\\ldots x_k)$ with $k$ steps.", "We can formulate MaxClear(R,T) restricted to ${\\cal S}_k$ using the following LP, which we denote $L_2^{(k)}$ .", "$\\text{max} \\quad & x_{k-1} + x_k & \\\\\\text{subject to} \\quad &x_1\\le \\rho \\\\&\\sum \\nolimits _{i=1}^{j+1} x_{i} \\le \\rho \\cdot x_j, &j \\in [1, k-2] \\\\&\\sum \\nolimits _{i=1}^{k} x_{i} \\le \\rho \\cdot x_{k-1} \\\\&2 \\sum \\nolimits _{i=1}^{k-1} x_{i} + x_k \\le T $ In this LP, constraints $(C_0)$ and $(C_1), \\ldots (C_{k-2})$ model the requirement for $(1+2\\rho )$ -competitiveness.", "$(C_0)$ models a target hiding at distance 1 from $O$ , whereas the remaining constraints model a target hiding right after the turn points of $x_1, \\ldots x_{k-2}$ , respectively.", "Constraint $(B)$ is the budget constraint.", "Last, constraint $(E_{k-1})$ models the extendability property, which on the line means remaining competitive for a target hiding just beyond the turn point of $x_{k-1}$ .", "Therefore, an optimal strategy is one of maximum objective value, among all feasible solutions to $L_2^{(k)}$ , for all $k\\ge 1$ .", "We will use this formulation to show that the optimal strategy has an intuitive statement.", "Let $Z=(z_i)$ be the aggressive strategy of competitive ratio $R$ .", "From $Z$ we derive the aggressive strategy with budget $T$, which is simply the maximal prefix of $Z$ that satisfies the budget constraint $(B)$ .", "We denote this strategy by $Z_T$ .", "Note that $Z_T$ may be wasteful, leaving a large portion of the budget unused, which suggests another intuitive strategy derived from $Z$ .", "Informally, one can “shrink” the search lengths of $Z$ in order to deplete the budget precisely at some turn point.", "Formally, we define the scaled aggressive strategy with budget $T$, denoted by $\\tilde{Z}_T$ as follows.", "Let $l$ be the minimum index such that $2\\sum _{i=1}^{l-1}z_i+z_l \\ge T$ , and define $\\gamma $ as $T/(2\\sum _{i=1}^{l-1}z_i+z_l)$ .", "Then $\\tilde{Z}_T$ is defined as $(\\tilde{z}_i)=(\\gamma \\cdot z_i)$ .", "We will prove that one of $Z_T$ , and $\\tilde{Z}_T$ is the optimal strategy.", "We can first argue about constraint tightness in an optimal solution to $L_2^{(k)}$ .", "Lemma 1 In any optimal solution to $L_2^{(k)}$ , at least one of the constraints $(C_0)$ and $(B)$ is tight, and all other constraints must be tight.", "By way of contradiction, let $X^=(x_i^)$ denote an optimal solution for the LP which does not obey the conditions of the lemma.", "Recall that we only consider solutions on the line which explore strictly farther each time they visit a side, i.e.", "$x_{i+2}>x_i$ .", "Suppose that a constraint $(C_j)$ is loose.", "Then we could decrease $x_j^$ by a small amount, say $\\delta $ , and increase $x_k^$ by $\\delta $ , maintaining feasibility, including the implicit constraint $x^_{j-2}<x^_j$ , and improving the objective, a contradiction.", "Similarly, if $(E_{k-1})$ is not tight, then we could decrease $x_{k-1}^$ by a small amount, say $\\delta $ , and increase $x_{k}^$ by $2\\delta $ , maintaining feasibility, including the implicit constraint $x^_{k-3}<x^_{k-1}$ , and improving the objective, a contradiction.", "It remains then to argue that one of the constraints $(C_0)$ and $(B)$ is tight.", "This is true because if they are both slack, then there would exist $\\alpha >1$ such that $(\\alpha \\cdot x_i^)$ is a feasible solution with a better objective value than $X^$ , a contradiction.", "Lemma REF shows that if $X^$ is optimal for $L_2^{(k)}$ , then one can subtract successive constraints from each other to obtain the linear recurrence relation $x^_{i+2} = \\rho (x^_{i+1}-x^_i)$ , with constraint $(C_1)$ giving an initial condition.", "So $X^$ , viewed as a point in $\\mathbb {R}^k$ , is on a line $\\Delta \\subset \\mathbb {R}^k$ , defined as the set of all points which satisfy $(C_1),\\dots ,(E_{k-1})$ with equality.", "This leaves us with two possibilities: either $X^=X_0^{(k)}$ the point on $\\Delta $ for which $(C_0)$ is tight, or $X^=X_B^{(k)}$ the point on $\\Delta $ for which $(B)$ is tight.", "Define now $\\mathcal {X}_0$ as the set of all feasible points $X_0^{(k)}$ and $\\mathcal {X}_B$ as the set of all feasible points $X_B^{(k)}$ .", "A point $X$ is optimal for one of these sets if its objective value is no worse than any point in that set.", "The following lemma is easy to see for $Z_T$ , and requires a little more effort for $\\tilde{Z}_T$ .", "Lemma 2 $Z_T$ is optimal for $\\mathcal {X}_0$ , and $\\tilde{Z}_T$ is optimal for $\\mathcal {X}_B$ .", "$X_0^{(k)}$ is simply a prefix $(z_1,\\dots ,z_k)$ of the aggressive strategy $Z$ , because the formulas defining them are identical.", "Because $z_i$ is increasing (see the formulas for $Z$ given above), the objective value of $X_0^{(k)}$ is increasing, and so $Z_T$ , which is the longest feasible prefix for $L_2^{(k)}$ , is optimal for $\\mathcal {X}_0$ .", "$X_B^{(k)}$ is a scaled version of $X_0^{(k)}$ (they both belong to the same line $\\Delta $ ), and so is given by $(\\gamma _k z_i)$ where $\\gamma _k = \\frac{T}{2 S_{k-1}^{(Z)} + z_k}$ .", "Denote ${\\bf clr}(X)$ the objective value, or clearance, of a strategy $X$ : we have ${\\bf clr}(X_B^{(k)}) =\\gamma _k (z_{k-1}+z_k)= \\frac{z_{k-1}+z_k}{2 \\rho z_{k-1}^{(Z)}-z_k}T$ , using the identity $(E_{k-1})$ , which holds because $X_B^{(k)}\\in \\Delta $ .", "We want to show that the clearance of $(X_B^{(k)})$ decreases with $k$ .", "A short calculation yields: ${\\bf clr}(X_B^{(k)})\\le {\\bf clr}(X_B^{(k-1)}) \\Leftrightarrow \\frac{z_k}{z_{k-1}} \\le \\frac{z_{k-1}}{z_{k-2}}.", "$ We now make use of the formulas for $z_i$ .", "For optimal $\\rho =\\rho _m^$ , we get $ \\frac{z_{i+1}}{z_i} = \\frac{m+i}{m+i-1}\\cdot \\frac{m}{m-1}, $ which is indeed decreasing, and for $\\rho >\\rho _2^$ a short calculation yields $ \\frac{z_{i+1}}{z_i}\\le \\frac{z_i}{z_{i-1}} \\Leftrightarrow \\zeta _1^2 + \\zeta _2^2 \\ge 2\\zeta _1\\zeta _2.$ Therefore for all $\\rho $ , $\\tilde{Z}_T$ is optimal for $\\mathcal {X}_B$ .", "From Lemma REF and REF we conclude that the better of the two strategies $Z_T$ and $\\tilde{Z}_T$ is optimal for Max(R,T) on the line.", "We call this strategy the mixed aggressive strategy." ], [ "Maximum Clearance on the Star", "We now move to the $m$ -ray star domain.", "We require that the strategy be $(1+2\\rho )$ -competitive, for some given $\\rho \\ge \\rho ^_m$ , where $\\rho ^_m=\\frac{m^m}{(m-1)^{m-1}}$ , and we are given a time budget $T$ ." ], [ "A first, but suboptimal approach", "An obvious first place to look is the space of geometric strategies.", "We wish the geometric strategy to have competitive ratio $1+2\\rho $ , so the strategy must have base $b \\in [\\zeta _1,\\zeta _2]$ , using the notation of the preliminaries.", "Since we want to maximize the clearance of our strategy, it makes sense to take $b=\\zeta _2$ .", "We define the scaled geometric strategy with budget T similarly to the scaled aggressive strategy: find the first step at which the budget $T$ is depleted, and scale down the geometric strategy so that it depletes $T$ precisely at the end of that step.", "The scaled geometric strategy represents the best known strategy prior to this work, but is suboptimal.", "For Maximum Clearance on the line, we proved that the optimal strategy is the best of the aggressive and the scaled aggressive strategies.", "One may ask then whether the optimal strategy in the star domain can also be expressed simply as the better of these two strategies.", "The answer is negative, as we show in the experimental evaluation." ], [ "Modeling as an LP", "As with the line, we first show how to formulate the problem using a family of LPs, denoted by $L^m_k$ , partitioning strategies according to their length $k$ .", "For a given step $j$ , we denote by $\\bar{\\jmath }$ the previous step for which the searcher visited the same ray, i.e, the maximum $\\bar{\\jmath } <j$ such that $r_{\\bar{\\jmath }}=r_j$ , assuming it exists, otherwise we set $x_{\\bar{\\jmath }}=1$ .", "We denote by $l_r$ the last step at which the searcher explores ray $r$ .", "Finally, we denote by $j_0$ the last step in which the searcher searches a yet unexplored ray, i.e., the largest step $j$ such that $\\bar{\\jmath }=0$ .", "This gives us: $\\text{max} &\\quad \\sum \\nolimits _{i=1}^m x_{l_i} & \\\\\\text{subject to}& \\quad \\sum \\nolimits _{i=1}^{j_0} x_i \\le \\rho & \\\\&\\sum \\nolimits _{i=1}^{j-1} x_i \\le \\rho \\cdot x_{\\bar{\\jmath }}, \\qquad \\; j \\in [j_0+1, k] &\\\\&\\sum \\nolimits _{i=1}^k x_i \\le \\rho \\cdot x_{l_r},\\; r \\in [1,m], l_r\\ne r_k & \\\\&2 \\sum \\nolimits _{i=1}^{k-1} x_i + x_k \\le T &$ Here, constraints $(C_0), (C_{j_0}), \\ldots (C_k)$ model the $(1+2\\rho )$ -competitiveness of the strategy, and constraint $(B)$ models the budget constraint.", "Constraints $(E_1), \\ldots ,(E_m)$ model the extendability property, by giving competitiveness constraints for targets placed just beyond the turn points at $x_{l_1},\\dots ,x_{l_r}$ .", "Details concerning the derivation of all constraints can be found in the Appendix.", "As is standard in star search problems, we can add some much-needed structure in the above formulation.", "Theorem 3 (Appendix) Any optimal solution $X^=(x^_i,r_i)$ to $L_m^{(k)}$ must be monotone and cyclic: $(x^_i)$ is increasing and $r_i = i \\mod {m}$ up to a permutation.", "This means that we can formulate the problem using a much simpler family of LPs which we denote by $P_m^{(k)}$ , where constraints $(M_i)$ model monotonicity.", "$\\text{max}& \\quad \\sum \\nolimits _{i=0}^{m-1} x_{k-i} &\\\\\\text{subj to}& \\quad \\sum \\nolimits _{i=1}^{m-1} x_i\\le \\rho & \\\\&\\sum \\nolimits _{i=1}^{j+m-1} \\!x_i\\le \\rho \\cdot x_{j}, \\qquad \\, j \\in [1, k-m] &\\\\&\\!\\!\\!\\!\\sum \\nolimits _{i=1}^k x_i \\le \\rho \\cdot x_j, \\;\\; j \\in [k-m+1,k-1] & \\\\&x_i \\le x_{i+1}, \\qquad \\qquad \\qquad \\quad \\; i \\in [1,k-1]&\\\\&2\\sum \\nolimits _{i=1}^{k-1} x_i+ x_k \\le T &$" ], [ "Solving $P_m^{(k)}$", "While proving cyclicality, we also prove that for any optimal solution to $L_m^{(k)}$ , most of the constraints are tight, similarly to Lemma REF .", "Applying this result to $P_m^{(k)}$ gives the following.", "Lemma 4 In an optimal solution to the LP $P_m^{(k)}$ , constraints $(M_i)$ are not necessarily tight, at least one of the constraints $(C_0)$ and $(B)$ is tight, and all other constraints must be tight.", "Subtracting $(C_i)$ from $(C_{i+1})$ and $(C_{k-m})$ from $(E_{k-m+1})$ gives a linear recurrence formula which any optimal solution $X^$ must satisfy: $ x_{i+m}^ = \\rho (x_{i+1}^-x_i^).", "\\quad i \\in [1,k-m] $ The constraints $(E_j)$ give us $m-1$ equations to help determine the solution: $\\rho x_{k-m+1}^=\\dots =\\rho x_{k-1}^=S_k$ .", "So $X^$ , viewed as a point in $\\mathbb {R}^k$ , is on a line $\\Delta _m^{(k)}\\subset \\mathbb {R}^k$ , defined as the set of all points which satisfy $(C_1),\\dots ,(E_{k-1})$ with equality.", "Lemma REF shows that the solution to $P_m^{(k)}$ is either the point $X_0^{(k)} \\in \\Delta _k^m$ for which constraint $(C_0)$ is tight, or the point $X_B^{(k)} \\in \\Delta _k^m$ for which constraint $(B)$ is tight.", "We can compute these two strategies efficiently for a fixed $k$ , as we will demonstrate for $X_B^{(k)}$ .", "We rewrite the conditions $X_B^{(k)}\\in \\Delta _k^m$ and “$(B)$ is tight” as a matrix equation: ${\\cal M}_{k,B}^m \\times X=\\begin{pmatrix}0 & \\cdots & 0 & T \\\\\\end{pmatrix}^\\top $ where ${\\cal M}_{k,B}^m$ is the following $k\\times k$ matrix: $\\begin{pmatrix}\\rho &-\\rho &0 &0 &\\cdots &1 &0 &\\cdots &0 &0 &0 \\\\0 &\\rho &-\\rho &0 &\\cdots &0 &1 &\\cdots &0 &0 &0 \\\\\\vdots &\\vdots &\\vdots &\\vdots &\\ddots &\\vdots &\\vdots &\\ddots &\\vdots &\\vdots &\\vdots \\\\0 &0 &0 &0 &\\cdots &0 &0 &\\cdots &\\rho &-\\rho &0 \\\\1 &1 &1 &1 &\\cdots &1 &1 &\\cdots &1 &1-\\rho &1 \\\\2 &2 &2 &2 &\\cdots &2 &2 &\\cdots &2 &2 &1 \\\\\\end{pmatrix}$ ${\\cal M}_{k,B}^m$ has a very nice structure, and is very sparse, as all coefficients are concentrated in three diagonals (numbered 1, 2, and $m+1$ ) and the last two lines.", "This is good for us: we can solve (REF ) in time $O(k)$ using Gaussian elimination.", "$X_0^{(k)}$ can be computed similarly, using the matrix ${\\cal M}_{k,0}^m$ , which is identical to ${\\cal M}_{k,B}^m$ except for the last line, which contains $(C_0)$ , and (REF ) becomes ${\\cal M}_{k,0}^m \\times X_0^{(k)} = (0 \\cdots 0 \\, \\rho )^\\top $ .", "When solving (REF ) we discarded the constraint $(C_0)$ , so we need to check whether $X_B^{(k)}$ is feasible for this constraint.", "Similarly, we need to check whether $X_0^{(k)}$ is feasible for $(B)$ ." ], [ "Finding the optimal strategy", "At this point, we have determined how to compute two families of strategies, the sets $\\mathcal {X}_0=\\lbrace X_0^{(k)}, k\\in \\mathbb {N}\\rbrace $ and $\\mathcal {X}_B = \\lbrace X_B^{(k)},k\\in \\mathbb {N}\\rbrace $ , and we have shown that any optimal strategy belongs to one of these two families.", "Define $k_0$ the highest $k$ for which $X_0^{(k)}$ is feasible, and $k_B$ the lowest $k$ for which $X_B^{(k)}$ is feasible.", "We conclude with our two main results.", "Theorem 5 (Appendix) $X_0^{(k)}$ is feasible if and only if $k\\le k_0$ , and $X_B^{(k)}$ is feasible if and only if $k\\ge k_B$ .", "Moreover, $X_0^{(k_0)}$ is optimal for $\\mathcal {X}_0$ , and $X_B^{(k_B)}$ is optimal for $\\mathcal {X}_B$ .", "Proof sketch.", "We show first that any point $(x_i)$ that is feasible for $P_m^{(k)}$ is positive: $\\forall i,x_i\\ge 0$ .", "Denote $X_0^{(k)}=(x_i)$ and $X_0^{(k-1)}=(y_i)$ .", "Using the convention $y_0=1$ , the strategy $D=(x_i-y_{i-1})$ is feasible for $P_m^{k}$ , therefore positive.", "This means that $X$ has a higher objective value than $Y$ , and also requires a larger budget: this shows that $k_0$ is well-defined and optimal.", "Because $X_0^{(k)}$ and $X_B^{(k)}$ are scaled versions of each other, we get $k_B = k_0$ or $k_0+1$ .", "Additional calculations show that the objective values of $X_B^{(k)}$ are decreasing.", "Theorem 6 The optimal strategy for the $m$ -ray star can be computed in time $O(m\\log (T)\\log (m\\log (T)))$ .", "Proof sketch.", "The scaled geometric strategy with base $b=\\frac{m}{m-1}$ is a feasible point for a certain $P_m^{(k_G)}$ , with $k_G=O(\\log _{b}(T))=O(m\\log (T))$ .", "This means that $X_B^{(k_G)}$ is feasible, and so $k_B\\le k_G$ gives us an upper bound.", "We can use binary search to find $k_B$ , solving (REF ) at each step at a cost of $O(k_G)$ .", "We know that $k_0$ is either $k_B$ or $k_B-1$ , so all that remains is to compare the two strategies, which gives us a total complexity of $O(m\\log (T)\\log (m\\log (T)))$ ." ], [ "Maximum Clearance in a Network", "In this section we study the setting in which the environment is a network, represented by an undirected, edge-weighted graph $Q=(V,E)$ , with a vertex $O$ designated as the root.", "Every edge has a non-negative length which represents the distance of the vertices incident to the edge.", "The target can hide anywhere along an edge, which means that the search strategy must be a traversal of all edges in the graph.", "We can think of the network $Q$ as being endowed with Lebesgue measure corresponding to the length.", "This allows as to define, for a given subset $A$ of the network, its measure $l(A)$ .", "Informally, $l(A)$ is the total length of all edges (partial or not) that belong in $A$ .", "Given a strategy $S$ and a target $t$ , the cost $d(S,t)$ and the distance $d(t)$ are well defined, and so is the competitive ratio according to (REF ).", "We will denote by $Q[r]$ the subnetwork that consists of all points in $Q$ within distance at most $r$ from $O$ .", "The exact competitive ratio of searching in a network is not known, and there are only $O(1)$ -approximations [27], [4] of the optimal competitive ratio.", "For this reason, as explained in the introduction, we interpret MaxClear(R,T) as a maximum clearance strategy with budget $T$ that is an $R$ -approximation of the optimal competitive ratio.", "The known approximations use searching based on iterative deepening, e.g.", "strategy Cpt(r), which in each round $i$ , searches $Q[r^i]$ using a Chinese Postman Tour (CPT) [14] of $Q[r^i]$ , for some suitably chosen value of $r$ .", "We could apply a similar heuristic to the problem of Maximum Clearance.", "However, searching using a CPT of $Q[r^i]$ is wasteful, since we repeatedly search parts of the network that have been explored in rounds $1 \\ldots i-1$ .", "Instead, we rely on heuristics for the Rural Postman Problem [15].", "In this problem, given an edge-weighted network $Q=(V,E)$ , and a subset $E_{\\text{req}} \\subseteq E$ of required edges, the objective is to find a minimum-cost traversal of all edges in $E_{\\text{req}}$ in $Q$ ; we call this tour RPT for brevity.", "Unlike the Chinese Postman Problem (CPP), finding an RPT is NP-hard.", "The best known approximation ratio is 1.5 [15], but several heuristics have been proposed, e.g. [12], [20].", "We thus propose the following strategy, which we call Rpt(r).", "For each round $i \\ge 1$ , let $R_{i-1} = Q[r^i]\\setminus Q[r^{i-1}]$ denote the part of the network that the searcher has not yet explored in the beginning of round $i$ (and needs to be explored).", "Compute both tours CPT$(Q[r^i])$ and RPT$(Q[r^i])$ , the latter with required set of edges the edge set of $R_{i-1}$ (using the 1.5-approximation algorithm), and choose the tour of minimum cost among them.", "This continues until the time budget $T$ is exhausted.", "It is very hard to argue from a theoretical standpoint that the use of RPT yields an improvement on the competitive ratio; nevertheless, the experimental evaluation shows that this is indeed beneficial to both competitiveness and clearance.", "Since Rpt(r) is at least as good as a strategy that is purely based on CPTs, we can easily show the following, which is proven analogously to the randomized strategies of [4].", "Proposition 7 For every $r>1$ , Rpt(r) is a $\\frac{r^2}{r-1}$ -approximation of the optimal competitive ratio.", "In particular, for $r=2$ , it is a 4-approximation.", "Let $\\ell _i$ denote the length of the optimal CPT in $G[r^i]$ .", "The competitive ratio of the strategy is at most $\\sup _{j \\ge 1} \\frac{\\sum _{i=1}^j \\ell _i}{r^{j-1}}.$ Let $R^$ denote the optimal (deterministic) competitive ratio.", "Then it holds that for every $i$ , $R^\\ge \\frac{l_i}{r^i}.$ This is because any deterministic strategy needs time at least $l_i$ to traverse $G[r^i]$ , and every point in $G[r^i]$ is at distance at most $b^i$ from $O$ .", "Combining the above inequalities, we obtain that the competitive ratio of Rpt(r) is at most $R^ \\cdot \\sup _{j \\ge 1} \\frac{\\sum _{i=1}^j r^i}{r^{j-1}} \\le R^* \\cdot \\sup _{j\\ge 1}\\frac{r^{j+1}-1}{r^{j-1}(r-1)} \\le R^* \\cdot \\frac{r^2}{r-1}.$ The last inequality implies that the best approximation factor is achieved for $r=2$ , and is equal to 4.", "Note that Rpt(r) is, by its statement, extendable, since it will always proceed to search beyond the boundary of round $i$ in round $i+1$ .", "Moreover, Rpt(r) is applicable to unbounded networks as well, provided that for any $D$ , the number of points in the network at distance $D$ from $O$ is bounded by a constant.", "This is necessary for the competitive ratio to be bounded [4]." ], [ "$m$ -ray star", "In this section we evaluate the performance of our optimal strategy against two other candidate strategies.", "The first candidate strategy is the scaled geometric strategy, with base $\\zeta _2, $ which we consider as the baseline for this problem prior to this work.", "The second candidate strategy is the mixed aggressive strategy.", "Recall that we defined both strategies at the beginning of the star section, and that all these strategies are defined for the same competitive ratio $R$ .", "Figure REF depicts the relative performance of the optimal strategy versus the performance of the other two strategies, for $m=4$ , and optimal competitive ratio $R=R^_4$ , for a range of budget values $T\\in [10,10^{15}]$ .", "Once the budget $T$ becomes meaningfully large (i.e, $T\\ge 50$ ), the optimal strategy dominates the other two, outperforming both by more than $20\\%$ .", "In contrast, the mixed aggressive strategy offers little improvement over the scaled geometric strategy for every reasonably large value of $T$ .", "Figure: Clearance ratios for m=4m=4 and R=R 4 R=R_4^, as function of TT.Figure: Clearance as function of mm, for T=10 8 T=10^8 and R=R m R=R_m^.Figure REF depicts the influence of the parameter $m$ on the clearance achieved by the three strategies, for a relatively large value of $T=10^8$ .", "For each value of $m$ in $[3,20]$ , we require that the strategies have optimal competitive ratio $R=R_m^$ .", "We observe that as $m$ increases, each strategies' clearance decreases, however the optimal strategy is far less impacted.", "This means that as $m$ increases, the relative performance advantage for the optimal strategy also increases, in comparison to the other two.", "Figure REF depicts the strategies' performance for $m=4$ , and $T=10^4$ , as a function of the competitive ratio $R\\ge R_4^$ .", "In particular, we consider $R\\in [R_4^, 3R_4^]$ .", "We observe that as $R$ increases, the mixed aggressive strategy is practically indistinguishable from the scaled geometric.", "The optimal strategy has a clear advantage over both strategies for all values of $R$ in that range.", "Figure: Clearance as function of RR, for m=4m=4 and T=10 4 T=10^4.More experimental results can be found in the Appendix." ], [ "Networks", "We tested the performance of Rpt(r) against the performance of Cpt(r).", "Recall that the former searches the network $Q[r^i]$ iteratively using the best among the two tours CPT$(Q[r^i])$ and RPT$(Q[r^i])$ , whereas the latter uses only the tour CPT($Q[r^i])$ .", "We found $r=2$ to be the value that optimizes the competitive ratio in practice, as predicted also by Proposition REF , so we chose this value for our experiments.", "We used networks obtained from the online library Transportation Network Test Problems [6], after making them undirected.", "This is a set of benchmarks that is very frequently used in the assessment of transportation network algorithms (see e.g. [23]).", "The size of the networks we chose was limited by the $O(n^3)$ time-complexity of Cpt(r) and Rpt(r) ($n$ is the number of vertices).", "For RPT we used the algorithm due to [15].", "Figures REF and REF depict the clearance achieved by each heuristic, as function of the budget $T$ , for a root chosen uniformly at random.", "The first network is a European city with no obvious grid structure, whereas the second is an American grid-like city.", "We observe that the clearance of Cpt(r) exhibits plateaus, which we expect must occur early in each round, since CPT must then traverse previously cleared ground.", "We also note that these plateaus become rapidly larger as the number of rounds increases, as expected.", "In contrast, Rpt(r) entirely avoids this problem, and performs significantly better, especially for large time budget.", "Figure: Comparison of the two strategies on the Berlin network (633 nodes, 1042 edges).Figure: Comparison of the two strategies on the Chicago network (933 nodes, 1475 edges).Figure REF depicts the ratio of the average clearance of Rpt(r) over the average clearance of Cpt(r) as a function of the time budget $T$ , calculated over 10 random runs of each algorithm on the Berlin network (each run with a root chosen uniformly at random).", "We observe that Rpt(r) consistently outperforms Cpt(r), by at least 8% for most values of $T$ , and up to 16% when $T$ is comparable to the total length of all edges in the graph (173299).", "At $T=250000$ , in most runs, Rpt(r) has cleared the entire network.", "Figure: Clearance ratio of Rpt(r) versus Cpt(r), for 10 randomly chosen roots, for the Berlin network.The average competitive ratios for these runs are 160 for Cpt(r) and 132 for Rpt(r), demonstrating a clear advantage.", "More experimental results can be found in the Appendix." ], [ "Extensions and conclusions", "One can define a problem “dual” to Maximum Clearance, which we call Earliest Clearance.", "Here, we are given a bound $L$ on the desired ground that we would like the searcher to clear, a required competitive ratio $R$ , and the objective is to design an $R$ -competitive strategy which minimizes the time to attain clearance $L$ .", "The techniques we use for Maximum Clearance can also apply to this problem, in fact Earliest Clearance is a simpler variant; e.g., for star search, optimal strategies suffice to saturate all but one constraint, instead of all but two (see Appendix).", "Maximum Clearance on a star has connections to the problem of scheduling contract algorithms with end guarantees [3].", "More precisely, our LP formulation has certain similarities with the formulation used in that work (see the LP $P_m$ , on page 5496 in [3]), and both works use the same general approach: first, a technique to solve the LP of index $k$ , and then a procedure for finding the optimal index $k^$ .", "However, there are certain significant differences.", "First, our formulations allow for any competitive ratio $\\rho \\ge \\rho _m^$ , whereas [3] only works for what is the equivalent of $\\rho _m^$ .", "Related to this, the solution given in that work is very much tied to the optimal performance ratios, and the same holds for the optimality proof which is quite involved and does not extend in an obvious way to any $\\rho $ .", "The theoretical worst-case runtime of the algorithm in [3] is $O(m^2 \\log L)$ , whereas the runtime of our algorithm has only an $O(m\\log m)$ dependency on $m$ , as guaranteed by Theorem REF .", "Given the conceptual similarities between the two problems, our techniques can be readily applicable to the scheduling problem as well, and provide the improvements we describe above.", "For clearance in networks, we demonstrated that RPT-based heuristics can have a significant impact on performance, in comparison to CPT-based heuristics.", "The RPT heuristic we implemented is from [15], but more complex and sophisticated heuristics are known [12].", "It would be interesting to further explore the impact of such heuristics in competitive search.", "Appendix" ], [ "Formulating the LPs, and extendability", "We introduce the shorthand notation $S_j^{(X)} = \\sum _{i=1}^j x_i$ .", "When it is obvious which strategy we are referring to, we will simply use the notation $S_j$ .", "For the line and star environments, it is clear that we can restrict ourselves to strategies where each step has positive length, and which go strictly further at each visit to a given ray.", "These conditions are implicit is our LP formulation." ], [ "Competitiveness constraints", "It is known that the worst-case competitive ratio corresponds to targets placed immediately after the turn points, and thus it suffices to enforce $R$ -competitiveness in those locations.", "So the total distance traveled by the searcher upon returning to a turn point for the first time must not exceed $R$ times the distance from the origin to this turn point.", "Using the notations we introduced at the beginning of the star section, we obtain: $ 2 S_{j-1} + x_{\\bar{\\jmath }} \\le (1+2\\rho ) x_{\\bar{\\jmath }} \\Leftrightarrow S_{j-1}\\le \\rho x_{\\bar{\\jmath }}, $ which yields the constraint $(C_j)$ .", "When searching a new ray for the first time, say on step $i$ , because we have assumed that the target is located at distance at least 1 from the origin, we obtain the constraint $2S_{i-1}+x_{i}\\le 1+2\\rho \\Leftrightarrow S_{i-1}\\le \\rho $ .", "Obviously we only need to keep the last such constraint, corresponding to step $j_0$ , which is the dominant constraint.", "Also, any competitiveness constraint before the step $j_0$ is superfluous, because the competitive factor is necessarily worse for points at the same distance but on ray $r_{j_0}$ .", "We thus showed how to obtain constraint $(C_0)$ .", "Constraint $(B)$ clearly reflects the budget requirement.", "It remains to explain the extendability constraints.", "We do so in detail in what follows." ], [ "Extendability constraints", "We begin with the line.", "As discussed in the main paper, in order to enforce the extendability property we consider targets placed just beyond the turn point at $x_{k-1}$ , and just beyond the end point at $x_k$ .", "For the end point $x_k$ , this property is satisfied by the strategy: the searcher can visit a point hiding infinitesimally beyond $x_k$ at an infinitesimally small aditional cost, and without changing the competitive ratio.", "For the turn point at $x_{k-1}$ , the extension of our strategy which gets there in the least time turns around at $x_k$ , goes through $O$ and reaches the turn point at $x_{k-1}$ , and thus we get the following constraint: $ 2 S_k + x_{k-1} \\le (1+2\\rho ) x_{k-1} \\Leftrightarrow S_k\\le \\rho x_{k-1}.", "$ For the star, the situation is analogous.", "For the end point $x_k$ , as for the line, the property is trivially satisfied; for the other points, by considering extensions which turn around at $x_k$ to explore each other ray, we get the family of constraints $ 2 S_k + x_{l_r} \\le (1+2\\rho ) x_{l_r} \\Leftrightarrow S_k\\le \\rho x_{l_r}, \\quad r\\ne r_k.", "$ In principle, we could apply this concepts in general environments, and we give the following formal definition: Definition 8 Let $S$ be a finite search strategy on an environment $E$ .We denote $S(E)$ the part of the environment which is explored by $S$ .", "We say that $S$ is $R$ -extendable if for any point $P$ along the boundary of $S(E)$ , there exist $S_P$ a strategy which extends $S$ (i.e.", "$S$ is a prefix of $S_P$ ) and $V_P$ a neighborhood of $P$ such that $V_P \\subset S_P(E)$ and ${\\tt {cr}} (S_P) \\le R$ .", "In other words, an $R$ -extendable strategy is an $R$ -competitive strategy which can be extended to explore infinitesimally farther beyond any point on the boundary of the area explored up to time $T$ , while keeping its competitive ratio below $R$ .", "Any prefix of an infinite strategy with bounded competitive ratio $R$ is $R$ -extendable; in particular prefixes of the geometric and aggressive strategies are extendable." ], [ "Computing the aggressive strategy on the star", "In this section we show that the aggressive strategy on the $m$ -ray star is well-defined for any competitive ratio $R\\le R^_m=1+2m^m/(m-1)^{m-1}$ , and we give an explicit formula for it.", "This aggressive strategy is a cyclic strategy $Z=(z_i)$ which successively maximizes the length searched at each step, within the competitive constraints.", "[24] show that this problem is well-defined, and that there is a strategy which satisfies the linear recurrence relation $ z_{i+m}=\\rho (z_{i+1}-z_i),$ with $R=1+2\\rho $ .", "They give a “canonical” solution for optimal $R=R_m^$ , which we prove is the only solution to this recurrence; we also provide a formula for $R>R^_m$ and prove its uniqueness.", "As noted by [24], there are two initial conditions that we can use to help determine the strategy: $ \\sum _{i=1}^{m-1} z_i = \\rho , \\text{ and } \\sum _{i=1}^m z_i = \\rho z_1.", "$ These correspond to the first two constraints which for finite strategies we denote $(C_0)$ and $(C_1)$ , and all other constraints serve in the recurrence relationship, obtained by subtracting $(C_i)$ from $(C_{i+1})$ .", "To our knowledge, no previous work has give an expression for $Z$ , for general $R$ and $m\\ge 3$ , and in this section we show how to derive it.", "The characteristic polynomial of the recurrence is $\\chi (t)=t^m-\\rho t+\\rho $ .", "If $\\chi $ has a root $\\zeta $ of order $n$ then $(a_0+a_1 i + \\dots + a_{n-1} i^{n-1})\\zeta ^i$ is a solution to the recurrence, for any $(a_0,\\dots ,a_{n-1})\\in \\mathbb {C}^n$ , and any solution is a linear combination of such terms.", "By Descartes' rule of signs, $\\chi $ has either two positive real roots (counting multiplicity) or none.", "Denote $r_m^=\\frac{m}{m-1}$ .", "For $\\rho \\ge \\rho _m^=\\frac{m^m}{(m-1)^{m-1}}$ we have $\\chi (r_m^)\\le 0$ , so $\\chi $ always has exactly two positive real roots, which we denote $\\zeta _1$ and $\\zeta _2$ .", "For $\\rho =\\rho _m^$ , $\\chi $ has a double root at $r^_m$ .", "First we study the case when $\\rho =\\rho _m^$ .", "We can factor $\\chi $ : $\\chi (t)=(t-r_m^)^2(t^{m-2}+2r_m^t^{m-3}+\\cdots +(m-1){r_m^}^{m-2})$ $ = \\frac{t^{m-2}}{r_m^}(t-r_m^)^2\\phi _m^{\\prime }\\left(\\frac{r^_m}{t}\\right), $ where $\\phi _m(t) = \\frac{t^m-1}{t-1}$ .", "$\\phi _m$ has $m-1$ distinct roots on the unit circle, so all roots of $\\phi ^{\\prime }$ are distinct, and inside the convex hull of the roots of $\\phi $ , therefore of norm $<1$ .", "This means that all roots of $\\chi $ which are not $r^_m$ are of norm $>r^_m$ , and as discussed above they must be negative or complex.", "Any meaningful solution to the recurrence formula must be positive, therefore these other roots cannot contribute to the solution.", "In conclusion, using the initialization constraints we obtain the following formula for $Z$ : $ z_i = \\frac{m+i-1}{m-1}\\left(\\frac{m}{m-1}\\right)^i.", "$ Now for the case when $\\rho >\\rho _m^$ .", "For $x\\in ]\\zeta _1,\\zeta _2[$ , we have $\\chi (x)<0 \\Leftrightarrow \\rho x> \\rho + x^m$ , and so Rouché's theorem tells us that there is exactly one root of norm $<\|\\zeta _2\|$ , which we know to be $\\zeta _1$ .", "Suppose that $\\zeta _2 e^{i\\theta }$ is a root of $\\chi $ .", "Then $ \\zeta _2^m e^{mi\\theta }-\\rho \\zeta _2 e^{i\\theta } +\\rho =0 \\text{ and } \\zeta _2^m = \\rho (\\zeta _2-1) $ $\\Rightarrow \\rho (\\zeta _2-1)e^{im\\theta } = \\rho (\\zeta _2 e^{i\\theta }-1)\\Leftrightarrow \\zeta _2 = \\frac{e^{im\\theta }-1}{e^{im\\theta }-e^{i\\theta }}\\in \\mathbb {R}$ $ \\Rightarrow \\frac{e^{im\\theta }-1}{e^{im\\theta }-e^{i\\theta }}=\\overline{\\frac{e^{im\\theta }-1}{e^{im\\theta }-e^{i\\theta }}}=\\frac{e^{im\\theta }-1}{e^{im\\theta }-e^{i\\theta }}e^{i\\theta } \\Rightarrow e^{i\\theta }=1.", "$ This shows that $\\zeta _2$ is the only root of $\\chi $ of that norm, so all other roots are of norm $>\|\\zeta _2\|$ , and being negative or complex they cannot contribute to the solution.", "In conclusion, using the initialization constraints we obtain the following formula for $Z$ : $ z_i = (1+\\alpha ) \\zeta _2^i -\\alpha \\zeta _1^i , \\text{ with } \\alpha = \\frac{\\zeta _1(\\zeta _2-1)}{\\zeta _2-\\zeta _1}.", "$ Computing $\\zeta _i$ can be done most efficiently with binary search using $1\\le \\zeta _1 \\le r_m^* \\le \\zeta _2 \\le \\rho ^{\\frac{1}{m-1}}$ ." ], [ "Cyclicality and monotonicity in $L_m^{(k)}$ ", "In this section we show that any optimal solution to $L_m^{(k)}$ corresponds to a cyclic and monotone strategy.", "The basic steps of the proof are similar to those found in [24].", "We begin with a tightness lemma similar to Lemma REF .", "Lemma 9 In any optimal solution to $L_m^{(k)}$ , at least one of $(C_0)$ and $(B)$ is tight.", "All other constraints $(C_j)$ and $(E_r)$ are tight.", "We extend the proof of Lemma REF to the case of the star.", "Suppose $X^=(x^_i)$ is an optimal solution to $L_m^{(k)}$ , which does not satisfy the conditions of the lemma.", "Recall that there are implicit conditions in the formulation of $L_m^{(k)}$ , namely $x_i>x_{\\bar{\\imath }}$ .", "If a constraint $(C_j)$ is not tight, then we can decrease $x^_{\\bar{\\jmath }}$ by a small quantity $\\delta $ and increase $x^_k$ by $\\delta $ in order to obtain a feasible solution with a higher objective value, which contradicts the optimality of $X^$ .", "If $(C_0)$ and $(B)$ are both loose, then we can scale up $X^$ by a factor $\\alpha >1$ , thus increasing the objective value, a contradiction.", "Finally, if a constraint $(E_r)$ is loose, then decreasing $x_{l_r}$ and increasing $x_k$ by a small quantity $\\delta $ creates a new feasible strategy which is also optimal, because it has the same objective.", "If $(C_k)$ exists, i.e.", "$j_0<k$ , then constraint $(C_k)$ becomes loose, and if not, then constraints $(C_0)$ and $(B)$ become simultaneously loose; either case provides a contradiction to the above.", "The following property is very intuitive and will be needed to show cyclicality.", "Similar properties are very often useful in star search problems.", "Property 10 Any optimal strategy visits, at each step, the ray which has been explored the least so far.", "First, we prove, by way of contradiction, that any optimal strategy begins by visiting each ray once.", "Let $X^$ be an optimal strategy, and recall that $j_0$ is the last step during which we explore a new ray.", "Suppose that $X^$ visits the same ray $r$ twice before step $j_0$ , say at steps $i_1$ and $i_2$ , with $i_2<j_0$ .", "Then we could simply halve the size of $x_{i_1}$ and obtain a new feasible strategy $\\hat{X}$ which has loose constraints: indeed, $x_{i_1}$ only shows up on the left-hand side of the inequalities in $L_m^{(k)}$ , so all constraints are loosened.", "But by lemma REF $\\hat{X}$ cannot be optimal, and neither can $X^$ , which has the same objective value, a contradiction.", "Now we look at the steps after $j_0$ .", "From Lemma REF , we get for any optimal strategy the set of equations $(S_{j-1}=\\rho x_{\\bar{\\jmath }})$ and $S_k = \\rho x_{l_r}, l_r\\ne r_k$ (Recall the definition of $S$ that we gave in the first line of this Appendix).", "This makes it clear that $(x_{\\bar{\\jmath }})_{j_0\\le j\\le k}$ is an increasing series, and that the final steps on each ray are the last $m$ steps.", "This is precisely the statement of our lemma: at each step $i$ , the length to which we had previously explored $r_i$ is increasing, or equivalently, at each step we visit the least explored ray.", "To see this more clearly, we give a proof by contradiction.", "Suppose there is a step $i_1$ where an optimal strategy $X^$ visits a ray $r_1$ which has been explored more than the least explored ray $r_0$ .", "$X^$ can never return to visit $r_1$ , because if it visits $r_0$ on step $i_0>i_1$ , then $x^_{\\bar{\\imath }_0}<x^_{\\bar{\\imath }_1}$ , a contradiction.", "But if $X^$ never returns to ray $r_0$ , then we have $x^_{l_{r_0}}<x^_{\\bar{\\imath }_1}$ , a contradiction.", "Now we can move on to the main result.", "Theorem 11 Any optimal solution to $L_m^{(k)}$ must be monotone and cyclic, that is $(x_i)$ must be increasing and $r_i = i \\mod {m}$ up to a permutation.", "Let $X=(x_i, r_i)$ be an optimal solution to $L_m^{(k)}$ .", "The proof of monotonicity borrows the swapping idea from [18].", "Suppose that $X$ is not monotone, i.e.", "$\\exists i_0, x_{i_0}>x_{i_0+1}$ .", "Define strategy $Y=(y_i,s_i)$ to be a modification of strategy $(x_i,r_i)$ where we swap the two steps $x_{i_0}$ and $x_{i_0+1}$ as well as the roles that rays $r_{i_0}$ and $r_{i_0+1}$ play after the swap.", "Formally, $y_i = x_i$ except for the swap $y_{i_0}=x_{i_0+1}, y_{i_0+1}=x_{i_0}$ , and $s_i = r_i$ , except when $i> i_0+1$ and we search $ r_{i_0}$ or $r_{i_0+1}$ : in this case $r_i=r_{i_0} \\Rightarrow s_i=r_{i_0+1}$ and $r_i=r_{i_0+1} \\Rightarrow s_i=r_{i_0}$ .", "Swapping does not increase the partial sums: $S_j^{(Y)} \\le S_j^{(X)}$ for all $j$ , so $(C_0)$ holds for $Y$ , as well as $(B)$ .", "Swapping does not change the set of the last steps on each ray: $\\lbrace x_{l_r},l_r\\rbrace =\\lbrace y_{l_r},l_r\\rbrace $ , and so if $i_0\\ne k-1$ , all constraints $(E_r)$ hold for $Y$ .", "Most importantly, swapping has the nice property that for all $j$ , $y_{\\bar{\\jmath }}=x_{\\bar{\\jmath }}$ .", "So for all $j$ , $(C_j)$ holds for $Y$ .", "Recall the tightness property (lemma REF ).", "$ \\rho y_{\\overline{i_0+1}} = \\rho x_{\\overline{i_0+1}} = S_{i_0}^{(X)} > S_{i_0}^{(Y)} $ so $(C_{i_0+1})$ is not tight for $Y$ , therefore $Y$ cannot be optimal according to lemma REF , and netiher can $X$ , which has the same objective value: a contradiction.", "We will address the case $i_0=k-1$ later.", "This does not impact the proof of cyclicality.", "Now for cyclicality.", "We showed above that any optimal strategy must be monotone (up to step $k-1$ ).", "Recall property REF .", "Take an optimal strategy: we can suppose that it begins by visiting the rays in order, 1 to $m$ .", "On step $m+1$ , it needs to visit the least visited ray so far, which is ray 1, because of monotonicity.", "Then ray 1 becomes the ray which has been visited the most so far; an immediate induction follows, proving that the strategy is cyclic.", "We left a piece of the monotonicity property hanging, the case where $i_0=k-1$ : we still need to prove that if $X=(x^_i)$ is an optimal strategy, then $x_k^\\ge x_{k-1}^$ .", "We can show this by applying algorithm (see definition in the proof of lemma REF on the next page) to the cyclic geometric strategy $G=(g_i)=({r_m^}^i)$ .", "We have the identity $g_k> g_{k-1}$ at the start of the algorithm, and at each step of the algorithm we increase $g_k$ and decrease $g_{k-1}$ , before scaling up by a factor $\\alpha $ , hence $x_k^\\ge x_{k-1}^$ ." ], [ "Finding the optimal values of $k$ for {{formula:3d0e3c4b-cf60-438c-963c-edf20009f007}}", "In this section we prove Theorems REF and REF .", "The main idea for the proof of Theorem REF is the same as for the line: we show that the terms in $X_0^{(k)}$ are increasing, therefore the largest feasible $k$ is optimal, and then we show that the objective values of $X_B^{(k)}$ are decreasing, therefore the smallest feasible $k$ is optimal.", "However, the proof is much more involved than the proof for the line, because $X_0^{(k)}$ is no longer simply a prefix of $X_0^{(k+1)}$ .", "Lemma REF is a technical result which allows us to prove Lemma REF , which results directly in the first part of the theorem; some more calculations give us the second half of the theorem in Lemma REF .", "In this whole section, we discard all monotonicity constraints $(M_i)$ , with the exception of the final one $x_k\\ge x_{k-1}$ , which we will relabel $(M)$ .", "We also discard the implicit constraints $x_i>0$ and $x_{i+m}>x_i$ , regarding $P_m^{(k)}$ as simply a set of equations.", "The following technical result is key to efficiently determining the optimal values of $k$ .", "Lemma 12 Any point $X=(x_i)$ which is satisfies all of the constraints in $P_m^{(k)}$ is positive, that is for all $i$ , $x_i\\ge 0$ .", "Also, $x^_k-x^_{k-1} \\ge x_k-x_{k-1}$ .", "Take $X=(x_i)$ a feasible point for $P_m^{(k)}$ .", "Using the methods from the proof of lemma REF , we can transform $X$ into the optimal strategy $X^$ by performing the process described in Algorithm .", "[htb!]", "Feasible to optimal Input: $X = (x_i)$ a feasible point for $P_m^{(k)}$ any constraint $(C_j)$ or $(E_j)$ is loose j=1,..., k-m $(C_j)$ is loose select $\\delta >0$ so that $(C_j)$ will become tight $x_j, x_k \\leftarrow x_j - \\delta ,\\,\\, x_k + \\delta $ j=k-m+1,...,k-1 $(E_j)$ is loose select $\\delta >0$ so that $(E_j)$ will become tight $x_j, x_k \\leftarrow x_j - \\delta ,\\,\\, x_k + \\delta $ select $\\alpha $ such that $(C_0)$ or $(B)$ will become tight $x_i \\leftarrow \\alpha x_i$ for all $i$ Output: $X=X^$ the optimal solution This algorithm has a purely conceptual value, because every time we tighten a constraint, we loosen at least one other, and thus it cannot finish in finite time.", "However, convergence is guaranteed by the fact that $x_k$ increases at each iteration and is bounded from above by $T$ , therefore it must converge.", "All other variables $x_i$ must also converge, because they are all decreasing, and each one has a total variation of less than $x^_k-x_k$ .", "The reason the output must be $X^$ is that all constraints are tight, witht the exception of $(M)$ and at most one of $(C_0)$ and $(B)$ .", "By running algorithm on $X$ , we decrease each $x_i,\\,i<k$ by a certain amount, then scale it up by some $\\alpha \\ge 1$ , and obtain $x^_i\\ge 0$ , hence necessarily $x_i\\ge 0$ .", "Using constraint $(M)$ we see that $x_k\\ge x_{k-1}\\ge 0$ .", "We call attention to a subtle detail: without constraint $(M)$ , we could have had $\\alpha <0$ , for example if we start from the negative version of the optimal solution $(-x^_i)$ .", "But constraint $(M)$ cannot be tight in the optimal solution, as shown by executing the algorithm on the geometric strategy $G=({r^_m}^i)$ : constraint $(M)$ starts out being non-tight and loosens progressively as the algorithm runs, therefore it cannot be tight for $X^$ .", "Because constraint $(M)$ is satisfied for $(x_i)$ all throughout the process, we cannot have $\\alpha <0$ , which would flip $(M)$ and violate it in $X^$ , a contradiction.", "Constraint $(M)$ can only get looser at each step of algorithm , which proves the second part of our lemma.", "If we remove constraint $(B)$ from $P_m^{(k)}$ , we get an LP $P_{m,0}^{(k)}$ for which the solution is $X_0^{(k)}$ , and similarly, by removing constraint $(C_0)$ from $P_m^{(k)}$ we get an LP $P_{m,B}^{(k)}$ , for which the solution is $X_B^{(k)}$ .", "Lemma REF and algorithm can be readily extended to show that any feasible point for $P_{k,0}^m$ or $P_{k,B}^m$ is positive, and that the inequality corresponding to constraint $(M)$ is valid.", "One would expect that as $k$ grows, giving $X_0^{(k)}$ more steps to explore the domain, it is able to explore farther; conversely, it seems reasonable, though not quite obvious, that once the time budget is used up, it is best to waste as little time as possible taking extra steps, which backtrack on previously covered ground, and so $X_B^{(k)}$ should perform best for smaller $k$ .", "We will show that this is indeed the case.", "Lemma 13 Denote $x_{0,i}^{(k)}$ the $i$ -th step in the strategy $X_0^{(k)}$ for each $k$ .", "For all $i$ , $(x_{0,k-i}^{(k)})_{k\\ge i}$ is increasing.", "Fix $k$ .", "In order to simplify notations, denote $X=(x_i)=X_0^{(k)}$ and $Y=(y_i) = X_0^{(k-1)}$ .", "It suffices to show that $\\forall i, x_i\\ge y_{i-1}$ .", "First we need to work to prove the following inequality: $x_k-x_{k-1} \\ge y_{k-1}-y_{k-2}.$ Set $r=m/(m-1)$ .", "Recall the characteristic polynomial $p(t) = t^m-\\rho t +\\rho $ .", "For the optimal $\\rho ^_m$ , we have $p(r)=0 \\Rightarrow r^m/(r-1) = \\rho ^_m$ , so in the general case $r^m/(r-1) \\le \\rho $ .", "This gives us the following identity: $ \\sum _{i=0}^{j+m-1} (r-1)r^i = r^{j+m} - 1 \\le \\rho (r-1)r^j - 1.", "$ Define $Z = (z_i)_{i\\le k}$ by $z_i = y_{i-1} + (r-1)r^{i-1}$ , using the convention $y_0=1$ .", "$Z$ is a feasible point for $P_{k,0}^m$ .", "Indeed, we verfiy each constraint: $S_{m-1}^{(Z)} &= S_{m-2}^{(Y)} + 1 + \\sum _{i=0}^{m-2} (r-1)r^i \\\\&\\le \\rho - y_{m-1} + \\frac{\\rho }{m} \\le \\rho , & (C_0)$ because $(r-1)/r = 1/m$ , and due to monotonicity (Theorem REF ), step $y_{m-1}$ needs to account for at least $1/(m-1)$ of the sum $S_{m-1}=\\rho $ , so $y_{m-1}\\ge \\rho /(m-1) \\ge \\rho /m$ .", "$S_{j+m-1}^{(Z)} &= S_{j+m-2}^{(Y)} + 1 + \\sum _{i=0}^{j+m-2} (r-1)r^i \\\\&\\le \\rho y_{j-1} + \\rho (r-1)r^{j-1} = \\rho z_j, & (C_j)$ and similarly each $(E_j)$ holds.", "Constraint $(M)$ also holds: $z_k = y_{k-1}+(r-1)r^{k-1}\\ge y_{k-2} + (r-1)r^{k-2} = z_{k-1}$ .", "We finish by applying the second half of lemma REF to $Z$ : $x_k - x_{k-1} &\\ge z_k-z_{k-1} = y_{k-1}-y_{k-2} + (r-1)^2 r^{k-2}\\\\&\\ge y_{k-1} - y_{k-2}.$ Now that we have (REF ), we can apply lemma REF to the difference of the strategies $X$ and $Y$ , in order to show that $X$ is “bigger” than $Y$ .", "Define $\\Delta = (\\delta _i)_{i\\le k}$ by $\\delta _i = x_i - y_{i-1}$ , with the convention $y_0=1$ .", "$\\Delta $ is a feasible point for $P_{k,0}^m$ .", "Indeed, we can verify each constraint: $ S_{m-1}^{(\\Delta )} = S_{m-1}^{(X)}-S_{m-2}^{(Y)} = y_{m-1} \\le \\rho , \\qquad (C_0) $ $ S_{j+m-1}^{(\\Delta )} = \\rho x_j - \\rho y_{j-1} = \\rho \\delta _{j}, \\quad j \\in [1,k-m+1] \\, (C_j) $ $ S_{k}^{(\\Delta )} = \\rho x_j - \\rho y_{j-1} = \\rho \\delta _{j}, \\quad j \\in [k-m+1,k-1] \\, (E_j) $ $ \\text{and finally } (\\ref {eq:delta.M}) \\Leftrightarrow \\delta _k \\ge \\delta _{k-1}.", "\\qquad (M) $ Using lemma REF we obtain that for all $i$ , $\\delta _i \\ge 0 \\Leftrightarrow x_i\\ge y_{i-1}$ , which concludes our proof.", "Corollary 14 Both the total length cleared and the time taken by strategy $X_0^{(k)}$ are increasing.", "The objective is $\\sum _{i=0}^{m-1} x_{0,k-i}^{(k)}$ which is a sum of increasing series; so is the time taken $S_{k-1} + S_k$ .", "Corollary 15 (First half of Theorem REF ) There is a critical value $k_0$ such that $X_0^{(k)}$ is feasible for $P_m^{(k)}$ if and only if $k\\le k_0$ .", "This critical value achieves the maximum clearance among all feasible strategies $X_0^{(k)}$ .", "For $X_B^{(k)}$ , we have the reverse situation, where the lowest feasible $k$ yields the optimal solution.", "The proof is a bit more difficult.", "Lemma 16 (Second half of Theorem REF ) There is a critical value $k_B$ such that $X_B^{(k)}$ is feasible for $P_m^{(k)}$ if and only if $k\\ge k_B$ ; either $k_B=k_0$ or $k_B=k_0+1$ .", "The optimal strategy among all feasible $X_B^{(k)}$ is $X_B^{(k_B)}$ .", "First, because $X_0^{(k)}$ and $X_B^{(k)}$ both belong to the same line $\\Delta _m^{(k)}$ , they are scaled versions of each other.", "We saw that the time taken by $X_0^{(k)}$ increases with $k$ , until constraint $(B)$ is surpassed for $k>k_0$ .", "Before this point, $X_B^{(k)}$ is infeasible for $P_m^{(k)}$ due to constraint $(C_0)$ .", "If $(B)$ is tight for $X_0^{(k_0)}$ , then $k_B=k_0$ and the two strategies $X_0^{(k_0)}$ and $X_B^{(k_B)}$ are identical.", "If not, then $k_B = k_0+1$ .", "Now we show that $X_B^{(k_B)}$ is optimal.", "In order to simplify notations, denote $X=(x_i)_{i\\le k} = X_0^{(k)}$ and $Y=(y_i)_{i\\le k-1} = X_0^{(k-1)}$ .", "Note that $(\\gamma x_i)_{i\\ge 2}$ is a feasible point for $P_m^{(k-1)}$ , for suitably small $\\gamma \\le 1$ , chosen to make constraint $(C_0)$ hold.", "Apply algorithm to $(\\gamma x_i)_{i\\ge 2}$ , and denote $X^{\\#}=(x_i^{\\#})_{i\\ge 2}$ the value taken by our strategy right before we scale it up by $\\alpha $ , i.e.", "the value of $X$ if we halt the algorithm at line 16.", "Considering constraint $(M)$ , which only gets looser as the algorithm runs, we have the following identity: $ \\gamma (x_{k}-x_{k-1}) \\le x_k^{\\#}-x_{k-1}^{\\#} = (y_{k-1}-y_{k-2})/\\alpha , $ and considering the penultimate step of our strategy: $ \\gamma x_{k-1} \\ge x_{k-1}^{\\#} = y_{k-2}/\\alpha .", "$ Dividing these two identities by each other, we obtain a key inequality: $\\frac{x_k-x_{k-1}}{x_{k-1}} \\le \\frac{y_{k-1}-y_{k-2}}{y_{k-2}} \\Leftrightarrow \\frac{x_k}{x_{k-1}} \\le \\frac{y_{k-1}}{y_{k-2}}.$ Denote the total area cleared by strategy $X$ by $ {\\bf clr}(X) = \\frac{x_{k-m+1}+\\dots +x_{k}}{2S_{k-1}^{(X)}+x+k}T=\\frac{(m-1)x_{k-1}+x_k}{2\\rho x_{k-1}-x_k}T. $ We conclude by showing ${\\bf clr}(X) \\le {\\bf clr}(Y)$ : $ \\frac{(m-1)x_{k-1}+x_k}{2\\rho x_{k-1}-x_k} \\le \\frac{(m-1)y_{k-2}+y_{k-1}}{2\\rho y_{k-2}-y_{k-1}} $ $ \\Leftrightarrow (2\\rho + m - 1) x_k y_{k-2} \\le (2\\rho + m - 1) x_{k-1} y_{k-1} $ (developing the cross-product and removing identical terms) $ \\Leftrightarrow \\frac{x_k}{x_{k-1}} \\le \\frac{y_{k-1}}{y_{k-2}} \\text{ which is the inequality (\\ref {eq:key}).", "}$ Lemma REF also provides the missing detail (the fact that $k_0$ is close to $k_B$ ) needed to complete the proof of Theorem REF ." ], [ "Implementation details", "We implemented the algorithms for both the star and the network in Python, and we run the experiments on a standard laptop.", "We implemented Cpt(r) and Rpt(r) using the NetworkX library (https://networkx.github.io).", "As stated in the main paper, we used networks from the online library Transportation Network Test Problems [6].", "We made the following minor modifications: we made the networks undirected, contracted nodes joined by edges of length 0, and then scaled each network so that the shortest edge has length 4: the last step is necessary because some networks have lengths in miles and others in meters.", "Table REF shows the sizes of the networks we used for our experiments.", "Table: Sizes of the networks used in our experiments." ], [ "Experiments on the star", "We observed that our optimal strategy has a strong relative advantage over the other two strategies (the mixed aggressive and the scaled geometric).", "Table REF demonstrates this advantage, for different values of $m$ and $R$ , and for a budget $T$ fixed to $T=10^{16}$ .", "As shown in Figure REF , for smaller $T$ we expect an even stronger advantage of the optimal strategy.", "From the same figure, we observe that as $T$ becomes even larger than $10^{16}$ , we expect the same asymptotic behavior as shown in Table REF .", "The relative advantage reaches $42\\%$ for large values of $m$ , and is significant for a wide range of values of $R$ .", "For much larger values of $R$ (i.e.", "$R\\ge 100 R_m^$ ), the relative advantage does eventually drop to 1, at which point the strategies are practically indistinguishable in terms of clearance.", "Table: Relative advantage of the optimal strategy over the other two strategies, for various values of mm andRR.", "Each entry is the ratio of the clearance achieved by the optimalstrategy over the clearance of the scaled aggressive strategy" ], [ "Experiments on networks", "We found that in practice, Rpt(r) never took longer than Cpt(r) to complete a tour: this is in part due to the fact that Rpt(r) performs its tour on a smaller subgraph than Cpt(r).", "We also added a small variation to Rpt(r): we do not require the RPT to return to the origin, and once all edges have been traversed, we use the current node as the starting point for the next tour.", "We present further experiments showing runs for other networks in our dataset (Figures REF and REF ).", "We see that Rpt(r) performs better than Cpt(r) even for smaller networks, though the results are more pronounced for the larger ones, as expected.", "Figure: Comparison of the two strategies on the Eastern-Massachussets network (74 nodes, 129 edges).Figure: Comparison of the two strategies on the Friedrichshain network (144 nodes, 240 edges).Figure REF depicts the influence of the parameter $r$ on the competitive ratios of Rpt(r) and Cpt(r), as run on the small Sioux Falls network, starting from a node located near the center of the network.", "We observe that there is indeed a minimum competitive ratio reached for $r\\approx 2$ .", "Interestingly, this is in accordance with Proposition REF , which shows that choosing $r=2$ yields the best approximation to the competitive ratio, for both Rpt(r) and Cpt(r).", "Figure: Competitive ratio of the strategies as a function of the parameter rr, calculated on the Sioux Falls network (24 nodes, 38 edges).Figure REF is analogous to Figure REF , but for 45 random runs on the Chicago network.", "We see that the relative advantage of Rpt(r) over Cpt(r) is even greater for a larger network.", "Figure: Clearance ratio of Rpt(r) versus Cpt(r), for 45 randomly chosen roots, for the Chicago network.Figure REF depicts the competitive ratios of each strategy over those 45 runs, sorted by increasing competitive ratio for Rpt(r).", "We see that Rpt(r) is consistently much more efficient than Cpt(r), and it is also much more stable, especially for those roots for which the algorithms yield larger competitive ratios.", "The average competitive ratio over these runs for Rpt(r) is 152, compared to 200 for Cpt(r); the standard deviations are 26 and 39 respectively.", "Figure: Competitive ratios of the strategies for 45 random chosen roots, for the Chicago network." ], [ "Solving the Earliest Clearance problem", "We give an overview about how the techniques we used in the context of the Maximum Clearance problem can help us solve this “dual” online search problem.", "Recall that the problem is defined in the last section of the main paper." ], [ "The line environment", "For the unbounded line, we have an LP formulation similar to $L_2^{(k)}$ , where we “exchange” the objective function and the final constraint: namely, we want to minimize $2\\sum _{i=1}^{k-1}+x_k$ , and add the constraint $x_k+x_{k-1}\\ge L$ .", "We can prove that in an optimal solution, all but one constraints must be tight, similarly to Lemma REF , though for this problem only the first constraint $(C_0)$ may be loose.", "We can argue that the scaled aggressive strategy is optimal, since the final constraint $x_k+x+{k-1}\\ge L$ is always tight." ], [ "The star environment", "We can formulate this problem using an LP similar to $L_m^{(k)}$ .", "First we can show a tightness result similar to Lemma REF , though this problem is easier: all constraints are tight except for possibly $(C_0)$ .", "The proof of monotonicity and cyclicality is identical.", "This allows us to consider the LP in cyclic form: $\\text{min}& \\quad 2\\sum \\nolimits _{i=1}^{k-1} x_i+ x_k &\\\\\\text{subj to}& \\quad \\sum \\nolimits _{i=1}^{m-1} x_i\\le \\rho & \\\\&\\sum \\nolimits _{i=1}^{j+m-1} \\!x_i\\le \\rho \\cdot x_{j}, \\qquad \\, j \\in [1, k-m] &\\\\&\\!\\!\\!\\!\\sum \\nolimits _{i=1}^k x_i \\le \\rho \\cdot x_j, \\;\\; j \\in [k-m+1,k-1] & \\\\&x_i \\le x_{i+1}, \\qquad \\qquad \\qquad \\quad \\; i \\in [1,k-1]&\\\\&\\sum \\nolimits _{i=0}^{m-1} x_{k-i} \\ge L &$ Each $P_m^{(k)}$ has a single solution which can be obtained in time $O(k)$ using Gaussian elimination on a matrix equation similar to (REF ).", "We can show by the same methods used in the proof of Theorem REF that the feasible solution with the fewest steps is the optimal solution, and with the geometric strategy giving an upper bound on this number of steps, we can use binary search to find the solution in time $O(m\\log T \\log (m\\log T))$ .", "The experimental results we observe are extremely similar to those for Maximum Clearance: in short, the optimal strategy dominates the scaled aggressive and geometric strategies, and the same dependencies on $m$ and $R$ are observed." ], [ "General networks", "For general networks, we use the same heuristic as for the Maximum Clearance problem: specifically, we run Rpt(r) until a total length $L$ has been cleared, using $r=2$ .", "Similar conclusions can be reached, and we can quantify the relative improvement of Rpt(r) over Cpt(r).", "For example, from Figure REF we can deduce for each value of clearance $L$ , the time it took the two heuristics to clear length $L$ .", "Figure REF is analogous to Figure REF , and depicts the average ratio between the time taken by Cpt(r) and the time taken by Rpt(r) as a function of the desired length $L$ .", "We observe the expected improvements, which get quite significant for large values of $L$ .", "Figure: Speed of Rpt(r) versus Cpt(r), for 45 randomly chosen roots, for the Chicago network." ] ]	2011.14144
[ [ "Neutrino cooling effect of primordial hot areas in dependence on its\n size" ], [ "Abstract We consider the temperature dynamics of hypothetical primordial hot areas in the Universe.", "Such areas can be produced by the primordial density inhomogeneities and can survive to the modern era, in particular due to primordial black hole (PBH) cluster of size $R \\gtrsim 1$ pc and more.", "Here we concentrate on the neutrino cooling effect which is realized due to reactions of weak $p\\leftrightarrow n$ transitions and $e^{\\pm}$ annihilation.", "The given neutrino cooling mechanism is found to work in a wide range of parameters.", "For those parameters typical for PBH cluster considered, the cooling mechanism is quite valuable for the temperatures $T \\gtrsim 3$ MeV." ], [ "Introduction", "There are some observations [1] indicating the existence of local heated areas in the early Universe.", "Hypothetical nature of local heated areas was discussed earlier [2], [3], [4].", "Such areas can appear due to large primordial density fluctuations and can be related to the clusters of Primordial Black Holes (PBHs) [1], [5], [6].", "We assume that the baryonic matter has been captured by the gravitational forces of these regions at the early Universe.", "They would remain hot for a long time.", "At the same time, many processes can heat or cool the matter inside them during their formation after it.", "Short list of them is the neutrino cooling [6], nuclear reactions, radiation of the hot plasma and stars formed inside the region [8], gravitational dynamics of the system, shock waves, diffusion of matter, variation of the vacuum state while the region is born [9], energy transfer from collapsing walls [10], [11], [12], [13], accretion, the Hawking evaporation.", "The last mechanisms are relevant in the case of PBHs origin of the regions [5], [14], [15], [16].", "In this proceedings, we continue our consideration of neutrino cooling of such regions.", "It could be the most important reason for the temperature evolution within initial temperature range – keV$<T<10$ MeV.", "In this research, we follow the initial conditions taken from [5], [6], where the mass of trapped matter is in wide range $10^4$ –$10^8 M_{\\odot }$ .", "The main initial parameters are as follows: the size of the region is about $R\\sim 1$ pc, its mass $10^4 \\, M_{\\odot }$ , initial temperature is in the interval $T_0\\sim 1\\text{ keV}\\div 10$ MeV.", "This temperature of such regions could be reached in several ways.", "The region can start to be formed at higher temperature and finish to do it having cooled down to $T_0$ .", "Also, the region could be heated up during formation, e.g., in the framework of model with collapsing domain walls [7].", "Without specific assumptions, we show that effect of neutrino cooling is wide spread phenomena valid in wide range of parameters.", "The range of initial parameters is under consideration.", "Neutrino cooling effect can be suppressed at high temperatures and large sizes when the area becomes opaque to the neutrinos.", "Neutrinos are produced due to reactions of $p\\leftrightarrow n$ transition and $e^+e^-$ annihilation.", "The characteristic time for photons to escape the area is bigger than the modern Universe age, this indicates that the size of cluster is big enough not to lose photons.", "In the given proceedings we study the impact of the size of the region on the neutrino cooling effect.", "Mechanism of neutrino cooling rates for the main reactions of the neutrino production is considered in Section 2.", "The impact of the diffusive character of particle propagation inside the cluster is briefly discussed in Section 3." ], [ "Cooling Rates", "Let us consider the reactions of the neutrino production: $e^- + p \\rightarrow n + \\nu _e,$ $e^+ + n \\rightarrow p + \\bar{\\nu }_e,$ $e^+ + e^- \\rightarrow \\nu _{e,\\mu ,\\tau }+\\bar{\\nu }_{e,\\mu ,\\tau },$ $n \\rightarrow p+e^- + \\bar{\\nu }_e.", "$ The produced neutrinos leave the heated area if it is not very big.", "The energy inside the volume is decreased that leads to the temperature decreasing.", "The rates per unit volume, $\\gamma _i\\equiv \\Gamma _i/V$ , for reactions listed above are respectively $\\gamma _{ep}=n_{e^-}n_p\\sigma _{ep}v,\\;\\;\\;\\;\\;\\gamma _{en}=n_{e^+}n_n\\sigma _{en}v,\\\\\\gamma _{ee}=n_{e^-}n_{e^+}\\sigma _{ee}v,\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\;\\gamma _n=\\frac{n_n}{\\tau _n}.$ Here $n_i$ is the concentration of the respective species, $\\sigma _{ij}$ is the cross section (see e.g.", "[6]) of interacting particles $i\\, {\\rm and }\\,j$ and $\\tau _n\\approx 1000$ s is the neutron lifetime.", "We consider the relativistic plasma so that the relative velocity $v\\simeq 1$ .", "The backward reactions for Eqs.", "(REF )–(REF ) are suppressed if neutrinos freely escape the cluster.", "We consider all densities to be independent of the space coordinate inside the region.", "The number densities are roughly described by the following formulas, see [6], $n_{e^-}=n_{e^+}+\\Delta n_e,\\;\\;\\;\\;\\;n_{e^+}= n_e^{eq}(T) \\exp \\left(-\\frac{m_e}{T}\\right),\\\\n_B\\equiv n_p+n_n=g_B\\, \\eta n_{\\gamma }(T_0),\\;\\;\\;\\;\\; \\Delta n_e\\equiv n_{e^-}-n_{e^+}=n_p.$ which are slightly corrected for better adjustment to the non-relativistic limit.", "Here $\\eta =n_B/n_{\\gamma }\\approx 0.6\\cdot 10^{-9}$ is the baryon to photon ratio in the modern Universe, $g_B\\sim 1$ is the correction factor of that relation due to entropy re-distribution, $n_{\\gamma }(T)=\\frac{2\\zeta (3)}{\\pi ^2}T^3$ and $n_e^{eq}(T)=\\frac{3\\zeta (3)}{2\\pi ^2}T^3$ are the equilibrium photon and relativistic electron number densities respectively.", "Note that $n_{\\gamma }(T_0)$ defines baryon density which is supposed to be unchanged starting from initial temperature $T_0$ contrary to that of $e^{\\pm }$ and $\\gamma $ .", "Number of $e^{\\pm }$ (along with $\\gamma $ ) changes due to $e-\\nu $ -conversion processes (reactions Eqs.", "(REF ) – (REF )).", "The temperature of the system decreases due to neutrino escape.", "Number densities of the electrons and photons fall down with temperature as $\\sim T^3$ ." ], [ "Escaping Time", "The escape time of neutrinos from the region of the size $R$ with temperature $T$ can be calculated as: $t_{esc}\\sim \\frac{R^2}{D} \\sim R^2\\cdot n_e \\sigma _{\\nu }$ in diffusion approximation.", "Here the diffusion coefficient is $D=\\frac{\\lambda _{\\nu }\\cdot v}{3}$ [17], velocity $v=1$ , the neutrino mean free path is $\\lambda _{\\nu }= 1/n_e \\sigma _{\\nu }$ and $e-\\nu $ interaction cross section was roughly taken as $\\sigma _{\\nu } \\sim G_F^2 \\cdot T^2$ .", "The electron number density $n_e\\sim n_{e^-}+n_{e^+}\\sim n_{e^+}$ is given by Eq.", "(REF ) which is $\\sim T^3$ at $T>m_e$ .", "One can conclude from inequality $t_{esc}\\sim R^2 G_F^2 T^5<t_U$ that the neutrino cooling effect is significant up to the present time $t_U$ for the region of the size $R< 35 \\cdot (T/{\\rm MeV})^{-5/2}\\,{\\rm pc}.$ Here conditions $T\\gtrsim m_e$ and $n_e\\sim T^3$ are assumed.", "Figure: Left: The relation between escaping time of neutrino and temperature of the area and the blue line is the modern age of the Universe.", "(Behaviour of the curves at t esc ∼10 -20 t_{esc}\\sim 10^{-20} Gyr reflects the fact that n e n_e becomes ∼Δn e \\sim \\Delta n_e, i.e.", "constant.)", "Right: The relation between size and temperature of the area.", "Neutrino cooling effect plays a prominent role below the thick red dot-dashed curve.", "The black solid line corresponds to the dependence of the Universe horizon from its temperature (R=10 -7 (MeV/T) 2 pcR=10^{-7} (\\text{MeV}/T)^2 \\text{ pc}).Neutrino cooling effect due to reactions of weak $p\\leftrightarrow n$ transitions and $e^{\\pm }$ annihilation are represented in Figure REF where the escaping time of neutrinos in dependence on temperature is shown.", "As seen, at the temperature $T\\lesssim 3$ MeV the escaping time for the most of considered cluster sizes is less than the modern Universe age, thus neutrino cooling works.", "Note, that at $T\\ll m_e$ the curves start to fall until the number density of electrons becomes $n_e \\sim \\Delta n_e$ .", "Dependence $R(T)$ is shown in Figure REF , right panel, which follows from Figure REF , left panel.", "The region above the red line relates to the case when neutrino cooling is suppressed (neutrinos do not run away from the region during the Universe age).", "Black line shows the horizon size of the Universe in dependence on the matter temperature.", "One can see, horizon size is much smaller than the maximal size of region at the same temperature when neutrino cooling effect is, shown by the red line.", "Therefore, the neutrino cooling effect holds under usual conditions, and can be suppressed in extreme cases.", "The region can start its formation at very high temperature so that it could be cooled to the considered temperature during its detachment from Hubble flow and virialization.", "Also, the region could be heated up additionally during its formation, e.g.", "due to wall collapsing [7].", "During detachment and virialization, the region could expand to some extent and hence, cool down." ], [ "Conclusions", "In earlier work [6], we have shown that due to neutrino emission (at a fixed size) the primordial hot areas are cooling down to the temperature value $\\sim $ 0.01 $\\div $ $0.1$ MeV.", "Here we just investigated the neutrino cooling mechanism of the heated region and dependence on its size.", "Considering the result of equation (REF ) for escaping time, we can find the size-changing more effectively with temperature.", "At the temperature $T > 3$ MeV, the diffusive character of particle propagation makes the time of escaping or time of cooling more than the modern Universe age.", "This result is obtained at the definite initial region parameters (size and temperature, relevant for PBH cluster model [5]) that could be slightly varied.", "It illustrates general property for such possible primordial inhomogeneities.", "It is seen that neutrino cooling effect should take place for a wide reasonable size/temperature range of parameter.", "Extreme heating up of the area while it has being formed could change situation.", "There are a variety of mechanisms that can be responsible for the area heating.", "Additional heating during their creation is also possible.", "As was mentioned in the Introduction, the area could be heated by the collapsing walls - the scalar field kinks.", "The fermion reflection on kinks was studied in [7].", "It was shown that the reflection weakly depends on the fermion mass.", "Therefore the kinks could prevent the neutrinos from escaping.", "More detailed analysis is necessary to clarify this effect." ], [ "Acknowledgements", "The work of K.B.", "was supported by the Ministry of Science and Higher Education of the Russian Federation by project No 0723-2020-0040 “Fundamental problems of cosmic rays and dark matter” and of S.R.", "by the project \"Fundamental properties of elementary particles and cosmology\" No 0723-2020-0041.", "The work of S.R.", "is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University and RFBR grant 19-02-00930." ] ]	2011.14221
[ [ "Towards Robust Partially Supervised Multi-Structure Medical Image\n Segmentation on Small-Scale Data" ], [ "Abstract The data-driven nature of deep learning (DL) models for semantic segmentation requires a large number of pixel-level annotations.", "However, large-scale and fully labeled medical datasets are often unavailable for practical tasks.", "Recently, partially supervised methods have been proposed to utilize images with incomplete labels in the medical domain.", "To bridge the methodological gaps in partially supervised learning (PSL) under data scarcity, we propose Vicinal Labels Under Uncertainty (VLUU), a simple yet efficient framework utilizing the human structure similarity for partially supervised medical image segmentation.", "Motivated by multi-task learning and vicinal risk minimization, VLUU transforms the partially supervised problem into a fully supervised problem by generating vicinal labels.", "We systematically evaluate VLUU under the challenges of small-scale data, dataset shift, and class imbalance on two commonly used segmentation datasets for the tasks of chest organ segmentation and optic disc-and-cup segmentation.", "The experimental results show that VLUU can consistently outperform previous partially supervised models in these settings.", "Our research suggests a new research direction in label-efficient deep learning with partial supervision." ], [ "Introduction", "Convolutional Neural Networks (CNNs) have been a game-changer for the task of semantic segmentation [1], [2], [3], as they can learn pixel-level mappings from the image space to the label space via end-to-end training.", "To learn these complex mappings, state-of-the-art CNNs usually leverage a large number of parameters and require the availability of large-scale fully labeled datasets, which are often unavailable for real-life tasks.", "In the medical domain, where annotations require substantial efforts from clinical experts, obtaining these datasets can be challenging.", "This has led to an increasing interest in learning from partially labeled data, when fully labeled data is not available.", "Partially supervised learning (PSL) is still an open research question in medical image segmentation [4], [5], [6], [7], [8].", "From the perspective of multi-task learning (MTL) [9], a semantic segmentation task can be decomposed into multiple sub-tasks corresponding to each semantic class of interest, which provides the theoretical foundations of learning from partial ground truth.", "Given a medical image segmentation task with multiple classes of interest, it is common to collect and merge several available, smaller but relevant datasets into a larger dataset under the challenges of small-scale data, dataset shift, and class imbalance.", "These smaller datasets were originally labeled for sub-tasks, such that only the objects related to the specific sub-task are annotated, while other objects are merged into the background.", "In other words, the training images do not have complete annotations for all classes of interest but are partially labeled.", "For example, in the task of abdominal organ segmentation, a pancreas dataset and a liver dataset might be available separately, where only the pancreas and the liver are labeled, respectively.", "A key challenge, leading to poor segmentation performance when considering multiple partially labeled datasets, is that the semantic classes of one dataset could be categorized as the background for another dataset that was annotated for a different purpose.", "Traditional semantic segmentation models [1], [2], [3] can therefore not be directly applied and trained end-to-end in a supervised fashion.", "Further, given the small amount of partially labeled data, deep learning (DL) models are prone to overfitting.", "Recent studies in PSL [4], [5], [6], [10], [7], [8] all assume that, for each class of interest, enough training examples are accessible.", "However, considering the data scarcity in most practical medical tasks, usually, only few training examples might be available, making previous approaches impractical.", "To bridge the methodological gaps when only small-scale partially labeled data is available, we propose a simple yet efficient framework Vicinal Labels Under Uncertainty (VLUU) by exploring the statistical similarity of human structures (e.g.", "shape, size, location) among different patients.", "See Fig.", "REF for an illustration of such a similarity.", "The proposed framework is motivated by vicinal risk minimization (VRM) [11], where the fully labeled vicinal examples are generated by linearly combining randomly sampled partial labels with a weight randomly sampled from a Dirichlet distribution.", "These vicinal examples allow us to transform the partially supervised problem into a fully supervised one.", "That is to say, we can utilize any existing supervised segmentation networks and loss functions to solve partially supervised problems.", "The generated vicinal labels contain uncertainty regions where classes of interest could potentially overlap.", "We utilize these uncertainties in the training process to improve the robustness of DL models.", "Recent studies have shown that VRM can consistently improve the performance of CNNs for image classification tasks [12], [13].", "However, there is a lack of definition of VRM for dense prediction tasks with incomplete labels, e.g.", "[12] and [13] can not be directly applied on partially supervised semantic segmentation tasks.", "Instead, we revisit VRM, a long-ignored but particularly efficient approach, to tackle this problem.", "Specifically, by defining a generic vicinity distribution, VLUU learns a mapping from a sequence of images to a vicinal label which is generated by statistically mixing up the corresponding partial labels of the input images.", "Figure: Annotated examples of different type of medical images (first row): (a) a posteroanterior X-ray image with the ground truth annotation of the left lung; (b) a sagittal MRI image with the ground truth annotation of the left ventricle; (c) an axial CT image with the ground truth annotation of the right atrium.", "The label distributions (normalized density heatmap) of the corresponding organs in public datasets (second row): (d) the left lungs in the JSRT dataset ; (e) the left ventricles in the MRI-WHS dataset ; (f) the right atriums in the CT-WHS dataset .We perform the first systematic study of partially supervised methods under data scarcity challenges, such as small-scale data, domain shift or dataset shift [17], and class imbalance, on two representative medical image segmentation tasks, namely chest organ segmentation and optic disc-and-cup segmentation.", "The experiments show that VLUU is more robust than previous partially supervised methods under these settings.", "The proposed framework has five advantages over previous methods: (1) it is easy to implement without relying on complex loss functions, network architectures, and optimization procedures; (2) it can be trained end-to-end in supervised settings with common segmentation networks and loss functions; (3) it does not require any fully labeled images in the training data; (4) it can efficiently reduce the risk of overfitting for small-scale data; and (5) it can be easily extended to adversarial training.", "Our main contributions can be summarized as follows: We propose a simple yet robust framework for partially supervised medical image segmentation, which is robust when there is only limited partially labeled data.", "We provide theoretical interpretations for the proposed framework based on vicinal risk minimization and multi-task learning.", "We systematically evaluate the robustness of partially supervised methods and show that the proposed framework can outperform state-of-the-art partially supervised methods under various data scarcity challenges.", "The rest of this paper is organized as follows.", "Sec.", "reviews the relevant literature.", "Sec.", "and Sec.", "describe the proposed framework and its properties.", "Sec.", "describes the proposed benchmark task and provides experimental results and analysis.", "Sec.", "summarizes this work.", "In machine learning, semi-supervised learning (SSL) falls between supervised learning (SL), where only fully labeled training data are available, and unsupervised learning (UL), where no labels are available.", "In semi-supervised learning, the training set consists of both labeled and unlabeled data.", "The robust state-of-the-art semi-supervised methods include label propagation (LP) [18], graph neural networks [19], [20], and cross consistency training [21].", "Most semi-supervised methods can not be applied to PSL problems directly as they are required to minimize a supervised loss, however, among these seminal SSL methods, LP [22] can be applied to tackle partially labeled data directly.", "With LP, pseudo-labels are generated based on prior information (partially labeled data).", "Then, the pseudo-labels are fine-tuned iteratively toward convergence [23].", "LP is computationally expensive and the quality of the pseudo-labels is highly dependent on the number of training data.", "[6] has demonstrated that LP is a powerful solution to PSL with fully labeled datasets as prior.", "As a robust method tested by time, LP is a strong baseline in this work." ], [ "Partially Supervised Learning", "Closely related to SSL, partially supervised learning (PSL), or the partial labels problem, describes the situation where each example has an incomplete label (e.g.", "only one semantic class is annotated out of a few classes of interest).", "Concretely, given a collection of multiple small partially labeled datasets, each dataset may only contain annotations for a proper subset of classes of interest and these subsets are disjoint.", "In such a case, the images in the collection are partially labeled.", "A more rigorous formulation of the problem is presented in Sec.", "REF .", "PSL is a topic of active research as the perfect fully labeled training datasets tend to be only available for specific research tasks.", "In recent studies, several methods have been proposed to address semantic segmentation with partial labels from different aspects.", "[24] treats a grid of image patches as nodes and uses conditional random fields to propagate information.", "However, as a result, the predicted segmentation masks will be unnatural due to the patch-wise prediction.", "In DL, a common approach is to treat the missing labels as the background.", "This approach can be viewed as a naive form of noisy labels [25] and only works when the pixels of missing classes take up a much smaller portion of the images, compared with the pixels of the background.", "For benchmark datasets in computer vision such as PASCAL VOC [26] and MS COCO [27], there are only a few classes present in each image or the objects can be very small.", "Thus, merging unlabeled pixels into the background might be an efficient solution for these datasets.", "In contrast, for common medical datasets, multiple classes can be present in each image and the objects of interest (e.g.", "organs) may take up the majority of the pixels.", "Another common approach in DL is to ignore the cross entropy of the missing classes during backpropagation [4], [5].", "The limitation of this approach is that abandoning the pixel information of missing classes means that the learners (CNNs) will receive much less supervision during the learning process, both from the image space and the label space.", "A direct result is that the learner can not discriminate the classes of interest against the background.", "Recently, PaNN [6] proposes a complex Expectation-Maximization (EM) algorithm with a primal-dual optimization procedure.", "However, PaNN requires the availability of fully labeled images as prior, which is often unavailable.", "To address general semantic segmentation [26], [28], [10] proposes to use a complex encoder-decoder architecture to condition the partial information within the CNN, which requires a large dataset to comply with the large number of parameters.", "PIPO-FAN [7] proposes a complex pyramid feature fusion mechanism and a target adaptive loss (TAL).", "Unlike the other methods, PIPO-FAN has a demanding requirement in the training process, i.e.", "the examples with the same partial labels must be trained together.", "It is worth mentioning that TAL also treats the missing labels as the background.", "Recently, a state-of-the-art work [8] tackles PSL by proposing a marginal loss and an exclusion loss, which are designed for partially supervised medical image segmentation.", "From the perspective of DL, [8] tries to address PSL at the last step of feed-forward propagation, while this work addresses PSL at the data preparation step, which is before the feed-forward propagation process.", "To sum up, all of these methods are only applicable when substantial partially labeled images or fully labeled images are available.", "In addition, previous studies do not consider the practical situations such as dataset shift and class imbalance.", "A detailed empirical analysis is provided in Sec.", "REF ." ], [ "Multi-Task Learning", "By leveraging task-specific information, multi-task learning (MTL) [9] can improve the model generalization when the tasks of interest are somewhat related.", "In the era of DL, we aim to use a neural network (NN) to map the input to the output, given a task.", "In contrast to single-task learning, where each task is handled by an independent NN, MTL can reduce the memory footprint, increase overall inference speed, and improve the model performance.", "When the associated tasks contain complementary information, MTL can regularize each single task.", "For dense prediction tasks, a good example is semantic segmentation, where we always assume that the classes of interest are mutually exclusive.", "Depending on the data modality, task affinity [29] between sub-tasks and task fusion strategy, there are various types of MTL.", "We depict several common MTL workflows that are related to our work in Fig.", "REF .", "Semantic segmentation falls into the category Fig REF (d).", "As pointed out by [30], pixel-level tasks in visual understanding often have similar characteristics, which can be potentially used to boost the performance by MTL.", "We argue that PSL problems can be reformulated as MTL problems by utilizing human structure similarity.", "Figure: Common MTL workflows for dense prediction tasks.", "The data modalities of the input are different: (a) The different tasks have separate networks, which are linked through soft parameter sharing.", "Note, without soft parameter sharing, (a) depicts the standard multiple single-task learning.", "(b) The different tasks have independent encoders and decoders but share the same network backbone (in purple), which is also known as hard parameter sharing.", "The data modalities of the input are identical: (c) Each task has independent output, which requires an independent decoder.", "(d) The tasks can share the same decoder." ], [ "Preliminaries", "In SL, given a training dataset $S = \\lbrace X, Y\\rbrace $ with images $X = \\lbrace x_i\\rbrace _{i=1}^n$ and ground truth labels $Y = \\lbrace y_i\\rbrace _{i=1}^n$ , the empirical risk is defined as $\\mathcal {R}(h) = \\frac{1}{n}\\sum _{i}^{n} L(h(x_i), y_i),$ where $L(\\cdot , \\cdot )$ is the loss function and $h \\in \\mathcal {H}$ is the hypothesis.", "In this work, we assume that $L$ and $h$ are universal as they can be any loss function and model in a standard supervised setting.", "For example, for a popular choice of semantic segmentation, $L$ could be the cross entropy and $h$ could be a CNN.", "The minimization of the empirical risk $\\mathcal {R}(h)$ is also known as Empirical Risk Minimization (ERM) in statistical learning literature [31]." ], [ "Problem Formulation", "Assume there are $K > 1$ mutually exclusive semantic classes of interest present in the same image, i.e.", "there is no hierarchical relationship between classes and all classes are present.", "In this work, we focus on the challenging situation that each image is annotated for only one semantic class.", "For partially labeled images, we can always split $S$ into $K$ sub-datasets where each sub-dataset contains label information of only one class.", "Here, the $K$ datasets are mutually exclusive in terms of both images and classes.", "Mathematically, we have $S = \\bigcup _{j}^K S_j$ , where $S_j = \\lbrace X_j, Y_j\\rbrace $ denotes the partially labeled dataset with label information of semantic class $j$ .", "In $S_j$ , $X_j = \\lbrace x^{j}_{i}\\rbrace _{i=1}^{n_j}$ is the image set of the images with label information of the semantic class $j$ and $Y_j = \\lbrace y^{j}_{i}\\rbrace _{i=1}^{n_j}$ contains the corresponding partial labels.", "In addition, we define $S_j \\subset \\mathcal {D}_j$ , where $\\mathcal {D}_j$ denotes the source domain for $S_j$ , and we define $d(\\mathcal {D}_{j_1}, \\mathcal {D}_{j_2}) \\ne 0 \\ \\forall j_1 \\ne j_2$ , where $d(\\cdot , \\cdot )$ measures the distributional discrepancy between two distribution.", "That is to say, dataset shift exists.", "As a comparison, previous studies usually fail to validate this assumption when using one fully labeled dataset to simulate the partially labeled datasets.", "Note, the problem formulation here describes the most general case as all other cases are trivial extensions.", "For example, when an image has annotations for more than one semantic class, duplicate image copies could exist in multiple datasets and the above mathematical formulation still holds.", "Figure: Illustration of the standard training pipeline.", "Here, we use the chest organ segmentation task as an example.", "Assume there are three classes of interest, which are left lung, heart, and right lung.", "And there are three corresponding partially labeled sub-datasets, denoted as S 1 S_1, S 2 S_2 and S 3 S_3.", "{(x 1 ,y 1 ),(x 2 ,y 2 ),(x 3 ,y 3 )}\\lbrace (x^1, y^1), (x^2, y^2), (x^3, y^3)\\rbrace are randomly sampled from S 1 S_1, S 2 S_2 and S 3 S_3, respectively.", "The vicinal example pair (x ˜,y ˜)(\\tilde{x}, \\tilde{y}) is generated by Eq.", "and Eq.", "with K=3K=3.", "The segmentation network could be any standard segmentation network such as FCN or U-Net .", "For simplicity, the background mask is not shown in the figure and we use grayscale images to visualize the vicinal labels." ], [ "Vicinal Labels Under Uncertainty", "In a fully supervised setting, introducing statistical randomness [11] and using the convex combination of the training data [12], [13] are two efficient methods to improve the robustness of DL models.", "However, as none of these methods can address the missing class information, they have been ignored in multi-class semantic segmentation with partial supervision for a long time.", "In this work, we integrate and extend these two simple ideas.", "Instead of designing complex networks [10], [7] or loss functions [8], we utilize the partial labels in a multi-task fashion.", "A naive solution is to decompose the partially supervised multi-class segmentation task into multiple binary segmentation tasks.", "As both the input and the output share the same characteristics, we want to use a shared encoder and decoder, similar to Fig.", "REF (d).", "However, unlike semantic segmentation, where there is only a single image as input and the corresponding label is based on the same image, we now have images and labels from different partially labeled datasets.", "We propose to fuse the tasks based on the human structure similarity.", "Let $x$ be a 2D medical image with size $H \\times W$ , represented by a 2D array, which has been pre-processed via instance normalization and optional spatial alignment.", "So $y$ is the corresponding partial label with one semantic class annotated, represented by a 3D array ($H \\times W \\times (K+1)$ ), where the last dimension corresponds to the semantic classes.", "For each pixel in $x$ , the corresponding element in $y$ is a $(K+1)$ -element one-hot vector for the background and $K$ semantic classes.", "For simplicity, we use $y[k]$ to denote the binary label map for class $k \\le K$ ($k = 0$ denotes the background), which is the $(k+1)^{th}$ semantic channel of $y$ .", "Let $(x^{j}, y^{j})$ be a random sample from $S_j$ , and so $\\lbrace (x^{j}, y^{j})\\rbrace _{j=1}^K$ is a $K$ -element tuple of such samples.", "We define $\\tilde{x} = concat(\\lbrace x^{j}\\rbrace _{j=1}^K)$ $\\tilde{y} ={\\left\\lbrace \\begin{array}{ll}\\frac{w_k y^{k}[k]}{\\sum _{j=1}^K w_j y^{j}[j] + \\epsilon } & k > 0\\\\1 - \\sum _{j=1}^K \\tilde{y}[j] & k = 0\\end{array}\\right.", "}$ , where $concat$ is the concatenate operation that concatenate $\\lbrace x^{j}\\rbrace _{j=1}^K$ along a new dimension.", "We have $\\mathbf {w} = (w_1, ..., w_K)\\sim \\text{Dirichlet}(\\mathbf {\\alpha })$ with $\\mathbf {\\alpha } = (\\alpha _1, ..., \\alpha _K) \\in (0, \\infty )^K$ and $\\epsilon > 0$ is a small number to ensure numeric stability, e.g.", "$\\epsilon = 10^{-3}$ .", "Without prior information over the true label distributions, we setup $\\mathbf {\\alpha }$ as a constant vector, i.e.", "$\\alpha _k = \\alpha \\ \\forall 1 \\le k \\le K$ .", "Given $(\\tilde{x}, \\tilde{y})$ , we transform a partially supervised problem into a fully supervised one and we can utilize any existing supervised segmentation network and loss function.", "See Fig.", "REF for the illustration of the training pipeline.", "In each class channel of the vicinal label, the continuous probabilities are transformed into grayscale pixels for visualization.", "There are two origins of uncertainty for generating the vicinal labels when there is an overlap between partial labels.", "First, the sampling of input images is stochastic.", "Second, $\\mathbf {w}$ is randomly sampled from a Dirichlet distribution (e.g.", "$\\mathbf {w} = (0.33, 0.41, 0.26)$ used in Fig.", "REF ).", "See the upper right corner in Fig.", "REF for visual examples intuitively, where $y_2$ and $y_3$ have an overlapping region." ], [ "Theoretical Interpretation", "The proposed solution can be interpreted from two aspects, namely vicinal risk minimization (VRM) [11] and MTL respectively.", "In VRM, a vicinity distribution $\\mathcal {V}$ is defined as the probability distribution for the virtual image-label pair (also known as vicinal example) $(\\tilde{x}, \\tilde{y})$ in the vicinity of $(x, y)$ .", "The vicinal risk is defined as $\\mathcal {R}_{\\mathcal {V}}(h) = \\frac{1}{n}\\sum _{i}^{n} L(h(\\tilde{x_i}), \\tilde{y_i}).$ Eq.", "REF factually defines a non-parametric anatomical prior for the label distribution.", "In state-of-the-art VRM works for image classification [12], [13], the vicinal image is usually defined as the convex combination of real images, where the parameters for the convex combination are sampled from statistical distributions.", "As a comparison, we utilize a CNN ($h$ in Eq.", "REF ) to learn this parametric convex combination jointly with semantic segmentation.", "Eq.", "REF and the CNN jointly play the role of $\\tilde{x}$ in Eq.", "REF .", "By combining Eq.", "REF and Eq.", "REF , we inexplicitly define a generic $\\mathcal {V}$ .", "On the other hand, given $K$ sub-tasks, we are using a CNN to learn a $K \\mapsto K$ task mapping.", "Eq.", "REF is a task-fusion process that fuses different but related task knowledge.", "We want to maximally share the network architecture from a MTL perspective.", "To achieve this, the novelty here is that we utilize the human structure similarity to mix up the partial labels.", "Meanwhile, the uncertainty regions in the vicinal labels, caused by the stochastic convex combination of partial labels, can reduce the risk of overfitting and improve the robustness when the training data is small.", "Figure: Illustration of adversarial training pipeline.", "(x ˜,y ˜)(\\tilde{x}, \\tilde{y}) is generated by the Eq.", "and Eq. .", "Same as Fig.", ", the background mask is not shown in the figure and we use grayscale images to visualize the vicinal labels.", "The segmentation network is trained with (x ˜,y ˜)(\\tilde{x}, \\tilde{y}) in a supervised fashion.", "y pred y_{pred} is the output of the segmentation network, which is the concatenation of (K+1)(K+1) probability maps.", "An auxiliary discriminator is trained to identify whether y pred y_{pred} is sampled from the vicinal distribution, i.e.", "discriminate y pred y_{pred} against y ˜\\tilde{y}.", "The segmentation network and the discriminator are trained alternatively.", "See Eq.", "and Eq.", "for details." ], [ "Extension to Adversarial Training", "Compared with previous works in PSL [4], [5], [6], [10], [7], [8], VLUU can be potentially further improved through adversarial training.", "Adversarial training was first proposed by [32] and several breakthroughs have been made through adversarial training in medical image segmentation [33], [34], [35], [36].", "However, adversarial training for semantic segmentation is ill-defined when the ground truth labels are missing [37].", "As VLUU can transform the partially supervised problem into a fully supervised one, it is natural to consider incorporating VLUU and adversarial training.", "Note, having complete labels during training gives VLUU unparalleled advantages in utilizing some well-known properties of adversarial training, which is difficult for most partially supervised methods.", "In standard adversarial training, the segmentation network and the discriminator play a zero-sum game.", "The discriminator is trained to discriminate the prediction masks produced by the segmentation network from the ground truth masks.", "Meanwhile, the segmentation network is trained to confuse the discriminator by producing realistic prediction masks.", "Adversarial training benefits from the human structure similarity as it makes the unknown true label distributions easier to be caught by the discriminator than for general objects [38].", "In other words, there is smaller instance-wise variation in the size, shape, and location of human organs (or structures), as shown in Fig.", "REF , than for general objects.", "Assume the segmentation network is parameterized by $f_{\\theta }$ and the discriminator is parameterized by $g_{\\phi }$ .", "Given $\\phi $ fixed, $\\theta $ is updated by minimizing $\\mathcal {L}_\\theta = \\mathcal {L}_{seg}(f_{\\theta }(\\tilde{x}), \\tilde{y}) - \\lambda \\log g_{\\phi }(f_{\\theta }(\\tilde{x})),$ where $\\mathcal {L}_{seg}$ is the multi-class cross-entropy loss for standard supervised semantic segmentation and $\\lambda $ controls the weight of the adversarial loss.", "Given $\\theta $ fixed, $\\phi $ is updated by minimizing $\\mathcal {L}_\\phi = -\\log g_{\\phi }(\\tilde{y}) - \\log (1 - g_{\\phi }(f_{\\theta }(\\tilde{x})).$ See Fig.", "REF for the illustration of adversarial training with the vicinal examples.", "We denote VLUU with adversarial training as VLUU-ADV.", "Further, continuous vicinal labels have a built-in advantage in stabilizing adversarial training.", "They alleviate the problem that there commonly is a clear discrepancy between the discrete distribution of the ground truth and the continuous distribution of the pixel-wise predictions, which can be easily caught by the discriminator [37] and destabilize training, leading to oscillating parameters [39].", "Last but not least, with adversarial training, VLUU can further utilize unlabeled data in addition to the partially labeled data.", "For the interested readers, the problem formulation and application of adversarial training for SSL can be found in [40]." ], [ "Theoretical Analysis", "In this section, we will discuss the theoretical advantages and limitations of the proposed framework." ], [ "Enlarged Sample Space", "One of the main challenges for DL is overfitting caused by data scarcity.", "In this work, there are two aspects of data scarcity: 1) each image has an incomplete label, and 2) each $S_i$ has only a small number of images.", "For 1), Eq.", "(REF ) and Eq.", "(REF ) generate fully labeled vicinal example pairs, thus traditional end-to-end training techniques in supervised learning can finally be applied.", "For 2), with limited training data, state-of-the-art CNN architectures can easily overfit to the training data.", "Let us first isolate the randomness effect caused by the Dirichlet distribution by setting $w_i = \\frac{1}{K}$ .", "The proposed framework enlarges the sample space from $\\sum _i n_i$ partially labeled examples to $\\prod _i n_i$ fully labeled example pairs.", "In fact, given $\\lbrace (x_{i}, y_{i})\\rbrace _{i=1}^K$ , $\\text{Dirichlet}(\\mathbf {\\alpha })$ can theoretically generate an infinite number of $\\tilde{y}$ determined by $\\mathbf {w}$ .", "We efficiently mitigate the overfitting problem by enlarging the sample space of $\\tilde{S}$ ." ], [ "Label Smoothing", "In semantic segmentation tasks, labels usually follow a discrete distribution, while Eq.", "(REF ) defines a continuous distribution.", "Even though the application of continuous label distributions is rare in semantic segmentation, they have led to recent breakthroughs in image classification [41], [12].", "We expect Eq.", "(REF ) can improve the robustness of the model as suggested by recent theoretical analysis of continuous label distributions [42]." ], [ "Computational Cost", "The training process of the proposed framework is almost identical to the training process for a fully supervised task, i.e.", "given a segmentation network, there is no additional optimization cost such as multi-stage training [6].", "Similarly, the proposed method utilizes the same memory footprint in terms of CNN weights.", "As a comparison, a semi-supervised method such as label propagation and knowledge transfer will require the training of multiple segmentation networks to generate pseudo-labels.", "For the proposed method, the major overheads arising from the data generation process are the random sampling and the element-wise operations on low-dimensional arrays, which are negligible compared to the backpropagation cost.", "Eq.", "(REF ) and Eq.", "(REF ) can be easily implemented by any scientific computing frameworks supporting broadcasting, such as NumPy, PyTorch, and TensorFlow." ], [ "Limitations", "The main purpose of the proposed framework is to train DL-based segmentation models with partial labels in an efficient way.", "As discussed in Sec.", "REF , the design of Eq.", "(REF ) and Eq.", "(REF ) makes a strong assumption that all classes of interest are present in each image and there is no hierarchical relationship between the semantic classes, i.e.", "the classes of interest are mutually exclusive, e.g.", "organs in the same body part or sub-structures under the same structure.", "The situation where the semantic classes have a hierarchical structure, e.g.", "liver and liver tumor, is beyond the scope of discussion.", "Note, the proposed framework is designed for DL tasks on only a few images without complete annotations.", "When fully labeled data is available, state-of-the-art supervised and semi-supervised methods have obvious advantages over the proposed framework.", "However, the proposed framework fills the gap when supervised and semi-supervised methods fail." ], [ "Empirical Analysis", "The purposes of the experimental design are threefold.", "First, there is no known empirical study of PSL with limited data.", "We want to investigate the impact of limited partial labels on DL.", "Second, we want to systematically evaluate the robustness of the representative partially supervised methods in a controlled environment.", "Third, we want to demonstrate the effectiveness of VLUU in situations where only a few partially labeled images are available.", "Thus, the choice of the network backbone or loss function is independent of the proposed learning framework.", "In addition, the simulated experiments are solely to demonstrate the challenges of data scarcity in a controllable environment.", "We consider two medical image segmentation tasks, chest organ segmentation and optic disc-and-cup segmentation." ], [ "Chest Organ Segmentation", "The task of chest organ segmentation is a simple benchmark task in medical image segmentation.", "In this task, we consider three semantic classes, namely left lung, right lung, and heart.", "We can easily control the environment to get an insight into the impact of the limited partial labels on various representative partially supervised methods and the efficiency of VLUU.", "Without specification, the experimental comparison is conducted in such a way that different models use the same network backbone, loss function, training strategy, and the set of hyperparameters.", "We use two public datasets to simulate the realistic situations that each partially labeled dataset is annotated for a different semantic class and is collected from an independent source.", "Unlike [8], which only consider partially labeled datasets, we use two fully labeled datasets to better understand the influence of partial labels.", "The JSRT dataset, released by the Japanese Society of Radiological Technology (JSRT), is a benchmark dataset for chest organ segmentation [14].", "JSRT contains 247 grayscale CXRs with pixel-wise annotations of lungs and hearts.", "Each CXR has a size of $2048 \\times 2048$ .", "The Wingspan dataset was collected by Wingspan Technology for the study of transfer learning and unsupervised domain adaptation in chest organ segmentation [40].", "Wingspan contains 221 grayscale CXRs with pixel-wise annotations of lungs and hearts.", "The CXRs were collected from 6 hospitals with different imaging protocols.", "Wingspan expresses a large variety in the data modalities including brightness, contrast, position, and size.", "We use three partially labeled datasets as the training set and one fully labeled as the test set, where the four datasets are collected from four different sources.", "We choose this setup to simulate the practical scenarios where dataset shift exists, which is a challenging situation for DL models.", "We use the JSRT dataset as the left lung dataset, denoted as L. We use a subset of the Wingspan dataset containing 18 CXRs as the right lung dataset, denoted as R. We use another subset of the Wingspan dataset containing 18 CXRs as the right lung dataset, denoted as H. We use the rest of the Wingspan dataset as the fully labeled test set, which contains 185 CXRs, and denote it as T. The visual comparison of the data modalities of the four sets can be viewed in Fig.", "REF .", "Note, all four sets are collected from 4 different sources (hospitals with different imaging protocols).", "Figure: Visual comparison of three partially labeled training sets and one fully labeled test set with corresponding annotations." ], [ "Baseline Models", "For a fair comparison, we use the same segmentation network for all methods, which is a FCN [1] with a ResNet18 [43] backbone.", "Considering the data scarcity situation, we choose ResNet-FCN as it can both achieve promising results on chest organ segmentation tasks [40] and avoid overfitting.", "We choose the following representative approaches as the baseline models.", "Fully Supervised Learning Approach To illustrate the effect of limited partial labels on DL models, we consider two practical approaches in computer vision that are commonly used during large-scale training.", "As discussed in Sec.", "REF , two methods can be used to train end-to-end methods in a supervised fashion.", "The first one is to categorize the uncertain (missing) classes as the background in the training, which can be considered as a naive solution with noisy labels.", "We denote the first baseline as MBG because we mix uncertain pixels with the background pixels.", "The second baseline is to ignore the cross-entropy of the missing classes during the backpropagation.", "This method is motivated by the nature of multi-task learning for neural networks.", "We denote this method as IMBP.", "It is worth mentioning that MBG and IMBP further motivate many recently proposed methods for PSL [4], [5], [7].", "Semi-Supervised Learning Approach We adopt a strong SSL baseline, label propagation (LP) [18], to solve PSL problem.", "LP is not an end-to-end method as there are multiple training stages.", "It first generates noisy pseudo-labels for the unlabeled classes based on the partially labeled data.", "Then the pseudo-labels and ground truth labels are trained together to make the final prediction.", "However, the quality of the noisy pseudo-labels is highly dependent on the quality of the partially labeled examples and noisy labels might harm the later fine-tuning stage.", "In this work, we use $K$ independent binary segmentation networks to generate the initial pseudo-labels.", "Multi-Task Learning Approach A classical way to address MTL problems is to fuse knowledge extracted from each individual sub-task [44], which is also known as knowledge transfer (KT) in the transfer learning literature.", "We train $K$ binary segmentation networks with a shared ResNet feature extractor but independent deconvolutional layers.", "We alternatively optimize $K$ binary segmentation networks on the corresponding $K$ partially labeled datasets.", "The final prediction masks is generated by fusing $K$ binary prediction masks.", "For each pixel, if all classes of interest have probabilities less than the threshold 0.5, we treat it as the background.", "Otherwise, the pixel is categorized as the class with the highest probability.", "Partially Supervised Learning Approach We consider the state-of-the-art partially supervised method exclusion loss (EL) [8], which is designed for the same problem formulation in Sec.", "REF .", "EL has shown superior performance over recent partially supervised methods, such as PaNN [6] and PIPO-FAN [7], in all aspects.", "Unlike EL, recent partially supervised methods rely on either large training data [4], [5], [10], [7] or fully labeled data as a prior [6], which are not applicable for some situations.", "Similar to our approach, EL can be applied to any existing segmentation networks.", "So they can be compared with VLUU in a fair setting." ], [ "Implementation", "The image size is fixed to be $256 \\times 256$ .", "We pre-process the raw images by instance normalization.", "Given an image $x$ , we obtain the normalized image $\\hat{x}$ by $\\hat{x}^{ij} = \\frac{x^{ij} - \\mu (x)}{\\sigma (x)}$ , where $(i,j)$ is the position of the pixel in a $256 \\times 256$ image, and $\\mu $ and $\\sigma $ are the mean and standard deviation of the pixels of $x$ .", "In this study, we do not apply other pre-processing techniques as there is no obvious difference in the relative position of objects in each image and the proposed framework is robust against slight misalignment.", "In practice, when partially labeled datasets are acquired from different imaging protocols, pre-processing techniques such as registration, resizing, and cropping are necessary.", "There are no fully labeled images in the training set and we consider the setting where each training image only has an annotation of one semantic class, as described in Sec.", "REF .", "All experiments are implemented in PyTorch on an NVIDIA Tesla V100.", "For a fair comparison, all the networks are initialized with the same random seed and trained from scratch.", "We use a standard multi-class cross-entropy as the loss function for all the experiments.", "The batch size is 8.", "The models are trained to converge with an Adam [45] optimizer and a fixed learning rate of $10^{-3}$ .", "The performance metric in this study is the mean Intersection-Over-Union (mIOU) between the prediction masks and ground truth masks over the three classes of interest.", "For VLUU, we set $\\alpha = 0.1$ ." ], [ "Comparison Under Small-Scale Data", "Because the partially labeled datasets are collected from different sources, we will focus on the challenges of data scarcity and class imbalance.", "As we want to examine how the size of the partially labeled datasets affects the DL models, we only include $n$ examples of each partially labeled dataset for a quantitative comparison.", "We provide the performance of the segmentation networks trained on the same training data but with complete annotations as an Oracle to provide a reference for the performance.", "The results are shown in Table REF .", "Supervised methods fail to address the partial labels due to overfitting.", "As shown in Fig.", "REF , MBG tends to predict every pixel as the background while IMBP fails to identify the background, which follows the discussion in Sec.", "REF .", "LP, KT, and EL mitigate the partial labels problem from different perspectives and achieve much better performance than supervised methods.", "However, these seminal methods suffer from the limited training data and multi-source domain shift.", "Among the baseline methods, LP is the most computationally expensive method as it requires considerably more training time and memory footprint than all other methods.", "In addition, LP is more sensitive to the size of the training set.", "In practice, semi-supervised models expect a large set of unlabeled data, which is not aligned with the problem formulation in this work.", "Compared with semi-supervised methods, MTL methods usually consume a much smaller memory footprint depending on the number of shared layers.", "By comparing KT and VLUU, we can see that VLUU has more shared neural architectures than KT, which can reduce the memory footprint and substantially improve the model performance.", "As the state-of-the-art partially supervised method, EL purely relies on using a modified loss function to extract knowledge from the training.", "When there is not enough training data, EL performs worse than KT and VLUU.", "In contrast to the baseline methods, VLUU achieves the best performance on small-scale data.", "Without acquiring any new supervision, VLUU incorporating a coarse anatomical knowledge by uniquely utilizing human structure similarity.", "Table: Quantitative comparison (mIOU) on partially supervised chest organ segmentation with small-scale data.", "The segmentation network is ResNet-FCN.", "nn denotes the number of images in each partially labeled dataset.It is worth mentioning that, MBG, IMBP, EL, and VLUU are end-to-end methods, i.e.", "they do not require any auxiliary NNs or multi-stage training procedures.", "We provide the qualitative comparison of end-to-end methods in Fig.", "REF .", "VLUU tends to output more realistic masks than the STOA method EL in terms of the location and shape.", "Figure: Qualitative comparison of end-to-end methods on partially supervised chest organ segmentation with n=15n=15.", "GT denotes the ground truth.", "The segmentation network is ResNet-FCN.", "nn denotes the number of images in each partially labeled dataset.", "Traditional training strategies for supervised learning, such as (b) MBG and (c) IMBP, fail for PSL.", "Compared with (d) EL, (e) VLUU generates more realistic organ masks." ], [ "Comparison Under Class Imbalance", "Considering the availability of the medical data and the difficulty of annotating certain organs or structures, we simulate the class imbalance situations in PSL.", "Here, we use $\\eta $ to control the class imbalance.", "As the heart is more difficult to annotate than the two lungs [33], we set the partially labeled dataset for the heart (H) to have $n=5$ and the partially labeled datasets for the two lungs (L and R) to both have $\\eta n$ examples.", "The results are shown in Table REF .", "Compared with Table REF , the class imbalance does have a severe negative impact on the baseline methods MBG, IMBP, and KT, as more training data could even decrease the performance.", "While LP, EL, and VLUU could benefit from more training data, LP achieves much lower performance than EL and VLUU.", "VLUU can generally achieve comparable performance with EL while outperforming EL by a large margin with small $n$ .", "Compared with the baseline methods, VLUU mitigates the class imbalance by utilizing human structure similarity to generate a balanced vicinal label distribution.", "Table: Quantitative comparison (mIOU) of methods on chest organ segmentation with class imbalance.", "The segmentation network is ResNet-FCN.", "η\\eta denotes the ratio of the number of images in the dataset L or R to the number of images in the dataset H." ], [ "Ablation Studies", "Impact of Network Complexity Under the data scarcity challenge, the complexity of the segmentation network will usually play an important role.", "The network complexity is determined by the number of parameters and the network architecture.", "For supervised tasks, U-Net should outperform ResNet-FCN because U-Net has more parameters than ResNet-FCNU-Net has 38.8M parameters and FCN with a ResNet18 backbone has 13.3M parameters.", "and a better network architecture design for medical image segmentation tasks [2].", "Clearly, there is a trade-off in the network selection between the network complexity and network performance when the partially labeled datasets are small.", "Here, we evaluate VLUU with both FCN and U-Net, and results are shown in Table REF .", "We hypothesize that complex networks have a negative impact on VLUU when there is only limited data.", "Given a small amount of training data, complex networks could have both performance gain due to more parameters and delicate architectures, and performance drop due to overfitting, depending on the amount of training data.", "Table: The impact of network complexity on VLUU with ResNet-FCN as the segmentation network.", "nn denotes the number of images in each partially labeled dataset.Sensitivity to $\\mathbf {\\alpha }$ The performance of a ResNet-FCN trained by VLUU with different $\\alpha $ is shown in Fig.", "REF .", "Overall, VLUU is not sensitive to $\\alpha $ as there are only small differences between the performance for different $\\alpha $ values.", "Note, $\\text{Dirichlet}(\\mathbf {\\alpha })$ is asymptotically close to a uniform distribution when $\\alpha \\rightarrow \\infty $ , i.e.", "$w_i = \\frac{1}{K}$ .", "In addition, there is a trade-off in selecting the optimal $\\alpha $ .", "Small $\\alpha $ indicates a larger variation in the label distribution, which means larger uncertainty.", "So, for tasks such as chest organ segmentation where the organs have relatively fixed locations and similar shapes, a large $\\alpha $ might help.", "However, a small $\\alpha $ should be more robust as it introduces more uncertainty when $K$ is large.", "In this work, we use $\\alpha = 0.1$ for consistency.", "Figure: Sensitivity of α\\alpha to nn.", "Overall, VLUU is robust for various α\\alpha .Effect of Random Initiation To examine the sensitivity of the proposed framework to the effect of random initiation, we repeat the experiments in Table REF for EL and VLUU for 5 times each.", "This time, the backbone network is randomly initiated at each time.", "Unlike the results in Table REF , which are the highest mIOU, we report the mean and standard deviation of mIOUs in Table REF .", "Compared with the loss-based partially supervised method EL, the label-based partially supervised method VLUU is more robust with smaller standard deviation.", "Table: Robustness of VLUU under different random initiations.", "The performance (mean mIOU ±\\pm standard deviation) of VLUU is more stable than the performance of EL.Adversarial Training For VLUU-ADV, we use a standard ResNet binary classifier as the discriminator as we use a ResNet-FCN as the segmentation network.", "In fact, the choice of the discriminator is a research question in its own right [38].", "[46] shows that having the same backbones for the segmentation network and the discriminator can increase the stability of adversarial training.", "We follow the training scheme in Sec.", "REF , where the adversarial loss [37] in Eq.", "(REF ) is weighted by $\\lambda = 0.001$ .", "We report the results of VLUU and VLUU-ADV in Table REF , where VLUU-ADV shows slightly better results than VLUU.", "We conclude that ADV can be used as an add-on module for VLUU with appropriate $\\alpha $ and delicate design of the network architecture for the discriminator.", "Table: Quantitative comparison (mIOU) between VLUU and VLUU-ADV with ResNet-FCN as the segmentation network.", "nn denote the number of images in each partially labeled dataset." ], [ "Optic Disc-and-Cup Segmentation", "In addition to chest organ segmentation, another task where all classes of interests are present in each image is the optic disc-and-cup segmentation.", "As an important step of early screening of glaucoma, optic disc-and-cup segmentation on the fundus images localizes the optic disc-and-cup for the analysis of the optical nerve head [47].", "An increase in the optic cup-to-disc ratio could be an indicator of the presence of glaucoma [48].", "The annotation of the optic disc is more difficult than that of the optic cup.", "In addition, the optic disc and optic cup have a unique geometric property that the optic cup is always enclosed by the optic disc.", "That is to say, if we want to annotate the optic disc, we have to annotate the optic cup first.", "Although this is not the standard problem formulation, VLUU can be applied to this situation directly as discussed in Sec.", "REF ." ], [ "Datasets", "We use the REFUGE datasethttps://refuge.grand-challenge.org to simulate the experiments for optic disc-and-cup segmentation.", "As there are two classes of interest, there should be at least two partially labeled datasets.", "However, as explained before, it is less practical to have a partially labeled dataset for optic disc.", "Instead, we have one larger partially labeled dataset for optic cup (denoted as P) and one smaller fully labeled dataset (denoted as F) as the training set.", "This motivation behind is twofold.", "First, the annotation of optic cup requires less human effort and is much cheaper to acquire than the annotation of optic disc.", "Second, we want to introduce the class imbalance.", "As REFUGE is collected from multiple sources, we create two sub-datasets from two sources to simulate the dataset shift in the training set.", "We use the validation set of REFUGE as the test set (denoted as T), which contains 400 fundus images.", "As REFUGE is collected from multiple sources, the fundus images have various image size.", "The images are pre-processed by registration, cropping, and resizing to have a fixed resolution of $256 \\times 256$ .", "So the pre-processed images contain the whole region of the optical nerve head.", "See Fig.", "REF for examples of the training set and the test set.", "Figure: Visual comparison of the fundus images in the training set and the test set.", "The training set consists of a partially labeled dataset for optic cup only and a fully labeled dataset for both optic disc and optic cup.", "(a) A fundus image from the dataset P. (b) The corresponding ground truth mask of (a) with the optic cup annotated as black.", "(c) A fundus image from the dataset F. (d) The corresponding ground truth mask of (c) with the optic disc annotated as grey and the optic cup annotated as black.", "(e) A fundus image from the test set T. (f) The corresponding ground truth mask of (e).", "Note, there are clear dataset shifts among the three datasets." ], [ "Implementation", "Based on the results in the previous section, we only compare EL and VLUU, as EL and VLUU consistently outperform other methods.", "In addition, we use a new baseline PaNN [6].", "PaNN requires that there is a small fully labeled dataset in the training set to learn the prior, which fits our task setup in Sec.", "REF perfectly.", "Again, for a fair comparison, we use a ResNet-FCN as the network backbone and use the same set of hyperparamters in Sec.", "REF .", "The performance metric is the mIOU between the unprocessedIn practice, the prediction masks could be further improved by image processing techniques.", "prediction masks and ground truth masks on optic disc and optic cup.", "In contrast to CXRs, the fundus images are color images with RGB channels.", "To generate a vicinal image, we concatenate two sampled images from the two partially labeled datasets along the RGB channels, i.e.", "the vicinal images now have 6 ($3K$ where $K = 2$ ) channels.", "Eq.", "REF and Eq.", "REF still hold.", "In the training of VLUU, we rearrange the training data as two partially labeled datasets.", "The small fully labeled dataset is split into two sub-datasets containing the same images, where one sub-dataset only contains labels for the optic disc and is treated as the new partially labeled dataest for the optic disc.", "The other sub-dataset with only labels for the optic cup is added into the partially labeled dataset for the optic cup.", "Table: Quantitative comparison (mIOU) of PSL methods on partially supervised optic disc-and-cup segmentation with class imbalance.", "The segmentation network is ResNet-FCN.", "nn denotes the number of images with optic disc annotated.Figure: Qualitative comparison on partially supervised optic disc-and-cup segmentation with n=3n=3.", "GT denotes the ground truth.", "The segmentation network is ResNet-FCN.", "nn denotes the number of images with optic disc annotated.", "A FCN trained with VLUU and partial labels can generate prediction masks which are qualitatively comparable with the masks predicted by a FCN trained with complete labels." ], [ "Results", "Compared with the experiments in Sec.", "REF , we use a more extreme setting to test the limit of partially supervised methods.", "We use only 10 images from P (i.e.", "10 images with optic cup annotated) and $n$ images from F (i.e.", "$n$ images with both optic disc and optic cup annotated).", "There is a severe class imbalance here, as the ratio of the number of labels for cup to the number of labels for disc is $\\frac{10 + n}{n}$ .", "The results measured in mIOU between the prediction masks and ground truth masks on optic disc and optic cup are presented in Table REF .", "With much smaller data size than before, EL fails.", "Besides, as EL is not designed for fully labeled datasets, the images with complete labels (from F) actually have a negative influence on the training.", "Meanwhile, PaNN cannot easily learn the image prior based on only a few fully labeled images.", "VLUU outperforms EL and PaNN by a large margin.", "Essentially, EL and PaNN do not solve the data scarcity problem, while VLUU can generate new vicinal examples.", "Moreover, a segmentation network trained with VLUU can even achieve comparable performance with the same network trained with complete labels (i.e.", "more supervision).", "Considering the existence of class imbalance and dataset shift, we conclude that VLUU is more robust on small-scale data.", "The visual comparison between PaNN, VLUU and Oracle is shown in Fig.", "REF .", "It can be seen that PaNN generates unrealistic shapes for the optic disc and optic cup if not enough fully labeled data is available learn a reasonable image prior.", "Note, although VLUU can achieve comparable performance with Oracle in numerical results, there are artifacts caused by the uncertainty of the vicinal labels, e.g.", "as shown in Fig.", "REF , VLUU may generate optic cup predictions outside the optic disc." ], [ "Conclusion", "In this paper, we discuss the robustness issue of partially supervised methods under the challenge of data scarcity.", "We present VLUU, an easy-to-implement framework, for medical image segmentation tasks with only small partially labeled data.", "Compared with previous methods, VLUU efficiently utilizes the human structure similarity.", "The experimental results show that VLUU is more robust than state-of-the-art partially supervised methods under various data scarcity situations.", "Our research suggests a new research direction in label-efficient DL with partial supervision by tackling the problem from the perspective of VRM." ], [ "Acknowledgment", "The authors would also like to thank Huawei, Amazon, and Google for providing cloud computing service for this study.", "This work was partially funded by the Research Council of Norway grants no.", "315029, 309439, and 303514." ] ]	2011.14164
[ [ "Global well-posedness and scattering of the energy critical\n Maxwell-Klein-Gordon system in the Lorenz gauge" ], [ "Abstract We study initial value problem of the $(1+4)$-dimensional Maxwell-Klein-Gordon system (MKG) in the Lorenz gauge.", "Since (MKG) in the Lorenz gauge does not possess an obvious null structure, it is not easy to handle the nonlinearity.", "To overcome this obstacle, we impose an additional angular regularity.", "In this paper, we prove global well-posedness and scattering of (MKG) for small data in a scale-invariant space which has extra weighted regularity in the angular variables.", "Our main improvement is to attain the scaling critical regularity exponent and prove global existence of solutions to (MKG) in the Lorenz gauge." ], [ "Introduction", "In this paper, we investigate global well-posedness and scattering of the $(1+4)$ -dimensional Maxwell-Klein-Gordon system in the Lorenz gauge.", "The Maxwell-Klein-Gordon (MKG) system describes a physical phenomena of a spin-zero particle in an electromagnetic field.", "The (MKG) system is obtained by coupling the Klein-Gordon scalar field $\\phi :\\mathbb {R}^{1+4}\\rightarrow \\mathbb {C}$ with an elctromagnetic field $F$ .", "To be precise, we consider the covariant form of the (MKG) system: $\\begin{aligned}\\partial ^\\nu F_{\\mu \\nu } & = \\textrm {Im}(\\phi \\overline{\\mathcal {D}_\\mu \\phi }), \\\\\\mathcal {D}^\\mu \\mathcal {D}_\\mu A & = m^2\\phi ,\\end{aligned}$ where $F=dA$ is the associated curvature 2-form given by $F_{\\mu \\nu }=\\partial _\\mu A_\\nu -\\partial _\\nu A_\\mu $ and $A=(A_\\mu ),\\ \\mu =0,1,\\cdots ,4$ is a real-valued 1-form on $\\mathbb {R}^{1+4}$ .", "We write $\\mathcal {D}^\\mu =\\partial ^\\mu -iA^\\mu $ for the connection.", "Here the constant $m\\ge 0$ is a mass.", "Since we study the initial value problem of (MKG) in a scale-invariant Besov space, we put $m=0$ in the sequel.", "In other words, we consider the mass-less (MKG) system.", "The (MKG) in $\\mathbb {R}^{1+4}$ obeys the law of conservation of energy.", "The conserved energy of a solution $(A,\\phi )$ at time $t$ is defined as $\\mathcal {E}_{MKG}(\\phi ,A):=\\frac{1}{2}\\int _{\\mathbb {R}^4}\|F_{\\mu \\nu }(t)\|^2+\|\\mathcal {D}_\\mu \\phi (t)\|^2\\,dx.", "$ We also note that the (MKG) system is invariant under the scaling $(A,\\phi )\\mapsto (\\lambda ^{-1}A,\\lambda ^{-1}\\phi )(\\lambda ^{-1}t,\\lambda ^{-1}x),$ and hence the critical Sobolev space for the system (REF ) is $\\dot{H}^1$ .", "Consequently, the Cauchy problem of the $(1+4)$ -dimensional (MKG) system can be regarded as energy-critical problem.", "We remark that the gauge potential $A_\\mu $ is not necessarily unique.", "Indeed, for a sufficiently smooth real-valued function $\\Lambda $ on $\\mathbb {R}^{1+4}$ , the (MKG) system is invariant under the transform $(A,\\phi )\\mapsto (A-d\\Lambda ,e^{i\\Lambda }\\phi )$ .", "Hence we have gauge freedom and it is reasonable to choose a representative, which is suitable for our purpose.", "(See also [19].)" ], [ "Maxwell-Klein-Gordon equations in the Lorenz gauge", "Imposing the Lorenz gauge $\\partial ^\\mu A_\\mu =0$ we obtain the following nonlinear wave equations after some computation $\\begin{aligned}\\Box \\phi = 2iA^\\mu \\partial _\\mu \\phi +A^\\mu A_\\mu \\phi , \\\\\\Box A = -\\textrm {Im}(\\phi \\overline{\\partial \\phi })-A\|\\phi \|^2,\\end{aligned}$ where $\\Box =\\partial _t^2-\\Delta $ is the d'Alembertian.", "The Maxwell-Klein-Gordon system has been extensively studied [8], [19].", "Typical choices of gauge are the Lorenz gauge $\\partial ^\\mu A_\\mu =0$ and the Coulomb gauge $\\partial ^jA_j=0$ .", "Keel-Roy-Tao [6] proved global existence below the energy space and almost optimal well-posedness is proven by Machedon-Sterbenz [12].", "In the Lorenz gauge, Pecher [17] showed almost critcal well-posedness in the Fourier-Lebesgue spaces.", "From now on we exclusively consider $(1+4)$ -dimensional setting.", "The (MKG) system is relatively well-understood in the Coulomb gauge.", "Selberg [18] proved well-posedness in $H^{1+}$ , which is almost optimal and Krieger-Sterbenz-Tataru [10] showed global well-posedness with small $H^1$ norm.", "Then Oh-Tataru [13], [14], [15] obtained global well-posedness with arbitrarily large data.", "Concerning the Lorenz gauge, Pecher [16] proved local well-posedness for $H^{\\frac{7}{6}+}$ data.", "Thus well-posedness for critical regularity data in the Lorenz gauge is still open.", "We briefly discuss the technical difficulty in proving low regularity well-posedness.", "When we are concerned with quadratic nonlinearity, the most difficult type of interaction is when the two inputs give rise to an output which is close to the light cone in the space-time Fourier space.", "However, if nonlinearity has a certain cancellation property, it would be possible to obtain better regularity properties.", "Such cancellation typically given by null structure plays a crucial role to attain the critical regularity.", "We refer the readers to [7], [4], [9], [11] for the study on the null form estimates and their applications.", "In the Lorenz gauge, however, the main obstacle is that the null structure inside the equations (REF ) is not enough to attain the critical regularity as we are concerned with a low dimensional setting.", "By not enough, we mean that the cancellation property of parallel interaction given by the null structure is only the angle $\\angle (\\xi _1,\\xi _2)$ between the two input frequencies.", "If the null structure is enough, for instance, it yields more than $\\angle (\\xi _1,\\xi _2)$ , such as the type of null form $Q_0$ , defined by $Q_0(u,v)=\\partial _tu\\partial _tv-\\partial _ju\\partial ^jv$ which gives $\\angle (\\xi _1,\\xi _2)^2$ , we could overcome this problem.", "Unfortunately, we cannot expect such an enough null structure in this case and hence it is not easy to handle the parallel interactions.", "Inspired by the work of Sterbenz [23], we expect that the rotation generators $\\Omega _{ij}=x_i\\partial _j-x_j\\partial _i$ plays a crucial role since they eliminate such delicate interactions.", "Furthermore, we will get a significant gain over the classical Strichartz estimates for the wave equation [23], [3], and hence obtain a crucial improvement at the level of multilinear estimates.", "See also [1], [2], [24], [27] and references therein for the study on the Dirac-Klein-Gordon and Yang-Mills systems via angular regularity.", "In this paper, we study global well-posedness and scattering of the mass-less (MKG) in a scale-invariant Besov space with additional angular regularity.", "For the purpose, we consider the following function space $\\Vert f\\Vert _{\\dot{B}^{s,\\sigma }_\\Omega } := \\Vert \\langle \\Omega \\rangle ^\\sigma f\\Vert _{\\dot{B}^s}, $ where $\\langle \\Omega \\rangle ^\\sigma := (1-\\Delta _{\\mathbb {S}^3})^\\frac{\\sigma }{2}$ .", "Here, $\\dot{B}^s=\\dot{B}^s_{2,1}$ is the usual homogeneous Besov space and $\\Delta _{\\mathbb {S}^3}$ is the Laplace-Beltrami operator on the unit sphere $\\mathbb {S}^3$ .", "We consider our system (REF ) with the initial data $(\\phi ,\\partial _t\\phi )\|_{t=0}&:=(\\phi _0,\\phi _1)\\in \\dot{B}^{1,1}_\\Omega \\times \\dot{B}^{0,1}_\\Omega , \\\\(A_0,\\mathbf {A})\|_{t=0}&:=(a_0,\\mathbf {a})\\in \\dot{B}^{1,1}_\\Omega .$ We also write $\\partial _tA_\\mu \|_{t=0}=\\dot{a_\\mu }\\in \\dot{B}^{0,1}_\\Omega $ .", "Then we consider the following constraints for data: $a_0 = \\dot{a_0} = 0, \\\\\\partial ^ka_k =0 .", "$ We need the condition (REF ) because otherwise the Lorenz gauge condition does not determine the potential uniquely.", "The condition () follows from the Lorenz condition in connection with (REF ).", "(See also [16].)", "We state our main result.", "Theorem 1.1 (Global well-posedness) Suppose that the initial data $(\\phi _0,\\phi _1,\\mathbf {a},\\dot{\\mathbf {a}})$ satisfy the smallness condition $\\Vert (\\phi _0,\\phi _1,\\mathbf {a},\\dot{\\mathbf {a}})\\Vert _{\\dot{B}^{1,1}_\\Omega \\times \\dot{B}^{0,1}_\\Omega \\times \\dot{B}^{1,1}_\\Omega \\times \\dot{B}^{0,1}_\\Omega } \\le \\epsilon _0,$ and the constraints (REF ) and ().", "Then there exists a global solution $(\\phi (t),A(t))$ to the mass-less (MKG) system with the Lorenz gauge, which satisfies the following stability condition: $\\sup _{-\\infty <t<+\\infty }\\Vert (\\phi (t),\\partial _t\\phi (t), A(t),\\partial _tA(t))\\Vert _{\\dot{B}^{1,1}_\\Omega \\times \\dot{B}^{0,1}_\\Omega \\times \\dot{B}^{1,1}_\\Omega \\times \\dot{B}^{0,1}_\\Omega } \\le C.$ Furthermore, the solution depends smoothly on the initial data.", "Our main improvement is to attain the scaling critical regularity exponent and prove global existence of solutions to (MKG) using a scale-invariance of function space.", "Then scattering is followed by Theorem REF .", "Theorem 1.2 (Scattering) For any given initial data $(\\phi _0,\\phi _1,\\mathbf {a},\\dot{\\mathbf {a}})\\in \\dot{B}^{1,1}_\\Omega \\times \\dot{B}^{0,1}_\\Omega \\times \\dot{B}^{1,1}_\\Omega \\times \\dot{B}^{0,1}_\\Omega $ satisfying the conditions (REF ), (), and (REF ), there exist unique functions $(\\phi _0^+,\\phi _1^+,\\mathbf {a}^+,\\dot{\\mathbf {a}}^+)$ and $(\\phi _0^-,\\phi _1^-,\\mathbf {a}^-,\\dot{\\mathbf {a}}^-)$ such that $\\begin{aligned}\\lim _{t\\rightarrow \\pm \\infty }\\big ( &\\Vert \\phi (t)-\\phi ^\\pm (t)\\Vert _{\\dot{B}^{1,1}_\\Omega }+\\Vert \\partial _t\\phi (t)-\\partial _t\\phi ^\\pm (t)\\Vert _{\\dot{B}^{0,1}_\\Omega } \\\\&\\qquad \\qquad \\qquad \\qquad +\\Vert A(t)-A^\\pm (t)\\Vert _{\\dot{B}^{1,1}_\\Omega }+\\Vert \\partial _tA(t)-\\partial _tA^\\pm (t)\\Vert _{\\dot{B}^{0,1}_\\Omega } \\big ) = 0.", "\\end{aligned}$ Moreover, the scattering operator which maps $(\\phi _0,\\phi _1,\\mathbf {a},\\dot{\\mathbf {a}})$ to $(\\phi _0^\\pm ,\\phi _1^\\pm ,\\mathbf {a}^\\pm ,\\dot{\\mathbf {a}}^\\pm )$ is a local diffeomorphism in $\\dot{B}^{1,1}_\\Omega \\times \\dot{B}^{0,1}_\\Omega \\times \\dot{B}^{1,1}_\\Omega \\times \\dot{B}^{0,1}_\\Omega $ ." ], [ "Strategy of proof", "We follow the approach due to the works of Sterbenz and Wang [24], [27].", "That is, we shall define the appropriate function spaces and use bilinear decomposition for angles and estimate the nonlinearity in the function spaces.", "As the author of [24] mentioned, the estimate of cubic terms is very straightforward.", "Indeed, we will only use Hölder's inequality and Strichartz estimates, and hence estimates of bilinear forms in (REF ) will be the crucial part of this paper.", "To do this, we consider all possible frequency interactions such as High$\\times $ High and Low$\\times $ High interactions.", "By the mercy of an extra weighted regularity in angular variables, we enjoy the improved space-time Strichartz estimates and hence the High$\\times $ High interaction can be treated rather easier than the Low$\\times $ High case.", "(See Section REF .)", "We remark that in the Low$\\times $ High interaction, the situation that the low frequency controls the angular regularity becomes more difficult case, since we cannot exploit the angular concentration estimate.", "(See Remark REF .)", "Organisation.", "The rest of this paper is organised as follows.", "In Section 2, we recall the Strichartz estimates and reveal null structure of $A^\\mu \\partial _\\mu \\phi $ .", "We construct the function spaces via the Littlewood-Paley projections in Section 3.", "We introduce the angular decompositions of bilinear form in Section 4.", "Then Section 5 will be the main part of this paper, devoted to the proof of our main result.", "Notations.", "As usual different positive constants, which are independent of dyadic numbers $\\mu ,\\lambda $ , and $d$ are denoted by the same letter $C$ , if not specified.", "$A \\lesssim B$ and $A \\gtrsim B$ means that $A \\le CB$ and $A \\ge C^{-1}B$ , respectively for some $C>0$ .", "$A \\approx B$ means that $A \\lesssim B$ and $A \\gtrsim B$ , i.e., $\\frac{1}{C}B \\le A\\le CB $ for some absolute constant $C$ .", "We also use the notation $A\\ll B$ if $A\\le \\frac{1}{C}B$ for some large constant $C$ .", "Thus for quantites $A$ and $B$ , we can consider three cases: $A\\approx B$ , $A\\ll B$ and $A\\gg B$ .", "In fact, $A\\lesssim B$ means that $A\\approx B$ or $A\\ll B$ .", "We shall use the notation $A\\pm $ which means that for small positive $\\epsilon >0$ , we may replace $A\\pm $ by $A\\pm \\epsilon $ .", "For example, we shall write the improved Strichartz estimates with additional angular regularity as $\\Vert e^{\\mp it\|\\nabla \|}f_1\\Vert _{L^2_tL^{3+}_x} \\lesssim \\Vert \\langle \\Omega \\rangle ^\\frac{1}{2} f_1\\Vert _{L^2_x}.$ The spatial and space-time Fourier transforms are defined by $\\widehat{f}(\\xi ) &= \\int _{\\mathbb {R}^4} e^{-ix\\cdot \\xi }f(x)\\,dx, \\\\\\widetilde{u}(\\tau ,\\xi ) &= \\int _{\\mathbb {R}^{1+4}}e^{-i(t\\tau +x\\cdot \\xi )}u(t,x)\\,dtdx.$ We also write $\\mathcal {F}_x(f)=\\widehat{f}$ and $\\mathcal {F}_{t,x}(u)=\\widetilde{u}$ .", "We denote the backward and forward wave propagation of a functiom $f$ on $\\mathbb {R}^4$ by $e^{\\mp it\|\\nabla \|}f = \\int _{\\mathbb {R}^4}e^{ix\\cdot \\xi }e^{\\mp it\|\\xi \|}\\widehat{f}(\\xi )\\,d\\xi .$ The notation $V$ denotes the parametrix for the inhomogeneous wave equation with zero initial data, that is, $u=Vf$ if and only if $\\Box u=f, \\quad u(0,\\cdot ) = 0, \\quad \\partial _tu(0,\\cdot ) = 0.$ Let $E$ be any fundamental solution to the homogeneous wave equation, that is, $\\Box E=\\delta $ , where $\\delta $ is the Dirac delta distribution.", "Then we can represent the parametrix operator $V$ via the following formula: $V(f) = Ef-W(Ef),$ where, for any smooth well-defined function $g=g(t,x)$ , $W(g)$ denotes the solution of linear homogeneous wave equation with initial data $(g,\\partial _tg)$ .", "Finally, we recall some basic facts from harmonic analysis on the sphere.", "The most of the ingredients here can be found in [24].", "We also refer the readers to [21] for the systematic introduction to the analysis on the sphere.", "We denote the standard infinitesimal generators of the rotations on $\\mathbb {R}^4$ by $\\Omega _{ij}=x_i\\partial _j-x_j\\partial _i$ .", "Then the Laplace-Beltrami operator $\\Delta _{\\mathbb {S}^3}$ can be written as $\\Delta _{\\mathbb {S}^3} = \\sum _{i<j}\\Omega _{ij}^2.$ We define the fractional order of the spherical Laplacian $\|\\Omega \|^\\sigma = (-\\Delta _{\\mathbb {S}^3})^\\frac{\\sigma }{2}.$ We also use the inhomogeneous form of $\\Delta _{\\mathbb {S}^3}$ : $\\langle \\Omega \\rangle ^\\sigma f = f_0+\|\\Omega \|^\\sigma f,$ where $f_0$ is the radial part of $f$ , given by $f_0(r) = \\frac{1}{\|\\mathbb {S}^3\|}\\int _{\\mathbb {S}^3}f(r\\omega )\\,d\\omega .$ An important fact of the operators $\\langle \\Omega \\rangle ^\\sigma $ is that they are commutative with the Fourier transform: $\\mathcal {F}[\\langle \\Omega \\rangle ^\\sigma f] = \\langle \\Omega \\rangle ^\\sigma \\mathcal {F}(f).$ Then the homogeneous Besov space with additional angular regularity is defined as $\\Vert f\\Vert _{\\dot{B}^{s,\\sigma }_\\Omega } := \\Vert \\langle \\Omega \\rangle ^\\sigma f\\Vert _{\\dot{B}^s},$ where $\\Vert f\\Vert _{\\dot{B}^s}=\\sum _{\\lambda \\in 2^{\\mathbb {Z}}}\\lambda ^s\\Vert P_\\lambda f\\Vert _{L^2_x}$ , and $P_\\lambda $ is the Littlewood-Paley prejection onto the set $\\lbrace \\xi \\in \\mathbb {R}^4 : \|\\xi \|\\approx \\lambda \\rbrace $ .", "Remark 1.3 Strictly speaking, the following Leibniz rule is not true: $\\langle \\Omega \\rangle (f g) = (\\langle \\Omega \\rangle f)g+f(\\langle \\Omega \\rangle g).$ This is because obviously the operator $\|\\Omega \|$ is a non-local operator.", "On the other hand, $\\Omega _{ij}$ is clearly a local operator and the Leibniz rule holds.", "However, for convenience, by abuse of notation we replace any instance of an single $\\Omega _{ij}$ with the operator $\\langle \\Omega \\rangle $ and assume the above Leibniz rule for $\\langle \\Omega \\rangle $ is true.", "See also [24], [27]." ], [ "Strichartz estimates", "We first introduce the classical Strichartz estimates for the homogeneous wave equations, based soley on translation invariant derivatives of the initial data.", "(See also [25], [5].)", "Proposition 2.1 Let $n=4$ be the number of spatial dimensions, and let $\\sigma =\\frac{3}{2}$ be the corresponding Strichartz admissibility exponent.", "If $f$ is any function of the spatial variable only, denote by $f_1=P_1f$ its unit frequency projection.", "Then one has the following family estimates for $2\\le q$ : $\\Vert e^{\\mp it\|\\nabla \|}f_1\\Vert _{L^q_tL^r_x} \\lesssim \\Vert f_1\\Vert _{L^2_x},$ where $\\frac{1}{q}+\\frac{\\sigma }{r}\\le \\frac{\\sigma }{2}$ .", "In sequel, we only use a few selected subset of admissible pair $(q,r)$ , namely, $\\Vert e^{\\mp it\|\\nabla \|}f_1\\Vert _{L^\\infty _tL^2_x} & \\lesssim \\Vert f_1\\Vert _{L^2_x}, \\\\\\Vert e^{\\mp it\|\\nabla \|}f_1\\Vert _{L^2_tL^\\infty _x} & \\lesssim \\Vert f_1\\Vert _{L^2_x}, \\\\\\Vert e^{\\mp it\|\\nabla \|}f_1\\Vert _{L^2_tL^6_x} & \\lesssim \\Vert f_1\\Vert _{L^2_x}.$ The sharpness of the above space-time estimates is proven by Knapp type counterexample, which does not have radial symmetry.", "Hence it is natural to expect that the Strichartz estimates would be improved if one impose spherical symmetry.", "We refer to [23], [3] for the details and other differential operators.", "Proposition 2.2 Let $n=4$ be the number of spatial dimensions, and let $\\sigma _\\Omega =3$ denote the four dimensional angular Strichartz admissible exponent.", "Let $f_1$ be a unit frequency function of the spatial variable only.", "Then for indices $(q,r)$ such that $\\frac{1}{q}+\\frac{\\sigma }{r}\\ge \\frac{\\sigma }{2}$ and $\\frac{1}{q}+\\frac{\\sigma _\\Omega }{r}<\\frac{\\sigma _\\Omega }{2}$ , and for every $0<\\epsilon $ , there is a $C_\\epsilon $ which depends only on $\\epsilon $ such that the following estimates hold: $\\Vert e^{\\mp it\|\\nabla \|}f_1\\Vert _{L^q_tL^r_x} \\lesssim C_\\epsilon \\Vert \\langle \\Omega \\rangle ^sf_1\\Vert _{L^2_x}, $ where $s=(1+\\epsilon )(\\frac{3}{r}+\\frac{2}{q}-\\frac{3}{2})$ .", "In particular, we shall use often the following space-time estimate: $\\Vert e^{\\mp it\|\\nabla \|}f_1\\Vert _{L^2_tL^{3+}_x} &\\lesssim \\Vert \\langle \\Omega \\rangle ^\\frac{1}{2} f_1\\Vert _{L^2_x}.$" ], [ "Null form of $A^\\mu \\partial _\\mu \\phi $", "In this section, we derive the null form in $A^\\mu \\partial _\\mu \\phi $ .", "See also [16].", "We begin with the standard $Q$ -type null form introduced in [7]: $Q_{0j}(u,v) & = \\partial _tu\\partial _jv-\\partial _ju\\partial _tv, \\\\Q_{jk}(u,v)& = \\partial _ju\\partial _kv-\\partial _ku\\partial _jv.$ We decompose the spatial part $\\mathbf {A}$ of the gauge potential $A$ into the divergence-free and curl-free parts $\\mathbf {A} = \\mathbf {A}^{\\rm df}+\\mathbf {A}^{\\rm cf},$ where $A^{\\rm df}_j := R^k(R_jA_k-R_kA_j),\\quad A^{\\rm cf}_j := -R_jR_kA^k.$ Here $R_j=\|\\nabla \|^{-1}\\partial _j$ is the Riesz transform.", "Then we see that $A^\\mu \\partial _\\mu \\phi &= A^0\\partial _t\\phi +\\mathbf {A}\\cdot \\nabla \\phi \\\\& = -A_0\\partial _t\\phi +\\mathbf {A}^{\\rm cf}\\cdot \\nabla \\phi +\\mathbf {A}^{\\rm df}\\cdot \\nabla \\phi \\\\& =: \\mathcal {N}_1+\\mathcal {N}_2.$ By the Lorenz gauge condition $\\partial _tA_0=\\partial ^jA_j$ , we get $\\mathcal {N}_1 & = -A_0\\partial _t\\phi - R_jR_kA^k\\partial ^j\\phi \\\\& = -A_0\\partial _t\\phi - (\|\\nabla \|^{-1}\\partial _tA_0)\\cdot \\nabla \\phi \\\\& = \\partial _j(\|\\nabla \|^{-1}R^jA_0)\\partial _t\\phi - \\partial _t(\|\\nabla \|^{-1}R_jA_0)\\partial ^j\\phi \\\\& = -Q_{0j}(\|\\nabla \|^{-1}R^jA_0,\\phi ).$ We also have $\\mathcal {N}_2 & = R^k(R_jA_k-R_kA_j)\\partial ^j\\phi \\\\& = \\langle \\nabla \\rangle ^{-2} \\partial ^k\\partial _jA_k\\partial ^j\\phi +A_j\\partial ^j\\phi \\\\& = -\\frac{1}{2}\\left( \\langle \\nabla \\rangle ^{-2}(\\partial _j\\partial ^jA_k-\\partial _j\\partial _kA^j)\\partial ^k\\phi - \\langle \\nabla \\rangle ^{-2}(\\partial ^k\\partial _jA_k-\\partial ^k\\partial _kA_j)\\partial ^j\\phi \\right) \\\\& = -\\frac{1}{2}\\left( \\partial _j\\langle \\nabla \\rangle ^{-1}(R^jA_k-R_kA^j)\\partial ^k\\phi - \\partial ^k\\langle \\nabla \\rangle ^{-1}(R_jA_k-R_kA_j)\\partial ^j\\phi \\right) \\\\& = -\\frac{1}{2} Q_{jk}(\\langle \\nabla \\rangle ^{-1}(R^jA^k-R^kA^j),\\phi ).$ We shall denote the Fourier symbols of $Q_{0j}$ and $Q_{jk}$ by $q_{0j}(\\xi ,\\eta )$ and $q_{jk}(\\xi ,\\eta )$ , respectively.", "Then the symbols satisfy the following estimates: $\|q_{0j}(\\xi ,\\eta )\|,\|q_{jk}(\\xi ,\\eta )\| \\lesssim \|\\xi \|\|\\eta \|\\angle (\\xi ,\\eta ),$ where $\\angle (\\xi ,\\eta )=\\arccos (\\frac{\\xi \\cdot \\eta }{\|\\xi \|\|\\eta \|})$ is the angle between $\\xi $ and $\\eta $ .", "(See [20].)", "The above notations seem too complicated.", "However, an important point here is that the bilinear form $A^\\mu \\partial _\\mu \\phi $ contributes an additional angle between two input frequencies.", "In other words, it is not necessary to distinguish the null forms $Q_{0j}$ and $Q_{jk}$ .", "In the sequel, we will by abuse of notation simply write $Q(\\varphi ,\\phi )$ for the bilinear form $A^\\mu \\partial _\\mu \\phi $ .", "That is, we even ignore the vector components of $A^\\mu $ and denote it by $\\varphi $ shortly." ], [ "Function spaces", "This section is devoted to the introduction of preliminary setup to be used in the proof of Theorem REF .", "We shall establish the function spaces in this section.", "The readers can find the following notations in [24], [27]." ], [ "Multipliers", "Let $\\beta $ be a smooth bump function given by $\\beta (s)=1$ for $\|s\|\\le 1$ and $\\beta (s)=0$ for $\|s\|\\ge 2$ .", "We define the dyadic scaling of $\\beta $ by $\\beta _\\lambda (s)=\\beta (\\lambda ^{-1}s)$ for $\\lambda \\in 2^{\\mathbb {Z}}$ .", "Now we define the Fourier multipliers which give localisation with respect to the spatial frequency, space-time frequency, distance to the cone (modulation), distance to the lower cone and the upper cone as follows: $p_\\lambda (\\xi ) & = \\beta _{2\\lambda }(\|\\xi \|)-\\beta _{\\lambda /2}(\|\\xi \|), \\quad s_\\lambda (\\tau ,\\xi ) = \\beta _{2\\lambda }(\|(\\tau ,\\xi )\|)-\\beta _{\\lambda /2}(\|(\\tau ,\\xi )\|), \\\\c_d(\\tau ,\\xi ) & = \\beta _{2d}(\|\\tau \|-\|\\xi \|)-\\beta _{d/2}(\|\\tau \|-\|\\xi \|), \\\\c_d^+(\\tau ,\\xi ) & = \\beta _{2d}(\\tau +\|\\xi \|)-\\beta _{d/2}(\\tau +\|\\xi \|),\\quad c_d^-(\\tau ,\\xi ) = \\beta _{2d}(\\tau -\|\\xi \|)-\\beta _{d/2}(\\tau -\|\\xi \|).$ Then we define the corresponding Fourier projection operators.", "For example, $\\mathcal {F}_x(P_\\lambda f)=p_\\lambda \\mathcal {F}_x(f)$ and $\\mathcal {F}_{t,x}(S_\\lambda u)=s_\\lambda \\mathcal {F}_{t,x}(u)$ .", "We also write $S_{\\lambda ,d}=S_\\lambda C_d$ , $S^\\pm _{\\lambda ,d}=S_\\lambda C^\\pm _d$ , and denote $S_{\\lambda ,\\cdot \\le d} = \\sum _{\\delta \\le d}S_{\\lambda ,\\delta },\\quad S^\\pm _{\\lambda ,\\cdot \\le d} = \\sum _{\\delta \\le d}S^\\pm _{\\lambda ,\\delta },$ and the projection $S_{\\lambda ,d\\le \\cdot }$ is defined in the obvious way.", "For any given $0<\\eta \\lesssim 1$ , we define $\\mathcal {C}_\\eta $ to be a collection of finitely overlapping caps of size $\\eta $ in the unit sphere $\\mathbb {S}^3$ in $\\mathbb {R}^4$ .", "Then we denote a smooth partition of unity subordinate to the angular sectors $\\lbrace \\xi \\ne 0 : \\frac{\\xi }{\|\\xi \|}\\in \\omega \\rbrace $ by $\\lbrace b^\\omega _\\eta \\rbrace _{\\omega \\in \\mathcal {C}_\\eta }$ .", "The corresponding angular localisation operator is denoted by $B^\\omega _\\eta $ .", "For simplicity, we allow the abuse of notation and identify the angular sectors $\\omega \\in \\mathcal {C}_\\eta $ and their centre.", "We define $S^\\omega _{\\lambda ,d} = B^\\omega _{(\\frac{d}{\\lambda })^\\frac{1}{2}}P_\\lambda S_{\\lambda ,d}, \\quad S^\\omega _{\\lambda ,\\cdot \\le d} = B^\\omega _{(\\frac{d}{\\lambda })^\\frac{1}{2}}P_\\lambda S_{\\lambda ,\\cdot \\le d}.$ We have the following lemma concerning boundedness of the above localisation operators.", "(See also [24], [27].)", "Lemma 3.1 The following multipliers are given by $L^1_{t}L^1_x$ kernels and are uniformly bounded in $L^1_{t}L^1_x$ : $\\lambda ^{-1}\\nabla S_\\lambda ,\\ B^\\omega _{(\\frac{d}{\\lambda })^\\frac{1}{2}}P_\\lambda , \\ S^\\omega _{\\lambda ,d}, \\ (\\lambda d)VS^\\omega _{\\lambda ,d}$ and also those operators are bounded in mixed Lebesgue spaces $L^q_tL^r_x$ .", "The following multipliers are uniformly bounded in the $L^q_tL^2_x$ spaces for $1\\le q\\le +\\infty $ : $S_{\\lambda ,d}, \\ S_{\\lambda ,\\cdot \\le d}$ ." ], [ "Function spaces", "Now we establish the function spaces with space-time frequency localised in the support of $s_\\lambda (\\tau ,\\xi )$ .", "We let $\\Vert u\\Vert _{X^{\\frac{1}{2},p}_\\lambda }^p := \\sum _{d;d\\in 2^\\mathbb {Z}} d^\\frac{p}{2} \\Vert S_{\\lambda ,d}u\\Vert _{L^2_{t}L^2_x}^p,$ and $\\Vert u\\Vert _{Y_\\lambda } := \\lambda ^{-1}\\Vert \\Box S_\\lambda u\\Vert _{L^1_tL^2_x}.$ We also define the space $Z_{\\Omega ,\\lambda }$ by $\\Vert f\\Vert _{Z_{\\Omega ,\\lambda }} := \\lambda ^{-1}\\sum _{d\\lesssim \\lambda }\\int \\sup _\\omega \\Vert S^\\omega _{\\lambda ,d}f\\Vert _{L^\\infty _x}\\,dt.$ Then we have $\\Vert f\\Vert _{Z_{\\Omega ,\\lambda }} \\lesssim \\lambda ^{-1}\\Vert S_\\lambda \\Box \\langle \\Omega \\rangle f\\Vert _{L^1_tL^2_x} \\lesssim \\Vert S_\\lambda \\langle \\Omega \\rangle f\\Vert _{Y_\\lambda },$ and hence $\\langle \\Omega \\rangle ^{-1}Y_\\lambda \\subset Z_{\\Omega ,\\lambda }$ .", "We also have $\\sup _{d\\lesssim \\lambda }\\lambda ^{-1}\\left\\Vert \\sup _\\omega \\Vert S^\\omega _{\\lambda ,d}f\\Vert _{L^\\infty _x}\\right\\Vert _{L^1_t} \\lesssim \\Vert f\\Vert _{F_{\\Omega ,\\lambda }}.$ Finally, we define the function space $F_{\\Omega ,\\lambda } = \\langle \\Omega \\rangle ^{-1}(X^{\\frac{1}{2},1}_\\lambda +Y_\\lambda )\\cap S_\\lambda (L^\\infty _tL^2_x)\\cap Z_{\\Omega ,\\lambda }.$ As $X^{\\frac{1}{2},1}_\\lambda \\subset S_\\lambda (L^\\infty _tL^2_x)$ , naturally we have $\\Box X^{\\frac{1}{2},1}_\\lambda \\subset \\Box S_\\lambda (L^\\infty _tL^2_x)$ , and by duality, we obtain $VS_\\lambda (L^1_tL^2_x)\\subset (\\Box X^{\\frac{1}{2},1}_\\lambda )^{\\prime } = (\\lambda X^{-\\frac{1}{2},1}_\\lambda )^{\\prime } = \\lambda ^{-1}X^{\\frac{1}{2},\\infty }_\\lambda ,$ and recall an obvious relation $X^{\\frac{1}{2},1}_\\lambda \\subset X^{\\frac{1}{2},\\infty }_\\lambda $ , hence we have $d^\\frac{1}{2} \\Vert S_{\\lambda ,d}f\\Vert _{L^2_{t,x}} & \\lesssim \\Vert \\langle \\Omega \\rangle ^{-1}f\\Vert _{F_{\\Omega ,\\lambda }}, \\quad \\textrm { for } d\\in 2^{\\mathbb {Z}}, \\quad 0<d\\le \\lambda , \\\\d^\\frac{1}{2} \\Vert S_{\\lambda ,d\\le \\cdot }f\\Vert _{L^2_{t,x}} & \\lesssim \\Vert \\langle \\Omega \\rangle ^{-1}f\\Vert _{F_{\\Omega ,\\lambda }}.$ Now we define the Besov type function space, which we will iterate to prove our main theorem: $\\Vert u\\Vert _{F_\\Omega } := \\sum _{\\lambda \\in 2^{\\mathbb {Z}}}\\lambda \\Vert S_\\lambda u\\Vert _{F_{\\Omega ,\\lambda }}.", "$ We list several space-time estimates to be used throughout this paper.", "$\\Vert S_\\mu \\langle \\Omega \\rangle u\\Vert _{L^\\infty _tL^2_x} & \\lesssim \\Vert u\\Vert _{F_{\\Omega ,\\mu }}, \\\\\\Vert S_\\mu \\langle \\Omega \\rangle u\\Vert _{L^2_tL^\\infty _x} & \\lesssim \\Vert u\\Vert _{F_{\\Omega ,\\mu }}, \\\\\\Vert S_\\mu \\langle \\Omega \\rangle u\\Vert _{L^2_tL^6_x} & \\lesssim \\mu ^\\frac{5}{6}\\Vert u\\Vert _{F_{\\Omega ,\\mu }}, \\\\\\Vert S_\\mu \\langle \\Omega \\rangle ^\\frac{1}{2}u\\Vert _{L^2_tL^{3+}_x} & \\lesssim \\mu ^{\\frac{1}{6}+}\\Vert u\\Vert _{F_{\\Omega ,\\mu }} .$ We end this section with angular concentration estimates and Sobolev embedding estimates.", "Lemma 3.2 (Lemma 5.2 of [24]) Let $2\\le n$ be a given integer.", "Then for any test function $u$ on $\\mathbb {R}^n$ , and any $2\\le p<\\infty $ , one has the following estimate: $\\sup _\\omega \\Vert B^\\omega _\\eta u\\Vert _{L^p} \\lesssim \\eta ^s\\Vert \\langle \\Omega \\rangle ^su\\Vert _{L^p},$ where $0\\le s<\\dfrac{n-1}{p}$ .", "Lemma 3.3 Let $f$ be a test function on $\\mathbb {R}^4$ .", "Then one has the following frequency localised estimate: $\\Vert B^\\omega _\\eta P_1f\\Vert _{L^p_x} \\lesssim \\eta ^{3(\\frac{1}{r}-\\frac{1}{p})}\\Vert f\\Vert _{L^r_x},$ and also by scaling argument, one has $\\Vert B^\\omega _\\eta P_\\lambda f\\Vert _{L^p_x} \\lesssim \\eta ^{3(\\frac{1}{r}-\\frac{1}{p})}\\lambda ^{4(\\frac{1}{r}-\\frac{1}{p})}\\Vert f\\Vert _{L^r_x}.$ We observe that in Lemma REF , the function $f$ is not required to be localised in an angular sector.", "Thus we can use Lemma REF first and then apply Lemma REF .", "For example, we write $\\left\\Vert \\sup _{\\omega }\\Vert S^\\omega _{\\mu ,d}u\\Vert _{L^\\infty _x}\\right\\Vert _{L^2_t} & \\lesssim \\mu ^{\\frac{4}{3}-}\\left(\\frac{d}{\\mu }\\right)^{1-}\\left\\Vert \\sup _\\omega \\Vert S^\\omega _{\\mu ,d}u\\Vert _{L^{3+}_x}\\right\\Vert _{L^2_t} \\\\& \\lesssim \\mu ^{\\frac{4}{3}-}\\left(\\frac{d}{\\mu }\\right)^{\\frac{5}{4}-}\\Vert S_{\\mu ,d}\\langle \\Omega \\rangle ^\\frac{1}{2}u\\Vert _{L^2_tL^{3+}_x} \\lesssim \\mu ^{\\frac{4}{3}-}\\left(\\frac{d}{\\mu }\\right)^{\\frac{5}{4}-}\\Vert S_\\mu u\\Vert _{F_{\\Omega ,\\mu }}.$" ], [ "Bilinear decompositions for angles", "We shall discuss various bilinear decompositions for frequency localised products of the form: $S_{\\lambda _0}(S_{\\lambda _1}u\\, S_{\\lambda _2}v).$ By the standard Littlewood-Paley trichotomy, the localised product (REF ) vanishes unless $\\min \\lbrace \\lambda _0,\\lambda _1,\\lambda _2 \\rbrace \\lesssim \\textrm {med}\\lbrace \\lambda _0,\\lambda _1,\\lambda _2 \\rbrace \\approx \\max \\lbrace \\lambda _0,\\lambda _1,\\lambda _2\\rbrace .$ We focus on two important interactions in (REF ), namely, the High$\\times $ High and Low$\\times $ High frequency interactions: $\\lambda _0\\lesssim & \\, \\lambda _1\\approx \\lambda _2, \\\\\\lambda _1\\lesssim & \\, \\lambda _0\\approx \\lambda _2.", "$ In what follows, we give the bilinear decompositions for angles for the type (REF ), ().", "We refer the reader to [22], [24], [27] for more details.", "Lemma 4.1 (Lemma 6.1 of [24]) For the following localised products of the form: $S_{\\mu ,d}(S_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }u\\, S_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }v),$ one has the following angular decomposition: $s_{\\mu ,d}^\\pm (s^-_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }s^+_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }) & = \\sum _{\\begin{array}{c} \\omega _1,\\omega _3 \\\\ \|\\omega _1\\mp \\omega _3\|\\approx (\\frac{d}{\\mu })^\\frac{1}{2} \\end{array}}s_{\\mu ,d}^{\\omega _1,\\pm }(s^-_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }b^{\\omega _3}_{(\\frac{d}{\\mu })^\\frac{1}{2}}s^+_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }) \\\\& = \\sum _{\\begin{array}{c} \\omega _1,\\omega _2,\\omega _3 \\\\ \|\\omega _1\\mp \\omega _3\|\\approx (\\frac{d}{\\mu })^\\frac{1}{2} \\\\ \|\\omega _2+\\omega _3\|\\approx (\\frac{d}{\\mu })^\\frac{1}{2} \\end{array}}s_{\\mu ,d}^{\\omega _1,\\pm }(b^{\\omega _2}_{(\\frac{d}{\\mu })^\\frac{1}{2}}s^-_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }b^{\\omega _3}_{(\\frac{d}{\\mu })^\\frac{1}{2}}s^+_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }),$ for the convolution of the associated cutoff functions in Fourier side.", "There is a similar decomposition for the terms $S_{\\mu ,\\cdot \\le d}(S_{\\lambda ,\\cdot \\le d}u\\, S_{\\lambda ,d}v)$ and $S_{\\mu ,\\cdot \\le d}(S_{\\lambda ,d}u\\, S_{\\lambda ,\\cdot \\le d}v)$ , where $d$ is in the range $d<c\\mu $ and $c\\ll 1$ is the small number fixed above.", "We also have the angular decomposition for the interaction of type ().", "Lemma 4.2 (Lemma 6.2 of [24]) For the following localised products of the form: $S_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }(S_{\\mu ,d}u\\, S_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }v),$ one has the following angular decomposition: $s^+_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }(s^\\pm _{\\mu ,d}s^+_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }) = \\sum _{\\begin{array}{c} \\omega _1,\\omega _2,\\omega _3 \\\\ \|\\omega _1\\mp \\omega _2\|\\approx (\\frac{d}{\\mu })^\\frac{1}{2} \\\\ \|\\omega _1-\\omega _3\|\\approx (\\frac{d}{\\mu })^\\frac{1}{2} \\end{array}} b^{\\omega _1}_{(\\frac{d}{\\mu })^\\frac{1}{2}}s^+_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }(s^{\\omega _2,\\pm }_{\\mu ,d}b^{\\omega _3}_{(\\frac{d}{\\mu })^\\frac{1}{2}}s^+_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }),$ for the convolution of the associated cutoff functions in Fourier side.", "There is a similar decomposition for the terms $S_{\\lambda ,\\cdot \\le d}(S_{\\mu ,\\cdot \\le d}u\\, S_{\\lambda ,d}v)$ and $S_{\\lambda ,d}(S_{\\mu ,\\cdot \\le d}u\\, S_{\\lambda ,\\cdot \\le d}v)$ in the range $d<c\\mu $ , where $c\\ll 1$ is a fixed small number.", "We note that the angular sectors involved in the summation on the right-hand side of the decompositions in Lemma REF and Lemma REF are essentially diagonal.", "To avoid verbatim, we write angular decompositions throughout this paper as follows: $\\begin{aligned}&S_{\\mu ,d}(S_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\varphi \\, S_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi )\\\\& \\qquad \\qquad = \\sum _\\omega S^{\\pm \\omega }_{\\mu ,d}(B^{-\\omega }_{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\varphi \\, B^\\omega _{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi ), \\end{aligned}$ $\\begin{aligned}&S_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }(S_{\\mu ,d}\\varphi \\, S_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi ) \\\\&\\qquad \\qquad = \\sum _\\omega B^{\\omega }_{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }(S^{\\pm \\omega }_{\\mu ,d}\\varphi \\, B^{\\omega }_{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{\\lambda ,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi ).", "\\end{aligned}$ We also need the following angular decomposition, which will be only used in Section REF .", "Lemma 4.3 (Lemma 6.4 of [24]) For the following expression: $S^\\omega _{\\lambda ,d}(S_\\mu u\\, S_{\\lambda ,\\cdot <c\\mu }v),$ one has the following angular restriction: $s^{\\omega _1,+}_{\\lambda ,d}(s^\\pm _\\mu s^+_{\\lambda ,\\cdot <c\\mu }) = s^{\\omega _1,+}_{\\lambda ,d}(s^\\pm _{\\mu }*b^{\\omega _3}_{(\\frac{d}{c\\lambda })^\\frac{1}{2}}s^+_{\\lambda ,\\cdot <c\\mu })$ for the convolution of the associated cutoff functions in Fourier space.", "Here the angles are restricted to the range $\|\\omega _1-\\omega _3\|\\approx \\left(\\dfrac{d}{c\\lambda }\\right)^\\frac{1}{2}$ ." ], [ "Proof of main Theorem", "This section is devoted to the proof of our main result.", "The proof of scattering is followed by Theorem REF .", "See [22], [24], [27] for details.", "Now we focus on the proof of Theorem REF .", "In view of Duhamel's principle and Picard's iteration, we need to show the following nonlinear estimates: $\\Vert VQ(\\varphi ,\\phi )\\Vert _{F_\\Omega } & \\lesssim \\Vert \\varphi \\Vert _{F_\\Omega }\\Vert \\phi \\Vert _{F_\\Omega }, \\\\\\Vert V(\\phi _1\\partial \\phi _2)\\Vert _{F_\\Omega } & \\lesssim \\Vert \\phi _1\\Vert _{F_\\Omega }\\Vert \\phi _2\\Vert _{F_\\Omega }, \\\\\\Vert V(\\phi \\varphi \\psi )\\Vert _{F_\\Omega } & \\lesssim \\Vert \\phi \\Vert _{F_\\Omega }\\Vert \\varphi \\Vert _{F_\\Omega }\\Vert \\psi \\Vert _{F_\\Omega }.", "$ The bilinear estimates () is already known.", "(See [24].)", "The treatment of cubic terms is very straightforward.", "It suffices to show that $\\Vert S_\\mu (S_1\\langle \\Omega \\rangle uS_1vS_1w)\\Vert _{L^1_tL^2_x} & \\lesssim \\mu ^{0+}\\Vert S_1u\\Vert _{F_\\Omega }\\Vert S_1v\\Vert _{F_\\Omega }\\Vert S_1w\\Vert _{F_\\Omega }, \\\\\\Vert S_1(S_\\mu \\langle \\Omega \\rangle uS_1vS_1w)\\Vert _{L^1_tL^2_x} & \\lesssim \\mu \\Vert S_\\mu u\\Vert _{F_{\\Omega ,\\mu }}\\Vert S_1v\\Vert _{F_\\Omega }\\Vert S_1w\\Vert _{F_\\Omega }.", "$ We simply use the Bernstein's inequality, Hölder's inequality and the Strichartz estimates.", "Indeed, for the proof of (REF ), we see that $\\Vert S_\\mu (S_1\\langle \\Omega \\rangle uS_1vS_1w)\\Vert _{L^1_tL^2_x} & \\lesssim \\mu ^{2-} \\Vert S_\\mu (S_1\\langle \\Omega \\rangle uS_1vS_1w)\\Vert _{L^1_tL^{1+}_x} \\\\& \\lesssim \\mu ^{2-}\\Vert S_1\\langle \\Omega \\rangle u\\Vert _{L^\\infty _tL^2_x}\\Vert S_1v\\Vert _{L^2_tL^6_x}\\Vert S_1w\\Vert _{L^{3+}_x} \\\\& \\lesssim \\Vert S_1u\\Vert _{F_\\Omega }\\Vert S_1v\\Vert _{F_\\Omega }\\Vert S_1w\\Vert _{F_\\Omega }.$ To prove (), we write $\\Vert S_1(S_\\mu \\langle \\Omega \\rangle uS_1vS_1w)\\Vert _{L^1_tL^2_x} & \\lesssim \\Vert S_\\mu \\langle \\Omega \\rangle u\\Vert _{L^2_tL^\\infty _x}\\Vert S_1vS_1w\\Vert _{L^2_tL^2_x} \\\\& \\lesssim \\mu ^{\\frac{4}{6}}\\Vert S_\\mu \\langle \\Omega \\rangle u\\Vert _{L^2_tL^6_x}\\Vert S_1v\\Vert _{L^2_tL^\\infty _x}\\Vert S_1w\\Vert _{L^\\infty _tL^2_x} \\\\& \\lesssim \\mu ^\\frac{3}{2} \\Vert S_\\mu u\\Vert _{F_{\\Omega ,\\mu }}\\Vert S_1v\\Vert _{F_\\Omega }\\Vert S_1w\\Vert _{F_\\Omega },$ where we used $\\Vert S_\\mu \\langle \\Omega \\rangle u\\Vert _{L^2_tL^6_x}\\lesssim \\mu ^\\frac{5}{6}\\Vert u\\Vert _{F_{\\Omega ,\\mu }}$ .", "Fron now on we exclusively consider the bilinear estimates (REF ).", "We first apply the dyadic decomposition on the space-time frequency of the bilinear form $Q(\\varphi ,\\phi )$ .", "It suffices to treat the High$\\times $ High and Low$\\times $ High interactions as follows: $\\sum _{\\mu \\lesssim \\max \\lbrace \\lambda _1,\\lambda _2\\rbrace }\\mu \\Vert VQ(S_{\\lambda _1}\\varphi ,S_{\\lambda _2}\\phi )\\Vert _{F_{\\Omega ,\\mu }} & \\lesssim \\lambda _1\\lambda _2\\Vert \\varphi \\Vert _{F_{\\Omega ,\\lambda _1}}\\Vert \\phi \\Vert _{F_{\\Omega ,\\lambda _2}}, \\\\\\Vert VQ(S_{\\mu }\\varphi ,S_\\lambda \\phi )\\Vert _{F_{\\Omega ,\\lambda } } & \\lesssim \\mu \\Vert \\varphi \\Vert _{F_{\\Omega ,\\mu }}\\Vert \\phi \\Vert _{F_{\\Omega ,\\lambda }}.$ As we are concerned with a scale-invariant function space, it is reasonable to assume that the high frequency $\\lambda =\\lambda _1=\\lambda _2=1$ and the low frequency $\\mu \\lesssim 1$ .", "In consequence, our aim is to prove the following bilinear estimates: $\\mu \\Vert VQ(S_{1}\\varphi ,S_1\\phi )\\Vert _{F_{\\Omega ,\\mu }} & \\lesssim \\mu ^{0+}\\Vert S_1\\phi \\Vert _{F_\\Omega }\\Vert S_1\\phi \\Vert _{F_\\Omega }, \\\\\\Vert VQ(S_\\mu \\varphi ,S_1\\phi )\\Vert _{F_{\\Omega ,1}} & \\lesssim \\mu \\Vert S_\\mu \\varphi \\Vert _{F_{\\Omega ,\\mu }}\\Vert S_1\\phi \\Vert _{F_\\Omega }.", "$ We present the main scheme of the proof of (REF ) and ().", "We first note that the Fourier projection operator $S_\\mu $ and the parametrix $V$ do not commute when $\\mu \\lesssim 1$ .", "Indeed, we have (see [24] for derivation,) $S_\\mu VQ(S_1\\varphi ,S_1\\phi ) = S_\\mu VS_\\mu Q(S_1\\varphi ,S_1\\phi ) - \\sum _{\\mu \\lesssim \\sigma \\lesssim 1}W(P_\\mu S_{\\sigma ,\\sigma }VQ(S_1\\varphi ,S_1\\phi )).", "$ For the second term of the right-handside of (REF ), we shall prove the following: $\\sum _\\mu \\mu \\left\\Vert \\sum _{\\mu \\lesssim \\sigma \\lesssim 1}P_\\mu S_{\\sigma ,\\sigma }VQ\\langle \\Omega \\rangle (S_1\\varphi ,S_1\\phi )\\right\\Vert _{L^\\infty _tL^2_x} \\lesssim \\Vert S_1\\varphi \\Vert _{F_\\Omega }\\Vert S_1\\phi \\Vert _{F_\\Omega },$ which is very straightforward.", "Then we further decompose $S_\\mu Q(S_1\\varphi ,S_1\\phi )$ into the space-time frequency which is away from the light cone and near the cone, respectively.", "For this purpose, we write $\\begin{aligned}S_\\mu Q(S_1\\varphi ,S_1\\phi ) & = S_\\mu Q(S_1\\varphi , S_{1,c\\mu \\le \\cdot }\\phi ) + S_\\mu Q(S_{1,c\\mu \\le \\cdot }\\varphi ,S_{1,\\cdot \\le c\\mu }\\phi ) \\\\& \\qquad \\qquad +S_\\mu Q(S_{1,\\cdot \\le c\\mu }\\varphi ,S_{1,\\cdot \\le c\\mu }\\phi ) \\\\& \\quad =: \\mathcal {H}\\mathcal {H}^1+\\mathcal {H}\\mathcal {H}^2+\\mathcal {H}\\mathcal {H}^3.\\end{aligned}$ The first and second terms $\\mathcal {H}\\mathcal {H}^1,\\mathcal {H}\\mathcal {H}^2$ are rather easier than the third term $\\mathcal {H}\\mathcal {H}^3$ , which is near the cone.", "We will use the angular decomposition (REF ) to $\\mathcal {H}\\mathcal {H}^3$ .", "Then we apply the Hölder inequality, Sobolev embedding, and angular concentration estimates and the Strichartz estimates to obtain the required estimates.", "In the proof of (), we simply get $S_1VQ(S_1\\varphi ,S_1\\phi ) = S_1VS_1Q(S_1\\varphi ,S_1\\phi )$ and we write $\\begin{aligned}S_1Q(S_\\mu \\varphi , S_1\\phi ) & = S_1Q(S_\\mu \\varphi ,S_{1,c\\mu \\le \\cdot }\\phi ) + S_{1,c\\mu \\le \\cdot }Q(S_\\mu \\varphi ,S_{1,\\cdot \\le c\\mu }\\phi ) \\\\& \\qquad \\qquad + S_{1,\\cdot \\le c\\mu }Q(S_\\mu \\varphi ,S_{1,\\cdot \\le c\\mu }\\phi ) \\\\& \\quad =: \\mathcal {L}\\mathcal {H}^1+\\mathcal {L}\\mathcal {H}^2+\\mathcal {L}\\mathcal {H}^3.\\end{aligned}$ The Low$\\times $ High interaction is more difficult than the High$\\times $ High interaction.", "This is because of the regularity $\\dot{B}^1$ of the scalar field $\\varphi $ .", "We will lose $\\mu $ from $\\varphi $ , and hence we must gain more.", "To overcome this problem, we first observe that the angular decomposition as Lemma REF will be very large when the modulation $d$ is close to the low frequency $\\mu $ .", "Thus we shall divide the Low$\\times $ High interaction into two cases: $(\\frac{d}{\\mu })^\\frac{1}{2}\\ll \\mu $ and $\\mu \\lesssim (\\frac{d}{\\mu })^\\frac{1}{2}$ .", "We still use Lemma REF for the first case.", "In the second case, instead of Lemma REF , we use a smaller sector with size $\\mu $ .", "In this case, the range of $d$ is given by $\\mu ^3\\lesssim d\\lesssim \\mu $ and hence the summation on $d$ makes no problem.", "We refer the readers to [27] for the change of weight between $\\mu $ and $(\\frac{d}{\\mu })^\\frac{1}{2}$ .", "We introduce the outline of the remainder of this section.", "We first treat the proof of (REF ) in Section REF .", "Then the estimates of the High$\\times $ High and Low$\\times $ High interaction away from the light cone is given in Section REF and REF , which are rather easier than the frequency near the cone.", "Section REF and REF are devoted to the estimates of the High$\\times $ High and Low$\\times $ High interaction near the cone." ], [ "High$\\times $ High interaction including Commutator term", "We simply use the Bernstein's inequality, the boundedness of multipliers (Lemma REF ), and Hölder's inequality.", "$\\sum _{\\mu \\lesssim 1}\\mu \\left\\Vert \\sum _{\\mu \\lesssim \\sigma \\lesssim 1}P_\\mu S_{\\sigma ,\\sigma }VQ(S_1\\varphi ,S_1\\phi )\\right\\Vert _{L^\\infty _tL^2_x} & \\lesssim \\sum _{\\mu \\lesssim 1}\\mu \\sum _{\\mu \\lesssim \\sigma \\lesssim 1}\\Vert P_\\mu S_{\\sigma ,\\sigma }VQ(S_1\\varphi ,S_1\\phi )\\Vert _{L^\\infty _tL^2_x} \\\\& \\lesssim \\sum _{\\mu \\lesssim 1}\\mu \\sum _{\\mu \\lesssim \\sigma \\lesssim 1}\\mu ^2\\Vert S_{\\sigma ,\\sigma }VQ(S_1\\varphi ,S_1\\phi )\\Vert _{L^\\infty _tL^1_x} \\\\& \\lesssim \\sum _{\\mu \\lesssim 1}\\mu \\sum _{\\mu \\lesssim \\sigma \\lesssim 1}\\left(\\frac{\\mu }{\\sigma }\\right)^2\\Vert S_1\\varphi S_1\\phi \\Vert _{L^\\infty _tL^1_x} \\\\& \\lesssim \\sum _{\\mu \\lesssim 1}\\mu \\Vert S_1\\varphi \\Vert _{L^\\infty _tL^2_x}\\Vert S_1\\phi \\Vert _{L^\\infty _tL^2_x} \\\\& \\lesssim \\Vert S_1\\varphi \\Vert _{F_\\Omega }\\Vert S_1\\phi \\Vert _{F_\\Omega }.$" ], [ "High$\\times $ High interaction away from cone ", "We recall the definition of the $Y$ space.", "We first observe that $\\sum _\\mu \\Vert S_\\mu (S_1\\varphi S_1\\phi )\\Vert _{L^1_tL^2_x} \\lesssim \\sum _{\\mu \\lesssim 1}\\sum _{\\sigma \\lesssim \\mu }\\Vert P_\\sigma S_\\mu (S_1\\varphi S_1\\phi )\\Vert _{L^1_tL^2_x}.$ We use the Bernstein's inequality to gain $\\sigma ^\\frac{2}{3}$ .", "The Hölder inequality and Strichartz estimates give the desired estimates as follows.", "$\\mu \\Vert V\\mathcal {H}\\mathcal {H}^1\\Vert _{F_{\\Omega ,\\mu }} \\lesssim \\mu \\Vert V\\mathcal {H}\\mathcal {H}^1\\Vert _{\\langle \\Omega \\rangle ^{-1}Y_{\\mu }} & \\lesssim \\sum _{\\mu \\lesssim 1}\\sum _{\\sigma \\lesssim \\mu }\\sigma ^\\frac{2}{3}\\Vert \\langle \\Omega \\rangle (S_1\\varphi S_{1,c\\mu \\le \\cdot }\\phi )\\Vert _{L^1_tL^\\frac{3}{2}_x} \\\\& \\lesssim \\sum _{\\mu \\lesssim 1}\\sum _{\\sigma \\lesssim \\mu }\\sigma ^\\frac{2}{3}\\Vert S_1\\langle \\Omega \\rangle \\varphi \\Vert _{L^2_tL^6_x}\\Vert S_{1,c\\mu \\le \\cdot }\\langle \\Omega \\rangle \\phi \\Vert _{L^2_tL^2_x} \\\\& \\lesssim \\sum _{\\mu \\lesssim 1}\\sum _{\\sigma \\lesssim \\mu }\\sigma ^\\frac{2}{3} (c\\mu )^{-\\frac{1}{2}}\\Vert S_1\\varphi \\Vert _{F_\\Omega }\\Vert S_1\\phi \\Vert _{F_\\Omega } \\\\& \\lesssim \\sum _{\\mu \\lesssim 1}c^{-\\frac{1}{2}}\\mu ^\\frac{1}{6}\\Vert S_1\\varphi \\Vert _{F_\\Omega }\\Vert S_1\\phi \\Vert _{F_\\Omega } \\\\& \\lesssim c^{-\\frac{1}{2}}\\Vert S_1\\varphi \\Vert _{F_\\Omega }\\Vert S_1\\phi \\Vert _{F_\\Omega },$ where we used $\\mu ^\\frac{1}{2}\\Vert S_{1,\\mu \\le \\cdot }\\langle \\Omega \\rangle u\\Vert _{L^2_tL^2_x} \\lesssim \\Vert S_1u\\Vert _{F_{\\Omega }}$ .", "The estimate of $\\mathcal {H}\\mathcal {H}^2$ is very similar.", "Indeed, $\\mu \\Vert V \\mathcal {H}\\mathcal {H}^2\\Vert _{F_{\\Omega ,\\mu }} & \\lesssim \\sum _{\\mu \\lesssim 1}\\sum _{\\sigma \\lesssim \\mu }\\Vert P_\\sigma S_\\mu Q\\langle \\Omega \\rangle (S_{1,c\\mu \\le \\cdot }\\varphi ,S_{1,\\cdot \\le c\\mu }\\phi )\\Vert _{L^1_tL^2_x} \\\\& \\lesssim \\sum _{\\mu \\lesssim 1}\\sum _{\\sigma \\lesssim \\mu }\\sigma ^\\frac{2}{3}\\Vert \\langle \\Omega \\rangle (S_{1,c\\mu \\le \\cdot }\\varphi S_{1,\\cdot \\le c\\mu }\\phi )\\Vert _{L^1_tL^\\frac{3}{2}_x} \\\\& \\lesssim \\sum _{\\mu \\lesssim 1}\\sum _{\\sigma \\lesssim \\mu }\\sigma ^\\frac{2}{3} (c\\mu )^{-\\frac{1}{2}}\\Vert S_1\\varphi \\Vert _{F_\\Omega }\\Vert S_1\\phi \\Vert _{F_\\Omega } \\\\& \\lesssim c^{-\\frac{1}{2}}\\mu ^\\frac{1}{6} \\Vert S_1\\varphi \\Vert _{F_\\Omega }\\Vert S_1\\phi \\Vert _{F_\\Omega }.$" ], [ "Low$\\times $ High interaction away from cone", "We first use the Hölder's inequality with respect to $t$ and then $x$ to get $\\Vert V\\mathcal {L}\\mathcal {H}^1\\Vert _{F_{\\Omega ,1}} \\lesssim \\Vert V\\mathcal {L}\\mathcal {H}^1\\Vert _{\\langle \\Omega \\rangle ^{-1}Y_1} & = \\Vert S_1Q\\langle \\Omega \\rangle (S_\\mu \\varphi ,S_{1,c\\mu \\le \\cdot }\\phi )\\Vert _{L^1_tL^2_x} \\\\& \\lesssim \\Vert S_\\mu \\langle \\Omega \\rangle \\varphi \\Vert _{L^2_tL^\\infty _x}\\Vert S_{1,c\\mu \\le \\cdot }\\langle \\Omega \\rangle \\phi \\Vert _{L^2_tL^2_x} \\\\& \\lesssim \\mu ^\\frac{2}{3}\\Vert S_\\mu \\varphi \\Vert _{L^2_tL^6_x}\\Vert S_{1,c\\mu \\le \\cdot }\\phi \\Vert _{L^2_tL^2_x} \\\\& \\lesssim \\mu ^\\frac{2}{3}\\mu ^\\frac{5}{6} \\Vert S_\\mu \\varphi \\Vert _{F_{\\Omega ,\\mu }}(c\\mu )^{-\\frac{1}{2}}\\Vert S_1\\phi \\Vert _{F_\\Omega } \\\\& \\lesssim c^{-\\frac{1}{2}}\\mu \\Vert S_\\mu \\varphi \\Vert _{F_{\\Omega ,\\mu }}\\Vert S_1\\phi \\Vert _{F_\\Omega }.$ Here we also used $\\Vert S_\\mu \\langle \\Omega \\rangle \\varphi \\Vert _{L^2_tL^6_x}\\lesssim \\mu ^\\frac{5}{6}\\Vert S_\\mu \\varphi \\Vert _{F_{\\Omega ,\\mu }}$ .", "The estimate of $\\mathcal {L}\\mathcal {H}^2$ in the $X$ space is quite similar.", "$\\Vert V\\mathcal {L}\\mathcal {H}^2\\Vert _{F_\\Omega } \\lesssim \\Vert V\\mathcal {L}\\mathcal {H}^2\\Vert _{\\langle \\Omega \\rangle ^{-1}X^{\\frac{1}{2},1}_1} & \\lesssim \\sum _{c\\mu <d}d^{-\\frac{1}{2}}\\Vert S_{1,d}Q\\langle \\Omega \\rangle (S_\\mu \\varphi , S_{1,\\cdot \\le c\\mu }\\phi )\\Vert _{L^2_tL^2_x} \\\\& \\lesssim (c\\mu )^{-\\frac{1}{2}}\\Vert S_\\mu \\langle \\Omega \\rangle \\varphi \\Vert _{L^2_tL^\\infty _x}\\Vert S_{1,\\cdot \\le c\\mu }\\langle \\Omega \\rangle \\phi \\Vert _{L^\\infty _tL^2_x} \\\\& \\lesssim c^{-\\frac{1}{2}}\\mu \\Vert S_\\mu \\varphi \\Vert _{F_{\\Omega ,\\mu }}\\Vert S_1\\phi \\Vert _{F_\\Omega }.", "$ We need to estimate $\\mathcal {L}\\mathcal {H}^2$ in the $Z$ space.", "To do this, we apply Lemma REF , Hölder inequality with respect to $t$ and the Bernstein's inequality and then Lemma REF .", "$\\Vert V\\mathcal {L}\\mathcal {H}^2\\Vert _{Z_{\\Omega ,1}} & = \\sum _{c\\mu <d}d^{-1}\\int \\sup _\\omega \\Vert B^\\omega _{d^\\frac{1}{2}}S_{1,d}Q(S_\\mu , S_{1,\\cdot \\le c\\mu }\\phi )\\Vert _{L^\\infty _x}dt \\\\& \\lesssim \\sum _{c\\mu <d}d^{-1}\\int \\sup _{\\begin{array}{c}\\omega _1,\\omega _2 \\\\ \|\\omega _1-\\omega _2\|\\approx (\\frac{d}{c})^\\frac{1}{2}\\end{array}}\\Vert B^{\\omega _1}_{d^\\frac{1}{2}}S_{1,d}Q(S_\\mu \\varphi ,B^{\\omega _2}_{(\\frac{d}{c})^\\frac{1}{2}}S_{1,\\cdot \\le c\\mu }\\phi )\\Vert _{L^\\infty _x}dt \\\\& \\lesssim \\sum _{c\\mu <d}d^{-1}\\Vert S_\\mu \\varphi \\Vert _{L^2_tL^\\infty _x}\\left\\Vert \\sup _\\omega \\Vert B^\\omega _{(\\frac{d}{c})^\\frac{1}{2}}S_{1,\\cdot \\le c\\mu }\\phi \\Vert _{L^\\infty _x}\\right\\Vert _{L^2_t} \\\\& \\lesssim \\sum _{c\\mu <d}d^{-\\frac{1}{4}-}\\mu ^{\\frac{4}{3}-}\\Vert S_\\mu \\varphi \\Vert _{L^2_tL^{3+}_x}\\Vert S_1\\langle \\Omega \\rangle ^{\\frac{1}{2}}\\phi \\Vert _{L^2_tL^{3+}_x} \\\\& \\lesssim \\mu ^{\\frac{13}{12}-}\\Vert S_\\mu \\varphi \\Vert _{F_{\\Omega ,\\mu }}\\Vert S_1\\phi \\Vert _{F_\\Omega }.$" ], [ "High$\\times $ High interaction near cone", "We further decompose $\\mathcal {H}\\mathcal {H}^3$ as follows: $S_\\mu Q(S_{1,\\cdot \\le c\\mu }\\varphi ,S_{1,\\cdot \\le c\\mu }\\phi ) & = \\sum _{d<c\\mu }S_{\\mu ,\\cdot \\le d}Q(S_{1,\\cdot \\le d}\\varphi ,S_{1,d}\\phi )+ \\sum _{d<c\\mu }S_{\\mu ,\\cdot \\le d}Q(S_{1,d}\\varphi ,S_{1,\\cdot \\le d}\\phi ) \\\\&\\quad + \\sum _{d\\lesssim \\mu }S_{\\mu ,d}Q(S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\varphi ,S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi ) \\\\& =: \\mathcal {H}\\mathcal {H}^3_1+\\mathcal {H}\\mathcal {H}^3_2+\\mathcal {H}\\mathcal {H}^3_3.", "$ We estimate the term $\\mathcal {H}\\mathcal {H}^3_1$ in the $Y$ space.", "We apply in order the angular decomposition (Lemma REF ), Sobolev embedding estimate (Lemma REF ), Hölder inequality, and then Strichartz estimates.", "$\\mu \\Vert V\\mathcal {H}\\mathcal {H}^3_1\\Vert _{\\langle \\Omega \\rangle ^{-1}Y_{\\mu }} & \\lesssim \\sum _{d<c\\mu }\\left\\Vert \\left(\\sum _{\\omega }\\Vert S^{\\omega }_{\\mu ,\\cdot \\le d}\\langle \\Omega \\rangle Q(B^{-\\omega }_{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,\\cdot \\le d}\\varphi ,B^\\omega _{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,d}\\varphi )\\Vert _{L^2_x}^2 \\right)^\\frac{1}{2}\\right\\Vert _{L^1_t} \\\\& \\lesssim \\sum _{d<c\\mu }\\mu ^\\frac{2}{3} \\left(\\frac{d}{\\mu }\\right)^\\frac{1}{4}\\left(\\frac{d}{\\mu }\\right)^\\frac{1}{2} \\left\\Vert \\left(\\sum _{\\omega }\\Vert \\langle \\Omega \\rangle (B^{-\\omega }_{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,\\cdot \\le d}\\varphi ,B^\\omega _{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,d}\\varphi )\\Vert _{L^\\frac{3}{2}_x}^2 \\right)^\\frac{1}{2}\\right\\Vert _{L^1_t} \\\\& \\lesssim \\sum _{d<c\\mu }\\mu ^{-\\frac{1}{12}}d^\\frac{3}{4}\\left\\Vert \\sup _\\omega \\Vert B^{-\\omega }_{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,\\cdot \\le d}\\langle \\Omega \\rangle \\varphi \\Vert _{L^6_x} \\left(\\sum _\\omega \\Vert B^\\omega _{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,d}\\langle \\Omega \\rangle \\phi \\Vert _{L^2_x}^2\\right)^\\frac{1}{2}\\right\\Vert _{L^1_t} \\\\& \\lesssim \\sum _{d<c\\mu }\\mu ^{-\\frac{1}{12}}d^\\frac{3}{4}\\Vert S_{1,\\cdot \\le d}\\langle \\Omega \\rangle \\varphi \\Vert _{L^2_tL^6_x}\\Vert S_{1,d}\\langle \\Omega \\rangle \\phi \\Vert _{L^2_tL^2_x} \\\\& \\lesssim \\mu ^\\frac{1}{6} \\Vert S_1\\varphi \\Vert _{F_\\Omega }\\Vert S_1\\phi \\Vert _{F_\\Omega }.$ The estimate of $\\mathcal {H}\\mathcal {H}^3_2$ is very similar.", "We omit the details.", "The treatment of $\\mathcal {H}\\mathcal {H}^3_3$ is also similar.", "Indeed, we apply the angular decomposition, Sobolev estimates and Hölder inequality.", "$\\mu \\Vert V\\mathcal {H}\\mathcal {H}^3_3\\Vert _{\\langle \\Omega \\rangle ^{-1}X^{\\frac{1}{2},1}_\\mu } & = \\sum _{d\\lesssim \\mu }d^{-\\frac{1}{2}}\\Vert S_{\\mu ,d}\\langle \\Omega \\rangle Q(S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\varphi ,S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi )\\Vert _{L^2_tL^2_x} \\\\& \\lesssim \\sum _{d\\lesssim \\mu }d^{-\\frac{1}{2}}\\mu ^{\\frac{5}{12}}d^\\frac{1}{4} \\left(\\frac{d}{\\mu }\\right)^\\frac{1}{2} \\\\&\\qquad \\times \\left\\Vert \\left(\\sum _\\omega \\Vert \\langle \\Omega \\rangle (B^{-\\omega }_{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\varphi S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi )\\Vert ^2_{L^\\frac{3}{2}_x}\\right)^\\frac{1}{2}\\right\\Vert _{L^2_t} \\\\& \\lesssim \\sum _{d\\lesssim \\mu }\\mu ^{-\\frac{1}{12}}d^\\frac{1}{4}\\Vert S_1\\langle \\Omega \\rangle \\varphi \\Vert _{L^2_tL^6_x}\\Vert S_1\\langle \\Omega \\rangle \\phi \\Vert _{L^\\infty _tL^2_x} \\\\& \\lesssim \\mu ^\\frac{1}{6} \\Vert S_1\\varphi \\Vert _{F_\\Omega }\\Vert S_1\\phi \\Vert _{F_\\Omega }.$ We need to estimate the term $\\mathcal {H}\\mathcal {H}^3_3$ in the $Z$ space also.", "We first use Lemma REF and then the following step is quite similar.", "$\\mu \\Vert V\\mathcal {H}\\mathcal {H}^3_3\\Vert _{Z_{\\Omega ,\\mu }} & \\lesssim \\sum _{d\\lesssim \\mu }(d\\mu )^{-1}\\int \\sup _\\omega \\Vert S^\\omega _{\\mu ,d}Q(S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\varphi ,S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi )\\Vert _{L^\\infty _x}dt \\\\& \\lesssim \\sum _{d\\lesssim \\mu }(d\\mu )^{-1}\\mu ^{\\frac{8}{3}-}\\left(\\frac{d}{\\mu }\\right)^{1-}\\left(\\frac{d}{\\mu }\\right)^{\\frac{1}{2}}\\int \\Vert S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\varphi ,S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi \\Vert _{L^{\\frac{3}{2}+}_x}dt \\\\& \\lesssim \\sum _{d\\lesssim \\mu }\\mu ^{\\frac{1}{6}-}d^{\\frac{1}{2}-}\\int \\Vert S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\varphi \\Vert _{L^{3+}_x}\\Vert S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi \\Vert _{L^{3+}_x}dt \\\\& \\lesssim \\mu ^{\\frac{2}{3}-}\\Vert S_1\\varphi \\Vert _{L^2_tL^{3+}_x}\\Vert S_1\\phi \\Vert _{L^2_tL^{3+}_x} \\\\& \\lesssim \\mu ^{\\frac{2}{3}-}\\Vert S_1\\varphi \\Vert _{F_\\Omega }\\Vert S_1\\phi \\Vert _{F_\\Omega }.$" ], [ "Low$\\times $ High interaction near cone", "As the previous section, we further decompose $\\mathcal {L}\\mathcal {H}^3$ as follows: $S_{1,\\cdot \\le c\\mu }Q(S_{\\mu }\\varphi ,S_{1,\\cdot <c\\mu }\\phi ) & = \\sum _{d<c\\mu }S_{1,\\cdot \\le d}Q(S_{\\mu ,\\cdot \\le d}\\varphi ,S_{1,d}\\phi )+ \\sum _{d<c\\mu }S_{1,d}Q(S_{\\mu ,\\cdot \\le d}\\varphi ,S_{1,\\cdot \\le d}\\phi ) \\\\& \\quad + \\sum _{d\\lesssim \\mu }S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }Q(S_{\\mu ,d}\\varphi ,S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi )\\\\& =: \\mathcal {L}\\mathcal {H}^3_1+\\mathcal {L}\\mathcal {H}^3_2+\\mathcal {L}\\mathcal {H}^3_3.", "$ To deal with the term $\\mathcal {L}\\mathcal {H}^3_1$ and $\\mathcal {L}\\mathcal {H}^3_2$ , we first apply Lemma REF .", "Then we use in order the Hölder inequality with respect to $t$ , and then $x$ .", "Then we use Lemma REF and Strichartz estimates to obtain the required estimates.", "The explicit treatment of $\\mathcal {L}\\mathcal {H}^3_1,\\mathcal {L}\\mathcal {H}^3_2$ is as follows.", "$\\Vert V\\mathcal {L}\\mathcal {H}^3_1\\Vert _{\\langle \\Omega \\rangle ^{-1}Y_1} & \\lesssim \\sum _{d<c\\mu }\\left(\\frac{d}{\\mu }\\right)^\\frac{1}{2} \\left\\Vert \\sup _\\omega \\Vert S^\\omega _{\\mu ,\\cdot \\le d}\\langle \\Omega \\rangle \\varphi \\Vert _{L^\\infty _x}\\right\\Vert _{L^2_t}\\left\\Vert \\left(\\sum _\\omega \\Vert B^\\omega _{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,d}\\langle \\Omega \\rangle \\phi \\Vert _{L^2_x}^2\\right)^\\frac{1}{2}\\right\\Vert _{L^2_t} \\\\& \\lesssim \\sum _{d<c\\mu }\\mu ^{-\\frac{1}{12}}d^\\frac{3}{4} \\Vert S_{\\mu }\\langle \\Omega \\rangle \\varphi \\Vert _{L^2_tL^6_x}d^{-\\frac{1}{2}}\\Vert S_1\\phi \\Vert _{F_\\Omega } \\\\& \\lesssim \\sum _{d<c\\mu }\\mu ^{-\\frac{1}{12}}d^\\frac{1}{4}\\mu ^\\frac{5}{6}\\Vert S_\\mu \\varphi \\Vert _{F_{\\Omega ,\\mu }}\\Vert S_1\\phi \\Vert _{F_\\Omega } \\\\& \\lesssim \\mu \\Vert S_\\mu \\varphi \\Vert _{F_{\\Omega ,\\mu }}\\Vert S_1\\phi \\Vert _{F_\\Omega },$ $\\Vert V\\mathcal {L}\\mathcal {H}^3_2\\Vert _{F_{\\Omega ,1}} \\lesssim \\Vert V\\mathcal {L}\\mathcal {H}^3_2\\Vert _{\\langle \\Omega \\rangle ^{-1}X^{\\frac{1}{2},1}_1}& \\lesssim \\sum _{d<c\\mu }d^{-\\frac{1}{2}}\\left(\\frac{d}{\\mu }\\right)^\\frac{1}{2} \\\\&\\qquad \\times \\left\\Vert \\sup _\\omega \\Vert S^\\omega _{\\mu ,\\cdot \\le d}\\langle \\Omega \\rangle \\varphi \\Vert _{L^\\infty _x} \\left(\\sum _\\omega \\Vert B^\\omega _{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,\\cdot \\le d}\\langle \\Omega \\rangle \\phi \\Vert _{L^2_x}^2\\right)^\\frac{1}{2}\\right\\Vert _{L^2_t} \\\\& \\lesssim \\sum _{d<c\\mu }\\mu ^\\frac{1}{6}\\left(\\frac{d}{\\mu }\\right)^\\frac{1}{4} \\Vert S_{\\mu }\\langle \\Omega \\rangle \\varphi \\Vert _{L^2_tL^6_x}\\Vert S_1\\langle \\Omega \\rangle \\phi \\Vert _{L^\\infty _tL^2_x} \\\\& \\lesssim \\mu \\Vert S_\\mu \\varphi \\Vert _{F_{\\Omega ,\\mu }}\\Vert S_1\\phi \\Vert _{F_\\Omega }.$ We also need to estimate the term $\\mathcal {L}\\mathcal {H}^3_2$ in the $Z$ space.", "Fron now on, the proof is quite different.", "$\\Vert V\\mathcal {L}\\mathcal {H}^3_2\\Vert _{Z_{\\Omega ,1}} & = \\sum _{d\\lesssim \\mu }\\int \\sup _\\omega \\Vert B^\\omega _{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,d}V\\mathcal {L}\\mathcal {H}^3_2\\Vert _{L^\\infty _x}dt \\\\& \\lesssim \\sum _{d\\lesssim \\mu }d^{-1}\\int \\sup _\\omega \\Vert B^\\omega _{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,d}Q(S_{\\mu ,\\cdot \\le d}\\varphi ,S_{1,\\cdot \\le d}\\phi )\\Vert _{L^\\infty _x}dt \\\\& \\lesssim \\sum _{d\\lesssim \\mu }(d\\mu )^{-\\frac{1}{2}}\\int \\sup _\\omega \\Vert B^\\omega _{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,d}(S^\\omega _{\\mu ,\\cdot \\le d}\\varphi B^\\omega _{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,\\cdot \\le d})\\Vert _{L^\\infty _x}dt \\\\& \\lesssim \\sum _{d\\lesssim \\mu }(d\\mu )^{-\\frac{1}{2}}\\int \\sup _\\omega \\Vert S^\\omega _{\\mu ,\\cdot \\le d}\\varphi \\Vert _{L^\\infty _x}\\sup _\\omega \\Vert B^\\omega _{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,\\cdot \\le d}\\phi \\Vert _{L^\\infty _x}dt \\\\& \\lesssim \\sum _{d\\lesssim \\mu }(d\\mu )^{-\\frac{1}{2}} \\mu ^{\\frac{4}{3}-}\\left(\\frac{d}{\\mu }\\right)^{1-} \\int \\Vert S_{\\mu }\\varphi \\Vert _{L^{3+}_x}\\Vert S_1\\phi \\Vert _{L^{3+}_x}dt \\\\&\\lesssim \\sum _{d\\lesssim \\mu }\\mu ^{-\\frac{1}{6}}d^{\\frac{1}{2}-}\\Vert S_\\mu \\varphi \\Vert _{L^2_tL^{3+}_x}\\Vert S_1\\phi \\Vert _{L^2_tL^{3+}_x}.$ In this manner, we can obtain the required estimate only if $d\\ll \\mu ^3$ .", "On the other hand, for $\\mu ^3\\lesssim d<c\\mu $ , we have $\\mu \\lesssim (\\frac{d}{\\mu })^\\frac{1}{2}$ , and hence instead of Lemma REF , we make the use of a smaller angular decomposition with size $\\mu $ for output frequency and high input frequency.", "$& \\sum _{\\mu ^3\\lesssim d<c\\mu }(d\\mu )^{-\\frac{1}{2}}\\int \\sup _\\omega \\Vert B^{\\omega }_{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,d}(S_{\\mu ,\\cdot \\le d}\\varphi S_{1,\\cdot \\le d}\\phi )\\Vert _{L^\\infty _x}dt \\\\& \\lesssim \\sum _{\\mu ^3\\lesssim d<c\\mu }(d\\mu )^{-\\frac{1}{2}}\\int \\sup _\\omega \\Vert B^\\omega _\\mu S_{1,d}(S_{\\mu ,\\cdot \\le d}\\varphi S_{1,\\cdot \\le d}\\phi )\\Vert _{L^\\infty _x}dt \\\\& \\lesssim \\sum _{\\mu ^3\\lesssim d<c\\mu }(d\\mu )^{-\\frac{1}{2}}\\int \\sup _{\\omega _1}\\sum _{\\begin{array}{c}\|\\omega _1+\\omega _3\|\\approx \\mu \\\\ \|\\omega _2\\mp \\omega _3\|\\approx (\\frac{d}{\\mu })^\\frac{1}{2}\\end{array}}\\Vert B^{\\omega _1}_\\mu S_{1,d}(S^{\\omega _2}_{\\mu ,\\cdot \\le d} B^{\\omega _3}_\\mu S_{1,\\cdot \\le d}\\phi )\\Vert _{L^\\infty _x}dt \\\\& \\lesssim \\sum _{\\mu ^3\\lesssim d<c\\mu }\\sum _{\\omega _2,\\omega _3}(d\\mu )^{-\\frac{1}{2}}\\int \\Vert S^{\\omega _2}_{\\mu ,\\cdot \\le d}\\varphi \\Vert _{L^\\infty _x}\\Vert B^{\\omega _3}_\\mu S_{1,\\cdot \\le d}\\phi \\Vert _{L^\\infty _x}dt \\\\& \\lesssim \\sum _{\\mu ^3\\lesssim d<c\\mu }\\sum _{\\omega _2,\\omega _3}(d\\mu )^{-\\frac{1}{2}}\\mu ^{\\frac{4}{3}-}\\left(\\frac{d}{\\mu }\\right)^{\\frac{1}{2}-}\\mu ^{1-} \\Vert S^{\\omega _2}_{\\mu ,\\cdot \\le d}\\varphi \\Vert _{L^2_tL^{3+}_x}\\Vert B^{\\omega _3}_\\mu S_{1,\\cdot \\le d}\\phi \\Vert _{L^2_tL^{3+}_x} \\\\& \\lesssim \\mu ^{\\frac{4}{3}-}\\Vert S_\\mu \\varphi \\Vert _{F_{\\Omega ,\\mu }}\\Vert S_1\\phi \\Vert _{F_\\Omega }.$ Here, the summation on $d$ makes only the loss of $\\log \\mu $ and hence we get the desired estiates.", "The term $\\mathcal {L}\\mathcal {H}^3_3$ is the most crucial part of our proof.", "We write $\\Vert V\\mathcal {L}\\mathcal {H}^3_3\\Vert _{F_{\\Omega ,1}} & \\lesssim \\sum _{d\\lesssim \\mu }\\Vert S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }Q(S_{\\mu ,d}\\varphi ,S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\langle \\Omega \\rangle \\phi )\\Vert _{L^1_tL^2_x} \\\\& \\quad + \\sum _{d\\lesssim \\mu }\\Vert S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }Q(S_{\\mu ,d}\\langle \\Omega \\rangle \\varphi ,S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi )\\Vert _{L^1_tL^2_x} \\\\& =: \\mathcal {J}_{1}+\\mathcal {J}_{2}.", "$ The term $\\mathcal {J}_1$ is rather easier than $\\mathcal {J}_2$ .", "We simply recall the property of the $Z$ space (REF ).", "$\\mathcal {J}_{1} & \\lesssim \\sum _{d\\lesssim \\mu }\\left(\\frac{d}{\\mu }\\right)^\\frac{1}{2}\\left\\Vert \\sup _\\omega \\Vert S^\\omega _{\\mu ,d}\\varphi \\Vert _{L^\\infty _x}\\right\\Vert _{L^1_t}\\Vert S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\langle \\Omega \\rangle \\phi \\Vert _{L^\\infty _tL^2_x} \\\\& \\lesssim \\mu \\Vert S_\\mu \\varphi \\Vert _{F_{\\Omega ,\\mu }}\\Vert S_1\\phi \\Vert _{F_\\Omega }.$ Now we are left to consider the $\\mathcal {J}_2$ .", "We further decompose the range of $d $ into $d\\ll \\mu ^3$ and $\\mu ^3\\lesssim d\\lesssim \\mu $ .", "If $d\\ll \\mu ^3$ , then we apply Lemma REF , Hölder inequality and then Sobolev estimates.", "$\\mathcal {J}_{2}^{d\\ll \\mu ^3} & \\lesssim \\sum _{d\\ll \\mu ^3}\\left(\\frac{d}{\\mu }\\right)^\\frac{1}{2} \\left\\Vert \\left(\\sum _\\omega \\Vert S^\\omega _{\\mu ,d}\\langle \\Omega \\rangle \\varphi \\Vert _{L^4_x}^2\\right)^\\frac{1}{2}\\right\\Vert _{L^2_t}\\left\\Vert \\sup _\\omega \\Vert B^\\omega _{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi \\Vert _{L^4_x}\\right\\Vert _{L^2_t} \\\\& \\lesssim \\sum _{d\\ll \\mu ^3}\\left(\\frac{d}{\\mu }\\right)^\\frac{1}{2}\\mu \\left(\\frac{d}{\\mu }\\right)^\\frac{3}{8}\\Vert S_{\\mu ,d}\\langle \\Omega \\rangle \\varphi \\Vert _{L^2_tL^2_x}\\left(\\frac{d}{\\mu }\\right)^{\\frac{1}{8}-}\\left\\Vert \\sup _\\omega \\Vert B^\\omega _{(\\frac{d}{\\mu })^\\frac{1}{2}}S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi \\Vert _{L^{3+}_x}\\right\\Vert _{L^2_t} \\\\& \\lesssim \\sum _{d\\ll \\mu ^3}\\mu ^\\frac{1}{2}\\left(\\frac{d}{\\mu }\\right)^{\\frac{1}{2}-}\\Vert S_\\mu \\varphi \\Vert _{F_{\\Omega ,\\mu }}\\Vert S_1\\phi \\Vert _{L^2_tL^{3+}_x}.$ Thus we can get the desired estimate for $d\\ll \\mu ^3$ .", "On the other hand, if $\\mu ^3\\lesssim d\\lesssim \\mu $ , we write $\\mathcal {J}_{2}^{\\mu ^3\\lesssim d\\lesssim \\mu } = \\sum _{d\\lesssim \\mu }\\left\\Vert \\sum _{\\begin{array}{c}\|\\omega _1+\\omega _3\|\\approx \\mu \\\\ \|\\omega _2\\mp \\omega _3\|\\approx (\\frac{d}{\\mu })^\\frac{1}{2}\\end{array}}B^{\\omega _1}_\\mu S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }Q(S^{\\omega _2}_{\\mu ,d}\\langle \\Omega \\rangle \\varphi ,B^{\\omega _3}_\\mu S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi )\\right\\Vert _{L^1_tL^2_x}.$ We rewrite the $L^2_x$ norm via duality and then use Hölder inequality.", "$\\Vert \\cdots \\Vert _{L^2_x} \\lesssim \\sup _{\\Vert h\\Vert _{L^2_x}=1}\\left(\\frac{d}{\\mu }\\right)^\\frac{1}{2}\\sum _{\\omega _2}\\Vert S^{\\omega _2}_{\\mu ,d}\\langle \\Omega \\rangle \\varphi \\Vert _{L^2_x}\\left\\Vert \\sum _{\\omega _1,\\omega _2,\\omega _3}B^{\\omega _2}_{(\\frac{d}{\\mu })^\\frac{1}{2}}P_{\\cdot \\lesssim \\mu }(B^{\\omega _3}_\\mu S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi B^{\\omega _1}_\\mu P_{\\cdot \\lesssim 1}h)\\right\\Vert _{L^2_x}.$ Then $\\mathcal {J}_{2}^{\\mu ^3\\lesssim d\\lesssim \\mu } & \\lesssim \\sum _{d\\lesssim \\mu }\\left(\\frac{d}{\\mu }\\right)^\\frac{1}{2}d^{-\\frac{1}{2}}d^{\\frac{1}{2}}\\Vert S_{\\mu ,d}\\langle \\Omega \\rangle \\varphi \\Vert _{L^2_tL^2_x}\\mu ^{\\frac{4}{3}-}\\left(\\frac{d}{\\mu }\\right)^{\\frac{1}{2}-}\\left\\Vert \\sup _\\omega \\Vert B^{\\omega }_\\mu S_{1,\\cdot \\le \\min \\lbrace d,c\\mu \\rbrace }\\phi \\Vert _{L^{3+}_x}\\right\\Vert _{L^2_t} \\\\& \\lesssim \\sum _{d\\lesssim \\mu }\\mu ^{\\frac{5}{6}-}\\left(\\frac{d}{\\mu }\\right)^{\\frac{1}{2}-}\\mu ^\\frac{1}{2}\\Vert S_1\\langle \\Omega \\rangle ^\\frac{1}{2}\\phi \\Vert _{L^2_tL^{3+}_x}\\Vert S_\\mu \\varphi \\Vert _{F_{\\Omega ,\\mu }} \\\\& \\lesssim \\mu ^{\\frac{4}{3}-}\\Vert S_\\mu \\varphi \\Vert _{F_{\\Omega ,\\mu }}\\Vert S_1\\phi \\Vert _{F_\\Omega }.", "$ Remark 5.1 In the Low$\\times $ High regime, the most difficult interaction is when the low frequency controls the angular regularity.", "In this case, we cannot exploit the angular concentration estimates to gain some positive power of $\\mu $ .", "Hence we only use the Sobolev embedding estimates." ], [ "Acknowledgements", "The author is supported by NRF-2018R1D1A3B07047782 and NRF-2016K2A9A2A13003815." ] ]	2011.14095
[ [ "A Morley-Wang-Xu element method for a fourth order elliptic singular\n perturbation problem" ], [ "Abstract A Morley-Wang-Xu (MWX) element method with a simply modified right hand side is proposed for a fourth order elliptic singular perturbation problem, in which the discrete bilinear form is standard as usual nonconforming finite element methods.", "The sharp error analysis is given for this MWX element method.", "And the Nitsche's technique is applied to the MXW element method to achieve the optimal convergence rate in the case of the boundary layers.", "An important feature of the MWX element method is solver-friendly.", "Based on a discrete Stokes complex in two dimensions, the MWX element method is decoupled into one Lagrange element method of Poisson equation, two Morley element methods of Poisson equation and one nonconforming $P_1$-$P_0$ element method of Brinkman problem, which implies efficient and robust solvers for the MWX element method.", "Some numerical examples are provided to verify the theoretical results." ], [ "Introduction", "In this paper, we shall apply the Morley-Wang-Xu (MWX) element [25], [36] to discretize the fourth order elliptic singular perturbation problem ${\\left\\lbrace \\begin{array}{ll}\\varepsilon ^2\\Delta ^{2}u-\\Delta u=f \\quad \\;\\; \\textrm {in}~\\Omega , \\\\u=\\partial _nu=0 \\quad \\quad \\quad \\textrm {on}~\\partial \\Omega ,\\end{array}\\right.", "}$ where $\\Omega \\subset \\mathbb {R}^d$ with $d\\ge 2$ is a convex and bounded polytope, $f\\in L^2(\\Omega )$ , $n$ is the unit outward normal to $\\partial \\Omega $ , and $\\varepsilon $ is a real small and positive parameter.", "The MWX element is the simplest finite element for fourth order problems, as it has the fewest number of degrees of freedom on each element.", "The generalization of the MWX element to any star-shaped polytope in any dimension is recently developed in the context of the virtual element in [12].", "However it is divergent to discretize problem (REF ) by the MWX element in the following standard way when $\\varepsilon $ is very close to 0 [26], [32]: find $u_{h0} \\in V_{h0} $ such that $\\varepsilon ^2 (\\nabla _h^{2}u_{h0}, \\nabla _h^{2}v_h)+(\\nabla _h u_{h0}, \\nabla _h v_h)=(f, v_h) \\quad \\forall ~v_h \\in V_{h0},$ where $V_{h0}$ is the global MWX element space.", "To this end, a modified MWX element method was advanced in [38], [33] to deal with this divergence by replacing $(\\nabla _h u_{h0}, \\nabla _h v_h)$ with $(\\nabla \\Pi _h u_{h0}, \\nabla \\Pi _h v_h)$ , where $\\Pi _h$ is an interpolation operator from $V_{h0}$ to some lower-order $H^1$ -conforming finite element space.", "Instead of introducing the interpolation operator, the combination of the MWX element and the interior penalty discontinuous Galerkin formulation [2] is proposed to discretize problem (REF ) in [39].", "Both modified Morley element methods in [38], [33], [39] uniformly converge with respect to the parameter $\\varepsilon $ .", "Apart from the MWX element, there are many other $H^2$ -nonconforming elements constructed to design robust numerical methods for problem (REF ), including $C^0$ $H^2$ -nonconforming elements in [26], [30], [21], [31], [8], [7], [40], [9], [34], [35] and fully $H^2$ -nonconforming elements in [14], [13], [31].", "And a $C^0$ interior penalty discontinuous Galerkin (IPDG) method with the Lagrange element space was devised for problem (REF ) in [3], [17].", "We refer to [29], [1], [42] for the $H^2$ -conforming finite element methods of problem (REF ), which usually suffer from large number of degrees of freedom.", "To design a simple finite element method for problem (REF ), we still employ the MWX element space and the standard discrete bilinear formulation as the left hand side of the discrete method (REF ) in this paper.", "We simply replace the right hand side $(f, v_h)$ by $(f, P_hv_h)$ , where $P_h$ is the $H^1$ -orthogonal projector onto the $H^1$ -conforming $\\ell $ th order Lagrange element space $W_h$ with $\\ell =1,2$ .", "In a word, we propose the following robust MWX element method for problem (REF ): find $u_{h0} \\in V_{h0} $ such that $\\varepsilon ^2 (\\nabla _h^{2}u_{h0}, \\nabla _h^{2}v_h)+(\\nabla _h u_{h0}, \\nabla _h v_h)=(f, P_hv_h) \\quad \\forall ~v_h \\in V_{h0}.$ The stiffness matrix of the discrete method (REF ) can be assembled in a standard way, which is sparser than that of the discontinuous Galerkin methods, such as [39].", "After establishing the interpolation error estimate and consistency error estimate, the optimal convergence rate $O(h)$ of the energy error is achieved.", "And the discrete method (REF ) possesses the sharp and uniform convergence rate $O(h^{1/2})$ of the energy error in consideration of the boundary layers.", "An important feature of the discrete method (REF ) is solver-friendly.", "First the discrete method (REF ) is equivalent to find $w_h\\in W_h$ and $u_{h0}\\in V_{h0}$ such that $(\\nabla w_h, \\nabla \\chi _h) & =(f, \\chi _h) \\quad \\quad \\quad \\quad \\; \\forall ~\\chi _h\\in W_h, \\\\\\varepsilon ^2 a_h(u_{h0},v_h)+b_h(u_{h0},v_h) & =(\\nabla w_h, \\nabla _h v_h) \\quad \\;\\; \\forall ~v_h\\in V_{h0}.", "$ Especially in two dimensions, thanks to the relationship between the Morley element space $V_{h0}$ and the vectorial nonconforming $P_1$ element space $V_{h0}^{CR}$ (cf.", "[16]), the discrete method () can be decoupled into two Morley element methods of Poisson equation and one nonconforming $P_1$ -$P_0$ element method of Brinkman problem, i.e., find $(z_h, \\phi _h, p_h, w_h)\\in V_{h0}\\times V_{h0}^{CR}\\times \\mathcal {Q}_{h}\\times V_{h0}$ such that $(\\operatorname{curl}_h z_h, \\operatorname{curl}_h v_h) & =(\\nabla w_h, \\nabla _h v_h)\\quad \\, \\forall ~v_h\\in V_{h0}, \\\\(\\phi _h, \\psi _h)+\\varepsilon ^2(\\nabla _h\\phi _h, \\nabla _h\\psi _h) + (\\operatorname{div}_h\\psi _h, p_h) & =(\\operatorname{curl}_h z_h, \\psi _h)\\quad \\forall ~\\psi _h\\in V_{h0}^{CR}, \\\\(\\operatorname{div}_h\\phi _h, q_h) & = 0\\quad \\quad \\quad \\quad \\quad \\quad \\, \\forall ~q_h\\in \\mathcal {Q}_{h}, \\\\(\\operatorname{curl}_h u_{h0}, \\operatorname{curl}_h \\chi _h) & = (\\phi _h, \\operatorname{curl}_h\\chi _h) \\quad \\forall ~\\chi _h\\in V_{h0}.", "$ When $\\varepsilon $ is small, the discrete method () can be easily solved by the the conjugate gradient (CG) method with the auxiliary space preconditioner [41].", "The decoupling (REF )-() will induce efficient and robust solvers for the MWX element method (REF ) for large $\\varepsilon $ .", "The Lagrange element method of Poisson equation (REF ), and the Morley element methods of Poisson equation (REF ) and () can be solved by the CG method with the auxiliary space preconditioner, in which the $H^1$ conforming linear element discretization on the same mesh for the Poisson equation can be adopted as the auxiliary problem.", "And the algebraic multigrid (AMG) method is used to solve the auxiliary problem.", "As for the nonconforming $P_1$ -$P_0$ element methods of Brinkman problem ()-(), we can use the block-diagonal preconditioner in [27], [24], [6] or the approximate block-factorization preconditioner in [10], which are robust with respect to the mesh size $h$ .", "The resulting fast solver of the MWX element method (REF ) also works for the shape-regular unstructured meshes.", "When $\\varepsilon $ is close to zero, the uniform convergence rate $O(h^{1/2})$ of the energy error of the discrete method (REF ) is sharp but not optimal, where the optimal convergence rate should be $O(h^{\\ell })$ for $\\ell =1,2$ .", "To promote the convergence rate in the case of the boundary layers, we apply the Nitsche's technique in [21] to the discrete method (REF ), i.e.", "impose the boundary condition $\\partial _nu=0$ weakly by the penalty technique [2].", "The optimal error analysis is present for the resulting discrete method, whose convergence rate is uniform with respect to the perturbation parameter $\\varepsilon $ when $\\varepsilon $ approaches zero.", "Similarly, as (REF )-(), the discrete method with Nitsche's technique on the boundary can also be decoupled into one Lagrange element method of Poisson equation, two Morley element methods of Poisson equation and one nonconforming $P_1$ -$P_0$ element method with Nitsche's technique of Brinkman problem, which is solver-friendly.", "The rest of this paper is organized as follows.", "Some notations, connection operators and interpolation operators are shown in section 2.", "In section 3, we develop and analyze the MWX element method, and the MWX element method with Nitsche's technique is devised and analyzed in section 4.", "Section 5 focuses on the equivalent decoupling of the MWX element methods.", "Finally, some numerical results are given in section 6 to confirm the theoretical results." ], [ "Notation", "Given a bounded domain $G\\subset \\mathbb {R}^{d}$ and a non-negative integer $m$ , let $H^m(G)$ be the usual Sobolev space of functions on $G$ , and $H^m(G;\\mathbb {X})$ the Sobolev space of functions taking values in the finite-dimensional vector space $\\mathbb {X}$ for $\\mathbb {X}$ being $\\mathbb {R}^d$ or $\\mathbb {M}$ , where $\\mathbb {M}$ is the space of all $d\\times d$ tensors.", "The corresponding norm and semi-norm are denoted respectively by $\\Vert \\cdot \\Vert _{m,G}$ and $\|\\cdot \|_{m,G}$ .", "Let $(\\cdot , \\cdot )_G$ be the standard inner product on $L^2(G)$ or $L^2(G;\\mathbb {X})$ .", "If $G$ is $\\Omega $ , we abbreviate $\\Vert \\cdot \\Vert _{m,G}$ , $\|\\cdot \|_{m,G}$ and $(\\cdot , \\cdot )_G$ by $\\Vert \\cdot \\Vert _{m}$ , $\|\\cdot \|_{m}$ and $(\\cdot , \\cdot )$ , respectively.", "Let $H_0^m(G)$ be the closure of $C_{0}^{\\infty }(G)$ with respect to the norm $\\Vert \\cdot \\Vert _{m,G}$ .", "Let $\\mathbb {P}_m(G)$ stand for the set of all polynomials in $G$ with the total degree no more than $m$ , and $\\mathbb {P}_m(G;\\mathbb {R}^d)$ the vectorial version of $\\mathbb {P}_m(G)$ .", "As usual, $\|G\|$ denotes the measure of a given open set $G$ .", "For any finite set $\\mathcal {S}$ , denote by $\\#\\mathcal {S}$ the cardinality of $\\mathcal {S}$ .", "We partition the domain $\\Omega $ into a family of shape regular simplicial grids $\\mathcal {T}_h$ (cf.", "[4], [15]) with $h:=\\max \\limits _{K\\in \\mathcal {T}_h}h_K$ and $h_K:=\\mbox{diam}(K)$ .", "Let $\\mathcal {F}_h$ be the union of all $d-1$ dimensional faces of $\\mathcal {T}_h$ , $\\mathcal {F}^i_h$ the union of all interior $d-1$ dimensional faces of the triangulation $\\mathcal {T}_h$ , and $\\mathcal {F}^{\\partial }_h:=\\mathcal {F}_h\\backslash \\mathcal {F}^i_h$ .", "Similarly, let $\\mathcal {E}_h$ be the union of all $d-2$ dimensional faces of $\\mathcal {T}_h$ , $\\mathcal {E}^i_h$ the union of all interior $d-2$ dimensional faces of the triangulation $\\mathcal {T}_h$ , and $\\mathcal {E}^{\\partial }_h:=\\mathcal {E}_h\\backslash \\mathcal {E}^i_h$ .", "Set $\\mathcal {F}^i(K):=\\lbrace F\\in \\mathcal {F}^i_h: F\\subset \\partial K\\rbrace , \\quad \\mathcal {F}^{\\partial }(K):=\\lbrace F\\in \\mathcal {F}_h^{\\partial }: F\\subset \\partial K\\rbrace ,$ $\\mathcal {E}(K):=\\lbrace e\\in \\mathcal {E}_h: e\\subset \\partial K\\rbrace .$ For each $K\\in \\mathcal {T}_h$ , denote by $n_K$ the unit outward normal to $\\partial K$ .", "Without causing any confusion, we will abbreviate $n_K$ as $n$ for simplicity.", "For each $F\\in \\mathcal {F}_h$ , denote by $h_F$ its diameter and fix a unit normal vector $n_F$ such that $n_F=n_K$ if $F\\in \\mathcal {F}^{\\partial }(K)$ .", "In two dimensions, i.e.", "$d=2$ , we use $t_F$ to denote the unit tangential vector of $F$ if $F\\in \\mathcal {F}^{\\partial }_h$ , and abbreviate it as $t$ for simplicity.", "For $s\\ge 1$ , define $H^s(\\mathcal {T}_h):=\\lbrace v\\in L^2(\\Omega ): v\|_K\\in H^s(K)\\quad \\forall ~K\\in \\mathcal {T}_h\\rbrace .$ For any $v\\in H^s(\\mathcal {T}_h)$ , define the broken $H^s$ norm and seminorm $\\Vert v\\Vert _{s,h}^2:=\\sum _{K \\in \\mathcal {T}_h} \\Vert v\\Vert _{s, K}^2, \\quad \|v\|_{s,h}^2:=\\sum _{K \\in \\mathcal {T}_h} \|v\|_{s, K}^2.$ For any $v\\in H^2(\\mathcal {T}_h)$ , introduce some other discrete norms $\\interleave v\\interleave _{2,h}^2:= \|v\|_{2, h}^2+\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }} h_F^{-1} \\Vert \\partial _nv\\Vert _{0,F}^2 ,\\\\\\Vert v \\Vert _{\\varepsilon ,h}^2:=\\varepsilon ^2 \|v\|_{2,h}^2+\|v\|_{1,h}^2, \\quad \\interleave v\\interleave _{\\varepsilon ,h}^2:=\\varepsilon ^2\\interleave v\\interleave _{2,h}^2+\|v\|_{1,h}^2 .$ For any face $F\\in \\mathcal {F}_h$ , set $\\partial ^{-1}F :=\\lbrace K\\in \\mathcal {T}_h: F\\subset \\partial K\\rbrace , \\quad \\omega _F:=\\textrm {interior}\\left(\\bigcup _{K\\in \\partial ^{-1}F}\\overline{K}\\right).$ For any simplex $K\\in \\mathcal {T}_h$ , denote $\\mathcal {T}_K:=\\lbrace K^{\\prime }\\in \\mathcal {T}_h: \\overline{K^{\\prime }}\\cap \\overline{K}\\ne \\varnothing \\rbrace , \\quad \\omega _K:=\\textrm {interior}\\left(\\bigcup _{K^{\\prime }\\in \\mathcal {T}_K}\\overline{K^{\\prime }}\\right),$ $\\omega _{K}^2:=\\textrm {interior}\\left(\\bigcup \\lbrace \\overline{\\omega _{K^{\\prime }}}\\in \\mathcal {T}_h: \\overline{K^{\\prime }}\\cap \\overline{K}\\ne \\varnothing \\rbrace \\right).$ Discrete differential operators $\\nabla _h$ , $\\operatorname{curl}_h$ and $\\operatorname{div}_h$ are defined as the elementwise counterparts of $\\nabla $ , $\\operatorname{curl}$ and $\\operatorname{div}$ associated with $\\mathcal {T}_h$ respectively.", "Throughout this paper, we also use “$\\lesssim \\cdots $ \" to mean that “$\\le C\\cdots $ \", where $C$ is a generic positive constant independent of $h$ and the parameter $\\varepsilon $ , which may take different values at different appearances.", "Moreover, we introduce averages and jumps on $d-1$ dimensional faces as in [22].", "Consider two adjacent simplices $K^+$ and $K^-$ sharing an interior face $F$ .", "Denote by $n^+$ and $n^-$ the unit outward normals to the common face $F$ of the simplices $K^+$ and $K^-$ , respectively.", "For a scalar-valued or vector-valued function $v$ , write $v^+:=v\|_{K^+}$ and $v^-:=v\|_{K^-}$ .", "Then define the average and jump on $F$ as follows: $\\lbrace v\\rbrace :=\\frac{1}{2}(v^++v^-),\\quad \\llbracket v\\rrbracket :=v^+n_F\\cdot n^++v^-n_F\\cdot n^-.$ On a face $F$ lying on the boundary $\\partial \\Omega $ , the above terms are defined by $\\lbrace v\\rbrace :=v, \\quad \\llbracket v\\rrbracket :=vn_F\\cdot n.$ Associated with the partition $\\mathcal {T}_h$ , the global Morley-Wang-Xu (MWX) element space $\\widetilde{V}_h$ consists of all piecewise quadratic functions on $\\mathcal {T}_h$ such that, their integral average over each $(d-2)$ -dimensional face of elements in $\\mathcal {T}_h$ are continuous, and their normal derivatives are continuous at the barycentric point of each $(d-1)$ -dimensional face of elements in $\\mathcal {T}_h$ (cf.", "[36], [25], [37]).", "And define $V_{h} :=\\left\\lbrace v\\in \\widetilde{V}_{h}: \\int _{e}v\\,{\\rm d}s=0\\quad \\forall \\,e\\in \\mathcal {E}^{\\partial }_h\\right\\rbrace ,$ $V_{h0} :=\\left\\lbrace v\\in V_{h}: \\int _{F}\\partial _nv\\,{\\rm d}s=0\\quad \\forall \\,F\\in \\mathcal {F}^{\\partial }_h\\right\\rbrace .$ Notice that we do not impose the boundary condition $\\int _{F}\\partial _nv\\,{\\rm d}s=0$ in the finite element space $V_h$ .", "Due to Lemma 4 in [36], we have $\\int _{F}\\llbracket \\nabla v_h\\rrbracket \\,{\\rm d}s=0\\quad \\forall ~v_h\\in V_h, F\\in \\mathcal {F}_h^i,$ $\\int _{F}\\llbracket \\nabla _F v_h\\rrbracket \\,{\\rm d}s=0\\quad \\forall ~v_h\\in V_h, F\\in \\mathcal {F}_h,$ $\\int _{F}\\llbracket \\nabla v_h\\rrbracket \\,{\\rm d}s=0\\quad \\forall ~v_h\\in V_{h0}, F\\in \\mathcal {F}_h,$ where the surface gradient $\\nabla _F v_h:=\\nabla v_h - \\partial _{n_F}v_hn_F$ ." ], [ "Connection operators", "In this subsection we will introduce some operators to connect the Lagrange element space and the MWX element space for analysis.", "Let the Lagrange element space $W_{h} :=\\left\\lbrace v\\in H_0^{1}(\\Omega ): v\|_K\\in P_{\\ell }(K)\\quad \\forall \\,K\\in \\mathcal {T}_h\\right\\rbrace $ with $\\ell =1$ or 2.", "Define a connection operator $E_h^L: V_h\\rightarrow W_h$ with $\\ell =2$ as follows: Given $v_h\\in V_h$ , $E_h^Lv_h\\in W_h$ is determined by $N(E_h^Lv_h):=\\frac{1}{\\#\\mathcal {T}_N}\\sum _{K\\in \\mathcal {T}_N}N(v_h\|_K)$ for each interior degree of freedom $N$ of the space $W_h$ , where $\\mathcal {T}_N\\subset \\mathcal {T}_h$ denotes the set of simplices sharing the degree of freedom $N$ .", "By the weak continuity of $V_{h0}$ and $V_h$ and the techniques adopted in [32], [5], we have for any $s=1, 2$ , $0\\le m\\le s$ and $j=0,1,2$ that $\|v_h-E_h^Lv_h\|_{m,K}\\lesssim h_K^{s-m}\|v_h\|_{s, \\omega _K}\\quad \\forall ~v_h\\in V_{h0},$ $\\Vert v_h-E_h^Lv_h\\Vert _{0,K}+h_K\|v_h-E_h^Lv_h\|_{1,K}\\lesssim h_K\|v_h\|_{1, \\omega _K}\\quad \\forall ~v_h\\in V_{h},$ $\|v_h-E_h^Lv_h\|_{j,K}\\lesssim h_K^{2-j}\\Big (\|v_h\|_{2, \\omega _K}+\\sum _{K^{\\prime }\\in \\mathcal {T}_K}\\sum _{F \\in \\mathcal {F}^{\\partial }(K^{\\prime })} h_F^{-1/2} \\Vert \\partial _nv_h\\Vert _{0,F}\\Big )\\;\\;\\forall ~v_h\\in V_{h}$ for each $K\\in \\mathcal {T}_h$ .", "Then we get $\|v_h-E_h^Lv_h\|_{1,h}\\lesssim \\min \\lbrace \|v_h\|_{1,h},h^{1/2}\|v_h\|_{1,h}^{1/2}\|v_h\|_{2,h}^{1/2}, h\|v_h\|_{2,h}\\rbrace \\quad \\forall ~v_h\\in V_{h0},$ $\|v_h-E_h^Lv_h\|_{1,h}\\lesssim \\min \\lbrace \|v_h\|_{1,h},h^{1/2}\|v_h\|_{1,h}^{1/2}\\interleave v\\interleave _{2,h}^{1/2}, h\\interleave v\\interleave _{2,h}\\rbrace \\quad \\forall ~v_h\\in V_h.$ To define interpolation operators later, we also need another two connection operators $E_h: W_h\\rightarrow V_{h}$ and $E_{h0}: W_h\\rightarrow V_{h0}$ .", "For any $v_h\\in W_h$ , $E_hv_h\\in V_{h}$ is determined by $\\int _e E_hv_h\\,{\\rm d}s=\\int _ev_h\\,{\\rm d}s\\quad \\forall ~e\\in \\mathcal {E}_h^i,$ $\\int _F\\partial _{n_F}( E_hv_h)\\,{\\rm d}s=\\int _F\\lbrace \\partial _{n_F}v_h\\rbrace \\,{\\rm d}s\\quad \\forall ~F\\in \\mathcal {F}_h.$ And $E_{h0}v_h\\in V_{h0}$ is determined by $\\int _e E_{h0}v_h\\,{\\rm d}s=\\int _ev_h\\,{\\rm d}s\\quad \\forall ~e\\in \\mathcal {E}_h^i,$ $\\int _F\\partial _{n_F}( E_{h0}v_h)\\,{\\rm d}s=\\int _F\\lbrace \\partial _{n_F}v_h\\rbrace \\,{\\rm d}s\\quad \\forall ~F\\in \\mathcal {F}_h^i.$" ], [ "Interpolation operators", "Let $I_h^{SZ}$ be the Scott-Zhang interpolation operator [28] from $H_0^1(\\Omega )$ onto $W_h$ with $\\ell =2$ .", "For any $1\\le s\\le 3$ and $0\\le m\\le s$ , it holds (cf.", "[28]) $\|v-I_h^{SZ}v\|_{m,K}\\lesssim h_K^{s-m}\|v\|_{s, \\omega _K}\\quad \\forall ~v\\in H_0^1(\\Omega )\\cap H^s(\\Omega ),\\; K\\in \\mathcal {T}_h.$ Then define two quasi-interpolation operators $I_h: H_0^1(\\Omega )\\rightarrow V_{h}$ and $I_{h0}: H_0^1(\\Omega )\\rightarrow V_{h0}$ as $I_h:=E_hI_h^{SZ},\\quad I_{h0}:=E_{h0}I_h^{SZ}.$ Next we will derive the error estimates of the interpolation operators $I_h$ and $I_{h0}$ following the argument in [21].", "Lemma 2.1 Let $2\\le s\\le 3$ and $0\\le m\\le s$ .", "We have $\|v-I_{h0}v\|_{m,h}\\lesssim h^{s-m}\|v\|_s\\quad \\forall ~v\\in H^s(\\Omega )\\cap H_0^2(\\Omega ),$ $\|v-I_{h0}v\|_{1, h}^2\\lesssim h\|v\|_1\|v\|_2\\quad \\forall ~v\\in H_0^2(\\Omega ),$ $\|v-I_hv\|_{m,h}\\lesssim h^{s-m}\|v\|_s\\quad \\forall ~v\\in H^s(\\Omega )\\cap H_0^1(\\Omega ),$ $\|v-I_hv\|_{1, h}^2\\lesssim h\|v\|_1\|v\|_2\\quad \\forall ~v\\in H^2(\\Omega )\\cap H_0^1(\\Omega ),$ $\|v-I_hv\|_{1, h}\\lesssim \|v\|_1\\quad \\forall ~v\\in H_0^1(\\Omega ).$ We only prove the inequalities (REF )-(REF ).", "The inequalities (REF )-(REF ) and (REF ) can be achieved by the same argument.", "Take any $K\\in \\mathcal {T}_h$ .", "By the definition of $E_h$ , we have $\\int _e(I_h^{SZ}v-E_hI_h^{SZ}v)\|_K\\,{\\rm d}s=0\\quad \\forall ~e\\in \\mathcal {E}(K),$ $\\int _F\\partial _{n_F}((I_h^{SZ}v-E_hI_h^{SZ}v)\|_K)\\,{\\rm d}s=\\frac{n_F\\cdot n_K}{2}\\int _F\\llbracket \\partial _{n_F}(I_h^{SZ}v)\\rrbracket \\,{\\rm d}s\\quad \\forall ~F\\in \\mathcal {F}^i(K),$ $\\int _F\\partial _{n_F}((I_h^{SZ}v-E_hI_h^{SZ}v)\|_K)\\,{\\rm d}s=0\\quad \\forall ~F\\in \\mathcal {F}^{\\partial }(K).$ Applying the inverse inequality, scaling argument and Cauchy-Schwarz inequality, it follows $\|I_h^{SZ}v-E_hI_h^{SZ}v\|_{m, K} & \\le h_K^{-m}\\Vert I_h^{SZ}v-E_hI_h^{SZ}v\\Vert _{0, K} \\\\& \\lesssim h_K^{2-d/2-m}\\sum _{F\\in \\mathcal {F}^i(K)}\\left\|\\int _F\\llbracket \\partial _{n_F}(I_h^{SZ}v)\\rrbracket \\,{\\rm d}s\\right\| \\\\& \\lesssim h_K^{3/2-m}\\sum _{F\\in \\mathcal {F}^i(K)}\\left\\Vert \\llbracket \\partial _{n_F}(I_h^{SZ}v-v)\\rrbracket \\right\\Vert _{0,F},$ which together with the trace inequality and (REF ) implies $\|I_h^{SZ}v-E_hI_h^{SZ}v\|_{m, K} & \\lesssim h_K^{1-m} \\sum _{F\\in \\mathcal {F}^i(K)}\\sum _{K^{\\prime }\\in \\partial ^{-1}F}(\|v-I_h^{SZ}v\|_{1,K^{\\prime }}+h_K\|v-I_h^{SZ}v\|_{2,K^{\\prime }}) \\\\& \\lesssim h_K^{s-m}\|v\|_{s,\\omega _{K}^2}.$ Employing (REF ), we get $\|v-I_hv\|_{m, K}\\le \|v-I_h^{SZ}v\|_{m, K}+\|I_h^{SZ}v-E_hI_h^{SZ}v\|_{m, K}\\lesssim h_K^{s-m}\|v\|_{s,\\omega _{K}^2},$ which indicates (REF ).", "Let $\\hat{K}$ be the reference element of $K$ , then it holds (cf.", "[19]) $\\Vert w\\Vert _{0,\\partial \\hat{K}}^2\\lesssim \\Vert w\\Vert _{0,\\hat{K}}\\Vert w\\Vert _{1,\\hat{K}}\\quad \\forall ~w\\in H^1(\\hat{K}).$ We obtain from (REF ), scaling argument and (REF ) that $& \\quad \\;\\;\|I_h^{SZ}v-E_hI_h^{SZ}v\|_{1, K}^2\\lesssim h_K\\sum _{F\\in \\mathcal {F}^i(K)}\\left\\Vert \\llbracket \\partial _{n_F}(I_h^{SZ}v-v)\\rrbracket \\right\\Vert _{0,F}^2 \\\\& \\lesssim \\sum _{F\\in \\mathcal {F}^i(K)}\\sum _{K^{\\prime }\\in \\partial ^{-1}F}\|v-I_h^{SZ}v\|_{1,K^{\\prime }}(\|v-I_h^{SZ}v\|_{1,K^{\\prime }}+h_K\|v-I_h^{SZ}v\|_{2,K^{\\prime }}) \\\\& \\lesssim h_K\|v\|_{1,\\omega _{K}^2}\|v\|_{2,\\omega _{K}^2}.$ Finally we achieve (REF ) from the last inequality, triangle inequality and (REF ).", "Let $u^0\\in H_0^1(\\Omega )$ be the solution of the Poisson equation ${\\left\\lbrace \\begin{array}{ll}-\\Delta u^0=f \\quad \\;\\; \\textrm {in}~\\Omega , \\\\u^0=0 \\quad \\quad \\quad \\; \\textrm {on}~\\partial \\Omega .\\end{array}\\right.", "}$ Since the domain $\\Omega $ is convex, we have the following regularities [26], [21], [19] $\|u\|_2+\\varepsilon \|u\|_3\\lesssim \\varepsilon ^{-1/2}\\Vert f\\Vert _0,$ $\|u-u^0\|_1\\lesssim \\varepsilon ^{1/2}\\Vert f\\Vert _0,$ $\\Vert u^0\\Vert _2\\lesssim \\Vert f\\Vert _0.$ Lemma 2.2 We have $\\interleave u-I_hu\\interleave _{\\varepsilon ,h}\\lesssim (\\varepsilon h+ h^{2})\|u\|_3.$ If $u^0\\in H_0^1(\\Omega )\\cap H^s(\\Omega )$ with $2\\le s\\le 3$ , it holds $\\interleave u-I_hu\\interleave _{\\varepsilon ,h}\\lesssim \\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}\\rbrace \\Vert f\\Vert _0 +h^{s-1}\|u^0\|_{s}.$ The inequality (REF ) follows from (REF ) and the trace inequality.", "Next we prove (REF ).", "By (REF ), we have $\|u-u^0-I_h(u-u^0)\|_{1,h}\\lesssim \|u-u^0\|_1.$ Due to (REF ), it follows $\|u-u^0-I_h(u-u^0)\|_{1,h}^2\\lesssim h\|u-u^0\|_1\|u-u^0\|_2.$ Combining the last two inequality, we get from (REF )-(REF ) that $\|u-u^0-I_h(u-u^0)\|_{1,h}^2 & \\lesssim \\min \\lbrace \|u-u^0\|_1^2, h\|u-u^0\|_1\|u-u^0\|_2\\rbrace \\\\& \\lesssim \\min \\lbrace \\varepsilon , h\\rbrace \\Vert f\\Vert _0^2.$ Then using the triangle inequality and (REF ), we acquire $\|u-I_hu\|_{1,h} & \\le \|u-u^0-I_h(u-u^0)\|_{1,h}+\|u^0-I_hu^0\|_{1,h} \\\\& \\lesssim \\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}\\rbrace \\Vert f\\Vert _0 +h^{s-1}\|u^0\|_{s}.$ Applying (REF ) and (REF ) again, we obtain $\|u-I_hu\|_{2,h}\\lesssim \|u\|_2\\lesssim \\varepsilon ^{-1/2}\\Vert f\\Vert _0,\\; \|u-I_hu\|_{2,h}^2\\lesssim h\|u\|_2\|u\|_3\\lesssim h\\varepsilon ^{-2}\\Vert f\\Vert _0^2.$ Thus $\\varepsilon \|u-I_hu\|_{2,h}\\lesssim \\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}\\rbrace \\Vert f\\Vert _0.$ By the trace inequality, (REF ) and (REF ), $\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}h_F^{-1}\\Vert \\partial _{n}(u-I_hu)\\Vert _{0,F}^2 & \\lesssim \\sum _{K\\in \\mathcal {T}_h}(h_K^{-2}\|u-I_hu\|_{1,K}^2+\|u-I_hu\|_{2,K}^2) \\\\& \\lesssim \\min \\lbrace \|u\|_2^2, h\|u\|_2\|u\|_3\\rbrace \\lesssim \\varepsilon ^{-2}\\min \\lbrace \\varepsilon , h\\rbrace \\Vert f\\Vert _0^2.$ Finally we derive (REF ) from (REF )-(REF ) and the last inequality.", "Lemma 2.3 We have $\\Vert u-I_{h0}u\\Vert _{\\varepsilon ,h}\\lesssim (\\varepsilon h+ h^{2})\|u\|_3,$ $\\Vert u-I_{h0}u\\Vert _{\\varepsilon ,h}\\lesssim h^{1/2}\\Vert f\\Vert _0.$ The inequality (REF ) is the immediate result of (REF ).", "Applying the trace inequality (REF ), (REF ) and (REF )-(REF ), $\\sum _{F\\in \\mathcal {F}_h^{\\partial }}h_F\\Vert \\partial _n(I_{h}^{SZ}(u-u^0)-(u-u^0))\\Vert _{0,F}^2\\lesssim h\|u-u^0\|_{1}\|u-u^0\|_{2}\\lesssim h\\Vert f\\Vert _0^2.$ Using (REF ) and (REF ) again, $& \\sum _{F\\in \\mathcal {F}_h^{\\partial }}h_F\\Vert \\partial _n(I_{h}^{SZ}u-u)\\Vert _{0,F}^2 \\\\\\lesssim & \\sum _{F\\in \\mathcal {F}_h^{\\partial }}h_F\\Vert \\partial _n(I_{h}^{SZ}(u-u^0)-(u-u^0))\\Vert _{0,F}^2 + \\sum _{F\\in \\mathcal {F}_h^{\\partial }}h_F\\Vert \\partial _n(I_{h}^{SZ}u^0-u^0)\\Vert _{0,F}^2 \\\\\\lesssim & h\\Vert f\\Vert _0^2+h^2\|u^0\|_2^2\\lesssim h\\Vert f\\Vert _0^2.$ By the definitions of $I_{h0}$ and $I_{h}$ , it follows $\|I_{h}u-I_{h0}u\|_{1,h}^2 & \\lesssim \\sum _{F\\in \\mathcal {F}_h^{\\partial }}h_F\\Vert \\partial _n(I_{h}^{SZ}u)\\Vert _{0,F}^2=\\sum _{F\\in \\mathcal {F}_h^{\\partial }}h_F\\Vert \\partial _n(I_{h}^{SZ}u-u)\\Vert _{0,F}^2\\lesssim h\\Vert f\\Vert _0^2.$ On the other side, we get from (REF ) and (REF ) that $\|I_{h}u-I_{h0}u\|_{2,h}^2 & \\lesssim \\sum _{F\\in \\mathcal {F}_h^{\\partial }}h_F^{-1}\\Vert \\partial _n(I_{h}^{SZ}u)\\Vert _{0,F}^2=\\sum _{F\\in \\mathcal {F}_h^{\\partial }}h_F^{-1}\\Vert \\partial _n(I_{h}^{SZ}u-u)\\Vert _{0,F}^2 \\\\& \\lesssim h\|u\|_2\|u\|_3\\lesssim \\varepsilon ^{-2}h\\Vert f\\Vert _0^2.$ Thus we obtain from the last two inequalities $\\Vert I_{h}u-I_{h0}u\\Vert _{\\varepsilon ,h}\\lesssim h^{1/2}\\Vert f\\Vert _0,$ which combined with (REF ) indicates (REF )." ], [ "Morley-Wang-Xu Element Method", "We will propose an MWX element method for the fourth order elliptic singular perturbation problem (REF ) in this section." ], [ "Morley-Wang-Xu Element Method", "To present the MWX element method, we need the $H^1$ -orthogonal projection $P_h: H^1(\\mathcal {T}_h)\\rightarrow W_h$ : given $v_h\\in H^1(\\mathcal {T}_h)$ , $P_hv_h\\in W_h$ is determined by $(\\nabla P_hv_h, \\nabla \\chi _h)= (\\nabla _hv_h, \\nabla \\chi _h)\\quad \\forall ~\\chi _h\\in W_h.$ It is well-known that for $s\\ge 1$ (cf.", "[15], [4]) $\|v-P_hv\|_1\\lesssim h^{\\min \\lbrace s-1,\\ell \\rbrace }\\Vert v\\Vert _s\\quad \\forall ~v\\in H_0^1(\\Omega )\\cap H^s(\\Omega ).$ We propose the following MWX element method for problem (REF ): find $u_{h0} \\in V_{h0} $ such that $\\varepsilon ^2 a_h(u_{h0},v_h)+b_h(u_{h0},v_h)=(f,P_hv_h) \\quad \\forall ~v_h \\in V_{h0},$ where $a_{h}(u_{h0}, v_h):= (\\nabla _h^{2}u_{h0}, \\nabla _h^{2}v_h),\\quad b_{h}(u_{h0}, v_h):= (\\nabla _h u_{h0}, \\nabla _h v_h).$ We use the simplest MWX element to approximate the exact solution in the discrete method (REF ).", "Compared the standard nonconforming finite element method, we only replace the right hand side term $(f, v_h)$ by $(f, P_hv_h)$ , thus the MWX element method (REF ) possesses a sparser stiffness matrix than those of the discontinuous Galerkin methods." ], [ "Error Estimates", "Using Cauchy-Schwarz inequality and (REF )-(REF ), we have following error estimates for $I_{h0}$ .", "Lemma 3.1 We have for any $v_h\\in V_{h0}$ $\\varepsilon ^2 a_h(I_{h0}u-u,v_h)+b_h(I_{h0}u-u,v_h)\\lesssim (\\varepsilon h+ h^{2})\|u\|_3\\Vert v_h\\Vert _{\\varepsilon , h},$ $\\varepsilon ^2 a_h(I_{h0}u-u,v_h)+b_h(I_{h0}u-u,v_h)\\lesssim h^{1/2}\\Vert f\\Vert _0\\Vert v_h\\Vert _{\\varepsilon , h}.$ Lemma 3.2 We have for any $v_h\\in V_{h0}$ $\\varepsilon ^2 a_h(u,v_h)+\\varepsilon ^2(\\operatorname{div}\\nabla ^2u, \\nabla E_h^{L}v_h)\\lesssim \\varepsilon \\min \\lbrace \\varepsilon , (\\varepsilon h)^{1/2}, h\\rbrace \|u\|_3\\Vert v_h\\Vert _{\\varepsilon , h},$ $\\varepsilon ^2 a_h(u,v_h)+\\varepsilon ^2(\\operatorname{div}\\nabla ^2u, \\nabla E_h^{L}v_h)\\lesssim \\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}\\rbrace \\Vert f\\Vert _0\\Vert v_h\\Vert _{\\varepsilon , h}.$ We get from integration by parts and (REF ) that $& a_h(u,v_h)+(\\operatorname{div}\\nabla ^2u, \\nabla _hv_h) \\\\= & \\sum _{K\\in \\mathcal {T}_h}((\\nabla ^2u)n, \\nabla _hv_h)_{\\partial K}=\\sum _{F\\in \\mathcal {F}_h}((\\nabla ^2u)n_F, \\llbracket \\nabla _hv_h\\rrbracket )_F \\\\= & \\sum _{F\\in \\mathcal {F}_h}((\\nabla ^2u)n_F-Q_0^F((\\nabla ^2u)n_F), \\llbracket \\nabla _hv_h\\rrbracket )_F \\\\= & \\sum _{F\\in \\mathcal {F}_h}((\\nabla ^2u)n_F-Q_0^F((\\nabla ^2u)n_F), \\llbracket \\nabla _hv_h\\rrbracket -Q_0^F(\\llbracket \\nabla _hv_h\\rrbracket ))_F,$ where $Q_0^F$ is the $L^2$ -orthogonal projection onto the constant space on face $F$ .", "By the error estimate of $Q_0^F$ (cf.", "[15], [4]) and the inverse inequality, we have $a_h(u,v_h)+(\\operatorname{div}\\nabla ^2u, \\nabla _hv_h)\\lesssim \|u\|_3\\min \\left\\lbrace \|v_h\|_{1,h}, h^{1/2}\|v_h\|_{1,h}^{1/2}\|v_h\|_{2,h}^{1/2}, h\|v_h\|_{2,h}\\right\\rbrace .$ On the other side, it follows from (REF ) that $(\\operatorname{div}\\nabla ^2u, \\nabla _h(E_h^{L}v_h-v_h)) & \\lesssim \|u\|_3\|E_h^Lv_h-v_h\|_{1,h} \\\\& \\lesssim \|u\|_3\\min \\left\\lbrace \|v_h\|_{1,h}, h^{1/2}\|v_h\|_{1,h}^{1/2}\|v_h\|_{2,h}^{1/2}, h\|v_h\|_{2,h}\\right\\rbrace .$ Combining the last two inequalities gives $a_h(u,v_h)+(\\operatorname{div}\\nabla ^2u, \\nabla E_h^{L}v_h)\\lesssim \|u\|_3\\min \\left\\lbrace \|v_h\|_{1,h}, h^{1/2}\|v_h\|_{1,h}^{1/2}\|v_h\|_{2,h}^{1/2}, h\|v_h\|_{2,h}\\right\\rbrace .$ Finally we derive (REF )-(REF ) from (REF ).", "Lemma 3.3 It holds for any $v_h\\in V_{h0}$ $b_h(u, v_h-E_h^Lv_h)- (f, P_hv_h-E_h^Lv_h)\\lesssim (\\varepsilon ^{-1/2}+1)h\\Vert f\\Vert _0\\Vert v_h\\Vert _{\\varepsilon , h}.$ If $u^0\\in H^s(\\Omega )$ with $2\\le s\\le 3$ , it holds for any $v_h\\in V_{h0}$ $& b_h(u, v_h-E_h^Lv_h)- (f, P_hv_h-E_h^Lv_h) \\\\\\lesssim & \\left(\\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}\\rbrace \\Vert f\\Vert _0 +h^{\\min \\lbrace s-1,\\ell \\rbrace }\\Vert u^0\\Vert _{s}\\right)\\Vert v_h\\Vert _{\\varepsilon , h}.", "$ Since $P_hv_h-E_h^Lv_h\\in H_0^1(\\Omega )$ , we get from (REF ), integration by parts and the definition of $P_h$ that $(f, P_hv_h-E_h^Lv_h) & =(\\nabla u^0, \\nabla (P_hv_h-E_h^Lv_h))=(\\nabla u^0, \\nabla P_hv_h)-(\\nabla u^0, \\nabla E_h^Lv_h) \\\\& =(\\nabla P_hu^0, \\nabla _hv_h)-(\\nabla u^0, \\nabla E_h^Lv_h) \\\\& =(\\nabla (P_hu^0-u^0), \\nabla _h v_h)+(\\nabla u^0, \\nabla _h(v_h-E_h^Lv_h)).$ Thus we have $& b_h(u, v_h-E_h^Lv_h)-(f, P_hv_h-E_h^Lv_h) \\\\= & (\\nabla (u-u^0), \\nabla _h(v_h-E_h^Lv_h))+(\\nabla (u^0-P_hu^0), \\nabla _h v_h).$ Adopting Cauchy-Schwarz inequality and (REF ), it holds $& b_h(u, v_h-E_h^Lv_h)-(f, P_hv_h-E_h^Lv_h) \\\\\\lesssim & \|u-u^0\|_{1}\|v_h-E_h^Lv_h\|_{1,h}+\|u^0-P_hu^0\|_1\|v_h\|_{1,h} \\\\\\lesssim & \\varepsilon ^{1/2}\\Vert f\\Vert _0\|v_h-E_h^Lv_h\|_{1,h}+\|u^0-P_hu^0\|_1\|v_h\|_{1,h}.$ Thanks to (REF ), we get $\|v_h-E_h^Lv_h\|_{1,h}\\lesssim \\varepsilon ^{-1/2}\\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}, \\varepsilon ^{-1/2}h\\rbrace \\Vert v_h\\Vert _{\\varepsilon , h}.$ Hence we obtain $& b_h(u, v_h-E_h^Lv_h)-(f, P_hv_h-E_h^Lv_h) \\\\\\lesssim & \\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}, \\varepsilon ^{-1/2}h\\rbrace \\Vert f\\Vert _0\\Vert v_h\\Vert _{\\varepsilon , h}+\|u^0-P_hu^0\|_1\\Vert v_h\\Vert _{\\varepsilon , h}.$ Therefore we acquire (REF )-(REF ) from (REF ).", "Theorem 3.4 Let $u\\in H_0^2(\\Omega )$ be the solution of problem (REF ), and $u_{h0}\\in V_{h0}$ be the discrete solution of the MWX element method (REF ).", "Assume $u^0\\in H^s(\\Omega )$ with $2\\le s\\le 3$ .", "We have $\\Vert u-u_{h0}\\Vert _{\\varepsilon , h}\\lesssim \\varepsilon ^{1/2}\\Vert f\\Vert _0 +h^{\\min \\lbrace s-1,\\ell \\rbrace }\\Vert u^0\\Vert _{s}+h(\\varepsilon +h)\|u\|_3,$ $\\Vert u-u_{h0}\\Vert _{\\varepsilon , h}\\lesssim h\\left((\\varepsilon ^{-1/2}+1)\\Vert f\\Vert _0+(\\varepsilon +h)\|u\|_3\\right),$ $\\Vert u-u_{h0}\\Vert _{\\varepsilon , h}\\lesssim h^{1/2}\\Vert f\\Vert _0,$ $\\Vert u^0-u_{h0}\\Vert _{\\varepsilon , h}\\lesssim \\left(\\varepsilon ^{1/2}+h^{1/2}\\right)\\Vert f\\Vert _0.$ Let $v_h=I_{h0}u-u_{h0}$ .", "We obtain from (REF ) that $-\\varepsilon ^2(\\operatorname{div}\\nabla ^2u, \\nabla E_h^{L}v_h) + (\\nabla u, \\nabla E_h^{L}v_h)=(f, E_h^{L}v_h).$ Then it follows from (REF ) that $& \\varepsilon ^2 a_h(u-u_{h0},v_h)+b_h(u-u_{h0},v_h) \\\\= & \\varepsilon ^2 a_h(u,v_h)+b_h(u,v_h)-(f, P_hv_h) \\\\= & \\varepsilon ^2 a_h(u,v_h)+\\varepsilon ^2(\\operatorname{div}\\nabla ^2u, \\nabla E_h^{L}v_h)+b_h(u,v_h-E_h^{L}v_h)-(f, P_hv_h-E_h^{L}v_h).$ Hence we acquire from (REF )-(REF ) and (REF )-(REF ) that $\\varepsilon ^2 a_h(u-u_{h0},v_h)+b_h(u-u_{h0},v_h)\\lesssim h\\left((\\varepsilon ^{-1/2}+1)\\Vert f\\Vert _0+\\varepsilon \|u\|_3\\right)\\Vert v_h\\Vert _{\\varepsilon , h},$ $& \\varepsilon ^2 a_h(u-u_{h0},v_h)+b_h(u-u_{h0},v_h) \\\\\\lesssim & \\left(\\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}\\rbrace \\Vert f\\Vert _0 +h^{\\min \\lbrace s-1,\\ell \\rbrace }\\Vert u^0\\Vert _{s}\\right)\\Vert v_h\\Vert _{\\varepsilon , h}.$ Using the triangle inequality and (REF )-(REF ), we get $& \\varepsilon ^2 a_h(I_{h0}u-u_{h0},v_h)+b_h(I_{h0}u-u_{h0},v_h) \\\\\\lesssim & \\left(\\varepsilon ^{1/2}\\Vert f\\Vert _0 +h^{\\min \\lbrace s-1,\\ell \\rbrace }\\Vert u^0\\Vert _{s}+h(\\varepsilon +h)\|u\|_3\\right)\\Vert v_h\\Vert _{\\varepsilon , h},$ $& \\varepsilon ^2 a_h(I_{h0}u-u_{h0},v_h)+b_h(I_{h0}u-u_{h0},v_h) \\\\\\lesssim & h\\left((\\varepsilon ^{-1/2}+1)\\Vert f\\Vert _0+(\\varepsilon +h)\|u\|_3\\right)\\Vert v_h\\Vert _{\\varepsilon , h},$ $\\varepsilon ^2 a_h(I_{h0}u-u_{h0},v_h)+b_h(I_{h0}u-u_{h0},v_h)\\lesssim h^{1/2}\\Vert f\\Vert _0\\Vert v_h\\Vert _{\\varepsilon , h}.$ Thus $\\Vert I_{h0}u-u_{h0}\\Vert _{\\varepsilon , h}\\lesssim \\varepsilon ^{1/2}\\Vert f\\Vert _0 +h^{\\min \\lbrace s-1,\\ell \\rbrace }\\Vert u^0\\Vert _{s}+h(\\varepsilon +h)\|u\|_3,$ $\\Vert I_{h0}u-u_{h0}\\Vert _{\\varepsilon , h}\\lesssim h\\left((\\varepsilon ^{-1/2}+1)\\Vert f\\Vert _0+(\\varepsilon +h)\|u\|_3\\right),$ $\\Vert I_{h0}u-u_{h0}\\Vert _{\\varepsilon , h}\\lesssim h^{1/2}\\Vert f\\Vert _0.$ Finally we get (REF )-(REF ) by combining the last three inequalities and (REF )-(REF ).", "The estimate (REF ) is a direct result of (REF ) and (REF )-(REF )." ], [ "Imposing Boundary Condition Using Nitsche's Method", "In the consideration of the boundary layer of problem (REF ), we will adjust the MWX element method (REF ) by using Nitsche's method to impose the boundary condition $\\partial _nu=0$ weakly in this section, as in [21]." ], [ "Discrete Method", "Through applying the Nitsche's technique, the MWX element method with weakly imposing the boundary condition is to find $u_h \\in V_{h} $ such that $\\varepsilon ^2 \\tilde{a}_h(u_h,v_h)+b_h(u_h,v_h)=(f,P_hv_h) \\quad \\forall ~v_h \\in V_{h},$ where $\\tilde{a}_{h}(u_h, v_h):= & (\\nabla _h^{2}u_h, \\nabla _h^{2}v_h) - \\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}(\\partial _{nn}^2u_h, \\partial _{n}v_h)_F- \\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}(\\partial _{n}u_h, \\partial _{nn}^2v_h)_F \\\\& +\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}\\frac{\\sigma }{h_{F}}(\\partial _{n}u_h, \\partial _{n}v_h)_F$ with $\\sigma $ being a positive real number.", "Lemma 4.1 There exists a constant $\\sigma _0>0$ depending only on the shape regularity of $\\mathcal {T}_h$ such that for any fixed number $\\sigma \\ge \\sigma _0$ , it holds $\\interleave v_h\\interleave _{2,h}^2\\lesssim \\tilde{a}_h(v_h, v_h)\\quad \\forall ~v_h\\in V_h.$ Due to the Cauchy-Schwarz inequality and inverse inequality, there exists a constant $C>0$ such that $2\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}(\\partial _{nn}^2v_h, \\partial _{n}v_h)_F & \\le 2\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}\\Vert \\partial _{nn}^2v_h\\Vert _{0,F}\\Vert \\partial _{n}v_h\\Vert _{0,F} \\\\& \\le C\|v_h\|_{2,h}\\left(\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}h_{F}^{-1}\\Vert \\partial _{n}v_h\\Vert _{0,F}^2\\right)^{1/2} \\\\& \\le \\frac{1}{2}\|v_h\|_{2,h}^2+\\frac{1}{2}C^2\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}h_{F}^{-1}\\Vert \\partial _{n}v_h\\Vert _{0,F}^2.$ Hence $\\tilde{a}_{h}(v_h, v_h) & = \|v_h\|_{2,h}^2 - 2\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}(\\partial _{nn}^2v_h, \\partial _{n}v_h)_F+\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}\\frac{\\sigma }{h_{F}}\\Vert \\partial _{n}v_h\\Vert _{0,F}^2 \\\\& \\ge \\frac{1}{2}\|v_h\|_{2,h}^2+\\left(\\sigma -\\frac{1}{2}C^2\\right)\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}h_{F}^{-1}\\Vert \\partial _{n}v_h\\Vert _{0,F}^2.$ The proof is finished by choosing $\\sigma _0=\\frac{1}{2}C^2+1$ .", "By (REF ), we have $\\interleave v_h \\interleave _{\\varepsilon ,h}^2\\lesssim \\varepsilon ^2 \\tilde{a}_h(v_h,v_h)+b_h(v_h,v_h)\\ \\ \\forall ~v_h \\in V_{h}.$ It is obvious that $\\varepsilon ^2\\tilde{a}_h(\\chi _h, v_h)+b_h(\\chi _h, v_h)\\lesssim \\interleave \\chi _h\\interleave _{\\varepsilon ,h}\\interleave v_h\\interleave _{\\varepsilon ,h} \\quad \\forall ~\\chi _h, v_h\\in V_h.$ The last two inequalities indicate the wellposedness of the MWX element method (REF )." ], [ "Error Estimates", "In this subsection we will present the error analysis for the discrete method (REF ).", "Lemma 4.2 It holds $\\varepsilon ^2 \\tilde{a}_h(I_{h}u-u,v_h)+b_h(I_{h}u-u,v_h)\\lesssim (\\varepsilon h+ h^{2})\|u\|_3\\interleave v_h\\interleave _{\\varepsilon , h}\\quad \\forall ~v_h\\in V_{h}.$ If $u^0\\in H_0^1(\\Omega )\\cap H^s(\\Omega )$ with $2\\le s\\le 3$ , we have $& \\varepsilon ^2 \\tilde{a}_h(I_{h}u-u,v_h)+b_h(I_{h}u-u,v_h)\\\\\\lesssim & (\\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}\\rbrace \\Vert f\\Vert _0 +h^{s-1}\|u^0\|_{s})\\interleave v_h\\interleave _{\\varepsilon , h}\\quad \\forall ~v_h\\in V_{h}.", "$ According to the trace inequality, (REF ) and (REF ), it follows $\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}h_F\\Vert \\partial _{nn}^2(I_{h}u-u)\\Vert _{0,F}^2 & \\lesssim \|I_{h}u-u\|_{2,h}(\|I_{h}u-u\|_{2,h}+h\|I_{h}u-u\|_{3,h}) \\\\& \\lesssim \\min \\lbrace h\|u\|_2\|u\|_3, h^2\|u\|_3^2\\rbrace \\lesssim \\min \\lbrace \\varepsilon ^{-2}h\\Vert f\\Vert _0^2, h^2\|u\|_3^2\\rbrace .$ Then we get $- \\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}(\\partial _{nn}^2(I_{h}u-u), \\partial _{n}v_h)_F\\le & \\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}\\Vert \\partial _{nn}^2(I_{h}u-u)\\Vert _{0,F}\\Vert \\partial _{n}v_h\\Vert _{0,F} \\\\\\lesssim & \\interleave v_h\\interleave _{2,h}\\left(\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}h_F\\Vert \\partial _{nn}^2(I_{h}u-u)\\Vert _{0,F}^2\\right)^{1/2} \\\\\\lesssim & \\varepsilon ^{-2}\\interleave v_h\\interleave _{\\varepsilon ,h}\\min \\lbrace h^{1/2}\\Vert f\\Vert _0, \\varepsilon h\|u\|_3\\rbrace .$ Using the inverse inequality, (REF ) and (REF ), we obtain $- \\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}(\\partial _{nn}^2(I_{h}u-u), \\partial _{n}v_h)_F\\le & \\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}\\Vert \\partial _{nn}^2(I_{h}u-u)\\Vert _{0,F}\\Vert \\partial _{n}v_h\\Vert _{0,F} \\\\\\lesssim & \|v_h\|_{1,h}\\left(\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}h_F^{-1}\\Vert \\partial _{nn}^2(I_{h}u-u)\\Vert _{0,F}^2\\right)^{1/2} \\\\\\lesssim & \|v_h\|_{1,h}\|u\|_3\\lesssim \\varepsilon ^{-3/2}\\interleave v_h\\interleave _{\\varepsilon ,h}\\Vert f\\Vert _0.$ Hence it follows from the last two inequalities that $- \\varepsilon ^{2}\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}(\\partial _{nn}^2(I_{h}u-u), \\partial _{n}v_h)_F\\lesssim \\interleave v_h\\interleave _{\\varepsilon ,h}\\min \\lbrace \\varepsilon ^{1/2}\\Vert f\\Vert _0, h^{1/2}\\Vert f\\Vert _0, \\varepsilon h\|u\|_3\\rbrace .$ Since $& \\varepsilon ^2 \\tilde{a}_h(I_{h}u-u,v_h)+b_h(I_{h}u-u,v_h) \\\\\\lesssim & \\interleave I_{h}u-u\\interleave _{\\varepsilon ,h}\\interleave v_h\\interleave _{\\varepsilon ,h} - \\varepsilon ^{2}\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}(\\partial _{nn}^2(I_{h}u-u), \\partial _{n}v_h)_F,$ we acquire (REF ) from (REF ) and (REF ), and (REF ) from (REF ) and (REF ).", "Applying the same argument as in Lemma REF , from (REF )-(REF ) and (REF ) we obtain the following estimates $\\varepsilon ^2 \\tilde{a}_h(u,v_h)+\\varepsilon ^2(\\operatorname{div}\\nabla ^2u, \\nabla E_h^{L}v_h)\\lesssim \\varepsilon \\min \\lbrace \\varepsilon , (\\varepsilon h)^{1/2}, h\\rbrace \|u\|_3\\interleave v_h\\interleave _{\\varepsilon , h},$ $\\varepsilon ^2 \\tilde{a}_h(u,v_h)+\\varepsilon ^2(\\operatorname{div}\\nabla ^2u, \\nabla E_h^{L}v_h)\\lesssim \\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}\\rbrace \\Vert f\\Vert _0\\interleave v_h\\interleave _{\\varepsilon , h}$ for any $v_h\\in V_{h}$ .", "Applying the same argument as in Lemma REF , we acquire $b_h(u, v_h-E_h^Lv_h)- (f, P_hv_h-E_h^Lv_h)\\lesssim (\\varepsilon ^{-1/2}+1)h\\Vert f\\Vert _0\\interleave v_h\\interleave _{\\varepsilon , h}$ for any $v_h\\in V_{h}$ .", "And if $u^0\\in H^s(\\Omega )$ with $2\\le s\\le 3$ , it holds $& b_h(u, v_h-E_h^Lv_h)- (f, P_hv_h-E_h^Lv_h) \\\\\\lesssim & \\left(\\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}\\rbrace \\Vert f\\Vert _0 +h^{\\min \\lbrace s-1,\\ell \\rbrace }\\Vert u^0\\Vert _{s}\\right)\\interleave v_h\\interleave _{\\varepsilon , h} \\quad \\forall ~v_h\\in V_{h}.", "$ Theorem 4.3 Let $u\\in H_0^2(\\Omega )$ be the solution of problem (REF ), and $u_{h}\\in V_{h}$ be the discrete solution of the MWX element method (REF ).", "Assume $u^0\\in H_0^1(\\Omega )\\cap H^s(\\Omega )$ with $2\\le s\\le 3$ .", "We have $\\interleave u-u_{h}\\interleave _{\\varepsilon , h}\\lesssim h\\left((\\varepsilon ^{-1/2}+1)\\Vert f\\Vert _0+(\\varepsilon +h)\|u\|_3\\right),$ $\\interleave u-u_{h}\\interleave _{\\varepsilon , h}\\lesssim \\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}\\rbrace \\Vert f\\Vert _0 +h^{\\min \\lbrace s-1,\\ell \\rbrace }\\Vert u^0\\Vert _{s},$ $\\Vert u^0-u_h\\Vert _{\\varepsilon , h}\\lesssim \\varepsilon ^{1/2}\\Vert f\\Vert _0 +h^{\\min \\lbrace s-1,\\ell \\rbrace }\\Vert u^0\\Vert _{s}.$ Let $v_h=I_{h}u-u_{h}$ .", "Adopting the similar argument as in the proof of Theorem REF , we get from (REF )-(REF ) and (REF )-(REF ) that $\\varepsilon ^2 \\tilde{a}_h(u-u_{h},v_h)+b_h(u-u_{h},v_h)\\lesssim h\\left((\\varepsilon ^{-1/2}+1)\\Vert f\\Vert _0+\\varepsilon \|u\|_3\\right)\\interleave v_h\\interleave _{\\varepsilon , h},$ $\\varepsilon ^2 \\tilde{a}_h(u-u_{h},v_h)+b_h(u-u_{h},v_h)\\lesssim \\left(\\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}\\rbrace \\Vert f\\Vert _0 +h^{\\min \\lbrace s-1,\\ell \\rbrace }\\Vert u^0\\Vert _{s}\\right)\\interleave v_h\\interleave _{\\varepsilon , h}.$ Together with (REF )-(REF ), we have $& \\varepsilon ^2\\tilde{a}_h(I_{h}u-u_{h},v_h)+b_h(I_{h}u-u_{h},v_h) \\\\\\lesssim & h\\left((\\varepsilon ^{-1/2}+1)\\Vert f\\Vert _0+(\\varepsilon +h)\|u\|_3\\right)\\interleave v_h\\interleave _{\\varepsilon , h},$ $& \\varepsilon ^2\\tilde{a}_h(I_{h}u-u_{h},v_h)+b_h(I_{h}u-u_{h},v_h) \\\\\\lesssim & \\left(\\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}\\rbrace \\Vert f\\Vert _0 +h^{\\min \\lbrace s-1,\\ell \\rbrace }\\Vert u^0\\Vert _{s}\\right)\\interleave v_h\\interleave _{\\varepsilon , h}.$ Then we obtain from (REF ) that $\\interleave I_{h}u-u_{h}\\interleave _{\\varepsilon , h}\\lesssim h\\left((\\varepsilon ^{-1/2}+1)\\Vert f\\Vert _0+(\\varepsilon +h)\|u\|_3\\right),$ $\\interleave I_{h}u-u_{h}\\interleave _{\\varepsilon , h}\\lesssim \\min \\lbrace \\varepsilon ^{1/2}, h^{1/2}\\rbrace \\Vert f\\Vert _0 +h^{\\min \\lbrace s-1,\\ell \\rbrace }\\Vert u^0\\Vert _{s}.$ Therefore we conclude (REF ) and (REF ) from (REF )-(REF ), and (REF ) from (REF ) and (REF )-(REF )." ], [ "Equivalent Formulations", "In this section, we will show some equivalent solver-friendly formulations of the MWX element methods (REF ) and (REF ).", "Lemma 5.1 The MWX element method (REF ) is equivalent to find $w_h\\in W_h$ and $u_{h0}\\in V_{h0}$ such that $(\\nabla w_h, \\nabla \\chi _h) & =(f, \\chi _h) \\quad \\quad \\quad \\quad \\; \\forall ~\\chi _h\\in W_h, \\\\\\varepsilon ^2 a_h(u_{h0},v_h)+b_h(u_{h0},v_h) & =(\\nabla w_h, \\nabla _h v_h) \\quad \\;\\; \\forall ~v_h\\in V_{h0}.", "$ By the definition of the $H^1$ -orthogonal projection $P_h$ and (REF ) with $\\chi _h= P_hv_h$ , the right hand side of (), it follows $(\\nabla w_h, \\nabla _h v_h)=(\\nabla w_h, \\nabla P_hv_h)=(f, P_hv_h).$ Therefore the MWX element method (REF ) is equivalent to the discrete method (REF )-().", "Similarly, we have the equivalent formulation of the MWX element method (REF ).", "Lemma 5.2 The MWX element method (REF ) is equivalent to find $w_h\\in W_h$ and $u_{h}\\in V_h$ such that $(\\nabla w_h, \\nabla \\chi _h) & =(f, \\chi _h) \\quad \\quad \\quad \\quad \\; \\forall ~\\chi _h\\in W_h, \\\\\\varepsilon ^2 \\tilde{a}_h(u_h,v_h)+b_h(u_h,v_h) & =(\\nabla w_h, \\nabla _h v_h) \\quad \\;\\; \\forall ~v_h\\in V_h.", "$ In two dimensions, we can further decouple the discrete methods () and () into the discrete methods of two Poisson equations and one Brinkman problem.", "To this end, define the vectorial nonconforming $P_1$ element space $V_h^{CR}:=\\Big \\lbrace v\\in L^2(\\Omega ;\\mathbb {R}^2): & ~v\|_K\\in \\mathbb {P}_1(K;\\mathbb {R}^2)\\textrm { for each } K\\in \\mathcal {T}_h, \\int _F\\llbracket v\\rrbracket \\,{\\rm d}s=0 \\textrm { for } \\\\& \\quad \\; \\textrm { each } F\\in \\mathcal {F}_h^i, \\textrm { and } \\int _Fv\\cdot n\\,{\\rm d}s=0 \\textrm { for each } F\\in \\mathcal {F}_h^{\\partial }\\Big \\rbrace ,$ $V_{h0}^{CR}:=\\Big \\lbrace v\\in V_h^{CR}: \\int _Fv\\,{\\rm d}s=0 \\textrm { for each } F\\in \\mathcal {F}_h^{\\partial }\\Big \\rbrace .$ And let $\\mathcal {Q}_h\\subset L_0^2(\\Omega )$ be the piecewise constant space with respect to $\\mathcal {T}_h$ , where $L_0^2(\\Omega )$ is the subspace of $L^2(\\Omega )$ with vanishing mean value.", "Due to Theorem 4.1 in [16], we have the following relationship between Morley element spaces and vectorial Crouzeix-Raviart element spaces $\\operatorname{curl}_h V_{h0}=\\lbrace v_h\\in V_{h0}^{CR}: \\operatorname{div}_h v_h=0\\rbrace ,$ $\\operatorname{curl}_h V_h=\\lbrace v_h\\in V_h^{CR}: \\operatorname{div}_h v_h=0\\rbrace .$ Lemma 5.3 In two dimensions, the discrete method () can be decoupled into two Morley element methods of Poisson equation and one nonconforming $P_1$ -$P_0$ element method of Brinkman problem, i.e., find $(z_h, \\phi _h, p_h, w_h)\\in V_{h}\\times V_{h0}^{CR}\\times \\mathcal {Q}_{h}\\times V_{h}$ such that $(\\operatorname{curl}_h z_h, \\operatorname{curl}_h v_h) & =(\\nabla w_h, \\nabla _h v_h)\\quad \\, \\forall ~v_h\\in V_{h}, \\\\(\\phi _h, \\psi _h)+\\varepsilon ^2(\\nabla _h\\phi _h, \\nabla _h\\psi _h) + (\\operatorname{div}_h\\psi _h, p_h) & =(\\operatorname{curl}_h z_h, \\psi _h)\\quad \\forall ~\\psi _h\\in V_{h0}^{CR}, \\\\(\\operatorname{div}_h\\phi _h, q_h) & = 0\\quad \\quad \\quad \\quad \\quad \\quad \\, \\forall ~q_h\\in \\mathcal {Q}_{h}, \\\\(\\operatorname{curl}_h u_{h0}, \\operatorname{curl}_h \\chi _h) & = (\\phi _h, \\operatorname{curl}_h\\chi _h) \\quad \\forall ~\\chi _h\\in V_{h}.", "$ Thanks to (REF ), it follows from ()-() that $\\phi _h=\\operatorname{curl}_h u_{h0} \\quad \\textrm { and } \\quad u_{h0}\\in V_{h0}.$ For any $v_h\\in V_{h0}$ , it is apparent that $(\\nabla _h\\operatorname{curl}_h u_{h0}, \\nabla _h\\operatorname{curl}_h v_h)=a_h(u_{h0}, v_h).$ Then replacing $\\phi _h$ with $\\operatorname{curl}_h u_{h0}$ and $\\psi _h$ with $\\operatorname{curl}_h v_h$ in (), we achieve $\\varepsilon ^2 a_h(u_{h0},v_h)+b_h(u_{h0},v_h) =(\\operatorname{curl}_h z_h, \\operatorname{curl}_h v_h),$ which combined with (REF ) induces ().", "Similarly, we get the decoupling of the discrete method () based on (REF ).", "Lemma 5.4 In two dimensions, the discrete method () can be decoupled into two Morley element methods of Poisson equation and one nonconforming $P_1$ -$P_0$ element method of Brinkman problem, i.e., find $(z_h, \\phi _h, p_h, w_h)\\in V_{h}\\times V_h^{CR}\\times \\mathcal {Q}_{h}\\times V_{h}$ such that $(\\operatorname{curl}_h z_h, \\operatorname{curl}_h v_h) & =(\\nabla w_h, \\nabla _h v_h)\\quad \\;\\; \\forall ~v_h\\in V_{h}, \\\\(\\phi _h, \\psi _h)+\\varepsilon ^2c_h(\\phi _h, \\psi _h) + (\\operatorname{div}_h\\psi _h, p_h) & =(\\operatorname{curl}_h z_h, \\psi _h)\\quad \\;\\, \\forall ~\\psi _h\\in V_h^{CR}, \\\\(\\operatorname{div}_h\\phi _h, q_h) & = 0\\quad \\quad \\quad \\quad \\quad \\quad \\;\\; \\forall ~q_h\\in \\mathcal {Q}_{h}, \\\\(\\operatorname{curl}_h u_h, \\operatorname{curl}_h \\chi _h) & = (\\phi _h, \\operatorname{curl}_h\\chi _h) \\quad \\; \\forall ~\\chi _h\\in V_{h}, $ where $c_h(\\phi _h, \\psi _h):= & (\\nabla _h\\phi _h, \\nabla _h\\psi _h) - \\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}(\\partial _{n}(\\phi _h\\cdot t), \\psi _h\\cdot t)_F - \\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}(\\phi _h\\cdot t, \\partial _{n}(\\psi _h\\cdot t))_F \\\\& +\\sum _{F \\in \\mathcal {F}_{h}^{\\partial }}\\frac{\\sigma }{h_{F}}(\\phi _h\\cdot t, \\psi _h\\cdot t)_F.$ Combining Lemma REF and Lemma REF yields an equivalent discrete method of the MWX element method (REF ).", "And combining Lemma REF and Lemma REF yields an equivalent discrete method of the MWX element method (REF ).", "Theorem 5.5 In two dimensions, the MWX element method (REF ) can be decoupled into (REF ) and (REF )-().", "That is, the MWX element method (REF ) can be decoupled into one Lagrange element method of Poisson equation, two Morley element methods of Poisson equation and one nonconforming $P_1$ -$P_0$ element method of Brinkman problem.", "Theorem 5.6 In two dimensions, the MWX element method (REF ) can be decoupled into one Lagrange element method of Poisson equation (REF ), two Morley element methods of Poisson equation and one nonconforming $P_1$ -$P_0$ element method of Brinkman problem (REF )-().", "The decoupling of the fourth order elliptic singular perturbation problem (REF ) into two Poisson equations and one Brinkman problem in the continuous level have been developed in [11], [18].", "The decoupling of the Morley element method of the biharmonic equation into two Morley element methods of Poisson equation and one nonconforming $P_1$ -$P_0$ element method of Stokes equation was firstly discovered in [23].", "When $\\varepsilon $ is very small, the stiffness matrix of the MWX element method (REF ) is very close to the stiffness matrix of the MWX element method for the Poisson equation, which can be efficiently solved by the CG method with AMG as the preconditioner.", "When $\\varepsilon \\eqsim 1$ , the equivalences in Theorems REF -REF will induce efficient and robust Poisson based solvers for the MWX element method (REF ) and (REF ).", "The Lagrange element methods of Poisson equation (REF ) and (REF ), the Morley element methods of Poisson equation (REF ), (), (REF ) and () can be solved by CG method with the auxiliary space preconditioner [41], in which the $H^1$ conforming linear element discretization on the same mesh for the Poisson equation can be adopted as the auxiliary problem.", "And the AMG method is used to solve the auxiliary problem.", "If $\\ell =1$ , the Lagrange element methods of Poisson equation (REF ) and (REF ) are the linear Lagrange element methods, which can be solved efficiently by CG method using the classical AMG method as the preconditioner.", "As for the nonconforming $P_1$ -$P_0$ element methods of Brinkman problem ()-() and ()-(), we can use the block-diagonal preconditioner in [27], [24], [6] or the approximate block-factorization preconditioner in [10], which are robust with respect to the parameter $\\varepsilon $ and mesh size $h$ .", "We will adopt the following approximate block-factorization preconditioner in the numerical part $\\begin{pmatrix}A_h-B_h^T\\widetilde{M}_h^{-1}B_h & B_h^T \\\\B_h & -\\widetilde{M}_h\\end{pmatrix}^{-1} & =\\begin{pmatrix}I & 0 \\\\\\widetilde{M}_h^{-1}B_h & -I\\end{pmatrix}^{-1}\\begin{pmatrix}A_h & -B_h^T \\\\0 & \\widetilde{M}_h\\end{pmatrix}^{-1} \\\\& =\\begin{pmatrix}I & 0 \\\\\\widetilde{M}_h^{-1}B_h & -I\\end{pmatrix}\\begin{pmatrix}A_h & -B_h^T \\\\0 & \\widetilde{M}_h\\end{pmatrix}^{-1}, $ where $\\begin{pmatrix}A_h & B_h^T \\\\B_h & O\\end{pmatrix}$ is the stiffness matrix of the Brinkman equation ()-(), and $\\widetilde{M}_h=\\alpha \\varepsilon ^{-2}M_h$ with $\\alpha >0$ and $M_h$ being the mass matrix for the pressure.", "Since all these solvers are based on the solvers of the Poisson equation and we use AMG method to solve the discrete methods of the Poisson equation, the designed fast solver of the MWX element method (REF ) (the MWX element method (REF )) based on the discrete method (REF ) and (REF )-() (the discrete method (REF ) and (REF )-()) also works for the shape-regular unstructured meshes." ], [ "Numerical Results", "In this section, we will provide some numerical examples to verify the theoretical convergence rates of the MWX element method (REF ) and (REF ), and test the efficiency of a solver based on the decoupled method (REF )-().", "Let $\\Omega $ be the unit square $(0,1)^2$ , and we use the uniform triangulation of $\\Omega $ .", "All the experiments are implemented with the scikit-fem library [20].", "Example 6.1 We first test the MWX element method (REF ) with the exact solution $u(x,y)=\\sin ^2(\\pi x)\\sin ^2(\\pi y).$ The right hand side $f$ is computed from (REF ).", "Notice that the solution $u$ does not have boundary layers.", "Take $\\ell =1$ .", "The energy error $\\Vert u-u_{h0}\\Vert _{\\varepsilon , h}$ with different $\\varepsilon $ and $h$ is shown in Table REF .", "From Table REF we observe that $\\Vert u-u_{h0}\\Vert _{\\varepsilon , h}=O(h)$ for $\\varepsilon =1, 10^{-1}, 10^{-2}$ , which agrees with the theoretical convergence result (REF ).", "While numerically $\\Vert u-u_{h0}\\Vert _{\\varepsilon , h}=O(h^2)$ for $\\varepsilon =10^{-3}, 10^{-4}, 10^{-5}$ in Table REF , which is superconvergent and one order higher than the theoretical convergence result (REF ).", "Then examine the efficiency of solvers for the MWX element method (REF ).", "The stop criterion in our iterative algorithms is the relative residual is less than $10^{-8}$ .", "And the initial guess is zero.", "First, we solve the MWX element method (REF )-() using the CG method with AMG method (AMG-CG) as the preconditioner.", "According to the iteration steps listed in the third column of Tables REF -REF , equation (REF ) is highly efficiently solved by the AMG-CG solver.", "From Table REF we can see that the AMG-CG solver is very efficient for $\\varepsilon =10^{-4}, 10^{-5}$ , and the iteration steps of the AMG-CG solver is also acceptable for $\\varepsilon =10^{-2}, 10^{-3}$ .", "However the AMG-CG solver deteriorates for $\\varepsilon =1, 10^{-1}$ , which means the AMG-CG solver doesn't work for large $\\varepsilon $ .", "To deal with this, we adopt a solver for the MWX element method () based on the equivalent decoupling (REF )-().", "To be specific, we adopt the AMG-CG solver to solve equations (REF ) and (), and the GMRES method with the preconditioner (REF ), in which the parameter $\\alpha =2$ , the restart in the GMRES is 20, and AMG is used to approximate the inverse of $A_h$ .", "By the iteration steps listed in Tables REF -REF , the AMG-CG solver is highly efficient for solving equations (REF ) and ().", "And the preconditioned GMRES algorithm is also very efficient and robust for the nonconforming $P_1$ -$P_0$ element method of Brinkman problem ()-() for $\\varepsilon =1, 10^{-1}$ .", "In summary, to efficiently solve the MWX method (REF ) we can employ the AMG-CG solver for small $\\varepsilon $ and the GMRES method with the preconditioner (REF ) for large $\\varepsilon $ .", "Table: Error ∥u-u h0 ∥ ε,h \\Vert u-u_{h0}\\Vert _{\\varepsilon , h} of the MWX method () for Example with different ε\\varepsilon and hhTable: Iteration steps of the MWX methods () for Example with different ε\\varepsilon amd hhTable: Iteration steps of the decoupled methods ()-() for Example with ε=1\\varepsilon =1 and different hhTable: Iteration steps of the decoupled methods ()-() for Example with ε=10 -1 \\varepsilon =10^{-1} and different hhExample 6.2 Next we verify the convergence of the MWX methods (REF ) and (REF ) for problem (REF ) with boundary layers.", "Let the exact solution of the Poisson equation (REF ) be $u^0(x,y)=\\sin (\\pi x)\\sin (\\pi y).$ Then the right hand term for both problems (REF ) and (REF ) is set to be $f (x, y) = -\\Delta u^0 = 2\\pi ^2\\sin (\\pi x)\\sin (\\pi y).$ The explicit expression solution $u$ for problem (REF ) with this right hand term is unknown.", "The solution $u$ possesses strong boundary layers when $\\varepsilon $ is very small.", "Here we choose $\\varepsilon =10^{-6}$ .", "Errors $\\Vert u^0-u_{h}\\Vert _{0}$ , $\|u^0-u_{h}\|_{1, h}$ , $\|u^0-u_{h}\|_{2, h}$ and $\\Vert u^0-u_{h}\\Vert _{\\varepsilon , h}$ of the discrete method (REF ) for $\\ell =1$ and $\\ell =2$ are present in Table REF and Table REF respectively, from which we can see that $\\Vert u^0-u_{h}\\Vert _{0}=O(h^{1.5})$ , $\|u^0-u_{h}\|_{1, h}=O(h^{0.5})$ , $\|u^0-u_{h}\|_{2, h}=O(h^{-0.5})$ and $\\Vert u^0-u_{h}\\Vert _{\\varepsilon , h}=O(h^{0.5})$ .", "The numerical convergence rate of error $\\Vert u^0-u_{h}\\Vert _{\\varepsilon , h}$ coincides with (REF ).", "After applying the Nitsche's technique with the penalty constant $\\sigma =5$ , errors $\\Vert u^0-u_{h}\\Vert _{0}$ , $\|u^0-u_{h}\|_{1, h}$ , $\\interleave u^0-u_{h}\\interleave _{2, h}$ and $\\interleave u^0-u_{h}\\interleave _{\\varepsilon , h}$ of the discrete method (REF ) for $\\ell =1$ and $\\ell =2$ are present in Table REF and Table REF respectively.", "When $\\ell =1$ , numerically $\\Vert u^0-u_{h}\\Vert _{0}=O(h^{2})$ , $\|u^0-u_{h}\|_{1, h}=O(h^{1.5})$ , $\\interleave u^0-u_{h}\\interleave _{2, h}=O(h^{0.5})$ and $\\interleave u^0-u_{h}\\interleave _{\\varepsilon , h}=O(h^{1.5})$ .", "All these convergence rates are optimal.", "And the convergence rates of $\|u^0-u_{h}\|_{1, h}$ and $\\interleave u^0-u_{h}\\interleave _{\\varepsilon , h}$ are half order higher than the optimal rates, as indicated by (REF ).", "For $\\ell =2$ , it is observed from Table REF that $\\Vert u^0-u_{h}\\Vert _{0}=O(h^{3})$ , $\|u^0-u_{h}\|_{1, h}=O(h^{2})$ , $\\interleave u^0-u_{h}\\interleave _{2, h}=O(h)$ and $\\interleave u^0-u_{h}\\interleave _{\\varepsilon , h}=O(h^{2})$ .", "Again all these convergence rates are optimal, and the convergence rate of $\\interleave u^0-u_{h}\\interleave _{\\varepsilon , h}$ is in coincidence with (REF ).", "Table: Errors of the discrete method () for Example with different hh when ε=10 -6 \\varepsilon =10^{-6} and ℓ=1\\ell =1Table: Errors of the discrete method () for Example with different hh when ε=10 -6 \\varepsilon =10^{-6} and ℓ=2\\ell =2Table: Errors of the discrete method () for Example with different hh when ε=10 -6 \\varepsilon =10^{-6} and ℓ=1\\ell =1Table: Errors of the discrete method () for Example with different hh when ε=10 -6 \\varepsilon =10^{-6} and ℓ=2\\ell =2" ] ]	2011.14064
[ [ "Fundamental Schemes for Efficient Unconditionally Stable Implicit\n Finite-Difference Time-Domain Methods" ], [ "Abstract This paper presents the generalized formulations of fundamental schemes for efficient unconditionally stable implicit finite-difference time-domain (FDTD) methods.", "The fundamental schemes constitute a family of implicit schemes that feature similar fundamental updating structures, which are in simplest forms with most efficient right-hand sides.", "The formulations of fundamental schemes are presented in terms of generalized matrix operator equations pertaining to some classical splitting formulae, including those of alternating direction implicit, locally one-dimensional and split-step schemes.", "To provide further insights into the implications and significance of fundamental schemes, the analyses are also extended to many other schemes with distinctive splitting formulae.", "Detailed algorithms are described for new efficient implementations of the unconditionally stable implicit FDTD methods based on the fundamental schemes.", "A comparative study of various implicit schemes in their original and new implementations is carried out, which includes comparisons of their computation costs and efficiency gains." ], [ "Introduction", "The finite-difference time-domain (FDTD) method has been widely used to obtain the numerical solutions of Maxwell's equations for investigating electromagnetic wave radiation, propagation and scattering problems [1].", "For the conventional explicit FDTD method [2], the computational efficiency is restricted by the Courant-Friedrichs-Lewy (CFL) stability condition, which imposes a maximum constraint on the time step size depending on the spatial mesh sizes.", "To remove the CFL condition, the unconditionally stable FDTD method based on the alternating direction implicit (ADI) technique has been developed [3], [4].", "The success of ADI-FDTD method has brought about a resurgence of interest in the unconditionally stable schemes not only within the electromagnetics community, but also among other scientists and mathematicians at large.", "With its unconditional stability advantage, the ADI-FDTD method has been extensively analyzed, improved and extended for many applications.", "Since the literature in these aspects is so voluminous, we could have easily missed out many references that are not directly relevant.", "For more comprehensive survey, we refer the readers to the bibliography in [1], [5], [6] (see also those cited in the author's previous works).", "The idea of ADI scheme that has been adapted in the recent celebrated ADI-FDTD, can be traced back to the early classic works by Peaceman and Rachford [7].", "Apart from the ADI scheme, many alternative implicit schemes have also been introduced by researchers in applied mathematics to deal with various parabolic, elliptic and hyperbolic partial differential systems.", "Some of such schemes that are more common and closely related to the present scope include locally one-dimensional (LOD) and Crank-Nicolson schemes [8]-[11].", "These schemes have been adapted as well for FDTD solutions of Maxwell's equations, leading to unconditionally stable LOD-FDTD method [12]-[13], split-step FDTD approach [14]-[15] and other Crank-Nicolson-based approximation methods [16].", "Most of these unconditionally stable FDTD methods have been built upon the classical implicit schemes by adopting their respective splitting formulae directly.", "Despite the successful adaptation of various classical schemes, continuing efforts are underway to devise new stable methods that are more efficient and simpler to implement.", "Recently, a new efficient algorithm has been presented for the ADI-FDTD method [17].", "The algorithm involves updating equations whose right-hand sides are much simpler and more concise than the conventional implementation [3], [4].", "This leads to substantial reduction in the number of arithmetic operations required for their computations.", "While the underlying principle of the new algorithm has helped make the ADI-FDTD simpler and more efficient, it actually has greater significance in its own right.", "This paper aims to clarify and extend such principle to arrive at a series of new efficient algorithms for various other unconditionally stable implicit FDTD methods, including some of those mentioned above.", "Moreover, it will be found that the resultant algorithms constitute a family of implicit schemes, all of which feature similar fundamental updating structures that are in simplest forms with most efficient right-hand sides.", "The organization of this paper is as follows.", "Section presents the formulations of fundamental schemes in terms of generalized matrix operator equations pertaining to some classical splitting formulae, including those of ADI, LOD and split-step schemes.", "To provide further insights into the implications and significance of fundamental schemes, the analyses are also extended to many other schemes with distinctive splitting formulae.", "In Section , detailed algorithms are described based on the fundamental schemes for new efficient implementations of the unconditionally stable implicit FDTD methods, e.g.", "ADI-FDTD and LOD-FDTD.", "In Section , a comparative study of various implicit schemes in their original and new implementations is carried out, which includes comparisons of their computation costs and efficiency gains.", "The fundamental nature of new implementations will also become evident through the comparisons and discussions." ], [ "Generalized Formulations", "In this section, we present the generalized formulations of fundamental schemes for implicit finite-difference methods.", "Starting from some classical implicit schemes, their generalized matrix operator equations are revisited and reformulated in the simplest and most efficient forms.", "These new forms feature convenient matrix-operator-free right-hand sides with least number of terms, which will lead to coding simplification in their algorithm implementations.", "For simplicity and clarity, we omit the nonhomogeneous terms that appear merely as vectors (without involving matrices)." ], [ "ADI", "The ADI scheme, originated by Peaceman and Rachford [7], is one of the most popular implicit finite-difference schemes in use today.", "This scheme calls for generalized splitting formulae in the form $\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+\\frac{1}{2}}= \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n}\\\\\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n+1}= \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+\\frac{1}{2}}.$ For many decades, such splitting formulae form the basis of many other numerical methods, which include the recent unconditionally stable ADI-FDTD method [3], [4].", "Note that the specific matrix operators $\\mathbf {A} {}$ and $\\mathbf {B} {}$ of [3], [4] (for 3-D Maxwell) are different from those of [7] (for 2-D parabolic), even though they conform to the same generalized form (REF ).", "In this section, we shall let these matrices be general but would caution that one must choose their operators properly for a particular scheme to stay unconditionally stable [16].", "To implement the ADI algorithm, it is more convenient to introduce auxiliary variables for denoting the right-hand sides of implicit equations.", "This allows us to rewrite the original algorithm as $\\mathbf {v} {}^{n}= \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n}\\\\\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+\\frac{1}{2}}= \\mathbf {v} {}^{n}\\\\\\mathbf {v} {}^{n+\\frac{1}{2}}= \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+\\frac{1}{2}}\\\\\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n+1}= \\mathbf {v} {}^{n+\\frac{1}{2}}$ where the $\\mathbf {v} {}$ 's serve as the auxiliary variables.", "If one exploits these auxiliary variables, the original algorithm above can be modified into one of the more efficient scheme.", "In particular, based on () at one time step backward: $\\mathbf {v} {}^{n-\\frac{1}{2}}= \\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n},$ it follows that $\\mathbf {v} {}^{n}$ of (REF ) is reducible to $\\mathbf {v} {}^{n}& = \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n}\\\\& = 2 \\mathbf {u} {}^{n} - \\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n}\\\\& = 2 \\mathbf {u} {}^{n} - \\mathbf {v} {}^{n-\\frac{1}{2}}.$ Furthermore, upon recognizing (), $\\mathbf {v} {}^{n+\\frac{1}{2}}$ of () is also reducible to $\\mathbf {v} {}^{n+\\frac{1}{2}}& = \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+\\frac{1}{2}}\\\\& = 2 \\mathbf {u} {}^{n+\\frac{1}{2}} - \\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+\\frac{1}{2}}\\\\& = 2 \\mathbf {u} {}^{n+\\frac{1}{2}} - \\mathbf {v} {}^{n}.$ With (REF ) and (REF ), algorithm (REF ) becomes more efficient as $\\mathbf {v} {}^{n}= 2 \\mathbf {u} {}^{n} - \\mathbf {v} {}^{n-\\frac{1}{2}}\\\\\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+\\frac{1}{2}}= \\mathbf {v} {}^{n}\\\\\\mathbf {v} {}^{n+\\frac{1}{2}}= 2 \\mathbf {u} {}^{n+\\frac{1}{2}} - \\mathbf {v} {}^{n}\\\\\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n+1}= \\mathbf {v} {}^{n+\\frac{1}{2}}.$ Through a simple re-definition of field variables $\\mathbf {\\tilde{u}} {}^{n} = 2 \\mathbf {u} {}^{n}, \\quad \\mathbf {\\tilde{u}} {}^{n+\\frac{1}{2}} = 2 \\mathbf {u} {}^{n+\\frac{1}{2}},$ we may reduce the algorithm further into [17] $\\mathbf {v} {}^{n}= \\mathbf {\\tilde{u}} {}^{n} - \\mathbf {v} {}^{n-\\frac{1}{2}}\\\\\\Big ( \\frac{1}{2}\\mathbf {I} {}- \\frac{ \\Delta t}{4} \\mathbf {A} {}\\Big ) \\mathbf {\\tilde{u}} {}^{n+\\frac{1}{2}}= \\mathbf {v} {}^{n}\\\\\\mathbf {v} {}^{n+\\frac{1}{2}}= \\mathbf {\\tilde{u}} {}^{n+\\frac{1}{2}} - \\mathbf {v} {}^{n}\\\\\\Big ( \\frac{1}{2}\\mathbf {I} {}- \\frac{ \\Delta t}{4} \\mathbf {B} {}\\Big ) \\mathbf {\\tilde{u}} {}^{n+1}= \\mathbf {v} {}^{n+\\frac{1}{2}}.$ This algorithm proceeds in terms of $\\mathbf {\\tilde{u}} {}$ 's for its main iterations, and only when the field output is required, one may retrieve the data from $\\mathbf {u} {}^{n+1} = \\frac{1}{2}\\mathbf {\\tilde{u}} {}^{n+1}.$ Equations (REF )-() constitute the most efficient ADI scheme that has the simplest right-hand sides without involving any explicit matrix operator.", "For nonzero initial fields, the algorithm takes the input initialization $\\mathbf {v} {}^{-\\frac{1}{2}} = \\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{0}.$ Note that such initialization will not degrade the accuracy of ADI scheme since it simply corresponds to (REF ) with $n=0$ or () with $n=-1$ .", "Furthermore, (REF ) along with (REF )-() are just (REF )-() or (REF ).", "Hence the equivalence of both present and original (generalized) Peaceman-Rachford schemes becomes evident here." ], [ "LOD1/SS1", "The LOD scheme, introduced in the early Russian literature (cf.", "[8], [10]), is another classical scheme that has been used extensively.", "This scheme calls for generalized splitting formulae in the form $\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+\\frac{1}{2}}= \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n}\\\\\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n+1}= \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n+\\frac{1}{2}}.$ Such splitting formulae have been adopted to develop the recent unconditionally stable LOD-FDTD method [12] and split-step approach [14].", "When $\\mathbf {A} {}$ and $\\mathbf {B} {}$ do not commute, the scheme is accurate to first order in time.", "Thus, the resultant LOD-FDTD method and split-step approach may be denoted by LOD1 and SS1 respectively.", "For most efficiency, the original LOD scheme can be modified into simplest form with matrix-operator-free right-hand sides.", "From (REF ), we have $\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+\\frac{1}{2}}& = \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n}\\\\& = 2 \\mathbf {u} {}^{n} - \\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n}.$ This can be manipulated readily to give $\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\Big ( \\mathbf {u} {}^{n+\\frac{1}{2}} + \\mathbf {u} {}^{n} \\Big )= 2 \\mathbf {u} {}^{n}$ where the vector terms in bracket may be denoted by auxiliary variable $\\mathbf {v} {}^{n+\\frac{1}{2}} = \\mathbf {u} {}^{n+\\frac{1}{2}} + \\mathbf {u} {}^{n}.$ Similar manipulation applies to () which leads to auxiliary variable $\\mathbf {v} {}^{n+1}$ .", "Combining all auxiliary and field variables, we arrive at $\\Big ( \\frac{1}{2}\\mathbf {I} {}- \\frac{ \\Delta t}{4} \\mathbf {A} {}\\Big ) \\mathbf {v} {}^{n+\\frac{1}{2}}= \\mathbf {u} {}^{n}\\\\\\mathbf {u} {}^{n+\\frac{1}{2}}= \\mathbf {v} {}^{n+\\frac{1}{2}} - \\mathbf {u} {}^{n}\\\\\\Big ( \\frac{1}{2}\\mathbf {I} {}- \\frac{ \\Delta t}{4} \\mathbf {B} {}\\Big ) \\mathbf {v} {}^{n+1}= \\mathbf {u} {}^{n+\\frac{1}{2}}\\\\\\mathbf {u} {}^{n+1}= \\mathbf {v} {}^{n+1} - \\mathbf {u} {}^{n+\\frac{1}{2}}.$ In these equations, all their right-hand sides are seen to be in the most convenient matrix-operator-free form.", "Furthermore, there is no special input initialization required and the output field solution is directly available from ()." ], [ "SS2", "While the former SS1 approach is only first-order accurate in time, the general split-step approach actually permits simple extensions to achieve higher-order temporal accuracy [14].", "Consider the split-step approach of second-order accuracy denoted by SS2.", "It is based on the Strang splitting formulae [9] and involves three updating procedures $\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{4} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+\\frac{1}{4}}= \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{4} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n}\\\\\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n+\\frac{3}{4}}= \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n+\\frac{1}{4}}\\\\\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{4} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+1}= \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{4} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+\\frac{3}{4}}.$ In line with the previous subsection, the algorithm implementation can be made much simpler along with improved efficiency by resorting to $\\Big ( \\frac{1}{2}\\mathbf {I} {}- \\frac{ \\Delta t}{8} \\mathbf {A} {}\\Big ) \\mathbf {v} {}^{n+\\frac{1}{4}}= \\mathbf {u} {}^{n}\\\\\\mathbf {u} {}^{n+\\frac{1}{4}}= \\mathbf {v} {}^{n+\\frac{1}{4}} - \\mathbf {u} {}^{n}\\\\\\Big ( \\frac{1}{2}\\mathbf {I} {}- \\frac{ \\Delta t}{4} \\mathbf {B} {}\\Big ) \\mathbf {v} {}^{n+\\frac{3}{4}}= \\mathbf {u} {}^{n+\\frac{1}{4}}\\\\\\mathbf {u} {}^{n+\\frac{3}{4}}= \\mathbf {v} {}^{n+\\frac{3}{4}} - \\mathbf {u} {}^{n+\\frac{1}{4}}\\\\\\Big ( \\frac{1}{2}\\mathbf {I} {}- \\frac{ \\Delta t}{8} \\mathbf {A} {}\\Big ) \\mathbf {v} {}^{n+1}= \\mathbf {u} {}^{n+\\frac{3}{4}}\\\\\\mathbf {u} {}^{n+1}= \\mathbf {v} {}^{n+1} - \\mathbf {u} {}^{n+\\frac{3}{4}}.$ Again, we have the right-hand sides of these equations cast in the most convenient matrix-operator-free form.", "However, since there are three implicit equations to be dealt with, the scheme still involves more arithmetic operations than those having merely two implicit equations with matrix-operator-free right-hand sides." ], [ "LOD2", "An alternative LOD-FDTD method denoted by LOD2 has been presented, which does not exhibit the non-commutativity error and preserves the second-order temporal accuracy [13].", "In essence, the main iterations proceed with two updating procedures in the same way like LOD1, cf.", "(REF )-().", "However, the time indices are associated with advancement of time steps from $n+1/4$ to $n+3/4$ and from $n+3/4$ to $n+1(1/4)$ as $\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+\\frac{3}{4}}= \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+\\frac{1}{4}}\\\\\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n+1\\frac{1}{4}}= \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n+\\frac{3}{4}}.$ To relate the fields to those normally at integer time steps, such association will prompt for additional processing for the input field data $\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{4} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{\\frac{1}{4}}= \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{4} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{0}$ as well as for the output field data $\\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{4} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n+1}= \\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{4} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n+1\\frac{1}{4}}.$ Note that the input processing is to be invoked only once at the beginning, while for the output processing at $n+1$ , it may be performed separately and independently in parallel without disrupting the main iterations.", "Furthermore, one often does not need to frequently output all field components, except probably one or two of interest at few desired observation points after certain (fairly long) interval periodically.", "The LOD2 algorithm above, with its facilitation for parallel/reduced/infrequent output processing, has been found to be more efficient than those of ADI and SS2 methods in their original implementations [13].", "To improve the efficiency further, we perform simplifications like previous subsections to arrive at matrix-operator-free right-hand sides for the main iterations as $\\Big ( \\frac{1}{2}\\mathbf {I} {}- \\frac{ \\Delta t}{4} \\mathbf {A} {}\\Big ) \\mathbf {v} {}^{n+\\frac{3}{4}}= \\mathbf {u} {}^{n+\\frac{1}{4}}\\\\\\mathbf {u} {}^{n+\\frac{3}{4}}= \\mathbf {v} {}^{n+\\frac{3}{4}} - \\mathbf {u} {}^{n+\\frac{1}{4}}\\\\\\Big ( \\frac{1}{2}\\mathbf {I} {}- \\frac{ \\Delta t}{4} \\mathbf {B} {}\\Big ) \\mathbf {v} {}^{n+1\\frac{1}{4}}= \\mathbf {u} {}^{n+\\frac{3}{4}}\\\\\\mathbf {u} {}^{n+1\\frac{1}{4}}= \\mathbf {v} {}^{n+1\\frac{1}{4}} - \\mathbf {u} {}^{n+\\frac{3}{4}}.$ If desired, similar simplifications may be performed for the input processing $\\Big ( \\frac{1}{2}\\mathbf {I} {}- \\frac{ \\Delta t}{8} \\mathbf {B} {}\\Big ) \\mathbf {v} {}^{\\frac{1}{4}}= \\mathbf {u} {}^{0}\\\\\\mathbf {u} {}^{\\frac{1}{4}}= \\mathbf {v} {}^{\\frac{1}{4}} - \\mathbf {u} {}^{0}$ and output processing $\\Big ( \\frac{1}{2}\\mathbf {I} {}+ \\frac{ \\Delta t}{8} \\mathbf {B} {}\\Big ) \\mathbf {v} {}^{n+1}= \\mathbf {u} {}^{n+1\\frac{1}{4}}\\\\\\mathbf {u} {}^{n+1}= \\mathbf {v} {}^{n+1} - \\mathbf {u} {}^{n+1\\frac{1}{4}}.$ Equations (REF )-() constitute the most efficient LOD2 scheme which is comparable to that of (REF )-()." ], [ "Fundamental Implications and Significance", "It turns out that the new algorithm implementations presented above represent some very fundamental ones for implicit finite-difference methods.", "As is evident from (REF ) and (REF ) or (REF ), these schemes merely involve similar updating structures, namely implicit updating with (simplest) matrix-operator-free right-hand sides and explicit updating via subtraction of two vectors.", "Their main slight difference might just be the sequence of implicit and explicit updatings.", "Moreover, one may convert from one scheme to another simply by proper initialization and swapping the roles of field and auxiliary variables.", "For instance, treating $\\mathbf {v} {}^{n}$ as the field variables (uncontaminated to second order) in (REF ) and using $\\mathbf {u} {}^{n}= \\Big ( \\frac{1}{2}\\mathbf {I} {}+ \\frac{ \\Delta t}{4} \\mathbf {B} {}\\Big ) \\mathbf {v} {}^{n}$ to initialize its iterations, one may obtain the $\\mathbf {v} {}^{n+1}$ field solutions from () that correspond to the $\\mathbf {u} {}^{n+1}$ field solutions from (REF )-(REF ).", "Such interesting but subtle link between LOD and ADI schemes has been pointed out earlier [18].", "Through the fundamental schemes herein, the link becomes particularly obvious with their similar updating structures and with the aid of variables that are otherwise not defined in the original schemes, e.g.", "$\\mathbf {v} {}$ is not directly seen in (REF ).", "The implications of the fundamental schemes above are actually more far-reaching.", "In particular, we note that many other classical and recent non-dissipative splitting schemes, such as D'Yakonov scheme, delta formulation, Crank-Nicolson direct-splitting method etc.", "may be cast into the same simplest form of (REF ) featuring matrix-operator-free right-hand sides.", "Such analyses when extended to these schemes with distinctive splitting formulae, will give further insights into the significance of fundamental schemes.", "Let us illustrate this point using the scheme with splitting formulae (cf.", "generalized matricization of D'Yakonov or Beam-Warming scheme) $\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{}= \\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big )\\Big ( \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n}\\\\\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n+1}= \\mathbf {u} {}^{}.$ Such scheme may be reduced to an efficient one with the iterations algorithm read as $\\mathbf {v} {}^{n}= \\mathbf {u} {}^{n} - \\mathbf {v} {}^{n-\\frac{1}{2}}\\\\\\Big ( \\frac{1}{2}\\mathbf {I} {}- \\frac{ \\Delta t}{4} \\mathbf {A} {}\\Big ) \\mathbf {u} {}^{n+\\frac{1}{2}}= \\mathbf {v} {}^{n}\\\\\\mathbf {v} {}^{n+\\frac{1}{2}}= \\mathbf {u} {}^{n+\\frac{1}{2}} - \\mathbf {v} {}^{n}\\\\\\Big ( \\frac{1}{2}\\mathbf {I} {}- \\frac{ \\Delta t}{4} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n+1}= \\mathbf {v} {}^{n+\\frac{1}{2}}$ along with input initialization $\\mathbf {v} {}^{-\\frac{1}{2}} = \\Big ( \\frac{1}{2}\\mathbf {I} {}- \\frac{ \\Delta t}{4} \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{0}.$ Notice that the form of (REF ) is just that of (REF ), while noting also the relation $\\mathbf {u} {}^{} = 2 \\mathbf {v} {}^{n+\\frac{1}{2}}.$ Another illustration is the splitting formulae (cf.", "generalized matricization of 2D Douglas-Gunn scheme or delta formulation) $\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {A} {}\\Big ) \\Delta \\mathbf {u} {}^{}= \\Delta t \\Big ( \\mathbf {A} {}+ \\mathbf {B} {}\\Big ) \\mathbf {u} {}^{n}\\\\\\Big ( \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\mathbf {B} {}\\Big ) \\Delta \\mathbf {u} {}^{}= \\Delta \\mathbf {u} {}^{}\\\\\\mathbf {u} {}^{n+1} = \\mathbf {u} {}^{n} + \\Delta \\mathbf {u} {}^{}.$ Again, this can be turned into the previous fundamental scheme, specifically (REF )-(REF ) and with $\\Delta \\mathbf {u} {}^{} = \\mathbf {\\tilde{u}} {}^{n+\\frac{1}{2}} - \\mathbf {\\tilde{u}} {}^{n}\\\\\\Delta \\mathbf {u} {}^{} = \\frac{1}{2}\\Big ( \\mathbf {\\tilde{u}} {}^{n+1} - \\mathbf {\\tilde{u}} {}^{n} \\Big ).$ Meanwhile, the fundamental scheme of (REF )-() forms the basis of simplification for many split-step methods of higher order accuracy in [14].", "This will lead to their convenient algorithms all with matrix-operator-free right-hand sides following the same way that underpins ()-(), (REF )-() and (REF )-(), etc.", "Even for the classical Crank-Nicolson scheme having the original unfactorized form $\\bigg [ \\mathbf {I} {}- \\frac{ \\Delta t}{2} \\Big ( \\mathbf {A} {}+ \\mathbf {B} {}\\Big ) \\bigg ] \\mathbf {u} {}^{n+1}= \\bigg [ \\mathbf {I} {}+ \\frac{ \\Delta t}{2} \\Big ( \\mathbf {A} {}+ \\mathbf {B} {}\\Big ) \\bigg ] \\mathbf {u} {}^{n},$ one may based on similar manipulation to devise $\\bigg [ \\frac{1}{2}\\mathbf {I} {}- \\frac{ \\Delta t}{4} \\Big ( \\mathbf {A} {}+ \\mathbf {B} {}\\Big ) \\bigg ] \\mathbf {u} {}^{n+\\frac{1}{2}}= \\mathbf {u} {}^{n}\\\\\\mathbf {u} {}^{n+1}= \\mathbf {u} {}^{n+\\frac{1}{2}} - \\mathbf {u} {}^{n}.$ This obviously constitutes the simpler algorithm than that of the original scheme.", "It should be mentioned that the extent of simplification and actual improvement of computation efficiency for various fundamental schemes depend much on the particular matrix operators and algorithm implementation details (to be described next).", "Although the scope of this paper is mainly about the unconditionally stable implicit FDTD methods in electromagnetics, it is evident that all fundamental schemes discussed can be extended readily to many other finite-difference schemes in computational physics, even conditionally stable or parabolic/elliptic ones, etc.", "Some numerical methods that bear resemblance to the schemes above such as implicit integrations or iterative solutions of certain control problems may be made simpler and more efficient in the similar manner." ], [ "Efficient Implementations", "In this section, we describe the detailed algorithms for new efficient implementations of the unconditionally stable implicit FDTD methods based on the fundamental schemes.", "For simplicity, consider the lossless isotropic medium with permittivity $\\epsilon $ and permeability $\\mu $ .", "The splitting matrix operators of Maxwell's equations may be selected as $\\mathbf {A} {}=\\begin{bmatrix}0 & 0 & 0 & 0 & 0 & \\frac{1}{\\epsilon } \\partial _y\\\\0 & 0 & 0 & \\frac{1}{\\epsilon } \\partial _z & 0 & 0\\\\0 & 0 & 0 & 0 & \\frac{1}{\\epsilon } \\partial _x & 0\\\\0 & \\frac{1}{\\mu } \\partial _z & 0 & 0 & 0 & 0\\\\0 & 0 & \\frac{1}{\\mu } \\partial _x & 0 & 0 & 0\\\\\\frac{1}{\\mu } \\partial _y & 0 & 0 & 0 & 0 & 0\\end{bmatrix}\\\\\\mathbf {B} {}=\\begin{bmatrix}0 & 0 & 0 & 0 & \\frac{-1}{\\epsilon } \\partial _z & 0\\\\0 & 0 & 0 & 0 & 0 & \\frac{-1}{\\epsilon } \\partial _x\\\\0 & 0 & 0 & \\frac{-1}{\\epsilon } \\partial _y & 0 & 0\\\\0 & 0 & \\frac{-1}{\\mu } \\partial _y & 0 & 0 & 0\\\\\\frac{-1}{\\mu } \\partial _z & 0 & 0 & 0 & 0 & 0\\\\0 & \\frac{-1}{\\mu } \\partial _x & 0 & 0 & 0 & 0\\end{bmatrix}$ where $\\partial _x$ , $\\partial _y$ , $\\partial _z$ are the spatial difference operators for the first derivatives along $x$ , $y$ , $z$ directions respectively.", "The vector components of field and auxiliary variables can be written as $\\mathbf {u} {}=\\begin{bmatrix}E_x \\\\ E_y \\\\ E_z \\\\ H_x \\\\ H_y \\\\ H_z\\end{bmatrix}, \\quad \\mathbf {\\tilde{u}} {}=\\begin{bmatrix}\\tilde{E}_x \\\\ \\tilde{E}_y \\\\ \\tilde{E}_z \\\\ \\tilde{H}_x \\\\ \\tilde{H}_y \\\\ \\tilde{H}_z\\end{bmatrix}, \\quad \\mathbf {v} {}=\\begin{bmatrix}e_x \\\\ e_y \\\\ e_z \\\\ h_x \\\\ h_y \\\\ h_z\\end{bmatrix}.$ For convenience, we also introduce the notations $b = \\frac{ \\Delta t}{2 \\epsilon }, \\quad d = \\frac{ \\Delta t}{2 \\mu }.$" ], [ "ADI", "Let us illustrate the implementation of fundamental ADI scheme (REF ).", "While equation (REF ) can be implemented directly as $e_\\xi ^{n}= \\tilde{E}_\\xi ^{n} - e_\\xi ^{n-\\frac{1}{2}}, \\quad \\xi =x,y,z\\\\h_\\xi ^{n}= \\tilde{H}_\\xi ^{n} - h_\\xi ^{n-\\frac{1}{2}}, \\quad \\xi =x,y,z,$ equation () may find more convenience through implicit updating $\\frac{1}{2}\\tilde{E}_x^{n+\\frac{1}{2}} - \\frac{b d}{2} \\partial _y^2 \\tilde{E}_x^{n+\\frac{1}{2}}= e_x^{n} + b \\partial _y h_z^{n}\\\\\\frac{1}{2}\\tilde{E}_y^{n+\\frac{1}{2}} - \\frac{b d}{2} \\partial _z^2 \\tilde{E}_y^{n+\\frac{1}{2}}= e_y^{n} + b \\partial _z h_x^{n}\\\\\\frac{1}{2}\\tilde{E}_z^{n+\\frac{1}{2}} - \\frac{b d}{2} \\partial _x^2 \\tilde{E}_z^{n+\\frac{1}{2}}= e_z^{n} + b \\partial _x h_y^{n},$ followed by explicit updating $\\tilde{H}_x^{n+\\frac{1}{2}}= 2 h_x^{n} + d \\partial _z \\tilde{E}_y^{n+\\frac{1}{2}}\\\\\\tilde{H}_y^{n+\\frac{1}{2}}= 2 h_y^{n} + d \\partial _x \\tilde{E}_z^{n+\\frac{1}{2}}\\\\\\tilde{H}_z^{n+\\frac{1}{2}}= 2 h_z^{n} + d \\partial _y \\tilde{E}_x^{n+\\frac{1}{2}}.$ Similar arguments apply to equations () and (), where the former is implemented directly as $e_\\xi ^{n+\\frac{1}{2}}= \\tilde{E}_\\xi ^{n+\\frac{1}{2}} - e_\\xi ^{n}, \\quad \\xi =x,y,z\\\\h_\\xi ^{n+\\frac{1}{2}}= \\tilde{H}_\\xi ^{n+\\frac{1}{2}} - h_\\xi ^{n}, \\quad \\xi =x,y,z,$ while the latter comprises implicit updating $\\frac{1}{2}\\tilde{E}_x^{n+1} - \\frac{b d}{2} \\partial _z^2 \\tilde{E}_x^{n+1}= e_x^{n+\\frac{1}{2}} - b \\partial _z h_y^{n+\\frac{1}{2}}\\\\\\frac{1}{2}\\tilde{E}_y^{n+1} - \\frac{b d}{2} \\partial _x^2 \\tilde{E}_y^{n+1}= e_y^{n+\\frac{1}{2}} - b \\partial _x h_z^{n+\\frac{1}{2}}\\\\\\frac{1}{2}\\tilde{E}_z^{n+1} - \\frac{b d}{2} \\partial _y^2 \\tilde{E}_z^{n+1}= e_z^{n+\\frac{1}{2}} - b \\partial _y h_x^{n+\\frac{1}{2}},$ and explicit updating $\\tilde{H}_x^{n+1}= 2 h_x^{n+\\frac{1}{2}} - d \\partial _y \\tilde{E}_z^{n+1}\\\\\\tilde{H}_y^{n+1}= 2 h_y^{n+\\frac{1}{2}} - d \\partial _z \\tilde{E}_x^{n+1}\\\\\\tilde{H}_z^{n+1}= 2 h_z^{n+\\frac{1}{2}} - d \\partial _x \\tilde{E}_y^{n+1}.$ When the difference operators above represent specifically the second-order central-differencing operators on Yee cells, equations (REF )-(REF ) correspond to the efficient algorithm delineated in [17].", "In particular, the solution $\\tilde{E}_\\xi ^{n+1}$ from (REF ) coincides with $E_\\xi ^\|^{n+1}$ from [17], while the solution $\\tilde{H}_\\xi ^{n+1}$ from (REF ) is half of $H_\\xi ^\|^{n+1}$ from [17].", "The reason for such magnetic field relation is that certain variables and coefficients in (REF )-(REF ) have been re-scaled in the corresponding auxiliary and explicit updating equations of [17].", "This helps save the scaling of final magnetic field solution that is otherwise necessary, cf.", "(REF ).", "Moreover, additional savings of operations can be achieved by combining (REF ) and (REF ) as $h_x^{n+\\frac{1}{2}}= h_x^{n} + d \\partial _z \\tilde{E}_y^{n+\\frac{1}{2}}\\\\h_y^{n+\\frac{1}{2}}= h_y^{n} + d \\partial _x \\tilde{E}_z^{n+\\frac{1}{2}}\\\\h_z^{n+\\frac{1}{2}}= h_z^{n} + d \\partial _y \\tilde{E}_x^{n+\\frac{1}{2}}.$ When the final magnetic field data is not needed frequently, (REF ) and (REF ) (at $n+1$ time step) may also be combined for higher efficiency: $h_x^{n+1}= h_x^{n+\\frac{1}{2}} - d \\partial _y \\tilde{E}_z^{n+1}\\\\h_y^{n+1}= h_y^{n+\\frac{1}{2}} - d \\partial _z \\tilde{E}_x^{n+1}\\\\h_z^{n+1}= h_z^{n+\\frac{1}{2}} - d \\partial _x \\tilde{E}_y^{n+1}.$ Other implementation details may be referred to [17] including for-looping, tridiagonal system solving, memory reuse etc.", "In actuality, there are many possibilities as far as detailed implementations are concerned, such as different definition and scaling of variables, different implicit and explicit updatings, different difference operators, e.g.", "higher order, parameter optimized, compact etc.", "[19]-[22].", "The updating equations presented above still leave much room and generality for exploring into these possibilities." ], [ "LOD2", "For subsequent discussions and comparisons, we also describe the algorithm implementation of fundamental LOD2 scheme (REF ) in more detail and yet general.", "Adopting the same notations of (REF )-(REF ), equation (REF ) may be split into implicit updating $\\frac{1}{2}e_x^{n+\\frac{3}{4}} - \\frac{b d}{2} \\partial _y^2 e_x^{n+\\frac{3}{4}}= E_x^{n+\\frac{1}{4}} + b \\partial _y H_z^{n+\\frac{1}{4}}\\\\\\frac{1}{2}e_y^{n+\\frac{3}{4}} - \\frac{b d}{2} \\partial _z^2 e_y^{n+\\frac{3}{4}}= E_y^{n+\\frac{1}{4}} + b \\partial _z H_x^{n+\\frac{1}{4}}\\\\\\frac{1}{2}e_z^{n+\\frac{3}{4}} - \\frac{b d}{2} \\partial _x^2 e_z^{n+\\frac{3}{4}}= E_z^{n+\\frac{1}{4}} + b \\partial _x H_y^{n+\\frac{1}{4}}$ and explicit updating $h_x^{n+\\frac{3}{4}}= 2 H_x^{n+\\frac{1}{4}} + d \\partial _z e_y^{n+\\frac{3}{4}}\\\\h_y^{n+\\frac{3}{4}}= 2 H_y^{n+\\frac{1}{4}} + d \\partial _x e_z^{n+\\frac{3}{4}}\\\\h_z^{n+\\frac{3}{4}}= 2 H_z^{n+\\frac{1}{4}} + d \\partial _y e_x^{n+\\frac{3}{4}},$ while () is directly $E_\\xi ^{n+\\frac{3}{4}}= e_\\xi ^{n+\\frac{3}{4}} - E_\\xi ^{n+\\frac{1}{4}}, \\quad \\xi =x,y,z\\\\H_\\xi ^{n+\\frac{3}{4}}= h_\\xi ^{n+\\frac{3}{4}} - H_\\xi ^{n+\\frac{1}{4}}, \\quad \\xi =x,y,z.$ Likewise, equation () consists of implicit updating $\\frac{1}{2}e_x^{n+1\\frac{1}{4}} - \\frac{b d}{2} \\partial _z^2 e_x^{n+1\\frac{1}{4}}= E_x^{n+\\frac{3}{4}} - b \\partial _z H_y^{n+\\frac{3}{4}}\\\\\\frac{1}{2}e_y^{n+1\\frac{1}{4}} - \\frac{b d}{2} \\partial _x^2 e_y^{n+1\\frac{1}{4}}= E_y^{n+\\frac{3}{4}} - b \\partial _x H_z^{n+\\frac{3}{4}}\\\\\\frac{1}{2}e_z^{n+1\\frac{1}{4}} - \\frac{b d}{2} \\partial _y^2 e_z^{n+1\\frac{1}{4}}= E_z^{n+\\frac{3}{4}} - b \\partial _y H_x^{n+\\frac{3}{4}}$ and explicit updating $h_x^{n+1\\frac{1}{4}}= 2 H_x^{n+\\frac{3}{4}} - d \\partial _y e_z^{n+1\\frac{1}{4}}\\\\h_y^{n+1\\frac{1}{4}}= 2 H_y^{n+\\frac{3}{4}} - d \\partial _z e_x^{n+1\\frac{1}{4}}\\\\h_z^{n+1\\frac{1}{4}}= 2 H_z^{n+\\frac{3}{4}} - d \\partial _x e_y^{n+1\\frac{1}{4}},$ whereas () is simply $E_\\xi ^{n+1\\frac{1}{4}}= e_\\xi ^{n+1\\frac{1}{4}} - E_\\xi ^{n+\\frac{3}{4}}, \\quad \\xi =x,y,z\\\\H_\\xi ^{n+1\\frac{1}{4}}= h_\\xi ^{n+1\\frac{1}{4}} - H_\\xi ^{n+\\frac{3}{4}}, \\quad \\xi =x,y,z.$ Table: Comparisons of Unconditionally Stable FDTD Methods with Second-order Spatial Central DifferenceAs before, there are many choices for the difference operators in equations (REF )-(REF ).", "For being more specific in the next section, we select the second-order central-differencing operators on Yee cells.", "To minimize the number of for-loops in the previous algorithm, some of their updating equations may be incorporated in the same loops.", "Moreover, when the auxiliary magnetic variables $h_\\xi $ are not to be output, the explicit updating equations (REF ) and () can be combined to read $H_x^{n+\\frac{3}{4}}= H_x^{n+\\frac{1}{4}} + d \\partial _z e_y^{n+\\frac{3}{4}}\\\\H_y^{n+\\frac{3}{4}}= H_y^{n+\\frac{1}{4}} + d \\partial _x e_z^{n+\\frac{3}{4}}\\\\H_z^{n+\\frac{3}{4}}= H_z^{n+\\frac{1}{4}} + d \\partial _y e_x^{n+\\frac{3}{4}},$ while (REF ) and () become $H_x^{n+1\\frac{1}{4}}= H_x^{n+\\frac{3}{4}} - d \\partial _y e_z^{n+1\\frac{1}{4}}\\\\H_y^{n+1\\frac{1}{4}}= H_y^{n+\\frac{3}{4}} - d \\partial _z e_x^{n+1\\frac{1}{4}}\\\\H_z^{n+1\\frac{1}{4}}= H_z^{n+\\frac{3}{4}} - d \\partial _x e_y^{n+1\\frac{1}{4}}.$ Meanwhile, to minimize the memory storage requirement, some of the variables may occupy the same memory spaces.", "For instance, one can choose to reuse the spaces of those variables grouped within curly braces as ($\\xi =x,y,z$ ): $\\lbrace E_\\xi ^{n+\\frac{1}{4}},e_\\xi ^{n+1\\frac{1}{4}},E_\\xi ^{n+1\\frac{1}{4}}\\rbrace ;\\quad \\lbrace e_\\xi ^{n+\\frac{3}{4}},E_\\xi ^{n+\\frac{3}{4}}\\rbrace ;\\\\\\lbrace H_\\xi ^{n+\\frac{1}{4}},h_\\xi ^{n+1\\frac{1}{4}},H_\\xi ^{n+1\\frac{1}{4}}\\rbrace ;\\quad \\lbrace h_\\xi ^{n+\\frac{3}{4}},H_\\xi ^{n+\\frac{3}{4}}\\rbrace .$ In a computer program, these variables could just share the same names to be assigned with new values successively.", "Note that in conjunction with (REF )-(REF ), one may absorb $h_\\xi $ altogether and reuse the $H_\\xi $ spaces.", "Following the similar arguments as above, the detailed algorithms for the fundamental LOD1/SS1 and SS2 schemes may be written and coded with reference to (REF ) and (REF ) accordingly." ], [ "Discussions and Comparisons", "Having systematically addressed various implicit schemes and their new algorithm implementations, we are now ready to make a comparative study.", "Of particular interest are the computation efficiency gains as compared to the original schemes, or more specifically over the prevailing ADI-FDTD implementation [3], [4].", "Furthermore, the fundamental nature of new algorithm implementations, as has been pointed out in Section REF , will become evident once again through the comparisons and discussions.", "In what follows, we acquire the floating-point operations (flops) count for the main iterations of ADI, LOD1/SS1, SS2 and LOD2 schemes.", "Based on the right-hand sides (RHS) of their respective updating equations using second-order central-differencing operators, cf.", "Sections and , the number of multiplications/divisions (M/D) and additions/subtractions (A/S) required for one complete time step are determined.", "Each complete step comprises two/three procedures whose implicit and explicit updating codes are assumed to have been arranged in order to achieve minimum number of for-loops.", "It is also assumed that all multiplicative factors have been precomputed and stored, while the number of electric and magnetic field components in all directions have been taken to be the same.", "Our results are summarized in Table REF , which lists the flops count for both original and new algorithm implementations of each scheme.", "For convenience, the pertaining equations involved in the main iterations are also indicated, along with those that may be needed for input initialization and output processing.", "When there is no input or output equation shown, it means that no special input or output treatment is required for the particular implementation.", "For the main iterations variables, their memory storage requirements are also specified in Table REF .", "By properly reusing the spaces of those variables grouped within the same table entity, only two field arrays need to be allocated for all.", "Note that judicious reuse of memory space is important so that the iterations do not often invoke virtual memory (e.g.", "via hard disk).", "Otherwise, they may lead to slower computations than those iterations with more flops but do not need virtual memory.", "From Table REF , it is clear that the total flops count (M/D+A/S) for all implicit schemes has been reduced considerably using the new implementations as compared to the original counterparts.", "Most notably, the conventional ADI-FDTD method based on (REF ) takes 102 flops [3], [4], whereas the new one based on (REF ) merely takes 42 flops, cf.", "Section REF (assuming no need for frequent output of magnetic field data).", "Notice that the LOD1/SS1 and LOD2 schemes, which take 72 flops in their original implementations [13], also take the same 42 flops using the new algorithms, cf.", "Section REF .", "Moreover, the number and type of flops for respective implicit and explicit updatings are seen to be identical for all these new implicit schemes.", "This is simply a manifestation of their aforementioned fundamental updating structures, which can be found to take 21 flops per updating procedure.", "Applying such argument for the SS2 scheme with three updating procedures, it is easy to deduce that its new algorithm will take 63 flops, in contrast to the 108 flops required when using its original implementation.", "Based on the flops count reduction for the right-hand sides of updating equations, Table REF has listed the efficiency gains over the conventional ADI-FDTD method.", "The new fundamental ADI, LOD1/SS1 and LOD2 schemes all feature the same efficiency gain of 2.43 with their same reduced total flops count.", "Since there is also the cost for solving tridiagonal systems, we estimate about $5N$ flops for a system of order $N$ using precomputed bidiagonally factorized elements.", "Taking this cost into account, all these new implementations still achieve an overall efficiency gain of 1.83 in flops count reduction over the conventional ADI-FDTD.", "It is interesting to find that even for the SS2 scheme with three updating procedures, the overall efficiency may still be higher (gain $\\sim 1.22$ ) by using the new algorithm implementation.", "Note that although we have carefully taken into account various flops count, there still exist other factors that may affect the actual computation efficiency.", "These factors are often dependent on the particular computer platform, hardware configuration, operating system, compiler software, program code arrangement, etc.", "[23].", "Besides the flops count, the for-loops count is also given in Table REF .", "Each for-loop is to perform the whole sweep of $i$ , $j$ , $k$ indices along $x$ , $y$ , $z$ directions for one field component.", "For both original and new implementations of ADI, LOD1/SS1 and LOD2 schemes, there are 12 for-loops for all updating equations in both procedures.", "For the SS2 scheme, there are 18 for-loops altogether corresponding to its three updating procedures.", "Meanwhile, the order of temporal accuracy is also summarized for various implicit schemes in Table REF .", "As mentioned in Section REF , even though the LOD1/SS1 scheme is only first-order accurate in time, it may be converted to other schemes of second-order accuracy by exploiting their similar fundamental updating structures.", "Although not included in Table REF , many other implicit schemes discussed in Section REF can be found to take the same reduced flops count (42) and the same for-loops count (12) when using the new implementations with (REF )-(REF ).", "This is anticipated because they all simply converge to the same fundamental schemes with matrix-operator-free right-hand sides just like those addressed in the table, even though their original splitting formulae are distinctive, cf.", "(REF )-(REF ).", "Here again the significance of these fundamental schemes accrues when one is equipped with a general methodical approach for improving the computation efficiency of their original counterparts.", "Moreover, their updating structures that are of fundamental nature make them well-suited to serve as the basis and benchmark for construction and development of future implicit schemes, e.g.", "with new $\\mathbf {A} {}$ and $\\mathbf {B} {}$ or different time-stepping procedures, etc." ], [ "Conclusion", "This paper has presented the generalized formulations of fundamental schemes for efficient unconditionally stable implicit FDTD methods.", "The formulations have been presented in terms of generalized matrix operator equations pertaining to classical splitting formulae of ADI, LOD and split-step schemes.", "To provide further insights into the implications and significance of fundamental schemes, the analyses have also been extended to many other schemes with distinctive splitting formulae.", "It has been noted that all the fundamental schemes feature similar fundamental updating structures that are in simplest forms with most efficient right-hand sides.", "Detailed algorithms have been described for new efficient implementations of the unconditionally stable implicit FDTD methods based on the fundamental schemes.", "A comparative study of various implicit schemes in their original and new implementations has been carried out, which includes comparisons of their computation costs and efficiency gains.", "The fundamental schemes presented in this paper will be of much usefulness and significance not only in electromagnetics, but also in many other areas that may be adopting various classical implicit schemes.", "With their simplest forms featuring the most efficient right-hand sides, these fundamental schemes will lead to coding simplification and efficiency improvement in algorithm implementations.", "Furthermore, their fundamental updating structures invite further investigations into their properties and subsequent extensions for their applications.", "They will also serve aptly as the basis and benchmark for construction and development of future implicit schemes." ] ]	2011.14043
[ [ "The ion-acoustic turbulence in the skin layer of the inductively coupled\n plasma" ], [ "Abstract The theory of the nonmodal ion-acoustic instability in the skin layer of the inductively coupled plasma (ICP) is developed.", "This instability has time dependent growth rate and is driven by the current formed in the skin layer by the accelerated motion of electrons relative to ions under the action of the ponderomotive force.", "It is found that the development of the ion acoustic turbulence (IAT) in the skin layer and the scattering of electrons by IAT are basic nonlinear channels of the nonlinear absorption of the RF energy in the skin layer." ], [ "Introduction", "The inductively coupled plasma (ICP) is a widespread and dominant plasma source for materials processing applications and operated in the low gas pressure regime of a few mTorr range [1].", "In this regime, the electron mean-free-path is comparable to the characteristic size of plasma devices, and the electron collision frequency with neutrals is much less than the driving frequency, $\\omega _{0}$ , of the applied electromagnetic (EM) wave.", "The absorption of the RF energy by electrons in this regime occurs due to the breaking of the phase coherence between the velocity of the individual thermal electron and the spatially nonuniform EM wave with frequency $\\omega _{0}$ much less than the electron plasma frequency $\\omega _{pe}$ .", "The RF electric field in ICPs with such a frequency is non-propagating and localized mainly in the skin layer near the plasma boundary adjacent to the RF antenna, where the main interaction of EM fields with plasma electrons takes place.", "This phenomenon involving the spatially inhomogeneous RF fields and nonlocal electron kinetics is well-known in plasma physics as the Landau damping effect.", "This regime of the ICP operation is also named as the regime of the anomalous skin effect[2], [3], [4] or nonlocal regime.", "The theory of the anomalous skin effect[2], [3], [4] is based on the solution of the boundary value problem for the combination of the linearized Vlasov equation for the electron distribution function and the Maxwell equations.", "This linear theory is developed for the limit of the weak RF field applied to the plasma boundary for which the quiver velocity of the plasma electrons in this spatially inhomogeneous RF field was assumed to be negligibly small comparing with the electron thermal velocity, $v_{Te}$ , and has been neglected.", "It is well known, however, that the relative motion of electrons and ions in a plasma under the action of EM waves is the source of the development of numerous parametric and current driven instabilities[5], [6], [7].", "The development of these instabilities is an alternative channel of the absorption of the EM wave and heating of plasma components, which is not included in the linear theory of the anomalous skin effect.", "It was found that the uniformly accelerated motion of electrons occurs in the skin layer, which stems from the ponderomotive force formed by the coupled action of the electric and the magnetic component of EM waves [8].", "The kinetic stability theory of the plasma with accelerated electrons [8] found that the electrons accelerated in skin layer triggers the current-driven instabilities.", "The analytical treatment of the instabilities driven by a current with spatially inhomogeneous or time dependent current velocity can't be investigated by employing the normal mode analysis, which assumes that the plasma perturbations have a structure $\\sim \\exp \\left(i\\mathbf {kr}-i\\omega t\\right)$ of a plane wave.", "We employed the nonmodal approach[8], which starts with the transformation of the position $\\mathbf {r}$ and the velocity $\\mathbf {v}$ variables of the Vlasov equation for the distribution function $F_{\\alpha }\\left(\\mathbf {v}, \\mathbf {r}, t\\right)$ of species $\\alpha $ (ions or electrons) determined in the laboratory frame to the variables $\\mathbf {r}_{\\alpha }$ and $\\mathbf {v}_{\\alpha }$ determined in the frame moving with spatially inhomogeneous time-dependent velocity $\\mathbf {V}_{\\alpha }\\left(\\mathbf {r}, t\\right)$ .", "In our approach, this spatially inhomogeneous velocity is determined by the Euler equation for the ideal fluid of particles species $\\alpha $ immersed in the spatially inhomogeneous non-stationary EM field.", "The Vlasov equation in the moving coordinates $\\mathbf {r}_{\\alpha }$ and $\\mathbf {v}_{\\alpha }$ has the same form as that in a plasma without an external EM field at a finite time interval during which a particle does not move into appreciably different regions of the EM field.", "The solution of this Euler equation was derived in our previous study[8] for the case of the high frequency RF wave, for which the force by the RF electric field acting on electrons in the skin layer prevails over the Lorentz force by the RF magnetic field.", "It was found that the electron velocity $\\mathbf {V}_{e}$ in this case is equal to the electron accelerated velocity under the action of the ponderomotive force.", "For the frequency range of the applied RF wave corresponding to the classical skin effect[3], [4], it was revealed[8] that the accelerated electrons leave the skin layer in a short time for the strong RF field for which the ponderomotive current velocity $U$ is larger than the electron thermal velocity $v_{Te}$ .", "This time appears to be insufficient for the development of the Buneman instability in the skin layer, which excites when the current velocity is larger than the electron thermal velocity.", "It was also found that electrons accelerated in the skin layer can trigger the ion-acoustic (IA) instability in the bulk plasma past the skin layer when the quasi-steady electron current velocity exceeds the IA velocity $v_{s}=\\left(T_{e}/m_{i}\\right)^{1/2}$ in this region[8].", "In this paper, we apply our nonmodal approach to the theory of the IA instability driven by the weak RF field, for which the accelerated electron current velocity in the skin layer is less than the electron thermal velocity, but exceeds the IA velocity.", "Under this condition, the accelerated electrons can develop the IA instability in the skin layer.", "In Sec.", ", we present basic transformations of the system of the Vlasov-Poisson equations, employed in the developed theory.", "We introduce a more general and simple solution of the Euler equation, presented in Appendix, for the electron velocity $\\mathbf {V}_{e}(\\mathbf {r}, t)$ in the decaying EM field without an initial assumption of the stronger electric force than the Lorentz force acting on electrons in the skin layer.", "In Sec.", ", we present the theory of the IA instability in the skin layer driven by the accelerated electrons.", "The nonlinear theory of the IA turbulence in the skin layer is presented in Sec.", "followed by conclusions in Sec.", "." ], [ "The nonmodal approach to the theory of the instabilities driven by the accelerated current in the skin layer", "In this paper, we consider the effect of the relative motion of plasma species on the development of the short-scale electrostatic perturbations in the skin layer under the condition of the classical skin effect.", "The skin effect is classified as classical or normal when the frequency $\\omega _{0}$ of the RF field belongs to the frequency range[3], [4] $&\\displaystyle \\omega _{pe}\\frac{v_{Te}}{c}\\ll \\omega _{0}\\ll \\omega _{pe},$ where $\\omega _{pe}$ is the electron plasma frequency, and $v_{Te}$ is the electron thermal velocity.", "We consider a plasma occupying region $z\\geqslant 0$ .", "The RF antenna which launches the RF wave with a frequency $\\omega _{0}$ is assumed to exist to the left of the plasma boundary $z=0$ .", "In the frequency range of Eq.", "(REF ), the electric, $\\mathbf {E}_{0}$ , and the magnetic, $\\mathbf {B}_{0}$ fields of the RF wave are exponentially decaying with $z$ and sinusoidally varying with time, $&\\displaystyle \\mathbf {E}_{0}\\left(z,t\\right)= \\mathbf {e}_{y} E_{0y}e^{-\\kappa z}\\sin \\omega _{0}t,$ and $&\\displaystyle \\mathbf {B}_{0}\\left(z,t\\right)= \\mathbf {e}_{x} E_{0y}\\frac{c\\kappa }{\\omega _{0}}e^{-\\kappa z}\\cos \\omega _{0}t ,$ where $\\mathbf {E}_{0}$ and $\\mathbf {B}_{0}$ satisfy the Faraday’s law, $\\partial E_{0}/\\partial z=\\partial B_{0}/c\\partial t$ , and $\\kappa ^{-1}=L_{s}$ is the skin depth for the classical skin effect[3], [4], $&\\displaystyle L_{s}=\\frac{c}{\\omega _{pe}}.$ Our theory bases on the Vlasov equations for the velocity distribution functions $F_{\\alpha }$ of species $\\alpha $ ($\\alpha =e$ for electrons and $\\alpha =i$ for ions), $&\\displaystyle \\frac{\\partial F_{\\alpha }}{\\partial t}+\\mathbf {v}\\cdot \\frac{\\partial F_{\\alpha }}{\\partial \\mathbf {r}}+\\frac{e_{\\alpha }}{m_{\\alpha }}\\left(\\mathbf {E}_{0}\\left(z,t\\right)+\\frac{1}{c}\\left[\\mathbf {v}\\times \\mathbf {B}_{0}\\left(z,t \\right)\\right] \\right.\\nonumber \\\\&\\displaystyle -\\nabla \\varphi \\left(\\mathbf {r},t\\right)\\left)\\cdot \\frac{\\partial F_{\\alpha }}{\\partial \\mathbf {v}}\\right.", "=0,$ and the Poisson equation for the electrostatic potential $\\varphi \\left(\\mathbf {r},t\\right)$ of the electrostatic plasma perturbations, $&\\displaystyle \\nabla ^2 \\varphi \\left(\\mathbf {r},t\\right)=-4\\pi \\sum _{\\alpha =i,e} e_{\\alpha }\\int f_{\\alpha }\\left(\\mathbf {v},\\mathbf {r}, t \\right)d\\textbf {v}_{\\alpha }.", "$ Here, $f_{\\alpha }$ is the perturbation of the equilibrium distribution function $F_{0\\alpha }$ .", "That is to say, $F_{\\alpha }=F_{0\\alpha }+f_{\\alpha }$ .", "In this paper, we employ the nonmodal approach to the solution of the Vlasov equation (REF ) with RF fields (REF ) and (REF ), developed in Ref.", "8.", "The first step in this approach is the transformation of the velocity $\\mathbf {v}$ and the position $\\mathbf {r}$ coordinates determined in the laboratory frame in Eq.", "(REF ) to the coordinates $\\mathbf {v}_{\\alpha }$ and $\\mathbf {r}_{\\alpha }$ , $&\\displaystyle \\mathbf {v}_{\\alpha }=\\mathbf {v}-\\mathbf {V}_{\\alpha }\\left(\\mathbf {r},t \\right) ,\\nonumber \\\\&\\displaystyle \\mathbf {r}_{\\alpha }=\\mathbf {r}-\\mathbf {R}_{\\alpha }\\left(\\mathbf {r},t \\right)= \\mathbf {r} -\\int \\limits ^{t} \\mathbf {V}_{\\alpha }\\left(\\mathbf {r},t_{1} \\right)dt_{1},$ determined in the moving frame of references with a velocity $ \\mathbf {V}_{\\alpha }\\left(\\mathbf {r},t \\right)$ .", "With new coordinates, the Vlasov equation for electrons becomes $&\\displaystyle \\frac{\\partial F_{e}\\left(\\mathbf {v}_{e},\\mathbf {r}_{e},t \\right) }{\\partial t}+\\mathbf {v}_{e}\\frac{\\partial F_{e}}{\\partial \\mathbf {r}_{e}}-v_{ej}\\int \\limits ^{t}_{t_{0}}\\frac{\\partial V_{ek}\\left( \\mathbf {r},t_{1}\\right) }{\\partial r_{j}}dt_{1}\\frac{\\partial F_{e}}{\\partial r_{ek}}\\nonumber \\\\&\\displaystyle -v_{ej}\\frac{\\partial V_{ek}\\left( \\mathbf {r}_{e},t\\right)}{\\partial r_{ej}}\\frac{\\partial F_{e}}{\\partial v_{ek}}-V_{ej}\\left( \\mathbf {r}_{e},t\\right)\\int \\limits ^{t}_{t_{0}}\\frac{\\partial V_{ek}\\left( \\mathbf {r},t_{1}\\right) }{\\partial r_{j}}dt_{1}\\frac{\\partial F_{e}}{\\partial r_{ek}}\\nonumber \\\\&\\displaystyle +\\frac{e}{m_{e}}\\left(\\nabla \\varphi \\left( \\mathbf {r},t\\right)-\\frac{1}{c}\\Big [\\mathbf {v}_{e}\\times \\mathbf {B}_{0}\\left(z,t \\right) \\Big ]\\right)\\cdot \\frac{\\partial F_{e}}{\\partial \\mathbf {v}_{e}}\\nonumber \\\\&\\displaystyle -\\left\\lbrace \\frac{\\partial \\mathbf {V}_{e}\\left(z, t\\right) }{\\partial t}+V_{ez}\\left(z, t\\right)\\frac{\\partial \\mathbf {V}_{e}\\left(z, t\\right) }{\\partial z}\\right.\\nonumber \\\\&\\displaystyle \\left.+ \\frac{e}{m_{e}}\\left(\\mathbf {E}_{0y}\\left(z,t \\right)+\\frac{1}{c}\\Big [\\mathbf {V}_{e}\\left(z, t\\right)\\times \\mathbf {B}_{0}\\left(z, t \\right) \\Big ]\\right)\\right\\rbrace \\nonumber \\\\&\\displaystyle \\cdot \\frac{\\partial F_{e}\\left(\\mathbf {v}_{e},\\mathbf {r}_{e},t \\right)}{\\partial \\mathbf {v}_{e}}=0.$ Velocity $\\mathbf {V}_{e}\\left(z, t\\right)$ in our approach is determined by the equation vanishing the expression in braces in Eq.", "(REF ).", "This equation is the Euler equation for the velocity of the ideal fluid in the EM field.", "For the electric field (REF ) and the magnetic field (REF ), the equations for $V_{ey}\\left(z, t\\right)$ and $V_{ez}\\left(z, t\\right)$ are $&\\displaystyle \\frac{\\partial V_{ey}\\left(z, t\\right)}{\\partial t}+V_{ez}\\left(z, t\\right)\\frac{\\partial V_{ey}\\left(z, t\\right)}{\\partial z}\\nonumber \\\\&\\displaystyle =-\\frac{e}{m_{e}}\\left(E_{0y}\\left(z,t \\right)+\\frac{1}{c}V_{ez}\\left(z, t\\right)B_{0x}\\left(z,t \\right)\\right),\\\\&\\displaystyle \\frac{\\partial V_{ez}\\left(z,t\\right)}{\\partial t}+V_{ez}\\left(z, t\\right)\\frac{\\partial V_{ez}\\left(z, t\\right)}{\\partial z}\\nonumber \\\\&\\displaystyle =\\frac{e}{m_{e}c}V_{ey}\\left(z, t\\right)B_{0x}\\left(z, t\\right).$ In Appendix, we present the detailed solution of these equations for $V_{ey}\\left(z,t\\right)$ and $V_{ez}\\left(z,t\\right)$ .", "The rest of Eq.", "(REF ) contains only the spatial derivatives of $\\mathbf {V}_{e}\\left(z, t\\right)$ .", "Therefore, Eq.", "(REF ) in convected variables $\\mathbf {v}_{e}$ and $\\mathbf {r}_{e}$ has a form as in plasma without an EM field for the spatially uniform time-dependent EM field (so-called dipole approximation).", "For the case of the spatially inhomogeneous EM field, Eq.", "(REF ) becomes very suitable for the investigation of the short scale perturbations with a wavelength less then the spatial inhomogeneity of the EM field.", "For the considered problem of the skin layer stability, the solution for $F_{e}\\left(\\mathbf {v}_{e}, \\mathbf {r}_{e},t\\right)$ may be derived in the form of power series in the small parameter $\\kappa \\delta r_{e}\\ll 1$ , where $\\delta r_{e}$ is the amplitude of the displacement of electron in the RF field.", "With velocities $V_{ey}\\left(z_{e},t\\right)$ and $V_{ez}\\left(z_{e},t\\right)$ , determined by Eqs.", "(REF ) and (REF ), the Vlasov equation (REF ) becomes $&\\displaystyle \\frac{\\partial F_{e}\\left(\\mathbf {v}_{e},\\mathbf {r}_{e}, t\\right)}{\\partial t}+\\left[v_{ey}+v_{ez}\\frac{\\omega _{ce}}{\\omega _{0}}\\sin \\omega _{0}t e^{-\\frac{1}{4}\\omega ^{2}_{ce}t^{2}}\\right]\\frac{\\partial F_{e}}{\\partial y_{e}}\\nonumber \\\\&\\displaystyle +v_{ez}\\left[1+\\left(\\frac{1}{2}\\omega ^{2}_{ce}t^{2}-\\frac{1}{4}\\frac{\\omega ^{2}_{ce}}{\\omega _{0}^{2}}\\right)e^{-\\frac{1}{2}\\omega ^{2}_{ce}t^{2}}\\right]\\frac{\\partial F_{e}}{\\partial z_{e}}\\nonumber \\\\&\\displaystyle -\\omega _{ce}v_{ez}\\left[1-\\cos \\omega _{0}t e^{-\\frac{1}{4}\\omega ^{2}_{ce}t^{2}}\\right]\\frac{\\partial F_{e}}{\\partial v_{ey}}\\nonumber \\\\&\\displaystyle +\\omega _{ce}\\left[v_{ey}+v_{ez}e^{-\\frac{1}{2}\\omega ^{2}_{ce}t^{2}}\\left(\\omega _{ce}t+\\frac{1}{2}\\frac{\\omega _{ce}}{\\omega _{0}}\\sin 2\\omega _{0}t\\right)\\right]\\frac{\\partial F_{e}}{\\partial v_{ey}}\\nonumber \\\\&\\displaystyle +\\frac{e}{m_{e}}\\nabla \\varphi \\left(\\mathbf {r}, t\\right)\\cdot \\frac{\\partial F_{e}}{\\partial \\mathbf {v}_{e}}=0.$ In this equation, $\\omega _{ce}$ is the electron cyclotron frequency formed by the magnetic field of the RF wave, which is determined by Eq.", "(REF ).", "Equation (REF ) is applicable for the treatment of the processes in the skin layer for $\\kappa z_{e}\\lesssim 1$ as well as those outside of the skin layer for $\\kappa z_{e}> 1$ , which occur during a time limited by the interval $\\omega _{0}^{-1}< t < \\omega _{ce}^{-1}$ .", "For the real experimental conditions, this time interval is sufficiently wide.", "For example, for a plasma with electron density $n_{0e}=10^{11}$ cm$^{-3}$ in the electric field $E_{0y}=1$ V/cm at $z_{e}=0$ with a frequency $\\omega _{0}=10^{-2}\\omega _{pe}= 1.7\\times 10^{8}$ s$^{-1}$ , and a skin depth $L_{s}=\\kappa ^{-1}=1.7$ cm, the electron cyclotron frequency $\\omega _{ce}$ is equal to $6\\times 10^{6}\\approx 3.5\\times 10^{-2}\\omega _{0}$ .", "The presence of the small parameter $\\omega _{ce}/\\omega _{0}\\ll 1$ in Eq.", "(REF ) gives a possibility to simplify Eq.", "(REF ) greatly." ], [ "Ion-acoustic instability of the skin layer driven by the accelerated electrons", "In this section, we present the theory of the ion-acoustic instability in the skin layer, driven by the accelerated electrons.", "The growth rate $\\gamma $ of this instability, presented below by Eq.", "(REF ), is of the order as $\\gamma \\sim kv_{s}\\left(m_{e}/m_{i}\\right)^{1/2}$ .", "The IA instability can develop and saturate in the skin layer when this growth rate is larger than $\\omega _{ci}$ .", "This occurs for the short wavelength perturbations with $k\\rho _{i}> v_{Te}/v_{Ti}$ .", "For the argon plasma, $\\left(m_{Ar}/m_{e}\\right)^{1/2}=278$ , with an argon ion temperature $T_{i}\\approx 0.026$ eV and an electron temperature $T_{e}=2$ eV the wave number $k$ of the unstable IA perturbations should be larger than $10^{2}$ cm$^{-1}$ .", "For the time $t\\sim \\gamma ^{-1}<\\omega _{ce}^{-1}$ , Eq.", "(REF ) has a form as in the uniform steady plasma for the electron distribution function $F_{e}$ in the electron moving frame and for $F_{i}$ in the ion frame, which in this problem almost coincides with a laboratory frame.", "Therefore, we select functions $F_{e0}\\left(\\mathbf {v}_{e}\\right)$ and $F_{i0}\\left(\\mathbf {v}_{i}\\right)$ as the Maxwellian distributions in the convective coordinates, $&\\displaystyle F_{\\alpha 0}\\left(v_{\\alpha y},v_{\\alpha z}\\right) = \\frac{n_{0\\alpha }}{2\\pi v^{2}_{T\\alpha } }\\exp \\left[-\\frac{v^{2}_{\\alpha z}+v^{2}_{\\alpha y}}{2v^{2}_{Te}}\\right].$ As displayed in Ref.", "8, the single manifestation of the RF wave on the plasma in this case is the accelerated motion of the electrons relative to the practically unmovable ions.", "The equation for the perturbed electrostatic potential $\\varphi \\left(\\mathbf {k}, t\\right)$ , Fourier-transformed in the ion (laboratory) frame, is determined by the Fourier transformation of the Poisson equation in the laboratory frame.", "This equation for the time $t\\gg \\omega ^{-1}_{0}$ has a form $&\\displaystyle \\varphi \\left(\\mathbf {k}, t\\right)+\\frac{1}{k^{2}\\lambda ^{2}_{Di}}\\int \\limits ^{t}_{t_{0}}dt_{1}\\varphi \\left(\\mathbf {k}, t_{1}\\right)\\frac{d}{dt_{1}}\\left(e^{-\\frac{1}{2}k^{2}v^{2}_{Ti}\\left(t-t_{1}\\right)^{2}}\\right)\\nonumber \\\\&\\displaystyle +\\frac{1}{k^{2}\\lambda ^{2}_{De}}\\int \\limits ^{t}_{t_{0}}dt_{1}\\varphi \\left(\\mathbf {k}, t_{1}\\right)e^{-\\frac{i}{2}k_{z}a_{ie}\\left(t^{2}-t^{2}_{1}\\right)}\\nonumber \\\\&\\displaystyle \\times \\frac{d}{dt_{1}}\\left(e^{-\\frac{1}{2}k^{2}v^{2}_{Te}\\left(t-t_{1}\\right)^{2}}\\right)=0,$ where $\\lambda _{Di(e)}$ is the ion (electron) Debye length, and $a_{ie}=\\omega ^{2}_{ce}/2\\kappa $ .", "For the adiabatic electrons, the approximation $e^{-\\frac{1}{2}k^{2}v^{2}_{Te}\\left(t-t_{1}\\right)^{2}}\\approx e^{-k^{2}v^{2}_{Te}t\\left(t-t_{1}\\right)}$ may be used for the most fast varying function in the electron term.", "For $T_{e}\\gg T_{i}$ , the electron term which contains the nonmodal time dependence is much less than the ion one.", "Therefore, we are looking for the solution to Eq.", "(REF ) in the form $&\\displaystyle \\varphi \\left(\\mathbf {k}, t\\right)= \\varphi \\left(\\mathbf {k}\\right)e^{-i\\omega \\left(t\\right)t},$ where $\\omega \\left(\\mathbf {k}, t\\right)$ slowly changes on the time scale $\\sim \\omega ^{-1}$ .", "Then, the partial integration of Eq.", "(REF ) with $\\varphi \\left(\\mathbf {k}, t\\right)$ in a form of Eq.", "(REF ) gives the following equation for $\\omega \\left(\\mathbf {k}, t\\right)$ , $&\\displaystyle \\varepsilon \\left(\\mathbf {k}, t\\right) \\equiv 1+\\frac{1}{k^{2}\\lambda ^{2}_{Di}}\\left(1+i\\sqrt{\\frac{\\pi }{2}}z_{i}\\left(t\\right)W\\left(z_{i}\\left(t\\right)\\right)\\right) +\\nonumber \\\\&\\displaystyle \\frac{1}{k^{2}\\lambda ^{2}_{De}}\\left(1+i\\sqrt{\\frac{\\pi }{2}}z_{e}\\left(t\\right)W\\left(z_{e}\\left(t\\right)\\right)\\right)=Q\\left(\\mathbf {k}, t, t_{0}\\right),$ where $W\\left(z\\right)=e^{-z^{2}}\\left(1 +\\left(2i/ \\sqrt{\\pi } \\right)\\int \\limits _{0}^{z} e^{t^{2}}dt \\right)$ is the complex error function, $z_{i}\\left(t\\right)=\\omega \\left(t\\right)/kv_{Ti}$ , and $z_{e}\\left(t\\right)=\\left(\\omega \\left(t\\right)-k_{z}a_{ie}t\\right)/kv_{Te}$ .", "The function $Q\\left(\\mathbf {k}, t, t_{0}\\right)$ , $&\\displaystyle Q\\left(\\mathbf {k}, t, t_{0}\\right)=\\frac{1}{k^{2}\\lambda ^{2}_{Di}}e^{i\\omega \\left(t\\right)\\left(t-t_{0}\\right)-\\frac{1}{2}k^{2}v^{2}_{Ti}\\left(t-t_{0}\\right)^{2}}\\nonumber \\\\&\\displaystyle +\\frac{1}{k^{2}\\lambda ^{2}_{De}}e^{i\\left(\\omega \\left(t\\right)-k_{z}a_{ie}t\\right)\\left(t-t_{0}\\right)-\\frac{1}{2}k^{2}v^{2}_{Te}\\left(t-t_{0}\\right)^{2}},$ determines the input from the $t=t_{0}$ limit of the integration of Eq.", "(REF ) by parts.", "For the unstable solutions of Eq.", "(REF ) for $\\omega \\left(t\\right)$ , function $Q\\left(\\mathbf {k}, t, t_{0}\\right)$ is exponentially small and may be neglected.", "The solution of Eq.", "(REF ) for the IA instability is $&\\displaystyle \\omega \\left(t\\right)=\\omega _{s}\\left(k\\right)+i\\gamma \\left(\\mathbf {k}, t\\right).$ Here, the IA frequency $\\omega _{s}\\left(k\\right)$ is determined as $&\\displaystyle \\omega ^{2}_{s}\\left(k\\right)=\\frac{k^{2}v^{2}_{s}}{1+k^{2}\\lambda ^{2}_{De}},$ where $v_{s}=(T_{e}/m_{i})^{1/2}$ is the ion sound velocity.", "The growth/damping rate of the IA instability is $\\gamma \\left(\\mathbf {k}, t\\right)=\\gamma _{e}\\left(\\mathbf {k}, t\\right)+\\gamma _{i}\\left(\\mathbf {k}\\right)$ , where $&\\displaystyle \\gamma _{i}\\left(\\mathbf {k}\\right)=-\\frac{\\omega _{s}\\left(k\\right)}{\\left(1+k^{2}\\lambda ^{2}_{De}\\right)^{3/2}}\\sqrt{\\frac{\\pi }{8}}\\left(\\frac{T_{e}}{T_{i}}\\right)^{3/2}\\nonumber \\\\&\\displaystyle \\times \\exp \\left(-\\frac{T_{e}}{2T_{i}\\left(1+k^{2}\\lambda ^{2}_{De}\\right)}\\right)$ stems from the ion Landau damping of the IA waves, and $&\\displaystyle \\gamma _{e}\\left(\\mathbf {k}, t\\right)=\\sqrt{\\frac{\\pi m_{e}}{8 m_{i}}}\\frac{\\omega _{s}\\left(k\\right)}{\\left(1+k^{2}\\lambda ^{2}_{De}\\right)^{3/2}}\\left(\\frac{k_{z}a_{ie}t}{\\omega _{s}\\left(k\\right)}-1\\right)\\nonumber \\\\&\\displaystyle =\\gamma _{s}\\left(\\mathbf {k}, t\\right)\\left(\\frac{k_{z}a_{ie}t}{\\omega _{s}\\left(k\\right)}-1\\right).$ For ICP plasmas with $T_{e} \\gg T_{i}$ , $\\gamma _{i}\\left(\\mathbf {k}\\right)$ is negligible small.", "The growth rate (REF ) corresponds to the initial linear stage of the IA instability development, at which electrons are uniformly accelerated under the action of the homogeneous electric field.", "In our case, electron acceleration occurs under the action of the effective electric field $&\\displaystyle E_{\\rm eff}=\\frac{1}{2}\\frac{e\\kappa }{m_{e}}\\frac{E^{2}_{0y}}{\\omega ^{2}_{0}}e^{-2\\kappa z_{e}}= E_{0y}\\frac{\\omega _{ce}}{2\\omega _{0}}e^{-2\\kappa z_{e}},$ formed by the ponderomotive force.", "$E_{\\rm eff}=1.75\\times 10^{-2}$ V/cm for the numerical sample considered in this paper with $E_{0y}=1$ V/cm at $z_{e}=0$ , $\\omega _{0}=10^{-2}\\omega _{pe}= 1.7\\times 10^{8}$ s$^{-1}$ , $L_{s}=\\kappa ^{-1}=1.7$ cm, $\\omega _{ce}=6\\times 10^{6}$ s$^{-1}$ $\\approx 3.5\\times 10^{-2}\\omega _{0}$ .", "The instability develops due to the inverse electron Landau damping of the IA waves at time $t>t_{th}=\\omega _{s}\\left(k\\right)/k_{z}a_{ie}$ .", "At that time, the temporal evolution of the IA spectral energy density $W\\left(\\mathbf {k}, t\\right)$ , $&\\displaystyle W\\left(\\mathbf {k}, t\\right)=k^{2}\\left\|\\varphi \\left(\\mathbf {k}\\right)\\right\|^{2}\\omega _{s}\\left(\\mathbf {k}\\right)\\frac{\\partial \\varepsilon \\left(\\mathbf {k}, t\\right)}{\\partial \\omega _{s}\\left(\\mathbf {k}\\right)}\\nonumber \\\\&\\displaystyle \\approx \\frac{\\omega ^{2}_{pi}}{\\omega ^{2}}k^{2}\\frac{1}{4\\pi }\\left\|\\varphi \\left(\\mathbf {k}, t\\right)\\right\|^{2},$ is determined by the equation $&\\displaystyle \\frac{\\partial W\\left(\\mathbf {k}, t\\right)}{\\partial t}=2\\gamma \\left(\\mathbf {k}, t\\right)W\\left(\\mathbf {k}, t\\right)$ and grows with time as $\\sim \\exp \\left(\\gamma _{s}k_{z}a_{ei}\\left(t-t_{th}\\right)^{2}/\\omega _{s}\\right)$ .", "The acceleration of the electron current velocity and the growth of the IA spectral intensity occur during a limited time until the effect of the ion acoustic turbulence (IAT) on the electron current velocity becomes negligibly small.", "At longer times, the temporal evolution of the IAT is determined by the nonlinear interaction of the electrons and ions with random electric fields of the IAT[16]." ], [ "The nonlinear evolution of the IA instability and the anomalous heating of electrons by the ion-acoustic turbulence", "Because the growth rate (REF ) and the growth rate of the conventional IA instability[9] are much less than the IA frequency $\\omega _{s}$ , the nonlinear IAT theory is based on the methods of the weak turbulence theory.", "The conventional IAT theory[16] involves the theory of the quasilinear relaxation of the electrons on the IA pulsation jointly with the theory of the induced scattering of the IA waves by the ions.", "The quasilinear equation for the ensemble averaged electron distribution function $\\bar{F}_{e}\\left(\\mathbf {v}_{e}, \\mathbf {r}_{e}, t\\right)$ in the electron frame is derived easily from Eq.", "(REF ) for time $t<\\omega _{ce}^{-1}$ .", "It is equal to $&\\displaystyle \\frac{\\partial \\bar{F}_{e}\\left(\\mathbf {v}_{e}, \\mathbf {r}_{e}, t\\right)}{\\partial t}=\\left\\langle \\nabla \\varphi \\left(\\mathbf {r},t\\right)\\frac{\\partial f_{e}\\left(\\mathbf {v}_{e}, \\mathbf {r}_{e}, t\\right)}{\\partial \\mathbf {v}_{e}}\\right\\rangle .$ where the angle brackets $\\left\\langle ...\\right\\rangle $ indicate the ensemble averaging of the expression in it.", "Employing the relation $&\\displaystyle \\varphi \\left(\\mathbf {k}, t\\right)= \\varphi _{e}\\left(\\mathbf {k}, t\\right)e^{-\\frac{i}{2}k_{z}a_{ie}t^{2}}$ between the Fourier transform $\\varphi _{e}\\left( \\mathbf {k}, t\\right)$ of the potential $\\varphi \\left( \\mathbf {r}, t\\right)$ over $\\mathbf {r}$ and the Fourier transform $\\varphi _{e}\\left(\\mathbf {k}, t\\right)$ of the potential $\\varphi _{e}\\left(\\mathbf {r}_{e}, t\\right)$ over $\\mathbf {r}_{e}$ , we derive the quasilinear equation $&\\displaystyle \\frac{\\partial \\bar{F}_{e}}{\\partial t}=\\pi \\frac{e^{2}}{m_{e}^{2}}\\int d\\mathbf {k}\\mathbf {k}\\frac{\\partial }{\\partial \\mathbf {v}_{e}}\\left\| \\varphi \\left(\\mathbf {k}\\right)\\right\|^{2}\\nonumber \\\\&\\displaystyle \\times \\delta \\left(\\omega \\left(\\mathbf {k}, t\\right)-\\mathbf {k}\\mathbf {v}_{e}-k_{z}a_{ie}t\\right)\\mathbf {k}\\frac{\\partial \\bar{F}_{e}}{\\partial \\mathbf {v}_{e}},$ which determines the temporal evolution of the distribution function $\\bar{F}_{e}\\left(\\mathbf {v}_{e}, \\mathbf {r}_{e}, t\\right)$ of the accelerated electrons under the action of IAT.", "By multiplying $\\mathbf {v}_{e}$ on Eq.", "(REF ) and integrating it over $\\mathbf {v}_{e}$ , we derive the equation $&\\displaystyle \\frac{d V_{ez}}{\\partial t}=-\\nu _{\\rm eff}V_{ez}\\nonumber \\\\&\\displaystyle =-\\frac{1}{n_{0e}m_{e}}\\int d\\mathbf {k} \\frac{k_{z}}{\\omega _{s}\\left(k\\right)}\\gamma _{e}\\left(\\mathbf {k}, t\\right)W\\left(\\mathbf {k}, t\\right),$ which determines slowing down of electrons due to their interactions with the IAT.", "The temporal evolution of $\\nu _{\\rm eff}$ depends on the temporal evolution of the growth rate $\\gamma \\left(\\mathbf {k}, t\\right)$ resulted from the quasilinear distortion of the electron distribution function by the IAT and on the temporal evolution of the IAT spectrum $W\\left(\\mathbf {k}, t\\right)$ caused by the induced scattering the of IA waves on ions.", "The theory of the IAT, which simultaneously takes into account both these processes, was developed in Ref.[17].", "It was found [16] that the effects of these processes on the nonlinear evolution of IAT spectrum depends greatly on the value of the applied electric field [16].", "When the electric field $E_{\\rm eff}$ is less than $E_{\\rm nl}$[16], [17], where $&\\displaystyle E_{\\rm nl}=\\frac{m_{e} v_{s}\\omega _{pi}}{6\\pi \|e\|}\\frac{T_{e}}{T_{i}},$ the quasilinear effects should be accounted for in the balance equation, which includes the growth rate of the IA instability and the nonlinear damping rate resulted from the induced scattering of IA waves on ions.", "For the numerical data used in this paper ($T_{e}=2$ eV, $T_{i}=0.026$ eV, $n_{0e}=10^{11}$ cm$^{-3}$ ), $E_{\\rm nl}=0.033$ V/cm which is almost two times larger than $E_{\\rm eff}=1.75\\times 10^{-2}$ V/cm of our sample.", "In this case, the level $W=\\int W\\left(\\mathbf {k}\\right)d\\mathbf {k}$ of the total energy density of the IAT determined by this process does not depend on the magnitude of the applied electric field $E_{0}$ and was estimated[16] by the expression $&\\displaystyle \\frac{W}{n_{0e}T_{e}} \\sim 0.1\\frac{\\omega _{pi}}{\\omega _{pe}}\\frac{\\lambda ^{2}_{De}}{\\lambda ^{2}_{Di}},$ where $\\omega _{pi}$ is the ion plasma frequency.", "In the approach employed in our paper, the ion dynamics in the skin layer is not affected by the EM field.", "Therefore, the theory of the induced scattering of the IA waves on ions developed in Ref.", "14 is completely applicable to the IA instability driven by the ponderomotive current for the time $t< \\omega _{ce}^{-1}$ , considered here.", "For the numerical parameters presented above the level (REF ) is estimated as $&\\displaystyle \\frac{W}{n_{0e}T_{e}} \\sim 2.8\\times 10^{-2}.$ In estimate (REF ), it should be accounted for that the directed accelerated velocity $V_{ez}\\left(t\\right)$ is slowed down to the almost threshold velocity of the order of IA velocity[16] at the nonlinearly established steady state of the IAT.", "On this level, the effective electron collision frequency with the IAT, $\\nu _{\\rm eff}$ , determined by Eq.", "(REF ), is estimated as $&\\displaystyle \\nu _{\\rm eff}\\sim \\frac{W}{n_{0e}m_{e}V_{ez}}\\frac{\\gamma _{s}}{v_{s}}\\sim 2.8\\times 10^{-2}\\omega _{s}\\left(\\frac{m_{i}}{m_{e}}\\right)^{1/2}.$ Because the induced scattering of ions redistributes the spectral maximum of the IAT to the longer IA waves, we use a value $k=10^{2}$ cm$^{-1}$ discussed above for the IA waves in the skin layer for the estimation of $\\nu _{\\rm eff}$ .", "We found from Eq.", "(REF ) that the magnitude of $\\nu _{\\rm eff}\\sim 1.7\\times 10^{8}$ s$^{-1}$ for this case is of the order of the frequency $\\omega _{0}=1.8\\times 10^{8}$ s$^{-1}$ used in our estimates.", "At this case of the weak electric field, $E_{\\rm eff} < E_{\\rm nl}$ , a quasi-stationary state for the IAT is established mainly due to the quasilinear relaxation of the electron distribution function.", "Ohm's law for the electron current density $j$ , derived for this case[16], [15], $&\\displaystyle j\\simeq 2.14\|e\|n_{0e}v_{s}=\\sigma _{A} E_{\\rm eff},$ predicts the dependence of $\\sigma _{A}\\sim E^{-1}_{\\rm eff}$ for the anomalous conductivity $\\sigma _{A}$ .", "By multiplying $m_{e}\\mathbf {v}^{2}_{e}/2$ on Eq.", "(REF ) and integrating it over $\\mathbf {v}_{e}$ , we derive the equation $&\\displaystyle n_{e0}\\frac{d T_{e}}{\\partial t}=\\int d\\mathbf {k} \\frac{\\left(k_{z}V_{ez}\\left(t\\right)-\\omega _{s}\\left(k\\right)\\right)}{\\omega _{s}\\left(k\\right)}\\gamma _{e}\\left(\\mathbf {k}, t\\right)W\\left(\\mathbf {k}\\right)\\nonumber \\\\&\\displaystyle \\sim \\gamma _{s}\\left(\\mathbf {k}_{0}\\right)\\frac{W}{n_{0e}T_{e}}T_{e},$ which determines the turbulent heating rate $\\nu _{Te}\\sim \\gamma W/n_{e0}T_{e}$ of the electrons due to their interaction with the IAT.", "For the numerical data, used above, $\\nu _{Te}\\sim 2.8 \\times 10^{-2}\\gamma _{s}$ .", "When the applied electric field $E_{\\rm eff}$ is above $E_{\\rm nl}$ , quasilinear effects are weak and the dominant nonlinear process is the induced scattering of IA waves on ions.", "In this case, the steady state level of the IAT[16], $&\\displaystyle \\frac{W}{n_{0e}T_{e}} \\sim 0.1\\frac{\\omega _{pi}}{\\omega _{pe}}\\frac{T_{e}}{T_{i}}\\sqrt{\\frac{E_{\\rm eff}}{E_{\\rm nl}}},$ grows with $E_{\\rm eff}$ growth.", "The Ohm's law for electron current density $j$ for this case, $&\\displaystyle j=\\frac{4.48}{\\pi }\|e\|n_{0e}v_{s}\\sqrt{\\frac{E_{\\rm eff}}{E_{\\rm nl}}}=\|e\|U_{0}n_{0e},$ and the anomalous conductivity $\\sigma _{A}$ , [16], [17] $&\\displaystyle \\sigma _{A}\\simeq 0.4\\omega _{pe}\\frac{\\lambda _{Di}}{\\lambda _{De}}\\left(\\frac{8\\pi n_{e0}T_{e}}{E_{\\rm eff}^{2}}\\right)^{1/4} \\sim E_{\\rm eff}^{-1/2}.$ were derived in Ref.", "14.", "The effective collision frequency $\\nu _{\\rm eff}$ corresponding to this case is given by the Sagdeev equation [18], $&\\displaystyle \\nu _{\\rm eff}=2.5\\times 10^{-2}\\omega _{pi}\\frac{U_{0}}{v_{s}}\\frac{T_{e}}{T_{i}},$ where $U_{0}$ is determined from Eq.", "(REF ).", "The regime with $E_{\\rm eff}>E_{\\rm nl}$ occurs for $E_{0y}=1.3$ V/cm with the same other parameters considered above.", "For this electric field $\\omega _{ce}/\\omega _{0} =4.6\\times 10^{-2}$ and $E_{\\rm eff}\\approx 5.89\\times 10^{-2}$ V/cm $> E_{\\rm nl}=3.3\\times 10^{-2}$ V/cm, that gives $&\\displaystyle \\frac{W}{n_{0e}T_{e}} \\sim 0.05,$ $U_{0}=2.57 v_{s}$ and $\\nu _{\\rm eff}=8.35\\times 10^{8}\\,{\\text{s}}^{-1}>\\omega _{0}=1.8\\times 10^{8}\\,{\\text{s}}^{-1}$ .", "The derived estimates for $\\nu _{\\rm eff}$ reveal that the effective electron collision frequency with the IAT pulsations is of the order of the RF driving frequency and is much larger than the electron-ion and electron-neutral collision frequencies in the mTorr range of gas pressure for the considered numerical parameters corresponding to the experimental conditions of ICP sources." ], [ "Conclusions", "In this paper, we present the theory of IA instability of the skin layer of ICP sources driven by the accelerated electrons, which move relative to ions under the ponderomotive force.", "This theory reveals that on the linear stage of the IA instability, driven by the steady electric field, always develops as the nonmodal instability with the growth rate growing with time.", "At the finite time interval $\\delta t< \\omega _{ce}^{-1}$ , the analysis of the nonlinear stage of the IA instability driven by the ponderomotive force is similar to the analysis of the IA instability and the IAT driven by the steady electric field.", "The accelerated electron velocity in the steady electric field decelerates due to the scattering of electrons by the IAT.", "This velocity approaches a particular steady value or continues to be accelerating[18] depending on the balance of the nonmodal growth with nonlinear processes: the quasilinear relaxation of the electron distribution function and the induced scattering of ions.", "The effect of these processes on the nonlinear evolution of the IAT depends on the relative value of the driving electric field with respect to $E_{\\rm nl}$ .", "The acting electric field in the skin layer application of the IAT theory is the effective field $E_{\\rm eff}$ determined by Eq.", "(REF ), which is strongly inhomogeneous across the skin layer in ICPs.", "It is found that the effective electron collision frequency $\\nu _{\\rm eff}$ with the IAT is of the order of or is larger than $\\omega _{0}$ at all considered regimes of the IAT evolution.", "The derived results prove that the development of the IAT in the skin layer and scattering of electrons by IAT are the primary channels of the nonlinear absorption of the RF wave energy in the skin layer.", "This result is also valid for the case of the strong RF field considered in Ref.", "8, for which the oscillatory velocity in the skin layer is larger than the electron thermal velocity, and the IA instability develops in the bulk of plasma past the skin layer.", "This work was supported by National R&D Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grants Nos.", "NRF–2018R1D1A1B07050372 and NRF–2019R1A2C1088518) and BK21 PLUS, the Creative Human Resource Education and Research Programs for ICT Convergence in the 4th Industrial Revolution.", "DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request." ], [ "Solutions to Eqs. (", "Here, we present the solutions to Eqs.", "(REF ) and () for $V_{ey}$ and $V_{ez}$ , alternative to those presented in Ref.", "8.", "These solutions do not require the usually employed assumption[12] in the calculation of the electron velocity in EM wave that the force by the RF electric field acting on electrons in the EM wave prevails over the Lorentz force by the RF magnetic field.", "With new variables $z_{e}$ and $t^{\\prime }$ determined by the relations[11], [8] $&\\displaystyle z= z_{e}+\\int \\limits ^{t^{\\prime }}_{0}V_{ez}\\left(z_{e},t^{\\prime }_{1} \\right) dt^{\\prime }_{1}, \\quad t=t^{\\prime }.$ Eqs.", "(REF ) and () becomes $&\\displaystyle \\frac{\\partial V_{ey}\\left( z_{e},t^{\\prime }\\right)}{\\partial t^{\\prime }} =-\\frac{\\omega _{0}\\omega _{ce}}{\\kappa }e^{-\\kappa \\int \\limits ^{t^{\\prime }}_{0}V_{ez}\\left(z_{e},t^{\\prime }_{1} \\right)dt^{\\prime }_{1}}\\sin \\omega _{0}t^{\\prime }\\nonumber \\\\&\\displaystyle -\\omega _{ce}\\cos \\omega _{0}t^{\\prime }e^{-\\kappa \\int \\limits ^{t^{\\prime }}_{0}V_{ez}\\left(z_{e},t^{\\prime }_{1} \\right)dt^{\\prime }_{1}}V_{ez}\\left(z_{e},t^{\\prime } \\right),$ $&\\displaystyle \\frac{\\partial V_{ez}\\left( z_{e},t^{\\prime }\\right) }{\\partial t^{\\prime }}=\\omega _{ce}e^{-\\kappa \\int \\limits ^{t^{\\prime }}_{0}V_{ez}\\left(z_{e},t^{\\prime }_{1} \\right) dt^{\\prime }_{1}}\\nonumber \\\\&\\displaystyle \\times V_{ey}\\left(z_{e},t^{\\prime } \\right)\\cos \\omega _{0}t^{\\prime },$ where $&\\displaystyle \\omega _{ce}=\\frac{e\\kappa E_{0y} e^{-\\kappa z_{e}}}{m_{e}\\omega _{0}}$ is the electron cyclotron frequency formed by the RF magnetic field (REF ) at $z_{e}$ .", "Now we derive the approximate solution to Eqs.", "(REF ) and (REF ) for the finite time interval, at which $&\\displaystyle \\left\|\\kappa \\int \\limits ^{t^{\\prime }}_{0}V_{ez}\\left(z_{e},t^{\\prime }_{1} \\right)dt^{\\prime }_{1}\\right\|\\ll 1.$ In the zero approximation, Eqs.", "(REF ) and (REF ) are reduced to the ordinary differential equation $&\\displaystyle \\frac{d U_{e}}{d t^{\\prime }}-i\\omega _{ce}\\cos \\omega _{0}t \\,U_{e}=-\\frac{\\omega _{0}\\omega _{ce}}{\\kappa }\\sin \\omega _{0}t,$ in which variable $z_{e}$ becomes a parameter, and $&\\displaystyle U_{e}=U_{e}\\left(z_{e},t\\right)= V_{ey}\\left(z_{e},t\\right)+iV_{ez}\\left(z_{e},t\\right).$ The solution to Eq.", "(REF ) for the initial value $U_{e}\\left(z_{e},t=0\\right)=0$ , $&\\displaystyle U_{e}\\left(z_{e},t\\right)= -\\frac{\\omega _{0}\\omega _{ce}}{\\kappa }\\exp \\left( i\\frac{\\omega _{ce}}{\\omega _{0}}\\sin \\omega _{0}t\\right)\\nonumber \\\\&\\displaystyle \\times \\int \\limits ^{t}_{0}dt_{1}\\sin \\omega _{0}t_{1}\\exp \\left( -i\\frac{\\omega _{ce}}{\\omega _{0}}\\sin \\omega _{0}t_{1}\\right) ,$ is simply presented in the explicit, however cumbersome, form for any values of the $\\omega _{ci}/\\omega _{0}$ ratio.", "The focus of our paper is on the weak high-frequency RF field for which $\\omega _{ce}/\\omega _{0}\\ll 1$ .", "By using the approximation for the exponent in Eq.", "(REF ), $&\\displaystyle \\exp \\left(\\pm i\\frac{\\omega _{ce}}{\\omega _{0}}\\sin \\omega _{0}t\\right)\\approx 1\\pm i\\frac{\\omega _{ce}}{\\omega _{0}}\\sin \\omega _{0}t$ which is valid for any values of $\\omega _{0}t$ , we derive the approximate solution for $U_{e}\\left(z_{e},t\\right)$ from Eq.", "(REF ) in the form $&\\displaystyle U_{e}\\left(z_{e},t\\right)= \\frac{\\omega _{ce}}{\\kappa }\\cos \\omega _{0}t+i\\frac{\\omega ^{2}_{ce}t}{2\\kappa }+i\\frac{\\omega ^{2}_{ce}}{4\\omega _{0}\\kappa }\\sin 2\\omega _{0}t.$ Equation (REF ) gives $&\\displaystyle V_{ey}\\left(z_{e},t\\right)= \\mathrm {Re}\\,U_{e}\\left(z_{e},t\\right)= \\frac{\\omega _{ce}}{\\kappa }\\cos \\omega _{0}t$ and $&\\displaystyle V_{ez}\\left(z_{e},t\\right)=\\mathrm {Im}\\,U_{e}\\left(z_{e},t\\right)\\nonumber \\\\&\\displaystyle =\\frac{\\omega ^{2}_{ce}t}{2\\kappa }+\\frac{\\omega ^{2}_{ce}}{4\\omega _{0}\\kappa }\\sin 2\\omega _{0}t.$ Solutions (REF ) and (REF ) are identical to solutions for $V_{ey}$ and $V_{ez}$ , derived in Ref.", "8, where other procedure was developed by the iterative solution of Eqs.", "(REF ) and ().", "It was based on the calculation of the ponderomotive motion of an electron in the spatially inhomogeneous electromagnetic field, assuming that the force by the RF electric field acting on electrons in the skin layer prevails over the Lorentz force by the RF magnetic field.", "The procedure developed in Ref.", "8 used small parameter $\\kappa \\xi _{e}\\ll 1$ where $\\xi _{e}=eE_{0y}\\left(z_{e}\\right)/m_{e}\\omega ^{2}_{0}$ is the amplitude of the displacement of an electron along the coordinate $y$ .", "This parameter is identically equal to ${\\omega _{ce}}/{\\omega _{0}}$ .", "For the time at which $\\omega _{0}t\\gg 1$ , $&\\displaystyle V_{ez}\\left(z_{e},t\\right)\\approx \\frac{\\omega _{ce}^{2}}{2\\kappa }t$ and $&\\displaystyle -\\kappa \\int \\limits ^{t^{\\prime }}_{0}V_{ez}\\left(z_{e},t^{\\prime }_{1} \\right)dt_{1}^{\\prime }=-\\frac{1}{4}\\omega _{ce}^{2}t^{2}.$ It follows from Eq.", "(REF ) that solutions (REF ) and (REF ) are valid for the time $t<\\omega _{ce}^{-1}$ .", "By employing the method of successive approximations to the solution of the nonlinear Eqs.", "(REF ) and (REF ) with $V_{ez}\\left(z_{e},t \\right)$ determined by Eq.", "(REF ) as the initial approximation, we obtain the following solutions to these equations for $V_{ey}\\left(z_{e},t \\right)$ , $&\\displaystyle V_{ey}\\left(z_{e},t\\right)= \\frac{\\omega _{ce}}{\\kappa }e^{-\\frac{1}{4}\\omega ^{2}_{ce}t^{2}}\\cos \\omega _{0}t,$ and for $V_{ez}\\left(z_{e},t\\right)$ , $&\\displaystyle V_{ez}\\left(z_{e},t\\right)=\\frac{\\omega ^{2}_{ce}t}{2\\kappa }e^{-\\frac{1}{2}\\omega ^{2}_{ce}t^{2}}\\nonumber \\\\&\\displaystyle +\\frac{\\omega ^{2}_{ce}}{4\\omega _{0}\\kappa }e^{-\\frac{1}{2}\\omega ^{2}_{ce}t^{2}}\\sin 2\\omega _{0}t +O\\left(\\frac{\\omega ^{2}_{ce}}{\\omega ^{2}_{0}}\\right),$ which are valid for the time $t\\gg \\omega ^{-1}_{0}$ .", "It follows from Eq.", "(REF ) that the maximum of the accelerating velocity $V_{ez}\\left(z_{e},t\\right)$ attains for $t_{\\ast }=\\sqrt{2}/\\omega _{ce}$ at which $\\omega ^{2}_{ce}t^{2}=2$ .", "At the time $t>2t_{\\ast }$ , an electron which was in $z_{e}=0$ at the time $t=0$ will cover the distance of the order of the skin depth.", "Therefore, the time admissible for the development and saturation of any instability in the skin layer driven by the accelerated electron current in the case of the high operation frequency for which $\\omega _{0}\\gg \\omega _{ce}$ is limited by time $t\\lesssim t_{\\ast }\\sim \\omega _{ce}^{-1}$ .", "In the case of the low frequency $\\omega _{0}$ , large amplitude RF wave, the electron cyclotron frequency $\\omega _{ce}$ may be larger than $\\omega _{0}$ [13], [14].", "The approximate solution to Eq.", "(REF ) can be derived in this case for the limited time interval $t\\ll \\omega _{0}^{-1}$ , by employing the simplest approximation $\\sin \\omega _{0}t \\approx \\omega _{0}t \\ll 1$ .", "In this time interval, the approximate solution to Eq.", "(REF ) is $&\\displaystyle U_{e}\\left(z_{e},t\\right)= -i\\frac{\\omega ^{2}_{0}t}{\\kappa }-\\frac{\\omega ^{2}_{0}}{\\kappa \\omega _{ce}\\left(z_{e}\\right)}\\nonumber \\\\&\\displaystyle +\\frac{\\omega ^{2}_{0}}{\\kappa \\omega _{ce}\\left(z_{e}\\right)}e^{i\\omega _{ce}\\left(z_{e}\\right)t}.$ It follows from Eq.", "(REF ) that $&\\displaystyle V_{ey}\\left(z_{e},t\\right)= \\frac{\\omega ^{2}_{0}}{\\kappa \\omega _{ce}\\left(z_{e}\\right)}\\left(1-\\cos \\omega _{ce}\\left(z_{e}\\right)t\\right)$ and $&\\displaystyle V_{ez}\\left(z_{e},t\\right)= -\\frac{\\omega ^{2}_{0}t}{\\kappa }+\\frac{\\omega ^{2}_{0}}{\\kappa \\omega _{ce}\\left(z_{e}\\right)}\\sin \\omega _{ce}\\left(z_{e}\\right)t\\nonumber \\\\&\\displaystyle \\approx -\\frac{\\omega ^{2}_{0}t}{\\kappa }.$ Note, that condition (REF ) with velocity $V_{ez}\\left(z_{e},t\\right)$ determined by Eq.", "(REF ) is valid at time $t\\ll \\omega _{0}^{-1}$ .", "The derived solution (REF ) can be easily improved by using the expansion $\\sin \\omega _{0}t\\approx \\omega _{0}t-\\frac{1}{6}\\left(\\omega _{0}t\\right)^{3}$ .", "In this case, the solution for $V_{ez}\\left(z_{e},t\\right)$ becomes equal to $&\\displaystyle V_{ez}\\left(z_{e},t\\right)\\approx -\\frac{\\omega _{0}}{\\kappa }\\left(1-3\\frac{\\omega ^{2}_{0}}{\\omega ^{2}_{ce}}\\right)\\omega _{0}t.$" ] ]	2011.14079
[ [ "Calculating the Mandel parameter for an oscillator-like system generated\n by generalized Chebyshev polynomials" ], [ "Abstract In this paper, we calculate the Mandel parameter $Q_M$ for an oscillator-like system generated by generalized Chebyshev polynomials \\cite{01}, \\cite{02}, \\cite{03}.", "The sign of the Mandel parameter $Q_M$ characterizes the deviation of the excitation statistics from the Poisson one.", "This work is a continuation of our works \\cite{04}, \\cite{05}." ], [ "Introduction", "The Mandel parameter was introduced in [6].", "For a standard harmonic oscillator in Fock space $\\mathcal {H}_F$ it is calculated by the formula $Q=\\frac{\\langle (\\Delta (n))^2\\rangle -\\langle n\\rangle }{\\langle n\\rangle },$ where $\\Delta (n)=\\sqrt{\\langle n^2\\rangle -\\langle n\\rangle ^2}$ , and $n=a^+a$ is the operator of the number of particles (excitations).", "The sign of the Mandel parameter determines the nature of the deviation of the excitation statistics from the Poisson one.", "For coherent states, $Q=0$ (Poisson statistics), $Q>0$ for ordinary (classical) states (super-Poisson case) and $Q<0$ for non-classical States (sub-Poisson case).", "When $Q<0$ , the phenomenon of anti-bunching occurs $\\langle (\\Delta (n))^2\\rangle <\\langle n\\rangle $ .", "For physical aspect of bunching and anti-bunching of photons in quantum optics, see [7].", "Detailed overview non-classical states in quantum optics, is given in [8].", "Where a large class of non-classical states: coherent and squeezed states, states with minimal uncertainty, intelligent states, binomial states, deformed coherent states, etc.", "are considered.", "In [5], we showed that for coherent states of oscillator-like systems, generated by known classes of orthogonal polynomials, the Mandel parameter can take both positive and negative values.", "In particular, for oscillators generated by Charlier polynomials [9], as well as for the standard oscillator (Hermite polynomials) the Mandel parameter takes a null value ($Q_M=0;$ Poisson statistics).", "For Laguerre and Meixner polynomials $Q_M<0$ (sub-Poisson statistics).", "For Chebyshev and Legendre polynomials $Q_M>0$ (super-Poisson statistics).", "In the case of Gegenbauer and Kravchuk polynomials, the sign of the Mandel parameter depends on the size of the eigenvalue of the annihilation operator corresponding to the coherent state.", "In the case of deformed $q$ -Hermite polynomials the sign $Q_M$ is determined by the value of the deformation parameter $q.$ Namely, $Q_M<0$ for $0<q<1,$ a for $q>1$ the Mandel parameter is positive ($Q_M>0$ ).", "Note that in [5] we used Barut - Girardello coherent states [10], which are defined as eigenstates of annihilation operator (in the case of coherent states for groups - as eigenstates of destruction operator).", "These states can also be defined for generalized oscillators associated with classical orthogonal polynomials and their $q$ -analogues, in the case when the corresponding the Hilbert space is infinite-dimensional.", "Similar results are valid also for coherent states of the Klauder - Gaseau type [11].", "In this paper we show on example of a generalized Chebyshev oscillator [1] that even a one-dimensional perturbation of the Jacobi matrix for Chebyshev polynomials can change the statistics of coherent states (from super-Poisson to sub-Poisson).", "For the simplest case of generalized Chebyshev polynomials $Ch_{n}(z;1;a)$ we prove that critical value of the perturbation parameter $a=\\frac{1}{\\sqrt{2}}.$ For values of $a<\\frac{1}{\\sqrt{2}}$ , the statistics can become sub-Poisson." ], [ "Generalized Chebyshev polynomials", "Generalized Chebyshev polynomials $Ch_{n}(z;k;a),\\, k\\ge 1,$ are defined by the recurrence relations: $b_{n}Ch_{n+1}(z;k;a)+b_{n-1}Ch_{n-1}(z;k;a)=zCh_{n}(z;k;a),\\quad n\\ge 0,\\\\Ch_{0}(z;k;a)=1,\\quad Ch_{-1}(z;k;a)=0,\\qquad \\qquad \\qquad \\qquad $ where $b_{n}=1,$ for $n\\ne {k-1}$ and $b_{k-1}=a$ .", "Using the expression obtained in [5] for polynomials related to with the relation (REF ) (and with the corresponding Jacobi matrix), we have $Ch_{n}(z;k;a)=\\!\\!\\!\\sum _{m=0}^{Ent(\\frac{n}{2})}\\frac{(-1)^m}{\\sqrt{[n]!", "}}b_{0}^{2m-n}\\beta _{2m-1,n-1}z^{n-m},$ where $\\beta _{-1,n-1}=1,\\, n\\ge 0,$ and $\\beta _{2m-1,n-1}=\\sum _{k_1=2m-1}^{n-1}\\!\\!\\!", "[k_1]!\\sum _{k_2=2m-3}^{k_{1}-2}[k_2]!\\cdots \\sum _{k_m=1}^{k_{m-1}-2}[k_m]!$ for all $m\\ge 1.$ Here $[s]=\\displaystyle \\frac{b_{s-1}^2}{b_{0}^2},$ and $Ent(x)$ — the integer part of $x$ .", "As an example, we give the last formulas in the case of $k=1$ , (denoting $\\Psi _{n}(z)=Ch_{n}(z;1;a)$ ): $\\qquad \\qquad \\qquad \\qquad \\qquad \\Psi _{0}(z)=1,\\quad \\Psi _{1}(z)=\\frac{z}{a},\\\\\\Psi _{n}(z)=\\frac{z^n}{a}-\\frac{n+(a^{2}-2)}{a}z^{n-2}+\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\\\\\sum _{m=2}^{Ent(\\frac{n}{2})}(-1)^{m}\\frac{(n-m-1)!", "(n+m(a^{2}-2))}{(n-2m)!m!a}z^{n-2m},\\quad n\\ge 2.$ Jacobi matrix $J_k$ associated with generalized Chebyshev polynomials$Ch_{n}(z;k;a)$ has the following form.", "All its elements $j^k_{i, j}$ are equal to zero except for the elements on the first over-diagonal and the first under-diagonal, which are equal to $\\left\\lbrace \\begin{aligned}j^k_{i,i+1}=j^k_{i+1,i}=&1,\\quad i\\ne k;\\\\j^k_{k,k+1}=j^k_{k+1,k}=&a,\\quad i= k,\\end{aligned}\\right.$ In other words, the parameter $a$ stands in the over-diagonal and under-diagonal at the $k$ -th place from the top, and all other elements are equal to 1.", "As an example, we give $J_k$ for $k=1$ and $k=4$ $J_1=\\begin{bmatrix}0&a&0&0&0&0&\\cdots &\\cdots \\\\a&0&1&0&0&0&\\ddots &\\cdots \\\\0&1&0&1&0&0&\\ddots &\\cdots \\\\0&0&1&0&1&0&\\ddots &\\cdots \\\\0&0&0&1&0&1&\\ddots &\\cdots \\\\\\dots &\\dots &\\dots &\\dots &\\dots &\\dots \\end{bmatrix}\\qquad J_4=\\begin{bmatrix}0&1&0&0&0&0&\\cdots &\\cdots \\\\1&0&1&0&0&0&\\ddots &\\cdots \\\\0&1&0&1&0&0&\\ddots &\\cdots \\\\0&0&1&0&a&0&\\ddots &\\cdots \\\\0&0&0&a&0&1&\\ddots &\\cdots \\\\\\dots &\\dots &\\dots &\\dots &\\dots &\\dots \\end{bmatrix} .$ In this paper, we restrict ourselves to the cases of $k=1$ and $k=2$ ." ], [ "Generalized Chebyshev oscillator (in the case of $\\mathbf {k=1}$ )", "Let $a>0$ , $\\mathcal {H}_a=L^2(\\mathbb {R};\\mu _{a})$ — a fixed Hilbert space, and $\\lbrace \\varphi _n(x)\\rbrace _{n=0}^\\infty $ be a system of polynomials orthonormal with respect to the measure $\\mu _a,$ where $d\\mu _{a}(x)=\\frac{1}{2\\pi }\\left\\lbrace \\begin{aligned} & \\frac{a^2\\sqrt{4-x^2}}{a^4-(a^2-1)x^2}dx,\\quad \\text{if}\\quad \|x\|\\le 2,\\\\&\\quad 0,\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\,\\,\\text{if}\\quad \|x\|>2.\\end{aligned}\\right.$ Then, as follows from [2] (see also [3]), the polynomials $\\varphi _n(x)$ are generalized Chebyshev polynomials $\\Psi _{n}(x)==Ch_{n}(x;1;a)$ (for the case $k=1$ ) and recurrent relations (REF ) take the form: $a\\Psi _{1}(x)=x\\Psi _{0}(x),\\quad \\Psi _{2}(x)+a\\Psi _{0}(x)=x\\Psi _{1}(x),\\nonumber \\\\\\Psi _{n+1}(x)+\\Psi _{n-1}(x)=x\\Psi _{n}(x),\\quad n\\ge 2,\\\\\\Psi _{0}(x)=1,\\quad \\Psi _{-1}(x)=0,\\nonumber $ The first few polynomials are equal $\\Psi _0=1;&\\qquad \\Psi _3=\\frac{x}{a}\\left(x^2-(a+1)\\right);\\\\\\Psi _1=\\frac{x}{a};&\\qquad \\Psi _4=\\frac{x^4-(2+a)x^2+a}{a};\\\\\\Psi _2=\\frac{x^2-a}{a};&\\qquad \\Psi _5=\\frac{x}{a}\\left(x^4-(3+a)x^2+(1+2a)\\right);$ $\\Psi _6=&\\frac{x^6-(4+a)x^4+3(a+1)x^2-a}{a};\\\\\\Psi _7=&\\frac{x}{a}\\left(x^6-(5+a)x^4+(6+4a)x^2-(1+3a)\\right);\\\\\\Psi _8=&\\frac{x^8-(6+a)x^6+(10+5a)x^4-(4+6a)x^2+a}{a};\\\\\\Psi _9=&\\frac{x}{a}\\left(x^8-(a+7)x^6+(15+6a)x^4-10(1+a)x^2+(1+4a)\\right);\\\\\\Psi _{10}=&\\frac{x^{10}-(8+a)x^8+(21+7a)x^6-(20+15a)x^4+(5+10a)x^2-a}{a}.$ In [12], a method for constructing an oscillator-like algebra $\\mathfrak {A}_{\\Psi }$ corresponding to this system of polynomials was proposed.", "Polynomials $\\lbrace \\Psi _n(x)\\rbrace $ $0\\le n<\\infty $ form the Fock basis for this algebra, $\\mathfrak {A}_{\\Psi }$ in the Fock space $\\mathcal {H}_a$ .", "Generators $a_{\\mu _a}^{+},\\, a_{\\mu _a}^{-},\\, N_{\\Psi }$ of algebra $\\mathcal {A}_\\Psi $ in this Fock representation acts as follows $a_{\\mu _a}^{+}\\Psi _n=\\sqrt{2}b_n\\Psi _{n+1},\\quad a_{\\mu _a}^{-}\\Psi _n=\\sqrt{2}b_{n-1}\\Psi _{n-1},\\quad N_{\\Psi }\\Psi _n=n\\Psi _n,$ where $b_{-1}=0,\\quad b_0=a,\\quad b_n=1,\\quad n\\ge 1.$ Let $I$ — be a unit operator in the Hilbert space $\\mathcal {H}_a.$ Define $B_{\\Psi }(N_{\\Psi })$ as an operator-valued function defined by the equalities $B_{\\Psi }(N_{\\Psi })\\Psi _{n}=b_{n-1}^{2}\\Psi _{n},\\quad B_{\\Psi }(N_{\\Psi }+I)\\Psi _{n}=b_{n}^{2}\\Psi _{n},\\quad n\\ge 0.$ Then the algebra of the generalized Chebyshev oscillator $\\mathfrak {A}_{\\Psi }$ is generated by the operators $a_{\\Psi }^{\\pm },$ $N_{\\Psi }$ and $I$ satisfying the relations $a_{\\mu _a}^{-}a_{\\mu _a}^{+}\\Psi _{n}=2B_{\\Psi }(N_{\\Psi }+I),\\quad a_{\\mu _a}^{+}a_{\\mu _a}^{-}\\Psi _{n}=2B_{\\Psi }(N_{\\Psi }),\\quad \\\\[N_{\\Psi },a_{\\mu _a}^{\\pm }]=\\pm a_{\\mu _a}^{\\pm },$ and by the commutators of these operators.", "Similarly, we can define the algebra of the generalized Chebyshev oscillator $\\mathfrak {A}_k$ corresponding to generalized Chebyshev polynomials $Ch_{n}(z;k;a)$ for an arbitrary integer $ k\\ge 1.$ Clearly, $\\mathfrak {A}_{\\Psi }=\\mathfrak {A}_1.$ For the algebra $\\mathfrak {A}_k$ we define coherent states of the Barut-Girardello type as usual [10] (see also [5]) $a_{k}^{-}\|z\\rangle =z\|z\\rangle ,\\quad \\\\\|z\\rangle =N^{-\\frac{1}{2}}(\|z\|^2)\\sum _{n=0}^{\\infty }\\frac{z^n}{(\\sqrt{2}b_{n-1})!", "}Ch_{n}(z;k;a),$ where the normalizing factor has the form $N(\|z\|^2)=\\sum _{n=0}^{\\infty }\\frac{\|z\|^{2n}}{(2b_{n-1}^2)!", "}.$ For computing the Mandel parameter in coherent states $\|z\\rangle $ for in [5], the following formula was obtained $Q_M(x)=x\\left(\\frac{N^{^{\\prime \\prime }}(x)}{N^{^{\\prime }}(x)}-\\frac{N^{^{\\prime }}(x)}{N(x)}\\right), \\quad (\|z\|^2=x).$ Using the values of the coefficients $b_n$ of recurrent relations (REF ) for generalized Chebyshev polynomials $Ch_{n}(z;k;a),\\, k\\ge 1,$ and the formulas (REF ) and (REF ), we calculate the Mandel parameter $Q_M(x;k;a)$ for coherent states of an algebra $\\mathfrak {A}_k$ of the generalized Chebyshev oscillator.", "A further task is to determine the sign-constant regions of the Mandel parameter $Q_M(x; k; a)$ depending on the values of the perturbation parameter $a$ of the Jacobi matrix for generalized Chebyshev polynomials." ], [ "Computation of the Mandel parameter (in the case of $\\mathbf {k=1}$ )", "We want to calculate the Mandel parameter $Q_M(x;1;a)$ for coherent states of the algebra $\\mathfrak {A}_1$ of the generalized Chebyshev oscillator.", "Let's start by calculating the normalizing factor $N_1.$ From the formula (REF ) it follows that $N_1(x)=\\sum _{n=0}^{\\infty }\\frac{x^{n}}{(2b_{n-1}^2)!", "},$ where $b_{-1}=0,\\, b_0=a,\\, b_n=1 , n\\ge 1.$ Expression $(2b_{n-1}^2)!$ is understood as a factorial by index, that is $(2b_{n-1}^2)!=(2b_0)(2b_1)...(2b_{n-1})$ and $(2b_{-1})!=0!=1$ because $b_{-1}=0.$ Then we get $N_1(x)=a^{-2}\\left(a^2 -1 +\\frac{2}{2-x}\\right), \\quad 0<x<2.$ The restriction $0<x<2$ is due to the convergence region of the power series in (REF ).", "Differentiating (REF ), we find $N_1^{\\prime }(x)=2a^{-2}(2-x)^{-2},\\quad N_1^{^{\\prime \\prime }}(x)=4a^{-2}(2-x)^{-3}.$ From (REF ),(REF ) and (REF ) we get the desired formula $Q_M(x;1;a)=\\frac{2x}{2-x}\\left(1- \\frac{1}{x+a^2(2-x)}\\right),\\quad 0<x<2.$ Let us study the sign of the Mandel parameter for different values of the perturbation parameter $a.$ Note that for $0<x<2$ , the multiplier before the bracket in the right hand part of the equality (REF ) is positive.", "Therefore, $\\text{sign}(Q_M(x;1;a))=\\text{sign}\\left(1- \\frac{1}{x+a^2(2-x)}\\right), \\quad 0<x<2.$ In addition, for $0<x<2$ , the denominator of the fraction, standing on the right hand side of the equality (REF ) is also positive.", "Therefore, $\\text{sign}(Q_M(x;1;a))=\\text{sign}\\left(x(1-a^2)+2a^2-1\\right), \\quad 0<x<2.$ Let us show that $Q_M(x;1;a)>0$ for $a^2\\ge \\frac{1}{2}$ for all $x\\in (0;2)$ , and when $a^2<\\frac{1}{2}$ if $x>\\displaystyle \\frac{1-2a^2}{1-a^2}$ .", "For $\\frac{1}{2}\\le a^2\\le 1$ ($0<x<2$ ) it is obvious that the inequality $x(1-a^2)>1-2a^2.$ hold true.", "For $a^2>1$ , we rewrite the inequality (REF ) as equivalent inequality $2a^2-1>x(a^2-1),$ which in turn is equivalent to the inequality $x<\\frac{2a^2-1}{a^2-1}=2+ \\frac{1}{a^2-1}.$ The last inequality obviously holds for $a^2\\ge 1$ and $0<x<2$ .", "Hence, the fairness of the inequality $Q_M(x;1;a)>0$ for $a^2\\ge \\frac{1}{2}$ ($0<x<2$ ) is proved.", "If $a^2<\\frac{1}{2}$ , then it follows from the inequality (REF ) that $Q_M(x;1;a)>0$ , when $x>\\frac{1-2a^2}{1-a^2}.$ Hence, $Q_M(x;1;a)<0$ for $a^2<\\frac{1}{2}$ and $0<x<\\frac{1-2a^2}{1-a^2}.$ As an illustration, we present the graphs of $Q_M(x;1;a)$ at $a=0.5$ and $a=0.65$ (on the left) and at $a=1$ and $a=2$ (on the right).", "Figure: Plots of the functions Q M (x;1;a)Q_M(x;1;a)The left graph shows that for $a<\\frac{1}{\\sqrt{2}}$ , when $a$ decreases, the area with the negative value of the Mandel parameteriz increases, and for $a>\\frac{1}{\\sqrt{2}}$ , there is no such area." ], [ "Calculating the Mandel parameter for $\\mathbf {k\\ge \\,1}$", "For further consideration, it is convenient to slightly transform the formula (REF ).", "Let's start by calculating the normalizing factor $N_k(x;a^2)$ .", "According to the formula (REF ) we have $N_k(x;a^2)=\\sum _{n=0}^{\\infty }\\frac{x^n}{(2b_{n-1}^{\\,\\,2})!", "}\\qquad k\\ge 1,$ where $x=\|z\|^2>0$ , $b_{k-1}=a>0$ , $b_n=1$ for $n\\ne (k-1)$ and $b_{-1}=0$ .", "The convergence condition of an infinite geometric progression leads to the restriction $0<x<2$ .", "Under this condition, the expression $N_k(x;a^2)$ can be rewritten as $\\qquad \\qquad N_k(x;a^2)=\\sum _{n=0}^{k-1}\\frac{x^n}{2^n}+\\frac{1}{a^2}\\sum _{n=k}^{\\infty }\\frac{x^n}{2^n}=\\\\\\frac{1}{a^2}\\,\\frac{1}{1-\\frac{x}{2}}+\\left(1-\\frac{1}{a^2}\\right)\\,\\frac{1-\\left(\\frac{x}{2}\\right)^k}{1-\\frac{x}{2}}\\qquad \\qquad \\qquad .$ Denoting $x=2t,\\,\\, a^2=\\tau $ , we rewrite this formula as $N_k(t;\\tau )=\\frac{1}{\\tau (1-t)}\\left(\\tau +(1-\\tau )t^k\\right),\\qquad (k\\ge 1,\\, 0<t<1,\\, \\tau >0).$ Substituting (REF ) in the formula (REF ) for the Mandel parameter (and taking into account that $\\displaystyle \\frac{d}{dx}=\\frac{1}{2}\\,\\frac{d}{dt}$ ) we get $Q_k(t;\\tau )=t\\left(\\frac{(N_k(t;\\tau ))_{t^2}^{\\prime \\prime }}{(N_k(t;\\tau ))_{t}^{\\prime }}-\\frac{(N_k(t;\\tau ))_{t}^{\\prime }}{N_k(t;\\tau )}\\right).$ For the sake of brevity, we will not specify the dependency on $\\tau $ in the future.", "Let us note that $N_k(t)\\,N_k^{\\prime }(t)=\\frac{1}{2} \\left(N_k^{2}(t)\\right)_t^{\\prime }$ and denote $q_k(t)=N_k^{\\prime \\prime }(t)N_k(t)-\\left(N_k^{\\prime }(t)\\right)^2.$ Then (REF ) can be rewritten in the form $Q_k(t)=\\frac{2t}{(N_k^2)^\\prime _t}\\,q_k(t).$ Recall that we are interested in the sign of $Q_k (t)$ for $0<t<1,\\, \\tau >0$ .", "We show that this sign coincides with the sign of $q_k(t)$ .", "To do this, it is sufficient to prove that $(N_k^2)^\\prime _t>0$ .", "We show that the function $N_k(t;\\tau )$ increases monotonically on the interval $0<t<1$ for all $\\tau >0$ .", "Indeed, the formula (REF ) can be rewritten as $N_k(t;\\tau )=\\frac{1-t^k}{1-t}+\\frac{1}{\\tau }\\,\\frac{t^k}{1-t}=(1+t+\\ldots +t^{k-1})+\\frac{1}{\\tau }\\,\\frac{t^k}{1-t}.$ The first term on the right side of (REF ) increases monotonically with grow of $t$ , and in the second term, for any $\\tau >0$ , the numerator monotonically increases, and the denominator monotonically decreases with increasing $t$ in the interval $(0;1)$ .", "Therefore, the function $N_k(t;\\tau )$ , and hence $N_k^2(t;\\tau )$ , increases monotonically over the interval $(0;1)$ for any $\\tau >0$ .", "It follows that $(N_k^2)^\\prime _t>0$ .", "So we have shown that $\\text{sign}(Q_k(t;\\tau ))=\\text{sign}(q_k(t;\\tau )), \\qquad 0<t<1,\\, \\tau >0.$ Thus, we need to calculate $q_k(t;\\tau )$ .", "From the formula (REF ) we have $(N_k)^{\\prime }_t(t)=\\frac{1}{\\tau (1-t)^2}\\left(\\tau +(1-\\tau )(kt^{k-1}+(1-k)t^k)\\right),$ $(N_k)^{\\prime \\prime }_t(t)=\\frac{1}{\\tau (1-t)^3}\\times \\\\\\left(2\\tau +(1-\\tau )(k(k-1)-2k(k-2)t+(k-1)(k-2)t^2)t^{k-2}\\right).$ Then from (REF )-(REF ) follows $q_k(t;\\tau )=\\frac{1}{\\tau ^2(1-t)^4}\\,P_k(t;\\tau )$ where $P_k(t;\\tau )=\\left[2\\tau +(1-\\tau )t^{k-2}\\left((k-1)(k-2)t^2-2k(k-2)t+k(k-1)\\right)\\right]\\times \\\\(\\tau +(1-\\tau )t^k)-\\left(\\tau +(1-\\tau )t^{k-1}(k+(1-k)t)\\right)^2.$ It follows from (REF ) that $\\text{sign}(q_k(t;\\tau ))=\\text{sign}(P_k(t;\\tau ))$ when $0<t<1,\\,\\, \\tau >0$ .", "Therefore, the problem was reduced to determining the sign of the polynomial $P_k(t;\\tau )$ , which is convenient to write as $P_k(t;\\tau )=\\tau ^2+\\tau (1-\\tau )k(k-1)t^{k-2}-2\\tau (1-\\tau ) k(k-1)t^{k-1}+\\\\\\tau (1-\\tau )(k(k-1)+2)t^k-(1-\\tau )^2kt^{2k-2}+\\\\2(1-\\tau )^2kt^{2k-1}-(1-\\tau )^2(k-1)t^{2k}.$ So, to determine the sign of the Mandel parameter $Q_M(t;\\tau )$ in the band $0<t<1,\\,\\, \\tau >0,$ we need to find the roots and areas of the constant sign for the polynomial $P_k(t;\\tau )$ in this band .", "We do not consider the solution of this general problem in this paper, but we will discuss the technical difficulties that arise using the example of $P_2(t;\\tau )$ ." ], [ "Investigation of the sign of the Mandel parameter for $k=2$", "For $k=2$ , the formula (REF ) takes the form $P_2(t;\\tau )=\\tau (2-\\tau )+4\\tau (\\tau -1)t+2(1-\\tau )(3\\tau -1)t^2+4(\\tau -1)^2t^3-(\\tau -1)^2t^4.$ Our task is to investigate $\\text{sign}(P_2(t;\\tau ))$ in the region $\\Pi :0<t<1,\\,\\, \\tau >0$ .", "We divide the $\\Pi $ region into three parts $\\Pi =\\Pi _1\\,\\cup \\,\\Pi _2\\,\\cup \\,\\Pi _3,$ where $\\Pi _1&=&\\lbrace 0<t<1,\\,\\, 0<\\tau \\le 1\\rbrace \\nonumber \\\\\\Pi _2&=&\\lbrace 0<t<1,\\,\\, 1<\\tau \\le 2\\rbrace \\\\\\Pi _3&=&\\lbrace 0<t<1,\\,\\, \\tau >2\\rbrace .", "\\nonumber $ Let us first consider the case $\\Pi _2$ (which also includes the classical Chebyshev polynomials of the 1st and 2nd kind).", "We prove that in the region $\\Pi _2$ the Mandel parameter is positive, i.e.", "$\\text{sign}(P_2(t;\\tau ))>0\\quad \\text{}\\quad 0<t<1,\\, 1<\\tau \\le 2.$ To prove this, we divide the terms on the right hand side of the equation into the following parts: $\\sigma _1&=&\\tau (2-\\tau )=1-(\\tau -1)^2; \\\\\\sigma _2&=&4\\tau (\\tau -1)t; \\\\\\sigma _3&=&2(1-\\tau )(3\\tau -1)t^2=-2(\\tau -1)^2t^2+4\\tau (1-\\tau )t^2=\\sigma _{31}+\\sigma _{32}; \\\\\\sigma _4&=&4(\\tau -1)^2t^3=3(\\tau -1)^2t^3+(\\tau -1)^2t^3=\\sigma _{41}+\\sigma _{42}; \\\\\\sigma _5&=&-(\\tau -1)^2t^4.$ Next, we prove that the following inequalities hold in the region $\\Pi _2$ : $\\gamma _1&=&(\\sigma _1+\\sigma _{31}+\\sigma _{41})>0; \\\\\\gamma _2&=&(\\sigma _2+\\sigma _{32})>0; \\\\\\gamma _3&=&(\\sigma _{42}+\\sigma _5)>0.$ The validity of the inequalities $\\gamma _2$ and $\\gamma _3$ is quite obvious.", "Really, $\\gamma _2=4\\tau (\\tau -1)t(1-t)>0\\quad \\text{and}\\quad \\gamma _3=(\\tau -1)^2(1-t)t^3>0,$ since in the domain under consideration, all factors are positive.", "Consider now $\\gamma _1=\\gamma _1(t;\\tau )$ .", "We have $\\gamma _1(t;\\tau )=1-(\\tau -1)^2-2(\\tau -1)^2t^2+3(\\tau -1)^2t^3.$ Let's denote $\\xi =\\tau -1$ .", "Then $0<\\xi <1$ , $\\gamma _1(t;\\tau )=\\gamma _1(t;\\xi )=1-\\xi ^2(3t^3-2t^2-1).$ To prove that $\\gamma _1(t;\\tau )$ is positive, it is sufficient to show that the function $\\psi (t)=3t^3-2t^2-1$ satisfies the inequality $\\psi (t)\\le 1$ on the interval $0<t<1$ .", "The latter inequality holds, since a simple analysis on the extremum shows that in fact, even the stronger inequality $\\psi (t)\\le 0$ is true on this interval.", "So we proved that in the domain $\\Pi _2$ , the Mandel parameter is positive.", "Unfortunately, in the regions $\\Pi _1$ and $\\Pi _3$ , the situation is much more complicated — the boundaries of the sections in which the Mandel parameter has a constant sign can be set for fixed values of the parameter $a$ only approximately, using numerical methods.", "The tables below allows to determine the nature of the location of the sign-constant regions of the Mandel parameter.", "In these tables, we return to the original variables $x$ and $a$ in which the Mandel parrameter for $k=2$ has the form $\\qquad Q_M(x;2;a)=\\\\-\\frac{x(a^4(x-2)^4+x^2(8-8 x+x^2)-2a^2(16-16x+16x^2-8x^3+x^4))}{(x-2)(a^2(x-2)^2-(x-4)x)(a^2(x^2-4)-x^2)}.$ For the area $\\Pi _3$ with $a>\\sqrt{2}$ , we have Table: NO_CAPTIONFrom the results presented in this table, it follows that when the parameter $a$ increases, the interval in which $Q_M$ is negative increases, the minimum value of $Q^{min}$ is shifted to the right end of the interval, and its absolute value $\|Q^{min}\|$ increases.", "In the band $\\Pi _1:0<a<1$ , the situation is somewhat different, as can be seen from the following table.", "Table: NO_CAPTIONAs $a$ increases, the interval in which $Q_M$ is negative and the absolute value $\|Q^{min}\|$ decreases.", "For $0.3<a<1$ , the Mandel parameter is positive.", "Acknowledgements Authors are grateful to I. K. Litskevich for assistance in performing some calculations." ] ]	2011.14140
[ [ "On Performance Comparison of Multi-Antenna HD-NOMA, SCMA and PD-NOMA\n Schemes" ], [ "Abstract In this paper, we study the uplink channel throughput performance of a proposed novel multiple-antenna hybrid-domain non-orthogonal multiple access (MA-HD-NOMA) scheme.", "This scheme combines the conventional sparse code multiple access (SCMA) and power-domain NOMA (PD-NOMA) schemes in order to increase the number of users served as compared to conventional NOMA schemes and uses multiple antennas at the base station.", "To this end, a joint resource allocation problem for the MA-HD-NOMA scheme is formulated that maximizes the sum rate of the entire system.", "For a comprehensive comparison, the joint resource allocation problems for the multi-antenna SCMA (MA-SCMA) and multi-antenna PD-NOMA (MA-PD-NOMA) schemes with the same overloading factor are formulated as well.", "Each of the formulated problems is a mixed-integer non-convex program, and hence, we apply successive convex approximation (SCA)- and reweighted $\\ell_1$ minimization-based approaches to obtain rapidly converging solutions.", "Numerical results reveal that the proposed MA-HD-NOMA scheme has superior performance compared to MA-SCMA and MA-PD-NOMA." ], [ "Introduction", "The non-orthogonal multiple access (NOMA) technique has been identified as one of the spectrally efficient technologies to deal with the current spectrum crisis[1], [2].", "This technique allows multiple users to share the same time-frequency resource for transmission; and consequently, the spectral-efficiency (SE) of the system improves.", "Several NOMA schemes have been developed, two notable ones being power-domain NOMA (PD-NOMA)[3], and sparse code multiple access (SCMA) [4].", "They employ successive interference cancellation (SIC) and the message passing algorithm (MPA), respectively, to remove interference resulting from the other users.", "To further improve the SE of the PD-NOMA and SCMA schemes, several works have recently emerged [5], [6], [7], [8].", "In [5], a novel power-domain SCMA (PSMA) scheme for downlink channels is developed by adopting both conventional PD-NOMA and SCMA schemes.", "This scheme enhances the SE at the cost of increased decoding complexity.", "In [6], [7], for uplink channels, a joint codebook assignment and power allocation problem for the SCMA scheme to enhance the SE is discussed.", "In [8], PD-NOMA and SCMA schemes are combined followed by user grouping for improving the downlink channel capacity.", "Previous works have restrictions on the number of users in the system.", "However, for the many applications of the Internet of Things (IoT) it is desired to support a large number of users in the system which is also important from the perspective of 5G and beyond networks[9].", "Further, NOMA when coupled with multiple-antenna technology, which is not considered in the aforementioned works, can enhance the SE of the entire system.", "In [10], the authors have proposed a novel single-antenna hybrid-domain NOMA (HD-NOMA) scheme from the perspective of supporting a large number of uplink users.", "This scheme combines the SCMA and PD-NOMA schemes where the base station (BS) first categorizes the users into strong user (SU) and weak user (WU) groups based on their channel path losses and then encodes the data of each group according to the SCMA scheme followed by the superposition of the data of the two groups according to the PD-NOMA scheme.", "At the BS, the SU group is decoded via the MPA and subtracted from the received signal followed by the decoding of the WU group via the MPA.", "Unlike the single-antenna HD-NOMA scheme in [10], in this work we consider a general and complete system model by employing multiple antennas at the BS and joint per-user power and subcarrier allocation to achieve a larger SE.", "The problem is intractable and challenging in its original from, and hence, a few equivalent transformations and approximations are employed to exploit hidden convexity and improve tractability.", "The main contributions of this paper are as follows: A novel multi-antenna HD-NOMA (MA-HD-NOMA) scheme is proposed that supports a larger number of uplink users compared to the previous schemes.", "For the MA-HD-NOMA and its comparison with the multi-antenna SCMA (MA-SCMA) and multi-antenna PD-NOMA (MA-PD-NOMA) schemes, three joint subcarrier and power allocation design problems are formulated that maximize the sum rate of the entire system.", "Successive convex approximation (SCA)- and reweighted $\\ell _1$ minimization- based approaches to handle the non-convex and sparsity constraints, respectively, are used to develop a rapidly converging algorithm.", "The paper is organized as follows.", "Section II presents the system model of the conventional and proposed schemes.", "Section III presents the problem formulations.", "Section IV discusses the proposed solution and the algorithm.", "Section V presents and discusses numerical simulation results and Section VI concludes the paper." ], [ "System Model", "We consider a single-cell uplink NOMA system where a cell-centered BS equipped with $N_R$ antennas is simultaneously accessed by a set $\\mathcal {J}$ of $\|\\mathcal {J}\|=J$ single antenna users using a set $\\mathcal {K}$ of $\|\\mathcal {K}\|=K$ orthogonal subcarriers." ], [ "MA-CD-NOMA", "In this scheme, a total of $J = \\tbinom{K}{d_f}$ users are allowed to access the BS, where $d_f$ denotes the number of subcarriers used by each user out of $K$ .", "Further, at most $d_v=\\tbinom{K-1}{d_f-1}$ users are allowed to transmit simultaneously on the same subcarrier.", "Therefore, user $j$ maps its $\\log _2M$ bits onto an $d_f$ -sparse $K$ -dimensional codeword $\\mathbf {x}_{j}=[f_{j,1}s_{j,1}, \\ldots , f_{j,K}s_{j,K}]^T$ using its user-specific $M$ -elements codebook, where $s_{j,k}$ is the $j$ th user symbol transmitted using the $k$ th subcarrier, with average power $p_{j,k}=\\mathbb {E}\\lbrace \|s_{j,k}\|^2\\rbrace $ , and $f_{j,k}=1$ indicates that the user $j$ uses the subcarrier $k$ otherwise it is zero.", "The complete user-subcarrier association for the SCMA scheme is represented by a factor graph matrix $\\mathbf {F}\\in \\mathbb {B}^{J\\times K}$ .", "Assuming that the transmission is synchronized, the received signal vector $\\mathbf {y}\\in \\mathbb {C}^{N_RK\\times 1}$ at the BS is given as $\\mathbf {y} = \\sum _{j=1}^{J}\\mathbf {H}_j\\mathbf {x}_j+\\mathbf {n}_{j}$ , where $\\mathbf {n}_j\\sim \\mathcal {CN}(\\mathbf {0}, \\sigma _n^2\\mathbf {I})$ is the additive white Gaussian noise (AWGN) vector, and $\\mathbf {H}_j = [\\text{diag}(\\mathbf {h}_{j}[1]), \\ldots , \\text{diag}(\\mathbf {h}_{j}[N_R])]^T\\in \\mathbb {C}^{N_RK\\times K}$ is the $j$ th user channel matrix with $\\mathbf {h}_{j}[n]=[h_{j,1}[n], \\ldots , h_{j,K}[n]]^T$ ; $h_{j,k}[n]= g_{j,k}[n]/\\text{PL}_j$ , where $\\text{PL}_j$ is the path loss for user $j$ and $g_{j,k}[n]\\sim \\mathcal {CN}(0,1)$ denotes the Rayleigh fading channel between the user $j$ and the BS antenna $n$ on the subcarrier $k$ .", "To decode the users' data, the BS employs the MPA decoder, which essentially removes the resulting interference.", "Thus, the achievable sum rate of the MA-SCMA scheme is given as [7] $\\text{SR}=\\sum _{k=1}^{K}\\log _2 \\Bigg (1+\\frac{\\sum _{j=1}^J\|\\mathbf {h}_{j,k}\|^2f_{j,k}p_{j,k}}{\\sigma _n^2}\\Bigg ),$ where $\\mathbf {h}_{j,k} = [h_{j,k}[1],\\ldots , h_{j,k}[N_R]]^T$ ." ], [ "MA-PD-NOMA", "In this scheme, the uplink users with different channel gains superimpose their data and transmit simultaneously to communicate with the BS.", "In this scheme, for a system with $J$ users and $K$ subcarriers, we assume that each user uses only one subcarrier, i.e., $d_f=1$ , and each subcarrier can be accessed by more than one users, i.e., $d_v>=1$ .", "Again assuming synchronized transmission, the received signal vector $\\mathbf {y}\\in \\mathbb {C}^{N_RK\\times 1}$ at the BS is given as $\\mathbf {y} = \\sum _{j=1}^{J}\\mathbf {H}_j\\mathbf {x}_j+\\mathbf {n}_{j}$ .", "At the BS, the strongest user is decoded first followed by SIC-based decoding to detect the rest of the users that are assumed to be arranged in descending order of their channel gains.In this work, we assume the following order: J>(J-1)>...>2>1.", "The achievable sum rate of the MA-PD-NOMA system under the condition that the SIC decoding is successful is given as [11] $\\text{SR}=\\sum _{k=1}^{K}\\log _2 \\Bigg (1+\\frac{\\sum _{j=1}^J\|\\mathbf {h}_{j,k}\|^2f_{j,k}p_{j,k}}{\\sigma _n^2}\\Bigg ).$ Although the sum rate expressions for the MA-SCMA and MA-PD-NOMA schemes are the same, the resulting sum rates for the two systems are different as they use different factor graph matrices.", "In MA-HD-NOMA, the conventional SCMA and PD-NOMA schemes are combined with the aim to support more number of users than supported by the conventional schemes.", "In order to achieve this, we consider $2J$ users in the set $\\mathcal {J}$ and categorized them into two smaller and equal-sized sets $\\mathcal {J}_s$ and $\\mathcal {J}_w$ of $\|\\mathcal {J}_s\|=J_s$ SUs and $\|\\mathcal {J}_w\|=J_w$ WUs, respectively, according to the users' large-scale channel fading gains.", "In this scheme, the users within each group are encoded according to the SCMA scheme whereas the data of the groups are superimposed according to the PD-NOMA scheme because the two groups have different large-scale channel gains.", "Again, assuming synchronized transmission, the received signal vector $\\mathbf {y}\\in \\mathbb {C}^{N_RK\\times 1}$ at the BS is given as $\\mathbf {y} = \\sum _{i=1}^{J_s}\\mathbf {H}^s_i\\mathbf {x}^s_i+\\sum _{j=1}^{J_w}\\mathbf {H}^w_j\\mathbf {x}^w_j+\\mathbf {n}_{j}$ , where $\\mathbf {H}^s_i = [\\text{diag}(\\mathbf {h}^s_i[1]),\\ldots , \\text{diag}(\\mathbf {h}^s_i[N_R])]^T$ and $\\mathbf {H}^w_j = [\\text{diag}(\\mathbf {h}^w_j[1]),\\ldots , \\text{diag}(\\mathbf {h}^w_j[N_R])]^T$ , where $\\mathbf {h}^s_i$ and $\\mathbf {h}^w_j$ are defined similarly as $\\mathbf {h}_j$ .", "At the BS, a two-step decoding process is employed to decode the users' data.", "In the first step, the MPA is applied on the received signal to decode the users of the SU group while treating the data that belongs to the WU group as noise.", "In the second step, SIC is performed to remove the data of the SU group from the received signal followed by the MPA on it to decode the users of the WU group.", "Thus, the achievable sum rate for the MA-HD-NOMA scheme under the condition that the SU group is decoded correctly is given as [10] ll SRHD-NOMA= k=1K2 (1+i=1Js\|hsi,k\|2fsi,kpsi,k+j=1Jw\|hwj,k\|2fwj,kpwj,kn2), where $f^s_{i,k}$ and $f^w_{j,k}$ denote the $(i,k)$ th and $(j,k)$ th elements of the factor graph matrix corresponding to the SU and WU groups, respectively.", "$p^s_{i,k}$ and $p^w_{j,k}$ denote the power allocated to the SU $i$ and the WU $j$ on the subcarrier $k$ , respectively.", "We assume that the distances and the channel coefficients from the users to the BS remain constant during the transmission interval.", "Next, we formulate the problem of joint subcarrier and power allocation design so as to maximize the sum rate of the MA-HD-NOMA system.", "For comparison purposes, we also formulate the joint subcarrier and power allocation problems for the MA-SCMA and MA-PD-NOMA systems with the same number of users as in the MA-HD-NOMA case, i.e, the overloading factors $\\delta = 2J/K$ of all the three systems are assumed to be the same.", "Further, it is assumed that the BS decodes the SU group before the WU group and has full channel state information of all the users to solve the problems in a centralized manner." ], [ "MA-HD-NOMA", "The joint subcarrier and power allocation design problem for the MA-HD-NOMA system is written as follow: lcl F,P SRHD-NOMA * s.t.", "k=1K 2(1+\|hwi,k\|2fwi,kpwi,kj=1i-1\|hwj,k\|2fwj,kpwj,k+n2) Rwi iJw, k=1K2(1+i=1Js\|hsi,k\|2fsi,kpsi,kj=1Jw\|hwj,k\|2fwj,kpwj,k+n2) Rs, k=1Kfsi,k dsv iJs, k=1Kfwj,k dwv jJw * i=1Jsfsi,k dsf kK, j=1Jwfwj,k dwf kK * k=1Kfsi,kpsi,k P iJs, k=1Kfwj,kpwi,k P jJw fsi,kB i Js k K, fwj,kB j Jw k K, where $\\mathbf {F}$ includes matrices $\\lbrace \\mathbf {F}^s, \\mathbf {F}^w\\rbrace $ , and $\\mathbf {P}$ includes matrices $\\lbrace \\mathbf {P}^s, \\mathbf {P}^w\\rbrace $ .", "$\\mathbf {F}^s=[[f^s_{1,1},\\ldots ,f^s_{J_s,1}]^T \\cdots [f^s_{1,K},\\ldots ,f^s_{J_s,K}]^T]$ and $\\mathbf {P}^s=[[p^s_{1,1},\\ldots ,p^s_{J_s,1}]^T \\cdots [p^s_{1,K},\\ldots ,p^s_{J_s,K}]^T]$ and similarly $\\mathbf {F}^w$ and $\\mathbf {P}^w$ are defined.", "$\\bar{R}^w_i$ , $\\bar{R}^s$ , and $\\bar{P}$ denote the minimum rate of the user $i$ in the WU group, the total sum rate of the users in the SU group, and the maximum transmit power of each user, respectively.", "Constraint (REF ) ensures that the user $i$ in the WU group receives a minimum rate of $\\bar{R}^w_i$ whereas the constraint (REF ) ensures that the SIC decoding is successful at the BS if the users in the SU group achieve the sum rate of $\\bar{R}^s$ b/s/Hz or more.", "Constraint (REF ) ensures that not more than $d_f$ subcarriers are used by a user in both groups whereas (REF ) ensures that not more than $d_v$ users occupy one subcarrier.", "Constraints (REF ) and (REF ) denote the restrictions on user's maximum transmit power and on each element of the factor graph matrix to be binary for both SU and WU groups, respectively." ], [ "MA-SCMA", "In this system, the SU group and the WU group are individually encoded according to the conventional SCMA scheme as previously described and allowed to transmit simultaneously.", "At the BS, group decoding is performed by treating the other group's data as noise.", "Hence, the sum rate for this MA-SCMA scheme is given as llc SRSCMA= k=1K(2 (1+j=1Js\|hsj,k\|2fsj,kpsj,ki=1Jw\|hwi,k\|2fwi,kpwi,k+n2) + 2 (1+i=1Jw\|hwi,k\|2fwi,kpwi,kj=1Js\|hsj,k\|2fsj,kpsj,k+n2)), and the achievable rate of the user $i$ in the WU group is given as ll Rwi=k=1K 2(1+\|hwi,k\|2fwi,kpwi,kj=1Js\|hsj,k\|2fsj,kpsj,k +m=1i-1\|hwm,k\|2fwm,kpwm,k+n2).", "The joint subcarrier and power allocation design problem for the MA-SCMA scheme is written as lcl F,P {SRSCMARwi Rwi iJw,(REF )-(REF )}." ], [ "MA-PD-NOMA", "In this system, the users from the SU and the WU groups are allowed to form a pair to superimpose their signals on a subcarrier.", "At the BS, the users are arranged in descending order of their channel gains and are decoded using the SIC technique on each subcarrier.", "Unlike the MA-SCMA, in this system, each user is restricted to use only one subcarrier.", "Since the BS performs SIC-based decoding, the sum rate expression of the MA-PD-NOMA system is the same as that for MA-HD-NOMA.", "Therefore, we use (REF ) to represent the sum rate of the MA-PD-NOMA system.", "Furthermore, the achievable rate of the user $i$ in the WU group is given as ll Rwi=k=1K 2(1+\|hwi,k\|2fwi,kpwi,kj=1i-1\|hwj,k\|2fwj,kpwj,k+n2), and the achievable sum rate of the SU group is given as ll Rs=k=1Ki=1Js 2(1+\|hsi,k\|2fsi,kpsi,kj=1i-1\|hsj,k\|2fsj,kpsj,k +m=1Jw\|hwm,k\|2fwm,kpwm,k+n2).", "The joint subcarrier and power allocation design problem for the MA-PD-NOMA system with $d^s_f=d^w_f=1$ and $d^s_v+d^w_v=3$ is written as lcl F,P {(REF )Rwi Rwi iJw,Rs Rs,(REF )-(REF )}.", "Note that, (REF ), (REF ), and (REF ) are mixed integer non-convex programs due to the presence of non-convex constraints and binary variables and obtaining a rapidly converging optimal solution to each of them is difficult.", "Hence, we seek suboptimal solutions by developing a rapidly converging algorithm in the following section.", "The proposed solution method is developed for the MA-HD-NOMA system but is applicable to the other two problems as well." ], [ "Proposed Solution", "In this section, we invoke two steps in an attempt to efficiently solve (REF ).", "The first step is about identifying the hidden convexity and achieve tractability by performing some equivalent transformations.", "Problem (REF ) is non-convex due to the non-linearity of type $xy$ in (REF )-(REF ) and (REF ) and $x/y$ type in (REF ) and (REF ).", "To tackle these issues, we introduce a set of positive slack variables $\\mathcal {V}\\in \\lbrace z_k, t^s_{i,k},t^w_{j,k},\\gamma ^{w}_{j,k}, \\gamma ^s_k,\\beta ^{w}_{j,k}, \\beta ^s_k\\rbrace ,\\forall i\\in \\mathcal {J}_s,\\forall j\\in \\mathcal {J}_w,\\forall k\\in \\mathcal {K}$ and rewrite (REF ) equivalently as: lcl F,P,V k=1Kzk * s.t.", "zk -1i=1Js\|hsi,k\|2tsi,k+j=1Jw\|hwj,k\|2twj,kn2, kK, tsi,k fsi,kpsi,k iJs, kK, twj,k fwj,kpwj,k jJw, kK, $\\prod _{k=1}^K (1+\\gamma ^w_{j,k})$ 2Rwj jJw, wj,kwj,k\|hwj,k\|2twj,k jJw, kK, wj,k j=1i-1\|hwj,k\|2twj,k+n2 jJw, kK, k=1K (1+sk) 2Rs, * sksk i=1Js\|hsi,k\|2tsi,k kK, * sk j=1Jw\|hwj,k\|2twj,k+n2 kK, k=1Ktsi,kP i Js, k=1Ktwj,kP j Jw, (REF ),(REF ), and (REF ), where () is due to the fact that logarithm is a monotonically non-decreasing function.", "Problem () is equivalent to (REF ) due to the fact that at the optimal point, all the constraints ()-() are satisfied as equalities.", "Hence, the optimal solution to () is also optimal for (REF ).", "After the equivalent transformations, () now becomes tractable, however, it is still a non-convex problem due to non-convex and binary constraints present in (), (), (), (), and (REF ), respectively.", "Next, we invoke the second step, which is about relaxing the binary constraints, applying SCA on the non-convex constraints, and developing an iterative algorithm for solving the problem.", "Inspired by the reweighted $\\ell _1$ -minimization method [12], we relax the binary constraint (REF ) and reformulate the objective function with a penalty term as presented in the following problem at the $n$ th iteration: lcl F,P,V ()-k=1K(i=1Jsws,(n)i,kfsi,k+j=1Jwww,(n)j,kfwj,k)* s.t.", "k=1Kfsi,k dv iJs, k=1Kfwj,k dv jJw i=1Jsfsi,k df kK, j=1Jwfwj,k df kK 0fsi,k, fwj,k1, iJs,jJw, kK ()-(), where $w^{s,(n)}_{i,k}=1/(\|f^{s,(n-1)}_{i,k}\|+\\epsilon )$ and $w^{w,(n)}_{j,k}=1/(\|f^{w,(n-1)}_{j,k}\|+\\epsilon )$ $\\forall i\\in \\mathcal {J}_s, \\forall j\\in \\mathcal {J}_w, \\forall k\\in \\mathcal {K}$ , with a very small value $\\epsilon $ , and constant $\\lambda $ is the weight of the penalty term.", "Next, to deal with (), (),() and (), we replace the nonlinear term of type $xy$ by its equivalent difference-of-convex functions, i.e., $xy=0.25((x+y)^2-(x-y)^2)$ followed by linearization of the term $(x+y)^2$ for () and (), and the term $(x-y)^2$ for () and () around the point $(x^{(n)},y^{(n)})$ at $n$ th iteration.", "Hence, the final approximate problem we solve at the $n$ th iteration is: lcl F,P,V ()* s.t.", "tsi,k - (fs,(n)i,k-ps,(n)i,k)(fsi,k - fs,(n)i,k+ psi,k - ps,(n)i,k)2- (fs,(n)i,k+ps,(n)i,k)24 + (fsi,k-psi,k)24 0, iJs, kK, twi,k - (fw,(n)i,k-pw,(n)i,k)(fwi,k - fw,(n)i,k+ pwi,k - pw,(n)i,k)2- (fw,(n)i,k+pw,(n)i,k)24 + (fwi,k-pwi,k)24 0, iJw, kK, (wi,k +wi,k )24 - (w,(n)i,k(n) -w,(n)i,k )24 - \|hwi,k\|2twi,k - (w,(n)i,k -w,(n)i,k )(wi,k - w,(n)i,k -wi,k + w,(n)i,k)20, iJw, kK,* (sk +k )24 - (s,(n)k -k(n) )24 - i=1Js\|hsi,k\|2tsi,k- (s,(n)k -k(n) )(sk - s,(n)k -k + k(n))2 0, kK, (),(),(),()-(),()-(), Problem () is convex and can be solved iteratively until the value of the objective function converges.", "The pseudo code that solves the optimization problem is presented in Algorithm 1.", "Figure: NO_CAPTIONThe objective function in problem () is upper bounded due to the power constraints, and Algorithm 1 generates a monotonic non-decreasing sequence of objective function values, and hence, Algorithm 1 converges to the Karush-Kuhn-Tucker (KKT) point of () [13].", "Since () admits the second-order conic program (SOCP) due to (), (), and (), Algorithm 1 solves the SOCP in each iteration.", "The worst case of the complexity is regulated by the SOCP [14] and is given as $\\mathcal {O}(3I(K-1)^3(2+J_w)^2)$ , where $I$ is the number of iterations required by the algorithm to converge.", "Figure: The average sum rate of MA-HD-NOMA versus the number of iterations for N R =4N_R=4 and P ¯=24\\bar{P}= 24 dBm.Figure: The average sum rate versus the maximum transmit power of the users for different number of antennas N R ={1,2,3,4}N_R=\\lbrace 1,2,3,4\\rbrace ." ], [ "Simulation Results", "In this section, we numerically evaluate the sum rate performances of the MA-HD-NOMA, MA-SCMA and MA-PD-NOMA schemes obtained through Algorithm 1 using YALMIP parser and MOSEK as the internal solver.", "The users are uniformly distributed within a circular cell of radius 100 m with the BS at the center.", "The users that lie within the circular area of radius $d_s\\le 50$ m are referred to as the strong users while the others are referred to as the weak users, i.e., $d_w>50$ m. The value of the parameters for the system are set as follows: $\\text{BW}=1\\,\\text{MHz},\\,N_R=\\lbrace 1,2,3,4\\rbrace ,\\, K=4,\\, J_s=6,\\, J_w=6,\\, \\epsilon =0.001$ .", "Path loss of the user $j$ is denoted as $\\text{PL}_j=-145.4+37.5\\log _{10}(d)$ dB, where $d$ (in kms.)", "is the propagation distance of the signal, and noise power is $\\sigma _n^2=-204+10\\log _{10}(\\text{BW}/K)$ .", "$d^s_f=d^w_f=2,\\, d^s_f=d^w_f=3$ for the MA-HD-NOMA and MA-SCMA schemes, and $d^s_f=d^w_f=1,\\, d^s_f+d^w_f=3$ is set for the MA-PD-NOMA scheme.", "In Fig.", "REF , we plot the average sum rate of the MA-HD-NOMA scheme versus the number of iterations required to obtain a stabilized sum rate for a given random channel realization.", "It can be observed that Algorithm 1 converges after a few iterations under two different random channel realizations.", "In Fig.", "REF , we compare the achieved sum rate versus user transmit power of the MA-HD-NOMA scheme to those of the MA-SCMA and MA-PD-NOMA schemes for different values of the number of receiver antennas of the BS.", "We have three observations from the figure.", "First, the achieved sum rate of the proposed MA-HD-NOMA system is higher than that of the MA-SCMA and MA-PD-NOMA schemes.", "Second, the achievable sum rate of all three schemes increases with the number of receiver antennas.", "Lastly, the performance of the MA-PD-NOMA scheme is better than the MA-SCMA scheme, which is due to the fact that the MA-SCMA scheme does not perform SIC decoding between the groups at the BS, i.e., the SU and WU group are decoded separately while considering the data from the other group as noise.", "The optimal solution obtained through a branch-and-bound algorithm is also plotted to compare the performance of the proposed algorithm.", "For $N_R=1$ , we observe that the performance of the MA-HD-NOMA system is quite close to the optimal one." ], [ "Conclusion", "In this letter, we have proposed the MA-HD-NOMA scheme, which has the potential to support more users than MA-SCMA and MA-PD-NOMA, and compared its achievable sum rate performance with these alternatives.", "The MA-HD-NOMA scheme is a combination of the SCMA and PD-NOMA schemes, and hence benefits from the encoding and MPA-based decoding schemes of the SCMA, and the SIC-based decoding scheme of the PD-NOMA.", "The subcarrier and power allocation design problem is formulated by maximizing the achievable sum rate for all three systems for a comprehensive comparison.", "Since the resulting optimization problems are mixed integer non-convex programs, which are difficult to solve, we have proposed a low-complexity approximation of the original problem that rapidly converges to a suboptimal solution.", "However, the performance of the algorithm is quite close to that of the optimal one.", "Further, under the same overloading factor, the MA-HD-NOMA scheme outperforms both MA-SCMA and MA-PD-NOMA schemes.", "Evaluating the performance of the proposed MA-HD-NOMA scheme in multiple cell scenarios is the next logical and viable research direction." ] ]	2011.14073
[ [ "High order asymptotic preserving discontinuous Galerkin methods for gray\n radiative transfer equations" ], [ "Abstract In this paper, we will develop a class of high order asymptotic preserving (AP) discontinuous Galerkin (DG) methods for nonlinear time-dependent gray radiative transfer equations (GRTEs).", "Inspired by the work \\cite{Peng2020stability}, in which stability enhanced high order AP DG methods are proposed for linear transport equations, we propose to pernalize the nonlinear GRTEs under the micro-macro decomposition framework by adding a weighted linear diffusive term.", "In the diffusive limit, a hyperbolic, namely $\\Delta t=\\mathcal{O}(h)$ where $\\Delta t$ and $h$ are the time step and mesh size respectively, instead of parabolic $\\Delta t=\\mathcal{O}(h^2)$ time step restriction is obtained, which is also free from the photon mean free path.", "The main new ingredient is that we further employ a Picard iteration with a predictor-corrector procedure, to decouple the resulting global nonlinear system to a linear system with local nonlinear algebraic equations from an outer iterative loop.", "Our scheme is shown to be asymptotic preserving and asymptotically accurate.", "Numerical tests for one and two spatial dimensional problems are performed to demonstrate that our scheme is of high order, effective and efficient." ], [ "Introduction", "Radiative transfer equations (RTEs) are a type of kinetic scale modeling equations, which are used to describe the time evolution of radiative intensity and energy transfer of a radiation field with its background material [5], [37].", "The system has many applications in astrophysics, inertial confinement fusion (ICF), plasma physics and so on.", "It has attracted a lot of attention for numerical studies due to its importance but high complexity.", "Gray radiative transfer equations (GRTEs) are a type of simplified RTEs for gray photons and coupled to the background with the material temperature.", "Due to its high dimensionality and the photons are traveling in the speed of light, a popular numerical method for simulating the GRTEs in literature is the implicit Monte Carlo method, see [10], [11], [7], [8], [41], [40] and references there in.", "On the other hand, to avoid numerical noises from the Monte Carlo method, deterministic methods are also developed.", "However, several difficulties arise.", "Firstly, for the GRTEs with a high or thick opacity background material, there are severe interactions between radiation and material, in which case, the photon mean free path is approaching zero and the diffusive radiation behavior becomes to dominate.", "Numerical methods for streaming transport equations in the low or thin opacity material will suffer from very small time step restriction in this diffusive regime.", "Secondly, the photons are traveling in a speed of light, so that the system also deserves implicit treatments for the transport term.", "Thirdly, the time evolution of the radiative intensity is coupled with the background material temperature.", "Its local thermal equilibrium is a Planckian, which is a nonlinear function of the material temperature.", "In the diffusive limit as the photon mean free path vanishing, it converges to a nonlinear diffusive equation.", "Due to the high dimensionality and strong coupled nonlinearity, fully implicit schemes are very difficult to solve, which require huge computational cost unaffordable even on modern computers.", "An efficient approach to deal with these difficulties is the asymptotic preserving (AP) scheme.", "AP schemes were first studied in the numerical solution of steady neutron transport problems [27], [26], [20], [21], and later applied to nonstationary transport problems [23], [22].", "For a scheme to be AP, in the diffusive scaling as we considered here, it means that when the spatial mesh size $h$ and time step $\\Delta t$ are fixed, as the photon mean free path approaching zero, the scheme automatically becomes a consistent discretization for the limiting diffusive equation, see [18], [19] for more concrete definitions of AP schemes for multi-scale kinetic equations.", "Recently AP schemes are further developed under different scenorios.", "For linear transport equations in the diffusive scaling, a class of AP schemes coupled with discontinuous Galerkin discretizations in space and globally stiffly accurate implicit-explicit (IMEX) schemes were developed in [16], [17], [34], [36] under the micro-macro decomposition framework [29], while in space using finite difference discretizations under the same framework was developed in [24].", "Crestetto et.", "al.", "recently proposed to decompose the solution of the linear transport equation in the diffusive regime in the time evoluation of an equilibrium state plus an perturbation, and a Monte Carlo solver for the perturbation with an Eulerian method for the equilibrium part was designed [6], which is asymptotically complexity diminishing.", "AP schemes are also coupled with gas-kinetic schemes for linear kinetic model [32] and nonlinear RTEs [42], [44], [43], [39], [48], etc.", "However, to the best of our knowledge, high order schemes for time-dependent nonlinear RTEs, which is AP for the vanishing photon mean free path, is rare in literature, except [14] which is based on multiscale high order/low order (HOLO) algorithms [4] and [30] which is based on high order S-stable diagonally implicit Runge-Kutta (SDIRK) methods and linearization on the Planck function, as some of such few examples.", "In this paper, inspired by the work [34], [17], we try to develop a class of high order AP DG-IMEX schemes for the GRTEs.", "Here in space, we use DG discretizations.", "The first DG scheme is developed for neutron transport equation by Reed and Hill [38].", "Later DG has been widely used to study stationary and non-stationary transport equations, e.g.", "[1], [12], [17], [49], [33], [24], [34], [45], [13], [50].", "In time, globally stiffly accurate IMEX schemes [2] are used.", "The same space and time discretizations have been proposed for linear transport equations in the diffusive scaling in [17] and analized in [16].", "Recently, Peng et.", "al.", "[34] proposed a class of stability enhanced schemes by adding a weighted diffusive term on both sides but numerically discretizing them differently in time.", "The schemes have been analized in [35], which show that the stability enhanced approach has a hyperbolic time step condition, namely $\\Delta t = {\\mathcal {O}}(h)$ where $\\Delta t$ and $h$ are the time step size and mesh size respectively, instead of a parabolic time step condition $\\Delta t={\\mathcal {O}}(h^2)$ in [16].", "Further, to avoid the ad-hoc weighted diffusive term, Peng and Li proposed a class of AP IMEX-DG-S schemes based on the Schur complement [36].", "Here, the stability-enhanced approach is borrowed, which is more straightforward than the scheme based on the Schur complement, when applied to the nonlinear GRTEs.", "The new challenge for GRTEs here is that the radiative intensity is nonlinearly coupled to the background material temperature.", "To effectively and efficiently deal with the nonlinearity appearred in the Planck function, we use a Picard iteration with a predictor-corrector procedure as is used in [46] for the capturing the right front propogation for GRTEs.", "The global nonlinear system is decoupled to a linear system with nonlinear algebraic equations only in each element from an outer iterative loop.", "We will formally prove that our scheme for GRTEs is AP and also asymptotically accurate (AA).", "Numerically we will show it is high order in space and in time, and can effectively and efficiently solve the nonlinear GRTEs.", "The rest of the paper is organized as follows.", "In Section 2, the GRTEs are revisited and reformulated by a micro-macro decomposition.", "DG space and IMEX time discretizations are described in Section 3.", "Formal AP and AA analyses are given in Section 4.", "We will perform some numerical tests in Section 5, followed by a conclusion in Section 6." ], [ "Gray radiative transfer equation", "Here we consider the gray radiative transfer equations in the scaled form [42], [37] $\\left\\lbrace \\begin{array}{l}{c}{\\partial t}+\\varepsilon \\,\\vec{\\Omega }\\cdot \\nabla I =\\sigma \\left(\\Phi -I\\right), \\\\ \\, \\\\\\varepsilon ^2\\,C_v {\\partial t}\\equiv \\varepsilon ^2{\\partial t}=\\sigma \\, \|{\\vec{\\Omega }}\| \\,\\left({\\langle {I}\\rangle }-\\Phi \\right),\\end{array}\\right.$ This system describes the radiative transfer and energy exchange between the radiation and the material.", "Here $t$ is the time variable, ${\\vec{r}}$ is the position variable and ${\\vec{\\Omega }}$ is the direction of traveling of the photons.", "$I(t,{\\vec{r}},{\\vec{\\Omega }})$ is the radiative intensity in the direction of ${\\vec{\\Omega }}$ , $T(t,{\\vec{r}})$ is the material temperature, $\\sigma (t,{\\vec{r}})$ is the opacity, $C_v(t,{\\vec{r}})$ is the scaled heat capacity and $U(t,{\\vec{r}})$ is the material energy density.", "$\\varepsilon > 0$ is the Knudsen number defined as the ratio of the photon mean free path over the characteristic length of space.", "We denote $\\langle \\cdot \\rangle $ as the integral average over the angular ${\\vec{\\Omega }}$ , namely ${\\langle {\\cdot }\\rangle } = \\frac{1}{\|{\\vec{\\Omega }}\|}\\int \\cdot \\,d{\\vec{\\Omega }},$ and $\\Phi =acT^4/\|{\\vec{\\Omega }}\|$ , where $a$ is the radiation constant, $c$ is the scaled speed of light.", "For simplicity, the internal source and scattering are omitted.", "The spatial position vector ${\\vec{r}}$ is usually presented by the Cartesian coordinate with ${\\vec{r}}=(x,y,z)$ , while the direction ${\\vec{\\Omega }}=(\\zeta ,\\eta ,\\mu )$ is described by a polar angle $\\theta $ measured with respect to any fixed direction in space (e.g., the $z$ axis) and a corresponding azimuthal angle $\\phi $ , then $\\mu = \\cos \\theta , \\quad \\zeta =\\sin \\theta \\cos \\phi , \\quad \\eta = \\sin \\theta \\sin \\phi ,$ and $d{\\vec{r}}=dx\\,dy\\,dz, \\quad d{\\vec{\\Omega }}=\\sin \\theta \\,d\\theta \\,d\\phi =-d\\mu \\,d\\phi ,$ In the one-dimensional case, (REF ) reduces to $\\left\\lbrace \\begin{array}{l}{c}{\\partial t}+\\varepsilon \\,\\mu \\, I_x =\\sigma \\left(\\Phi -I\\right), \\\\ \\, \\\\\\varepsilon ^2 C_v {\\partial t}\\equiv \\varepsilon ^2{\\partial t}=2\\sigma \\left({\\langle {I}\\rangle }-\\Phi \\right),\\end{array}\\right.$ with $\\mu \\in [-1,1]$ .", "While in the two dimensional case, it becomes $\\left\\lbrace \\begin{array}{l}{c}{\\partial t}+\\varepsilon \\,(\\zeta \\, I_x + \\eta \\, I_y)=\\sigma \\left(\\Phi -I\\right), \\\\ \\, \\\\\\varepsilon ^2 C_v {\\partial t}\\equiv \\varepsilon ^2{\\partial t}=2\\pi \\,\\sigma \\left({\\langle {I}\\rangle }-\\Phi \\right),\\end{array}\\right.$ with $\\zeta =\\sqrt{1-\\mu ^2}\\,\\cos \\phi \\in [-1,1], \\quad \\eta =\\sqrt{1-\\mu ^2}\\,\\sin \\phi \\in [-1,1],\\quad \\mu \\in [0,1],\\quad \\phi \\in [0,2\\pi ].$ (REF ) is a relaxation model for the radiative intensity to its local thermodynamic equilibrium, at where the emission source is a Planckian at the local material temperature $\\sigma {\|{\\vec{\\Omega }}\|}$ .", "The material temperature $T(t,{\\vec{r}})$ is related to the material energy density $U(t,{\\vec{r}})$ by ${\\partial T}=C_v > 0,$ where $C_v$ is the heat capacity.", "It has been shown in [25] that as the Knudsen number $\\varepsilon \\rightarrow 0$ , away from boundaries and initial times, the intensity $I$ will approach to the Planckian at the local temperature equilibrium, $I^{(0)}={\|{\\vec{\\Omega }}\|}ac(T^{(0)})^4,$ and the corresponding local temperature $T^{(0)}$ satisfies the following nonlinear diffusive equation ${\\partial t}U(T^{(0)})+{\\partial t}a(T^{(0)})^4=\\nabla \\cdot {3\\sigma }\\nabla ac(T^{(0)})^4.$" ], [ "Micro-macro decomposition", "We consider to reformulate the radiative transfer equations (REF ) by a micro-macro decomposition [29], [17].", "We start to split the radiative intensity $I$ as $I = \\rho + \\frac{\\varepsilon }{{\\sqrt{\\sigma _0}}} \\, g,$ where $\\rho = \\Pi \\, I=\\langle I \\rangle $ , $\\varepsilon \\,g/{\\sqrt{\\sigma _0}}=({\\mathbb {I}}-\\Pi )I$ is the perturbation.", "${\\mathbb {I}}$ is the identity and $\\Pi $ is an orthogonal projection.", "$\\sigma _0$ is a constant which is defined as a referred opacity.", "${\\sqrt{\\sigma _0}}$ is added in the consideration that when $\\sigma $ is relatively large, (REF ) will also be close to the thermodynamic equilibrium, which is different from [17], [34].", "If we integrate over the travel direction ${\\vec{\\Omega }}$ of photons, subtracting it from (REF ), we will obtain the following micro-macro decomposition system $\\left\\lbrace \\begin{array}{l}{\\partial t}+\\frac{c}{{\\sqrt{\\sigma _0}}}\\,\\nabla \\cdot \\langle {\\vec{\\Omega }}\\, g \\rangle ={\\varepsilon ^2/\\sigma _0}\\left(\\Phi -\\rho \\right), \\\\ \\, \\\\\\frac{\\varepsilon ^2}{\\sigma _0} C_v {\\partial t}={\\tilde{\\sigma }}\\, \|{\\vec{\\Omega }}\|(\\rho -\\Phi ), \\\\ \\, \\\\g_t+c̑{\\varepsilon }({\\mathbb {I}}-\\Pi )\\nabla \\cdot ({\\vec{\\Omega }}\\, g)+{\\varepsilon ^2}\\nabla \\cdot ({\\vec{\\Omega }}\\,\\rho )=-{\\varepsilon ^2/\\sigma _0}g,\\end{array}\\right.$ where ${\\tilde{\\sigma }}=\\sigma /\\sigma _0$ is a relative opacity compared to the reference opacity $\\sigma _0$ , and $\\sigma _0$ will be specified later.", "In (REF ), if we assume all variables, such as $\\rho $ and $g$ , are of $\\mathcal {O}(1)$ , formally if $\\varepsilon /{\\sqrt{\\sigma _0}}\\rightarrow 0$ , in the leading order we get $\\rho =\\Phi $ and $g=-{\\sigma }\\nabla \\cdot ({\\vec{\\Omega }}\\,\\rho )$ from the second and third equations respectively.", "Substituting them into the first equation, replacing the right hand side by $-c\\,\\frac{C_v}{\|{\\vec{\\Omega }}\|}{\\partial t}$ from the second equation, the nonlinear diffusive equation (REF ) is directly implied, which shows the advantage of this micro-macro reformulation.", "A more rigorous asymptotic analysis will be given in the Section .", "For more discussions on micro-macro decomposition, see [29], [17].", "In [16], [17], a family of high order asymptotic preserving (AP) schemes was proposed and analyzed for the linear transport equations (corresponding to $\\Phi =\\rho $ in the second equation of (REF ), namely, the temperature is at the equilibrium which is time independent).", "This type of scheme is based on nodal discontinuous Galerkin (DG) spatial discretization and globally stiffly accurate implicit-explicit (IMEX) temporal discretization.", "However, the scheme as $\\varepsilon \\rightarrow 0$ in the diffusive limit is intrinsically an explicit time discretization for the limiting heat equation and numerical stability requires a time step size $\\Delta t\\sim h^2$ when $\\varepsilon \\ll 1$ , where $h$ is the spatial mesh size.", "This time step condition is quite stringent for the consideration of computational efficiency.", "Recently, Peng et.", "al.", "[34], [35] proposed a stability enhanced approach by adding a weighted diffusive term, based on the knowledge of the diffusive limit, which can improve the time step condition up to $\\Delta t\\sim h$ for stability and the scheme becomes much more efficient.", "Now let us follow the idea in [34] to further reformulate (REF ) by adding a linear diffusive term on both sides in the first equation, with the referred opacity $\\sigma _0$ $\\left\\lbrace \\begin{array}{l}{\\partial t}+c\\,\\left(\\frac{1}{{\\sqrt{\\sigma _0}}}\\nabla \\cdot \\langle {\\vec{\\Omega }}\\, g \\rangle +{3\\sigma _0}\\Delta \\rho \\right)={{\\tilde{\\varepsilon }}^2}\\left(\\Phi -\\rho \\right)+c\\,{3\\sigma _0}\\Delta \\rho , \\\\ \\, \\\\{\\tilde{\\varepsilon }}^2 \\tilde{C}_v{\\partial t}={\\tilde{\\sigma }}(\\rho -\\Phi ), \\\\ \\, \\\\g_t+{{\\tilde{\\varepsilon }}}({\\mathbb {I}}-\\Pi )\\nabla \\cdot ({\\vec{\\Omega }}\\, g)+{{\\tilde{\\varepsilon }}^2}\\nabla \\cdot ({\\vec{\\Omega }}\\,\\rho )=-{{\\tilde{\\varepsilon }}^2}g,\\end{array}\\right.$ where $\\tilde{C}_v=C_v/\|{\\vec{\\Omega }}\|$ and ${\\tilde{\\varepsilon }}=\\varepsilon /{\\sqrt{\\sigma _0}}$ are taken for shorthand notation.", "$\\omega $ is a non-negative weighted function of ${\\tilde{\\varepsilon }}$ and it is bounded and independent of ${\\vec{r}}$ .", "It suggests to take $\\omega =\\exp (-\\varepsilon /h)$ or $\\omega =\\exp (-\\varepsilon ^2/\\Delta t)$ in [34].", "In this work, we take $\\omega =\\exp (-{\\tilde{\\varepsilon }}/h)$ .", "The referred value $\\sigma _0$ can be seen as a linear diffusive pernalization coefficient, e.g., as suggested in [47], we may take $1/\\sigma _0\\ge 0.54 \\max (1/\\sigma )$ .", "The two added linear diffusive terms will be numerically treated differently, one is explicit and the other is implicit.", "Before we continue to numerical discretizations for (REF ), we first mention the angular approximation.", "For the angular direction ${\\vec{\\Omega }}$ , we simply use the discrete ordinate method, also known as $S_N$ method [28].", "In the one dimensional case, ${\\vec{\\Omega }}$ simply becomes $\\mu \\in [-1,1]$ , a Gauss-Legendre quadrature rule with weights $w_m$ and nodes $\\mu _m$ is used, for $m=1,\\ldots ,N_{g}$ .", "In the two dimensional case, a Legendre-Chebyshev quadrature rule with weights $w_m$ and nodes ${\\vec{\\Omega }}_m=(\\zeta _m,\\eta _m)$ , for $m=1,\\ldots ,N_g=N_lN_c$ is used.", "The nodes ${\\vec{\\Omega }}_m=(\\zeta _m,\\eta _m)$ are given by $\\zeta _m=\\sqrt{1-\\mu ^2_i}\\,\\cos \\phi _j, \\quad \\eta _m=\\sqrt{1-\\mu ^2_i}\\,\\sin \\phi _j, \\quad m=(i-1)N_c+j,$ where $\\mu _i$ , $i=1,\\ldots ,N_l$ , denote the roots of the Legendre polynomial of degree $N_l$ .", "$\\phi _j=(2j-1)\\pi /N_c$ , $j=1,\\ldots ,N_c$ are the nodes based on a Chebyshev polynomial.", "In the following, for easy presentation, we still keep ${\\vec{\\Omega }}$ continuous and focus on the numerical discretizations in time and space.", "Remark 2.1 We notice that taking the micro-macro decomposition in (REF ), which is different from [17], [34], it has the right diffusive limit both as $\\varepsilon \\rightarrow 0$ and $\\sigma $ relatively large.", "Besides, it is better for numerical boundary treatments, e.g., the inflow-outflow close-loop boundary condition for the Marshak wave problem in Section ." ], [ "Numerical methods", "In this section, we will describe the discontinuous Galerkin (DG) finite element spatial discretization for the system (REF ), where local DG is used for the diffusive terms.", "We take the two-dimensional case as an example, and the one-dimensional problem can be formulated similarly.", "In time, the scheme is coupled with high order globally stiffly accurate explicit-implicit (IMEX) methods.", "The resulting scheme is nonlinear only for the macroscopic variables $\\rho $ and $T$ .", "We use a Picard iteration with a predictor-corrector procedure, so that the nonlinear system is decoupled to a linear system, coupled with algebraic nonlinear equations restricted in each element, so that robust and fast convergence can be obtained." ], [ "Semi-discrete Discontinuous Galerkin scheme", "We consider a partition of the space computational domain $\\mathbb {D}$ with a set of non-overlapping elements $\\mathcal {T}_h=\\lbrace K\\rbrace $ , which can completely cover the domain $\\mathbb {D}$ .", "$h$ is the maximum edge size of these elements.", "For simplicity, in this work, we consider a rectangular computational domain $\\mathbb {D}$ with a partition of square elements $K=I_{ij}=I_i\\times I_j$ .", "Here $I_i=[{x_{i-\\frac{1}{2}}},{x_{i+\\frac{1}{2}}}]$ , $I_j=[{y_{j-\\frac{1}{2}}},{y_{j+\\frac{1}{2}}}]$ and $(x_i,y_j)$ is the center of each element $K$ .", "Given any non-negative integer vector ${\\bf k}=(k_1,k_2)$ , we define a finite dimensional discrete piecewise polynomial space as follows $V_h^{\\bf k}=\\left\\lbrace u\\in L^2(\\mathbb {D}): \\,\\, u\|_K\\in \\mathcal {Q}^{\\bf k}(K), \\,\\,\\forall \\, K\\in \\mathcal {T}_h\\right\\rbrace ,$ where $\\mathcal {Q}^{\\bf k}(K)$ consists of tensor product polynomials of degree not exceeding $k_i$ along the $i$ -th direction on the element $K$ , for $i=1,2$ .", "Here $\\mathcal {Q}^{\\bf k}(K)$ is chosen as it has better $p$ -adaptivity along each spatial direction on a rectangular domain.", "The $\\mathcal {P}^k(K)$ local space which consists of polynomials of degree not exceeding $k$ can be used for the following scheme too.", "For convenience, on each edge $e$ of $\\mathcal {T}_h$ , we follow [9] to define a unit normal vector ${\\bf n}^e$ in the following way.", "If $e\\in \\partial \\mathbb {D}$ , ${\\bf n}^e$ is defined as the unit normal vector pointing outside of $\\mathbb {D}$ .", "For an interior edge $e=\\partial K^+\\cap \\partial K^-$ , we first denote ${\\bf n}^+$ and ${\\bf n}^-$ as the outward unit normal vectors of e taken from the elements $K^+$ and $K^-$ , respectively.", "We fix ${\\bf n}^e$ as one of ${\\bf n}^\\pm $ .", "If we use the notations $u^+$ and $u^-$ to denote the values of a function $u$ on $e$ taken from $K^+$ and $K^-$ , respectively, then the jump notation $[u]$ over an edge $e$ for a scalar valued function $u$ is defined as $[u]\|_e = -(u^+{\\bf n}^+ + u^-{\\bf n}^-)\\cdot {\\bf n}^e,$ and for a vector-valued function ${\\bf q}$ , the jump $[{\\bf q}\\cdot {\\bf n}]$ is defined as $[{\\bf q}\\cdot {\\bf n}]\|_e = -({\\bf q}^+\\cdot {\\bf n}^+ + {\\bf q}^-\\cdot {\\bf n}^-).$ Accordingly, the averages of $u$ and ${\\bf q}\\cdot {\\bf n}$ are defined as $\\lbrace u\\rbrace \|_e = -\\frac{1}{2}(u^+{\\bf n}^+ - u^-{\\bf n}^-)\\cdot {\\bf n}^e,\\;\\;\\lbrace {\\bf q}\\cdot {\\bf n}\\rbrace \|_e = -\\frac{1}{2}({\\bf q}^+\\cdot {\\bf n}^+ - {\\bf q}^-\\cdot {\\bf n}^-).$ Taking square elements as an example, along the $x$ direction, $K^-$ and $K^+$ are the left and right elements associated to the edge $e$ respectively.", "We may fix ${\\bf n}^e={\\bf n}^-$ , then $[u]\|_e=u^+-u^-,\\;\\; [{\\bf q}\\cdot {\\bf n}]\|_e={\\bf q}^+-{\\bf q}^-, \\;\\;\\lbrace u\\rbrace \|_e=\\frac{1}{2}(u^++u^-),\\;\\; \\lbrace {\\bf q}\\cdot {\\bf n}\\rbrace \|_e=\\frac{1}{2}({\\bf q}^++{\\bf q}^-).$ With these notations, let ${\\bf q}=\\nabla \\rho $ , a semi-discrete (local) DG scheme for (REF ) is given below.", "Looking for $\\rho _h(t,{\\vec{r}})$ , $g_h(t,{\\vec{r}},{\\vec{\\Omega }})$ , $T_h(t,{\\vec{r}}) \\in V_h^{{\\bf k}}$ and ${\\bf q}_h(t,{\\vec{r}})=(q_{1,h}(t,{\\vec{r}}),q_{2,h}(t,{\\vec{r}}))^T$ where $q_{i,h}(t,{\\vec{r}}) \\in V_h^{{\\bf k}}$ for $i=1,2$ , such that $\\forall \\, {\\alpha }({\\vec{r}})$ , ${\\beta }({\\vec{r}}), {\\gamma }({\\vec{r}}) \\in V_h^{{\\bf k}}$ , and ${\\bf Z}({\\vec{r}})=(z_1({\\vec{r}}),z_2({\\vec{r}}))^T$ where $z_i({\\vec{r}}) \\in V_h^{{\\bf k}}$ for $i=1,2$ , we have $\\left\\lbrace \\begin{array}{l}(\\partial _t \\rho _h, {\\alpha })+\\frac{c}{{\\sqrt{\\sigma _0}}}\\,a_h(\\langle {\\vec{\\Omega }}\\,g_h \\rangle , {\\alpha }) + c\\,\\frac{\\omega }{3\\sigma _0} \\,d_h({\\bf q}_h,{\\alpha }) = -\\frac{c}{{\\tilde{\\varepsilon }}^2}({\\tilde{\\sigma }}\\,(\\rho _h-\\Phi _h),{\\alpha })+ c\\,\\frac{\\omega }{3\\sigma _0} \\,d_h({\\bf q}_h,{\\alpha }), \\\\ \\, \\\\\\end{array}{\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,(\\partial _t T_h, {\\gamma }) = ({\\tilde{\\sigma }}\\,(\\rho _h-\\Phi _h),{\\gamma }), \\\\ \\, \\\\({\\bf q}_h,{\\bf Z}) = G_h(\\rho _h,{\\bf Z}), \\\\ \\, \\\\(\\partial _t g_h, {\\beta })+ \\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}}\\, b_h({\\vec{\\Omega }}\\,g_h, {\\beta })+\\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}^2} r_h({\\vec{\\Omega }}\\,\\rho _h, {\\beta }) =-\\frac{c}{{\\tilde{\\varepsilon }}^2}({\\tilde{\\sigma }}\\,g_h, {\\beta }), \\right.$ where $\\left\\lbrace \\begin{array}{l}a_h(\\langle {\\vec{\\Omega }}\\,g_h \\rangle ,{\\alpha })=-\\sum _{K\\in \\mathcal {T}_h} \\int _K \\langle {\\vec{\\Omega }}\\,g_h \\rangle {\\nabla \\cdot }{\\alpha }({\\vec{r}}) d{\\vec{r}}+ \\sum _{K\\in \\mathcal {T}_h} \\int _{\\partial K} {\\bf n}^e\\cdot \\widehat{\\langle {\\vec{\\Omega }}\\,g_h\\rangle }\\,{\\alpha }({\\vec{r}})\\,ds,\\\\ \\, \\\\d_h({\\bf q}_h, {\\alpha })=-\\sum _{K\\in \\mathcal {T}_h} \\int _K {\\bf q}_h {\\nabla \\cdot }{\\alpha }({\\vec{r}}) d{\\vec{r}}+ \\sum _{K\\in \\mathcal {T}_h} \\int _{\\partial K} {\\bf n}^e\\cdot \\widehat{{\\bf q}_h}\\,{\\alpha }({\\vec{r}})\\,ds,\\\\ \\, \\\\b_h({\\vec{\\Omega }}\\,g_h,{\\beta })=(({\\mathbb {I}}-\\Pi ){\\mathcal {D}}_h(g_h; {\\vec{\\Omega }}), {\\beta }({\\vec{r}})) =({\\mathcal {D}}_h(g_h; {\\vec{\\Omega }}) - \\langle {\\mathcal {D}}_h(g_h; {\\vec{\\Omega }})\\rangle , {\\beta }({\\vec{r}})),\\\\ \\, \\\\r_h({\\vec{\\Omega }}\\,\\rho _h, {\\beta })=-\\sum _{K\\in \\mathcal {T}_h} \\int _K \\rho _h\\,{\\vec{\\Omega }}\\, {\\nabla \\cdot }{\\beta }({\\vec{r}}) d{\\vec{r}}+ \\sum _{K\\in \\mathcal {T}_h} \\int _{\\partial K} {\\bf n}^e\\cdot {\\vec{\\Omega }}\\,\\widehat{\\rho _h}\\,{\\beta }({\\vec{r}})\\,ds,\\\\ \\, \\\\G_h(\\rho _h, {\\bf Z})=-\\sum _{K\\in \\mathcal {T}_h} \\int _K \\rho _h \\nabla {\\bf Z}({\\vec{r}}) d{\\vec{r}}+ \\sum _{K\\in \\mathcal {T}_h} \\int _{\\partial K} \\widehat{\\rho _h}\\,{\\bf Z}({\\vec{r}})\\,ds.\\end{array}\\right.$ Here and below, the standard inner product $(\\cdot , \\cdot )$ for the $L^2(\\mathbb {D})$ space is used, see e.g.", "in (REF ), the first term in the first equation and two terms in the second equation.", "The function ${\\mathcal {D}}_h(g_h; {\\vec{\\Omega }})$ in the third equation of (REF ) is defined as an upwind discretization of ${\\nabla \\cdot }({\\vec{\\Omega }}\\,g)$ within the DG framework, $({\\mathcal {D}}_h(g_h; {\\vec{\\Omega }}), {\\beta })=-\\sum _{K\\in \\mathcal {T}_h} \\int _K {\\vec{\\Omega }}\\,g_h {\\nabla \\cdot }{\\beta }({\\vec{r}}) d{\\vec{r}}+ \\sum _{K\\in \\mathcal {T}_h} \\int _{\\partial K} {\\bf n}^e\\cdot \\left(\\widetilde{{\\vec{\\Omega }}\\,g_h}{\\beta }({\\vec{r}})\\right)\\,ds,$ with $\\widetilde{{\\vec{\\Omega }}\\,g}$ being an upwind numerical flux consistent to ${\\vec{\\Omega }}\\,g$ , $\\widetilde{{\\vec{\\Omega }}\\,g}:=\\left\\lbrace \\begin{array}{ll}{\\vec{\\Omega }}\\, g^-,&\\mbox{if}\\; {\\vec{\\Omega }}\\cdot {\\bf n}^e\\,>0,\\\\{\\vec{\\Omega }}\\, g^+,&\\mbox{if}\\; {\\vec{\\Omega }}\\cdot {\\bf n}^e\\,<0.\\end{array}\\right.$ Remember that ${\\bf n}^e$ is a fixed unit normal vector of ${\\bf n}^-$ and ${\\bf n}^+$ associated to two elements $K^-$ and $K^+$ and $e=\\partial K^+\\cap \\partial K^-$ .", "$g^\\pm $ are the values of $g$ on $e$ taken from $K^\\pm $ respectively.", "${\\bf n}^e\\cdot \\widehat{\\langle {\\vec{\\Omega }}\\,g_h\\rangle }$ , ${\\bf n}^e\\cdot \\widehat{{\\bf q}_h}$ and $\\widehat{\\rho _h}$ are also numerical fluxes, and they are consistent to the physical fluxes ${\\bf n}^e\\cdot \\langle {\\vec{\\Omega }}\\,g\\rangle $ , ${\\bf n}^e\\cdot {\\bf q}$ and $\\rho $ , respectively.", "The following choices could be taken: ${\\textrm {alternating left-right:}} &\\;\\; {\\bf n}^e\\cdot \\widehat{\\langle {\\vec{\\Omega }}\\,g\\rangle } = {\\bf n}^-\\cdot {\\langle {\\vec{\\Omega }}\\,g\\rangle }^-,\\; {\\bf n}^e \\cdot \\widehat{{\\bf q}}={\\bf n}^- \\cdot \\widehat{{\\bf q}}^-, \\; \\hat{\\rho } = {\\rho }^+; \\; \\\\\\textrm {alternating right-left:} &\\;\\; {\\bf n}^e\\cdot \\widehat{\\langle {\\vec{\\Omega }}\\,g\\rangle } = -{\\bf n}^+\\cdot {\\langle {\\vec{\\Omega }}\\,g\\rangle }^+,\\; {\\bf n}^e \\cdot \\widehat{{\\bf q}}=-{\\bf n}^+ \\cdot \\widehat{{\\bf q}}^+, \\; \\hat{\\rho } = {\\rho }^-; \\;\\\\{\\textrm {central:}} &\\;\\;{\\bf n}^e\\cdot \\widehat{\\langle {\\vec{\\Omega }}\\,g\\rangle } = \\lbrace {\\bf n}\\cdot {\\langle {\\vec{\\Omega }}\\,g\\rangle }\\rbrace ,\\; {\\bf n}^e \\cdot \\widehat{{\\bf q}}=\\lbrace {\\bf n}\\cdot \\widehat{{\\bf q}}\\rbrace , \\; \\hat{\\rho } = \\lbrace \\rho \\rbrace \\;.$" ], [ "Temporal discretization", "To ensure the correct asymptotic property as ${\\tilde{\\varepsilon }}\\rightarrow 0$ , the semi-discrete method in (REF ) is further coupled with globally stiffly accurate IMEX Runge-Kutta methods in time [2], [3].", "We first give a fully discrete scheme with a first order IMEX method, termed as DG-IMEX1.", "Given $\\rho ^n_h(\\cdot )$ , $g^n_h(\\cdot ,{\\vec{\\Omega }})$ , $T^n_h(\\cdot ) \\in V_h^{{\\bf k}}$ and ${\\bf q}^n_h(\\cdot )=(q^n_{1,h}(\\cdot ),q^n_{2,h}(\\cdot ))^T$ where $q^n_{i,h}(\\cdot ) \\in V_h^{{\\bf k}}$ for $i=1,2$ , which approximate the solution $\\rho $ , $g$ , $T$ and ${\\bf q}$ at $t=t^n$ , we look for $\\rho ^{n+1}_h(\\cdot )$ , $g^{n+1}_h(\\cdot ,{\\vec{\\Omega }})$ , $T^{n+1}_h(\\cdot ) \\in V_h^{{\\bf k}}$ and ${\\bf q}^{n+1}_h(\\cdot )=(q^{n+1}_{1,h}(\\cdot ),q^{n+1}_{2,h}(\\cdot ))^T$ where $q^{n+1}_{i,h}(\\cdot ) \\in V_h^{{\\bf k}}$ , such that $\\forall \\, {\\alpha }({\\vec{r}})$ , ${\\beta }({\\vec{r}}), {\\gamma }({\\vec{r}}) \\in V_h^{{\\bf k}}$ , and ${\\bf Z}({\\vec{r}})=(z_1({\\vec{r}}),z_2({\\vec{r}}))^T$ where $z_i({\\vec{r}}) \\in V_h^{{\\bf k}}$ for $i=1,2$ , we have $ &\\left(\\frac{\\rho ^{n+1}_h-\\rho ^n_h}{\\Delta t}, {\\alpha }\\right)+\\frac{c}{{\\sqrt{\\sigma _0}}}\\,a_h(\\langle {\\vec{\\Omega }}\\,g^n_h \\rangle , {\\alpha }) + c\\,\\frac{\\omega }{3\\sigma _0} \\,d_h({\\bf q}^n_h,{\\alpha }) = \\\\ \\, \\nonumber \\\\ &\\hspace{142.26378pt}-\\frac{c}{{\\tilde{\\varepsilon }}^2}({\\tilde{\\sigma }}\\,(\\rho ^{n+1}_h-\\Phi ^{n+1}_h),{\\alpha })+c\\,\\frac{\\omega }{3\\sigma _0} \\,d_h({\\bf q}^{n+1}_h,{\\alpha }), \\nonumber \\\\ \\,\\nonumber \\\\ &{\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\left(\\frac{T^{n+1}_h-T^n_h}{\\Delta t}, {\\gamma }\\right) = \\left({\\tilde{\\sigma }}\\,(\\rho ^{n+1}_h-\\Phi ^{n+1}_h),{\\gamma }\\right), \\\\ \\,\\nonumber \\\\ &({\\bf q}^{n+1}_h,{\\bf Z}) = G_h(\\rho ^{n+1}_h,{\\bf Z}),\\\\ \\, \\nonumber \\\\ &\\left(\\frac{g^{n+1}_h-g^n_h}{\\Delta t}, {\\beta }\\right)+ \\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}}\\, b_h({\\vec{\\Omega }}\\,g^n_h, {\\beta })+\\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}^2}\\, r_h({\\vec{\\Omega }}\\,\\rho ^{n+1}_h, {\\beta })=-\\frac{c}{{\\tilde{\\varepsilon }}^2}\\,({\\tilde{\\sigma }}\\,g^{n+1}_h, {\\beta }).", "$ As we can see that the most stiff terms, in both the convective and collisional terms, with a scale of $\\frac{1}{{\\tilde{\\varepsilon }}^2}$ , are treated implicitly.", "The two added diffusive terms, one is treated explicitly and the other is treated implicitly.", "For the scheme (REF ), we may first solve the first three equations to update $\\rho ^{n+1}_h$ , $T^{n+1}_h$ and ${\\bf q}^{n+1}_h$ , then we substitute $\\rho ^{n+1}_h$ and $T^{n+1}_h$ to the last equation to update $g^{n+1}_h$ .", "Notice that ${\\tilde{\\sigma }}$ might depend on $T$ nonlinearly.", "In this case, the only unknown $g^{n+1}_h$ involving the angular direction ${\\vec{\\Omega }}$ is updated explicitly, which is another main point for the design of asymptotic preserving scheme under the micro-macro decompsition framework [17].", "However, we would note that due to the nonlinear relation $\\Phi =acT^4/\|{\\vec{\\Omega }}\|$ , the first three equations of (REF ) still appears to be a fully nonlinear coupled system for the unknowns $\\rho ^{n+1}_h$ and $T^{n+1}_h$ , this is the main difference from the linear radiative transport equations in [17], [34].", "Before we talk about how to deal with this nonlinearity, let us finish our high order temporal discretization.", "To achieve high order accuracy in time, we adopt the globally stiffly accurate multi-stage IMEX Runge-Kutta (RK) schemes [3], [17], [34].", "A multi-stage IMEX RK scheme can be represented by a double Butcher tableau $\\begin{array}{c\|c}\\tilde{c} & \\tilde{\\mathcal {A}}\\\\\\hline \\vspace{-7.11317pt}\\\\& \\tilde{b^T} \\end{array}, \\ \\ \\ \\ \\ \\begin{array}{c\|c}{c} & {\\mathcal {A}}\\\\\\hline \\vspace{-7.11317pt}\\\\& {b^T} \\end{array},$ where both $\\tilde{\\mathcal {A}} = (\\tilde{a}_{ij})$ and $\\mathcal {A} = (a_{ij})$ are $(s+1)\\times (s+1)$ matrices, with $\\tilde{\\mathcal {A}}$ being lower triangular with zero diagonal entries.", "The coefficients $\\tilde{c}$ and $c$ are given by the usual relation $\\tilde{c}_i = \\sum _{j=0}^{i-1} \\tilde{a}_{ij}$ , $c_i = \\sum _{j=0}^{i} a_{ij}$ , and vectors $\\tilde{b} = (\\tilde{b}_j)$ and $b = (b_j)$ provide the quadrature weights to combine internal stages of the RK method.", "The IMEX RK scheme is said to be globally stiffly accurate [3] if $c_s = \\tilde{c}_s = 1, \\;\\;\\textrm {and}\\;\\; a_{sj} = b_j, \\;\\; \\tilde{a}_{sj} = \\tilde{b}_j, \\;\\;\\forall j=0, \\cdots , s.$ This double Butcher tableau corresponds to explicit and implicit time discretization respectively.", "In this work, the type ARS IMEX RK scheme is used, where $\\mathcal {A}$ in (REF ) has the following structure, $\\left[\\begin{array}{cc}0 & 0 \\\\0 & \\hat{\\mathcal {A}}\\end{array}\\right]$ where the diagonal entries of $\\hat{\\mathcal {A}}$ are nonzero.", "Now we will apply a general globally stiffly accurate IMEX RK method, represented by (REF ) with the property (REF ), to the semi-discrete DG scheme (REF ).", "For a $p$ -th order IMEX RK method, our scheme is termed as “DG-IMEXp\".", "This is the same implicit-explicit strategy as in the first order case.", "Given $\\rho ^n_h(\\cdot )$ , $g^n_h(\\cdot ,{\\vec{\\Omega }})$ , $T^n_h(\\cdot ) \\in V_h^{{\\bf k}}$ and ${\\bf q}^n_h(\\cdot )=(q^n_{1,h}(\\cdot ),q^n_{2,h}(\\cdot ))^T$ where $q^n_{i,h}(\\cdot ) \\in V_h^{{\\bf k}}$ for $i=1,2$ that approximate the solution $\\rho $ , $g$ , $T$ and ${\\bf q}$ at $t=t^n$ , we look for $\\rho ^{n+1}_h(\\cdot )$ , $g^{n+1}_h(\\cdot ,{\\vec{\\Omega }})$ , $T^{n+1}_h(\\cdot ) \\in V_h^{{\\bf k}}$ and ${\\bf q}^{n+1}_h(\\cdot )=(q^{n+1}_{1,h}(\\cdot ),q^{n+1}_{2,h}(\\cdot ))^T$ where $q^{n+1}_{i,h}(\\cdot ) \\in V_h^{{\\bf k}}$ , such that $\\forall \\, {\\alpha }({\\vec{r}})$ , ${\\beta }({\\vec{r}}), {\\gamma }({\\vec{r}}) \\in V_h^{{\\bf k}}$ , and ${\\bf Z}({\\vec{r}})=(z_1({\\vec{r}}),z_2({\\vec{r}}))^T$ where $z_i({\\vec{r}}) \\in V_h^{{\\bf k}}$ for $i=1,2$ , we have $&\\left(\\frac{\\rho ^{n+1}_h-\\rho ^n_h}{\\Delta t}, {\\alpha }\\right)+\\frac{c}{{\\sqrt{\\sigma _0}}}\\,\\sum _{\\ell =0}^s \\tilde{b}_\\ell \\,a_h(\\langle {\\vec{\\Omega }}\\,g^{(\\ell )}_h \\rangle , {\\alpha }) + c\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{\\ell =0}^s \\tilde{b}_\\ell \\,d_h({\\bf q}^{(\\ell )}_h,{\\alpha }) = \\\\ & \\hspace{142.26378pt} -\\frac{c}{{\\tilde{\\varepsilon }}^2}\\sum _{\\ell =0}^s b_\\ell \\, ({\\tilde{\\sigma }}\\,(\\rho ^{(\\ell )}_h-\\Phi ^{(\\ell )}_h),{\\alpha })+c\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{\\ell =0}^s \\,b_\\ell \\,d_h({\\bf q}^{(\\ell )}_h,{\\alpha }), \\nonumber \\\\ \\, \\nonumber \\\\&{\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\left(\\frac{T^{n+1}_h-T^n_h}{\\Delta t}, {\\gamma }\\right) = \\sum _{\\ell =0}^s \\,b_\\ell \\,\\left({\\tilde{\\sigma }}\\,(\\rho ^{(\\ell )}_h-\\Phi ^{(\\ell )}_h),{\\gamma }\\right), \\\\ \\, \\nonumber \\\\&({\\bf q}^{n+1}_h,{\\bf Z}) = G_h(\\rho ^{n+1}_h,{\\bf Z}),\\\\ \\, \\nonumber \\\\&\\left(\\frac{g^{n+1}_h-g^n_h}{\\Delta t}, {\\beta }\\right)+ \\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}} \\,\\sum _{\\ell =0}^s \\tilde{b}_\\ell \\,b_h({\\vec{\\Omega }}\\,g^{(\\ell )}_h, {\\beta })+\\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}^2} \\,\\sum _{\\ell =0}^s b_\\ell \\,r_h({\\vec{\\Omega }}\\,\\rho ^{(\\ell )}_h, {\\beta })=\\\\ &\\hspace{142.26378pt}-\\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}^2}\\,\\sum _{\\ell =0}^s b_\\ell \\,({\\tilde{\\sigma }}\\,g^{(\\ell )}_h, {\\beta }).", "\\nonumber $ The approximations at the internal stages $\\rho _h^{(\\ell )}(\\cdot )$ , $g_h^{(\\ell )}(\\cdot ,{\\vec{\\Omega }})$ , $T_h^{(\\ell )}(\\cdot ) \\in V_h^{\\bf k}$ and ${\\bf q}_h^{(\\ell )}(\\cdot )=(q_{1,h}^{(\\ell )}(\\cdot ),q_{2,h}^{(\\ell )}(\\cdot ))$ where $q_{i,h}^{(\\ell )}(\\cdot )\\in V^{\\bf k}_h$ for $i=1,2$ , with $\\ell =1, \\cdots , s$ , would satisfy $&\\left(\\frac{\\rho ^{(\\ell )}_h-\\rho ^n_h}{\\Delta t}, {\\alpha }\\right)+\\frac{c}{{\\sqrt{\\sigma _0}}}\\,\\sum _{j=0}^{\\ell -1} \\tilde{a}_{\\ell ,j} \\,a_h(\\langle {\\vec{\\Omega }}\\,g^{(j)}_h \\rangle , {\\alpha }) + c\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{j=0}^{\\ell -1} \\tilde{a}_{\\ell ,j} \\,d_h({\\bf q}^{(j)}_h,{\\alpha }) = \\\\ &\\hspace{142.26378pt} -\\frac{c}{{\\tilde{\\varepsilon }}^2}\\sum _{j=0}^{\\ell } a_{\\ell ,j} \\, ({\\tilde{\\sigma }}\\, (\\rho ^{(j)}_h-\\Phi ^{(j)}_h),{\\alpha })+c\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{j=0}^{\\ell } \\,a_{\\ell ,j}\\,d_h({\\bf q}^{(j)}_h,{\\alpha }), \\nonumber \\\\ \\, \\nonumber \\\\&{\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\left(\\frac{T^{(\\ell )}_h-T^n_h}{\\Delta t}, {\\gamma }\\right) = \\sum _{j=0}^{\\ell } \\,a_{\\ell ,j}\\,\\left({\\tilde{\\sigma }}\\,(\\rho ^{(j)}_h-\\Phi ^{(j)}_h),{\\gamma }\\right), \\\\ \\, \\nonumber \\\\&({\\bf q}^{(\\ell )}_h,{\\bf Z}) = G_h(\\rho ^{(\\ell )}_h,{\\bf Z}),\\\\ \\, \\nonumber \\\\&\\left(\\frac{g^{(\\ell )}_h-g^n_h}{\\Delta t}, {\\beta }\\right)+ \\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}} \\,\\sum _{j=0}^{\\ell -1} \\tilde{a}_{\\ell ,j} \\,b_h({\\vec{\\Omega }}\\,g^{(j)}_h, {\\beta })+\\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}^2} \\,\\sum _{j=0}^{\\ell } a_{\\ell ,j} \\,r_h({\\vec{\\Omega }}\\,\\rho ^{(j)}_h, {\\beta })= \\\\ &\\hspace{142.26378pt}-\\frac{c}{{\\tilde{\\varepsilon }}^2}\\,\\sum _{j=0}^{\\ell } a_{\\ell ,j} \\,({\\tilde{\\sigma }}\\,g^{(j)}_h, {\\beta }).", "\\nonumber $ Notice that for $\\ell =0$ , $\\rho ^{(0)}_h=\\rho ^n_h,\\;g^{(0)}=g^n_h,\\;T^{(0)}_h=T^n_h$ and ${\\bf q}^{(0)}_h={\\bf q}^n_h$ .", "This high order temporal discretization (REF )-(REF ) can be solved similarly as the first order scheme (REF ) in a stage-by-stage manner.", "That is, in each internal stage, from (REF ), we first solve the first three equations to update $\\rho _h^{(\\ell )}(\\cdot )$ , $T_h^{(\\ell )}(\\cdot )$ and ${\\bf q}_h^{(\\ell )}(\\cdot )$ , and then substituting $\\rho _h^{(\\ell )}(\\cdot )$ and $T_h^{(\\ell )}(\\cdot )$ into the fourth equation to get $g_h^{(\\ell )}(\\cdot ,{\\vec{\\Omega }})$ .", "After all stage values are obtained, the numerical solutions $\\rho ^{n+1}_h(\\cdot )$ , $g^{n+1}_h(\\cdot ,{\\vec{\\Omega }})$ , $T^{n+1}_h(\\cdot ) $ and ${\\bf q}^{n+1}_h(\\cdot )$ at the next time level $t^{n+1}$ can be accumulated from (REF ).", "For a globally stiffly accurate IMEX RK scheme, from (REF ), the solutions at $t^{n+1}$ concide with the last stage values, that is, $\\rho ^{n+1}_h=\\rho ^{(s)}_h, \\quad g^{n+1}_h=g^{(s)}_h, \\quad T^{n+1}_h=T^{(s)}_h \\; \\textrm { and } \\; \\Phi ^{n+1}=ac(T^{n+1}_h)^4/\|{\\vec{\\Omega }}\|.$ However, numerically we find that due to (REF )-() is solved iteratively (as described in the next section), the exact conservation is lost and we cannot get the right front propagation of the material temperature.", "Keeping $g^{n+1}_h=g^{(s)}_h$ , combining (REF ) and () we propose to update $\\rho ^{n+1}_h$ and $T^{n+1}_h$ again with $&\\left(\\frac{\\rho ^{n+1}_h-\\rho ^n_h}{\\Delta t}, {\\alpha }\\right)+\\tilde{C}_v\\,\\left(\\frac{T^{n+1}_h-T^n_h}{\\Delta t} , {\\alpha }\\right) = \\\\ & \\hspace{56.9055pt} +\\frac{c}{{\\sqrt{\\sigma _0}}}\\,\\sum _{\\ell =0}^s \\tilde{b}_\\ell \\,a_h(\\langle {\\vec{\\Omega }}\\,g^{(\\ell )}_h \\rangle , {\\alpha }) + c\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{\\ell =0}^s \\tilde{b}_\\ell \\,d_h({\\bf q}^{(\\ell )}_h,{\\alpha })+c\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{\\ell =0}^s \\,b_\\ell \\,d_h({\\bf q}^{(\\ell )}_h,{\\alpha }), \\nonumber \\\\ \\, \\nonumber \\\\&{\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\left(\\frac{T^{n+1}_h-T^n_h}{\\Delta t}, {\\gamma }\\right) - b_s\\,\\left({\\tilde{\\sigma }}\\,(\\rho ^{n+1}_h-\\Phi ^{n+1}_h),{\\gamma }\\right) = \\sum _{\\ell =0}^{s-1} \\,b_\\ell \\,\\left({\\tilde{\\sigma }}\\,(\\rho ^{(\\ell )}_h-\\Phi ^{(\\ell )}_h),{\\gamma }\\right).$ The idea is that spatial derivatives are discretized in a conservative flux difference form for updating $\\rho ^{n+1}_h$ .", "With (REF ), both conservation and $\\rho ^{n+1}_h=\\Phi ^{n+1}_h+{\\mathcal {O}}({\\tilde{\\varepsilon }})$ when ${\\tilde{\\varepsilon }}\\ll 1$ can be ensured." ], [ "Picard iteration with a predictor-corrector procedure", "As mentioned in the last subsection, the first three equations in the scheme (REF ) or (REF ) are fully coupled nonlinear system.", "To avoid solving the nonlinear system directly, we empoly a Picard iteration with a predictor-corrector procedure.", "This is motivated from the work in [46], where this procedure is designed for an accurate front capturing asymptotic preserving scheme.", "The idea is that, to avoid a coupled global nonlinear system, we need to seperate the spatial differential operator $\\Delta \\rho $ from the nonlinear term $\\Phi =acT^4/\|{\\vec{\\Omega }}\|$ with respect to $T$ .", "We start with the first order in time scheme (REF ).", "We focus on the first three equations and we rewrite them in the following way $&(\\rho ^{n+1}_h, {\\alpha })+c\\,\\tilde{C}_v\\,\\left(T^{n+1}_h, {\\alpha }\\right) -c\\,\\Delta t\\,\\frac{\\omega }{3\\sigma _0} \\,d_h({\\bf q}^{n+1}_h,{\\alpha })\\;=\\;(\\textrm {RHS}_1^n,{\\alpha }), \\\\ \\,\\nonumber \\\\&{\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\left(T^{n+1}_h, {\\gamma }\\right) -\\Delta t\\, \\left({\\tilde{\\sigma }}\\,(\\rho ^{n+1}_h-\\Phi ^{n+1}_h),{\\gamma }\\right)\\;=\\;(\\textrm {RHS}_2^n,{\\gamma }), \\\\ \\,\\nonumber \\\\&({\\bf q}^{n+1}_h,{\\bf Z}) - G_h(\\rho ^{n+1}_h,{\\bf Z})\\;=\\; 0.", "$ We have combined (REF ) and () to obtain (REF ).", "Besides, we have put all unknown values at the time level $t^{n+1}$ on the left, while all values at the time level $t^n$ on the right, where $(\\textrm {RHS}_1^n,{\\alpha })=(\\rho ^{n}_h, {\\alpha })+c\\,\\tilde{C}_v\\,\\left(T^{n}_h, {\\alpha }\\right) -c\\,\\Delta t\\,\\frac{\\omega }{3\\sigma _0} \\,d_h({\\bf q}^{n}_h,{\\alpha })-\\frac{c}{{\\sqrt{\\sigma _0}}}\\,\\Delta t\\,a_h(\\langle {\\vec{\\Omega }}\\,g^n_h\\rangle ,{\\alpha }),$ and $(\\textrm {RHS}_2^n,{\\gamma })={\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\left(T^{n}_h, {\\gamma }\\right).$ Now the Picard iteration for solving (REF ) is defined as follows: for the iterative number $m$ starting at $m=0$ , where $\\rho ^{n+1,(0)}_h=\\rho ^n_h$ , $T^{n+1,(0)}_h=T^n_h$ and ${\\bf q}^{n+1,(0)}={\\bf q}^n_h$ , we update the unknowns $\\rho ^{n+1,(m+1)}_h$ , $T^{n+1,(m+1)}_h$ and ${\\bf q}^{n+1,(m+1)}$ iteratively by the following two steps: Step 1: we first solve the following linear system to obtain $\\rho ^{n+1,(m+1)}_h$ and ${\\bf q}^{n+1,(m+1)}$ $\\left\\lbrace \\begin{array}{l}(\\rho ^{n+1,(m+1)}_h, {\\alpha }) -c\\,\\Delta t\\,\\frac{\\omega }{3\\sigma _0} \\,d_h({\\bf q}^{n+1,(m+1)}_h,{\\alpha })\\;=\\;(\\textrm {RHS}_1^n,{\\alpha })-c\\,\\tilde{C}_v\\,\\left(T^{n+1,(m)}_h, {\\alpha }\\right), \\\\ \\,\\\\({\\bf q}^{n+1,(m+1)}_h,{\\bf Z}) - G_h(\\rho ^{n+1,(m+1)}_h,{\\bf Z})\\;=\\; 0.", "\\end{array}\\right.$ Notice that $d_h({\\bf q}^{n+1,(m+1)}_h,{\\alpha })$ combined with $({\\bf q}^{n+1,(m+1)}_h,{\\bf Z})$ gives the local DG discretization for the Laplacian $\\Delta \\rho $ , which results in a linear system; Step 2: with $\\rho ^{n+1,(m+1)}_h$ and ${\\bf q}^{n+1,(m+1)}$ obtained from Step 1, we now solve $\\left\\lbrace \\begin{array}{l}(\\rho ^{n+1,(m+1)}_h, {\\alpha })+c\\,\\tilde{C}_v\\,\\left(T^{n+1,(m+1)}_h, {\\alpha }\\right) \\;=\\;(\\textrm {RHS}_1^n,{\\alpha })+c\\,\\Delta t\\,\\frac{\\omega }{3\\sigma _0} \\,d_h({\\bf q}^{n+1,(m+1)}_h,{\\alpha }), \\\\ \\, \\\\{\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\left(T^{n+1,(m+1)}_h, {\\gamma }\\right) -\\Delta t\\, \\left({\\tilde{\\sigma }}\\,(\\rho ^{n+1,(m+1)}_h-\\Phi ^{n+1,(m+1)}_h),{\\gamma }\\right)\\;=\\;(\\textrm {RHS}_2^n,{\\gamma }).", "\\end{array}\\right.$ $d_h({\\bf q}^{n+1,(m+1)}_h,{\\alpha })$ can be precomputed from ${\\bf q}^{n+1,(m+1)}_h$ .", "Instead of using $\\rho ^{n+1,(m+1)}_h$ obtained from step 1, here we express it by $T_h^{n+1,(m+1)}$ and $\\Phi _h^{n+1,(m+1)}$ from the second equation of (REF ), and substitute it into the first equation.", "In this way, the system (REF ) becomes a local algebraic nonlinear system for $T_h^{n+1,(m+1)}$ within each element $K$ , so that solving a global nonlinear system is avoid.", "The Newton iteration can be simply used to solve this algebraic nonlinear system.", "In Step 2, if we assume ${\\tilde{\\sigma }}$ is simply a positive constant, from the second equation, we have $(\\rho ^{n+1,(m+1)}_h,{\\gamma })\\;=\\;(\\Phi ^{n+1,(m+1)}_h,{\\gamma })-{\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\left(T^{n+1,(m+1)}_h, {\\gamma }\\right)/({\\tilde{\\sigma }}\\,\\Delta t)-(\\textrm {RHS}_2^n,{\\gamma })/({\\tilde{\\sigma }}\\,\\Delta t).$ substituting it into the first equation, it yields $(\\Phi ^{n+1,(m+1)}_h,{\\alpha })+c\\,\\tilde{C}_v\\,\\left(T^{n+1,(m+1)}_h, {\\alpha }\\right)-{\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\left(T^{n+1,(m+1)}_h, {\\alpha }\\right)/({\\tilde{\\sigma }}\\,\\Delta t) \\;=\\;(\\textrm {RHS}_3^n,{\\alpha }),$ where $(\\textrm {RHS}_3^n,{\\alpha })=(\\textrm {RHS}_1^n,{\\alpha })+c\\,\\Delta t\\,\\frac{\\omega }{3\\sigma _0} \\,d_h({\\bf q}^{n+1,(m+1)}_h,{\\alpha })+(\\textrm {RHS}_2^n,{\\alpha })/({\\tilde{\\sigma }}\\,\\Delta t).$ (REF ) in the diffusive limit as ${\\tilde{\\varepsilon }}\\rightarrow 0$ , it turns out to be $\\begin{array}{l}(\\Phi ^{n+1,(m+1)}_h,{\\alpha })+c\\,\\tilde{C}_v\\,\\left(T^{n+1,(m+1)}_h, {\\alpha }\\right) \\;=\\;(\\rho ^{n}_h, {\\alpha })+c\\,\\tilde{C}_v\\,\\left(T^{n}_h, {\\alpha }\\right) -c\\,\\Delta t\\,\\frac{\\omega }{3\\sigma _0} \\,d_h({\\bf q}^{n}_h,{\\alpha }) \\\\ \\hspace{184.9429pt} -\\frac{c}{{\\sqrt{\\sigma _0}}}\\,\\Delta t\\,a_h(\\langle {\\vec{\\Omega }}\\,g^n_h\\rangle ,{\\alpha })+c\\,\\Delta t\\,\\frac{\\omega }{3\\sigma _0} \\,d_h({\\bf q}^{n+1,(m+1)}_h,{\\alpha }).\\end{array}$ Since $\\rho ^n_h$ approaches $\\Phi ^n_h$ from (), combining with () in the limit ${\\tilde{\\varepsilon }}\\rightarrow 0$ , (REF ) is a consistent discretization for the diffusive equation (REF ) (for more details, see the asymptotic analysis in the following section).", "Namely (REF ) is mimicking a numerical discretization for the diffusive limiting equation (REF ).", "The two steps are solved with the iterative number $m$ untill convergent, where the stop criteria is defined as $\\Vert \\rho ^{n+1,(m+1)}_h-\\rho ^{n+1,(m)}_h\\Vert _2 < \\delta .$ The $L^2$ norm is used and we take $\\delta =10^{-9}$ in our numerical tests.", "This procedure can be similarly applied to high order IMEX RK time discretizations (REF ) at each stage.", "That is, for the first three equations in (REF ), at the stage $\\ell $ ($\\ell =1,\\cdots ,s$ ), we rewrite them as $&(\\rho ^{(\\ell )}_h, {\\alpha })+c\\,\\tilde{C}_v\\,\\left(T^{(\\ell )}_h, {\\alpha }\\right) -c\\,a_{\\ell \\ell }\\,\\Delta t\\,\\frac{\\omega }{3\\sigma _0} \\,d_h({\\bf q}^{(\\ell )}_h,{\\alpha })\\;=\\;(\\textrm {RHS}_1,{\\alpha }), \\\\ \\,\\nonumber \\\\&{\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\left(T^{(\\ell )}_h, {\\gamma }\\right) -a_{\\ell \\ell }\\,\\Delta t\\, \\left({\\tilde{\\sigma }}\\,(\\rho ^{(\\ell )}_h-\\Phi ^{(\\ell )}_h),{\\gamma }\\right)\\;=\\;(\\textrm {RHS}_2,{\\gamma }), \\\\ \\,\\nonumber \\\\&({\\bf q}^{(\\ell )}_h,{\\bf Z}) - G_h(\\rho ^{(\\ell )}_h,{\\bf Z})\\;=\\; 0.", "$ Similarly we have combined (REF ) and () to obtain (REF ).", "All known values before stage $\\ell $ are put on the right, where $(\\textrm {RHS}_1,{\\alpha })=(\\rho ^{n}_h, {\\alpha })+c\\,\\tilde{C}_v\\,\\left(T^{n}_h, {\\alpha }\\right) -\\frac{c}{{\\sqrt{\\sigma _0}}}\\,\\Delta t\\,\\sum _{j=1}^{\\ell -1} \\tilde{a}_{\\ell ,j} \\,a_h(\\langle {\\vec{\\Omega }}\\,g^{(j)}_h \\rangle , {\\alpha }) - c\\,\\Delta t\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{j=0}^{\\ell -1} \\left(\\tilde{a}_{\\ell ,j}-a_{\\ell ,j}\\right) \\,d_h({\\bf q}^{(j)}_h,{\\alpha }),$ and $(\\textrm {RHS}_2,{\\gamma })={\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\left(T^{n}_h, {\\gamma }\\right)+\\Delta t\\,\\sum _{j=0}^{\\ell -1} \\,a_{\\ell ,j}\\,\\left({\\tilde{\\sigma }}\\,(\\rho ^{(j)}_h-\\Phi ^{(j)}_h),{\\gamma }\\right).$ Notice that (REF ) is in the same form as (REF ), so that it can be solved similarly with the Picard iteration described above.", "The solutions $\\rho ^{n+1}_h$ and $T^{n+1}_h$ are updated by (REF ) following the same procedure in Step 2.", "We omit the details here to save space." ], [ "Formal asymptotic preserving analysis", "In this section, we will perform formal asymptotic preserving (AP) and asymptotically accurate (AA) analyses for the proposed schemes by assuming ${\\tilde{\\varepsilon }}\\ll 1$ , while the mesh parameters $h$ and $\\Delta t$ are fixed.", "We will show that our DG-IMEX1 scheme (REF ) is AP in the sence that when ${\\tilde{\\varepsilon }}\\rightarrow 0$ , it becomes a consistent discretization for the limiting diffusive equation (REF ).", "On the other hand, our DG-IMEXp scheme (REF )-(REF ) is asymptotically accurate, namely, the schemes of (REF )-(REF ) as ${\\tilde{\\varepsilon }}\\rightarrow 0$ , maintains its order of temporal accuracy for the limiting equation (REF ).", "The analyses mainly follow the notations and assumptions as used in [36].", "We first assume the initial data is well-prepared without considering the initial layers, namely, $I(t,{\\vec{r}},{\\vec{\\Omega }})=\\rho (t,{\\vec{r}})+{\\tilde{\\varepsilon }}\\,g(t,{\\vec{r}},{\\vec{\\Omega }})$ and $\\rho (t,{\\vec{r}})=\\mathcal {O}(1), g(t,{\\vec{r}},{\\vec{\\Omega }})=\\mathcal {O}(1)$ .", "We further make the assumptions that the spatial derivatives of $\\rho $ and $g$ at $t=0$ , have comparable scales as $\\Vert \\rho (0,\\cdot )\\Vert $ and $\\Vert g(0,\\cdot ,{\\vec{\\Omega }})\\Vert $ with respect to ${\\tilde{\\varepsilon }}$ , that is, they all are $\\mathcal {O}(1)$ .", "Under theses assumptions, all spatial derivative approximations in (REF ) also all are $\\mathcal {O}(1)$ .", "For the small parameters ${\\tilde{\\varepsilon }}$ , $h$ and $\\Delta t$ , it is assumed ${\\tilde{\\varepsilon }}^2 \\ll \\Delta t$ , ${\\tilde{\\varepsilon }}\\le \\Delta t\\le 1$ and $\\Delta t/h=\\mathcal {O}(1)$ to avoid the explicit dependence on $\\Delta t$ and $h$ of the hidden constant in the $\\mathcal {O}$ notation.", "We begin to write our scheme in a strong form by defining the following linear operators for (REF ) $({\\mathcal {D}}^g_h(\\langle {\\vec{\\Omega }}\\,g_h \\rangle ), {\\alpha })=a_h(\\langle {\\vec{\\Omega }}\\,g_h \\rangle ,{\\alpha }),\\qquad ({\\mathcal {D}}^\\rho _h({\\vec{\\Omega }}\\,\\rho _h), {\\beta })=-r_h({\\vec{\\Omega }}\\,\\rho _h, {\\beta }),$ and $({\\mathcal {D}}^{grad}_h\\rho _h, {\\beta })=G_h(\\rho _h, {\\beta }), \\qquad ({\\mathcal {D}}^{div}_h{\\bf q}_h, {\\alpha })=d_h({\\bf q}_h, {\\alpha }).$ They are well-defined bounded operators following the Riesz representation, and determined entirely by the discrete space $V_h^{\\bf k}$ and the involved numerical fluxes." ], [ "Asymptotic preserving (AP) analysis for DG-IMEX1", "The DG-IMEX1 scheme (REF ) in the strong form becomes $ &\\frac{\\rho ^{n+1}_h-\\rho ^n_h}{\\Delta t}+\\frac{c}{{\\sqrt{\\sigma _0}}}\\,{\\mathcal {D}}^g_h(\\langle {\\vec{\\Omega }}\\,g^n_h \\rangle ) + c\\,\\frac{\\omega }{3\\sigma _0} \\,{\\mathcal {D}}^{div}_h({\\bf q}^n_h) = -\\frac{c}{{\\tilde{\\varepsilon }}^2}{\\tilde{\\sigma }}\\,(\\rho ^{n+1}_h-\\Phi ^{n+1}_h)+c\\,\\frac{\\omega }{3\\sigma _0} \\,{\\mathcal {D}}^{div}_h({\\bf q}^{n+1}_h), \\\\ \\,\\nonumber \\\\ &{\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\frac{T^{n+1}_h-T^n_h}{\\Delta t} = {\\tilde{\\sigma }}\\,(\\rho ^{n+1}_h-\\Phi ^{n+1}_h), \\\\ \\,\\nonumber \\\\ &{\\bf q}^{n+1}_h = {\\mathcal {D}}^{grad}_h(\\rho ^{n+1}_h),\\\\ \\, \\nonumber \\\\ &\\frac{g^{n+1}_h-g^n_h}{\\Delta t}+ \\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}}\\, ({\\mathbb {I}}-\\Pi ){\\mathcal {D}}_h(g^n_h;\\,{\\vec{\\Omega }})-\\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}^2}\\, {\\mathcal {D}}^\\rho _h({\\vec{\\Omega }}\\,\\rho ^{n+1}_h)=-\\frac{c}{{\\tilde{\\varepsilon }}^2}\\,{\\tilde{\\sigma }}\\,g^{n+1}_h.", "$ We notice that ${\\mathcal {D}}^{div}_h{\\mathcal {D}}^{grad}_h$ is an approximation of the Laplacian operator $\\Delta $ .", "Combining (REF ) and (), we have $\\frac{\\rho ^{n+1}_h-\\rho ^n_h}{\\Delta t}+\\frac{c}{{\\sqrt{\\sigma _0}}}\\,{\\mathcal {D}}^g_h(\\langle {\\vec{\\Omega }}\\,g^n_h \\rangle ) + c\\,\\frac{\\omega }{3\\sigma _0} \\,{\\mathcal {D}}^{div}_h({\\bf q}^n_h) = -\\tilde{C}_v\\,\\frac{T^{n+1}_h-T^n_h}{\\Delta t}+c\\,\\frac{\\omega }{3\\sigma _0} \\,{\\mathcal {D}}^{div}_h({\\bf q}^{n+1}_h).$ We start from the exact solutions at time level $t^n$ and we assume a formal ${\\tilde{\\varepsilon }}$ -expansion of the quantities $(\\rho _h, g_h, {\\bf q}_h, T_h, \\Phi _h)$ , that is $&\\rho _h = \\rho _{0,h}+{\\tilde{\\varepsilon }}\\,\\rho _{1,h}+\\cdots , \\\\&g_h = g_{0,h}+{\\tilde{\\varepsilon }}\\,g_{1,h}+\\cdots , \\\\&{\\bf q}_h = {\\bf q}_{0,h}+{\\tilde{\\varepsilon }}\\,{\\bf q}_{1,h}+\\cdots , \\\\&T_h = T_{0,h}+{\\tilde{\\varepsilon }}\\,T_{1,h}+\\cdots , \\\\&\\Phi _h = \\Phi _{0,h}+{\\tilde{\\varepsilon }}\\,\\Phi _{1,h}+\\cdots ,$ Substituting them into (REF ), first from (), we can see that $\\rho _{0,h}^{n+1}=\\Phi ^{n+1}_{0,h}+{\\mathcal {O}}({\\tilde{\\varepsilon }})$ , and from (), we have $g^{n+1}_{0,h}=-\\frac{{\\sqrt{\\sigma _0}}}{\\sigma }{\\mathcal {D}}^\\rho _h({\\vec{\\Omega }}\\rho ^{n+1}_{0,h})+{\\mathcal {O}}({{\\tilde{\\varepsilon }}})$ .", "Substituting them with () into (REF ), we obtain $&\\frac{\\Phi ^{n+1}_{0,h}-\\rho ^n_{0,h}}{\\Delta t}+c\\,\\tilde{C}_v\\,\\frac{T^{n+1}_{0,h}-T^n_{0,h}}{\\Delta t} =c\\,{\\mathcal {D}}^g_h(\\langle {\\vec{\\Omega }}\\,\\frac{1}{\\sigma }{\\mathcal {D}}^\\rho _h({\\vec{\\Omega }}\\,\\rho ^n_{0,h}) \\rangle ) \\\\ &\\hspace{162.18062pt} +c\\,\\frac{\\omega }{3\\sigma _0} \\,{\\mathcal {D}}^{div}_h{\\mathcal {D}}^{grad}_h(\\rho ^{n+1}_{0,h}-\\rho ^{n}_{0,h})+{\\mathcal {O}}({\\tilde{\\varepsilon }}).", "\\nonumber $ With the fluxes defined in (REF ), from (REF ) we have ${\\mathcal {D}}^g_h(\\langle {\\vec{\\Omega }}\\,g^n_h \\rangle )={\\mathcal {D}}^{div}(\\langle {\\vec{\\Omega }}\\,g^n_h \\rangle ), \\quad {\\mathcal {D}}^\\rho _h({\\vec{\\Omega }}\\,\\rho ^n_h) = {\\vec{\\Omega }}\\cdot {\\mathcal {D}}^{grad}(\\rho ^n_h).$ Since $\\langle {\\vec{\\Omega }}{\\vec{\\Omega }}\\rangle =\\frac{1}{3}{\\mathbb {I}}$ , here ${\\mathbb {I}}$ is the $2\\times 2$ identity matrix, we find that ${\\mathcal {D}}^g_h(\\langle {\\vec{\\Omega }}\\,\\frac{1}{\\sigma }{\\mathcal {D}}^\\rho _h({\\vec{\\Omega }}\\,\\rho ^n_{0,h}) \\rangle )$ is a local DG approximation to $\\nabla \\cdot \\frac{1}{3\\sigma }\\nabla \\rho $ at time level $t^n$ , that is ${\\mathcal {D}}^g_h(\\langle {\\vec{\\Omega }}\\,\\frac{1}{\\sigma }{\\mathcal {D}}^\\rho _h({\\vec{\\Omega }}\\,\\rho ^n_{0,h}) \\rangle )={\\mathcal {D}}^{div}_h\\left(\\frac{1}{3\\sigma }{\\mathcal {D}}^{grad}_h\\left(\\rho ^n_{0,h}\\right)\\right).$ Due to well-prepared initial conditions, we also have $\\rho ^n_{0,h}=\\Phi ^n_{0,h}+{\\mathcal {O}}({\\tilde{\\varepsilon }})$ .", "Combining (REF ) and (REF ), and replacing $\\Phi _{0,h}$ by $acT_{0,h}^4/\|{\\vec{\\Omega }}\|$ , we obtain $&\\frac{a(T^{n+1}_{0,h})^4-a(T^n_{0,h})^4}{\\Delta t}+ C_v \\,\\frac{T^{n+1}_{0,h}-T^n_{0,h}}{\\Delta t} ={\\mathcal {D}}^{div}_h\\left(\\frac{1}{3\\sigma }{\\mathcal {D}}^{grad}_h\\left(ac(T^n_{0,h})^4\\right)\\right) \\\\ &\\hspace{142.26378pt} +\\frac{\\omega }{3\\sigma _0} \\,{\\mathcal {D}}^{div}_h{\\mathcal {D}}^{grad}_h(ac(T^{n+1}_{0,h})^4-ac(T^{n}_{0,h})^4)+{\\mathcal {O}}({\\tilde{\\varepsilon }}), \\nonumber $ where both sides have been divided by $c/\|{\\vec{\\Omega }}\|$ and $\\tilde{C}_v$ is replaced by $C_v$ .", "Since the second term on the right hand side is of order ${\\mathcal {O}}(\\Delta t)$ , we can clearly see (REF ) in the leading order of the unknowns is a consistent discretization for the diffusive limiting equation (REF ), so that the AP property is guaranteed.", "Remark 4.1 We should notice that in the limit scheme (REF ), on the right hand side, the second term appears as a linear pernalization to the first nonlinear diffusive term, the stability condition is $\\Delta t={\\mathcal {O}}(h)$ instead of $\\Delta t={\\mathcal {O}}(h^2)$ .", "For more discussions, see [47]." ], [ "Asymptotically accurate (AA) analysis for DG-IMEXp.", "Now let us prove that our DG-IMEXp scheme, as ${\\tilde{\\varepsilon }}\\rightarrow 0$ , is a consistent and high order discretization of the limiting diffusive equation (REF ), especially, the order $p$ of temporal accuracy is maintained without reduction.", "Similarly, we write the high order DG-IMEXp scheme in a strong form, $&\\frac{\\rho ^{n+1}_h-\\rho ^n_h}{\\Delta t}+\\frac{c}{{\\sqrt{\\sigma _0}}}\\,\\sum _{\\ell =0}^s \\tilde{b}_\\ell \\,{\\mathcal {D}}^g_h(\\langle {\\vec{\\Omega }}\\,g^{(\\ell )}_h \\rangle ) + c\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{\\ell =0}^s \\tilde{b}_\\ell \\,{\\mathcal {D}}^{div}_h({\\bf q}^{(\\ell )}_h) = \\\\ & \\hspace{142.26378pt} -\\frac{c}{{\\tilde{\\varepsilon }}^2}\\sum _{\\ell =0}^s b_\\ell \\, {\\tilde{\\sigma }}\\,(\\rho ^{(\\ell )}_h-\\Phi ^{(\\ell )}_h)+c\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{\\ell =0}^s \\,b_\\ell \\,{\\mathcal {D}}^{div}_h({\\bf q}^{(\\ell )}_h), \\nonumber \\\\ \\, \\nonumber \\\\&{\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\frac{T^{n+1}_h-T^n_h}{\\Delta t} = \\sum _{\\ell =0}^s \\,b_\\ell \\,{\\tilde{\\sigma }}\\,(\\rho ^{(\\ell )}_h-\\Phi ^{(\\ell )}_h), \\\\ \\, \\nonumber \\\\&{\\bf q}^{n+1}_h = {\\mathcal {D}}^{grad}_h(\\rho ^{n+1}_h),\\\\ \\, \\nonumber \\\\&\\frac{g^{n+1}_h-g^n_h}{\\Delta t}+ \\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}} \\,\\sum _{\\ell =0}^s \\tilde{b}_\\ell \\,({\\mathbb {I}}-\\Pi ){\\mathcal {D}}_h(g^{(\\ell )}_h;\\,{\\vec{\\Omega }}) -\\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}^2} \\,\\sum _{\\ell =0}^s b_\\ell \\,{\\mathcal {D}}^\\rho _h({\\vec{\\Omega }}\\,\\rho ^{(\\ell )}_h)= \\\\ & \\hspace{142.26378pt} -\\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}^2}\\,\\sum _{\\ell =0}^s b_\\ell \\,{\\tilde{\\sigma }}\\,g^{(\\ell )}_h, \\nonumber $ with internal stages satisfying $&\\frac{\\rho ^{(\\ell )}_h-\\rho ^n_h}{\\Delta t} +\\frac{c}{{\\sqrt{\\sigma _0}}}\\,\\sum _{j=0}^{\\ell -1} \\tilde{a}_{\\ell ,j} \\,{\\mathcal {D}}^g_h(\\langle {\\vec{\\Omega }}\\,g^{(j)}_h \\rangle ) + c\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{j=0}^{\\ell -1} \\tilde{a}_{\\ell ,j} \\,{\\mathcal {D}}^{div}_h({\\bf q}^{(j)}_h) = \\\\ &\\hspace{142.26378pt} -\\frac{c}{{\\tilde{\\varepsilon }}^2}\\sum _{j=0}^{\\ell } a_{\\ell ,j} \\, ({\\tilde{\\sigma }}\\, (\\rho ^{(j)}_h-\\Phi ^{(j)}_h))+c\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{j=0}^{\\ell } \\,a_{\\ell ,j}\\,{\\mathcal {D}}^{div}_h({\\bf q}^{(j)}_h), \\nonumber \\\\ \\, \\nonumber \\\\&{\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\frac{T^{(\\ell )}_h-T^n_h}{\\Delta t} = \\sum _{j=0}^{\\ell } \\,a_{\\ell ,j}\\,{\\tilde{\\sigma }}\\,(\\rho ^{(j)}_h-\\Phi ^{(j)}_h), \\\\ \\, \\nonumber \\\\&{\\bf q}^{(\\ell )}_h = {\\mathcal {D}}^{grad}_h(\\rho ^{(\\ell )}_h),\\\\ \\, \\nonumber \\\\&\\frac{g^{(\\ell )}_h-g^n_h}{\\Delta t}+ \\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}} \\,\\sum _{j=0}^{\\ell -1} \\tilde{a}_{\\ell ,j} \\,({\\mathbb {I}}-\\Pi ){\\mathcal {D}}_h(g^{(j)}_h;\\,{\\vec{\\Omega }})-\\frac{c/{\\sqrt{\\sigma _0}}}{{\\tilde{\\varepsilon }}^2} \\,\\sum _{j=0}^{\\ell } a_{\\ell ,j} \\,{\\mathcal {D}}^\\rho _h({\\vec{\\Omega }}\\,\\rho ^{(j)}_h)=\\\\ &\\hspace{142.26378pt}-\\frac{c}{{\\tilde{\\varepsilon }}^2}\\,\\sum _{j=0}^{\\ell } a_{\\ell ,j} \\,{\\tilde{\\sigma }}\\,g^{(j)}_h.", "\\nonumber $ We use the mathematical induction, first to prove the AA property for the internal stages $\\ell =1,\\cdots ,s$ , by assuming a formal ${\\tilde{\\varepsilon }}$ -expansion (REF ) for all inner stage values of $(\\rho ^{(\\ell )}_h, g^{(\\ell )}_h, {\\bf q}^{(\\ell )}_h, T^{(\\ell )}_h, \\Phi ^{(\\ell )}_h)$ .", "Similarly, first we combine (REF ) - () to get $&\\frac{\\rho ^{(\\ell )}_h-\\rho ^n_h}{\\Delta t} +\\frac{c}{{\\sqrt{\\sigma _0}}}\\,\\sum _{j=0}^{\\ell -1} \\tilde{a}_{\\ell ,j} \\,{\\mathcal {D}}^g_h(\\langle {\\vec{\\Omega }}\\,g^{(j)}_h \\rangle ) + c\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{j=0}^{\\ell -1} \\tilde{a}_{\\ell ,j} \\,{\\mathcal {D}}^{div}_h{\\mathcal {D}}^{grad}(\\rho ^{(j)}_h) = \\\\ &\\hspace{142.26378pt} -\\tilde{C}_v\\,\\frac{T^{(\\ell )}_h-T^n_h}{\\Delta t}+c\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{j=0}^{\\ell } \\,a_{\\ell ,j}\\,{\\mathcal {D}}^{div}_h{\\mathcal {D}}^{grad}(\\rho ^{(j)}_h).", "\\nonumber $ For the first stage $\\ell = 0$ , we have $\\rho ^{(0)}_{h}=\\rho ^n_h,\\, g^{(0)}_{h}=g^n_h,\\,{\\bf q}^{(0)}_{h}={\\bf q}^n_h,\\, T^{(0)}_{h}=T^n_h$ and $\\Phi ^{(0)}_{h}=\\Phi ^n_h$ .", "Then for $\\ell =1$ , it is the same AP analysis for the DG-IMEX1 scheme (REF ), we have $\\rho ^{(1)}_{0,h}=\\Phi ^{(1)}_{0,h}+{\\mathcal {O}}({\\tilde{\\varepsilon }})=ac(T^{(1)}_{0,h})^4/\|{\\vec{\\Omega }}\|+{\\mathcal {O}}({\\tilde{\\varepsilon }}), \\quad g^{(1)}=-\\frac{{\\sqrt{\\sigma _0}}}{\\sigma }{\\mathcal {D}}^\\rho _h({\\vec{\\Omega }}\\rho ^{(1)}_{0,h})+{\\mathcal {O}}({\\tilde{\\varepsilon }}), \\quad \\Phi ^{(1)}_{0,h}=ac(T^{(1)}_{0,h})^4,$ and $T^{(1)}_{0,h}$ satisfies $&\\frac{a(T^{(1)}_{0,h})^4-a(T^n_{0,h})^4}{\\Delta t}+ C_v \\,\\frac{T^{(1)}_{0,h}-T^n_{0,h}}{\\Delta t} =\\tilde{a}_{10}\\,{\\mathcal {D}}^{div}_h\\left(\\frac{1}{3\\sigma }{\\mathcal {D}}^{grad}_h\\left(ac(T^n_{0,h})^4\\right)\\right) \\\\ &\\hspace{142.26378pt} +\\frac{\\omega }{3\\sigma _0} \\,{\\mathcal {D}}^{div}_h{\\mathcal {D}}^{grad}_h\\left(a_{11}ac(T^{(1)}_{0,h})^4-\\tilde{a}_{10}ac(T^{n}_{0,h})^4\\right)+{\\mathcal {O}}({\\tilde{\\varepsilon }}).", "\\nonumber $ Now we assume for inner stages $j=1$ up to $j=\\ell -1$ , there holds $\\rho ^{(j)}_{0,h}=\\Phi ^{(j)}_{0,h}+{\\mathcal {O}}({\\tilde{\\varepsilon }})=ac(T^{(j)}_{0,h})^4/\|{\\vec{\\Omega }}\|+{\\mathcal {O}}({\\tilde{\\varepsilon }}), \\quad g^{(j)}=-\\frac{{\\sqrt{\\sigma _0}}}{\\sigma }{\\mathcal {D}}^\\rho _h({\\vec{\\Omega }}\\rho ^{(j)}_{0,h})+{\\mathcal {O}}({\\tilde{\\varepsilon }}), \\quad \\Phi ^{(j)}_{0,h}=ac(T^{(j)}_{0,h})^4,$ and $T^{(j)}_{0,h}$ satisfies $&\\frac{a(T^{(j)}_{0,h})^4-a(T^n_{0,h})^4}{\\Delta t}+ C_v \\,\\frac{T^{(j)}_{0,h}-T^n_{0,h}}{\\Delta t} =\\sum _{j=0}^{\\ell -2} \\tilde{a}_{\\ell -1,j}\\,{\\mathcal {D}}^{div}_h\\left(\\frac{1}{3\\sigma }{\\mathcal {D}}^{grad}_h\\left(ac(T^{(j)}_{0,h})^4\\right)\\right) \\\\ &\\hspace{113.81102pt} +\\frac{\\omega }{3\\sigma _0} \\,{\\mathcal {D}}^{div}_h{\\mathcal {D}}^{grad}_h\\left(\\sum _{j=0}^{\\ell -1} a_{\\ell -1,j}ac(T^{(j)}_{0,h})^4-\\sum _{j=0}^{\\ell -2} \\tilde{a}_{\\ell -1,j}ac(T^{(j)}_{0,h})^4\\right)+{\\mathcal {O}}({\\tilde{\\varepsilon }}).", "\\nonumber $ By induction, we prove them also holding for the next stage $\\ell $ .", "First from () and (), by the definition of $\\Phi =acT^4/\|{\\vec{\\Omega }}\|$ , we easily get $\\rho ^{(\\ell )}_{0,h}=\\Phi ^{(\\ell )}_{0,h}+{\\mathcal {O}}({\\tilde{\\varepsilon }})=ac(T^{(\\ell )}_{0,h})^4/\|{\\vec{\\Omega }}\|+{\\mathcal {O}}({\\tilde{\\varepsilon }}), \\quad g^{(\\ell )}=-\\frac{{\\sqrt{\\sigma _0}}}{\\sigma }{\\mathcal {D}}^\\rho _h({\\vec{\\Omega }}\\rho ^{(\\ell )}_{0,h})+{\\mathcal {O}}({\\tilde{\\varepsilon }}), \\quad \\Phi ^{(\\ell )}_{0,h}=ac(T^{(\\ell )}_{0,h})^4.$ Substituting them with () into (REF ), we get $&\\frac{a(T^{(\\ell )}_{0,h})^4-a(T^n_{0,h})^4}{\\Delta t}+ C_v \\,\\frac{T^{(\\ell )}_{0,h}-T^n_{0,h}}{\\Delta t} =\\sum _{j=0}^{\\ell -1} \\tilde{a}_{\\ell ,j}\\,{\\mathcal {D}}^{div}_h\\left(\\frac{1}{3\\sigma }{\\mathcal {D}}^{grad}_h\\left(ac(T^{(j)}_{0,h})^4\\right)\\right) \\\\ &\\hspace{113.81102pt} +\\frac{\\omega }{3\\sigma _0} \\,{\\mathcal {D}}^{div}_h{\\mathcal {D}}^{grad}_h\\left(\\sum _{j=0}^{\\ell } a_{\\ell ,j}ac(T^{(j)}_{0,h})^4-\\sum _{j=0}^{\\ell -1} \\tilde{a}_{\\ell ,j}ac(T^{(j)}_{0,h})^4\\right)+{\\mathcal {O}}({\\tilde{\\varepsilon }}).", "\\nonumber $ By letting $\\ell =s$ , we have proved that (REF ) and (REF ) hold for all inner stages from $\\ell =0$ to $s$ .", "For updating the solution at the time level $t^{n+1}$ , (REF ) is used, which in the strong form can be written as $&\\frac{\\rho ^{n+1}_h-\\rho ^n_h}{\\Delta t}+\\tilde{C}_v\\,\\frac{T^{n+1}_h-T^n_h}{\\Delta t} = \\\\ & \\hspace{28.45274pt} +\\frac{c}{{\\sqrt{\\sigma _0}}}\\,\\sum _{\\ell =0}^s \\tilde{b}_\\ell \\,{\\mathcal {D}}^g_h(\\langle {\\vec{\\Omega }}\\,g^{(\\ell )}_h \\rangle ) + c\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{\\ell =0}^s \\tilde{b}_\\ell \\,{\\mathcal {D}}^{div}_h({\\bf q}^{(\\ell )}_h,{\\alpha })+c\\,\\frac{\\omega }{3\\sigma _0} \\,\\sum _{\\ell =0}^s \\,b_\\ell \\,{\\mathcal {D}}^{div}_h({\\bf q}^{(\\ell )}_h,{\\alpha }), \\nonumber \\\\ \\, \\nonumber \\\\&{\\tilde{\\varepsilon }}^2\\,\\tilde{C}_v\\,\\frac{T^{n+1}_h-T^n_h}{\\Delta t} - b_s\\,{\\tilde{\\sigma }}\\,(\\rho ^{n+1}_h-\\Phi ^{n+1}_h) = \\sum _{\\ell =0}^{s-1} \\,b_\\ell \\,{\\tilde{\\sigma }}\\,(\\rho ^{(\\ell )}_h-\\Phi ^{(\\ell )}_h).$ Submitting (REF ) and (REF ) into (REF ), first from (), we get $\\rho ^{n+1}_{0,h} = \\Phi ^{n+1}_{0,h}+{\\mathcal {O}}({\\tilde{\\varepsilon }}),$ then with (REF ), we obtain $&\\frac{a(T^{n+1}_{0,h})^4-a(T^n_{0,h})^4}{\\Delta t}+ C_v \\,\\frac{T^{n+1}_{0,h}-T^n_{0,h}}{\\Delta t} =\\sum _{\\ell =0}^{s} \\tilde{b}_{\\ell }\\,{\\mathcal {D}}^{div}_h\\left(\\frac{1}{3\\sigma }{\\mathcal {D}}^{grad}_h\\left(ac(T^{(\\ell )}_{0,h})^4\\right)\\right) \\\\ &\\hspace{96.73918pt} +\\frac{\\omega }{3\\sigma _0} \\,{\\mathcal {D}}^{div}_h{\\mathcal {D}}^{grad}_h\\left(\\sum _{\\ell =0}^{s} b_{\\ell }ac(T^{(\\ell )}_{0,h})^4-\\sum _{\\ell =0}^{s} \\tilde{b}_{\\ell }ac(T^{(\\ell )}_{0,h})^4\\right)+{\\mathcal {O}}({\\tilde{\\varepsilon }}).", "\\nonumber $ (REF ) with inner stages (REF ), as ${\\tilde{\\varepsilon }}\\rightarrow 0$ , becomes a high order IMEX RK discretization for the limiting diffusive equation (REF ) and the temporal order $p$ from the high order IMEX RK scheme is maintained, namely, our DG-IMEXp scheme is AA." ], [ "Numerical examples", "In this section, some numerical tests will be performed to validate the high order accuracy, AP and AA properties of our proposed scheme.", "In the angular direction, the discrete coordinate $S_N$ method is used.", "In 1D for $\\mu \\in [-1,1]$ , the 8-point Gauss quadrature rule is used.", "while in 2D, we take the Gauss-Chebyshev quadrature rule, with 8 Gauss quadrature points and 4 Chebyshev quadrature points.", "In space, we use the nodal DG scheme [15] with the Larangian bases, and $k$ Gaussian quadrature points are used in each space direction for a $k$ -th order method.", "In time, we take the 3rd order type ARS(4,4,3) IMEX RK scheme [2], with the double Butcher table given by $\\begin{array}{c\|c c c c c}0 & 0&0&0& 0 & 0\\\\1/2 &1/2&0&0& 0 &0\\\\2/3 &11/18&1/18&0& 0 &0\\\\1/2 &5/6&-5/6&1/2& 0 &0\\\\1 &1/4&7/4&3/4& -7/4 &0\\\\\\hline &1/4&7/4&3/4& -7/4 &0\\\\\\end{array}, \\ \\ \\ \\ \\ \\begin{array}{c\|c c c c c}0 & 0&0&0& 0 & 0\\\\1/2 &0&1/2&0&0 &0\\\\2/3 &0&1/6&1/2& 0 &0\\\\1/2 &0&-1/2&1/2& 1/2 &0\\\\1 &0&3/2&-3/2& 1/2 &1/2\\\\\\hline &0&3/2&-3/2& 1/2 &1/2\\\\\\end{array}.$ The time step is chosen to be $\\Delta t = 0.01\\,h$ for the 3rd order scheme.", "In the following, $C_v$ and $\\varepsilon $ instead of $\\tilde{C}_v$ and ${\\tilde{\\varepsilon }}$ are specified.", "In the computation, the unit of the length is taken to be centimeter ($cm$ ), the mass unit is gramme ($g$ ), the time unit is nanosecond ($ns$ ), the temperature unit is kilo electron-volt ($keV$ ) and the energy unit is $10^9$ Joules ($GJ$ ).", "Under these units, the light speed $c$ is $29.98\\,cm/ns$ and the radiation constant $a$ is $0.01372\\,GJ/cm^3/keV^4$ , unless otherwise specified." ], [ "Accuracy test in 1D", "We first consider a 1D example with smooth initial conditions at the equilibrium, which are given by $\\rho (0,x)=(b_0+b_1\\,\\sin (x))^4, \\quad T(0,x)=b_0+b_1\\,\\sin (x), \\quad g(0,x,\\mu )=-4\\mu b_1\\,\\cos (x)(b_0+b_1\\,\\sin (x))^6,$ where $g=-\\mu \\rho _x/\\sigma $ and $\\sigma =1/T^3$ .", "For simplicity, all constant parameters, such as $a$ , $c$ and $C_v$ , are all taken to be 1.", "We choose $b_0=0.8$ and $b_1=0.1$ in this test.", "$\\sigma _0$ is taken to be 1.", "Periodic boundary condition is used.", "We run the solution with 3rd order method in time.", "While in space, the nodal DG scheme with $k=2$ to 4 Gaussian quadrature points are considerred, resulting 2nd, 3rd or 4th order respectively.", "$\\varepsilon =1, 10^{-2}, 10^{-6}$ in three different regimes are taken.", "Since the exact solutions are not known, we compute the numerical errors by comparing the solutions on two consecutive mesh sizes.", "In Table REF , we show the errors and orders in $L^1$ and $L^\\infty $ norms for different $k$ 's and different $\\varepsilon $ 's.", "Due to the small time step condition, the spatial errors are dominate, and we can observe $k$ th order for the three $\\varepsilon $ 's, which verfies that our scheme is asymptotic preserving and asymptotically accurate.", "Table: Numerical errors and orders of L 1 L^1 and L ∞ L^\\infty norms for accuracy test in 1D.", "T=0.2T=0.2.", "kk Gaussian quadrature points for k=2,3,4k=2,3,4 in space are used." ], [ "Marshak wave", "The second example is the Marshak wave problem in 1D [42].", "We first take the absorption/emission coefficient to be $\\sigma =300/T^3/cm$ for the Marshak wave-2B problem.", "The specific heat is $0.1\\,GJ/g/keV$ , the density is $3.0\\,g/cm^3$ and $\\varepsilon =1$ .", "The initial temperature $T$ is set to be $10^{-6}\\,keV$ everywhere.", "A constant isotropic incident radiation intensity with a Planckian distribution at $1\\,keV$ is set at the left boundary.", "For this problem, the inflow-outflow close-loop boundary condition [17], [34] is used.", "We take the 3rd method both in space and in time with $N=80$ .", "For the 3rd order nodal DG scheme, a double minmod slope limiter [31] is used to control the numerical oscillations at the propagation front.", "For the Marshak wave-2B problem, we take $\\sigma _0=300$ as the reference opacity.", "Due to $\\sigma _0$ is relatively large, the solution to the GRTEs would be very close to the solution for the limiting diffusive equation (REF ).", "In Fig.", "REF , we show the solutions obtained from our 3rd order AP scheme, which is denoted as “AP3” and compared to the limiting scheme for the diffusive equation (REF ) (directly implemented for (REF )) denoted as “D”.", "We can observe that the two sets of solutions are very close at $t=10, 30, 50, 74$ nano seconds.", "The front positions also match those in [42] (Fig.", "4 & 5), which show that our scheme has the right diffusive limit.", "Then we take the absorption/emission coefficient is taken to be $\\sigma =30/T^3/cm$ for the Marshak wave-2A problem.", "Other settings are the same as the Marshak wave-2B problem and the same boundary condition is used.", "Here $\\sigma _0=30$ .", "For this case, we should observe that the solution of the AP scheme would deviate from the solution in the diffusive limit.", "In Fig.", "REF , we show the solution from the AP3 scheme at $t=0.2, 0.4, 0.6, 0.8$ nano seconds, and compare them to the solution at $t=1ns$ from the D scheme.", "As we can see that, the solution of AP3 scheme propagates slower than that from the D scheme.", "Also our solutions have comparable propagation fronts as those in [42] (Fig.", "6 & 7), including the solution from the D scheme.", "Figure: Marshak wave-2B problem.", "3rd order scheme with N=80N=80.Figure: Marshak wave-2A problem.", "3rd order scheme with N=80N=80.." ], [ "Accuracy test in 2D", "Now we test the order of convergence in the 2D case.", "We consider smooth initial conditions at the equilibrium similar to the 1D problem, which are given by $\\rho (0,x,y) = \\left((a1+b1\\sin (x))(a2+b2\\sin (y)\\right)^4, \\quad T(0,x,y)=(a1+b1\\sin (x))(a2+b2\\sin (y)),$ and $g(0,x,y)=-\\vec{\\Omega }\\cdot \\nabla \\rho (0,x,y)/\\sigma $ .", "Here $\\sigma =1$ .", "Similarly all constant parameters, such as $a$ , $c$ and $C_v$ , are all taken to be 1.", "We choose $a1=a2=0.8$ and $b1=b2=0.1$ in this test.", "Periodic boundary conditions are used in both space directions.", "We run the solution with 3rd order method in time.", "While in space, the nodal DG scheme with $k=2$ to 4 Gaussian quadrature points in the two space directions are considerred, resulting 2nd, 3rd or 4th order respectively.", "$\\varepsilon =1, 10^{-2}, 10^{-6}$ in three different regimes are taken.", "Similarly we compute the numerical errors by comparing the solutions on two consecutive mesh sizes.", "In Table REF , we show the errors and orders in $L^1$ and $L^\\infty $ norms for different $k$ 's and different $\\varepsilon $ 's.", "Similar results as the 1D problem are obtained.", "Table: Numerical errors and orders of L 1 L^1 and L ∞ L^\\infty norms for accuracy test in 2D.", "T=0.01T=0.01.", "kk Gaussian quadrature points for k=2,3,4k=2,3,4 in both two space directions are used." ], [ "Tophat test", "This is a 2D gray radiative transfer test problem, which has already been studied in [11], [42], [14], [41].", "The computational domain is $[0,7]\\times [-2,2]$ .", "The dense and opaque material with density $10\\,g/cm^3$ and opacity $\\sigma =2000\\,cm^{-1}$ is located in the following regions: $(3,4)\\times (-1,1), (0,2.5)\\times (-2,-0.5), (0,2.5)\\times (0.5,2),(4.5,7)\\times (-2,-0.5), (4.5,7)\\times (0.5,2),(2.5,4.5)\\times (-2,-1.5),(2.5,4.5)\\times (1.5,2)$ .", "The pipe, which has density $0.01\\,g/cm^3$ and opacity $\\sigma =0.2\\,cm^{-1}$ , occupies all other regions.", "The heat capacity is $0.1\\,GJ/g/keV$ and $\\varepsilon =1$ .", "Initially, the material has a temperature $0.05\\,keV$ everywhere, and the radiation and material temperature are in equilibrium.", "A heating source with a fixed temperature $0.5\\,keV$ is located on the left boundary for $-0.5<y<0.5$ .", "Others are outflow boundary conditions.", "Five probes (A,B,C,D,E) are placed at $(0.25,0), (2.75,0), (3.5,1.25), (4.25,0), (6.75,0)$ to monitor the change of temperature in the thin opacity material.", "See Fig.", "REF for the initial configuration.", "For this problem, we take $\\sigma _0=10000$ .", "3rd order in space is used.", "For this long time simulation, we choose the first order IMEX scheme (REF ) in time.", "Besides, due to the rapid change of the opacity at the interface between the two materials, the simple numerical fluxes (REF ) and (REF ) cannot work properly.", "Instead, the opacity should be taken into account for computing the numerical fluxes at the interface.", "For our scheme (REF ) under the micro-macro decomposition, it mainly affects the fluxes related to $\\rho $ .", "Taking (REF ) as an example, instead of purely downwind for $\\hat{\\rho }=\\rho ^+$ , at the interface, we modify it to be $\\hat{\\rho }= (1-\\omega _1) \\rho ^+ + \\omega _1 \\rho ^-$ , where $\\omega _1=\\exp (-c{\\tilde{\\sigma }}^+\\,dt/{\\tilde{\\varepsilon }}^2)$ and ${\\tilde{\\sigma }}^+$ is taking from the same side as $\\rho ^+$ .", "This weight is the exponential function appeared in the time dependent evolution solution for the GRTEs (REF ), which is used to desigh a unified gas-kinetic scheme, e.g., see (3.6) in [42].", "We first take mesh size to be $112\\times 64$ and compute the solution upto $t=400ns$ .", "In Fig.", "REF and Fig.", "REF , we show the radiative temperature, which is defined as $T_r=(\|{\\vec{\\Omega }}\|\\langle I\\rangle /ac)^{1/4}$ , and the material temperature $T$ at $t=10$ and 100 nano seconds, respectively.", "We also show the change of the radiative temperature $T_r$ and the material temperature $T$ at the five probles on the mesh $56\\times 42$ in Fig.", "REF .", "It is similar to the results in [42] (Fig.", "10).", "We note that it is not easy to do a good comparison for the change of $T_r$ and $T$ , especially when the time is not long enough, e.g., see [14] (Fig.", "5).", "Here we compare $T_r$ and $T$ from our method on two different meshes $112\\times 64$ and $56\\times 32$ , see Fig.", "REF .", "Relative convergent results can be seen.", "Figure: Initial configuration for the Tophat test.", "In the pipe, ρ=0.01g/cm 3 \\rho =0.01\\,g/cm^3 and σ=0.2cm -1 \\sigma =0.2\\,cm^{-1}, and outer ρ=10g/cm 3 \\rho =10\\,g/cm^3 and opacity σ=2000cm -1 \\sigma =2000\\,cm^{-1}.", "Five probes are placed to monitor the change of temperature.Figure: Tophat test.", "The radiative temperature T r T_{r} and the material temperature TT at 10ns10\\,ns.", "Mesh is 112×64112\\times 64.Figure: Tophat test.", "The radiative temperature T r T_{r} and the material temperature TT at 100ns100\\,ns.", "Mesh 112×64112\\times 64.Figure: Tophat test.", "The change of the radiative temperature T r T_r and the material temperature TT at the five probes upto 1000ns1000\\,ns.", "Mesh is 56×3256\\times 32.Figure: Tophat test.", "A comparison of the radiative temperature T r T_r (left) and the material temperature TT (right) at the five probes upto 600ns600\\,ns.", "Mesh1 is 56×3256\\times 32 and mesh2 is 112×64112\\times 64." ], [ "Conclusion", "In this work, a class of high order asymptotic preserving DG IMEX schemes is developed for the gray radiative transfer equations.", "A weighted linear diffusive term is added for pernalization, so that the resulting scheme in the diffusive limit, requires only a hyperbolic type time step restriction.", "The scheme has been formally proved to be AP and AA for well-prepared initial conditions.", "Numerical results have validated the high order, AP and AA properties of the scheme, and good performance for the 1D Marshak wave and 2D Tophat test problems.", "The efficiency of the scheme as compared to some other schemes will be investigated in our future work." ], [ "Acknowledgement", "All the authors acknowledge support by the Science Challenge Project No.", "TZ2016002.", "T. Xiong also acknowledges support by NSFC grant No.", "11971025, NSF grant of Fujian Province No.", "2019J06002." ] ]	2011.14090
[ [ "Unsupervised Spoken Term Discovery Based on Re-clustering of\n Hypothesized Speech Segments with Siamese and Triplet Networks" ], [ "Abstract Spoken term discovery from untranscribed speech audio could be achieved via a two-stage process.", "In the first stage, the unlabelled speech is decoded into a sequence of subword units that are learned and modelled in an unsupervised manner.", "In the second stage, partial sequence matching and clustering are performed on the decoded subword sequences, resulting in a set of discovered words or phrases.", "A limitation of this approach is that the results of subword decoding could be erroneous, and the errors would impact the subsequent steps.", "While Siamese/Triplet network is one approach to learn segment representations that can improve the discovery process, the challenge in spoken term discovery under a complete unsupervised scenario is that training examples are unavailable.", "In this paper, we propose to generate training examples from initial hypothesized sequence clusters.", "The Siamese/Triplet network is trained on the hypothesized examples to measure the similarity between two speech segments and hereby perform re-clustering of all hypothesized subword sequences to achieve spoken term discovery.", "Experimental results show that the proposed approach is effective in obtaining training examples for Siamese and Triplet networks, improving the efficacy of spoken term discovery as compared with the original two-stage method." ], [ "Introduction", "Unsupervised speech modeling is the task of discovering and modeling speech units at various levels from audio recording without using any prior linguistic information.", "It is an interesting, challenging and impactful research problem as phonetic, lexical and even semantic information could be acquired without the process of transcribing and understanding the given speech data.", "The relevant technology is particularly important to facilitate data preparation especially in the scenarios where: 1) a large (even unlimited) amount of audio data are readily available online but they are untranscribed; 2) a large amount of audio recording is available for a smaller language which no structured linguistic knowledge or documentation can be found.", "Spoken term discovery is a representative task of unsupervised speech modeling.", "It aims to discover repetitively occurred words and/or phrases from untranscribed audio.", "The problem is commonly tackled with a two-stage approach.", "In the first stage, a set of subword units are automatically discovered from untranscribed speech data and these units in turn can be used to represent the speech data as a symbol sequence.", "In the second stage, variable-length sequence matching and clustering are performed on the subword sequence representations.", "One major drawback of this is that the subword decoding errors in the first stage would propagate to deteriorate the outcome of spoken term discovery in the second stage.", "The present study investigates the use of Siamese and Triplet networks in learning segment representations for spoken term discovery when no training labels are available.", "Siamese network has been commonly applied to pattern classification or matching problems when only weak labels are available.", "We propose to train a Siamese/Triplet network with a small dataset of matched and mismatched sequence pairs obtained and use the trained network to generate feature representations for unseen subword sequences.", "The training dataset is constructed based on hypothesized spoken term clusters from an baseline spoken term discovery system developed in our previous study.", "With the new feature representations learned by the Siamese/Triplet network, re-clustering of subword sequences is carried out to generate an improved set of discovered spoken terms." ], [ "Spoken term discovery", "The first attempt in discovering acoustic units is the acoustic segment model (ASM) [1], [2], where a self-derived acoustic model is trained on untranscribed audio to discover subword units.", "There are attempts that further extended to discover longer acoustic units such as spoken term segments in [3] through pattern matching of acoustic features, which is later known as spoken term discovery.", "Spoken term discovery aims to find and extract repetitively occurred sequential pattern from audio in an unsupervised manner.", "In general, a spoken term discovery system performs three tasks one after the other: segmentation, matching and clustering [4].", "There are mainly two approaches to spoken term discovery.", "In the first approach, pattern discovery is done directly with acoustic features.", "Word-level speech segments are matched using sequence matching algorithms like segmental-DTW.", "The matching could be based on conventional frame-level features [3] or fixed-dimension segment representations [5], [6].", "Another approach involves a two-stage process.", "The ASM is trained on untranscribed audio, resulting in symbolic representations known as the pseudo-transcription of speech.", "Sequential pattern discovery is then performed by local alignment or string matching and clustering of sequential patterns [7], [8], [9], [10].", "The results of clustering could be corresponded to the discovered spoken terms in the given audio dataset." ], [ "Siamese and Triplet networks", "Siamese neural network was proposed in [11].", "It consists of two identical sub-network components, which share the learnable parameters.", "Through the two sub-network components, Siamese neural network is trained to perform a designated classification task on a pair of data samples.", "The most common task is to determine whether the two input samples are from the same class or not.", "In other words, the exact class identities for individual training samples are not needed.", "The training of Siamese network requires relatively fewer training samples than conventional neural network classifiers [12].", "Siamese network is widely used in computer vision.", "It is shown to have the ability of comparing samples from unseen classes in the problem of one shot classification [12].", "Triplet network [13] is an extension of Siamese network.", "It consists of three identical sub-networks, which process 3 input samples in parallel, including one reference sample, one matched and one mismatched samples.", "The network is trained to capture the similarity between the matched sample and the reference and the dissimilarity between the mismatched sample and the reference." ], [ "Siamese network on spoken term detection/discovery", "It has been shown that Siamese network is able to learn new representations from audio signals, which facilitate spoken word classification [14].", "It is also able to generate effective representations for spoken term detection [15], [16].", "While existing work assumes matched pair and mismatched pairs for training the Siamese network are available, one challenge in unsupervised spoken term discovery is that no information is given to the system except the audio only.", "In order to apply Siamese network in learning segment representations, reliable matched and mismatched pairs are required for training the network.", "Relatively less work is done on unsupervised generation of matched and mismatched training pairs.", "There is work that identifies frame-level training samples.", "After segmentation, frames from same segments are treated as matched pairs, frames from adjacent segments are treated as mismatched pairs [17].", "There is also work that extracts training examples from available spoken term discovery system, with sampling based on distributions of speakers and matched/mismatched pairs [18]." ], [ "Proposed System", "To generate reliable matched and mismatched pair, we consider the approach of relaying information generated from a trained spoken term discovery system.", "Subwords and term clusters are learned in unsupervised manner, training pairs are identified by evaluating the discovered term clusters based on the discovered subword units.", "In our previous work [10], a two-stage spoken term discovery approach was investigated on recording of classroom lectures.", "The audio signals are first converted into frame-level bottleneck features using a multilingual deep neural network model.", "A set of subword-level speech units are discovered based on the bottleneck features.", "The discovered subword units are treated as phonemes to be the acoustic modeling units in a conventional ASR system.", "The audio signals are in turn decoded by the ASR system into pseudo-transcription.", "Sequential pattern matching is applied to the pseudo-transcription to obtain segments represented in subword sequences, follow by clustering of the subword sequences.", "The resulted clusters were shown to be strongly associated with keywords or key phrases that occur frequently in the audio signals.", "In particular, clusters formed by long subword sequences generally are able to represent meaningful whole words or phrases.", "Nevertheless, many of the resulted clusters, especially those formed by short sequences, do not provide much useful information for spoken term discovery." ], [ "Training data for Siamese network", "The intended problem of spoken term discovery assumes the absence of any kind of data labels for supervised model training.", "To address this issue, we adopt the Siamese/Triplet network, which can be trained with weakly labelled data to learn robust segment-level representation of speech.", "The required segments and their “weak” labels, which tell whether a pair of speech segments contain the same or different spoken terms, is obtained by leveraging the preliminary clustering results of the two-stage approach described above.", "Simply speaking, the clusters with high “purity” are used to provide the training data and their labels.", "Let $C$ denote a cluster initially determined by the two-stage approach.", "$C$ contains a number of speech segments that hypothetically correspond to the same word or phrase.", "Consider two segments $i$ and $j$ in $C$ , and let $lev(i, j)$ be the Levenshtein distance between the symbol representations of $i$ and $j$ .", "We calculate the mean and standard deviation of the Levenshtein distances of all pairs of segments in $C$ , i.e., $\\mu _s = \\sum _{i,j \\in C} {lev(i, j) \\over \|C\|^2}$ $\\sigma _s = \\sqrt{{\\sum _{i,j \\in C} (lev(i, j) - \\mu _s)^2 \\over \|C\|^2}}.$ A small value of $\\mu _s$ implies that members in $C$ have similar pseudo representations.", "A small value of $\\sigma _s$ means that the distances between different member pairs are similar.", "These two measures can be used to indicate the purity of $C$ .", "We propose to retain a set of clusters with $\\mu _s$ and $\\sigma _s$ below certain empirically determined thresholds, i.e., $C \\in T \\text{ if }\\mu _s < thres_{\\mu _s} \\cdot \\bar{C}\\text{ and }\\sigma _s < thres_{\\sigma _s} \\cdot \\bar{C},$ where $\\bar{C}$ denotes the average length of symbol sequences in $C$ .", "Let the collection of retained “pure” clusters be denoted by $T$ .", "Speech segments in the same cluster are believed to contain the same spoken term and therefore are used to form matching pairs for the training of Siamese/Triplet network.", "On the other hand, contrasting training pairs are formed by segments from contrasting clusters that have large inter-cluster distance.", "Consider clusters $C_1$ and $C_2$ in $T$ , and define $\\mu _d = \\sum _{i \\in C_1, j \\in C_2} {lev(i, j) \\over \|C_1\| \|C_2\|}$ $\\sigma _d = \\sqrt{{\\sum _{i \\in C_1, j \\in C_2} (lev(i, j) - \\mu _d)^2 \\over \|C_1\| \|C_2\|}}.$ $C_1$ and $C_2$ are selected as contrasting clusters if $\\mu _d > thres_{\\mu _d} \\cdot {(\\bar{C_1} + \\bar{C_2}) \\over 2}\\text{ and }\\sigma _d < thres_{\\sigma _d} \\cdot {(\\bar{C_1} + \\bar{C_2}) \\over 2}$" ], [ "Siamese/Triplet network", "The Siamese network consists of two identical convolution neural networks (CNN) with shared parameters.", "In the proposed model, the two CNN take in the bottleneck features from a pair of speech segments, denoted $x_0$ and $x_1$ , and their outputs are the respective learnt representations denoted as $f(x_0)$ and $f(x_1)$ .", "If $x_0$ and $x_1$ are a matched pair, the overall output of the Siamese network is trained to be 1.", "If they are a mismatched pair, the output is trained to be 0.", "Figure: Siamese network (left) and Triplet network (right)The network parameters are trained to minimize the contrastive loss function defined as $\\begin{aligned}L(x_0,& x_1,y) = {1 \\over 2} y \|\| f(x_0) - f(x_1)\|\|^2_2 + \\\\& {1 \\over 2} (1-y) \\lbrace max(0, m - \|\|f(x_0) - f(x_1)\|\|_2\\rbrace ^2\\end{aligned}$ The Triplet network is also composed of the same type of CNN components.", "It takes three segments $x_a$ , $x_p$ and $x_n$ as the input, where $x_p$ and $x_a$ are matched pair, and $x_n$ and $x_a$ are mismatched pair.", "The Triplet network aims at embedding matching samples closer and meanwhile keeping contrasting samples away in the representation space.", "The Triplet loss function is given as, $\\begin{aligned}L &(x_a,x_p,x_n) =\\\\& max(0, m+ \|\|f(x_a) - f(x_p)\|\|^2_2 -\|\| f(x_a) - f(x_n)\|\|^2_2)\\end{aligned}$ where $m$ denotes the the margin between matched and mismatched samples from $x_a$ ." ], [ "Segment representations clustering", "The Siamese/Triplet network is trained to learn segment representations that can be used to measure the similarity between segments.", "Our idea is to apply re-clustering on speech segments so as to achieve spoken term discovery.", "Hierarchical Density-based Spatial clustering of Applications with Noise (HDBSCAN) [19] is adopted.", "The clustering algorithm uses data samples to construct a minimum spanning tree of the distance-weighted graph.", "Each node of the tree represents a data sample, and the weight of edge connecting two nodes represents the distance between the data samples.", "A hierarchical level of clustering is built from the tree.", "The tree is then condensed based on minimal cluster size and finally stable clusters are obtained.", "In some cases, HDBSCAN may produce lot of micro-clusters on high-density region, so methods that combine the use of DBSCAN and HDBSCAN are introduced, such as introducing cluster selection epsilon that extracts DBSCAN results on region larger than the epsilon instead [20].", "This hybrid clustering approach is also considered in our implementation." ], [ "Dataset and evaluation metrics", "Different from the previous work where real life lecture recordings were used but hard to evaluate.", "Instead, we evaluate the systems with the 2017 Zerospeech Challenge dataset [21].", "The dataset consists of three languages: English, French and Mandarin, with duration of 45 hours, 24 hours and 2.5 hours respectively.", "The challenge provides varies evaluation metrics in the spoken term discovery track.", "The grouping, type, token and boundary scores are measured in 3 aspects: precision (P), recall (R) and F-score (F).", "Grouping scores compute the intrinsic quality in terms of the cluster phonetic composition.", "Type scores compare how the cluster boundaries match with the actual transcript with true lexicon.", "Token scores evaluate the word tokens that are correctly segmented.", "Boundary scores evaluate how many actual word boundaries are found.", "Normalized edit distance (NED) measures how similar the discovered segments are to the transcript.", "The smaller the value the more similar.", "Coverage (Cov) is the fraction of the corpus that are discovered.", "There are also metrics that give extra information about the system but are hard to evaluate individually.", "n-words is the number of system generated spoken term clusters, and n-pairs is the number of segment pairs generated.", "Detailed description of each of the metrics can be found here Evaluation metrics of the Zerospeech 2017 spoken term discovery track: https://zerospeech.com/2017/track_2.html." ], [ "Baseline spoken term discovery system", "The same system architecture is adopted as in [10], with slight modification on the system parameters.", "In the first stage, language dependent ASMs are individually trained on each language.", "The ASM are trained by first clustering segment-level 40 dimensional multilingual bottleneck features into 55 subwords with Bayesian Gaussian mixtures model (BGMM) [22].", "The bottleneck features and alignment information are obtained from the same multilingual model trained on 5 corpora.", "The initial pseudo-transcription of the audio are then trained iteratively with a DNN-HMM to generate the finalized pseudo-transcription.", "In the second stage, word or phrase level segments are obtained from the pseudo-transcription by finding matching subword sequences with local sequence alignment.", "Normalized Levenshtein distance is used as similarity metric.", "Different from the previous setup which discards very short segments, all the segments are kept for future use.", "The segments are then clustered into spoken term clusters using leader clustering.", "Ideally the system will run with various system parameters until the optimal language-dependent set is found.", "But assuming we have no prior knowledge on a zero resource language, a fixed parameters of $T = 0.4, a = 1.8, R= 3$ are used on the three languages.", "These parameters are determined by prior experience in previous work with slight adjustment.", "$T$ defines the radius of the cluster, $a$ defines the distance between the center of 2 clusters and $R$ is the minimum subword length of segments being considered for spoken term clustering." ], [ "Siamese and Triplet network training", "The training data for the Siamese and Triplet networks are created from baseline spoken term clusters as described in Section REF .", "The Siamese training pairs and Triplet training tuples are obtained by setting $thres_{\\mu _s}$ = 0.2, $thres_{\\sigma _s}$ = 0.2, and $thres_{\\mu _d}$ = 0.4, $thres_{\\sigma _d}$ = 0.2.", "From all the possible valid combinations, $600,000$ Siamese and Triplet training examples are sampled to train the networks respectively.", "A simple CNN network is used to construct the sub-network component in the Siamese/Triplet network.", "The CNN comprises 2 blocks of Conv-ReLU-Max Pooling followed by a Conv-ReLU connected to 2 fully-connected layers with a ReLU activation function.", "The output layer is a linear layer of 40 dimension.", "The input contains frame-level bottleneck features from speech segments.", "The variable-length feature sequences are zero-padded to derive fixed-length sequences for all segments.", "The Siamese and Triplet networks are trained for no more than 20 epochs until a reasonable loss value is attained.", "After training, the networks are used to transform all segments obtained from the sequence alignment step into fixed-dimension segment representations.", "Subsequently HDBSCAN is applied to cluster the representations into spoken term clusters.", "Both HDBSCAN and its hybrid extension are experimented, with cluster selection epsilon being set to $0.2$ ." ], [ "Results and Analysis", "Performance of the systems is listed in Table REF .", "The grouping scores for English are “NA”, due to the time limit for evaluation is hit and the scoring is not completed.", "First, we look at the coverage of the systems.", "The baseline system has varying performance on different languages, from 65% to 89.9%, showing it is highly language dependent and the optimal model parameters need to be identified individually.", "However the Siamese and Triplet systems are relatively consistent, with less difference of 75.6% to 88.4% on the best system (Triplet HDBSCAN) under the same parameters across all 3 languages.", "Similar coverage is achieved by Siamese and Triplet networks on both clustering algorithms, with Triplet network being slightly better.", "Even the Mandarin baseline system only discovers 65% of the words, the Siamese and Triplet networks are able to learn effective segment representations that can discover new terms which are not covered before, raising the coverage for at least 20%.", "However, the coverage of the Siamese and Triplet systems are slightly worse than the baseline by 5%-10% on English and French.", "The missing terms might be the difficult examples that are discovered in the baseline's less confident clusters, but are unable to be well-represented and discovered by the Siamese and Triplet systems.", "As a sanity check, experiments are also done by using the Siamese and Triplet systems to discover only the longer segments (with more than 3 subwords) on English and French, but the coverage is only degraded by at most 3% on both languages.", "This implies not much spoken terms can be discovered in shorter segments.", "One possible reason is that both English and French are more poly-phonetic than Mandarin where the words are likely to be formed by more phonemes.", "The effect of throwing away short segments is not obvious, beside reducing computation effort.", "The number of segment pairs (n-pairs) increases exponentially when data size increases for the baseline.", "From 25K in Mandarin to 100M-400M in English and French, including too many variations of similar terms.", "HDBSCAN produces the smallest n-pairs with similar coverage as other systems, which is more preferable.", "Hybrid generally produces larger n-pairs with similar n-words as HDBSCAN.", "One possible reason for the small n-pairs in English Triplet hybrid system is that the threshold to activate hybrid mode is not reached yet.", "The proposed systems achieve better cluster quality than the baseline in general, with higher grouping, token, type and boundary scores, especially on Mandarin.", "HDBSCAN and its hybrid extension have different strength and weakness in terms of grouping quality and NED.", "HDBSCAN produces lower NED, but gives limited improvement to the grouping scores.", "While the hybrid method gives a better grouping scores with the exchange of higher NED." ], [ "Conclusion", "In this work, the attempt of using Siamese and Triplet networks for spoken term discovery under a complete unsupervised scenario is made.", "The initial segmentation and cluster information is obtained from other spoken term discovery system.", "The clusters with high confidence are used to generate matched and mismatched pairs and tuples for training the Siamese and Triplet networks.", "The networks are used to generate representations for all the available segments, follow by HDBSCAN on the segment representations to obtain new set of spoken term clusters.", "It is shown that even the exact labels of the segments are unavailable, Siamese/Triplet network can still be trained when a small set of high confidence matched and mismatched data pairs are presented.", "Experiments on Zerospeech dataset show that the segment representations generated by Siamese and Triplet networks are effective in representing unseen segments.", "Triplet network is slightly better than Siamese network in learning segment representations for spoken term discovery.", "The proposed systems also provide better cluster quality and consistent coverage than the baseline two-stage model, but still have room for improvement in discovering hard examples." ] ]	2011.14062
[ [ "Markovian linearization of random walks on groups" ], [ "Abstract In operator algebra, the linearization trick is a technique that reduces the study of a non-commutative polynomial evaluated at elements of an algebra A to the study of a polynomial of degree one, evaluated on the enlarged algebra A x M r (C), for some integer r. We introduce a new instance of the linearization trick which is tailored to study a finitely supported random walk on a group G by studying instead a nearest-neighbor colored random walk on G x {1,.", ".. , r}, which is much simpler to analyze.", "As an application we extend well-known results for nearest-neighbor walks on free groups and free products of finite groups to colored random walks, thus showing how one can obtain explicit formulas for the drift and entropy of a finitely supported random walk." ], [ "Introduction", "Let $\\mathcal {A}$ be a complex unital algebra and consider a non-commutative polynomial $\\mathcal {P}=\\mathcal {P}(x_{1}, \\ldots , x_{n})$ in the variables $x_{1}, \\ldots , x_{n} \\in \\mathcal {A}$ .", "In many cases, a detailed study of the relevant properties of $\\mathcal {P}$ is possible only when the degree of $\\mathcal {P}$ is small, typically when $\\mathcal {P}$ is of degree one, in which case $\\mathcal {P}$ is just a linear combination of the $x_{i}$ 's.", "The linearization trick precisely consists in constructing another polynomial $\\tilde{\\mathcal {P}}$ of degree one, which can be related to the relevant properties of $\\mathcal {P}$ .", "It thus can be used to make computations that were not a priori possible for $\\mathcal {P}$ .", "The price to pay is an enlargement of the algebra: writing $\\mathcal {P}$ as a finite sum of monomials $\\mathcal {P}= \\sum _{i_{1}, \\ldots , i_{k}} \\alpha _{i_{1}\\ldots i_{k}} \\: x_{i_{1}} \\cdots \\ x_{i_{k}},$ with $\\alpha _{i_{1} \\ldots i_{k}} \\in \\mathbb {C}$ , $\\tilde{\\mathcal {P}}$ is generally constructed as $\\tilde{\\mathcal {P}} = \\sum _{i} \\tilde{\\alpha }_{i} \\otimes x_{i}$ where $\\tilde{\\alpha }_{i} \\in M_{r}(\\mathbb {C})$ are complex matrices.", "Therefore $\\tilde{\\mathcal {P}}$ is no longer in the algebra $\\mathcal {A}$ but in the algebra $M_{r}(\\mathbb {C}) \\otimes \\mathcal {A}$ for some integer $r$ that depends on $\\mathcal {P}$ (usually through its degree).", "Under various names, the linearization trick has been used in several domains such as electrical engineering, random matrices, operators algebras, automata theory, etc, we refer to the survey [14].", "A famous application of the linearization trick in operator algebra is the computation of the spectrum of a non-commutative polynomial.", "If $a \\in \\mathcal {A}$ , the spectrum of $a$ is defined as $\\sigma (a) := \\lbrace z \\in \\mathbb {C}, z I - a \\text{ is not invertible} \\rbrace $ .", "Given a non-commutative polynomial $\\mathcal {P}(x_{1}, \\ldots , x_{n}) \\in \\mathcal {A}$ , it is possible to construct a linear $\\tilde{\\mathcal {P}} \\in M_{r}(\\mathbb {C}) \\otimes \\mathcal {A}$ for some integer $r$ , such that $(z I -\\mathcal {P})$ is invertible in $\\mathcal {A}$ if and only if $(\\Lambda (z)\\otimes I_{\\mathcal {A}} - \\tilde{\\mathcal {P}})$ is invertible in $M_{r}(\\mathbb {C}) \\otimes \\mathcal {A}$ , where $\\Lambda (z)$ is the $r \\times r$ matrix with only zero elements except the $(1,1)$ entry which is $z$ .", "Moreover, $(z I -\\mathcal {P})^{-1}$ is then precisely the $(1,1)$ entry of $(\\Lambda (z) \\otimes I_{\\mathcal {A}} -\\tilde{\\mathcal {P}})^{-1}$ (seen as a $r\\times r$ matrix of elements in $\\mathcal {A}$ ).", "Such a construction can be found for instance in the monograph [23].", "As illustrated in this last example, we note that the relevant properties of $\\mathcal {P}$ (here its spectrum and its resolvent) dictates the linearization procedure.", "In this paper, we introduce a new linearization of the Markov kernel of a random walk with finite range on a group.", "This linearization is a new Markov kernel on an enlarged state space and it corresponds to a nearest-neighbor random walk on the group.", "Classical quantities such as speed or entropy rate of the original random walk can be read on the new random walk.", "We illustrate our method on free groups and free product of finite groups.", "Notably, we establish new formulas for drift and entropy of finite range random walks on these groups." ], [ "The linearization trick and colored random walks on groups", "We let $G$ be a finitely generated group with identity element $e$ and we fix a finite set of generators $S$ .", "We assume that $S$ is symmetric, meaning that $g \\in S$ implies $g^{-1} \\in S$ .", "Let $p=(p_{g})_{g \\in G}$ be a probability measure on $G$ .", "Consider the Markov chain $(X_{n})_{n \\ge 0}$ on $G$ with transition probabilities $\\mathbb {P}[X_{n}=h \\: \| \\: X_{n-1} = g] = p_{g^{-1}h},$ for all $g,h \\in G$ .", "Such a Markov chain is called a convolution random walk.", "The random walk is said to be finitely supported, or to have finite range, if the measure $p$ is finitely supported.", "It is a nearest-neighbor random walk if $p$ is supported on the set of generators $S$ .", "As for any Markov chain, the transition kernel of a right convolutional random walk can be seen as an operator acting on square integrable functions: $f=f(x) \\mapsto (\\mathcal {P}f)(x)=\\sum _{y} P(x,y) f(y)$ .", "In the present case, it can be written in terms of the left multiplication operators $\\lambda (g)$ , $g \\in G$ , defined as: for all $f \\in \\ell ^{2}(G)$ , $\\lambda (g) \\cdot f : x \\mapsto f(gx).$ Letting $\\mathcal {P}(g,h)=p_{g^{-1}h}$ it is then possible to write $\\mathcal {P}= \\sum _{g \\in G} p_{g} \\lambda (g).$ This sum is finite if and only if the random walk is finitely supported.", "Furthermore, $\\lambda $ is a group morphism so $\\lambda (gh) = \\lambda (g) \\lambda (h)$ for all $g,h \\in G$ ($\\lambda $ is the left regular representation).", "Writing each element $g$ in the support of the walk as a product of elements of $S$ , it is thus possible to write $\\mathcal {P}$ as a non-commutative polynomial in the operators $\\lambda (g), g \\in S$ .", "In other words, $\\mathcal {P}$ is an element of the left group algebra which is generated by the $\\lambda (g)$ 's, $g \\in G$ .", "The polynomial $\\mathcal {P}$ is of degree 1 if and only if the random walk is nearest neighbor.", "With this point of view, the linearization trick potentially allows to study any finitely supported random walk by considering instead a “nearest-neighbor operator” $\\tilde{\\mathcal {P}} = \\sum _{g \\in G} \\tilde{p}_{g} \\otimes \\lambda (g)$ for some well-chosen matrices $p_{g}\\in M_r(\\mathbb {C})$ such that $\\sum _{g} \\tilde{p}_{g}$ is a stochastic matrix, that is, the transition kernel of a Markov chain on $[r]$ where for $n \\ge 1$ integer, we set $[n] = \\lbrace 1, \\ldots , n\\rbrace $ .", "Then, it turns out that this operator $\\tilde{\\mathcal {P}}$ is the transition kernel of a Markov chain $(\\tilde{X}_{n})_{n}$ on the state space $G \\times [r]$ , which at each step moves from a pair $(g,u)$ to a pair $(h,v)$ with probability $\\tilde{p}_{g^{-1}h}(u,v)$ .", "Equivalently, if one interprets $[r]$ as a set of colors, $\\tilde{X}_{n}$ can at each step be multiplied by $g \\in G$ while the color is updated from $u$ to $v$ with probability $p_{g}(u,v)$ .", "Such a Markov chain will be called a colored random walk and can thus be defined similarly to a classical convolution random walk.", "Definition 1 Let $p=(p_{g})_{g \\in G}$ be a family of matrices with non-negative coefficients of $M_{r}(\\mathbb {R})$ such that $P = \\sum _{g \\in G} p_{g}$ is a stochastic matrix.", "A colored random walk is a Markov chain $(Y_{n})_{n}$ on the state space $G \\times [r]$ , with transition probabilities $\\mathbb {P}[Y_{n} = (h,v) \\: \| \\: Y_{n-1} = (g,u)] = p_{g^{-1}h}(u,v).$ The support of the random walk is the set of elements $g \\in G$ such that $p_{g} \\ne 0$ .", "The colored walk is finitely supported if the support is finite, nearest-neighbor if the support is included in $S$ .", "By looking only at the color coordinate, $(Y_{n})$ induces a Markov chain on the set of colors $[r]$ whose transition matrix is exactly the matrix $P$ in (REF )." ], [ "Irreducibility assumptions.", "For standard convolution random walks, it is generally assumed that the support generates the whole group, which then makes the random walk irreducible.", "For colored random walks, we will make the same assumption but suppose as well that the matrix $P$ defines an irreducible Markov chain on the colors.", "Note this assumption does not necessarily make the colored walk irreducible, for there may be unreachable pairs $(g,u)$ .", "Definition 2 A colored walk is quasi-irreducible if the marginal of its support on $G$ generates the whole group and the matrix $P$ is irreducible." ], [ "Reversibility.", "An important property satisfied by some Markov chains is reversibility.", "Recall that, if $X$ is a countable set, $\\nu $ a measure on $X$ and $Q$ is a Markov chain, then $Q$ is said to be reversible with respect to $\\nu $ if for all $x,y \\in X$ , $\\nu (x) Q(x,y) = \\nu (y) Q(y,x)$ .", "In other words, $Q$ seen as an operator on $\\ell ^2(X,\\nu )$ is self-adjoint.", "In our setting, $\\mathcal {P}$ defined in (REF ) is said to be reversible if it is reversible for the counting measure on $G$ .", "This is equivalent to the condition: for all $g \\in G$ , $p_g = p_{g^{-1}}$ .", "Similarly, consider a colored random walk $\\tilde{\\mathcal {P}}$ of the form (REF ).", "Assume that $P$ defined by (REF ) has invariant measure $\\pi $ on $[r]$ .", "Then $\\tilde{\\mathcal {P}}$ is reversible if it is reversible for the product of $\\pi $ and the counting measure on $G$ .", "This is equivalent to the condition, for all $g \\in G$ , $u,v \\in [r]$ , $\\pi (u) p_{g} (u,v) = \\pi (v) p_{g^{-1}} (v,u).$" ], [ "Linearizing random walks.", "The main contribution of this paper is to formalize a way to apply the linearization trick to random walks on groups.", "Definition 3 Let $G$ be a a group generated by a finite set $S$ , with identity element $e$ .", "Let $(X_{n})_{ n \\ge 0}$ be a random walk with kernel $\\mathcal {P}$ as in (REF ) and with finite support generating $G$ .", "Let $(Y_{n})_{n \\ge 0}$ on $G \\times [r]$ be a quasi-irreducible, nearest-neighbor colored random walk with kernel $\\tilde{\\mathcal {P}}$ as in (REF ).", "We say that $(Y_{n})_{n \\ge 0}$ linearizes $(X_{n})_{n \\ge 0}$ (or $\\tilde{\\mathcal {P}}$ linearizes $\\mathcal {P}$ ) if the following two property holds: $(i)$ if $Y_{0}=(e,1)$ there exists a sequence of stopping times $(\\tau _{n})_{n \\ge 0}$ with $\\tau _{0} = 0$ such that $(Y_{\\tau _{n}})_{n \\ge 0}$ is a realization of the random walk $(X_n)_{n\\ge 0}$ with initial condition $e$ , and $(ii)$ these stopping times are a renewal process, that is the variables $\\tau _{n+1} - \\tau _{n}$ are iid with finite mean.", "To be precise, in the above definition, when saying that $Y_{\\tau _{n}}$ is a convolution random walk, we of course identify $Y_{\\tau _{n}}$ with its $G$ coordinate and forget about the color, which in this case is constant equal to 1.", "We remark also that because of a transitivity property, namely one can always translate the starting point to $e \\in G$ by left multiplication, there is no loss of generality in supposing that the walks are started at $e \\in G$ .", "More details on this are given in Section .", "Theorem 1 Let $G$ be a a group generated by a finite set $S$ , with identity element $e$ .", "Consider a random walk with kernel $\\mathcal {P}$ as in (REF ) with finite support generating $G$ .", "Then there exists $r \\ge 1$ and a colored random walk on $G \\times [r]$ with kernel $\\tilde{\\mathcal {P}}$ as in (REF ) which linearizes $\\mathcal {P}$ .", "Moreover, if $\\mathcal {P}$ is reversible, then $\\tilde{\\mathcal {P}}$ can be chosen to be also reversible.", "Theorem REF has been stated in a non-constructive manner for ease of notation.", "The proof of Theorem REF in Section will exhibit two simple linearization constructions which have a vector $p = (p_g)_{g \\in G}$ as input and gives as output the integer $r$ and the family of matrices $(\\tilde{p}_g)_{g \\in S}$ .", "There is one construction in the general case and one construction which preserves reversibility.", "We refer to Remark REF for the number of colors $r$ needed in both constructions.", "We note also that the previous spectral linearization tricks reviewed in [14] did not preserve the Markov property and could not be used to prove Theorem REF .", "There are possible extensions of the Markovian linearization trick.", "It is possible with the same proof techniques to linearize a colored random walk on $G \\times [r]$ with finite range and obtain a colored nearest-neighbor colored random walk on $G \\times [r^{\\prime }]$ with $r^{\\prime } \\ge r$ .", "In another direction, it is possible to linearize random walks on $G$ which do not have a finite range provided that we allow a countable set of colors, see Remark REF below.", "Finally, in this paper we focus on groups only but our first linearization construction applies to random walks on monoids as well." ], [ "Application to the computation of drift and entropy.", "Initially, our goal was to use the linearization trick to try to compute explicit formulas for various invariant quantities of interest, mainly entropy and drift.", "Elements of $G$ can be written as products of elements in $S$ , which will be called words.", "A word $a_{1} \\cdots a_{k}$ with the $a_{i} \\in S$ such that $g=a_{1} \\cdots a_{k}$ is called a representative of the element $g \\in G$ .", "The length with respect to $S$ of $g \\in G$ is $\|g\| := \\min \\lbrace k, g = a_{1} \\cdots a_{k}, a_{i} \\in S \\rbrace .$ Consider a convolution random walk $(X_{n})_{n \\ge 0}$ on a finitely generated group $G$ , started at $e$ .", "Assuming finite moment $\\mathbb {E}\\left[\\left\|X_{1}\\right\|\\right] < \\infty $ , which is always satisfied if the walk is finitely supported, the sequence $(\\mathbb {E}\\left[\\left\|X_{n}\\right\|\\right])_{n \\ge 0}$ is easily shown to be sub-additive (see Section ), which implies the existence of the limit $\\gamma := \\lim _{n \\rightarrow \\infty } \\frac{\\mathbb {E}\\left\|X_{n}\\right\|}{n}.$ The scalar $\\gamma $ is called the drift, or rate of escape, of the random walk.", "The drift $\\gamma $ is in fact also an a.s. and $L^1$ limit, as can be shown using for instance Kingman's sub-additive theorem.", "Therefore $\\gamma := \\lim _{n \\rightarrow \\infty } \\frac{\\left\|X_{n}\\right\|}{n} \\quad \\text{a.s. and in $L^{1}$}.$ Similarly, from the definition of the entropy of $X_{n}$ as $H(X_{n}) := - \\sum _{g} \\mathbb {P}\\left[X_{n}=g\\right] \\log \\mathbb {P}\\left[X_{n} =g\\right]$ one can define $h := \\lim _{n \\rightarrow \\infty } \\frac{H(X_{n})}{n}.$ as an $L^{1}$ limit or as an a.s. limit: $h = \\lim _{n \\rightarrow \\infty } - \\frac{ \\log \\mathcal {P}^{n}(e, X_{n})}{n} \\quad \\text{a.s. and in $L^{1}$}$ where $\\mathcal {P}^{n}$ the $n$ -th power of the transition kernel of the random walk.", "$h$ is called the Avez entropy, or entropy rate, or asymptotic entropy, of the random walk.", "In fact, this entropy can be interpreted as the drift for the Green pseudo-metric, see [1].", "In this paper it will often simply be referred to as the entropy of the random walk.", "We refer the reader to Woess [16] for a general reference on the topic.", "These notions can be extended to colored random walks: it is straightforward to extend the definition of $H(X_{n})$ , while for the drift we set $\\left\|x\\right\| := \\left\|g\\right\|$ for all $x =(g,u) \\in G \\times [r]$ .", "Proposition 2 Let $(X_{n})_{n \\ge 0}$ be a quasi-irreducible colored random walk on a group $G$ .", "The following limits exist and do not depend on the starting color: $\\gamma &= \\lim _{n \\rightarrow \\infty } \\frac{\\left\|X_{n}\\right\|}{n} \\quad \\text{a.s. and in $L^{1}$,} \\\\h &= \\lim _{n \\rightarrow \\infty } - \\frac{ \\log \\mathcal {P}^{n}((e,u), X_{n})}{n} \\quad \\text{a.s. and in $L^{1}$}.$ for all starting color $u \\in [r]$ .", "The proof is given to Section .", "The a.s. convergence of the entropy and drift combined with the law of large numbers yield the following corollary.", "Corollary 3 Let $(X_{n})_{n \\ge 0}$ be a finitely supported random walk on a group $G$ and $(Y_{n})_{n \\ge 0}$ a colored random walk that linearizes $(X_{n})_{n \\ge 0}$ in the sense of Definition REF .", "The drift $\\tilde{\\gamma }$ and entropy $\\tilde{h}$ of $(Y_{n})_{n \\ge 0}$ can be related to the drift $\\gamma $ and entropy $h$ of $(X_{n})_{n \\ge 0}$ by: $&\\gamma = \\mathbb {E}[\\tau _{1}] \\tilde{\\gamma },\\\\&h = \\mathbb {E}[\\tau _{1}] \\tilde{h}.$ The expected time $\\mathbb {E}[\\tau _1]$ has a simple expression in the two linearization constructions given in Section , see Remark REF and Remark REF .", "In this paper, we have chosen to focus on drift and entropy but there are other numerical invariants associated to a random walk such as the spectral radius of $\\mathcal {P}$ or the index of exponential decay of the return probability: $\\lim _{n \\rightarrow \\infty } \\mathcal {P}^{2n}(e,e)^{1/(2n)}$ (they coincide for reversible walks).", "Our linearization technique could also be used to study these last two quantities but this is less novel since these quantities can be read on the resolvent operator and previous linearization techniques allow to compute resolvent of operators of the form (REF ), see [14] for such linearization and [20], [6] for examples of computation of resolvent operators of the form (REF )." ], [ "Entropy and drift for plain groups.", "Most of the time the exact computation of the drift and entropy is not accessible.", "However for nearest-neighbor random walks on groups with an underlying tree structure like free groups [16], [19] and free products of finite groups [15], [22], [9], [7], there exist techniques that yield explicit formulas.", "This paper shows how to extend extend some of these techniques to the colored setting.", "Combined with the linearization trick, our results give explicit formulas for finitely supported random walks.", "By 'explicit', we mean that we express drift and entropy in terms of the unique solutions of some finite dimensional fixed point equations.", "Following [15], we combine the case of free groups and free product of finite groups.", "These groups are called plain groups in [15], [13].", "Definition 4 A plain group is a free product of a finitely generated free group and a finite family of finite groups.", "Write $\\mathbb {F}_{d}$ for the free group on $d$ generators $a_{1}, \\ldots a_{d}$ and let $G_{1}, \\ldots , G_{m}$ be finite groups.", "Consider the plain group $G = \\mathbb {F}_{d}\\ast G_{1} \\ast \\cdots \\ast G_{m}$ , with the set of generators $S := \\bigcup _{i=1}^{d} \\lbrace a_{i}, a_{i}^{-1}\\rbrace \\bigcup \\left( \\bigsqcup _{j=1}^{m} S_{j} \\right).$ In the above expression, for all $j = 1, \\ldots , m$ , $S_{j} := G_{j} \\setminus \\lbrace e\\rbrace $ is the set of elements of $G_{j}$ distinct from the identity element, and $\\bigsqcup $ denotes a disjoint union.", "Introducing the notation from [15], $\\forall g \\in S, \\quad \\mathrm {Next}(g) := \\left\\lbrace \\begin{array}{l l}S \\setminus \\lbrace g^{-1}\\rbrace & \\text{if $g \\in \\mathbb {F}_{d}$} \\\\S \\setminus S_{i} & \\text{if $g \\in S_{i}$}\\end{array}\\right.,$ we see that every element $g \\in G$ writes as a word $g=g_{1} \\cdots g_{n}$ with $n = \\left\|g\\right\|$ and $g_{i+1} \\in \\mathrm {Next}(g_{i})$ for all $i=1, \\ldots , n-1$ .", "If we set aside the trivial cases where $G$ is isomorphic to $\\mathbb {Z}$ or $\\mathbb {Z}/ 2 \\mathbb {Z}\\ast \\mathbb {Z}/ 2 \\mathbb {Z}$ , $G$ is a non-amenable group which implies any irreducible convolution random walk on $G$ is transient (see for instance [26]).", "Therefore if $X_{n}$ is a convolution random walk on $G$ , $\\left\|X_{n}\\right\| \\rightarrow \\infty $ and $X_{n}$ converges in some sense to an infinite word $X_{\\infty }$ made of letters in $S$ .", "The law $p^{\\infty }$ of $X_{\\infty }$ is called the harmonic measure and it provides much information on the asymptotic properties of $X_{n}$ .", "It can in particular be used to compute its entropy and drift.", "Given $g \\in S$ , let $\\mu _{g} := p^{\\infty }\\left[\\xi _{1} = g\\right]$ be the probability under $p^{\\infty }$ that an infinite word starts with letter $g$ .", "The most explicit formulas for nearest-neighbor random walks on plain groups express the entropy and drift in terms of the parameters $p_{g}$ and $\\mu _{g}$ .", "In this paper we extend this technique to the colored setting.", "It had already been shown that such techniques could be adapted to nearest-neigbhor random walks on more general free products, see notably [24], [10], [11].", "Here, the non-commutativity of $M_r(\\mathbb {C})$ creates new difficulties.", "We consider a natural generalization of the harmonic measure $p^{\\infty }_{u}$ , $u \\in [r]$ being the starting color, and a matrix version of the parameters $\\mu _{g}$ , which satisfy $p^{\\infty }_{u}(\\xi _{1} = g) = \\sum _{v \\in [r]} \\mu _{g}(u,v), \\: \\quad \\forall u \\in [r].$ The family of parameters $(\\mu _g)_{g \\in S}$ is uniquely characterized by a family of matrix relations (REF ), called the traffic equations, which arrives as consequences of the stationarity.", "The traffic equations can be solved quite explicitly for the free group.", "The precise definitions and properties of the harmonic measure are given in Section .", "We start with a formula for the drift.", "In the sequel, $\\mathbb {1}$ denotes the vector $\\mathbb {R}^{r}$ with all coordinates equal to 1 and $\\mathbb {1}_{u}$ the indicator at $u \\in [r]$ , that is the vector with zero coordinates except the $u$ entry equal to 1.", "The usual scalar product is denoted by $\\left\\langle \\: , \\: \\right\\rangle $ .", "Theorem 4 Let $(Y_{n})_{n \\ge 0}$ be a nearest-neighbor quasi-irreducible colored random walk on $G \\times [r]$ defined by a family of matrices $(p_{g})_{g \\in S}$ .", "Let $\\pi $ be the unique invariant probability measure of the stochastic matrix $P = \\sum _{g} p_{g}$ .", "The drift of $(Y_{n})_{n \\ge 0}$ is given by $\\begin{split}\\gamma &= \\sum _{g \\in S} \\big \\langle {\\mathbb {1}} , {\\pi p_{g} \\big ( -\\mu _{g^{-1}} + \\sum _{h \\in \\mathrm {Next}(g)} \\mu _{h} \\big ) }\\big \\rangle .\\\\&= \\sum _{g \\in S} \\sum _{u,v,w \\in [r]} \\pi (u) p_{g}(u,v) \\big ( -\\mu _{g^{-1}}(v,w) + \\sum _{h \\in \\mathrm {Next}(g)} \\mu _{h}(v,w) \\big ).\\end{split}$ Theorem REF is the exact counterpart of the already known formulas for colorless walks (corresponding to $r=1$ ).", "For the entropy, we first provide an integral formula in Theorem REF which is again the counterpart of a known formula in the colorless setting.", "Theorem 5 Let $(Y_{n})_{n \\ge 0}$ be a nearest-neighbor quasi-irreducible colored random walk on $G\\times [r]$ defined by a family of matrices $(p_{g})_{g \\in S}$ .", "Let $\\pi $ be the unique invariant probability measure of the stochastic matrix $P = \\sum _{g} p_{g}$ .", "The asymptotic entropy of $(Y_{n})_{n \\ge 0}$ is given by $h = - \\sum _{\\begin{array}{c}g \\in G \\\\ u,v \\in [r]\\end{array}} \\pi (u) p_{g}(u,v) \\int \\log \\left(\\frac{d g^{-1} p^{\\infty }_{u}}{d p^{\\infty }_{v}} \\right) d p^{\\infty }_{v}.$ For nearest-neighbor convolution random walks on plain groups, it is then possible to derive an explicit expression from this integral formula.", "For colored random walks, adapting the computation naturally leads to the problem of determining the limit of infinite products of random matrices, which in turn leads to a formula of the entropy in terms of the parameters $(\\mu _g)_{g \\in S}$ and the law of a family indexed by $[r]$ of random probability measures on $\\mathbb {R}^{r}$ whose law is uniquely characterized by some invariance equation.", "To state our result more precisely, we first introduce the hitting probabilities defined as follows.", "Given $g \\in G, u \\in [r]$ , let $\\tau _{(g,u)} := \\inf \\lbrace n : X_{n} = (g,u)\\rbrace $ be the hitting time of a pair $(g,u)$ , so that $\\tau _{g}:= \\min _{u} \\tau _{(g,u)}$ is the hitting time of $g$ .", "For $g \\in S$ , set $q_{g}(u,v) := \\mathbb {P}[ \\tau _{g} < \\infty \\text{ and } \\tau _{g} = \\tau _{(g,v)} ].$ As we will check, the matrices $q_g$ and $\\mu _g$ satisfy a simple relation (REF ) and the hitting probabilities $(q_g)_{g \\in S}$ are characterized by a quadratic equation (REF ) which can be solved quite explicitly for the free group.", "In the statement below, $\\mathcal {P}_+$ denotes the open simplex of probability measures $[r]$ : $\\mathcal {P}^{+}:= \\left\\lbrace x \\in \\mathbb {R}^{r} : \\sum _{i \\in [r]} x_{i} =1 \\text{ and } x_{i} > 0, \\: \\forall i \\in [r]\\right\\rbrace .$ Theorem 6 Suppose the matrices $(q_{g})_{g \\in S}$ satisfy (REF ) defined in Section .", "Then the entropy of the colored random walk is given by $\\begin{split} h = - \\sum _{g \\in S} \\sum _{u,v,w\\in [r]} \\pi (u) p_{g}(u,v) \\Bigg [ \\mu _{g^{-1}}(v,w) \\int \\log \\left( \\frac{\\left\\langle \\mathbb {1}_{v} \\: , \\: z \\right\\rangle }{\\left\\langle q_{g^{-1}}(u,\\cdot ) \\: , \\: z \\right\\rangle } \\right) \\: d \\nu _{w}(z) \\\\+ \\sum _{ h \\in \\mathrm {Next}(g) } \\mu _{h}(v,w) \\int \\log \\left( \\frac{\\left\\langle q_{g}(v,\\cdot ) \\: , \\: q_{h} z \\right\\rangle }{\\left\\langle \\mathbb {1}_{u} \\: , \\: q_{h} z \\right\\rangle } \\right)\\: d \\nu _{w}(z) \\\\+ \\sum _{ h \\in S : gh \\in S } \\mu _{h}(v,w) \\int \\log \\left( \\frac{\\left\\langle q_{gh}(v,\\cdot ) \\: , \\: z \\right\\rangle }{\\left\\langle q_{h}(u,\\cdot ) \\: , \\: z \\right\\rangle } \\right) \\: d \\nu _{w}(z) \\Bigg ] ,\\end{split}$ where $(\\nu _{u})_{u \\in [r]}$ is the unique family of probability measures on $\\mathcal {P}^{+}$ satisfying $\\int f(z) \\: d \\nu _{u}(z) = \\sum _{\\begin{array}{c}g \\in S \\\\ v \\in [r]\\end{array}} \\int f( q_{g} z) \\mu _{g}(u,v) \\: d \\nu _{v}(z), \\quad \\forall u \\in [r],$ for all bounded measurable functions $f$ on $\\mathcal {P}^{+}$ .", "The technical condition (REF ) is described in Section .", "Let us simply note that this condition is automatically satisfied for colored random walks arising as a linearization of a finite range random walk on $G$ (this is the content of Proposition REF below).", "Applying Theorem REF to a nearest-neighbor walk for which $r= 1, \\pi = 1, \\nu = \\delta _1$ , we get back formula (22) in [15]: $h = - \\sum _{g \\in S} p_{g} \\left( \\mu _{g^{-1}} \\log \\frac{1}{q_{g^{-1}}} + \\sum _{h \\in \\mathrm {Next}(g)} \\mu _{h} \\log q_{g} + \\sum _{h \\in S : \\ gh \\in S} \\mu _{h} \\log \\frac{q_{gh}}{q_{h}} \\right).$ As alluded to above, in Theorem REF , the measure $\\nu _g$ on $\\mathcal {P}_+$ will arise as the convergence in direction toward a random rank one projector of the product of matrices $q_{g_1}q_{g_2}\\cdots q_{g_n}$ where $g_1,g_2, \\ldots $ in $S$ are the successive letters of $X_\\infty $ with law $p^\\infty $ conditioned to start with $g_1 = g$ .", "For finite range random walks on plain groups, fine qualitative results on drift and entropy were already available such as [17], [18] and even on general hyperbolic groups [12], [8] and references therein.", "As pointed in [19], [8], the computations for drift and entropy for nearest-neighbor random walk on the free group have been known for around 50 years but it was however unexpected that these explicit formulas could be extended to finite range random walks.", "To our knowledge, Theorem REF and Theorem REF provide the first formulas for finite range random walks on plain groups, see Figure REF for a numerical illustration.", "Figure: Numerical simuation of a finite range random walk (X t ) t≥0 (X_t)_{t \\ge 0} on 𝔽 2 =〈a,b〉\\mathbb {F}_2 = \\langle a,b \\rangle with 12 generators of length 1,21,2 and 3, and with random transition probabilities.", "The linearization procedure gives r=16r= 16.", "On the left, an average of \|X t \|\|X_t\| over n=20000n = 20000 trajectories compared to the line γt\\gamma t. On the right, an average of H(X t )H(X_t) over nn trajectories compared to the line hth t. For computational efficiency, the entropy H(X t )H(X_t) is approximated with the entropy of a random walk on [n][n] where λ(g),g∈{a,b},\\lambda (g), g \\in \\lbrace a, b\\rbrace , are replaced by an independent uniform permutation matrix of size nn (in other words, we consider a random action of 𝔽 2 \\mathbb {F}_2 on a set of size nn).", "This approximation is valid as long as ht≤lnnh t \\le \\ln n, see ." ], [ "Organization of the paper.", "In Section we propose a general construction of linearized colored walks, with some variations, that lead to Theorem REF .", "Section is devoted to the proof of Proposition REF .", "In Section we define the harmonic measure for a random walk on a plain group.", "Mimicking results for standard random walks, we prove the Markovian structure of the harmonic measure, which is the main ingredient for computing drift and entropy.", "The computations leading to Theorem REF and Theorem REF and additionnal formulas for the entropy are given in section .", "Finally Section provides a modest example illustrating the use of the linearization trick." ], [ "Acknowledgments.", "Part of this work was performed during a visit of both authors to Kyoto University.", "Both authors gratefully acknowledge the support of JSPS and Kyoto University.", "CB was supported by the research grant ANR-16-CE40-0024-01." ], [ "Linearization: proof of Theorem ", "Let $(X_{n})_{n \\ge 0}$ be a finitely supported random walk on $G$ .", "In this section we construct a nearest-neighbor colored random walk $(Y_{n})_{n \\ge 0}$ from $(X_{n})_{n \\ge 0}$ in such a way the entropy and drift can easily be related.", "What seems the most natural is to directly look for $Y_{n}$ as the walk that follows the same trajectory as $X_{n}$ but decomposes a transition in several steps in order to be nearest-neighbor.", "What follows formalizes this idea and adds two possible improvements.", "The first improvement allows to reduce the dimension of the matrices, the second is a linearization that preserves reversibility.", "From now on, let $a_{1}, \\ldots , a_{d}$ be the elements of the generating set $S$ , and $a_{0} := e$ .", "Let $K$ denote the support of the probability measure $p$ and $\\ell _K$ be the maximal length of an element of $K$ .", "For each $g \\in K$ choose a representative $s(g)$ , which can always be assumed to be of length $\|g\|$ .", "We will generally identify $g$ with its representative and write simply $g=a_{i_{1}} \\cdots a_{i_{k}}$ to refer to the fact that $s(g) = a_{i_{1}} \\cdots a_{i_{k}}$ , for some $i_{1}, \\ldots , i_{k} \\in [d]$ .", "Similarly, set $p_{i_{1} \\cdots i_{k}} := p_{g}$ , with in particular $p_{0} := p_{e}$ and $p_{i} := p_{a_{i}}$ , whether $a_{i} \\in K$ or not.", "Since we look for a nearest-neighbor colored random walk, the colored random walk $(Y_{n})_{n \\ge 0}$ is defined by the $r\\times r$ matrices $\\tilde{p}_{i}, i \\in [0:d]$ ." ], [ "Initial construction.", "As mentioned previously, one can first construct $Y_{n}$ as the walk which, when $X_{n}$ is multiplied by $g=a_{i_{1}} \\cdots A_{i_{k}} \\in K$ , visits each $a_{i}$ consecutively.", "Colors can be used to impose $Y_{n}$ to follow the trajectory of the initial random walk $X_{n}$ .", "For instance, consider a neutral color, written 1 and given $g=a_{i_{1}} \\cdots a_{i_{k}} \\in K$ , consider $k-1$ different colors $u_{1}(g), \\ldots , u_{k-1}(g)$ and set $\\tilde{p}_{i_{1}}(1,u_{1}(g)) := p_{g}, \\tilde{p}_{i_{2}}(u_{1}(g),u_{2}(g)) := 1, \\ldots , \\tilde{p}_{i_{k}}(u_{k}(g),1) = 1$ .", "Here the color 1 is to be interpreted as a neutral color: supposing the colored walk $Y_{n}$ is at $(e,1)$ , it goes to $a_{i_{1}}$ with probability $p_{i_{1} \\cdots i_{k}}$ , and is then forced to go through $a_{i_{2}} \\cdots a_{i_{k}}$ until it is back at color 1.", "Except for the color 1, all the colors serve only for the intermediary steps.", "If one considers now another element $h \\in K, h=a_{i_{1}^{\\prime }} \\cdots a_{i_{k^{\\prime }}^{\\prime }}$ , one can take $k^{\\prime }-1$ new colors to distinguish between the intermediary steps which lead to $g$ and those which lead to $h$ .", "Repeating the process with each generator in the support of the walk, one can construct a colored random walk $(Y_{n})_{n \\ge 0}$ such that $Y_{\\tau _{n}} \\stackrel{(d)}{=} X_{n}$ for all $n$ , where $\\tau _{n}$ are the successive hitting times of the color 1." ], [ "Linearization in the non-reversible case", "We now describe more precisely a construction which reduces the number of colors needed in the initial construction by noticing that elements with common prefixes can use the same colors when visiting these prefixes.", "Recall that a representative of minimal length has been fixed for each element of $K$ .", "These representatives are called the representatives of $K$ .", "Given $g,h \\in K, g = a_{i_{1}} \\cdots a_{i_{k}}, h = a_{j_{1}} \\cdots a_{j_{k^{\\prime }}}$ , the prefix of $g$ and $h$ is defined as $g \\wedge h = a_{i_{1}} \\cdots a_{i_{m}}$ where $m := \\max \\lbrace n \\ge 0: i_{1} = j_1, \\ldots , i_n = j_n \\rbrace $ .", "$h$ is a strict prefix of $g$ if $g \\wedge h = h$ and $\\left\|g\\right\| > \\left\|h\\right\|$ .", "For $k \\in [\\ell _K]$ and $i_{1} \\ldots , i_{k} \\in [d]$ , let $[i_{1} \\cdots i_{k}] := \\lbrace g \\in K, g \\wedge a_{i_{1}} \\cdots a_{i_{k}} = a_{i_{1}} \\cdots a_{i_{k}}, \\left\|g\\right\| \\ge k+1 \\rbrace $ and $p_{[i_{1} \\cdots i_{k}]} := \\sum _{g \\in [i_{1} \\cdots i_{k}]} p_{g}$ be the cumulative mass under $p$ of all words in $K$ for which $a_{i_{1}} \\cdots a_{i_{k}}$ is a strict prefix.", "Then for $k \\in [\\ell _K]$ , we set $q_{i_{1} \\cdots i_{k}} = \\frac{p_{[i_{1} \\cdots i_{k}]}}{p_{[i_{1} \\cdots i_{k-1}]}}$ where the denumerator is to be taken as equal to one in the case $k=1$ .", "Here we will make use of the operator point of view in order to write all the matrices defining $(Y_{n})_{n \\ge 0}$ at once.", "As $M_{r}(\\mathbb {C}) \\otimes \\mathcal {A}$ is isomorphic to $M_{r}(\\mathcal {A})$ for any unital algebra $\\mathcal {A}$ , the transition kernel $\\tilde{P}$ can be written as one matrix whose coefficients are operators on $\\ell ^{2}(G)$ .", "In this case, the matrix has coefficients which are linear combinations of the multiplication operators.", "The matrices $p_{i}, i \\in [0,d]$ can easily be deduced: to obtain $p_{i}$ it suffices to replace $\\lambda (a_i)$ by 1 and the $\\lambda (a_j), j \\ne i$ by 0.", "For all $k \\in [\\ell _K]$ , let $r_{k} := \\mathrm {Card} \\ \\lbrace (i_{1}, \\ \\ldots \\ , i_{k}) \\in S^{k}, [i_{1} \\cdots i_{k}] \\ne \\emptyset \\rbrace $ be the number of strict prefixes of elements of $K$ that have length $k$ .", "Then, we define the following matrices: $C(k)$ is the $r_{k-1} \\times 1$ column matrix $C(k) := \\left( \\sum _{j \\in [d]} \\frac{p_{i_{1} \\cdots i_{k-1} j}}{p_{[i_{1} \\cdots i_{k-1}]}} \\:\\lambda (a_j) \\right)_{i_{1} \\cdots i_{k-1}},$ which is indexed by strict prefixes of $K$ of length $k-1$ .", "Given a representative $i_{1} \\cdots i_{k-1}$ , define the row matrix $L(i_{1}, \\ldots , i_{k-1}) := \\left( q_{i_{1} \\cdots i_{k-1} j} \\: \\lambda (a_{j}) \\right)_{j}$ indexed by all $j$ such that $i_{1} \\cdots i_{k-1} j$ is the strict prefix of a representative of $K$ .", "Then use these row matrices to form the $r_{k-1} \\times r_{k}$ diagonal block matrix $D(k)$ , whose $i_{1} \\cdots i_{k-1}$ diagonal entry is $L(i_{1}, \\cdots , i_{k-1})$ : $D(k) = \\begin{pmatrix}\\ddots & & \\\\& L(i_{1}, \\ldots , i_{k-1}) & \\\\& & \\ddots \\end{pmatrix}.$ Finally combine all the preceding matrices to construct: $\\tilde{\\mathcal {P}} := \\begin{pmatrix}\\sum _{i \\in [0:d]} p_{i} \\lambda (a_{i}) & D(1) & 0 & & & 0\\\\C(2) & 0 & D(2) & & &\\\\C(3) & 0 & 0 & D(3) & &\\\\\\vdots & & & & \\ddots &\\\\C(\\ell _K-1) & & & & & D(\\ell _K-1) \\\\C(\\ell _K) & & & & & 0\\end{pmatrix}$ Example 1 The construction will certainly be clearer on a concrete example.", "Consider a random walk on the group $G$ given by the presentation $\\langle a,b \\: \| \\: ab=ba \\rangle $ , which is in fact isomorphic to $\\mathbb {Z}^{2}$ .", "Suppose the random walk is supported by all words in $a$ and $b$ of length less than 3, that is $K = \\lbrace e, a,b, a^{2}, ab, b^{2},a^{3}, a^{2}b, ab^{2},b^{3} \\rbrace .$ To avoid large matrices, we forget about the inverses $a^{-1}, b^{-1}$ but they should not be discarded in a general case.", "Because of the relation $ab=ba$ , there can be several ways to write an element of $G$ as words in $a$ and $b$ , for instance $bab = ab^{2}$ .", "Therefore we fix a representative for each element of the support.", "In the present case, group elements can be written uniquely as $a^{k}b^{l}$ so it is natural to choice these words as representatives.", "Applying the preceding construction, one eventually obtains the following operator matrix: $\\begin{pmatrix}p_{e} \\lambda (e) + p_{a} \\lambda (a) + p_{b} \\lambda (b) & q_{a} \\lambda (a) & q_{b} \\lambda (b) & 0 & 0 & 0\\\\\\frac{p_{a^{2}}}{p_{[a]}} \\lambda (a) + \\frac{p_{ab}}{p_{[a]}} \\lambda (b) & 0 & 0 & q_{a^{2}} \\lambda (a) & q_{ab}\\lambda (b) & 0 \\\\\\frac{p_{b^{2}}}{p_{[b]}} \\lambda (b) & 0 & 0 & 0 & 0 & q_{b^{2}} \\lambda (b) \\\\\\frac{p_{a^{3}}}{p_{[a^{2}]}} \\lambda (a) + \\frac{p_{a^{2}b}}{p_{[a^{2}]}} \\lambda (b) & 0 & 0 & 0 & 0 & 0 \\\\\\frac{p_{ab^{2}}}{p_{[ab]}} \\lambda (b) & 0 & 0 & 0 & 0 & 0 \\\\\\frac{p_{b^{3}}}{p_{[b^{2}]}} \\lambda (b) & 0 & 0 & 0 & 0 & 0\\end{pmatrix}$ where $ \\begin{array}{l l l}q_{a} = p_{[a]} = p_{a^{2}}+p_{ab} + p_{a^{3}} + p_{a^{2}b} + p_{ab^{2}}, & q_{b} = p_{[b]} = p_{b^{2}} + p_{b^{3}}, & q_{a^{2}} = p_{[a^{2}]}/p_{[a]} = (p_{a^{3}} + p_{a^{2}b})/p_{[a]}\\end{array}$ $\\begin{array}{l l}q_{ab} = p_{[ab]}/p_{[a]} = p_{ab^{2}}/p_{[a]}, & q_{b^{2}} = p_{[b^{2}]}/p_{[b]} = p_{b^{3}}/p_{[b]}.\\end{array}$ [Proof of Theorem REF (non-reversible case)] On row $i_{1} \\cdots i_{k}$ , the sum of entries of the matrix $P= \\sum \\tilde{p}_{i}$ is $\\sum _{j \\in [d]} \\frac{p_{i_{1} \\cdots i_{k-1} j}}{p_{[i_{1} \\cdots i_{k-1}]}} + \\sum _{j \\in [d]} q_{i_{1} \\cdots i_{k-1}j} = \\frac{1}{p_{[i_{1} \\cdots i_{k-1}]}} \\sum _{j \\in [d]} (p_{i_{1} \\cdots i_{k-1} j} + p_{[i_{1} \\cdots i_{k-1} j]}) = 1.$ Thus $P$ is stochastic and $\\tilde{\\mathcal {P}}$ defines indeed a colored random walk $(Y_{n})_{n \\ge 0}$ .", "Suppose now $Y_{n} = (g_{n},u_{n})$ is started at color 1.", "Then define $(\\tau _{n})_{n \\ge 0}$ as the successive return times at the first color: $\\tau _{0} := 0$ and for $n \\ge 1$ $\\tau _{n} := \\inf \\lbrace m > \\tau _{n-1} , u_{m}=1\\rbrace $ By the Markov property, the random variables $\\tau _{n} - \\tau _{n-1}$ are iid with the same law as $\\tau _{1}$ .", "For all $g = i_{1} \\cdots i_{k}$ in $K$ , the probability that $Y_{\\tau _{1}} = g$ is $\\mathbb {P}\\left[Y_{\\tau _{1}} = g\\right] &= q_{i_{1}} \\, q_{i_{1} i_{2}} \\, \\cdots \\ q_{i_{1} \\cdots i_{k-1}} \\ \\frac{p_{i_{1} \\cdots i_{k}}}{p_{[i_{1} \\cdots i_{k-1}]}} \\\\&= p_{[i_{1}]} \\frac{p_{[i_{1} i_{2}]}}{p_{[i_{1}]}} \\cdots \\frac{p_{[i_{1} \\cdots i_{k-1}]}}{p_{[i_{1} \\cdots i_{k-2}]}} \\frac{p_{i_{1} \\cdots i_{k}}}{p_{[i_{1} \\cdots i_{k-1}]}} \\\\&= p_{i_{1} \\cdots i_{k}} = p_{g}.$ By Markov property and the fact that the increments are iid one easily deduce that $Y_{\\tau _{n}} \\stackrel{(d)}{=} X_{n}$ for all $n \\ge 0$ .", "Finally, the irreducibility of $X_{n}$ implies the quasi-irreducibility of $Y_{n}$ .", "Remark 1 The expectation of the hitting time $\\tau _{1}$ is very simple to compute: $Y_{1}$ can go through each $g \\in K$ with probability $p_{g}$ , in which case it needs $\|g\|$ steps to get back to the color 1.", "Hence the expectation of $\\tau _{1}$ is just the average length of elements in $K$ : $\\mathbb {E}\\left[\\tau _{1}\\right] = p_{e} + \\sum _{g \\in K} p_{g} \|g\| = p_{e} + \\mathbb {E}[ \|X_1\| ].$ Remark 2 If $\\mathcal {P}$ is not of finite range, then the construction produces a countable set of colors.", "The construction could nevertheless be useful if the expected time $\\mathbb {E}[\\tau _1] = p_{e} + \\mathbb {E}[\|X_1\|]$ is finite." ], [ "Linearization in the reversible case", "In the previous constructions, it is impossible for the colored random walk to backtrack and hence to be reversible.", "To correct this, we propose the following variation of the initial construction.", "We assume that for all $g \\in K$ , the representative of $g = a_{i_1}\\cdots a_{i_n}$ is chosen such that the representative of $g^{-1}$ is $a_{i_n}^{-1}\\cdots a_{i_1}^{-1}$ .", "We start with the neutral color 1 and, for each pair $(g,g^{-1})$ with $\|g\| = \|g^{-1}\| \\ge 2$ , we add $\\left\|g\\right\| - 1$ new colors $u_{1}(g), \\ldots , u_{\\left\|g\\right\|-1}(g)$ and we set $u_{k} ( g^{-1} ) = u_{\|g\|-k}(g)$ for all $1\\le k \\le \|g\|-1$ .", "Suppose $g$ is written as $g=a_{i_{1}} \\cdots a_{i_{k}}$ .", "For all $h \\in G$ , the transition probability to go from $(h,1)$ to $(h a_{i_{1}}, u_{1}(g))$ is set to some value $\\alpha _{g}$ to be determined, such that $\\alpha _{g} = \\alpha _{g^{-1}}$ and $\\sum _{g \\in S} \\alpha _{g} =1$ .", "All other transition probabilities on the segment joining $(x,1)$ to $(xg,1)$ are set to $1/2$ , see Figure REF .", "Figure: The linearizing reversible random walkIn matrix form, the construction is as follows.", "Given $\\alpha \\in \\mathbb {R}_{+}$ and $g=a_{i_{1}} \\cdots a_{i_{k}}, k \\ge 3$ , consider the following operator-valued matrix of size $k \\times k$ $A_{g}(\\alpha ) := \\begin{pmatrix}0 & \\alpha \\lambda (a_{i_{1}}) & 0 & \\cdots & 0 & \\alpha \\lambda (a_{i_{k}}^{-1}) \\\\1/2 \\: \\lambda (a_{i_{1}}^{-1}) & 0 & 1/2 \\: \\lambda (a_{i_{2}}) & 0 & \\cdots & 0 \\\\0 & 1/2 \\: \\lambda (a_{i_{2}}^{-1}) & 0 & \\ddots & & \\vdots \\\\\\vdots & 0 & \\ddots & 0 & & 0 \\\\0 & \\vdots & & 1/2 \\: \\lambda (a_{i_{k-2}}^{-1}) & 0 & 1/2 \\: \\lambda (a_{i_{k-1}}) \\\\1/2 \\: \\lambda (a_{i_{k}}) & 0 & \\cdots & 0 & 1/2 \\: \\lambda (a_{i_{k-1}}^{-1}) & 0\\end{pmatrix}.$ If $k=2$ , let $A_{a_{i_1}a_{i_2}}(\\alpha ) := \\begin{pmatrix}0 & \\alpha \\lambda (a_{i_{1}}) + \\alpha \\lambda (a_{i_{2}}^{-1}) \\\\1/2 \\: \\lambda (a_{i_{2}}) + 1/2 \\: \\lambda (a_{i_{1}}^{-1}) & 0\\end{pmatrix}.$ If $k =1 $ , let $A_{a_i}(\\alpha )=(\\alpha \\lambda (a_{i}) + \\alpha \\lambda (a_{i}^{-1}))$ and finally for $k =0$ set $A_{e}(\\alpha ) := (\\alpha \\lambda (e))$ .", "In the case $k \\ge 2$ , one can write $A_{g}(\\alpha ) = \\begin{pmatrix}0 & L_{i_{1},i_{k}}(\\alpha ) \\\\C_{i_{1},i_{k}} & \\tilde{A}_{i_{2} \\cdots i_{k-1}}\\end{pmatrix},$ where $L_{i_{1},i_{k}}(\\alpha ), C_{i_{1},i_{k}}$ are respectively a $1 \\times (k-1)$ row matrix and a $(k-1) \\times 1$ column matrix, while $\\tilde{A}_{i_{2} \\cdots i_{k-1}}$ is a tri-diagonal $(k-1) \\times (k-1)$ matrix depending only $i_{2}, \\ldots , i_{k-1}$ , with zero diagonal.", "Now given a square matrix $M$ of matrix $m \\times m$ , use $A_{g}(\\alpha )$ to form the matrix of size $m+k-1$ $M \\cup A_{g}(\\alpha ) := \\begin{pmatrix}M & \\begin{matrix}L_{i_{1},i_{k}}(\\alpha ) \\\\0\\end{matrix} \\\\\\begin{matrix} C_{i_{1},i_{k}} & 0 \\end{matrix} & \\tilde{A}_{i_{2}, \\ldots , i_{k-1}}\\end{pmatrix},$ where 0 is the null matrix of the appropriate size.", "If $k \\in \\lbrace 0,1\\rbrace $ , simply set $M \\cup A_{g}(\\alpha ) := M + \\mathrm {diag}(A_{g}(\\alpha ),0, \\ldots , 0)$ .", "The transition kernel eventually considered is $\\tilde{\\mathcal {P}} := \\bigcup _{g \\in K} A_{g}(\\alpha _{g}).$ Thanks to the constraint $\\sum _{g \\in G} \\alpha _g = 1$ , $ \\tilde{\\mathcal {P}} $ is indeed a Markov transition kernel.", "However, the colored walk can now make steps that would not have been possible in the initial construction.", "For instance, it can go from $(e,1)$ to some $(g,u)$ and then come back to $(e,1)$ , even if $p_{e} = 0$ for the initial walk.", "Therefore, the stopping times cannot be taken directly as the return times at color 1.", "Instead define, if $Y_n = (g_n,u_n)$ has transition kernel $\\tilde{\\mathcal {P}}$ , $\\tau _{n} := \\left\\lbrace \\begin{array}{l l}\\inf \\lbrace m \\ge \\tau _{n-1} : u_{m}=1, g_{m} \\ne g_{\\tau _{n-1}} \\rbrace & \\text{if $g_{\\tau _{n-1}+1} \\ne g_{\\tau _{n-1}}$} \\\\\\tau _{n-1}+1 & \\text{if $g_{\\tau _{n-1}+1} = g_{\\tau _{n-1}}$}\\end{array} \\right.", ".$ Theorem REF in the reversible case is now contained in the following lemma.", "Lemma 7 Suppose $X_{n}$ is a finitely supported random walk on $G$ defined by a probability vector $p$ and started at $e$ .", "Setting for each $g \\in K$ $\\alpha _{g} := \\left\\lbrace \\begin{array}{l l}\\frac{(1-p_{e})\|g\| p_{g}}{\\sum _{h \\in K} \|h\| p_{h}} & \\text{if $g \\ne e$} \\\\p_{e} & \\text{if $g = e$}\\end{array}\\right.$ the preceding construction yields an operator $\\tilde{\\mathcal {P}}$ which defines a reversible colored random walk $(Y_{n})_{n \\ge 0}$ satisfying $Y_{\\tau _{n}} \\stackrel{(d)}{=} X_{n}$ for all $n \\ge 0$ if started at $(e,1)$ .", "As already pointed, the construction define a colored random walk as long as $\\sum _{g \\in S} \\alpha _{g} =1$ , which is true for $\\alpha _{g}$ defined as in (REF ).", "Furthermore such a colored walk is reversible: for all $(g,u) \\in G \\times [r]$ , letting $\\mu (g,u) = \\left\\lbrace \\begin{array}{l l}1 & \\text{if $u=1$} \\\\2 \\, \\alpha _{h} & \\text{if $u=u_{h}(k)$ for some $k < \\left\|h\\right\|$}.\\end{array}\\right.$ defines a reversible measure for the random walk, as it can be checked directly.", "Consider now the colored walk $(Y_{n})_{n \\ge 0}$ defined by (REF ), started at $Y_{0}=(e,1)$ .", "Suppose first that $p_{e}=0$ , so that $\\tau _{1}$ is the hitting time of the set $(K \\setminus \\lbrace e\\rbrace ) \\times \\lbrace 1\\rbrace $ .", "At step 1 the colored walk necessarily enters the segment joining $(e,1)$ to $(h,1)$ for some $h \\in K$ .", "If the walk escapes this segment at $h$ , then $Y_{\\tau _{1}} = h$ .", "Otherwise, it necessarily goes back to the starting state $(e,1)$ .", "Now on each of these segments, the colored walk is simply performing a simple random walk so the escape probabilities are given by the standard gambler's ruin problem.", "Namely the simple random walk on $[0:n]$ reaches $n$ before it gets to 0 with probability $k/n$ when started at $k \\in [0:n]$ .", "Therefore by Markov's property $\\mathbb {P}_{(e,1)}\\left[Y_{\\tau _{1}} = g\\right]= \\alpha _{g}/\\left\|g\\right\| + \\sum _{h \\in K} \\alpha _{h}(1-1/\\left\|h\\right\|) \\mathbb {P}_{(e,1)}\\left[Y_{\\tau _{1}}=g\\right].$ We deduce that $\\mathbb {P}_{(e,1)}\\left[Y_{\\tau _{1}} = g\\right] = \\frac{\\alpha _{g} / \\left\|g\\right\|}{\\sum _{h \\in K} \\alpha _{h} / \\left\|h\\right\|} = p_{g}.$ For the general case $p_{e} \\ne 0$ , $Y_{\\tau _{1}} = (e,1)$ if and only if $Y_{1} = (e,1)$ which occurs with probability $\\alpha _{e} = p_{e}$ .", "For $g \\ne 0$ , consider the random walk conditionned to move at every step, written $(Y^{\\prime }_{n})_{n \\ge 0}$ .", "If $q(x,y)$ is the transition probability for $Y_{n}$ between two states $x$ and $y$ , then the transition probability for $Y^{\\prime }_{n}$ is 0 if $y=x$ and $q(x,y)/(1-q(x,x))$ otherwise.", "The previous argument apply to this walk, so $\\mathbb {P}_{(e,1)}\\left[Y^{\\prime }_{\\tau ^{\\prime }_{1}} = g\\right] = \\frac{\\alpha _{g} / ((1-p_{e})\\left\|g\\right\|)}{\\sum _{h \\in K \\setminus \\lbrace e\\rbrace } \\alpha _{h} / ((1-p_{e})\\left\|h\\right\|)} = \\frac{p_{g}}{1-p_{e}},$ with $\\tau ^{\\prime }_{1}$ is the obvious extension of $\\tau _{1}$ to $Y^{\\prime }_{n}$ .", "Coming back to $Y_{\\tau _{1}}$ : to reach $g \\ne e$ it it is necessary that $Y_{1} \\ne e$ , which occurs with probability $(1-p_{e})$ .", "Conditional on that event it no longer matters whether the random walk comes back at $e$ and possibly stays there, so one can reason with $Y^{\\prime }_{n}$ instead.", "Hence $\\mathbb {P}_{(e,1)}\\left[Y_{\\tau _{1}} = g\\right] = (1-p_{e}) \\mathbb {P}_{(e,1)}\\left[Y^{\\prime }_{\\tau ^{\\prime }_{1}} = g\\right] = p_{g}.$ The conclusion follows.", "Remark 3 The expected time $\\mathbb {E}\\left[\\tau _{1}\\right]$ , although not as simple as in the non-reversible case, can nonetheless be computed quite easily using for instance the electric network paradigm.", "Using [21], we found in the case $p_{e} = 0$ $\\mathbb {E}\\left[\\tau _{1}\\right] = \\sum _{g \\in K} p_{g} \\left\|g\\right\|^{2} = \\mathbb {E}\\left\|X_{1}\\right\|^{2}.$ Remark 4 In the initial construction, we have that the total number of colors is $r = 1 + \\sum _{g \\in K} (\|g\| -1).$ In the reversible construction, there is a factor $1/2$ on front of the sum (because we use the same colors for $g$ and $g^{-1}$ ).", "In the improved construction of Subsection REF , this is an upper bound, the actual value is $r = 1 + \\mathrm {Card}(K^{\\prime }) - \\mathrm {Card}(K)$ where $K^{\\prime }$ is the set of words which are a suffix of some element in $K$ (in the chosen representatives).", "In a concrete application, it is often possible to design linearization procedures which are more economical in terms of number of colors, see Section for an example." ], [ "Entropy and drift for colored random walks", "In this section we extend the notions of entropy and drift to colored random walks, proving Proposition REF ." ], [ "Transitivity properties.", "If $(X_{n})_{n \\ge 0}$ is a Markov chain with transition kernel $Q$ on a discrete state space $V$ , $(X_{n})$ is said to be transitive if for all $x,y \\in V$ there exists a bijection $\\varphi : V \\rightarrow V$ such that $\\phi (x) = y$ and for all $z \\in V$ , $Q(x,z) = Q(y, \\phi (z))$ .", "Transitivity has the consequence that the Markov chain can be translated in order to start on a specific state.", "Sometimes a weaker notion of transitivity is satisfied: there exists a finite partion of the state space such that a bijections as above exist only between states that are in the same set of the partition.", "In this case the Markov chain is said to be quasi-transitive.", "Transitivity is an essential property of convolution random walks on groups, bijections being simply given by the left multiplications.", "It is the property that makes the sequences $(\\left\|X_{n}\\right\|)_{n \\ge 0}$ and $(H(X_{n}))_{n \\ge 0}$ sub-additive and hence allows to define drift and entropy.", "On the other hand, for colored random walks it is possible that no bijection as above exists between pairs $(g,u)$ and $(h,v) \\in G \\times [r]$ , but there is one between $(g,u)$ and $(h,u)$ for all $g,h \\in G, u \\in [r]$ , given by the left multiplication by $hg^{-1}$ .", "Hence colored random walks are only quasi-transitive.", "For this reason it is slightly less straightforward to define entropy and drift for colored random walks than it is for convolution walks.", "Let $(X_{n})_{n \\ge 0}$ be a colored random walk defined by a family of matrices $(p_{g})_{g \\in G}$ .", "Thanks to quasi-transitivity it can be without loss of generality to be started at $e \\in G$ .", "Write $\\mathbb {P}_{u} = \\mathbb {P}_{(e,u)}$ for the law of this colored chain when started at $(e,u)$ .", "The definitions of entropy and drift can naturally be extended to colored random walks: recall that if $X_{n} = (g_{n}, u_{n}) \\in G \\times [r]$ , $\\left\|X_{n}\\right\| := \\left\|g_{n}\\right\|$ and set $H_{u}(X_{n}) := - \\sum _{g,v} \\mathbb {P}_{u} \\left[X_{n}=(g,v)\\right] \\log \\mathbb {P}_{u} \\left[X_{n} =(g,v)\\right].$ Let $\\pi $ be the unique invariant probability measure of the stochastic matrix $P$ .", "Let $\\mathbb {P}_{\\pi }$ denote the law of the colored Markov chain started at $e$ with the starting color being chosen according to $\\pi $ , that is $\\mathbb {P}_{\\pi } = \\sum _{u \\in [r]} \\pi (u) \\mathbb {P}_{u}.$ Then $H_{\\pi }(X_{n}) = \\sum _{u \\in [r]} \\pi (u) H_{u}(X_{n})$ forms a sub-additive sequence.", "Indeed, for $k \\le n$ , the Markov property yields $H_{u}(X_{n} \\: \| \\: X_{k}) &= \\sum _{g,v} \\mathbb {P}_{(e,u)}(X_{k}=(g,v)) H_{v}(X_{n-k}) \\\\&= \\sum _{v} P^{k}(u,v) H_{v}(X_{n-k}).$ If we multiply by $\\pi (u)$ and sum over $u \\in [r]$ , this proves that $H_{\\pi }(X_{n} \\: \| \\: X_{k}) = H_{\\pi }(X_{n-k})$ thanks to the invariance of $\\pi $ .", "Therefore one can bound $H_{\\pi }(X_{n+m}) \\le H_{\\pi }(X_{n+m}, X_{m}) = H_{\\pi }(X_{m}) + H_{\\pi }(X_{n+m} \| X_{m}) = H_{\\pi }(X_{m}) + H_{\\pi }(X_{n}).$ and prove the existence of the limit $h_{\\pi } := \\lim _{n \\rightarrow \\infty } \\frac{H_{\\pi }(X_{n})}{n}.$ The previous computation is very simple but only give limits in expectation.", "By applying Kingman's subadditive theorem one can obtain $L^{1}$ as well as a.s. limits.", "Because the color set $[r]$ is finite, one can also expect that the limits do not depend on the starting color in the case $P$ is irreducible.", "This is exactly the content of Proposition REF .", "To prove this proposition, the following lemma will be needed: Lemma 8 ([21]) Let $(X_{n})_{n}$ be a Markov chain on a state space $V$ and let $f : V^{\\mathbb {N}} \\rightarrow \\mathbb {R}$ be a Borel function.", "If the law of $f=f(X_{0}, X_{1}, \\ldots )$ does not depend on $X_{0}$ and $f(X_{1}, X_{2}, \\ldots ) \\le f(X_{0}, X_{1}, \\ldots )$ a.s., then $f$ is a.s. constant.", "[Proof of Proposition REF ] We make use of the following explicit construction of the probability space.", "Let $\\Omega := (G \\times [r])^{\\mathbb {N}}$ , let $\\mathcal {F}$ be the product sigma-algebra and $\\theta : (\\omega _{n})_{n \\ge 0} \\mapsto (\\omega _{n+1})_{n\\ge 0}$ the shift operator.", "The coordinates of $\\omega _{i} \\in G \\times [r]$ are written here $\\omega _{i}=(g_{i},u_{i})$ .", "Consider the measure $\\mathbb {P}_{g,u}$ on $\\Omega $ defined on cylinders by $\\mathbb {P}_{g,u} \\left[\\omega _{0}=(g_{0},u_{0}), \\omega _{1}=(g_{1},u_{1}), \\ldots , \\omega _{n} = (g_{n},u_{n})\\right]= \\mathbb {1}_{g_{0}=g, u_{0} = u} \\ p_{g_{1}}(u,u_{1}) \\cdots p_{g_{n}}(u_{n-1},u_{n}).$ Then $X_{n} := (g_{0} g_{1} \\cdots g_{n}, u_{n})$ is a realization of the colored random walk defined by $p$ and $\\mathbb {P}_{g,u}$ is indeed the law of the random walk started at $(g,u)$ where $\\omega _{n}$ is the pair generator-color chosen by the walk at time $n$ .", "Thanks to the invariance of $\\pi $ with respect to $P$ , the measure $\\mathbb {P}_{\\pi p} := \\sum _{g,u} \\pi (u) p_{g}(u,v) \\mathbb {P}_{g,v}$ is invariant by the shift $\\theta $ .", "Let $\\mathbb {P}^{n}_{g,u}$ denote the law of $X_{n}$ under $\\mathbb {P}_{g,u}$ and consider the function $f_{n}$ on $\\Omega $ defined by $f_{n} = - \\log \\mathbb {P}_{X_{0}}^{n}\\left[X_{n}\\right].$ By the Markov property, $\\mathbb {P}_{X_{0}}^{m+n}[X_{m+n}] \\ge \\mathbb {P}_{X_{0}}^{m}[X_{m}] \\mathbb {P}_{X_{m}}^{n}[X_{n+m}]$ , so $f_{n}$ is sub-additive in the sense that $f_{n+m} \\le f_{m} + f_{n} \\circ \\theta ^{m}$ .", "Because the walk is finitely supported, $f_{1}$ is integrable so Kingman's sub-additive theorem ensures that $(f_{n}/n)$ converges both a.s. and in $L^{1}$ to a function $f \\in L^{1}(\\mathbb {P}_{\\pi p})$ , invariant under $\\theta $ and such that $\\int f \\: d\\mathbb {P}_{\\pi p} = \\lim _{n} \\int f_{n}/n \\: d\\mathbb {P}_{\\pi p} = \\inf _{n} \\int f_{n}/n \\: d\\mathbb {P}_{\\pi p}$ .", "The last step is to use the lemma to prove that $f$ is constant.", "By abuse of notation let us write $f=f(X_{0},X_{1}, \\ldots )$ .", "Since $\\mathbb {P}_{X_{0}}^{n}[X_{n}] \\ge \\mathbb {P}_{X_{0}}[X_{1}] \\mathbb {P}_{X_{1}}^{n-1}[X_{n}]$ , by taking logarithms and dividing by $n$ , we get that a.s. $f(X_{1}, X_{2}, \\ldots ) \\le f(X_{0}, X_{1}, \\ldots )$ .", "It then remains to see that the law of $f$ does not depend on $X_{0}$ .", "Thanks to quasi-transitivity, the law of $f$ can only depend on the starting color.", "On the other hand the matrix $P$ is irreducible so for all starting color $u$ there exists a.s. a random integer $m$ such that $X_{m}$ has color $v$ .", "Suppose $m$ is the first time color $v$ is visited.", "Then, by the strong Markov property, for all $n \\ge 0$ $\\mathbb {P}_{e,u}^{n+m}[X_{n+m}] &\\ge \\mathbb {P}_{e,u}^{m}[X_{m}] \\mathbb {P}_{g_{m},v}^{n}[X_{n+m}] \\\\&= \\mathbb {P}_{e,u}^{m}[X_{m}] \\mathbb {P}_{e,v}^{n}[\\tilde{X}_{n}],$ with $\\tilde{X}_{n} = \\omega _{m+1} \\cdots \\omega _{n}$ .", "Now take logarithms, divide by $n$ and take $n \\rightarrow \\infty $ .", "The left-hand side converges a.s. to $f((e,u), \\ldots )$ while the right-hand side converges to $f((e,v), \\ldots )$ .", "Thus a.s. we have $f((e,u), \\ldots ) \\ge f((e,v), \\ldots )$ .", "By symmetry the converse inequality also holds true so there is in fact equality, which proves $f$ is constant.", "Hence the entropy is well defined on almost all trajectory and does not depend on the starting color.", "The result is proved similarly for the drift." ], [ "Colored random walks on plain groups", "Closed formulas for the drift and entropy of nearest-neighbor random walks on free groups were obtained in [16], [5], [19].", "Similar approaches can be carried out for free products of finite groups, of monoids, of finite alphabets [9], [7], [22].", "The case of free groups and free products of finite groups is very similar in nature and they can be handled together as done in [15].", "As a preliminary step for closed formulas, in this section, we define and study the harmonic measure for colored random walks on plain groups.", "Let $G_{1}, \\ldots , G_{m}$ be finite groups and consider the plain group $G = \\mathbb {F}_{d}\\ast G_{1} \\ast \\cdots \\ast G_{m}$ , with the set of generators $S := \\bigcup _{i=1}^{d} \\lbrace a_{i}, a_{i}^{-1}\\rbrace \\bigcup \\left( \\bigsqcup _{j=1}^{m} S_{j} \\right),$ where for all $j = 1, \\ldots , m$ , $S_{j} := G_{j} \\setminus \\lbrace e\\rbrace $ .", "Recall the definition of the map $\\mathrm {Next}$ in (REF ).", "Every element $g \\in G$ writes uniquely as a word $g=g_{1} \\cdots g_{n}$ with $n = \\left\|g\\right\|$ and $g_{i} \\in S, g_{i+1} \\in \\mathrm {Next}(g_{i})$ for all $i$ .", "Such words will be called reduced.", "If for all $j = 1 , \\ldots , m$ , $G_j = \\mathbb {Z}/2 \\mathbb {Z}$ , then we say, with a slight abuse of vocabulary, that $G = \\mathbb {F}_{d}\\ast \\mathbb {Z}/ 2 \\mathbb {Z}\\ast \\cdots \\ast \\mathbb {Z}/ 2 \\mathbb {Z}$ is a free group (two generators are free if there are not inverse)." ], [ "The harmonic measure on the boundary", "Consider a colored random walk $(X_{n})_{n \\ge 0}$ on $G \\times [r]$ defined by a family $(p_g)$ of matrices.", "We assume that the walk is quasi-irreducible and nearest-neighbor.", "We assume furthermore that $G$ is not isomorphic to $\\mathbb {Z}$ or $\\mathbb {Z}/2\\mathbb {Z}\\ast \\mathbb {Z}/2\\mathbb {Z}$ .", "This implies the colored walk is transient, as will be proved in particular in Proposition REF below.", "Define the boundary $\\partial G$ as $\\partial G := \\left\\lbrace \\xi _{0}\\xi _{1} \\cdots \\xi _{n} \\cdots , \\forall i \\ge 0, \\xi _{i} \\in G, \\xi _{i+1} \\in \\mathrm {Next}(\\xi _{i}) \\right\\rbrace .$ The multiplication action by $G$ on itself can be extended to $\\partial G$ : given $g \\in G$ , $\\xi = \\xi _{0} \\xi _{1} \\cdots \\in \\partial G$ , define $g \\xi := \\left\\lbrace \\begin{array}{l l}g \\xi _{0} \\xi _{1} \\cdots & \\text{if $\\xi _{0} \\in \\mathrm {Next}(g)$} \\\\(g \\xi _{0}) \\xi _{1} \\cdots & \\text{if $g \\xi _{0} \\in S$} \\\\\\xi _{1} \\xi _{2} \\cdots & \\text{if $g = \\xi _{1}^{-1}$}\\end{array}\\right.", ".$ The boundary $\\partial G$ , which can be seen as a subset of $G^{\\mathbb {N}}$ , is equipped with the product topology and the corresponding $\\sigma $ -algebra.", "Given a measure $\\nu $ on $\\partial G$ , let $g \\cdot \\nu $ be the image measure of $\\nu $ under the mutiplication by $g$ , that is the measure defined by the fact that $\\int f(\\xi ) \\: d (g \\cdot \\nu )(\\xi ) := \\int f( g \\xi ) \\: d \\nu (\\xi ),$ for all bounded measurable function $f$ on $\\partial G$ .", "Definition 5 A colored measure is a family $\\nu = (\\nu _{u})_{u \\in [r]}$ of probability measures on $\\partial G$ indexed by colors.", "A colored measure $\\nu $ is stationary if for all $u \\in [r]$ , $\\nu _{u} = \\sum _{g \\in G, v \\in [r]} p_{g}(u,v) \\: g \\cdot \\nu _{v}.$ The following result extends Theorem 1.12 in Ledrappier [19] to colored random walks.", "The proof is exactly the same and is reproduced below.", "Proposition 9 There exists a random variable $X_{\\infty } \\in \\partial G$ such that $X_{n}$ converges a.s. to $X_{\\infty }$ .", "The law of $X_{\\infty }$ is called the harmonic measure and is the unique stationary colored measure on the boundary $\\partial G$ .", "It will be denoted $(p^{\\infty }_{u})_{u \\in [r]}$ where the index $u$ is to be interpreted as the starting color of $(X_{n})_{n \\ge 0}$ .", "Consider the topology on $\\partial G$ defined by the distance $d(\\xi ,\\xi ^{\\prime }) = e^{-\\left\|\\xi \\wedge \\xi ^{\\prime }\\right\|},$ with $\\left\|\\xi \\wedge \\xi ^{\\prime }\\right\|$ being the length of the prefix common to $\\xi $ and $\\xi ^{\\prime }$ .", "This topology makes $\\partial G$ a compact set so by the Lévy-Prokhorov theorem, the set $\\mathcal {P}(\\partial G)$ of probability measures on $\\partial G$ is a non-empty convex and compact Hausdorff set when embedded with the weak-* topology.", "Let $\\zeta : \\nu _{u} \\mapsto \\sum _{g \\in G, v \\in [r]} p_{g}(u,v) (g \\cdot \\nu _{v})$ .", "The multiplication in $\\partial G$ by an element of $G$ being continuous, $\\zeta $ is also a continuous map from $\\mathcal {P}(\\partial G)$ to itself.", "By the Schauder-Tychonoff theorem, $\\zeta $ admits a fixed point, which is exactly a stationary measure.", "Consider now a stationary measure $\\nu $ .", "Since $G$ is not isomorphic to $\\mathbb {Z}$ or $\\mathbb {Z}/2\\mathbb {Z}\\times \\mathbb {Z}/ 2\\mathbb {Z}$ the orbits under multiplication are dense in $\\partial G$ , that is for all $\\xi , \\xi ^{\\prime } \\in \\partial G$ and $\\epsilon \\in (0,1)$ there exists $g \\in G$ such that $d(g \\xi , \\xi ^{\\prime }) < \\epsilon $ .", "From this we deduce that the only closed subsets of $\\partial G$ that are invariant under multiplication by $G$ are the empty set and $\\partial G$ itself.", "In particular the support of $\\nu $ must be $\\partial G$ and $\\nu $ must be continuous: otherwise the set of points with the largest measure would be invariant and finite, so distinct from $\\partial G$ .", "Stationarity implies that the sequence $(X_{n} \\cdot \\nu )$ has the martingale property, in the sense that for every measure set $A$ the sequence $(X_{n} \\cdot \\nu (A))$ is a martingale.", "This martingale is positive and bounded, so $(X_{n} \\cdot \\nu )$ converges a.s. and in $L^{1}$ to a random measure $Z$ .", "This implies in particular that a.s.$\|X_{n}\| \\rightarrow \\infty $ .", "Indeed, one could otherwise extract a subsequence on which $X_{n} = g$ for some $g \\in G$ , with positive probability.", "Hence $Z = g \\cdot \\nu $ and because $\\nu $ is continuous and has support $\\partial G$ , this would imply $X_{n} \\rightarrow g$ and consequently $X_{n}$ would be stationary with positive probability.", "By the quasi-irreducibility assumption, the support of the colored walk is not reduced to $e$ , so Borel-Cantelli lemma shows this is not possible.", "Note that $\\left\|X_{n}\\right\| \\rightarrow \\infty $ shows in particular that $X_{n}$ is transient.", "We prove now that $Z$ must be a Dirac mass.", "Since $\|X_{n}\| \\rightarrow \\infty $ and $\\nu $ is continuous the measure under $X_{n} \\cdot \\nu $ of any cylinder has to tend to 0.", "Therefore the limit measure $Z(\\omega )$ is exactly a Dirac mass on a point $X_{\\infty }(\\omega )$ .", "This proves a.s. $X_{n} \\rightarrow X_{\\infty }$ .", "Markov property then shows that the law of $X^{\\infty }$ is indeed stationary.", "Finally by the martingale property and the $L^{1}$ convergence : $\\nu = \\mathbb {E}_{e}[X_{n} \\nu ] \\rightarrow \\mathbb {E}_{e}[ \\mathbb {1}_{X_{\\infty }} ] = \\mathbb {P}_{e}[ X_{\\infty } \\in \\cdot ].$ Hence $\\nu $ is also equal to the law $X_{\\infty }$ , which proves uniqueness of the stationary measure." ], [ "Markovian structure of the harmonic measure", "A measure on $\\partial G$ is uniquely determined by its mass on cylinders.", "Given a measure $\\nu $ , we will write $\\nu (\\xi _{1} \\cdots \\xi _{n})$ for the mass of the cylinder containing all infinite words which start with the prefix $\\xi _{1} \\cdots \\xi _{n}$ .", "In the colorless case, the tree structure of the group $G$ implies the harmonic measure is Markovian.", "It can be computed entirely from the solutions of a set of equations derived from the stationarity of the harmonic measure with the following interpretation.", "For all $k \\ge 0$ , let $X_{\\infty }^{(k)}$ be the restriction of $X_{\\infty }$ to the first $k$ -th letters.", "Thus for all $k \\ge 1$ the mass under the harmonic measure of cylinders of size $k$ is given by the law of $X_{\\infty }^{(k)}$ .", "On the other hand $(X_{\\infty }^{(k)})_{k \\ge 1}$ is a non-backtracking walk which one can interpret as the loop-erased random walk formed from $(X_{n})_{n \\ge 0}$ .", "In the standard colorless setting, the tree structure of the group makes the loop-erased random walk a Markov chain whose transition probabilities can be computed easily.", "In the colored setting, one can expect to have similar properties but the loop-erased random walk of the process is no longer being a Markov chain and this is not the right process to consider.", "Instead let $\\tau _{g} := \\inf \\lbrace n \\ge 0, X_{n} = ( g, \\cdot ) \\rbrace $ be the hitting time of $g$ by the random walk $X_n$ and $u_{k}$ the color at time $\\tau _{X_{\\infty }^{(k)}}$ .", "Thus $u_{k}$ is the first color seen by the random walk $X_{n}$ when it visits the same element as the $k$ -th step of $X_{\\infty }$ , but it is not necessary the color visited at the step $X_{n}$ where the $k$ -th letter of walk starts coinciding with $X_{\\infty }^{(k)}$ forever.", "Given $g \\in S$ , set $\\mu _{g}(u,v) := \\mathbb {P}_{(e,u)}[X_{\\infty }^{(1)} = g, X_{\\tau _{g}}=(g,v)].$ In words, $\\mu _{g}(u,v)$ is the probability that the random walk, starting from $(e,u)$ , visits $g$ , with color $v$ for the first time, and later escapes at infinity in direction $g$ .", "The process considered which is the equivalent of the loop erased random walk in the colored setting is $(X_{\\infty }^{(k)},u_{k})_{k \\ge 1}$ , which is a colored Markov chain with increment distribution $\\mu $ .", "As we shall check in Lemma REF , the family of matrices $\\mu = (\\mu _{g})_{g \\in S}$ is solution of the following set of matrix equations: $x_{g} = p_{g} \\Delta (x)_{g} + \\sum _{\\begin{array}{c}h,h^{\\prime } \\in S \\\\ hh^{\\prime } = g\\end{array}} p_{h} x_{h^{\\prime }} + \\sum _{h \\in \\mathrm {Next}(g)} p_{h^{-1}} x_{h} \\Delta (x)_{h} ^{-1} x_{g}, \\quad \\forall g \\in S,$ where $\\Delta (x)_{g}$ is the diagonal matrix with entries, for $u \\in [r]$ , $\\Delta (x)_{g}(u,u) := \\sum _{h \\in \\mathrm {Next}(g)} \\sum _{v \\in [r]} x_{h}(u,v).$ In the sequel, $\\Delta (\\mu )_{g}$ will be simply written $\\Delta _{g}$ .", "These equations generalize the so-called traffic equations of Mairesse [15].", "Beware that the products are to be understood as matrix products.", "In particular one has to be careful not to change the order of the different terms when considering products.", "Lemma 10 The family $\\mu = (\\mu _{g})_{g \\in S}$ is the unique family of matrices with non-negative entries which sum to a stochastic matrix and which is solution of Equation (REF ).", "Moreover, for all starting color $u \\in [r]$ and all cylinder $\\xi _{1} \\cdots \\ \\xi _{n}$ , $\\mathbb {P}_{u} \\left[X_{\\infty }^{(n)} = \\xi _{1} \\cdots \\xi _{n}, u_{n} = v\\right] = \\mathbb {1}_{u}^{\\top } \\mu _{\\xi _{1}} \\Delta _{\\xi _1} ^{-1} \\mu _{\\xi _{2}} \\cdots \\Delta _{\\xi _{n-1}} ^{-1} \\mu _{\\xi _{n}} \\mathbb {1}_{v},$ and $p^{\\infty }_{u}(\\xi _{1} \\cdots \\ \\xi _{n}) = \\mathbb {1}_{u}^{\\top } \\mu _{\\xi _{1}} \\Delta _{\\xi _1}^{-1} \\mu _{\\xi _{2}} \\cdots \\Delta _{\\xi _{n-1}} ^{-1} \\mu _{\\xi _{n}} \\mathbb {1}.$ For $n=1$ equation (REF ) is a consequence of the definition of the $\\mu _{g}$ .", "Now if $n \\ge 2$ , the probability that $X_{\\infty }^{(n)} = \\xi _{1} \\cdots \\xi _{n}, u_{n} =v$ conditional on $X_{\\infty }^{(n-1)} = \\xi _{1} \\cdots \\xi _{n-1}, u_{n-1} = u$ is just the probability the random walk escapes at infinity in direction $\\xi _{n}$ , conditionned on the fact it cannot backtrack (in the group), that is $\\frac{\\mu _{\\xi _{n}}(u,v)}{\\sum _{h \\in \\mathrm {Next}(\\xi _{n-1})} \\sum _{w \\in [r]} \\mu _{h}(u,w)} = \\Delta _{\\xi _{n-1}} ^{-1} \\mu _{\\xi _{n}} (u,v).$ An immediate induction yields (REF ).", "Summing over colors then gives (REF ).", "Let us now prove now that the matrices $\\mu _{g}$ are characterized by Equation (REF ).", "Consider any family $(\\nu _{g})_{g \\in S}$ of non-negative matrices, solutions of (REF ) and such that $\\sum _{g \\in S} \\nu _{g}$ is a stochastic matrix.", "Then for all $u \\in [r]$ define the measure $\\nu ^{\\infty }_{u}$ on $\\partial G$ by setting for each cylinder $\\xi _{1} \\cdots \\ \\xi _{n}$ $\\nu ^{\\infty }_{u} = \\mathbb {1}_{u} \\nu _{\\xi _{1}} \\Delta (\\nu )_{\\xi _1} ^{-1} \\nu _{\\xi _{2}} \\cdots \\Delta (\\nu )_{\\xi _{n-1}} ^{-1} \\nu _{\\xi _{n}} \\mathbb {1}$ The fact that $\\sum _{g \\in S} \\nu _{g}$ is a stochastic matrix proves that $\\nu ^{\\infty }_{u}$ is indeed a probability measure on $\\partial G$ .", "Let us prove that the hypotheses on $\\nu _{g}$ necessarily imply $\\nu ^{\\infty }_{u} = p^{\\infty }_{u}$ for all $u \\in [r]$ .", "By uniqueness of the harmonic measure, it suffices to show that $\\nu ^{\\infty }$ is stationary.", "From (REF )-(REF ), to prove stationarity, we need to show that for all cylinder $\\xi _{1} \\cdots \\ \\xi _{n}$ , for all $u \\in [r]$ , $\\nu ^{\\infty }_{u}(\\xi _{1} \\cdots \\ \\xi _{n}) & = & \\sum _{v \\in [r]} p_{\\xi _{1}}(u,v) \\nu ^{\\infty }_{v}(\\xi _{2} \\cdots \\xi _{n}) + \\sum _{\\begin{array}{c}g,h \\in S \\\\ gh = \\xi _{1}\\end{array}} p_{g}(u,v) \\nu ^{\\infty }_{v}(h \\xi _{2} \\cdots \\ \\xi _{n}) \\\\& & \\quad \\quad \\quad + \\sum _{\\begin{array}{c}g \\in S\\\\ g^{-1} \\xi _1 \\notin S \\cup \\lbrace e\\rbrace \\end{array}} p_{g}(u,v) \\nu ^{\\infty }_{v}(g^{-1} \\xi _{1} \\cdots \\ \\xi _{n}).$ We use the expression of $\\nu ^{\\infty }$ in this equation and notice there are matrices appearing in both sides of the equation.", "Furthermore, the case where there exist $g,h \\in S$ such that $gh = \\xi _{1}$ can only occur if $g,h$ and $\\xi _{1}$ belong to the same finite group $G_{i}$ , in which case $\\mathrm {Next}(h)= \\mathrm {Next}(\\xi _{1})$ .", "Combining these two observations, we find that a sufficient condition for stationarity is $\\nu _{\\xi _{1}} = p_{\\xi _{1}} \\Delta (\\nu )_{\\xi _{1}} + \\sum _{\\begin{array}{c}g,h \\in S \\\\ gh = \\xi _{1}\\end{array}} p_{g} \\nu _{h} + \\sum _{\\begin{array}{c}g \\in S\\\\ g^{-1} \\xi _1 \\notin S \\cup \\lbrace e\\rbrace \\end{array}} p_{g} \\nu _{g^{-1}} \\Delta (\\nu )_{g^{-1}} ^{-1} \\nu _{\\xi _{1}}.$ This is precisely ensured by the fact that the $\\nu _{g}$ are solutions of (REF ).", "Indeed, by construction, for $a \\in S$ , $\\mathrm {Next}(a) = \\lbrace g \\in S : a g \\notin S \\cup \\lbrace e\\rbrace \\rbrace $ and thus $\\lbrace g \\in S : g^{-1} \\xi _1 \\notin S \\cup \\lbrace e\\rbrace \\rbrace = \\mathrm {Next}(\\xi _1 ^{-1})$ .", "Thus we just proved that for all $g \\in S, u \\in [r]$ , $p^{\\infty }_{u}(g) = \\sum _{v \\in [r]} \\mu _{g}(u,v) = \\sum _{v \\in [r]} \\nu _{g}(u,v)$ .", "In particular, this yields that $\\Delta _{g} = \\Delta (\\nu )_{g}$ for all $g \\in S$ .", "Consequently the matrices $\\mu _{g}$ and $\\nu _{g}$ can be seen as the fixed points of a map $f: M=(M_{g})_{g \\in S} \\mapsto (f(M)_{g})_{g \\in S}$ from the set of non-negative matrices to itself, where $f(M)_{g} := A_{g} + \\sum _{\\begin{array}{c}h,h^{\\prime } \\in S \\\\ hh^{\\prime } = g\\end{array}} p_{h} M_{h^{\\prime }} + \\sum _{h \\in \\mathrm {Next}(g)} p_{h^{-1}} M_{h} B_{g} M_{g},$ for all $g \\in S$ , and $A_{g}, B_{g}$ are non-negative matrices that do not depend on $M$ .", "One can now argue as in the proof of [15] to deduce that $\\mu _g = \\nu _g$ .", "Let $\\le $ be the coordinate-wise ordering for matrices: $M \\le N$ if and only if $M(u,v) \\le N(u,v)$ for all $u,v \\in [r]$ .", "Consider the matrices $m_{g}$ defined by $m_{g}(u,v) := \\mu _{g}(u,v) \\wedge \\nu _{g}(u,v)$ As each map $f_{g} : M \\mapsto f(M)_{g}$ is obviously non-decreasing with respect to the coordinate-wise ordering, one must have $f(m)_{g} \\le f(\\nu )_{g} = \\nu _{g}$ and $f(m)_{g} \\le f(\\mu )_{g} = \\mu _{g}$ for all $g \\in S$ , so $f(m) \\le m$ .", "By Brouwer's fixed point theorem, $f$ must therefore admit another fixed point $m^{\\prime } \\le m$ .", "with the same properties as $\\mu $ and $\\nu $ .", "This $m^{\\prime }$ is thus another solution of (REF ) and so must satisfy $\\sum _{v \\in [r]} m^{\\prime }_{g}(u,v) = \\sum _{v \\in [r]} \\mu _{g}(u,v)$ for all $u \\in [r]$ , but since $m^{\\prime } \\le m \\le \\mu $ we deduce $\\mu = m^{\\prime } = \\nu $ ." ], [ "Hitting probabilities.", "Let us discuss here another way to compute the harmonic measure through hitting probabilities.", "It is less direct but we believe it makes the computation more natural in several examples such as free groups.", "Given $g \\in G, u \\in [r]$ , recall that $\\tau _{g}$ is the hitting time of $g$ .", "Let $\\tau _{(g,u)} := \\inf \\lbrace n : X_{n} = (g,u)\\rbrace $ be the hitting time of a pair $(g,u)$ , so that $\\tau _{g}:= \\min _{u} \\tau _{(g,u)}$ .", "For $g \\in S$ , set $q_{g}(u,v) := \\mathbb {P}[ \\tau _{g} < \\infty \\text{ and } \\tau _{g} = \\tau _{(g,v)} ].$ Conditioning on $X_{1}$ and applying Markov property, one obtains that the family of matrices $q = (q_{g})_{g \\in S}$ satisfy the matrix equation $x_{g} = p_{g} + \\sum _{\\begin{array}{c}h,h^{\\prime } \\in S \\\\ h h^{\\prime } = g\\end{array}} p_{h}x_{h^{\\prime }} + \\sum _{h \\in \\mathrm {Next}(g)} p_{h^{-1}} x_{h} x_{g} .$ In the case where $G= \\mathbb {F}_{d}\\ast \\mathbb {Z}/ 2 \\mathbb {Z}\\ast \\cdots \\ast \\mathbb {Z}/ 2 \\mathbb {Z}$ is a free group .", "This equation can again be found for classical nearest-neighbor random walks in [19], [15], [22] but here the equation is a matrix equation.", "Lemma 11 The family $q = (q_{g})_{g \\in S}$ is the unique solution to Equation (REF ) among family of matrices $(m_g)_{g\\in S}$ such that $\\mu _{g}(u,v) \\le m_{g}(u,v) \\le 1$ for all $u,v \\in [r]$ .", "One can argue as in the proof of lemma REF or the proof of [15]: solutions of Equation (REF ) can be seen as fixed points of quadratic maps that are non-increasing with respect to the coordinate-wise ordering, which cannot have two fixed points.", "The relation between the matrices $q_{g}$ and $\\mu _{g}$ is the following.", "Suppose the random walk is started at $u$ .", "In order to have $X_{\\infty }^{(1)}=g, u_{1} = v$ , the random walk has to visit $g$ with first color $v$ , which occurs with probability $q_{g}(u,v)$ .", "Arrived at $(g,v)$ , it has to escape at infinity in some direction $h \\in \\mathrm {Next}(g)$ .", "Hence by Markov property, $\\mu _{g}(u,v) = q_{g}(u,v) \\Delta _{g}(v,v), \\: \\quad \\forall g \\in S.$ This gives yet another way to write equations (REF ) and (REF ): $\\mathbb {P}_{u} \\left[X_{\\infty }^{(k)} = \\xi _{1} \\cdots \\xi _{k}, u_{k} = v\\right] & = &\\left( q_{\\xi _{1}} \\cdots q_{\\xi {k-1}} \\mu _{\\xi _{k}} \\right)(u,v), \\\\p_{u}^{\\infty }(\\xi _{1} \\cdots \\xi _{k}) & = & \\sum _{v} \\left( q_{\\xi _{1}} \\cdots q_{\\xi {k-1}} \\mu _{\\xi _{k}} \\right)(u,v) .$" ], [ "General case.", "We have seen in the previous section that the harmonic measure is fully described by the family of matrices $\\mu = (\\mu _g)_{g \\in S}$ on $[r]$ defined by (REF ).", "By Lemma REF , these matrices are uniquely characterized by the traffic equation (REF ).", "It is possible to evaluate numerically $\\mu $ by iterating the map defining the traffic equations.", "The hitting probabilities $q = (q_g)_{g \\in S}$ satisfy a simpler quadratic equation (REF ) than the traffic equation of $\\mu $ .", "As explained in the proof of Lemma REF , it is possible to evaluate numerically from above and from below $q$ by iterating the map $f$ defined in (REF ) such that $q = f(q)$ .", "Moreover, once the hitting probabilities are computed, it is easy to compute $\\mu $ .", "Indeed, using (REF ), the traffic equation reads (REF ) $\\mu _{g} = p_{g} \\Delta _{g} + \\sum _{\\begin{array}{c}h,h^{\\prime } \\in S \\\\ hh^{\\prime } = g\\end{array}} p_{h} \\mu _{h^{\\prime }} + \\sum _{h \\in \\mathrm {Next}(g)} p_{h^{-1}} q_{h} \\mu _{g}, \\quad \\forall g \\in S.$ Observe that $y_g := \\Delta _g \\mathbb {1}= \\mu _g \\mathbb {1}$ and $y_g \\in \\mathbb {R}^r$ is equal to the diagonal of the diagonal matrix $\\Delta _{g}$ .", "By construction, we have that $\\sum _g y_g = \\mathbb {1}$ .", "If we evaluate the matrix equation (REF ) on the vector $\\mathbb {1}$ , we obtain a linear equation for $y = (y_g)_{g\\in S}$ , seen as a vector with coordinates in $\\mathbb {R}^r$ , of the form $T y = 0$ where $T$ is matrix on $S \\times S$ with matrix-valued coefficients in $M_r(\\mathbb {R})$ .", "Finally, once, $\\Delta _g$ is known, Equation (REF ) becomes linear in $\\mu $ seen as a vector of matrices." ], [ "Case of a free group.", "In some special cases, one can find additional relations allowing to write the matrix $\\mu _{g}$ as an explicit function of the matrices $q_{g}$ .", "This case occurs for instance when $G= \\mathbb {F}_{d}\\ast \\mathbb {Z}/ 2 \\mathbb {Z}\\ast \\cdots \\ast \\mathbb {Z}/ 2 \\mathbb {Z}$ is a free group.", "Given $g \\in S$ , let $d_{g}$ be the diagonal matrix with diagonal entries $d_{g}(u,u):= \\sum _{w} q_{g}(u,w).$ In words, $d_{g}(u,u)$ is the probability to ever reach $g$ , starting from the pair $(e,u)$ .", "Proposition 12 When $G= \\mathbb {F}_{d}\\ast \\mathbb {Z}/ 2 \\mathbb {Z}\\ast \\cdots \\ast \\mathbb {Z}/ 2 \\mathbb {Z}$ is a free group, for all $g \\in S$ the matrix $(I-q_{g^{-1}} q_{g})$ is invertible and $\\mu _{g}(u,v) = q_{g}(u,v) \\sum _{w \\in [r]} (I-q_{g^{-1}} q_{g})^{-1} (I-d_{g^{-1}})(v,w) \\quad \\forall g \\in S, u,v \\in [r].$ In the case of a free group, for all $g \\in S, u \\in [r]$ , $\\mu _{g}(u,v) = q_{g}(u,v) \\left( 1-\\sum _{w \\in [r]} q_{g^{-1}}(v,w) + \\sum _{w,z \\in [r]} q_{g^{-1}}(v,w) \\mu _{g}(w,z) \\right).$ In particular $q_{g}(u,v) = 0$ implies $\\mu _{g}(u,v) = 0$ .", "Otherwise, rewrites this equation as $\\frac{\\mu _{g}(u,v)}{q_{g}(u,v)} = 1 - d_{g^{-1}}(v,v) + \\sum _{w,z \\in [r]} q_{g^{-1}}(v,w) q_{g}(w,z) \\frac{\\mu _{g}(w,z)}{q_{g}(w,z)}.$ The right-hand being independant of $u$ , so is the left-hand side.", "Consequently let $x_{v} := \\mu _{g}(u,v) / q_{g}(u,v)$ .", "Then one can again rewrite the above equation as the matrix linear equation $x = (I-d_{g^{-1}}) \\mathbb {1}+ q_{g^{-1}} q_{g} x$ Provided $I - q_{g^{-1}} q_{g}$ is invertible, solving this equation yields the desired expression for $\\mu _{g}(u,v)$ .", "Therefore we are left to prove that $I-q_{g^{-1}} q_{g}$ is invertible.", "This can be justified by the fact that $\\sum _{v \\in [r]} \\sum _{n \\ge 0} (q_{g^{-1}} q_{g})^{n}(u,v)$ is the average number of times the walk goes to $g^{-1}$ and comes back to $e$ when starting at color $u$ .", "By transience of the walk, this number must be finite.", "Hence the sum $\\sum _{n \\ge 0} (q_{g^{-1}} q_{g})^{n}$ is convergent and the inverse of this matrix is precisely $I - q_{g^{-1}} q_{g}$ .", "We now further study the matrix equation (REF ) satisfied by the hitting probabilities $(q_g)_{g \\in S}$ in the case where $G= \\mathbb {F}_{d}\\ast \\mathbb {Z}/ 2 \\mathbb {Z}\\ast \\cdots \\ast \\mathbb {Z}/ 2 \\mathbb {Z}$ is a free group.", "As already pointed, in this case, we have for all $g \\in S$ , $q_{g} = p_{g} + \\sum _{h \\ne g^{-1}} p_{h^{-1}} q_{h} q_{g}.$ In the colorless case, it is possible to reduce Equation (REF ) to a scalar equation, see for example [19].", "We extend this computation to the colored case.", "We define the matrix in $M_r(\\mathbb {R})$ , $z = I - \\sum _{g \\in S} p_{g^{-1}} q_{g}.$ We now express $q_g$ as a function of $(z,p_g,p_{g^{-1}})$ and find a closed equation satisfied by $z$ .", "For simplicity, we assume that for all $g \\in G$ , the matrix $p_g$ is invertible.", "If this is not the case, a similar argument holds but one should be careful with pseudo-inverses.", "We define the matrices in $M_{2r} (\\mathbb {R})$ : $P_g = \\begin{pmatrix}0 & p_g \\\\p_{g^{-1}} & 0\\end{pmatrix} \\; , \\quad Z = \\begin{pmatrix}z & 0 \\\\0 & z\\end{pmatrix} \\quad \\hbox{ and } \\quad Q_g = \\begin{pmatrix}0 & q_{g} \\\\q_{g^{-1}} & 0\\end{pmatrix}.$ From our assumption, $P_g$ is invertible and $P_{g}^{-1} = \\begin{pmatrix}0 & (p_{g^{-1}})^{-1} \\\\p_{g}^{-1} & 0\\end{pmatrix}.$ Applying (REF ) to $g$ and $g^{-1}$ , we find that, for all $g \\in S$ , $zq_g = p_g - p_g q_{g^{-1}} q_g$ and thus $Z Q_g = P_g - P_g Q_g^2.$ We set $Z_g = P_g^{-1} Z$ .", "The above equation rewrites $(Z_{g} + Q_{g}) Q_{g} = I.$ In particular $Z_{g} = Q_{g}^{-1} - Q_{g}$ so $Z_{g}$ and $Q_{g}$ commute.", "We may thus solve the quadratic equation $Q_g^2 + Z_{g} Q_{g} - I = 0$ with $Q_g$ as unknown as in the scalar case.", "Completing the square yields $(Q_g + Z_g /2)^2 = I + Z_{g}^{2} / 4.$ Therefore, for some proper choice of the matrix square root function, we get $Q_g = \\frac{1}{2} \\left( \\@root * \\of { 4 I + (P_g^{-1} Z)^2 } - P_g^{-1} Z \\right),$ where $\\@root * \\of {\\cdot }$ is a notation to stress that the choice of the square root is unknown.", "First, as $Z_{g}$ and $Q_{g}$ commute, the eigenvalues completely determine the square root.", "Also, since $Q_g$ is a block antidiagonal matrix, for every of its eigenvalue $\\lambda $ , $- \\lambda $ must also be an eigenvalue with the same multiplicity (algebraic and geometric).", "In particular, we are left with at most $2^r$ choices for the square root to pick in (REF ) (a choice of sign for each eigenvalue pair $(\\lambda ,-\\lambda )$ ).", "There is one useful property to further determine the square root.", "Consider $R_g = Q_g P_g^{-1} = \\begin{pmatrix}q_g p_{g}^{-1} & 0 \\\\0 & q_{g^{-1}} p_{g^{-1}}^{-1}\\end{pmatrix}.$ We observe from (REF ) that $q_g p_g^{-1} = (I - \\sum _{h \\ne g^{-1}} p_{h^{-1}} q_h)^{-1}$ .", "The matrix $\\sum _{h \\ne g^{-1}} p_{h^{-1}} q_h$ is sub-stochastic: it has non-negative entries and the sum over each row is less or equal than one (from a starting color, it corresponds to the probability that the colored walk killed when visiting $g$ comes back to $e$ after some time).", "In particular, the matrix $\\sum _{h \\ne g^{-1}} p_{h^{-1}} q_h$ has spectral radius less than one.", "Thus all eigenvalues of $q_g p_g^{-1}$ and $R_g$ have positive real part.", "Using (REF ), the same argument shows that $z$ and $Z$ have all their eigenvalues with positive real parts.", "Finally, from Equation (REF ), we have $Z = I - \\sum _g P_g Q_g = I - \\sum _g P_g R_g P_g$ .", "We deduce that $Z = I - \\frac{1}{2} \\sum _{g \\in S} \\left( P_g \\@root * \\of { 4 I + ( P_g^{-1} Z ) ^2 } - Z\\right).$ Up to this issue of square root, we thus have found a fixed point equation satisfied by $z$ (in Equation (REF )) and expressed $q_g$ as a function of $(z,p_{g},p_{g^{-1}})$ (in Equation (REF )).", "If $r=1$ , we can retrieve a known formula for colorless random walks.", "We note that Equation (REF ) should be compared to Proposition 3.1 in Lehner [20] where a related formula is derived in a self-adjoint case (it can be checked that $2z$ is the inverse of the diagonal term of the Green function $(I - \\tilde{\\mathcal {P}})^{-1} (e,e)$ where $(I - \\tilde{\\mathcal {P}})^{-1} $ is seen as an infinite matrix indexed by $G \\times G$ with coefficients in $M_r(\\mathbb {C})$ )." ], [ "Computation of the drift: proof of Theorem ", "A proof of Theorem REF in the colorless case is given in [19] for the free group.", "The proof can be adapted for more general free products, as mentionned in [15], [22].", "It is also presented in a slightly different way in [9].", "This is the latter proof that we here extend in the colored case.", "[Proof of Theorem REF ] We aim to compute the limit of $\\mathbb {E}\\left\|X_{n}\\right\|/n$ .", "This can be seen the Cesàro limit of the sequence $( \\mathbb {E}\\left\|X_{n+1}\\right\| - \\mathbb {E}\\left\|X_{n}\\right\|)_{n \\ge 0}$ .", "Here we deliberately omitted to precise the starting color, as the final result does not depend on it by Proposition REF .", "In particular one can choose the starting color to be distributed according to the measure $\\pi $ .", "By Markov property and the invariance of $\\pi $ , for all $n \\ge 0$ $\\mathbb {E}_{\\pi } \\left\|X_{n+1}\\right\| - \\mathbb {E}_{\\pi } \\left\|X_{n}\\right\| &= \\sum _{u,v \\in [r]} \\pi (u) p_{g}(u,v) \\mathbb {E}_{v} \\left\|gX_{n}\\right\| - \\sum _{u} \\pi (u) \\mathbb {E}_{u} \\left\|X_{n}\\right\| \\\\&= \\sum _{u,v \\in [r]} \\pi (u) p_{g}(u,v) \\left( \\mathbb {E}_{v} \\left\|g X_{n}\\right\| - \\mathbb {E}_{v} \\left\|X_{n}\\right\| \\right) \\\\&= \\sum _{u,v \\in [r]} \\pi (u) p_{g}(u,v) \\left( \\mathbb {E}_{v} \\left[ \\left\|gX_{n}\\right\| - \\left\|X_{n}\\right\|\\right] \\right).$ We fix a color $v$ .", "The random variable $Z_{n} := \\left\|gX_{n}\\right\| - \\left\|X_{n}\\right\|$ takes values in $\\lbrace -1,0,1\\rbrace $ .", "On the other hand, $X_{n}$ converges to the random variable $X_{\\infty }$ as $n$ goes to infinity.", "Consequently $g X_{n}$ converges to $g X_{\\infty } \\in \\partial G$ and $Z_{n}$ converges to a random variable $Z_{\\infty } \\in \\lbrace -1,0,1\\rbrace $ .", "It is now easy to see what happens: $Z_{\\infty } = \\left\\lbrace \\begin{array}{l l}-1 & \\text{if $X_{\\infty }^{(1)} = g^{-1}$} \\\\1 & \\text{if $X_{\\infty }^{(1)} \\in \\mathrm {Next}(g)$} \\\\0 & \\text{otherwise}\\end{array}\\right.", ".$ Hence, by the dominated convergence theorem, we find $\\mathbb {E}_{v} \\left[Z_{n}\\right] \\rightarrow \\mathbb {E}_{v} \\left[Z_{\\infty }\\right] &= \\int Z_{\\infty } \\: d p^{\\infty }_{v} \\\\&= - \\sum _{w \\in [r]} \\mu _{g^{-1}}(v,w) + \\sum _{h \\in \\mathrm {Next}(g)} \\sum _{w \\in [r]} \\mu _{h}(v,w),$ as requested." ], [ "Entropy: : proofs of Theorem ", "The following computation is originally due to Kaimanovich and Vershik [16] and is here adapted to the matrix case.", "Given a discrete random variable $X$ and a $\\sigma $ -algebra $\\mathcal {F}$ , the conditional entropy of $X$ with respect to $\\mathcal {F}$ is defined as $H(X \\: \| \\: \\mathcal {F}) = \\mathbb {E}\\left[ - \\sum _{x} \\mathbb {P}(X=x \\: \| \\: \\mathcal {F}) \\log \\mathbb {P}(X=x \\: \| \\: \\mathcal {F})\\right].$ Let $\\mathcal {T}:= \\bigcap _{n \\ge 0} \\sigma \\lbrace X_{k}, k \\ge n \\rbrace $ be the tail $\\sigma $ -algebra.", "As one can check easily, the tail $\\sigma $ -algebra is also the $\\sigma $ -algebra generated by $X_{\\infty }$ .", "Therefore a function is $\\mathcal {T}$ -measurable if and only if it writes as a function of $X_{\\infty }$ .", "Proposition 13 The entropy of a colored random walk satisfies $h = H_{\\pi }(X_{1}) - H_{\\pi }(X_{1} \\: \| \\: \\mathcal {T}).$ By the Markov property for all $n \\ge 1$ $H_{\\pi }(X_{1} \\: \| \\: X_{n}, X_{n+1}, \\ldots ) = H_{\\pi }(X_{1} \\: \| \\: X_{n})$ When $n \\rightarrow \\infty $ the left-hand side converges to $H_{\\pi }(X_{1} \| \\mathcal {T})$ .", "On the other hand, $H_{\\pi }(X_{1} \\: \| \\: X_{n}) &= H_{\\pi }(X_{1}, X_{n}) - H_{\\pi }(X_{n}) \\\\&= H_{\\pi }(X_{1}) + H_{\\pi }(X_{n} \\: \| \\: X_{1}) - H_{\\pi }(X_{n}) \\\\&= H_{\\pi }(X_{1}) - \\left(H_{\\pi }(X_{n}) - H_{\\pi }(X_{n-1}) \\right).$ In the last line we have used equation (REF ).", "Since $H_{\\pi }(X_{n})/n$ converges to $ h$ as $n$ goes to infinity, one must also have that $H_{\\pi }(X_{n}) - H_{\\pi }(X_{n-1})$ converges to $h$ , hence the result.", "Given a $\\sigma $ -algebra $\\mathcal {F}$ and $g,h \\in G, u,v \\in [r]$ , let $\\mathbb {P}\\left[(g,u),(h,v) \\: \| \\: \\mathcal {F}\\right]$ be the conditional transition probability with respect to $\\mathcal {F}$ , that is $\\mathbb {P}\\left[(g,u),(h,v) \\: \| \\: \\mathcal {F}\\right] := \\mathbb {P}_{g,v} \\left[ X_{1} = (h,v) \\: \| \\: \\mathcal {F}\\right] .$ The stationarity of the harmonic measure $p^{\\infty }_{u} = \\sum _{\\begin{array}{c} g \\in G \\\\ v \\in [r]\\end{array}} p_{g}(u,v) \\: g \\cdot p^{\\infty }_{v}$ implies in particular that the measures $g \\cdot p^{\\infty }_{v}$ are absolutely continous with respect to $p^{\\infty }_{u}$ if $p_g(u,v) >0$ .", "Proposition 14 The conditional probabilities with respect to $\\mathcal {T}$ are given by: $\\mathbb {P}\\left[(g,u), (h,v) \\: \| \\: \\mathcal {T}\\right] = p_{g^{-1} h}(u,v) \\frac{d \\, h p^{\\infty }_{v}}{d \\, g p^{\\infty }_{u}} \\quad \\text{a.s..}$ [Sketch of proof] The result is true when replacing $\\mathcal {T}$ with the invariant $\\sigma $ -algebra $\\mathcal {I}$ which is the $\\sigma $ -algebra generated by all bounded invariant functions, that is, all bounded functions $f : (G \\times [r])^{\\mathbb {N}} \\rightarrow \\mathbb {R}$ such that $f(x_{0}, x_{1}, \\ldots ) = f(x_{1,}, x_{2}, \\ldots )$ .", "This is the content of [21].", "Generally, the invariant and tail $\\sigma $ -algebras do not coincide but they do for transitive Markov chains, up to negligible sets.", "The proof goes basically like this: there is an equivalence between the set of bounded invariant functions and the set of bounded harmonic functions.", "There is a similar equivalence between bounded tail functions and bounded space-time harmonic functions.", "Using a zero-two law, one then shows that space-time harmonic functions do not actually depend on the time parameter and hence are classical harmonic functions.", "The equivalences between functions thus prove the equivalence of the $\\sigma $ -algebras.", "See [21] for details.", "For colored Markov chains, the arguments extend naturally, using quasi-irreducibility and quasi-transitivity.", "Therefore the above formula is correct with $\\mathcal {T}$ instead of $\\mathcal {I}$ .", "[Proof of Theorem REF ] By Proposition REF and Equation (REF ), $h &= H_{\\pi }(X_{1}) - H_{\\pi }(X_{1} \\: \| \\: \\mathcal {T}) = H_{\\pi }(X_{1}) + \\sum _{u} \\pi (u) \\int \\log \\mathbb {P}_{(e,u)}[X_{1} \\: \| \\: \\mathcal {T}](X_{\\infty }) \\: d\\mathbb {P}_{u} \\\\&= H_{\\pi }(X_{1}) + \\sum _{u} \\pi (u) \\sum _{\\begin{array}{c}g \\in S \\\\ v \\in [r]\\end{array}} \\int p_{g}(u,v) \\log \\mathbb {P}_{(e,u)}[X_{1}=(g,v) \\: \| \\: \\mathcal {T}](g \\xi ) \\: dp^{\\infty }_{v}(\\xi ) \\\\&= H_{\\pi }(X_{1}) + \\sum _{u \\in [r]} \\pi (u) \\sum _{\\begin{array}{c}g \\in S \\\\ v \\in [r]\\end{array}} p_{g}(u, v) \\int \\log \\mathbb {P}_{(g^{-1},u)}[X_{1}=(e,v) \\: \| \\: \\mathcal {T}](\\xi ) \\: dp^{\\infty }_{v}(\\xi )$ Now, using Proposition REF , we find $h &=H_{\\pi }(X_{1}) + \\sum _{u} \\pi (u) \\sum _{g \\in \\mathbb {F}_{d}, v \\in [r]} p_{g}(u, v) \\int \\log \\left(p_{g}(u,v) \\frac{d p^{\\infty }_{v}}{d g^{-1} p^{\\infty }_{u}}(\\xi ) \\right) dp^{\\infty }_{v}(\\xi ) \\\\&= - \\sum _{u} \\pi (u) \\sum _{g \\in \\mathbb {F}_{d}, v \\in [r]} p_{g}(u, v) \\int \\log \\left(\\frac{d g^{-1} p^{\\infty }_{u}}{d p^{\\infty }_{v}}(\\xi ) \\right) dp^{\\infty }_{v}(\\xi ).$ It concludes the proof.", "Equation (REF ) is the matrix version of the known formula for colorless random walks [16], [19].", "It shows that the computation of the entropy ultimately comes down to the computation of some Radon-Nikodym derivatives.", "In the colorless case, this computation goes basically as follows.", "Consider a cylinder $\\xi _{1} \\cdots \\xi _{n}$ .", "Compute $g \\cdot p^{\\infty }(\\xi _{1} \\cdots \\xi _{n})$ distinguishing the different cases occurring: if $g = \\xi _{1}$ then $g \\cdot p^{\\infty }(\\xi _{1} \\cdots \\xi _{n}) = p^{\\infty }(\\xi _{2} \\cdots \\xi _{n})$ if $\\xi _{1} \\in G_{i}, g \\ne \\xi _{1}$ and there exists $h \\in G_{i}$ such that $gh = \\xi _{1}$ , then $g \\cdot p^{\\infty }(\\xi _{1} \\cdots \\xi _{n}) = p^{\\infty }(h \\xi _{2} \\cdots \\xi _{n})$ otherwise, $g \\cdot p^{\\infty }(\\xi _{1} \\cdots \\xi _{n}) = p^{\\infty }(g^{-1} \\xi _{1} \\cdots \\xi _{n}).$ As one can check, the last case occurs if and only if $\\xi _{1} \\in \\mathrm {Next}(g^{-1})$ .", "Expand now the expressions above into products using from () that $p^{\\infty }(\\xi _{1} \\cdots \\xi _{n}) = q_{\\xi _{1}} \\cdots q_{\\xi _{n-1}} \\mu _{\\xi _{n}}$ to get $\\frac{g \\cdot p^{\\infty }(\\xi _{1} \\cdots \\xi _{n})}{p^{\\infty }(\\xi _{1} \\cdots \\xi _{n})} =\\left\\lbrace \\begin{array}{l l}1/q_{g} & \\text{if $\\xi _{1} = g$,} \\\\q_{g^{-1}} & \\text{if $\\xi _{1} \\in \\mathrm {Next}(g^{-1})$,} \\\\q_{h}/q_{\\xi _{1}} = q_{g^{-1} \\xi _{1}}/q_{\\xi _{1}} & \\text{if $h \\in S$ and $gh = \\xi _{1}$.", "}\\end{array}\\right.$ As will be proved after, the left-hand side converges to the Radon-Nikodym derivative as $n \\rightarrow \\infty $ .", "One can then easily express the integral (REF ) in terms of the scalars $p_{g}, q_{g}$ and $\\mu _{g}$ and obtain the formulas given in [19], [22].", "For colored random walks, the same computations can be made with matrices except that the cancellation between products no longer takes place.", "Instead, one has to deal with an infinite product of random matrices.", "For example, as one might guess from (REF )-(), it is true that $\\frac{d g p^{\\infty }_{v}}{d p^{\\infty }_{u}}(\\xi ) = \\lim _{n \\rightarrow \\infty } \\frac{ \\mathbb {1}_{v}^{\\top } \\ q_{g^{-1}} q_{\\xi {1}} \\cdots q_{\\xi {n-1}} \\mu _{\\xi {n}} \\mathbb {1}}{\\mathbb {1}_{u}^{\\top } \\ q_{\\xi {1}} \\cdots q_{\\xi {n-1}} \\mu _{\\xi {n}} \\mathbb {1}} \\quad \\text{a.s.},$ if $\\xi _{1} \\in \\mathrm {Next}(g^{-1}) $ .", "In the subsection, we prove that this kind of convergence holds and that the limit can be expressed thanks to random probability measure on $[r]$ which is uniquely characterized by an invariance property." ], [ "Convergence in direction for inhomogeneous products of matrices ", "In this final subsection, we prove Theorem REF .", "The computation of the Radon-Nikodym derivatives in the integral formula of the entropy requires to investigate infinite products of random, non-negative matrices (that is, with non-negative entries).", "We use first use results for non-negative, deterministic matrices [25] to justify limits of infinite matrix products.", "This yields formulas like (REF ) and a first expression for the entropy (REF ).", "Then we use results for products of random matrices [3], [4] to describe the law of the limits obtained, which yields eventually Proposition REF ." ], [ "Convergence of the Radon-Nikodym derivatives.", "Instead of the general results in [25], it turns out that we can directly use a result from Lalley who already had to inquire about infinite matrix products in his study of finite range random walks [17].", "It does not seem however that his methods can be interpreted as an application of the linearization trick.", "Let $X$ be a finite set and $Y$ a subset of $X \\times X$ such that for all $x \\in X$ , the set $\\lbrace y \\in X, (x,y) \\in T \\rbrace $ is non empty.", "Let $\\Sigma := X^{\\mathbb {Z}}$ be the space of doubly infinite sequences $(\\xi _{n})_{n \\in \\mathbb {Z}}$ with values in $X$ , such that $(\\xi _{n},\\xi _{n+1}) \\in Y$ for all $n \\in \\mathbb {Z}$ .", "Let $\\sigma : (\\xi _{n})_{n \\in \\mathbb {Z}} \\mapsto (\\xi _{n+1})_{n \\in \\mathbb {Z}}$ denote the standard shift on $\\Sigma $ and given a function $f : \\Sigma \\rightarrow \\mathbb {C}$ and $n \\in \\mathbb {N}$ , write $S_{n}f := f + f \\circ \\sigma + \\ldots f \\circ \\sigma ^{n-1}.$ Recall the definition of $\\mathcal {P}^+$ in (REF ).", "Proposition 15 ([17]) Let $(M_{x})_{x \\in X}$ be a family of $r \\times r$ matrices with non-negative entries.", "Assume there exists integers $m \\ge 0, k \\ge 1$ and a function $B : X^{k} \\rightarrow 2^{[r]} \\setminus \\lbrace \\emptyset \\rbrace $ (the set of non-empty subsets of $[r]$ ), such that for every $n \\ge m$ and every family $x_{1}, \\ldots , x_{n+k} \\in X$ with $(x_{i},x_{i+1}) \\in Y$ for all $i$ , $(M_{x_{1}} M_{x_{2}} \\cdots \\ M_{x_{n+k}})_{u,v} > 0 \\Leftrightarrow v \\in B(x_{n+1}, \\ldots , x_{n+k}) \\qquad \\forall u,v \\in [r]$ Then there exist constants $C > 0$ and $0 < \\alpha < 1$ , maps $\\varphi , \\gamma : \\Sigma \\rightarrow \\mathbb {R}$ and $V,W : \\Sigma \\rightarrow \\mathcal {P}^{+}$ such that for all $\\xi \\in \\Sigma $ , $\\left\\Vert e^{- S_{n} \\varphi (\\xi )} M_{\\xi _{1}} M_{\\xi _{2}} \\cdots \\ M_{\\xi _{n}} - \\gamma (\\sigma _{n} \\xi ) V(\\xi ) W(\\sigma ^{n} \\xi )^{\\top } \\right\\Vert \\le C \\alpha ^{n} ,$ where $V=V(\\xi _{1}, \\xi _{2}, \\ldots )$ depends only on the “forward coordinates“, while $W = W(\\xi _{0}, \\xi _{-1}, \\ldots )$ depends only on the ”backward coordinates”.", "Furthermore, $M_{\\xi _{1}} V(\\sigma \\xi ) = e^{\\varphi (\\xi )} V(\\xi ).$ Proposition REF states shows under condition (REF ), the product $M_{\\xi _{1}} M_{\\xi _{2}} \\cdots \\ M_{\\xi _{n}}$ tends up to renormalization factor to a rank one matrix whose range is spanned by the vector $V(\\xi )$ .", "This result is well known for powers of a matrix with positive entries, in which case the vector $V$ is nothing but the Perron-Frobenius eigenvector.", "Proposition REF is thus a generalization of the Perron Frobenius theory to inhomogeneous products of non-negative matrices with possibly zero columns (condition (REF )).", "We refer to [17] for additional properties of the functions $\\varphi , \\gamma $ with the vectors $V,W$ which will not be needed here.", "To apply Proposition REF in context of linearized random walks, take $X = S$ and $(x,y) \\in Y$ if and only if $y \\in \\mathrm {Next}(g)$ .", "Corollary 16 Suppose the matrices $(q_{g})_{g \\in S}$ satisfy the hypothesis of Proposition REF and let $V: \\partial G \\rightarrow \\mathcal {P}^{+}$ be the corresponding map.", "Then for all $u,v \\in [r]$ , $\\xi \\in \\partial G$ , $\\frac{d g p^{\\infty }_{v}}{d p^{\\infty }_{u}}( \\xi ) = \\left\\lbrace \\begin{array}{l l}\\frac{\\left\\langle \\mathbb {1}_{v} \\: , \\: V(\\sigma \\xi ) \\right\\rangle }{\\left\\langle q_{g}(u,\\cdot ) \\: , \\: V(\\sigma \\xi ) \\right\\rangle } & \\text{if $\\xi _{1} = g$,} \\vspace{4.0pt} \\\\\\frac{\\langle q_{g^{-1}}(v,\\cdot ) , V(\\xi ) \\rangle }{\\left\\langle \\mathbb {1}_{u} \\: , \\: V(\\xi ) \\right\\rangle } & \\text{if $\\xi _{1} \\in \\mathrm {Next}(g^{-1})$,} \\vspace{4.0pt}\\\\\\frac{\\left\\langle q_{h}(v, \\cdot ) \\: , \\: V(\\sigma \\xi ) \\right\\rangle }{\\left\\langle q_{gh}(u, \\cdot ) \\: , \\: V(\\sigma \\xi ) \\right\\rangle } & \\text{if $h \\in S$ and $gh = \\xi _{1}$.", "}\\end{array}\\right.$ From (REF )-(), for any cylinder $\\xi _{1} \\cdots \\xi _{n}$ $g \\cdot p^{\\infty }_{v}(\\xi _{1} \\cdots \\xi _{n}) = \\left\\lbrace \\begin{array}{l l}\\mathbb {1}_{v}^{\\intercal } q_{\\xi _{2}} \\cdots q_{\\xi _{n-1}} \\mu _{\\xi _{n}} \\mathbb {1}& \\text{if $\\xi _{1} = g$,} \\\\\\mathbb {1}_{v}^{\\intercal } q_{g^{-1}} q_{\\xi _{1}} \\cdots q_{\\xi _{n-1}} \\mu _{\\xi _{n}} \\mathbb {1}& \\text{if $\\xi _{1} \\in \\mathrm {Next}(g^{-1})$,} \\\\\\mathbb {1}_{v}^{\\intercal } q_{h} q_{\\xi _{2}} \\cdots \\ \\mu _{\\xi _{n}} \\mathbb {1}& \\text{if $h \\in S$ and $gh =\\xi _{1}$.", "}\\end{array}\\right.$ Let $\\xi \\in \\partial G$ .", "Combining the previous computation with Proposition REF , there exist $\\lambda _{n} >0$ , uniformly lower bounded, and $\\alpha < 1$ such that if $\\xi _{1} = g$ , $\\frac{g \\cdot p^{\\infty }_{v}(\\xi _{1} \\cdots \\xi _{n})}{p^{\\infty }_{u}(\\xi _{1} \\cdots \\xi _{n})} &= \\frac{\\mathbb {1}_{v}^{\\top } \\left( q_{\\xi _{2}} \\cdots q_{\\xi _{n-1}} \\right) \\left( \\mu _{\\xi _{n}} \\mathbb {1}\\right)}{q_{g}(u, \\cdot )^{\\top } \\left( q_{\\xi _{2}} \\cdots q_{\\xi _{n-1}} \\right) \\left( \\mu _{\\xi _{n}} \\mathbb {1}\\right)} \\\\&= \\frac{\\mathbb {1}_{v}^{\\top } V( \\sigma \\xi ) \\lambda _{n} + O(\\alpha ^{n})}{q_{g}(u, \\cdot )^{\\top } V(\\sigma \\xi ) \\lambda _{n} + O(\\alpha ^{n})} \\\\&\\xrightarrow[n \\rightarrow \\infty ]{} \\frac{ \\left\\langle \\mathbb {1}_{v} \\: , \\: V(\\sigma \\xi ) \\right\\rangle }{\\left\\langle q_{g}(u, \\cdot ) \\: , \\: V(\\sigma \\xi ) \\right\\rangle }.$ The other cases are treated similarly.", "On the other hand $\\frac{g \\cdot p^{\\infty }_{v}(\\xi _{1} \\cdots \\xi _{n})}{p^{\\infty }_{u}(\\xi _{1} \\cdots \\xi _{n})} = \\int _{\\partial G} \\frac{d g p^{\\infty }_{v}}{p^{\\infty }_{u}} \\frac{\\mathbb {1}_{\\xi _{1} \\cdots \\xi _{n}}}{p^{\\infty }_{u}(\\xi _{1} \\cdots \\xi _{n})} \\ d p^{\\infty }_{u}$ but as $n \\rightarrow \\infty $ the measure $\\frac{\\mathbb {1}_{\\xi _{1} \\cdots \\xi _{n}}}{p^{\\infty }_{u}(\\xi _{1} \\cdots \\xi _{n})} \\ d p^{\\infty }_{u}$ converges to the Dirac mass at $\\xi $ : for all integer $k$ and $\\epsilon > 0$ , the mass of all $k$ -cylinders except the cylinder $\\xi _{1} \\cdots \\ \\xi _{k}$ is bounded by $\\epsilon $ for $n$ large enough.", "Thus the above limits give indeed the Radon-Nikodym derivative.", "The next proposition asserts that for the colored random walks that we are mainly interested in, Corollary REF applies.", "Proposition 17 Suppose the colored random walk $(X_{n})_{n \\ge 0}$ is a linearized random walk obtained from a finite range random walk $(Y_{n})_{n \\ge 0}$ via one of the procedures presented in Section .", "If $(Y_{n})_{n \\ge 0}$ is irreducible, then the matrices $(q_{g})_{g \\in S}$ , satisfy the hypothesis of Proposition REF .", "Beware that the notations are the contrary to those of Section : here $(X_{n})_{n \\ge 0}$ is the colored walk linearizing the finite range walk $(Y_{n})_{n \\ge 0}$ .", "Let us first review briefly a relevant argument in [17].", "Lalley proves that the hypothesis of Proposition REF is satisfied by matrices $H_{g}$ which are related to the walk $(Y_{n})_{n \\ge 0}$ .", "These matrices are defined as follows.", "Letting $K$ denote the support of the finite range random walk $(Y_{n})_{n \\ge 0}$ , define $L := \\max \\lbrace \\left\|g\\right\|, g \\in K \\rbrace $ and $\\mathcal {B}:= \\lbrace g \\in G, \\left\|g\\right\| \\le L \\rbrace $ .", "The matrices $H_{g}$ will be taken in $\\mathbb {R}^{\\mathcal {B}\\times \\mathcal {B}}$ .", "Given $g \\in G$ , define $T_{g} := \\inf \\lbrace n \\ge 0, Y_{n} \\in g \\mathcal {B}\\rbrace ,$ that is $T_{g}$ is the first time the representative of $Y_{n}$ writes as $g$ times an element in $\\mathcal {B}$ .", "When $G$ is a plain group, representatives are unique so $T_{g}$ is well defined.", "For all $a,b \\in \\mathcal {B}$ , the $(a,b)$ -entry of $H_{g}$ is then defined as $H_{g}(a,b) := \\mathbb {P}\\left[T_{g} < \\infty , Y_{T_{g}} = gb \\: \| \\: Y_{0} = a\\right]$ In [17], Lalley only considers the case of a free group.", "The following properties can however be easily extended to plain groups.", "The matrices $H_{g}$ , $g \\in S$ have the property that if $g= g_{1} \\cdots \\ g_{n}$ , then $H_{g} = H_{g_{1}} \\cdots H_{g_{n}}$ , see [17].", "Thus proving that the family $(H_{g})_{g \\in S}$ satisfies condition (REF ) is equivalent to prove that there exists $k \\ge 1$ and $B : S^{k} \\rightarrow 2^{\\mathcal {B}}\\setminus \\lbrace \\emptyset \\rbrace $ such that for all $g \\in G$ with length $\\left\|g\\right\|$ large enough, $H_{g}(a,b) > 0 \\Leftrightarrow b \\in B$ where $B$ is a subset of $\\mathcal {B}$ that depends only on the last $k$ letters of $g$ .", "This is exactly what is done in [17], under the condition that $(Y_{n})_{n \\ge 0}$ is irreducible.", "We now come back to the proof of Proposition REF .", "As for the matrix $H_{g}$ , if $g$ writes as a reduced word $g = g_{1} \\cdots \\ g_{n}$ , then we set $q_{g} = q_{g_{1}} \\cdots \\ q_{g_{n}}$ , so one can argue as for the $H_{g}$ , namely prove that $q_{g}(u,v) > 0$ for $\\left\|g\\right\|$ large enough if and only if $v$ is in some subset of $[r]$ that depends only on the last letters of $g$ .", "The two constructions of the linearized random walks considered are made so that if $Y_{0} = e$ and $X_{0} = (e,1)$ , one can couple $(X_{n})_{n \\ge 0}$ and $(Y_{n})_{n \\ge 0}$ to have $Y_{n} = X_{\\tau _{n}}$ where $\\tau _{n}$ always corresponds to a return time at color 1.", "Moreover, colors are associated to elements of $K$ : for all $v \\in [r], v \\ne 1$ , the colored walk $X_{n}$ , when at color 1, needs to be multiplied by specific elements of $K$ in order to pass through the color $v$ .", "More precisely there exists (at least one) $h \\in K$ with representative $h=h_{1} \\cdots h_{k} h_{k+1} \\cdots h_{l}$ , with the property that when at color 1, the random walk $(X_{n})$ reaches color $v$ when being multiplied by $h_{1} \\cdots h_{k}$ .", "The main difference about the two constructions is the number of such $h$ that one can associate to the color $v$ , but this has no consequence for what follows.", "Let $K(v)$ be the set of such elements $h$ .", "For each $h = h_{1} \\cdots h_{k} h_{k+1} \\cdots h_{l} \\in K(v)$ , set $p(h) := h_{1} \\cdots h_{k}$ and $s(h) := h_{k+1} \\cdots h_{l}$ .", "By what precedes, to have color $v$ when visiting $g$ , $X_{n}$ has first to get to $(g p(h)^{-1},1)$ for some $h \\in K(v)$ , and then be multiplied by $p(h)$ .", "Arrived at $g$ , it has no other choice then than to be multiplied by $s(h)$ .", "The same goes for the starting color $u$ : if the walk is at $(e,u)$ , then it is bound to arrive at $(s(h^{\\prime }),1)$ for some $h^{\\prime } \\in K(u)$ .", "Thus using that $X_{\\tau _{n}} = Y_{n}$ , one deduces that the colored random walk can go from $(e,u)$ to $(g,v)$ if and only if the finite range random walk can go from one of the $s(h^{\\prime }), h^{\\prime } \\in K(u)$ to $g p(h)^{-1}$ for some $h \\in K(v)$ and then to $g s(h)$ .", "Hence for any $h \\in K(v), h^{\\prime } \\in K(u)$ , $q_{g}(u,v) \\ge H_{g}(s(h^{\\prime }), s(h)).$ Provided $\\left\|g\\right\|$ is large enough, Equation (REF ) precisely states that the right-hand side is positive if and only if $s(h)$ is in some subset of $\\mathcal {B}$ that only depends on the last letters of $g$ .", "Consequently the positivity of $q_{g}(u,v)$ does not depend on $u$ for $\\left\|g\\right\|$ large enough, which proves the result." ], [ "Law of Radon-Nikodym derivatives.", "The previous result is purely deterministic and does not take into account that the process $(X_{\\infty }^{(n)},u_{n})_{n \\ge 0}$ is a Markov chain.", "Our goal in this paragraph is to determine the law of the probability vector $V$ appearing in Corollary REF .", "The monograph [3] (see also [4]) state results for products of iid random matrices which, as we shall see, apply to Markovian products as well.", "Remploying the general framework of Proposition REF , consider a Markov chain $(Z_{n})_{ n \\ge 0}$ on the finite state space $X$ with transition matrix $Q$ and $Y:= \\lbrace (x,y) \\in X \\times X, Q(x,y) > 0\\rbrace $ .", "Let $(M_{x})_{x \\in X}$ be a family of $r \\times r$ non-negative matrices, such that condition (REF ) holds.", "Consider the sequence defined by $Y_{n} := M_{Z_{n}}, n \\ge 1$ .", "By Proposition REF , a.s. the product $Y_{1} \\cdots Y_{n}$ is non-zero and converges in direction to a rank one matrix spanned by a random vector $V \\in \\mathcal {P}^{+}$ .", "If one multiplies the product on the left by another matrix $M_{x}$ , one still has a products of matrices $M_{x}$ , so one can expect the law of $V$ to satisfy some invariance property.", "For all non-negative matrix $A \\in M_{r}(\\mathbb {R})$ and all $z \\in \\mathcal {P}^{+}$ , we define $A z$ to be the normalized image of $z$ (identified with a vector in $\\mathbb {R}^{r}$ ) that makes it a probability vector, provided $A z \\ne 0$ .", "The latter case will not be an issue: condition (REF ) implies in particular that matrices $M_{x}$ have no zero row, and vectors of $\\mathcal {P}^{+}$ have no zero coordinate so $M_{x} z$ is well defined for all $x \\in X, z \\in \\mathcal {P}^{+}$ .", "Since $Z_n$ is a Markov chain, the law of $V$ may depend on the starting state of $Z_{n}$ .", "Hence it is natural to think of it as a colored measure (colors being here the states $x \\in X$ ).", "Definition 6 A family $\\nu =(\\nu _{x})_{x \\in X}$ of probability measure on $\\mathcal {P}^{+}$ is called a colored measure.", "Let $Q \\ast \\nu $ denote the colored measure defined by $\\int f(z) \\: d (Q \\ast \\nu )_{x}(z) = \\sum _{y} \\int f(M_{y} z) Q(x,y) \\: d \\nu _{y}(z)$ for all bounded measurable function $f$ on $\\mathcal {P}^{+}$ and $x \\in X$ .", "The colored measure $\\nu $ is said to be invariant if $Q \\ast \\nu = \\nu $ .", "Lemma 18 Let $\\nu _{x}$ be the law of $V$ when $Z_{n}$ is started at $x$ , the colored measure $(\\nu _{x})_{x \\in X}$ is the unique colored measure which is invariant with respect to $Q$ .", "We use Markov property together with (REF ) to obtain the invariance of the law of $V$ .", "The proof of uniqueness is similar to the proof for the harmonic measure in Proposition REF .", "Consider another invariant colored measure $(\\rho _{x})_{x \\in X}$ .", "By invariance, for all bounded measurable function $f$ on $\\mathcal {P}^{+}$ the sequence $M_{n} := \\int f( Y_{1} \\cdots \\ Y_{n} w) \\: d \\rho _{Z_{n}}(w), \\quad n \\ge 0$ is a bounded martingale with respect to the filtration $\\mathcal {F}_{n} := \\sigma ( Z_{0}, Z_{1}, \\ldots , Z_{n})$ , so $M_{n}$ converges a.s. and in $L^{1}$ .", "This being true for all bounded measurable function, this implies the measures $Y_{1} \\cdots \\ Y_{n} \\ast \\rho _{Z_{n}}$ converges weakly to a measure.", "Now, because the product $Y_{1} \\cdots Y_{n}$ converges in direction to $V$ , the limit is necessarily the Dirac mass at $V$ .", "On the other hand, the martingale property with the $L^{1}$ convergence gives that for all bounded measurable functions $f$ on $\\mathcal {P}^{+}$ , for all $x \\in X$ , $\\mathbb {E}_{x} \\left[ f(V) \\right] = \\mathbb {E}_{x} \\left[ \\int f \\: \\delta _{V} \\right] = \\int f \\: d \\rho _{x} .$ Therefore $\\rho _{x}$ must be the law of $V$ when $Z_0 =x$ , that is $\\rho _x = \\nu _{x}$ , and is consequently unique.", "We are finally ready to prove Theorem REF .", "[Proof of Theorem REF ] We apply Lemma REF in the context of colored random walks, with $X = S \\times [r]$ and $Q((g,u)(h,v)) = \\mu _{h}(u,v)$ .", "Due to the form of the transition probabilities the law of $V$ is only indexed by colors.", "It remains to use Corollary REF in conjonction with the integral formula of the entropy given by Theorem REF ." ], [ "Application", "We now give an example of how the linearization trick can be used in order to compute the drift of a random walk.", "The example is simple enough so that the linearization trick is actually not needed.", "However it is a good illustration of its usefulness: whereas the direct computation requires some thought and can appear tedious, the linearization trick allows to hide all the technical aspects into matrix multiplications.", "The computation in the end is the same, but it can be handled quite smoothly with the linearization trick.", "Let $G_{1}, G_{2}$ be finite groups and consider the group $G = G_{1} \\ast G_{2}$ with the usual set of generators $S = G_{1} \\bigsqcup G_{2} \\setminus \\lbrace e \\rbrace $ .", "Elements $g$ of $G$ can be written as words of length $\\left\|g\\right\|$ with letters alternatively taking values in $G_{1}$ or $G_{2}$ .", "Let $k_{i}$ denote the cardinality of $G_{i} \\setminus \\lbrace e \\rbrace $ , $i =1,2$ , $L \\le 1$ be a fixed integer and consider the following random walk $(X_{t})_{t \\ge 0}$ supported on the set of words of length smaller than $L$ .", "At each step the walk is multiplied on the right by an element chosen uniformly among words of length $n \\le L$ that start with $i \\in \\lbrace 1,2\\rbrace $ with probability $p_{i}(n)$ .", "In other words for all $g=g_{i_{1}} \\cdots g_{i_{n}}$ of length $n \\le L$ , with $i_{1}, \\ldots , i_{n} \\in \\lbrace 1,2\\rbrace , i_{1} \\ne i_{2} \\cdots \\ne i_{n}$ , $p_{g} = \\frac{p_{i_{1}}(n)}{k_{i_{1}} \\cdots k_{i_{n}}},$ where $p=(p_{i}(n))_{n \\le L, i=1,2}$ is a probability vector : $\\sum _{i,n} p_{i}(n)=1$ .", "Of course in the denumerator above, one could regroup powers of $k_{1}, k_{2}$ together.", "This random walk is the generalization of an example given in [22] for which the drift is explicitly computed in the nearest-neighbor case $L = 1$ .", "As seen in the proof of Theorem REF , the drift is given by $\\gamma = \\lim _{t \\rightarrow \\infty } \\frac{1}{t} \\sum _{k=0}^{t-1} \\mathbb {E}\\left[ \\left\|X_{k+1}\\right\| - \\left\|X_{k}\\right\|\\right].$ In this model, the knowledge of what group the last letters of $X_{t}$ belongs to is enough to determine the quantity $Z_{t} := \\left\|X_{t+1}\\right\| - \\left\|X_{t}\\right\|$ .", "Neglecting the case where $X_{t}$ or $X_{t+1}$ og-f length smaller than $L$ , $Z_{t}$ can thus be determined from a Markov chain on the state space $\\lbrace 1, 2\\rbrace $ .", "The drift of $X_{t}$ can then be easily deduced from the stationary measure of this Markov chain.", "This is the computation done in [22] for $L=1$ .", "This argument remains valid both for the finite support case $L > 1$ and for colored random walks.", "Hence the computation can be made with or without the linearization trick." ], [ "Direct computation.", "Given $g \\in G$ , let $\\underline{g}$ , resp.", "$\\overline{g}$ denote the first, resp.", "last letter of $g$ .", "If $g = e, \\underline{g} = \\overline{g} = 1$ .", "Suppose that at time $t$ , $\\left\|X_{t}\\right\| > L$ so that neither in $X_{t}$ nor $X_{t+1}$ can equal $e$ .", "At time $t+1$ , $X_{t}$ is multiplied by some element $g$ .", "Here are the different cases occuring.", "If $\\underline{g}$ is not in the same group as $\\overline{X_{t}}$ , there is no cancellation when concatenating $X_{t}$ and $g$ , so $X_{t+1} = X_{t} g$ ends with the same letter as $g$ , ie $\\overline{X_{t+1}} = \\overline{g}$ .", "Otherwise there might be some cancellations and $\\overline{X_{t+1}} = \\overline{g}$ if and only if there is no complete cancellation between $X_{t}$ and $g$ .", "By complete cancellation we mean that $g$ is exactly the inverse of the word formed by the $\\left\|g\\right\|$ last letters of $X_{t}$ .", "If complete cancellation occurs, then $\\overline{X_{t+1}}$ is bound not to be in the same group as $\\overline{g}$ .", "Since there are only two groups, $\\overline{g}=\\underline{g}$ if and only if $\\left\|g\\right\|$ is odd.", "Thanks to the simple form of the transition probabilities in this model, one can deduce thus deduce that the probability the last letter of $X_{t}$ goes from $G_{1}$ to $G_{2}$ is $Q_{1,2} = \\sum _{n \\ge 0} p_{2}(2n+1) + \\sum _{n \\ge 0} \\frac{p_{1}(2n+1)}{k_{1}^{n+1} k_{2}^{n}} + \\sum _{n \\ge 1} p_{1}(2n)(1-\\frac{1}{k_{1}^{n}k_{2}^{n}}),$ and the probability to stay in $G_{1}$ $Q_{1,1}=\\sum _{n \\ge 2n+1} p_{1}(2n+1)(1-\\frac{1}{k_{1}^{n+1}k_{2}^{n}}) + \\sum _{n \\ge 1} \\frac{p_{1}(2n)}{k_{1}^{n} k_{2}^{n}} + \\sum _{n \\ge 1} p_{2}(2n).$ The sums above are finite as $p_{i}(n) = 0$ if $n > L$ .", "By symmetry, exchange the roles of 1 and 2 to get $Q_{2,1}$ and $Q_{2,2}$ .", "In the end, one obtains a $2 \\times 2$ stochastic matrix $Q$ which is the transition matrix of the Markov chain induced by the last letter of $X_{t}$ in the sense that $\\mathbb {P}\\left[ \\overline{X_{t+1}} \\in G_{j} \\: \| \\: \\overline{X_{t}} \\in G_{i}, \\left\|X_{t}\\right\| > L\\right] = Q_{i,j}$ for $i,j \\in \\lbrace 1,2\\rbrace $ .", "Since $X_{t}$ is transient it spends only a finite amount of time in the ball $\\lbrace g \\in G, \\left\|g\\right\| \\le L \\rbrace $ so the ergodic theorem for Markov chains shows that the average amount of time $\\overline{X_{t}}$ is in $G_{i}$ converges to converges to $\\pi (i)$ , where $\\pi $ is the unique invariant measure of the transition matrix $Q$ .", "Now what is $Z_{t} := \\left\|X_{t+1}\\right\| - \\left\|X_{t}\\right\|$ ?", "Consider the cases previously identified.", "If $\\underline{g}$ and $\\overline{X_{t}}$ belong to different group, then $Z_{t} = \\left\|g\\right\|$ .", "Otherwise, each cancellation occuring when concatenating $X_{t}$ and $g$ make the distance decrease by one.", "Hence in case of complete cancellation $Z_{t} = - \\left\|g\\right\|$ .", "Otherwise, once the cancellations have been dealt with, the first step simply updates $\\overline{X_{t}}$ and consequently does not increase the distance, while the other remaining steps increase it by one.", "Therefore the contribution of $g$ is $-\\left\|g\\right\|$ in case of complete cancellation and $\\left\|g\\right\| - 1 -2i$ otherwise, where $i$ is the number of cancellations occurring when concatenating $X_{t}$ and $g$ .", "Combining all these observations, the change in distance when going from group $G_{1}$ to group $G_{2}$ is $\\tilde{Q}_{1,2} = \\sum _{n \\ge 0} (2n+1) p_{2}(2n+1) - \\sum _{n \\ge 0} (2n+1) \\frac{p_{1}(2n+1)}{k_{1}^{n+1} k_{2}^{n}} \\\\ + \\sum _{n \\ge 1} p_{1}(2n) \\sum _{i = 1}^{2n} \\left(2n - 1 -2(i-1)\\right) \\frac{1}{k_{1} \\cdots k_{i-1}} \\left(1-\\frac{1}{k_{i}} \\right)$ and when staying in $G_{1}$ : $\\tilde{Q}_{1,1}=\\sum _{n \\ge 2n+1} \\sum _{i=1}^{2n+1} (2n - 2(i-1)) p_{1}(2n+1)\\frac{1}{k_{1} \\cdots k_{i-1}} \\left(1-\\frac{1}{k_{i}}\\right)\\\\ - \\sum _{n \\ge 1} (2n) \\frac{p_{1}(2n)}{k_{1}^{n} k_{2}^{n}} + \\sum _{n \\ge 1} (2n) p_{2}(2n).$ As before, $\\tilde{Q}_{2,1}, \\tilde{Q}_{2,2}$ can be obtained by exchanging the roles of 1 and 2.", "In the end the drift of $X_{t}$ is given by $\\gamma = \\pi (1) (\\tilde{Q}_{1,1} + \\tilde{Q}_{1,2}) + \\pi (2) (\\tilde{Q}_{2,1} + \\tilde{Q}_{2,2}).$" ], [ "Computation with the linearization trick.", "Since we know specifically the random walk $(X_{t})$ , we do not follow exactly the construction of the linearized random walk in Section but adapt it to the example.", "What seems quite natural is the following: consider matrices $\\tilde{p}_{g}=(\\tilde{p}_{g}(i_{1}\\cdots i_{n}))_{i_{1} \\cdots i_{n}}$ indexed by reduced words $i_{1} \\ne i_{2} \\cdots \\ne i_{n}$ of length $n \\le L-1$ , including the case $n = 0$ of the empty word, written exceptionnally 0.", "For all $g \\in S$ and all reduced word $i_{1} \\cdots i_{n}$ of length $n \\le L-1$ , set $\\begin{split}\\tilde{p}_{g}(0,i_{1} \\cdots i_{n}) &= \\left\\lbrace \\begin{array}{l l}\\frac{p_{i_{0}}(n+1)}{k_{i_{0}}} & \\text{if $g \\in G_{i_{0}}$ and $i_{0} \\ne i_{1} $} \\\\0 & \\text{otherwise}\\end{array} \\right.\\\\\\tilde{p}_{g}(i_{1} \\cdots i_{n},i_{2} \\cdots i_{n}) &= \\left\\lbrace \\begin{array}{l l}\\frac{1}{k_{i_{1}}} & \\text{if $g \\in G_{i_{1}}$} \\\\0 & \\text{otherwise.", "}\\end{array} \\right.\\end{split}$ It is easy to see that $\\sum _{g \\in S} \\tilde{p}_{g}$ is a stochastic matrix.", "Thus the matrices $\\tilde{p}_{g}$ can be used to define a colored random walk $(Y_{t})_{t \\ge 0}$ .", "In Equation (REF ), the color 0 is to be interpreted as a neutral color, while $i_{1} \\cdots i_{n}$ is to be interpreted as the “remaining step” for $Y_{t}$ , meaning that $Y_{t}$ will be mutliplied by successive elements of $G_{i_{1}}, \\ldots , G_{i_{n}}$ before it arrives at the color 0.", "For instance the probability to go from $(e,0)$ to $(g,212)$ with $g \\in G_{1}$ is $p_{1}(4)/k_{1}$ .", "From $(g,212)$ , the walk then goes to some $(gh,12)$ , $h \\in G_{2}$ with probability $1/k_{2}$ , then to $(ghg^{\\prime },1), g^{\\prime } \\in G_{1}$ with probability $1/k_{1}$ and finally to $(ghg^{\\prime }h^{\\prime },0)$ with probability $1/k_{2}$ .", "Hence with probability $p_{1}(4)$ it moved between two states with color 0, from $e$ to a uniform element of length 4 starting in group $G_{1}$ .", "This should make clear that that $Y_{t}$ linearizes the initial random walk $X_{t}$ in the sense of Theorem REF : if $Y_{t}$ starts at $(e,0)$ , then it behaves exactly like $X_{t}$ when evaluated at the $t$ -th return time at color 0.", "Let us now apply the previous argument to this colored random walk.", "Let $r$ be the number of colors used in the linearization and for $i=1,2$ set $\\tilde{P}_{i} := \\sum _{g \\in G_{i}} \\tilde{p}_{g}$ .", "When $Y_{t}$ is far from $e$ , it induces a colored Markov chain on the state space $\\lbrace 1,2\\rbrace \\times [r]$ .", "Because the random walk is nearest-neighbor, we are spared of all the considerations necessary to deal with finite support.", "The transition matrix we are looking for is simply $Q = \\begin{pmatrix}(1-1/k_{1}) P_{1} & P_{1} / k_{1} + P_{2} \\\\P_{1} + P_{2}/k_{2} & (1-1/k_{2}) P_{2}\\end{pmatrix}$ which is the exact block analog of the matrix in [22].", "Likewise the change in distance is simply given by the matrix: $\\tilde{Q}= \\begin{pmatrix}0 & -P_{1} / k_{1} + P_{2} \\\\P_{1} - P_{2}/k_{2} & 0\\end{pmatrix}.$ The matrix $Q$ defining an irreducible Markov chain, it admits a unique stationary measure $\\pi $ .", "As before, the ergodic theorem gives the drift of $Y_{t}$ as $\\sum _{v \\in [r]} (\\pi \\tilde{Q})(v)$ and by Theorem REF the drift of $X_{t}$ is $\\gamma = \\left(\\sum _{n=1}^{L} p_{1}(n) + p_{2}(n)\\right) \\sum _{v \\in [r]} (\\pi \\tilde{Q})(v).$" ] ]	2011.14127
[ [ "Reciprocal maximum likelihood degrees of diagonal linear concentration\n models" ], [ "Abstract We show that the reciprocal maximal likelihood degree (rmld) of a diagonal linear concentration model $\\mathcal L \\subseteq \\mathbb{C}^n$ of dimension $r$ is equal to $(-2)^r\\chi_M( \\textstyle\\frac{1}{2})$, where $\\chi_M$ is the characteristic polynomial of the matroid $M$ associated to $\\mathcal L$.", "In particular, this establishes the polynomiality of the rmld for general diagonal linear concentration models, positively answering a question of Sturmfels, Timme, and Zwiernik." ], [ "Introduction", "Let $\\mathbb {S}^n$ be the space of (real or complex) $n\\times n$ symmetric matrices, and $\\mathbb {S}^n_{>0}$ the subset consisting of real positive definite symmetric matrices.", "For a fixed $S\\in \\mathbb {S}^n_{>0}$ , the log-likelihood function $\\ell _S: \\mathbb {S}^n_{>0} \\rightarrow \\mathbb {R}$ is defined by $\\ell _S(K) := \\log \\det K - \\operatorname{trace}(S\\cdot K).$ For a subvariety $\\mathcal {L}\\subseteq \\mathbb {S}^n$ , the maximum likelihood (ML) degree $\\operatorname{mld}(\\mathcal {L})$ is the number of invertible complex critical points of $\\ell _S$ on the smooth locus of $\\mathcal {L}$ , counted with multiplicity, for a general choice of $S$ .", "Writing $\\mathcal {L}^{-1} \\subseteq \\mathbb {S}^n$ for the subvariety obtained as the closure of $\\lbrace K^{-1} \\in \\mathbb {S}^n \\mid K\\in \\mathcal {L}\\text{ invertible}\\rbrace $ , one defines the reciprocal maximum likelihood degree $\\operatorname{rmld}(\\mathcal {L})$ as the number of invertible complex critical points of $\\ell _S$ on the smooth loci of $\\mathcal {L}^{-1}$ , counted with multiplicity.", "Computing (reciprocal) ML degrees arises in statistical applications, where $\\mathbb {S}^n_{>0}$ is often considered as the set of concentration matrices of multivariate normal distributions [17].", "We caution that the terminology here regarding reciprocal vs. non-reciprocal ML degree is the opposite of that in [16], [1], where $\\mathbb {S}^n_{>0}$ is considered as the set of covariance matrices (inverses of concentration matrices).", "In particular, our $\\operatorname{rmld}$ is the ML degree of a linear covariance model.", "Our convention here agrees with [17], [10], [5], [8].", "Let $[n] = \\lbrace 1, \\ldots , n\\rbrace $ .", "A diagonal linear concentration model is a linear subspace $\\mathcal {L}\\subseteq \\mathbb {C}^{[n]}$ , where $\\mathbb {C}^{[n]}$ is identified with the space of diagonal matrices in $\\mathbb {S}^n$ .", "Let $M$ be the matroid on $[n]$ whose independent subsets are $I\\subseteq [n]$ such that the composition $\\mathbb {C}^I \\hookrightarrow \\mathbb {C}^{[n]} \\twoheadrightarrow \\mathcal {L}^\\vee $ is injective.", "Without loss of generality, we always assume that $\\mathcal {L}$ is not contained in a coordinate hyperplane, or equivalently, that $M$ is loopless, since otherwise $\\operatorname{rmld}(\\mathcal {L}) = 0$ from the definition.", "Our main result is the formula for the reciprocal ML degree of $\\mathcal {L}$ in terms of $M$ .", "Let $\\mathcal {L}\\subseteq \\mathbb {C}^{[n]}$ be a diagonal linear concentration model of dimension $r$ , and $M$ the associated matroid of rank $r$ on $[n]$ .", "Then we have $\\operatorname{rmld}(\\mathcal {L}) = (-2)^r\\chi _M( \\textstyle \\frac{1}{2}),$ where $\\chi _M$ is the characteristic polynomial of $M$ .", "In [17], [16], the (non-reciprocal) ML degree of $\\mathcal {L}$ was shown to be $\|\\chi _{M}(0)\|$ .", "Computing the reciprocal ML degree presents fundamentally new challenges; see Remark for a comparison.", "From computational experiments, the authors of [16] asked whether the reciprocal ML degree of a general diagonal linear concentration model of dimension $r$ in $\\mathbb {C}^{[n]}$ is a polynomial in $n$ of degree $r-1$ .", "Evaluating our Theorem at uniform matroids answers their question positively.", "Let $\\mathcal {L}\\subseteq \\mathbb {C}^{[n]}$ be a general linear concentration model of dimension $r$ .", "Then we have $\\operatorname{rmld}(\\mathcal {L}) = \\sum _{i=1}^{r} \\textstyle \\binom{n-i-1}{r-i}2^{r-i}.$ For instance, when $r = 3$ we have $2n^2 - 8n + 7$ , and when $r = 4$ we have $4/3n^3 - 10n^2 + 68/3n - 15$ , as predicted in [16] from numerical computations.", "To prove Theorem , we use the following alternate description of the reciprocal ML degree, obtained by a standard computation in multivariable calculus.", "Let $\\mathcal {L}^\\perp $ denote the orthogonal complement of a subspace $\\mathcal {L}\\subseteq \\mathbb {C}^{[n]}$ under the standard pairing $\\langle (x_1, \\ldots , x_n), (y_1, \\ldots , y_n) \\rangle := \\sum _{i=1}^n x_i y_i.$ [16] The reciprocal ML degree of a linear subspace $\\mathcal {L}\\subseteq \\mathbb {C}^{[n]}$ is equal to the number of solutions $(x_1,\\ldots ,x_n) \\in (\\mathbb {C}^)^{[n]}$ , counted with multiplicity, to the following system of equations, where $s_1,\\ldots ,s_n \\in \\mathbb {C}$ are generic parameters: $(x_1^{-1},\\ldots ,x_n^{-1}) \\in \\mathcal {L}\\quad \\text{and}\\quad (s_1x_1^2-x_1,\\ldots ,s_nx_n^2-x_n) \\in \\mathcal {L}^{\\perp }.$ Thus, we prove Theorem by establishing the following generalization.", "For an $r$ -dimensional linear subspace $\\mathcal {L}\\subseteq \\mathbb {C}^{[n]}$ , a generic choice of parameters $s_1, \\ldots , s_n \\in \\mathbb {C}$ , and any integer $d\\ge 1$ , the number of solutions $(x_1, \\ldots , x_n) \\in (\\mathbb {C}^)^{[n]}$ , counted with multiplicity, to the system of equations $(x_1^{-1},\\ldots ,x_n^{-1}) \\in \\mathcal {L}\\quad \\text{and}\\quad (s_1x_1^d-x_1,\\ldots ,s_nx_n^d-x_n) \\in \\mathcal {L}^{\\perp }\\qquad \\mathrm {(\\dagger )}$ is equal to $(-d)^r\\chi _M( \\textstyle \\frac{1}{d}),\\quad \\text{or equivalently,} \\quad d^r T_M(1- \\frac{1}{d},0),$ where $\\chi _M$ is the characteristic polynomial and $T_M$ is the Tutte polynomial of the matroid $M$ associated to $\\mathcal {L}$ .", "For an $r$ -dimensional subspace $\\mathcal {L}\\subseteq \\mathbb {C}^{[n]}$ , let $U(\\mathcal {L}) := \\mathcal {L}\\cap (\\mathbb {C}^)^{[n]}$ be the hyperplane arrangement complement, and $M$ the associated matroid.", "Then the Poincaré polynomial of $U(\\mathcal {L})$ $P_{U(\\mathcal {L})}(q) := \\sum _{i\\ge 0} \\big (\\operatorname{rank} H_i(U(\\mathcal {L});\\mathbb {Z})\\big )q^i$ coincides with polynomial $(-q)^r\\chi _M(-\\frac{1}{q})$ [11].", "In particular, Theorem states that $\\operatorname{rmld}(\\mathcal {L}) = (-1)^rP_{U(\\mathcal {L})}(-2)$ .", "This echoes the result of [7], which showed that, for a different log-likelihood function (from discrete statistical models), the ML degree of a smooth very affine variety $U$ is its signed topological Euler characteristic $(-1)^{\\dim U}P_U(-1)$ .", "However, ML degrees in our case are not topological invariants of very affine varieties: Observe that $\\mathcal {L}^{-1}\\cap (\\mathbb {C}^)^{[n]} \\simeq \\mathcal {L}\\cap (\\mathbb {C}^)^{[n]}$ but in general $\\operatorname{rmld}(\\mathcal {L}^{-1}) = \\operatorname{mld}(\\mathcal {L}) = \\chi _M(0) \\ne (-2)^r\\chi _M(\\frac{1}{2}) = \\operatorname{rmld}(\\mathcal {L})$ .", "It may still be interesting to find other families of subvarieties $\\mathcal {L}\\subseteq \\mathbb {S}^n$ such that $\\operatorname{rmld}(\\mathcal {L}) = (-1)^{\\dim \\mathcal {L}}P_{U(\\mathcal {L})}(-2)$ , where $U(\\mathcal {L}) := \\lbrace K\\in \\mathcal {L}\\mid K \\text{ invertible}\\rbrace $ .", "For example, general pencils of conics form one such family [3], [5].", "Outline.", "In Section we review properties of reciprocal linear spaces $\\mathcal {L}^{-1}$ , and introduce score varieties, which together with $\\mathcal {L}^{-1}$ encode the system of equations (REF ).", "After establishing two key technical lemmas in Section , in Section we compute the number of solutions to the system of equations (REF ) in $\\mathbb {C}^{[n]}$ , instead of in $(\\mathbb {C}^)^{[n]}$ , in two different ways: One is a Bézout-like computation, and the other is a summation, with each summand corresponding to a set of solutions with specified support (non-zero coordinates).", "An inclusion-exclusion argument in Section then yields the proof of Theorem ." ], [ "Notation", "For an affine subvariety $X \\subseteq \\mathbb {C}^n$ , we write $\\overline{X} \\subseteq \\mathbb {P}^n$ for its projective closure.", "If $X\\subseteq \\mathbb {C}^n$ is defined by a homogeneous ideal, then we write $\\mathbb {P}X \\subseteq \\mathbb {P}^{n-1}$ for its projectivization.", "For a point $p\\in X\\subseteq \\mathbb {C}^{[n]}$ , we write $TC_pX \\subseteq \\mathbb {C}^{[n]}$ for the tangent cone of $X$ at $p$ .", "For $p = (p_1, \\ldots , p_n) \\in \\mathbb {C}^{[n]}$ , we write $\\operatorname{supp}(p) := \\lbrace i\\in [n] \\mid p_i \\ne 0\\rbrace $ for its support, and for $I \\subseteq [n]$ , write $p_{\|I}$ for the projection of $p$ onto $\\mathbb {C}^I \\subseteq \\mathbb {C}^{[n]}$ ." ], [ "Reciprocal linear spaces and score varieties", "We set notations concerning matroids associated to linear subspaces, and review necessary facts about reciprocal linear spaces.", "We assume familiarity with matroid theory, and refer to [19], [12] as standard references.", "Let us fix a linear subspace $\\mathcal {L}\\subseteq \\mathbb {C}^{[n]}$ of dimension $r$ .", "Let $A$ be an $r\\times n$ matrix whose row-span equals $\\mathcal {L}$ .", "We will often use the fact that the minimal sets among supports of elements in the row-span of $A$ form the cocircuits of $M$ .", "For a subset $I\\subseteq [n]$ , let $\\mathcal {L}_{\|I} \\subseteq \\mathbb {C}^I$ be the image of $\\mathcal {L}$ under the coordinate projection $\\mathbb {C}^{[n]} \\twoheadrightarrow \\mathbb {C}^I$ , and let $\\mathcal {L}_{/I}$ be the intersection of $\\mathcal {L}$ with the coordinate subspace $\\lbrace 0\\rbrace ^I \\times \\mathbb {C}^{[n]\\setminus I}$ , considered as a subspace of $ \\mathbb {C}^{[n]\\setminus I}$ .", "The matroid of $\\mathcal {L}_{\|I}$ is the restriction $M\|I$ , whereas the matroid of $\\mathcal {L}_{/I}$ is the contraction $M/I$ .", "The reciprocal linear space $\\mathcal {L}^{-1}$ of $\\mathcal {L}$ is the Zariski closure in $\\mathbb {C}^{[n]}$ of $\\lbrace (x_1, \\ldots , x_n) \\in (\\mathbb {C}^)^{[n]} \\mid (x_1^{-1}, \\ldots , x_n^{-1})\\in \\mathcal {L}\\rbrace $ .", "Note that $\\mathcal {L}^{-1} \\cap (\\mathbb {C}^)^{[n]}$ is smooth, being isomorphic to $\\mathcal {L}\\cap (\\mathbb {C}^)^{[n]}$ .", "For $I\\subseteq [n]$ , we write $\\mathcal {L}^{-1}_{\|I}$ for $(\\mathcal {L}_{\|I})^{-1}$ , and likewise write $\\mathcal {L}^{-1}_{/I} = (\\mathcal {L}_{/I})^{-1}$ .", "We collect together in the following theorem the known properties of $\\mathcal {L}^{-1}$ that we will need.", "Let $\\mathcal {L}^{-1} \\subseteq \\mathbb {C}^{[n]}$ be the reciprocal linear space of $\\mathcal {L}\\subseteq \\mathbb {C}^{[n]}$ .", "[13] The ideal of $\\mathcal {L}^{-1}$ is homogeneous, and $\\mathbb {P}\\mathcal {L}^{-1}$ has degree $\|\\mu (M)\|$ , where $\\mu (M) := \\chi _M(0)$ is the Möbius invariant of the matroid $M$ .", "[13] The ideal of $\\mathcal {L}^{-1}$ is Cohen-Macaulay, with any basis of $\\mathcal {L}^\\perp $ forming a system of parameters, i.e.", "$\\mathcal {L}^{-1} \\cap \\mathcal {L}^\\perp = \\lbrace \\mathbf {0}\\rbrace $ .", "[13] The intersection $\\mathcal {L}^{-1} \\cap ((\\mathbb {C}^)^F \\times \\lbrace 0\\rbrace ^{[n]\\setminus F})$ is nonempty if and only if $F\\subseteq [n]$ is a flat of $M$ , and in that case, one has $\\mathcal {L}^{-1} \\cap ((\\mathbb {C}^)^F \\times \\lbrace 0\\rbrace ^{[n]\\setminus F}) = (\\mathcal {L}_{\|F}^{-1} \\cap (\\mathbb {C}^)^F) \\times \\lbrace 0\\rbrace ^{[n]\\setminus F}.$ [14] For a flat $F\\subseteq [n]$ and a point $p\\in \\mathcal {L}^{-1}$ with $\\operatorname{supp}(p) = F$ , the tangent cone of $\\mathcal {L}^{-1}$ at $p$ is the product $TC_p\\mathcal {L}^{-1} = TC_{p_{\|F}}\\mathcal {L}_{\|F}^{-1} \\times \\mathcal {L}_{/F}^{-1} \\simeq \\mathcal {L}_{\|F} \\times \\mathcal {L}_{/F}^{-1}.$ All four statements in Theorem can be derived easily from the Gröbner basis for the defining ideal of $\\mathcal {L}^{-1}$ computed in [13].", "In [14], the statement of Theorem .REF originally reads $TC_p\\mathcal {L}^{-1} = p_{\|F}^2 \\mathcal {L}_{\|F} \\times \\mathcal {L}_{/F}^{-1}$ , where $p^2 \\mathcal {L}$ denotes the linear subspace $\\lbrace (p_1^2 x_1, \\ldots , p_n^2 x_n) \\mid (x_1, \\ldots , x_n) \\in \\mathcal {L}\\rbrace $ .", "It is straightforward to verify that $TC_{p_{\|F}}\\mathcal {L}_{\|F}^{-1} = p_{\|F}^2 \\mathcal {L}_{\|F}$ .", "Theorem .REF also follows from [18], which expressed the Hilbert series of the ideal of $\\mathcal {L}^{-1}$ in terms of the characteristic polynomial $\\chi _M$ .", "The reciprocal linear space $\\mathcal {L}^{-1}$ encodes the left half of the system of equations in Equation (REF ).", "Let us now consider the variety encoding the condition $(s_1x_1^d - x_1, \\ldots , s_nx_n^d - x_n) \\in \\mathcal {L}^\\perp $ .", "For an integer $d\\ge 1$ and a parameter $s = (s_1, \\ldots , s_n) \\in \\mathbb {C}^n$ , we define the score variety as $Y(\\mathcal {L}, s, d) := \\lbrace (x_1, \\ldots , x_n) \\in \\mathbb {C}^{[n]} \\mid (s_1x_1^d-x_1, \\ldots , s_nx_n^d - x_n) \\in \\mathcal {L}^\\perp \\rbrace .$ We will simply write $Y$ when we trust that no confusion will arise.", "We note here that score varieties are smooth for a generic choice of $s\\in \\mathbb {C}^n$ .", "For $d \\ge 1$ and a generic choice of $(s_1, \\ldots , s_n)\\in \\mathbb {C}^n$ , the score variety $Y$ is smooth.", "If $d=1$ , then $Y$ is linear, so suppose $d\\ge 2$ .", "Let $A$ be the $r\\times n$ matrix whose row span equals $\\mathcal {L}$ , and let $g_1, \\ldots , g_r$ be the polynomials obtained by multiplying the rows of $A$ with $(s_1x_1^d - x_1, \\ldots , s_nx_n^d - x_n)^T$ .", "These minimally generate the defining ideal $I_Y\\subseteq \\mathbb {C}[x_1, \\ldots , x_n]$ of $Y$ .", "The Jacobian matrix with respect to these minimal generators is $\\operatorname{Jac}(x) = A \\cdot \\operatorname{diag}(ds_1x_1^{d-1} - 1, \\ldots , ds_nx_n^{d-1} -1),$ i.e.", "matrix $A$ whose $i$ -th column is scaled by $ds_ix_i^{d-1} -1$ for each $1\\le i \\le n$ .", "Suppose now that $\\operatorname{Jac}(x)$ has rank $< r$ for some $x\\in \\mathbb {C}^n$ , that is, the restriction $M\|I$ of the matroid $M$ to the set $I = \\lbrace i\\in [n] \\mid ds_ix_i^{d-1} - 1 \\ne 0\\rbrace $ has rank $<r$ .", "This happens if and only if $I$ is contained in a hyperplane flat of $M$ , or equivalently, the subset $J := [n]\\setminus I$ contains a cocircuit of $M$ .", "As the minimal supports of the row-space of $A$ constitute the cocircuits of $M$ , let $v = (v_1, \\ldots , v_n) \\in \\mathbb {C}^n$ be the element in the row-space of $A$ whose support $C^* = \\operatorname{supp}(v) \\subseteq [n]$ is a cocircuit of $M$ contained in $J$ .", "Then we have $v \\cdot (s_1x_1^d - x_1, \\ldots , s_nx_n^d - x_n)^T = \\sum _{i\\in C^} v_i (s_ix_i^d - x_i) = (\\textstyle {\\frac{1}{d}} - 1) \\displaystyle \\sum _{i\\in C^} v_i x_i$ where last equality follows from $ds_ix_i^{d-1} -1 = 0$ for $i\\in J$ .", "This quantity needs to be zero if $x\\in Y$ .", "We claim that for a general choice of $(s_1, \\ldots , s_n)$ this quantity can never be zero: Consider the set $Z := \\lbrace (\\zeta _1, \\ldots , \\zeta _n) \\in \\mathbb {C}^n \\mid \\zeta _i \\text{ is a $(d-1)$-th root of } \\textstyle {\\frac{1}{ds_i}} \\text{ if $i\\in C^$}\\rbrace .$ For a general choice of $(s_1,\\ldots ,s_n)$ , no element of $Z$ satisfies $\\sum _{i\\in C^} v_ix_i=0$ ." ], [ "Two genericity lemmas", "We now present the two key technical lemmas for our future intersection multiplicity computations.", "Both make essential use of the fact that the parameter $s\\in \\mathbb {C}^n$ can be chosen generically, and the second lemma uses that $\\mathbb {C}$ has characteristic zero.", "To state the first lemma, let us define a subscheme of $\\mathbb {C}^{[n]}$ $Y_\\infty (\\mathcal {L},s,d) := \\lbrace (x_1, \\ldots , x_n)\\in \\mathbb {C}^{[n]} \\mid (s_1x_1^d, \\ldots , s_nx_n^d) \\in \\mathcal {L}^\\perp \\rbrace .$ For a generic choice of $s\\in \\mathbb {C}^n$ and any integer $d\\ge 1$ , one has $\\mathcal {L}^{-1}\\cap Y_\\infty (\\mathcal {L},s,d) = \\lbrace \\mathbf {0}\\rbrace .$ Let us define a subscheme $V \\subseteq \\mathbb {C}^{[n]}\\times \\mathbb {C}^{[n]}$ by $V = \\lbrace (x,s) \\in \\mathbb {C}^{[n]}\\times \\mathbb {C}^{[n]} \\mid x \\in \\mathcal {L}^{-1} \\text{ and } (s_1x_1^d, \\ldots , s_nx_n^d) \\in \\mathcal {L}^\\perp \\rbrace .$ For any point $x$ in the dense open loci $\\mathcal {L}^{-1}\\cap (\\mathbb {C}^)^{[n]}$ of $\\mathcal {L}^{-1}$ , the set $\\lbrace s \\in \\mathbb {C}^{[n]}\\mid (s_1x_1^d, \\ldots , s_nx_n^d) \\in \\mathcal {L}^\\perp \\rbrace $ is a linear subspace of dimension $n-r$ , so the dimension of $V$ is $r + (n-r) = n$ .", "Moreover, the subscheme $V$ is bi-homogeneous, and thus the bi-projectivization $\\overline{V} \\subseteq \\mathbb {P}^{n-1} \\times \\mathbb {P}^{n-1}$ has dimension $n-2$ .", "Writing $\\pi _2$ for the projection of $\\overline{V}$ to the second $\\mathbb {P}^{n-1}$ , we hence find that the loci $\\mathbb {P}^{n-1}\\setminus \\pi _2(\\overline{V})$ is nonempty and open in $\\mathbb {P}^{n-1}$ .", "That is, the affine cone over $\\mathbb {P}^{n-1}\\setminus \\pi _2(\\overline{V})$ is dense open in $\\mathbb {C}^{[n]}$ , and $\\mathcal {L}^{-1} \\cap Y_\\infty (\\mathcal {L},s,d) = \\lbrace \\mathbf {0}\\rbrace $ for any $s\\in \\mathbb {C}^{[n]}$ in the affine cone.", "When $s = (1,\\ldots , 1)$ and $d=1$ , Lemma is the second half of Theorem .REF , which was established by an explicit Gröbner basis computation.", "For $d\\ge 2$ however, the lemma fails in general with $s = (1, \\ldots , 1)$ .", "For a generic choice of $s\\in \\mathbb {C}^n$ and any integer $d\\ge 1$ , the intersection $\\mathcal {L}^{-1} \\cap Y(\\mathcal {L}, s, d) \\cap (\\mathbb {C}^)^{[n]}$ is either empty or smooth of dimension 0.", "Without loss of generality, we assume that the $r\\times n$ matrix $A$ whose row-span equals $\\mathcal {L}$ is of the form $[I_r \\ \| \\ A^{\\prime }]$ , where $I_r$ is the $r\\times r$ identity matrix.", "For $1\\le i \\le r$ , let $a^{\\prime }_i$ be the $i$ -th row of $A^{\\prime }$ .", "The ideal of $Y$ is minimally generated by $\\mathfrak {d} = \\lbrace (s_ix_i^d - x_i) - a^{\\prime }_i \\cdot (s_{r+1}x_{r+1}^d-x_{r+1}, \\ldots , s_nx_n^d-x_n)^T\\mid 1\\le i \\le r\\rbrace .$ Fixing a generic choice of $s_{r+1}, \\ldots , s_n$ , and letting $s_1, \\ldots , s_r$ vary freely, for each $i = 1, \\ldots , r$ we may consider $(s_ix_i^d - x_i) - a^{\\prime }_i \\cdot (s_{r+1}x_{r+1}^d-x_{r+1}, \\ldots , s_nx_n^d-x_n)^T$ as a pencil $\\mathfrak {d}_i$ of hypersurfaces in $\\mathbb {C}^{[n]}$ .", "Note that the union of the base loci of $\\mathfrak {d}_1, \\ldots , \\mathfrak {d}_r$ is contained in the union of the coordinate hyperplanes.", "Thus, we obtain a map $\\mathcal {L}^{-1}\\cap (\\mathbb {C}^)^{[n]} \\rightarrow (\\mathbb {P}^1)^r$ of smooth varieties.", "By generic smoothness [6], the general fiber, which is the intersection $\\mathcal {L}^{-1} \\cap Y \\cap (\\mathbb {C}^)^{[n]}$ for a general choice of $s\\in \\mathbb {C}^n$ , is either empty or smooth of dimension 0.", "Let us now denote $\\begin{split}\\mathcal {D}(\\mathcal {L},d) :=& \\text{ the degree of the (empty or 0-dimensional) subscheme} \\\\&\\ \\mathcal {L}^{-1} \\cap Y(\\mathcal {L},s,d) \\cap (\\mathbb {C}^)^{[n]} \\subset \\mathbb {C}^{[n]}\\end{split}$ for a generic choice of $s\\in \\mathbb {C}^n$ , which is equal to the number of points in the intersection since it is smooth by Lemma .", "Theorem is now equivalently stated as $\\mathcal {D}(\\mathcal {L},d) = (-d)^r\\chi _M(\\frac{1}{d})$ , and Proposition states that $\\mathcal {D}(\\mathcal {L},2) = \\operatorname{rmld}(\\mathcal {L})$ ." ], [ "Total intersection multiplicity", "We now compute the degree of the intersection $\\mathcal {L}^{-1} \\cap Y(\\mathcal {L},s,d)$ as a subscheme of $\\mathbb {C}^{[n]}$ in two different ways.", "First, we have a Bézout-like computation.", "For a generic choice of $s\\in \\mathbb {C}^n$ , the intersection $\\mathcal {L}^{-1} \\cap Y(\\mathcal {L},s,d)$ is a 0-dimensional scheme of degree $d^r\|\\mu (M)\|$ .", "For $i = 1, \\ldots , r$ , let $\\overline{f}_i \\in \\mathbb {C}[x_0, x_1, \\ldots , x_n]$ be the homogeneous polynomial obtained as $i$ -th row of $A$ times $(s_1x_1^d - x_1x_0^{d-1}, \\ldots , s_nx_n^d - x_nx_0^{d-1})^T$ .", "We first claim that $(\\overline{f}_1, \\ldots , \\overline{f}_r)$ forms a regular sequence on the projective closure $\\overline{\\mathcal {L}^{-1}} \\subset \\mathbb {P}^n$ .", "Since the projective variety $\\overline{\\mathcal {L}^{-1}}$ is (arithmetically) Cohen-Macaulay by Theorem .REF , it suffices to show that the intersection $\\overline{\\mathcal {L}^{-1}} \\cap V(\\overline{f}_1, \\ldots , \\overline{f}_r)$ is 0-dimensional, as every system of parameters in a standard graded Cohen-Macaulay ring is a regular sequence [2].", "At the hyperplane at infinity, the intersection $V(x_0) \\cap \\overline{\\mathcal {L}^{-1}} \\cap V(\\overline{f}_1, \\ldots , \\overline{f}_r)$ is isomorphic to $\\mathbb {P}\\mathcal {L}^{-1} \\cap \\mathbb {P}Y_\\infty $ , which is empty for a generic $s\\in \\mathbb {C}^n$ by Lemma .", "On the complement of the hyperplane at infinity, the intersection is equal to $\\mathcal {L}^{-1} \\cap Y$ , since the dehomogenizations of the polynomials $\\overline{f}_1, \\ldots , \\overline{f}_r$ give the defining equations of $Y$ .", "From Theorem .REF and the definition of $Y(\\mathcal {L},s,d)$ , it follows that $\\mathcal {L}^{-1}\\cap Y \\cap ((\\mathbb {C}^)^F \\times \\lbrace 0\\rbrace ^{[n]\\setminus F}) = (\\mathcal {L}_{\|F}^{-1} \\cap Y(\\mathcal {L}_{\|F},s_{\|F},d) \\cap (\\mathbb {C}^)^F ) \\times \\lbrace 0\\rbrace ^{[n]\\setminus F}$ if $F\\subseteq [n]$ is a flat or empty otherwise, and thus Lemma applied to each flat $F$ implies that $\\mathcal {L}^{-1} \\cap Y$ is 0-dimensional.", "Thus, the degree $d$ polynomials $(\\overline{f}_1, \\ldots , \\overline{f}_r)$ form a regular sequence on $\\overline{\\mathcal {L}^{-1}}$ , and hence the degree of $\\overline{\\mathcal {L}^{-1}} \\cap V(\\overline{f}_1, \\ldots , \\overline{f}_r)$ is $d^r \\deg (\\overline{\\mathcal {L}^{-1}})$ .", "As the ideal of $\\mathcal {L}^{-1}$ homogeneous, the degrees of $\\mathbb {P}\\mathcal {L}^{-1}$ and $\\overline{\\mathcal {L}^{-1}}$ are equal, with the value being $\|\\mu (M)\|$ by Theorem .REF .", "Lastly, since $\\overline{\\mathcal {L}^{-1}} \\cap V(\\overline{f}_1, \\ldots , \\overline{f}_r)$ is empty at the hyperplane at infinity, the degree of the intersection is equal to the degree of $\\mathcal {L}^{-1} \\cap Y(\\mathcal {L},s,d)$ .", "We now compute the degree of $\\mathcal {L}^{-1} \\cap Y(\\mathcal {L},s,d)$ as the sum of contributions from the various strata of $\\mathcal {L}^{-1}$ .", "First, we need the following notation.", "Note that a 0-dimensional subscheme $X \\subset \\mathbb {C}^{[n]}$ is a union $\\bigcup _{\\alpha } X_\\alpha $ of irreducible (possibly non-reduced) components $X_\\alpha $ , each of which is topologically a point $(X_\\alpha )_{red}$ in $\\mathbb {C}^{[n]}$ .", "For a subset $I\\subseteq [n]$ , we write $X^F$ to be the subscheme of $X$ defined as the union of components of $X$ whose support is $F$ , i.e.", "$X^F := \\bigcup \\lbrace X_\\alpha \\mid (X_\\alpha )_{red} \\in (\\mathbb {C}^)^F \\times \\lbrace 0\\rbrace ^{[n]\\setminus F}\\rbrace .$ Moreover, recall the notation that $\\mathcal {D}(\\mathcal {L},d)$ denotes the degree of the subscheme $\\mathcal {L}^{-1} \\cap Y(\\mathcal {L},s,d) \\cap (\\mathbb {C}^)^{[n]} \\subset \\mathbb {C}^{[n]}$ for a generic choice of $s\\in \\mathbb {C}^n$ .", "For a generic choice of $s\\in \\mathbb {C}^n$ , and for a flat $F\\subseteq [n]$ of $M$ , the degree of $(\\mathcal {L}^{-1} \\cap Y(\\mathcal {L},s,d))^F \\subset \\mathbb {C}^{[n]}$ is equal to $\\mathcal {D}(\\mathcal {L}_{\|F}, d) \\cdot \|\\mu (M/F)\|$ .", "As topological spaces, the subscheme $(\\mathcal {L}^{-1} \\cap Y)^F$ is equal to the intersection $\\mathcal {L}^{-1} \\cap Y \\cap ((\\mathbb {C}^)^F \\times \\lbrace 0\\rbrace ^{[n]\\setminus F})$ , which is by Theorem .REF isomorphic to $\\mathcal {L}_{\|F}^{-1} \\cap Y(\\mathcal {L}_{\|F}, s_{\|F}, d) \\cap (\\mathbb {C}^)^F$ , which as a scheme is a disjoint union of $\\mathcal {D}(\\mathcal {L}_{\|F},d)$ many smooth points by Lemma .", "It remains only to show that if $\\widetilde{p}$ is an irreducible component in $(\\mathcal {L}^{-1} \\cap Y)^F$ , then the degree of $\\widetilde{p}$ is $\|\\mu (M/F)\|$ .", "For this end, we recall [4]: Suppose two Cohen-Macaulay subvarieties $X$ and $X^{\\prime }$ of complementary dimensions in a smooth variety $Z$ intersect dimensionally properly.", "Then, the degree of the intersection $X\\cap X^{\\prime }$ at a point $q$ in the intersection is equal to the product of the degrees of projectivized tangent cones $\\mathbb {P}TC_qX$ and $\\mathbb {P}TC_qX^{\\prime }$ , provided that $\\mathbb {P}TC_qX$ and $\\mathbb {P}TC_qX^{\\prime }$ are disjoint in $\\mathbb {P}T_qZ$ .", "We apply this to $\\mathcal {L}^{-1}$ and $Y$ , which are Cohen-Macaulay respectively by Theorem .REF and Lemma , and they intersect dimensionally properly by Proposition .", "Topologically $\\widetilde{p}$ is a point $p\\in (\\mathbb {C}^)^F \\times \\lbrace 0\\rbrace ^{[n]\\setminus F}$ .", "Combining Theorem .REF with Theorem .REF , one has that the degree of $\\mathbb {P}TC_p\\mathcal {L}^{-1}$ is equal to $\|\\mu (M/F)\|$ .", "The degree of $\\mathbb {P}TC_pY$ is 1 since $Y$ is smooth (Lemma ).", "Thus, we are done once we show that $\\mathbb {P}TC_p\\mathcal {L}^{-1}$ and $\\mathbb {P}TC_pY$ are disjoint.", "This is done in the following lemma.", "Let $p$ be a point in $\\mathcal {L}^{-1} \\cap Y \\cap ((\\mathbb {C}^)^F\\times \\lbrace 0\\rbrace ^{[n]\\setminus F})$ for a generic choice of $s\\in \\mathbb {C}^n$ , and for a flat $F\\subseteq [n]$ .", "Then we have $\\mathbb {P}TC_p\\mathcal {L}^{-1} \\cap \\mathbb {P}TC_pY = \\emptyset $ .", "Let $A$ be an $r\\times n$ matrix whose row-span is $\\mathcal {L}$ .", "As computed in the proof of Lemma in Equation (REF ), the tangent cone $TC_pY$ is equal to $\\ker \\big (A \\cdot \\operatorname{diag}(ds_1p_1^{d-1}-1, \\ldots , ds_np_n^{d-1}-1)\\big ) \\subset \\mathbb {C}^{[n]}.$ Since $p_i = 0$ for $i\\in [n]\\setminus F$ , and since the cocircuits of the matroid $M/F$ are cocircuits of $M$ contained in $[n]\\setminus F$ , if $x\\in TC_pY$ then $x_{\|[n]\\setminus F} \\in \\mathcal {L}_{/F}^\\perp \\subset \\mathbb {C}^{[n]\\setminus F}$ .", "On the other hand, by Theorem .REF we have $TC_p\\mathcal {L}^{-1} = TC_{p_{\|F}}\\mathcal {L}_{\|F}^{-1} \\times \\mathcal {L}_{/F}^{-1}.$ Let us now consider $x \\in TC_p\\mathcal {L}^{-1}\\cap TC_pY$ .", "We have $x_{\|[n]\\setminus F} \\in \\mathcal {L}_{/F}^\\perp \\cap \\mathcal {L}_{/F}^{-1} = \\lbrace \\mathbf {0}\\rbrace $ , where the equality follows from Theorem .REF , and thus $x = x_{\|F}\\times \\mathbf {0}$ .", "That $x\\in TC_pY$ now implies that $x_{\|F} \\in TC_{p_{\|F}}Y(\\mathcal {L}_{\|F},s_{\|F},d)$ .", "Thus, we conclude $x_{\|F}=0$ , since by Lemma the intersection $\\mathcal {L}_{\|F}^{-1} \\cap Y(\\mathcal {L}_{\|F},s_{\|F},d)$ is smooth, and in particular transversal at $p_{\|F}$ , i.e.", "$TC_{p_{\|F}}\\mathcal {L}_{\|F}^{-1} \\cap TC_{p_{\|F}}Y(\\mathcal {L}_{\|F},s_{\|F},d) = \\lbrace \\mathbf {0}\\rbrace $ .", "Combining Propositions and yields the following.", "For a generic choice of $s\\in \\mathbb {C}^n$ , we have $d^r\|\\mu (M)\| = \\deg \\big ( \\mathcal {L}^{-1} \\cap Y(\\mathcal {L},s,d)\\big ) = \\sum _{{F\\subseteq [n]\\\\ \\text{a flat of $M$}}} \\mathcal {D}(\\mathcal {L}_{\|F},d) \|\\mu (M/F)\|.$" ], [ "Inclusion-exclusion", "We now finish the proof of the main theorem by combining Corollary with an inclusion-exclusion argument.", "For the facts regarding the lattice of flats of a matroid and the Möbius invariant used here, see [21].", "[Proof of Theorem ] Write $\\operatorname{rk}: 2^{[n]} \\rightarrow \\mathbb {Z}$ for the rank function of the matroid $M$ associated to $\\mathcal {L}$ .", "Let us recall that for a matroid $M^{\\prime }$ of rank $r^{\\prime }$ , one has $(-1)^{r^{\\prime }} \\mu (M^{\\prime }) = \|\\mu (M^{\\prime })\|$ .", "Then, Corollary states that $(-d)^{\\operatorname{rk} [n]} \\mu (M) = \\sum _{{F\\subseteq [n]\\\\ \\text{a flat of $M$}}} (-1)^{\\operatorname{rk} [n] - \\operatorname{rk} F}\\mathcal {D}(\\mathcal {L}_{\|F},d) \\mu (M/F),$ or more generally, one has, for any flat $F\\subseteq [n]$ of $M$ , $(-d)^{\\operatorname{rk}F} \\mu (M\|F) = \\sum _{{F^{\\prime }\\subseteq F\\\\ \\text{a flat of $M$}}} (-1)^{\\operatorname{rk} F - \\operatorname{rk} F^{\\prime }}\\mathcal {D}(\\mathcal {L}_{\|F^{\\prime }},d) \\mu (M\|F/F^{\\prime }).$ As $\\mu (M\|F/F^{\\prime })$ is the value of the Möbius function $\\mu (F^{\\prime },F)$ on the lattice of flats of $M$ , applying the Möbius inversion formula [15] (with $f(F)=d^{\\operatorname{rk}F}\\mu ({M}\|{F})$ and $g(F)=(-1)^{\\operatorname{rk}F}\\mathcal {D}(\\mathcal {L}_{\|F},d)$ ) yields $(-1)^{\\operatorname{rk}F}\\mathcal {D}(\\mathcal {L}_{\|F},d) = \\sum _{{F^{\\prime }\\subseteq F\\\\ \\text{a flat of $M$}}}d^{\\operatorname{rk}F^{\\prime }}\\mu ({M}\|{F^{\\prime }}).$ Now, letting $F = [n]$ and noting $\\operatorname{rk} [n] = r$ , we have $(-1)^{r} \\mathcal {D}(\\mathcal {L},d) = d^{r} \\sum _{{F^{\\prime }\\subseteq [n]\\\\ \\text{a flat of $M$}}} ( \\textstyle {\\frac{1}{d}})^{r - \\operatorname{rk}F^{\\prime }}\\mu (M\|F^{\\prime }) =d^r \\chi _M(\\frac{1}{d}),$ so that $\\mathcal {D}(\\mathcal {L},d) = (-d)^r\\chi _M(\\frac{1}{d}) = d^rT_M(1-\\frac{1}{d},0)$ as desired.", "For the non-reciprocal ML degree $\\operatorname{mld}(\\mathcal {L})$ , a standard computation similar to the one that gives Proposition (see [17], [16]) yields $\\operatorname{mld}(\\mathcal {L}) = \\deg \\big (\\mathcal {L}^{-1} \\cap Y(\\mathcal {L},s,d=0) \\cap (\\mathbb {C}^)^{[n]} \\big ) = \\mathcal {D}(\\mathcal {L}, d=0).$ One can hence recover [17], which states $\\operatorname{mld}(\\mathcal {L}) = \|\\mu (M)\|$ , by minor modifications of our arguments here.", "This case is in fact simpler, with no need for the consideration of tangent cones as was done in Proposition , because the intersection $\\mathcal {L}^{-1} \\cap Y(\\mathcal {L},s,d)$ lies entirely in $(\\mathbb {C}^)^{[n]}$ when $d=0$ .", "We emphasize that $\\mathcal {L}^{-1} \\cap Y(\\mathcal {L},s,d)$ never lies entirely in $(\\mathbb {C}^)^{[n]}$ when $d\\ge 1$ .", "Combining Theorem with the “recipe formula” for Tutte-Grothendieck invariants of matroids (see for instance [20]), one has that $\\mathcal {D}(\\mathcal {L},d)$ for $d\\ge 1$ satisfies the following deletion-contraction relation given an element $e\\in [n]$ : $\\mathcal {D}(\\mathcal {L},d) ={\\left\\lbrace \\begin{array}{ll}0 &\\text{if $e$ is a loop,}\\\\(d-1) \\cdot \\mathcal {D}(\\mathcal {L}_{/e},d) & \\text{if $e$ is a coloop,}\\\\\\mathcal {D}(\\mathcal {L}_{\\setminus e},d) + d\\cdot \\mathcal {D}(\\mathcal {L}_{/e},d) & \\text{if $e$ neither loop nor coloop,}\\end{array}\\right.", "}$ where the base cases are $\\mathcal {D}(\\mathcal {L}= \\mathbb {C}^{[1]},d) = 1$ and $\\mathcal {D}(\\mathcal {L}= \\lbrace 0\\rbrace \\subset \\mathbb {C}^{[1]},d) = 0$ .", "From this deletion-contraction relation, one can verify that the following statements are equivalent: $\\mathcal {D}(\\mathcal {L},2) = \\operatorname{rmld}(\\mathcal {L}) = 1$ , $M$ is a partition matroid (i.e.", "every component of $M$ has rank 1), $\\operatorname{mld}(\\mathcal {L}) = \|\\mu (M)\| = 1$ , and $\\mathcal {L}^{-1}$ is linear.", "It may be interesting to find a proof of Theorem that directly reflects the deletion-contraction relation above geometrically.", "[Proof of Corollary ] For an $r$ -dimensional general subspace $\\mathcal {L}\\subseteq \\mathbb {C}^{[n]}$ , the associated matroid is the uniform matroid $U_{r,n}$ , for which the Tutte polynomial has the following formula (see for instance [9]) $T_{U_{r,n}}(x,y) = \\sum _{i=1}^{r}\\textstyle \\binom{n-i-1}{r-i} x^i + \\displaystyle \\sum _{j=1}^{n-r} \\textstyle \\binom{n-j-1}{r-1} y^j.$ Thus, for a general $\\mathcal {L}$ , Theorem implies that $D(\\mathcal {L},d) = d^r T_{U_{r,n}}\\left( \\textstyle {1-\\frac{1}{d}},0\\right) = \\displaystyle d^r\\sum _{1=1}^{r}\\textstyle \\binom{n-i-1}{r-i}\\left(1-\\frac{1}{d} \\right)^{i}.$ Evaluating at $d=2$ gives us $\\operatorname{rmld}(\\mathcal {L}) = 2^r\\sum _{i=1}^{r} \\textstyle \\binom{n-i-1}{r-i}\\left(\\frac{1}{2} \\right)^{i} \\displaystyle = \\sum _{i=1}^{r} \\textstyle \\binom{n-i-1}{r-i}2^{r-i},$ as desired.", "It is interesting to note that, for diagonal linear concentration models, our formula Theorem implies that $\\operatorname{rmld}(\\mathcal {L})$ is always odd unless it is zero.", "The same pattern seems to persist for reciprocal ML degrees of general linear concentration models [16]." ], [ "Acknowledgements", "We thank Yairon Cid Ruiz and Bernd Sturmfels for helpful conversations, and we thank the referee for simplifying the proof of Lemma 3.1.", "We also thank the organizers of the Linear Spaces of Symmetric Matrices working group at MPI MiS Leipzig.", "C.E.", "is partially supported by the US National Science Foundation (DMS-2001854)." ] ]	2011.14182
[ [ "Cutting and tearing thin elastic sheets: two novel single-period cracks\n and the first period-doubling crack" ], [ "Abstract Two novel single-period cracks were observed in experiments of cutting a folded sheet with a blunt object and tearing a thin brittle sheet under the guidance of a meterstick.", "Additionally, we observed a period-doubling crack in the tearing experiment.", "We cut and tore the sheet in different directions.", "The experimental results suggested that the anisotropy of the thin sheet played an important role in the formation of these two types of saw-tooth cracks.", "We demonstrated that the formation of the period-doubling crack was closely correlated with the changing of the contact region between the sheet and the meterstick.", "We also showed that the growth process of crack made by cutting was a logistic growth process (S-curve), while the cracks made by tearing propagated in the form of approximate power-law function." ], [ "Cutting and tearing thin elastic sheets: two novel single-period cracks and the first period-doubling crack Chuang-Shi Shen 1,2 Chun-Lin Du 2 Huan-Fang Wang 2 Chao Zhang 2 1 School of Civil Aviation, Zhengzhou University of Aeronautics, Zhengzhou 450015, China 2 School of Aeronautics, Northwestern Polytechnical University, Xi'an 710072, China Two novel single-period cracks were observed in experiments of cutting a folded sheet with a blunt object and tearing a thin brittle sheet under the guidance of a meterstick.", "Additionally, we observed a period-doubling crack in the tearing experiment.", "We cut and tore the sheet in different directions.", "The experimental results suggested that the anisotropy of the thin sheet played an important role in the formation of these two types of saw-tooth cracks.", "We demonstrated that the formation of the period-doubling crack was closely correlated with the changing of the contact region between the sheet and the meterstick.", "We also showed that the growth process of crack made by cutting was a logistic growth process (S-curve), while the cracks made by tearing propagated in the form of approximate power-law function.", "The classical fracture theories initially proposed by Griffith [1] fall short of predicting the path of a crack as it propagates through a solid [2].", "The study of the fracture path and the associated instabilities has been the subject of many research attempts.", "A body of unstable cracks has been observed in the laboratory including branching cracks and cracks with a rough surface [3], [4], [5], [6], [7], oscillatory cracks [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], shark-fin-like cracks [16], [17], [18], [19], [20], [21], spiral cracks [22], [23], [24], [25], [26], crescent cracks [24], tongue-like cracks [27], [28], [29], [30], zigzag cracks [31], [32], en passant cracks [33], [34], [35], helical cracks [36], [37], sideway cracks [38], [39], etc.", "Thin sheets are ubiquitous [40] such that almost everyone experienced tearing and cutting a sheet, such as opening an envelope or tearing a piece of paper in half.", "Many studies have been carried out regarding these common experiences, and a number of crack growth paths were revealed [41].", "Most previous studies assumed that a thin sheet was isotropic.", "However, most thin sheets are anisotropic because of their inherent micro-structure and their manufacturing process.", "This raises the question of whether there are any undiscovered crack propagation paths in the fracture of an anisotropic sheet.", "Moreover, as the previously discovered oscillatory cracks were all single-period cracks [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], the question remained whether there are any period-doubling cracks [42], [43].", "Here, we studied the cutting of a folded sheet with a blunt tool and the tearing of a sheet under the guidance of a meterstick.", "In both experiments, we employed anisotropic sheets of bi-axially oriented polypropylene (widely applied in fields of product packaging) of thickness $t$ =$40\\mu m$ , $53\\mu m$ with a strength that varied from 120 to 180MPa depending on the direction.", "All experiments were performed quasi-statically.", "Fig.", "REF shows a schematic of these two experiments.", "After tearing and cutting, the sheets were digitized using a scanner, and the morphology of the fracture path was measured.", "The first experiment [Fig.", "REF (a)] was performed to scrutinize the crack propagation in the process of cutting a folded thin sheet with a tool, which mimicked the process of opening an envelope with a knife.", "The cutting process was as follows.", "First, a thin sheet was folded in half.", "No crease was formed in the sheet during the folding process.", "Second, this folded sheet was placed on a platform.", "Third, a plane was put on it and an external force was applied on the top surface of this plane, and an initial notch (orange) was cut with a knife.", "Finally, a blunt object (red) was utilized to cut this sheet.", "Two types of tools were used in terms of shape: rectangular and circular.", "The angle between the tool and the sheet was $\\alpha {\\rm { = 3}}{{\\rm {5}}^ \\circ } \\pm {{\\rm {5}}^ \\circ }$ .", "Figure: (color online) Schematic of the experimental set up: (a) cutting; (b) tearing.Figure: (color online) (a) Scanned photographs of fracture path cut by a rectangular tool with a thickness of 2.2 mm (from top to bottom, β≈60 ∘ ,50 ∘ ,40 ∘ ,30 ∘ ,20 ∘ ,10 ∘ ,0 ∘ \\beta \\approx {60^ \\circ },{50^ \\circ },{40^ \\circ },{30^ \\circ },{20^ \\circ },{10^ \\circ },{0^ \\circ }) and a circular tool with a thickness of 3 mm (from top to bottom, β≈50 ∘ ,40 ∘ ,30 ∘ ,20 ∘ ,10 ∘ ,0 ∘ \\beta \\approx {50^ \\circ },{40^ \\circ },{30^ \\circ },{20^ \\circ },{10^ \\circ },{0^ \\circ } ), the sheets with thickness of 40um.", "(b) The phase diagram of the two kind of cracks.", "The line (β∝a(1-e (-s/b) ){\\beta } \\propto a(1 - {e^{( - s/b)}}), up: a=57a=57, b=0.45b=0.45; down: a=47a=47, b=0.32b=0.32), drawn to guide the eye, indicates the phase boundary between the straight and non-straight cracks.", "(c) The wavelength of the cracks cut by rectangle (0.4, 1, 2.2mm) and circular (0.5, 1.5, 3mm) tool as function of β\\beta .", "The sheets with thickness of 40μm40\\mu m (red) and 53μm53\\mu m (green).", "The symbols ◯\\bigcirc and ♦\\diamondsuit represent the circular and rectangular tool, respectively.Figure: (color online) Sequence of the crack propagation for β≈0 ∘ \\beta \\approx {0^ \\circ }.", "(a) Initial symmetrical crack.", "The crack grows (b) along an upward inclined direction, (c) along a downward inclined direction, and (d) along an upward inclined direction.Examples of the crack formed by cutting with two kinds of tools are shown in Fig.", "REF (a).", "The crack path was related to the shape of the tool, which was different from the fracture path in clamped thin sheets [16], [17], [18], [19], [20], [21].", "The fracture path is a straight line when $\\beta > {\\beta ^ * }$ (${\\beta }$ was the angle between $x$ -direction and the direction of the least strength of sheet).", "The value of ${\\beta ^ * }$ was associated with the size $s$ (thickness or diameter) of the tool.", "As shown in Fig.", "REF (b), ${\\beta ^ * }$ increased dramatically at the small size, but the rate of increase slowed down gradually with a further increase of the tool size.", "It was expected that ${\\beta ^ * }$ would reach a saturation value when the thickness exceeded a certain threshold.", "The transition boundary between the straight crack and the non-straight crack followed ${\\beta } \\approx a(1 - {e^{( - s/b)}}$ .", "For sufficiently thin tools, ${\\beta ^ * } = 0$ , the crack path was a straight line, as expected.", "The amplitude (the peak-to-valley distance in the $y$ -direction) of the fracture path was approximately the same as the thickness/diameter of the tool.", "The wavelength $\\lambda $ (the peak-to-peak distance in the $x$ -direction) of the fracture path was dependent on the ${\\beta }$ , and $\\lambda \\propto {1 \\mathord {\\left\\bad.", "{\\vphantom{1 {\\cos \\beta }}} \\right.\\hspace{0.0pt}} {\\cos \\beta }}$ [Fig.REF (c)].", "These results suggested the formation of no-straight cracks have something to do with the anisotropy of the sheets.", "A possible explanation is as follows.", "For smaller $\\beta $ , such as $\\beta \\approx {0^ \\circ }$ (the whole system was not completely symmetrical, it was difficult to ensure ${\\beta ^ * } = 0$ ), as the tool moved toward an initial symmetrical crack [Fig.", "REF (a), also see Supplementary Movies [44]], the crack opening displacement had to be increased to accommodate the tool motion.", "Hence, the sheet was subjected to two forces (in-plane force ${F^I}$ and out-of-plane force ${F^O}$ ).", "Because $\\sigma _{\\rm {w}}^ * < \\sigma _{\\rm {s}}^ * $ ($\\sigma _s^ * $ , $\\sigma _w^ * $ denote the strength of the strong and weak direction of the sheet, respectively), the crack tended to grow along an upward-inclined direction rather than the cutting direction (perpendicular to the strongest direction) [Fig.", "REF (b)].", "The out-of-plane force applied by the tool to the upper part of the sheet was decreased to zero when the crack propagated toward the upward edge of the tool.", "The crack began to propagate along the cutting direction under the action of in-plane tensile force, and the out-of-plane force applied on the lower part of the sheet showed an increasing trend during this process.", "When the out-of-plane force increased to a critical value, the crack began to grow along a downward-inclined direction [Fig.", "REF (c)].", "Then it reached the downward edge of the tool and ultimately propagated along the cutting direction.", "Then, the above processes were repeated until the cutting process was completed.", "However, for larger $\\beta $ , such as $\\beta \\approx {90^ \\circ }$ , although the sheet was also subjected to the out-of-plane force of the tool, the crack was not affected by this force since $\\sigma _{\\rm {w}}^ * < \\sigma _{\\rm {s}}^ * $ .", "The crack in this case always followed a straight path parallel to the cutting direction [see Supplementary Movies [44]].", "There were four stages in each fracture period: I, II, III, IV [Fig.", "REF (a)].", "Among them, the propagation direction of the crack in stages I and III had a certain angle with the cutting direction.", "For $\\beta > 0$ , the crack in stage I basically propagated along the orthogonal direction of the cutting, while the crack in stage III did not.", "This was because ${\\theta }$ (the angle between the propagation direction of crack and the direction of the least strength of sheets) in these two stages was different: ${\\theta _{III}} < {90^ \\circ }$ in stage III, and ${\\theta _I} > {90^ \\circ }$ in stage I.", "The absolute value of the slope $S$ of the crack in stage III was negatively correlated with $\\beta $ .", "In stages II and IV, the crack propagated along the cutting direction.", "However, for larger $\\beta $ , the length $l$ of the crack in stage IV was significantly longer than that in stage II.", "This could be attributed to the different crack paths in stages I and III.", "The crack path in stage I was nearly perpendicular to the cutting direction, i.e.", "${S_I} > {S_{III}}$ .", "Thus, the out-of-plane force applied on the crack of stage I was larger than that on the crack of stage III, i.e.", "$F_I^o > F_{III}^o$ , thus ${l_{II}} < {l_{IV}}$ .", "The slope of the crack in stage I was not sensitive to $\\beta $ , hence, the length of the crack in stage II changed a little with $\\beta $ [Fig.", "REF (a)].", "Figure: (color online) (a) Details of the cutting crack path (β≈30 ∘ \\beta \\approx {\\rm {3}}{0^ \\circ }, t=40μmt = 40\\mu m).", "(b) The growth parameter kk for different β\\beta .The formation process of the cutting cracks could be explained by using the logistic growth model.", "The formation of the non-straight crack was related to the out-of-plane force ${F^o}$ .", "Therefore, the growth rate of the crack ${{dy}\\mathord {\\left\\bad.", "{\\vphantom{{dy} {dx}}} \\right.\\hspace{0.0pt}} {dx}}$ was positively correlated with ${F^o}$ , i.e.", "${{dy} \\mathord {\\left\\bad.", "{\\vphantom{{dy} {dx}}} \\right.\\hspace{0.0pt}} {dx}} \\sim {F^o}$ .", "${F^o}$ increased initially and then diminished during the formation of the crack in stage I and stage III, as did ${{dy} \\mathord {\\left\\bad.", "{\\vphantom{{dy} {dx}}} \\right.\\hspace{0.0pt}} {dx}}$ .", "Here we assumed ${{dy} \\mathord {\\left\\bad.", "{\\vphantom{{dy} {dx}}} \\right.\\hspace{0.0pt}} {dx}}$ followed the logistic growth model as follows: $\\frac{{dy}}{{dx}} = ky(1 - \\frac{y}{s})$ where $k$ is the growth parameter.", "The size $s$ of the tool is the limit value of the function of the crack path $y$ .", "The solution of Eq.", "(1) is $y = \\frac{s}{{1 + {e^{ - k(x - {x^ * })}}}}$ The growth parameter $k$ should be related to $\\beta $ .", "However, because the cutting process was highly nonlinear, it was difficult for us to accurately construct a formula to describe the relationship between $k$ and $\\beta $ .", "The growth parameter $k$ was estimated by fitting Eq.", "(2) to experimental data.", "Fig.", "REF (a) shows that the prediction of the shape of the cutting crack, $y$ (the solid line) by logistic growth model, was in good agreement with the experiments (the green scatter).", "For stage II to stage IV, the fitting results suggested that the parameter $k$ (absolute value) reduced with the increase of $\\beta $ [Fig.", "REF (b)], as expected.", "However, for stage IV to stage II (black), $k$ first increased and then decreased with the increase of $\\beta $ .", "The physical reasoning of this phenomenon still needs to be uncovered.", "Figure: (color online) (a) Typical scanned photographs of the tearing cracks (From top to bottom, β≈40 ∘ ,30 ∘ ,20 ∘ ,15 ∘ ,0 ∘ \\beta \\approx {40^ \\circ },{30^ \\circ },{20^ \\circ },{15^ \\circ },{0^ \\circ }), period-doubling cracks (Light blue region), the sheet with thickness of 53um.", "(b) The average wavelength and the average amplitude of the crack paths.", "(c) The ratio of wavelength to amplitude as a function of β\\beta .The second experiment [Fig.", "REF (b)] was set up to examine the crack propagation during the tearing of a sheet under the guidance of a meterstick.", "The tearing process was as follows: first, a thin sheet was placed on a platform.", "Then a meterstick ($l$ =160mm, $w$ =150mm) was set on the sheet and an external force was applied on each end of this meterstick to ensure that the sheet did not slide out during the tearing process.", "Then an initial notch (orange) was cut with a knife.", "Ultimately, this sheet was torn along a line (red dashed line).", "The angle $\\varphi $ between this line and the edge of the sheet was about ${\\rm {4}}{{\\rm {5}}^ \\circ }$ .", "Fig.", "REF (a) presents typically scanned photographs of the tearing cracks.", "It can be observed that the cracks formed by the tearing process were completely different from the cracks formed by cutting [Fig.", "2(a)].", "Depending on the angle $\\beta $ , the crack grew straight ($\\beta > {55^ \\circ }$ ) or non-straight around the meterstick.", "For $\\beta < {20^ \\circ }$ , the cracks in thick sheets ($53\\mu m$ ) followed a period-doubling (one small and one large [42], [43]) path when the tearing distance exceeded a critical value.", "In contrast, period-doubling cracks were rarely formed in thin sheets ($40\\mu m$ ).", "During the last 40 years, several different types of single-period cracks have been reported, including the oscillating fracture in thermal quenching experiments [8], bi-axially stretched rubber [10], pure uniaxial tension of thin brittle gels [10], clamped thin sheets [16], [17], [18], [19], [20], [21] and thin coatings [24].", "However, period-doubling cracks have never been reported in the literature.", "The cracks presented here are the first period-doubling cracks.", "We measured the average wavelength and amplitude of the fracture paths.", "As shown in Fig.", "REF (b), the wavelength/amplitude increased during the tearing process because of the changing of the loading direction (the crack tip to the loading point).", "The relation between the wavelength/amplitude and $\\beta $ was not monotonic function.", "We had no good explanation for this observation.", "Additional research is needed to confirm and explain this observation.", "However, the ratio of wavelength to amplitude increased exponentially with the increase of $\\beta $ [Fig.", "REF (c)], i.e.", "${\\lambda \\mathord {\\left\\bad.", "{\\vphantom{\\lambda A}} \\right.\\hspace{0.0pt}} A} \\approx c + d{e^{Rx}}$ ,the value of $c$ , $d$ and $R$ were related to the thickness of the sheet ($40\\mu m$ : $c = 2.12$ , $d = 0.3$ and $R = 17.15$ ; $53\\mu m$ : $c = 2.10$ , $d = 0.347$ and $R = 20$ ).", "Figure: (color online) (a) The crack first propagated along the meterstick in a period, and (b) then propagated along the vertical direction of the meterstick.", "The moment when the large tooth (c) and the small tooth (d) formedFigure: (color online) Details of the tearing-crack path (t=53μmt = 53\\mu m).Like the cutting configuration, the formation of the crack formed by tearing was also related to the anisotropy of the material.", "Its formation process could also be divided into four stages: I, II, III, and IV [Fig.", "REF (a) and Fig.", "REF ].", "However, the formation process of the tearing and cutting cracks were entirely different.", "The tearing cracks in stage I and stage IV were formed on the side of the meterstick ($x$ -$z$ plane), while the cracks in stage II and stage III were formed on the upper surface of the meterstick ($x$ -$y$ plane).", "For smaller $\\beta $ , such as $\\beta \\approx {{\\rm {0}}^ \\circ }$ , the crack (mode I+II) in stage II propagated along the meterstick [Fig.", "REF (a), also see Supplemental Movies [44]] under two in-plane forces ${F_x}$ and ${F_y}$ (the crack tended to grow along X (Y) direction under ${F_x}$ (${F_y}$ )).", "During this process, the tooth gradually became larger.", "As a result, the stress ${\\sigma _x}$ in the crack tip gradually increased, i.e.", "${\\sigma _x} \\propto \\int _{\\rm {0}}^L {{\\tau _x}} dx \\propto L$ .", "When the stress ${\\sigma _x}$ exceeded a certain critical value, the crack (mode I+III) then propagated (stage III) along the vertical direction of the meterstick [Fig.", "REF (b), also see Supplemental Movies [44]] due to $\\sigma _{\\rm {w}}^ * < \\sigma _{\\rm {s}}^ * $ , mainly under in-plane force ${F_x}$ , since the out-of-plane force ${F^O}$ applied by the meter-stick to the crack tooth was small because the sheet was very thin.", "However, for larger $\\beta $ , such as $\\beta \\approx {\\rm {9}}{{\\rm {0}}^ \\circ }$ , the crack always followed a straight path parallel to the meterstick for the same reason [see Supplemental Movies [44]].", "The shape of the tearing cracks [Fig.", "REF ] was well expressed by the following equation: $y = \\left\\lbrace \\begin{array}{l}A{\\left\| {x - {x^ * }} \\right\|^{{p_{_l}}}}\\begin{array}{{20}{c}}{}&{x < {x^ }}\\end{array}\\\\A{\\left\| {x - {x^ * }} \\right\|^{{p_{_{r}}}}}\\begin{array}{{20}{c}}{}&{x > {x^ }}\\end{array}\\end{array} \\right.$ where ${x^ * }$ is the valley of the tearing crack, which indicates that the tearing cracks grow in the form of power-law [45], rather than S-curve.", "To understand why the period-doubling cracks were formed, we recorded the formation process of the tearing cracks [see Supplemental Movies [44]].", "The deformation morphology of the sheet when the large tooth and the small tooth had just been formed is shown in Fig.", "REF (c) and REF (d), respectively.", "The red line is a boundary.", "On its left side, the sheet was attached to the meterstick.", "On its right side, we found that the boundaries in Fig.", "REF (c) and REF (d) were very different.", "In Fig.", "REF (c), the boundary was a straight line.", "Conversely, the boundary was a curve in Fig.", "REF (d).", "We watched the recorded video repeatedly.", "We found that the straight boundary did not appear in the formation process of the single-period cracks.", "It seemed that the straight boundary appeared only after the formation of the large tooth.", "A small tooth was always formed after its appearance.", "In conclusion, we have reported two new period-single cracks and a period-doubling crack in cutting and tearing thin elastic sheets.", "We showed that they stemmed from the anisotropy of the sheet and the interaction between the sheet and the tool or meterstick.", "The experimental results of the present study can be a motivation for the development of new theoretical models to accurately predict the growth process of these types of cracks and to understand how the anisotropy of material affects the fracture path [46], [47], [48].", "These robust cracks can be employed as good test cases for theoretical models that couple anisotropy of material and fracture." ] ]	2011.14103
[ [ "Differential rotation of the chromosphere in the He I absorption line" ], [ "Abstract Differential rotation is the basis of the solar dynamo theory.", "Synoptic maps of He I intensity from Carrington rotations 2032 to 2135 are utilized to investigate the differential rotation of the solar chromosphere in the He I absorption line.", "The chromosphere is surprisingly found to rotate faster than the photosphere below it.", "The anomalous heating of the chromosphere and corona has been a big problem in modern astronomy.", "It is speculated that the small-scale magnetic elements with magnetic flux in the range of $(2.9 - 32.0)\\times 10^{18}$ Mx which are anchored in the leptocline, heat the quiet chromosphere to present the anomalous temperature increase, causing it to rotate at the same rate as the leptocline.", "The differential of rotation rate in the chromosphere is found to be strengthened by strong magnetic fields, but in stark contrast, at the photosphere strong magnetic fields repress the differential of rotation rate.", "A plausible explanation is given for these findings." ], [ "Introduction", "The sun is magnetic, and solar activity and variation are both related to solar magnetic fields (Fang et al.", "2008).", "Solar differential rotation is a classical research topic in solar physics due to its close relationship with solar magnetic fields (Babcock 1961), and measurements of differential rotation in the solar atmosphere and especially in the solar interior can provide some clues for solar dynamo theory.", "Two kinds of methods have been utilized to measure the differential rotation in the solar atmosphere.", "One is direct observation measurements, such as positioning and tracking of solar long-life structures (tracers) and Doppler spectrum observations, and the other is the periodicity analyses of long-term solar activity indexes modulated by solar rotation (Howard 1984; Javaraiah 2003; Chandra, Vats, $\\&$ Iyer 2010; Vats $\\&$ Chandra 2011; Bhatt et al.", "2017; Li et al.", "2019; Xiang, Ning, $\\&$ Li 2020).", "Additionally, helioseismic measurements have recently used to detect interior motions of the sun, such as the inner zonal and meridional flows, and the flows are the foundation for understanding the differential rotation of the solar atmosphere (Howe 2009).", "Sunspots are the most classical tracer to measure the rotation rate of the solar atmosphere.", "Generally magnetic elements in quiet regions rotate faster than sunspots in active regions (Howard 1984; Xiang et al.", "2014; Xu $\\&$ Gao 2016), and they both rotate faster than the photospheric atmosphere (Stenflo 1989; Beck 2000; Lamb 2017; Li et al.", "2019; Bertello et al.", "2020).", "Recently Sudar et al.", "(2016) and Li et al.", "(2019) found that the corona anomalously rotates faster than the photosphere below it, although temperature and rotation rate should theoretically decrease (actually increase) from the bottom of the atmosphere, the photosphere to a higher layer of the solar atmosphere, because thermal energy and material of the solar atmosphere flow outward from the photosphere.", "Combined with the abnormal heating of the upper solar atmosphere is abnormally the rapid rotation of the coronal atmosphere.", "The chromosphere is daily observed in the He I $10830Å$ line at National Solar Observatory (NSO)/Kitt Peak (Livingston et al.", "1976).", "In this study, synoptical maps observed in the line at NSO are utilized to determine the rotation rate of the chromosphere." ], [ "Differential rotation of the solar quiet chromosphere", "The solar chromosphere had been routinely surveyed in the He I $10830Å$ line by the Vacuum Telescope at NSO/Kitt Peak from July 2005 to March 2013 (Livingston et al.", "1976; Harvey $\\&$ Livingston 1994), and correspondingly, synoptic maps of He I intensity ($I_{He}$ ) from Carrington rotations 2032 to 2135 were obtained.", "They can be downloaded from the NSO's web site: ftp://nispdata.nso.edu/kpvt/synoptic/, and the web site, https://solis.nso.edu/0/vsm/aboutmaps.html introduced how to create a synoptic map from daily observations.", "Each synoptic map of He I intensity was measured at 360 equidistant longitudes from $1^{\\circ }$ to $360^{\\circ }$ and 180 latitudes ($\\varphi $ s), and these measurement latitudes have 180 equal steps in the sine of the latitudes ($sin(\\varphi )$ ) ranging from -1 (the solar south pole) to +1 (the north pole).", "Here Figure 1 shows all synoptic maps and an individual map, the synoptic map at Carrington rotation 2057 as an example.", "The unit of He I line intensity in these synoptic maps is arbitrary but constant.", "The He I line is in absorption at the solar disk (Fleck 1994; Harvey 1994; Brajsa et al.", "1996).", "Observations of the quiet chromosphere disk are unique in the He I line, that is, structures with strong magnetic field, such as general network bright points, bright calcium and hydrogen flocculi are dark in the line, displaying absorption features with relatively low intensities, instead of bright structures in the chromosphere generally observed by other chromosphere lines, such as $H_{\\alpha }$ , Ca II, Mg II, and so on (Harvey $\\&$ Sheeley 1977; Golub et al.", "1989; Fleck et al.", "1994; Brajsa et al.", "1996).", "Therefore the quiet chromosphere can be approximated through removing low-intensity values, and next in order to achieve this goal, we will count the intensity distribution of these synoptic maps to evaluate the proportion of low intensity values in the entire data.", "Resultantly, Figure 2 shows the distribution of the original data points of $I_{He}$ intensity in the range of $-200 \\sim 100$ , and those points beyond the range account only for $ 0.54\\%$ of the total points.", "We don't know why the range of $I_{He}$ intensity is very wide, from the minimum of -2398.1 to the maximum of 3034.9.", "When $I_{He}$ is less respectively than -50, -30, and -10, the ratio of the number of data points to the total is $4.70\\%$ , $8.31\\%$ , and $17.35\\%$ , correspondingly.", "When $I_{He}$ s are in the range of $-200 \\sim 100$ , and meanwhile just those latitudes which are not larger than $30^{\\circ }$ are considered, the distribution of $I_{He}$ s is counted up, which is shown in the figure as well, and then when $I_{He}$ is less respectively than -50, -30, and -10, the ratio of the number of data points to the total is is $5.49\\%$ , $8.57\\%$ , and $17.84\\%$ , correspondingly.", "Therefore active regions of sunspots with strong magnetic fields are basically excluded, if those $I_{He}$ s which are less than -10 are not considered, and then the quiet chromosphere can be approximated.", "Because dark structures in the He I chromosphere are usually observed/measured in more than one day, we can derive their rotation rates.", "The classical autocorrelation analysis method for a time series is used to determine the rotation period of the series (Xu $\\&$ Gao 2016; Li $\\&$ Feng 2019).", "For a time series $X_{i}$ (i=1, 2,..., N), its autocorrelation coefficients is 1 when there is no relative shift, where N is the number of data points of the series.", "First, the series is one-point shifted with respect to itself, and the unpaired endpoints of the two involved series are deleted.", "At this time, one series is $X_{i}$ (i=1, 2, ..., N-1), the other is $X_{i}$ (i=2, 3, ..., N), and the correlation coefficient ($CC$ ) can be obtained for these two series.", "Next, the original series is two-point shifted with respect to itself, and the unpaired endpoints of the two series are deleted.", "At this time, one series is $X_{i}$ (i=1, 2, ..., N-2), the other is $X_{i}$ (i=3, 4, ..., N), and a new correlation coefficient can be obtained then.", "The above process is repeated again and again, until relative phase shift is 500 data points.", "If there are no observations of $I_{He}$ s for a Carrington rotation available, $I_{He}$ s for the Carrington rotation are given to a very large negative value.", "When the negative value appears in the calculation of a correlation coefficient ($CC$ ), it and its paired value are ignored, and just the rest data are involved in the calculation.", "For a time series at a certain latitude, $CC$ is calculated to vary with relative phase shift of the series vs itself, and the shift which corresponds to the local maximum $CC$ around shifts of 27 days is regarded as the rotation period of the series.", "As an example Figure 3 shows the calculation results ($CC$ varying with relative phase shift) of 6 time series.", "Finally, 180 rotation periods are obtained at the corresponding 180 measurement latitudes, whose calculation results are given in an animation.", "In Figure 3, CR is the length of a Carrington synodic rotation period, and 1 CR = 27.275 days, therefore the resolution of relative phase shifts is $27.275/360\\approx 0.076$ (days).", "After the synodic rotation period ($P$ ) of a series being obtained, its rotation rate can be then gotten to be $P\\times 13.199$ (in degrees/day, Shi $\\&$ Xie 2013; Deng et al.", "2020), which is shown in Panel I of Figure 4, where 13.199 degrees/day is the rotation rate corresponding to the Carrington synodic period.", "Meanwhile Panel I of Figure 5 shows $CC$ corresponding to each obtained rotation period, and all $CC$ s in the panel are significant at the $99\\%$ confidence level.", "Differential rotation rate ($\\Omega (\\varphi )$ ) is generally expressed as $\\Omega (\\varphi )=A + B sin^{2}(\\varphi ) + C sin^{4}(\\varphi ).$ The expression is used to fit those rotation rates whose latitudes are not higher respectively than $30^{\\circ }$ , $35^{\\circ }$ , and $40^{\\circ }$ .", "Resultantly, $CC$ s for the fitting lines are 0.946, 0.963, and 0.938 in turn, which are all significant at the $99\\%$ confidence level.", "Similarly, the autocorrelation analysis method is also used to determine rotation periods of those $I_{He}$ s which are not less respectively than -50, -30, and -10, and resultantly, Panels II, III, and IV of Figure 4 show the obtained rotation rate at each measurement latitude in turn.", "Correspondingly, Panels II, III, and IV of Figure 5 display $CC$ at each measurement latitude in turn.", "For the sake of brevity, the rotation rates determined from $I_{He}$ s which are not less respectively than -50, -30, and -10 are called RR II, RR III, and RR IV in turn, and those determined from the complete original data are called RR I.", "The expression is used to fit those rotation rates whose latitudes are not higher respectively than $30^{\\circ }$ , $35^{\\circ }$ , and $40^{\\circ }$ , and the obtained fitting lines are shown in Figure 4.", "These fitting lines are all highly significant.", "The fitting lines respectively for RR I to RR IV at latitudes which are not higher than $30^{\\circ }$ are put together at the left top panel of Figure 6.", "Figure 6 also shows the fitting lines respectively to RR I to RR IV at latitudes which are not higher than $35^{\\circ }$ at its left bottom panel, and at the right panel of the figure displayed are the fitting lines respectively to RR I to RR IV at latitudes which are not higher than $40^{\\circ }$ .", "As Figure 6 displays, rotation slightly becomes fast at low latitudes after eliminating the influence of strong magnetic fields of active regions, and this result is hardly affected by the low-latitude range considered.", "The Doppler rotation rate of the photosphere is also drawn in Figure 6, which was given by Snodgrass et al.", "(1984).", "The Doppler rotation rate is less than the rotation rate of the quiet chromosphere, and the (quiet) chromosphere rotates faster than the photosphere overall.", "RR I whose latitudes are not higher respectively than $60^{\\circ }$ , $70^{\\circ }$ , and $80^{\\circ }$ are fitted with the following expression of rotation rate, $\\Omega (\\varphi )=A + B sin^{2}(\\varphi ) + C sin^{4}(\\varphi ) + D sin^{6}(\\varphi )$ , and resultantly, the obtained fitting lines are shown in Panel I of Figure 7.", "These fitting lines are all highly significant.", "Due to low rotation rates around latitudes of about $50^{\\circ }$ , the fitting line to the latitudes which are not higher than $60^{\\circ }$ does not show a monotonous increase from low to high latitudes, but the other two fitting lines do.", "Similarly, the expression is used to respectively fit RR II, RR III, and RR IV, at latitudes which are not higher respectively than $60^{\\circ }$ , $70^{\\circ }$ , and $80^{\\circ }$ , and resultantly, Panels II, III, and IV of Figure 7 correspondingly display the fitting lines in turn.", "All these fitting lines show a clear decrease trend of rotation rate from low to high latitudes.", "Therefore for the quiet chromosphere, rotation rate decreases from the equator to high latitudes of about $70^{\\circ } \\sim 80^{\\circ }$ on the whole, and at latitudes of about $40^{\\circ }$ onwards, rotation rate fluctuates greatly." ], [ "Conclusions and Discussion", "Synoptic maps of He I intensity, which are measured in the interval of Jul.", "2005 to Mar.", "2013, are used to determine the rotation rate of the full-disk chromosphere observed in the He I $10830Å$ absorption line.", "In the two cases of subtracting and not subtracting large magnetic-field values in active regions from the original data, the chromosphere is surprisingly found to rotate faster than the photosphere beneath it.", "Energy and material flow from the solar interior, through the photosphere to the chromosphere.", "Since the rotation of the chromosphere is obviously different from the photosphere rotation, the driving force for the chromosphere to rotate in this way is certainly from the solar interior, not from the photosphere.", "Helioseismic measurements indicate that, at and just at the depths (the so-called leptocline) of $0.99 R_{\\odot }$ to near $1 R_{\\odot }$ of the solar interior, the long-term variation of the solar seismic radius is in anti-phase with the Schwabe cycle, and the in-situ magnetic fields are generated (Lefebvre et al.", "2009), here $R_{\\odot }$ is the solar radius.", "The quiet chromosphere is found to be also in anti-phase with the Schwabe cycle (Li et al.", "2020).", "Therefore, the driving force for the quiet chromosphere to rotate in such a way is believed to come from the leptocline in all probability, since the two present the same cycle phase.", "The small-scale magnetic elements ($SME$ s) whose magnetic flux spans $(2.9 - 32.0)\\times 10^{18}$ Mx are in anti-phase with the Schwabe cycle and distributed all over the solar disk (Jin et al.", "2011; Jin $\\&$ Wang 2012).", "They are thought to be the “in-situ magnetic fields\" generated in the leptocline, due to their same cycle phase.", "It is these small-scale magnetic elements that heat the chromosphere to lead to its formation, so that its rotation is traced by the rotation of the leptocline.", "Therefore, the quiet chromosphere, $SME$ s, and the leptocline are all in anti-phase with the solar cycle.", "Temperature changes from the outward decreasing in the photosphere to the outward increasing in the chromosphere.", "Studies of modern observations, theory, and statistics have showed that the quiet chromosphere is heated mainly by small-scale magnetic activities (Parker 1991; De Pontieu et al.", "2017; Li et al.", "2018), supporting the primary role of the magnetic elements in the heating and formation of the quiet chromosphere.", "The magnetic elements are anchored in the solar interior, connecting the chromosphere to the leptocline and transmitting the long-term evolution characteristics of the leptocline to the quiet chromosphere.", "Small-scale magnetic activities not only cause the quiet chromosphere to be abnormally heated, but also cause it to abnormally rotate faster than the photosphere plasma.", "The rotation rate of small-scale magnetic elements is generally larger than that of sunspots, and they both are larger than that of plasma in the photosphere (Stenflo 1989; Howard 1996; Xiang et al.", "2014; Xu $\\&$ Gao 2016; Lamb 2017).", "This is the reason why the rotation rate of the quiet chromosphere is found here to be larger than that of the photosphere.", "When strong magnetic fields are not eliminated, the rotation rate of the chromosphere decreases from the equator to middle latitudes faster than that when they are eliminated, and further, the more data points of strong magnetic fields are eliminated, the slower the rotation rate of the chromosphere decreases from the equator to middle latitudes.", "This means that strong magnetic fields should strengthen the differential of rotation rate.", "However in sharp contrast, strong magnetic fields are found to repress the differential of rotation rate in the photosphere (Brajsa et al.", "2006; Li et al.", "2103).", "Sunspots generally appear at low latitudes ($8^{\\circ } \\sim 35^{\\circ }$ ), and their rotation rates are shown in Figure 6, which come from Howard (1996).", "As the figure shows, the rotation rate of sunspots is lower than that of the quiet chromosphere at the low latitudes, and thus the appearance of sunspots at a certain latitude should increase the difference of rotation rate between the latitude and the equator.", "Therefore, the differential of rotation rate in the chromosphere is strengthened by strong magnetic fields.", "The rotation rate of sunspots is larger than that of photosphere plasma, and thus the appearance of sunspots at a certain latitude should decrease the difference of rotation rate between the equator and the latitude.", "Therefore, the differential of rotation rate in the photosphere is repressed by strong magnetic fields.", "Like an “intruder\", active regions of sunspots form an invasion zone into the quiet chromosphere, namely the so-called “butterfly diagram\", and the quiet chromosphere superimposed with the butterfly pattern is the observed chromosphere (Li et al 2020), as Figure 1 displays here.", "Similarly, active regions of sunspots, which take along the general rotation characteristics of the tachocline beneath the lepocline, invade the background flow field of the leptocline, forming a low-speed band with the “butterfly diagram\" migration, which is the observed zonal flow at the leptocline.", "At the same time, a high-speed band with the “butterfly diagram\" migration (the so-called torsional oscillations) appears in the photosphere (Howard $\\&$ LaBonte 1980).", "Therefore in the photosphere, the rotation rate of sunspots is larger than that of the photosphere plasma, and the differential of rotation rate is depressed by the rotation of sunspots.", "All in all, rotation activities in the solar atmosphere are highly related to the magnetic activities anchored in the solar interior, and $SME$ s are inferred to originate from the leptocline, while sunspots should come from deeper layers of the interior, probably from the tachocline.", "Among the used data of 104 rotations, 7 rotations are missed (Carrington rotations 2041, 2042, and 2091 to 2095), accounting for $6.7\\%$ of the total.", "When the missing data are filled with adjacent Carrington rotations, the obtained result is almost the same as that when missing rotations are left out of the analysis.", "Therefore, such the small data loss does not have a significant impact on the obtained conclusion.", "Synoptic maps of He I intensity are affected by both the projection around the solar poles and the limb darkening, so rotation rates are of lower reliability at high latitudes.", "This is inferred to be one of the reasons for a large scatter of the chromosphere rotation rates at high latitudes.", "Zhang, Zirin, $\\&$ Marquette (1997) also found a great fluctuation in the rotation rates at high latitudes of over $55^{\\circ }$ .", "At high latitudes, the magnetic fields are largely unipolar (Stenflo 1989), and their dependence on time is not explicit (Zhang, Zirin, $\\&$ Marquette 1997), that is, it is relatively difficult to distinguish the appearance and disappearance of a single magnetic field in the time series of magnetic fields, so the determination of the rotation rate of magnetic fields is of greatly uncertainty.", "This is the main reason why the autocorrelation profile of a time series of magnetic fields is wider at high latitudes than that at low latitudes, whose peak overlays a slowly decaying background (Stenflo 1989), and a reason why a large scatter of the chromosphere rotation rates appears here at high latitudes.", "In this study, latitudes higher than $80^{\\circ }$ are not considered, and three latitude bands ($0 - 60^{\\circ }$ , $0 - 70^{\\circ }$ , and $0 - 80^{\\circ }$ ) are separately considered, in order to somewhat present the influence of these factors on the obtained results.", "In these three cases, rotation rate for the quiet chromosphere is found to present a decrease trend from the equator to high latitudes as well, but this finding does not have high credibility and need be demonstrated by the precise observations of solar polar regions in the future.", "acknowledgments: We thank the anonymous referee for careful reading of the manuscript and constructive comments which improved the original version of the manuscript.", "The full-disk synoptic maps of He I intensity are courtesy to be publicly downloaded from the NSO's web site.", "The authors would like to express their deep thanks to the staffs of the web site.", "This work is supported by the National Natural Science Foundation of China (11973085, 11903077, 11803086, 11703085, and 11633008), the Yunnan Ten-Thousand Talents Plan (the Yunling-Scholar Project), the national project for large scale scientific facilities (2019YFA0405001), the CAS “Light of West China\" Program, and the Collaborating Research Program of CAS Key Laboratory of Solar Activity (KLSA201912,KLSA202012), and the Chinese Academy of Sciences." ] ]	2011.14055
[ [ "Transverse momentum dependent forward neutron single spin asymmetries in\n transversely polarized $p$$+$$p$ collisions at $\\sqrt{s}=200$ GeV" ], [ "Abstract In 2015, the PHENIX collaboration has measured very forward ($\\eta>6.8$) single-spin asymmetries of inclusive neutrons in transversely polarized proton-proton and proton-nucleus collisions at a center of mass energy of 200 GeV.", "A previous publication from this data set concentrated on the nuclear dependence of such asymmetries.", "In this measurement the explicit transverse-momentum dependence of inclusive neutron single spin asymmetries for proton-proton collisions is extracted using a bootstrapping-unfolding technique on the transverse momenta.", "This explicit transverse-momentum dependence will help improve the understanding of the mechanisms that create these asymmetries." ], [ "Introduction", "At the beginning of the era of polarized proton collisions at the relativistic heavy ion collider (RHIC), a dedicated experiment based on a prototype zero-degree calorimeter (ZDC) [1] was set up to initially study very forward neutral pion asymmetries in transversely polarized proton collisions in relation to earlier results that showed nonzero results [2].", "Instead of finding a neutral pion asymmetry, that was only recently discovered at low transverse momentum by the RHICf experiment [3], a sizable neutron asymmetry was found in the forward direction of the transversely polarized proton beam [4].", "Earlier theoretical studies related very forward neutron production to the one-pion-exchange (OPE) model [5], [6], [7] in which the exchange of one pion between the proton and another colliding particle can create the outgoing neutron.", "Such a model was reasonably successful in describing unpolarized, very forward neutron production as previously observed at the ISR [8].", "However, a simple pion exchange model would not be able to describe any spin dependence of the observed neutron distributions.", "To accommodate that, an interference with another particle exchange would be necessary to have helicity-flip and nonflip amplitudes available that can create a single spin left-right asymmetry.", "Within the general framework of Regge theory [9], such an interference could be accomplished when adding also a scalar meson exchange and the resulting pseudoscalar-scalar meson interference would then create the asymmetry.", "Recent calculations of such an OPE based description of very forward neutron single spin asymmetries [10] are able to qualitatively describe the RHIC measurements which, by now, include transversely polarized proton-proton collisions at $\\sqrt{s}= $ 62 GeV, 200 GeV, as well as 500 GeV [11], although the transverse momentum information enters only indirectly via the different collision energies.", "The very different asymmetries observed in proton-nucleus collisions [12], with different sign and much larger magnitude, indicate that at high impact parameters and at least for high-Z nuclei ultraperipheral collisions (UPC) [13] also contribute to these asymmetries in a very different way.", "This data together with the recent very forward nonzero neutral pion result [3] may provide crucial information to the underlying mechanisms that create these asymmetries.", "So far, none of these results have been extracted with an explicit transverse momentum dependence while the different collision energies provide some indirect information on it.", "Obtaining it can directly test the proposed mechanism and the dependence that results from its theory calculation [10].", "Therefore, extracting the actual transverse momentum dependence is the focus of this publication.", "A substantial understanding is required of the transverse momentum smearing in the PHENIX ZDCs.", "Also needed are determinations of systematic uncertainties in unfolding transverse momenta, which were studied via a Monte Carlo (MC) bootstrap method as described later.", "In the following sections, the detector description, analyzed data sets and the forward neutron selection are covered.", "Next are described the procedure for unfolding the neutron single spin asymmetries as a function of the reconstructed transverse momenta to obtain the true transverse momentum dependence.", "Then, the final results are presented before summarizing." ], [ "Data sets", "In 2015, the PHENIX experiment recorded polarized proton-proton and proton-nucleus collision data at a center of mass energy of 200 GeV.", "In the proton-proton collision data, the beams were transversely polarized with the spin direction pointing vertically up or down with respect to the plane defined by the accelerator ring.", "Inclusive neutrons were detected with the ZDC, which comprise 3 modules of Cu-W alloy absorbers layered with optical fibers of 1.7 nuclear interaction lengths each (51 radiation lengths per module), covering a projected area of 10 cm by 10 cm transverse to the beam direction.", "The absorber layers of the ZDC are tilted 45 degrees upward to maximize the light yield from Čerenkov light.", "The location of the ZDCs is 18 m up and downstream of the PHENIX beam interaction point, thus covering a range of pseudorapidity $\\eta > 6.8$ .", "The ZDC is used to measure the energy of forward neutrons, and its energy resolution is about 20% for neutron energies of 100 GeV.", "Between the first and the second module, approximately at the position of the maximal hadronic shower are located scintillator strip detectors with a projected width of 15 mm horizontally and vertically.", "This shower max detector (SMD) is used to determine the position of the neutrons that are selected by calculating the weighted average of the deposited energy for all strip positions.", "The position resolution of the SMD for neutrons is $\\approx 1$ cm.", "Additionally, the SMD is used for local polarimetry of the polarized beams by making use of the nonzero neutron asymmetries in proton-proton collisions and allowed to track the transverse spin orientation or confirm the spin orientation to be rotated in the longitudinal direction.", "See Ref.", "[14] for a more detailed detector description.", "Collision events were selected for this result by a logical OR of north and south ZDCs that require approximately an energy deposit of more than 15 GeV on either detector.", "Within the proton collisions were accumulated about 35M neutron events that were triggered by the ZDCs." ], [ "Event and particle selection criteria", "Neutron candidates in the north ZDC were selected by requiring more than 3% of the total deposited energy to be in the second ZDC module.", "This effectively rejects photon candidates, that deposited their energy in the first module due to being electromagnetic showers.", "Also nonzero hits in both horizontal and vertical SMDs are required to reliably estimate the neutron position and transverse momentum.", "Furthermore, the reconstructed neutron energies were selected between 40 and 120 GeV.", "The hit position as defined by the SMDs has to be within 0.5 to 4 cm in radius from the nominal beam position.", "Additionally, data under stable running conditions with no problems in the polarized beam diagnostics were selected.", "The transverse momentum $P_T$ is reconstructed from the neutron energy $E$ , the radius of the average hit position $r$ and the distance from the interaction point $z_{ZDC}$ : $P_{T} = \\frac{r}{z_{ZDC} }E.$ The events that fulfill the above conditions are then binned in 4 transverse momentum bins of [0.01, 0.06, 0.11, 0.16, 0.21] GeV/$c$ and 6 equidistant azimuthal angular bins that cover full azimuth around the polarized beam direction.", "The two spin states are kept separated for the unfolding, but to obtain the asymmetries needed in the bootstrap method of the MC, they are also directly calculated here as: $A_N (\\phi ) = \\frac{1}{\\langle P \\rangle }\\frac{N^+(\\phi ) - \\mathcal {R} N^-(\\phi )}{N^+(\\phi ) +\\mathcal {R}N^-(\\phi )},$ where $\\langle P\\rangle $ is the average beam polarization (for this running period 52% [15]) and $N^{\\pm }$ are the yields of neutrons in the up/down spin state as a function of azimuthal angle $\\phi $ that is defined relative to the spin-up direction.", "$\\mathcal {R}$ is the ratio of accumulated luminosities for the down and up spin states, and is close to unity in this analysis.", "The actual $A_N$ is then calculated by fitting a sine modulation to it with magnitude and phase as free parameters.", "As systematic uncertainties, the amount of charged particle background (dominated by protons) and the uncertainty of the beam center position need to be evaluated.", "Unlike other years, no charge veto counter in front of the ZDC was installed in this running period.", "This resulted in a rather asymmetric charged hadron background predominantly from protons that are swept into the ZDC by the dipole magnet which joins and separates the two beams.", "The fraction of charged hadron background was statistically subtracted on the spin dependent yield level by applying the background fractions that were obtained in the 2008 running period when the charge veto counter was installed.", "The statistical uncertainties on these background fractions were then assigned as systematic uncertainties on the resulting raw asymmetries.", "The central beam position relative to the ZDC also cannot be perfectly determined due to the large lever arm as well as varying beam conditions.", "As such, the assigned beam position was artificially varied by 1 cm horizontally and 0.5 cm vertically around the nominal beam position, respectively.", "All neutron positions, transverse momenta, and azimuthal angles were recalculated before evaluating the asymmetries.", "These variations were motivated by the uncertainties based on two independent methods of reconstructing the beam center positions using the ZDCs.", "The combined uncertainties on the asymmetries from charged background and beam position are then used as a basis for variation of the bootstrap MC method of unfolding the asymmetries as discussed in the next section.", "The uncertainties due to charged background and beam position remain negligible compared to the large systematic uncertainties this unfolding introduces." ], [ "Transverse momentum unfolding", "As hadronic showers develop a substantial size and the segmentation of the SMDs is limited, the reconstructed neutron energy and in particular the transverse momentum are smeared.", "The transverse momentum dependence of the single spin asymmetries is however of much interest for the understanding of the mechanism that creates these asymmetries.", "We have performed detailed MC simulations using 5 different types of event generators as input to full geant3 [16] simulations of the forward region of PHENIX [17], [18] including the ZDCs, SMDs, the beam-pipe as well as the dipole magnet that merges and separates the incoming and outgoing beams and is responsible for an asymmetric spray from charged particles.", "These geant simulations have been shown to describe these effects, as well as differences between top and bottom that originate from the light collection and back scattering in the top part of the ZDC.", "The composition, energy and momentum distributions of particles in the far forward region are not very well understood in general, and therefore different types of generators were used to gauge the impact of these differences on the unfolded asymmetries.", "The three full generators pythia6.1 [19], pythia8.2 [20] and dpmjet3.1 [21] were applied, where in particular diffractive processes are handled very differently.", "Additionally, an empirical distribution of forward neutrons in longitudinal and transverse momentum was used to mimic an OPE model.", "In this case, a pion that balances the energy and momentum between the incoming proton and the thrown neutron (i.e.", "$p \\rightarrow \\pi ^+ + n$ ) was collided with the other beam using pythia8 again (i.e.", "$\\pi +p$ ).", "Moreover, as the forward pA results have indicated [12], ultraperipheral collisions can also play a role in forward neutron production although that will be more prominent in proton-nucleus collisions.", "Therefore, the yield and distribution of photons from the other beam was simulated using starlight [22] and collided with the proton beam using pythia8 again.", "As none of these generators is intrinsically spin dependent, spin effects ($w$ in the following expression) were simulated by reweighting generated events as a function of true transverse momentum and azimuthal angle where the spin state was randomly assigned.", "Three different functional forms were used in the reweighting to provide as much flexibility as possible for the true transverse momentum dependence of the single spin asymmetries.", "The most general parameterization is a 3rd order polynomial in the transverse momentum with a vanishing constant term due to the requirement for the asymmetry to vanish at zero transverse momentum: $w = \\left(a \\cdot P_{T,g} + b \\cdot P_{T,g}^2+ c \\cdot P_{T,g}^3\\right) \\sin (\\phi _g + \\lambda \\cdot \\pi ),$ where $P_{T,g}$ and $\\phi _g$ are the true transverse momenta and azimuthal angles, respectively and $\\lambda $ ($\\pm 1$ ) is the spin state while $a$ , $b$ and $c$ are free parameters that are varied.", "A second functional form (with $a$ and $b>0$ free parameters) is based on a power-law behavior: $w = \\left(a \\cdot P_{T,g}^b\\right) \\sin (\\phi _g + \\lambda \\cdot \\pi ),$ and the last parameterization (with $a$ and $b$ free parameters) follows an exponential form that eventually reaches an asymptotic constant: $w = a \\left( 1- {\\rm e}^{ b\\cdot P_{T,g}}\\right)\\sin (\\phi _g + \\lambda \\cdot \\pi ).$ In the power law parameterization, only positive powers are allowed to avoid unphysical nonzero asymmetries at zero transverse momentum.", "For each set of parameters, functional form, and MC generator, the single spin asymmetries were extracted from the reconstructed kinematic variables that included these weights based on the true variables.", "In a first step, the reconstructed asymmetries that were obtained from the data are compared to the reconstructed asymmetries from MC for a large number of variations of the parameters.", "The quality of a set of parameters, functional form and MC generator in reproducing the data asymmetries was evaluated by calculating the $\\chi ^2$ between the actual data points and the smeared asymmetry points.", "While the MC statistics are generally large enough, many functional forms can describe the data within the experimentally measured uncertainties.", "Figure: Neutron transverse single spin asymmetries as a function ofthe true transverse momentum.", "The data points represent the unfoldedasymmetries obtained via the average over all parameterizations and MCgenerators.", "The uncertainty boxes represent the systematic uncertaintiesdue to the parameterization, functional form, MC generator and unfoldingprocedure.Figure REF displays the functional forms probed for the OPE motivated MC generator.", "The regions where a parametrization results in a $\\chi ^2$ below 10 units is also visualized to highlight the range of reasonable parameterizations.", "Despite the different transverse momentum distributions of forward neutrons in the different MC generators, their best asymmetry distributions are very similar for each set of functional forms.", "In all functional forms, a rapidly rising asymmetry is preferred at small $P_T$ , while at intermediate transverse momentum (within the second and third data points) no large variation in the asymmetries is visible.", "The exponential function generally prefers the asymptotic value to be reached already at these transverse momenta.", "At higher transverse momenta above 0.2 GeV/$c$ , the sensitivity is very limited for all functional forms despite a nonzero smearing into the observed range.", "As such, the slightly rising, constant or even diminishing asymmetries at high transverse momenta can describe the measured data reasonably well.", "In a second step, the spin-dependent data yields that are two-dimensional in reconstructed transverse momentum and azimuthal angle are unfolded using the spin-dependent, weighted smearing matrices obtained for each set of parameters, functional form and MC generator set.", "For the unfolding itself, the TSVDUnfold package of root [23] based on a regularized singular value decomposition [24] was used.", "After the unfolding, the asymmetries are calculated from the unfolded yields as described above.", "The best parameterization for each functional form and MC generator is used to obtain the central point of the unfolded asymmetries and statistical uncertainties.", "The impact of the variation of parameters is evaluated by obtaining the root mean square (RMS) of the spread of unfolded asymmetries which are weighted by the inverse of their respective $\\chi ^2$ to take the quality of a parameter set into account.", "These uncertainties are also displayed in Fig.", "REF , together with the unfolded asymmetries using the OPE generator.", "The variation of the unfolded asymmetries is displayed in Fig.", "REF for each transverse momentum bin and parameterization, while spreads from the different MC generators had been combined equally.", "The central values from these distributions have been taken as the final asymmetry values while the rms value is taken as the uncertainty due to the various parameterizations.", "In addition to these uncertainties, further systematic uncertainties are studied by varying the regularization parameter in the TSVDUnfold method as well as the uncertainties due to the statistical uncertainties on the smearing matrices themselves.", "However, most of these values are within the boundaries of the uncertainties obtained from the variation of parameters and functional forms.", "Only those systematic contributions that exceed the aforementioned uncertainties have been added in quadrature." ], [ "Results", "The inclusive neutron asymmetries obtained from the average of all parameterizations and MC generators are displayed in Fig.", "REF as a function of the true transverse momentum.", "The final results are tabulated in Table REF .", "The absolute values of the asymmetries are consistent with an increase with transverse momentum but show an indication of leveling off at higher transverse momenta.", "A simple linear dependence as suggested by [10], as well as the central values of transverse momentum integrated asymmetries at different collision energies [18], [4], seems not necessarily to be preferred by the data.", "However, a simple linear dependence cannot be excluded within uncertainties either.", "From the MC reweighting exercise no substantial differences between the different MC generators have been seen.", "Taking into account the indication of very different asymmetries in ultraperipheral collisions in proton-nucleus collisions [12], [13] and in particular a different sign, it appears that the UPC contribution to the proton-proton collisions is limited in this $p_T$ region.", "This is expected given the electromagnetic nature of the interaction being proportional with $Z^2$ .", "However, in these inclusive results some contribution from UPC events may remain, which could alter the transverse momentum behavior in comparison to the purely hadronic theoretical calculations.", "Table: Neutron single spin asymmetries as a function of transversemomentum after unfolding transverse-momentum and azimuthal-angularsmearing.", "ΔA N \\Delta A_N corresponds to the statistical uncertaintieswhile the last two columns specify the upper and lower systematicuncertainties δA N \\delta A_N." ], [ "Summary", "In summary, the PHENIX experiment has measured for the first time the transverse momentum dependence of very forward neutron single spin asymmetries in proton-proton collisions at a center of mass energy of 200 GeV.", "With these measurements the first reliable tests of the suggested mechanisms producing such forward neutron asymmetries can be performed.", "While the uncertainties from the unfolding are very sizable, a simple linear transverse momentum dependence as suggested in [10] is not inconsistent; however, the asymmetries appear to level off at higher transverse momenta.", "Instead, a much slower rise of the asymmetries or even a turnaround at larger transverse momenta is favored when considering the best parameterizations.", "To understand the mechanisms in even more detail, the correlations with other detector activity will be useful.", "We thank the staff of the Collider-Accelerator and Physics Departments at Brookhaven National Laboratory and the staff of the other PHENIX participating institutions for their vital contributions.", "We acknowledge support from the Office of Nuclear Physics in the Office of Science of the Department of Energy, the National Science Foundation, Abilene Christian University Research Council, Research Foundation of SUNY, and Dean of the College of Arts and Sciences, Vanderbilt University (U.S.A), Ministry of Education, Culture, Sports, Science, and Technology and the Japan Society for the Promotion of Science (Japan), Conselho Nacional de Desenvolvimento Científico e Tecnológico and Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil), Natural Science Foundation of China (People's Republic of China), Croatian Science Foundation and Ministry of Science and Education (Croatia), Ministry of Education, Youth and Sports (Czech Republic), Centre National de la Recherche Scientifique, Commissariat à l'Énergie Atomique, and Institut National de Physique Nucléaire et de Physique des Particules (France), Bundesministerium für Bildung und Forschung, Deutscher Akademischer Austausch Dienst, and Alexander von Humboldt Stiftung (Germany), J. Bolyai Research Scholarship, EFOP, the New National Excellence Program (ÚNKP), NKFIH, and OTKA (Hungary), Department of Atomic Energy and Department of Science and Technology (India), Israel Science Foundation (Israel), Basic Science Research and SRC(CENuM) Programs through NRF funded by the Ministry of Education and the Ministry of Science and ICT (Korea).", "Physics Department, Lahore University of Management Sciences (Pakistan), Ministry of Education and Science, Russian Academy of Sciences, Federal Agency of Atomic Energy (Russia), VR and Wallenberg Foundation (Sweden), the U.S.", "Civilian Research and Development Foundation for the Independent States of the Former Soviet Union, the Hungarian American Enterprise Scholarship Fund, the US-Hungarian Fulbright Foundation, and the US-Israel Binational Science Foundation." ] ]	2011.14187
[ [ "Counterbalancing steps at random in a random walk" ], [ "Abstract A random walk with counterbalanced steps is a process of partial sums $\\check S(n)=\\check X_1+ \\cdots + \\check X_n$ whose steps $\\check X_n$ are given recursively as follows.", "For each $n\\geq 2$, with a fixed probability $p$, $\\check X_n$ is a new independent sample from some fixed law $\\mu$, and with complementary probability $1-p$, $\\check X_n= -\\check X_{v(n)}$ counterbalances a previous step, with $v(n)$ a uniform random pick from $\\{1, \\ldots, n-1\\}$.", "We determine the asymptotic behavior of $\\check S(n)$ in terms of $p$ and the first two moments of $\\mu$.", "Our approach relies on a coupling with a reinforcement algorithm due to H.A.", "Simon, and on properties of random recursive trees and Eulerian numbers, which may be of independent interest.", "The method can be adapted to the situation where the step distribution $\\mu$ belongs to the domain of attraction of a stable law." ], [ "Introduction", "In short, the purpose of the present work is to investigate long time effects of an algorithm for counterbalancing steps in a random walk.", "As we shall first explain, our motivation stems from a nearest neighbor process on the integer lattice, know as the elephant random walk.", "The elephant random walk is a stochastic process with memory on $\\mathbb {Z}$ , which records the trajectory of an elephant that makes steps with unit length left or right at each positive integer time.", "It has been introduced by Schütz and Trimper [32] and triggered a growing interest in the recent years; see, for instance, [4], [5], [16], [17], [23], [24], and also [2], [6], [8], [15], [21], [27] for related models.", "The dynamics depend on a parameter $q\\in [0,1]$ and can be described as follows.", "Let us assume that the first step of the elephant is a Rademacher variable, that is equals $+1$ or $-1$ with probability $1/2$ .", "For each time $n\\ge 2$ , the elephant remembers a step picked uniformly at random among those it made previously, and decides either to repeat it with probability $q$ , or to make the opposite step with complementary probability $1-q$ .", "Obviously, each step of the elephant has then the Rademacher law, although the sequence of steps is clearly not stationary.", "Roughly speaking, it seems natural to generalize these dynamics and allow steps to have an arbitrary distribution on $\\mathbb {R}$ , say $\\mu $ .", "In this direction, Kürsten [24] pointed at an equivalent way of describing the dynamics of the elephant random walk which makes such generalization non trivialNote that merely replacing the Rademacher distribution for the first step of the elephant by $\\mu $ would not be interesting, as one would then just get the evolution of the elephant random walk multiplied by some random factor with law $\\mu $ .. Let $p\\in [0,1]$ , and imagine a walker who makes at each time a step which is either, with probability $p$ , a new independent random variable with law $\\mu $ , or, with probability $1-p$ , a repetition of one of his preceding steps picked uniformly at random.", "It is immediately checked that when $\\mu $ is the Rademacher distribution, then the walker follows the same dynamics as the elephant random walk with parameter $q=1-p/2$ .", "When $\\mu $ is an isotropic stable law, this is the model referred to as the shark random swim by Businger [15], and more generally, when $\\mu $ is arbitrary, this is the step reinforced random walk that has been studied lately in e.g.", "[9], [10], [11], [12].", "The model of Kürsten yields an elephant random walk only with parameter $q\\in [1/2,1]$ ; nonetheless the remaining range can be obtained by a simple modification.", "Indeed, let again $p\\in [0,1]$ and imagine now an averse walker who makes at each time a step which is either, with probability $p$ , a new independent random variable with law $\\mu $ , or, with probability $1-p$ , the opposite of one of his previous steps picked uniformly at random.", "When $\\mu $ is the Rademacher distribution, we simply get the dynamics of the elephant random walk with parameter $q=p/2\\in [0,1/2]$ .", "More formally, we consider a sequence $(X_n)$ of i.i.d.", "real random variables with some given law $\\mu $ and a sequence $(\\varepsilon _n)_{n\\ge 2}$ of i.i.d.", "Bernoulli variables with parameter $p\\in [0,1]$ , which we assume furthermore independent of $(X_n)$ .", "We construct a counterbalanced sequence $(\\check{X}_n)$ by interpreting each $\\lbrace \\varepsilon _n=0\\rbrace $ as a counterbalancing event and each $\\lbrace \\varepsilon _n=1\\rbrace $ as an innovation event.", "Specifically, we agree that $\\varepsilon _1=1$ for definiteness and denote the number of innovations after $n$ steps by ${\\mathrm {i}}(n) \\sum _{j=1}^n\\varepsilon _j\\qquad \\text{for }n\\ge 1.$ We introduce a sequence $(v(n))_{n\\ge 2}$ of independent variables, where each $v(n)$ has the uniform distribution on $\\lbrace 1, \\ldots , n-1\\rbrace $ , and which is also independent of $(X_n)$ and $(\\varepsilon _n)$ .", "We then define recursively $\\check{X}_n \\left\\lbrace \\begin{matrix}- \\check{X}_{v(n)}&\\text{ if }\\varepsilon _n=0, \\\\X_{{\\mathrm {i}}(n)}&\\text{ if }\\varepsilon _n=1.\\\\\\end{matrix}\\right.$ Note that the same step can be counterbalanced several times, and also that certain steps counterbalance previous steps which in turn already counterbalanced earlier ones.", "The process $\\check{S}(n)\\check{X}_1+ \\cdots + \\check{X}_n, \\qquad n\\ge 0$ which records the positions of the averse walker as a function of time, is called here a random walk with counterbalanced steps.", "Note that for $p=1$ , i.e.", "when no counterbalancing events occur, $\\check{S}$ is just a usual random walk with i.i.d.", "steps.", "In short, we are interested in understanding how counterbalancing steps affects the asymptotic behavior of random walks.", "We first introduce some notation.", "Recall that $\\mu $ denotes the distribution of the first step $X_1=\\check{X}_1$ and f we write $m_k\\int _\\mathbb {R}x^k \\mu (\\mathrm {d} x)$ for the moment of order $k\\ge 1$ of $\\mu $ whenever $\\int _\\mathbb {R}\|x\|^k \\mu (\\mathrm {d} x)<\\infty $ .", "To start with, we point out that if the first moment is finite, then the algorithm (REF ) yields the recursive equation $\\mathbb {E}(\\check{S}(n+1)) =pm_1 + (1-(1-p)/n) \\mathbb {E}(\\check{S}(n)), \\qquad n\\ge 1,$ with the initial condition $\\mathbb {E}(\\check{S}(1))=m_1$ .", "It follows easily that $\\mathbb {E}(\\check{S}(n))&\\sim \\frac{p}{2-p}\\, m_1 n\\qquad \\text{as }n\\rightarrow \\infty ;$ see e.g.", "Lemma 4.1.2 in [19].", "Our first result about the ballistic behavior should therefore not come as a surprise.", "Proposition Let $p\\in [0,1]$ .", "If $\\int _\\mathbb {R}\|x\| \\mu (\\mathrm {d} x)<\\infty $ , then there is the convergence in probability $\\lim _{n\\rightarrow \\infty } \\frac{\\check{S}(n)}{n} = \\frac{p}{2-p} m_1.$ We see in particular that counterbalancing steps reduces the asymptotic velocity of a random walk by a factor $p/(2-p)<1$ .", "The velocity is smaller when the innovation rate $p$ is smaller (i.e.", "when counterbalancing events have a higher frequency), and vanishes as $p$ approaches $0+$ .", "The main purpose of this work is to establish the asymptotic normality when $\\mu $ has a finite second moment.", "Theorem Let $p\\in (0,1]$ .", "If $\\int _\\mathbb {R}\|x\|^2 \\mu (\\mathrm {d} x)<\\infty $ , then there is the convergence in distribution $\\lim _{n\\rightarrow \\infty } \\frac{\\check{S}(n)-\\frac{p}{2-p}m_1n }{\\sqrt{n}} = \\mathcal {N}\\left(0, \\frac{m_2 -\\left(\\frac{p}{2-p}m_1\\right)^2 }{3-2p} \\right) ,$ where the right-hand side denotes a centered Gaussian variable parametrized by mean and variance.", "It is interesting to observe that the variance of the Gaussian limit depends linearly on the square $m_1^2$ of the first moment and the second moment $m_2$ of $\\mu $ only, although not just on the variance $m_2-m_1^2$ (except, of course, for $p=1$ ).", "Furthermore, it is not always a monotonous functionFor instance, in the simplest case when $\\mu $ is a Dirac point mass, i.e.", "$m_2=m_1^2$ , then the variance is given by $\\frac{4(1-p)m_2}{(3-2p)(2-p)^2}$ and reaches its maximum for $p=(9-\\sqrt{17})/8\\approx 0.6$ .", "At the opposite, when $\\mu $ is centered, i.e.", "$m_1=0$ , the variance is given by $m_2/(3-2p)$ and hence increases with $p$ .", "of the innovation rate $p$ , and does not vanish when $p$ tends to 0 either.", "Actually, our proofs of Proposition and Theorem provide a much finer analysis than what is encapsulated by those general statements.", "Indeed, we shall identify the main actors for the evolution of $\\check{S}$ and their respective contributions to its asymptotic behavior.", "In short, we shall see that the ballistic behavior stems from those of the variables $X_j$ that have been used just once by the algorithm (REF ) (in particular, they have not yet been counterbalanced), whereas the impact of variables that occurred twice or more regardless of their signs $\\pm $ , is asymptotically negligible as far as only velocity is concerned.", "Asymptotic normality is more delicate to analyze.", "We shall show that, roughly speaking, it results from the combination of, on the one hand, the central limit theorem for certain centered random walks, and on the other hand, Gaussian fluctuations for the asymptotic frequencies of some pattern induced by (REF ).", "Our analysis relies on a natural coupling of the counterbalancing algorithm (REF ) with a basic linear reinforcement algorithm which has been introduced a long time ago by H.A.", "Simon [33] to explain the occurrence of certain heavy tailed distributions in a variety of empirical data.", "Specifically, Simon defined recursively a sequence denoted here by $(\\hat{X}_n)$ (beware of the difference of notation between $\\hat{X}$ and $\\check{X}$ ) via $\\hat{X}_n \\left\\lbrace \\begin{matrix}\\hat{X}_{v(n)}&\\text{ if }\\varepsilon _n=0, \\\\X_{{\\mathrm {i}}(n)}&\\text{ if }\\varepsilon _n=1.\\\\\\end{matrix}\\right.$ We stress that the same Bernoulli variables $\\varepsilon _n$ and the same uniform variables $v(n)$ are used to run both Simon's algorithm (REF ) and (REF ); in particular either $\\check{X}_n= \\hat{X}_n$ or $\\check{X}_n= -\\hat{X}_n$ .", "It might then seem natural to refer to (REF ) and (REF ) respectively as negative and positive reinforcement algorithms.", "However, in the literature, negative reinforcement usually refers to a somehow different notion (see e.g.", "[30]), and we shall thus avoid using this terminology.", "A key observation is that (REF ) can be recovered from (REF ) as follows.", "Simon's algorithm naturally encodes a genealogical forest with set of vertices $\\mathbb {N}=\\lbrace 1, \\ldots ,\\rbrace $ and edges $(v(j),j)$ for all $j\\ge 2$ with $\\varepsilon _j=0$ ; see the forthcoming Figure 1.", "Then $\\check{X}_n=\\hat{X}_n$ if the vertex $n$ belongs to an even generation of its tree component, and $\\check{X}_n=-\\hat{X}_n$ if $n$ belongs to an odd generation.", "On the other hand, the statistics of Simon's genealogical forest can be described in terms of independent random recursive trees (see e.g.", "Chapter 6 in [18] for background) conditionally on their sizes.", "This leads us to investigate the difference $\\Delta (\\mathbb {T}_k)$ between the number of vertices at even generations and the number of vertices at odd generations in a random recursive tree $\\mathbb {T}_k$ of size $k\\ge 1$ .", "The law of $\\Delta (\\mathbb {T}_k)$ can be expressed in terms of Eulerian numbers, and properties of the latter enable us either to compute explicitly or estimate certain quantities which are crucial for the proofs of Proposition and Theorem .", "It is interesting to compare asymptotic behaviors for counterbalanced steps with those for reinforced steps.", "If we write $\\hat{S}(n)=\\hat{X}_1+ \\cdots + \\hat{X}_n$ for the random walk with reinforced steps, then it is known that the law of large numbers holds for $\\hat{S}$ , namely $\\hat{S}(n)/n\\rightarrow m_1$ a.s. when $\\int _{\\mathbb {R}}\|x\|\\mu (\\mathrm {d}x)<\\infty $ , independently of the innovation parameter $p$ .", "Further, regarding fluctuations when $\\int _{\\mathbb {R}}\|x\|^2\\mu (\\mathrm {d}x)<\\infty $ , a phase transition occurs for the critical parameter $p_c=1/2$ , in the sense that $\\hat{S}$ is diffusive for $p>1/2$ and superdiffusive for $p<1/2$ ; see [11], [12].", "Despite of the natural coupling between (REF ) and (REF ), there are thus major differencesThis should not come as a surprise.", "In the simplest case when $\\mu =\\delta _1$ is the Dirac mass at 1, one has $\\hat{S}(n)\\equiv n$ whereas $\\check{S}$ is a truly stochastic process, even for $p=0$ when there is no innovation.", "between the asymptotic behaviors of $\\check{S}$ and of $\\hat{S}$ : Proposition shows that the asymptotic speed of $\\check{S}$ depends on $p$ , and Theorem that there is no such phase transition for counterbalanced steps and $\\check{S}$ is always diffusive.", "The phase transition for step reinforcement when $\\mu $ has a finite second moment can be explained informally as follows; for the sake of simplicity, suppose also that $\\mu $ is centered, i.e.", "$m_1=0$ .", "There are $\\mathrm {i}(n)\\sim pn$ trees in Simon's genealogical forest, which are overwhelmingly microscopic (i.e.", "of size $O(1)$ ), whereas only a few trees reach the size $O(n^{1-p})$ .", "Because $\\mu $ is centered, the contribution of microscopic trees to $\\hat{S}(n)$ is of order $\\sqrt{n}$ , and that of the few largest trees of order $n^{1-p}$ .", "This is the reason why $\\hat{S}(n)$ grows like $\\sqrt{n} \\gg n^{1-p}$ when $p>1/2$ , and rather like $n^{1-p}\\gg \\sqrt{n}$ when $p<1/2$ .", "For counterbalanced steps, we will see that, due to the counterbalancing mechanism, the contribution of a large tree with size $\\ell \\gg 1$ is now only of order $\\sqrt{\\ell }$ .", "As a consequence, the contribution to $\\check{S}(n)$ of the largest trees of Simon's genealogical forest is only of order $O(n^{(1-p)/2})$ .", "This is always much smaller than the contribution of microscopic trees which remain of order $\\sqrt{n}$ .", "We further stress that, even though only the sizes of the trees in Simon's genealogical forest are relevant for the analysis of the random walk $\\hat{S}$ with reinforced steps, the study of the random walk $\\check{S}$ with counterbalanced steps is more complex and requires informations on the fine structure of those trees, not merely their sizes.", "The rest of this text is organized as follows.", "Section 2 focusses on the purely counterbalanced case $p=0$ .", "In this situation, for each fixed $n\\ge 1$ , the distribution of $\\check{S}(n)$ can be expressed explicitly in terms of Eulerian numbers.", "Section 3 is devoted to the coupling between the counterbalancing algorithm (REF ) and H.A.", "Simon's algorithm (REF ), and to the interpretation of the former in terms of a forest of random recursive trees induced by the latter.", "Proposition and Theorem are proved in Section 4, where we analyze more finely the respective contributions of some natural sub-families.", "Last, in Section 5, we present a stable version of Theorem when $\\mu $ belongs to the domain of attraction (without centering) of an $\\alpha $ -stable distribution for some $\\alpha \\in (0,2)$ ." ], [ "Warm-up: the purely counterbalanced case", "This section is devoted to the simpler situationObserve that this case without innovation has been excluded in Theorem .", "when $p=0$ .", "So $\\varepsilon _n\\equiv 0$ for all $n\\ge 2$ , meaning that every step, except of course the first one, counterbalances some preceding step.", "The law $\\mu $ then only plays a superficial role as it is merely relevant for the first step.", "For the sake of simplicity, we further focus on the case when $\\mu =\\delta _1$ is the Dirac mass at 1.", "The dynamics are entirely encoded by the sequence $(v(n))_{n\\ge 2}$ of independent uniform variables on $\\lbrace 1,\\ldots , n-1\\rbrace $ ; more precisely the purely counterbalanced sequence of bits is given by $\\check{X}_1= 1 \\quad \\text{and} \\quad \\check{X}_n=-\\check{X}_{v(n)}\\quad \\text{for all }n\\ge 2.$ The random algorithm (REF ) points at a convenient representation in terms of random recursive trees.", "Specifically, the sequence $(v(n))_{n\\ge 2}$ encodes a random tree ${\\mathbb {T}}_{\\infty }$ with set of vertices $\\mathbb {N}$ and set of edges $\\lbrace (v(n),n): n\\ge 2\\rbrace $ .", "Roughly speaking, ${\\mathbb {T}}_{\\infty }$ is constructed recursively by incorporating vertices one after the other and creating an edge between each new vertex $n$ and its parent $v(n)$ which is picked uniformly at random in $\\lbrace 1, \\ldots , n-1\\rbrace $ and independently of the other vertices.", "If we view 1 as the root of ${\\mathbb {T}}_{\\infty }$ and call a vertex $j$ odd (respectively, even) when its generation (i.e.", "its distance to the root in ${\\mathbb {T}}_{\\infty }$ ) is an odd (respectively, even) number, then $\\check{X}_n= \\left\\lbrace \\begin{matrix} 1 & \\text{ if $n$ is an even vertex in $\\mathbb {T}_{\\infty }$,}\\\\-1 & \\text{ if $n$ is an odd vertex in $\\mathbb {T}_{\\infty }$.", "}\\end{matrix} \\right.$ Let us now introduce some notation with that respect.", "For every $n\\ge 1$ , we write ${\\mathbb {T}}_n$ for the restriction of ${\\mathbb {T}}_{\\infty }$ to the set of vertices $\\lbrace 1,\\ldots , n\\rbrace $ and refer to ${\\mathbb {T}}_n$ as a random recursive tree with size $n$ .", "We also write $\\mathrm {Odd}({\\mathbb {T}}_n)$ (respectively, $\\mathrm {Even}({\\mathbb {T}}_n)$ ) for the number of odd (respectively, even) vertices in ${\\mathbb {T}}_n$ and set $\\Delta ({\\mathbb {T}}_n) \\mathrm {Even}({\\mathbb {T}}_n) - \\mathrm {Odd}({\\mathbb {T}}_n) = n-2 \\mathrm {Odd}({\\mathbb {T}}_n).$ Of course, we can also express $\\Delta ({\\mathbb {T}}_n)= \\check{X}_1+ \\cdots + \\check{X}_n,$ which is the trajectory of a completely averse elephant random walk (i.e.", "for the parameter $q=0$ ).", "The main observation of this section is that law of the number of odd vertices is readily expressed in terms of the Eulerian numbers.", "Recall that $\\langle {}^n_k\\rangle $ denotes the number of permutations $\\varsigma $ of $\\lbrace 1, \\ldots ,n\\rbrace $ with $k$ descents, i.e.", "such that $\\#\\lbrace 1\\le j<n: \\varsigma (j) > \\varsigma (j+1)\\rbrace =k$ .", "Obviously $\\langle {}^n_k\\rangle \\ge 1$ if and only if $0\\le k <n$ , and one has $\\sum _{k=0}^{n-1} {\\displaystyle \\left\\langle {n \\atop k}\\right\\rangle }=n!.$ The linear recurrence equation ${\\displaystyle \\left\\langle {n \\atop k}\\right\\rangle }=(n-k){\\displaystyle \\left\\langle {{n-1} \\atop {k-1}}\\right\\rangle }+(k+1){\\displaystyle \\left\\langle {{n-1} \\atop k}\\right\\rangle }$ is easily derived from a recursive construction of permutations (see Theorem 1.3 in [31]); we also mention the explicit formula (see Corollary 1.3 in [31]) ${\\displaystyle \\left\\langle {n \\atop k}\\right\\rangle }=\\sum _{j=0}^{k}(-1)^{j}{\\binom{n+1}{j}}(k+1-j)^{n}.$ Lemma For every $n\\ge 1$ , we have $\\mathbb {P}(\\mathrm {Odd}({\\mathbb {T}}_n)=\\ell )= \\frac{1}{(n-1)! }", "{\\displaystyle \\left\\langle {{n-1} \\atop {\\ell -1}}\\right\\rangle },$ with the conventionNote that this convention is in agreement with the linear recurrence equation (REF ).", "that $\\langle {}^{\\ 0}_{-1}\\rangle =1$ in the right-hand side for $n=1$ and $\\ell =0$ .", "Consider $n\\ge 1$ and note from the very construction of random recursive trees that there is the identity $ \\mathbb {P}(\\mathrm {Odd}({\\mathbb {T}}_{n+1})=\\ell ) = \\frac{\\ell }{n}\\, \\mathbb {P}(\\mathrm {Odd}({\\mathbb {T}}_n)=\\ell ) + \\frac{n+1-\\ell }{n}\\, \\mathbb {P}(\\mathrm {Odd}({\\mathbb {T}}_n)=\\ell -1).$ Indeed, the first term of the sum in the right-hand side accounts for the event that the parent $v(n+1)$ of the new vertex $n+1$ is an odd vertex (then $n+1$ is an even vertex), and the second term for the event that $v(n+1)$ is an even vertex (then $n+1$ is an odd vertex).", "In terms of $A(n,k) n!", "\\mathbb {P}(\\mathrm {Odd}({\\mathbb {T}}_{n+1})=k+1)$ , this yields $ A(n,k) = (k+1) A(n-1,k) + (n-k) A(n-1,k-1),$ which is the linear recurrence equation (REF ) satisfied by the Eulerian numbers.", "Since plainly $A(1,0)=\\mathbb {P}(\\mathrm {Odd}(2)=1)=1=\\langle {}^1_0\\rangle $ , we conclude by iteration that $A(n,k)=\\langle {}^n_k\\rangle $ for all $n\\ge 1$ and $0\\le k<n$ .", "Last, the formula in the statement also holds for $n=1$ since $\\mathrm {Odd}(1)=0$ .", "Remark Lemma is implicit in Mahmoud [25]Beware however that the definition of Eulerian numbers in [25] slightly differs from ours, namely $\\langle {}^n_k\\rangle $ there corresponds to $\\langle {}^{\\ n}_{k-1}\\rangle $ here..", "Indeed $\\mathrm {Odd}({\\mathbb {T}}_n)$ can be viewed as the number blue balls in an analytic Friedman's urn model started with one white ball and replacement scheme $\\left({}^{0\\ 1}_{1\\ 0}\\right)$ ; see Section 7.2.2 in [25].", "In this setting, Lemma is equivalent to the formula for the number of white balls at the bottom of page 127 in [25].", "Mahmoud relied on the analysis of the differential system associated to the replacement scheme via a Riccati differential equation and inversion of generating functions.", "The present approach based on the linear recurrence equation (REF ) is more direct.", "Lemma is also a closed relative to a result due to Najock and Heyde [28] (see also Theorem 8.6 in Mahmoud [25]) which states that the number of leaves in a random recursive tree with size $n$ has the same distribution as that appearing in Lemma .", "We next point at a useful identity related to Lemma which goes back to Laplace (see Exercise 51 of Chapter I in Stanley [34]) and is often attributed to Tanny [35].", "For every $n\\ge 0$ , there is the identity in distribution $\\mathrm {Odd}({\\mathbb {T}}_{n+1})\\,{\\overset{\\mathrm {(d)}}{=}}\\, \\lceil U_1+\\cdots + U_n\\rceil ,$ where in the right-hand side, $U_1, U_2, \\ldots $ is a sequence of i.i.d.", "uniform variables on $[0,1]$ and $\\lceil \\cdot \\rceil $ denotes the ceiling function.", "We now record for future use the following consequences.", "Corollary (i) For every $n\\ge 2$ , the variable $\\Delta ({\\mathbb {T}}_n)$ is symmetric, i.e.", "$\\Delta ({\\mathbb {T}}_n)\\,{\\overset{\\mathrm {(d)}}{=}} \\, -\\Delta ({\\mathbb {T}}_n)$ , and in particular $\\mathbb {E}(\\Delta ({\\mathbb {T}}_n))= 0$ .", "(ii) For all $n\\ge 3$ , one has $ \\mathbb {E}(\\Delta ({\\mathbb {T}}_n)^2)= n/3$ .", "(iii) For all $n\\ge 1$ , one has $\\mathbb {E}(\|\\Delta ({\\mathbb {T}}_n)\|^4) \\le 6n^2$ .", "(i) Equivalently, the assertion claims that in a random recursive tree of size at least 2, the number of odd vertices and the number of even vertices have the same distribution.", "This is immediate from (REF ) and can also be checked directly from the construction.", "(ii) This has been already observed by Schütz and Trimper [32] in the setting of the elephant random walk; for the sake of completeness we present a short argument.", "The vertex $n+1$ is odd (respectively, even) in ${\\mathbb {T}}_{n+1}$ if and only if its parent is an even (respectively, odd) vertex in ${\\mathbb {T}}_{n}$ .", "Hence one has $\\mathbb {E}(\\Delta ({\\mathbb {T}}_{n+1})-\\Delta ({\\mathbb {T}}_{n})\\mid {\\mathbb {T}}_{n})= -\\frac{1}{n}\\Delta ({\\mathbb {T}}_{n}),$ and since $\\Delta ({\\mathbb {T}}_{n+1})-\\Delta ({\\mathbb {T}}_{n})= \\pm 1$ , this yields the recursive equation $\\mathbb {E}(\\Delta ({\\mathbb {T}}_{n+1})^2)= (1-2/n)\\mathbb {E}(\\Delta ({\\mathbb {T}}_{n})^2) +1.$ By iteration, we conclude that $ \\mathbb {E}(\\Delta ({\\mathbb {T}}_n)^2)= n/3$ for all $n\\ge 3$ .", "(iii) Recall that the process of the fractional parts $\\lbrace U_1+\\cdots + U_n\\rbrace $ is a Markov chain on $[0,1)$ whose distribution at any fixed time $n\\ge 1$ is uniform on $[0,1)$ .", "Writing $V_n= 1-2U_n$ and $W_n=2\\lbrace U_1+\\cdots + U_n\\rbrace -1$ , we see that $V_1, V_2, \\ldots $ is a sequence of i.i.d.", "uniform variables on $[-1,1]$ and that $W_n$ has the uniform distribution on $[-1,1]$ too.", "The characteristic function of the uniform variable $V_j$ is $\\mathbb {E}(\\exp ({i}\\theta V_j))=\\theta ^{-1}\\sin (\\theta ) = 1-\\frac{\\theta ^2}{6} + \\frac{\\theta ^4}{120} + O(\\theta ^6)\\qquad \\text{as }\\theta \\rightarrow 0,$ and therefore for every $n\\ge 1$ , $\\mathbb {E}(\\exp ({i}\\theta (V_1+\\cdots +V_n)))&=\\left(1-\\frac{\\theta ^2}{6} + \\frac{\\theta ^4}{120} + O(\\theta ^6)\\right)^n\\\\&=1-\\frac{n}{6}\\theta ^2 + \\left(\\frac{n}{120}+ \\frac{n(n-1)}{72}\\right) \\theta ^4+ O(\\theta ^6).$ It follows that $\\mathbb {E}( (V_1+\\cdots +V_n)^4)= 24 \\left(\\frac{n}{120}+ \\frac{n(n-1)}{72}\\right) \\le n^2/3.$ We can rephrase (REF ) as the identity in distribution $\\Delta ({\\mathbb {T}}_{n+1})\\,{\\overset{\\mathrm {(d)}}{=}}\\, V_1+\\cdots +V_n + W_n.$ Since $\\mathbb {E}(W_n^4)=1/3$ , the proof is completed with the elementary bound $(a+b)^4\\le 16(a^4+b^4)$ .", "We now conclude this section with an application of (REF ) to the asymptotic normality of $\\Delta ({\\mathbb {T}}_n)$ .", "Since $\\mathbb {E}(U)=1/2$ and $\\mathrm {Var}(U)=1/12$ , the classical central limit theorem immediately yields the following.", "Corollary Assume $p=0$ and $\\mu =\\delta _1$ .", "One has $\\lim _{n\\rightarrow \\infty } \\frac{ \\Delta ({\\mathbb {T}}_n)}{\\sqrt{n}} = {\\mathcal {N}}(0,1/3)\\qquad \\text{in distribution.", "}$ Corollary goes back to [28] in the setting of the number of leaves in random recursive trees; see also [4], [5], [16], [17] for alternative proofs in the framework of the elephant random walk." ], [ "Genealogical trees in Simon's algorithm", "From now on, $\\mu $ is an arbitrary probability law on $\\mathbb {R}$ and we also suppose that the innovation rate is strictly positive, $p\\in (0,1)$ .", "Recall the construction of the sequence $(\\hat{X}_n)$ from Simon's reinforcement algorithm (REF ).", "Simon was interested in the asymptotic frequencies of variables having a given number of occurrences.", "Specifically, for every $n,j\\in \\mathbb {N}$ , we write ${N}_j(n)\\#\\lbrace \\ell \\le n: \\hat{X}_{\\ell }=X_j\\rbrace $ for the number of occurrences of the variable $X_j$ until the $n$ -th step of the algorithm (REF ), and $\\nu _k(n)\\#\\lbrace 1\\le j \\le \\mathrm {i}(n): {N}_j(n)=k\\rbrace , \\qquad k\\in \\mathbb {N}$ for the number of such variables that have occurred exactly $k$ times.", "Observe also that the number of innovations satisfies the law of large numbers ${\\mathrm {i}}(n)\\sim p n$ a.s. Lemma For every $k\\ge 1$ , we have $\\lim _{n\\rightarrow \\infty } \\frac{\\nu _k(n)}{pn}=\\frac{1}{1-p}{\\mathrm {B}}(k,1+1/(1-p)) \\qquad \\text{in probability,}$ where ${\\mathrm {B}}$ denotes the beta function.", "Lemma is essentially due to H.A.", "Simon [33], who actually only established the convergence of the mean value.", "The strengthening to convergence in probability can be obtained as in [14] from a concentration argument based on the Azuma-Hoeffding's inequality; see Section 3.1 in [29].", "The right-hand side in the formula is a probability mass on $\\mathbb {N}$ known as the Yule-Simon distribution with parameter $1/(1-p)$ .", "We record for future use a couple of identities which are easily checked from the integral definition of the beta function: $ \\frac{1}{1-p} \\sum _{k=1}^{\\infty } {\\mathrm {B}}(k,1+1/(1-p)) =1$ and $ \\frac{1}{1-p} \\sum _{k=1}^{\\infty } k {\\mathrm {B}}(k,1+1/(1-p)) = \\frac{1}{p}.$ For $k=1$ , Lemma reads $\\lim _{n\\rightarrow \\infty } n^{-1}\\nu _1(n)= \\frac{p}{2-p}\\qquad \\text{in probability.", "}$ We shall also need to estimate the fluctuations, which can be derived by specializing a Gaussian limit theorem for extended Pólya urns due to Bai et al.", "[1].", "Lemma There is the convergence in distribution $\\lim _{n\\rightarrow \\infty } \\frac{\\nu _1(n)- np/(2-p)}{\\sqrt{n}} = \\mathcal {N}\\left(0,\\frac{2p^3-8p^2+6p}{(3-2p)(2-p)^2} \\right).$ The proof relies on the observation that Simon's algorithm can be coupled with a two-color urn governed by the same sequences of random bits $(\\varepsilon _n)$ and of uniform variables $(v(n))$ as follows.", "Imagine that we observe the outcome of Simon's algorithm at the $n$ -step and that for each $1\\le j \\le n$ , we associate a white ball if the variable $\\hat{X}_j$ appears exactly once, and a red ball otherwise.", "At the initial time $n=1$ , the urn contains just one white ball and no red balls.", "At each step $n\\ge 2$ , a ball picked uniformly at random in the urn (in terms of Simon's algorithm, this is given by the uniform variable $v(n)$ ).", "If $\\varepsilon _n=1$ , then the ball picked is returned to the urn and one adds a white ball (in terms of Simon's algorithm, this corresponds to an innovation and $\\nu _1(n)=\\nu _1(n-1)+1$ ).", "If $\\varepsilon _n=0$ , then the ball picked is removed from the urn and one adds two red balls (in terms of Simon's algorithm, this corresponds to a repetition and either $\\nu _1(n)=\\nu _1(n-1)-1$ if the ball picked is white, or $\\nu _1(n)=\\nu _1(n-1)$ if the ball picked is red).", "By construction, the number $W_n$ of white balls in the urn coincides with the number $\\nu _1(n)$ of variables that have appeared exactly once in Simon's algorithm (REF ).", "We shall now check our claim in the setting of [1] by specifying the quantities which appear there.", "The evolution of number of white balls in the urn is governed by Equation (2.1) in [1], viz.", "$W_n=W_{n-1}+ I_nA_n+(1-I_n)C_n,$ where $I_n=1$ if a white ball is picked and $I_n=0$ otherwise.", "In our framework, we further have $A_n=2\\varepsilon _n-1$ and $C_n=\\varepsilon _n$ .", "If we write ${\\mathcal {F}}_n$ for the natural filtration generated by the variables $(A_k,C_k, I_k)_{k\\le n}$ , then $A_n$ and $C_n$ are independent of ${\\mathcal {F}}_{n-1}$ with $\\mathbb {E}(A_n)= 2p-1, \\quad \\mathbb {E}(C_n)= p, \\quad \\mathrm {Var}(A_n)= 4(p-p^2), \\quad \\mathrm {Var}(C_n)= p-p^2.$ This gives in the notation (2.2) of [1]: $\\sigma ^2_M&= \\frac{p}{2-p}4(p-p^2)+ \\left(1-\\frac{p}{2-p}\\right) (p-p^2) + (p-1)^2 \\frac{p}{2-p} \\left(1-\\frac{p}{2-p}\\right)\\\\&= \\frac{2p^3-8p^2+6p}{(2-p)^2},$ and finally $\\sigma ^2=\\frac{2p^3-8p^2+6p}{(3-2p)(2-p)^2}.$ Our claim can now be seen as a special case of Corollary 2.1 in [1].", "We shall also need a refinement of Lemma in which one does not only record the number of occurrences of the variable $X_j$ , but more generally the genealogical structure of these occurrences.", "We need to introduce first some notation in that respect.", "Fix $n\\ge 1$ and let some $1\\le j \\le {\\mathrm {i}}(n)$ (i.e.", "the variable $X_j$ has already appeared at the $n$ -th step of the algorithm).", "Write $\\ell _1< \\ldots < \\ell _k \\le n$ for the increasing sequence of steps of the algorithm at which $X_j$ appears, where $k={N}_j(n)\\ge 1$ .", "The genealogy of occurrences of the variable $X_j$ until the $n$ -th step is recorded as a tree $T_j(n)$ on $\\lbrace 1, \\ldots , k\\rbrace $ such that for every $1\\le a < b \\le k$ , $(a,b)$ is an edge of $T_j(n)$ if and only if $v(\\ell _b)=\\ell _a$ , that is, if and only if the identity $\\hat{X}_{\\ell _b}=X_j$ actually results from the fact that the algorithm repeats the variable $\\hat{X}_{\\ell _a}$ at its $\\ell _b$ -th step.", "Plainly, $T_j(n)$ is an increasing tree with size $k$ , meaning a tree on $\\lbrace 1, \\ldots , k\\rbrace $ such that the sequence of vertices along any branch from the root 1 to a leaf is increasing.", "In this direction, we recall that there are $(k-1)!$ increasing trees with size $k$ and that the uniform distribution of the set increasing trees with size $k$ coincides with the law of the random recursive tree of size $k$ , ${\\mathbb {T}}_k$ .", "See for instance Section 1.3.1 in Drmota [18].", "Figure: Example of a genealogical forest representation of Simon's algorithm () after 18 steps.", "The dotted edges account for innovation events, i.e.", "ε j =1\\varepsilon _j=1and the four genealogical trees are rooted at 1,5,7,141,5,7,14.", "In each subtree, vertices at even generations are colored in green and vertices at odd generation in white.", "For instance the genealogical tree T 2 (18)T_2(18) is rooted at 5, it has 3 even vertices and 2 odd vertices.More generally, the distribution of the entire genealogical forest given the sizes of the genealogical trees can be described as follows.", "Lemma Fix $n\\ge 1$ , $1\\le k\\le n$ , and let $n_1, \\ldots , n_k\\ge 1$ with sum $n_1+ \\cdots + n_k=n$ .", "Then conditionally on ${N}_j(n)=n_j$ for every $j=1, \\ldots , k$ , the genealogical trees $T_1(n), \\ldots , T_k(n)$ are independent random recursive trees with respective sizes $n_1, \\ldots , n_k$ .", "Recall that the set $\\lbrace (v(j),j)$ for $1\\le j\\le n\\rbrace $ is that of the edges of the random recursive tree with size $n$ , ${\\mathbb {T}}_n$ .", "The well-known splitting property states that removing a given edge, say $(v(j),j)$ for some fixed $j$ , from ${\\mathbb {T}}_n$ produces two subtrees which in turn, conditionally on their sizes, are two independent random recursive trees.", "This has been observed first by Meir and Moon [26]; see also [3] and references therein for more about this property.", "The genealogical trees $T_1(n), \\ldots , T_k(n)$ result by removing the edges $(v(j),j)$ in ${\\mathbb {T}}_n$ for which $\\varepsilon _j=1$ and enumerating in each subtree component their vertices in the increasing order.", "Our statement is now easily seen by applying iteratively this splitting property.", "We shall also need for the proofs of Proposition and Theorem an argument of uniform integrability that relies in turn on the following lemma.", "Recall that if $T$ is a rooted tree, $\\Delta (T)$ denotes the difference between the number of vertices at even distance from root and that at odd distance.", "Lemma For every $1<\\beta <2\\wedge \\frac{1}{1-p}$ , one has $\\sup _{n\\ge 1} \\frac{1}{n} \\sum _{j=1}^{n}\\mathbb {E}({N}_j(n)^{\\beta })< \\infty $ and $\\sup _{n\\ge 1} \\frac{1}{n}\\mathbb {E}\\left( \\sum _{j=1}^{{\\mathrm {i}}(n)} \|\\Delta (T_j(n) ) \|^{2\\beta } \\right) <\\infty .$ The first claim is a consequence of Lemma 3.6 of [10] which states that for $\\beta \\in (1,1/(1-p))$ [beware that the parameter denoted by $p$ in [10] is actually $1-p$ here], there exists numerical constants $c>0$ and $\\eta \\in (0,1)$ such that $\\mathbb {E}({N}_j(n)^{\\beta })\\le c(n/j)^{\\eta }$ for all $1\\le j \\le n$ .", "Next, combining Jensen's inequality with Corollary (iii), we get that for $k\\ge 2$ $\\mathbb {E}( \|\\Delta ({\\mathbb {T}}_k)\|^{2\\beta }) \\le \\mathbb {E}(\|\\Delta ({\\mathbb {T}}_k)\|^4)^{\\beta /2}\\le 6k^{\\beta }.$ Then recall that conditionally on ${N}_j(n)=k\\ge 1$ , $T_j(n) $ has the law of the random recursive tree with size $k$ , ${\\mathbb {T}}_k$ , and hence $\\mathbb {E}\\left( \\sum _{j=1}^{{\\mathrm {i}}(n)} \|\\Delta (T_j(n) ) \|^{2\\beta } \\right)&= \\sum _{j=1}^{n} \\left(\\sum _{k=1}^{n} \\mathbb {E}( \|\\Delta ({\\mathbb {T}}_k)\|^{2\\beta }) \\mathbb {P}({N}_j(n)=k)\\right)\\\\&\\le 6\\sum _{j=1}^{n} \\left(\\sum _{k=1}^{n} k^{\\beta } \\mathbb {P}({N}_j(n)=k)\\right).$ We know from the first part that this last quantity is finite, and the proof is complete." ], [ "Proofs of the main results", "As its title indicates, the purpose of this section is to establish Proposition and Theorem .", "The observation that for every $n\\ge 1$ and $1\\le j \\le {{\\mathrm {i}}(n)}$ , the variable $X_j$ appears exactly $\\mathrm {Even}(T_j(n))$ times and its opposite $-X_j$ exactly $\\mathrm {Odd}(T_j(n))$ times until the $n$ -step of the algorithm (REF ), yields the identity $\\check{S}(n)\\sum _{i=1}^n\\check{X}_i = \\sum _{j=1}^{{\\mathrm {i}}(n)} \\Delta (T_j(n)) X_j,$ which lies at the heart of our approach.", "We stress that in (REF ) as well as in related expressions that we shall use in the sequel, the sequence of i.i.d.", "variables $(X_n)$ and the family of genealogical trees $(T_j(n))$ are independent, because the latter are constructed from the sequences $(\\varepsilon _n)$ and $(v(n))$ only.", "Actually, our proof analyzes more precisely the effects of the counterbalancing algorithm (REF ) by estimating specifically the contributions of certain sub-families to the asymptotic behavior of $\\check{S}$ .", "Specifically, we set for every $k\\ge 1$ , $ \\check{S}_{k}(n) \\sum _{j=1}^{{\\mathrm {i}}(n)} \\Delta (T_j(n)) X_j \\mathbb {1}_{N_j(n)=k},$ so that $\\check{S}(n) = \\sum _{k=1}^{n} \\check{S}_k(n).$" ], [ "Proof of Proposition ", "The case $p=1$ (no counterbalancing events) of Proposition is just the weak law of large numbers, and the case $p=0$ (no innovations) is a consequence of Corollary .", "The case $p\\in (0,1)$ derives from the next lemma which shows more precisely that the variables $X_j$ that have appeared in the algorithm (REF ) but have not yet counterbalanced determine the ballistic behavior of $\\check{S}$ , whereas those that have appeared twice or more (i.e.", "such that $N_j(n)\\ge 2$ ) have a negligible impact.", "Lemma Assume that $\\int _\\mathbb {R}\|x\| \\mu (\\mathrm {d} x)<\\infty $ and recall that $m_1=\\int _{\\mathbb {R}} x \\mu (\\mathrm {d} x)$ .", "Then the following limits hold in probability: $\\lim _{n\\rightarrow \\infty } n^{-1} \\check{S}_1(n) =m_1p/(2-p)$ , $\\lim _{n\\rightarrow \\infty } n^{-1} \\sum _{k=2}^n \\left\| \\check{S}_{k}(n) \\right\| = 0$ .", "(i) Recall the notation (REF ) and that the sequence of i.i.d.", "variables $(X_j)$ is independent of the events $\\lbrace {N}_j(n)=1\\rbrace $ .", "We see that there is the identity in distribution $\\check{S}_1(n) \\,{\\overset{\\mathrm {(d)}}{=}}\\, S_1(\\nu _1(n)),$ where $S_1(n)=X_1+\\cdots + X_n$ is the usual random walk.", "The claim follows readily from the law of large numbers and (REF ).", "(ii) We first argue that for each fixed $k\\ge 2$ , $\\lim _{n\\rightarrow \\infty } n^{-1} \\check{S}_{k}(n) = 0 \\qquad \\text{almost surely.", "}$ Indeed, recall that $\\nu _k(n)$ denotes the number of genealogical trees $T_j(n)$ with size $k$ .", "It follows from Lemma that conditionally on $\\nu _k(n)=\\ell $ , the sub-family of such $T_j(n)$ enumerated in the increasing order of the index $j$ , is given by $\\ell $ i.i.d.", "copies of the random recursive tree ${\\mathbb {T}}_k$ .", "Hence, still conditionally on $\\nu _k(n)=\\ell $ , enumerating the elements of the sub-family $\\lbrace X_j \\Delta (T_j(n)): {N}_j(n)=k\\rbrace $ in the increasing order of $j$ yields $\\ell $ independent variables, each being distributed as $X_1 \\Delta ({\\mathbb {T}}_k)$ with $X_1$ and $ \\Delta ({\\mathbb {T}}_k)$ independent.", "We deduce from Corollary (i) that the variable $X_1 \\Delta ({\\mathbb {T}}_k)$ symmetric, and since it is also integrable, it is centered.", "Since $\\nu _k(n)\\le n$ , this readily entails (REF ) by an application of the law of large numbers.", "The proof can be completed by an argument of uniform integrability.", "In this direction, fix an arbitrarily large integer $\\ell $ and write by the triangular inequality $ \\frac{1}{n} \\sum _{k=\\ell }^n \\left\| \\check{S}_{k}(n) \\right\| &\\le \\frac{1}{n} \\mathbb {E}\\left( \\sum _{j=1}^{{\\mathrm {i}}(n)} \|X_j \| {N}_j(n) \\mathbb {1}_{{N}_j(n)\\ge \\ell } \\right)\\\\&=\\frac{ \\mathbb {E}(\|X_1\|)}{n} \\sum _{j=1}^{n} \\mathbb {E}\\left( {N}_j(n) \\mathbb {1}_{{N}_j(n)\\ge \\ell } \\right)\\\\&\\le \\ell ^{1-\\beta } \\frac{ \\mathbb {E}(\|X_1\|)}{n}\\sum _{j=1}^{n} \\mathbb {E}\\left( {N}_j(n)^{\\beta } \\right),$ where the last inequality holds for any $\\beta >1$ .", "We see from Lemma that the right-hand side converges to 0 as $\\ell \\rightarrow \\infty $ uniformly in $n\\ge 1$ , and the rest of the proof is straightforward." ], [ "Proof of Theorem ", "For $p=1$ (no counterbalancing events), Theorem just reduces to the classical central limit theorem, so we assume $p\\in (0,1)$ .", "The first step of the proof consists in determining jointly the fluctuations of the components $\\check{S}_k$ defined in (REF ).", "Lemma Assume that $\\int _\\mathbb {R}\|x\|^2 \\mu (\\mathrm {d} x)<\\infty $ and recall that $m_2=\\int _{\\mathbb {R}} x^2 \\mu (\\mathrm {d} x)$ .", "Then as $n\\rightarrow \\infty $ , the sequences of random variables $ \\frac{ \\check{S}_1(n) -pm_1/(2-p)}{\\sqrt{n}} \\qquad \\text{(for $k=1$)}$ and $ \\frac{ \\check{S}_{k}(n)}{\\sqrt{n}}\\qquad \\text{(for $k\\ge 2$)}$ converge jointly in distribution towards a sequence $\\left( \\mathcal {N}_{k}(0,\\sigma ^2_{k})\\right)_{k\\ge 1}$ of independent centered Gaussian variables, where $ \\sigma _1^2\\frac{p m_2}{2-p} - \\frac{p^2m_1^2}{(3-2p)(2-p)^2},$ $\\sigma ^2_20$ , and $\\sigma ^2_{k}\\frac{ k p m_2}{3(1-p)} \\mathrm {B}(k,1+1/(1-p)) \\qquad \\text{for $k\\ge 3$}.$ For each $k\\ge 1$ , let $(Y_k(n))_{n\\ge 1}$ be a sequence of i.i.d.", "copies of $\\Delta (\\mathbb {T}_k) X$ , where $X$ has the law $\\mu $ and is independent of the random recursive tree $\\mathbb {T}_k$ .", "We further assume that these sequences are independent.", "Taking partial sums yields a sequence indexed by $k$ of independent random walks $S_k(n)= Y_k(1)+ \\cdots + Y_k(n), \\qquad n\\ge 0.", "$ For each $n\\ge 1$ , the family of blocks $B_{k}(n)\\lbrace j\\le {\\mathrm {i}}(n): N_j(n)=k\\rbrace \\qquad \\text{for }1\\le k\\le {\\mathrm {i}}(n)$ forms a random partition of $\\lbrace 1, \\ldots , {\\mathrm {i}}(n)\\rbrace $ which is independent of the $X_j$ 's.", "Recall that we are using the notation $\\nu _k(n)=\\#B_k(n)$ , and also from Lemma , that conditionally on the $N_j(n)$ 's, the genealogical trees $T_j(n)$ are independent random recursive trees.", "We now see from the very definition (REF ) that for every fixed $n\\ge 1$ , there is the identity in distribution $\\left( \\check{S}_k(n)\\right)_{k\\ge 1} \\,{\\overset{\\mathrm {(d)}}{=}}\\, \\left( S_k(\\nu _k(n))\\right)_{k\\ge 1},$ where in the right-hand side, the random walks $(S_k)_{k\\ge 1}$ are independent of the sequence of block sizes $(\\nu _k(n))_{k\\ge 1}$ .", "Next we write, first for $k=1$ , $S_1(\\nu _1(n)) - \\frac{pn}{2-p} m_1 = S_1\\left(\\left\\lfloor \\frac{pn}{2-p}\\right\\rfloor \\right) - \\frac{pn}{2-p}m_1 + \\sum _{j=\\lceil pn/(2-p)\\rceil }^{\\nu _1(n)} Y_1(j) ,$ second $S_2\\equiv 0$ (since $\\Delta (\\mathbb {T}_2)\\equiv 0$ ) for $k=2$ , and third, for $k\\ge 3$ , $S_k(\\nu _k(n)) = S_k\\left(\\left\\lfloor \\frac{pn}{1-p}{\\mathrm {B}}(k,1+1/(1-p))\\right\\rfloor \\right) + \\sum _{j=\\lceil \\frac{pn}{1-p}{\\mathrm {B}}(k,1+1/(1-p))\\rceil }^{\\nu _k(n)} Y_k(j) ,$ with the usual summation convention that $\\sum _{j=a}^b = - \\sum _{j=b}^a$ when $b<a$ .", "Since the i.i.d.", "variables $Y_1(\\cdot )$ have mean $m_1$ and variance $m_2-m_1^2$ , the central limit theorem ensures that there is the convergence in distribution $ \\lim _{n\\rightarrow \\infty } n^{-1/2}\\left( S_1\\left(\\left\\lfloor \\frac{pn}{2-p}\\right\\rfloor \\right) - \\frac{pn}{2-p}m_1\\right) = \\mathcal {N}_1\\left( 0,\\frac{p (m_2-m_1^2)}{2-p}\\right).$ Similarly, for $k\\ge 3$ , each $Y_k(n)$ is centered with variance $km_2/3$ (by Corollary (i-ii)) and hence, using the notation in the statement, there is the convergence in distribution $ \\lim _{n\\rightarrow \\infty } n^{-1/2}S_k\\left(\\left\\lfloor \\frac{pn}{1-p}{\\mathrm {B}}(k,1+1/(1-p))\\right\\rfloor \\right) & = \\mathcal {N}_k\\left( 0,\\sigma _k^2 \\right)$ Plainly, the weak convergences (REF ) and (REF ) hold jointly when we agree that the limits are independent Gaussian variables.", "Next, from Lemma and the fact that for $k\\ge 3$ , the i.i.d.", "variables $Y_k(j)$ are centered with finite variance, we easily get $\\lim _{n\\rightarrow \\infty } n^{-1/2} \\left\| \\sum _{j=\\lceil \\frac{pn}{1-p}{\\mathrm {B}}(k,1+1/(1-p))\\rceil }^{\\nu _k(n)} Y_k(j) \\right\|=0 \\qquad \\text{in }L^2(\\mathbb {P}).$ Finally, for $k=1$ , we write $ \\sum _{j=\\lceil pn/(2-p)\\rceil }^{\\nu _1(n)} Y_1(j)= m_1(\\nu _1(n) - \\lfloor pn/(2-p)\\rfloor ) + \\sum _{j=\\lceil pn/(2-p)\\rceil }^{\\nu _1(n)} (Y_1(j)-m_1).$ On the one hand, we have from the same argument as above that $\\lim _{n\\rightarrow \\infty } n^{-1/2} \\left\| \\sum _{j=\\lceil pn/(2-p)\\rceil }^{\\nu _1(n)} (Y_1(j)-m_1)\\right\|=0 \\qquad \\text{in }L^2(\\mathbb {P}).$ On the other hand, we already know from Lemma that there is the convergence in distribution $\\lim _{n\\rightarrow \\infty } m_1\\frac{\\nu _1(n) - \\lfloor pn/(2-p)\\rfloor }{\\sqrt{n}} = \\mathcal {N}\\left( 0,\\frac{2p^3-8p^2+6p}{(3-2p)(2-p)^2} m_1^2\\right).$ Obviously, this convergence in law hold jointly with (REF ) and (REF ), where the limiting Gaussian variables are independent.", "Putting the pieces together, this completes the proof.", "The final step for the proof of Theorem is the following lemma.", "Lemma We have $\\lim _{K\\rightarrow \\infty } \\sup _{n\\ge 1} n^{-1} \\mathbb {E}\\left( \\left\| \\sum _{k\\ge K} \\check{S}_{k}(n)\\right\|^2 \\right) =0.$ We write $\\sum _{k\\ge K} \\check{S}_k(n)= \\sum _{j=1}^{n} X_j \\Delta (T_j(n)) \\mathbb {1}_{{N}_j(n)\\ge K}.$ Since the $X_j$ are independent of the $T_j(n)$ , we get $ \\mathbb {E}\\left( \\left\| \\sum _{k\\ge K} \\check{S}_k(n)\\right\|^2 \\right)&= \\mathbb {E}\\left( \\sum _{j,{j^{\\prime }}=1}^{n} X_j X_{j^{\\prime }} \\Delta (T_j(n)) \\mathbb {1}_{{N}_{j^{\\prime }}(n)\\ge K} \\Delta (T_{j^{\\prime }}(n)) \\mathbb {1}_{{N}_{j^{\\prime }}(n)\\ge K}\\right)\\\\&\\le m_2 \\sum _{j,{j^{\\prime }}=1}^{n} \\mathbb {E}\\left( \\Delta (T_j(n)) \\mathbb {1}_{{N}_{j^{\\prime }}(n)\\ge K} \\Delta (T_{j^{\\prime }}(n)) \\mathbb {1}_{{N}_{j^{\\prime }}(n)\\ge K}\\right).$ We evaluate the expectation in the right-hand side by conditioning first on $N_j(n)=k$ and $N_{j^{\\prime }}(n)=k^{\\prime }$ with $k,k^{\\prime }\\ge 3$ .", "Recall from Lemma that the genealogical trees $T_j(n)$ and $T_{j^{\\prime }}(n)$ are then two random recursive trees with respective sizes $k$ and $k^{\\prime }$ , which are further independent when $j\\ne j^{\\prime }$ .", "Thanks to Corollary (i-ii) we get $ &\\mathbb {E}\\left( \\Delta (T_j(n)) \\mathbb {1}_{{N}_{j^{\\prime }}(n)\\ge K} \\Delta (T_{j^{\\prime }}(n)) \\mathbb {1}_{{N}_{j^{\\prime }}(n)\\ge K}\\right)\\\\&= \\left\\lbrace \\begin{matrix}\\frac{1}{3} \\mathbb {E}(N_j(n)\\mathbb {1}_{{N}_j(n)\\ge K}) &\\text{ if }j=j^{\\prime }\\,\\\\0 &\\text{ if }j\\ne j^{\\prime }.\\end{matrix}\\right.$ We have thus shown that $\\mathbb {E}\\left( \\left\| \\sum _{k\\ge K} \\check{S}_k(n)\\right\|^2 \\right) \\le \\frac{m_2 }{3}\\sum _{j=1}^{n} \\mathbb {E}(N_j(n)\\mathbb {1}_{{N}_j(n)\\ge K} ),$ which yields our claim just as in the proof of Lemma REF (ii).", "The proof of Theorem is now easily completed by combining Lemmas REF and REF .", "Indeed, the identity $\\frac{pm_2}{2-p}+\\sum _{k=2}^{\\infty }\\sigma ^2_{k} = \\frac{m_2}{3-2p}$ is easily checked from (REF )." ], [ "A stable central limit theorem", "The arguments for the proof of Theorem when the step distribution $\\mu $ has a finite second moment can be adapted to the case when $\\mu $ belongs to some stable domaine of attraction; for the sake of simplicity we focus on the situation without centering.", "Specifically, let $(a_n)$ be a sequence of positive real numbers that is regularly varying with exponent $1/\\alpha $ for some $\\alpha \\in (0,2)$ , in the sense that $\\lim _{n\\rightarrow \\infty } a_{\\lfloor rn\\rfloor }/a_n= r^{1/\\alpha }$ for every $r>0$ , and suppose that $ \\lim _{n\\rightarrow \\infty } \\frac{X_1+\\cdots + X_n}{a_n} = Z \\qquad \\text{in distribution},$ where $Z$ is some $\\alpha $ -stable random variable.", "We refer to Theorems 4 and 5 on p. 181-2 in [20] and Section 6 of Chapter 2 in [22] for necessary and sufficient conditions for (REF ) in terms of $\\mu $ only.", "We write $\\varphi _{\\alpha }$ for the characteristic exponent of $Z$ , viz.", "$\\mathbb {E}\\left(\\exp ({i}\\theta Z)\\right) = \\exp (-\\varphi _{\\alpha }(\\theta ))\\qquad \\text{for all }\\theta \\in \\mathbb {R};$ recall that $\\varphi _{\\alpha }$ is homogeneous with exponent $\\alpha $ , i.e.", "$\\varphi _{\\alpha }(\\theta )= \|\\theta \|^{\\alpha } \\varphi _{\\alpha }(\\textrm {sgn}(\\theta ))\\qquad \\text{for all }\\theta \\ne 0.$ Recall the definition and properties of the Eulerian numbers $\\langle {}^{n}_{k}\\rangle $ from Section 2, and also the Pochhammer notation $(x)^{(k)} \\frac{\\Gamma (x+k)}{\\Gamma (x)} =\\prod _{j=0}^{k-1} (x+j), \\qquad x>0, k\\in \\mathbb {N},$ for the rising factorial, where $\\Gamma $ stands for the gamma function.", "We can now claim: Theorem Assume (REF ).", "For each $p\\in (0,1)$ , we have $\\lim _{n\\rightarrow \\infty } \\frac{\\check{S}(n)}{a_n} = \\check{Z} \\qquad \\text{in distribution},$ where $\\check{Z}$ is an $\\alpha $ -stable random variable with characteristic exponent $\\check{\\varphi }_{\\alpha }$ given by $\\check{\\varphi }_{\\alpha }(\\theta ) = \\frac{p }{(1-p)} \\sum _{k=1}^{\\infty } \\sum _{\\ell =0}^{k-1}\\frac{ \\varphi _{\\alpha }((k-2\\ell )\\theta )}{ (1+1/(1-p))^{(k)}} {\\displaystyle \\left\\langle {{k-1} \\atop {\\ell -1}}\\right\\rangle }, \\qquad \\theta \\in \\mathbb {R}.$ The proof of Theorem relies on a refinement of Simon's result (Lemma ) to the asymptotic frequencies of genealogical trees induced by the reinforcement algorithm (REF ).", "We denote by ${{\\mathcal {T}}^{\\uparrow }}$ the set of increasing trees (of arbitrary finite size), and for any $\\tau \\in {\\mathcal {T}}^{\\uparrow }$ , we write $\|\\tau \|$ for its the size (number of vertices) and $\\Delta (\\tau )$ for the difference between its numbers of even vertices and of odd vertices.", "Refining (REF ), we also define $\\nu _{\\tau }(n) \\sum _{j= 1}^{{\\mathrm {i}}(n)} \\mathbb {1}_{\\lbrace T_j(n) =\\tau \\rbrace }, \\qquad \\tau \\in {\\mathcal {T}}^{\\uparrow }.$ Lemma We have $\\sum _{\\tau \\in {{\\mathcal {T}}^{\\uparrow }}}\\frac{ \|\\tau \|+ \|\\Delta (\\tau )\|^2}{(1+1/(1-p))^{(\|\\tau \|)}}= \\frac{4p}{3(1-p)},$ and there is the convergence in probability $\\lim _{n\\rightarrow \\infty } \\sum _{\\tau \\in {{\\mathcal {T}}^{\\uparrow }}} (\|\\tau \|+ \|\\Delta (\\tau )\|^2)\\left\| \\frac{\\nu _{\\tau }(n)}{n} -\\frac{ p}{(1-p) (1+1/(1-p))^{(\|\\tau \|)}} \\right\| =0.$ We start by claiming that for every $k\\ge 1$ , and every tree $\\tau \\in {{\\mathcal {T}}^{\\uparrow }}$ with size $\|\\tau \|=k$ , we have $\\lim _{n\\rightarrow \\infty } \\frac{\\nu _{\\tau }(n)}{n} =\\frac{ p}{(1-p) (1+1/(1-p))^{(k)}} \\qquad \\text{in probability.", "}$ Indeed, the distribution of the random recursive tree $\\mathbb {T}_k$ of size $k$ is the uniform probability measure on the set of increasing trees with size $k$ , which has $(k-1)!$ elements.", "We deduce from Lemma and the law of large numbers that $\\nu _{\\tau }(n) \\sim \\nu _k(n)/(k-1)!.$ The claim (REF ) now follows from Lemma and the identity $ {\\mathrm {B}}(k,1+/(1-p)) = \\frac{ (k-1)!", "}{ (1+1/(1-p))^{(k)}}.$ We now have to prove that (REF ) holds in $L^1(\|\\tau \|+ \|\\Delta (\\tau )\|^2,{\\mathcal {T}}^{\\uparrow })$ .", "On the one hand, one has obviously for every $n\\ge 1$ $\\sum _{\\tau \\in {{\\mathcal {T}}^{\\uparrow }}} \|\\tau \|\\nu _{\\tau }(n)= n.$ On the other hand, there are $(k-1)!$ increasing trees with size $k$ and hence $\\frac{p}{1-p} \\sum _{\\tau \\in {{\\mathcal {T}}^{\\uparrow }}} \\frac{\|\\tau \|}{(1+1/(1-p))^{(\|\\tau \|)}}= \\frac{p}{1-p} \\sum _{k=1}^{\\infty } k{\\mathrm {B}}(k,1+1/(1-p))=1,$ where the second equality is (REF ).", "We deduce from Scheffé's Lemma and (REF ) that there is the convergence in probability $\\lim _{n\\rightarrow \\infty } \\sum _{\\tau \\in {{\\mathcal {T}}^{\\uparrow }}}\|\\tau \|\\left\| \\frac{\\nu _{\\tau }(n)}{n} -\\frac{ p}{(1-p) (1+1/(1-p))^{(\|\\tau \|)}} \\right\| =0.$ Similarly, we deduce from Corollary (ii) and Lemma that, for every $n\\ge 0$ $\\mathbb {E}\\left( \\sum _{\\tau \\in {{\\mathcal {T}}^{\\uparrow }}} \\Delta (\\tau )^2 \\nu _{\\tau }(n)\\right) & = \\mathbb {E}\\left(\\sum _{j=1}^{{\\mathrm {i}}(n)} \\Delta (T_j(n))^2\\right)= \\frac{1}{3} \\mathbb {E}\\left(\\sum _{j=1}^{{\\mathrm {i}}(n)} \|T_j(n)\|\\right) = n/3,$ and further, since there are $(k-1)!$ increasing trees with size $k$ and $\\mathbb {T}_k$ has the uniform distribution on the set of such trees, $\\frac{p}{1-p} \\sum _{\\tau \\in {{\\mathcal {T}}^{\\uparrow }}} \\frac{\\Delta (\\tau )^2}{(1+1/(1-p))^{(\|\\tau \|)}}&= \\frac{p}{1-p} \\sum _{k=1}^{\\infty } \\mathbb {E}(\\Delta (\\mathbb {T}_k)^2){\\mathrm {B}}(k,1+1/(1-p))\\\\&= \\frac{p}{1-p} \\sum _{k=1}^{\\infty } \\frac{k}{3}{\\mathrm {B}}(k,1+1/(1-p))=\\frac{1}{3}.$ We conclude again from Scheffé's Lemma that $\\lim _{n\\rightarrow \\infty } \\sum _{\\tau \\in {{\\mathcal {T}}^{\\uparrow }}}\\Delta (\\tau )^2\\left\| \\frac{\\nu _{\\tau }(n)}{pn} -\\frac{ p}{(1-p) (1+1/(1-p))^{(\|\\tau \|)}} \\right\| =0,$ and the proof is complete.", "We now establish Theorem .", "[Proof of Theorem ] We denote the characteristic function of $\\mu $ by $\\Phi (\\theta )=\\int _{\\mathbb {R}} {\\mathrm {e}}^{\\mathrm {i} \\theta x} \\mu (\\mathrm {d} x) \\qquad \\text{for }\\theta \\in \\mathbb {R}.$ Fix $r>0$ small enough so that $\|1-\\Phi (\\theta )\|<1$ whenever $\|\\theta \|\\le r$ , and then define the characteristic exponent $\\varphi :[-r,r]\\rightarrow \\mathbb {C}$ as the continuous determination of the logarithm of $\\Phi $ on $[-r,r]$ .", "In words, $\\varphi $ is the unique continuous function on $[-r,r]$ with $\\varphi (0)=0$ and such that $\\Phi (\\theta )= \\exp (-\\varphi (\\theta ))$ for all $\\theta \\in [-r,r]$ .", "For definitiveness, we further set $\\varphi (\\theta )=0$ whenever $\|\\theta \|>r$ .", "Next, observe from the Markov's inequality that for any $1<\\beta <2\\wedge (1-p)^{-1}$ and any $a>0$ $\\mathbb {P}\\left( \\exists j\\le {\\mathrm {i}}(n): \|\\Delta (T_j(n))\|\\ge a\\sqrt{n}\\right) \\le a^{-2\\beta } n^{-\\beta } \\mathbb {E}\\left(\\sum _{j=1}^{{\\mathrm {i}}(n)} \|\\Delta (T_j(n))\|^{2\\beta }\\right),$ so that, thanks to Lemma , $\\lim _{n\\rightarrow \\infty } \\frac{1}{\\sqrt{n}} \\max _{1\\le j \\le {\\mathrm {i}}(n)} \|\\Delta (T_j(n))\|=0\\qquad \\text{ in probability.", "}$ In particular, since the sequence $(a_n)$ is regularly varying with exponent $1/\\alpha >1/2$ , for every $\\theta \\in \\mathbb {R}$ , the events $\\Lambda (n,\\theta )\\lbrace \|\\theta \\Delta (T_j(n))/a_n\|< r\\text{ for all } j=1, \\ldots {\\mathrm {i}}(n) \\rbrace , \\qquad n\\ge 1$ occur with high probability as $n\\rightarrow \\infty $ , in the sense that $\\lim _{n\\rightarrow \\infty } \\mathbb {P}(\\Lambda (n,\\theta ))=1$ .", "We then deduce from (REF ) and the fact that the variables $X_j$ are i.i.d.", "with law $\\mu $ that for every $\\theta \\in \\mathbb {R}$ , $\\mathbb {E}(\\exp (\\mathrm {i} \\theta \\check{S}(n)/a_n)\\mathbb {1}_{\\Lambda (n,\\theta )}) =\\mathbb {E}\\left(\\exp \\left( - \\frac{1}{n} \\sum _{j=1}^{{\\mathrm {i}}(n)} n\\varphi \\left( \\theta a_n^{-1} \\Delta (T_j(n))\\right)\\right)\\mathbb {1}_{\\Lambda (n,\\theta )} \\right).$ We then write, in the notation of Lemma , $\\frac{1}{n} \\sum _{j=1}^{{\\mathrm {i}}(n)} n\\varphi \\left( \\theta a_n^{-1} \\Delta (T_j(n))\\right)= \\sum _{\\tau \\in {\\mathcal {T}}^{\\uparrow }}n\\varphi \\left( \\theta a_n^{-1} \\Delta (\\tau )\\right) \\frac{\\nu _{\\tau }(n)}{n}.$ Recall that we assume (REF ).", "According to Theorem 2.6.5 in Ibragimov and Linnik [22], $\\varphi $ is regularly varying at 0 with exponent $\\alpha $ , and since $\\alpha <2$ , the Potter bounds (see Theorem 1.5.6 in [13]) show that for some constant $C$ : $n\\varphi \\left( \\theta a_n^{-1} \\Delta (\\tau )\\right)\\le C \|\\theta \\Delta (\\tau )\|^2.$ We deduce from Lemma that for every fixed $\\theta \\in \\mathbb {R}$ , there is the convergence in probability $\\lim _{n\\rightarrow \\infty } \\sum _{\\tau \\in {\\mathcal {T}}^{\\uparrow }}n\|\\varphi \\left( \\theta a_n^{-1} \\Delta (\\tau )\\right) \| \\left\| \\frac{\\nu _{\\tau }(n)}{n} -\\frac{ p}{(1-p) (1+1/(1-p))^{(\|\\tau \|)}} \\right\|=0 .$ Furthermore, still from Theorem 2.6.5 in Ibragimov and Linnik [22], we have $\\lim _{n\\rightarrow \\infty }n\\varphi (\\theta /a_n) = \\varphi _{\\alpha }(\\theta ), \\qquad \\text{for every }\\theta \\in \\mathbb {R},$ and we deduce by dominated convergence, using Lemma and (REF ), that $\\lim _{n\\rightarrow \\infty } \\sum _{\\tau \\in {\\mathcal {T}}^{\\uparrow }}n\|\\varphi \\left( \\theta a_n^{-1} \\Delta (\\tau )\\right) -\\varphi _{\\alpha }(\\theta \\Delta (\\tau ))\|\\frac{ p}{(1-p) (1+1/(1-p))^{(\|\\tau \|)}} =0 .$ Putting the pieces together, we have shown that $\\lim _{n\\rightarrow \\infty } \\mathbb {E}(\\exp (\\mathrm {i} \\theta \\check{S}(n)/a_n)) =\\sum _{\\tau \\in {{\\mathcal {T}}^{\\uparrow }}} \\varphi _{\\alpha }(\\theta \\Delta (\\tau ))\\frac{ p}{(1-p) (1+1/(1-p))^{(\|\\tau \|)}}.$ It only remains to check that the right-hand side above agrees with the formula of the statement.", "This follows from Lemma and the fact that for every $k\\ge 1$ , $\\mathbb {T}_k$ has the uniform distribution on $\\lbrace \\tau \\in {{\\mathcal {T}}^{\\uparrow }}: \|\\tau \|=k\\rbrace $ ." ] ]	2011.14069
[ [ "Engineering analog quantum chemistry Hamiltonians using cold atoms in\n optical lattices" ], [ "Abstract Using quantum systems to efficiently solve quantum chemistry problems is one of the long-sought applications of near-future quantum technologies.", "In a recent work, ultra-cold fermionic atoms have been proposed for these purposes by showing us how to simulate in an analog way the quantum chemistry Hamiltonian projected in a lattice basis set.", "Here, we continue exploring this path and go beyond these first results in several ways.", "First, we numerically benchmark the working conditions of the analog simulator, and find less demanding experimental setups where chemistry-like behaviour in three-dimensions can still be observed.", "We also provide a deeper understanding of the errors of the simulation appearing due to discretization and finite size effects and provide a way to mitigate them.", "Finally, we benchmark the simulator characterizing the behaviour of two-electron atoms (He) and molecules (HeH$^+$) beyond the example considered in the original work." ], [ "Introduction", "Solving quantum chemistry problems, such as obtaining the electronic structure of complex molecules or understanding chemical reactions, is an extremely challenging task.", "Even if one considers the nuclei positions $\\lbrace \\mathbf {R}_\\alpha \\rbrace _{\\alpha =1}^{N_n}$ fixed due to their larger mass (Born-Oppenheimer approximation), and focus only on the electronic degrees of freedom, these problems still involve many electrons interacting through Coulomb forces, whose associated Hilbert space grows exponentially with the number of electrons ($N_e$ ).", "One way of circumventing this exponential explosion [2] consists in using the electron density instead of the wavefunction, like in density-functional methods [3], [4], where the complexity is hidden in the choice of exchange-correlation density functionals.", "Educated guesses of such functionals have already allowed us to study the properties of large molecules [5].", "Unfortunately, there is no unambiguous path for improving these functionals [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], which are known to fail in certain regimes [6].", "A complementary route consists in projecting the quantum chemistry Hamiltonian in a basis set [18], [19] with a finite number of elements $N_o$ .", "The typical choices for the basis are linear combinations of atomic orbitals with Slater- or gaussian-type radial components.", "These methods generally provide good accuracies with small $N_o$ .", "However, the quality of the solution ultimately depends on the basis choice.", "And on top of that, the Hilbert space of the projected Hamiltonian still grows exponentially with $N_o$ , which complicates their solution if large basis sets are required, especially for non-equilibrium situations.", "In parallel to these developments, the last few years have witnessed the emergence of an alternative route to study these problems based on using quantum systems to perform the computation.", "This idea was first proposed by Feynman as a way of preventing the exponential explosion of resources of quantum many-body problems [2], formalized later by Lloyd [20], and first exported into the quantum chemistry realm by Aspuru-Guzik et al [21].", "First algorithms used Gaussian orbital sets and phase-estimation methods to obtain ground-state molecular energies [22], [23].", "Despite the initial pessimistic scaling of the gate complexity with the number of orbitals (polynomial but with a large exponent), recent improvements through the use of more efficient algorithms [24] or different basis sets, e.g., plane-waves [25], [26], [27], have reduced significantly the gate scaling complexity.", "Since these algorithms typically assume fault-tolerant quantum computers that will not be available in the near-future, in the last years there has also been an intense effort on hybrid variational approaches more suitable for current noisy quantum computers [28], [29], [30].", "However, these will be ultimately limited by the available ansätze that can be obtained with current devices, as well as on the optimization procedure [31], [32].", "The previously described efforts (see Ref.", "[33] for an updated review) fall into what is called the digital quantum simulation framework, in which the fermionic problem is mapped into qubits and the Hamiltonian evolution is performed stroboscopically.", "In a recent work [1], the authors and co-workers opened a complementary route to study these problems showing how to simulate in an analog way the quantum chemistry Hamiltonian using a discretized space (or grid) basis representation [34].", "These representations have been generally less used in the literature due to the large basis sets required to obtain accurate results.", "However, they have recently experienced a renewed interest [35], [36], [37] due to their better suitability for DMRG methods [38].", "In our case [1], we use this grid representation because it is well suited for describing fermions trapped in optical lattice potentials, where the fermionic space is naturally discretized in the different trapping minima of the potential.", "Then, as explained in Ref.", "[1], fermionic atoms with two internal atomic states can hop around the lattice, playing the role of electrons, spatially-shaped laser beams simulate the nuclear attraction, while an additional auxiliary atomic specie mediates an effective repulsion between the fermionic atoms that mimics the Coulomb repulsion between electrons.", "In this work, we continue exploring the path opened by Ref.", "[1] and extend its results.", "The manuscript is structured as follows: In Section we introduce the different parts of the quantum chemistry Hamiltonians projected in finite basis sets.", "We discuss both the grid basis representation that we use in our analog simulation and the widely-used linear combination of atomic orbitals, emphasizing the similarities and the differences between these two approaches.", "In Section we review how to obtain the single-particle parts of the quantum chemistry Hamiltonian, that are, the electron kinetic energy and the nuclear attraction, as proposed in Ref. [1].", "Besides, we extend the previous analysis with a deeper understanding of the discretization and finite size errors of the simulation, which allows us to introduce an extrapolation method that mitigates the limitations imposed by these errors for a given lattice size.", "In Section we analyze the role of the different ingredients introduced in Ref.", "[1] to obtain an effective pair-wise and Coulomb-like repulsion between the fermionic atoms.", "This analysis enables us to present less demanding simplified experimental setups to simulate chemistry-like behaviour in three-dimensional systems.", "Besides, we numerically benchmark the parameter regimes where the analog simulator works beyond the perturbative analysis of Ref. [1].", "In Section , we put all the ingredients together and benchmark our simulator beyond the example considered in Ref.", "[1], i.e., considering two-electron atoms (He) and molecules (He$^+$ -H).", "Finally, in Section we summarize our findings and point to further directions of work." ], [ "Quantum chemistry Hamiltonians in discrete basis sets: atomic orbitals ", "The typical problems in quantum chemistry are either calculating the electronic structure of a complex molecule in equilibrium, $\\hat{H}_e\\left\| \\Psi \\right\\rangle _e=E_e\\left\| \\Psi \\right\\rangle _e$ , or its time-evolution in an out-of-equilibrium situation: $i\\partial _t\\left\| \\Psi (t)\\right\\rangle _e=\\hat{H}_e\\left\| \\Psi (t)\\right\\rangle _e$ .", "These problems are generally calculated using the Born-Oppenheimer approximation (BOA), that is, treating each nuclei classically as a fixed particle of charge $Z_\\alpha e$ .", "Along the manuscript, we will use atomic units $m_e=e=\\hbar =(4\\pi \\varepsilon _0)^{-1}\\equiv 1$ , such that the natural unit of length will be given by the Bohr Radius $\\text{a}_0=(4\\pi \\varepsilon _0 \\hbar ^2)/(m_e e^2)\\equiv 1$ , and the unit of energy is the Hartree-Energy $E_h=\\hbar ^2/(m_e \\text{a}_0^2)\\equiv 1$ (twice the Rydberg energy (Ry)).", "Using these units, the BOA-electronic Hamiltonian for a molecule with $N_e$ electrons and a given nuclei configuration $\\lbrace \\mathbf {R}_\\alpha \\rbrace _{\\alpha =1}^{N_n}$ reads: $\\hat{H}_e=&\\sum _{j=1}^{N_e}\\left[-\\frac{1}{2}\\hat{\\nabla }_j^2-\\sum _{\\alpha =1}^{N_n}Z_\\alpha \\hat{V}_c(\\hat{\\mathbf {r}}_j,\\mathbf {R}_\\alpha )\\right]\\\\&+\\frac{1}{2}\\sum _{i\\ne j=1}^{N_e} \\hat{V}_c(\\hat{\\mathbf {r}}_i,\\hat{\\mathbf {r}}_j)=\\\\&=\\sum _{j=1}^{N_e} \\hat{H}_1(\\hat{\\mathbf {r}}_j)+\\frac{1}{2}\\sum _{i\\ne j=1}^{N_e} \\hat{V}_c(\\hat{\\mathbf {r}}_i,\\hat{\\mathbf {r}}_j)\\,,$ where bold letters indicate three-dimensional vectors and $V_c(\\mathbf {r}_1,\\mathbf {r}_2)=\\frac{\\text{a}}{\|\\mathbf {r}_1-\\mathbf {r}_2\|}$ is the pair-wise Coulomb potential between the charged particles (electrons and nuclei).", "The expression inside the brackets of Eq.", "(REF ) is labeled as the single-electron part of the Hamiltonian ($\\hat{H}_1(\\hat{\\mathbf {r}}_j)$ ) and contains both the electron kinetic energy ($\\hat{T}_e=-\\sum _{j}\\nabla _j^2/2$ ) and the electron-nuclei attraction ($\\hat{H}_{n-e}=-\\sum _{\\alpha =1}^{N_n}Z_\\alpha \\hat{V}_c(\\hat{\\mathbf {r}}_j,\\mathbf {R}_\\alpha ))$ , whereas Eq.", "() corresponds to the electron-electron repulsion ($\\hat{V}_{e-e}$ ).", "Since the molecular electrons are indistinguishable (up to the spin degree of freedom), for computational purposes it is typically more convenient to write a second-quantized version of the Hamiltonian that already takes into account the fermionic statistics of the particle.", "There is a general recipe to do it [39], [40]: first, one needs a set of single-particle states $\\mathcal {B}=\\lbrace \\left\| \\phi _i\\right\\rangle \\rbrace $ , that can be used to define an abstract Hilbert space of states $\\left\| n_1,n_2,\\dots \\right\\rangle $ , denoting that there are $n_i$ electrons occupying the $i$ -th single-particle states.", "With these states, one can then define annihilation (creation) operators $\\hat{c}_{i}^{(\\dagger )}$ that denote the creation/destruction of a fermionic particle in the $i$ -th single-particle state.", "This labelling already accounts for the different spin states and the fermionic statistics of the particle through their anticommutation rules: $\\lbrace \\hat{c}_{i},\\hat{c}_{j}^\\dagger \\rbrace =\\delta _{ij}$ , and $\\lbrace \\hat{c}_{i},\\hat{c}_{j}\\rbrace =\\lbrace \\hat{c}^\\dagger _{i},\\hat{c}_{j}^\\dagger \\rbrace =0$ .", "With these operators, one can define the field operators: $\\hat{\\Psi }(\\mathbf {r})&=\\sum _{i} \\phi _i(\\mathbf {r}) \\hat{c}_{i}\\,,\\\\\\hat{\\Psi }^{\\dagger }(\\mathbf {r})&=\\sum _{i} \\phi ^{}_i(\\mathbf {r}) \\hat{c}_{i}^{\\dagger }\\,,$ that can be used to write the Hamiltonian in the following form: $\\hat{H}_e&=\\int d\\mathbf {r}\\hat{\\Psi }^\\dagger (\\mathbf {r})\\left[-\\frac{1}{2}\\hat{\\nabla }^2-\\sum _{\\alpha =1}^{N_n} \\frac{Z_\\alpha }{\|\\mathbf {r}-\\mathbf {R}_\\alpha \|}\\right]\\hat{\\Psi }(\\mathbf {r}) \\\\&+\\frac{1}{2}\\iint d\\mathbf {r}d\\mathbf {r}^{\\prime }\\hat{\\Psi }^\\dagger (\\mathbf {r})\\hat{\\Psi }^\\dagger (\\mathbf {r}^{\\prime })\\frac{1}{\|\\mathbf {r}-\\mathbf {r}^{\\prime }\|} \\hat{\\Psi }(\\mathbf {r}^{\\prime })\\hat{\\Psi }(\\mathbf {r})\\,.$ If the basis $\\mathcal {B}$ of single-particle states is complete (i.e., it is infinite dimensional), the mapping between the first quantized Hamiltonian of Eqs.", "(REF )-() and the second quantized one of Eqs.", "(REF )-() would be exact.", "However, this is generally not practical since the associated Hilbert space will still be infinite.", "For those reasons, the typical approach consists in projecting the Hamiltonian in the subspace spanned by the tensor product of a finite-dimensional discrete basis set, $\\mathcal {B}_t$ , and solving the problem within that subspace.", "The prototypical bases chosen are built out of (linear combinations) of atomic orbitals centered around the nuclei position, labeled as linear combination of atomic orbitals (LCAO) basis sets [18].", "However, for our analog quantum chemistry simulation it will be more adequate to use an alternative representation based on a grid discretization of the continuum in a finite set of points.", "In what follows, we discuss how the second quantized electronic Hamiltonian looks in both cases, and highlight their main differences.", "Linear combination of atomic orbitals (LCAO).", "Here, the basis set is composed of $N_o (>N_e)$ single-particle (orthonormal) atomic orbitals, $\\mathcal {B}_t=\\lbrace \\left\| \\phi _i\\right\\rangle \\rbrace _{i=1}^{N_o}$ , with which the Hamiltonian reads: $\\hat{H}_e=\\sum _{i,j=1}^{N_o} t_{ij}\\hat{c}^\\dagger _{i}\\hat{c}_{j}+\\sum _{i,j.k,l=1}^{N_o} \\frac{V_{ijkl}}{2} \\hat{c}^\\dagger _{i}\\hat{c}^\\dagger _{j}\\hat{c}_{l}\\hat{c}_{k}\\,,$ where the parameters of the discrete Hamiltonian $t_{ij}$ and $V_{ijkl}$ can be computed using the real space representation of the orbitals, $\\phi _i(\\mathbf {r})=\\langle \\mathbf {r}\|\\phi _i\\rangle $ , as follows: $t_{ij}&=\\int d\\mathbf {r}\\phi ^_i(\\mathbf {r})\\left[-\\frac{\\nabla ^2}{2}-\\sum _{\\alpha }Z_\\alpha V_c(\\mathbf {r},\\mathbf {R}_\\alpha )\\right]\\phi _j(\\mathbf {r})\\,,\\\\V_{ijkl}&=\\iint d\\mathbf {r}d\\mathbf {r}^{\\prime }\\phi ^_i(\\mathbf {r}) \\phi ^_j(\\mathbf {r}^{\\prime })\\frac{1}{\|\\mathbf {r}-\\mathbf {r}^{\\prime }\|}\\phi _l(\\mathbf {r}^{\\prime }) \\phi _k(\\mathbf {r})\\,,$ The number of $t_{ij}$ - and $V_{ijkl}$ -parameters scales with the size of the $\\mathcal {B}_t$ -basis as $N_o^2$ and $N_o^4$ , respectively, while their value depends on the particular states chosen.", "Convenient choices widely-used in quantum chemistry are linear combinations of Gaussian or exponential type-orbitals localized around the nuclei [18], [19].", "The former are particularly appealing since the properties of Gaussian functions can simplify substantially the calculations of $t_{ij}, V_{ijkl}$ , which can become a bottleneck if large basis sets are required.", "An advantage of this approach is that the number of orbitals required typically scales proportionally with $N_e$ .", "Besides, it is a variational method that provides an unambiguous path to reach to the true ground state energy by increasing $N_o$ .", "This is why these representations have been the most popular ones in most current approaches for digital quantum simulation [33].", "On the down side, the accuracy of the solution will depend on the particular molecular structure since the basis sets are composed of functions with fixed asymptotic decays that might not be suitable, e.g., to describe diffuse molecules [41], [42], [43], [44], [45], [46].", "Local or grid-discretized basis.", "This option consists in writing the continuum Hamiltonian $\\hat{H}_e$ in grid points $\\mathbf {n}=(n_x,n_y,n_z)\\text{a}$ , where $\\text{a}$ is the spacing between the discretized points, $n_i\\in \\mathbb {Z}$ , and $N=N_x N_y N_z$ being the total number of points.", "To do it, one can approximate the derivatives of the kinetic energy term in Eq.", "REF by finite-differences, and evaluate the potentials at the grid points.", "This ultimately results in a second quantized Hamiltonian with the following shape [34], [19]: $\\hat{H}_e&=-\\sum _{\\mathbf {n},\\mathbf {m},\\sigma }J_{\\mathbf {n},\\mathbf {m}}\\hat{c}^\\dagger _{\\mathbf {n},\\sigma }\\hat{c}_{\\mathbf {m},\\sigma }\\\\&-\\sum _{\\alpha =1}^{N_n}\\sum _{\\mathbf {n},\\sigma }{V}_\\mathbf {n}(\\mathbf {R}_{\\alpha })\\hat{c}^\\dagger _{\\mathbf {n},\\sigma }\\hat{c}_{\\mathbf {n},\\sigma }\\\\&+\\frac{1}{2}\\sum _{\\mathbf {n}, \\mathbf {m},\\sigma , \\sigma ^{\\prime }}{V}_\\mathrm {el}(\\mathbf {n},\\mathbf {m}) \\hat{c}^\\dagger _{\\mathbf {n},\\sigma }\\hat{c}^\\dagger _{\\mathbf {m},\\sigma ^{\\prime }}\\hat{c}_{\\mathbf {n},\\sigma }\\hat{c}_{\\mathbf {m},\\sigma ^{\\prime }} \\,,$ where $\\hat{c}^{(\\dagger )}_{\\mathbf {n},\\sigma }$ are now the local operators creating an electron with spin $\\sigma $ at site position $\\mathbf {n}$ , satisfying $\\lbrace \\hat{c}_{\\mathbf {m},\\sigma },\\hat{c}_{\\mathbf {n},\\sigma ^{\\prime }}^\\dagger \\rbrace =\\delta _{\\mathbf {m}\\mathbf {n}}\\delta _{\\sigma ,\\sigma ^{\\prime }}$ .", "The kinetic energy coefficients [in Eq.", "(REF )] $J_{\\mathbf {n},\\mathbf {m}}$ depend on the expansion order chosen to approximate the Laplacian, and decay with the separation between sites $\|\\mathbf {n}-\\mathbf {m}\|$ .", "For this manuscript, we will use the simplest finite difference formula for the second order derivative: $\\frac{d^2 f(x)}{d x^2}\\approx \\frac{f(x+\\text{a})-2 f(x)+f(x-\\text{a})}{\\text{a}^2}\\,,$ which means that only nearest neighbour hoppings (and on-site energy) will appear in the kinetic energy term of Eq.", "(REF ), and $J_{\\mathbf {n},\\mathbf {m}}\\equiv 0$ for the rest of the hopping terms.", "The nuclei-attraction term [Eq.", "()] induces a position-dependent energy shift on the discretized electron orbitals coming from the attraction of the nuclei.", "Finally, the electron-electron repulsion [Eq.", "()] translates into long-range density-density interactions between the localized fermionic states.", "In the limit where $N\\rightarrow \\infty $ and $\\text{a}\\rightarrow 0$ , the Hamiltonian of Eqs.", "(REF )-() converges to the continuum one.", "This method typically requires larger basis sets to obtain accurate results [34] compared to LCAO ones.", "However, the number of interaction terms ${V}_\\mathrm {el}(\\mathbf {n},\\mathbf {m})$ scales quadratically with the size of the basis because only density-density interaction terms appear.", "This can yield dramatic improvements when applying tensor-network methods, which motivates the renewed interest they have experienced in the last years [35], [36], [37].", "Besides, for the analog quantum simulation perspective such density-density interactions appear more naturally than the four-index interactions appearing in LCAO approaches.", "A potential disadvantage is that these methods are generally not variational.", "That is, increasing $N_o$ might sometimes yield a larger energy than the one of smaller basis sets.", "This has been identified as a problem of underestimation of the kinetic energy when using the finite-difference approximation of the derivates [Eq.", "(REF )] [47].", "However, there are constructive ways of making the kinetic operator variational using different approximations of the kinetic energy [47].", "Along this manuscript, however, we will stick to the simple finite-difference formula of Eq.", "(REF ) because of its simplicity.", "Besides, we will also provide a way of mitigating such discretization errors using an extrapolation method that we discuss in section REF .", "In what follows, we explain how to simulate the different parts of the quantum chemistry Hamiltonian projected in a grid basis using ultra-cold atoms trapped in optical lattices, as initially proposed in Ref. [1].", "The reason for choosing this platform is that fermionic atoms with (at least) two internal atomic states can be used to describe electrons without the need to encode these operators into qubits, simplifying the Hamiltonian simulation, as already pointed out in earlier proposals [48], [49], [50].", "We start by considering the single-particle part of the Hamiltonian in Section , and then explain how to obtain the electron repulsion in Section ." ], [ "Simulating single-particle Hamiltonian with ultra-cold atoms in optical lattices: kinetic and nuclear energy terms", "The dynamics of ultra-cold fermionic atoms trapped in optical lattices is described by the following first-quantized Hamiltonian: $\\hat{H}_{f}=\\hat{T}_f+\\hat{V}_{\\mathrm {per}}(\\hat{\\mathbf {r}})+\\hat{V}_{\\mathrm {aux}}(\\hat{\\mathbf {r}})\\,,$ which contains three terms: The kinetic energy of the fermionic atoms, which reads: $\\hat{T}_f=-\\frac{\\hbar ^2}{2 M_f}\\sum _{j=1}^{N_f}\\hat{\\nabla ^2}_j\\,,$ where $N_f$ is number of fermions of the system (that should be equal to the number of electrons we want to simulate), and $M_f$ their mass.", "The optical lattice potential $\\hat{V}_{\\mathrm {per}}(\\mathbf {r})$ .", "This is typically generated by retro-reflected laser beams with a wavelength $\\lambda $ that is off-resonant with a given atomic optical transition.", "These lasers form standing-waves which generate a spatially periodic energy shift [51], whose amplitude can be controlled through the laser intensity and/or detuning from the optical transition.", "Assuming a cubic geometry for the lasers, the optical potential reads [52]: $V_{\\mathrm {per}}(\\mathbf {r})=-V_D\\sum _{\\alpha =x,y,z} \\sin ^2\\left(\\frac{2\\pi \\alpha }{\\lambda }\\right)\\,,$ where $V_D$ is the trapping potential depth, that we assume to be equal for the three directions.", "The lattice constant of such potential is $\\text{a}=\\lambda /2$ , and it imposes a maximum kinetic energy of the fermions $E_R=\\hbar ^2\\pi ^2/(2 M_f \\text{a}^2)$ , typically labeled as recoil energy, which is the natural energy scale of these systems.", "Note that because of the larger mass of the fermions compared to the real electron systems, the dynamics will occur at a much slower timescale (ms) compared to electronic systems (fs).", "This will facilitate the observation of real-time dynamics of the simulated chemical processes, something very difficult to do in real chemistry systems.", "Finally, we also included $V_\\mathrm {aux}(\\mathbf {r})$ that takes into account all possible atomic potential contributions which are not periodic, as it will be the case of the nuclear attractive potential.", "Note that when writing only these three contributions for the fermionic Hamiltonian $\\hat{H}_f$ , we are already assuming to be in the regime where inter-atomic interactions between the fermions are negligible.", "This regime can be obtained, e.g., tuning the scattering length using Feshbach resonances [53].", "As for the quantum chemistry case, here it is also convenient to write a second-quantized version of the Hamiltonian $\\hat{H}_f$ .", "For that, we need to first find an appropriate set of single-particle states to define the field operators $\\hat{\\Psi }_f(\\mathbf {r})$ as in Eq.", "(REF ), and afterward write the second-quantized Hamiltonian using them.", "In what follows, we explain the steps and approximations in the canonical approach to do it, whose details can be found in many authoritative references [54], [52], [55].", "First, it is useful to characterize the band-structure emerging in the single-particle sector due to the potential $\\hat{V}_\\mathrm {per}(\\mathbf {r})$ .", "Since $\\hat{V}_\\mathrm {per}(\\mathbf {r}+\\mathbf {R})=\\hat{V}_\\mathrm {per}(\\mathbf {r})$ for $\\mathbf {R}=\\sum _{i=1}^3 n_i \\mathbf {a}_i$ , with $\\mathbf {a}_{1,2,3}=\\mathrm {a}\\hat{x},\\mathrm {a}\\hat{y},\\mathrm {a}\\hat{z}$ and $n_i\\in \\mathbb {Z}$ , we can use the Bloch theorem to write the single-particle eigenstates of $\\hat{H}_f$ as follows: $\\psi _{n,\\mathbf {q}}(\\mathbf {r})=u_{n,\\mathbf {q}}(\\mathbf {r}) e^{i\\mathbf {q}\\cdot \\mathbf {r}}\\,,$ where $\\mathbf {q}$ is the quasimomentum in the reciprocal space, $u_{n,\\mathbf {q}}(\\mathbf {r})$ is a function with the same periodicity than $V_\\mathrm {per}(\\mathbf {r})$ , and $n$ is denoting the index of the energy band $E_{n}(\\mathbf {q})$ .", "In the limit where the trapping potential depth is much larger than the recoil energy ($V_D/E_R\\gg 1$ ), the atomic wavefunctions become localized in the potential minima.", "This is why, in that limit, it is useful to adopt a description based on Wannier functions localized in each potential minima, instead of the Bloch states $\\phi _{\\mathbf {q},n}(\\mathbf {r})$ of Eq.", "(REF ).", "The Wannier function of a site $\\mathbf {j}$ for the $n$ -th band can be obtained from $\\phi _{n,\\mathbf {q}}(\\mathbf {r})$ as follows: $W_{n,\\mathbf {j}}(\\mathbf {r})=\\frac{1}{\\sqrt{N}}\\sum _{\\mathbf {q}\\in \\mathrm {BZ}}\\phi _{n,\\mathbf {q}}(\\mathbf {r})e^{-i\\mathbf {j}\\mathbf {q}} \\,,$ where $N$ is the total number of sites of the optical potential.", "In the strong-confinement limit, $V_D\\gg E_R$ , the atoms only probe the positions close to the minima where the periodic potential can be expanded to $V_\\mathrm {per}(\\mathbf {r})=V_D \\pi ^2 r^2/\\text{a}^2$ , where $r=\|\\mathbf {r}\|$ .", "This allows one to obtain an analytical expression for the Wannier functions in this limit in terms of the eigenstates of an harmonic potential with trapping frequency: $\\omega _{t,\\alpha }=\\frac{\\sqrt{4 V_D E_R}}{\\hbar }\\,,$ with $\\alpha =x,y,z$ , which provides an energy estimate for the energy separation between the different bands appearing in the structure.", "Now, to write the field operator $\\hat{\\Psi }_f(\\mathbf {r})$ , one typically assumes that the atoms are prepared in the motional ground state of each trapping minimum, and that interband transitions are negligible [54], [52], [55].", "With these assumptions, $\\hat{\\Psi }_f(\\mathbf {r})$ can be expanded only in terms of states of the lowest energy band: $\\hat{\\Psi }_f(\\mathbf {r})=\\sum _{\\mathbf {j},\\sigma } W_{\\mathbf {j}}(\\mathbf {r}) \\hat{f}_{\\mathbf {j},\\sigma }\\,,$ where we drop the band-index $n$ , and where we define the annihilation (creation) operators $\\hat{f}^{(\\dagger )}_{\\mathbf {j},\\sigma }$ of a fermionic state with spin $\\sigma $ at site $\\mathbf {j}\\in \\mathbb {Z}^3$ We assume that the lattice and nuclei positions are normalized to the lattice constant $d$ , such that $V_0$ has units of energy., which also obey anti-commutation rules $\\lbrace \\hat{f}_{\\mathbf {i},\\sigma },\\hat{f}^\\dagger _{\\mathbf {j},\\sigma ^{\\prime }}\\rbrace =\\delta _{\\mathbf {i},\\mathbf {j}}\\delta _{\\sigma ,\\sigma ^{\\prime }}$ .", "With these operators, the second quantized fermionic Hamiltonian, $\\hat{H}_f$ , reads: $\\hat{H}_f=-\\sum _{\\mathbf {i},\\mathbf {j},\\sigma } t_{\\mathbf {i},\\mathbf {j}}\\hat{f}_{\\mathbf {i},\\sigma }^\\dagger \\hat{f}_{\\mathbf {j},\\sigma }+\\sum _{\\mathbf {i},\\sigma } \\varepsilon _\\mathbf {j}\\hat{f}_{\\mathbf {i},\\sigma }^\\dagger \\hat{f}_{\\mathbf {j},\\sigma }\\,,$ where $t_{\\mathbf {i},\\mathbf {j}}$ is the tunneling between the sites $\\lbrace \\mathbf {i},\\mathbf {j}\\rbrace $ induced by the kinetic energy of the atoms, and $\\varepsilon _\\mathbf {j}$ a position dependent energy shift coming from $\\hat{V}_\\mathrm {aux}(\\mathbf {r})$ .", "Note that these terms resemble the single-particle part of the quantum chemistry Hamiltonian $\\hat{H}_e$ of Eqs.", "(REF )-().", "In what follows, we analyze in detail both terms, and explain how to make them match exactly those of Eqs.", "(REF )-()." ], [ "Electron kinetic energy", "Equivalently to Eq.", "(REF ), the strength of tunneling amplitude matrix $t_{\\mathbf {i},\\mathbf {j}}$ of Eq.", "(REF ) is given by The on-site tunneling term $t_{\\mathbf {i},\\mathbf {i}}=\\hbar \\omega _t$ .", "Since this is a constant energy term in all lattice sites this is typically taken as the energy reference and set to 0: $t_{\\mathbf {i},\\mathbf {j}}=\\int d\\mathbf {r}W_\\mathbf {i}^*(\\mathbf {r})\\left[-\\frac{\\hbar ^2}{2 M_f}\\nabla ^2+V_\\mathrm {per}(\\mathbf {r})\\right]W_\\mathbf {j}(\\mathbf {r})\\,.$ In the strong-confinement limit that we are interested in, the Wannier functions are strongly localized around the minima (as Gaussians), such that in practical terms only nearest neighbour contributions appear.", "The strength of the nearest neighbour hopping terms can be estimated within this limit calculating the overlap between the Wannier functions as [52]: $t_{\\left\\langle \\mathbf {i},\\mathbf {j}\\right\\rangle }\\equiv t_f\\approx E_R\\sqrt{\\frac{4}{\\pi }}\\left(\\frac{V_D}{E_R}\\right)^{3/4} e^{-2\\sqrt{V_D/E_R}}\\,,$ where $\\left\\langle \\mathbf {i},\\mathbf {j}\\right\\rangle $ denotes nearest-neighbor positions in the lattice.", "The kinetic part of the ultra-cold fermionic atoms in optical lattices is then approximated by: $\\hat{T}_f=-t_f\\sum _{\\left\\langle \\mathbf {i},\\mathbf {j}\\right\\rangle ,\\sigma } \\hat{f}_{\\mathbf {i},\\sigma }^\\dagger \\hat{f}_{\\mathbf {j},\\sigma }\\,.$ This gives exactly the electron kinetic energy terms of Eq.", "(REF ) using the finite-difference approximation of the derivative of Eq.", "(REF ), up to a constant energy shift $2t_f$ that commutes with the complete Hamiltonian." ], [ "Nuclear attraction", "The nuclear attraction term of Eq.", "() can be simulated by the position dependent shift $\\varepsilon _{\\mathbf {j}}$ , whose expression in terms of the Wannier functions reads as: $\\varepsilon _\\mathbf {j}=\\int d\\mathbf {r}\|W_\\mathbf {j}(\\mathbf {r})\|^2 V_\\mathrm {aux}(\\mathbf {r})\\approx V_\\mathrm {aux}(\\mathbf {j}) \\,.$ Thus, in order to match the nuclear attraction term of the quantum chemistry Hamiltonian of Eq.", "(), we just require that $V_\\mathrm {aux}(\\mathbf {r})$ has the shape of the nuclear Coloumb attraction, at least, at the optical lattice minima $\\mathbf {j}$ where the fermions can hop.", "To obtain that, one can add a red-detuned spatially shaped electric field beam, $\\mathbf {E}_\\alpha (\\mathbf {r})$ , for each of the nuclei we want to simulate, such that the induced light-shift generates an optical potential with the shape: $V_\\mathrm {aux}(\\mathbf {j})=-\\sum _{\\alpha =1}^{N_n}\\frac{\|\\mathbf {E}_\\alpha (\\mathbf {j})\|^2}{\\delta _\\alpha }\\approx -\\sum _{\\alpha =1}^{N_n} \\frac{Z_\\alpha V_0}{\|\\mathbf {j}-\\mathbf {R}_{\\alpha }\|}\\,,$ with $V_0$ being the overall energy scale of the potential controlled by the intensity of the laser and/or detuning $\\delta _\\alpha $ .", "For consistency of our model, the maximum energy difference between the different sites, that is of the order $\\Delta \\varepsilon _\\mathrm {max}=Z_\\mathrm {max} V_0$ , should be much smaller than the trapping depth of the overall potential $\\Delta \\varepsilon _\\mathrm {max}\\ll V_D$ , such that the tunnelings $t_f$ are not affected by it, and also $\\Delta \\varepsilon _\\mathrm {max}\\ll \\hbar \\omega _t$ so that it does not create interband transitions (see [58], [59], [60], [61] where similar effects were considered due to the existence of confinement potentials).", "Both limits can be satisfied in the regime of parameters we are interested in.", "The non-trivial part here consists in obtaining the electric fields $\\mathbf {E}_\\alpha (\\mathbf {r})$ with the desired intensity pattern, $\|\\mathbf {E}_\\alpha (\\mathbf {r})\|^2$ .", "One option could be harnessing the advances in 3D holographic techniques that allow to shape the electromagnetic field in a given volume by imprinting complex phase patterns in a two-dimensional grid and using Fourier optics to propagate them to the position of the optical lattice [63].", "These 3D holograms have already enabled, for example, trapping Rydberg atoms in exotic three-dimensional (3D) configurations [62].", "The idea of holographic traps is conceptually simple (see Fig.", "REF a): one impinges a monochromatic laser beam with wavelength $\\lambda _f$ into a spatial light-modulator (SLM) that imprints a non-uniform phase pattern in a grid with $N_f\\times N_f$ pixels.", "The reflected laser field is then focused with a high-numerical aperture lens to generate the desired 3D holographic intensity pattern [63] that depends on the imprinted phases.", "The minimum spatial resolution ($\\text{a}_f$ ) in which the electric field can be modulated depends on both the optical setup and the wavelength of the incident laser $\\lambda _f$ , but it will be always lower bounded by the diffraction limit of light $\\text{a}_f\\ge \\lambda _f/2$ .", "This motivates the use of high numerical aperture lenses [64], [65], [66] to reduce the waist of the holographic beam.", "We will label as $R_f=\\text{a}/\\text{a}_f$ to the ratio between the inter-atomic distance in the optical lattice and the spatial resolution of the hologram.", "The first step to design the 3D holograms consists in finding the appropriate phase pattern that should be imprinted in the $N_f\\times N_f$ grid of the SLM to obtain the desired electric field intensity.", "Fortunately, there are many constructive algorithms of doing it [67], [68], [69].", "Inspired by the original Gerchberg-Saxton (G-S) algorithm [70], [71], [72], [73], here we follow the one of Ref.", "[73] adapted for modulating 3D electric fields in discrete points of space.", "This algorithm initially starts by a random set of phases in the $N_f\\times N_f$ , and then iteratively looks for a solution that both approximates a given intensity pattern at the fermionic positions, $V_\\mathbf {j}^0$ , with the only restriction of satisfying Maxwell's equation.", "That is, the $\\mathbf {k}-$ components of the beam of monochromatic light $\\lambda _f$ lie in the Ewald sphere of radius $k_f=2\\pi /\\lambda _f$ .", "The convergence of the solution can be monitored using the dimensionless factor: $\\epsilon = \\sum _\\mathbf {j}\\left\| \\frac{V_\\mathbf {j}-V^0_\\mathbf {j}}{V^0_\\mathbf {j}}\\right\|\\,,$ where $V_\\mathbf {j}$ is the electric field intensity in position $\\mathbf {j}$ obtained at an iteration of the algorithm and $V_\\mathbf {j}^0$ the targetted one.", "A key element for the convergence of the algorithm is the number of sampling points $N_f$ of the grid, which for simplicity we will assume to be proportional to the number of optical lattice positions $N_f=R_f N$ , choosing $R_f$ as the proportionality factor.", "Like this, if $R_f>1$ the hologram can find solutions where the electric field intensity is modulated also within the fermionic positions, which will facilitate the convergence of the algorithm.", "In Fig.", "REF (b) we plot the result of applying this algorithm [73] for several (integer) $R_f$ to the case of having a single nucleus at the origin position, that is, when $V_\\mathbf {j}^0$ should have a Coulomb shape potential around the origin We always consider that the nuclei are centered in a position separated half-a-lattice constant away from the lattice sites to avoid the divergent behaviour.", "This will introduce an error in our simulation, as we will consider more in detail in the next subsection.. We apply the G-S algorithm to find the phase mask for a given (integer) $R_f$ until the improvement of the relative error $\\epsilon $ from one iteration to the next is below $10^{-4}$ .", "Then, we plot in the main panel a linear cut of the 3D electric field amplitude at the final iteration $V_\\mathbf {j}$ (with markers) compared to the targeted one $V^0_\\mathbf {j}$ (in solid green line), and its corresponding relative error $\\epsilon $ in the inset.", "In purple squares we plot the case $R_f=1$ where we observe that the agreement with the desired potential leads to a final error above 10%.", "However, as we increase the number of sampling points $N_f$ using larger $R_f$ , the algorithm is able to find better solutions, as clearly indicated by the decrease of the final relative error as $R_f$ increases (see inset of Fig.", "REF b).", "For example, with $R_f=4$ (blue crosses), the potential finally obtained captures very well the desired intensity profile at the positions of the fermions, obtaining a normalized relative error of $\\epsilon \\approx 0.02$ .", "The most obvious way to increase $R_f$ consists in either increasing the optical lattice period or using smaller wavelengths for the focused laser.", "One option is to use Alkaline-Earth atoms which have a level structure that combines telecom transitions ($1.4$ $\\mu $ m) with ultra-violet (400 nm) ones [76], although we recognize that going to large $R_f$ will be experimentally challenging and will require the use of innovative ideas, e.g., developing novel tweezers techniques [66].", "For this reason, in Section REF we will discuss the impact of imperfect potential in the precision of the simulators, showing how already $\\varepsilon \\sim 0.1$ can provide energy errors smaller than 1$\\%$ for the simplest case of atomic hydrogen." ], [ "Errors: discretization, finite-size, and mitigation strategies", "Up to now, we have shown that the dynamics of ultra-cold fermionic atoms in deep optical lattices ($V_D\\gg E_R$ ), and with an appropriate shaping of $V_\\mathrm {aux}(\\mathbf {r})$ can mimic the single-particle part of the quantum chemistry Hamiltonian: $\\hat{H}_f=-t_f \\sum _{\\left\\langle \\mathbf {i},\\mathbf {j}\\right\\rangle ,\\sigma } \\hat{f}^\\dagger _{\\mathbf {i},\\sigma }\\hat{f}_{\\mathbf {j},\\sigma }-\\sum _{\\alpha =1}^{N_n}\\sum _{\\mathbf {j},\\sigma }\\frac{Z_\\alpha V_0}{\|\\mathbf {j}-\\mathbf {R}_\\alpha \|}\\hat{f}^\\dagger _{\\mathbf {j},\\sigma }\\hat{f}_{\\mathbf {j},\\sigma }\\,,$ Before showing how to simulate the electron repulsion part of the quantum chemistry Hamiltonian (Eq.", "), in this section we will provide intuition on how the chemistry energies and length scales translate into the cold-atom simulation, and which are the errors appearing due to two competing mechanisms: discretization and finite size effects, taking as a case of study the Hydrogen atom.", "The reason for choosing this case is two fold: first, it can be simulated directly using the Hamiltonian of Eq.", "(REF ) imposing $N_n=1$ and $Z_1=1$ , since one does not require the electron interactions; second, it is fully understood analytically in the continuum limit, that will allow us to easily benchmark our results and define the natural units of our system.", "Identifying the discretized and continuum Hamiltonians, one can obtain the following correspondence: $Ry&=\\frac{V_0^2}{4t_f}\\,, \\\\\\frac{\\text{a}_0}{\\text{a}}&=\\frac{2 t_f}{V_0}\\,,$ where $Ry$ is the Rydberg energy of the simulated Hydrogen, and $\\text{a}_0/\\text{a}$ is the effective Bohr radius in units of the lattice constant.", "Since one can control the ratio $t_f/V_0$ at will with the lasers creating the optical potentials, one can effectively choose the Bohr radius of the discrete Hydrogen atom and, consequently, of the simulated molecules when more nuclei are present.", "This will be an important asset of our simulation toolbox since it will allow one to minimize the errors coming from discretization and finite-size effects.", "In order to illustrate it, we plot in Fig.", "REF the lowest energy spectrum of the discrete Hydrogen as a function of the effective Bohr radius $\\text{a}_0/\\text{a}$ defined in Eq.", "(REF ).", "The black dashed lines are the expected energies in the continuum Hamiltonian, i.e., $E_n=Ry/n^2$ , whereas in the different colors are the different numerical energies for a fixed system size of $N=100^3$ sites.", "From this calculation, we observe several features of the grid discretized basis that we are choosing to represent the quantum chemistry Hamiltonian.", "For example, when $\\text{a}_0/\\text{a}\\lesssim 1$ , all the states deviate from the expected energy.", "This is not surprising because in this regime, all the fermionic density is expected to concentrate around one trapping minimum, such that discretization effects become large.", "In the opposite regime, when the Bohr radius becomes comparable with system size, $\\text{a}_0/\\text{a}\\gtrsim N^{1/3}$ , the energies also deviate from the continuum result, since the discrete Hydrogen atom does not fit in our system.", "Only in the intermediate regime one can minimize both errors and approximate well the correct energy.", "Note, however, that the optimal range of $\\text{a}_0/\\text{a}$ depends on the particular orbital considered.", "For example, the ground state $s$ orbital ($n=1$ ) is more sensitive to discretization effects since it has a larger fraction of atomic density close to the nucleus, while larger orbitals are more sensitive to finite size effects because their spatial extension grows with $n$ .", "This dependence of the convergence to the continuum limit on the particular atomic and molecular orbital will be commonplace in this method, and it also occurs for other basis representations [19].", "In spite of this, by analyzing the sources of errors one can extract some general conclusions that can provide valuable information when performing the experiments: Discretization effects.", "Analyzing numerically the convergence to the correct result in Rydberg units: $\\Delta E=\|E-E_\\infty \|$ (see inset in Fig.", "REF ), we found an heuristic scaling of the error given by: $\\frac{\\Delta E_{\\mathrm {dis}}}{Ry}\\propto \\left(\\frac{\\text{a}}{\\text{a}_0}\\right)^2\\propto \\left(\\frac{V_0}{t_f}\\right)^2\\,,$ where the proportionality factor depends on the particular orbital studied.", "This scaling can be justified by considering the errors introduced by the discretization of the derivative and the integrals in the kinetic and potential energy term, respectively, leading both to the same scaling presented in Eq.", "(REF ) (see Appendix ).", "Finite size effects.", "These errors can be associated to the part of the electron density that can not be fitted within our system size.", "Since the Hydrogen orbitals decay exponentially with the principal quantum number $e^{-r /(n\\text{a}_0)}$ , one can estimate the errors due to finite size effect become exponentially smaller with the ratio between the system size and the orbital size, i.e.., $\\propto e^{- N^{1/3}\\text{a}/(n\\text{a}_0)}$ .", "Figure: (a) Ground state of the discretized Atomic Hamiltonian depicted in Fig.", ", for different values of the effective Bohr radius and N=250 3 N=250^3.", "Inset: example of the fitting of the energy difference to the continuum value ΔE=E-E ∞ \\Delta E=E-E_\\infty , following a polynomial regression of the form ΔE/Ry=c·(a 0 /a) b \\Delta E/Ry=c \\cdot (\\text{a}_0/\\text{a})^b for fitting parameters, cc and bb.", "In this example we have chosen the continuum value E ∞ E_\\infty to be 1Ry1Ry.", "The best candidate for the value in the continuum needs to be calculated, as one generally does not know it a priori.", "(b) More systematically, here we show the standard deviation σ\\sigma in the determination of the fitting parameter bb for different candidate values of E guess E_\\text{guess}.", "We identify the best choice of E guess /RyE_\\text{guess}/Ry as the one with the smallest deviation, E ∞ E_\\infty .", "In this simple example, one extrapolates the numerical result E ∞ =0.9998RyE_\\infty =0.9998 Ry (see inset), for the energy of 1s1s orbital.", "This is, a precision of 0.01%0.01\\% for this simple scenario, gaining one order of magnitude with respect to the precision one can directly achieve before finite-size effects appear.Even though these estimates were done based on numerical evidence of the Hydrogen atom, one can already extract important conclusions for the simulation of larger molecules.", "On the one hand, one can estimate the error scaling with electron density.", "Since each level with principal quantum number $n$ can fit $2 n^2$ electrons, an atom/molecule with $N_e$ electrons is expected to occupy a maximum quantum number $n_m\\propto N_e^{1/3}$ , such that its estimated size will be $L\\propto N_e^{1/3} \\text{a}_0/\\text{a}$ .", "Thus, following Eq.", "(REF ), the discretization errors for such distances will scale with $\\Delta E_\\mathrm {dis}/Ry\\propto \\rho _e^{2/3}$ , with $\\rho _e=N_e/N$ the electron density.", "On the other hand, we can design an extrapolation method to obtain the energies with accuracies beyond the particular system size chosen and, importantly, without an a priori knowledge of the exact result.", "We illustrate the method in Fig.", "REF for the ground state of Hydrogen, although in Section we apply it as well to the case of multi-electron systems.", "The key steps go as follows: first, one calculates (or measures in the case of an experiment) the ground state energy for a fixed system size $N$ and for several ratios $\\text{a}_0/\\text{a}$ (panel a).", "Then, one defines $\\Delta E/Ry=(E-E_\\mathrm {guess})/Ry$ for several values of $E_\\mathrm {guess}$ (panel b) and fit the resulting function to a polynomial regression $\\Delta E/Ry=c (\\text{a}_0/\\text{a})^b$ , with free fitting parameters $b$ and $c$ .", "We identify the right choice of the guess energy as the one with smallest standard deviation $\\sigma $ (panel c), that we will say it is the one of the continuum limit $E_\\infty \\equiv E_\\mathrm {guess}(\\sigma _\\mathrm {min})$ .", "Using this procedure for a system size $N=250^3$ , we obtain $E_\\infty =0.9998Ry$ , which is one order of magnitude better than the result one would obtain without extrapolation (i.e., directly looking at the minimum value of $E$ for that system size).", "For completeness, we also check that this value of $E_\\infty $ also leads to an exponent factor compatible with $b=-2$ (not shown), which is the error scaling consistent with Eq.", "(REF ).", "In Section , this will be the criterion used to identify the best estimation for atomic and molecular energies beyond the discretization of the lattice.", "Figure: Ground-state energy of the 3D Hydrogen-like Hamiltonian for increasing values of the average normalized error of the nuclear potential, ε\\varepsilon .", "30 random iterations are considered for each choice of ε\\varepsilon , showing the average value (dashed line) and standard deviation (coloured region).", "Inset: the relative error in the resulting energy follows a linear scaling with the normalized nuclear potential error, ΔE/E≈0.06ε\\Delta E/E\\approx 0.06 \\varepsilon (red line).", "Parameters: N=60N=60, t f /V 0 =1t_f/V_0=1.Beyond this error caused by discretization and finite size effects, in Fig.", "REF we further benchmark energy deviations in the Hydrogen ground-sate caused by a relative normal error $\\varepsilon $ in the induced nuclear potential (see Sec.", "REF ).", "For lattice size $N=60$ , we observe that $\\varepsilon \\sim 0.1$ (compatible with $R_f=2$ in Fig.", "REF ), already provides an accuracy in the retrieved energy of order $1\\%$ ." ], [ "Simulating electron repulsion in optical lattices", "In this section, we explain how to obtain the interacting part of the quantum chemistry Hamiltonian as given by Eq. .", "This is the most complicated part of the simulation since it requires to describe long-range density-density interactions between ultra-cold fermionic atoms, whose interactions are typically local.", "That is, they only interact when their wavefunctions overlap significantly (i.e., same site).", "As proposed in Ref.", "[1], the key idea consists in using an auxiliary atomic species trapped together with the fermions such that the long-range interactions are effectively mediated by it.", "For concreteness, we assume this auxiliary atom to be a boson, although this will not play a big role for the physics that will be discussed along this manuscript.", "These auxiliary atoms need to be trapped in an optical lattice of similar wavelength It would be enough that the period of the auxiliary atom lattice is commensurate than the one of the fermions, and it should be able to interact locally with the fermions through the following Hamiltonian: $\\hat{H}_{\\mathrm {f}-\\mathrm {aux}}=U \\sum _{\\mathbf {j},\\sigma } \\hat{f}^\\dagger _{\\mathbf {j},\\sigma } \\hat{f}_{\\mathbf {j},\\sigma } \\hat{b}^\\dagger _\\mathbf {j}\\hat{b}_\\mathbf {j}\\,,$ where $\\hat{b}^\\dagger _\\mathbf {j}(\\hat{b}_\\mathbf {j})$ are the creation/destruction operators associated to the bosonic atoms at the $\\mathbf {j}$ site.", "We consider that the bosonic optical lattice can have a different size than the fermionic one, i.e., having $N_M$ lattice sites.", "These atoms will have an internal dynamics described by a Hamiltonian $\\hat{H}_\\mathrm {aux}$ that will depend on the particular optical lattice configuration chosen, and that will ultimately determine the effective interactions induced in the fermionic atoms.", "Thus, the idea consists in properly engineering $\\hat{H}_\\mathrm {aux}$ such that the effective fermionic interactions give rise to the desired pair-wise, Coulomb potential.", "In what follows, in Section REF we first explain the general formalism that we will use to analyze this problem.", "Unlike the proposal of Ref.", "[1], here we introduce two simplified setups that we analyze in Sections REF -REF , and that will result in slightly different repulsive potentials from the targetted one.", "The complete proposal of Ref.", "[1] is discussed in Section REF , where we also numerically benchmark that the perturbative working conditions that were derived in Ref.", "[1] are correct.", "The motivation for this incremental discussion is two-fold: first, it allows one to understand the role of all the ingredients required in the final proposal; second, even though the models discussed in Sections REF and REF do not provide a fully scalable Coulomb-like interaction, they can be used as simpler, but still meaningful, experiments that can simulate chemistry-like behaviour and guide the way to the full proposal.", "Figure: (a) Schemes presented in this work, to induce repulsion among fermionc cold atoms (white) trapped in a three-dimensional optical lattice (red, pictorically represented in a lower dimension for simplicity).", "In Scheme I (Sec.", "), repulsion is induced by a single mediating atom (green) in a given internal state a\\left\| a\\right\\rangle .", "This mediating atom tunnels with rate t a t_a, and experiences an on-site interaction with occupying the same lattice site as the fermionic atom.", "This induces a non-Coulomb and non-scalable repulsion for more than two fermionic atoms.", "In Scheme II (Sec.", "), one relies on a second internal level of the mediating atom to reach the Coulomb-like potential.", "Both levels a\\left\| a\\right\\rangle and b\\left\| b\\right\\rangle are coupled with rate gg, and tunneling rate depends on their state.", "In Scheme III (Sec. )", "we use a different approach to achieve the Coulomb repulsion among an arbitrary number of atoms.", "Here, rather than a single atom, the mediating species is in a Mott phase, with exactly one atom occupation per site (translucent gray).", "While these atoms are fixed, an atomic excitation (coloured blue) A\\left\| A\\right\\rangle can propagate to a neighboring site through magnetic exchange.", "They are also coupled with rate gg to another internal level B\\left\| B\\right\\rangle , that is subject to a Raman-assisted cavity-mediated collective interaction, J c J_c, and a on-site repulsion UU with fermionic atoms.", "(b) Configurations corresponding to the simulation of atomic He, where a nuclear potential centered in the lattice is induced using holographic techniques , and two fermionic atoms trapped in the lattice play the role of two electrons.", "(c) The configuration is modified to simulate HeH + ^+, where one tailors the attraction due to to the distinct Hydrogen (left) and Helium (right) nuclear charges, separated, in this scheme, by d/a=3d/\\text{a}=3 sites." ], [ "General formalism", "Our approach to calculate the effective fermionic interactions will be based on the separation of energy scales between the fermionic dynamics ($\\hat{H}_f$ ) and the rest ($\\hat{H}_{f-\\mathrm {aux}}+\\hat{H}_\\mathrm {aux}$ ).", "In particular, we will assume that $\|\|\\hat{H}_f\|\|\\ll \|\|\\hat{H}_{f-\\mathrm {aux}}+\\hat{H}_\\mathrm {aux}\|\|$ , such that we can consider the fermions fixed in the auxiliary atomic timescales.", "Thus, if we have $N_e$ electrons placed in positions $\\left\\lbrace \\mathbf {j}\\right\\rbrace \\equiv \\mathbf {j}_1,\\dots ,\\mathbf {j}_{N_e}$ we can make the following ansatz for the full atomic mixture wavefunction: $\\left\| \\Psi _{f-\\mathrm {aux}}\\left( \\left\\lbrace \\mathbf {j}\\right\\rbrace \\right)\\right\\rangle = \\left\| \\mathbf {j}_1,\\dots ,\\mathbf {j}_{N_e}\\right\\rangle _f \\otimes \\left\| \\varphi (\\mathbf {j}_1,\\dots ,\\mathbf {j}_{N_e})\\right\\rangle _\\mathrm {aux}\\,.$ In this way, one can first solve the problem for the auxiliary atoms degrees of freedom within a fixed fermionic configuration $\\lbrace \\mathbf {j}_i\\rbrace _{i=1}^{N_e}$ : $&\\left[\\hat{H}_{f-\\mathrm {aux}}(\\lbrace \\mathbf {j}\\rbrace )+\\hat{H}_\\mathrm {aux}\\right]\\left\| \\varphi _m(\\lbrace \\mathbf {j}\\rbrace \\right\\rangle _\\mathrm {aux}=\\nonumber \\\\&=E_{m,\\mathrm {aux}}(\\lbrace \\mathbf {j}\\rbrace )\\left\| \\varphi _m(\\lbrace \\mathbf {j}\\rbrace \\right\\rangle _\\mathrm {aux}\\,,$ where the index $m$ denotes the possible eigenstates within the same fermionic configuration, and where $\\hat{H}_{f-\\mathrm {aux}}(\\lbrace \\mathbf {j}\\rbrace $ reads: $\\hat{H}_{f-\\mathrm {aux}}(\\lbrace \\mathbf {j}\\rbrace =U\\sum _{\\lbrace \\mathbf {j}\\rbrace } \\hat{b}^\\dagger _{\\mathbf {j}_i} \\hat{b}_{\\mathbf {j}_i}\\,.$ Note that $\\sum _{\\lbrace \\mathbf {j}\\rbrace }$ indicates a sum now only over the fermionic positions.", "Once the auxiliary atom problem of Eq.", "(REF ) is solved, we divide the bosonic Hilbert space for each fermionic configuration distinguishing between the contribution of one of the eigenstates, $\\left\| \\varphi _s(\\lbrace \\mathbf {j}\\rbrace \\right\\rangle _\\mathrm {aux}$ with eigenenergy $E_{s,\\mathrm {aux}}(\\lbrace \\mathbf {j}\\rbrace )$ , and the rest of states, that we label as $\\left\| \\varphi ^\\perp _{m}(\\lbrace \\mathbf {j}\\rbrace \\right\\rangle _\\mathrm {aux}$ , with $m=1,\\dots , N_M-1$ .", "Then, one can calculate what is the effective fermionic Hamiltonian resulting from the dressing of such particular eigenstate $\\left\| \\varphi _s(\\lbrace \\mathbf {j}\\rbrace \\right\\rangle _\\mathrm {aux}$ by projecting in this space all possible fermionic configurations.", "The resulting effective Hamiltonian for the fermions reads $\\hat{H}_\\mathrm {eff}&=\\sum _{\\mathrm {all}\\lbrace \\mathbf {j}\\rbrace }\\left\\langle \\varphi _s(\\lbrace \\mathbf {j}\\rbrace \\right\|\\left[\\hat{H}_f+\\hat{H}_{f-\\mathrm {aux}}+\\hat{H}_\\mathrm {aux}\\right]\\left\| \\varphi _s(\\lbrace \\mathbf {j}\\rbrace \\right\\rangle _\\mathrm {aux}\\nonumber \\\\&\\approx -t_f \\mathcal {F} \\sum _{\\left\\langle \\mathbf {i},\\mathbf {j}\\right\\rangle ,\\sigma } \\hat{f}^\\dagger _{\\mathbf {i},\\sigma }\\hat{f}_{\\mathbf {j},\\sigma }-\\sum _{\\alpha =1}^{N_n}\\sum _{\\mathbf {j},\\sigma }\\frac{Z_\\alpha V_0}{\|\\mathbf {j}-\\mathbf {R}_\\alpha \|}\\hat{f}^\\dagger _{\\mathbf {j},\\sigma }\\hat{f}_{\\mathbf {j},\\sigma }+ \\\\& +\\sum _{\\mathrm {all}\\lbrace \\mathbf {j}\\rbrace } E_{s,\\mathrm {aux}}(\\lbrace \\mathbf {j}\\rbrace ) \\left\| \\lbrace \\mathbf {j}\\rbrace \\right\\rangle \\left\\langle \\lbrace \\mathbf {j}\\rbrace \\right\|\\,,$ where we see how the auxiliary atomic state has two effects over the fermionic Hamiltonian.", "First, it renormalizes the fermionic kinetic energy through the Franck-Condon coefficient: $\\mathcal {F}=\\langle \\varphi _s(\\lbrace \\mathbf {j}\\rbrace \\left\| \\varphi _s(\\lbrace \\mathbf {i}\\rbrace \\right\\rangle _\\mathrm {aux} \\,,$ that is the overlap between the bosonic states for two fermionic configurations $\\lbrace \\mathbf {j}\\rbrace ,\\lbrace \\mathbf {i}\\rbrace $ in which all the fermions have the same position, except one that is displaced to a nearest neighbour position.", "As we will see, $\\mathcal {F}$ can be considered independent of the particular position occupied by the fermions.", "Since the only effect of this term is to renormalize the kinetic energy, in what follows we can assume $t_f \\mathcal {F}\\rightarrow t_f$ , and still write the single particle part of $\\hat{H}_\\mathrm {eff}$ like the $\\hat{H}_f$ of Eq.", "(REF ).", "Second, and more importantly, a position-dependent energy-term which, in principle, depends on all the fermion positions, being therefore $2N_e$ -body operator.", "However, when $E_{s,\\mathrm {aux}}(\\lbrace \\mathbf {j}\\rbrace )$ can be written as a sum of pairwise contributions, i.e., $E_{s,\\mathrm {aux}}(\\lbrace \\mathbf {j}\\rbrace )=\\sum _{\\begin{array}{c}m,n=1,\\\\ m\\ne n\\end{array}}^{N_e} V(\\mathbf {j}_m-\\mathbf {j}_n)\\,,$ the term of Eq.", "(REF ) reduces to a density-density operator $\\hat{H}_\\mathrm {eff}\\approx \\hat{H}_f+ \\sum _{\\begin{array}{c}\\sigma \| m,n=1,\\\\ m\\ne n\\end{array}}^{N_e} V(\\mathbf {j}_m-\\mathbf {j}_n) \\hat{f}^\\dagger _{\\mathbf {j}_m,\\sigma }\\hat{f}^\\dagger _{\\mathbf {j}_n,\\sigma }\\hat{f}_{\\mathbf {j}_m,\\sigma }\\hat{f}_{\\mathbf {j}_n,\\sigma }\\,,$ that will mimic that of the quantum chemistry Hamiltonian of Eq.", "() if $V(\\mathbf {j}_m-\\mathbf {j}_n)= V_0\\text{a}/\|\\mathbf {j}_m-\\mathbf {j}_n\|$ , with $V_0>0$ in order to be repulsive.", "To conclude, let us summarize then the conditions to achieve the fermionic repulsion: There should be one eigenstate $\\left\| \\varphi _s(\\lbrace \\mathbf {j}\\rbrace \\right\\rangle _\\mathrm {aux}$ from the Hamiltonian of Eq.", "(REF ), whose energy can be written as: $E_{s,\\mathrm {aux}}(\\lbrace \\mathbf {j}\\rbrace )=\\sum _{\\begin{array}{c}m,n=1,\\\\ m\\ne n\\end{array}}^{N_e} \\frac{V_0 \\text{a}}{\|\\mathbf {j}_m-\\mathbf {j}_n\|}\\,,$ where $V_0>0$ determines the strength of the effective repulsion between the electrons.", "For self-consistency, $\\left\| \\varphi _s(\\lbrace \\mathbf {j}\\rbrace \\right\\rangle _\\mathrm {aux}$ needs to be the dominant state of the Hilbert space of the auxiliary atoms dressing the fermionic configuration $\\lbrace \\mathbf {j}\\rbrace $ , so that the total state writes as $\\left\| \\Psi \\right\\rangle = \\sum _{\\left\\lbrace \\mathbf {j}\\right\\rbrace } \\psi (\\left\\lbrace \\mathbf {j}\\right\\rbrace ) \\left\| \\lbrace \\mathbf {j}\\rbrace \\right\\rangle _f \\left\| \\varphi _s(\\lbrace \\mathbf {j}\\rbrace \\right\\rangle _\\mathrm {aux}$ .", "For consistency, the different parts of the Hamiltonian ($\\hat{H}_f$ , $\\hat{H}_\\text{f,aux}$ ,$\\hat{H}_\\text{aux}$ ) should not couple significantly these state to the orthogonal ones $\\left\| \\varphi ^\\perp _m(\\lbrace \\mathbf {r}\\rbrace \\right\\rangle _\\mathrm {aux}$ of any given fermionic configuration.", "This means that, if any of the Hamiltonian parts $\\hat{H}_\\alpha $ connect $\\left\| \\varphi _s(\\lbrace \\mathbf {j}\\rbrace \\right\\rangle _\\mathrm {aux}$ to an state $\\left\| \\varphi ^\\perp _m(\\lbrace \\mathbf {r}\\rbrace \\right\\rangle _\\mathrm {aux}$ , the transition should be prevented by a large enough energy gap between them, denoted by $\\Delta _{m,\\mathbf {r}}$ .", "Like this, we can upper-bound the error introduced by such couplings using perturbation theory: $\\varepsilon _\\alpha =\\sum _{\\mathrm {all}\\lbrace \\mathbf {r}\\rbrace }\\sum _{m}\\left\|\\frac{_f\\left\\langle \\lbrace \\mathbf {r}\\rbrace \\right\|_\\mathrm {aux}\\left\\langle \\varphi ^\\perp _{m,\\lbrace \\mathbf {r}\\rbrace }\\right\|\\hat{H}_\\alpha \\left\| \\Psi \\right\\rangle }{\\Delta _{m,\\mathbf {r}}}\\right\|^2\\,,$ which should of course satisfy: $\\varepsilon _\\alpha \\ll 1\\,,$ for all $\\alpha =\\mathrm {f,f-aux,aux}$ .", "This provides a second working condition for the dynamics to be governed by the effective fermionic repulsion Hamiltonian of Eq.", "(REF ).", "In what follows, we will introduce sequentially the different schemes for the interaction of the mediating atoms in 3D, $\\hat{H}_\\mathrm {aux}$ , until we obtain the desired repulsive, pair-wise, Coulomb potential between the fermionic atoms.", "For notation simplicity from now on, we will omit the fermionic spin degree of freedom in $\\hat{f}_\\mathbf {j}$ , but since the fermion-auxiliary atom interactions in Eq.", "(REF ) are assumed to be equal for both spin states, so will be the effective fermionic repulsion." ], [ "Scheme I: Repulsion mediated by single atoms: non-Coulomb & non-scalable", "Let us assume initially the simplest level configuration for the auxiliary atomic state, that is, it has only a single ground state level subject to an optical potential with the same geometry as the fermionic one, but with different amplitude [see Fig.", "REF (a)].", "The resulting auxiliary Hamiltonian in this case reads as: $\\hat{H}_\\text{I,aux}=-t_b \\sum _{\\left\\langle \\mathbf {i},\\mathbf {j}\\right\\rangle } \\hat{b}^\\dagger _\\mathbf {i}\\hat{b}_\\mathbf {j}\\,,$ where $\\hat{b}_\\mathbf {j}^{(\\dagger )}$ represents the annihilation (creation) of auxiliary atoms at positions $\\mathbf {j}$ , and $t_b$ their effective tunneling amplitude to the nearest neighbouring site.", "Note that this Hamiltonian can be easily diagonalized in momentum space by introducing periodic boundary conditions, where $\\hat{H}_\\text{I,aux}$ reads: $\\hat{H}_\\text{I,aux}=\\sum _\\mathbf {k}\\omega _\\mathbf {k}\\hat{b}_\\mathbf {k}^\\dagger \\hat{b}_\\mathbf {k}\\,$ being $\\hat{b}^{\\dagger }_\\mathbf {k}=\\frac{1}{\\sqrt{N_M}}\\sum _\\mathbf {j}\\hat{b}_\\mathbf {j}^\\dagger e^{i \\mathbf {k}\\cdot \\mathbf {j}}$ , and $\\hat{b}_\\mathbf {k}=\\frac{1}{\\sqrt{N_M}}\\sum _\\mathbf {j}\\hat{b}_\\mathbf {j}e^{-i \\mathbf {k}\\cdot \\mathbf {j}}$ , the atomic creation and annihilation operators in momentum space, and $\\omega _\\mathbf {k}=-2 t_b\\sum _{\\alpha =x,y,z}\\cos (k_\\alpha )$ their corresponding eigenenergies for a given momentum vector $\\mathbf {k}=(k_x,k_y,k_z)$ , with $k_\\alpha \\text{a}\\in \\left\\lbrace 2j\\pi /N_M \\text{ for }j=1\\ldots N_M\\right\\rbrace $ , $\\alpha \\in \\left\\lbrace x,y,z\\right\\rbrace $ , and $N_M$ the total number of sites of the auxiliary atom potential along one direction.", "For the purpose of this section, we will focus on a single auxiliary atom living in the lattice.", "In this case, one can write an ansatz for the wavefunction of the auxiliary atom $\\left\| \\phi _m(\\lbrace \\mathbf {j}\\rbrace )\\right\\rangle _\\mathrm {aux}=\\sum _\\mathbf {k}\\phi _{m,\\mathbf {k}}(\\lbrace \\mathbf {j}\\rbrace ) \\hat{b}_\\mathbf {k}^\\dagger \\left\| \\mathrm {vac}\\right\\rangle $ that can be used to find their corresponding eigenenergies: $\\left[U\\sum _{\\lbrace \\mathbf {j}\\rbrace }\\hat{b}^\\dagger _\\mathbf {j}\\hat{b}_\\mathbf {j}+\\hat{H}_\\text{I,aux}\\right]\\left\| \\phi _m(\\lbrace \\mathbf {j}\\rbrace )\\right\\rangle _\\mathrm {aux}=E_m(\\lbrace \\mathbf {j}\\rbrace ) \\left\| \\phi _m(\\lbrace \\mathbf {j}\\rbrace )\\right\\rangle _\\mathrm {aux}\\,.$ In what follows, we analyze first the case of having a single fermion in the system where we will see the emergence of a bound state of the auxiliary atom around the fermionic position [78], [79], [80].", "Then, we will see how this bound state can mediate a repulsive interaction when two or more fermions are hopping in the lattice that, unfortunately, does not have the correct spatial dependence presented in Eq.", "(REF )." ], [ "Single fermion", "If only a single fermion is present at the system at position $\\mathbf {j}_0$ , then Eq.", "(REF ) leads to the following equation: $U^{-1}=\\frac{1}{N_M} \\sum _\\mathbf {k}\\frac{1}{E_\\text{I,B}-\\omega _\\mathbf {k}}\\,,$ which has a bound-state solution for the auxiliary atom whose energy $E_\\text{I,B}$ lies above the scattering spectrum, i.e., $E_\\text{I,B}>6t_b$ .", "Its associated wavefunction in the position representation, $\\left\| \\varphi _{B,\\mathbf {j}_0)}\\right\\rangle =\\sum _\\mathbf {r}\\varphi _{B,\\mathbf {j}_0}(\\mathbf {r})\\, \\hat{b}_\\mathbf {r}^\\dagger $ reads as: $\\varphi _{B,\\mathbf {j}_0}(\\mathbf {r})=\\frac{1}{\\sqrt{\\mathcal {N}_B}\\, N_M} \\sum _\\mathbf {k}\\frac{e^{i\\mathbf {k}\\cdot (\\mathbf {r}-\\mathbf {j}_0)}}{E_\\text{I,B}-\\omega _\\mathbf {k}}\\,,$ with $\\mathcal {N}_B=\\frac{1}{N_M} \\sum _\\mathbf {k}\\frac{1}{\\left( E_B-\\omega _\\mathbf {k}\\right)^2}$ .", "Taking the continuum limit, $N_M\\rightarrow \\infty $ , to replace the summation by an integral, and making a quadratic expansion of the energy dispersion around the band-edges (see Appendix ), one can obtain an analytical expression for the wavefunction that reads as: $\\varphi _{B,\\mathbf {j}_0}(\\mathbf {r})\\propto \\frac{e^{-\\left\| \\mathbf {r}-\\mathbf {j}_0\\right\|/L_\\text{I}}}{r}\\,.$ That is, a Yukawa-type localization around the fermionic position $\\mathbf {j}_0$ with a localization length given by $L_\\text{I}=\\text{a}\\sqrt{E_\\text{I,B}/t_b-6}$ which, to leading order in $t_b/U$ , reads as (see Appendix ): $L_\\text{I}=\\text{a}\\left[ 3.176-4\\pi t_b/U\\right]^{-1}\\,.$ Figure: Radial wavefunction of the mediating auxiliary atom placed at 𝐣 0 /a=N M 1/3 /2[1,1,1]\\mathbf {j}_0/\\text{a}=\\left\\lfloor N_M^{1/3}/2\\right\\rfloor [1,1,1] for lattice size N M =100 3 ,200 3 N_M=100^3,200^3 and U=4t b U=4t_b.", "Displacements are studied in the xx-axis, 𝐣=𝐣 0 +[r,0,0]\\mathbf {j}=\\mathbf {j}_0+[r,0,0].", "Dashed line follows the Yukawa potential in Eq. ().", "Shaded region indicates the region r/a≤L I r/\\text{a}\\le L_\\text{I} where Coulomb potential applies.", "Here, L I L_I is calculated using Eq.", "().Interestingly, this localization length can be tuned with the experimental parameters, i.e., changing $U/t_b$ , and be made very large.", "In particular, the bound state can display a $1/r$ shape over the whole fermionic lattice as long as $L_\\text{I}\\gg N^{1/3}$ .", "These analytical formulae can be numerically benchmarked by solving Eq.", "(REF ) for the case of a single fermion and a finite lattice size.", "This is done in Fig.", "REF where we plot the spatial dependence of the numerically obtained wavefunction for two different system sizes: $N_M=100$ and $N_M=200$ , represented with filled and empty circles, respectively, together with the Yukawa shape (in dashed black) predicted from Eq.", "(REF ).", "We have chosen $U/t_b=4$ such that the expected length is $L_\\text{I}/\\text{a}\\approx 29$ , indicated by the shaded red region of the figure.", "From this figure we can extract two conclusions: first, the spatial wavefunction displays, as expected from Eq.", "(REF ), an approximate $1/r$ decay for short distances, i.e., $r<L_\\text{I}$ (shaded red area).", "Second, for larger distances the spatial wavefunction follows the Yukawa shape of Eq.", "(REF ) until it becomes closer to the border.", "Thus, to observe the $1/r$ decay for the whole fermionic space we require that $N^{1/3}\\ll L_\\text{I}\\ll N_M^{1/3}$ .", "An additional condition comes from reducing the coupling to non-orthogonal states, i.e., the condition of Eq.", "(REF ).", "In this case, only $\\hat{H}_f$ contributes as follows: $\\varepsilon _f&=\\sum _m \\left\| \\frac{t_f \\langle \\varphi ^\\perp _{m,\\mathbf {j}_0+1}\|\\varphi _{B,\\mathbf {j}_0}\\rangle }{\\Delta _{m,\\mathbf {j}_0+1,\\mathbf {j}_0}}\\right\|^2\\le \\frac{t_f^2}{\\mathcal {N}_B N_M} \\sum _{\\mathbf {k}} \\frac{1}{\\left( E_\\text{I,B}-\\omega _\\mathbf {k}\\right)^4}$ such that the condition $\\varepsilon _f\\ll 1$ translates into: $t_f/t_b\\ll \\left( a/L_\\text{I}\\right)^2\\,,$ providing the working condition that guarantees the separation of energy scales between the fermionic and auxiliary atom dynamics.", "Here, $\\mathbf {j}_0+1$ denotes a nearest-neighbor of $\\mathbf {j}_0$ , and we have made use of the calculations in Appendix .", "This energy separation guarantees that the auxiliary atom will immediately follow the fermion as it hops through the lattice.", "As we already explained in the previous section, this auxiliary atom dressing renormalizes as well the fermion hopping by the Franck-Condon coefficient (see Eq.", "(REF )).", "For the nearest-neighbour hoppings, that are the only non-negligible ones in this case, this coefficient reads $\\mathcal {F}_\\text{I}= \\langle \\varphi _{B,\\mathbf {j}_0+1}\|\\varphi _{B,\\mathbf {j}_0}\\rangle \\approx e^{-\\text{a}/L_\\text{I}}$ so that the fermionic hopping is less affected the more delocalized the auxiliary atom wavefunction is.", "Let us now explain what occurs in the case where two fermions are placed at positions $\\lbrace \\mathbf {j}_1,\\mathbf {j}_2\\rbrace $ .", "Solving the time-independent Schrödinger equation Eq.", "(REF ), one can find that now it has two, not one, bound-state solutions, i.e., with energies $E_{\\pm } (\\lbrace \\mathbf {j}\\rbrace )>6t_b$ , and whose wavefunction in momentum space reads: $\\phi _{\\pm ,\\mathbf {j}}(\\mathbf {k}) \\propto \\frac{e^{-i\\mathbf {k}\\mathbf {j}_{1}}\\pm e^{-i\\mathbf {k}\\mathbf {j}_{2}}}{E_{\\pm } (\\lbrace \\mathbf {j}\\rbrace )-\\omega _{\\mathbf {k}}}\\,.$ When transforming these expressions into real space, one can see that they correspond to the combination of excitations states localized around the fermionic positions $\\lbrace \\mathbf {j}\\rbrace $ .", "However, as explained in the previous section (see Eq.", "(REF )), what governs the effective induced interaction between fermions is the spatial dependence of the eigenenergies $E_{\\pm } (\\lbrace \\mathbf {j}\\rbrace )$ , which is given by, $U^{-1}=\\frac{1}{N_M}\\sum _{\\mathbf {k}}\\frac{1 \\pm e^{i\\mathbf {k}\\cdot \\mathbf {j}_{12}}}{E_{\\pm }(\\lbrace \\mathbf {j}\\rbrace )-\\omega _{\\mathbf {k}}}\\,,$ where $\\mathbf {j}_{12}=\\mathbf {j}_1-\\mathbf {j}_2$ .", "The shape of this energy dependence depends on both the symmetric/antisymmetric character of the wavefunction, and whether the solution is found above/below the scattering spectrum ($\\omega _\\mathbf {k}$ ), that can be tuned by modifying $U/t_b$ .", "By numerical inspection, we observe that to obtain a repulsive interaction we must use the symmetric state and tune the parameters such that $ E_{+}(\\lbrace \\mathbf {j}\\rbrace )>\\omega _\\mathbf {k}$ .", "In that case, it can be shown how the energy of the symmetric state can be written as $E_{+}(\\lbrace \\mathbf {j}\\rbrace )\\approx E_\\text{I,B} + V_\\text{I}(\\mathbf {j}_{12})$ where $E_\\text{I,B}$ is the bound-state energy of a single fermion, and where the spatial dependence $V_\\text{I}(\\mathbf {j}_{12})$ is given by: $V_\\text{I}(\\mathbf {j}_{12})\\approx \\frac{1}{\\mathcal {N}_B N_M} \\sum _\\mathbf {k}\\frac{e^{i\\mathbf {k}\\mathbf {j}_{12}}}{E_\\text{I,B}-\\omega _\\mathbf {k}}\\,,$ that is the term that induces a position-dependent interaction between the fermions (see Eq.", "(REF )).", "Note as well the similarity between $V_\\text{I}(\\mathbf {j}_{12})$ and the bound-state wavefunction of the single-fermion case (Eq.", "(REF )).", "Thus, we can also take the continuum limit of this expression to transform the sums into integrals and make a parabolic expansion of $\\omega _\\mathbf {k}$ to obtain an analytical formula of $V_\\text{I}(\\mathbf {j}_{12})$ .", "In the long-distance limit, that is, when $\|\\mathbf {j}_{12}\|\\gg L_\\text{I}$ , the potential shows the same Yukawa shape: $V_{\\text{I},>}(\\mathbf {j}_{12})=V_{\\text{I},>}(\|\\mathbf {j}_{12}\|)&\\approx \\frac{2\\text{a}^2 t_b}{\|\\mathbf {j}_{12}\|\\cdot L_\\text{I}} e^{-\|\\mathbf {j}_{12}\|/L_\\text{I}}\\,.$ Unfortunately, in the opposite limit, i.e., $\|\\mathbf {j}_{12}\|\\ll L_\\text{I}$ , where the shape should display the desired $1/\|\\mathbf {j}_{12}\|$ shape, Eq.", "(REF ) induces an additional correction which yields (see Appendix ) $V_{\\text{I},<}(\\mathbf {j}_{12})/t_b&\\approx \\frac{0.322 \\text{a}^2}{\|\\mathbf {j}_{12}\|^2}+\\frac{0.724 \\text{a}^2}{\|\\mathbf {j}_{12}\|\\cdot L_\\text{I}}\\,.$ These analytical expressions are numerically benchmarked by solving the bosonic Hamiltonian (REF ) in a finite system for two fermions separated by an increasing number of sites, and two different values $L_\\text{I}$ , as shown in Fig.", "REF .", "There, we observe how the energy spatial decay never displays the desired $1/\|\\mathbf {j}_{12}\|$ scaling but rather the $1/\|\\mathbf {j}_{12}\|^2$ predicted by Eq.", "(REF ).", "The intuition behind this limitation is that we do not have enough tunable parameters since $U/t_b$ controls both the strength and the range of the interaction ($L_\\text{I}$ ).", "Thus, when $L_\\text{I}$ is tuned to be large enough, the correction to the energy $E_{\\pm }(\\lbrace \\mathbf {j}\\rbrace )$ is so strong that it induces a different spatial dependence from the $1/r$ shape.", "Let us also mention here that when $t_b>0$ , there is an additional checkerboard phase pattern in the spatial dependence $V_\\text{I}(\\mathbf {j}_{12})$ that appears because the closer energy modes of the upper band-edge of $\\omega _\\mathbf {k}$ have $\\pm \\pi $ -momenta.", "Therefore, if one wants the fermion not to be sensitive to it, it is needed that the periodicity of the auxiliary atom lattice is half the one of the fermions.", "Another option consists on working with an excited energy band that shows $t_b<0$ [81], so that this checkerboard phase pattern does not appear.", "Figure: Comparison between the numerical calculation of V I (d)V_\\text{I}(d) obtained from equating Eqs.", "() and () (round markers), as compared to the analytical expansions () (dashed lined) and () (dotted-dashed line), valid in the regimes d/L I ≪1d/L_\\text{I}\\ll 1 (coloured region) and d/L I ≫1d/L_\\text{I}\\gg 1, respectively.", "Parameters: N M =120 3 N_M=120^3.For completeness, let us also mention here that as the two fermions separate, the auxiliary atom wavefunction approximates a superposition of the single-boson density of Eq.", "(REF ) centered at each position, such that the Franck-Condon coefficient of Eq.", "(REF ) approximates as $\\mathcal {F}_\\text{I}\\approx 0.5\\left( 1+e^{-\\text{a}/L_\\text{I}}\\right)\\,.$ Additionally to the error in Eq.", "(REF ) caused by the coupling to states in the band due to $\\hat{H}_f$ , the condition on $t_f/t_b$ derived for the single fermion case now includes an additional contribution given by Eq.", "(REF ) due to the coupling to the antisymmetric bound-state.", "This additional contribution reads as, $\\left\| \\frac{t_f \\langle \\varphi _{-\\left( \\mathbf {j}_1+1,\\mathbf {j}_2\\right)}\|\\varphi _{+\\left( \\mathbf {j}_1,\\mathbf {j}_2\\right)}\\rangle }{E_+\\left( \\mathbf {j}_1,\\mathbf {j}_2\\right)-E_-\\left( \\mathbf {j}_1+1,\\mathbf {j}_2\\right)}\\right\|^2\\approx \\left\| \\frac{t_f\\left( 1-\\mathcal {F}_\\text{I}\\right)}{4V_\\text{I}(d)}\\right\|^2\\ll 1\\,,$ where, for $d/\\text{a}\\gg 1$ , we have approximated $\\langle \\varphi _-\\left( \\mathbf {j}_1+1,\\mathbf {j}_2\\right)\|\\varphi _+\\left( \\mathbf {j}_1,\\mathbf {j}_2\\right)\\rangle \\approx 0.5\\left( 1-\\mathcal {F}_\\text{I}\\right)$ .", "From the definition of the Franck-Condon coefficient (REF ), in the limit $L_\\text{I}/\\text{a}\\gg 1$ , this can be approximated as $0.5 \\text{a}/L_\\text{I}$ .", "We observe that since the gap between the symmetric and antisymmetric state is given by $2V_\\text{I}(\|\\mathbf {j}_{12}\|)$ , the condition becomes more demanding as the two fermions separate, since $V_\\text{I}(\|\\mathbf {j}_{12}\|\\rightarrow \\infty )\\rightarrow 0$ .", "Ensuring that the symmetry of state is preserved irrespective of the fermionic positions will be one of the main motivations to introduce the cavity-assisted hoppings required for the model discussed in Section REF .", "Although we already showed in the previous section that this auxiliary atom configuration will not be able to deliver the desired Coulomb potential for two fermions, let us here consider the general situation with $N_f$ fermions to see that an additional complication arises, that is, that the eigenenergy $E_+(\\lbrace \\mathbf {j}\\rbrace )$ does not correspond to a pair-wise sum.", "Instead, the auxiliary atomic excitation tends to localize more strongly around those fermions closer to each other, making the proposal non-scalable.", "To illustrate this effect, in Fig.", "REF we plot an example of a numerically calculated energy $E_{+}(\\lbrace \\mathbf {j}\\rbrace )$ when three fermions are placed in a triangular disposition and move the distance of one of them such that it goes from an equilateral configuration to an isosceles one.", "We plot the ratio between the population in the fermionic sites at the apex of the triangle, compared to one of the positions of the base ($\\eta $ in the figure).", "There, we observe that the population only becomes equal in the equilateral superposition.", "Figure: Asymmetry between the population of the bound state of Hamiltonian for a configuration of three simulating atoms.", "Here we choose an isosceles distribution, of fixed base s/as/\\text{a}, and variable height d/ad/\\text{a} (see inserted scheme).", "Continuous line follow the ground state obtained with exact diagonalization for different values of d/ad/a.", "η\\eta denotes the ratio between the population of the bath in the apex site, and one of the vertices of the base.", "A value 1, indicating the desired symmetric superposition between the three atoms, is only achieved in the equilateral configuration (red cross).", "Panels show axial cuts of the population in bath bb for geometrical configurations d/a=6,14,d/\\text{a}=6,14, and 30 (from left to right).", "Qualitatively, one observes that the symmetric superposition at the three vertices is less enforced as triangle sides become more unequal.", "(Parameters: L I /a=10L_\\text{I}/\\text{a}=10, s/a=6s/\\text{a}=6, N M =120 3 N_M=120^3).Having identified the problems with this simple auxiliary atom configuration, in the next subsections we will show how by adding complexity to the internal dynamics of the auxiliary atom, one can solve these problems." ], [ "Scheme II: Repulsion mediated by atoms subject to state-dependent potentials: Coulomb but non-scalable", "One of the problems of the previous proposal is the impossibility of independently tuning the strength and range of the interactions, since there is only a single tunable parameter ($U/t_b$ ).", "Here, we will show how to harness the latest advances in state-dependent optical lattices [82], [83], [84], [85], [86] to gain that tunability.", "The idea consists of assuming that one can engineer two very different potentials for two long-lived states of the auxiliary atoms that we label as $a$ and $b$ (see Fig.", "REF (b) for a scheme), such that when the atoms are in $b$ , they tunnel at a much slower rate, $t_b$ , than when they are in $a$ , i.e., $t_a\\gg t_b$ .", "These states can be either the hyperfine states of an Alkali specie, or the metastable excited states of Alkaline-Earth ones.", "What is important is that these states can be coherently coupled either through a two-photon Raman transition or a direct one with effective coupling amplitude $g$ and detuning $\\Delta $ .", "Like this, the global internal dynamics for the auxiliary atom is described by the following Hamiltonian: $\\begin{split}\\hat{H}_\\text{II,aux} =&\\Delta \\sum _\\mathbf {j}\\hat{b}_\\mathbf {j}^\\dagger \\hat{b}_\\mathbf {j}-t_a \\sum _{\\langle \\mathbf {i}, \\mathbf {j}\\rangle }\\hat{a}_\\mathbf {i}^\\dagger \\hat{a}_\\mathbf {j}+ g\\sum _\\mathbf {j}(\\hat{b}_\\mathbf {j}^\\dagger \\hat{a}_\\mathbf {j}+\\text{H.c.})\\\\&-t_b \\sum _{\\langle \\mathbf {i}, \\mathbf {j}\\rangle }\\hat{b}_\\mathbf {i}^\\dagger \\hat{b}_\\mathbf {j}\\,.\\end{split}$ Using this Hamiltonian, one can solve again Eq.", "(REF ) for two fermions in a configuration $\\lbrace \\mathbf {j}\\rbrace $ , but now replacing $H_\\text{I,aux}\\rightarrow H_\\text{II,aux}$ .", "One can write the following ansatz for the auxiliary atom wavefunction: $\\left\| \\phi _\\text{II,m}(\\lbrace \\mathbf {j}\\rbrace )\\right\\rangle _\\mathrm {aux}=\\sum _\\mathbf {k}\\left(\\phi ^a_{m,\\left\\lbrace \\mathbf {j}\\right\\rbrace }(\\mathbf {k}) \\hat{a}_\\mathbf {k}^\\dagger +\\phi ^b_{m,\\left\\lbrace \\mathbf {j}\\right\\rbrace }(\\mathbf {k}) \\hat{b}_\\mathbf {k}^\\dagger \\right) \\left\| \\mathrm {vac}\\right\\rangle \\,.$ Under these conditions, we find that there is again a symmetric bound state in bath $b$ localized around the fermions, whose associated eigenenergy $E_\\text{II,+}(\\mathbf {j})$ leads to repulsive spatially-dependent interactions.", "Since $t_b\\ll t_a$ , the spatial dependence is dominated by the hopping in the $a$ -bath.", "In order to obtain an analytical expression for $E_\\text{II,+}(\\mathbf {j})$ , we will further assume that $g\\ll t_a$ and that $t_b=0$ .", "Note that even if one takes originally $t_b= 0$ , one does still obtain an effective tunnelling through the $a$ bath given by $t_b\\approx g^2t_a/\\Delta ^2$ , that we will neglect to get the analytical expression.", "These assumptions allow us to obtain $E_\\text{II,+}(\\mathbf {j})$ using second-order perturbation theory, which yields: $E_\\text{II,+}(\\lbrace \\mathbf {j}\\rbrace )&\\approx E^{(2)}_\\text{II,B}+\\frac{g^2}{N_M} \\sum _{\\mathbf {k}} \\frac{e^{i\\mathbf {k}\\cdot \\mathbf {j}_{12}}}{E_\\text{II,B}^{(0)}-\\omega _{\\text{II},\\mathbf {k}}}\\,,$ where $\\omega _{\\text{II},\\mathbf {k}}=-2 t_a\\sum _{\\alpha =x,y,z}\\cos (k_\\alpha )$ is the energy-dispersion ruling the propagation of the $a$ modes, and $E_\\text{II,B}^{(0)}=U+\\Delta $ , $E_\\text{II,B}^{(2)}=E_\\text{II,B}^{(0)}+ \\frac{g^2}{\\mathcal {N}}\\sum _{\\mathbf {k}} \\frac{1}{E_\\text{II,B}^{(0)}-\\omega _{\\text{II},\\mathbf {k}}}$ are the bound-state energies for the single-fermion case in this atomic configuration calculated to 0-th/2-nd order, respectively (see Appendix for more details on the calculation).", "As we did for $E_\\text{I,+}(\\lbrace \\mathbf {j}\\rbrace )$ one can obtain a formula for the spatial dependence taking the continuum limit to transform the sums into integrals, and expanding $\\omega _\\mathbf {k}$ around its band-edges, yielding $E_\\text{II,+}(\\lbrace \\mathbf {j}\\rbrace )-E^{(2)}_\\text{II,B}&\\approx V_\\text{II}\\frac{\\text{a} e^{-\|\\mathbf {j}_{12}\|/L^{(0)}_\\text{II}}}{\|\\mathbf {j}_{12}\|}\\,,$ with $V_\\text{II}=g^2/(4\\pi t_a)$ being the strength of the repulsive interaction, and $L_\\text{II}^{(0)}=\\text{a}\\cdot \\left( E_\\text{II,B}^{(0)}/t_a-6\\right)^{-1/2}$ its range calculated using the 0-th order energy.", "The latter can also be calculated exactly obtaining a value $L_\\text{II}$ that should ideally satisfy $L_\\text{II}\\approx L_\\text{II}^{(0)}$ (see Fig.", "REF and the discussion around it).", "From Eq.", "(REF ) we can already see that this atomic configuration solves one of the problems of the previous proposal of section REF , that is, that now one can tune independently the strength $V_\\text{II}$ and its range $L_\\text{II}$ .", "This enables going to a regime where $L_\\text{II}$ is bigger than the fermionic system size, i.e., $L_\\text{II}\\gg N^{1/3}$ , while still keeping the $1/r$ -dependence such that the two-fermion repulsion has a truly Coulomb-like shape in all space.", "Now, let us see the working conditions, based on the discussion around Eq.", "(REF ), where this effective repulsion works.", "Let us first bound the corrections introduced by the fermion hopping Hamiltonian $\\hat{H}_f$ .", "Focusing on the two-fermion case, these contributions are: $\\varepsilon _f&=\\left\| \\frac{t_f}{U}\\right\|^2+\\left\| \\frac{\\text{a}}{L_\\text{II}}\\frac{t_f }{4V_\\text{II}(d)}\\right\|^2\\ll 1$ where the first term corresponds to the coupling to states $\\hat{b}_\\mathbf {j}^\\dagger \\left\| \\text{vac}\\right\\rangle $ in positions not occupied by a fermion, and the second term corresponds to the antysimmetric state whose population in level $b$ is approximately $\\left( \\hat{b}_{\\mathbf {j}_1}^\\dagger -\\hat{b}_{\\mathbf {j}_2}^\\dagger \\right)/\\sqrt{2} \\left\| \\text{vac}\\right\\rangle $ , analogously to Eq.", "(REF ).", "As it occurred in the previous model, ensuring the right symmetry for the mediating state becomes more demanding as the two fermions separate.", "From the definition of the Bohr-radius (REF ), larger orbital sizes require to increase the effective length of the Yukawa potential so that, in the worst-case-scenario where the fermions are maximally separated, $\\left( \\text{a}_0/\\text{a}\\right)\\left( N/L_\\text{II}\\right)\\ll 1$ is still satisfied.", "The correction introduced by $H_\\text{II,aux}$ can be bounded by (see Appendix for details): $\\varepsilon _\\mathrm {aux}\\le \\frac{V_\\text{II}L_\\text{II}}{t_a\\text{a}} \\ll 1 \\,,$ that guarantees that the population in the $a$ modes remains small, such that the second-order expansion used in Eq.", "(REF ) holds.", "Besides, as aforementioned, it is desirable that the localization length $L_\\text{II}$ is independent on the particular fermionic configuration.", "However, by solving numerically Eq.", "(REF ) with $H_\\text{II,aux}$ for a single fermion, we find that the length of the bound state that will afterwards mediate the interaction can depend on the ratio $g/t_a$ , and thus on $V_\\text{II}(d)$ .", "This is shown explicitly in Fig.", "REF where we plot the $L_\\text{II}$ obtained by a numerical fitting of the bound-state shape as a function of $g/t_a$ and for several $U/t_a$ , and compare it with $L_\\text{II}^{(0)}$ (dashed black lines).", "There, we observe how indeed $L_\\text{II}^{(0)}$ matches well the numerically obtained value until a critical $g/t_a$ where it starts to deviate significantly.", "We numerically observe that $L_\\text{II}$ deviates significantly from $L_\\text{II}^{(0)}$ , when the population in $a$ -mode deviates from its first-order expansion terms (in dashed black).", "Using that intuition, we can then estimate the conditions for the $L_\\text{II}$ -independence of parameters by imposing that the higher-order terms in the $a$ -modes are smaller than the first order ones, which yields the following inequality (see Appendix ) $V_\\text{II}/t_a \\ll \\left( \\text{a}/L_\\text{II}\\right)^2$ From an energy perspective, we see that this bound obtained from the population of atoms in level $ a$ , dictates that the mediated repulsion, $E_\\text{II,+}(\\lbrace \\mathbf {j}\\rbrace )-E^{(2)}_\\text{II,B}$ , needs to be smaller than the energy-gap, $E^{(2)}_\\text{II,B}/t_a-6$ , defining $L_\\text{II}^{(0)}$ .", "This condition also ensures that the higher-order corrections to the bound-state energy dependent on the fermionic configuration can be neglected.", "Figure: Effective Yukawa-type length obtained from fitting the exponentially decaying wavefunction in bath aa (shaded marker), and the state energy, E II,B /t a -6 -1/2 \\left( E_\\text{II,B}/t_a - 6\\right)^{-1/2}, (contoured marker) for different values of on-site interaction U/t a U/t_a.", "Dashed lines follow the leading-order approximation L II (0) /a=E II,B (0) /t a -6 -1/2 L_\\text{II}^{(0)}/\\text{a}= \\left( E_\\text{II,B}^{(0)}/t_a-6\\right)^{-1/2}, and the dotted line indicates the critical coupling strength g/t a g/t_a when second-order corrections in the population aa-modes are 1%\\% larger than the leading order.", "Here, N M =100 3 N_M=100^3.Under these conditions, this experimental setup allows us to simulate faithfully a quantum chemistry interaction for two-electron (fermion) problems.", "Unfortunately, this proposal inherits the same problems of scalability than the previous one: when more than two fermions are present, the bound state tends to localize more strongly in the position of the closest ones (remember Fig.", "REF ), and $E_\\text{II,+}$ can not be written as a pairwise potential." ], [ "Scheme III: Repulsion mediated by atomic spin excitations and cavity assisted transitions", "For completeness of this manuscript, we finally review here the proposal introduced in Ref.", "[1] with all the ingredients required to obtain the repulsive, pair-wise, $1/r$ , potential needed for quantum chemistry simulation.", "The goal is two-fold: on the one hand, the previous analysis of the simplified setups will allow us for a more intuitive understanding of the role of the different elements.", "On the other hand, we will numerically benchmark through exact calculations the working conditions of the simulator derived perturbatively in Ref. [1].", "This proposal requires (see Fig.", "REF (a)): Three long-lived states that we label as $a,b,c$ , subject to different state-dependent potentials, such that they can only hop when they are in the $a$ state.", "The auxiliary atoms should be initialized in a Mott-insulating state $\\left\| \\mathrm {Mott}\\right\\rangle =\\prod _{\\mathbf {i}}\\hat{c}^\\dagger _\\mathbf {i}\\left\| \\mathrm {vac}\\right\\rangle $ with unit filling.", "Like this, instead of working with atomic excitations directly like we did in the previous two subsections, the second-quantized operators $\\hat{A}_\\mathbf {j},\\hat{B}_\\mathbf {j}$ will denote single-spin excitations over the Mott-state, i.e., $\\hat{A}_\\mathbf {j}^\\dagger /\\hat{B}_\\mathbf {j}^\\dagger \\left\| \\mathrm {Mott}\\right\\rangle =\\left(\\prod _{\\mathbf {i}\\ne \\mathbf {j}}\\hat{c}^\\dagger _\\mathbf {i}\\right) \\hat{a}^\\dagger _\\mathbf {j}/\\hat{b}^\\dagger _\\mathbf {j}\\left\| \\mathrm {vac}\\right\\rangle \\,,$ We also demand controllable cavity-assisted transitions that can be engineered to transfer excitations between levels $c$ and $b$ [87], [88], [89].", "These transitions induce a long-range interaction term, $J_c/N_M$ , where we already include explicitly the inverse volume dependence of the cavity-assisted couplings.", "Besides, we still keep the local Raman assisted transitions between the $a$ and $b$ levels already used in section REF , with strength $g$ and detuning $\\Delta $ .", "Summing up all these ingredients, the internal dynamics of the auxiliary atoms will be ruled by the following Hamiltonian: $\\begin{split}\\hat{H}_\\text{III,aux}&=\\frac{J_c}{N_M}\\sum _{\\mathbf {i},\\mathbf {j}}\\hat{B}_\\mathbf {i}^\\dagger \\hat{B}_\\mathbf {j}+ \\Delta \\sum _\\mathbf {j}\\hat{B}_\\mathbf {j}^\\dagger \\hat{B}_\\mathbf {j}\\\\&+J_A\\sum _{\\left\\langle \\mathbf {i},\\mathbf {j}\\right\\rangle }\\hat{A}^\\dagger _\\mathbf {i}\\hat{A}_\\mathbf {j}+g\\sum _\\mathbf {j}(\\hat{A}_\\mathbf {j}^\\dagger \\hat{B}_\\mathbf {j}+\\mathrm {H.c.})\\,,\\end{split}$ where $J_A$ is the super-exchange coupling strengths, that can be tuned from positive to negative [90], [91], and that we will consider here to be $J_A>0$ .", "Note that, apart from the first term describing cavity-assisted transitions, this Hamiltonian for the spin excitation is formally identical to the mediating Hamiltonian $H_\\text{II,aux}$ of Eq.", "(REF ).", "To show the scalability of the proposal, we study directly the case when $N_e$ fermions are present in the system with positions $\\lbrace \\mathbf {j}\\rbrace =\\lbrace \\mathbf {j}_1,\\dots ,\\mathbf {j}_{N_e}\\rbrace $ .", "Inspired by the previous sections, we study the fermion interaction induced when only a single spin excitation is present in the system, which is initially symmetrically distributed among all fermionic positions: $\\left\| \\phi ^{(0)}_{+}\\right\\rangle =\\frac{1}{\\sqrt{N_e}}\\sum _{\\lbrace \\mathbf {j}\\rbrace }\\hat{B}^\\dagger _{\\mathbf {j}}\\left\| \\mathrm {Mott}\\right\\rangle \\,.$ From Eq.", "(REF ), it can be proven that $[\\hat{H}_\\text{III,aux},\\sum _{\\mathbf {i}}\\left(\\hat{B}^\\dagger _\\mathbf {i}B_\\mathbf {i}+\\hat{A}^\\dagger _\\mathbf {i}A_\\mathbf {i}\\right)]=0$ , such that the number of spin excitations in this Hamiltonian is conserved, allowing us to work in the single excitation subspace of the Hamiltonian $\\hat{H}_\\text{III,aux}$ .", "Thus, all the possible wavefunctions are captured by the following ansatz: $\\left\| \\phi _\\text{III,m}(\\lbrace \\mathbf {j}\\rbrace )\\right\\rangle _\\mathrm {aux}=\\sum _\\mathbf {k}\\left(\\phi ^A_{m,\\left\\lbrace \\mathbf {j}\\right\\rbrace }(\\mathbf {k}) \\hat{A}_\\mathbf {k}^\\dagger +\\phi ^B_{m,\\left\\lbrace \\mathbf {j}\\right\\rbrace }(\\mathbf {k}) \\hat{B}_\\mathbf {k}^\\dagger \\right) \\left\| \\mathrm {Mott}\\right\\rangle \\,.$ Then, in order to obtain an analytical expression of the energy of the symmetric configuration $E_\\text{III,+}(\\lbrace \\mathbf {j}\\rbrace )$ including the energy shift of the fermions, $\\hat{H}_{f-\\mathrm {aux}}(\\lbrace \\mathbf {j}\\rbrace )+\\hat{H}_\\text{III,aux}$ , we apply perturbation theory using: $\\hat{H}_0=\\Delta \\sum _{\\mathbf {j}}\\hat{B}_\\mathbf {j}^\\dagger \\hat{B}_\\mathbf {j}+ U\\sum _{\\lbrace \\mathbf {j}\\rbrace } \\hat{B}_\\mathbf {j}^\\dagger \\hat{B}_\\mathbf {j}\\,,$ as the unperturbed Hamiltonian.", "At this level, there is a degeneracy of the order of the number of fermions, that the cavity will break.", "Then, we include $\\hat{H}_\\text{cav} &=\\frac{J_c}{N_M}\\sum _{\\mathbf {i},\\mathbf {j}}\\hat{B}_\\mathbf {i}^\\dagger \\hat{B}_\\mathbf {j}\\,,\\\\\\hat{H}_A &=J_A\\sum _{\\left\\langle \\mathbf {i},\\mathbf {j}\\right\\rangle }\\hat{A}^\\dagger _\\mathbf {i}\\hat{A}_\\mathbf {j}+g\\sum _\\mathbf {j}(\\hat{A}_\\mathbf {j}^\\dagger B_\\mathbf {j}+\\mathrm {H.c.})\\,,$ as the two perturbations over it.", "Using perturbation theory, we find that the eigenenergy of the unperturbed state $\\left\| \\phi ^{(0)}_{+}\\right\\rangle $ , with unperturbed energy $\\hat{H}_0\\left\| \\phi ^{(0)}_{+}\\right\\rangle = E_\\text{III,B}^{(0)} \\left\| \\phi ^{(0)}_{+}\\right\\rangle =(U+\\Delta )\\left\| \\phi ^{(0)}_{+}\\right\\rangle $ , is perturbed to first order by the $\\hat{H}_c$ leading to: $E_\\text{III,B}^{(1)}=U+\\Delta +\\rho _M J_c$ , where $\\rho _M=N_e/N_M$ is the fermionic density, because the cavity breaks the degeneracy between the symmetric/antisymmetric wavefunctions, creating an energy difference $\\rho _M J_c$ between them.", "In the next order, $\\hat{H}_A$ leads to an additional correction of the energy which introduces the desired spatial dependence: $E_\\text{III,+}^{(2)}(\\lbrace \\mathbf {j}\\rbrace )&\\approx E_\\text{III,B}^{(1)}+\\frac{g^2}{N_e}\\frac{1}{N_M}\\sum _\\mathbf {k}\\frac{\\left\| e^{i\\mathbf {k}\\cdot \\mathbf {j}_1}+\\ldots +e^{i\\mathbf {k}\\cdot \\mathbf {j}_{N_e}}\\right\|^2}{ E_\\text{III,B}^{(1)}-\\omega _{\\text{III},\\mathbf {k}}}\\,,$ where $\\omega _{\\text{III},\\mathbf {k}}=2 J_A\\sum _{\\alpha =x,y,z}\\cos (k_\\alpha )$ is the eigenenergy of the Hamiltonian $J_A\\sum _{\\left\\langle \\mathbf {i},\\mathbf {j}\\right\\rangle }\\hat{A}^\\dagger _\\mathbf {i}\\hat{A}_\\mathbf {j}$ using periodic boundary conditions.", "In that equation, we observe that $\\hat{H}_A$ delocalizes the auxiliary $A$ -spin excitations providing the position dependent part of $E^{(2)}_\\text{III,+}(\\lbrace \\mathbf {j}\\rbrace )$ , which can be broken into a constant and a sum of pair-wise contributions which have the same shape as the one in Eq.", "(REF ).", "In the continuum limit, $N_M\\rightarrow \\infty $ , the pair-wise contributions can again be written as an integral that yields: $E^{(2)}_\\text{III,+}(\\lbrace \\mathbf {j}\\rbrace )&\\approx E_\\text{III,B}^{(2)}+V_\\text{III}\\sum _{\\mathbf {i},\\mathbf {j}} \\frac{\\text{a}e^{-\|\\mathbf {i}-\\mathbf {j}\|/L_\\text{III}}}{\|\\mathbf {i}-\\mathbf {j}\|}\\,,$ where, to second order, $E^{(2)}_\\text{III,B}= E_\\text{III,B}^{(1)} +\\frac{g^2}{\\mathcal {N}} \\sum _\\mathbf {k}\\frac{1}{E_\\text{III,B}^{(0,1)}-\\omega _\\mathbf {k}} $ .", "Here, $L_\\text{III}$ is the effective length of the Yukawa-Type potential, now given by, $L_\\text{III}=\\text{a}\\left( E_\\text{III,B}^{(1)}/J_A-6\\right)$ , and $V_\\text{III}=\\frac{g^2}{2\\pi J_A N_e}$ its strength.", "For self-consistency in the derivation of $E^{(2)}_\\text{III,+}(\\lbrace \\mathbf {j}\\rbrace )$ , in Eq.", "(REF ), we must impose that the corrections to the unperturbed state, $\\left\| \\phi ^{(0)}_{+}\\right\\rangle $ , due to different elements of the Hamiltonian are small [Eq.", "(REF )].", "Deriving these contributions one by one, Figure: Exact diagonalization calculation of the bound state associated to Hamiltonian () for three fermions in the isosceles configuration illustrated in Fig. .", "We represent the ratio η\\eta between the population in bath aa of the fermion at the apex of the triangle, and one vertix of the base, as a function of the cavity strength.The no-cavity limit (J c /J A =0J_c/J_A=0) is determined by the fermionic geometry and effective length L III /aL_\\text{III}/a.As the cavity interaction increases, inequality () defines the lower cavity strength limit at which the population of the bath is equal for each of the fermionic positions.", "The black dotted line shows the total population of atoms in level AA at the position of the three vertices (WW as defined in the main text), which is close to 1 for J c <UJ_c<U, and quickly decays to an uniform distribution among all sites when the cavity interaction dominates the on-site interaction with the fermion J c >UJ_c>U.Inset: population of bath a for different values of cavity interaction.", "Note in the last inset that all the bath is equally populated when the cavity strength overpasses the on-site interaction UU.Parameters: U=2J A U=2J_A, Δ c =10J A \\Delta _c=10J_A, N M =160 3 N_M=160^3, s/a=6s/\\text{a}=6, d/a=24d/\\text{a}=24.", "The cavity $\\hat{H}_c$ Hamiltonian tends to delocalize the auxiliary atomic excitations beyond the fermion positions, which does not occur when the fermion-auxiliary atom interaction is large enough.", "Using Eq.", "(REF ), we find that the cavity-mediated population of other symmetric states rather than $\\left\| \\phi _{+}^{(0)}\\right\\rangle $ is upper bounded by: $\\varepsilon _\\text{cav}=\\left\| \\frac{\\sqrt{\\rho _M}J_c}{U-J_c}\\right\|^2\\,.$ such that one sufficient condition to satisfy $\\varepsilon _\\text{cav}\\ll 1$ is: $J_c\\ll U\\,.$ This is numerically confirmed in Fig.", "REF , where we study a three fermion configuration discussed in Fig.", "REF using now the Hamiltonian $\\hat{H}_{f-\\mathrm {aux}}(\\lbrace \\mathbf {j}\\rbrace )+\\hat{H}_\\text{III,aux}$ .", "For illustration, we plot the weight of the wavefunctions in the fermionic positions, i.e., $W=\\sum _{\\mathbf {i}\\in \\lbrace \\mathbf {j}\\rbrace }\|\\phi ^A_{m,\\left\\lbrace \\mathbf {j}\\right\\rbrace }(\\mathbf {i})\|^2$ (black dashed line in Fig.", "REF ), as a function of $J_c/J_A$ for a fixed $U$ and for several values of $g/J$ .", "There, we observe that $W\\approx 1$ when Eq.", "(REF ) is satisfied, irrespective of the particular choice of the rest of the parameters.", "As it occurred in subsection REF , the hoppings in $\\hat{H}_A$ connect $\\left\| \\phi _{+}^{(0)}\\right\\rangle $ with two different set of states: (i) it dresses it with some population in the $A$ -modes; and (ii) it takes it out of the symmetric sector.", "One can upper bound the corrections due to these two processes by $\\varepsilon _A=\\varepsilon _{A,i}+\\varepsilon _{A,ii}$ , where: $\\begin{split}\\varepsilon _{A,i} &= \\frac{g^2}{N_e N_M} \\sum _\\mathbf {k}\\left\| \\frac{e^{i\\mathbf {k}\\cdot \\mathbf {j}_1}+\\ldots +e^{i\\mathbf {k}\\cdot \\mathbf {j}_{N_e}}}{E_\\text{II,B}^{(0)}-\\omega _{II,\\mathbf {k}}}\\right\|^2 \\\\& \\le \\frac{g^2L_\\text{III}}{8\\pi \\text{a} J_A^2}N_e \\ll 1\\,,\\end{split}$ assuming the desired condition $d/L_\\text{III}\\ll 1$ for any pair of fermions, so that the Coulomb scaling prevails over the exponential decay.", "One observes that the final inequality scales as $g^2/J_A^2 \\lesssim a/L_\\text{III}$ , similarly to the two-fermion condition we encountered in the previous scheme (see Eq.", "(REF )).", "The other contribution coming from the antisymmetric states is prevented by the energy gap between the symmetric/antisymmetric sector induced by the cavity-assisted transitions ($\\rho _M J_c$ ), and it can be upper bounded by: $\\varepsilon _{A,ii}=\\left(\\frac{V_\\text{III}}{\\rho _M J_c }\\right)^2 G(\\lbrace \\mathbf {j}\\rbrace )\\,,$ where $G(\\lbrace \\mathbf {j}\\rbrace )$ is a function that solely depends on the particular fermionic configuration (see Appendix ).", "Interestingly, $G(\\lbrace \\mathbf {j}\\rbrace )\\equiv 0$ in the case where all the fermions are equally spaced or when there are only two fermions, while in general it can always be upper-bounded by $\\left\| G(\\lbrace \\mathbf {j}\\rbrace )\\right\| \\le \\left( N_e/2-1\\right)$ .", "Then, the inequality to be satisfied when many fermions are present reads as: $\\varepsilon _{A,ii}\\le \\left(\\frac{V_\\text{III}}{\\rho _M J_c }\\right)^2 \\frac{N_e}{2} \\ll 1 \\,.$ This condition is also numerically benchmarked for the case triangular configuration of three fermions represented in Fig.", "REF .", "As in Fig.", "REF , we plot the ratio of the weight of the wavefunction in the basis positions compared to the apex ($\\eta $ , see scheme in Fig.", "REF ), showing how they only become equal in the limit when Eq.", "(REF ) is satisfied.", "Besides, an extra condition appears to avoid that $\\hat{H}_f$ connects the mediating state with the rest of the subspace (see Eqs.", "(REF )-(REF )).", "We can upper bound this contribution coming from the antysimmetric distribution of spin excitations at the fermionic positions by (see Appendix ): $\\begin{split}&\\varepsilon _{f} \\approx \\frac{\\text{a}}{L_\\text{III}N_e}\\left( \\frac{t_f}{\\rho _M J_c}\\right)^2\\ll 1 \\,.\\end{split}$ Testing this inequality numerically in a three-dimensional model is an outstanding challenge as it involves the three-dimensional Hilbert space of both the fermion and spin excitations in the $a$ and $b$ levels.", "Instead, in Fig.", "REF , we test Eq.", "(REF ) in a minimal model of two fermions hopping in a 1D lattice for different values of the cavity coupling $J_c$ .", "We observe a qualitative good agreement with the scaling $\\propto (t_f/J_c)^2$ before the error introduced by an excessive cavity strength appears (Eq.", "(REF )).", "Also as it occurred in the previous section, there is an additional condition to force that $L_\\text{III}$ does not vary depending on the fermionic configuration, as this will imply that the effective repulsive potential will change as the fermions hop to the lattice.", "Making an energy argument analogous to the derivation used in Eq.", "(REF ), one would desire $\\left\| E_\\text{III,+}-E_\\text{III,B}^{(1)}\\right\|\\ll (a/L_\\text{III})^2J_A$ .", "This bound will highly depend on the particular fermionic configuration.", "An (unrealistic) upper bound for electronic repulsion would correspond to the case where all fermions are as close to each other as they can be, while respecting their fermionic character.", "In the limit of many simulated electrons, this scales as $V_\\text{III}N_e^{5/3}\\ll (a/L_\\text{III})^2J_A$ .", "This, however, does not make use of the entire allowed space for the fermions.", "A more realistic bound, taking into consideration that fermions will distribute in the entire lattice for the optimal simulation, one can approximate the repulsive energy as See Eq.", "(6.6.19) of Ref.", "[98] $\\left\| E_\\text{III,+}-E_\\text{III,B}^{(1)}\\right\|\\approx \\frac{V_\\text{III}}{2^{1/3}}\\left( N_e-1\\right)^{2/3}\\sum _\\mathbf {j}\\rho (\\mathbf {j})^{4/3}$ , where $\\rho (\\mathbf {j})$ denotes the fermionic density in site $\\mathbf {j}$ .", "This estimation can then be particularized for any atomic/molecular level of interest.", "As a back-of-the-envelope calculation, considering an homogeneous density $\\rho (\\mathbf {r})\\approx \\rho _e$ , one obtains, $\\frac{V_\\text{III}}{2^{1/3}}\\left( N_e-1\\right)^{2/3}N_e^{4/3}/N^{1/3} \\ll (\\text{a}/L_\\text{III})^2J_A\\,.$ The left-hand side of this estimation corresponds to the repulsion of an homogeneous distribution of $N_e$ atoms in a cubic lattice of $N$ sites.", "Intuitively, one can see that its scaling $\\propto N_e^2/N^{1/3}$ corresponds to the previous unrealistic scaling $N_e^{5/3}$ of a cubic array of distance 1, corrected by the new characteristic length $\\left( N_e/N\\right)^{1/3}$ when all the lattice is occupied Note that there is an errata in Eq.", "(10) of the printed version of Ref.", "[1].", "There, an additional transformation $V_0\\rightarrow V_0 \\rho _e^{1/3}$ was assumed to account for the characteristic size of the molecule.", "This effect is already accounted without further assumptions when all inequalities are satisfied..", "Finally, there is an additional condition that only involves the localization length $L_\\text{III}$ of the Yukawa-type potential and the sizes of the fermionic/auxiliary atom lattice, that is, $N^{1/3}\\ll L_\\text{III}/\\text{a} \\ll N_M^{1/3}\\,,$ whose intuition is clear: the length of the Yukawa potential has to be larger than then number of sites of the fermionic optical lattice ($N$ ), such that the fermions repel with a $1/r$ -scaling, but smaller than the auxiliary atomic optical lattice, in order not to be distorted by finite size effects.", "In particular, one can relax condition (REF ), aimed to ensure that the effective length in the Yukawa potential is constant regardless the fermionic configuration, and impose instead that the smallest and largest of them are contained within the range $\\left[ N^{1/3},N_M^{1/3}\\right]$ .", "Figure: Population of the mediating atom ground-state on contributions not corresponding to the leading-order ground-state, B ^ 𝐣 1 +B ^ 𝐣 2 f ^ 𝐣 1 f ^ 𝐣 2 /2\\left( \\hat{B}_{\\mathbf {j}_1}+\\hat{B}_{\\mathbf {j}_2}\\right)\\hat{f}_{\\mathbf {j}_1}\\hat{f}_{\\mathbf {j}_2}/\\sqrt{2}, for any combination of 𝐣 1 ,𝐣 2 \\mathbf {j}_1, \\mathbf {j}_2 (contoured marker), see main text.", "Here we use an exact diagonalization of the ground-state energy for a minimal model where two fermions are hopping on a 1D lattice, are attracted by two nuclei separated 8 lattice sites, repel each other with an effective potential V 0 /dV_0/d (being dd the interfermionic separation) and experience an on-site interaction UU with a bosonic species connected to a cavity mode (i.e.", "the terms associated to the fermionic dynamics and population in state bb of the Hamiltonian ()).", "Plus markers show the total population of the antisymmetric states of the form B ^ 𝐣 1 -B ^ 𝐣 2 f ^ 𝐣 1 f ^ 𝐣 2 /2\\left( \\hat{B}_{\\mathbf {j}_1}-\\hat{B}_{\\mathbf {j}_2}\\right)\\hat{f}_{\\mathbf {j}_1}\\hat{f}_{\\mathbf {j}_2}/\\sqrt{2} and coloured markers the population of sites not occupied by the fermions, B ^ 𝐫 f ^ 𝐣 1 f ^ 𝐣 2 \\hat{B}_{\\mathbf {r}}\\hat{f}_{\\mathbf {j}_1}\\hat{f}_{\\mathbf {j}_2} for 𝐫∉𝐣 1 ,𝐣 2 \\mathbf {r}\\notin \\left\\lbrace \\mathbf {j}_1, \\mathbf {j}_2\\right\\rbrace .", "These are compared to the scaling predicted by the first-order analytical predictions given by Eq.", "() (red dashed line) and Eq.", "() (dotted line), respectively.", "Note that the dynamics in level aa is not included in this minimal model.", "Parameters: N=50 3 ,V 0 =1/4J A N=50^3,\\,V_0=1/4J_A , U=4000J A U=4000J_A." ], [ "Benchmarking the simulator with two electron atoms (He) and molecules (HeH$^+$ )", "After having explained how to simulate all the elements of the quantum chemistry Hamiltonian in a grid basis representation [Eqs.", "(REF )-()], here we illustrate the performance of our simulator for two-electron systems beyond the H$_2$ molecule considered in Ref. [1].", "In particular, we study in detail the simulation of the He atom (in subsection REF ), that we will use to illustrate how to explore the physics of different spin symmetry sectors; and the HeH$^{+}$ molecule (in subsection REF ) to illustrate molecular physics for the case of unequal nuclei charges.", "For simplicity, we will use directly the $\\hat{H}_\\mathrm {eff}$ of Eq.", "(REF ) with a $V(\\mathbf {i}-\\mathbf {j})=V_0 \\text{a}/\|\\mathbf {i}-\\mathbf {j}\|$ , assuming that all the simulator conditions are satisfied.", "Despite the apparent simplicity of the problem, obtaining the ground state energy of $\\hat{H}_\\mathrm {eff}$ for two fermions with, e.g., exact diagonalization methods, poses already an outstanding challenge since the number of single-particle states in a grid basis scale with the number of fermionic lattice sites $N$ .", "To obtain the results that we will show in the next subsections, we have then adopted an approach reminiscent of Hartree Fock methods, where the Hamiltonian is projected in a basis that combines atomic states calculated from the single-particle problem, together with electronic orbitals that interact with an average charge caused by the rest of electrons (see Appendix for more details).", "Let us remark that these are just numerical limitations to benchmark the simulator, that would be free from these calculations details." ], [ "He atom", "The first system we consider is the case of the He atom.", "This corresponds to a system with a single nucleus with $Z_1=2$ , such that one just requires a single spatially-shaped laser beam to mimic the nuclear potential, and two simulated electrons.", "Let us note that since $\\hat{H}_\\mathrm {eff}$ does not couple the position and spin degree of freedom, one can solve independently the problems where the spin degrees of freedom are in a singlet (antisymmetric) or triplet states (symmetric), that will result in spatially symmetric/antisymmetric wavefunctions, traditionally labeled as para- and orthohelium, respectively.", "In Fig.", "REF (a) we plot the ground state energy of the He atom as a function of the effective Bohr radius presented in Eq.", "(REF ).", "Note that there is no closed solution already for this very simple system, and our simulator will be compared to numerical results with no relativistic or QED corrections [94].", "Furthermore, we use the extrapolation strategy explained in Section REF based on the scaling of the error $\\Delta E/Ry\\propto (a_0/d)^{-2}$ to obtain the expected energy that will come out from the simulation, yielding $E_{\\mathrm {para,He}}^\\infty =-5.79$ Ry, and $E_{\\mathrm {ortho,He}}^\\infty =-4.31$ Ry.", "Their relative error to the respected tabulated values [94], $-4.3504$ Ry and $-5.8074$ Ry, is therefore of $0.3\\%$ and $0.9\\%$ respectively for the benchmarking done with a system $N=100^3$ .", "Note that the bigger error corresponds to orthohelium, whose orbitals are larger and thus more affected by the discretization of the lattice.", "Figure: Ground state energy energy of the discretized Hamiltonian associated to atomic He, in the ortho (a), and parahelium sectors (b).", "Following the extrapolation method, dashed lines indicate the values E para , He ∞ =-5.79E_{\\mathrm {para,He}}^\\infty =-5.79 Ry and E ortho , He ∞ =-4.31E_{\\mathrm {ortho,He}}^\\infty =-4.31 Ry, respectively, for which the scaling of the energy error as (a 0 /a) -2 (\\text{a}_0/\\text{a})^{-2} is observed (insets).", "Round (crossed) markers correspond to N=100 3 N=100^3 (N=75 3 N=75^3)." ], [ "HeH$^+$ molecule", "Here, we study the two-electron molecule He$^{+}$ -H, which has two nuclei, one with charge $Z_1=1$ (the one corresponding to the H atom) and another one with $Z_2=2$ (the one corresponding to the He cation).", "Thus, the simulator requires two spatially-shaped laser beams, one with double the intensity than the other, such that its induced potential is twice as big.", "One of the magnitudes of interest in molecular physics is the molecular potential, that is, how the ground state energy of the molecule varies as a function of the distance $d$ between the nuclei.", "This curve already provides useful information, such as its equilibrium molecular position (if any) as well as its dissociation energy.", "In our simulator, in order to always maintain the nuclei half a site away from the nodes of the lattice (to avoid a divergent value of the potential) we choose integer values of $d/\\text{a}$ .", "In Fig.", "REF we plot the molecular potential that could be obtained with our simulator for two different system sizes $N=75^3$ and $N=100^3$ .", "As we did before, for each value of $d/\\text{a}_0$ , we choose the optimal discrete Bohr radius, $\\text{a}_0/\\text{a}$ , using the extrapolation strategy explained in section REF .", "Notice that this molecular potential needs to include nuclear repulsion, and its minimum corresponds to the distance at equilibrium.", "The energy at this point, $E_{\\mathrm {min,HeH^+}}^\\infty =-5.95$ Ry, which is in agreement with the numerical value $-5.95740408$ Ry reported in [95].", "As the separation increases, we observe that the error of the finite simulator increases, since the finiteness of the lattice is more restrictive when more sites need to separate the nuclear position.", "We also observe that the continuum result obtained with the mitigation approach still tends to the dissociation limit corresponding to ortho-Helium, discussed in the previous sections.", "Figure: Molecular potential of HeH + \\text{HeH}^+ as a function of the lattice size calculated for a finite lattice of N=75 3 N=75^3 (coloured markers) and N=100 3 N=100^3 (contoured markers).", "Crossed markers follow the mitigation strategy using these sizes, and described in Sec. .", "Black dashed line follows the molecular potential beyond discretization numerically calculated .", "Inset zooms around the position of the minimum." ], [ "Conclusion and outlook", "Summing up, we have expanded the analysis of the original proposal of Ref.", "[1] on how to simulate quantum chemistry Hamiltonians in an analog fashion using ultra-cold fermionic atoms in optical lattices.", "In particular, this work provides several original results, such as: i) A discussion of the physics of the holographic potentials required to obtain the nuclear attraction term.", "ii) The introduction of two simplified setups to obtain fermionic repulsion.", "Although the emergent interactions are not fully Coulomb-like, these simpler setups can already be used as intermediate, but meaningful, experiments to observe chemistry-like behaviour [96], and to benchmark existing numerical algorithms.", "iii) An extrapolation strategy which allows us to obtain the expected energies in the continuum limit beyond the limitations imposed by the finite size of the simulator and, importantly, without an a priori knowledge of the expected energy.", "This approach could also guide other systems simulating chemistry problems in a lattice.", "iv) A numerical benchmark of the working conditions of the simulator.", "v) Finally, an illustration of the simulator capabilities for two-electron systems like the He atom and the HeH$^+$ molecule.", "Taking this work as basis, there are many interesting directions that one can pursue.", "A particularly appealing one in the near-term is to continue simplifying the ingredients required for the proposal, even at the cost of not simulating real chemistry [96].", "Another one would be the study of dynamical processes, e.g., chemical reactions or photo-assisted chemistry, that is typically very hard numerically, and where the slower timescales of our simulator and the excellent imaging techniques can provide real-time access to the wavefunction properties.", "Finally, given our ability to tune the effective fermion interaction, one can use a different bound state to mediate attractive interactions.", "This would allow us to simulate chemistry beyond Born-Oppenheimer approximation by including another atomic specie that plays the role of the nuclei.", "Beyond the chemistry simulation, we also envision that the method to engineer non-local interactions in ultra-cold atoms can be exported to explore other phenomena where that type of interactions play a role, e.g., like in long-range enhanced topological superconductors [97]." ], [ "Acknowledgements", "The authors acknowledge very insightful discussions and feedback from J. I. Cirac and P. Zoller, with whom they worked in the original proposal of Ref. [1].", "J.A.-L. acknowledges support from 'la Caixa' Foundation (ID 100010434) through the fellowship LCF/BQ/ES18/11670016, the Spanish Ministry of Economy and Competitiveness through the 'Severo Ochoa' program (CEX2019-000910-S), Fundació Privada Cellex, Fundació Mir-Puig, and Generalitat de Catalunya through the CERCA program and QuantumCat (001-P-001644).", "T. S. acknowledges the support from NSFC 11974363.", "A. G.-T. acknowledges support from the Spanish project PGC2018-094792-B-100 (MCIU/AEI/FEDER, EU) and from the CSIC Research Platform on Quantum Technologies PTI-001." ], [ "Discretization error scaling", "This discretization inherent to our lattice approach gives rise to certain errors that need to be considered and vanish in the infinite-size limit.", "Here we estimate the errors related to (i) the disretization of the integral appearing int the Coulomb term, and (ii) the discretization of the Laplacian appearing in the kinetic term." ], [ "Discretization of the integrals.", "The calculation of expected energies over the continuum are based on integrals on the entire real space.", "The discretization of the lattice, however, transforms these integrals into a finite sum of terms, introducing an error that vanishes in the limit of infinite sites.", "This effect is closely related to the definition of a Riemann integral evaluated at mid-point values, $\\mathbf {m}_\\mathbf {j}=\\mathbf {j}+(0.5,0.5,0.5)$ , written in the units of the lattice spacing.", "Its error is given by, $\\Delta _\\textrm {Int} & =\\left\| \\int \\ d\\mathbf {s}f(\\mathbf {s})-\\text{a}^3\\sum _\\mathbf {j}f(\\mathbf {m}_\\mathbf {j})\\right\| \\approx \\frac{\\text{a}^2}{24}\\int d\\mathbf {s}\\left[ f_{xx}(\\mathbf {s})+f_{yy}(\\mathbf {s})+f_{zz}(\\mathbf {s})\\right]\\,,$ where $f_{\\alpha \\beta }(\\mathbf {x})=\\partial _\\alpha \\partial _\\beta f(\\mathbf {x})$ .", "A back-of-the-envelope calculation could be illustrative in this case.", "We focus on the integrals for the Coulomb potential, $f(\\mathbf {r})=\|\\psi _{n\\ell }(\\mathbf {r})\|^2\\cdot V(\\mathbf {r})$ , for Hydrogen atomic orbitals, $\\psi _{n\\ell }(\\mathbf {r}) \\propto g_{n\\ell }(r/a_0)\\cdot \\exp (-nr/\\text{a}_0)$ ; being $g_{n\\ell }$ an ($n-1$ )-degree polynomial.", "Rescaling coordinates the lattice units, $r\\rightarrow r\\text{a}_0$ , one has $\\Delta ^{(C)}_\\textrm {Int} \\equiv V_0 \\frac{a^3\\int d\\mathbf {r}\\; \\frac{\\partial ^2}{\\partial x^2}\\left[ g_{n\\ell }^2(r)\\cdot \\exp (-2nr)/r\\right]}{a_0^3\\int d\\mathbf {r}\\; g_{n\\ell }^2(r)\\cdot \\exp (-2nr)}\\propto V_0\\left( \\text{a}/\\text{a}_0\\right)^3\\,.$ Expressing this in Rydberg units, one gets the scaling, $\\Delta ^{(C)}_\\textrm {Int}/Ry\\propto \\left( V_0/t_F\\right)^2\\,.$ The precise constant accompanying this scaling is a geometrical factor, characteristic of each atomic orbital.", "In the discrete Hamiltonian, the kinetic term is approximated as a first-neighbor hopping term.", "One can estimate the error in this approximation from the next order terms of the expansion of $\\nabla ^2 f$ , that correspond to $\\frac{a^4}{12}\\left[ \\partial _x^4 f(\\mathbf {r})+\\partial _y^4 f(\\mathbf {r})+\\partial _z^4 f(\\mathbf {r})\\right]$ .", "Again, one can make an estimation on how this error of the kinetic term scales with the atomic units, and therefore with the size of the system.", "Using the Hydrogen wavefunctions used before, one gets, $\\Delta ^{(L)}_\\textrm {lin} \\equiv t_F \\frac{a^4\\int d\\mathbf {s}\\; g_{n\\ell }(r)\\exp (-r/n)\\cdot \\frac{\\partial ^4}{\\partial x^4}\\left[ g_{n\\ell }(r) \\exp (-nr)\\right]}{a_0^4\\int d\\mathbf {r}\\; g_{n\\ell }^2(r)\\cdot \\exp (-2nr)}\\propto t_F(\\text{a}/\\text{a}_0)^4\\,.$ Expressing this result in atomic units, one obtains the leading correction for the final error in energies, $\\Delta ^{(L)}_\\textrm {lin}/Ry\\propto \\left( V_0/t_F\\right)^2\\,.$ Interestingly, both effects lead to the heuristic scaling $\\propto \\left( V_0/t_F\\right)^2$ for discretization error that we numerically observe in Fig.", "REF ." ], [ "Details on the full perturbation theory analysis of Section ", "Here, we complete the details on the derivation of the bounds presented in the main text.", "In our derivation, one is interested in finding the mediating species in the state providing repulsion.", "Following the approach introduced in Eq.", "(REF ), our bound will arise from the coupling of this state to other orthogonal ones, and the energy gap between them, $ \\varepsilon _\\alpha =\\sum _{\\mathrm {all}\\lbrace \\mathbf {r}\\rbrace }\\sum _{m}\\bar{\\varepsilon }_\\alpha \\left( \\varphi ^\\perp _{m,\\lbrace \\mathbf {r}\\rbrace }\\right)$ , with $\\bar{\\varepsilon }_\\alpha \\left( \\varphi ^\\perp _{m,\\lbrace \\mathbf {r}\\rbrace }\\right)=\\left\|\\frac{_f\\left\\langle \\lbrace \\mathbf {j}\\rbrace \\right\|_\\mathrm {aux}\\left\\langle \\varphi ^\\perp _{m,\\lbrace \\mathbf {r}\\rbrace }\\right\|\\hat{H}_\\alpha \\left\| \\Psi \\right\\rangle }{\\Delta _{m,\\mathbf {r},\\mathbf {j}}}\\right\|^2\\,.$" ], [ "Useful analytical expressions", "Here, we first derive the analytical expressions of certain integrals that appear several times in the calculations of the error bounds.", "These are expressions of the form: $\\Sigma (z,\\mathbf {r})=\\frac{1}{(2\\pi )^3}\\int _D d\\mathbf {k}\\frac{e^{i\\mathbf {k}\\cdot \\mathbf {r}}}{z-\\omega (\\mathbf {k})}\\,,$ for $D=\\left[ -\\pi ,\\pi \\right]^{\\otimes 3}$ , and $\\omega (\\mathbf {k})=2t\\left[ \\cos (k_x)+\\cos (k_y)+\\cos (k_z)\\right]$ (we assume $t\\equiv 1$ from now on), that, for example, governs the shape of the single-fermion bound-state wavefunction.", "Other expressions that appear are of the type: $g(z,\\mathbf {r})=\\frac{1}{(2\\pi )^3}\\int _D d\\mathbf {k}\\frac{e^{i\\mathbf {k}\\cdot \\mathbf {r}}}{\\left[ z-\\omega (\\mathbf {k})\\right]^2}\\,,$ that governs the Franck-Condon coefficient in the same situation.", "Note that the latter is related to $\\Sigma (z,\\mathbf {r})$ by a derivative: $g(z,\\mathbf {r})=-\\partial _z \\Sigma (z,\\mathbf {r})$ .", "Remarkably, in the limit $\\mathbf {k}\\cdot \\mathbf {r}\\gg 1$ , one can expand the dispersion relation around their band-edges, $\\omega (\\mathbf {k})\\approx 6-\\mathbf {k}^2$ , and extend the integration domains to infinite to obtain an analytical expression: $\\begin{split}\\Sigma (z,\\mathbf {r})&=\\frac{1}{(2\\pi )^2} \\int _0^{\\pi } d\\theta \\int _0^\\infty dk \\, \\frac{e^{ik\\cdot r\\cos \\theta }}{(z-6)+k^2}\\, k^2\\sin \\theta =\\frac{1}{(2\\pi )^2} \\int _{-1}^{1} ds \\int _0^\\infty dk \\, \\frac{e^{ik\\cdot rs}}{(z-6)+k^2}\\, k^2 \\\\&=\\frac{-i}{r(2\\pi )^2} \\int _0^\\infty dk \\, \\frac{e^{ik\\cdot r}-e^{-ik\\cdot r}}{(z-6)+k^2}\\, k=\\frac{-i}{r(2\\pi )^2} \\int _{-\\infty }^\\infty dk \\, \\frac{e^{ik\\cdot r}}{(z-6)+k^2} \\\\&= \\frac{1}{4\\pi r}e^{-r\\sqrt{z-6 }} \\,.\\end{split}$ Note that for $\\mathbf {r}=[0,0,0]$ the integral does not converge, because we have artificially introduced a divergence by expanding the domain of integration to infinite.", "A way of renormalizing consists in artificially introducing an exponential cut-off $e^{-k\\Lambda }$ with $\\Lambda \\rightarrow 0$ , such that: $\\begin{split}\\Sigma (z,0)&=\\frac{1}{2\\pi ^2} \\int _0^\\infty dk \\, \\frac{k^2 e^{-k\\Lambda }}{(z-6)+k^2} = \\frac{1}{2\\pi ^2\\Lambda }-\\frac{\\sqrt{z-6}}{4\\pi }\\,.\\end{split}$ However, in lattice systems this cutoff appears naturally, and one can analytically obtain an expression for $\\Sigma (z,0)$ [99], [100]: $\\begin{split}\\Sigma (z,0)\\approx 0.253-\\frac{\\sqrt{z-6}}{4\\pi } \\,.\\end{split}$ Once we have the analytical expansions of $\\Sigma (z,\\mathbf {r})$ it is straightforward to obtain the higher order terms, e.g., $g(z,\\mathbf {r})$ as follows: $g(z,r)=-\\partial _z \\Sigma (z,\\mathbf {r})=\\frac{1}{8\\pi \\sqrt{z-6}} e^{-r\\sqrt{z-6} }\\,.$ In the single-fermion case of Scheme I, the maximum ratio between the hopping of the fermionic and mediating species was obtained in Eq.", "(REF ) from the coupling to the scattering states of the mediating atom when the fermion hops: $\\sum _m \\bar{\\varepsilon }_f\\left( \\varphi ^\\perp _{m,\\mathbf {j}_0+1}\\right) \\le t_f^2 \\sum _\\mathbf {k}\\left\| \\frac{ \\hat{b}_\\mathbf {k}\\left\| \\varphi _{B,\\mathbf {j}_0}\\right\\rangle }{E_\\text{I,B}-\\omega _\\mathbf {k}}\\right\|^2$ .", "Note that in the three-dimensional lattice, the sum to nearest neighbors introduces a factor 6, that we have omitted along the text to focus on the scalings.", "Without loss of generality, we can consider $\\mathbf {j}_0$ to be the origin of coordinates.", "Replacing the wavefunction (REF ) in momentum-space, one obtains $\\left\| \\hat{b}_\\mathbf {k}\\left\| \\varphi _{B,\\mathbf {j}_0}\\right\\rangle \\right\|^2=1/\\left[ \\mathcal {N}_B N_M\\left( E_\\text{I,B}-\\omega _\\mathbf {k}\\right)^2\\right]$ .", "Replacing the expression (REF ), and its second derivative, one obtains the inequality (REF ).", "As we see, moving to momentum space simplifies the calculation of $\\mathcal {F}_1$ in Eq.", "(REF ), $\\langle \\varphi _{B,\\mathbf {j}_0+1}\|\\varphi _{B,\\mathbf {j}_0}\\rangle =\\frac{1}{\\mathcal {N}_B N_M}\\sum _\\mathbf {k}\\frac{e^{ik_z}}{\\left( E_\\text{I,B}-\\omega _\\mathbf {k}\\right)^2}=e^{-a/L_\\text{I}}\\,,$ where we have made use of Eq.", "(REF ).", "Moving now to the two-fermion case, one can relate Eqs.", "(REF ) and (REF ) to obtain, $\\frac{1}{N_M} \\sum _\\mathbf {k}\\frac{1}{E_\\text{I,B}-\\omega _\\mathbf {k}}=\\frac{1}{N_M}\\sum _{\\mathbf {k}}\\frac{1 + e^{i\\mathbf {k}\\cdot \\mathbf {j}_{12}}}{E_+(\\lbrace \\mathbf {j}\\rbrace )-\\omega _{\\mathbf {k}}}\\,,$ and then replacing the expression (REF ) and (REF ) leads to: $-\\sqrt{E_\\text{I,B}/t_b-6}=-\\sqrt{E_+(\|\\mathbf {j}_{12}\|)/t_b-6}+\\frac{\\text{a}e^{-\|\\mathbf {j}_{12}\|\\sqrt{E_+(d)/t_b-6}}}{\|\\mathbf {j}_{12}\|}\\,.$ Here, we need to separate the discussion in two different regimes.", "In the case $\|\\mathbf {j}_{12}\|\\gg L_\\text{I}/a$ the latter term in (REF ) is dominated by the exponential decay, and one can expand to lowest order the effective repulsive potential $V_{\\text{I},>}(\\mathbf {j}_{12})$ in Eq.", "(REF ) by replacing $E_+(d)\\approx E_\\text{I,B}$ in the exponential.", "In the regime $\|\\mathbf {j}_{12}\|\\ll L_\\text{I}/a$ that simplification is, however, not possible.", "A general expansion of Eq.", "(REF ) in this regime corresponds to, $V_{\\text{I},<}(\\mathbf {j}_{12})=\\frac{\\gamma ^2\\text{a}^2}{\|\\mathbf {j}_{12}\|^2} + \\frac{2\\gamma }{1+\\gamma }\\frac{\\text{a}^2}{\|\\mathbf {j}_{12}\|L_\\text{I}}+\\frac{\\text{a}^2\\mathcal {O} \\left[ \\left( \|\\mathbf {j}_{12}\|/L_\\text{I}\\right)^2\\right]}{\|\\mathbf {j}_{12}\|^2} \\,,$ with $\\gamma \\approx 0.567$ .", "This leads to Eq.", "(REF ).", "In this exponential regime, we now need to bound the undesirable coupling to the antisymmetric state due to the fermionic hopping, $\\bar{\\varepsilon }_f\\left( \\varphi _-\\left( \\mathbf {j}_1+1,\\mathbf {j}_2\\right)\\right)$ .", "From the bound-state wavefunction of Eq.", "(REF ), neglecting terms exponentially suppressed by the distance, we obtain: $\\langle \\varphi _{-,\\left( \\mathbf {j}_1+1,\\mathbf {j}_2\\right)}\|\\varphi _{+,\\left( \\mathbf {j}_1,\\mathbf {j}_2\\right)}\\rangle \\approx \\frac{1}{2\\mathcal {N}_B N_M}\\sum _\\mathbf {k}\\frac{1-e^{ik_z}}{\\left( E_\\text{I,B}-\\omega _\\mathbf {k}\\right)^2}\\approx \\frac{1-\\mathcal {F}_1}{2}\\,.$ Within this regime, we can replace in the denominator $E_+\\left( \\mathbf {j}_1,\\mathbf {j}_2\\right)-E_-\\left( \\mathbf {j}_1+1,\\mathbf {j}_2\\right)\\approx 2V_\\text{I}\\left( \|\\mathbf {j}_{12}\|\\right)$ , which leads to the result stated in Eq.", "(REF ).", "As detailed in the main text, in this scheme to gain tunability we included a second level in the mediating species.", "To analyze its effect, we can separate the total Hamiltonian into the unperturbed $(\\hat{H}_0)$ and perturbed $(\\hat{H}_1)$ terms: $\\begin{split}\\hat{H}_0 =&\\Delta \\sum _\\mathbf {j}\\hat{b}_\\mathbf {j}^\\dagger \\hat{b}_\\mathbf {j}+ U \\sum _{\\mathbf {j}_i\\in \\lbrace \\mathbf {j}\\rbrace }\\hat{b}_{\\mathbf {j}_i}^\\dagger \\hat{b}_{\\mathbf {j}_i} -t_a \\sum _{\\langle \\mathbf {i}, \\mathbf {j}\\rangle }\\hat{a}_\\mathbf {i}^\\dagger \\hat{a}_\\mathbf {j}-t_b \\sum _{\\langle \\mathbf {i}, \\mathbf {j}\\rangle }\\hat{b}_\\mathbf {i}^\\dagger \\hat{b}_\\mathbf {j}\\,,\\\\\\hat{H}_1 =&g\\sum _\\mathbf {j}(\\hat{b}_\\mathbf {j}^\\dagger \\hat{a}_\\mathbf {j}+\\text{H.c.})\\,.\\end{split}$ To lowest order, and assuming that $t_b$ is negligible, the ground state of $\\hat{H}_0$ corresponds to $\\left\| \\varphi _\\text{II,+}^{(0)}\\right\\rangle =(\\hat{b}_{\\mathbf {j}_1}^\\dagger +\\hat{b}_{\\mathbf {j}_2}^\\dagger )/\\sqrt{2}\\left\| \\text{vac}\\right\\rangle $ , with energy $E_\\text{II,+}^{(0)}=U+\\Delta $ .", "The effective repulsion enters then as a second-order contribution in perturbation theory, $E_\\text{II,+}^{(2)}=\\frac{\\left\| \\langle \\text{vac}\|\\hat{a}_\\mathbf {k}\\hat{H}_1\|\\varphi _\\text{II,+}^{(0)}\\rangle \\right\|^2}{E_\\text{II,+}^{(0)}-\\omega _{\\text{II},\\mathbf {k}}}=\\frac{g^2}{2N_M}\\sum _{\\mathbf {k}}\\frac{\\left\| e^{i\\mathbf {k}\\cdot \\mathbf {j}_1}+e^{i\\mathbf {k}\\cdot \\mathbf {j}_2}\\right\|^2}{E_\\text{II,+}^{(0)}-\\omega _{\\text{II},\\mathbf {k}}}=\\frac{g^2}{N_M}\\sum _{\\mathbf {k}}\\frac{1}{E_\\text{II,+}^{(0)}-\\omega _{\\text{II},\\mathbf {k}}}+\\frac{g^2}{N_M}\\sum _{\\mathbf {k}}\\frac{e^{i\\mathbf {k}(\\mathbf {j}_1-\\mathbf {j}_2)}}{E_\\text{II,+}^{(0)}-\\omega _{\\text{II},\\mathbf {k}}}\\,,$ which corresponds to Eq.", "(REF ).", "To higher order, $\\left\| \\varphi _\\text{II,+}\\right\\rangle $ will also have contribution in level $a$ of the form, $\\left\| \\varphi _\\text{II,+}\\right\\rangle =\\left\| \\varphi _\\text{II,+}^{(0)}\\right\\rangle +\\alpha _\\mathbf {k}\\hat{a}_\\mathbf {k}\\left\| \\text{vac}\\right\\rangle $ .", "To make this perturbative expansion valid, Eq.", "(REF ) bounds the first-order contributions as follows: $\\varepsilon _\\mathrm {aux}=\\sum _\\mathbf {k}\\left\| \\alpha _\\mathbf {k}^{(0)}\\right\|^2= \\sum _\\mathbf {k}\\left\| \\frac{\\langle \\text{vac}\|\\hat{a}_\\mathbf {k}\\hat{H}_1\|\\varphi _\\text{II,+}^{(0)}\\rangle }{E_\\text{II,+}^{(0)}-\\omega _{\\text{II},\\mathbf {k}}}\\right\|^2 \\approx \\frac{g^2}{2 N_M}\\sum _\\mathbf {k}\\left\| \\frac{e^{i\\mathbf {k}\\cdot \\mathbf {j}_1}+e^{i\\mathbf {k}\\cdot \\mathbf {j}_2}}{E_\\text{II,+}^{(0)}-\\omega _{\\text{II}\\mathbf {k}}}\\right\|^2\\le \\frac{2g^2}{N_M}\\sum _\\mathbf {k}\\frac{1}{\\left( E_\\text{II,+}^{(0)}-\\omega _{\\text{II}\\mathbf {k}}\\right)^2}=\\frac{V_\\text{II} L_\\text{II}}{t_a \\text{a}}\\,.$ While non-dominant, this population is relevant as it is responsible for the induced repulsion.", "To make the effective length independent on the particular fermionic configuration, in the derivation of Eq.", "(REF ), we have bounded the next-order contribution to this population.", "Exploring the next non-negligible order, we obtain: $\\sum _\\mathbf {k}&\\left\| \\alpha _\\mathbf {k}^{(0)}+\\alpha _\\mathbf {k}^{(3)}\\right\|^2=\\sum _\\mathbf {k}\\left\| \\frac{\\langle \\text{vac}\|\\hat{a}_\\mathbf {k}\\hat{H}_1\|\\varphi _\\text{II,+}^{(0)}\\rangle }{E_\\text{II,+}^{(0)}-\\omega _{\\text{II},\\mathbf {k}}}+\\sum _\\mathbf {q}\\frac{\\left\| \\langle \\text{vac}\|\\hat{a}_\\mathbf {q}\\hat{H}_1\|\\varphi _\\text{II,+}^{(0)}\\rangle \\right\|^2\\langle \\text{vac}\|\\hat{a}_\\mathbf {k}\\hat{H}_1\|\\varphi _\\text{II,+}^{(0)}\\rangle }{\\left( E_\\text{II,+}^{(0)}-\\omega _{\\text{II},\\mathbf {k}}\\right)\\left( E_\\text{II,+}^{(0)}-\\omega _{\\text{II},\\mathbf {q}}\\right)}\\left( \\frac{1}{E_\\text{II,+}^{(0)}-\\omega _{\\text{II},\\mathbf {k}}}+\\frac{1}{2\\left( E_\\text{II,+}^{(0)}-\\omega _{\\text{II},\\mathbf {q}}\\right)}\\right)\\right\|^2 \\\\\\approx & \\varepsilon _{aux}+\\sum _\\mathbf {k}\\frac{\\left\| \\langle \\text{vac}\|\\hat{a}_\\mathbf {k}\\hat{H}_1\|\\varphi _\\text{II,+}^{(0)}\\rangle \\right\|^2}{\\left( E_\\text{II,+}^{(0)}-\\omega _{\\text{II},\\mathbf {k}}\\right)^3}\\sum _\\mathbf {q}\\frac{\\left\| \\langle \\text{vac}\|\\hat{a}_\\mathbf {q}\\hat{H}_1\|\\varphi _\\text{II,+}^{(0)}\\rangle \\right\|^2}{\\left( E_\\text{II,+}^{(0)}-\\omega _{\\text{II},\\mathbf {q}}\\right)}+\\ldots \\approx \\frac{V_\\text{II} L_\\text{II}}{t_a \\text{a}} + \\frac{V_\\text{II} L_\\text{II}}{t_a \\text{a}} \\left( \\frac{g L_\\text{II}}{4t_a \\text{a}}\\right)^2+\\ldots \\,,$ where, in the right hand side, we have applied Eqs.", "(REF ) and (REF ); and omitted higher order terms in $\\left( \\frac{g L_\\text{II}}{4t_a \\text{a}}\\right)^2$ .", "In Fig.", "REF we observed that imposing now that the second term in Eq.", "(REF ) is smaller than the first one, e.g., using a ratio $\\left( \\frac{g L_\\text{II}}{4t_a \\text{a}}\\right)^2=0.01$ (dotted line), allows for a constant definition of $L_\\text{II}$ for any fermionic configuration.", "In this final scheme, we include a cavity interaction to ensure a pairwise effective repulsion when more than two fermions are included in the system.", "There are several errors one should account for: The cavity also couples the unperturbed state $\\left\| \\phi ^{(0)}_{+}\\right\\rangle =\\frac{1}{\\sqrt{N_e}}\\sum _{\\lbrace \\mathbf {j}\\rbrace }\\hat{B}^\\dagger _{\\mathbf {j}}\\left\| \\mathrm {Mott}\\right\\rangle $ , to the other symmetric state in positions not occupied by the fermions, $\\frac{1}{\\sqrt{N_M-N_e}}\\sum _{\\mathbf {r}\\notin \\lbrace \\mathbf {j}\\rbrace }\\hat{B}^\\dagger _{\\mathbf {r}}\\left\| \\mathrm {Mott}\\right\\rangle $ .", "The coupling between them has intensity $J_c \\sqrt{N_e\\left( N_M-N_e\\right)}/{N_M}$ , and the energy difference is $U-\\left( 1-2N_e/N_M\\right)J_c$ .", "Therefore, the error of Eq.", "(REF ) one needs to bound is, $\\varepsilon _\\text{cav}=\\left\| \\frac{J_c\\sqrt{N_e\\left( N_M-N_e\\right)}/{N_M}}{U-\\left( 1-2N_e/N_M\\right)J_c}\\right\|^2 \\approx \\left\| \\frac{J_c\\sqrt{N_e/N_M}}{U-J_c}\\right\|^2$ for $N_e\\ll N_M$ .", "Even if the cavity does not couple this state with other antisymmetric ones, this can still occur as a consequence of coupling $\\hat{H}_A$ .", "For $J_c\\ll U$ , the relevant energy gap corresponds to $U$ , which separates the excitation of state $B$ in atoms placed at fermionic position, against unoccupied positions.", "Therefore, now we focus on the $N_e-1$ orthogonal states, that are also orthogonal to $\\left\| \\phi ^{(0)}_{+}\\right\\rangle $ , and can be written as $ \\left\| \\phi ^{(0)}_{\\bot ,m}\\right\\rangle =\\sum _{\\mathbf {r}\\in \\lbrace \\mathbf {j}\\rbrace }\\lambda _{m,\\mathbf {r}}\\hat{B}^\\dagger _{\\mathbf {r}}\\left\| \\mathrm {Mott}\\right\\rangle $ , with energy $U+\\Delta $ , and satisfying $\\sum _{\\mathbf {r}\\in \\left\\lbrace \\mathbf {j}\\right\\rbrace }{\\lambda _{m,\\mathbf {r}}}=0$ and $\\sum _{\\mathbf {r}\\in \\left\\lbrace \\mathbf {j}\\right\\rbrace }\\left\| \\lambda _{m,\\mathbf {r}}\\right\|^2=1$ for every $m=1\\ldots N_e-1$ .", "The error due to the coupling to these states reads as, $\\varepsilon _{A,ii}=\\sum _m & \\bar{\\varepsilon }_A\\left( \\left\| \\phi ^{(0)}_{\\bot ,m}\\right\\rangle \\right)=\\sum _m\\left\| \\sum _{\\mathbf {k}} \\frac{\\langle \\phi ^{(0)}_{\\bot ,m}\|\\hat{H}_A \\hat{A}^\\dagger _{\\mathbf {k}}\| \\mathrm {Mott} \\rangle \\langle \\mathrm {Mott}\|\\hat{A}_{\\mathbf {k}} \\hat{H}_A\| \\phi ^{(0)}_{+} \\rangle }{\\left( E_\\text{III,B}^{(1)}-\\omega _{\\text{III},\\mathbf {k}}\\right)\\left( E_\\text{III,B}^{(1)}-U-\\Delta \\right)}\\right\|^2 \\nonumber \\\\& = \\left( \\frac{g^2}{\\rho _M J_c}\\right)^2 \\frac{1}{N_e}\\, \\sum _{m}\\left\| \\frac{1}{N_M}\\sum _\\mathbf {k}\\frac{f_\\mathbf {k}(m)}{E_\\text{III,B}^{(1)}-\\omega _{\\text{III},\\mathbf {k}}}\\right\|^2 \\,.$ where $f_\\mathbf {k}(m)=\\sum _{\\mathbf {s},\\mathbf {r}\\in \\left\\lbrace \\mathbf {s}\\right\\rbrace ; \\mathbf {r}\\ne \\mathbf {s}} \\lambda _{m,\\mathbf {s}} e^{i\\mathbf {k}(\\mathbf {j}-\\mathbf {r})}$ accounts for the relative distances weighted by the components of the states involved.", "To upper-bound this sum, it translates after integration into, $\\sum _{m}\\left\| \\frac{1}{N_M}\\sum _\\mathbf {k}\\frac{f_\\mathbf {k}(m)}{E_s-\\omega _\\mathbf {k}}\\right\|^2\\approx \\sum _{m} \\left\| \\sum _{\\mathbf {s}\\in \\left\\lbrace \\mathbf {j}\\right\\rbrace } \\sum _{\\mathbf {r}\\in \\left\\lbrace \\mathbf {j}\\right\\rbrace ; \\mathbf {r}\\ne \\mathbf {s}} \\lambda _{m,\\mathbf {s}} x_{\\mathbf {s}\\mathbf {r}}\\right\|^2\\,,$ where $x_{\\mathbf {s}\\mathbf {r}}=\\frac{1}{4\\pi J_A}\\frac{\\text{a}}{\\left\| \\mathbf {s}-\\mathbf {r}\\right\|}\\in \\frac{1}{4\\pi J_A} (0,1]$ .", "To give a base-independent argument, one can simply reformulate the sum to express it in terms of the symmetric state which, in the basis $\\left\\lbrace \\hat{B}^\\dagger _{\\mathbf {j}_1}\\left\| \\mathrm {Mott}\\right\\rangle ,\\ldots , \\hat{B}^\\dagger _{\\mathbf {j}_{N_e}}\\left\| \\mathrm {Mott}\\right\\rangle \\right\\rbrace $ , writes as $\\lambda _s=\\left( 1\\ldots 1\\right)/\\sqrt{N_e}$ .", "Then, $N_e \\Delta ^2[y]&=\\sum _{m} \\left\| \\sum _{\\mathbf {s}\\in \\left\\lbrace \\mathbf {j}\\right\\rbrace } \\sum _{\\mathbf {r}\\in \\left\\lbrace \\mathbf {j}\\right\\rbrace ; \\mathbf {r}\\ne \\mathbf {s}} \\lambda _{m,\\mathbf {s}} x_{\\mathbf {s}\\mathbf {r}}\\right\|^2 = \\sum _{m} \\left\| \\langle {1}\|X\| \\lambda _m \\rangle \\right\|^2=\\left\|X\\left\| {1}\\right\\rangle \\right\|^2 - \\left\| \\langle {1}\|X\| \\lambda _{s} \\rangle \\right\|^2 \\\\&=\\sum _{\\mathbf {s}\\in \\left\\lbrace \\mathbf {j}\\right\\rbrace } \\left( \\sum _{\\mathbf {r}\\in \\left\\lbrace \\mathbf {j}\\right\\rbrace ; \\mathbf {r}\\ne \\mathbf {s}} x_{\\mathbf {s}\\mathbf {r}}\\right)^2-\\frac{1}{N_e}\\left( \\sum _{\\mathbf {s}\\in \\left\\lbrace \\mathbf {j}\\right\\rbrace } \\sum _{\\mathbf {r}\\in \\left\\lbrace \\mathbf {j}\\right\\rbrace ; \\mathbf {r}\\ne \\mathbf {s}} x_{\\mathbf {s}\\mathbf {r}}\\right)^2 \\,,$ where $(X)_{\\mathbf {s}\\mathbf {r}}=x_{\\mathbf {s}\\mathbf {r}}$ , and $\\lambda _m=\\left( \\lambda _{m,\\mathbf {j}_1}\\ldots \\lambda _{m,\\mathbf {j}_{N_e}}\\right)$ .", "The right hand side of the previous equation corresponds to $N_e$ times the variance of an homogeneous distribution of variables $y_\\mathbf {s}=\\sum _{\\mathbf {r}\\in \\left\\lbrace \\mathbf {j}\\right\\rbrace ; \\mathbf {r}\\ne \\mathbf {s}} x_{\\mathbf {s}\\mathbf {r}}$ , with $\\mathbf {s}\\in \\left\\lbrace \\mathbf {j}\\right\\rbrace $ .", "It is therefore null when all fermions are equidistant, and the upper-bound is reached in the worst-case scenario of two fermions at distance 1 and the rest at infinite separation from each other.", "In this most-unfavourable situation, the latter expression reads as, $\\left( 2-4/N_e\\right)\\left( 4\\pi J_A\\right)^{-2}\\approx \\left[ 2\\left( 2\\pi J_A\\right)^{2}\\right]^{-1}$ .", "This contribution is therefore null for the two-fermion case.", "For many fermions it reduces the condition to $\\varepsilon _{A,ii}=\\left(\\frac{V_\\text{III}}{\\rho _M J_c }\\right)^2 G(\\lbrace \\mathbf {j}\\rbrace )\\ll 1$ with, $G(\\lbrace \\mathbf {j}\\rbrace )=\\left( 2\\pi J_A N_e\\right)^2\\Delta ^2[y]\\ll N_e/2\\,,$ as used in Eq.", "(REF ).", "The population of antisymmetric states in positions not occupied by the fermions is already bounded with these conditions.", "One should also note that the fermionic hopping can also couple to symmetric to the antisymmetric states.", "This leads to an additional error that can be bounded by: $\\varepsilon _f &=\\sum _m \\bar{\\varepsilon }_f\\left( \\left\| \\phi ^{(0)}_{\\bot ,m}\\right\\rangle \\right)\\approx \\sum _{m} \\left\| t_F \\frac{\\langle \\phi ^{(0)}_{\\bot ,m,\\left\\lbrace \\mathbf {j}+1\\right\\rbrace }\|\\phi ^{(0)}_{+,\\left\\lbrace \\mathbf {j}\\right\\rbrace }\\rangle }{\\rho _M J_c}\\right\|^2\\\\&=\\left( \\frac{t_F}{\\rho _M J_c}\\right)^2\\left( 1-\\left\| \\langle \\phi ^{(0)}_{+,\\left\\lbrace \\mathbf {j}+1\\right\\rbrace }\|\\phi ^{(0)}_{+,\\left\\lbrace \\mathbf {j}\\right\\rbrace }\\rangle \\right\|^2\\right)=\\left( \\frac{t_F}{\\rho _M J_c}\\right)^2\\left( 1-\\frac{N_e-1+\\mathcal {F}_\\text{III}}{N_e}\\right)\\approx \\left( \\frac{t_F}{\\rho _M J_c}\\right)^2\\frac{\\text{a}}{L_\\text{III}N_e}$ which corresponds to inequality (REF ).", "Note that we have assumed that the nearest neighbor in $\\left\\lbrace \\mathbf {j}+1\\right\\rbrace $ is not occupied by a fermion, which is valid in the limit $\\rho _M\\ll 1$ .", "In the last approximation, we have focused on the regime $L/_\\text{III}/\\text{a}\\gg 1$ where the Coulomb interaction dominates the Yukawa potential.", "This allows for a more relaxed condition than the one obtained when the effect of the Franck-Condon coefficient is neglected, as considered in [1]." ], [ "Numerical methods for multi-electronic systems.", "To numerically capture both the geometry of the atom[molecule] and the interactions with other electrons in our analog simulator, we project the discretized fermionic Hamiltonian $\\hat{H}_e$ in Eqs.", "(REF )-(), on a set of atomic[molecular] orbitals $\\lbrace \\phi _i\\rbrace _{i=1}^{n}$ composed of two types of orbitals: Single-electron orbitals, corresponding to the $n$ first lowest energy eigestates of a single electron attracted to the same nuclear configuration.", "That means only the terms in Eqs.", "(REF ) and () in $\\hat{H}_e$ .", "Electronic orbitals that interact with an average-charge caused by the rest of electrons.", "For the case of two-electrons that we benchmark in this work, these Hartree-Fock-like orbitals are iteratively calculated by adding to the nuclear and kinetic terms in Eqs.", "(REF ) and () the repulsion due to the lowest-energy state obtained in the previous iteration given by Eq. ().", "Combining both sets of orbitals, the total basis is orthogonalized using Gram-Schmidt algorithm.", "The projected fermionic Hamiltonian then reads as, $\\hat{H}_e=\\sum _{i,j,r,s=1}^{n} h_{ijrs}\\left\| \\phi _i\\;\\phi _j\\right\\rangle \\left\\langle \\phi _r\\; \\phi _s\\right\|\\,,$ where $h_{ijrs}=\\left\\langle \\phi _i\\; \\phi _j\\right\| \\hat{H}_e\\left\| \\phi _r \\; \\phi _s\\right\\rangle $ , and $\\left\| \\phi _i\\;\\phi _j\\right\\rangle $ denotes the product of the single-fermion states $\\left\| \\phi _i\\right\\rangle \\otimes \\left\| \\phi _j\\right\\rangle $ .", "A complete set would include $N$ independent orbitals in the basis, while we are computationally restricted to a dozen of orbitals when calculating the ground state.", "This limitation does not have any impact on the experiment, but it is desirable to estimate the imprecision made in this numerical benchmark.", "The success of this strategy then depends on how accurately the interactions in the Hamiltonian are captured by the orbitals included in this set and, therefore on the number and type of orbitals that we include in the truncated basis.", "In the following, we illustrate the application of this approach to the species benchmarked in the main text: atomic Helium, and molecular HeH$^+$ ." ], [ "Numerical benchkmarking of atomic Helium", "In the case of Helium, one can explore the ortho-, and parahelium ground-states by restricting the projected Hamiltonian (REF ) to the corresponding symmetry sectors.", "As para(ortho)helium is characterized by a(n) (anti)symmetric spin configuration, their spatial configuration needs to be antisymmetric(symmetric) due to their fermionic character.", "While this will be naturally ensured by the fermionic nature of our atomic simulator, the computation cost of the numerical calculation gets simplified by imposing these symmetries.", "In particular, one can define the reduced Hamiltonian $\\left.", "\\hat{H}_e \\right\|_\\text{para(ortho)}=\\sum _{i,r}^{n}\\sum _{j\\ge i, s\\ge r}^{n} h_{ijrs}^\\text{para(ortho)}\\left\| \\left\| \\phi _i\\;\\phi _j\\right\\rangle \\right\\rangle ^\\text{para(ortho)} \\left\\langle \\left\\langle \\phi _r\\; \\phi _s\\right\|\\right\|^\\text{para(ortho)} \\,,$ where $\\left\| \\left\| \\phi _i\\;\\phi _j\\right\\rangle \\right\\rangle ^\\text{para(ortho)}=\\left[ \\left\| \\phi _i\\;\\phi _j\\right\\rangle +(-)\\left\| \\phi _j\\;\\phi _i\\right\\rangle \\right]/\\sqrt{2}$ , and $h_{ijrs}^\\text{para(ortho)}=h_{ijrs}+(-)h_{jirs}$ , where we have used the identity, $h_{ijrs}=h_{jisr}$ .", "We should emphasize that this projection on a single-particle basis is just a numerical strategy that enables us to numerically benchmark the model, but does not have any impact on the experimental implementation of the proposed analog simulator.", "In Fig.", "REF (a), we analyze the convergence of result by calculating plot the lowest energy of He atoms as a function of the type and number of orbitals included in the basis.", "As expected, orbitals obtained using the HF approach (coloured round markers) diminish more easily the ground-state energy than single electron orbitals (crossed markers).", "A combination of both basis (contoured round markers) show the greatest reduction.", "For the convergence of the results shown along the text, we have chosen 30 orbitals: 15 coming from the single electron calculation, and 15 obtained with the described HF method, which show energy variations smaller than the energy error provided.", "Figure: (a) Numerical benchmarking for the ground-state energy of He for an increasing number of orbitals in the projective basis for the example of tunneling rate, t f /V 0 =6t_f/V_0=6 and N=75 3 N=75^3.", "In particular, we compare the lowest energy of the projected Hamiltonian for basis composed of the nn lowest-energy single-electron orbitals of He + ^+ (crossed markers), Hartree-Fock orbitals constructed as described in this section (coloured round markers), or a combination of the n-th first of them (contoured round markers).", "(b) We repeat this analysis for the lowest-energy of HeH + ^+, for an internuclear distance d/a 0 =1.5d/a_0=1.5 simulated with a separation of d/a=15d/\\text{a}=15 sites, and N=75 3 N=75^3.In this case, the chosen Bohr-radius $\\text{a}_0/\\text{a}$ modifies the effective internuclear separation $d/\\text{a}_0$ .", "To explore the effect of discretization, for a given physical distance $d/\\text{a}_0$ , we then modify the nuclear separation $d/\\text{a}$ taking integer values, and adjust the effective Bohr-radius $\\text{a}_0/\\text{a}$ accordingly.", "In Fig.", "REF , this process is repeated for lattice sizes $N=75^3$ and $N=100^3$ , and the extrapolation method is then used to extract the best estimation of the ground-state energy in the continuum from our Hamiltonian in the lattice for each value of $d/\\text{a}_0$ .", "As it occurred in the case of He, a single-electron base obtained from a Hartree-Fock approach is used to solve the discretized molecular Hamiltonian in a a projected basis.", "In Fig.", "(REF )(b) we benchmark its convergence with the number of orbitals, observing energy variations smaller than the energy error for the choice of 15 orbitals using the single electron calculation, and 15 obtained with the described HF method." ] ]	2011.14113
[ [ "Hijack-GAN: Unintended-Use of Pretrained, Black-Box GANs" ], [ "Abstract While Generative Adversarial Networks (GANs) show increasing performance and the level of realism is becoming indistinguishable from natural images, this also comes with high demands on data and computation.", "We show that state-of-the-art GAN models -- such as they are being publicly released by researchers and industry -- can be used for a range of applications beyond unconditional image generation.", "We achieve this by an iterative scheme that also allows gaining control over the image generation process despite the highly non-linear latent spaces of the latest GAN models.", "We demonstrate that this opens up the possibility to re-use state-of-the-art, difficult to train, pre-trained GANs with a high level of control even if only black-box access is granted.", "Our work also raises concerns and awareness that the use cases of a published GAN model may well reach beyond the creators' intention, which needs to be taken into account before a full public release.", "Code is available at https://github.com/a514514772/hijackgan." ], [ "Introduction", "Generative Adversarial Nets (GANs) [14] have achieved remarkable success in many applications, such as image synthesis [20], [19], [21] and image translation [8], [10], [9], [33].", "By learning a mapping between noise and images, the models are skilled to produce photo-realistic images from random noise.", "However, as the architectures become sophisticated [20], [19], [7], training modern GANs often requires massive data and computation resources.", "For example, it takes one high-quality face dataset, 8 V100 GPUs, and one week to train a single StyleGAN [1] model.", "In light of this trend, it is crucial to reuse existing pre-trained GANs, such as those being released by researchers or industry, for building green AI systems, in which the critical factor is to achieve other tasks beyond the original intention of GANs.", "To reuse GANs for other tasks, prior works have shown that semantic manipulation can be realized by vector arithmetic [23] or moving along constant attribute vectors [13], [28], [25], [12] in latent spaces.", "For instance, InterfaceGAN [25] demonstrates that facial attribute manipulation can be achieved by moving noise close to or away from the linear decision boundary of the desired attribute.", "Although these methods reveal the potential that pre-trained GANs could go beyond unconditional generation, they highly rely on the assumption of linear manifolds, thereby ignoring the nature of highly non-linear latent spaces (e.g., $\\mathcal {Z}$ -space of StyleGAN [25], [34]).", "This strong assumption could be harmful, especially for rare attributes, and lead to ineffective manipulation.", "Inspired by this observation, we propose a novel framework, which gains high-level control over unconditional image generation by iteratively traversing the non-linear latent spaces.", "Specifically, we first train a proxy model that bypasses the gradients from one pretrained GAN and other fixed task models, and then dynamically decides the moving direction in each step, thus producing smoother and more effective attribute transition.", "Next, we propose an orthogonal constraint to solely edit the attribute of interest while retaining others in images.", "Despite only black-box access, we show that our method can achieve various unintended tasks, including manipulation over facial attributes, head poses, and landmarks.", "As a result of our experiments, we find that our framework with pre-trained GANs not only facilitates other vision tasks but raises concerns regarding further usage.", "Even without access to model parameters, the models can still be applied to unintended tasks potentially for malicious purposes.", "The owners of GANs should be aware and cautious about the potential risks before releasing their models.", "Overall, our contributions are summarized below.", "We propose a framework which leverages off-the-shelf GANs to approach unintended vision tasks in a black-box setting.", "We propose a constraint that helps our framework solely edit the attribute of interest while retaining others in images.", "Extensive results show that our method can produce smoother and more effective manipulation while preserving non-target attributes better, as compared to prior work.", "We also shed light on the potential risks of unintended usage by gaining control over facial attributes, head poses, and landmarks." ], [ "Related Work", "Generative Adversarial Nets (GANs).", "Since GAN [14] was proposed, it has advanced many applications such as image synthesis [1], [19], [21] and image translation [8], [10], [9], [33].", "The rationale behind it is to map noise drawn from a simple distribution (e.g., Gaussian) to a real data distribution (e.g., images) by non-linear networks.", "Many works [20], [7], [19], [21], [30], [4] have been proposed to improve image quality while the demands on resources also significantly increase.", "In this work, we consider two state-of-the-art GANs, PGGAN [19] and StyleGAN [20], as our generator backbones.", "The former one takes a vector of noise as input and progressively upsamples features to generate images.", "The later one adopts a similar strategy but first embed the noise vector by neural networks and treat them as style representations.", "As observed by Shen [25], the $z$ -space of StyleGAN is expected to be more entangled than PGGAN.", "In general, GAN models for new tasks are often trained from scratch, making them inefficient and difficult to scale up.", "With our framework, we show that pre-trained GANs can well approach a range of applications beyond their original purposes.", "Study on Latent Spaces of GANs.", "A considerable amount of works has attempted to understand the latent space of GANs.", "In particular, [23], [28] show that semantic manipulation is achievable by vector arithmetic on latent code; [20] show that style transfer can be achieved by mixing two latent represents.", "Some studies have attempted to identity semantically meaningful directions by self-supervised learning [29] or PCA on latent spaces [15].", "They are able to perform transformations by moving noise toward the direction.", "However, since they approach it in an unsupervised fashion, they may be limited to generalize to new tasks beyond simple transformations.", "Recent works further indicate that various semantic meanings may be implicitly encoded in the latent space of pre-trained GANs even though the concepts are not specified in the training set, ranging from memorability [13], transformation [18], to facial attributes [25], [12], [27].", "For instance, InterfaceGAN [25] uses the normal vector of the SVM decision boundary to edit facial attributes by moving noise along the vector.", "Notably, most of the works mentioned above assume that the manifolds expand linearly in the latent space, ignoring that the underlying manifold could be extremely non-linear.", "In contrast to the prior works, we propose a general framework that can be applied to various vision applications by traversing the non-linear manifolds in an iterative scheme.", "Study on Leveraging Pre-trained GANs.", "Recently, several studies have investigated how to leverage existing GANs to achieve new tasks.", "One way is to treat pre-trained GANs as strong image priors, which can be achieved by either exploiting GANs as a component of their networks [22], [5], [6], [17], [3] or directly editing the latent code to find feasible solutions [1], [2], [11].", "These methods often require access to model parameters or additional training data.", "In contrast, this work aims to study the potential usage of black-box pretrained GANs, advising the owners to be careful about the risk before releasing models." ], [ "Hijack-GAN", "In this section, we first formalize the problem and then describe how to reuse models by our framework, followed by a proposed orthogonal constraint to improve disentanglement ability.", "Lastly, we discuss the practical considerations.", "The overall architecture is shown in Figure REF .", "Figure: An overview of the proposed framework.", "The framework takes two steps to reuse GANs: (a) train a proxy model to distill information from pre-trained models, circumventing accessing the gradients of pre-trained models; (b) identify and iteratively traverse a non-linear trajectory under the guidance of gradients." ], [ "Problem Statement", "We aim to study the possibility that pre-trained GANs can be used for unintended applications beyond unconditional generation.", "Specifically, we consider a victim generator $G:z \\rightarrow I$ that maps noise $z\\sim p(z)$ to realistic images $I$ , and one or multiple victim task models $\\mathcal {M}:I \\rightarrow A$ that map images $I$ to the attributes space $A$ .", "We aim to find a trajectory $\\mathcal {T}$ in the latent space $p(z)$ such that, as traversing along the path, the desired task can be gradually achieved, which can be further expressed as follows: $\\mathcal {L} \\circ \\mathcal {M} \\circ G(z^{(i+1)}) \\le \\mathcal {L} \\circ \\mathcal {M} \\circ G(z^{(i)}), \\forall z^{(i)} \\in \\mathcal {T},$ where $\\mathcal {L}$ is the loss function of unintended tasks.", "Note that the parameters and training data for the task are inaccessible, making it impossible to be directly solved by optimization methods since they require the gradients from the models." ], [ "Non-linear Traversal", "To identify a meaningful trajectory in the black-box setting, we have to address the following two issues: (1) the gradients from the GAN and the task models are impassable.", "(2) we need to identify a trajectory that precisely describes the highly non-linear manifolds.", "Since the gradients are impassable, we first train a proxy model $\\mathcal {P}$ to distill the information from the models.", "We assume that only $z$ -space is available and synthesize data pairs $(z, \\mathcal {M} \\circ G(z))$ to train the proxy model such that it can map input noise to the attribute space; therefore, we circumvent directly accessing the gradients while remaining informed about the relation between attributes and noise.", "Next, we compute the Jacobian matrix of the proxy model $\\mathcal {P}$ with respect to the input noise $z \\in \\mathbb {R}^n$ .", "$\\mathbf {J} = \\begin{bmatrix}\\frac{\\partial \\mathcal {P}_1}{\\partial z_1} & \\cdots & \\frac{\\partial \\mathcal {P}_1}{\\partial z_n} \\\\\\vdots & \\ddots & \\vdots \\\\\\frac{\\partial \\mathcal {P}_m}{\\partial z_1} & \\cdots & \\frac{\\partial \\mathcal {P}_m}{\\partial z_n} \\\\\\end{bmatrix},$ where $\\mathcal {P}_j$ denotes j-th attributes predicted by the proxy.", "Each row vector of the matrix $\\mathbf {J}$ , termed $\\mathbf {J}_j$ for simplicity, can be interpreted as the direction in which the corresponding attribute changes most quickly; meanwhile, this vector also gives us a hint to identify a meaningful non-linear trajectory.", "In light of this, we design an algorithm that iteratively updates the position of noise under the guidance of the vector $\\mathbf {J}_j$ : $z^{(i+1)} = z^{(i)} - \\lambda \\mathbf {J}^{(i)}_j,$ where $\\lambda $ is a hyper-parameter deciding moving speed and $\\mathbf {J}^{(i)}_j$ is associated with the attribute of interest at step i.", "Note that we normalize the $\\mathbf {J}^{(i)}_j$ to better control the level of changes.", "By repeatedly computing Eq.", "REF in each step, the target attribute in the generated image $G(z)$ would be gradually modified, granting us high-level control over image generation despite the black-box access." ], [ "Constraint for Disentanglement", "In many cases, attributes may be entangled with each other, meaning that other non-target attributes would be changed if we solely follow the steepest direction in Eq.", "REF .", "To alleviate the problem, we additionally propose a constraint to encourage disentanglement.", "Since each row vector $\\mathbf {J}_j$ in Eq.", "REF represents one direction that affects certain attributes most, we aim to find a vector which have the maximum inner product with the target direction $\\mathbf {J}_j$ while being orthogonal to other non-target directions $\\mathbf {J}_{k \\ne j}$ .", "We formulate the constraint as a linear program as follows.", "$\\begin{aligned}& \\underset{n}{\\text{maximize}} && \\mathbf {J}_{j}^Tn \\\\& \\text{subject to} && An = 0,\\end{aligned}$ where $n$ is the direction vector of interest, and each row of $A$ consists of the attribute vector $\\mathbf {J}_{k \\ne j}$ on which we want to condition.", "By substituting $n$ for $\\mathbf {J}^{(i)}_j$ in Eq.", "REF , we can solely edit the attribute of interest while retaining other non-target attributes in the image.", "Note that since Eq.", "REF is evaluated at each iteration, we are still capable of capturing the non-linear manifolds." ], [ "Implementation", "We consider two state-of-the-art GANs, PGGAN [19] and StyleGAN [20], as the victim generators.", "The input of both models is 512-D noise drawn from a standard normal distribution.", "Following Shen [25], we do not normalize the input noise for both models.", "The proxy models are implemented by a stack of fully-connected layers.", "Except for the last layer, each layer is followed by a ReLU activation function and a Dropout [26] layer with a rate of 0.2.", "We empirically find that a proxy model with 3 layers works sufficiently well for PGGAN, while it takes 8 layers to work on StyleGAN.", "It is also observed that the Dropout function plays a critical role to prevent over-fitting on StyleGAN.", "These observations imply that the $z$ -space of StyleGAN may be highly entangled and non-linear.", "For attribute manipulation, we find it important to train the proxy models with balanced datasets.", "We synthesize datasets for every attribute, each of which involves 100k positive and 100k negative samples.", "Since the annotations are unavailable, we adopt a ResNet-50 [16] classifier pre-trained on CelebAHQ [19] to annotate the generated images.", "We discard data with confidence lower than 0.9 to reduce the ambiguity.", "For other regression tasks, we generate random samples over the latent space and discard those with lower confidence.", "It is also observed that the proxy model on StyleGAN benefits from more training data, while the effect is marginal on PGGAN.", "In this section, we provide analysis of our non-linear iterative scheme by applying it to attribute manipulation, head pose manipulation, and landmark editing, showing that our method benefits from the non-linearity and demonstrates the possibility of unintended usage." ], [ "Experiment Setup", "Settings.", "We take two state-of-the-art GANs, StyleGAN [20] and PGGAN [19], as our backbone generators.", "Both models are pre-trained on CelebAHQ [19].", "We adopt different task models according to the applications.", "We use ResNet-50 [16] pre-trained on CelebAHQ for attribute manipulation, HopeNet [24] pre-trained on 300W-LP [35] for head pose manipulation, and MTCNN [31] to crop faces and detect landmarks for landmark editing.", "Note that in all experiments, the model parameters and additional training data are unavailable.", "Baselines.", "We consider two baselines here.", "First, following Denton [12], we employ the same classifier as our framework to compute the gradient with respect to the initial point and take it as a constant attribute vector $v$ .", "The attribute manipulation is realized as below, $z^{(i+1)} = z^{(i)} + \\lambda v.$ We refer to this baseline as Linear.", "The key difference is that the proposed framework recomputes gradients in every step.", "Similarly, the second baseline, InterfaceGAN [25], adopts the same linear strategy but derive the vector $v$ from the normal vector of the decision boundary of a linear SVM, which can be viewed as a common direction that changes the attribute most.", "InterfaceGAN achieves conditional manipulation by pairwisely computing an orthogonal vector for the normal vectors.", "Note that, instead of using a common direction for all data, we derive moving directions for specific input." ], [ "Attribute Manipulation", "We show that our framework can be applied to attribute manipulation.", "We consider unconditional and conditional settings, respectively.", "The former one is achieved by solely applying Eq.", "REF ; The latter one adopts Eq.", "REF to ensure the preservation of non-target attributes.", "Figure: Unconditional attribute manipulation on PGGAN (left) and StyleGAN (right) with respect to Eyeglasses, Gender, Smile, and Age.", "We compare our method to Linear and InterfaceGAN.Unconditional Manipulation.", "In the unconditional setting, we expect to see an effective transition as compared to the baselines since our method dynamically decides the moving direction in each step and follows the steepest direction.", "Note that although the main objective here is to change the target attribute, other attributes might change arbitrarily.", "Figure REF compares our method to two baselines, Linear and InterfaceGAN, on 4 attributes.", "On PGGAN, all three methods successfully edit the attributes, while our method produces transition with much less distortion, e.g., smile and age.", "Also, we observe that our method and InterfaceGAN perform similarly on eyeglasses and gender.", "It may imply that for some attributes, the underlying manifold expands linearly as is derived by InterfaceGAN.", "On the other hand, our method behaves clearly differently from the two linear baselines on StyleGAN, showing that the manifold may be highly non-linear.", "We verify the benefit of the iterative scheme by comparing our method to the baselines.", "First, Linear appears to fail to edit smiling even if it uses the same classifier as our framework.", "When comparing to InterfaceGAN, our method preserves more non-target contents than InterfaceGAN although the goal is not to preserve attributes.", "For example, age changes when editing eyeglasses, and hair color changes when editing smiles.", "Figure: Logit changes over steps on StyleGAN.", "From left to right: Eyeglasses, Gender, Smile, Age, Narrow Eyes, Blond Hair, and Pale Skin.", "The solid lines represent the predictions of the target attributes while the dot lines represent the mean values over all the other non-target attribute predictions.", "Zoom in for better visualization.Figure: Unconditional rare attribute manipulation on StyleGAN with respect to Bald, Narrow Eyes, Pale Skin, and Blond Hair.", "Each set consists of input (left), our method (middle), and InterfaceGAN (right).Effectiveness.", "To further verify the effectiveness of the proposed method, we compare our method to InterfaceGAN on logit changes over steps.", "We experiment on StyleGAN since it tends to be more non-linear as observed in Figure REF .", "Specifically, we choose 7 attributes and sample 2500 trajectories for each attribute, each of which consists of 40 steps with the step size equal to 0.2.", "Next, we compute the mean values over trajectories and report the target and non-target logit changes, respectively.", "Figure REF shows that our method can rapidly change the target attribute on most attributes, especially on rare attributes such as blond hair, pale skin, and narrow eyes.", "The result aligns with our hypothesis that not all attributes in the latent space distribute linearly; thus, our framework can benefit from the non-linearity.", "We additionally visualize the manipulation over the rare attributes in Figure REF , confirming the effectiveness of our method.", "In addition to effective manipulation, it is also seen that the non-target attributes of all methods are changed during the manipulation.", "It is thus crucial to limit such an undesired effect.", "Figure: Conditional attribute manipulation on PGGAN (left) and StyleGAN (right) with respect to Eyeglasses, Gender, Smile, and Age.", "We compare our method to InterfaceGAN.Conditional Manipulation.", "We present that our framework along with the proposed constraint in Eq.", "REF can preserve the non-target attributes better.", "In this experiment, we consider 1 of 4 attributes to edit while conditioned on the other three.", "Figure REF presents the results on PGGAN and StyleGAN, respectively.", "On PGGAN, our method strongly preserves the non-target attributes and effectively modify the target attribute.", "InterfaceGAN appears to be ineffective when it has to maintain non-target attributes.", "For example, the smiling is changed when editing eyeglasses.", "On StyleGAN, both methods succeed to edit the attributes while our method looks closer to the input.", "These results suggest that our method with the constraint can effectively edit the target attribute while retaining others.", "Figure: Attribute preservation ratio.", "We compare our method (orange) to InterfaceGAN (mist blue) in unconditional (non-stripe) and conditional (stripe) settings.", "The higher values, the better attributes are preserved during attribute manipulation.Attribute Preservation.", "To inspect how well our method can preserve the non-target attributes, we follow the same sampling strategy as in Figure REF and measure the ratio of target prediction changes to non-target prediction changes on StyleGAN.", "Higher values indicate that the subject can more effectively edit target attributes with lower perturbation to non-target attributes.", "We consider 7 attributes to edit.", "The former four attributes are conditioned on each other except themselves, and the latter three attributes are conditioned on all the former four.", "This design makes the latter three more challenging since they have to edit rare attributes while maintaining 4 attributes simultaneously.", "Figure REF compares our method to InterfaceGAN in both conditional (Eq.", "REF ) and unconditional (Eq.", "REF ) settings.", "It is immediately observed that our method with the proposed constraint achieves the highest ratio on almost all attributes, validating that the non-linear iterative scheme not only advances the attribute manipulation but also help the attribute preservation.", "Interestingly, the ratio of our unconditional method on the last three attributes are higher than InterfaceGAN in the conditional setting.", "It could be attributed to inefficiency of linear methods.", "As observed in Figure REF , linear methods are less activate on rare attributes.", "Moreover, it uses the same direction for every step, which may be sub-optimal for the points far away from the initial point.", "Both two reasons together lead to the phenomenon." ], [ "Analysis", "We conduct two analysis to provide more insights from the perspective of smoothness and function approximation.", "Table: Modified Perceptual Path Length measured on PGGAN and StyleGAN.", "Lower is better.Smoothness.", "We compare the trajectories generated by our method to those by linear methods such as InterfaceGAN in terms of smoothness.", "Previously, Karras [20] propose a metric, Perceptual Path Length (PPL), to measure smoothness over the whole latent space.", "However, we only focus on smoothness of trajectories; thus we propose to use modified PPL (mPPL) that only consider noise sampled from the generated trajectories.", "Formally, we have $\\begin{aligned}l_z = {E}_{z^{(i)} \\sim \\mathcal {T}, \\mathcal {T} \\sim p(\\mathcal {T})}[\\frac{1}{\\epsilon ^2}d(& G(\\text{lerp}(z^{(i)}, z^{(i+1)}, t)), \\\\ &G(\\text{lerp}(z^{(i)}, z^{(i+1)}, t+\\epsilon )))],\\end{aligned}$ where $\\mathcal {T}$ is a generated trajectory, $z^{(i)}$ is the noise of i-th step in the trajectory, $G$ is the generator, $\\text{lerp}$ is linear interpolation function with factor $t$ drawn from uniform distribution, $\\epsilon $ is a small displacement, and $d(\\cdot )$ is LPIPS metric [32].", "For each attribute, we sample 1000 trajectories and each consists of 600 steps with step size 0.01, leading to 600k images in total.", "Following Karras [20], we set $\\epsilon $ to 1e-4 and report the mean value over all samples.", "Lower values mean smoother trajectories.", "Table REF presents the evaluation results on both PGGAN and StyleGAN.", "In the unconditional setting, both methods produce comparably smooth trajectories.", "It also happens in the condition setting on PGGAN.", "It could be attributed to the relatively linear manifolds.", "Note that smoothness does not mean the effectiveness.", "One could generate images with no change to gain smoother transition.", "Lastly, our method surpasses the baseline on StyleGAN in the conditional setting by a large margin since it can strongly retain the non-target attributes, leading to smoother transition.", "Table: Errors of first order Taylor approximation.", "Errors are reported according to the L2-norm between the initial point and the estimated points.", "ff consists of the generator and the task model.Figure: Pose manipulation on StyleGAN with respect to yaw (top) and pitch (bottom).Figure: Landmark editting on StyleGAN with respect to Nose up/down, Nose left/right, Left corner of mouth, Right corner of mouth.Function Approximation.", "We show that our iterative scheme can better traverse the highly non-linear latent space from the perspective of function approximation.", "We first recall the first order Taylor approximation, $f(x) = f(a) + f^\\prime (a)(x-a).$ If a method models the underlying manifold better, it is expected to have lower error between the real logit $f(z^{(i+1)})$ and the estimated logit $f(z^{(i)})+f^\\prime (z^{(i)})(z^{(i+1)} - z^{(i)})$ .", "Note that the real gradient $f^\\prime $ in Eq.", "REF is unavailable for all baselines and our model.", "Instead, all methods use estimated gradients, namely $\\mathbf {J}_j$ (Eq.", "REF ) for our method and $v$ (see Section REF ) for InterfaceGAN.", "Since InterfaceGAN uses normal vectors of SVM decision boundaries as guiding signals, we expect it to work well in a local region while becoming less accurate in the farther distance.", "To verify this assumption, we measure the errors and categorize them according to the L2 distance between the estimated and initial points.", "Table REF compares our method to InterfaceGAN.", "It shows that our method approximates the underlying function more accurately on average.", "As for the smiling, since we discard the data with lower confidence (see Section REF ), it may make the proxy model less robust at the region where many ambiguous samples happen.", "Lastly, due to the iterative scheme, our method achieves lower errors even when the estimated points moves far away from the initial points." ], [ "Head Pose Manipulation", "To further study the possibility of unintended usage of pretrained GANs, we employ our framework to achieve head pose manipulation by leveraging pre-trained HopeNet [24] as the task model.", "In particular, we attempt to gain control over the yaw and pitch axes since the training dataset of the pre-trained GANs, CelebAHQ, is well-aligned by eyes; thus it is almost impossible to control the roll axis.", "It also suggests that the framework could be still limited by the training dataset of pretrained GANs even though we are capable of gaining control over the generation process.", "To generate fine-grained transition, we choose small step size 0.01 and produces 1000 steps in total.", "Due to the proposed constraint in Eq.", "REF , our framework can solely edit one axis by conditioning on four attributes and the other axis.", "As shown in Figure REF , we smoothly interpolate the images for yaw and pitch axes.", "We also note that some artifacts or undesired changes happen when forcing the pose to go beyond certain degrees.", "With our framework, we can generate faces with arbitrary poses if they remain feasible in the training data of pre-trained GANs." ], [ "Landmark Editing", "By utilizing pre-trained MTCNN [31], we take one step further to consider landmark editing, which is a relatively challenging application.", "The reasons are three-fold.", "First, a landmark is two coordinates, which are extremely local attributes as compared to, e.g., age or gender.", "Second, since landmarks are coordinates, it becomes very sensitive for annotation.", "The performance may be degraded if the accuracy of task models are not satisfactory.", "Lastly, landmarks are often correlated with other features.", "Based on the results of previous experiments, we edit landmarks while conditioning on 4 attributes, poses, and a subset of landmarks.", "Note that since the training dataset is well aligned, the landmark of eyes are not editable.", "Figure REF demonstrates the results when editing the coordinates of noses and mouths.", "Surprisingly, we find some attributes are highly entangled.", "For instance, mouth landmarks is highly correlated with smile.", "To measure the correlation, we additionally compute the cosine similarity between direction vectors (Eq.", "REF ).", "Higher similarity indicates higher entanglement.", "We analyze the correlation of mouth landmarks vs. smile (0.296) and mouth landmarks vs. gender (0.035).", "The former one is around 8.5 times larger than the latter.", "Additionally, nose landmarks vs. yaw are highly correlated by 0.5014.", "In this analysis, we do find measurable entanglement.", "In this paper, we demonstrate that publicly released state-of-the-art GANs can be applied for a wide range of applications beyond the creators' intention.", "To achieve this, we propose a framework to gain high-level control over unconditional image generation.", "Extensive results show that the framework is advantageous over non-iterative linear baselines in many aspects and can be readily applied to various unintended tasks.", "While this allows to re-use high quality GANs that are becoming increasingly costly to train for new applications, we hope that this work also raises awareness regarding potential unintended usage, urging the creators and owners to be cautious about the possible risks of their models before a full release." ], [ "Acknowledgment", "This work is partially funded by the Helmholtz Association within the project \"Protecting Genetic Data with Synthetic Cohorts from Deep Generative Models (PRO-GENE-GEN)\" (ZT-I-PF-5-23).", "KwToinForfordo:end Appendix" ], [ "Quality of Generated Images", "We follow the evaluation protocol of StyleGAN and report average FID scores.", "As shown in Table REF , we observe that both methods degrade the image quality of StyleGAN.", "As compared to InterfaceGAN, our method performs slightly better in the unconditional setting and significantly better in the conditional setting.", "The experiment confirms the effectiveness of our non-linear method on the quality of generated images.", "Note that following InterfaceGAN, we do not normalize latent codes and find the vanilla setting slightly worse than the one reported by StyleGAN (5.04).", "Table: FID evaluation on StyleGAN.", "Lower is better." ], [ "Detailed Iterative Algorithm", "We detail the proposed iterative framework in Algorithm .", "We first recall our proposed framework.", "The Jacobian matrix of the proxy model is computed as follows, $\\mathbb {J} = \\begin{bmatrix}\\frac{\\partial \\mathcal {P}_1}{\\partial z_1} & \\cdots & \\frac{\\partial \\mathcal {P}_1}{\\partial z_n} \\\\\\vdots & \\ddots & \\vdots \\\\\\frac{\\partial \\mathcal {P}_m}{\\partial z_1} & \\cdots & \\frac{\\partial \\mathcal {P}_m}{\\partial z_n} \\\\\\end{bmatrix},$ where $\\mathcal {P}_j$ denotes j-th attributes predicted by the proxy.", "We iteratively explore the latent space by dynamically discovering the most representative direction, which can be shown as below.", "$z^{(i+1)} = z^{(i)} - \\lambda \\mathbb {J}^{(i)}_j,$ where $\\lambda $ is a hyper-parameter deciding moving speed and $\\mathbb {J}^{(i)}_j$ is associated with the attribute of interest at step i.", "To encourage disentanglement, we further propose an orthogonal constraint.", "$\\begin{aligned}& \\underset{n}{\\text{maximize}} && \\mathbb {J}_{j}^Tn \\\\& \\text{subject to} && An = 0,\\end{aligned}$ where $n$ is the direction vector of interest, and each row of $A$ consists of the attribute vector $\\mathbb {J}_{k \\ne j}$ on which we want to condition.", "Note that we take decreasing logit values as an example for simplicity.", "One could achieve the opposite direction by gradient ascent.", "Trajectory length $L$ , step size $\\lambda $ , initial point $z^{(0)}$ , target attribute index $j$ , proxy model $\\mathcal {P}$ , condition index set $K$ Trajectory $\\mathcal {T}=\\lbrace z^{(1)}, \\ldots , z^{(N)}\\rbrace $ Initialize $\\mathcal {T} \\leftarrow \\emptyset $ i $\\lbrace 0, \\ldots , L-1\\rbrace $ Compute the Jacobian matrix $\\mathbb {J}^{(i)}$ by Eg.", "REF $K \\ne \\emptyset $ Construct matrix $A$ by vectors $\\mathbb {J}_{k \\in K}^{(i)}$ Solve orthogonal vector $n$ by Eq.", "REF $n \\leftarrow \\mathbb {J}_{j}^{(i)}$ $z^{(i+1)} \\leftarrow z^{(i)} - \\lambda n$ (Eq.", "REF ) Add $z^{(i+1)}$ to $\\mathcal {T}$ $\\mathcal {T}$ HijackGAN" ], [ "Visualization of Smoothness", "Although we have verified the smoothness of our method in terms of modified Perceptual Path Length in Table 1 of the main paper, we additionally provide qualitative results on PGGAN (Figure REF ) and StyleGAN (Figure REF ), respectively.", "All experiments are conducted in the conditional setting, meaning that we solely edit one attribute, and others should remain the same.", "In particular, we take 40 steps with step size 0.2 and show the images from every 5 steps.", "Note that, the closer to the right-hand side, the farther from the initial point.", "From Figure REF and Figure REF , we make two observation.", "First, our method preserves attributes better on both models.", "For example, our method can preserve smiling when editing eyeglasses, and preserve the smiling when editing age on both models.", "Second, on StyleGAN, our method can produce smoother transitions.", "For example, gender and eyeglasses.", "Figure: (Conditional) visualization of smoothness on PGGAN.", "All attributes should remain the same except the target one.Figure: (Conditional) visualization of smoothness on StyleGAN.", "All attributes should remain the same except the target one." ] ]	2011.14107
[ [ "Method of numerical simulation of interacting quantum gas kinetics on\n finite momentum lattice" ], [ "Abstract We present the efficient and universal numerical method for simulation of interacting quantum gas kinetics on a finite momentum lattice, based on the Boltzmann equation for occupation numbers.", "Usually, the study of models with two-particle interaction generates the excessive amount of terms in the equations essentially limiting the possible system size.", "Here we employ the original analytical transformation to decrease the scaling index of the amount of calculations.", "As a result, lattice sizes as large as 48x48x48 can be simulated, allowing to study realistic problems with complex interaction models.", "The method was applied to the simulation of weakly interacting Fermi and Bose gases where we calculated the relaxation times depending on the momentum and temperature." ], [ "Introduction", "There are many problems concerning the time evolution of complex interacting quantum systems in modern physics.", "Examples in solid state physics include the behaviour of nonequilibrium charge carriers in semiconductors [1], [2] or metals [3], [4], the relaxation of an excited state in a superconductor [6], [5], the condensation of exciton polaritons [8], [7], and the dynamics of a Bose-Enstein condensate (BEC) in a reconfigurable quantum simulator [9].", "To correctly describe such physical phenomena as formation of BEC state [10], [11], [12], [13], relaxation of excited state of electron subsystem created by a ultrashort laser pulse [3], [14] or after absorbing cosmic particle [6], and many other promising applications, the detailed theoretical approaches are required.", "Popular analytical approaches include Boltzmann equation [15], nonequilibrium Green functions [16], Liouville equation [17], Fermi liquid theory [18], and others [19], [20], [14].", "Application of purely analytical approaches to modern quantum problems, however, can be difficult due to complexity of the models and, therefore, numerical calculations are unavoidable [12].", "The Boltzmann equation is efficient for numerical study of weakly interacting systems not far from equilibrium [4], [15], [21].", "Here we present the efficient numerical method to study the kinetics of quantum systems of various statistics using the Boltzmann equation on a finite momentum lattice.", "The efficiency is achieved with the special analytical transformation, which is described in details.", "The method allows to take into account the energy exchange with thermal bath or nonequilibrium phonons as well as two-particle interaction, which usually requires high amount of calculations.", "The possibility to take into consideration arbitrary single-particle energies and level broadening factors enables to study extensive set of promising physical problems.", "The operation of the method is demonstrated by the calculation of relaxation times in weakly interacting Bose and Fermi gases." ], [ "Finite momentum lattice", "In this work, we present the numerical method to simulate the kinetics of a sufficiently small system, a finite atomic cluster or nanocrystal.", "For certainty, let consider a nanocrystal of L$\\times $ L$\\times $ L atoms with a simple cubic lattice (generalizations for other lattices are obvious).", "The number of points in the reciprocal lattice is also L$\\times $ L$\\times $ L with the step $\\Delta k = 2 \\pi /La$ , where $a$ is the lattice constant.", "The Hamiltonian of the system is taken in the form: $\\hat{H} = \\hat{H}_1 + \\hat{H}_{\\textrm {int}},\\\\\\hat{H}_1 = \\sum \\limits _{\\bf k} \\varepsilon _{\\bf k} \\hat{n}_{\\bf k},$ where $\\hat{H}_1$ and $\\hat{H}_{\\textrm {int}}$ are the single-particle Hamiltonian and interaction part, correspondingly, $\\varepsilon _{\\bf k}$ are particle energies, $\\hat{n}_{\\bf k}$ is the operator of occupation number.", "Index ${\\bf k}$ also includes a spin index in the case of fermions.", "For complicated problems, additional sorts of particles can be introduced by relevant terms in the Hamiltonian (REF ), for example, phonons to account for the interaction with the lattice.", "The kinetics of the system due to interaction is described by the Boltzmann equation [22] derived from Fermi's Golden rule: $\\frac{1}{\\tau _{\\bf k}} = \\frac{2 \\pi }{\\hbar } \\sum \\limits _{i,f}\\left<i\\right\|\\hat{H}_{\\textrm {int}}\\left\|f\\right>^2 \\delta (E_i - E_f),$ where $\\left<i\\right\|\\hat{H}_{\\textrm {int}}\\left\|f\\right>$ are the matrix elements of interaction operator between initial $i$ and final $f$ states changing the occupation $n_{\\bf k}$ .", "Later on, for simplicity, we use the time units where $2 \\pi / \\hbar = 1$ .", "The application of Boltzmann equation to continual problems comes down to performing integrations with the particle density on the energy axis [6], [4].", "In the case of a nanocluster with finite momentum step, however, the numerical summation of the original expressions should be used.", "On the other side, the number of terms rapidly grows with the lattice size $L$ and sizes $L > 8$ are practically untractable [23].", "In this section we present a convenient and efficient way to extend possible system sizes up to $ L \\sim 32 \\div 48$ using a special transformation.", "This allows simulations of large enough sizes for subsequent extrapolation to the continual limit $L \\rightarrow \\infty $ .", "As a result, new physical problems can be studied which could not be explored by the aforementioned analytical methods." ], [ "Interaction with the phonon subsystem.", "We begin with a simple case: a system of free particles in the presence of phonons (crystal lattice with given temperature $T$ ): $\\hat{H}_{\\mathrm {int,phon}} = M_0 \\sum \\limits _{\\bf k q}{ \\hat{a}^{\\dagger }_{\\bf k} \\hat{a}_{{\\bf k}-{\\bf q}}\\hat{b}_{\\bf q} } + H.c.,$ where $\\hat{a}_{\\bf k}$ , $\\hat{a}^{\\dagger }_{\\bf k}$ and $\\hat{b}_{\\bf q}$ , $\\hat{b}^{\\dagger }_{\\bf q}$ are operators of particles under consideration and phonons, correspondingly, and $M_0$ is the matrix element of interaction with phonons.", "In the case of Fermi particles, spin index is for simplicity included in the particle momentum ${\\bf k}$ .", "Figure: The processesfor the corresponding terms in Eq.", "()generated by the interaction with phonons:adding or removing particle with momentum 𝐤{\\bf k}and phonon with momentum 𝐪{\\bf q}.In Figure REF , we show the processes generated by this type of interaction that change the number of particles $n_{\\bf k}$ and phonons $n^{\\text{(phon)}}_{\\bf k}$ .", "The corresponding terms in the kinetic equations according to (REF ) in the case of Bose statistics have the form: $\\frac{d n_{\\bf k}}{d t} =P^{(+)}_{\\mathrm {in}} +P^{(+)}_{\\mathrm {out}}- P^{(-)}_{\\mathrm {in}}- P^{(-)}_{\\mathrm {out}} ,\\\\\\nonumber P^{(+)}_{\\mathrm {in}} = M^2_0 \\sum \\limits _{\\bf q}(n_{\\bf k} + 1) n_{{\\bf k}-{\\bf q}} n_{\\bf q}^{\\text{(phon)}}F( \\delta E ),\\\\\\nonumber P^{(+)}_{\\mathrm {out}} = M^2_0 \\sum \\limits _{\\bf q}(n_{\\bf k} + 1) n_{{\\bf k}+{\\bf q}} (n_{\\bf q}^{\\text{(phon)}} + 1)F( \\delta E ),\\\\\\nonumber P^{(-)}_{\\mathrm {in}} = M^2_0 \\sum \\limits _{\\bf q}n_{\\bf k} (n_{{\\bf k}+{\\bf q}} + 1 ) n_{\\bf q}^{\\text{(phon)}}F( \\delta E ),\\\\\\nonumber P^{(-)}_{\\mathrm {out}} = M^2_0 \\sum \\limits _{\\bf q}n_{\\bf k} (n_{{\\bf k}-{\\bf q}} + 1 ) (n_{\\bf q}^{\\text{(phon)}} + 1)F (\\delta E ),$ where phonon energies are denoted $\\tilde{\\varepsilon }_{\\bf q}$ and $\\delta E$ is the correspoding energy difference for incoming and outcoming particles.", "The factor $F (\\delta E)$ is introduced to account for the finite width of the energy levels.", "The four given terms correspond to the various types of interaction with phonon absorption or radiation, as shown in Figure REF .", "In the case of Fermi statistics, $(n+1)$ is replaced with $(1-n)$ .", "Depending on the problem under study, the phonon subsystem can be taken into account in two ways: considering phonons as equilibrium (thermal bath) or nonequilibrium.", "For equilibrium phonons, their occupation numbers are determined by the Bose-Einstein distribution function $n_{\\bf q}^{\\text{(phon,0)}} = f( \\tilde{\\varepsilon }_{\\bf q}, T)$ , here $T$ is the temperature.", "In the case of nonequilibrium phonons, the system (REF ) is supplemented with similar equations for the phonon numbers, and finite phonon lifetime $\\tau ^\\text{(phone)} $ related to their decay in the area of consideration is taken into account: $P^{\\mathrm {(phon)}}_{\\mathrm {dec}} = - \\frac{1 }{\\tau ^\\text{(phon)}}\\left( n_{\\bf q}^\\text{(phon)} - n_{\\bf q}^\\text{(phon,0)} \\right).$ The interaction with the phonon subsystem is crucial when the pair interaction is negligible, for example, at low particle density." ], [ "Two-particle interaction.", "However, the interaction with phonons alone is often not enough to correctly describe kinetic phenomena in complex systems, and it is necessary to take into account the processes due to the interaction between particles [4].", "We write the two-particle interaction as: $\\hat{H}_{\\mathrm {int,pair}} =\\sum \\limits _{\\bf k p q } U({\\bf q}){ \\hat{a}^{\\dagger }_{\\bf k} \\hat{a}^{\\dagger }_{\\bf p}\\hat{a}_{{\\bf p}+{\\bf q}} \\hat{a}_{{\\bf k}-{\\bf q}}}$ Next, for convenience, we denote the momenta of four particles involved in the interaction ${\\bf k}$ , ${\\bf p}$ , ${\\bf p}+{\\bf q}$ , ${\\bf k}-{\\bf q}$ as ${\\bf 1}$ , ${\\bf 2}$ , ${\\bf 3}$ , ${\\bf 4}$ , correspondingly, and show the conservation law explicitly with Kronecker delta: $\\hat{H}_{\\mathrm {int,pair}} =\\sum \\limits _{\\bf 1 2 3 4 } U_{{\\bf 3}-{\\bf 2}}{ \\hat{a}^{\\dagger }_{\\bf 1} \\hat{a}^{\\dagger }_{\\bf 2}\\hat{a}_{\\bf 3} \\hat{a}_{\\bf 4}\\delta _{{\\bf 1}+{\\bf 2},{\\bf 3}+{\\bf 4}}}$ Then, kinetic equations take the form: $\\frac{dn_{\\bf 1}}{dt}=\\sum \\limits _{\\bf 2 3 4 } \|U_{{\\bf 3}-{\\bf 2}}\|^2\\left[(n_{\\bf 1}+1) (n_{\\bf 2}+1+\\delta _{\\bf 12}) n_{\\bf 3} (n_{\\bf 4}-\\delta _{\\bf 34})\\right.-\\left.n_{\\bf 1} (n_{\\bf 2}-\\delta _{\\bf 12}) (n_{\\bf 3}+1) (n_{\\bf 4}+1+\\delta _{\\bf 34})\\right]\\delta _{\\Delta \\varepsilon , 0} \\delta _{{\\bf 1}+{\\bf 2},{\\bf 3}+{\\bf 4}}$ for Bose statistics and $\\frac{dn_{\\bf 1}}{dt}=\\sum \\limits _{\\bf 2 3 4 } \|U_{{\\bf 3}-{\\bf 2}}\|^2\\left[(1-n_{\\bf 1}) (1-n_{\\bf 2}-\\delta _{\\bf 12}) n_{\\bf 3} (n_{\\bf 4}-\\delta _{\\bf 34})\\right.-\\left.n_{\\bf 1} (n_{\\bf 2}-\\delta _{\\bf 12}) (1-n_{\\bf 3}) (1-n_{\\bf 4}-\\delta _{\\bf 34})\\right]\\delta _{\\Delta \\varepsilon , 0} \\delta _{{\\bf 1}+{\\bf 2},{\\bf 3}+{\\bf 4}}$ for Fermi statistics, respectively.", "We should note the corrections using $\\delta _{\\bf 12}$ , $\\delta _{\\bf 34}$ in the terms with coinciding momenta.", "While they are negligible in the continual case, for a finite system the exact form is essential.", "As we mentioned earlier, the practical use of the expressions (REF ), (REF ) on the momentum lattice $L \\times L \\times L$ is hindered by the necessity to calculate a large number of terms which can be estimated as $\\sim V^4=L^{12}$ , and as a result, the simulation of system with lattice size $L > 8$ is practically impossible [23], [24].", "However, in the model of the form (REF ), when the total momentum is conserved, we can use the analytical transformation working in the extended space $({\\bf k}, \\varepsilon )$ of size $L^3 \\times N_\\varepsilon \\sim L^5$ that dramatically reduces the amount of calculations to $ \\sim $ $L^5 \\ln {L}$ .", "It allows to greatly increase the affordable system sizes." ], [ "Free gas case.", "We demonstrate the principle of this transformation using the example of a free gas of Fermi or Bose particles with the energy spectrum $ \\varepsilon _{\\bf k} \\sim {\\bf k}^2$ and the contact interaction $U_{\\bf q} = const = U_0$ : $\\nonumber \\hat{H}_{\\mathrm {int,pair}} =U_0 \\sum \\limits _{\\bf 1 2 3 4 }{ \\hat{a}^{\\dagger }_{\\bf 1} \\hat{a}^{\\dagger }_{\\bf 2}\\hat{a}_{\\bf 3} \\hat{a}_{\\bf 4} }\\delta _{{\\bf 1}+{\\bf 2},{\\bf 3}+{\\bf 4}}$ First of all, we note that due to Kronecker delta $\\delta _{{\\bf 1}+{\\bf 2},{\\bf 3}+{\\bf 4}}$ , the expressions (REF ), (REF ) have the form of a discrete convolution $\\sum \\limits _{k^{\\prime }=1}^{L} f(k^{\\prime }) g(k-k^{\\prime }) \\equiv (fg)(k),$ which can be efficiently calculated using the Convolution Theorem: $(fg)(k) = \\frac{1}{L} \\,\\mathrm {FFT}\\,^{-1} \\left[ F(r) \\cdot G(r) \\right] (k),$ where $F(r)$ , $G(r)$ are the Fourier transforms of functions $f(k)$ , $g(k)$ .", "This allows us to calculate the sum of (REF ) with $ \\sim L \\ln L$ operations using the Fast Fourier transform (FFT).", "In the case of a space of dimension $d$ , the number of operations is $\\sim V \\ln V \\sim L^d \\ln L$ .", "Second, the same transformation can be performed on the energy axis, since the discrete particle energies in this model are proportional to integer numbers: $\\varepsilon _n = n \\varepsilon _1$ , where $n=0 \\dots N_\\varepsilon -1$ .", "To correctly use this scheme for non-periodic functions, the extended range of values $n = 0 \\dots N_{\\mathrm {m}ax}-1$ is employed, with the so-called 'zero-padding' [25] at $n \\ge N_\\varepsilon $ .", "Finally, we introduce functions in the extended space $({\\bf k}, \\varepsilon ) \\equiv {\\bf \\rho }$ and $({\\bf r}, \\gamma ) \\equiv {\\bf R}$ : $n_{{\\bf k} \\varepsilon } \\equiv n_{\\bf k} \\delta _{\\varepsilon ,\\varepsilon _{\\bf k}},\\\\s_{{\\bf k} \\varepsilon } \\equiv \\delta _{\\varepsilon ,\\varepsilon _{\\bf k}},\\\\N_{\\bf R} \\equiv \\,\\mathrm {FFT}\\,( n_{\\bf \\rho } ),\\\\S_{\\bf R} \\equiv \\,\\mathrm {FFT}\\,( s_{\\bf \\rho } ).$ Substituting them in the expressions (REF ), (REF ) and replacing Kronecker delta symbols with sums $\\delta _{\\Delta {\\bf k}, 0} =\\frac{1}{L^3} \\sum \\limits _{\\bf r} e^{i {\\Delta \\bf kr}}$ and $\\delta _{\\Delta \\varepsilon , 0} =\\frac{1}{N_\\varepsilon } \\sum \\limits _\\gamma e^{i \\Delta \\varepsilon \\gamma }$ , we obtain the final equation (the detailed derivation is given in Appendix): $\\frac{dn_{\\bf k}}{dt}= U_0^2 \\left([ p_{{\\bf k},\\varepsilon _{\\bf k}}+ \\widetilde{p}_{2{\\bf k},2\\varepsilon _{\\bf k}} ]+ n_{\\bf k}[ q_{{\\bf k},\\varepsilon _{\\bf k}}+ \\widetilde{q}_{2{\\bf k},2\\varepsilon _{\\bf k}} ]\\right),$ where $p_{\\bf \\rho } \\equiv \\,\\mathrm {FFT}\\,^{-1}( P_{\\bf R} ),\\\\\\nonumber q_{\\bf \\rho } \\equiv \\,\\mathrm {FFT}\\,^{-1}( Q_{\\bf R} ),\\\\\\nonumber \\widetilde{p}_{\\bf \\rho } \\equiv \\,\\mathrm {FFT}\\,^{-1}( \\widetilde{P}_{\\bf R} ),\\\\\\nonumber \\widetilde{q}_{\\bf \\rho } \\equiv \\,\\mathrm {FFT}\\,^{-1}( \\widetilde{Q}_{\\bf R} ).$ The four-dimensional matrices $P$ , $Q$ , $\\widetilde{P}$ , $\\widetilde{Q}$ in the case of Bose statistics are given by the expressions: $P_{\\bf R} =S_{- \\bf R} N^2_{\\bf R}+N_{- \\bf R} N^2_{\\bf R}-N_{- \\bf R} N_{2 \\bf R}-S_{- \\bf R} N_{2 \\bf R}-N_{\\bf R} \\mathbb {Z}_{\\bf R},\\\\\\nonumber \\widetilde{P}_{\\bf R} = N^2_{\\bf R},\\\\\\nonumber Q_{\\bf R} =S_{ - \\bf R} n^2_{\\bf R}-2 N_{ - \\bf R} N_{\\bf R} S_{\\bf R}-N_{- \\bf R} S^2_{\\bf R}-N_{- \\bf R} S_{2 \\bf R}-S_{- \\bf R} N_{2 \\bf R}-2 N_{- \\bf R} N_{2 \\bf R}+S_{\\bf R} \\mathbb {Z}_{\\bf R},\\\\\\nonumber \\widetilde{Q}_{\\bf R} =S^2_{\\bf R}+2 N_{\\bf R} S_{\\bf R}+2 N^2_{\\bf R}.$ and in the case of Fermi statistics (paying additional attention to spin indices): $P_{\\bf R} =S_{- \\bf R} N^2_{\\bf R}-N_{- \\bf R} N^2_{\\bf R}+N_{- \\bf R} N_{2 \\bf R}-S_{- \\bf R} N_{2 \\bf R}+N_{\\bf R} \\mathbb {Z}_{\\bf R},\\\\\\nonumber \\widetilde{P}_{\\bf R} = - N^2_{\\bf R},\\\\\\nonumber Q_{\\bf R} =-S_{ - \\bf R} N^2_{\\bf R}+2 N_{ - \\bf R} N_{\\bf R} S_{\\bf R}-N_{- \\bf R} S^2_{\\bf R}+N_{- \\bf R} S_{2 \\bf R}+S_{- \\bf R} N_{2 \\bf R}-2 N_{- \\bf R} N_{2 \\bf R}-S_{\\bf R} \\mathbb {Z}_{\\bf R},\\\\\\nonumber \\widetilde{Q}_{\\bf R} =S^2_{\\bf R}-2 N_{\\bf R} S_{\\bf R}+2 N^2_{\\bf R}.$ The function $\\mathbb {Z}_{\\bf R}$ is defined in Appendix.", "As we see, multiple sums in the expressions (REF ), (REF ) are converted to several more performance-efficient Fourier transforms.", "The achieved reduction in the amount of calculations makes it possible to increase the available size of the system to a relatively macroscopic $L \\sim 16 \\div 48$ .", "Later in the Chapter , we show the application of this method to the detailed study of relaxation times in weakly interacting Bose and Fermi systems.", "Note that more complex models with momentum-dependent interaction $U_{\\bf q} \\ne const$ can also use this transformation in almost identical way but the expressions are slightly more complicated.", "For example, the second term in the Eq.", "(REF ) takes the form: $U^2_0 n_{- \\bf R} n^2_{\\bf R}\\;\\rightarrow \\;N_{{\\bf r} \\gamma } \\cdot \\sum \\limits _{{\\bf r}^{\\prime }}{ N_{{\\bf r}^{\\prime },-\\gamma } N_{-{\\bf r}^{\\prime },\\gamma }u_{{\\bf r}^{\\prime } + {\\bf r}} }$ where $u_{\\bf r} \\equiv \\,\\mathrm {FFT}\\,( U^2_{\\bf q} )$ .", "The sum can again be recognized as a convolution and calculated with appropriate sequence of Fourier transforms." ], [ "The generalization to arbitrary energy levels.", "Now we show how to extend this approach to the case of arbitrary energy levels $\\varepsilon _{\\bf k}$ , i.e.", "for systems with non-parabolic dispersion law (electrons far from the band edge, exciton polaritons, Bogoliubov quasiparticles, etc.).", "If the discrete energy levels are not proportional to integer numbers, the precise Fourier transform on the energy axis used in transformation (REF ), (REF ) is not possible.", "Using the smaller step on the energy axis we can make the grid values closer to the actual single-particle energy levels.", "The necessary accuracy of the approximation is determined by the width of the levels in the physical problem.", "In the equations (REF ), (REF ) we replace $\\delta _{\\Delta \\varepsilon , 0}$ to broadening factor $f(\\varepsilon _1 + \\varepsilon _2 - \\varepsilon _3 - \\varepsilon _4)$ , which describes the accuracy of the energy conservation.", "Depending on the problem under consideration, it can be taken in the form of Lorentzian or Gaussian function [26].", "The introducing of the broadening factor alongwith the reasonable choice of smaller energy step allow us to consider systems with an arbitrary not equidistant single-particle spectrum.", "We can notice that typical terms that have received the additional multiplier (broadening factor) can still be represented by multiple convolution: $\\sum \\limits _{ \\begin{array}{c} {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 \\\\\\varepsilon _2 \\varepsilon _3 \\varepsilon _4 \\end{array}}(\\dots )\\delta _{\\Delta \\varepsilon ,0}\\rightarrow \\sum \\limits _{ \\begin{array}{c} {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 \\\\\\varepsilon _2 \\varepsilon _3 \\varepsilon _4 \\end{array}}(\\dots )f(\\Delta \\varepsilon ).$ As a result, the expressions in (REF ) become $p_{{\\bf k}\\varepsilon } \\equiv \\,\\mathrm {FFT}\\,^{-1}( P_{{\\bf r}\\gamma } F(\\gamma ) ) ,\\\\\\nonumber q_{{\\bf k}\\varepsilon } \\equiv \\,\\mathrm {FFT}\\,^{-1}( Q_{{\\bf r}\\gamma } F(\\gamma ) ),\\\\\\nonumber \\widetilde{p}_{{\\bf k}\\varepsilon } \\equiv \\,\\mathrm {FFT}\\,^{-1}( \\widetilde{P}_{{\\bf r}\\gamma } F(\\gamma ) ),\\\\\\nonumber \\widetilde{q}_{{\\bf k}\\varepsilon } \\equiv \\,\\mathrm {FFT}\\,^{-1}( \\widetilde{Q}_{{\\bf r}\\gamma } F(\\gamma ) ),$ where $F(\\gamma )$ is the Fourier transform of the factor $f(\\varepsilon )$ .", "To verify the correctness of the transformation, the results of calculation using Eqs.", "(REF )–(REF ) were compared with the direct summation of the original expressions (REF ), (REF ) for several small systems with dimensions 4$\\times $ 4$\\times $ 4, 8$\\times $ 8$\\times $ 8 and various particle statistics.", "The resulting numbers were equal with the precision of at least 13 digits for all the problem parameters, proving the validity of the presented method.", "As a conclusion, in this Chapter we reported the universal approach for numerical simulation of the kinetics in interacting quantum system with arbitrary energy spectrum on a finite momentum lattice of relatively large dimensions." ], [ "Relaxation time in finite Fermi and Bose systems", "Generally, the study of kinetics by numerical solution of equations (REF ),(REF ),(REF ) is quite laborous.", "For problems with occupation numbers near the equilibrium $n_{\\bf k} \\simeq n^{(0)}_{\\bf k} \\equiv f( \\varepsilon _{\\bf k}, \\mu , T)$ , a simple model based on characteristic relaxation times is often used: $\\frac{dn_{\\bf k}}{dt} = - \\frac{1}{\\tau _{\\bf k}}\\left( n_{\\bf k} - n^{(0)}_{\\bf k} \\right).$ Values of $\\tau _{\\bf k}$ are estimated from existing analytical expressions or experimental data.", "There exists a well-known expression in the Fermi-liquid theory [4] for the lifetime of quasiparticles near Fermi surface: $\\tau (\\varepsilon ) \\sim \\frac{1}{ (\\pi k_B T)^2 + (\\varepsilon - F)^2 },$ where $F$ is the Fermi level.", "For the Bose gas, analytical results for lifetime of quasiparticles in some limit cases are available, namely, the so-called Beliaev damping [27], [28] and Landau damping [28], [29].", "Detailed data on the momentum dependency of characteristic times $\\tau _{\\bf k}$ would make it possible to improve the quality of model (REF ).", "Using numerical calculation with the method reported in Chapter , we can obtain the values of relaxation times for the complete range of momenta in the system.", "In this Chapter, we present the calculation details and the results for relaxation times of weakly interacting Fermi and Bose gases on a finite momentum lattice.", "Figure: Relaxationof nonequilibrium occupation Δn 𝐤 (t) \\Delta n_{\\bf k}(t)for Fermi gas on the lattice 8×8×88 \\times 8 \\times 8,Fermi energy F=2.0F=2.0,Fermi surface radius k F =π/2ak_F=\\pi /2a." ], [ "Calculation details.", "In order to determine the momentum dependency of relaxation time $\\tau _{\\bf k}$ , the evolution of the equilibrium system excited at the corresponding momenta was simulated.", "The excitation at the given momentum ${\\bf k}$ was done by increasing the occupation number $n_{\\bf k}$ by a small value $ \\Delta n_{\\bf k}(t=0) \\sim 10^{-3}$ .", "The typical time dependency of nonequilibrium occupations $\\Delta n_{\\bf k}(t)$ demonstrating the exponential-like decay is shown in Figure REF .", "The dependency $\\Delta n_{\\bf k}(t) = A e^{-t/\\tau _{\\bf k}}$ allows to use the relation for the relaxation time $\\tau _{\\bf k} = -\\frac{d n_{\\bf k}/dt}{d^2 n_{\\bf k}/dt^2},$ where the derivatives are calculated using a finite-difference scheme.", "The scale of relaxation time is determined by energy units.", "In this Chapter, we put $U_0=1$ ." ], [ "Relaxation time in Fermi gas.", "First we calculate the relaxation times $\\tau _{\\bf k}$ in the weakly interacting Fermi gas where analytical result (REF ) exists, so that the verification of the simulation method can be made.", "The Hamiltonian of the system is: $\\hat{H} =\\sum \\limits _{{\\bf k}\\sigma }\\varepsilon _{\\bf k} \\hat{n}_{{\\bf k}\\sigma }+ U_0\\sum \\limits _{ \\bf k p q }{ \\hat{a}^{\\dagger }_{{\\bf k}\\uparrow }\\hat{a}^{\\dagger }_{{\\bf p}\\downarrow }\\hat{a}_{{\\bf p}+{\\bf q}\\downarrow } \\hat{a}_{{\\bf k}-{\\bf q}\\uparrow } },$ where $ \\varepsilon _{\\bf k} = \\frac{\\varepsilon _1}{\\Delta k^2} {\\bf k}^2$ , $\\Delta k=\\frac{2 \\pi }{La}$ is the discreteness of the momentum in the Brillouin zone, determined by the size of the crystal.", "Figure: Relaxation timeτ 𝐤 \\tau _{\\bf k} calculated forthe Fermi gas on the lattice 32×32×3232 \\times 32 \\times 32,F=2.0F=2.0,k F =π/2ak_F=\\pi /2a, k B T=0.7k_B T = 0.7,as a function of momentum 𝐤=(k x ,k y ,0){\\bf k}=(k_x, k_y, 0).Figure: Relaxation timeas a function of momentum 𝐤=(k x ,0,0){\\bf k}=(k_x, 0, 0),calculated forthe Fermi gas on the lattice 32×32×3232 \\times 32 \\times 32,F=2.0F=2.0,k F =π/2ak_F=\\pi /2a,for several temperatures.Lines are to guide the eye.Figure: The calculated relaxation timecompared with the Fermi-liquid theory, Eq.", "().The parameters of the system are the same as for Figure ,temperature k B T=0.2k_B T=0.2.In the Figures REF , REF the calculated relaxation time is shown as a function of momentum for the Fermi system at several temperatures $k_B T=0.2$ , $0.3$ , $0.5$ , $0.7$ , $1.0$ .", "We see the typical Fermi-liquid behaviour reaching maximum values near the Fermi surface and proportional to $T^{-2}$ .", "To demonstrate the overall agreement with the theory, the values calculated for temperature $k_B T = 0.2$ are shown in the Figure REF alongwith the analytical dependence (REF ).", "We can conclude that the application of the reported method allows to calculate the momentum dependency of the relaxation time in the weakly interacting quantum gases." ], [ "Relaxation time in Bose gas.", "Next we apply this approach to the weakly interacting Bose gas.", "The Hamiltonian is $\\hat{H} =\\sum \\limits _{{\\bf k}}\\varepsilon _{\\bf k} \\hat{n}_{\\bf k}+ U_0\\sum \\limits _{ \\bf k p q }{ \\hat{a}^{\\dagger }_{{\\bf k}}\\hat{a}^{\\dagger }_{{\\bf p}}\\hat{a}_{{\\bf p}+{\\bf q}} \\hat{a}_{{\\bf k}-{\\bf q}} }.$ Figure: Relaxation timeas a function of momentum 𝐤=(k x ,0,0){\\bf k}=(k_x, 0, 0),calculated forthe Bose gas on the lattice 32×32×3232 \\times 32 \\times 32,with total particle number N=10 3 N=10^3,for several temperaturesk B T=0.2k_B T=0.2, 0.30.3, 0.50.5, 0.70.7, 1.01.0.Figure: Relaxation timein Bose gasfor momentum 𝐤=0{\\bf k}=0as a function of temperature,calculated for various lattice sizes LL=32 (a), 16 (b), 48 (c),with the particle density large enough to show the BEC transition(total particle number NN=1000, 125, 3375, correspondingly).For reference, the occupation N 0 (T)N_0(T) is plotted.The calculated relaxation times as a function of momentum for Bose system with the total particle number N=10$^3$ is shown in Figure REF .", "We see that the relaxation becomes slower as the energy $\\varepsilon _{\\bf k}$ decreases.", "The temperature dependency of the relaxation time $\\tau _0$ at the central momentum ${\\bf k}=0$ is given in Figure REF (a).", "The parameters of the problem were chosen such that the Bose-Einsten condensation occurs at high enough temperature ($T_c$ =0.2236).", "The dependence $\\tau _0(T)$ shows the pronounced peak near the transition point.", "At low temperatures the second feature is visible.", "We associate it to the increasing difficulty of the energy transfer to the discrete spectrum caused by the finite momentum lattice.", "To reveal the effect of lattice size, we performed the calculations for different sizes $L$ =16, 48 with the same particle density ($N$ =125, 3375, correspondingly).", "The functions $\\tau _0(T)$ are shown in Figures REF (b) and (c).", "As we see, the low temperature feature in relaxation time becomes relatively weaker with increasing system size, while the qualitative behaviour at large temperatures $T \\gtrsim T_c$ remains almost the same with minor change of the peak sharpness as the phase transition becomes narrower." ], [ "Conclusion", "We presented a universal and efficient method for numerical simulation of kinetics of weakly interacting quantum systems on a finite momentum lattice, using the original transformation to improve the efficiency and to increase the affordable system sizes.", "This approach can be applied to the wide range of models of various statistics with arbitrary single-particle spectrum $\\varepsilon _{\\bf q}$ and two-particle interaction obeying the momentum conservation.", "The system size can be as large as $\\sim 50^3$ , which can help to obtain properties of continual systems using appropriate extrapolation.", "As a demonstration, the momentum dependence of relaxation time was calculated for the weakly interacting Fermi gas on the momentum lattice 32$\\times $ 32$\\times $ 32, in overall agreement with Fermi-liquid theory predictions (Figure REF ).", "Similar calculation for weakly interacting Bose gas shows the temperature dependence with features near to the BEC transition point and at low temperatures due to discrete energy spectrum (Figure REF ).", "The numerical method reported in this article can be applied to various time-dependent problems, such as relaxation processes in superconductors, behaviour of nonequilibrium carriers in semiconductors [21] and metals [30], kinetics of atomic gases in magneto-optical traps [31], etc.", "The work was supported by the Ministry of Science and Higher Education of Russian Federation (state assignment project No.", "0723-2020-0036)." ], [ "Appendix: Transformation", "In this Appendix, we show how to convert equations (REF ), (REF ) to (REF ), (REF ), (REF ) in the case of Bose statistics.", "First, we expand the brackets in the equation (REF ): $\\frac{dn_{{\\bf k}_1}}{dt}=U_0^2\\sum \\limits _{ {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 }\\left\\lbrace [ n_{\\bf 2} n_{\\bf 3} n_{\\bf 4}+ n_{\\bf 3} n_{\\bf 4}+ n_{\\bf 1} (n_{\\bf 3} n_{\\bf 4}- n_{\\bf 2} n_{\\bf 3}- n_{\\bf 2} n_{\\bf 4}- n_{\\bf 2}]\\right.\\\\\\nonumber - \\delta _{\\bf 34} [(n_{\\bf 2} n_{\\bf 3} + n_{\\bf 3}) + n_{\\bf 1} (2 n_{\\bf 2} n_{\\bf 3} + n_{\\bf 2} + n_{\\bf 3})]\\\\\\nonumber + \\delta _{\\bf 12} [n_{\\bf 3} n_{\\bf 4} + n_{\\bf 1} (2 n_{\\bf 3} n_{\\bf 4} + n_{\\bf 3} + n_{\\bf 4} + 1)]\\\\\\nonumber \\left.+ \\delta _{\\bf 12} \\delta _{\\bf 34} (n_{\\bf 1} - n_{\\bf 3})\\right\\rbrace \\delta _{\\varepsilon _1+\\varepsilon _2, \\varepsilon _3+\\varepsilon _4}\\delta _{{\\bf 1}+{\\bf 2},{\\bf 3}+{\\bf 4}}$ Here we grouped the terms where the particle momenta coincide in the initial or final states ($\\delta _{\\bf 12}$ and $\\delta _{\\bf 34}$ , correspondingly)." ], [ "The first line", "in the expression (REF ) can be conveniently written as: $\\left(\\frac{dn_{{\\bf k}_1}}{dt}\\right)^{(1)}=U_0^2\\left[(A_{{\\bf k}_1} + B_{{\\bf k}_1})+n_{{\\bf k}_1}(B_{{\\bf k}_1} - 2C_{{\\bf k}_1} - D_{{\\bf k}_1})\\right],$ where $A_{{\\bf k}_1} \\equiv \\sum \\limits _{ {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 }n_{{\\bf k}_2} n_{{\\bf k}_3} n_{{\\bf k}_4}\\delta _{\\varepsilon _1+\\varepsilon _2, \\varepsilon _3+\\varepsilon _4}\\delta _{{{\\bf k}_1}+{{\\bf k}_2},{{\\bf k}_3}+{{\\bf k}_4}}, \\\\B_{{\\bf k}_1} \\equiv \\sum \\limits _{ {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 }n_{{\\bf k}_3} n_{{\\bf k}_4}\\delta _{\\varepsilon _1+\\varepsilon _2, \\varepsilon _3+\\varepsilon _4}\\delta _{{{\\bf k}_1}+{{\\bf k}_2},{{\\bf k}_3}+{{\\bf k}_4}},\\\\C_{{\\bf k}_1} \\equiv \\sum \\limits _{ {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 }n_{{\\bf k}_2} n_{{\\bf k}_3}\\delta _{\\varepsilon _1+\\varepsilon _2, \\varepsilon _3+\\varepsilon _4}\\delta _{{{\\bf k}_1}+{{\\bf k}_2},{{\\bf k}_3}+{{\\bf k}_4}},\\\\D_{{\\bf k}_1} \\equiv \\sum \\limits _{ {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 }n_{{\\bf k}_2}\\delta _{\\varepsilon _1+\\varepsilon _2, \\varepsilon _3+\\varepsilon _4}\\delta _{{{\\bf k}_1}+{{\\bf k}_2},{{\\bf k}_3}+{{\\bf k}_4}}.$ We rewrite the expression (REF ) in the expanded space $({\\bf k}, \\varepsilon ) \\equiv {\\bf \\rho }$ , employ the notation (REF )-(), and replace Kronecker delta symbols with sums: $A_{{\\bf k}_1 \\varepsilon _1} =\\frac{1}{L^3 N_\\varepsilon } \\sum \\limits _{\\begin{array}{c} {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 \\\\\\varepsilon _2 \\varepsilon _3 \\varepsilon _4\\end{array}}n_{{\\bf k}_2 \\varepsilon _2} n_{{\\bf k}_3 \\varepsilon _3} n_{{\\bf k}_4 \\varepsilon _4}\\\\ \\nonumber \\times \\sum \\limits _{{\\bf r}\\gamma }e^{i ({{\\bf k}_1}+{{\\bf k}_2}-{{\\bf k}_3}-{{\\bf k}_4}) {\\bf r} } \\,e^{i (\\varepsilon _1 + \\varepsilon _2 - \\varepsilon _3 - \\varepsilon _4) \\gamma }$ Using the relation $n_{{\\bf k} \\varepsilon } = \\sum \\limits _{{\\bf r}\\gamma } N_{{\\bf r} \\gamma }e^{i ({\\bf k}{\\bf r} + \\varepsilon \\gamma ) }$ , we can write: $A_{{\\bf k}_1 \\varepsilon _1} =\\frac{1}{L^3 N_\\varepsilon }\\sum \\limits _{ \\begin{array}{c} {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 \\\\\\varepsilon _2 \\varepsilon _3 \\varepsilon _4\\end{array}}\\sum \\limits _{ \\begin{array}{c} {\\bf r}_2 {\\bf r}_3 {\\bf r}_4 \\\\\\gamma _2 \\gamma _3 \\gamma _4\\end{array}}N_{{\\bf r}_2 \\gamma _2} N_{{\\bf r}_3 \\gamma _3} N_{{\\bf r}_4 \\gamma _4}\\\\ \\nonumber \\times \\sum \\limits _{{\\bf r}\\gamma }e^{i ({\\bf k}_2 {\\bf r}_2 +{\\bf k}_3 {\\bf r}_3+{\\bf k}_4 {\\bf r}_4)}e^{i (\\varepsilon _2 \\gamma _2 + \\varepsilon _3 \\gamma _3 + \\varepsilon _4 \\gamma _4) }\\\\ \\nonumber \\times e^{i ({{\\bf k}_1}+{{\\bf k}_2}-{{\\bf k}_3}-{{\\bf k}_4}) {\\bf r} }e^{i (\\varepsilon _1 + \\varepsilon _2 - \\varepsilon _3 - \\varepsilon _4) \\gamma }\\\\ \\nonumber = \\frac{1}{L^3 N_\\varepsilon }\\sum \\limits _{{\\bf r}\\gamma }N_{-{\\bf r}, -\\gamma } N_{{\\bf r}\\gamma } N_{{\\bf r}\\gamma }e^{i ({\\bf k}_1 {\\bf r} + \\varepsilon _1 \\gamma ) },$ which takes in the form of the Fourier transform.", "The functions $B$ , $C$ , $D$ can be converted in the same way.", "As a result, we obtain: $A_{{\\bf r} \\gamma } = N_{-{\\bf r}, -\\gamma } (N_{{\\bf r} \\gamma })^2,\\\\B_{{\\bf r} \\gamma } = S_{{-{\\bf r}, -\\gamma }} (N_{{\\bf r} \\gamma })^2,\\\\C_{{\\bf r} \\gamma } = N_{{-{\\bf r}, -\\gamma }} N_{{\\bf r} \\gamma } S_{{\\bf r} \\gamma }, \\\\D_{{\\bf r} \\gamma } = N_{{-{\\bf r}, -\\gamma }} (S_{{\\bf r} \\gamma })^2,$ where $S_{{\\bf r} \\gamma }$ is the Fourier transform of the previously introduced function $s_{{\\bf k} \\varepsilon } \\equiv \\delta _{\\varepsilon ,\\varepsilon _{\\bf k}}$ .", "The function $A_{{\\bf r} \\gamma }$ , $\\dots $ , $D_{{\\bf r} \\gamma }$ are turned into $A_{{\\bf k} \\varepsilon }$ , $\\dots $ , $D_{{\\bf k} \\varepsilon }$ using the inverse Fourier transform, after that the desired values $A_{\\bf k}$ , $\\dots $ , $D_{\\bf k}$ for the expression (REF ) can be obtained with $\\varepsilon =\\varepsilon _{\\bf k}$ ." ], [ "The second line", "in the expression (REF ) corresponds to the case when ${\\bf k}_3={\\bf k}_4$ .", "It can be written as: $\\left(\\frac{dn_{{\\bf k}_1}}{dt}\\right)^{(2)}= - U_0^2\\left[( E_{{\\bf k}_1} + F_{{\\bf k}_1} )+n_{{\\bf k}_1}(2 E_{{\\bf k}_1} + F_{{\\bf k}_1} + G_{{\\bf k}_1})\\right],$ where $E_{{\\bf k}_1} \\equiv \\sum \\limits _{ {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 }n_{{\\bf k}_2} n_{{\\bf k}_3}\\delta _{{{\\bf k}_3},{{\\bf k}_4}}\\delta _{\\varepsilon _1+\\varepsilon _2, \\varepsilon _3+\\varepsilon _4}\\delta _{{{\\bf k}_1}+{{\\bf k}_2},{{\\bf k}_3}+{{\\bf k}_4}}, \\\\F_{{\\bf k}_1} \\equiv \\sum \\limits _{ {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 }n_{{\\bf k}_3}\\delta _{{{\\bf k}_3},{{\\bf k}_4}}\\delta _{\\varepsilon _1+\\varepsilon _2, \\varepsilon _3+\\varepsilon _4}\\delta _{{{\\bf k}_1}+{{\\bf k}_2},{{\\bf k}_3}+{{\\bf k}_4}},\\\\G_{{\\bf k}_1} \\equiv \\sum \\limits _{ {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 }n_{{\\bf k}_2}\\delta _{{{\\bf k}_3},{{\\bf k}_4}}\\delta _{\\varepsilon _1+\\varepsilon _2, \\varepsilon _3+\\varepsilon _4}\\delta _{{{\\bf k}_1}+{{\\bf k}_2},{{\\bf k}_3}+{{\\bf k}_4}}.$ A reasoning similar to the previous paragraph gives the expressions: $E_{{\\bf r} \\gamma } = N_{-{\\bf r}, -\\gamma } N_{2{\\bf r} 2\\gamma },\\\\F_{{\\bf r} \\gamma } = S_{{-{\\bf r}, -\\gamma }} N_{2{\\bf r} 2\\gamma },\\\\G_{{\\bf r} \\gamma } = N_{{-{\\bf r}, -\\gamma }} S_{2 {\\bf r} 2\\gamma }.$" ], [ "The third line", "in the expression (REF ) corresponds to the case when ${\\bf k}_1={\\bf k}_2$ .", "Taking into consideration the same role of ${\\bf k}_3$ and ${\\bf k}_4$ in the sums, this line can be written as: $\\left(\\frac{dn_{{\\bf k}_1}}{dt}\\right)^{(3)}= U_0^2\\left[H_{{\\bf k}_1}+n_{{\\bf k}_1}(2 H_{{\\bf k}_1} + 2 I_{{\\bf k}_1} + J_{{\\bf k}_1})\\right],$ where $H_{{\\bf k}_1} \\equiv \\sum \\limits _{ {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 }n_{{\\bf k}_3} n_{{\\bf k}_4}\\delta _{{{\\bf k}_1},{{\\bf k}_2}}\\delta _{\\varepsilon _1+\\varepsilon _2, \\varepsilon _3+\\varepsilon _4}\\delta _{{{\\bf k}_1}+{{\\bf k}_2},{{\\bf k}_3}+{{\\bf k}_4}}, \\\\I_{{\\bf k}_1} \\equiv \\sum \\limits _{ {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 }n_{{\\bf k}_3}\\delta _{{{\\bf k}_1},{{\\bf k}_2}}\\delta _{\\varepsilon _1+\\varepsilon _2, \\varepsilon _3+\\varepsilon _4}\\delta _{{{\\bf k}_1}+{{\\bf k}_2},{{\\bf k}_3}+{{\\bf k}_4}}, \\\\J_{{\\bf k}_1} \\equiv \\sum \\limits _{ {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 }\\delta _{{{\\bf k}_1},{{\\bf k}_2}}\\delta _{\\varepsilon _1+\\varepsilon _2, \\varepsilon _3+\\varepsilon _4}\\delta _{{{\\bf k}_1}+{{\\bf k}_2},{{\\bf k}_3}+{{\\bf k}_4}}.$ Repeating the transformation made in (REF ), we convert the equation (REF ) to the form: $H_{{\\bf k}_1 \\varepsilon _1} =\\frac{1}{L^3 N_\\varepsilon }\\sum \\limits _{{\\bf r}\\gamma }N_{{\\bf r} \\gamma } N_{{\\bf r}\\gamma } e^{i (2 {\\bf k}_1 {\\bf r} + 2 \\varepsilon _1 \\gamma ) }= \\tilde{H}_{2{\\bf k}_1, 2\\varepsilon _1}$ i.e.", "the required values of $H_{{\\bf k}_1 \\varepsilon _1}$ are calculated using the Fourier transform of the auxiliary function $\\tilde{H}_{{\\bf r} \\gamma } = ( N_{{\\bf r} \\gamma } )^2.$ The auxiliary functions for variables $I_{{\\bf k}_1 \\varepsilon _1}$ and $J_{{\\bf k}_1 \\varepsilon _1}$ are obtained similarly: $\\tilde{I}_{{\\bf r} \\gamma } = N_{{\\bf r} \\gamma } S_{{\\bf r} \\gamma },\\\\\\tilde{J}_{{\\bf r} \\gamma } = ( S_{{\\bf r} \\gamma } )^2.$" ], [ "The fourth line", "in the expression (REF ) corresponds to the case when ${\\bf k}_1={\\bf k}_2$ and ${\\bf k}_3={\\bf k}_4$ .", "It can be written as: $\\left(\\frac{dn_{{\\bf k}_1}}{dt}\\right)^{(4)}= U_0^2\\left[- X_{{\\bf k}_1}+n_{{\\bf k}_1} Y_{{\\bf k}_1}\\right],$ where $X_{{\\bf k}_1} \\equiv \\sum \\limits _{ {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 }n_{{\\bf k}_3}\\delta _{{{\\bf k}_1},{{\\bf k}_2}}\\delta _{{{\\bf k}_3},{{\\bf k}_4}}\\delta _{\\varepsilon _1+\\varepsilon _2, \\varepsilon _3+\\varepsilon _4}\\delta _{{{\\bf k}_1}+{{\\bf k}_2},{{\\bf k}_3}+{{\\bf k}_4}}, \\\\Y_{{\\bf k}_1} \\equiv \\sum \\limits _{ {\\bf k}_2 {\\bf k}_3 {\\bf k}_4 }\\delta _{{{\\bf k}_1},{{\\bf k}_2}}\\delta _{{{\\bf k}_3},{{\\bf k}_4}}\\delta _{\\varepsilon _1+\\varepsilon _2, \\varepsilon _3+\\varepsilon _4}\\delta _{{{\\bf k}_1}+{{\\bf k}_2},{{\\bf k}_3}+{{\\bf k}_4}}.$ Taking into consideration the equalities ${\\bf k}_1={\\bf k}_2$ , ${\\bf k}_3={\\bf k}_4$ , the relation ${\\bf k}_1+{\\bf k}_2={\\bf k}_3+{\\bf k}_4$ can be rewritten as $2{\\bf k}_1 = 2{\\bf k}_3 + {\\mathbb {G}}$ .", "It means that the momenta ${\\bf k}_1$ and ${\\bf k}_3$ are either equal or differ by a half of the unit vector of the reciprocal lattice.", "To obtain all different vectors ${\\bf k}_3$ for a given ${\\bf k}_1$ , we should use all half-vectors of reciprocal lattice (including zero vector) contained in the first Brillouin zone.", "We denote them as ${\\mathbb {G}}/2$ .", "In this case, we get the expressions similar to (REF ) with extra factor $e^{i ({\\bf k}_1 - {\\bf k}_3) {\\bf r}} = e^{i {\\bf r \\mathbb {G}}/2}$ : $X_{{\\bf k}_1 \\varepsilon _1} =\\frac{1}{L^3 N_\\varepsilon }\\sum \\limits _{{\\bf r}\\gamma }e^{i ( {{\\bf k}_1} {\\bf r} + \\varepsilon _1 \\gamma ) }N_{{\\bf r} \\gamma }\\sum \\limits _{{\\bf \\mathbb {G}}/2}e^{i {\\bf r} {\\bf \\mathbb {G}}/2 }, \\\\Y_{{\\bf k}_1 \\varepsilon _1} =\\frac{1}{L^3 N_\\varepsilon }\\sum \\limits _{{\\bf r}\\gamma }e^{i ( {{\\bf k}_1} {\\bf r} + \\varepsilon _1 \\gamma ) }S_{{\\bf r} \\gamma }\\sum \\limits _{{\\bf \\mathbb {G}}/2}e^{i {\\bf r} {\\bf \\mathbb {G}}/2 }.$ The factor $\\sum \\limits _{{\\bf \\mathbb {G}/2}} e^{i {\\bf r} {\\bf \\mathbb {G}}/2 }\\equiv \\mathbb {Z}_{\\bf r}$ contains $2^d$ terms (where $d$ is the dimension of space) and is equal to either 0 or $2^d$ , depending on the components of the vector ${\\bf r}$ .", "As a result, we obtain: $\\tilde{X}_{{\\bf r} \\gamma } = N_{{\\bf r} \\gamma } {\\mathbb {Z}}_{\\bf r},\\\\\\tilde{Y}_{{\\bf r} \\gamma } = S_{{\\bf r} \\gamma } {\\mathbb {Z}}_{\\bf r}.$ The relations for Fermi statistics are derived using the same reasoning.", "Combining all the terms $A_{{\\bf r} \\gamma }$ , $\\dots $ , $Y_{{\\bf r} \\gamma }$ , we get the expressions (REF ), (REF )." ] ]	2011.14173
[ [ "Strongly Lensed Supernovae in Well-Studied Galaxy Clusters with the Vera\n C. Rubin Observatory" ], [ "Abstract Strong lensing by galaxy clusters can be used to significantly expand the survey reach, thus allowing observation of magnified high-redshift supernovae that otherwise would remain undetected.", "Strong lensing can also provide multiple images of the galaxies that lie behind the clusters.", "Detection of strongly lensed Type Ia supernovae (SNe Ia) is especially useful because of their standardizable brightness, as they can be used to improve either cluster lensing models or independent measurements of cosmological parameters.", "The cosmological parameter, the Hubble constant, is of particular interest given the discrepancy regarding its value from measurements with different approaches.", "Here, we explore the feasibility of the Vera C. Rubin Observatory Legacy Survey of Space and Time (LSST) of detecting strongly lensed SNe in the field of five galaxy clusters (Abell 1689 and Hubble Frontier Fields clusters) that have well-studied lensing models.", "Considering the 88 systems composed of 268 individual multiple images in the five cluster fields, we find that the LSST will be sensitive to SNe~Ia (SNe~IIP) exploding in 41 (23) galaxy images.", "The range of redshift of these galaxies is between $1.01 < z < 3.05$.", "During its 10 years of operation, LSST is expected to detect $0.2\\pm0.1$ SN~Ia and $0.9\\pm0.3$ core collapse SNe.", "However, as LSST will observe many more massive galaxy clusters, it is likely that the expectations are higher.", "We stress the importance of having an additional observing program for photometric and spectroscopic follow-up of the strongly lensed SNe detected by LSST." ], [ "Introduction", "Supernovae (SNe) have proved to be invaluable tools for various astrophysical and cosmological applications.", "The strong gravitational lensing effect is another powerful tool; it occurs when a foreground mass distribution is located along the line of sight to a background source.", "The observer can see multiple images of the background source appearing around the foreground lens (see, e.g., in [1] for a review).", "As the images have taken different paths through space before reaching us, the time differences between the images are sensitive to the expansion rate of the universe, parametrized by the Hubble constant, H$_0$ [2].", "These time delays are sensitive to other cosmological parameters, albeit to a much lesser extent [3].", "Using time delay of strongly lensed variable sources for measuring H$_0$ is also known as the Refsdal method as it was suggested by the author more than 50 years ago [4].", "The time delays depend also on the lensing potential; therefore, strongly lensed SNe can be used for both cosmological and cluster lens studies [5], [6], [7], [8].", "In recent years, a discrepancy at least at the $4.4\\sigma $ level is emerging between the value of H$_0$ inferred from the indirect measurement from the cosmic microwave background [9] and the value derived from the local distance scale composed of classical Cepheid variable stars and SNe Type Ia (SNe Ia) [10], [11], [12].", "The Refsdal method provides a third independent alternative to measure H$_0$, and as strongly lensed SNe are rare events, it has been used on quasars (see, e.g., in [13], [3]).", "The H0LiCOW collaboration measured H$_0$ to 2.4% accuracy, combining six lensed quasars, each with 6–10% precision [14].", "However, quasar light curves are stochastic and unpredictable; therefore, they require long observations spanning over years and even more than a decade (see, e.g., in [13]).", "Relative to quasars, lightcurves of SNe Ia form a homogeneous group (see, e.g., in [15], [16]).", "Knowing the absolute brightness of SNe Ia allows to estimate the absolute magnification of SNe Ia, and therefore to break the so-called mass-sheet degeneracy of gravitational lenses [17].", "Thus, they could be used to put constraints on the lensing potential, if a cosmological model is assumed (see, e.g., in [18], [19], [20]).", "Systematic searches for SNe in background galaxies behind clusters have been performed with both space- and ground-based instruments [21], [22], [19], [20], [23], [24].", "Very large Hubble Space telescope (HST) programs with more than thousand orbits have been targeting galaxy clusters, to improve cluster lensing models and search for SNe [25], [26].", "Remarkably, they resulted in the discovery of the first multiply-imaged SN (dubbed “SN Refsdal”) behind a galaxy cluster [27], [28] and it was most probably a core-collapse (CC) type of explosion [28].", "Several teams predicted the reappearance of SN Refsdal almost a year later, which allowed testing of their lens models [29], [30].", "By measuring the time delays of SN Refsdal and having a high-quality strong lensing model of the galaxy cluster, it was shown that it is possible to measure H$_0$ with 6% total uncertainty [31].", "Dedicated ground-based searches for lensed SNe behind galaxy clusters have been performed using near-infrared instruments at the Very Large Telescope [32], [22], [23], [24].", "Most notably, they reported the discovery of one of the most distant CC SN ever found, at redshift $z = 1.703$ with a lensing magnification factor of $4.3 \\pm 0.3$ [33].", "Furthermore, thanks to the power of the lensing cluster, it was possible to estimate the volumetric CC SN rates for $0.4\\le z< 2.9$ and compare it with the predictions from cosmic star formation history [23].", "The first resolved multiply-imaged SN Ia, iPTF16geu, with four resolved images arranged symmetrically around the lens galaxy, was magnified by a galaxy lens [34].", "Before the discovery of iPTF16geu, only one other strongly lensed SN Ia by galaxy lens was known, PS1-10afx, but only few years later it was realized that it was a SN Ia magnified $\\sim 30$ times by the foreground galaxy [35].", "PS1-10afx most probably had multiple images, which could not be resolved with the available data [36], [35].", "The multiple images of iPTF16geu were well resolved by HST and ground-based images with adaptive optics, but the estimated time delays were about half a day, which is difficult to measure with SNe Ia light curves [37].", "Cluster lensing timescales are much longer, typically at the order of months and years [7], [23], [24], [38].", "Therefore, the microlensing effects, i.e., the stellar-scale lensing which is important at micro-arcsecond scales and causes fluctuations on timescales of weeks to months [39], become subdominant.", "These two considerations could make cluster lens time delays measurement potentially more feasible, in the cases where the lens potential is well studied, so the predicted time delays have small uncertainties [8].", "In this work, we focus on prospects of observing SNe which are strongly lensed by well-studied galaxy clusters with the upcoming Vera C. Rubin Observatory (Rubin Observatory, hereafter).", "The Rubin Observatory will scan the transient sky through its Legacy Survey for Space and Time (LSST) with a 8.4 m telescope, targeting at least 18,000 deg$^2$ of the southern hemisphere with a field of view of 9.6 deg$^2$ [40].", "Its duration is set to at least ten years and it will use set of six broad photometric bands, $ugrizy$ .", "The LSST observing strategy has not been finalized [40], which means that distribution of observations to a field within a year and the distribution between filters have not been decided.", "Choosing an observing strategy is an important and challenging task as the main science goals of the LSST are quite broad, ranging from constraining dark energy to exploring the Solar System, the Milky Way, and the transient optical sky.", "There will be a main “wide-fast-deep” (WFD) survey which will consume $\\sim 80\\%$ of the whole survey time, covering the equatorial declination range $-62< \\delta < +2 $ , and excluding the central portion of the Galactic plane.", "In the WFD survey, two visits in either the same or different filters are acquired each night, to allow identification of moving objects and rapidly varying transients.", "These pairs of visits are repeated every three to four nights throughout the period the field is visible in each year.", "Currently, the official LSST observation is a pair of 15 s exposures, which helps to reject cosmic rays, but there is a possibility that it could be switched a single 30 s exposure because of a potential gain of 7% efficiency and improved image quality [41].", "Here, we consider the baseline observing strategyhttps://cadence-hackathon.readthedocs.io/en/)which uses 2 $\\times $ 15 s exposures, simulated with the LSST Operations Simulator (OpSim) [40], based on a model of the observatory (including telescope) and historical data of observational conditions [42].", "We use the latest baseline OpSim simulation released in August 2020 http://astro-lsst-01.astro.washington.edu:8080/?runId=16." ], [ "Estimating the Expected Number of Strongly Lensed Supernovae Observable by the Rubin Observatory", "We consider the six galaxy clusters from the Hubble Frontier Fields (HFF) [26] program with the HST.", "These are Abell 2744, MACS J0416.1-2403, MACS J0717.5+3745, MACS J1149.5+2223, Abell S1063, and Abell 370.", "We add to this list the well-studied galaxy cluster Abell 1689.", "The reason for this choice is the existence of reliable lensing models for these clusters, thus well constrained magnification maps, based on good quality data, such as deep HST images and Multi-Unit Spectroscopic Explorer (MUSE) spectroscopy.", "Using recently published lensing models of the HFF clusters [43], [44], [45], the predicted magnifications and time delays between the images were presented in [24], [38].", "In the same studies, by using multi-band HST photometry, the global properties of the multiply-imaged galaxies were inferred.", "These properties are the stellar mass, star formation rate (SFR), and SN rate, which depend on the luminosity of the galaxy corrected for the predicted magnification.", "This means that the inferred galaxy properties depend on the lensing model.", "From these seven galaxy clusters, two (MACS J1149.5+2223 and MACS J0717.5+3745) are not visible from the latitude of the Rubin Observatory.", "The final list is given in the Table REF together with number of the background galaxies, number of multiple images that these galaxies have, and their redshifts.", "An example of one galaxy cluster with the background galaxies and its 67 multiple images are shown in Figure REF .", "We only consider the multiply imaged galaxies that have a spectroscopic redshift.", "Considering the redshifts of the SNe from Table REF , the most important filters are at the longest wavelengths: $i$ , $z$ , and $y$ .", "The reason for this is because most of the light of nearby SNe is in the optical bands, so their light is redshifted to the longer wavelengths when higher redshifts are considered.", "An example of typical SN Ia and SN IIP at redshift that can be observed in the galaxy cluster fields considered here is given in Figure REF , together with the LSST filter response system.", "Most of the pairs of images that have time delays that are shorter than ten years (see column 3 of Table REF from the work in [24] and column 3 of Table REF from the work in [38]), which is the planned duration of LSST.", "In Table REF , we also include the number of visits to the clusters that the LSST will dedicate in ten years.", "In the baseline observing strategy considered here, there is no large difference between the number of visits per filter or per cluster because LSST is designed to observe at least 825 times each field over the duration of the WFD survey.", "However, the image depth differs between filters; it is the highest in the $i$ and the lowest in the $y$ band.", "The average $5\\sigma $ image depth of the selected cluster fields in $i$ band is $\\sim 23.4$ , $\\sim 22.7$ in $z$ , and $\\sim 22.0$ in $y$ band.", "Table: The galaxy clusters considered in this work.", "The number of unique galaxies behind the cluster is given in column 2, and the number of their multiple images of these galaxies in column 3.", "The redshift range of these galaxies is given in column 4.", "Columns 5–7 show the numbers of observations that the cluster field will have in the 10 years of the LSST, divided by filter izyizy.Figure: Galaxy cluster Abell 370 overplotted with the positions of all the 21 background galaxies with 67 multiple images with expected time delays presented in .", "The images that belong to the same system are shown with the same color.", "The predicted magnifications for these galaxies from the lensing model are approximately ∼\\sim 1–4 magnitudes.", "As an example, a zoomed-in view of multiply-imaged galaxies 1.3 and 1.1 is shown, which are magnified by 2.81±0.082.81\\pm 0.08 and 2.80±0.072.80\\pm 0.07 mag, respectively.", "Adopted from the work in .Figure: The filter response system that will be on the Rubin Observatory overplotted with a typical spectra of SN Ia and SN IIP.", "The redshift of the SN spectra is at 1.28, as the examples from Figure where multiply-imaged galaxies in Abell 370 are shown at which the survey will be sensitive, considering the image depth and the magnification from the galaxy cluster.We use the procedures outlined in [23] to calculate the sensitivity of LSST to detect strongly lensed SNe in the multiply lensed background galaxies of the five galaxy clusters.", "First, we estimate the control time for each galaxy image, $T_i$ , which indicates the amount of time the survey is sensitive of detecting a SN candidate [46].", "The control time above the detection threshold, is a function of the SN light curve, absolute intrinsic SN brightness, the image depth, dates of observation, filter, extinction by dust and the lensing magnification.", "The peak V band brightness and its one standard deviation is assumed as $19.3\\pm 0.3$ for SNe Ia, while for CC SNe we use the values compiled by the authors of [47] in their Table REF .", "The probability distribution of the absolute intrinsic brightness is assumed to be Gaussian.", "As in [23], [24], the values for the color excess were drawn from a positive Gaussian distribution with a mean $E(B-V)=0.15$ and $\\sigma _E=0.02$ justified by extinction studies of high-redshift galaxies behind Abell 1689 [48].", "The dates of observation, filter, and image depth are taken from the simulated survey strategy generated with OpSim.", "LSST can detect a SN in those galaxies that have positive control time.", "Few examples of simulated SN Ia light curves that could be observed with LSST are shown in Figure REF .", "The synthetic light curves in the observer filters for redshift $z$ were obtained by applying cross-filter k-corrections [49].", "Furthermore, to obtain the “observed” magnitude in the chosen LSST filter, the lensing magnification from the galaxy cluster is also taken into consideration.", "For example, the multiple images (labeled 5.1, 5.2, 5.3, and 5.4 in the last column in Figure REF ) of galaxy source 5 at redshift z $ =1.28$ , are magnified by $2.99\\pm 0.08$ , $4.09\\pm 0.09$ , $4.27\\pm 0.12$ , and $4.21\\pm 0.13$ mag, respectively (Table REF from the work in [24]).", "With this magnification, typical SNe are detectable with the LSST, even at this high redshift.", "Furthermore, the estimated time delays between 5.4 and the other three images is $94\\pm 4$ , $104\\pm 8$ , and $135\\pm 7$ days, respectively (Table REF from the work in [24]).", "The length of these time delays is within the duration of LSST, so if a SN explodes in this galaxy, there is a possibility that it will be detected in the other images during the survey, as the median revisit of the same field is 1 to 8 days, depending on the filter.", "However, there are gaps of observations up to 200–300 days, due to, for example, visibility issues.", "Therefore, it is not guaranteed that LSST will observe that specific field when the other SN images appear, and observations with another photometric and spectroscopic program should be scheduled when the reappearance is expected.", "Figure: Examples of simulated light curves of SNe Ia for pairs of images of strongly lensed galaxies behind Abell 370.", "The first column are in the LSST ii band, the second in the yy, while the third is in zz band.", "If a SN explodes in one of the galaxies, the explosion should appear in the corresponding images, thus allowing for scheduling observations accordingly.", "Without the magnification from the galaxy cluster, the SNe Ia exploding in these galaxies would hardly be observable with the Rubin Observatory.Table: Sensitivity of LSST to SNe Ia exploding in the the multiply-imaged galaxies in HFF galaxy clusters + Abell 1689 fields.", "Columns 2 (4) show the number of galaxy images in which LSST could detect a SN Ia (SN IIP) explosion.", "In columns 3 and 5 the largest redshift of these galaxies is reported .", "Table: Expectations for the number of lensed SNe in the multiply-imaged galaxies that could be detected behind the HFF galaxy clusters + Abell 1689 in the 10 years of the Rubin Observatory.", "The errors in the N Ia _{Ia} and N CC _{CC} originate from the propagated uncertainty in the SFR.", "Second, we calculate the expected number of SNe in each galaxy $N_i$ , by multiplying the SN rate $R_i$ and the control time $T_i$ , $N_i=R_i\\cdot T_i,$ where $i$ indicates the individual galaxy.", "As the control time is dependent on the SN type, it is computed separately for each SN type.", "The total CC control time is obtained by weighting the contributions for the CC SN types.", "We use the values for the relative CC SN fractions from [47], compiled in their Table 1.", "The expected SN Ia rate in a galaxy, $R_i$ , depends on the SFR and the stellar mass through an empirical relation [50], while the CC SN rate scales with the SFR.", "We used the SFR, stellar masses and SN rates estimates from the works in [24], [38] for the HFF clusters and from the work in [7] for Abell 1689.", "To obtain the total expected number of SNe Ia and CC SNe over all the systems, we sum the expectations over the individual galaxies." ], [ "Results", "First, considering the five well-studied galaxy clusters and their background strongly lensed galaxies with multiple images, we find that the LSST will be sensitive to SNe Ia exploding in 41 galaxies, while for the most common CC SNe, SNe IIP, the number is lower, 23.", "The main reason for this is that SNe IIP are simply fainter.", "The galaxies for which LSST is sensitive of detecting SNe have redshift between $0.73 < z < 1.95$ for SNe Ia, while for CC SNe $0.73 < z < 3.05$ .", "SNe IIP are visible to LSST at higher redshifts because they emit much more in the rest-frame UV, so at high redshift that region is shifted and it is still visible with the LSST filters at the longest wavelengths (see Figure REF ).", "As SNe Ia are not bright at rest-frame UV, they are practically not observable at $z\\gtrsim 2$ with an optical survey.", "The sensitivity of LSST for each galaxy cluster is shown in Table REF .", "Second, we show the estimated number of expected strongly lensed SNe in the selected galaxy cluster fields in 10 years of the Rubin Observatory, presented in Table REF .", "The observability of a SN in the multiply-imaged galaxies is sensitive to the SN properties such as light curve, absolute intrinsic SN brightness; galaxy properties such as redshift, star formation rate, extinction in the line of site and magnification of the galaxy, but also on the survey parameters such as image depth, cadence and filter.", "Considered the baseline observing strategy of the 10 years of operation, LSST is expected to detect $0.2\\pm 0.1$ SN Ia and $0.9\\pm 0.3$ CC SN in the five cluster fields.", "However, since we have considered only SNe in multiply-imaged galaxies with spectroscopic redshift of five galaxy clusters, this is a lower limit.", "As there are more galaxies with photometric redshift only, it is likely that the expectations are higher.", "We note that, as the LSST official observing strategy might be adjusted in the future, our forecast will also change." ], [ "Discussion", "The Rubin Observatory is not optimal for the discovery of strongly lensed SNe in galaxy cluster fields, given that it is a ground-based observatory with filters which do not extend to the near-infrared and infrared, where most of the light is emitted of SNe at redshifts considered here (see Figure REF ).", "The problem is especially accentuated for SNe Ia.", "Therefore, for this purpose, a space-based observatory with extended near-infrared and infrared filters such as the upcoming James Webb Space Telescope (JWST [51]) or the Nancy Grace Roman Space Telescope [52] would be more appropriate.", "By considering only four visits of one hour which can be taken in one year in the broad near-infrared filter F150W to the clusters considered here, the JWST/NIRCam instrument is expected to find $0.19\\pm 0.07$ SNe Ia [38].", "When CC SNe are considered, the expected number is much higher, $1.7\\pm 0.5$ .", "SN rates at high redshifts are dominated by CC SNe, so it is likely that the strongly lensed SNe by galaxy clusters will by dominated by the CC type [23], [38].", "Obtaining observing time for a dedicated multi-year search with JWST relatively small field of view of $2.2^{\\prime } \\times 2.2^{\\prime }$ will most-likely be competitive with high over-subscription rates.", "On the other hand, LSST observations of the galaxy cluster fields will be part of the nominal program and the data can be searched for strongly lensed SNe immediately.", "However, the Rubin Observatory will serve to detect the strongly lensed SNe, but it is better to have additional follow-up by other photometric and spectroscopic instruments for detailed study of the lensed SNe and their time delays.", "One way to improve LSST sensitivity to high-redshift SNe is to obtain better image depth in the reddest bands ($izy$ ) through co-adding images from visits closely separated in time.", "Improving the image depth means that the survey becomes sensitive to detecting SNe in more multiply-imaged galaxies.", "This can be done through a dedicated program which will focus on monitoring the galaxy cluster fields.", "A similar strategy is currently implemented by the Zwicky Transient Facility [53] for searching for strongly lensed SNe Ia by lens galaxies [54], [55], [56].", "As we noted in the previous section, the results presented in this work are lower limits, as we have only considered five galaxy clusters and the multiply-imaged galaxies in the field with spectroscopic redshift.", "It is expected that LSST will observe $\\sim 70$ galaxy clusters with Einstein radii larger than $\\theta _E >20 \"$ [57].", "More systems behind galaxy clusters will be measured with spectroscopic campaigns with instruments such as MUSE at the Very Large Telescope (see, e.g., in [58], [43], [45]), which not only will increase the number of multiply-imaged galaxies with spectroscopic redshifts in the clusters considered here, but will also allow to have more galaxy clusters with high quality lensing models.", "For the purpose of inferring H$_0$ with high precision, it would be most useful to measure time delays of SNe Ia, which have a well-established relationship between their optical peak luminosity and rate of brightness decline, which makes them excellent standardizable candles.", "However, detecting strongly lensed and magnified CC SNe, which are much more common at high redshifts [59], is also beneficial for several reasons.", "Despite the fact that SN Refsdal is a CC type of explosion, thanks to the high-quality strong lensing model of the galaxy cluster, it was shown that it is possible to measure H$_0$ with 6% total uncertainty [31].", "Furthermore, the discovery and the photometric and spectroscopic follow-up data that can be obtained could also help understand CC SNe at high redshifts.", "This would be interesting, as CC SNe are on average intrinsically fainter than SNe Ia and often embedded in dusty environments, making their studies at $z>0.4$ challenging (see, e.g., in [60]).", "Therefore, the magnification from galaxy clusters can provide an advance in redshift and can even allow the constraints of the CC SN rate at high redshifts [23].", "As both the area and the flux of background galaxies are magnified, it also allows for accurately studying the host environment, which would not be possible in unlensed scenario [33].", "Therefore, even if the LSST detects the more common case, strongly lensed CC SNe with significant magnification, but no multiple images, they can still be useful for the means of studying SNe and their host environment at unprecedented redshifts." ], [ "Conclusions", "As there is a current tension regarding the H$_0$ value, one of the priorities of the community is to provide an independent measure of H$_0$, which can be done by monitoring time delays of cluster lensed SNe.", "There is a further utility of magnified SNe by clusters.", "They can be used as test of galaxy cluster lens models, or in the cases of well-known lensing models, there is the opportunity to study SNe and their host environment in the early universe.", "In a previous work, we estimated the magnifications and the time delays of the multiply-imaged galaxies behind HFF clusters based on high-quality lens models by the authors of [43], [45], [44].", "We also inferred their properties such as SFRs, stellar masses, and SN rates by using very deep HST photometry.", "Here, we explored the feasibility of detecting strongly lensed SNe with the Rubin Observatory behind these galaxy clusters.", "We found that in the 10 years of LSST in the five galaxy cluster fields, the expectations are $0.2\\pm 0.1$ SN Ia and $0.9\\pm 0.3$ CC SN, though these are likely lower limits, as LSST will observe dozens more massive galaxy clusters.", "Given the redshift range of $0.73<z<5.75$ , the ground-based Rubin Observatory filter set is not optimal for detecting SNe in these galaxies.", "Next generation space telescopes will be much more appropriate for this task.", "Nevertheless, massive clusters as those considered in this study will work as precious gravitational telescopes providing concrete possibility for LSST to observe strongly SNe at high redshift.", "This research was funded by the Slovenian Research Agency (grants I0-0033, P1-0031, J1-8136 and Z1-1853).", "The author declares no conflict of interest.", "The following abbreviations are used in this manuscript: Table: NO_CAPTIONno References" ] ]	2011.14122
[ [ "Box and segment intersection graphs with large girth and chromatic\n number" ], [ "Abstract We prove that there are intersection graphs of axis-aligned boxes in $\\mathbb{R}^3$ and intersection graphs of straight lines in $\\mathbb{R}^3$ that have arbitrarily large girth and chromatic number." ], [ "Introduction", "Erdős [8] proved that there exist graphs with arbitrarily large girth and chromatic number.", "We prove that there exist such graphs that can be realised geometrically as intersection graphs of axis-aligned boxes in $\\mathbb {R}^3$ and of straight lines in $\\mathbb {R}^3$ .", "To do so, we make use of a surprising connection to arithmetic Ramsey theory.", "These are the first non-trivial examples of geometric intersection graphs with arbitrarily large girth and chromatic number.", "Theorem 1 There are intersection graphs of axis-aligned boxes in $\\mathbb {R}^3$ with arbitrarily large girth and chromatic number.", "Theorem 2 There are intersection graphs of straight lines in $\\mathbb {R}^3$ with arbitrarily large girth and chromatic number.", "In 1965, Burling [3] proved that there are axis-aligned boxes in $\\mathbb {R}^3$ whose intersection graphs are triangle-free and have arbitrarily large chromatic number.", "In light of Burling's construction, the problem of whether intersection graphs of axis-aligned boxes in $\\mathbb {R}^3$ with large girth have bounded chromatic number was raised by Kostochka and Perepelitsa [17].", "Later Kostochka [14] further speculated, for all positive integers $d$ , that intersection graphs of axis-aligned boxes in $\\mathbb {R}^d$ with girth at least 5 should have bounded chromatic number.", "Theorem REF extends Burling's result and shows that this is not the case even with any given larger girth in $\\mathbb {R}^3$ .", "We prove Theorem REF in the more restricted setting of “grounded square boxes” where the boxes intersect a given plane and their top and bottom faces are squares (see Theorem REF ).", "In the plane, the situation is different.", "In 1948, Bielecki [2] asked if triangle-free intersection graphs of axis-aligned rectangles have bounded chromatic number.", "This was answered positively by Asplund and Grünbaum [1], who proved the more general statement that axis-aligned rectangle intersection graphs with bounded clique number have bounded chromatic number, i.e., they are $\\chi $ -bounded.", "The best known $\\chi $ -bounding function for such graphs is $O(\\omega \\log \\omega )$ [4].", "The problem of whether intersection graphs of lines in $\\mathbb {R}^3$ are $\\chi $ -bounded had been raised explicitly by Pach, Tardos, and Tóth [23], but had been circulating in the community a few years prior.", "Norin [22] (see [24]) answered this in the negative by showing that double shift graphs are line intersection graphs.", "Since Norin's construction had only been communicated privately, we present it in Section before extending the result by proving Theorem REF .", "Intersection graphs of straight (infinite) lines are a subclass of the intersection graphs of straight line segments.", "Thus Theorem REF also applies to the class of intersection graphs of segments in $\\mathbb {R}^3$ .", "The significance of Burling's construction was highlighted when it was used much later to show that there are triangle-free intersection graphs of segments in the plane with arbitrarily large chromatic number [25].", "This disproved Scott's [30] conjecture that graphs not containing an induced subdivision of a given graph $H$ are $\\chi $ -bounded.", "In light of Theorem REF , segment intersection graphs are perhaps a more interesting setting for Burling graphs, indeed it is even conjectured [5] that Burling graphs might be the only obstruction to intersection graphs of segments in the plane being $\\chi $ -bounded.", "Kostochka and Nešetřil [15] proved that if the intersection graph of $n$ segments in the plane has girth at least 5, then the graph has only $O(n)$ edges, consequently such graphs have bounded chromatic number.", "Fox and Pach [11] asked if this result could be extended to segments in $\\mathbb {R}^3$ ; Theorem REF answers this question with an emphatic no.", "For $k\\ge 4$ , Chudnovsky, Scott, and Seymour [5] proved that intersection graphs of rectangles or segments in the plane with no induced cycle of length $k$ are $\\chi $ -bounded.", "Compared to this, Theorems REF and REF show that the situation in $\\mathbb {R}^3$ is much more dire, and that in a sense these graphs are much further away from being $\\chi $ -bounded.", "For more on $\\chi $ -boundedness, see the recent survey by Scott and Seymour [31].", "Tomon and Zakharov [34] recently showed that there exist bipartite box intersection graphs with girth at least 6 that have a super-linear number of edges.", "This resolved another problem of Kostochka [14].", "We extend this showing that even bipartite box intersection graphs with arbitrarily large girth can have a super-linear number of edges.", "Corollary 3 There are bipartite intersection graphs of axis-aligned boxes in $\\mathbb {R}^3$ with arbitrarily large girth and minimum degree.", "As one may suspect, it is possible to prove Corollary REF by modifying the construction we use to prove Theorem REF .", "We leave this as an exercise, and instead note that it can also be instantly deduced as a corollary of Theorem REF and a recent theorem of Kwan, Letzter, Sudakov, and Tran [19] that every triangle-free graph with minimum degree at least $c$ contains an induced bipartite subgraph of minimum degree at least $\\Omega (\\ln c / \\ln \\ln c)$ .", "We may also obtain an analogue of Corollary REF for lines in $\\mathbb {R}^3$ in the same way.", "One pleasing consequence of the connection to arithmetic Ramsey theory is that we may obtain Burling's classical result as an application of Van der Waerden's theorem [35] on arithmetic progressions.", "To prove Theorems REF and REF we require a certain strengthening of Van der Waerden's theorem which we introduce in the next section." ], [ "Preliminaries", "A homothetic map $f:\\mathbb {R}^d \\rightarrow \\mathbb {R}^d$ is one of the form $f(x)=sx+\\bar{c}$ for some $s\\in \\mathbb {R}_{>0}$ and $\\bar{c}\\in \\mathbb {R}^d$ .", "In other words, a homothetic map is a composition of uniform scaling and a translation.", "A set $T^{\\prime }\\subset \\mathbb {R}^d$ is a homothetic copy of a set $T\\subset \\mathbb {R}^d$ if there is a homothetic map $f$ such that $f(T)=T^{\\prime }$ .", "We say that a collection of distinct sets $T_1,\\dots , T_k$ form a cycle $C$ of length $k$ if there exist distinct elements $x_1,\\dots , x_k$ such that for all $i\\in \\lbrace 1,\\dots ,k-1\\rbrace $ we have $x_i \\in T_i \\cap T_{i+1}$ and $x_k \\in T_k \\cap T_1$ .", "A triangle-free intersection graph of a collection of sets $\\cal T$ has a cycle of length $k$ if and only if there is a collection of $k$ sets contained in $\\cal T$ that form a cycle of length $k$ .", "We often find it convenient to consider colourings and cycles of objects in $\\mathbb {R}^3$ directly rather than in their intersection graphs.", "We require a sparse version of Gallai's theorem [28] proven by Prömel and Voigt [26].", "The one dimensional version is sufficient for our purposes.", "Theorem 4 (Prömel and Voigt [26]) Let $T\\subset \\mathbb {R}$ be a finite subset containing at least three elements and let $g$ and $k$ be positive integers.", "Then there exists a finite set $X\\subset \\mathbb {R}$ such that every $k$ -colouring of $X$ contains a monochromatic homothetic copy $T^{\\prime }$ of $T$ and no set of at most $g-1$ homothetic copies of $T$ form a cycle.", "Prömel and Voigt [27] also proved sparse versions of other theorems including the Hales–Jewett theorem [13] and the Graham–Rothschild theorem [12].", "A special case of Theorem REF is also equivalent to a sparse Van der Waerden's theorem.", "The graphs we shall construct are based on a hypergraph variation of Tutte's [6], [7] construction of triangle-free graphs with arbitrarily large chromatic number that has been considered several times before [16], [33], [21].", "Although we need not make any mention of hypergraphs, their use is implicit in the application of Theorem REF , indeed this theorem essentially provides a very specialised hypergraph.", "A necessarily more relaxed variation of Tutte's construction was also used to show that there are axis-aligned boxes in $\\mathbb {R}^3$ with disjoint interiors whose intersection graph has arbitrarily large chromatic number [29].", "Magnant and Martin [20] further showed that there are such intersection graphs where the boxes only intersect on their top and bottom faces (and thus are triangle-free), and have arbitrarily large chromatic number.", "We remark that while Tutte's construction can be sparsified, these variations used for boxes with disjoint interiors cannot be; an intersection graph of $n$ axis-aligned boxes in $\\mathbb {R}^3$ with disjoint interiors, and with girth at least 5, has at most $24n$ edges.", "To show this, first observe that if two boxes with disjoint interiors intersect, then there is a plane that contains a pair of intersecting faces, one from each of the two boxes.", "Glebov [10] proved that every intersection graph of $r$ axis-aligned rectangles in the plane with girth at least 5 has at most $4r$ edges.", "Each of the 6 faces of a box is contained in a unique plane, so we may obtain the bound of $24n$ by applying Glebov's result to each plane contained in $\\mathbb {R}^3$ and the rectangular faces that they contain.", "It would be interesting to obtain improved bounds for both the number of edges and the chromatic number of these graphs." ], [ "Boxes", "Before proving Theorem REF , we need to introduce a more restricted setting to facilitate the inductive argument.", "Let $p_x:\\mathbb {R}^3 \\mapsto \\mathbb {R}$ denote the projection onto the $x$ -axis.", "Similarly for $p_y$ and $p_z$ .", "We say that an axis-aligned box $B\\subseteq \\mathbb {R}^3$ is a square box if its top and bottom faces are squares.", "Let $P_{x=y}\\subset \\mathbb {R}^3$ be the $x=y$ plane.", "We say that a square box $B\\subset \\mathbb {R}^3$ is grounded if $B$ intersects $P_{x=y}$ , and $p_x(a)\\ge p_y(a)$ for all $a\\in B$ .", "Notice that if $B$ is grounded, then $B \\cap P_{x=y}$ is a vertical line segment, and an edge of the box.", "A graph is a grounded square box graph if it is the intersection graph of a collection of grounded square boxes.", "We prove Theorem REF for the subclass of grounded square box graphs.", "While the grounded condition is used to ease the inductive argument, achieving square boxes rather than general grounded axis-aligned boxes is simply a bonus we get for free from the proof.", "Theorem 5 For every $g\\ge 3$ and $k\\ge 1$ , there exists a grounded square box graph with girth at least $g$ and chromatic number at least $k$ .", "This is trivially true for $k\\le 3$ , as all odd cycles are grounded square box graphs.", "So we fix $g$ and proceed inductively on $k$ .", "Let ${\\cal B}_{g,k}$ be a collection of grounded square boxes whose intersection graph has girth at least $g$ and chromatic number at least $k$ .", "We may assume that each box has a non-empty interior, and by possibly performing small uniform scaling and translations along $P_{x=y}$ to individual boxes, we may assume that the points $p_x(P_{x=y}\\cap B)$ and $p_x(P_{x=y}\\cap B^{\\prime })$ are distinct for each pair of distinct boxes $B, B^{\\prime }\\in {\\cal B}_{g,k}$ .", "Let $T=p_{x}(P_{x=y}\\cap {\\cal B}_{g,k})\\subset \\mathbb {R}$ .", "By Theorem REF there exists a finite set $X\\subset \\mathbb {R}$ such that every $k$ -colouring of $X$ contains a monochromatic homothetic copy $T^{\\prime }$ of $T$ and no set of at most $\\lfloor \\frac{g}{3} \\rfloor $ homothetic copies of $T$ contained in $X$ form a cycle.", "Let $\\cal T$ be the set of homothetic copies of $T$ in $X$ .", "Choose some $0<\\epsilon < \\min \\lbrace \|x-x^{\\prime }\| : x,x^{\\prime } \\in X, x \\ne x^{\\prime }\\rbrace $ .", "Now for each $x\\in X$ , let $B_x$ be the grounded square box $[x,x+\\epsilon ] \\times [x-\\epsilon ,x] \\times [0,1]$ .", "Let ${\\cal B}_{X}$ be the collection of all such grounded square boxes.", "Clearly no pair of these boxes intersect.", "For each $T^{\\prime }\\in {\\cal T}$ : let $I_{T^{\\prime }}$ be an interval contained in $[0,1]$ , such that the intervals $\\lbrace I_{T^{\\prime }}: T^{\\prime }\\in {\\cal T}\\rbrace $ are pairwise disjoint, let $s_{T^{\\prime }}$ be some homothetic map that maps the intervals of $p_z({\\cal B}_{g,k})$ to intervals contained in $I_{T^{\\prime }}$ , let $f_{T^{\\prime }}$ be the homothetic map that maps $T$ to $T^{\\prime }$ in $\\mathbb {R}$ , let $h_{T^{\\prime }}: \\mathbb {R}^3 \\mapsto \\mathbb {R}^3$ be the map such that $h_{T^{\\prime }}(x,y,z)=(f_{T^{\\prime }}(x),f_{T^{\\prime }}(y),s_{T^{\\prime }}(z))$ for all $(x,y,z) \\in \\mathbb {R}^3$ , and lastly let ${\\cal B}_{T^{\\prime }}=h_{T^{\\prime }}({\\cal B}_{g,k})$ .", "As $f_{T^{\\prime }}$ and $s_{T^{\\prime }}$ are both homothetic maps, ${\\cal B}_{T^{\\prime }}$ is a collection grounded square boxes whose intersection graph is isomorphic to that of ${\\cal B}_{g,k}$ .", "Furthermore each box of ${\\cal B}_{T^{\\prime }}$ intersects exactly one of the boxes $\\lbrace B_t: t\\in T^{\\prime }\\rbrace \\subset {\\cal B}_X$ , and no box of ${\\cal B}_{T^{\\prime }}$ intersects a box of ${\\cal B}_{T^}$ for any $T^\\in {\\cal T}\\backslash \\lbrace T^{\\prime }\\rbrace $ .", "Let ${\\cal B}_{g,k+1}$ be the union of the grounded square boxes ${\\cal B}_X$ and $({\\cal B}_{T^{\\prime }} : T^{\\prime }\\in {\\cal T})$ .", "It remains to show that the intersection graph of ${\\cal B}_{g,k+1}$ has girth at least $g$ and chromatic number at least $k+1$ .", "We handle the chromatic number first.", "Suppose for sake of contradiction that there is a $k$ -colouring of ${\\cal B}_{g,k+1}$ .", "Then by choice of $X$ , there is a $T^{\\prime }\\in {\\cal T}$ such that the boxes $\\lbrace B_t: t\\in T^{\\prime }\\rbrace $ are monochromatic.", "Every box of ${\\cal B}_{T^{\\prime }}$ intersects a box of $\\lbrace B_t: t\\in T^{\\prime }\\rbrace $ , so the boxes of ${\\cal B}_{T^{\\prime }}$ must be $(k-1)$ -coloured.", "But the intersection graph of ${\\cal B}_{T^{\\prime }}$ has chromatic number at least $k$ , a contradiction.", "Next we consider the girth.", "By the inductive assumption, a cycle $C$ of length less than $g$ in ${\\cal B}_{g,k+1}$ must contain at least one box $B$ of ${\\cal B}_X$ .", "Let $B_1,B_2$ be the boxes immediately after and before $B$ in the cycle $C$ .", "Then there exist distinct $T_1,T_2\\in {\\cal T}$ such that $B_1\\in {\\cal B}_{T_1}$ and $B_2\\in {\\cal B}_{T_2}$ .", "But then as $\\cal T$ contains no cycle of length less than $\\lceil \\frac{g}{3} \\rceil $ , we see that $C$ must contain at least $\\lceil \\frac{g}{3} \\rceil $ boxes of ${\\cal B}_X$ .", "Furthermore the boxes ${\\cal B}_X$ are pairwise disjoint and there is no box that intersects two boxes of ${\\cal B}_X$ .", "Hence all cycles must have length at least $3 \\lceil \\frac{g}{3} \\rceil \\ge g$ as required.", "It only takes a minor modification of this proof to obtain a proof of Burling's [3] classical result as an application of Van der Waerden's theorem.", "In fact, here we can already achieve a girth of 6.", "One just has to arrange at the start that in addition to the points $p_x(P_{x=y}\\cap B)$ and $p_x(P_{x=y}\\cap B^{\\prime })$ being distinct for each pair of distinct boxes $B, B^{\\prime }\\in {\\cal B}_{6,k}$ , we also have that $p_x(P_{x=y}\\cap B)\\in \\mathbb {Q}$ for each $B \\in {\\cal B}_{6,k}$ .", "Then there is some arithmetic progression in $\\mathbb {Q}$ that contains $p_x(P_{x=y}\\cap {\\cal B}_{6,k})$ , and so we may use Van der Waerden's theorem in place of Theorem REF .", "For larger girth we could also choose to use the sparse Van der Waerden's theorem [27] in place of Theorem REF .", "Another interesting proof of Burling's result was given by Krawczyk and Walczak [18].", "Their proof uses an online colouring approach and reproduces the same graphs constructed by Burling." ], [ "Lines", "Before proving Theorem REF , we first present Norin's [22] construction of double shift graphs as intersection graphs of lines in $\\mathbb {R}^3$ .", "For a positive integer $n$ and a finite set $S \\subset \\mathbb {R}$ with $\|S\|=n$ , the double shift graph $G_{n}$ is isomorphic to the graph where the vertex set is all the triples $(a,b,c)$ with $a,b,c\\in S$ and $a<b<c$ , and where two vertices $(a,b,c)$ and $(d,e,f)$ are adjacent when $b=d$ and $c=e$ .", "For a positive integer $n$ , the double shift graph $G_n$ can be constructed as an intersection graph of lines in $\\mathbb {R}^3$ as follows.", "Let $S\\subseteq \\mathbb {R}$ be an algebraically independent set with $\|S\|=n$ .", "Then for each triple $(a,b,c)$ with $a,b,c\\in S$ and $a<b<c$ , let $L_{(a,b,c)}(t)= (ab + bc + t, abc + bt, ab^2c + (ab + bc)t).$ Then by design, we have that $L_{(a,b,c)}(cd) = L_{(b,c,d)}(ab)$ , and using the fact that $S$ is algebraically independent, it can be shown that all the points of intersection between these lines are of this form.", "Thus the intersection graph of these lines is isomorphic to the double shift graph $G_n$ .", "Erdős and Hajnal [9] showed that $\\lim _{n \\rightarrow \\infty } \\chi (G_n) = \\infty $ , and so intersection graphs of lines in $\\mathbb {R}^3$ are not $\\chi $ -bounded.", "Another related and more general class that this applies to is the intersection graphs of unit length straight line segments in $\\mathbb {R}^3$ .", "Unlike segments of arbitrary length, in the plane these graphs are $\\chi $ -bounded [32].", "Next we prove Theorem REF , that there are intersection graphs of lines in $\\mathbb {R}^3$ with arbitrarily large girth and chromatic number.", "The proof is similar to that of Theorem REF , we just have to modify the geometric arguments for this setting.", "We restate Theorem REF slightly differently for convenience.", "Theorem 6 For every $g\\ge 3$ and $k\\ge 1$ , there exist lines in $\\mathbb {R}^3$ whose intersection graph has girth at least $g$ and chromatic number at least $k$ .", "The theorem is trivially true for $k\\le 3$ , as all odd cycles are intersection graphs of lines in $\\mathbb {R}^3$ .", "So we fix $g$ and proceed inductively on $k$ .", "Let ${\\cal S}_{g,k}$ be a collection of lines in $\\mathbb {R}^3$ whose intersection graph has girth at least $g$ and chromatic number at least $k$ .", "Choose a plane $P\\subset \\mathbb {R}^3$ such that: every line of ${\\cal S}_{g,k}$ intersects $P$ , and no pair of distinct lines of ${\\cal S}_{g,k}$ intersect $P$ at the same point.", "Let $D$ be the set of points of $P$ contained in a line of ${\\cal S}_{g,k}$ , in particular there is a one to one correspondence between points of $D$ and lines in ${\\cal S}_{g,k}$ .", "Now choose an auxiliary line $L$ contained in $P$ such that: no pair of distinct points in $D$ are contained in a single straight line segment $L^{\\prime }\\subset P$ that is perpendicular to $L$ , and there is no plane $Q$ such that $P\\cap Q$ is a line perpendicular to $L$ , the plane $Q$ contains a line $S$ of ${\\cal S}_{g,k}$ , and some translation of $Q$ contains a line $S^$ of ${\\cal S}_{g,k}$ that is not parallel to $S$ .", "Such a line $L$ can be chosen, since for non-parallel lines $S,S^\\in {\\cal S}_{g,k}$ , there is a unique plane $Q$ such that $Q$ contains $S$ and some translation of $Q$ contains $S^$ .", "For each $d\\in D$ , let $t_d$ be the point of $L$ such that there is a straight line segment contained in $P$ that is perpendicular to $L$ and contains both $t_d$ and $d$ .", "Now let ${T=\\lbrace t_d : d\\in D \\rbrace }$ .", "By choice of $L$ , there is a one to one correspondence between points of $T$ and lines of ${\\cal S}_{g,k}$ .", "By Theorem REF , there exists a finite set $X\\subset L$ such that every $k$ -colouring of $X$ contains a monochromatic homothetic copy $T^{\\prime }$ of $T$ and no set of at most $\\lfloor \\frac{g}{3} \\rfloor $ homothetic copies of $T$ contained in $X$ form a cycle.", "Let $\\cal T$ be the set of homothetic copies of $T$ in $X$ .", "For each $x\\in X$ , let $S_x$ be the line contained in $P$ that is perpendicular to $L$ and contains the point $x$ .", "Let ${\\cal S}_X = \\lbrace S_x : x\\in X\\rbrace $ .", "There is a one to one correspondence between lines of ${\\cal S}_{g,k}$ and lines of $\\lbrace S_t : t\\in T\\rbrace $ .", "Notice that each line $S$ of ${\\cal S}_{g,k}$ intersects its corresponding line $S_t$ and no other line contained in ${\\cal S}_X$ .", "Let ${\\cal S}_{T}= {\\cal S}_{g,k}$ .", "For each $T^{\\prime }\\in {\\cal T}\\backslash \\lbrace T\\rbrace $ , let ${\\cal S}_{T^{\\prime }}$ be a homothetic copy of ${\\cal S}_{T}$ with a homothetic map $f_{T^{\\prime }}$ such that: each line of ${\\cal S}_{T^{\\prime }}$ intersects its corresponding line of $\\lbrace S_{t^{\\prime }} : t^{\\prime } \\in T^{\\prime }\\rbrace $ and no other line of ${\\cal S}_X$ , and for distinct $T_1,T_2\\in {\\cal T}$ , no line of ${\\cal S}_{T_1}$ intersects a line of ${\\cal S}_{T_2}$ .", "For each $T^{\\prime }\\in {\\cal T}\\backslash {T}$ , the homethetic map $f_{T^{\\prime }}$ would be a composition of a homethetic map that maps $T$ to $T^{\\prime }$ in $\\mathbb {R}^3$ (ensuring the correct intersections between lines of ${\\cal S}_{T^{\\prime }}$ and $\\lbrace S_{t^{\\prime }} : t^{\\prime } \\in T^{\\prime }\\rbrace $ ) and some translation in the direction parallel to the lines of ${\\cal S}_X$ so as to avoid lines of ${\\cal S}_{T^{\\prime }}$ intersecting lines of another homothetic copy of ${\\cal S}_{T}$ .", "Such translations exist by the choice of $L$ , since for each $x\\in X$ and translated copies $S, S^$ of lines in ${\\cal S}_{T}= {\\cal S}_{g,k}$ such that $S_x, S , S^$ are coplanar, the lines $S$ and $S^$ must be parallel.", "Let ${\\cal S}_{g,k+1}$ be the union of the lines of ${\\cal S}_X$ and $({\\cal S}_T : T\\in {\\cal T})$ .", "The proof that the intersection graph of ${\\cal S}_{g,k+1}$ has girth at least $g$ and chromatic number at least $k+1$ is essentially the same as for the boxes in Theorem REF ." ], [ "Acknowledgments", "The author would like to thank Jim Geelen, Matthew Kroeker, and Rose McCarty for helpful discussions, which in particular improved the proof of Theorem REF .", "The author would also like to thank Tom Johnston for pointing out an issue in the proof of Theorem REF in a previous version of this manuscript.", "The author thanks Sergey Norin for allowing the inclusion of his construction of double shift graphs as line intersection graphs.", "Lastly the author thanks the anonymous referees for some helpful suggestions on the presentation of the paper.", "[jd] James Davies Department of Combinatorics and Optimization University of Waterloo Waterloo, Canada jgdaviesuwaterlooca" ] ]	2011.14174

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